ASTROPHYSICAL GYROKINETICS: KINETIC AND FLUID TURBULENT CASCADES IN MAGNETIZED WEAKLY COLLISIONAL PLASMAS

Suppose the average energy per unit time per unit volume that the system dissipates is ε. This energy has to be transferred from some (large) outer scale L at which it is injected to some (small) inner scale(s) at which the dissipation occurs (see Section 1.5). It is assumed that in the range of scales intermediate between the outer and the inner (the inertial range), the statistical properties of the turbulence are universal (independent of the macrophysics of injection or of the microphysics of dissipation), spatially homogeneous and isotropic and the energy transfer is local in scale space. The flux of kinetic energy through any inertial-range scale λ is independent of λ:

$$\begin{eqnarray} &&{u_{\lambda }^2\over \tau _\lambda }\sim \varepsilon = {\rm const}, \end{eqnarray} \tag{ 1 }$$

where the (constant) density of the medium is absorbed into ε, u_λ is the typical velocity fluctuation associated with the scale λ, and τ_λ is the cascade time.⁹ Since interactions are assumed local, τ_λ must be expressed in terms of quantities associated with scale λ. It is then dimensionally inevitable that τ_λ ∼ λ/u_λ (the nonlinear interaction time, or turnover time), so we get

$$\begin{eqnarray} &&u_{\lambda }\sim (\varepsilon \lambda)^{1/3}. \end{eqnarray} \tag{ 2 }$$

This corresponds to a k^−5/3 spectrum of kinetic energy.

That astronomical data appear to point to a ubiquitous nature of what, in its origin, is a dimensional result for the turbulence in a neutral fluid, might appear surprising. Indeed, the astrophysical plasmas in question are highly conducting and support magnetic fields whose energy is at least comparable to the kinetic energy of the motions. Let us consider a situation where the plasma is threaded by a uniform dynamically strong magnetic field B₀ (the mean, or guide, field; see Section 1.3 for a brief discussion of the validity of this assumption). In the presence of such a field, there is no dimensionally unique way of determining the cascade time τ_λ because besides the nonlinear interaction time λ/u_λ, there is a second characteristic time associated with the fluctuation of size λ, namely the Alfvén time l_∥λ/v_A, where v_A is the Alfvén speed and l_∥λ is the typical scale of the fluctuation along the magnetic field.

The first theories of magnetohydrodynamic (MHD) turbulence (Iroshnikov 1963; Kraichnan 1965; Dobrowolny et al. 1980) calculated τ_λ by assuming an isotropic cascade (l_∥λ ∼ λ) of weakly interacting Alfvén-wave packets (τ_λ ≫ l_∥λ/v_A) and obtained a k^−3/2 spectrum. The failure of the observed spectra to conform to this law (see references above) and especially the observational (see references at the end of this subsection) and experimental (Robinson & Rusbridge 1971; Zweben et al. 1979) evidence of anisotropy of MHD fluctuations led to the isotropy assumption being discarded (Montgomery & Turner 1981).

The modern form of MHD turbulence theory is commonly associated with the names of Goldreich & Sridhar (1995, 1997, henceforth GS). It can be summarized as follows. Assume that (a) all electromagnetic perturbations are strongly anisotropic, so that their characteristic scales along the mean field are much larger than those across it, l_∥λ ≫ λ, or, in terms of wavenumbers, k_∥ ≪ k_⊥; (b) the interactions between the Alfvén-wave packets are strong and the turbulence at sufficiently small scales always arranges itself in such a way that the Alfvén timescale and the perpendicular nonlinear interaction timescale are comparable to each other, i.e.,

$$\begin{eqnarray} &&\omega \sim k_{\parallel }v_A\sim k_{\perp }u_\perp, \end{eqnarray} \tag{ 3 }$$

where ω is the typical frequency of the fluctuations and u_⊥ is the velocity fluctuation perpendicular to the mean field. Taken scale by scale, this assumption, known as the critical balance, removes the dimensional ambiguity of the MHD turbulence theory. Thus, the cascade time is τ_λ ∼ l_∥λ/v_A ∼ λ/u_λ, whence

$$\begin{eqnarray} u_{\lambda }&\sim &(\varepsilon l_{\parallel \lambda }/v_A)^{1/2}\sim \left(\varepsilon \lambda \right)^{1/3}, \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} l_{\parallel \lambda }&\sim &l_0^{1/3}\lambda ^{2/3}, \end{eqnarray} \tag{ 5 }$$

where l₀ = v³_A/ε. The scaling relation (4) is equivalent to a k^−5/3_⊥ spectrum of kinetic energy, while Equation (5) quantifies the anisotropy by establishing the relationship between the perpendicular and parallel scales. Note that Equation (4) implies that in terms of the parallel wavenumbers, the kinetic-energy spectrum is ∼k⁻²_∥.

The above considerations apply to Alfvénic fluctuations, i.e., perpendicular velocities and magnetic-field perturbations from the mean given (at each scale) by $\delta B_\perp \sim u_\perp \sqrt{4\pi \rho _0}$, where ρ₀ is the mean mass density of the plasma (see Figure 1 and discussion in Section 8.1.1). Other low-frequency MHD modes—slow waves and the entropy mode—turn out to be passively advected by the Alfvénic component of the turbulence (this follows from the anisotropy; see Lithwick & Goldreich 2001, and Sections 2.4–2.6, 5.5, and 6.3 for further discussion of the compressive fluctuations).

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As we have mentioned above, the anisotropy was, in fact, incorporated into MHD turbulence theory already by Montgomery & Turner (1981). However, these authors' view differed from the GS theory in that they thought of MHD turbulence as essentially two dimensional, described by a Kolmogorov-like cascade (Fyfe et al. 1977), with an admixture of Alfvén waves having some spectrum in k_∥ unrelated to the perpendicular structure of the turbulence (note that Higdon 1984, while adopting a similar view, anticipated the scaling relation (5), but did not seem to consider it to be anything more than the confirmation of an essentially two-dimensional nature of the turbulence). In what we are referring to here as GS turbulence, the two-dimensional and Alfvénic fluctuations are not separate components of the turbulence. The turbulence is three dimensional, with correlations parallel and perpendicular to the (local) mean field related at each scale by the critical balance assumption.

Indeed, intuitively, we cannot have k_∥v_A ≪ k_⊥u_⊥: the turbulence cannot be any more two dimensional than allowed by the critical balance because fluctuations in any two planes perpendicular to the mean field can only remain correlated if an Alfvén wave can propagate between them in less than their perpendicular decorrelation time. In the opposite limit, weakly interacting Alfén waves with fixed k_∥ and ω = k_∥v_A ≫ k_⊥u_⊥ can be shown to give rise to an energy cascade towards smaller perpendicular scales where the turbulence becomes strong and Equation (3) is satisfied (Goldreich & Sridhar 1997; Galtier et al. 2000; Yousef et al. 2009). Thus, there is a natural tendency towards critical balance in a system containing nonlinearly interacting Alfvén waves. We will see in what follows that critical balance may, in fact, be taken as a general physical principle relating parallel scales (associated with linear propagation) and perpendicular scales (associated with nonlinear interaction) in anisotropic plasma turbulence (see Sections 7.5, 7.9.4, and 7.10.3).

We emphasize that, the anisotropy of astrophysical plasma turbulence is an observed phenomenon. It is seen most clearly in the spacecraft measurements of the turbulent fluctuations in the solar wind (Belcher & Davis 1971; Matthaeus et al. 1990; Bieber et al. 1996; Dasso et al. 2005; Bigazzi et al. 2006; Sorriso-Valvo et al. 2006; Horbury et al. 2005, 2008; Osman & Horbury 2007; Hamilton et al. 2008) and in the magnetosheath Sahraoui et al. (2006); Alexandrova et al. (2008b). In a recent key development, solar-wind data analysis by Horbury et al. (2008) approaches quantitative corroboration of the critical balance conjecture by confirming the scaling of the spectrum with the parallel wavenumber ∼k⁻²_∥ that follows from the first scaling relation in Equation (4). Anisotropy is also observed indirectly in the ISM (Wilkinson et al. 1994; Trotter et al. 1998; Rickett et al. 2002; Dennett-Thorpe & de Bruyn 2003), including recently in molecular clouds (Heyer et al. 2008), and, with unambiguous consistency, in numerical simulations of MHD turbulence (Shebalin et al. 1983; Oughton et al. 1994; Cho & Vishniac 2000; Maron & Goldreich 2001; Cho et al. 2002; Müller et al. 2003).¹⁰

In the discussion above, treating MHD turbulence as turbulence of Alfvénic fluctuations depended on assuming the presence of a mean (guide) field B₀ that is strong compared to the magnetic fluctuations, δB/B₀ ∼ u/v_A ≪ 1. We will also need this assumption in the formal developments to follow (see Sections 2.1, 3.1). Is it legitimate to expect that such a spatially regular field will be generically present? Kraichnan (1965) argued that in a generic situation in which all magnetic fields are produced by the turbulence itself via the dynamo effect, one could assume that the strongest field will be at the outer scale and that this field will play the role of an (approximately) uniform guide field for the Alfvén waves in the inertial range. Formally, this amounts to assuming that in the inertial range,

$$\begin{eqnarray} &&{\delta B\over B_0}\ll 1,\quad k_{\parallel }L\ll 1. \end{eqnarray} \tag{ 6 }$$

It is, however, by no means obvious that this should be true. When a strong mean field is imposed by some external mechanism, the turbulent motions cannot bend it significantly, so only small perturbations are possible and δB ≪ B₀. In contrast, without a strong imposed field, the energy density of the magnetic fluctuations is at most comparable to the kinetic-energy density of the plasma motions, which are then sufficiently energetic to randomly tangle the field, so δB ≫ B₀.

In the weak-mean-field case, the dynamically strong stochastic magnetic field is a result of saturation of the small-scale, or fluctuation, dynamo—amplification of magnetic field due to random stretching by the turbulent motions (see review by Schekochihin & Cowley 2007). The definitive theory of this saturated state remains to be discovered. Both physical arguments and numerical evidence (Schekochihin et al. 2004; Yousef et al. 2007) suggest that the magnetic field in this case is organized in folded flux sheets (or ribbons). The length of these folds is comparable to the outer scale, while the scale of the field-direction reversals transverse to the fold is determined by the dissipation physics: in MHD with isotropic viscosity and resistivity, it is the resistive scale.¹¹ Although Alfvén waves propagating along the folds may exist (Schekochihin et al. 2004; Schekochihin & Cowley 2007), the presence of the small-scale direction reversals means that there is no scale-by-scale equipartition between the velocity and magnetic fields: while the magnetic energy is small-scale dominated due to the direction reversals,¹² the kinetic energy should be contained primarily at the outer scale, with some scaling law in the inertial range.

Thus, at the current level of understanding we have to assume that there are two asymptotic regimes of MHD turbulence: anisotropic Alfvénic turbulence with δB ≪ B₀ and isotropic MHD turbulence with small-scale field reversals and δB ≫ B₀. In this paper, we shall only discuss the first regime. The origin of the mean field may be external (as, e.g., in the solar wind, where it is the field of the Sun) or due to some form of mean-field dynamo (rather than small-scale dynamo), as usually expected for galaxies (see, e.g., Shukurov 2007).

Note finally that the condition δB ≪ B₀ need not be satisfied at the outer scale and in fact is not satisfied in most space or astrophysical plasmas, where more commonly δB ∼ B₀ at the outer scale. This, however, is sufficient for the Kraichnan hypothesis to hold and for an Alfénic cascade to be set up, so at small scales (in the inertial range and beyond), the assumptions (6) are satisfied.

The GS theory of MHD turbulence (Section 1.2) allows us to make sense of the magnetized turbulence observed in cosmic plasmas exhibiting the same statistical scaling as turbulence in a neutral fluid (although the underlying dynamics are very different in these two cases!). However, there is an aspect of the observed astrophysical turbulence that undermines the applicability of any type of fluid description: in most cases, the inertial range where the Kolmogorov scaling holds extends to scales far below the mean free path deep into the collisionless regime. For example, in the case of the solar wind, the mean free path is close to 1 AU, so all scales are collisionless—an extreme case, which also happens to be the best studied, thanks to the possibility of in situ measurements (see Section 8).

The proper way of treating such plasmas is using kinetic theory, not fluid equations. The basis for the application of the MHD fluid description to them has been the following well known result from the linear theory of plasma waves: while the fast, slow and entropy modes are damped at the mean-free-path scale both by collisional viscosity (Braginskii 1965, see Section 6.1.2) and by collisionless wave–particle interactions (Barnes 1966, see Section 6.2.2), the Alfvén waves are only damped at the ion gyroscale. It has, therefore, been assumed that the MHD description, inasmuch as it concerns the Alfvén-wave cascade, can be extended to the ion gyroscale, with the understanding that this cascade is decoupled from the damped cascades of the rest of the MHD modes. This approach and its application to the turbulence in the ISM are best explained by Lithwick & Goldreich (2001).

While the fluid description may be sufficient to understand the Alfvénic fluctuations in the inertial range, it is certainly inadequate for everything else: the compressive fluctuations in the inertial range and turbulence in the dissipation range (below the ion gyroscale), where power-law spectra are also detected (e.g., Denskat et al. 1983; Leamon et al. 1998; Czaykowska et al. 2001; Smith et al. 2006; Sahraoui et al. 2006; Alexandrova et al. 2008a, 2008b; see also Figure 1). The fundamental challenge that a comprehensive theory of astrophysical plasma turbulence must meet is to give the full account of how the turbulent fluctuation energy injected at the outer scale is cascaded to small scales and deposited into particle heat. We shall see (Sections 3.4 and 3.5) that the familiar concept of an energy cascade can be generalized in the kinetic framework as the kinetic cascade of a single quantity that we call the generalized energy (see also Schekochihin et al. 2008b, and references therein). The small scales developed in the process are small scales both in the position and velocity space. The fundamental reason for this is the low collisionality of the plasma: since heating cannot ultimately be accomplished without collisions, large gradients in phase space are necessary for the collisions to be effective.

The idea of a generalized energy cascade in phase space as the engine of kinetic plasma turbulence is the central concept of this paper. In order to understand the physics of the kinetic cascade in various scale ranges, we derive in what follows a hierarchy of simplified, yet rigorous, reduced kinetic, fluid and hybrid descriptions. While the full kinetic theory of turbulence is very difficult to handle either analytically or numerically, the models we derive are much more tractable. For all, the regimes of applicability (scale/parameter ranges, underlying assumptions) are clearly stated. In each of these regimes, the kinetic cascade splits into several channels of energy transfer, some of them familiar (e.g., the Alfvénic cascade, Sections 5.3 and 5.4), others conceptually new (e.g., the kinetic cascade of collisionless compressive fluctuations, Section 6.2, or the entropy cascade, Sections 7.9–7.12).

So as to introduce this theoretical framework in a way that is both analytically systematic and physically intelligible, let us first consider the characteristic scales that are relevant to the problem of astrophysical turbulence (Section 1.5). The models we derive are previewed in Section 1.6, at the end of which the plan of further developments is given.

1.5.1. Outer Scale

1.5.2. Microscales

There are four microphysical scales that mark the transitions between distinct physical regimes:

Electron diffusion scale. At k_∥λ_mfpi(m_i/m_e)^1/2 ≫ 1, the electron response is isothermal (Section 4.4, Appendix A.4). At k_∥λ_mfpi(m_i/m_e)^1/2 ≪ 1, it is adiabatic (Section 4.8.4, Appendix A.3).

Mean free path. At k_∥λ_mfpi ≫ 1, the plasma is collisionless. In this regime, wave–particle interactions can damp compressive fluctuations via Barnes damping (Section 6.2.2), so kinetic description becomes essential. At k_∥λ_mfpi ≪ 1, the plasma is collisional and fluid-like (Section 6.1, Appendices A and D).

Ion gyroscale. At k_⊥ρ_i ≪ 1, ions (as well as the electrons) are magnetized and the magnetic field is frozen into the ion flow (the E × B velocity field). At k_⊥ρ_i ∼ 1, ions can exchange energy with electromagnetic fluctuations via wave–particle interactions (and ion heating eventually occurs via a kinetic ion-entropy cascade, see Sections 7.9–7.10). At k_⊥ρ_i ≫ 1, the ions are unmagnetized and have a Boltzmann response (Section 7.2). Note that the ion inertial scale $d_i= \rho _i/\sqrt{\beta _i}$ is comparable to the ion gyroscale unless the plasma beta β_i = 8πn_iT_i/B² is very different from unity. In the theories developed below, d_i does not play a special role except in the limit of T_i ≪ T_e, which is not common in astrophysical plasmas (see further discussion in Section 7.1 and Appendix E).

Electron gyroscale. At k_⊥ρ_e ≪ 1, electrons are magnetized and the magnetic field is frozen into the electron flow (Sections 4, 7, Appendix C). At k_⊥ρ_e ∼ 1, the electrons absorb the energy of the electromagnetic fluctuations via wave–particle interactions (leading to electron heating via a kinetic electron-entropy cascade, see Section 7.12).

Typical values of these scales and of several other key parameters are given in Table 1. In Figure 2, we show how the wavenumber space, (k_⊥, k_∥), is divided by these scales into several domains, where the physics is different. Further partitioning of the wavenumber space results from comparing k_⊥ρ_i and k_∥λ_mfpi (k_⊥ρ_i ≪ k_∥λ_mfpi is the limit of strong magnetization, see Appendix A.2) and, most importantly, from comparing parallel and perpendicular wavenumbers. As we explained above, observational and numerical evidence tells us that Alfvénic turbulence is anisotropic, k_∥ ≪ k_⊥. In Figure 2, we sketch the path the turbulent cascade is expected to take in the wavenumber space (we use the scalings of k_∥ with k_⊥ that follow from the GS argument for the Alfvén waves and an analogous argument for the kinetic Alfvén waves, reviewed in Sections 1.2 and 7.5, respectively).

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Table 1. Representative Parameters for Astrophysical Plasmas

Parameter	Solar wind at 1 AU(a)	Warm ionized ISM(b)	Accretion flow near Sgr A*(c)	Galaxy clusters (cores)(d)
ne = ni, cm−3	30	0.5	106	6 × 10−2
Te, K	∼Ti(e)	8000	1011	3 × 107
Ti, K	5 × 105	8000	∼1012(f)	?(e)
B, G	10−4	10−6	30	7 × 10−6
βi	5	14	4	130
vthi, km s−1	90	10	105	700
vA, km s−1	40	3	7 × 104	60
U, km s−1(f)	∼10	∼10	∼104	∼102
L, km(f)	∼105	∼1015	∼108	∼1017
(mi/me)1/2λmfpi, km	1010	2 × 108	4 × 1010	4 × 1016
λmfpi, km(g)	3 × 108	6 × 106	109	1015
ρi, km	90	1000	0.4	104
ρe, km	2	30	0.003	200

Notes. ^aValues for slow wind (mean flow speed V_sw = 350 km s⁻¹ in this case) measured by Cluster spacecraft and taken from Bale et al. (2005), except the value of T_e, which they do not report, but which is expected to be of the same order as T_i (Newbury et al. 1998). Note that the data interval studied by Bale et al. (2005) is slightly atypical, with β_i higher than usual in the solar wind (the full range of β_i variation in the solar wind is roughly between 0.1 and 10; see Howes et al. (2008a) for another, perhaps more typical, fiducial set of slow-wind parameters and Appendix A of the review by Bruno & Carbone (2005) for slow- and fast-wind parameters measured by Helios 2). However, we use their parameter values as our representative example because the spectra they report show with particular clarity both the electric and magnetic fluctuations in both the inertial and dissipation ranges (see Figure 1). See further discussion in Sections 8.1 and 8.2. ^bTypical values (see, e.g., Norman & Ferrara 1996; Ferrière 2001). See discussion in Section 8.4. ^cValues based on observational constraints for the radio-emitting plasma around the Galactic Center (Sgr A*) as interpreted by Loeb & Waxman (2007) (see also Quataert 2003). See discussion in Section 8.5. ^dValues for the core region of the Hydra A cluster taken from Enßlin & Vogt (2006); see Schekochihin & Cowley (2006) for a consistent set of numbers for the hot plasmas outside the cores. See discussion in Section 8.6. ^eWe assume T_i ∼ T_e for these estimates. ^fRough order-of-magnitude estimate. ^gDefined λ_mfpi = v_thi/ν_ii, where ν_ii is given by Equation (52).

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What is the correct analytical description of the turbulent plasma fluctuations along the (presumed) path of the cascade? As we promised above, it is going to be possible to simplify the full kinetic theory substantially. These simplifications can be obtained in the form of a hierarchy of approximations and as these emerge, specific physical mechanisms that control the turbulent cascade in various physical regimes become more transparent.

Gyrokinetics (Section 3). The starting point for these developments and the primary approximation in the hierarchy is gyrokinetics, a low-frequency kinetic theory resulting from averaging over the cyclotron motion of the particles. Gyrokinetics is appropriate for the study of subsonic plasma turbulence in virtually all astrophysically relevant parameter ranges (Howes et al. 2006). For fluctuations at frequencies lower than the ion cyclotron frequency, ω ≪ Ω_i, gyrokinetics can be systematically derived by making use of the following two assumptions, which also underpin the GS theory (Section 1.2): (a) anisotropy of the turbulence, so ∼ k_∥/k_⊥ is used as the small parameter, and (b) strong interactions, i.e., the fluctuation amplitudes are assumed to be such that wave propagation and nonlinear interaction occur on comparable timescales: from Equation (3), u_⊥/v_A ∼ . The first of these assumptions implies that fluctuations at Alfvénic frequencies satisfy ω ∼ k_∥v_A ≪ Ω_i even when their perpendicular scale is such k_⊥ρ_i ∼ 1. This makes gyrokinetics an ideal tool both for analytical theory and for numerical studies of astrophysical plasma turbulence; the numerical approaches are also made attractive by the long experience of gyrokinetic simulations accumulated in the fusion research and by the existence of publicly available gyrokinetic codes (Kotschenreuther et al. 1995; Jenko et al. 2000; Candy & Waltz 2003; Chen & Parker 2003). A concise review of gyrokinetics is provided in Section 3 (see Howes et al. 2006 for a detailed derivation). The reader is urged to pay particular attention to Sections 3.4 and 3.5, where the concept of the kinetic cascade of generalized energy is introduced and the particle heating in gyrokinetics is discussed (Appendix F introduces additional conservation laws that arise in two dimensions and sometimes also in three dimensions). This establishes the conceptual framework in which most of the subsequent physical arguments are presented. The region of validity of gyrokinetics is illustrated in Figure 3: it covers virtually the entire path of the turbulent cascade, except the largest (outer) scales, where one cannot assume anisotropy. Note that the two-fluid theory, which is the starting point for the MHD theory (see Appendix A), is not a good description at collisionless scales. It is important to mention, however, that the formulation of gyrokinetics that we adopt, while appropriate for treating fluctuations at collisionless scales, does nevertheless require a certain (weak) degree of collisionality (see discussion in Section 3.1.3 and an extended treatment of collisions in gyrokinetics in Appendix B).

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Isothermal electron fluid (Section 4). While gyrokinetics constitutes a significant simplification, it is still a fully kinetic description. Further progress towards simpler models is achieved by showing that, for parallel scales smaller than the electron diffusion scale, k_∥λ_mfpi ≫ (m_e/m_i)^1/2, and perpendicular scales larger than the electron gyroscale, k_⊥ρ_e ≪ 1, the electrons are a magnetized isothermal fluid while ions must be treated (gyro)kinetically. This is the secondary approximation in our hierarchy, derived in Section 4 via an asymptotic expansion in (m_e/m_i)^1/2 (see also Appendix C.1). The plasma is described by the ion gyrokinetic equation and two fluid-like equations that contain electron dynamics—these are summarized in Section 4.9. The region of validity of this approximation is illustrated in Figure 4: it does not capture the dissipative effects around the electron diffusion scale or the electron heating, but it remains uniformly valid as the cascade passes from collisional to collisionless scales and also as it crosses the ion gyroscale.

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In order to elucidate the nature of the turbulence above and below the ion gyroscale, we derive two tertiary approximations, one of which is valid for k_⊥ρ_i ≪ 1 (Sections 5 and 6) and the other for k_⊥ρ_i ≫ 1 (Section 7; see also Appendix C, which gives a non-rigorous, non-gyrokinetic, but perhaps more intuitive, derivation of the results of Sections 4 and 7.2).

Kinetic reduced MHD (Sections 5 and 6). On scales above the ion gyroscale, known as the "inertial range" we demonstrate that the decoupling of the Alfvén-wave cascade and its indifference to both collisional and collisionless damping are explicit and analytically provable properties. We show rigorously that the Alfvén-wave cascade is governed by a closed set of two fluid-like equations for the stream and flux functions—the reduced magnetohydrodynamics (RMHD)—independently of the collisionality (Sections 5.3 and 5.4; the derivation of RMHD from MHD and its properties are presented in Section 2). The cascade proceeds via interaction of oppositely propagating wave packets and is decoupled from the density and magnetic-field-strength fluctuations (the "compressive" modes; in the collisional limit, these are the entropy and slow modes; see Section 6.1 and Appendix D). The latter are passively mixed by the Alfvén waves, but, unlike in the fluid (collisional) limit, this passive cascade is governed by a (simplified) kinetic equation for the ions (Section 5.5). Together with RMHD, it forms a hybrid fluid-kinetic description of magnetized turbulence in a weakly collisional plasma, which we call kinetic reduced MHD (KRMHD). The KRMHD equations are summarized in Section 5.7. Their collisional and collisionless limits are explored in Sections 6.1 and 6.2, respectively. Whereas the Alfvén waves are undamped in this approximation, the compressive fluctuations are subject to damping both in the collisional (Braginskii 1965 viscous damping, Section 6.1.2) and collisionless (Barnes 1966 damping, Section 6.2.2) limits. In the collisionless limit, the compressive component of the turbulence is a simple example of an essentially kinetic turbulence, including such features as conservation of generalized energy despite collisionless damping and (parallel) phase mixing, possibly leading to ion heating (Sections 6.2.3–6.2.5). How strongly the compressive fluctuations are damped depends on the parallel scale of these fluctuations. Since the ion kinetic equation turns out to be linear along the moving field lines associated with the Alfvén waves, the compressive fluctuations do not, in the absence of finite-gyroradius effects, develop small parallel scales and their cascade may be only weakly damped above the ion gyroscale—this is discussed in Section 6.3.

Electron reduced MHD (Section 7). At the ion gyroscale, the Alfvénic and the compressive cascades are no longer decoupled and their energy is partially damped via collisionless wave–particle interactions (Section 7.1). This part of the energy is channeled into ion heat. The rest of it is converted into a cascade of kinetic Alfvén waves (KAW). This cascade extends through what is known as the "dissipation range" to the electron gyroscale, where its turn comes to be damped via wave–particle interaction and transferred into electron heat. The KAW turbulence is again anisotropic with k_∥ ≪ k_⊥. It is governed by a pair of fluid-like equations, also derived from gyrokinetics. We call them electron reduced MHD (ERMHD). In the high-beta limit, they coincide with the reduced (anisotropic) form of the previously known electron MHD (Kingsep et al. 1990). The ERMHD equations are derived in Section 7.2 (see also Appendix C.2) and the KAW cascade is considered in Sections 7.3–7.5. The fate of the inertial-range energy collisionlessly damped at the ion gyroscale is investigated in Sections 7.9–7.11; an analogous consideration for the KAW energy damped at the electron gyroscale is presented in Section 7.12. In these sections, we introduce the notion of the entropy cascade—a nonlinear phase-mixing process whereby the collisionless damping occurring at the ion and electron gyroscales is made irreversible and particles are heated. This part of the cascade is purely kinetic and its salient feature is the particle distribution functions developing small scales in the gyrokinetic phase space. Note that besides deriving rigorous sets of equations for the dissipation-range turbulence, Section 7 also presents a number of Kolmogorov-style scaling predictions—both for the KAW cascade (Section 7.5) and for the entropy cascade (Sections 7.9.2, 7.10.2, 7.10.4, 7.12).

Hall reduced MHD (Appendix E). The reduced (anisotropic) form of the popular Hall MHD system can be derived as a special limit of gyrokinetics (k_⊥ρ_i ≪ 1, T_i ≪ T_e, β_i ≪ 1). The resulting Hall reduced MHD (HRMHD) equations are a convenient model for some purposes because they simultaneously capture the cold-ion, low-beta limits of both the KRMHD and ERMHD systems. However, they are usually not strictly applicable in space and astrophysical plasmas of interest, where ions are rarely cold and β_i is not particularly low. The HRMHD equations are derived in Section E.1, the kinetic cascade of generalized energy in the Hall limit is discussed in Section E.2, and the circumstances under which the ion inertial and ion sound scales become important in theories of plasma turbulence are summarized in Section E.4. Theories of the dissipation-range turbulence based on Hall MHD are briefly discussed in Section 8.2.6.

The regions of validity of the tertiary approximations—KRMHD and ERMHD—are illustrated in Figure 2. In this figure, we also show the region of validity of the RMHD system derived from the standard compressible MHD equations by assuming anisotropy of the turbulence and strong interactions. This derivation is the fluid analog of the derivation of gyrokinetics. We present it in Section 2, before embarking on the gyrokinetics-based path outlined above, in order to make a connection with the conventional MHD treatment and to demonstrate with particular simplicity how the assumption of anisotropy leads to a reduced fluid system in which the decoupling of the cascades of the Alfvén waves and of the compressive modes is manifest (Appendix A extends this derivation to Braginskii (1965) two-fluid equations in the limit of strong magnetization; it also works out rigorously the transition from the fluid limit to the KRMHD equations).

The main formal developments of this paper are contained in Sections 3–7. The outline given above is meant to help the reader navigate these sections. In Section 8, we discuss at some length how our results apply to various astrophysical plasmas with weak collisionality: the solar wind and the magnetosheath, the ISM, accretion disks, and galaxy clusters (Sections 8.1 and 8.2 can also be read as an overall summary of the paper in light of the evidence available from space-plasma measurements). Finally, in Section 9, we provide a brief epilogue and make a few remarks about future directions of inquiry.

As we explained in the Introduction, observational and numerical evidence makes it safe to assume that the turbulence in such a system will be anisotropic with k_∥ ≪ k_⊥ (at scales smaller than the outer scale, k_∥L ≫ 1; see Sections 1.3 and 1.5.1). Let us, therefore, introduce a small parameter ∼ k_∥/k_⊥ and carry out a systematic expansion of Equations (7)–(10) in . In this expansion, the fluctuations are treated as small, but not arbitrarily so: in order to estimate their size, we shall adopt the critical-balance conjecture (3), which is now treated not as a detailed scaling prescription but as an ordering assumption. This allows us to introduce the following ordering:

$$\begin{eqnarray} &&{\delta \rho \over \rho _0} \sim {u_\perp \over v_A} \sim {u_\parallel \over v_A} \sim {\delta p\over p_0} \sim {\delta B_\perp \over B_0} \sim {\delta B_\parallel \over B_0} \sim {k_{\parallel }\over k_{\perp }} \sim \epsilon, \end{eqnarray} \tag{ 12 }$$

where $v_A=B_0/\sqrt{4\pi \rho _0}$ is the Alfvén speed. Note that this means that we order the Mach number

$$\begin{eqnarray} &&M\sim {u\over c_s} \sim {\epsilon \over \sqrt{\beta _i}}, \end{eqnarray} \tag{ 13 }$$

where c_s = (γp₀/ρ₀)^1/2 is the speed of sound and

$$\begin{eqnarray} &&\beta ={8\pi p_0\over B_0^2}={2\over \gamma }{c_s^2\over v_A^2} \end{eqnarray} \tag{ 14 }$$

is the plasma beta, which is ordered to be order unity in the expansion (subsidiary limits of high and low β can be taken after the expansion is done; see Section 2.4).

In Equation (12), we made two auxiliary ordering assumptions: that the velocity and magnetic-field fluctuations have the character of Alfvén and slow waves (δB_⊥/B₀ ∼ u_⊥/v_A, δB_∥/B₀ ∼ u_∥/v_A) and that the relative amplitudes of the Alfvén-wave-polarized fluctuations (δB_⊥/B₀, u_⊥/v_A), slow-wave-polarized fluctuations (δB_∥/B₀, u_∥/v_A) and density/pressure/entropy fluctuations (δρ/ρ₀, δp/p₀) are all the same order. Strictly speaking, whether this is the case depends on the energy sources that drive the turbulence: as we shall see, if no slow waves (or entropy fluctuations) are launched, none will be present. However, in astrophysical contexts, the outer-scale energy input may be assumed random and, therefore, comparable power is injected into all types of fluctuations.

We further assume that the characteristic frequency of the fluctuations is ω ∼ k_∥v_A (Equation (3)), meaning that the fast waves, for which ω ≃ k_⊥(v²_A + c²_s)^1/2, are ordered out. This restriction must be justified empirically. Observations of the solar-wind turbulence confirm that it is primarily Alfvénic (see, e.g., Bale et al. 2005) and that its compressive component is substantially pressure-balanced (Roberts 1990; Burlaga et al. 1990; Marsch & Tu 1993; Bavassano et al. 2004, see Equation (22) below). A weak-turbulence calculation of compressible MHD turbulence in low-beta plasmas (Chandran 2005b) suggests that only a small amount of energy is transferred from the fast waves to Alfvén waves with large k_∥. A similar conclusion emerges from numerical simulations (Cho & Lazarian 2002, 2003). As the fast waves are also expected to be subject to strong collisionless damping and/or to strong dissipation after they steepen into shocks, we eliminate them from our consideration of the problem and concentrate on low-frequency turbulence.

We start by observing that the Alfvén-wave-polarized fluctuations are two-dimensionally solenoidal: since, from Equation (7),

$$\begin{eqnarray} &&{\bm\nabla}\cdot {\bf u}= - {d\over dt}{\delta \rho \over \rho _0} = O(\epsilon ^2) \end{eqnarray} \tag{ 15 }$$

and ${\bm\nabla}\cdot \delta {\bf B}=0$ exactly, separating the O() part of these divergences gives ${\bm\nabla}_{\perp }\cdot {\bf u}_\perp =0$ and ${\bm\nabla}_{\perp }\cdot \delta {\bf B}_\perp =0$. To lowest order in the expansion, we may, therefore, express u_⊥ and δB_⊥ in terms of scalar stream (flux) functions:

$$\begin{eqnarray} &&{\bf u}_\perp = {\bf\hat{z}}\times {\bm\nabla}_{\perp }\Phi,\qquad {\delta {\bf B}_\perp \over \sqrt{4\pi \rho _0}} = {\bf\hat{z}}\times {\bm\nabla}_{\perp }\Psi. \end{eqnarray} \tag{ 16 }$$

Evolution equations for Φ and Ψ are obtained by substituting the expressions (16) into the perpendicular parts of the induction Equation (10) and the momentum Equation (8)—of the latter the curl is taken to annihilate the pressure term. Keeping only the terms of the lowest order, O(²), we get

$$\begin{eqnarray} {\partial \Psi \over \partial t} + \left\lbrace \Phi,\Psi \right\rbrace &=& v_A{\partial \Phi \over \partial z},\\ {\partial \over \partial t}\nabla _{\perp }^2\Phi + \left\lbrace \Phi,\nabla _{\perp }^2\Phi \right\rbrace &=& v_A{\partial \over \partial z}\nabla _{\perp }^2\Psi + \left\lbrace \Psi,\nabla _{\perp }^2\Psi \right\rbrace, \end{eqnarray} \tag{ 17 }$$

where $\left\lbrace \Phi,\Psi \right\rbrace ={\bf\hat{z}}\cdot ({\bm\nabla}_{\perp }\Phi \times {\bm\nabla}_{\perp }\Psi)$ and we have taken into account that, to lowest order,

$$\begin{eqnarray} {d\over dt} &=& {\partial \over \partial t} + {\bf u}_\perp \cdot {\bm\nabla}_{\perp }={\partial \over \partial t} + \left\lbrace \Phi,\cdots \right\rbrace,\\ {\bf\hat{b}}\cdot {\bm\nabla}&=& {\partial \over \partial z}+ {\delta {\bf B}_\perp \over B_0}\cdot {\bm\nabla}_{\perp }= {\partial \over \partial z}+ {1\over v_A}\left\lbrace \Psi,\cdots \right\rbrace. \end{eqnarray} \tag{ 19 }$$

Here ${\bf\hat{b}}={\bf B}/B_0$ is the unit vector along the perturbed field line.

Equations (17)–(18) are known as the reduced magnetohydrodynamics (RMHD). The first derivations of these equations (in the context of fusion plasmas) are due to Kadomtsev & Pogutse (1974) and to Strauss (1976). These were followed by many systematic derivations and generalizations employing various versions and refinements of the basic expansion, taking into account the non-Alfvénic modes (which we will do in Section 2.4), and including the effects of spatial gradients of equilibrium fields (e.g., Strauss 1977; Montgomery 1982; Hazeltine 1983; Zank & Matthaeus 1992; Kinney & McWilliams 1997; Bhattacharjee et al. 1998; Kruger et al. 1998). A comparative review of these expansion schemes and their (often close) relationship to ours is outside the scope of this paper. One important point we wish to emphasize is that we do not assume the plasma beta (defined in Equation (14)) to be either large or small.

Equations (17) and (18) form a closed set, meaning that the Alfvén-wave cascade decouples from the slow waves and density fluctuations. It is to the turbulence described by Equations (17)–(18) that the GS theory outlined in Section 1.2 applies.¹³ In Section 5.3, we will show that Equations (17) and (18) correctly describe inertial-range Alfvénic fluctuations even in a collisionless plasma, where the full MHD description (Equations (7)–(10)) is not valid.

The MHD Equations (7)–(10) in the incompressible limit (ρ = const) acquire a symmetric form if written in terms of the Elsasser fields ${\bf z}^\pm ={\bf u}\pm \delta {\bf B}/\sqrt{4\pi \rho }$ (Elsasser 1950). Let us demonstrate how this symmetry manifests itself in the reduced equations derived above.

We introduce Elsasser potentials ζ^± = Φ ± Ψ, so that ${\bf z}^\pm _\perp ={\bf\hat{z}}\times {\bm\nabla}_{\perp }\zeta ^\pm$. For these potentials, Equations (17)–(18) become

$$\begin{eqnarray} \nonumber {\partial \over \partial t}\nabla _{\perp }^2\zeta ^\pm \mp v_A{\partial \over \partial z}\nabla _{\perp }^2\zeta ^\pm &=& -{1\over 2}\left(\left\lbrace \zeta ^+,\nabla _{\perp }^2\zeta ^-\right\rbrace + \left\lbrace \zeta ^-,\nabla _{\perp }^2\zeta ^+\right\rbrace \right.\\ &&\left.\mp \nabla _{\perp }^2\left\lbrace \zeta ^+,\zeta ^-\right\rbrace \right).\quad \end{eqnarray} \tag{ 21 }$$

These equations show that the RMHD has a simple set of exact solutions: if ζ⁻ = 0 or ζ⁺ = 0, the nonlinear term vanishes and the other, non-zero, Elsasser potential is simply a fluctuation of arbitrary shape and magnitude propagating along the mean field at the Alfvén speed v_A: ζ^± = f^±(x, y, z ∓ v_At). These solutions are finite-amplitude Alfvén-wave packets of arbitrary shape. Only counterpropagating such solutions can interact and thereby give rise to the Alfvén-wave cascade (Kraichnan 1965). Note that these interactions are conservative in the sense that the "+" and "−" waves scatter off each other without exchanging energy.

Note that the individual conservation of the "+" and "−" waves' energies means that the energy fluxes associated with these waves need not be equal, so instead of a single Kolmogorov flux ε assumed in the scaling arguments reviewed in Section 1.2, we could have ε⁺ ≠ ε⁻. The GS theory can be generalized to this case of imbalanced Alfvénic cascades (Lithwick et al. 2007; Beresnyak & Lazarian 2008a; Chandran 2008), but here we will focus on the balanced turbulence, ε⁺ ∼ ε⁻. If one considers the turbulence forced in a physical way (i.e., without forcing the magnetic field, which would break the flux conservation), the resulting cascade would always be balanced. In the real world, imbalanced Alfvénic fluxes are measured in the fast solar wind, where the influence of initial conditions in the solar atmosphere is more pronounced, while the slow-wind turbulence is approximately balanced (Marsch & Tu 1990a; see also reviews by Tu & Marsch 1995; Bruno & Carbone 2005 and references therein).

In order to derive evolution equations for the remaining MHD modes, let us first revisit the perpendicular part of the momentum equation and use Equation (12) to order terms in it. In the lowest order, O(), we get the pressure balance

$$\begin{eqnarray} &&{\bm\nabla}_{\perp }\left(\delta p+ {B_0\delta B_\parallel \over 4\pi }\right) = 0 \Rightarrow {\delta p\over p_0} = -\gamma \,{v_A^2\over c_s^2}{\delta B_\parallel \over B_0}. \end{eqnarray} \tag{ 22 }$$

Using Equation (22) and the entropy Equation (9), we get

$$\begin{eqnarray} &&\hspace{25pt}{d\delta s\over dt} = 0,\quad {\delta s\over s_0} = {\delta p\over p_0} - \gamma {\delta \rho \over \rho _0} = - \gamma \left({\delta \rho \over \rho _0} + {v_A^2\over c_s^2}{\delta B_\parallel \over B_0}\right), \!\!\!\!\!\nonumber \\ \end{eqnarray} \tag{ 23 }$$

where s₀ = p₀/ρ^γ₀. Now, substituting Equation (15) for ${\bm\nabla}\cdot {\bf u}$ in the parallel component of the induction Equation (10), we get

$$\begin{eqnarray} &&{d\over dt}\left({\delta B_\parallel \over B_0} - {\delta \rho \over \rho _0}\right) - {\bf\hat{b}}\cdot {\bm\nabla}u_\parallel = 0. \end{eqnarray} \tag{ 24 }$$

Combining Equations (23) and (24), we obtain

$$\begin{eqnarray} {d\over dt}{\delta \rho \over \rho _0} &=& - {1\over 1+ c_s^2/v_A^2}\,{\bf\hat{b}}\cdot {\bm\nabla}u_\parallel,\\ {d\over dt}{\delta B_\parallel \over B_0} &=& {1\over 1 + v_A^2/c_s^2}\,{\bf\hat{b}}\cdot {\bm\nabla}u_\parallel. \end{eqnarray} \tag{ 25 }$$

Finally, we take the parallel component of the momentum Equation (8) and notice that, due to the pressure balance (22) and to the smallness of the parallel gradients, the pressure term is O(³), while the inertial and tension terms are O(²). Therefore,

$$\begin{eqnarray} &&{du_\parallel \over dt} = v_A^2{\bf\hat{b}}\cdot {\bm\nabla}{\delta B_\parallel \over B_0}. \end{eqnarray} \tag{ 27 }$$

Equations (26)–(27) describe the slow-wave-polarized fluctuations, while Equation (23) describes the zero-frequency entropy mode, which is decoupled from the slow waves.¹⁴ The nonlinearity in Equations (26)–(27) enters via the derivatives defined in Equations (19)–(20) and is due solely to interactions with Alfvén waves. Thus, both the slow-wave and the entropy-mode cascades occur via passive scattering/mixing by Alfvén waves, in the course of which there is no energy exchange between the cascades.

Note that in the high-beta limit, c_s ≫ v_A (see Equation (14)), the entropy mode is dominated by density fluctuations (Equation (23), c_s ≫ v_A), which also decouple from the slow-wave cascade (Equation (25), c_s ≫ v_A). and are passively mixed by the Alfvén-wave turbulence:

$$\begin{eqnarray} &&{d\delta \rho \over dt}=0. \end{eqnarray} \tag{ 28 }$$

The high-beta limit is equivalent to the incompressible approximation for the slow waves.

In Section 5.5, we will derive a kinetic description for the inertial-range compressive fluctuations (density and magnetic-field strength), which is more generally valid in weakly collisional plasmas and which reduces to Equations (26)–(27) in the collisional limit (see Appendix D). While these fluctuations will in general satisfy a kinetic equation, they will remain passive with respect to the Alfvén waves.

The original Elsasser (1950) symmetry was derived for incompressible MHD equations. However, for the "compressive" slow-wave fluctuations, we may introduce generalized Elsasser fields:

$$\begin{eqnarray} &&z_\parallel ^\pm = u_\parallel \pm {\delta B_\parallel \over \sqrt{4\pi \rho _0}}\left(1+{v_A^2\over c_s^2}\right)^{1/2}. \end{eqnarray} \tag{ 29 }$$

Straightforwardly, the evolution equation for these fields is

$$\begin{eqnarray} \nonumber {\partial z_\parallel ^\pm \over \partial t} &\mp & {v_A\over \sqrt{1+v_A^2/c_s^2}}{\partial z_\parallel ^\pm \over \partial z}=-{1\over 2}\left(1\mp {1\over \sqrt{1+v_A^2/c_s^2}}\right)\bigl \lbrace \zeta ^+,z_\parallel ^\pm \bigr \rbrace \\ &&{-}\,{1\over 2}\left(1\pm {1\over \sqrt{1+v_A^2/c_s^2}}\right)\bigl \lbrace \zeta ^-,z_\parallel ^\pm \bigr \rbrace. \end{eqnarray} \tag{ 30 }$$

In the high-beta limit (v_A ≪ c_s), the generalized Elsasser fields (29) become the parallel components of the conventional incompressible Elsasser fields. We see that only in this limit do the slow waves interact exclusively with the counterpropagating Alfvén waves, and so only in this limit does setting ζ⁻ = 0 or ζ⁺ = 0 gives rise to finite-amplitude slow-wave-packet solutions z^±_∥ = f^±(x, y, z ∓ v_At) analogous to the finite-amplitude Alfvén-wave packets discussed in Section 2.3.¹⁵ For general β, the phase speed of the slow waves is smaller than that of the Alfvén waves and, therefore, Alfvén waves can "catch up" and interact with the slow waves that travel in the same direction. All of these interactions are of scattering type and involve no exchange of energy.

The scaling of the passively mixed scalar fields introduced above is slaved to the scaling of the Alfvénic fluctuations. Consider for example the entropy mode (Equation (23)). As in Kolmogorov–Obukhov theory (see Section 1.1), one assumes a local-in-scale-space cascade of scalar variance and a constant flux ε_s of this variance. Then, analogously to Equation (1),

$$\begin{eqnarray} &&{v_{{\rm th}i}^2\over s_0^2}{\delta s_{\lambda }^2\over \tau _\lambda }\sim \varepsilon _{s}. \end{eqnarray} \tag{ 31 }$$

Since the cascade time is $\tau _\lambda ^{-1}\sim {\bf u}_\perp \cdot {\bm\nabla}_{\perp }\sim v_A/l_{\parallel \lambda }\sim \varepsilon /u_{\perp \lambda }^2$,

$$\begin{eqnarray} &&{\delta s_{\lambda }\over s_0}\sim \left(\varepsilon _{s}\over \varepsilon \right)^{1/2}{u_{\perp \lambda }\over v_{{\rm th}i}}, \end{eqnarray} \tag{ 32 }$$

so the scalar fluctuations have the same scaling as the turbulence that mixes them (Obukhov 1949; Corrsin 1951). In GS turbulence, the scalar-variance spectrum should, therefore, be k^−5/3_⊥ (Lithwick & Goldreich 2001). The same argument applies to all passive fields.

It is the (presumably) passive electron-density spectrum that provides the main evidence of the k^−5/3 scaling in the interstellar turbulence (Armstrong et al. 1981, 1995; Lazio et al. 2004, see further discussion in Section 8.4.1). The explanation of this spectrum in terms of passive mixing of the entropy mode, originally proposed by Higdon (1984), was developed on the basis of the GS theory by Lithwick & Goldreich (2001). The turbulent cascade of the compressive fluctuations and the relevant solar-wind data are discussed further in Section 6.3. In particular, it will emerge that the anisotropy of these fluctuations remains a non-trivial issue: is there an analog of the scaling relation (5)? The scaling argument outlined above does not invoke any assumptions about the relationship between the parallel and perpendicular scales of the compressive fluctuations (other than the assumption that they are anisotropic). Lithwick & Goldreich (2001) argue that the parallel scales of the Alfvénic fluctuations will imprint themselves on the passively advected compressive ones, so Equation (5) holds for the latter as well. In Section 6.3, we examine this conclusion in view of the solar-wind evidence and of the fact that the equations for the compressive modes become linear in the Lagrangian frame associated with the Alfvénic turbulence.

Thus, the anisotropy and critical balance (3) taken as ordering assumptions lead to a neat decomposition of the MHD turbulent cascade into a decoupled Alfvén-wave cascade and cascades of slow waves and entropy fluctuations passively scattered/mixed by the Alfvén waves. More precisely, Equations (23), (21), and (30) imply that, for arbitrary β, there are five conserved quantities:¹⁶

$$\begin{eqnarray} W_{\rm AW}^\pm &=& {1\over 2}\int d^3{\bf r}\,\rho _0 |{\bm\nabla}_{\perp }\zeta ^\pm |^2 \qquad {\rm (Alfven\ waves),}\\ W_{\rm sw}^\pm &=& {1\over 2}\int d^3{\bf r}\,\rho _0 |z_\parallel ^\pm |^2 \qquad \quad {\rm (slow\ waves),}\\ W_s&=& {1\over 2}\int d^3{\bf r}\,{\delta s^2\over s_0^2} \qquad \ \,{\rm (entropy\ fluctuations).} \end{eqnarray} \tag{ 33 }$$

W⁺_AW and W⁻_AW are always cascaded by interaction with each other, W_s is passively mixed by W⁺_AW and W⁻_AW, W^±_sw are passively scattered by W^∓_AW and, unless β ≫ 1, also by W^±_AW.

This is an example of splitting of the overall energy cascade into several channels (recovered as a particular case of the more general kinetic cascade in Appendix D.2)—a concept that will repeatedly arise in the kinetic treatment to follow.

The decoupling of the slow- and Alfvén-wave cascades in MHD turbulence was studied in some detail and confirmed in direct numerical simulations by Maron & Goldreich (2001, for β ≫ 1) and by Cho & Lazarian (2002, 2003, for a range of values of β). The derivation given in Sections 2.2 and 2.4 (cf. Lithwick & Goldreich 2001) provides a straightforward theoretical basis for these results, assuming anisotropy of the turbulence (which was also confirmed in these numerical studies).

It turns out that the decoupling of the Alfvén-wave cascade that we demonstrated above for the anisotropic MHD turbulence is a uniformly valid property of plasma turbulence at both collisional and collisionless scales and that this cascade is correctly described by the RMHD equations (17)–(18) all the way down to the ion gyroscale, while the fluctuations of density and magnetic-field strength do not satisfy simple fluid evolution equations anymore and require solving the kinetic equation. In order to prove this, we adopt a kinetic description and apply to it the same ordering (Section 2.1) as we used to reduce the MHD equations. The kinetic theory that emerges as a result is called gyrokinetics.

As in Section 2 we set up a static equilibrium with a uniform mean field, ${\bf B}_0=B_0{\bf\hat{z}}$, E₀ = 0, assume that the perturbations will be anisotropic with k_∥ ≪ k_⊥ (at scales smaller than the outer scale, k_∥L ≫ 1; see Sections 1.3 and 1.5.1), and construct an expansion of the kinetic theory around this equilibrium with respect to the small parameter ∼ k_∥/k_⊥. We adopt the ordering expressed by Equations (3) and (12), i.e., we assume the perturbations to be strongly interacting Alfvén waves plus electron density and magnetic-field-strength fluctuations.

Besides , several other dimensionless parameters are present, all of which are formally considered to be of order unity in the gyrokinetic expansion: the electron–ion mass ratio m_e/m_i, the charge ratio

$$\begin{eqnarray} &&Z={q_i/|q_e|}={q_i/e} \end{eqnarray} \tag{ 40 }$$

(for hydrogen, this is 1, which applies to most astrophysical plasmas of interest to us), the temperature ratio¹⁷

$$\begin{eqnarray} &&\tau ={T_i/ T_e}, \end{eqnarray} \tag{ 41 }$$

and the plasma (ion) beta

$$\begin{eqnarray} &&\beta _i={v_{{\rm th}i}^2\over v_A^2}={8\pi n_iT_i\over B_0^2} = \beta \left(1+{Z\over \tau }\right)^{-1}, \end{eqnarray} \tag{ 42 }$$

where v_thi = (2T_i/m_i)^1/2 is the ion thermal speed and the total β was defined in Equation (14) based on the total pressure p = n_iT_i + n_eT_e. We shall occasionally also use the electron beta

$$\begin{eqnarray} &&\beta _e={8\pi n_eT_e\over B_0^2}={Z\over \tau }\,\beta _i. \end{eqnarray} \tag{ 43 }$$

The total beta is β = β_i + β_e.

3.1.1. Wave Numbers and Frequencies

As we want our theory to be uniformly valid at all (perpendicular) scales above, at or below the ion gyroscale, we order

$$\begin{eqnarray} &&k_{\perp }\rho _i \sim 1, \end{eqnarray} \tag{ 44 }$$

where ρ_i = v_thi/Ω_i is the ion gyroradius, Ω_i = q_iB₀/cm_i the ion cyclotron frequency. Note that

$$\begin{eqnarray} &&\rho _e = {Z\over \sqrt{\tau }}\sqrt{{m_e\over m_i}}\,\rho _i. \end{eqnarray} \tag{ 45 }$$

Assuming Alfvénic frequencies implies

$$\begin{eqnarray} &&{\omega \over \Omega _i}\sim {k_{\parallel }v_A\over \Omega _i}\sim {k_{\perp }\rho _i\over \sqrt{\beta _i}}\,\epsilon. \end{eqnarray} \tag{ 46 }$$

Thus, gyrokinetics is a low-frequency limit that averages over the timescales associated with the particle gyration. Because we have assumed that the fluctuations are anisotropic and have (by order of magnitude) Alfvénic frequencies, we see from Equation (46) that their frequency remains far below Ω_i at all scales, including the ion and even electron gyroscale—the gyrokinetics remains valid at all of these scales and the cyclotron-frequency effects are negligible (cf. Quataert & Gruzinov 1999).

3.1.2. Fluctuations

Equation (3) allows us to order the fluctuations of the scalar potential: on the one hand, we have from Equation (3) u_⊥ ∼ v_A; on the other hand, the plasma mass flow velocity is (to the lowest order) the E × B drift velocity of the ions, u_⊥ ∼ cE_⊥/B₀ ∼ ck_⊥φ/B₀, so

$$\begin{eqnarray} &&{e\varphi \over T_e} \sim {\tau \over Z}{1\over k_{\perp }\rho _i\sqrt{\beta _i}}\,\epsilon. \end{eqnarray} \tag{ 47 }$$

All other fluctuations (magnetic, density, parallel velocity) are ordered according to Equation (12).

Note that the ordering of the flow velocity dictated by Equation (3) means that we are considering the limit of small Mach numbers:

$$\begin{eqnarray} &&M\sim {u\over v_{{\rm th}i}}\sim {\epsilon \over \sqrt{\beta _i}}. \end{eqnarray} \tag{ 48 }$$

This means that the gyrokinetic description in the form used below does not extend to large sonic flows that can be present in many astrophysical systems. It is, in principle, possible to extend the gyrokinetics to systems with sonic flows (e.g., in the toroidal geometry; see Artun & Tang 1994; Sugama & Horton 1997). However, we do not follow this route because such flows belong to the same class of non-universal outer-scale features as background density and temperature gradients, system-specific geometry etc.—these can all be ignored at small scales, where the turbulence should be approximately homogeneous and subsonic (as long as k_∥L ≫ 1, see discussion in Section 1.5.1).

3.1.3. Collisions

Finally, we want our theory to be valid both in the collisional and the collisionless regimes, so we do not assume ω to be either smaller or larger than the (ion) collision frequency ν_ii:

$$\begin{eqnarray} &&{\omega \over \nu _{ii}}\sim {k_{\parallel }\lambda _{{\rm mfp}i}\over \sqrt{\beta _i}} \sim 1, \end{eqnarray} \tag{ 49 }$$

where λ_mfpi = v_thi/ν_ii is the ion mean free path (this ordering can actually be inferred from equating the gyrokinetic entropy production terms to the collisional entropy production; see extended discussion in Howes et al. 2006). Note that the ordering (49) holds on the understanding that we have ordered k_⊥ρ_i ∼ 1 (Equation (44)) because the fluctuation frequency can depend on k_⊥ρ_i in the dissipation range (see Section 7.3).

Other collision rates are related to ν_ii via a set of standard formulae (see, e.g., Helander & Sigmar 2002), which will be useful in what follows:

$$\begin{eqnarray} \nu _{ei}&=& Z\nu _{ee}= {\tau ^{3/2}\over Z^2}\sqrt{m_i\over m_e}\,\nu _{ii},\\ \nu _{ie}&=& {8\over 3\sqrt{\pi }}{\tau ^{3/2}\over Z}\sqrt{m_e\over m_i}\,\nu _{ii},\\ \nu _{ii}&=& {\sqrt{2}\pi Z^4 e^4 n_i \ln \Lambda \over m_i^{1/2} T_i^{3/2}}, \end{eqnarray} \tag{ 50 }$$

where ln Λ is the Coulomb logarithm and the numerical factor in the definition of ν_ie has been inserted for future notational convenience (see Appendix A). We always define

$$\begin{eqnarray} &&\lambda _{{\rm mfp}i}={v_{{\rm th}i}\over \nu _{ii}},\quad \lambda _{{\rm mfp}e}={v_{{\rm th}e}\over \nu _{ei}} = \left({Z\over \tau }\right)^2\lambda _{{\rm mfp}i}. \end{eqnarray} \tag{ 53 }$$

The ordering of the collision frequency expressed by Equation (49) means that collisions, while not dominant as in the fluid description (Appendix A), are still retained in the version of the gyrokinetic theory adopted by us. Their presence is required in order for us to be able to assume that the equilibrium distribution is Maxwellian (Equation (54) below) and for the heating and entropy production to be treated correctly (Sections 3.4 and 3.5). However, our ordering of collisions and of the fluctuation amplitudes (Section 3.1.2) imposes certain limitations: thus, we cannot treat the class of nonlinear phenomena involving particle trapping by parallel-varying fluctuations, non-Maxwellian tails of particle distributions, plasma instabilities arising from the equilibrium pressure anisotropies (mirror, firehose) and their possible nonlinear evolution to large amplitudes (see discussion in Section 8.3).

The region of validity of the gyrokinetic approximation in the wavenumber space is illustrated in Figure 3—it embraces all of the scales that are expected to be traversed by the anisotropic energy cascade (except the scales close to the outer scale).

As we explained above, m_e/m_i, β_i, k_⊥ρ_i and k_∥λ_mfpi (or ω/ν_ii) are assigned order unity in the gyrokinetic expansion. Subsidiary expansions in small m_e/m_i (Section 4) and in small or large values of the other three parameters (Sections 5–7) can be carried out at a later stage as long as their values are not so large or small as to interfere with the primary expansion in . These expansions will yield simpler models of turbulence with more restricted domains of validity than gyrokinetics.

Given the gyrokinetic ordering introduced above, the expansion of the distribution function up to first order in can be written as

$$\begin{eqnarray} &&\hspace{-24pt}f_s(t,{\bf r},{\bf v}) = F_{0s}(v) - {q_s\varphi (t,{\bf r})\over T_{0s}}F_{0s}(v)+h_s(t,{\bf R}_s,v_\perp,v_\parallel). \end{eqnarray} \tag{ 54 }$$

To zeroth order, it is a Maxwellian:¹⁸

$$\begin{eqnarray} &&\hspace{-20pt}F_{0s}(v) = {n_{0s}\over (\pi v_{{\rm th}s}^2)^{3/2}}\exp \left(-{v^2\over v_{{\rm th}s}^2}\right), \quad v_{{\rm th}s}=\sqrt{2T_{0s}\over m_s},\quad \end{eqnarray} \tag{ 55 }$$

with uniform density n_0s and temperature T_0s and no mean flow. As will be explained in more detail in Section 3.5, F_0s has a slow time dependence via the equilibrium temperature, T_0s = T_0s(²t). This reflects the slow heating of the plasma as the turbulent energy is dissipated. However, T_0s can be treated as a constant with respect to the time dependence of the first-order distribution function (the timescale of the turbulent fluctuations). The first-order part of the distribution function is composed of the Boltzmann response (second term in Equation (54), ordered in Equation (47)) and the gyrocenter distribution function h_s. The spatial dependence of the latter is expressed not by the particle position r but by the position R_s of the particle gyrocenter (or guiding center)—the center of the ring orbit that the particle follows in a strong guide field:

$$\begin{eqnarray} &&{\bf R}_s = {\bf r}+ {{\bf v}_\perp \times {\bf\hat{z}}\over \Omega _s}. \end{eqnarray} \tag{ 56 }$$

Thus, some of the velocity dependence of the distribution function is subsumed in the R_s dependence of h_s. Explicitly, h_s depends only on two velocity-space variables: it is customary in the gyrokinetic literature for these to be chosen as the particle energy ε_s = m_sv²/2 and its first adiabatic invariant μ_s = m_sv²_⊥/2B₀ (both conserved quantities to two lowest orders in the gyrokinetic expansion). However, in a straight uniform guide field $B_0{\bf\hat{z}}$, the pair (v_⊥, v_∥) is a simpler choice, which will mostly be used in what follows (we shall sometimes find an alternative pair, v and ξ = v_∥/v, useful, especially where collisions are concerned). It must be constantly kept in mind that derivatives of h_s with respect to the velocity-space variables are taken at constant R_s, not at constant r.

The function h_s satisfies the gyrokinetic equation:

$$\begin{eqnarray} &&{\partial h_s\over \partial t} + v_\parallel {\partial h_s\over \partial z} + {c\over B_0}\big\lbrace \langle \chi \rangle _{{\bf R}_s},h_s\big\rbrace={q_sF_{0s}\over T_{0s}}\,{\partial \langle \chi \rangle _{{\bf R}_s}\over \partial t} + \left({\partial h_s\over \partial t}\right)_{\rm c},\nonumber\\ \end{eqnarray} \tag{ 57 }$$

where

$$\begin{eqnarray} &&\chi (t,{\bf r},{\bf v}) = \varphi - {v_\parallel A_\parallel \over c} - {{\bf v}_\perp \cdot {\bf A}_\perp \over c}, \end{eqnarray} \tag{ 58 }$$

the Poisson brackets are defined in the usual way:

$$\begin{eqnarray} &&\lbrace \langle \chi \rangle _{{\bf R}_s},h_s\rbrace = {\bf\hat{z}}\cdot \left({\partial \langle \chi \rangle _{{\bf R}_s}\over \partial {\bf R}_s}\times {\partial h_s\over \partial {\bf R}_s}\right), \end{eqnarray} \tag{ 59 }$$

and the ring average notation is introduced:

$$\begin{eqnarray} &&\hspace{-10pt}\langle \chi (t,{\bf r},{\bf v})\rangle _{{\bf R}_s} = {1\over 2\pi }\int _0^{2\pi }d\vartheta \, \chi \left(t,{\bf R}_s-{{\bf v}_\perp \times {\bf\hat{z}}\over \Omega _s},{\bf v}\right), \end{eqnarray} \tag{ 60 }$$

where ϑ is the angle in the velocity space taken in the plane perpendicular to the guide field $B_0{\bf\hat{z}}$. Note that, while χ is a function of r, its ring average is a function of R_s. Note also that the ring averages depend on the species index, as does the gyrocenter variable R_s. Equation (57) is derived by transforming the first-order kinetic equation to the gyrocenter variable (56) and ring averaging the result (see Howes et al. 2006, or the references given at the beginning of Section 3). The ring-averaged collision integral (∂h_s/∂t)_c is discussed in Appendix B.

To Equation (57), we must append the equations that determine the electromagnetic field, namely, the potentials φ(t, r) and A(t, r) that enter the expression for χ (Equation (58)). In the non-relativistic limit (v_thi ≪ c), these are the plasma quasi-neutrality constraint (which follows from the Poisson Equation (38) to lowest order in v_thi/c):

$$\begin{eqnarray} &&0 = \sum _s q_s\delta n_s = \sum _s q_s\left[-{q_s\varphi \over T_{0s}}n_{0s}+ \int d^3{\bf v}\langle h_s\rangle _{\bf r}\right] \end{eqnarray} \tag{ 61 }$$

and the parallel and perpendicular parts of Ampère's law (Equation (39) to lowest order in and in v_thi/c):

$$\begin{eqnarray} \nabla _{\perp }^2A_\parallel &=& - {4\pi \over c}\,j_\parallel = -{4\pi \over c}\sum _s q_s\int d^3{\bf v}\,v_\parallel \langle h_s\rangle _{\bf r},\\ \nabla _{\perp }^2\delta B_\parallel &=& {-}\,{4\pi \over c}\,{\bf\hat{z}}\cdot ({\bm\nabla}_{\perp }\times {\bf j}_\perp)\nonumber\\ &=& {-}\,{4\pi \over c}\,{\bf\hat{z}}\cdot \left[{\bm\nabla}_{\perp }\times \sum _s q_s\int d^3{\bf v}\langle {\bf v}_\perp h_s\rangle _{\bf r}\right], \end{eqnarray} \tag{ 62 }$$

where we have used $\delta B_\parallel = {\bf\hat{z}}\cdot \left({\bm\nabla}_{\perp }\times {\bf A}_\perp \right)$ and dropped the displacement current. Since field variables φ, A_∥ and δB_∥ are functions of the spatial variable r, not of the gyrocenter variable R_s, we had to determine the contribution from the gyrocenter distribution function h_s to the charge distribution at fixed r by performing a gyroaveraging operation dual to the ring average defined in Equation (60):

$$\begin{eqnarray} &&\langle h_s(t,{\bf R}_s,v_\perp,v_\parallel)\rangle _{{\bf r}}={1\over 2\pi }\!\int _0^{2\pi }\!\!d\vartheta \, h_s\left(t,{\bf r}+{{\bf v}_\perp \times {\bf\hat{z}}\over \Omega _s},v_\perp,v_\parallel \right)\!.\nonumber\\ \end{eqnarray} \tag{ 64 }$$

In other words, the velocity-space integrals in Equations (61)–(63) are performed over h_s at constant r, rather than constant R_s. If we Fourier transform h_s in R_s, the gyroaveraging operation takes a simple mathematical form:

$$\begin{eqnarray} \nonumber \langle h_s\rangle _{\bf r}&=& \sum _{{\bf k}}\langle e^{i{{\bf k}}\cdot {\bf R}_s}\rangle _{\bf r}h_{s{{\bf k}}}(t,v_\perp,v_\parallel)\\ \nonumber &=& \sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf r}} \left<\exp \left(i{{\bf k}}\cdot {{\bf v}_\perp \times {\bf\hat{z}}\over \Omega _s}\right)\right>_{\bf r}h_{s{{\bf k}}}(t,v_\perp,v_\parallel) \\ &=&\sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf r}}J_0(a_s)h_{s{{\bf k}}}(t,v_\perp,v_\parallel), \end{eqnarray} \tag{ 65 }$$

where a_s = k_⊥v_⊥/Ω_s and J₀ is a Bessel function that arose from the angle integral in the velocity space. In Equation (63), an analogous calculation taking into account the angular dependence of v_⊥ leads to

$$\begin{eqnarray} &&\delta B_\parallel = -{4\pi \over B_0}\sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf r}} \sum _s\int d^3{\bf v}\,m_sv_\perp ^2{J_1(a_s)\over a_s}\,h_{s{{\bf k}}}(t,v_\perp,v_\parallel).\nonumber\\ \end{eqnarray} \tag{ 66 }$$

Note that Equation (63) (and, therefore, Equation (66)) is the gyrokinetic equivalent of the perpendicular pressure balance that appeared in Section 2 (Equation (22)):

$$\begin{eqnarray} &&\nabla _{\perp }^2{B_0\delta B_\parallel \over 4\pi } = {\bm\nabla}_{\perp }\cdot \sum _s {q_sB_0\over c}\int d^3{\bf v}\langle {{\bf\hat{z}}\times {\bf v}_\perp }h_s\rangle _{\bf r}\nonumber\\ &&\quad= {\bm\nabla}_{\perp }\cdot \sum _s\Omega _s m_s\int d^3{\bf v}\,{\partial {\bf v}_\perp \over \partial \vartheta }\, h_s\left(t,{\bf r}+ {{\bf v}_\perp \times {\bf\hat{z}}\over \Omega _s},v_\perp,v_\parallel \right)\nonumber\\ &&\quad= -{\bm\nabla}_{\perp }{\bm\nabla}_{\perp }:\sum _s\int d^3{\bf v}\,m_s\langle {\bf v}_\perp {\bf v}_\perp \,h_s\rangle _{\bf r}= -{\bm\nabla}_{\perp }{\bm\nabla}_{\perp }:\delta {\bf P}_\perp,\nonumber\\ \end{eqnarray} \tag{ 67 }$$

where we have integrated by parts with respect to the gyroangle ϑ and used $\partial {\bf v}_\perp /\partial \vartheta = {\bf\hat{z}}\times {\bf v}_\perp$, ∂²v_⊥/∂ϑ² = −v_⊥ (cf. the Appendix of Roach et al. 2005).

Once the fields are determined, they have to be substituted into χ (Equation (58)) and the result ring averaged (Equation (60)). Again, we emphasize that φ, A_∥ and δB_∥ are functions of r, while $\langle \chi \rangle _{{\bf R}_s}$ is a function of R_s. The transformation is accomplished via a calculation analogous to the one that led to Equations (65) and (66):

$$\begin{eqnarray} \langle \chi \rangle _{{\bf R}_s}&=& \sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf R}_s}\langle \chi \rangle _{{\bf R}_s,{{\bf k}}},\\ \langle \chi \rangle _{{\bf R}_s,{{\bf k}}}&=& J_0(a_s)\left(\varphi _{{\bf k}}- {v_\parallel A_{\parallel {{\bf k}}}\over c}\right) + {T_{0s}\over q_s}{2v_\perp ^2\over v_{{\rm th}s}^2}{J_1(a_s)\over a_s}{\delta B_{\parallel {{\bf k}}}\over B_0}. \nonumber \\ \end{eqnarray} \tag{ 68 }$$

The last equation establishes a correspondence between the Fourier transforms of the fields with respect to r and the Fourier transform of $\langle \chi \rangle _{{\bf R}_s}$ with respect to R_s.

As promised in Section 1.4, the central unifying concept of this paper is now introduced.

If we multiply the gyrokinetic Equation (57) by T_0sh_s/F_0s and integrate over the velocities and gyrocenters, we find that the nonlinear term conserves the variance of h_s and

$$\begin{eqnarray} &&{d\over dt}\int d^3{\bf v}\int d^3{\bf R}_s\,{T_{0s}h_s^2\over 2F_{0s}} = \int d^3{\bf v}\int d^3{\bf R}_s\,q_s\,{\partial \langle \chi \rangle _{{\bf R}_s}\over \partial t}\,h_s\nonumber\\ &&\quad{+}\,\int d^3{\bf v}\int d^3{\bf R}_s\,{T_{0s}h_s\over F_{0s}}\left({\partial h_s\over \partial t}\right)_{\rm c}.\quad \end{eqnarray} \tag{ 70 }$$

Let us now sum this equation over all species. The first term on the right-hand side is

$$\begin{eqnarray} &&\sum _sq_s\int d^3{\bf v}\int d^3{\bf R}_s\,{\partial \langle \chi \rangle _{{\bf R}_s}\over \partial t}\,h_s\nonumber\\ &&\quad{=}\,\int d^3{\bf r}\sum _sq_s\int d^3{\bf v}\left<{\partial \chi \over \partial t}\,h_s\right>_{\bf r}\nonumber\\ &&\quad{=}\,\int d^3{\bf r}\left[{\partial \varphi \over \partial t}\sum _sq_s\int d^3{\bf v}\langle h_s\rangle _{\bf r}\right.\nonumber\\ &&\qquad\left.- {1\over c}{\partial {\bf A}\over \partial t}\cdot\sum _sq_s\int d^3{\bf v}\langle {\bf v}\,h_s\rangle _{\bf r}\right]\nonumber\\ &&\quad{=}\,{d\over dt}\int d^3{\bf r}\sum _s{q_s^2\varphi ^2n_{0s}\over 2T_{0s}} + \int d^3{\bf r}\,{\bf E}\cdot {\bf j}, \end{eqnarray} \tag{ 71 }$$

where we have used Equation (61) and Ampère's law (Equations (62)–(63)) to express the integrals of h_s. The second term on the right-hand side is the total work done on plasma per unit time. Using Faraday's law (Equation (37)) and Ampère's law (Equation (39)), it can be written as

$$\begin{eqnarray} &&\int d^3{\bf r}\,{\bf E}\cdot {\bf j}= - {d\over dt}\int d^3{\bf r}\,{|\delta {\bf B}|^2\over 8\pi } + P_{\rm ext}, \end{eqnarray} \tag{ 72 }$$

where P_ext ≡ −∫d³r E · j_ext is the total power injected into the system by the external energy sources (outer-scale stirring; in terms of the Kolmogorov energy flux ε used in the scaling arguments in Section 1.2, P_ext = Vm_in_0iε, where V is the system volume). Combining Equations (70)–(72), we find (Howes et al. 2006)

$$\begin{eqnarray} {dW\over dt}&\equiv & {d\over dt}\int\! d^3{\bf r}\left[\!\sum _s\!\left(\int d^3{\bf v}\,{T_{0s}\langle h_s^2\rangle _{\bf r}\over 2F_{0s}}\,{-}\,{q_s^2\varphi ^2n_{0s}\over 2T_{0s}}\right){+}\,{|\delta {\bf B}|^2\over 8\pi }\right]\nonumber\\ &=& P_{\rm ext}+ \sum _s\int d^3{\bf v}\int d^3{\bf R}_s\,{T_{0s}h_s\over F_{0s}}\left({\partial h_s\over \partial t}\right)_{\rm c}. \end{eqnarray} \tag{ 73 }$$

W is a positive definite quantity—this becomes explicit if we use Equation (61) to express it in terms of the total perturbed distribution function δf_s = −q_sφF_0s/T_0s + h_s (see Equation (54)):

$$\begin{eqnarray} &&W = \int d^3{\bf r}\left(\sum _s\int d^3{\bf v}\,{T_{0s}\delta f_s^2\over 2F_{0s}} + {|\delta {\bf B}|^2\over 8\pi }\right). \end{eqnarray} \tag{ 74 }$$

We will refer to W as the generalized energy. We use this term to emphasize the role of W as the cascaded quantity in gyrokinetic turbulence (see below). This quantity is, in fact, the gyrokinetic version of a collisionless kinetic invariant variously referred to as the generalized grand canonical potential (see Hallatschek 2004, who points out the fundamental role of this quantity in plasma turbulence simulations) or free energy (e.g., Fowler 1968; Scott 2007). The non-magnetic part of W is related to the perturbed entropy of the system (Krommes & Hu 1994; Sugama et al. 1996; Howes et al. 2006; Schekochihin et al. 2008b, see discussion in Section 3.5).¹⁹

Equation (73) is a conservation law of the generalized energy: P_ext is the source and the second term on the right-hand side, which is negative definite, represents collisional dissipation. This suggests that we might think of kinetic plasma turbulence in terms of the generalized energy W injected by the outer-scale stirring and dissipated by collisions. In order for the dissipation to be important, the collisional term in Equation (73) has to become comparable to P_ext. This can happen in two ways:

• 1.  
  At collisional scales (k_∥λ_mfpi ∼ 1) due to deviations of the perturbed distribution function from a local perturbed Maxwellian (see Section 6.1 and Appendix D);
• 2.  
  At collisionless scales (k_∥λ_mfpi ≫ 1) due the development of small scales in the velocity space—large gradients in v_∥ (see Section 6.2.4) or v_⊥ (which is accompanied by the development of small perpendicular scales in the position space; see Section 7.9.1).

Thus, the dissipation is only important at particular (small) scales, which are generally well separated from the outer scale. The generalized energy is transferred from the outer scale to the dissipation scales via a nonlinear cascade. We shall call it the kinetic cascade. It is analogous to the energy cascade in fluid or MHD turbulence, but a conceptually new feature is present: the small scales at which dissipation happens are small scales both in the velocity and position space. Whereas the large gradients in v_∥ are produced by the linear parallel phase mixing, whose role in the kinetic dissipation processes has been appreciated for some time (Landau 1946; Hammett et al. 1991; Krommes & Hu 1994; Krommes 1999; Watanabe & Sugama 2004, see Section 6.2.4), the emergence of large gradients in v_⊥ is due to an essentially nonlinear phase mixing mechanism (Section 7.9.1). At spatial scales smaller than the ion gyroradius, this nonlinear perpendicular phase mixing turns out to be a faster and, therefore, presumably the dominant way of generating small-scale structure in the velocity space. It was anticipated in the development of gyrofluid moment hierarchies by Dorland & Hammett (1993). Here we treat it for the first time as a phase-space turbulent cascade: this is done in Sections 7.9 and 7.10 (see also Schekochihin et al. 2008b).

In the sections that follow, we shall derive particular forms of W for various limiting cases of the gyrokinetic theory (Sections 4.7, 5.6, 6.2.5, 7.8, Appendices D.2 and E.2). We shall see that the kinetic cascade of W is, indeed, a direct generalization of the more familiar fluid cascades (such as the RMHD cascades discussed in Section 2) and that W contains the energy invariants of the fluid models in the appropriate limits. In these limits, the cascade of the generalized energy will split into several decoupled cascades, as it did in the case of RMHD (Section 2.7). Whenever one of the physically important scales (Section 1.5.2) is crossed and a change of physical regime occurs, these cascades are mixed back together into the overall kinetic cascade of W, which can then be split in a different way as it emerges on the "opposite side" of the transition region in the scale space. The conversion of the Alfvénic cascade into the KAW cascade and the entropy cascade at k_⊥ρ_i ∼ 1 is the most interesting example of such a transition, discussed in Section 7.

The generalized energy appears to be the only quadratic invariant of gyrokinetics in three dimensions; in two dimensions, many other invariants appear (see Appendix F).

In a stationary state, all of the turbulent power injected by the external stirring is dissipated and thus transferred into heat. Mathematically, this is expressed as a slow increase in the temperature of the Maxwellian equilibrium. In gyrokinetics, the heating timescale is ordered as ∼(²ω)⁻¹.

Even though the dissipation of turbulent fluctuations may be occurring "collisionlessly" at scales such that k_∥λ_mfpi ≫ 1 (e.g., via wave–particle interaction at the ion gyroscale; Section 7.1), the resulting heating must ultimately be effected with the help of collisions. This is because heating is an irreversible process and it is a small amount of collisions that make "collisionless" damping irreversible. In other words, slow heating of the Maxwellian equilibrium is equivalent to entropy production and Boltzmann's H-theorem rigorously requires collisions to make this possible. Indeed, the total entropy of species s is

$$\begin{eqnarray} S_s&=&{-}\,\int d^3{\bf r}\int d^3{\bf v}\,f_s\ln f_s\nonumber\\ &=&{-}\,\int d^3{\bf r}\int d^3{\bf v}\left(F_{0s}\ln F_{0s}\,{+}\,{\delta f_s^2\over 2F_{0s}}\right)\,{+}\,O(\epsilon ^3), \end{eqnarray} \tag{ 75 }$$

where we took ∫d³r δf_s = 0. It is then not hard to show that

$$\begin{eqnarray} &&{3\over 2}\,Vn_{0s}\,{1\over T_{0s}}{dT_{0s}\over dt}\,{=}\,\overline{dS_s\over dt}\,{=}\,{-}\,\overline{\int d^3{\bf v}\int d^3{\bf R}_s\,{T_{0s}h_s\over F_{0s}}\left({\partial h_s\over \partial t}\right)_{\rm c}},\nonumber\\ \end{eqnarray} \tag{ 76 }$$

where the overlines mean averaging over times longer than the characteristic time of the turbulent fluctuations ∼ω⁻¹ but shorter than the typical heating time ∼(²ω)⁻¹ (see Howes et al. 2006; Schekochihin et al. 2008b for a detailed derivation of this and related results on heating in gyrokinetics; see also earlier discussions of the entropy production in gyrokinetics by Krommes & Hu 1994; Krommes 1999; Sugama et al. 1996). We have omitted the term describing the interspecies collisional temperature equalization. Note that both sides of Equation (76) are order ²ω.

If we now time average Equation (73) in a similar fashion, the left-hand side vanishes because it is a time derivative of a quantity fluctuating on the timescale ∼ω⁻¹ and we confirm that the right-hand side of Equation (76) is simply equal to the average power $\overline{P_{\rm ext}}$ injected by external stirring. The import of Equation (76) is that it tells us that heating can only be effected by collisions, while Equation (73) implies that the injected power gets to the collisional scales in velocity and position space by means of a kinetic cascade of generalized energy.

The first term in the expression for the generalized energy (74) is −∑_sT_0sδS_s, where δS_s is the perturbed entropy (see Equation (75)). The second term in Equation (74) is magnetic energy. Collisionless damping of electromagnetic fluctuations can be thought of as a redistribution of the generalized energy, transferring the electromagnetic energy into entropy fluctuations, while the total W is conserved (a simple example of how that happens for collisionless compressive fluctuations in the inertial range is worked out in Section 6.2.3).

The contribution to the perturbed entropy from the gyrocenter distribution is the integral of −h²_s/2F_0s, whose evolution Equation (70) can be viewed as the gyrokinetic version of the H-theorem. The first term on the right-hand side of this equation represents the wave–particle interaction (collisionless damping). Under time average, it is related to the work done on plasma (Equation (71)) and hence to the average externally injected power $\overline{P_{\rm ext}}$ via time-averaged Equation (72).²⁰ In a stationary state, this is balanced by the second term in the right-hand side of Equation (70), which is the collisional-heating, or entropy-production, term that also appears in Equation (76). Thus, the generalized energy channeled by collisionless damping into entropy fluctuations is eventually converted into heat by collisions. The sub-gyroscale entropy cascade, which brings the perturbed distribution function h_s to collisional scales, will be discussed further in Sections 7.9 and 7.10 (see also Schekochihin et al. 2008b).

This concludes a short primer on gyrokinetics necessary (and sufficient) for adequate understanding of what is to follow. Formally, all further analytical derivations in this paper are simply subsidiary expansions of the gyrokinetics in the parameters we listed in Section 3.1: in Section 4, we expand in (m_e/m_i)^1/2, in Section 5 in k_⊥ρ_i (followed by further subsidiary expansions in large and small k_∥λ_mfpi in Section 6), and in Section 7 in 1/k_⊥ρ_i.

In view of Equation (77), a_e ≪ 1, so we can expand the Bessel functions arising from averaging over the electron ring motion:

$$\begin{eqnarray} &&\hspace{-20pt}J_0(a_e)\,{=}\,1\,{-}\,{1\over 4}\,{a_e^2}\,{+}\,\cdots,\quad {J_1(a_e)\over a_e}\,{=}\,{1\over 2}\left(1\,{-}\,{1\over 8}\,a_e^2\,{+}\,{\cdots}\right). \end{eqnarray} \tag{ 78 }$$

Keeping only the lowest-order terms of the above expansions in Equation (69) for $\langle \chi \rangle _{{\bf R}_e}$, then substituting this $\langle \chi \rangle _{{\bf R}_e}$ and q_e = −e in the electron gyrokinetic equation, we get the following kinetic equation for the electrons, accurate up to and including the first order in (m_e/m_i)^1/2 (or in k_⊥ρ_e):

$$\begin{eqnarray} \hspace{-20pt}&&\underbrace{{\partial h_e\over \partial t_{\vphantom{\rm the}}^{\vphantom{2}}}}_{\lefteqn{\bigcirc}{\,\,\,\bf 1}} + \underbrace{v_\parallel {\partial h_e\over \partial z_{\vphantom{\rm the}}^{\vphantom{2}}}}_{\lefteqn{\bigcirc }{\,\,\,\bf 0}} + {c\over B_0}\biggl \lbrace \underbrace{\varphi }_{\lefteqn{\bigcirc }{\,\,\,\bf 1}} - \underbrace{{v_\parallel A_\parallel \over c_{\vphantom{\rm the}}^{\vphantom{2}}}}_{\lefteqn{\bigcirc }{\,\,\,\bf 0}} - \underbrace{{T_{0e}\over e}{v_\perp ^2\over v_{{\rm th}e}^2}{\delta B_\parallel \over B_0}}_{\lefteqn{\bigcirc }{\,\,\,\bf 1}},h_e\biggr \rbrace\nonumber\\ \hspace{-20pt}&&=-{eF_{0e}\over T_{0e}}{\partial \over \partial t} \biggl (\underbrace{\varphi }_{\lefteqn{\bigcirc }{\,\,\,\bf 1}} - \underbrace{{v_\parallel A_\parallel \over c_{\vphantom{\rm the}}^{\vphantom{2}}}}_{\lefteqn{\bigcirc }{\,\,\,\bf 0}} - \underbrace{{T_{0e}\over e}{v_\perp ^2\over v_{{\rm th}e}^2}{\delta B_\parallel \over B_0}}_{\lefteqn{\bigcirc }{\,\,\,\bf 1}}\biggr) + \underbrace{\left({\partial h_e\over \partial t}\right)_{\rm c}}_{\lefteqn{\bigcirc }{\,\,\,\bf 0}}. \end{eqnarray} \tag{ 79 }$$

Note that φ, A_∥, δB_∥ in Equation (79) are taken at r = R_e. We have indicated the lowest order to which each of the terms enters if compared with v_∥∂h_e/∂z. In order to obtain these estimates, we have assumed that the physical ordering introduced in Section 3.1 holds with respect to the subsidiary expansion in (m_e/m_i)^1/2 as well as for the primary gyrokinetic expansion in , so we can use Equations (3) and (12) to order terms with respect to (m_e/m_i)^1/2. We have also made use of Equations (45), (47), and of the following three relations:

$$\begin{eqnarray} {k_{\parallel }v_\parallel \over \omega }&\sim & {v_{{\rm th}e}\over v_A}\sim \sqrt{\beta _i\over \tau }\sqrt{m_i\over m_e}, \end{eqnarray} \tag{ 80 }$$

$$\begin{eqnarray} \hspace{-15pt}{(v_\parallel /c)A_\parallel \over \varphi }&\sim & {v_{{\rm th}e}\delta B_\perp \over ck_{\perp }\varphi } \sim {1\over k_{\perp }\rho _e}{T_{0e}\over e\varphi }{\delta B_\perp \over B_0} \sim \sqrt{\beta _i\over \tau }\sqrt{m_i\over m_e},\quad \end{eqnarray} \tag{ 81 }$$

$$\begin{eqnarray} {T_{0e}\over e\varphi }{v_\perp ^2\over v_{{\rm th}e}^2}{\delta B_\parallel \over B_0} &\sim & {Z\over \tau }\,k_{\perp }\rho _i\sqrt{\beta _i}. \end{eqnarray} \tag{ 82 }$$

The collision term is estimated to be zeroth order because (see Equations (49) and (50))

$$\begin{eqnarray} &&{\nu _{ei}\over \omega } \sim {\tau ^{3/2}\sqrt{\beta _i}\over Z^2}\sqrt{m_i\over m_e} {1\over k_{\parallel }\lambda _{{\rm mfp}i}}. \end{eqnarray} \tag{ 83 }$$

The consequences of other possible orderings of the collision terms are discussed in Section 4.8. We remind the reader that all dimensionless parameters except k_∥/k_⊥ ∼ and (m_e/m_i)^1/2 are held to be order unity.

We now let h_e = h⁽⁰⁾_e + h⁽¹⁾_e + ⋅ ⋅ ⋅ and carry out the expansion to two lowest orders in (m_e/m_i)^1/2.

To zeroth order, the electron kinetic equation is

$$\begin{eqnarray} &&v_\parallel {\bf\hat{b}}\cdot {\bm\nabla}h_e^{(0)}= v_\parallel \,{eF_{0e}\over cT_{0e}}{\partial A_\parallel \over \partial t} + \left({\partial h_e^{(0)}\over \partial t}\right)_{\rm c}, \end{eqnarray} \tag{ 84 }$$

where we have assembled the terms in the left-hand side to take the form of the derivative of the distribution function along the perturbed magnetic field:

$$\begin{eqnarray} &&{\bf\hat{b}}\cdot {\bm\nabla}= {\partial \over \partial z}+ {\delta {\bf B}_\perp \over B_0}\cdot {\bm\nabla}= {\partial \over \partial z}- {1\over B_0}\left\lbrace A_\parallel,\cdots \right\rbrace. \end{eqnarray} \tag{ 85 }$$

We now multiply Equation (84) by h⁽⁰⁾_e/F_0e and integrate over v and r (since we are only retaining lowest-order terms, the distinction between r and R_e does not matter here). Since ${\bm\nabla}\cdot {\bf B}=0$, the left-hand side vanishes (assuming that all perturbations are either periodic or vanish at the boundaries) and we get

$$\begin{eqnarray} &&\hspace{-6pt}\int d^3{\bf r}\int d^3{\bf v}\, {h_e^{(0)}\over F_{0e}}\left({\partial h_e^{(0)}\over \partial t}\right)_{\rm c} = - {en_{0e}\over cT_{0e}}\int d^3{\bf r}{\partial A_\parallel \over \partial t}\,u_{\parallel e}^{(0)} = 0.\nonumber\\ \end{eqnarray} \tag{ 86 }$$

The right-hand side of this equation is zero because the electron flow velocity is zero in the zeroth order, u⁽⁰⁾_∥e = (1/n_0e)∫d³vv_∥h⁽⁰⁾_e = 0. This is a consequence of the parallel Ampére's law (Equation (62)), which can be written as follows:

$$\begin{eqnarray} &&u_{\parallel e}= {c\over 4\pi en_{0e}}{\bm\nabla}_{\perp }^2A_\parallel + u_{\parallel i}, \end{eqnarray} \tag{ 87 }$$

where

$$\begin{eqnarray} &&u_{\parallel i}= \sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf r}}{1\over n_{0i}}\int d^3{\bf v}\,v_\parallel J_0(a_i)h_{i{{\bf k}}}. \end{eqnarray} \tag{ 88 }$$

The three terms in Equation (87) can be estimated as follows

$$\begin{eqnarray} {u_{\parallel e}^{(0)}\over v_A} &\sim & {\epsilon v_{{\rm th}e}\over v_A} \sim \sqrt{\beta _i\over \tau }\sqrt{m_i\over m_e}\,\epsilon, \end{eqnarray} \tag{ 89 }$$

$$\begin{eqnarray} {u_{\parallel i}\over v_A} &\sim & \epsilon, \end{eqnarray} \tag{ 90 }$$

$$\begin{eqnarray} {c{\bm\nabla}_{\perp }^2A_\parallel \over 4\pi en_{0e}v_A} &\sim & {k_{\perp }\rho _i\over Z\sqrt{\beta _i}}\,\epsilon, \end{eqnarray} \tag{ 91 }$$

where we have used the fundamental ordering (12) of the slow waves (u_∥i ∼ v_A) and Alfvén waves (δB_⊥ ∼ B₀). Thus, the two terms in the right-hand side of Equation (87) are one order of (m_e/m_i)^1/2 smaller than u⁽⁰⁾_∥e, which means that to zeroth order, the parallel Ampère's law is u⁽⁰⁾_∥e = 0.

The collision operator in Equation (86) contains electron–electron and electron–ion collisions. To lowest order in (m_e/m_i)^1/2, the electron–ion collision operator is simply the pitch-angle scattering operator (see Equation (B20) in Appendix B and recall that u_∥i is first order). Therefore, we may then rewrite Equation (86) as follows:

$$\begin{eqnarray} \nonumber &&\int d^3{\bf r}\int d^3{\bf v}\, {h_e^{(0)}\over F_{0e}}\,C_{ee}[h_e^{(0)}]\\ &&- \int d^3{\bf r}\int d^3{\bf v}\, {\nu _D^{ei}(v)\over F_{0e}}{1-\xi ^2\over 2} \left({\partial h_e^{(0)}\over \partial \xi }\right)^2 = 0. \end{eqnarray} \tag{ 92 }$$

Both terms in this expression are negative definite and must, therefore, vanish individually. This implies that h⁽⁰⁾_e must be a perturbed Maxwellian distribution with zero mean velocity (this follows from the proof of Boltzmann's H-theorem; see, e.g., Longmire 1963), i.e., the full electron distribution function to zeroth order in the mass-ratio expansion is (see Equation (54)):

$$\begin{eqnarray} &&\hspace{-20pt}f_e = F_{0e}+ {e\varphi \over T_{0e}} + h_e^{(0)}= {n_e\over \left(2\pi T_e/m_e\right)^{3/2}}\exp \left(-{m_e v^2\over 2 T_e}\right), \end{eqnarray} \tag{ 93 }$$

where n_e = n_0e + δn_e, T_e = T_0e + δT_e. Expanding around the unperturbed Maxwellian F_0e, we get

$$\begin{eqnarray} &&h_e^{(0)}= \left[{\delta n_e\over n_{0e}} - {e\varphi \over T_{0e}} + \left({v^2\over v_{{\rm th}e}^2} - {3\over 2}\right){\delta T_e\over T_{0e}}\right]F_{0e}, \end{eqnarray} \tag{ 94 }$$

where the fields are taken at r = R_e. Now substitute this solution back into Equation (84). The collision term vanishes and the remaining equation must be satisfied at all values of v. This gives

$$\begin{eqnarray} {1\over c}{\partial A_\parallel \over \partial t} + {\bf\hat{b}}\cdot {\bm\nabla}\varphi &=& {\bf\hat{b}}\cdot {\bm\nabla}{T_{0e}\over e}{\delta n_e\over n_{0e}},\\ {\bf\hat{b}}\cdot {\bm\nabla}{\delta T_e\over T_{0e}} &=& 0. \end{eqnarray} \tag{ 95 }$$

The collision term is neglected in Equation (95) because, for h⁽⁰⁾_e given by Equation (94), it vanishes to zeroth order.

Equation (95) implies that the magnetic flux is conserved and magnetic-field lines cannot be broken to lowest order in the mass-ratio expansion. Indeed, we may follow Cowley (1985) and argue that the left-hand side of Equation (95) is minus the projection of the electric field on the total magnetic field (see Equation (37)), so we have

$$\begin{eqnarray} &&{\bf E}\cdot {\bf\hat{b}}= -{\bf\hat{b}}\cdot {\bm\nabla}\left({T_{0e}\over e}{\delta n_e\over n_{0e}}\right); \end{eqnarray} \tag{ 97 }$$

hence the total electric field is

$$\begin{eqnarray} &&{\bf E}= \left({\bf\hat{I}}- {\bf\hat{b}}{\bf\hat{b}}\right)\cdot \left({\bf E}+ {\bm\nabla}{T_{0e}\over e}{\delta n_e\over n_{0e}}\right) - {\bm\nabla}{T_{0e}\over e}{\delta n_e\over n_{0e}} \end{eqnarray} \tag{ 98 }$$

and Faraday's law becomes

$$\begin{eqnarray} {\partial {\bf B}\over \partial t} &=& -c{\bm\nabla}\times {\bf E}= {\bm\nabla}\times \left({\bf u}_{\rm eff}\times {\bf B}\right), \end{eqnarray} \tag{ 99 }$$

$$\begin{eqnarray} {\bf u}_{\rm eff}&=& {c\over B^2}\left({\bf E}+ {\bm\nabla}{T_{0e}\over e}{\delta n_e\over n_{0e}}\right)\times {\bf B}, \end{eqnarray} \tag{ 100 }$$

i.e., the magnetic field lines are frozen into the velocity field u_eff. In Appendix C.1, we show that this effective velocity is the part of the electron flow velocity u_e perpendicular to the total magnetic field B (see Equation (C6)).

The flux conservation is broken in the higher orders of the mass-ratio expansion. In the first order, Ohmic resistivity formally enters in Equation (95) (unless collisions are even weaker than assumed so far; if they are downgraded one order as is done in Section 4.8.3, resistivity enters in the second order). In the second order, the electron inertia and the finiteness of the electron gyroradius also lead to unfreezing of the flux. This can be seen formally by keeping second-order terms in Equation (79), multiplying it by v_∥ and integrating over velocities. The relative importance of these flux unfreezing mechanisms is evaluated in Section 7.7.

Equation (96) mandates that the perturbed electron temperature must remain constant along the perturbed field lines. Strictly speaking, this does not preclude δT_e varying across the field lines. However, we shall now assume δT_e = const (has no spatial variation), which is justified, e.g., if the field lines are stochastic. Assuming that no spatially uniform perturbations exist, we may set δT_e = 0. Equation (94) then reduces to

$$\begin{eqnarray} &&h_e^{(0)}= \left({\delta n_e\over n_{0e}} - {e\varphi \over T_{0e}}\right)F_{0e}(v), \end{eqnarray} \tag{ 101 }$$

or, using Equation (54),

$$\begin{eqnarray} &&\delta f_e= {\delta n_e\over n_{0e}}\,F_{0e}(v). \end{eqnarray} \tag{ 102 }$$

Hence follows the equation of state for isothermal electrons:

$$\begin{eqnarray} &&\delta p_e= {T_{0e}\delta n_e}. \end{eqnarray} \tag{ 103 }$$

We now integrate Equation (79) over the velocity space and retain the lowest (first) order terms only. Using Equation (101), we get

$$\begin{eqnarray} && \qquad\quad \hspace{-1.8pc}{\partial \over \partial t}\left({\delta n_e\over n_{0e}}-{\delta B_\parallel \over B_0}\right) + {c\over B_0}\left\lbrace \varphi,{\delta n_e\over n_{0e}}-{\delta B_\parallel \over B_0}\right\rbrace \nonumber\\ && \qquad\quad \hspace{-1.8pc}\quad{+}\,{\partial u_{\parallel e}\over \partial z} - {1\over B_0}\left\lbrace A_\parallel,u_{\parallel e}\right\rbrace + {cT_{0e}\over eB_0}\left\lbrace {\delta n_e\over n_{0e}},{\delta B_\parallel \over B_0}\right\rbrace = 0, \nonumber \\ \end{eqnarray} \tag{ 104 }$$

where the parallel electron velocity is first order:

$$\begin{eqnarray} &&u_{\parallel e}= u_{\parallel e}^{(1)}={1\over n_{0e}}\int d^3{\bf v}\,v_\parallel h_e^{(1)}. \end{eqnarray} \tag{ 105 }$$

The velocity-space integral of the collision term does not enter because it is subdominant by at least one factor of (m_e/m_i)^1/2: indeed, as shown in Appendix B.1, the velocity integration leads to an extra factor of k²_⊥ρ²_e, so that

$$\begin{eqnarray} \nonumber {1\over n_{0e}}\int d^3{\bf v}\left({\partial h_e\over \partial t}\right)_{\rm c} &\sim &\nu _{ei}k_{\perp }^2\rho _e^2\,{\delta n_e\over n_{0e}}\\ &\sim &\sqrt{\tau \,{m_e\over m_i}} k_{\perp }^2\rho _i^2\nu _{ii}\,{\delta n_e\over n_{0e}}, \end{eqnarray} \tag{ 106 }$$

where we have used Equations (45) and (50). The collision term is subdominant because of the ordering of the ion collision frequency given by Equation (49).

Using Equation (101) and q_i = Ze, n_0e = Zn_0i, T_0e = T_0i/τ, we derive from the quasi-neutrality Equation (61); see also Equation (65):

$$\begin{eqnarray} &&\hspace{-20pt}{\delta n_e\over n_{0e}} = {\delta n_i\over n_{0i}} = -{Ze\varphi \over T_{0i}} + \sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf r}}{1\over n_{0i}}\int d^3{\bf v}\,J_0(a_i)h_{i{{\bf k}}}, \end{eqnarray} \tag{ 107 }$$

and, from the perpendicular part of Ampère's law (Equation (66), using also Equation (107)),

$$\begin{eqnarray} \hspace{-15pt}{\delta B_\parallel \over B_0} &=& {\beta _i\over 2}\Biggl \lbrace \left(1+{Z\over \tau }\right){Ze\varphi \over T_{0i}} -\sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf r}}\Biggr.\nonumber\\ \hspace{-15pt}&&{\times}\,\Biggl.{1\over n_{0i}}\int d^3{\bf v}\,\left[{Z\over \tau }\,J_0(a_i) + {2v_\perp ^2\over v_{{\rm th}i}^2}{J_1(a_i)\over a_i}\right]h_{i{{\bf k}}}\Biggr \rbrace. \end{eqnarray} \tag{ 108 }$$

The parallel electron velocity, u_∥e, is determined from the parallel part of Ampère's law, Equation (87).

The ion distribution function h_i that enters these equations has to be determined by solving the ion gyrokinetic equation: Equation (57) with s = i.

The generalized energy (Section 3.4) for the case of isothermal electrons is calculated by substituting Equation (102) into Equation (74):

$$\begin{equation} W = \int d^3{\bf r}\left(\int d^3{\bf v}\,{T_{0i}\delta f_i^2\over 2F_{0i}} + {n_{0e}T_{0e}\over 2}{\delta n_e^2\over n_{0e}^2} + {|\delta {\bf B}|^2\over 8\pi }\right), \end{equation} \tag{ 109 }$$

where δf_i = h_i − (Zeφ/T_0i)F_0i (see Equation (54)).

Let us examine the range of spatial scales in which the equations derived above are valid. In carrying out the expansion in (m_e/m_i)^1/2, we ordered k_⊥ρ_i ∼ 1 (Equation (77)) and k_∥λ_mfpi ∼ 1 (Equation (83)). Formally, this means that the perpendicular and parallel wavelengths of the perturbations must not be so small or so large as to interfere with the mass ratio expansion. We now discuss the four conditions that this requirement leads to and whether any of them can be violated without destroying the validity of the equations derived above.

4.8.1. k_⊥ρ_i ≪ (m_i/m_e)^1/2

4.8.2. k_⊥ρ_i ≫ (m_e/m_i)^1/2

4.8.3. k_∥λ_mfpi ≪ (m_i/m_e)^1/2

Let us consider what happens if this condition is broken and k_∥λ_mfpi ≳ (m_i/m_e)^1/2. In this case, the collisions become even weaker and the expansion procedure must be modified. Namely, the collision term picks up one extra order of (m_e/m_i)^1/2, so it is first order in Equation (79). To zeroth order, the electron kinetic equation no longer contains collisions: instead of Equation (84), we have

$$\begin{eqnarray} &&v_\parallel {\bf\hat{b}}\cdot {\bm\nabla}h_e^{(0)}= v_\parallel \,{eF_{0e}\over cT_{0e}}{\partial A_\parallel \over \partial t}. \end{eqnarray} \tag{ 110 }$$

We may seek the solution of this equation in the form h⁽⁰⁾_e = H(t, R_e)F_0e + h⁽⁰⁾_e,hom, where H(t, R_e) is an unknown function to be determined and h⁽⁰⁾_e,hom is the homogeneous solution satisfying

$$\begin{eqnarray} &&{\bf\hat{b}}\cdot {\bm\nabla}h_{e,{\rm hom}}^{(0)}= 0, \end{eqnarray} \tag{ 111 }$$

i.e., h⁽⁰⁾_e,hom must be constant along the perturbed magnetic field. This is a generalization of Equation (96). Again assuming stochastic field lines, we conclude that h⁽⁰⁾_e,hom is independent of space. If we rule out spatially uniform perturbations, we may set h⁽⁰⁾_e,hom = 0. The unknown function H(t, R_e) is readily expressed in terms of δn_e and φ:

$$\begin{eqnarray} &&{\delta n_e\over n_{0e}} = {e\varphi \over T_{0e}} + {1\over n_{0e}}\int d^3{\bf v}h_e^{(0)} \Rightarrow H = {\delta n_e\over n_{0e}} - {e\varphi \over T_{0e}}, \end{eqnarray} \tag{ 112 }$$

so h⁽⁰⁾_e is again given by Equation (101), so the equations derived in Sections 4.2–4.6 are unaltered. Thus, the mass-ratio expansion remains valid at k_∥λ_mfpi ≳ (m_i/m_e)^1/2.

4.8.4. k_∥λ_mfpi ≫ (m_e/m_i)^1/2

If the parallel wavelength of the fluctuations is so long that this is violated, k_∥λ_mfpi ≲ (m_e/m_i)^1/2, the collision term in Equation (79) is minus first order. This is the lowest-order term in the equation. Setting it to zero obliges h⁽⁰⁾_e to be a perturbed Maxwellian again given by Equation (94). Instead of Equation (84), the zeroth-order kinetic equation is

$$\begin{eqnarray} &&v_\parallel {\bf\hat{b}}\cdot {\bm\nabla}h_e^{(0)}= v_\parallel \,{eF_{0e}\over cT_{0e}}{\partial A_\parallel \over \partial t} + \left({\partial h_e^{(1)}\over \partial t}\right)_{\rm c}. \end{eqnarray} \tag{ 113 }$$

Now the collision term in this order contains h⁽¹⁾_e, which can be determined from Equation (113) by inverting the collision operator. This sets up a perturbation theory that in due course leads to the reduced MHD version of the general MHD equations—this is what was considered in Section 2. Equation (96) no longer needs to hold, so the electrons are not isothermal. In this true one-fluid limit, both electrons and ions are adiabatic with equal temperatures (see Equation (115) below). The collisional transport terms in this limit (parallel and perpendicular resistivity, viscosity, heat fluxes, etc.) were calculated (starting not from gyrokinetics but from the general Vlasov–Landau Equation (36)) in exhaustive detail by Braginskii (1965). His results and the way RMHD emerges from them are reviewed in Appendix A.

In physical terms, the electrons can no longer be isothermal if the parallel electron diffusion time becomes longer than the characteristic time of the fluctuations (the Alfvén time):

$$\begin{equation} {1\over v_{{\rm th}e}\lambda _{{\rm mfp}i}k_{\parallel }^2} \gtrsim {1\over k_{\parallel }v_A} \quad \Leftrightarrow \quad k_{\parallel }\lambda _{{\rm mfp}i}\lesssim {1\over \sqrt{\beta _i}}\sqrt{m_e\over m_i}. \end{equation} \tag{ 114 }$$

Furthermore, under a similar condition, electron and ion temperatures must equalize: this happens if the ion–electron collision time is shorter than the Alfvén time,

$$\begin{eqnarray} &&{1\over \nu _{ie}}\lesssim {1\over k_{\parallel }v_A} \quad \Leftrightarrow \quad k_{\parallel }\lambda _{{\rm mfp}i}\lesssim \sqrt{\beta _i}\sqrt{m_e\over m_i}\quad \end{eqnarray} \tag{ 115 }$$

(see Lithwick & Goldreich 2001 for a discussion of these conditions in application to the ISM).

The original gyrokinetic description introduced in Section 3 was a system of two kinetic equations (Equation (57)) that evolved the electron and ion distribution functions h_e, h_i and three field equations (Equations (61)–(63)) that related φ, A_∥ and δB_∥ to h_e and h_i. In this section, we have taken advantage of the smallness of the electron mass to treat the electrons as an isothermal magnetized fluid, while ions remained fully gyrokinetic.

In mathematical terms, we solved the electron kinetic equation and replaced the gyrokinetics with a simpler closed system of equations that evolve 6 unknown functions: φ, A_∥, δB_∥, δn_e, u_∥e and h_i. These satisfy two fluid-like evolution Equations (95) and (104), three integral relations (107), (108), and (87) which involve h_i, and the kinetic Equation (57) for h_i. The system is simpler because the full electron distribution function has been replaced by two scalar fields δn_e and u_∥e. We now summarize this new system of equations: denoting a_i = k_⊥v_⊥/Ω_i, we have

$$\begin{eqnarray} {1\over c}{\partial A_\parallel \over \partial t} + {\bf\hat{b}}\cdot {\bm\nabla}\varphi &=& {\bf\hat{b}}\cdot {\bm\nabla}{T_{0e}\over e}{\delta n_e\over n_{0e}},\\ {d\over d t}\left({\delta n_e\over n_{0e}}-{\delta B_\parallel \over B_0}\right) +{\bf\hat{b}}\cdot {\bm\nabla}u_{\parallel e}&=& -{cT_{0e}\over eB_0}\left\lbrace {\delta n_e\over n_{0e}},{\delta B_\parallel \over B_0}\right\rbrace,\nonumber\\ \end{eqnarray} \tag{ 116 }$$

$$\begin{eqnarray} {\delta n_e\over n_{0e}} &=& -{Ze\varphi \over T_{0i}} + \sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf r}}{1\over n_{0i}}\int d^3{\bf v}\,J_0(a_i)h_{i{{\bf k}}},\\ u_{\parallel e}&=&{c\over 4\pi en_{0e}}{\bm\nabla}_{\perp }^2A_\parallel + \sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf r}}{1\over n_{0i}}\int d^3{\bf v}\,v_\parallel J_0(a_i)h_{i{{\bf k}}},\nonumber\\ \\ {\delta B_\parallel \over B_0} &=& {\beta _i\over 2}\Biggl \lbrace \left(1+{Z\over \tau }\right){Ze\varphi \over T_{0i}} -\sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf r}}\Biggr.\nonumber\\ &&{\times}\,\Biggl.{1\over n_{0i}}\int d^3{\bf v}\left[{Z\over \tau }J_0(a_i)+{2v_\perp ^2\over v_{{\rm th}i}^2}{J_1(a_i)\over a_i}\right]h_{i{{\bf k}}}\Biggr\rbrace, \end{eqnarray} \tag{ 117 }$$

and Equation (57) for s = i and ion–ion collisions only:

$$\begin{eqnarray} &&{\partial h_i\over \partial t} + v_\parallel {\partial h_i\over \partial z} + {c\over B_0}\left\lbrace \langle \chi \rangle _{{\bf R}_i},h_i\right\rbrace = {Ze\over T_{0i}}{\partial \langle \chi \rangle _{{\bf R}_i}\over \partial t}\,F_{0i}+ \left<C_{ii}[h_i]\right>_{{\bf R}_i},\nonumber\\ \end{eqnarray} \tag{ 121 }$$

where $\left<C_{ii}[\dots ]\right>_{{\bf R}_i}$ is the gyrokinetic ion–ion collision operator (see Appendix B) and the ion–electron collisions have been neglected to lowest order in (m_e/m_i)^1/2 (see Equation (51)). Note that Equations (116)–(117) have been written in a compact form, where

$$\begin{eqnarray} &&{d\over dt} = {\partial \over \partial t} + {\bf u}_E\cdot {\bm\nabla}= {\partial \over \partial t} + {c\over B_0}\left\lbrace \varphi,\cdots \right\rbrace \end{eqnarray} \tag{ 122 }$$

is the convective derivative with respect to the E × B drift velocity, ${\bf u}_E=-c{{\bm\nabla}_{\perp }\varphi \times {\bf\hat{z}}/B_0}$, and

$$\begin{eqnarray} &&{\bf\hat{b}}\cdot {\bm\nabla}= {\partial \over \partial z}+ {\delta {\bf B}_\perp \over B_0}\cdot {\bm\nabla}= {\partial \over \partial z}- {1\over B_0}\left\lbrace A_\parallel,\cdots \right\rbrace \end{eqnarray} \tag{ 123 }$$

is the gradient along the total magnetic field (mean field plus perturbation).

The generalized energy conserved by Equations (116)–(121) is given by Equation (109).

It is worth observing that the left-hand side of Equation (116) is simply minus the component of the electric field along the total magnetic field (see Equation (37)). This was used in Section 4.3 to prove that the magnetic flux described by Equation (116) is exactly conserved (see Section 7.7 for a discussion of scales at which this conservation is broken). Equation (116) is the projection of the generalized Ohm's law onto the total magnetic field—the right-hand side of this equation is the so-called thermoelectric term. This is discussed in more detail in Appendix C.1, where we also show that Equation (117) is the parallel part of Faraday's law and give a qualitative non-gyrokinetic derivation of Equations (116)–(117).

We will refer to Equations (116)–(121) as the equations of isothermal electron fluid. They are valid in a broad range of scales: the only constraints are that k_∥ ≪ k_⊥ (gyrokinetic ordering, Section 3.1), k_⊥ρ_e ≪ 1 (electrons are magnetized, Section 4.8.1) and k_∥λ_mfpi ≫ (m_e/m_i)^1/2 (electrons are isothermal, Section 4.8.4). The region of validity of Equations (116)–(121) in the wavenumber space is illustrated in Figure 4. A particular advantage of this hybrid fluid-kinetic system is that it is uniformly valid across the transition from magnetized to unmagnetized ions (i.e., from k_⊥ρ_i ≪ 1 to k_⊥ρ_i ≫ 1).

Let us formally split the ion gyrocenter distribution function into two parts:

$$\begin{eqnarray} \nonumber h_i&=& {Ze\over T_{0i}}\left\langle \varphi -{{\bf v}_\perp \cdot {\bf A}_\perp \over c}\right\rangle _{{\bf R}_i}F_{0i}+ g\\ &=& \sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf R}_i}\left[J_0(a_i){Ze\varphi _{{\bf k}}\over T_{0i}} + {2v_\perp ^2\over v_{{\rm th}i}^2}{J_1(a_i)\over a_i}{\delta B_{\parallel {{\bf k}}}\over B_0}\right]F_{0i}+ g.\nonumber\\ \end{eqnarray} \tag{ 124 }$$

Then g satisfies the following equation, obtained by substituting Equation (124) and the expression for ∂A_∥/∂t that follows from Equation (116) into the ion gyrokinetic Equation (121):

$$\begin{eqnarray} &&\underbrace{{\partial g\over \partial t} + v_\parallel {\partial g\over \partial z} + {c\over B_0}\left\lbrace \langle \chi \rangle _{{\bf R}_i},g\right\rbrace - \left<C_{ii}[g]\right>_{{\bf R}_i}}_{\lefteqn{\bigcirc }{\,\,\,\bf 0}}\nonumber\\ &&\quad=-{Ze\over T_{0i}}\,v_\parallel \left\langle \underbrace{{1\over B_0}\left\lbrace A_\parallel,\varphi -\langle \varphi \rangle _{{\bf R}_i}\right\rbrace }_{\lefteqn{\bigcirc }{\,\,\,\bf 1}}\right.\nonumber\\ &&\quad{+}\left. \underbrace{{\bf\hat{b}}\cdot {\bm\nabla}\left({T_{0e}\over e}{\delta n_e\over n_{0e}}- \left\langle {{\bf v}_\perp \cdot {\bf A}_\perp \over c}\right\rangle _{{\bf R}_i}\right)}_{\lefteqn{\bigcirc }{\,\,\,\bf 0}} \right\rangle{}_{{\bf R}_i}\,F_{0i}\nonumber\\ &&\quad{+}\,{Ze\over T_{0i}}\left<C_{ii}\biggl [\biggl \langle \underbrace{\varphi }_{\lefteqn{\bigcirc }{\,\,\,\bf 1}} -\underbrace{{{\bf v}_\perp \cdot {\bf A}_\perp \over c}}_{\lefteqn{\bigcirc }{\,\,\,\bf 0}}\biggr \rangle_{{\bf R}_i}\,F_{0i}\biggr ]\right>{}_{{\bf R}_i}. \end{eqnarray} \tag{ 125 }$$

In the above equation, we have used compact notation in writing out the nonlinear terms: e.g., $\left<\left\lbrace A_\parallel,\varphi -\langle \varphi \rangle _{{\bf R}_i}\right\rbrace \right\rangle _{{\bf R}_i} = \left<\left\lbrace A_\parallel ({\bf r}),\varphi ({\bf r})\right\rbrace \right\rangle _{{\bf R}_i} -\left\lbrace \langle A_\parallel \rangle _{{\bf R}_i},\langle \varphi \rangle _{{\bf R}_i}\right\rbrace$, where the first Poisson bracket involves derivatives with respect to r and the second with respect to R_i.

The field Equations (118)–(120) rewritten in terms of g are

$$\begin{eqnarray} &&\underbrace{{\delta n_{{{\bf k}}e}\over n_{0e}}}_{\lefteqn{\bigcirc }{\,\,\,\bf 0}} - \underbrace{\Gamma _1(\alpha _i){\delta B_{\parallel {{\bf k}}}\over B_0}}_{\lefteqn{\bigcirc }{\,\,\,\bf 0}} + \underbrace{\bigl [1-\Gamma _0(\alpha _i)\bigr ]{Ze\varphi _{{\bf k}}\over T_{0i}}}_{\lefteqn{\bigcirc }{\,\,\,\bf 1}}\nonumber \\ &&\quad=\underbrace{{1\over n_{0i}}\int d^3{\bf v}\,J_0(a_i)g_{{\bf k}}}_{\lefteqn{\bigcirc }{\,\,\,\bf 0}}, \end{eqnarray} \tag{ 126 }$$

$$\begin{eqnarray} &&\hspace{-10pt}\underbrace{u_{\parallel {{\bf k}}e}}_{\lefteqn{\bigcirc }{\,\,\,\bf 0}} + \underbrace{{c\over 4\pi en_{0e}}\,k_{\perp }^2A_{\parallel {{\bf k}}}}_{\lefteqn{\bigcirc }{\,\,\,\bf 1}} = \underbrace{{1\over n_{0i}}\int d^3{\bf v}\,v_\parallel J_0(a_i)g_{{\bf k}}}_{\lefteqn{\bigcirc }{\,\,\,\bf 0}} = u_{\parallel {{\bf k}}i},\nonumber\\ \end{eqnarray} \tag{ 127 }$$

$$\begin{eqnarray} &&\underbrace{{Z\over \tau }{\delta n_{{{\bf k}}e}\over n_{0e}}}_{\lefteqn{\bigcirc }{\,\,\,\bf 0}} + \underbrace{\left[\Gamma _2(\alpha _i)+{2\over \beta _i}\right]{\delta B_{\parallel {{\bf k}}}\over B_0}}_{\lefteqn{\bigcirc }{\,\,\,\bf 0}} - \underbrace{\bigl [1-\Gamma _1(\alpha _i)\bigr ]{Ze\varphi _{{\bf k}}\over T_{0i}}}_{\lefteqn{\bigcirc }{\,\,\,\bf 1}}\nonumber\\ &&\quad= -\underbrace{{1\over n_{0i}}\int d^3{\bf v}\,{2v_\perp ^2\over v_{{\rm th}i}^2}{J_1(a_i)\over a_i}g_{{\bf k}}}_{\lefteqn{\bigcirc }{\,\,\,\bf 0}}, \end{eqnarray} \tag{ 128 }$$

where a_i = k_⊥v_⊥/Ω_i, α_i = k²_⊥ρ²_i/2 and we have defined

$$\begin{eqnarray} \nonumber \Gamma _0(\alpha _i) &=& {1\over n_{0i}}\int d^3{\bf v}\,\left[J_0(a_i)\right]^2F_{0i}\\ &=& I_0(\alpha _i)\,e^{-\alpha _i} = 1-\alpha _i + \cdots,\\ \nonumber \Gamma _1(\alpha _i) &=& {1\over n_{0i}}\int d^3{\bf v}\,{2v_\perp ^2\over v_{{\rm th}i}^2}J_0(a_i) {J_1(a_i)\over a_i}F_{0i}= -\Gamma _0^{\prime }(\alpha _i)\\ &=& \left[I_0(\alpha _i)-I_1(\alpha _i)\right]\,e^{-\alpha _i} = 1 - {3\over 2}\,\alpha _i + \cdots,\\ \Gamma _2(\alpha _i) &=& {1\over n_{0i}}\int d^3{\bf v}\,\left[{2v_\perp ^2\over v_{{\rm th}i}^2} {J_1(a_i)\over a_i}\right]^2F_{0i}= 2\Gamma _1(\alpha _i). \end{eqnarray} \tag{ 129 }$$

Underneath each term in Equations (125)–(128), we have indicated the lowest order in k_⊥ρ_i to which this term enters.

In order to carry out a subsidiary expansion in small k_⊥ρ_i, we must order all terms in Equations (95)–(104) and (125)–(128) with respect to k_⊥ρ_i. Let us again assume, like we did when expanding the electron equation (Section 4), that the ordering introduced for the gyrokinetics in Section 3.1 holds also for the subsidiary expansion in k_⊥ρ_i. First note that, in view of Equation (47), we must regard Zeφ/T_0i to be minus first order:

$$\begin{eqnarray} &&{Ze\varphi \over T_{0i}}\sim {\epsilon \over k_{\perp }\rho _i\sqrt{\beta _i}}. \end{eqnarray} \tag{ 132 }$$

Also, as δB_⊥/B₀ ∼ (Equation (12)),

$$\begin{eqnarray} &&{(v_\parallel /c)A_\parallel \over \varphi }\sim {v_{{\rm th}i}\delta B_\perp \over ck_{\perp }\varphi } \sim {1\over k_{\perp }\rho _i}{T_{0i}\over Ze\varphi }{\delta B_\perp \over B_0} \sim \sqrt{\beta _i}, \end{eqnarray} \tag{ 133 }$$

so φ and (v_∥/c)A_∥ are same order.

Since u_∥ = u_∥i (electrons do not contribute to the mass flow), assuming that slow waves and Alfvén waves have comparable energies implies u_∥i ∼ u_⊥. As u_∥i is determined by the second equality in Equation (127), we can order g (using Equation (12)):

$$\begin{eqnarray} &&{g\over F_{0i}}\sim {u_\parallel \over v_{{\rm th}i}}\sim {u_\perp \over v_{{\rm th}i}} \sim {\epsilon \over \sqrt{\beta _i}}, \end{eqnarray} \tag{ 134 }$$

so g is zeroth order in k_⊥ρ_i. Similarly, δn_e/n_0e ∼ δB_∥/B₀ ∼ are zeroth order in k_⊥ρ_i—this follows directly from Equation (12).

Together with Equation (3), the above considerations allow us to order all terms in our equations. The ordering of the collision term involving φ is explained in Appendix B.2.

We shall now show that the RMHD Equations (17)–(18) hold in this approximation. There is a simple correspondence between the stream and flux functions defined in Equation (16) and the electromagnetic potentials φ and A_∥:

$$\begin{eqnarray} &&\Phi = {c\over B_0}\,\varphi,\quad \Psi = -{A_\parallel \over \sqrt{4\pi m_in_{0i}}}. \end{eqnarray} \tag{ 135 }$$

The first of these definitions says that the perpendicular flow velocity u_⊥ is the E × B drift velocity; the second definition is the standard MHD relation between the magnetic flux function and the parallel component of the vector potential.

5.3.1. Derivation of Equation (17)

Deriving Equation (17) is straightforward: in Equation (95), we retain only the lowest—minus first—order terms (those that contain φ and A_∥). The result is

$$\begin{eqnarray} &&{\partial A_\parallel \over \partial t} + c{\partial \varphi \over \partial z} - {c\over B_0}\left\lbrace A_\parallel,\varphi \right\rbrace = 0. \end{eqnarray} \tag{ 136 }$$

Using Equation (135) and the definition of the Alfvén speed, $v_A=B_0/\sqrt{4\pi m_in_{0i}}$, we get Equation (17). By the argument of Section 4.3, Equation (136) expresses the fact that that magnetic-field lines are frozen into the E × B velocity field, which is the mean flow velocity associated with the Alfvén waves (see Section 5.4).

5.3.2. Derivation of Equation (18)

As we are about to see, in order to derive Equation (18), we have to separate the first-order part of the k_⊥ρ_i expansion. The easiest way to achieve this, is to integrate Equation (125) over the velocity space (keeping r constant) and expand the resulting equation in small k_⊥ρ_i. Using Equations (126) and (127) to express the velocity-space integrals of g, we get

$$\begin{eqnarray} &&\underbrace{{\partial \over \partial t}\bigl [1-\Gamma _0(\alpha _i)\bigr ]{Ze\varphi _{{\bf k}}\over T_{0i}}}_{\lefteqn{\bigcirc }{\,\,\,\bf 1}} +\underbrace{{\partial \over \partial t}\biggl [{\delta n_{{{\bf k}}e}\over n_{0e}} - \Gamma _1(\alpha _i){\delta B_{\parallel {{\bf k}}}\over B_0}\biggr ]}_{\lefteqn{\bigcirc }{\,\,\,\bf 0}}\nonumber\\ &&\qquad+{\partial \over \partial z}\biggl (\underbrace{u_{\parallel {{\bf k}}e}}_{\lefteqn{\bigcirc }{\,\,\,\bf 0}} + \underbrace{{c\over 4\pi en_{0e}}\,k_{\perp }^2A_{\parallel {{\bf k}}}}_{\lefteqn{\bigcirc }{\,\,\,\bf 1}}\biggr)\nonumber\\ &&\qquad+\underbrace{{c\over B_0}{1\over n_{0i}}\int d^3{\bf v}\,J_0(a_i)\left\lbrace \langle \chi \rangle _{{\bf R}_i},g\right\rbrace _{{\bf k}}}_{\lefteqn{\bigcirc }{\,\,\,\bf 0}}\nonumber\\ &&\quad= {1\over n_{0i}}\int d^3{\bf v}\,J_0(a_i)\Biggl <C_{ii}\biggl [{Ze\over T_{0i}}\biggr \langle \underbrace{\varphi }_{\lefteqn{\bigcirc }{\,\,\,\bf 3}} -\underbrace{{{\bf v}_\perp \cdot {\bf A}_\perp \over c}}_{\lefteqn{\bigcirc }{\,\,\,\bf 2}}\biggl \rangle {}_{{\bf R}_i}\!F_{0i}\biggr.\Biggr.\nonumber\\ &&\qquad+\Biggl.\biggr. \underbrace{g}_{\lefteqn{\bigcirc }{\,\,\,\bf 2}} \biggr ]\Biggr >{}_{{\bf R}_i,{{\bf k}}}\,. \end{eqnarray} \tag{ 137 }$$

Underneath each term, the lowest order in k_⊥ρ_i to which it enters is shown. We see that terms containing φ are all first order, so it is up to this order that we shall retain terms. The collision term integrated over the velocity space picks up two extra orders of k_⊥ρ_i (see Appendix B.1), so it is second order and can, therefore, be dropped. As a consequence of quasi-neutrality, the zeroth-order part of the above equation exactly coincides with Equation (104), i.e., δn_i/n_0i = δn_e/n_0e satisfy the same equation. Indeed, neglecting second-order terms (but not first-order ones!), the nonlinear term in Equation (137) (the last term on the left-hand side) is

$$\begin{eqnarray} &&{c\over B_0}\left\lbrace \varphi,{1\over n_{0i}}\int d^3{\bf v}\,g\right\rbrace - {1\over B_0}\left\lbrace A_\parallel,{1\over n_{0i}}\int d^3{\bf v}\,v_\parallel g\right\rbrace \nonumber\\ &&\quad{+}\,{cT_{0i}\over Ze B_0}\left\lbrace {\delta B_\parallel \over B_0},{1\over n_{0i}}\int d^3{\bf v}\,{v_\perp ^2\over v_{{\rm th}i}^2}g\right\rbrace, \end{eqnarray} \tag{ 138 }$$

and, using Equations (126)–(128) to express velocity-space integrals of g in the above expression, we find that the zeroth-order part of the nonlinearity is the same as the nonlinearity in Equation (104), while the first-order part is

$$\begin{equation} -{c\over B_0}\left\lbrace \varphi,{1\over 2}\,\rho _i^2\nabla _{\perp }^2{Ze\varphi \over T_{0i}}\right\rbrace + {1\over B_0}\left\lbrace A_\parallel,{c\over 4\pi en_{0e}}\,\nabla _{\perp }^2A_\parallel \right\rbrace, \end{equation} \tag{ 139 }$$

where we have used the expansion (129) of Γ₀(α_i) and converted it back into x space.

Thus, if we subtract Equation (104) from Equation (137), the remainder is first order and reads

$$\begin{eqnarray} \qquad\quad \hspace{-10pt}&&{\partial \over \partial t}{1\over 2}\,\rho _i^2\nabla _{\perp }^2{Ze\varphi \over T_{0i}} + {c\over B_0}\left\lbrace \varphi,{1\over 2}\,\rho _i^2\nabla _{\perp }^2{Ze\varphi \over T_{0i}}\right\rbrace \nonumber\\ \qquad\quad \hspace{-10pt}&&\quad{+}\,{\partial \over \partial z}{c\over 4\pi en_{0e}}\nabla _{\perp }^2A_\parallel - {1\over B_0}\left\lbrace A_\parallel,{c\over 4\pi en_{0e}}\,\nabla _{\perp }^2A_\parallel \right\rbrace= 0. \nonumber \\ \end{eqnarray} \tag{ 140 }$$

Multiplying Equation (140) by 2T_0i/Zeρ²_i and using Equation (135), we get the second RMHD Equation (18).

We have established that the Alfvén-wave component of the turbulence is decoupled and fully described by the RMHD Equations (17) and (18). This result is the same as that in Section 2.2 but now we have proven that collisions do not affect the Alfvén waves and that a fluid-like description only requires k_⊥ρ_i ≪ 1 to be valid.

Let us write explicitly the distribution function of the ion gyrocenters (Equation (124)) to two lowest orders in k_⊥ρ_i:

$$\begin{eqnarray} &&h_i= {Ze\over T_{0i}}\langle \varphi \rangle _{{\bf R}_i}F_{0i}+ {v_\perp ^2\over v_{{\rm th}i}^2}{\delta B_\parallel \over B_0}\,F_{0i}+ g+ \cdots, \end{eqnarray} \tag{ 141 }$$

where, up to corrections of order k²_⊥ρ²_i, the ring-averaged scalar potential is $\langle \varphi \rangle _{{\bf R}_i} = \varphi ({\bf R}_i)$, the scalar potential taken at the position of the ion gyrocenter. Note that in Equation (141), the first term is minus first order in k_⊥ρ_i (see Equation (132)), the second and third terms are zeroth order (Equation (134)), and all terms of first and higher orders are omitted. In order to compute the full ion distribution function given by Equation (54), we have to convert h_i to the r space. Keeping terms up to zeroth order, we get

$$\begin{eqnarray} \nonumber {Ze\over T_{0i}}\langle \varphi \rangle _{{\bf R}_i} \simeq {Ze\over T_{0i}}\varphi ({\bf R}_i) &=& {Ze\over T_{0i}}\left[\varphi ({\bf r}) + {{\bf v}_\perp \times {\bf\hat{z}}\over \Omega _i}\cdot {\bm\nabla}\varphi ({\bf r}) + \cdots \right]\\ &=& {Ze\over T_{0i}}\,\varphi ({\bf r}) + {2{\bf v}_\perp \cdot {\bf u}_E\over v_{{\rm th}i}^2} + \cdots, \end{eqnarray} \tag{ 142 }$$

where ${\bf u}_E = -c{\bm\nabla}\varphi ({\bf r})\times {\bf\hat{z}}/B_0$, the E × B drift velocity. Substituting Equation (142) into Equation (141) and then Equation (141) into Equation (54), we find

$$\begin{eqnarray} &&f_i = F_{0i}+ {2{\bf v}_\perp \cdot {\bf u}_E\over v_{{\rm th}i}^2}\,F_{0i}+ {v_\perp ^2\over v_{{\rm th}i}^2}{\delta B_\parallel \over B_0}\,F_{0i}+ g+ \cdots. \end{eqnarray} \tag{ 143 }$$

The first two terms can be combined into a Maxwellian with mean perpendicular flow velocity u_⊥ = u_E. These are the terms responsible for the Alfvén waves. The remaining terms, which we shall denote $\delta \!\tilde{f}_i$, are the perturbation of the Maxwellian in the moving frame of the Alfvén waves—they describe the passive (compressive) component of the turbulence (see Section 5.5). Thus, the ion distribution function is

$$\begin{eqnarray} &&\hspace{-15pt}f_i = {n_{0i}\over (\pi v_{{\rm th}i}^2)^{3/2}}\,\exp \left[-{({\bf v}_\perp -{\bf u}_E)^2 + v_\parallel ^2\over v_{{\rm th}i}}\right] + \delta \!\tilde{f}_i. \end{eqnarray} \tag{ 144 }$$

This sheds some light on the indifference of Alfvén waves to collisions: Alfvénic perturbations do not change the Maxwellian character of the ion distribution. Unlike in a neutral fluid or gas, where viscosity arises when particles transport the local mean momentum a distance ∼λ_mfpi, the particles in a magnetized plasma instantaneously take on the local E × B velocity (they take a cyclotron period to adjust, so, roughly speaking, ρ_i plays the role of the mean free path). Thus, there is no memory of the mean perpendicular motion and, therefore, no perpendicular momentum transport.

Some readers may find it illuminating to notice that Equation (140) can be interpreted as stating simply ${\bm\nabla}\cdot {\bf j}=0$: the first two terms represent the divergence of the polarization current, which is perpendicular to the magnetic field;²² the last two terms are ${\bf\hat{b}}\cdot {\bm\nabla}j_\parallel$. No contribution to the current arises from the collisional term in Equation (137) as ion–ion collisions cause no particle transport to lowest order in k_⊥ρ_i.

The equations that describe the density (δn_e) and magnetic-field-strength (δB_∥) fluctuations follow immediately from Equations (125)–(128) if only zeroth-order terms are kept. In these equations, terms that involve φ and A_∥ also contain factors ∼k²_⊥ρ²_i and are, therefore, first-order (with the exception of the nonlinearity on the left-hand side of Equation (125)). The fact that $\left<C_{ii}[\langle \varphi \rangle _{{\bf R}_i}F_{0i}]\right>_{{\bf R}_i}$ in Equation (125) is first order is proved in Appendix B.2. Dropping these terms along with all other contributions of order higher than zeroth and making use of Equation (69) to write out $\langle \chi \rangle _{{\bf R}_i}$, we find that Equation (125) takes the form

$$\begin{eqnarray} &&{dg\over dt} + v_\parallel \,{\bf\hat{b}}\cdot {\bm\nabla}\left[g+ \left({Z\over \tau }{\delta n_e\over n_{0e}} + {v_\perp ^2\over v_{{\rm th}i}^2}{\delta B_\parallel \over B_0}\right)F_{0i}\right]\nonumber\\ &&\quad= \left<C_{ii}\left[g+{v_\perp ^2\over v_{{\rm th}i}^2}{\delta B_\parallel \over B_0}F_{0i}\right]\right>_{{\bf R}_i}, \end{eqnarray} \tag{ 145 }$$

where we have used definitions (122)–(123) of the convective time derivative d/dt and the total gradient along the magnetic field ${\bf\hat{b}}\cdot {\bm\nabla}$ to write our equation in a compact form. Note that, in view of the correspondence between Φ, Ψ and φ, A_∥ (Equation (135)), these nonlinear derivatives are the same as those defined in Equations (19)–(20). The collision term in the right-hand side of the above equation is the zeroth-order limit of the gyrokinetic ion–ion collision operator: a useful model form of it is given in Appendix B.3 (Equation (B18)).

To zeroth order, Equations (126)–(128) are

$$\begin{eqnarray} {\delta n_e\over n_{0e}} - {\delta B_\parallel \over B_0} &=& {1\over n_{0i}}\int d^3{\bf v}\,g,\\ u_\parallel &=& {1\over n_{0i}}\int d^3{\bf v}\,v_\parallel g,\\ {Z\over \tau }{\delta n_e\over n_{0e}} + 2\left(1+{1\over \beta _i}\right){\delta B_\parallel \over B_0} &=& - {1\over n_{0i}}\int d^3{\bf v}\,{v_\perp ^2\over v_{{\rm th}i}^2}\,g. \end{eqnarray} \tag{ 146 }$$

Note that u_∥ is not an independent quantity—it can be computed from the ion distribution but is not needed for the determination of the latter.

Equations (145)–(148) evolve the ion distribution function g, the "slow-wave quantities" u_∥, δB_∥, and the density fluctuations δn_e. The nonlinearities in Equation (145), contained in d/dt and ${\bf\hat{b}}\cdot {\bm\nabla}$, involve the Alfvén-wave quantities Φ and Ψ (or, equivalently, φ and A_∥) determined separately and independently by the RMHD Equations (17)–(18). The situation is qualitatively similar to that in MHD (Section 2.4), except now a kinetic description is necessary—Equations (145)–(148) replace Equations (25)–(27)—and the nonlinear scattering/mixing of the slow waves and the entropy mode by the Alfvén waves takes the form of passive advection of the distribution function g. The density and magnetic-field-strength fluctuations are velocity-space moments of g.

Another way to understand the passive nature of the compressive component of the turbulence discussed above is to think of it as the perturbation of a local Maxwellian equilibrium associated with the Alfvén waves. Indeed, in Section 5.4, we split the full ion distribution function (Equation (144)) into such a local Maxwellian and its perturbation

$$\begin{eqnarray} &&\delta \!\tilde{f}_i= g+ {v_\perp ^2\over v_{{\rm th}i}^2}{\delta B_\parallel \over B_0}\,F_{0i}. \end{eqnarray} \tag{ 149 }$$

It is this perturbation that contains all the information about the compressive component; the second term in the above expression enforces to lowest order the conservation of the first adiabatic invariant μ_i = m_iv²_⊥/2B. In terms of the function (149), Equations (145)–(148) take a somewhat more compact form (cf. Schekochihin et al. 2007):

$$\begin{eqnarray} &&{d\over dt}\left(\delta \!\tilde{f}_i- {v_\perp ^2\over v_{{\rm th}i}^2}{\delta B_\parallel \over B_0}\,F_{0i}\right) + v_\parallel {\bf\hat{b}}\cdot {\bm\nabla}\left(\delta \!\tilde{f}_i+ {Z\over \tau }{\delta n_e\over n_{0e}}\,F_{0i}\right)\nonumber\\ &&\quad= \langle C_{ii}[\delta \tilde{f}_i]\rangle_{{\bf R}_i}, \end{eqnarray} \tag{ 150 }$$

$$\begin{eqnarray} {\delta n_e\over n_{0e}} &=& {1\over n_{0i}}\int d^3{\bf v}\,\delta \!\tilde{f}_i,\\ {\delta B_\parallel \over B_0} &=& - {\beta _i\over 2}{1\over n_{0i}}\int d^3{\bf v}\,\left({Z\over \tau } + {v_\perp ^2\over v_{{\rm th}i}^2}\right)\delta \!\tilde{f}_i. \end{eqnarray} \tag{ 151 }$$

The generalized energy (Section 3.4) in the limit k_⊥ρ_i ≪ 1 is calculated by substituting into Equation (109) the perturbed ion distribution function $\delta f_i= 2{\bf v}_\perp \cdot {\bf u}_EF_{0i}/v_{{\rm th}i}^2 + \delta \!\tilde{f}_i$ (see Equations (143) and (149)). After performing velocity integration, we get

$$\begin{eqnarray} W &=& \int d^3{\bf r}\left[{m_in_{0i}u_E^2\over 2} + {\delta B_\perp ^2\over 8\pi } \right.\nonumber\\ &&+\ \left.{n_{0i}T_{0i}\over 2} \left({Z\over \tau }{\delta n_e^2\over n_{0e}^2} + {2\over \beta _i}{\delta B_\parallel ^2\over B_0^2} + {1\over n_{0i}}\int d^3{\bf v}\,{\delta \!\tilde{f}_i^2\over F_{0i}}\right)\right]\nonumber\\ &=& W_{\rm AW} + W_{\rm compr}. \end{eqnarray} \tag{ 153 }$$

We see that the kinetic energy of the Alfvénic fluctuations has emerged from the ion-entropy part of the generalized energy. The first two terms in Equation (153) are the total (kinetic plus magnetic) energy of the Alfvén waves, denoted W_AW. As we learned from Section 5.3, it cascades independently of the rest of the generalized energy, W_compr, which contains the compressive component of the turbulence (Section 5.5) and is the invariant conserved by Equations (150)–(152).

In terms of the potentials used in our discussion of RMHD in Section 2, we have

$$\begin{eqnarray} \nonumber W_{\rm AW} &=& \int d^3{\bf r}\,{m_in_{0i}\over 2}\left(|{\bm\nabla}_{\perp }\Phi |^2 + |{\bm\nabla}_{\perp }\Psi |^2\right)\\ \nonumber &=&\int d^3{\bf r}\,{m_in_{0i}\over 2}\left(|{\bm\nabla}_{\perp }\zeta ^+|^2 + |{\bm\nabla}_{\perp }\zeta ^-|^2\right)\\ &=& W_{\rm AW}^+ + W_{\rm AW}^- \end{eqnarray} \tag{ 154 }$$

where W⁺_AW and W⁻_AW are the energies of the "+" and "−" waves (Equation (33)), which, as we know from Section 2.3, cascade by scattering off each other but without exchanging energy.

Thus, the kinetic cascade in the limit k_⊥ρ_i ≪ 1 is split, independently of the collisionality, into three cascades: of W⁺_AW, W⁻_AW and W_compr. The compressive cascade is, in fact, split into three independent cascades—the splitting is different in the collisional limit (Appendix D.2) and in the collisionless one (Section 6.2.5). Figure 5 schematically summarizes both the splitting of the kinetic cascade that we have worked out so far and the upcoming developments.

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In Section 4, gyrokinetics was reduced to a hybrid fluid-kinetic system by means of an expansion in the electron mass, which was valid for k_⊥ρ_e ≪ 1. In this section, we have further restricted the scale range by taking k_⊥ρ_i ≪ 1 and as a result have been able to achieve a further reduction in the complexity of the kinetic theory describing the turbulent cascades. The reduced theory derived here evolves 5 unknown functions: Φ, Ψ, δB_∥, δn_e and g. The stream and flux functions, Φ and Ψ are related to the fluid quantities (perpendicular velocity and magnetic field perturbations) via Equation (16) and to the electromagnetic potentials φ, A_∥ via Equation (135). They satisfy a closed system of equations, Equations (17)–(18), which describe the decoupled cascade of Alfvén waves. These are the same equations that arise from the MHD approximations, but we have now proven that their validity does not depend on the assumption of high collisionality (the fluid limit) and extends to scales well below the mean free path, but above the ion gyroscale. The physical reasons for this are explained in Section 5.4. The density and magnetic-field-strength fluctuations (the "compressive" fluctuations, or the slow waves and the entropy mode in the MHD limit) now require a kinetic description in terms of the ion distribution function g (or $\delta \!\tilde{f}_i$, Equation (149)), evolved by the kinetic Equation (145) (or Equation (150)). The kinetic equation contains δn_e and δB_∥, which are, in turn calculated in terms of the velocity-space integrals of g via Equations (146) and (148) (or Equations (151) and (152)). The nonlinear evolution (turbulent cascade) of g, δB_∥ and δn_e is due solely to passive advection of g by the Alfvén-wave turbulence.

Let us summarize the new set of equations:

$$\begin{eqnarray} {\partial \Psi \over \partial t} &=& v_A{\bf\hat{b}}\cdot {\bm\nabla}\Phi,\\ {d\over dt}\nabla _{\perp }^2\Phi &=& v_A{\bf\hat{b}}\cdot {\bm\nabla}\nabla _{\perp }^2\Psi,\\ \nonumber {dg\over dt} + v_\parallel \,{\bf\hat{b}}\cdot {\bm\nabla}\left[g\right.&+&\left. \left({Z\over \tau }{\delta n_e\over n_{0e}} + {v_\perp ^2\over v_{{\rm th}i}^2}{\delta B_\parallel \over B_0}\right)F_{0i}\right] \end{eqnarray} \tag{ 155 }$$

$$\begin{eqnarray} &=& \left<C_{ii}\left[g+{v_\perp ^2\over v_{{\rm th}i}^2}{\delta B_\parallel \over B_0}F_{0i}\right]\right>_{{\bf R}_i}, \end{eqnarray} \tag{ 157 }$$

$$\begin{eqnarray} {\delta n_e\over n_{0e}} &=&\! {-}\left[\!{Z\over \tau } + 2\left(\!1 + {1\over \beta _i}\!\right)\!\right]^{-1}\!\!\! {1\over n_{0i}}\!\!\int\! d^3{\bf v}\!\left[\!{v_\perp ^2\over v_{{\rm th}i}^2} - 2\left(\!1+{1\over \beta _i}\!\right)\!\right]g\!,\nonumber\\ \\ {\delta B_\parallel \over B_0} &=&\! {-}\left[{Z\over \tau } + 2\left(1 + {1\over \beta _i}\right)\right]^{-1}\!\!\! {1\over n_{0i}}\int d^3{\bf v}\,\left({v_\perp ^2\over v_{{\rm th}i}^2} + {Z\over \tau }\right)g,\nonumber\\ \end{eqnarray} \tag{ 158 }$$

where

$$\begin{eqnarray} &&\qquad\quad {d\over dt} = {\partial \over \partial t} + \left\lbrace \Phi,\cdots \right\rbrace,\quad {\bf\hat{b}}\cdot {\bm\nabla}= {\partial \over \partial z}+ {1\over v_A}\left\lbrace \Psi,\cdots \right\rbrace. \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\nonumber \\ \end{eqnarray} \tag{ 160 }$$

An explicit form of the collision term in the right-hand side of Equation (157) is provided in Appendix B.3 (Equation (B18)).

The generalized energy conserved by Equations (155)–(159) is given by Equation (153). The kinetic cascade is split, the Alfvénic cascade proceeding independently of the compressive one (see Figure 5).

The decoupling of the Alfvénic cascade is manifested by Equations (155)–(156) forming a closed subset. As already noted in Section 4.9, Equation (155) is the component of Ohm's law along the total magnetic field, B · E = 0. Equation (156) can be interpreted as the evolution equation for the vorticity of the perpendicular plasma flow velocity, which is the E × B drift velocity.

We shall refer to the system of Equations (155)–(159) as kinetic reduced magnetohydrodynamics (KRMHD).²³ It is a hybrid fluid-kinetic description of low-frequency turbulence in strongly magnetized weakly collisional plasma that is uniformly valid at all scales satisfying k_⊥ρ_i ≪ min(1, k_∥λ_mfpi) (ions are strongly magnetized)²⁴ and k_∥λ_mfpi ≫ (m_e/m_i)^1/2 (electrons are isothermal), as illustrated in Figure 2. Therefore, it smoothly connects the collisional and collisionless regimes and is the appropriate theory for the study of the turbulent cascades in the inertial range. The KRMHD equations generalize rather straightforwardly to plasmas that are so collisionless that one cannot assume a Maxwellian equilibrium distribution function (Chen et al. 2009)—a situation that is relevant in some of the solar-wind measurements (see further discussion in Section 8.3).

KRMHD describe what happens to the turbulent cascade at or below the ion gyroscale—we shall move on to these scales in Section 7, but first we would like to discuss the turbulent cascades of density and magnetic-field-strength fluctuations and their damping by collisional and collisionless mechanisms.

6.1.1. Equations

In the collisional regime, k_∥λ_mfpi ≪ 1, the fluid limit is recovered by expanding Equations (155)–(159) in small k_∥λ_mfpi. The calculation that is necessary to achieve this is done in Appendix D (see also Appendix A.4). The result is a closed set of three fluid equations that evolve δB_∥, δn_e and u_∥:

$$\begin{eqnarray} {d\over dt}{\delta B_\parallel \over B_0} &=& {\bf\hat{b}}\cdot {\bm\nabla}u_\parallel + {d\over dt}{\delta n_e\over n_{0e}},\\ {du_\parallel \over dt} &=& v_A^2{\bf\hat{b}}\cdot {\bm\nabla}{\delta B_\parallel \over B_0} + \nu _{\parallel i}{\bf\hat{b}}\cdot {\bm\nabla}\left({\bf\hat{b}}\cdot {\bm\nabla}u_\parallel \right),\\ {d\over dt}{\delta T_i\over T_{0i}} &=& {2\over 3}{d\over dt}{\delta n_e\over n_{0e}} + \kappa _{\parallel i}{\bf\hat{b}}\cdot {\bm\nabla}\left({\bf\hat{b}}\cdot {\bm\nabla}{\delta T_i\over T_{0i}}\right), \end{eqnarray} \tag{ 161 }$$

where

$$\begin{equation} \left(1+ {Z\over \tau }\right){\delta n_e\over n_{0e}} = - {\delta T_i\over T_{0i}} - {2\over \beta _i}\left({\delta B_\parallel \over B_0} + {1\over 3v_A^2} \nu _{\parallel i}{\bf\hat{b}}\cdot {\bm\nabla}u_\parallel \right), \end{equation} \tag{ 164 }$$

and ν_∥i and κ_∥i are the coefficients of parallel ion viscosity and thermal diffusivity, respectively. The viscous and thermal diffusion are anisotropic because plasma is magnetized, λ_mfpi ≫ ρ_i (Braginskii 1965). The method of calculation of ν_∥i and κ_∥i is explained in Appendix D.3. Here we shall ignore numerical prefactors of order unity and give order-of-magnitude values for these coefficients:

$$\begin{eqnarray} &&\nu _{\parallel i}\sim \kappa _{\parallel i}\sim {v_{{\rm th}i}^2\over \nu _{ii}}\sim v_{{\rm th}i}\lambda _{{\rm mfp}i}. \end{eqnarray} \tag{ 165 }$$

If we set ν_∥i = κ_∥i = 0, Equations (161)–(164) are the same as the RMHD equations of Section 2 with the sound speed defined as

$$\begin{eqnarray} &&c_s = v_A\sqrt{{\beta _i\over 2}\left({Z\over \tau } + {5\over 3}\right)} = \sqrt{{ZT_{0e}\over m_i} + {5\over 3}{T_{0i}\over m_i}}. \end{eqnarray} \tag{ 166 }$$

This is the natural definition of c_s for the case of adiabatic ions, whose specific heat ratio is γ_i = 5/3, and isothermal electrons, whose specific heat ratio is γ_e = 1 (because δp_e = T_0eδn_e; see Equation (103)). Note that Equation (164) is equivalent to the pressure balance (Equation (22) of Section 2) with p = n_iT_i + n_eT_e and δp_e = T_0eδn_e.

As in Section 2, the fluctuations described by Equations (161)–(164) separate into the zero-frequency entropy mode and the left- and right-propagating slow waves with

$$\begin{eqnarray} &&\omega = \pm {k_{\parallel }v_A\over \sqrt{1+v_A^2/c_s^2}} \end{eqnarray} \tag{ 167 }$$

(see Equation (30)). All three are cascaded independently of each other via nonlinear interaction with the Alfvén waves. In Appendix D.2, we show that the generalized energy W_compr for this system, given in Section 5.6, splits into the three familiar invariants W⁺_sw, W⁻_sw, and W_s, defined by Equations (34)–(35) (see Figure 5).

6.1.2. Dissipation

The diffusion terms add dissipation to the equations. Because diffusion occurs along the field lines of the total magnetic field (mean field plus perturbation), the diffusive terms are nonlinear and the dissipation process also involves interaction with the Alfvén waves. We can estimate the characteristic parallel scale at which the diffusion terms become important by balancing the nonlinear cascade time and the typical diffusion time:

$$\begin{eqnarray} &&k_{\parallel }v_A \sim v_{{\rm th}i}\lambda _{{\rm mfp}i}k_{\parallel }^2 \quad \Leftrightarrow \quad k_{\parallel }\lambda _{{\rm mfp}i}\sim {1/\sqrt{\beta _i}}, \end{eqnarray} \tag{ 168 }$$

where we have used Equation (165).

Technically speaking, the cutoff given by Equation (168) always lies in the range of k_∥ that is outside the region of validity of the small-k_∥λ_mfpi expansion adopted in the derivation of Equations (161)–(163). In fact, in the low-beta limit, the collisional cutoff falls manifestly in the collisionless scale range, i.e., the collisional (fluid) approximation breaks down before the slow-wave and entropy cascades are damped and one must use the collisionless (kinetic) limit to calculate the damping (see Section 6.2.2). The situation is different in the high-beta limit: in this case, the expansion in small k_∥λ_mfpi can be reformulated as an expansion in small $1/\sqrt{\beta _i}$ and the cutoff falls within the range of validity of the fluid approximation. Equations (161)–(163) in this limit are

$$\begin{eqnarray} {d\over dt}{\delta B_\parallel \over B_0} &=& {\bf\hat{b}}\cdot {\bm\nabla}u_\parallel,\\ {du_\parallel \over dt} &=& v_A^2{\bf\hat{b}}\cdot {\bm\nabla}{\delta B_\parallel \over B_0} + \nu _{\parallel i}{\bf\hat{b}}\cdot {\bm\nabla}\left({\bf\hat{b}}\cdot {\bm\nabla}u_\parallel \right),\\ {d\over dt}{\delta n_e\over n_{0e}} &=& {1+Z/\tau \over 5/3 + Z/\tau }\,\kappa _{\parallel i}{\bf\hat{b}}\cdot {\bm\nabla}\left({\bf\hat{b}}\cdot {\bm\nabla}{\delta n_e\over n_{0e}}\right). \end{eqnarray} \tag{ 169 }$$

As in Section 2 (Equation (28)), the density fluctuations (Equation (171)) have decoupled from the slow waves (Equations (169)–(170)). The former are damped by thermal diffusion, the latter by viscosity. The corresponding linear dispersion relations are

$$\begin{eqnarray} \omega &=& -i{1+Z/\tau \over 5/3 + Z/\tau }\,\kappa _{\parallel i}k_{\parallel }^2,\\ \omega &=& \pm k_{\parallel }v_A\sqrt{1-\left({\nu _{\parallel i}k_{\parallel }\over 2v_A}\right)^2} - i\,{\nu _{\parallel i}k_{\parallel }^2\over 2}. \end{eqnarray} \tag{ 172 }$$

Equation (172) describes strong diffusive damping of the density fluctuations. The slow-wave dispersion relation (173) has two distinct regimes:

• 1.  
  When k_∥ < 2v_A/ν_∥i, it describes viscously-damped slow waves. In particular, in the limit $k_{\parallel }\lambda _{{\rm mfp}i}\ll 1/\sqrt{\beta _i}$, we have

  $$\begin{eqnarray} &&\omega \simeq \pm k_{\parallel }v_A - i\,{\nu _{\parallel i}k_{\parallel }^2\over 2}. \end{eqnarray} \tag{ 174 }$$
• 2.  
  For k_∥>2v_A/ν_∥i, both solutions become purely imaginary, so the slow waves are converted into aperiodic decaying fluctuations. The stronger-damped (diffusive) branch has ω ≃ −iν_∥ik²_∥, the weaker-damped one has

  $$\begin{eqnarray} &&\omega \simeq -i\,{v_A^2\over \nu _{\parallel i}} \sim -{i\over \beta _i}{v_{{\rm th}i}\over \lambda _{{\rm mfp}i}} \sim -{i\over \sqrt{\beta _i}}{v_A\over \lambda _{{\rm mfp}i}}. \end{eqnarray} \tag{ 175 }$$

  This damping effect is called viscous relaxation. It is valid until k_∥λ_mfpi ∼ 1, where it is replaced by the collisionless damping discussed in Section 6.2.2 (see Equation (190)).

The viscous and thermal-diffusive dissipation mechanisms described above lead, in the limits where they are efficient, to ion heating via the standard fluid (collisional) route, involving the development of small parallel scales in the position space, but not in velocity space (see Sections 3.4 and 3.5).

6.2.1. Equations

In the collisionless regime, k_∥λ_mfpi ≫ 1, the collision integral in the right-hand side of the kinetic Equation (157) can be neglected. The v_⊥ dependence can then be integrated out of Equation (157). Indeed, let us introduce the following two auxiliary functions:

$$\begin{eqnarray} G_n(v_\parallel) &=& -\left[{Z\over \tau } + 2\left(1 + {1\over \beta _i}\right)\right]^{-1}\nonumber\\ && \times \,{2\pi \over n_{0i}}\!\!\int _0^\infty dv_\perp \,v_\perp \left[{v_\perp ^2\over v_{{\rm th}i}^2} - 2\left(1+{1\over \beta _i}\right)\right]g,\nonumber\\ \\ G_B(v_\parallel) &=& -\left[{Z\over \tau } + 2\left(1 + {1\over \beta _i}\right)\right]^{-1}\nonumber\\ && \times \,{2\pi \over n_{0i}}\int _0^\infty dv_\perp \,v_\perp \left({v_\perp ^2\over v_{{\rm th}i}^2} + {Z\over \tau }\right)g. \end{eqnarray} \tag{ 176 }$$

In terms of these functions,

$$\begin{eqnarray} &&{\delta n_e\over n_{0e}} = \int dv_\parallel G_n,\quad {\delta B_\parallel \over B_0} = \int dv_\parallel G_B \end{eqnarray} \tag{ 178 }$$

and Equation (157) reduces to the following two coupled one-dimensional kinetic equations

$$\begin{eqnarray} \hspace{-10pt}\nonumber {dG_n\over dt} &+& v_\parallel {\bf\hat{b}}\cdot {\bm\nabla}G_n= -\left[{Z\over \tau } + 2\left(1 + {1\over \beta _i}\right)\right]^{-1}v_\parallel F_M(v_\parallel)\\ \hspace{-10pt}&&\times {\bf\hat{b}}\cdot {\bm\nabla}\left[{Z\over \tau }\left(1+{2\over \beta _i}\right){\delta n_e\over n_{0e}} + {2\over \beta _i}{\delta B_\parallel \over B_0}\right],\\ \hspace{-10pt}\nonumber {dG_B\over dt} &+& v_\parallel {\bf\hat{b}}\cdot {\bm\nabla}G_B= \left[{Z\over \tau } + 2\left(1 + {1\over \beta _i}\right)\right]^{-1}v_\parallel F_M(v_\parallel)\\ \hspace{-10pt}&&\times {\bf\hat{b}}\cdot {\bm\nabla}\left[{Z\over \tau }\left(1+{Z\over \tau }\right){\delta n_e\over n_{0e}} + \left(2 + {Z\over \tau }\right){\delta B_\parallel \over B_0}\right],\quad \end{eqnarray} \tag{ 179 }$$

where $F_M(v_\parallel) = (1/\sqrt{\pi }v_{{\rm th}i})\exp (-v_\parallel ^2/v_{{\rm th}i}^2)$ is a one-dimensional Maxwellian. This system can be diagonalized, so it splits into two decoupled equations

$$\begin{equation} {dG^\pm \over dt} + v_\parallel {\bf\hat{b}}\cdot {\bm\nabla}G^\pm = {v_\parallel F_M(v_\parallel)\over \Lambda ^\pm }\,{\bf\hat{b}}\cdot {\bm\nabla}\int _{-\infty }^{+\infty }dv_\parallel ^{\prime }\,G^\pm (v_\parallel ^{\prime }), \end{equation} \tag{ 181 }$$

where

$$\begin{eqnarray} &&\hspace{-20pt}\Lambda ^\pm = -{\tau \over Z} + {1\over \beta _i} \pm \sqrt{\left(1+{\tau \over Z}\right)^2 + {1\over \beta _i^2}} \end{eqnarray} \tag{ 182 }$$

and we have introduced a new pair of functions

$$\begin{eqnarray} &&G^+ = G_B+ {1\over \sigma }\left(1+{Z\over \tau }\right)G_n,\hspace{5pt} G^- \!= G_n+\! {1\over \sigma }{\tau \over Z}{2\over \beta _i}G_B,\nonumber\\ \end{eqnarray} \tag{ 183 }$$

where

$$\begin{eqnarray} &&\sigma = 1 + {\tau \over Z} + {1\over \beta _i} + \sqrt{\left(1+{\tau \over Z}\right)^2 + {1\over \beta _i^2}}. \end{eqnarray} \tag{ 184 }$$

Equation (181) describes two decoupled kinetic cascades, which we will discuss in greater detail in Sections 6.2.3–6.2.5.

6.2.2. Collisionless Damping

Fluctuations described by Equation (181) are subject to collisionless damping. Indeed, let us linearize Equation (181), Fourier transform in time and space, divide through by −i(ω − k_∥v_∥), and integrate over v_∥. This gives the following dispersion relation (the "−" branch is for G⁻, the "+" branch for G⁺)

$$\begin{eqnarray} &&\zeta _iZ\left(\zeta _i\right) = \Lambda ^\pm - 1, \end{eqnarray} \tag{ 185 }$$

where $\zeta _i=\omega /|k_{\parallel }|v_{{\rm th}i}=\omega /|k_{\parallel }|v_A\sqrt{\beta _i}$ and we have used the plasma dispersion function (Fried & Conte 1961)

$$\begin{eqnarray} &&Z\left(\zeta _i\right) = {1\over \sqrt{\pi }}\int _{-\infty }^\infty dx\,{e^{-x^2}\over x-\zeta _i} \end{eqnarray} \tag{ 186 }$$

(the integration is along the Landau contour). This function is not to be confused with the ion charge parameter Z = q_i/e.

Formally, Equation (185) has an infinite number of solutions. When β_i ∼ 1, they are all strongly damped with damping rates Im(ω) ∼ |k_∥|v_thi ∼ |k_∥|v_A, so the damping time is comparable to the characteristic timescale on which the Alfvén waves cause these fluctuations to cascade to smaller scales.

It is interesting to consider the high- and low-beta limits.

High-Beta Limit.

When β_i ≫ 1, we have in Equation (185)

$$\begin{eqnarray} \Lambda ^- -1 &\simeq & -2\left(1 + {\tau \over Z}\right), \quad G^-\simeq G_n,\\ \Lambda ^+ -1 &\simeq & {1\over \beta _i}, \qquad \qquad \ \ \,G^+\simeq G_B\!+\! {1\over 2}{Z\over \tau }\,G_n. \end{eqnarray} \tag{ 187 }$$

The "−" branch corresponds to the density fluctuations. The solution of Equation (185) has Im(ζ_i) ∼ 1, so these fluctuations are strongly damped:

$$\begin{eqnarray} &&\omega \sim -i|k_{\parallel }|v_A\sqrt{\beta _i}. \end{eqnarray} \tag{ 189 }$$

The damping rate is much greater than the Alfvénic rate k_∥v_A of the nonlinear cascade. In contrast, for the "+" branch, the damping rate is small: it can be obtained by expanding $Z(\zeta _i)=i\sqrt{\pi }+O\left(\zeta _i\right)$, which gives²⁵

$$\begin{eqnarray} &&\omega = - i\,{|k_{\parallel }|v_{{\rm th}i}\over \sqrt{\pi }\beta _i} = - i\,{|k_{\parallel }|v_A\over \sqrt{\pi \beta _i}}. \end{eqnarray} \tag{ 190 }$$

Since G_n is strongly damped, Equation (188) implies G⁺ ≃ G_B, i.e., the fluctuations that are damped at the rate (190) are predominantly of the magnetic-field strength. The damping rate is a constant (independent of k_∥) small fraction $\sim 1/\sqrt{\beta _i}$ of the Alfvénic cascade rate.

In Figure 6, we give a schematic plot of the damping rate of the magnetic-field-strength fluctuations (slow waves) connecting the fluid and kinetic limits for β_i ≫ 1.

Zoom In Zoom Out Reset image size

Low-Beta Limit.

When β_i ≪ 1, we have

$$\begin{eqnarray} \Lambda ^- -1 &\simeq & -\left(1+{\tau \over Z}\right), \qquad\!\! G^-\simeq G_n+ {\tau \over Z}\,G_B,\\ \Lambda ^+ -1 &\simeq & {2\over \beta _i}, \qquad \qquad \quad \,G^+\simeq G_B. \end{eqnarray} \tag{ 191 }$$

For the "−" branch, we again have Im(ζ_i) ∼ 1, so

$$\begin{eqnarray} &&\omega \sim -i|k_{\parallel }|v_A\sqrt{\beta _i}, \end{eqnarray} \tag{ 193 }$$

which now is much smaller than the Alfvénic cascade rate k_∥v_A. For the "+" branch (predominantly the field-strength fluctuations), we seek a solution with $\zeta =-i\tilde{\zeta }_i$ and $\tilde{\zeta }_i\gg 1$. Then Equation (185) becomes $\zeta _iZ(\zeta _i) \simeq 2\sqrt{\pi }\,\tilde{\zeta }_i\exp (\tilde{\zeta }_i) = 2/\beta _i$. Up to logarithmically small corrections, this gives $\tilde{\zeta }_i\simeq \sqrt{|\ln \beta _i|}$, whence

$$\begin{eqnarray} &&\omega \sim -i|k_{\parallel }|v_A\sqrt{\beta _i|\ln \beta _i|}. \end{eqnarray} \tag{ 194 }$$

While this damping rate is slightly greater than that of the "−" branch, it is still much smaller than the Alfvénic cascade rate.

6.2.3. Collisionless Invariants

Equation (181) obeys a conservation law, which is very easy to derive. Multiplying Equation (181) by G^±/F_M and integrating over space and velocities and performing integration by parts in the right-hand side, we get

$$\begin{eqnarray} \nonumber {d\over dt}&\int& d^3{\bf r}\,\int dv_\parallel {(G^\pm)^2\over 2F_M} \qquad \qquad \qquad \qquad \qquad \qquad \\ &=& -{1\over \Lambda ^\pm }\int d^3{\bf r}\,\left(\int dv_\parallel G^\pm \right) {\bf\hat{b}}\cdot {\bm\nabla}\int dv_\parallel v_\parallel G^\pm.\quad \end{eqnarray} \tag{ 195 }$$

On the other hand, integrating Equation (181) over v_∥ gives

$$\begin{eqnarray} &&{d\over dt}\int dv_\parallel G^\pm = - {\bf\hat{b}}\cdot {\bm\nabla}\int dv_\parallel v_\parallel G^\pm. \end{eqnarray} \tag{ 196 }$$

Using this to express the right-hand side of Equation (195) as a full time derivative, we find

$$\begin{eqnarray} &&{dW_{\rm compr}^\pm \over dt} = 0, \end{eqnarray} \tag{ 197 }$$

where the two invariants are

$$\begin{equation} W_{\rm compr}^\pm = \int d^3{\bf r}\,{n_{0i}T_{0i}\over 2}\left[\int dv_\parallel {(G^\pm)^2\over F_M} - {1\over \Lambda ^\pm }\left(\int dv_\parallel G^\pm \right)^2\right]. \end{equation} \tag{ 198 }$$

It is useful (and always possible) to split

$$\begin{eqnarray} &&G^\pm = F_M\int dv_\parallel G^\pm + \tilde{G}^\pm, \end{eqnarray} \tag{ 199 }$$

where $\int dv_\parallel \tilde{G}^\pm =0$ by construction. Then

$$\begin{eqnarray} \nonumber W_{\rm compr}^\pm &=& \int d^3{\bf r}\,{n_{0i}T_{0i}\over 2}\left[ \int dv_\parallel {(\tilde{G}^\pm)^2\over F_M} \right.\\ &&+ \left.\left(1- {1\over \Lambda ^\pm }\right)\left(\int dv_\parallel G^\pm \right)^2\right]. \end{eqnarray} \tag{ 200 }$$

Written in this form, the two invariants W^±_compr are manifestly positive definite quantities because Λ⁺>1 and Λ⁻ < 0. The invariants regulate the two decoupled kinetic cascades of compressive fluctuations in the collisionless regime. The collisionless damping derived in Section 6.2.2 leads to exponential decay of the density and field-strength fluctuations, or, equivalently, of ∫dv_∥G^±, while conserving W^±_compr. This means that the damping is merely a redistribution of the conserved quantity W^±_compr: the first term in Equation (200) grows to compensate for the decay of the second.

6.2.4. Linear Parallel Phase Mixing

6.2.5. Generalized Energy: Three Collisionless Cascades

We will now show how the generalized energy for compressive fluctuations in the collisionless regime incorporates the two invariants derived in Section 6.2.3.

Rewriting the compressive part of the KRMHD generalized energy (Equation (153)) in terms of the function g (see Equation (149)), we get

$$\begin{eqnarray} \qquad\quad &&\hspace{-1.8pc}W_{\rm compr}= {n_{0i}T_{0i}\over 2}\int d^3{\bf r}\,\left\lbrace {1\over n_{0i}}\int d^3{\bf v}\,{g^2\over F_{0i}}\right. \nonumber \\ \qquad\quad &&\hspace{-1.8pc}\quad\left.+\, {Z\over \tau }\left({\delta n_e\over n_{0e}} - {\delta B_\parallel \over B_0}\right)^2 - \left[{Z\over \tau } + 2\left(1+{1\over \beta _i}\right)\right]{\delta B_\parallel ^2\over B_0^2}\right\rbrace. \nonumber \\ \end{eqnarray} \tag{ 201 }$$

Using Equations (178) and (183), we can express δn_e and δB_∥ in terms of ∫dv_∥G^± as follows

$$\begin{eqnarray} {\delta n_e\over n_{0e}} &=& {1\over \kappa }\left(\sigma \int dv_\parallel G^- - {\tau \over Z}{2\over \beta _i} \int dv_\parallel G^+\right),\\ {\delta B_\parallel \over B_0} &=& {1\over \kappa }\left[\sigma \int dv_\parallel G^+ - \left(1+{Z\over \tau }\right) \int dv_\parallel G^-\right], \end{eqnarray} \tag{ 202 }$$

where σ was defined in Equation (184) and

$$\begin{eqnarray} &&\kappa = \sqrt{\left(1+{\tau \over Z}\right)^2 + {1\over \beta _i^2}}. \end{eqnarray} \tag{ 204 }$$

In order to express g in terms of G^±, we have to reconstruct the v_⊥ dependence of g, which we integrated out at the beginning of Section 6.2.1.

Let us represent the distribution function as follows

$$\begin{eqnarray} &&\qquad\quad g= {n_{0i}\over \pi v_{{\rm th}i}^2}\,e^{-x}\hat{g}(x,v_\parallel),\quad \hat{g}(x,v_\parallel) = \sum _{l=0}^\infty \!L_l(x) G_l(v_\parallel), \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\nonumber \\ \end{eqnarray} \tag{ 205 }$$

where x = v²_⊥/v²_thi and we have expanded $\hat{g}$ in Laguerre polynomials L_l(x) = (e^x/l!)(d^l/dx^l)x^le^−x. Since Laguerre polynomials are orthogonal, the first term in Equation (201) splits into a sum of "energies" associated with the expansion coefficients:

$$\begin{eqnarray} &&{1\over n_{0i}}\int d^3{\bf v}\,{g^2\over F_{0i}} = \sum _{l=0}^\infty \int dv_\parallel {G_l^2\over F_M}. \end{eqnarray} \tag{ 206 }$$

The expansion coefficients are determined via the Laguerre transform:

$$\begin{eqnarray} &&G_l(v_\parallel) = \int _0^\infty dx\, e^{-x} L_l(x)\hat{g}(x,v_\parallel). \end{eqnarray} \tag{ 207 }$$

As L₀ = 1 and L₁ = 1 − x, it is easy to see that δn_e and δB_∥ can be expressed as linear combinations of ∫dv_∥G₀ and ∫dv_∥G₁ (see Equations (176)–(178)). Using Equations (176), (177), and (183), we can show that

$$\begin{eqnarray} G_0&=&{-}\,{1\over \kappa }\left[\!\left(\sigma\,{-}\,{2\over \beta _i}\!\right)\!\Lambda ^+ G^+ \!+\! {Z\over \tau }\left(\sigma\,{-}\,1\,{-}\,{\tau \over Z}\right)\!\Lambda ^- G^-\!\right],\nonumber\\ \\ G_1 &=& {1\over \kappa }\left[\sigma \Lambda ^+ G^+ - \left(1+{Z\over \tau }\right)\Lambda ^- G^-\right], \end{eqnarray} \tag{ 208 }$$

where G^± satisfy Equation (181). As follows from Equation (157) (neglecting the collision integral), all higher-order expansion coefficients satisfy a simple homogeneous equation:

$$\begin{eqnarray} {d G_l\over dt} + v_\parallel {\bf\hat{b}}\cdot {\bm\nabla}G_l &=& 0,\quad l>1. \end{eqnarray} \tag{ 210 }$$

Thus, the distribution function can be explicitly written in terms of G^±:

$$\begin{eqnarray} &&\qquad\quad g= \left[\!G_0(v_\parallel) + \!\left(1-\!{v_\perp ^2\over v_{{\rm th}i}^2}\!\right)G_1(v_\parallel)\!\right]{n_{0i}\over \pi v_{{\rm th}i}^2}\,e^{-v_\perp ^2/v_{{\rm th}i}^2} + \tilde{g}, \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\nonumber \\ \end{eqnarray} \tag{ 211 }$$

where G₀ and G₁ are given by Equations (208)–(209) and $\tilde{g}$ comprises the rest of the Laguerre expansion (all G_l with l>1), i.e., it is the homogeneous solution of Equation (157) that does not contribute to either density or magnetic-field strength:

$$\begin{eqnarray} &&\qquad\quad {d\tilde{g}\over dt} + v_\parallel {\bf\hat{b}}\cdot {\bm\nabla}\tilde{g}= 0,\ \int d^3{\bf v}\,\tilde{g}= 0,\ \int \!\!d^3{\bf v}{v_\perp ^2\over v_{{\rm th}i}^2}\tilde{g}\!= 0. \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\nonumber \\ \end{eqnarray} \tag{ 212 }$$

Now substituting Equations (208) and (209) into Equation (206) and then substituting the result and Equations (202)–(203) into Equation (201), we find after some straightforward manipulations

$$\begin{eqnarray} \nonumber W_{\rm compr}&=& \int d^3{\bf r}\,\int d^3{\bf v}{T_{0i}\tilde{g}^2\over 2F_{0i}}\\ \nonumber && +\,\,4\left[1+{1\over \kappa }\left(1+{\tau \over Z}\right)\right] (\Lambda ^+)^2W_{\rm compr}^+\\ && +\,\, 2\,{Z^2\over \tau ^2}\left(1 + {1\over \kappa }{1\over \beta _i}\right) (\Lambda ^-)^2W_{\rm compr}^-, \end{eqnarray} \tag{ 213 }$$

where κ is defined by Equation (204) and W^±_compr are the two independent invariants that we derived in Section 6.2.3. Thus, the generalized energy for compressive fluctuations splits into three independently cascading parts: W^±_compr associated with the density and magnetic-field-strength fluctuations and a purely kinetic part given by the first term in Equation (213) (see Figure 5). The dynamical evolution of this purely kinetic component is described by Equation (212)—it is a passively mixed, undamped ballistic-type mode.

All three cascade channels lead to small perpendicular spatial scales via passive mixing by the Alfvénic turbulence and also to small scales in v_∥ via the parallel phase mixing process discussed in Section 6.2.4 (note that $\tilde{g}$ is subject to this process as well).

Let us return to the kinetic Equation (157) and transform it to the Lagrangian frame associated with the velocity field ${\bf u}_\perp ={\bf\hat{z}}\times {\bm\nabla}_{\perp }\Phi$ of the Alfvén waves: (t, r) → (t, r₀), where

$$\begin{eqnarray} &&{\bf r}(t,{\bf r}_0) = {\bf r}_0 + \int _0^t dt^{\prime }{\bf u}_\perp (t^{\prime },{\bf r}(t^{\prime },{\bf r}_0)). \end{eqnarray} \tag{ 214 }$$

In this frame, the convective derivative d/dt defined in Equation (160) turns into ∂/∂t, while the parallel spatial gradient ${\bf\hat{b}}\cdot {\bm\nabla}$ can be calculated by employing the Cauchy solution for the perturbed magnetic field $\delta {\bf B}_\perp ={\bf\hat{z}}\times {\bm\nabla}_{\perp }\Psi$:

$$\begin{eqnarray} &&{\bf\hat{b}}(t,{\bf r}) = {\bf\hat{z}}+ {\delta {\bf B}_\perp (t,{\bf r})\over B_0} = {\bf\hat{b}}(0,{\bf r}_0)\cdot {\bm\nabla}_0{\bf r}, \end{eqnarray} \tag{ 215 }$$

where r is given by Equation (214) and ${\bm\nabla}_0=\partial /\partial {\bf r}_0$. Then

$$\begin{eqnarray} &&\qquad\quad {\bf\hat{b}}\cdot {\bm\nabla}= {\bf\hat{b}}(0,{\bf r}_0)\cdot \bigl ({\bm\nabla}_0{\bf r}\bigr)\cdot {\bm\nabla}= {\bf\hat{b}}(0,{\bf r}_0)\cdot {\bm\nabla}_0 = {\partial \over \partial s_0}, \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\nonumber \\ \end{eqnarray} \tag{ 216 }$$

where s₀ is the arc length along the perturbed magnetic field taken at t = 0 (if δB_⊥(0, r₀) = 0, s₀ = z₀). Thus, in the Lagrangian frame associated with the Alfvénic component of the turbulence, Equation (157) is linear. This means that, if the effect of finite ion gyroradius is neglected, the KRMHD system does not give rise to a cascade of density and magnetic-field-strength fluctuations to smaller scales along the moving (perturbed) field lines, i.e., ${\bf\hat{b}}\cdot {\bm\nabla}\delta n_e$ and ${\bf\hat{b}}\cdot {\bm\nabla}\delta B_\parallel$ do not increase. In contrast, there is a perpendicular cascade (cascade in k_⊥): the perpendicular wandering of field lines due to the Alfvénic turbulence causes passive mixing of δn_e and δB_∥ in the direction transverse to the magnetic field (see Section 2.6 for a quick recapitulation of the standard scaling argument on the passive cascade that leads to a k^−5/3_⊥ in the perpendicular direction). Figure 7 illustrates this situation.²⁶

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We emphasize that this lack of nonlinear refinement of the scale of δn_e and δB_∥ along the moving field lines is a particular property of the compressive component of the turbulence, not shared by the Alfvén waves. Indeed, unlike Equation (157), the RMHD Equations (155)–(156), do not reduce to a linear form under the Lagrangian transformation (214), so the Alfvén waves should develop small scales both across and along the perturbed magnetic field.

Whether the density and magnetic-field-strength fluctuations develop small scales along the magnetic field has direct physical and observational consequences. Damping of these fluctuations, both in the collisional and collisionless regimes, discussed in Sections 6.1.2 and 6.2.2, respectively, depends precisely on their scale along the perturbed field: indeed, the linear results derived there are exact in the Lagrangian frame (214). To summarize these results, the damping rate of δn_e and δB_∥ at β_i ∼ 1 is

$$\begin{eqnarray} \gamma &\sim & v_{{\rm th}i}\lambda _{{\rm mfp}i}k_{\parallel 0}^2,\quad\ \ k_{\parallel 0}\lambda _{{\rm mfp}i}\ll 1,\\ \gamma &\sim & v_{{\rm th}i}k_{\parallel 0},\quad \qquad k_{\parallel 0}\lambda _{{\rm mfp}i}\gg 1, \end{eqnarray} \tag{ 217 }$$

where $k_{\parallel 0}\sim {\bf\hat{b}}\cdot {\bm\nabla}$ is the wavenumber along the perturbed field (i.e., if there is no parallel cascade, the wavenumber of the large-scale stirring).

Whether this damping cuts off the cascades of δn_e and δB_∥ depends on the relative magnitudes of the damping rate γ for a given k_⊥ and the characteristic rate at which the Alfvén waves cause δn_e and δB_∥ to cascade to higher k_⊥. This rate is ω_A ∼ k_∥Av_A, where k_∥A is the parallel wavenumber of the Alfvén waves that have the same k_⊥. Since the Alfvén waves do have a parallel cascade, assuming scale-by-scale critical balance (3) leads to (Equation (5))

$$\begin{eqnarray} &&k_{\parallel A}\sim k_{\perp }^{2/3}l_0^{-1/3}. \end{eqnarray} \tag{ 219 }$$

If, in contrast to the Alfvén waves, δn_e and δB_∥ have no parallel cascade, k_∥0 does not grow with k_⊥, so, for large enough k_⊥, k_∥0 ≪ k_∥A and γ ≪ ω_A. This means that, despite the damping, the density and field-strength fluctuations should have perpendicular cascades extending to the ion gyroscale.

The validity of the argument at the beginning of this section that ruled out the parallel cascade of δn_e and δB_∥ is not quite as obvious as it might appear. Lithwick & Goldreich (2001) argued that the dissipation of δn_e and δB_∥ at the ion gyroscale would cause these fluctuations to become uncorrelated at the same parallel scales as the Alfvénic fluctuations by which they are mixed, i.e., k_∥0 ∼ k_∥A. The damping rate then becomes comparable to the cascade rate, cutting off the cascades of density and field-strength fluctuations at k_∥λ_mfpi ∼ 1. The corresponding perpendicular cutoff wavenumber is (see Equation (219))

$$\begin{eqnarray} &&k_{\perp }\sim l_0^{1/2}\lambda _{{\rm mfp}i}^{-3/2}. \end{eqnarray} \tag{ 220 }$$

Asymptotically speaking, in a weakly collisional plasma, this cutoff is far above the ion gyroscale, k_⊥ρ_i ≪ 1. However, the relatively small value of λ_mfpi in the warm ISM, which was the main focus of Lithwick & Goldreich 2001, meant that the numerical value of the perpendicular cutoff scale given by Equation (220) was, in fact, quite close both to the ion gyroscale (see Table 1) and to the observational estimates for the inner scale of the electron-density fluctuations in the ISM (Spangler & Gwinn 1990; Armstrong et al. 1995). Thus, it was not possible to tell whether Equation (220), rather than k_⊥ ∼ ρ⁻¹_i, represented the correct prediction.

The situation is rather different in the nearly collisionless case of the solar wind, where the cutoff given by Equation (220) would mean that very little density or field-strength fluctuations should be detected above the ion gyroscale. Observations do not support such a conclusion: the density fluctuations appear to follow a k^−5/3 law at all scales larger than a few times ρ_i (Lovelace et al. 1970; Woo & Armstrong 1979; Celnikier et al. 1983, 1987; Coles & Harmon 1989; Marsch & Tu 1990b; Coles et al. 1991), consistently with the expected behavior of an undamped passive scalar field (see Section 2.6). An extended range of k^−5/3 scaling above the ion gyroscale is also observed for the fluctuations of the magnetic-field strength (Marsch & Tu 1990b; Bershadskii & Sreenivasan 2004; Hnat et al. 2005; Alexandrova et al. 2008a).

These observational facts suggest that the cutoff formula (220) does not apply. This does not, however, conclusively vitiate the Lithwick & Goldreich (2001) theory. Heuristically, their argument is plausible, although it is, perhaps, useful to note that in order for the effect of the perpendicular dissipation terms, not present in the KRMHD Equations (157)–(159), to be felt, the density and field-strength fluctuations should reach the ion gyroscale in the first place. Quantitatively, the failure of the compressive fluctuations in the solar wind to be damped could still be consistent with the Lithwick & Goldreich (2001) theory because of the relative weakness of the collisionless damping, especially at low beta (Section 6.2.2)—the explanation they themselves favor. The way to check observationally whether this explanation suffices would be to make a comparative study of the compressive fluctuations for solar-wind data with different values of β_i. If the strength of the damping is the decisive factor, one should always see cascades of both δn_e and δB_∥ at low β_i, no cascades at β_i ∼ 1, and a cascade of δB_∥ but not δn_e at high β_i (in this limit, the damping of the density fluctuations is strong, of the field-strength weak; see Section 6.2.2). If, on the other hand, the parallel cascade of the compressive fluctuations is intrinsically inefficient, very little β_i dependence is expected and a perpendicular cascade should be seen in all cases.

Obviously, an even more direct observational (or numerical) test would be the detection or non-detection of near-perfect alignment of the density and field-strength structures with the moving field lines (not with the mean magnetic field—see footnote ²⁶, but it is not clear how to measure this reliably. It is interesting, in this context, that in near-the-Sun measurements, the density fluctuations are reported to have the form of highly anisotropic filaments aligned with the magnetic field (Armstrong et al. 1990; Grall et al. 1997; Woo & Habbal 1997). Another intriguing piece of observational evidence is the discovery that the local structure of the magnetic-field-strength and density fluctuations at 1 AU is, in a certain sense, correlated with the solar cycle (Kiyani et al. 2007; Hnat et al. 2007; Wicks et al. 2009)—this suggests a dependence on initial conditions that is absent in the Alfvénic fluctuations and that presumably should also disappear in the compressive fluctuations if the latter are fully mixed both in the perpendicular and parallel directions.

The validity of the theory discussed in Sections 5 and 6 breaks down when k_⊥ρ_i ∼ 1. As the ion gyroscale is approached, the Alfvén waves are no longer decoupled from the rest of the plasma dynamics. All modes now contain perturbations of density and magnetic-field strength and can be collisionlessly damped. Because of the low-frequency nature of the Alfvén-wave cascade, ω ≪ Ω_i even at k_⊥ρ_i ∼ 1 (Equation (46)), so the ion cyclotron resonance (ω − k_∥v_∥ = ±Ω_i) is not important, while the Landau one (ω = k_∥v_∥) is. The linear theory of this collisionless damping in the gyrokinetic approximation is worked out in detail in Howes et al. (2006) (see also Gary & Borovsky 2008). Figure 8 shows the solutions of their dispersion relation that illustrate how the Alfvén wave becomes a dispersive kinetic Alfvén wave (KAW) (see Section 7.3) and collisionless damping becomes important as the ion gyroscale is reached.

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We stress that this transition occurs at the ion gyroscale, not at the ion inertial scale $d_i=\rho _i/\sqrt{\beta _i}$ (except in the limit of cold ions, τ = T_0i/T_0e ≪ 1; see Appendix E). This statement is true even when β_i is not order unity, as illustrated in Figure 8: for the three cases plotted there, k_⊥d_i = 1 corresponds to k_⊥ρ_i = 0.1, 1 and 10 for β_i = 0.01, 1 and 100, respectively, but there is no trace of the ion inertial scale in the solutions of the linear dispersion relation. Nonlinearly, in the limit β_i ≪ 1, we may consider the scales k_⊥d_i ∼ 1 and expand the gyrokinetics in $k_{\perp }\rho _i=k_{\perp }d_i\sqrt{\beta _i}\ll 1$ in a way similar to how it was done in Section 5 and obtain precisely the same results: Alfvénic fluctuations described by the RMHD equations and compressive fluctuations passively advected by them and satisfying the reduced kinetic equation derived in Section 5.5. Thus, even though d_i ≫ ρ_i at low beta, there is no change in the nature of the turbulent cascade until k_⊥ρ_i ∼ 1 is reached.

The nonlinear theory of what happens at k_⊥ρ_i ∼ 1 is very poorly understood. It is, however, possible to make progress by examining what kind of fluctuations emerge on the other side of the transition, at k_⊥ρ_i ≫ 1. As we will demonstrate below, it turns out that another turbulent cascade—this time of KAW—is possible in this so-called dissipation range. It can transfer the energy of KAW-like fluctuations down to the electron gyroscale, where electron Landau damping becomes important (see Howes et al. 2006). Some observational evidence of KAW is, indeed, available in the solar wind and the magnetosphere (Bale et al. 2005; Grison et al. 2005, see further discussion in Section 8.2.4). Below we derive the equations that describe KAW-like fluctuations in the scale range k_⊥ρ_i ≫ 1, k_⊥ρ_e ≪ 1 (Section 7.2) and work out a Kolmogorov-style scaling theory for this cascade (Section 7.5).

Because of the presence of the collisionless damping at the ion gyroscale, only a certain fraction of the turbulent power arriving there from the inertial range is converted into the KAW cascade, while the rest is Landau-damped. The damping leads to the heating of the ions, but the process of depositing the collisionlessly damped fluctuation energy into the ion heat is non-trivial because, as we explained in Section 3.5, collisions do need to play a role in order for true heating to occur. As we explained in Section 3.5 and will see specifically for the dissipation range in Section 7.8, the electromagnetic-fluctuation energy does not disappear as a result of the Landau damping but is converted into ion entropy fluctuations, while the generalized energy is conserved. Collisions are then accessed and ion heating achieved via a purely kinetic phenomenon: the ion entropy cascade in phase space (nonlinear phase mixing), for which a theory is developed in Sections 7.9 and 7.10. A similar process of conversion of the KAW energy into electron entropy fluctuations and then electron heat is treated in Section 7.12.

Figure 5 illustrates the routes energy takes from the ion gyroscale towards heating. Crucially, it is at k_⊥ρ_i ∼ 1 that it is decided how much energy would eventually go into the ions and how much into electrons.²⁷ How this distribution of energy depends on plasma parameters (β_i and T_0i/T_0e) is an open theoretical question²⁸ of considerable astrophysical interest: e.g., the efficiency of ion heating is a key unknown in the theory of advection-dominated accretion flows (Quataert & Gruzinov 1999, see discussion in Section 8.5) and of the solar corona (e.g., Cranmer & van Ballegooijen 2003); we will also see in Section 7.11 that it may determine the form of the observed dissipation-range spectra in space plasmas.

A short summary of this section is given in Section 7.14.

The derivation is straightforward: when a_i ∼ k_⊥ρ_i ≫ 1, all Bessel functions in Equations (118)–(120) are small, so the integrals of the ion distribution function vanish and Equations (118)–(120) become

$$\begin{eqnarray} {\delta n_e\over n_{0e}} &=& {-}{Ze\varphi \over T_{0i}} = - {2\over \sqrt{\beta _i}}{\Phi \over \rho _i v_A},\\ u_{\parallel e}&=& {c\over 4\pi en_{0e}}\nabla _{\perp }^2A_\parallel = - {\rho _i\nabla _{\perp }^2\Psi \over \sqrt{\beta _i}},\qquad u_{\parallel i}= 0,\\ {\delta B_\parallel \over B_0} &=& {\beta _i\over 2}\left(1+{Z\over \tau }\right){Ze\varphi \over T_{0i}} = \sqrt{\beta _i}\left(1+{Z\over \tau }\right){\Phi \over \rho _i v_A}, \end{eqnarray} \tag{ 221 }$$

where we used the definitions (135) of the stream and flux functions Φ and Ψ.

These equations are a reflection of the fact that, for k_⊥ρ_i ≫ 1, the ion response is effectively purely Boltzmann, with the gyrokinetic part h_i contributing nothing to the fields or flows (see Equation (54) with h_i omitted; h_i does, however, play an important role in the energy balance and ion heating, as explained in Sections 7.8–7.10 below). The Boltzmann response for ion density is expressed by Equation (221). Equation (222) states that the parallel ion flow velocity can be neglected. Finally, Equation (223) expresses the pressure balance for Boltzmann (and, therefore, isothermal) electrons (Equation (103)) and ions: if we write

$$\begin{equation} {B_0\delta B_\parallel \over 4\pi } = -\delta p_i- \delta p_e= -T_{0i}\delta n_i- T_{0e}\delta n_e, \end{equation} \tag{ 224 }$$

it follows that

$$\begin{equation} {\delta B_\parallel \over B_0} = -{\beta _i\over 2}\left(1+{Z\over \tau }\right){\delta n_e\over n_{0e}}, \end{equation} \tag{ 225 }$$

which, combined with Equation (221), gives Equation (223). We remind the reader that the perpendicular Ampère's law, from which Equation (223) was derived (Equation (66) via Equation (120)) is, in gyrokinetics, indeed equivalent to the statement of perpendicular pressure balance (see Section 3.3).

Substituting Equations (221)–(223) into Equations (116)–(117), we obtain the following closed system of equations

$$\begin{eqnarray} {\partial \Psi \over \partial t} &=& v_A\left(1+{Z\over \tau }\right){\bf\hat{b}}\cdot {\bm\nabla}\Phi,\\ {\partial \Phi \over \partial t} &=& -{v_A\over 2 + \beta _i\left(1+Z/\tau \right)}\,{\bf\hat{b}}\cdot {\bm\nabla}\left(\rho _i^2\nabla _{\perp }^2\Psi \right). \end{eqnarray} \tag{ 226 }$$

Note that, using Equation (223), Equations (226) and (227) can be recast as two coupled evolution equations for the perpendicular and parallel components of the perturbed magnetic field, respectively (Equations (C10) in Appendix C.2).

We shall refer to Equations (226)–(227) as electron reduced MHD (ERMHD). They are related to the electron magnetohydrodynamics (EMHD)—a fluid-like approximation that evolves the magnetic field only and arises if one assumes that the magnetic field is frozen into the electron flow velocity u_e, while the ions are immobile, u_i = 0 (Kingsep et al. 1990):

$$\begin{eqnarray} &&{\partial {\bf B}\over \partial t} = -{c\over 4\pi en_{0e}}{\bm\nabla}\times \left[\left({\bm\nabla}\times {\bf B}\right)\times {\bf B}\right]. \end{eqnarray} \tag{ 228 }$$

As explained in Appendix C.2, the result of applying the RMHD/gyrokinetic ordering (Sections 2.1 and 3.1) to Equation (228), where ${\bf B}=B_0{\bf\hat{z}}+ \delta {\bf B}$ and

$$\begin{eqnarray} &&{\delta {\bf B}\over B_0} = {1\over v_A}{\bf\hat{z}}\times {\bm\nabla}_{\perp }\Psi + {\bf\hat{z}}\,{\delta B_\parallel \over B_0}, \end{eqnarray} \tag{ 229 }$$

coincides with our Equations (226)–(227) in the effectively incompressible limits of β_i ≫ 1 or β_e = β_iZ/τ ≫ 1. When betas are arbitrary, density fluctuations cannot be neglected compared to the magnetic-field-strength fluctuations (Equation (225)) and give rise to perpendicular ion flows with ${\bm\nabla}\cdot {\bf u}_i\ne 0$. Thus, our ERMHD system constitutes the appropriate generalization of EMHD for low-frequency anisotropic fluctuations without the assumption of incompressibility.

A (more tenuous) relationship also exists between our ERMHD system and the so-called Hall MHD, which, like EMHD, is based on the magnetic field being frozen into the electron flow, but includes the ion motion via the standard MHD momentum equation (Equation (8)). Strictly speaking, Hall MHD can only be used in the limit of cold ions, τ = T_0i/T_0e ≪ 1 (see, e.g., Ito et al. 2004; Hirose et al. 2004, and Appendix E), in which case it can be shown to reduce to Equations (226)–(227) in the appropriate small-scale limit (Appendix E). Although τ ≪ 1 is not a natural assumption for most space and astrophysical plasmas, Hall MHD has, due to its simplicity, been a popular theoretical paradigm in the studies of space and astrophysical plasma turbulence (see Section 8.2.6). We have therefore devoted Appendix E to showing how this approximation fits into the theoretical framework proposed here: namely, we derive the anisotropic low-frequency version of the Hall MHD approximation from gyrokinetics under the assumption τ ≪ 1 and discuss the role of the ion inertial and ion sound scales, which acquire physical significance in this limit. However, outside this Appendix, we assume τ ∼ 1 everywhere and shall not use Hall MHD.

The validity of the ERMHD equations as a model for plasma dynamics in the dissipation range is further discussed in Section 7.6.

The linear modes supported by ERMHD are kinetic Alfvén waves (KAW) with frequencies

$$\begin{eqnarray} &&\omega _{{\bf k}}= \pm \sqrt{1+Z/\tau \over 2+\beta _i\left(1+Z/\tau \right)}\, k_{\perp }\rho _ik_{\parallel }v_A. \end{eqnarray} \tag{ 230 }$$

This dispersion relation is illustrated in Figure 8: note that the transition from Alfvén waves to dispersive KAW always occurs at k_⊥ρ_i ∼ 1, even when β_i ≪ 1 or β_i ≫ 1. In the latter case, there is a sharp frequency jump at the transition (accompanied by very strong ion Landau damping).

The eigenfunctions corresponding to the two waves with frequencies (230) are

$$\begin{equation} \Theta ^\pm _{{\bf k}}= \sqrt{\left(1+{Z\over \tau }\right)\left[2+\beta _i\left(1+{Z\over \tau }\right)\right]} \,{\Phi _{{\bf k}}\over \rho _i} \mp k_{\perp }\Psi _{{\bf k}}. \end{equation} \tag{ 231 }$$

Using Equations (229) and (223), the perturbed magnetic-field vector can be expressed as follows

$$\begin{equation} {\delta {\bf B}_{{\bf k}}\over B_0} = -i{\bf\hat{z}}\times {{{\bf k}}_{\perp }\over k_{\perp }}{\Theta ^+_{{\bf k}}- \Theta ^-_{{\bf k}}\over 2v_A} +{\bf\hat{z}}\,\sqrt{1+Z/\tau \over 2+\beta _i\left(1+Z/\tau \right)}\, {\Theta ^+_{{\bf k}}+ \Theta ^-_{{\bf k}}\over 2v_A}, \end{equation} \tag{ 232 }$$

so, for a single "+" or "−" wave (corresponding to Θ⁻_k = 0 or Θ⁺_k = 0, respectively), δB_k rotates in the plane perpendicular to the wave vector k_⊥ clockwise with respect to the latter, while the wave propagates parallel or antiparallel to the guide field (Figure 9).

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The waves are elliptically right-hand polarized. Indeed, using Equation (223), the perpendicular electric field is:

$$\begin{eqnarray} \nonumber {\bf E}_{\perp {{\bf k}}}&=& -i{{\bf k}}_{\perp }\varphi + {i\omega _{{\bf k}}\over c}\,{\bf A}_{\perp {{\bf k}}}\\ &=& \left[-i{{\bf k}}_{\perp }+ {\bf\hat{z}}\times {{\bf k}}_{\perp }{\omega _{{\bf k}}\over \Omega _i} {\beta _i\over k_{\perp }^2\rho _i^2}\left(1+{Z\over \tau }\right)\right]\varphi \end{eqnarray} \tag{ 233 }$$

(cf. Gary 1986; Hollweg 1999). The second term is small in the gyrokinetic expansion, so this is a very elongated ellipse (Figure 9).

As we are about to argue for a critically balanced KAW turbulence in a fashion analogous to the GS theory for the Alfvén waves (Section 1.2), it is a natural question to ask how similar the nonlinear properties of a putative KAW cascade will be to an Alfvén-wave cascade. As in the case of Alfvén waves, there are two counterpropagating linear modes (Equations (230) and (231)), and it turns out that certain superpositions of these modes (KAW packets) are also exact nonlinear solutions of Equations (226)–(227). Let us show that this is the case.

We might look for the nonlinear solutions of Equations (226)–(227) by requiring that the nonlinear terms vanish. Since ${\bf\hat{b}}\cdot {\bm\nabla}={\partial /\partial z} + (1/v_A)\lbrace \Psi,\cdots \rbrace$, this gives

$$\begin{eqnarray} \lbrace \Psi,\Phi \rbrace = 0 &\quad \Rightarrow \quad & \Psi = c_1\Phi, \end{eqnarray} \tag{ 234 }$$

$$\begin{eqnarray} \lbrace \Psi,\rho _i^2\nabla _{\perp }^2\Psi \rbrace = 0 &\quad \Rightarrow \quad & \rho _i^2\nabla _{\perp }^2\Psi = c_2\Psi, \end{eqnarray} \tag{ 235 }$$

where c₁ and c₂ are constants. Whether such solutions are possible is determined by substituting Equations (234) and (235) into Equations (226) and (227) and demanding that the two resulting linear equations be consistent with each other (both equations now just evolve Ψ). This is achieved if²⁹

$$\begin{eqnarray} &&c_1^2 = -{1\over c_2}\left(1+{Z\over \tau }\right)\left[2+\beta _i\left(1+{Z\over \tau }\right)\right], \end{eqnarray} \tag{ 236 }$$

so real solutions exist if c₂ < 0. In particular, wave packets consisting of KAW given by one of the linear eigenmodes (231) with an arbitrary shape in z but confined to a single shell |k_⊥| = k_⊥ = const, satisfy Equations (234)–(236) with c₂ = −k²_⊥ρ²_i. This outcome is, in fact, only mildly non-trivial: in gyrokinetics, the Poisson bracket nonlinearity (Equation (59)) vanishes for any monochromatic (in k_⊥) mode because the Poisson bracket of two modes with wavenumbers k_⊥ and k'_⊥ is $\propto {\bf\hat{z}}\cdot ({{\bf k}}_{\perp }\times {{\bf k}}_{\perp }^{\prime })$. Therefore, any monochromatic solution of the linearized equations is also an exact nonlinear solution. As we have shown above, a superposition of monochromatic KAW that have a fixed k_⊥, or, somewhat more generally, satisfy Equation (235) with a fixed c₂, is still an exact solution.

Note that a similar procedure applied to the RMHD Equations (17)–(18) returns the Elsasser solutions: perturbations of arbitrary shape that satisfy Φ = ±Ψ. The physical difference between these finite-amplitude Alfven-wave packets and the finite-amplitude KAW packets discussed above is that nonlinear interactions can occur not just between counterpropagating KAW but also between copropagating ones—a natural conclusion because KAW are dispersive (their group velocity along the guide field is ∝v_Ak_⊥ρ_i), so copropagating waves with different k_⊥ can "catch up" with each other and interact.³⁰

A scaling theory for the turbulence described by Equations (221)–(227) can be constructed along the same lines as the GS theory for the Alfvén-wave turbulence (Section 1.2). Namely, we shall assume that the turbulence below the ion gyroscale consists of KAW-like fluctuations with k_∥ ≪ k_⊥ (Quataert & Gruzinov 1999) and that the interactions between them are critically balanced (Cho & Lazarian 2004), i.e., that the propagation time and nonlinear interaction time are comparable at every scale. We stress that none of these assumptions are, strictly speaking, inevitable³¹ (and, in fact, neither were they inevitable in the case of Alfvén waves). Since we have derived Equations (226)–(227) from gyrokinetics, the anisotropy of the fluctuations described by these equations is hard-wired, but it is not guaranteed that the actual physical cascade below the ion gyroscale is indeed anisotropic, although analysis of solar-wind measurements does seem to indicate that at least a significant fraction of it is (see Leamon et al. 1998; Hamilton et al. 2008). Numerical simulations based on Equation (228) (Biskamp et al. 1996, 1999; Ghosh et al. 1996; Ng et al. 2003; Cho & Lazarian 2004; Shaikh & Zank 2005) have revealed that the spectrum of magnetic fluctuations scales as k^−7/3_⊥, the outcome consistent with the assumptions stated above. Let us outline the argument that leads to this scaling.

First assume that the fluctuations are KAW-like and that Θ⁺ and Θ⁻ (Equation (231)) have similar scaling. This implies

$$\begin{eqnarray} &&\Psi _{\lambda }\sim \sqrt{1+\beta _i}\,{\lambda \over \rho _i}\,{\Phi _{\lambda }} \end{eqnarray} \tag{ 237 }$$

(for the purposes of scaling arguments and order-of-magnitude estimates, we set Z/τ = 1, but keep the β_i dependence so low- and high-beta limits could be recovered if necessary). The fact that fixed-k_⊥ KAW packets, which satisfy Equation (237) with λ = 1/k_⊥, are exact nonlinear solutions of the ERMHD equations (Section 7.4) lends some credence to this assumption.

Assuming scale-space locality of interactions implies a constant-flux KAW cascade: analogously to Equation (1),

$$\begin{eqnarray} &&{(\Psi _{\lambda }/\lambda)^2\over \tau _{{\rm KAW}\lambda }} \sim {(1+\beta _i)(\Phi _{\lambda }/\rho _i)^2\over \tau _{{\rm KAW}\lambda }} \sim \varepsilon _{\rm KAW}= {\rm const}, \end{eqnarray} \tag{ 238 }$$

where τ_KAWλ is the cascade time and ε_KAW is the KAW energy flux proportional to the fraction of the total flux ε (or the total turbulent power $\overline{P_{\rm ext}}$; see Section 3.4) that was converted into the KAW cascade at the ion gyroscale.

Using Equations (226)–(227) and Equation (237), it is not hard to see that the characteristic nonlinear decorrelation time is λ²/Φ_λ. If the turbulence is strong, then this time is comparable to the inverse KAW frequency (Equation (230)) scale by scale and we may assume the cascade time is comparable to either:

$$\begin{eqnarray} &&\tau _{{\rm KAW}\lambda }\sim {\lambda ^2\over \Phi _{\lambda }} \sim {1\over \sqrt{1+\beta _i}}{\rho _i\over \lambda }{v_A\over l_{\parallel \lambda }}. \end{eqnarray} \tag{ 239 }$$

In other words, this says that $\partial /\partial z\sim (\delta {\bf B}_\perp /B_0)\cdot {\bm\nabla}_{\perp }$ and so δB_⊥λ/B₀ ∼ λ/l_∥λ (note that the last relation confirms that our scaling arguments do not violate the gyrokinetic ordering; see Sections 2.1 and 3.1). Equation (239) is the critical-balance assumption for KAW. As in the case of the Alfvén waves (Section 1.2), we might argue physically that the critical balance is set up because the parallel correlation length l_∥λ is determined by the condition that a wave can propagate the distance l_∥λ in one nonlinear decorrelation time corresponding to the perpendicular correlation length λ.

Combining Equations (238) and (239), we get the desired scaling relations for the KAW turbulence:

$$\begin{eqnarray} \Phi _{\lambda }&\sim & \left({\varepsilon _{\rm KAW}\over \varepsilon }\right)^{1/3} {v_A\over (1+\beta _i)^{1/3}}\,l_0^{-1/3}\rho _i^{2/3}\lambda ^{2/3}, \end{eqnarray} \tag{ 240 }$$

$$\begin{eqnarray} l_{\parallel \lambda }&\sim & \left({\varepsilon \over \varepsilon _{\rm KAW}}\right)^{1/3} {l_0^{1/3}\rho _i^{1/3}\lambda ^{1/3}\over (1+\beta _i)^{1/6}}, \end{eqnarray} \tag{ 241 }$$

where l₀ = v³_A/ε, as in Section 1.2. The first of these scaling relations is equivalent to a k^−7/3_⊥ spectrum of magnetic energy, the second quantifies the anisotropy (which is stronger than for the GS turbulence). Both scalings were confirmed in the numerical simulations of Cho & Lazarian (2004)—it is their detection of the scaling (241) that makes a particularly strong case that KAW turbulence is not weak and that the critical balance hypothesis applies.

For KAW-like fluctuations, the density (Equation (221)) and magnetic field (Equations (223) and (231)) have the same spectrum as the scalar potential, i.e., k^−7/3_⊥, while the electric field E ∼ k_⊥φ has a k^−1/3_⊥ spectrum. The solar-wind fluctuation spectra reported by Bale et al. (2005) indeed are consistent with a transition to KAW turbulence around the ion gyroscale: k^−5/3 magnetic and electric-field power spectra at kρ_i ≪ 1 are replaced, for kρ_i ≳ 1, with what appears to be consistent with a k^−7/3 scaling for the magnetic-field spectrum and a k^−1/3 for the electric one (see Figure 1). A similar result is recovered in fully gyrokinetic simulations with β_i = 1, τ = 1 (Howes et al. 2008b). However, not all solar-wind observations are quite as straightforwardly supportive of the notion of the KAW cascade and much steeper magnetic-fluctuation spectra have also been reported (e.g., Denskat et al. 1983; Leamon et al. 1998; Smith et al. 2006). Possible reasons for this will emerge in Sections 7.6 and 7.11 and the solar-wind data are further discussed in Sections 8.2.4 and 8.2.5.

The ERMHD equations derived in Section 7 are valid provided k_⊥ρ_i ≫ 1 and also provided it is sufficient to use the leading order in the mass-ratio expansion (isothermal electrons; see Section 4). In particular, this means that the electron Landau damping is neglected. Asymptotically speaking, this is a rigorous limit, but one must be cautious in applying it to real plasmas. Since the width of the scale range where k_⊥ρ_i ≫ 1 and k_⊥ρ_e ≪ 1 is only ∼(m_i/m_e)^1/2 ≃ 43, for some values of the plasma parameters (T_0i/T_0e and β_i) there may not be a very broad interval of scales where the electron Landau damping is truly negligible. Consider, for example, the low-beta limit, β_i ≪ 1. In this limit, the KAW frequency is ω ∼ k_⊥ρ_ik_∥v_A (Equation (230)). The electron Landau damping becomes important when ω ∼ k_∥v_the, or $k_{\perp }\rho _e\sim \sqrt{\beta _i}\ll 1$, so the ERMHD approximation breaks down and, consequently, the KAW cascade, if any, should be interrupted well before the electron gyroscale is reached. Figure 8 shows the solution of the full gyrokinetic dispersion relation (Howes et al. 2006) for small, unity and large β_i. One can judge for which scales and how well (or how badly) the ERMHD approximation holds from the precision with which the exact frequency follows the asymptotic solution Equation (230) and from the relative strength of the damping compared to the real frequency of the waves.

Non-negligible electron Landau damping may affect turbulence spectra because one can no longer assume a constant flux of KAW energy as we did in Section 7.5. To evaluate the consequences of this effect, Howes et al. (2008a) constructed a simple model of spectral energy transfer and concluded that Landau damping leads to steepening of the KAW spectra—one of several possible reasons for steep dissipation-range spectra observed in space plasmas (see also Section 7.11).

As ERMHD is a limit of the isothermal-electron-fluid system (Section 4), the magnetic-field lines remain unbroken (see Section 4.3). Within the orderings employed above (small mass ratio, ν_ii ∼ ω, β_i ∼ 1, τ ∼ 1), the flux unfreezes only in the vicinity of the electron gyroscale. It is interesting to evaluate somewhat more precisely the scale at which this happens as a function of plasma parameters.

Physically, there are three kinds of mechanisms by which the flux conservation is broken: electron inertia, the effects of finite electron gyroradius, and Ohmic resistivity. Let us take the v_∥ moment of the electron gyrokinetic equation (Equation (57), s = e, integration at constant r) and use Equation (222) to evaluate the inertial term in the resulting parallel electron momentum equation:

$$\begin{eqnarray} &&{cm_e\over e}{\partial u_{\parallel e}\over \partial t} = {\partial \over \partial t}\,d_e^2\nabla _{\perp }^2A_\parallel, \end{eqnarray} \tag{ 242 }$$

where $d_e=\rho _e/\sqrt{\beta _e}$ is the electron inertial scale and β_e = Zβ_i/τ. Comparing this with the ∂A_∥/∂t term in the right-hand side of the electron momentum equation, we see that the electron inertia becomes important when $k_{\perp }\rho _e\sim \sqrt{\beta _e}$. The finite-gyroradius effects enter when k_⊥ρ_e ∼ 1. Thus, at low β_e, the electron inertia becomes important above the electron gyroscale, whereas at high β_e, the finite-gyroradius effects enter first. Finally, the Ohmic resistivity comes from the collision term (see Appendix B.4):

$$\begin{equation} {cm_e\over e}{1\over n_{0e}}\int d^3{\bf v}\,v_\parallel \left({\partial h_e\over \partial t}\right)_{\rm c}\sim {cm_e\over e}\nu _{ei}u_{\parallel e}\sim \nu _{ei}k_{\perp }^2 d_e^2A_\parallel. \end{equation} \tag{ 243 }$$

Thus, resistivity starts to act when k_⊥d_e ∼ (ω/ν_ei)^1/2. Using the KAW frequency (Equation (230)) to estimate ω and assuming that τ is not small, we get

$$\begin{eqnarray} &&k_{\perp }\rho _e \sim k_{\parallel }\lambda _{{\rm mfp}i}\sqrt{\beta _i\over 1+\beta _i}{Z^2\over \tau ^2}. \end{eqnarray} \tag{ 244 }$$

Thus, the resistive scale can only be larger the electron gyroscale if the plasma is collisional (k_∥λ_mfpi ≪ 1) and/or electrons are much colder than ions (τ ≫ 1) and/or β_i ≪ 1. Note if only the last of these conditions is satisfied, the electron inertia still becomes important at larger scales than resistivity.

The generalized energy (Section 3.4) in the limit k_⊥ρ_i ≫ 1 is calculated by substituting Equations (221) and (223) into Equation (109):

$$\begin{eqnarray} \nonumber W &=& \int d^3{\bf r}\left\lbrace \int d^3{\bf v}\,{T_{0i}\langle h_i^2\rangle _{\bf r}\over 2F_{0i}} + {\delta B_\perp ^2\over 8\pi } \right.\\ \nonumber &&\left.+\ {n_{0i}T_{0i}\over 2}\left(1+{Z\over \tau }\right)\left[1+{\beta _i\over 2}\left(1+{Z\over \tau }\right)\right] \left({Ze\varphi \over T_{0i}}\right)^2\right\rbrace \\ &=& W_{h_i}+ W_{\rm KAW}. \end{eqnarray} \tag{ 245 }$$

Here the first term, $W_{h_i}$, is the total variance of h_i, which is proportional to minus the entropy of the ion gyrocenter distribution (see Section 3.5) and whose cascade to collisional scales will be discussed in Sections 7.9 and 7.10. The remaining two terms are the independently cascaded KAW energy:

$$\begin{eqnarray} \nonumber W_{\rm KAW} &=& \int d^3{\bf r}\,{m_in_{0i}\over 2}\Biggl \lbrace |{\bm\nabla}_{\perp }\Psi |^2 \Biggr.\\ \nonumber &&\Biggl. +\ \left(1+{Z\over \tau }\right)\left[1+{\beta _i\over 2}\left(1+{Z\over \tau }\right)\right]{\Phi ^2\over \rho _i^2}\Biggr \rbrace \\ &=& \int d^3{\bf r}\,{m_in_{0i}\over 2}\left(|\Theta ^+|^2 + |\Theta ^-|^2\right). \end{eqnarray} \tag{ 246 }$$

Although we can write W_KAW as the sum of the energies of the "+" and "−" linear KAW eigenmodes (Equation (231)), which are also exact nonlinear solutions (Section 7.4), the two do not cascade independently and can exchange energy. Note that the ERMHD equations also conserve ∫d³r ΨΦ, which is readily interpreted as the helicity of the perturbed magnetic field (see Appendix F.3). However, it does not affect the KAW cascade discussed in Section 7.5 because it can be argued to have a tendency to cascade inversely (Appendix F.6).

Comparing the way the generalized energy is split above and below the ion gyroscale (see Section 5.6 for the k_⊥ρ_i ≪ 1 limit), we interpret what happens at the k_⊥ρ_i ∼ 1 transition as a redistribution of the power that arrived from large scales between a cascade of KAW and a cascade of the (minus) gyrocenter entropy in the phase space (see Figure 5). The latter cascade is the way in which the energy diverted from the electromagnetic fluctuations by the collisionless damping (wave–particle interaction) can be transferred to the collisional scales and deposited into heat (Section 7.1). The concept of entropy cascade as the key agent in the heating of the plasma was introduced in Section 3.5, where we promised a more detailed discussion later on. We now proceed to this discussion.

The ion-gyrocenter distribution function h_i satisfies the ion gyrokinetic Equation (121), where ion–electron collisions are neglected under the mass-ratio expansion. At k_⊥ρ_i ≫ 1, the dominant contribution to $\langle \chi \rangle _{{\bf R}_i}$ comes from the electromagnetic fluctuations associated with KAW turbulence. Since the KAW cascade is decoupled from the entropy cascade, h_i is a passive tracer of the ring-averaged KAW turbulence in phase space. Expanding the Bessel functions in the expression for $\langle \chi \rangle _{{\bf R}_i,{{\bf k}}}$ (a_i ≫ 1 in Equation (69) with s = i) and making use of Equations (222)–(223) and of the KAW scaling Ψ ∼ Φ/k_⊥ρ_i (Equation (231)), it is not hard to show that

$$\begin{eqnarray} &&{Ze\over T_{0i}}\,\langle \chi \rangle _{{\bf R}_i,{{\bf k}}}\simeq {Ze\over T_{0i}}\,\langle \varphi \rangle _{{\bf R}_i,{{\bf k}}} = {2\over \sqrt{\beta _i}}{J_0(a_i)\Phi _{{\bf k}}\over \rho _i v_A}, \end{eqnarray} \tag{ 247 }$$

where

$$\begin{eqnarray} &&\qquad\quad\; J_0(a_i)\simeq \sqrt{2\over \pi a_i}\,\cos \left(a_i-{\pi \over 4}\right),\quad a_i=k_{\perp }\rho _i\,{v_\perp \over v_{{\rm th}i}}, \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\nonumber \\ \end{eqnarray} \tag{ 248 }$$

so h_i satisfies (Equation (121))

$$\begin{equation} {\partial h_i\over \partial t} + v_\parallel {\partial h_i\over \partial z} + \left\lbrace \langle \Phi \rangle _{{\bf R}_i},h_i\right\rbrace = {2\over \sqrt{\beta _i}\,\rho _i v_A} {\partial \langle \Phi \rangle _{{\bf R}_i}\over \partial t}\,F_{0i}+ \left<C_{ii}[h_i]\right>_{{\bf R}_i} \end{equation} \tag{ 249 }$$

with the conservation law (Equation (70), s = i)

$$\begin{eqnarray} \nonumber {1\over T_{0i}}{dW_{h_i}\over dt} &\equiv & {d\over dt}\int d^3{\bf v}\int d^3{\bf R}_i\,{h_i^2\over 2F_{0i}}\\ \nonumber &=& {2\over \sqrt{\beta _i}\,\rho _i v_A}\int d^3{\bf v}\int d^3{\bf R}_i\,{\partial \langle \Phi \rangle _{{\bf R}_i}\over \partial t}\,h_i\\ &&+\ \int d^3{\bf v}\int d^3{\bf R}_i\,{h_i\left<C_{ii}[h_i]\right>_{{\bf R}_i}\over F_{0i}}. \qquad \end{eqnarray} \tag{ 250 }$$

7.9.1. Nonlinear Perpendicular Phase Mixing

The wave–particle interaction term (the first term on the right hand sides of these two equations) will shortly be seen to be subdominant at k_⊥ρ_i ≫ 1. It represents the source of the invariant $W_{h_i}$ due to the collisionless damping at the ion gyroscale of some fraction of the energy arriving from the inertial range. In a stationary turbulent state, we should have $\overline{dW_{h_i}/dt} = 0$ and this source should be balanced on average by the (negative definite) collisional dissipation term (= heating; see Section 3.5). This balance can only be achieved if h_i develops small scales in the velocity space and carries the generalized energy, or, in this case, entropy, to scales in the phase space at which collisions are important. A quick way to see this is by recalling that the collision operator has two velocity derivatives and can only balance the terms on the left-hand side of Equation (249) if

$$\begin{eqnarray} &&\nu _{ii}v_{{\rm th}i}^2\left({\partial \over \partial {\bf v}}\right)^2\sim \omega \quad \Rightarrow \quad {\delta v\over v_{{\rm th}i}}\sim \Biggl ({\nu _{ii}\over \omega }\Biggr)^{1/2}, \end{eqnarray} \tag{ 251 }$$

where ω is the characteristic frequency of the fluctuations of h_i. If ν_ii ≪ ω, δv/v_thi ≪ 1. This is certainly true for k_⊥ρ_i ∼ 1: taking ω ∼ k_∥v_A and using k_∥λ_mfpi ≫ 1 (which is the appropriate limit at and below the ion gyroscale for most of the plasmas of interest; cf. footnote ²⁴), we have $\nu _{ii}/\omega \sim \sqrt{\beta _i}/k_{\parallel }\lambda _{{\rm mfp}i}\ll 1$.

The condition (251) means that the collision rate can be arbitrarily small—this will always be compensated by the sufficiently fine velocity-space structure of the distribution function to produce a finite amount of entropy production (heating) independent of ν_ii in the limit ν_ii → +0. The situation bears some resemblance to the emergence of small spatial scales in neutral-fluid turbulence with arbitrarily small but non-zero viscosity (Kolmogorov 1941). The analogy is not perfect, however, because the ion gyrokinetic Equation (249) does not contain a nonlinear interaction term that would explicitly cause a cascade in the velocity space. Instead, the (ring-averaged) KAW turbulence mixes h_i in the gyrocenter space via the nonlinear term in Equation (249), so h_i will have small-scale structure in R_i on characteristic scales much smaller than ρ_i. Let us assume that the dominant nonlinear effect is a local interaction of the small-scale fluctuations of h_i with the similarly small-scale component of $\langle \Phi \rangle _{{\bf R}_i}$. Since ring averaging is involved and k_⊥ρ_i is large, the values of $\langle \Phi \rangle _{{\bf R}_i}$ corresponding to two velocities v and v' will come from spatially decorrelated electromagnetic fluctuations if k_⊥v_⊥/Ω_i and k_⊥v'_⊥/Ω_i (the argument of the Bessel function in Equation (247)) differ by order unity, i.e., for

$$\begin{eqnarray} &&{\delta v_\perp \over v_{{\rm th}i}}={|v_\perp -v_\perp ^{\prime }|\over v_{{\rm th}i}}\sim {1\over k_{\perp }\rho _i} \end{eqnarray} \tag{ 252 }$$

(see Figure 10). This relation gives a correspondence between the decorrelation scales of h_i in the position and velocity space. Combining Equations (252) and (251), we see that there is a collisional cutoff scale determined by k_⊥ρ_i ∼ (ω/ν_ii)^1/2 ≫ 1.³² The cutoff scale is much smaller than the ion gyroscale. In the range between these scales, collisional dissipation is small. The ion entropy fluctuations are transferred across this scale range by means of a cascade, for which we will construct a scaling theory in Section 7.9.2 (and, for the case without the background KAW turbulence, in Section 7.10).

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It is important to emphasize that no matter how small the collisional cutoff scale is, all of the generalized energy channeled into the entropy cascade at the ion gyroscale eventually reaches it and is converted into heat. Note that the rate at which this happens is in general amplitude-dependent because the process is nonlinear, although we will argue in Section 7.9.4 (see also Section 7.10.3) that the nonlinear cascade time and the parallel linear propagation (particle streaming) time are related by a critical-balance-like condition (we will also argue there that the linear parallel phase mixing, which can generate small scales in v_∥, is a less efficient process than the nonlinear perpendicular one discussed above).

It is interesting to note the connection between the entropy cascade and certain aspects of the gyrofluid closure formalism developed by Dorland & Hammett (1993). In their theory, the emergence of small scales in v_⊥ manifested itself as the growth of high-order v_⊥ moments of the gyrocenter distribution function. They correctly identified this effect as a consequence of the nonlinear perpendicular phase mixing of the gyrocenter distribution function caused by a perpendicular-velocity-space spread in the ring-averaged E × B velocities (given by $\langle {\bf u}_E\rangle _{{\bf R}_i} = {\bf\hat{z}}\times {\bm\nabla}\langle \Phi \rangle _{{\bf R}_i}$ in our notation) arising at and below the ion gyroscale.

7.9.2. Scalings

Since entropy is a conserved quantity, we will follow the well trodden Kolmogorov path, assume locality of interactions in scale space and constant entropy flux, and conclude, analogously to Equation (1),

$$\begin{eqnarray} &&{v_{{\rm th}i}^8\over n_{0i}^2}{h_{i\lambda }^2\over \tau _{h\lambda }}\sim \varepsilon _{h}={\rm const}, \end{eqnarray} \tag{ 253 }$$

where ε_h is the entropy flux proportional to the fraction of the total turbulent power ε (or $\overline{P_{\rm ext}}$; see Section 3.4) that was diverted into the entropy cascade at the ion gyroscale, and τ_hλ is the cascade time that we now need to find.

By the critical-balance assumption, the decorrelation time of the electromagnetic fluctuations in KAW turbulence is comparable at each scale to the KAW period at that scale and to the nonlinear interaction time (Equation (239)):

$$\begin{equation} \tau _{{\rm KAW}\lambda }\sim {\lambda ^2\over \Phi _{\lambda }} \sim \left({\varepsilon \over \varepsilon _{\rm KAW}}\right)^{1/3}(1+\beta _i)^{1/3} {l_0^{1/3}\rho _i^{-2/3}\lambda ^{4/3}\over v_A}. \end{equation} \tag{ 254 }$$

The characteristic time associated with the nonlinear term in Equation (249) is longer than τ_KAWλ by a factor of (ρ_i/λ)^1/2 due to the ring averaging, which reduces the strength of the nonlinear interaction. This weakness of the nonlinearity makes it possible to develop a systematic analytical theory of the entropy cascade (Schekochihin & Cowley 2009). It is also possible to estimate the cascade time τ_hλ via a more qualitative argument analogous to that first devised by Kraichnan (1965) for the weak turbulence of Alfvén waves: during each KAW correlation time τ_KAWλ, the nonlinearity changes the amplitude of h_i by only a small amount:

$$\begin{eqnarray} &&\Delta h_{i\lambda }\sim (\lambda /\rho _i)^{1/2}h_{i\lambda }\ll h_{i\lambda }; \end{eqnarray} \tag{ 255 }$$

these changes accumulate with time as a random walk, so after time t, the cumulative change in amplitude is Δh_iλ(t/τ_KAWλ)^1/2; finally, the cascade time t = τ_hλ is the time after which the cumulative change in amplitude is comparable to the amplitude itself, which gives, using Equation (254),

$$\begin{equation} \tau _{h\lambda }\sim {\rho _i\over \lambda }\,\tau _{{\rm KAW}\lambda }\sim \left({\varepsilon \over \varepsilon _{\rm KAW}}\right)^{1/3}(1+\beta _i)^{1/3} {l_0^{1/3}\rho _i^{1/3}\lambda ^{1/3}\over v_A}. \end{equation} \tag{ 256 }$$

Substituting this into Equation (253), we get

$$\begin{equation} h_{i\lambda }\sim {n_{0i}\over v_{{\rm th}i}^3}\Biggl ({\varepsilon _{h}\over \varepsilon }\Biggr)^{1/2} \left({\varepsilon \over \varepsilon _{\rm KAW}}\right)^{1/6} {(1+\beta _i)^{1/6}\over \sqrt{\beta _i}}\, l_0^{-1/3}\rho _i^{1/6}\lambda ^{1/6}, \end{equation} \tag{ 257 }$$

which corresponds to a k^−4/3_⊥ spectrum of entropy.

In the argument presented above, we assumed that the scaling of h_i was determined by the nonlinear mixing of h_i by the ring-averaged KAW fluctuations rather than by the wave–particle-interaction term on the right-hand side of Equation (249). We can now confirm the validity of this assumption. The change in amplitude of h_i in one KAW correlation time τ_KAWλ due to the wave–particle-interaction term is

$$\begin{eqnarray} \nonumber \Delta h_{i\lambda }&\sim & {n_{0i}\over v_{{\rm th}i}^3}\left({\lambda \over \rho _i}\right)^{1/2} {\Phi _{\lambda }\over \sqrt{\beta _i}\,\rho _i v_A}\\ &\sim & {n_{0i}\over v_{{\rm th}i}^3}\left({\varepsilon _{\rm KAW}\over \varepsilon }\right)^{1/3} {1\over \sqrt{\beta _i}\,(1+\beta _i)^{1/3}}\, l_0^{-1/3}\rho _i^{-5/6}\lambda ^{7/6},\qquad \end{eqnarray} \tag{ 258 }$$

where we have used Equation (240). Comparing this with Equation (255) and using Equation (257), we see that Δh_iλ in Equation (258) is a factor of (λ/ρ_i)^1/2 smaller than Δh_iλ due to the nonlinear mixing.

7.9.3. Phase-Space Cutoff

To work out the cutoff scales both in the position and velocity space, we use Equations (251) and (252): in Equation (251), ω ∼ 1/τ_hλ, where τ_hλ is the characteristic decorrelation time of h_i given by Equation (256); using Equation (252), we find the cutoffs:

$$\begin{equation} {\delta v_\perp \over v_{{\rm th}i}}\sim {1\over k_{\perp }\rho _i}\sim (\nu _{ii}\tau _{\rho _i})^{3/5} = {\rm Do}^{-3/5}, \end{equation} \tag{ 259 }$$

where $\tau _{\rho _i}$ is the cascade time (Equation (256)) taken at λ = ρ_i. By a recently established convention, the dimensionless number ${\rm Do}=1/\nu _{ii}\tau _{\rho _i}$ is called the Dorland number. It plays the role of Reynolds number for kinetic turbulence, measuring the scale separation between the ion gyroscale and the collisional dissipation scale (Schekochihin et al. 2008b; Tatsuno et al. 2009a, 2009b).

7.9.4. Parallel Phase Mixing

Another assumption, which was made implicitly, was that the parallel phase mixing due to the second term on the left-hand side of Equation (249) could be ignored. This requires justification, especially because it is with this "ballistic" term that one traditionally associates the emergence of small-scale structure in the velocity space (e.g., Krommes & Hu 1994; Krommes 1999; Watanabe & Sugama 2004). The effect of the parallel phase mixing is to produce small scales in velocity space δv_∥ ∼ 1/k_∥t. Let us assume that the KAW turbulence imparts its parallel decorrelation scale to h_i and use the scaling relation (241) to estimate k_∥ ∼ l⁻¹_∥λ. Then, after one cascade time τ_hλ (Equation (256)), h_i is decorrelated on the parallel velocity scales

$$\begin{eqnarray} &&{\delta v_\parallel \over v_{{\rm th}i}}\sim {l_{\parallel \lambda }\over v_{{\rm th}i}\tau _{h\lambda }} \sim {1\over \sqrt{\beta _i(1+\beta _i)}}\sim 1. \end{eqnarray} \tag{ 260 }$$

We conclude that the nonlinear perpendicular phase mixing (Equation (259)) is more efficient than the linear parallel one. Note that up to a β_i-dependent factor Equation (260) is equivalent to a critical-balance-like assumption for h_i in the sense that the propagation time is comparable to the cascade time, or k_∥v_∥ ∼ τ⁻¹_hλ (see Equation (249)).

It is not currently known how one might determine analytically what fraction of the turbulent power arriving from the inertial range to the ion gyroscale is channeled into the KAW cascade and what fraction is dissipated via the kinetic ion-entropy cascade introduced in Section 7.9 (perhaps it can only be determined by direct numerical simulations). It is certainly a fact that in many solar-wind measurements, the relatively shallow magnetic-energy spectra associated with the KAW cascade (Section 7.5) fail to appear and much steeper spectra are detected (close to k⁻⁴; see Leamon et al. 1998; Smith et al. 2006). In view of this evidence, it is interesting to ask what would be the nature of electromagnetic fluctuations below the ion gyroscale if the KAW cascade failed to be launched, i.e., if all (or most) of the turbulent power were directed into the entropy cascade (i.e., if $W\simeq W_{h_i}$ in Section 7.8).

7.10.1. Equations

It is again possible to derive a closed set of equations for all fluctuating quantities.

Let us assume (and verify a posteriori; Section 7.10.4) that the characteristic frequency of such fluctuations is much lower than the KAW frequency (Equation (230)) so that the first term in Equation (116) is small and the equation reduces to the balance of the other two terms. This gives

$$\begin{eqnarray} &&{\delta n_e\over n_{0e}} = {e\varphi \over T_{0e}}, \end{eqnarray} \tag{ 261 }$$

meaning that the electrons are purely Boltzmann (h_e = 0 to lowest order; see Equation (101)). Then, from Equation (118),

$$\begin{equation} {Ze\varphi \over T_{0i}} \equiv {2\Phi \over \rho _iv_{{\rm th}i}} = \left(1+{\tau \over Z}\right)^{-1}\sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf r}} {1\over n_{0i}}\int d^3{\bf v}\,J_0(a_i)h_{i{{\bf k}}} \end{equation} \tag{ 262 }$$

Using Equation (262), we find from Equation (120) that the field-strength fluctuations are

$$\begin{equation} {\delta B_\parallel \over B_0} = -{\beta _i\over 2}\sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf r}} {1\over n_{0i}}\int d^3{\bf v}\,{2v_\perp ^2\over v_{{\rm th}i}^2}{J_1(a_i)\over a_i}\,h_{i{{\bf k}}}, \end{equation} \tag{ 263 }$$

which is smaller than Zeφ/T_0i by a factor of β_i/k_⊥ρ_i.

Therefore, we can neglect δB_∥/B₀ compared to δn_e/n_0e in Equation (117). Using Equation (261), we get what is physically the electron continuity equation:

$$\begin{eqnarray} &&{\partial \over \partial t}{e\varphi \over T_{0e}} + {\bf\hat{b}}\cdot {\bm\nabla}\left({c\over 4\pi en_{0e}}\,{\bm\nabla}_{\perp }^2A_\parallel + u_{\parallel i}\right) = 0, \end{eqnarray} \tag{ 264 }$$

$$\begin{eqnarray} &&u_{\parallel i}= \sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf r}}{1\over n_{0i}}\int d^3{\bf v}\,v_\parallel J_0(a_i)h_{i{{\bf k}}}. \end{eqnarray} \tag{ 265 }$$

Note that in terms of the stream and flux functions, Equation (264) takes the form

$$\begin{eqnarray} &&{\partial \over \partial z}\,\rho _i^2{\bm\nabla}_{\perp }^2\Psi = \sqrt{\beta _i}\left({2\tau \over Z}{1\over v_{{\rm th}i}}{\partial \Phi \over \partial t} + \rho _i\,{\partial u_{\parallel i}\over \partial z}\right), \end{eqnarray} \tag{ 266 }$$

where we have approximated ${\bf\hat{b}}\cdot {\bm\nabla}\simeq \partial /\partial z$, which will, indeed, be shown to be correct in Section 7.10.4.

Together with the ion gyrokinetic equation, which determines h_i, Equations (261)–(264) form a closed set. They describe low-frequency fluctuations of the density and electromagnetic field due solely to the presence of fluctuations of h_i below the ion gyroscale.

It follows from Equation (263) that δB_∥/B₀ contributes subdominantly to $\langle \chi \rangle _{{\bf R}_i}$ (Equation (69) with s = i and a_i ≫ 1). It will be verified a posteriori (Section 7.10.4) that the same is true for A_∥. Therefore, Equations (247) and (249) continue to hold, as in the case with KAW. This means that Equations (249) and (262) form a closed subset. Thus the kinetic ion-entropy cascade is self-regulating in the sense that h_i is no longer passive (as it was in the presence of KAW turbulence; Section 7.9) but is mixed by the ring-averaged "electrostatic" fluctuations of the scalar potential, which themselves are produced by h_i according to Equation (262).

The magnetic fluctuations are passive and determined by the electrostatic and entropy fluctuations via Equations (263) and (264).

7.10.2. Scalings

From Equation (262), we can establish a correspondence between Φ_λ and h_iλ (the electrostatic fluctuations and the fluctuations of the ion-gyrocenter distribution function):

$$\begin{equation} \Phi _{\lambda }\sim {\rho _iv_{{\rm th}i}}\left({\lambda \over \rho _i}\right)^{1/2} {h_{i\lambda }v_{{\rm th}i}^3\over n_{0i}}\left({\delta v_\perp \over v_{{\rm th}i}}\right)^{1/2} \sim {v_{{\rm th}i}^4\over n_{0i}}\,h_{i\lambda }\lambda, \end{equation} \tag{ 267 }$$

where the factor of (λ/ρ_i)^1/2 comes from the Bessel function (Equation (248)) and the factor of (δv_⊥/v_thi)^1/2 results from the v_⊥ integration of the oscillatory factor in the Bessel function times h_i, which decorrelates on small scales in the velocity space and, therefore, its integral accumulates in a random-walk-like fashion. The velocity-space scales are related to the spatial scales via Equation (252), which was arrived at by an argument not specific to KAW-like fluctuations and, therefore, continues to hold.

Using Equation (267), we find that the wave–particle-interaction term in the right-hand side of Equation (249) is subdominant: comparing it with ∂h_i/∂t shows that it is smaller by a factor of (λ/ρ_i)^3/2 ≪ 1. Therefore, it is the nonlinear term in Equation (249) that controls the scalings of h_iλ and Φ_λ.

We now assume again the scale-space locality and constancy of the entropy flux, so Equation (253) holds. The cascade (decorrelation) time is equal to the characteristic time associated with the nonlinear term in Equation (249): τ_hλ ∼ (ρ_i/λ)^1/2λ²/Φ_λ. Substituting this into Equation (253) and using Equation (267), we arrive at the desired scaling relations for the entropy cascade (Schekochihin et al. 2008b):

$$\begin{eqnarray} h_{i\lambda }&\sim & {n_{0i}\over v_{{\rm th}i}^3}\left({\varepsilon _{h}\over \varepsilon }\right)^{1/3} {1\over \sqrt{\beta _i}}\, l_0^{-1/3}\rho _i^{1/6}\lambda ^{1/6},\\ \Phi _{\lambda }&\sim & \left({\varepsilon _{h}\over \varepsilon }\right)^{1/3} {v_{{\rm th}i}\over \sqrt{\beta _i}}\,l_0^{-1/3}\rho _i^{1/6}\lambda ^{7/6},\\ \tau _{h\lambda }&\sim & \left({\varepsilon \over \varepsilon _{h}}\right)^{1/3} {\sqrt{\beta _i}\over v_{{\rm th}i}}\, l_0^{1/3}\rho _i^{1/3}\lambda ^{1/3}, \end{eqnarray} \tag{ 268 }$$

where l₀ = v³_A/ε, as in Section 1.2. Note that since the existence of this cascade depends on it not being overwhelmed by the KAW fluctuations, we should have ε_KAW ≪ ε and ε_h = ε − ε_KAW ≈ ε.

The scaling for the ion-gyrocenter distribution function, Equation (268), implies a k^−4/3_⊥ spectrum—the same as for the KAW turbulence (Equation (257)). The scaling for the cascade time, Equation (270), is also similar to that for the KAW turbulence (Equation (256)). Therefore the velocity- and gyrocenter-space cutoffs are still given by Equation (259), where $\tau _{\rho _i}$ is now given by Equation (270) taken at λ = ρ_i.

A new feature is the scaling of the scalar potential, given by Equation (269), which corresponds to a k^−10/3_⊥ spectrum (unlike the KAW spectrum, Section 7.5). This is a measurable prediction for the electrostatic fluctuations: the implied electric-field spectrum is k^−4/3_⊥. From Equation (261), we also conclude that the density fluctuations should have the same spectrum as the scalar potential, k^−10/3_⊥—another measurable prediction.

The scalings derived above for the spectra of the ion distribution function and of the scalar potential have been confirmed in the numerical simulations by Tatsuno et al. (2009a, 2009b), who studied decaying electrostatic gyrokinetic turbulence in two spatial dimensions. They also found velocity-space scalings in accord with Equation (252) (using a spectral representation of the correlation functions in the v_⊥ space based on the Hankel transform of the distribution function; see Plunk et al. 2009).

7.10.3. Parallel Cascade and Parallel Phase Mixing

We have again ignored the ballistic term (the second on the left-hand side) in Equation (249). We will estimate the efficiency of the parallel spatial cascade of the ion entropy and of the associated parallel phase mixing by making a conjecture analogous to the critical balance: assuming that any two perpendicular planes only remain correlated provided particles can stream between them in one nonlinear decorrelation time (cf. Sections 1.2 and 7.9.4), we conclude that the parallel particle-streaming frequency k_∥v_∥ should be comparable at each scale to the inverse nonlinear time τ⁻¹_hλ, so

$$\begin{eqnarray} &&k_{\parallel }v_{{\rm th}i}\tau _{h\lambda }\sim 1. \end{eqnarray} \tag{ 271 }$$

As we explained in Section 7.9.4, the parallel scales in the velocity space generated via the ballistic term are related to the parallel wavenumbers by δv_∥ ∼ 1/k_∥t. From Equation (271), we find that after one cascade time τ_hλ, the typical parallel velocity scale is δv_∥/v_thi ∼ 1, so the parallel phase mixing is again much less efficient than the perpendicular one.

Note that Equation (271) combined with Equation (270) means that the anisotropy is again characterized by the scaling relation k_∥ ∼ k^1/3_⊥, similarly to the case of KAW turbulence (see Equation (241) and Section 7.9.4).

7.10.4. Scalings for the Magnetic Fluctuations

The scaling law for the fluctuations of the magnetic-field strength follows immediately from Equations (263) and (269):

$$\begin{eqnarray} &&{\delta B_{\parallel \lambda }\over B_0}\sim \beta _i\,{\lambda \over \rho _i} {\Phi _{\lambda }\over \rho _iv_{{\rm th}i}}\sim \sqrt{\beta _i}\,l_0^{-1/3}\rho _i^{-11/6}\lambda ^{13/6}, \end{eqnarray} \tag{ 272 }$$

whence the spectrum of these fluctuations is k^−16/3_⊥.

The scaling of A_∥ (the perpendicular magnetic fluctuations) depends on the relation between k_∥ and k_⊥. Indeed, the ratio between the first and the third terms on the left-hand side of Equation (264) (or, equivalently, between the first and second terms on the right-hand side of Equation (266)) is ∼(k_∥v_thiτ_hλ)⁻¹. For a critically balanced cascade, this makes the two terms comparable (Equation (271)). Using the first term to work out the scaling for the perpendicular magnetic fluctuations, we get, using Equation (269),

$$\begin{equation} {\delta B_{\perp \lambda }\over B_0}\sim {1\over \lambda }{\Psi _{\lambda }\over v_A} \sim \beta _i\,{\lambda \over \rho _i}{\Phi _{\lambda }\over \rho _iv_{{\rm th}i}} \sim \sqrt{\beta _i}\,l_0^{-1/3}\rho _i^{-11/6}\lambda ^{13/6}, \end{equation} \tag{ 273 }$$

which is the same scaling as for δB_∥/B₀ (Equation (272)).

Using Equation (273) together with Equations (269) and (270), it is now straightforward to confirm the three assumptions made in Section 7.10.1 that we promised to verify a posteriori:

• 1.  
  In Equation (116), $\partial A_\parallel /\partial t\ll c{\bf\hat{b}}\cdot {\bm\nabla}\varphi$, so Equation (261) holds (the electrons remain Boltzmann). This means that no KAW can be excited by the cascade.
• 2.  
  δB_⊥/B₀ ≪ k_∥/k_⊥, so ${\bf\hat{b}}\cdot {\bm\nabla}\simeq \partial /\partial z$ in Equation (264). This means that field lines are not significantly perturbed.
• 3.  
  In the expression for $\langle \chi \rangle _{{\bf R}_i}$ (Equation (69)), v_∥A_∥/c ≪ φ, so Equation (249) holds. This means that the electrostatic fluctuations dominate the cascade.

The spectra of magnetic fluctuations obtained in Section 7.10.4 are very steep—steeper, in fact, than those normally observed in the dissipation range of the solar wind (Section 8.2.5). One might speculate that the observed spectra may be due to a superposition of the two cascades realizable below the ion gyroscale: a high-frequency cascade of KAW (Section 7.5) and a low-frequency cascade of electrostatic fluctuations due to the ion entropy fluctuations (Section 7.10). Such a superposition could happen if the power going into the KAW cascade is relatively small, ε_KAW ≪ ε. One then expects an electrostatic cascade to be set up just below the ion gyroscale with the KAW cascade superseding it deeper into the dissipation range. Comparing Equations (240) and (269), we can estimate the position of the spectral break:

$$\begin{eqnarray} &&k_{\perp }\rho _i \sim \left(\varepsilon /\varepsilon _{\rm KAW}\right)^{2/3}. \end{eqnarray} \tag{ 274 }$$

Since ρ_i/ρ_e ∼ (τm_i/m_e)^1/2/Z is not a very large number, the dissipation range is not very wide. It is then conceivable that the observed spectra are not true power laws but simply non-asymptotic superpositions of the electrostatic and KAW spectra with the observed range of "effective" spectral exponents due to varying values of the spectral break (274) between the two cascades.³³

The value of ε_KAW/ε specific to any particular set of parameters (β_i, τ, etc.) is set by what happens at k_⊥ρ_i ∼ 1 (Section 7.1; see Sections 8.2.2, 8.2.5, and 8.5 for further discussion).

Finally, let us consider what happens when k_⊥ρ_e ≫ 1. At these scales, we have to return to the full gyrokinetic system of equations. The quasi-neutrality (Equation (61)), parallel (Equation (62)) and perpendicular (Equation (66)) Ampère's law become

$$\begin{eqnarray} && {e\varphi \over T_{0e}} = -\left(1+{Z\over \tau }\right)^{-1}\sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf r}} {1\over n_{0e}}\int d^3{\bf v}\,J_0(a_e)h_{e{{\bf k}}},\\ &&{c\over 4\pi en_{0e}}\,\nabla _{\perp }^2A_\parallel = \sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf r}} {1\over n_{0e}}\int d^3{\bf v}\,v_\parallel J_0(a_e)h_{e{{\bf k}}},\\ &&{\delta B_\parallel \over B_0} = -{\beta _e\over 2}\sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf r}} {1\over n_{0e}}\int d^3{\bf v}\,{2v_\perp ^2\over v_{{\rm th}e}^2}{J_1(a_e)\over a_e}\,h_{e{{\bf k}}}, \end{eqnarray} \tag{ 275 }$$

where β_e = β_iZ/τ. We have discarded the velocity integrals of h_i both because the gyroaveraging makes them subdominant in powers of (m_e/m_i)^1/2 and because the fluctuations of h_i are damped by collisions (assuming the collisional cutoff given by Equation (259) lies above the electron gyroscale). To Equations (275)–(277), we must append the gyrokinetic equation for h_e (Equation (57) with s = e), thus closing the system.

The type of turbulence described by these equations is very similar to that discussed in Section 7.10. It is easy to show from Equations (275)–(277) that

$$\begin{eqnarray} &&{\delta B_\perp \over B_0}\sim {\delta B_\parallel \over B_0}\sim {\beta _e\over k_{\perp }\rho _e}\,{e\varphi \over T_{0e}}. \end{eqnarray} \tag{ 278 }$$

Hence the magnetic fluctuations are subdominant in the expression for $\langle \chi \rangle _{{\bf R}_e}$ (Equation (69) with s = e and a_e ≫ 1), so $\langle \chi \rangle _{{\bf R}_e}\simeq \langle \varphi \rangle _{{\bf R}_e}$. The electron gyrokinetic equation then is

$$\begin{eqnarray} &&{\partial h_e\over \partial t} + v_\parallel \,{\partial h_e\over \partial z} + {c\over B_0}\left\lbrace \langle \varphi \rangle _{{\bf R}_e},h_e\right\rbrace = \left({\partial h_e\over \partial t}\right)_{\rm c}, \end{eqnarray} \tag{ 279 }$$

where the wave–particle-interaction term in the right-hand side has been dropped because it can be shown to be small via the same argument as in Section 7.10.2.

Together with Equation (275), Equation (279) describes the kinetic cascade of electron entropy from the electron gyroscale down to the scale at which electron collisions can dissipate it into heat. This cascade the result of collisionless damping of KAW at k_⊥ρ_e ∼ 1, whereby the power in the KAW cascade is converted into the electron-entropy fluctuations: indeed, in the limit k_⊥ρ_e ≫ 1, the generalized energy is simply

$$\begin{eqnarray} &&W = \int d^3{\bf v}\int d^3{\bf R}_e\,{T_{0e}h_e^2\over 2F_{0e}} = W_{h_e} \end{eqnarray} \tag{ 280 }$$

(see Figure 5).

The same scaling arguments as in Section 7.10.2 apply and scaling relations analogous to Equations (268)–(270), and (272) duly follow:

$$\begin{eqnarray} h_{e\lambda }&\sim& {n_{0e}\over v_{{\rm th}e}^3} \left({\varepsilon _{\rm KAW}\over \varepsilon }\right)^{1/3}\left({1\over \beta _e}{m_e\over m_i}\right)^{1/2} l_0^{-1/3}\rho _e^{1/6}\lambda ^{1/6},\\ \Phi _{\lambda }&\sim& \left({\varepsilon _{\rm KAW}\over \varepsilon }\right)^{1/3}\left({1\over \beta _e}{m_e\over m_i}\right)^{1/2} v_{{\rm th}e}\,l_0^{-1/3}\rho _e^{1/6}\lambda ^{7/6},\\ \tau _{h\lambda }&\sim& \left({\varepsilon \over \varepsilon _{\rm KAW}}\right)^{1/3}\left({\beta _e}{m_i\over m_e}\right)^{1/2} {l_0^{1/3}\rho _e^{1/3}\lambda ^{1/3}\over v_{{\rm th}e}},\\ {\delta B_{\lambda }\over B_0}&\sim& \left({\varepsilon _{\rm KAW}\over \varepsilon }\right)^{1/3}\left(\beta _e{m_e\over m_i}\right)^{1/2} l_0^{-1/3}\rho _e^{-11/6}\lambda ^{13/6}, \end{eqnarray} \tag{ 281 }$$

where l₀ = v³_A/ε, as in Section 1.2. The formula for the collisional cutoffs in the wavenumber and velocity space is analogous to Equation (259):

$$\begin{eqnarray} &&{\delta v_\perp \over v_{{\rm th}i}}\sim {1\over k_{\perp }\rho _i}\sim (\nu _{ei}\tau _{\rho _e})^{3/5}, \end{eqnarray} \tag{ 285 }$$

where $\tau _{\rho _e}$ is the cascade time (283) taken at λ = ρ_e.

As the kinetic cascade takes the (generalized) energy to ever smaller scales, the frequency ω of the fluctuations increases. In applying the gyrokinetic theory, one must be mindful of the need for this frequency to stay smaller than Ω_i. Using the scaling formulae for the characteristic times of the fluctuations derived above (Equations (254), (270) and (283)), we can determine the conditions for ω ≪ Ω_i. Thus, for the gyrokinetic theory to be valid everywhere in the inertial range, we must have

$$\begin{eqnarray} &&k_{\perp }\rho _i\ll \beta _i^{3/4} \left({l_0\over \rho _i}\right)^{1/2} \end{eqnarray} \tag{ 286 }$$

at all scales down to k_⊥ρ_i ∼ 1, i.e., ρ_i/l₀ ≪ β^3/2_i, not a very stringent condition.

Below the ion gyroscale, the KAW cascade (Section 7.5) remains in the gyrokinetic regime as long as

$$\begin{equation} k_{\perp }\rho _i\ll \left({\varepsilon \over \varepsilon _{\rm KAW}}\right)^{1/4}\beta _i^{3/8}(1+\beta _i)^{1/4} \left({l_0\over \rho _i}\right)^{1/4} \end{equation} \tag{ 287 }$$

(we are assuming T_i/T_e ∼ 1 everywhere). The condition for this still to be true at the electron gyroscale is

$$\begin{equation} {\rho _i\over l_0}\ll {\varepsilon \over \varepsilon _{\rm KAW}}\,\beta _i^{3/2}(1+\beta _i) \left({m_e\over m_i}\right)^2. \end{equation} \tag{ 288 }$$

The ion entropy fluctuations passively mixed by the KAW turbulence (Section 7.9) satisfy Equation (287) at all scales down to the ion collisional cutoff (Equation (259)) if

$$\begin{equation} {\lambda _{{\rm mfp}i}\over l_0}\ll \left({\varepsilon \over \varepsilon _{\rm KAW}}\right)^{3/4}\beta _i^{9/8}(1+\beta _i)^{3/4} \left({\rho _i\over l_0}\right)^{1/4}. \end{equation} \tag{ 289 }$$

Note that the condition for the ion collisional cutoff to lie above the electron gyroscale is

$$\begin{equation} {\lambda _{{\rm mfp}i}\over l_0}\ll \left({\varepsilon \over \varepsilon _{\rm KAW}}\right)^{1/3}\sqrt{\beta _i}(1+\beta _i)^{1/3} \left({m_i\over m_e}\right)^{5/6}\left({\rho _i\over l_0}\right)^{2/3}. \end{equation} \tag{ 290 }$$

In the absence of KAW turbulence, the pure ion-entropy cascade (Section 7.10) remains gyrokinetic for

$$\begin{equation} k_{\perp }\rho _i\ll \beta _i^{3/2}{l_0\over \rho _i}. \end{equation} \tag{ 291 }$$

This is valid at all scales down to the ion collisional cutoff provided λ_mfpi/l₀ ≪ β³_i(l₀/ρ_i), an extremely weak condition, which is always satisfied. This is because the ion-entropy fluctuations in this case have much lower frequencies than in the KAW regime. The ion collisional cutoff lies above the electron gyroscale if, similarly to Equation (290),

$$\begin{equation} {\lambda _{{\rm mfp}i}\over l_0}\ll \sqrt{\beta _i}\left({m_i\over m_e}\right)^{5/6} \left({\rho _i\over l_0}\right)^{2/3}. \end{equation} \tag{ 292 }$$

If the condition (290) is satisfied, all fluctuations of the ion distribution function are damped out above the electron gyroscale. This means that below this scale, we only need the electron gyrokinetic equation to be valid, i.e., ω ≪ Ω_e. The electron-entropy cascade (Section 7.12), whose characteristic timescale is given by Equation (283), satisfies this condition for

$$\begin{equation} k_{\perp }\rho _e\ll \left({\varepsilon \over \varepsilon _{\rm KAW}}\right) \beta _e^{3/2} \left({m_i\over m_e}\right)^{3/2}{l_0\over \rho _e}. \end{equation} \tag{ 293 }$$

This is valid at all scales down to the electron collisional cutoff (Equation (285)) provided λ_mfpe/l₀ ≪ (ε/ε_KAW)²β³_e(m_i/m_e)³(l₀/ρ_e), which is always satisfied.

Within the formal expansion we have adopted (k_⊥ρ_i ∼ 1 and $k_{\parallel }\lambda _{{\rm mfp}i}\sim \sqrt{\beta _i}$), it is not hard to see that λ_mfpi/l₀ ∼ ² and ρ_i/l₀ ∼ ³. Since all other parameters (m_e/m_i, β_i, β_e etc.) are order unity with respect to , all of the above conditions for the validity of the gyrokinetics are asymptotically correct by construction. However, in application to real astrophysical plasmas, one should always check whether this construction holds. For example, substituting the relevant parameters for the solar wind shows that the gyrokinetic approximation is, in fact, likely to start breaking down somewhere between the ion and electron gyroscales (Howes et al. 2008a).³⁴ This releases a variety of high-frequency wave modes, which may be participating in the turbulent cascade around and below the electron gyroscale (see, e.g., the recent detailed observations of these scales in the magnetosheath by Mangeney et al. 2006; Lacombe et al. 2006 or the early measurements of high-frequency fluctuations in the solar wind by Denskat et al. 1983; Coroniti et al. 1982).

In this section, we have analyzed the turbulence in the dissipation range, which turned out to have many more essentially kinetic features than the inertial range.

At the ion gyroscale, k_⊥ρ_i ∼ 1, the kinetic cascade rearranged itself into two distinct components: part of the (generalized) energy arriving from the inertial range was collisionlessly damped, giving rise to a purely kinetic cascade of ion-entropy fluctuations, the rest was converted into a cascade of kinetic Alfvén waves (KAW) (Figure 5; see Sections 7.1 and 7.8).

The KAW cascade is described by two fluid-like equations for two scalar functions, the magnetic flux function $\Psi =-A_\parallel /\sqrt{4\pi m_in_{0i}}$ and the scalar potential, expressed, for continuity with the results of Section 5, in terms of the function Φ = (c/B₀)φ. The equations are (see Section 7.2)

$$\begin{eqnarray} {\partial \Psi \over \partial t} &=& v_A\left(1+{Z\over \tau }\right){\bf\hat{b}}\cdot {\bm\nabla}\Phi,\\ {\partial \Phi \over \partial t} &=& -{v_A\over 2 + \beta _i\left(1+Z/\tau \right)}\,{\bf\hat{b}}\cdot {\bm\nabla}\left(\rho _i^2\nabla _{\perp }^2\Psi \right), \end{eqnarray} \tag{ 294 }$$

where ${\bf\hat{b}}\cdot {\bm\nabla}=\partial /\partial z + (1/v_A)\lbrace \Psi,\cdots \rbrace$. The density and magnetic-field-strength fluctuations are directly related to the scalar potential:

$$\begin{eqnarray} &&{\delta n_e\over n_{0e}} = -{2\over \sqrt{\beta _i}}{\Phi \over \rho _i v_A},\quad {\delta B_\parallel \over B_0} = \sqrt{\beta _i}\left(1+{Z\over \tau }\right){\Phi \over \rho _i v_A}.\nonumber\\ \end{eqnarray} \tag{ 296 }$$

We call Equations (294)–(296) the electron reduced magnetohydrodynamics (ERMHD).

The ion-entropy cascade is described by the ion gyrokinetic equation:

$$\begin{equation} {\partial h_i\over \partial t} + v_\parallel {\partial h_i\over \partial z} + \left\lbrace \langle \Phi \rangle _{{\bf R}_i},h_i\right\rbrace = \left<C_{ii}[h_i]\right>_{{\bf R}_i}. \end{equation} \tag{ 297 }$$

The ion distribution function is mixed by the ring-averaged scalar potential and undergoes a cascade both in the velocity and gyrocenter space—this phase-space cascade is essential for the conversion of the turbulent energy into the ion heat, which can ultimately only be done by collisions (see Section 7.9).

If the KAW cascade is strong (its power ε_KAW is an order-unity fraction of the total injected turbulent power ε), it determines Φ in Equation (297), so the ion-entropy cascade is passive with respect to the KAW turbulence. Equations (294)–(295) and (297) form a closed system that determines the three functions Φ, Ψ, h_i, of which the latter is slaved to the first two. One can also compute δn_e and δB_∥, which are proportional to Φ (Equation (296)). The generalized energy conserved by these equations is given by Equation (245).

If the KAW cascade is weak (ε_KAW ≪ ε), the ion-entropy cascade dominates the turbulence in the dissipation range and drives low-frequency mostly electrostatic fluctuations, with a subdominant magnetic component. These are given by the following relations (see Section 7.10)

$$\begin{eqnarray} \Phi &=& {\rho _iv_{{\rm th}i}\over 2(1+\tau /Z)} \sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf r}} {1\over n_{0i}}\int d^3{\bf v}\,J_0(a_i)h_{i{{\bf k}}},\\ {\delta n_e\over n_{0e}} &=& {2Z\over \tau } {\Phi \over \rho _iv_{{\rm th}i}},\\ \Psi &=& \rho _i\sqrt{\beta _i} \sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf r}} \nonumber\\ &&{\times}\,{1\over n_{0i}} \int d^3{\bf v}\,\left({1\over 1+Z/\tau }{i\over k_{\parallel }}{\partial \over \partial t} - v_\parallel \right) \!\!{J_0(a_i)\over k_{\perp }^2\rho _i^2}\,h_{i{{\bf k}}}, \nonumber \\\\ {\delta B_\parallel \over B_0} &=& {-}{\beta _i\over 2}\sum _{{\bf k}}e^{i{{\bf k}}\cdot {\bf r}} {1\over n_{0i}}\int d^3{\bf v}\,{2v_\perp ^2\over v_{{\rm th}i}^2}{J_1(a_i)\over a_i}\,h_{i{{\bf k}}}, \end{eqnarray} \tag{ 298 }$$

where a_i = k_⊥v_⊥/Ω_i, Equations (297) and (298) form a closed system for Φ and h_i. The rest of the fields, namely δn_e, Ψ and δB_∥, are slaved to h_i via Equations (299)–(301).

The fluid and kinetic models summarized above are valid between the ion and electron gyroscales. Below the electron gyroscale, the collisionless damping of the KAW cascade converts it into a cascade of electron entropy, similar in nature to the ion-entropy cascade (Section 7.12).

The KAW cascade and the low-frequency turbulence associated with the ion-entropy cascade have distinct scaling behaviors. For the KAW cascade, the spectra of the electric, density and magnetic fluctuations are (Section 7.5)

$$\begin{equation} E_E(k_{\perp })\propto k_{\perp }^{-1/3},\quad E_n(k_{\perp })\propto k_{\perp }^{-7/3},\quad E_B(k_{\perp })\propto k_{\perp }^{-7/3}. \end{equation} \tag{ 302 }$$

For the ion- and electron-entropy cascades (Sections 7.9 and 7.12),

$$\begin{equation} E_E(k_{\perp })\propto k_{\perp }^{-4/3},\quad E_n(k_{\perp })\propto k_{\perp }^{-10/3},\quad E_B(k_{\perp })\propto k_{\perp }^{-16/3}. \end{equation} \tag{ 303 }$$

We argued in Section 7.11 that the observed spectra in the dissipation range of the solar wind could be the result of a superposition of these two cascades, although a number of alternative theories exist (Section 8.2.6).

In the inertial range, i.e., for k_⊥ρ_i ≪ 1, the solar-wind turbulence should be described by the reduced hybrid fluid-kinetic theory derived in Section 5 (KRMHD). Its applicability hinges on three key assumptions: (i) the turbulence is Alfvénic, i.e., consists of small (δB/B₀ ≪ 1) low-frequency (ω ∼ k_∥v_A ≪ Ω_i) perturbations of an ambient mean magnetic field and corresponding velocity fluctuations; (ii) it is strongly anisotropic, k_⊥ ≫ k_∥; (iii) the equilibrium distribution can be approximated or, at least, reasonably modeled by a Maxwellian without loss of essential physics (this will be discussed in Section 8.3). If these assumptions are satisfied, KRMHD (summarized in Section 5.7) is a rigorous set of dynamical equations for the inertial range, a set of Kolmogorov-style scaling predictions for the Alfvénic component of the turbulence can be produced (the GS theory, reviewed in Section 1.2), while to the compressive fluctuations, the considerations of Section 6 apply. So let us examine the observational evidence.

8.1.1. Alfvénic Nature of the Turbulence

8.1.2. Energy Spectrum

8.1.3. Anisotropy

8.1.4. Compressive Fluctuations

At scales approaching the ion gyroscale, k_⊥ρ_i ∼ 1, effects associated with the finite extent of ion gyroorbits start to matter. Observationally, this transition manifests itself as a clear break in the spectrum of magnetic fluctuations, with the inertial-range k^−5/3 scaling replaced by a steeper slope (see Figure 1). While the electrons at these scales can be treated as an isothermal fluid (as long as we are considering fluctuations above the electron gyroscale, k_⊥ρ_e ≪ 1; see Section 4), the fully gyrokinetic description (Section 3) has to be adopted for the ions. It is, indeed, to understand plasma dynamics at and around k_⊥ρ_i ∼ 1 that gyrokinetics was first designed in fusion plasma theory (Frieman & Chen 1982; Brizard & Hahm 2007). In order for gyrokinetics and further dissipation-range approximations that follow from it (Section 7) to be a credible approach in the solar wind and other space plasmas, it has to be established that fluctuations at and below the ion gyroscale are still strongly anisotropic, k_∥ ≪ k_⊥. If that is the case, then their frequencies (ω ∼ k_∥v_Ak_⊥ρ_i, see Section 7.3) will still be smaller than the cyclotron frequency in at least a part of the "dissipation range"³⁸—the range of scales k_⊥ρ_i ≳ 1 (see Section 7.13).

Note that additional information about the dissipation-range turbulence can be extracted from the measurements in the magnetosheath—while scales above the ion gyroscale are probably non-universal there, the dissipation range appears to display universal behavior, mostly similar to the solar wind (see, e.g., Alexandrova 2008). This complements the observational picture emerging from the solar-wind data and allows us to learn more as fluctuation amplitudes in the magnetosheath are larger and much smaller scales can be probed than in the solar wind (Mangeney et al. 2006; Lacombe et al. 2006; Alexandrova et al. 2008b).

8.2.1. Anisotropy

8.2.2. Transition at the Ion Gyroscale: Collisionless Damping and Heating

8.2.3. Ion Gyroscale vs. Ion Inertial Scale

8.2.4. KAW Turbulence

8.2.5. Variability of the Spectral Slope

8.2.6. Alternative Theories of the Dissipation Range

In rigorous theoretical terms, the weakest point of this paper is the use of a Maxwellian equilibrium. Formally, this is only justified when the collisions are weak but not too weak: we ordered the collision frequency as similar to the fluctuation frequency (Equation (49)). This degree of collisionality is sufficient to prove that a Maxwellian equilibrium distribution F_0s(v) does indeed emerge in the lowest order of the gyrokinetic expansion (Howes et al. 2006). This argument works well for plasmas such as the ISM (Section 8.4), where collisions are weak (λ_mfpi ≫ ρ_i) but non-negligible (λ_mfpi ≪ L). In space plasmas, the mean free path is of the order of 1 AU—the distance between the Sun and the Earth (see Table 1). Strictly speaking, in so highly collisionless a plasma, the equilibrium distribution does not have to be either Maxwellian or isotropic.

The conservation of the first adiabatic invariant, μ = v²_⊥/2B, suggests that temperature anisotropy with respect to the magnetic-field direction (T_0⊥ ≠ T_0∥) may exist. When the relative anisotropy is larger than (roughly) 1/β_i, it triggers several very fast growing plasma instabilities: most prominently the firehose (T_0⊥ < T_0∥) and mirror (T_0⊥>T_0∥) modes (e.g., Gary et al. 1976). Their growth rates peak around the ion gyroscale, thus giving rise to additional energy injection at k_⊥ρ_i ∼ 1.

No definitive analytical theory of how these fluctuations saturate, cascade and affect the equilibrium distribution has been proposed. It appears to be a reasonable expectation that the fluctuations resulting from temperature anisotropy will saturate by limiting this anisotropy. This idea has some support in solar-wind observations: while the degree of anisotropy of the core particle distribution functions varies considerably between data sets, the observed anisotropies do seem to populate the part of the parameter plane (T_0⊥/T_0∥, β_i) circumscribed in a rather precise way by the marginal stability boundaries for the mirror and firehose (Gary et al. 2001; Kasper et al. 2002; Marsch et al. 2004; Hellinger et al. 2006; Matteini et al. 2007).⁴¹

If we want to study turbulence in data sets that do not lie too close to these stability boundaries, assuming an isotropic Maxwellian equilibrium distribution (Equation (54)) is probably an acceptable simplification, although not an entirely rigorous one. Further theoretical work is clearly possible on this subject: thus, it is not a problem to formulate gyrokinetics with an arbitrary equilibrium distribution (Frieman & Chen 1982) and starting from that, once can generalize the results of this paper (for the KRMHD system, Section 5, this has been done by Chen et al. 2009). Treating the instabilities themselves might prove more difficult, requiring the gyrokinetic ordering to be modified and the expansion carried to higher orders to incorporate features that are not captured by gyrokinetics, e.g., short parallel scales (Rosin et al. 2009), particle trapping (Pokhotelov et al. 2008; Rincon et al. 2009), or nonlinear finite-gyroradius effects (Califano et al. 2008). Note that the theory of the dissipation-range turbulence will probably need to be modified to account for the additional energy injection from the instabilities and for the (yet unclear) way in which this energy makes its way to dissipation and into heat.

Besides the anisotropies, the particle distribution functions in the solar wind (especially the electron one) exhibit non-Maxwellian suprathermal tails (see Maksimovic et al. 2005; Marsch 2006, and references therein). These contain small (∼5% of the total density) populations of energetic particles. Both the origin of these particles and their effect on turbulence have to be modeled kinetically. Again, it is possible to formulate gyrokinetics for general equilibrium distributions of this kind and examine the interaction between them and the turbulent fluctuations, but we leave such a theory outside the scope of this paper.

Thus, much remains to be done to incorporate realistic equilibrium distribution functions into the gyrokinetic description of the solar wind plasma. In the meanwhile, we believe that the gyrokinetic theory based on a Maxwellian equilibrium distribution as presented in this paper, while idealized and imperfect, is nevertheless a step forward in the analytical treatment of the space-plasma turbulence compared to the fluid descriptions that have prevailed thus far.

While the solar wind is unmatched by other astrophysical plasmas in the level of detail with which turbulence in it can be measured, the interstellar medium (ISM) also offers an observer a number of ways of diagnosing plasma turbulence, which, in the case of the ISM, is thought to be primarily excited by supernova explosions (Norman & Ferrara 1996). The accuracy and resolution of this analysis are due to improve rapidly thanks to many new observatories, e.g., LOFAR,⁴² Planck (Enßlin et al. 2006), and, in more distant future, the SKA (Lazio et al. 2004).

The ISM is a spatially inhomogeneous environment consisting of several phases that have different temperatures, densities and degrees of ionization (Ferrière 2001).⁴³ We will use the Warm ISM phase (see Table 1) as our fiducial interstellar plasma and discuss briefly what is known about the two main observationally accessible quantities—the electron density and magnetic fields—and how this information fits into the theoretical framework proposed here.

8.4.1. Electron Density Fluctuations

8.4.2. Magnetic Fluctuations

Accretion of plasma onto a central black hole or neutron star is responsible for many of the most energetic phenomena observed in astrophysics (see, e.g., Narayan & Quataert 2005 for a review). It is now believed that a linear instability of differentially rotating plasmas—the magnetorotational instability (MRI)—amplifies magnetic fields and gives rise to MHD turbulence in astrophysical disks (Balbus & Hawley 1998). Magnetic stresses due to this turbulence transport angular momentum, allowing plasma to accrete. The MRI converts the gravitational potential energy of the inflowing plasma into turbulence at the outer scale that is comparable to the scale height of the disk. This energy is then cascaded to small scales and dissipated into heat—powering the radiation that we see from accretion flows. Fluid MHD simulations show that the MRI-generated turbulence in disks is subsonic and has β ∼ 10 − 100. Thus, on scales much smaller than the scale height of the disk, homogeneous turbulence in the parameter regimes considered in this paper is a valid idealization and the kinetic models developed above should represent a step forward compared to the purely fluid approach.

Turbulence is not yet directly observable in disks, so models of turbulence are mostly used to produce testable predictions of observable properties of disks such as their X-ray and radio emission. One of the best observed cases is the (presumed) accretion flow onto the black hole coincident with the radio source Sgr A* in the center of our Galaxy (see review by Quataert 2003).

Depending on the rate of heating and cooling in the inflowing plasma (which in turn depend on accretion rate and other properties of the system under consideration), there are different models that describe the physical properties of accretion flows onto a central object. In one class of models, a geometrically thin optically thick accretion disk (Shakura & Sunyaev 1973), the inflowing plasma is cold and dense and well described as an MHD fluid. When applied to Sgr A*, these models produce a prediction for its total luminosity that is several orders of magnitude larger than observed. Another class of models, which appears to be more consistent with the observed properties of Sgr A*, is called radiatively inefficient accretion flows (RIAFs; see Rees et al. 1982; Narayan & Yi 1995 and review by Quataert 2003 of the applications and observational constraints in Sgr A*). In these models, the inflowing plasma near the black hole is believed to adopt a two-temperature configuration, with the ions (T_i ∼ 10¹¹ − 10¹² K) hotter than the electrons (T_e ∼ 10⁹ − 10¹¹ K).⁴⁴ The electron and ion thermodynamics decouple because the densities are so low that the temperature equalization time ∼ν⁻¹_ie is longer than the time for the plasma to flow into the black hole. Thus, like the solar wind, RIAFs are macroscopically collisionless plasmas (see Table 1 for plasma parameters in the Galactic center; note that these parameters are so extreme that the gyrokinetic description, while probably better than the fluid one, cannot be expected to be rigorously valid; at the very least, it needs to be reformulated in a relativistic form). At the high temperatures appropriate to RIAFs, electrons radiate energy much more efficiently than the ions (by virtue of their much smaller mass) and are, therefore, expected to contribute dominantly to the observed emission, while the thermal energy of the ions is swallowed by the black hole. Since the plasma is collisionless, the electron heating by turbulence largely determines the thermodynamics of the electrons and thus the observable properties of RIAFs. The question of which fraction of the turbulent energy goes into ion and which into electron heating is, therefore, crucial for understanding accretion flows—and the answer to this question depends on the detailed properties of the small-scale kinetic turbulence (e.g., Quataert & Gruzinov 1999; Sharma et al. 2007), as well as on the linear properties of the collisionless MRI (Quataert et al. 2002; Sharma et al. 2003).

Since all of the turbulent power coming down the cascade must be dissipated into either ion or electron heat, it is really the amount of generalized energy diverted at the ion gyroscale into the ion entropy cascade (Sections 7.8–7.9) that decides how much energy is left to heat the electrons via the KAW cascade (Sections 7.2–7.5, 7.12). Again, as in the case of the solar wind (Sections 8.2.2 and 8.2.5), the transition around the ion gyroscale from the Alfvénic turbulence at k_⊥ρ_i ≪ 1 to the KAW turbulence at k_⊥ρ_i ≫ 1 emerges as a key unsolved problem.

Galaxy clusters are the largest plasma objects in the Universe. Like the other examples discussed above, the intracluster plasma is in the weakly collisional regime (see Table 1). Fluctuations of electron density, temperature and of magnetic fields are measured in clusters by X-ray and radio observatories, but the resolution is only just enough to claim that a fairly broad scale range of fluctuations exists (Schuecker et al. 2004; Vogt & Enßlin 2005). No power-law scalings have yet been established beyond reasonable doubt.

What fundamentally hampers quantitative modeling of turbulence and related effects in clusters is that we do not have a definite theory of the basic properties of the intracluster medium: its (effective) viscosity, magnetic diffusivity or thermal conductivity. In a weakly collisional and strongly magnetized plasma, all of these depend on the structure of the magnetic field (Braginskii 1965), which is shaped by the turbulence. If (or at scales where) a reasonable a priori assumption can be made about the field structure, further analytical progress is possible: thus, the theoretical models presented in this paper assume that the magnetic field is a sum of a slowly varying in space "mean field" and small low-frequency perturbations (δB ≪ B₀).

In fact, since clusters do not have mean fields of any magnitude that could be considered dynamically significant, but do have stochastic fields, the outer-scale MHD turbulence in clusters falls into the weak-mean-field category (see Section 1.3). The magnetic field should be highly filamentary, organized in long folded direction-reversing structures. It is not currently known what determines the reversal scale.⁴⁵ Observations, while tentatively confirming the existence of very long filaments (Clarke & Enßlin 2006), suggest that the reversal scale is much larger than the ion gyroscale: thus, the magnetic-energy spectrum for the Hydra A cluster core reported by Vogt & Enßlin (2005) peaks at around 1 kpc, compared to ρ_i ∼ 10⁵ km. Below this scale, an Alfvén-wave cascade should exist (as is, indeed, suggested by Vogt & Enßlin's spectrum being roughly consistent with k^−5/3 at scales below the peak). As these scales are collisionless (λ_mfpi ∼ 100 pc in the cores and ∼10 kpc in the bulk of the clusters), it is to this turbulence that the theory developed in this paper should be applicable.

Another complication exists, similar to that discussed in Section 8.3: pressure anisotropies could give rise to fast plasma instabilities whose growth rate peaks just above the ion gyroscale. As was pointed out by Schekochihin et al. (2005), these are, in fact, an inevitable consequence of any large-scale fluid motions that change the strength of the magnetic field. Although a number of interesting and plausible arguments can be made about the way the instabilities might determine the magnetic-field structure (Schekochihin & Cowley 2006; Schekochihin et al. 2008a; Rosin et al. 2009; Rincon et al. 2009), it is not currently understood how the small-scale fluctuations resulting from these instabilities coexist with the Alfvénic cascade.

The uncertainties that result from this imperfect understanding of the nature of the intracluster medium are exemplified by the problem of its thermal conductivity. The magnetic-field reversal scale in clusters is certainly not larger than the electron diffusion scale, (m_i/m_e)^1/2λ_mfpi, which varies from a few kpc in the cores to a few hundred kpc in the bulk. Therefore, one would expect that the approximation of isothermal electron fluid (Section 4) should certainly apply at all scales below the reversal scale, where δB ≪ B₀ presumably holds. Even this, however, is not absolutely clear. One could imagine the electrons being effectively adiabatic if (or in the regions where) the plasma instabilities give rise to large fluctuations of the magnetic field (δB/B₀ ∼ 1) at the ion gyroscale reducing the mean free path to λ_mfpi ∼ ρ_i (Schekochihin et al. 2008a; Rosin et al. 2009; Rincon et al. 2009). Such fluctuations cannot be described by the gyrokinetics in its current form. The current state of the observational evidence does not allow one to exclude either of these possibilities. Both isothermal (Fabian et al. 2006; Sanders & Fabian 2006) and non-isothermal (Markevitch & Vikhlinin 2007) coherent structures that appear to be shocks are observed. Disordered fluctuations of temperature can also be detected, which allows one to infer an upper limit for the scale at which the isothermal approximation can start being valid: thus, Markevitch et al. (2003) find temperature variations at all scales down to ∼100 kpc, which is the statistical limit that defines the spatial resolution of their temperature map. In none of these or similar measurements is the magnetic field data available that would make possible a pointwise comparison of the magnetic and thermal structure.