Fluid Theory of Coherent Magnetic Vortices in High-β Space Plasmas

Magnetic structures at ion break scale, commonly in the form of Alfvén vortices with diameter 10–30 times longer than the ion scales, have been observed in the solar wind and in the magnetosheaths of Earth and Saturn (Alexandrova et al. 2004, 2006; Alexandrova & Saur 2008; Alexandrova 2008; Lion et al. 2016; Perrone et al. 2016, 2017). Similar structures, but with the diameter comparable to the ion-acoustic Larmor radius and identified as the drift-Alfvén vortices, were observed in Earth's magnetospheric cusp region (Sundkvist et al. 2005), where the ratio of thermal and magnetic pressures is considerably smaller. A detailed statistical analysis of the diverse magnetic fluctuations in the turbulent cascade close to the ion spectral break, detected in the slow and fast solar wind streams by the multispacecraft Cluster mission, has been presented by Perrone et al. (2016, 2017). They have shown that the intermittency of the magnetic turbulence is due to the presence of coherent structures of various natures. The compressible structures observed in the slow wind are predominantly parallel perturbations of the magnetic field ($\delta {B}_{\parallel }\gg \delta {B}_{\perp }$) and have the form of magnetic holes, solitons, and shock waves. Coherent shear Alfvénic perturbations have been observed both in the slow and the fast solar wind, featuring β_i ≳ 1 and β_i ≲ 1, respectively, where $\beta =2p/{c}^{2}{\epsilon }_{0}{B}^{2}$ is the ratio between the plasma pressure p and the magnetic pressure. They appear either as current sheets or the vortex-like structures. Predominantly torsional Alfvénic vortices with $\delta {B}_{\perp }\gg \delta {B}_{\parallel }$, but with finite $\delta {B}_{\parallel }$, are commonly present both in the slow and the fast winds. Vortices with a larger compressional magnetic field component, $\delta {B}_{\perp }\gtrsim \delta {B}_{\parallel }$, have been observed only in the slow wind. The observed compressible component $\delta {B}_{\parallel }$ is usually well localized within the structure, while the torsional part $\delta {B}_{\perp }$ is more delocalized, extending itself outside of the vortex core.

The multipoint Cluster measurements have enabled the determination of the spatial and temporal properties of these structures, such as their propagation velocity, the direction of the normal, and the spatial scale along this normal. The normal is always perpendicular to the local magnetic field, indicating that the structures are strongly elongated in the direction of ${\boldsymbol{B}}$. The majority of the structures is convected by the wind, but in the slow wind, a significant fraction (∼25%) propagates in the perpendicular direction and with finite velocities relative to the plasma. In the fast wind, no structures propagating relative to the plasma could be verified because the propagation velocities that came out were smaller than the error of the measurements. Typical scales of the structures along the normal are two to five characteristic ion lengths, such as the ion Larmor radius ${v}_{{\mathrm{Ti}}_{\perp }}/{{\rm{\Omega }}}_{i}$, acoustic radius c_S/Ω_i, and ion inertial length c/ω_pi, which in a solar wind plasma are of the same order of magnitude. Similar features of plasma turbulence were observed previously in the magnetosheaths downstream of quasi-perpendicular bow shocks of Earth and Saturn by Alexandrova et al. (2004, 2006) and Alexandrova & Saur (2008), who detected coherent shear Alfvénic vortex structures in the form of current filaments slightly tilted relative to the magnetic field, ${{\rm{\nabla }}}_{\parallel }\ll {{\rm{\nabla }}}_{\perp }$, exhibiting only perpendicular magnetic perturbations, $\delta {B}_{\parallel }\to 0$. Recently, using the high-resolution in situ measurements from the Magnetospheric Multiscale (MMS) mission, Wang et al. (2019) presented a detailed analysis of the plasma features within an Alfvén vortex. They demonstrated that the quasi-monopolar, mostly torsional ${{\rm{\nabla }}}_{\parallel }\ll {{\rm{\nabla }}}_{\perp }$, Alfvén vortex with a radius of ∼10 proton Larmor radii observed in Earth's turbulent magnetosheath had the magnetic fluctuations $\delta {{\boldsymbol{B}}}_{\perp }$ anticorrelated with the velocity fluctuations $\delta {{\boldsymbol{V}}}_{\perp }$, while its compressive features were in qualitative agreement with the theory developed in the present paper.

Shear Alfvénic fluid vortices were predicted theoretically by Petviashvili & Pokhotelov (1992), who demonstrated that structures of this type with the transverse size bigger than the ion inertial length c/ω_pi, where ${\omega }_{\mathrm{pi}}={({n}_{0}{e}^{2}/{m}_{i}{\epsilon }_{0})}^{1/2}$ is the ion plasma frequency, could exist in plasmas with incompressible density (usually occurring when β is small). Under such conditions, nonlinear solutions are described by the standard Kadomtsev–Pogutse–Strauss reduced magnetohydrodynamic (MHD) equations (Kadomtsev & Pogutse 1973, 1974; Strauss 1976, 1977). Solutions can be nontraveling monopoles or propagating structures. A hydrodynamic vortex that moves relative to the plasma always has the form of a dipole, but also, a monopole can be superimposed on it. We will show below that such monopolar component is possible only in the absence of the compressibility of the magnetic field, which is realized if either the structures are (much) bigger than the ion scale, or the plasma β is sufficiently small. O. Alexandrova proposed (Alexandrova et al. 2006; Alexandrova 2008) that, within Kadomtsev–Pogutse–Strauss' reduced MHD description, such vortices might be created by the filamentation of the nonlinear slab-like structures arising from the saturation of the linearly unstable Alfvén ion cyclotron waves (Alexandrova et al. 2004) or, more likely, arising naturally as the intermittency of the turbulence (Alexandrova et al. 2006; Alexandrova 2008; Lion et al. 2016; Perrone et al. 2016, 2017; Roberts et al. 2016). When a spacecraft encounters a dipole, the recorded signal depends on the relative position of the dipole's axis and the satellite's trajectory. The satellite may observe either one "pole" of the dipole or both, and the detected signals superficially appear to be qualitatively different.

In this paper, we present a hydrodynamic theory of coherent vortices in space plasmas that can be characterized by anisotropic electron and ion temperatures, and with arbitrary plasma β. We generalize the classical shear Alfvén result (Petviashvili & Pokhotelov 1992) by including the diamagnetic and finite Larmor radius effects via Braginskii's collisionless stress tensor and the compressional magnetic component via a generalized pressure balance. We demonstrate that perturbations that are bigger than the ion inertial length are properly described by the Kadomtsev–Pogutse–Strauss equations of reduced MHD and that in plasmas with a modest β ∼ 1, such description remains valid also when the size of the structure is only slightly bigger than the ion inertial length. We find a general reduced MHD vortex solution, whose torsional magnetic field component is leaking outside of the vortex core while the compressional magnetic field is restricted to its interior, which is why the latter may remain undetected by a spacecraft. They also possess a finite density perturbation and parallel fluid velocity that are proportional to the vorticity and the compressional magnetic field. Furthermore, in plasmas with β ≲ 1 and on a characteristic length that belongs to the ion scale, we find two different particular solutions in the form of dipole vortices that can be regarded as the generalizations of the Kadomtsev–Pogutse–Strauss structures to smaller scales and of the nonlinear drift mode to the slow magnetoacoustic-mode structures in large-β plasmas, respectively. Our generalized Kadomtsev–Pogutse–Strauss dipoles possess all three components of the magnetic field perturbation, and their phase velocity component in the direction of the ambient magnetic field lies in the Alfvén and the acoustic ranges, ${u}_{\parallel }\sim ({c}_{{\rm{A}}},{c}_{S})$, while the nonpropagating monopoles have ${u}_{\parallel }\to \infty$. The slow magnetosonic dipoles that we are able to study analytically propagate much slower, ${u}_{\parallel }\ll {c}_{{\rm{A}}}$ (the range of permitted ${u}_{\parallel }$ is broadened in the presence of ion temperature anisotropy), and their magnetic field perturbation is mostly compressional, i.e., ${{\boldsymbol{B}}}_{\perp }\to 0$. The moving monopolar structures in the compressional magnetic field will be considered in a separate publication, as they require a (gyro)kinetic description due to their ability to trap particles in the parallel direction and to redistribute their parallel and perpendicular temperatures.

We consider a collisionless plasma with the unperturbed density n₀ immersed in the homogeneous magnetic field ${{\boldsymbol{e}}}_{z}{B}_{0}$. We assume that the electron and ion temperatures can be anisotropic, i.e., that the temperatures associated with the particles' random motions along and perpendicular to the magnetic field may be different, ${T}_{{j}_{\parallel }}\ne {T}_{{j}_{\perp }}$, where j = e, i and the subscripts $\parallel$ and $\perp$ denote the components parallel and perpendicular to the magnetic field, respectively. The hydrodynamic equations of continuity and momentum for each of the species have the form

$$\begin{eqnarray}&&\left(\displaystyle \frac{\partial }{\partial t}+{\boldsymbol{U}}\cdot {\rm{\nabla }}\right)n+n{\rm{\nabla }}\cdot {\boldsymbol{U}}=0,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\left(\displaystyle \frac{\partial }{\partial t}+{\boldsymbol{U}}\cdot {\rm{\nabla }}\right){\boldsymbol{U}}=\displaystyle \frac{q}{m}\left({\boldsymbol{E}}+{\boldsymbol{U}}\times {\boldsymbol{B}}\right)-\displaystyle \frac{1}{{mn}}{\rm{\nabla }}\cdot \left({\boldsymbol{P}}+{\boldsymbol{\pi }}\right),\end{eqnarray} \tag{ 2 }$$

where, for simplicity, we have omitted the subscripts e and i referring to the electrons and ions, respectively. In the above, n, ${\boldsymbol{U}}$, q, and m are the number density, fluid velocity, charge, and mass, respectively. The pressure ${\boldsymbol{P}}$ and the stress ${\boldsymbol{\pi }}$ are diagonal and off-diagonal tensors. Using the standard shorthand notation, the pressure tensor in the case of an anisotropic temperature is given by ${\boldsymbol{P}}={p}_{\perp }({\boldsymbol{I}}-{\boldsymbol{b}}\,{\boldsymbol{b}})+{p}_{\parallel }{\boldsymbol{b}}\,{\boldsymbol{b}}$, where ${\boldsymbol{I}}$ is a unit tensor, viz. ${I}_{\alpha ,\beta }={\delta }_{\alpha ,\beta }$ and ${\delta }_{\alpha ,\beta }$ is the Kronecker delta, and ${\boldsymbol{b}}\,{\boldsymbol{b}}$ stands for the dyadic product, whose components are given by ${({\boldsymbol{b}}{\boldsymbol{b}})}_{\alpha ,\beta }={b}_{\alpha }{b}_{\beta }$. Here, ${\boldsymbol{b}}$ is the unit vector parallel to the magnetic field, ${\boldsymbol{b}}={\boldsymbol{B}}/B$; B is the intensity of the magnetic field, $B=| {\boldsymbol{B}}|$; ${p}_{\parallel }\,=\,{{nT}}_{\parallel }$; and ${p}_{\perp }={{nT}}_{\perp }$. Likewise, the stress tensor is written as ${\boldsymbol{\pi }}={{\boldsymbol{e}}}_{\alpha }{{\boldsymbol{e}}}_{\beta }\,{\pi }_{\alpha ,\beta }$. These enable us to use the standard formula from the tensor algebra ${\rm{\nabla }}\cdot {\boldsymbol{q}}\,{\boldsymbol{r}}\,=({\rm{\nabla }}\cdot {\boldsymbol{q}}+{\boldsymbol{q}}\cdot {\rm{\nabla }}){\boldsymbol{r}}$ and to write the divergence of the pressure and the stress tensors as

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\nabla }}\cdot {\boldsymbol{P}} & = & {\rm{\nabla }}{p}_{\perp }+{\boldsymbol{b}}\left({\boldsymbol{b}}\cdot {\rm{\nabla }}\right)\left({p}_{\parallel }-{p}_{\perp }\right)\\ & & +\,\left({p}_{\parallel }-{p}_{\perp }\right)\left({\rm{\nabla }}\cdot {\boldsymbol{b}}+{\boldsymbol{b}}\cdot {\rm{\nabla }}\right){\boldsymbol{b}},\end{array}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\boldsymbol{\pi }}={{\boldsymbol{e}}}_{\beta }\left({{\boldsymbol{e}}}_{\alpha }\cdot {\rm{\nabla }}\right){\pi }_{\alpha ,\beta }+{\pi }_{\alpha ,\beta }\left({\rm{\nabla }}\cdot {{\boldsymbol{e}}}_{\alpha }+{{\boldsymbol{e}}}_{\alpha }\cdot {\rm{\nabla }}\right){{\boldsymbol{e}}}_{\beta }.\end{eqnarray} \tag{ 4 }$$

The endmost terms on the right-hand sides on the above equations arise from the curvature of the magnetic field lines. For later convenience, we introduce the notation

$$\begin{eqnarray}&&{\left({\rm{\nabla }}\cdot {\boldsymbol{\pi }}\right)}_{\mathrm{curv}}={\pi }_{\alpha ,\beta }\left({\rm{\nabla }}\cdot {{\boldsymbol{e}}}_{\alpha }+{{\boldsymbol{e}}}_{\alpha }\cdot {\rm{\nabla }}\right){{\boldsymbol{e}}}_{\beta }.\end{eqnarray} \tag{ 5 }$$

The chain of hydrodynamic equations is truncated by using the Braginskii (1965) collisionless (nongyrotropic) stress tensor, appropriate for perturbations weakly varying both on the timescale of the gyroperiod and on the perpendicular scale of the Larmor radius. Within the adopted large-scale limit, Braginskii's stress tensor has been generalized to anisotropic temperatures; see, e.g., classical works by Yajima (1966) and the more recent ones by Schekochihin et al. (2010) and Sulem & Passot (2012, 2015). Following these authors, who ignored the heat flux, we disregard the hydrodynamic equations for pressure and stress tensors, and use instead the generic equations of state:

$$\begin{eqnarray}&&{{dp}}_{\perp }/{p}_{\perp }={\gamma }_{\perp }\,{dn}/n,\quad \quad {{dp}}_{\parallel }/{p}_{\parallel }={\gamma }_{\parallel }\,{dn}/n,\end{eqnarray} \tag{ 6 }$$

in which the multipliers ${\gamma }_{\parallel }$ and ${\gamma }_{\perp }$ are some functionals of the plasma density, see also Belmont & Mazelle (1992), Passot et al. (2013) where the original (complex) polytropic indices for collisionless plasmas have been derived. We consider regimes in which the perturbations of the density and of the magnetic field are not too large—see Equation (12)—which permits us to make an estimate of the functionals ${\gamma }_{\parallel }$ and ${\gamma }_{\perp }$ from the linearized Vlasov equation. These are further simplified under the drift scaling (Equation (11)) and for the weak dependence along magnetic field lines, Equation (12). Under such conditions, the parallel functional ${\gamma }_{\parallel }$ reduces to a constant, viz. ${\gamma }_{\parallel }=3$, when the characteristic parallel velocity of propagation (i.e., the phase velocity u_z) is bigger than the parallel thermal velocity ${v}_{{T}_{\parallel }}$ and the process can be considered adiabatic, and to ${\gamma }_{\parallel }=1$ when ${u}_{z}\ll {v}_{{T}_{\parallel }}$, i.e., the process is isothermal. Likewise, the perpendicular functional γ_⊥ reduces to γ_⊥ = 1 for arbitrary ratios ${u}_{z}/{v}_{{T}_{\parallel }}$ if the characteristic perpendicular size of the solution is much bigger than the Larmor radius. Conversely, for solutions whose transverse scale approaches the ion scales, ${\gamma }_{\perp }$ can be approximated by a constant only in a limited number of cases, for which vortex solutions are found in Section 3. These are the shear Alfvén solution, whose parallel electric field is zero ${E}_{\parallel }=0$, decoupled from acoustic perturbations, ${u}_{z}\gt {v}_{{\mathrm{Ti}}_{\parallel ,\perp }}$; and the kinetic slow magnetosonic mode solution, whose torsion of the magnetic field is absent, $\delta {{\boldsymbol{B}}}_{\perp }=0$, realized in the regime ${c}_{{\rm{A}}}\gt {u}_{z}\gt {v}_{{\mathrm{Ti}}_{\parallel ,\perp }}$, when the coupling with acoustic perturbation is negligible. In both cases, we have ${\gamma }_{{i}_{\perp }}=2$. For more details, see Appendix C.

The system of equations is closed by Faraday's and Ampere's laws:

$$\begin{eqnarray}&&{\rm{\nabla }}\times {\boldsymbol{E}}=-\displaystyle \frac{\partial {\boldsymbol{B}}}{\partial t},\quad \quad {\rm{\nabla }}\times {\boldsymbol{B}}=\displaystyle \frac{1}{{c}^{2}}\left(\displaystyle \frac{\partial {\boldsymbol{E}}}{\partial t}+\displaystyle \frac{{\boldsymbol{j}}}{{\epsilon }_{0}}\right).\end{eqnarray} \tag{ 7 }$$

The latter is simplified on temporal scales that are slow compared to the electron plasma frequency ${\omega }_{\mathrm{pe}}=\sqrt{{n}_{0}{e}^{2}/{m}_{e}{\epsilon }_{0}}$ and spatial scales that are long compared to the electron Debye length ${\lambda }_{\mathrm{De}}={v}_{\mathrm{Te}}/{\omega }_{\mathrm{pe}}$, when we can ignore both the charge separation and the displacement current, yielding

$$\begin{eqnarray}&&{n}_{e}={n}_{i}\equiv n,\quad \quad \displaystyle \frac{n}{{n}_{0}}\left({{\boldsymbol{U}}}_{i}-{{\boldsymbol{U}}}_{e}\right)=\displaystyle \frac{e}{{m}_{e}}\displaystyle \frac{{c}^{2}}{{\omega }_{\mathrm{pe}}^{2}}\,{\rm{\nabla }}\times {\boldsymbol{B}}.\end{eqnarray} \tag{ 8 }$$

Here and in the rest of the paper, we consider single-charged ions, viz. q_i = −q_e = e.

For later reference, we write also the parallel momentum equation that is obtained when we multiply the momentum equation with ${\boldsymbol{b}}\,\cdot$, viz.

$$\begin{eqnarray}&&\begin{array}{l}\left(\displaystyle \frac{\partial }{\partial t}+{\boldsymbol{U}}\cdot {\rm{\nabla }}\right){U}_{\parallel }-\displaystyle \frac{q}{m}\,{\boldsymbol{b}}\cdot \left({\boldsymbol{E}}-\displaystyle \frac{1}{{qn}}\,{\rm{\nabla }}{p}_{\parallel }+\displaystyle \frac{{p}_{\parallel }-{p}_{\perp }}{{qnB}}\,{\rm{\nabla }}B\right)\\ \,+\,\displaystyle \frac{1}{{mn}}\left[\displaystyle \frac{\partial {\pi }_{\alpha ,b}}{\partial {x}_{\alpha }}+{\boldsymbol{b}}\cdot {\left({\rm{\nabla }}\cdot {\boldsymbol{\pi }}\right)}_{\mathrm{curv}}\right]={{\boldsymbol{U}}}_{\perp }\cdot \left(\displaystyle \frac{\partial }{\partial t}+{\boldsymbol{U}}\cdot {\rm{\nabla }}\right){\boldsymbol{b}},\end{array}\end{eqnarray} \tag{ 9 }$$

where ${U}_{\parallel }={\boldsymbol{b}}\cdot {\boldsymbol{U}}$. Likewise, multiplying the electron and ion momentum equations by ${m}_{e}{n}_{e}$ and m_in_i, respectively, adding, and taking the component perpendicular to the magnetic field, after some tedious but straightforward algebra, we obtain the perpendicular momentum equation for the plasma fluid:

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\nabla }}}_{\perp }\left({c}^{2}{\epsilon }_{0}\displaystyle \frac{{B}^{2}}{2}+{p}_{{e}_{\perp }}+{p}_{{i}_{\perp }}\right)={\boldsymbol{b}}\times \left\{{\boldsymbol{b}}\times \left[{m}_{e}{n}_{e}\left(\displaystyle \frac{\partial }{\partial t}+{{\boldsymbol{U}}}_{e}\cdot {\rm{\nabla }}\right){{\boldsymbol{U}}}_{e}+{m}_{i}{n}_{i}\left(\displaystyle \frac{\partial }{\partial t}+{{\boldsymbol{U}}}_{i}\cdot {\rm{\nabla }}\right){{\boldsymbol{U}}}_{i}\right]\right\}\\ \ +\,{\epsilon }_{0}\left[{{\boldsymbol{E}}}_{\perp }\left({\rm{\nabla }}\cdot {\boldsymbol{E}}\right)+B{\boldsymbol{b}}\times \displaystyle \frac{\partial {\boldsymbol{E}}}{\partial t}\right]+\left({c}^{2}{\epsilon }_{0}{B}^{2}-{p}_{{e}_{\parallel }}-{p}_{{i}_{\parallel }}+{p}_{{e}_{\perp }}+{p}_{{i}_{\perp }}\right)\left({\boldsymbol{b}}\cdot {\rm{\nabla }}\right){\boldsymbol{b}}-{\rm{\nabla }}\cdot \left({\pi }_{{e}_{\alpha ,\beta }}+{\pi }_{{i}_{\alpha ,\beta }}\right),\end{array}\end{eqnarray} \tag{ 10 }$$

which is in the quasi-neutrality regime, Equation (8) (i.e., on the adopted scales bigger than the Debye length λ_De), simplified by setting ${\rm{\nabla }}\cdot {\boldsymbol{E}}=e({n}_{i}-{n}_{e})/{\epsilon }_{0}=0$. Our Equations (1), (2), (9), and (10) are vastly simplified under the drift scaling:

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\rm{\Omega }}}\displaystyle \frac{\partial }{\partial t}\sim \displaystyle \frac{1}{{\rm{\Omega }}}\,{\boldsymbol{U}}\cdot {\rm{\nabla }}\sim \epsilon \ll 1\end{eqnarray} \tag{ 11 }$$

( being a small parameter) and in the regime of small perturbations of the density and of the magnetic field, and of the weak dependence along the magnetic field line:

$$\begin{eqnarray}&&\displaystyle \frac{\delta n}{n}\sim \displaystyle \frac{| \delta {\boldsymbol{B}}| }{| {\boldsymbol{B}}| }\sim \displaystyle \frac{{\boldsymbol{b}}\,\cdot {\rm{\nabla }}}{{{\rm{\nabla }}}_{\perp }}\sim \epsilon ,\end{eqnarray} \tag{ 12 }$$

provided the fluid motion is not predominantly 1D in the parallel direction, ${{\boldsymbol{U}}}_{\parallel }\,/\!\!\!\!\!\!\gg {{\boldsymbol{U}}}_{\perp }$. Here, Ω is the gyrofrequency, ${\rm{\Omega }}={qB}/m$, and δ denotes the deviation of a quantity from its unperturbed value. The explicit form of the collisionless stress tensor ${\boldsymbol{\pi }}$ in a plasma with anisotropic temperature can be seen, e.g., in Passot et al. (2013), Sulem & Passot (2012), and Yajima (1966), where it has been calculated under the scaling of Equation (11), with the accuracy to first order in the FLR (finite Larmor radius) corrections, and expressed in the local orthogonal coordinate system with the curvilinear b-axis along the magnetic lines of force:

$$\begin{eqnarray}\begin{array}{rcl}{\pi }_{m,m} & = & -{\pi }_{l,l}=\left({p}_{\perp }/2{\rm{\Omega }}\right)\left(\partial {U}_{l}/\partial {x}_{m}+\partial {U}_{m}/\partial {x}_{l}\right),\\ {\pi }_{l,m} & = & {\pi }_{m,l}=\left({p}_{\perp }/2{\rm{\Omega }}\right)\left(\partial {U}_{l}/\partial {x}_{l}-\partial {U}_{m}/\partial {x}_{m}\right),\\ {\pi }_{l,b} & = & {\pi }_{b,l}=-\left({p}_{\perp }/{\rm{\Omega }}\right)\left(\partial {U}_{b}/\partial {x}_{m}\right)\\ & & -\left[\left(2{p}_{\parallel }-{p}_{\perp }\right)/{\rm{\Omega }}\right]\left(\partial {U}_{m}/\partial {x}_{b}\right),\\ {\pi }_{m,b} & = & {\pi }_{b,m}=\left({p}_{\perp }/{\rm{\Omega }}\right)\left(\partial {U}_{b}/\partial {x}_{l}\right)\\ & & +\left[\left(2{p}_{\parallel }-{p}_{\perp }\right)/{\rm{\Omega }}\right]\left(\partial {U}_{l}/\partial {x}_{b}\right),\\ {\pi }_{b,b} & = & 0.\end{array}\end{eqnarray} \tag{ 13 }$$

Here, ${\boldsymbol{b}}$, ${{\boldsymbol{e}}}_{l}$, and ${{\boldsymbol{e}}}_{m}$ are three mutually perpendicular unit vectors and $\partial /\partial {x}_{\alpha }\equiv {{\boldsymbol{e}}}_{\alpha }\cdot {\rm{\nabla }}$, where α = l, m, b. We adopt the last two vectors as ${{\boldsymbol{e}}}_{l}={\boldsymbol{b}}\times ({{\boldsymbol{e}}}_{x}\times {\boldsymbol{b}})/| {{\boldsymbol{e}}}_{x}\times {\boldsymbol{b}}|$ and ${{\boldsymbol{e}}}_{m}={\boldsymbol{b}}\times {{\boldsymbol{e}}}_{l}$. In the regime of the weak curvature of the magnetic field lines, Equation (12), these unit vectors are properly approximated by the expressions given in Equation (30) below.

The perpendicular component of the fluid velocity is obtained after we multiply Equation (2) with ${\boldsymbol{b}}\times$ and can be readily written as the sum of the ${\boldsymbol{E}}\times {\boldsymbol{B}}$, diamagnetic, anisotropic temperature, and polarization drifts, together with the leading part and the curvature correction of the stress-related (or the FLR) drift:

$$\begin{eqnarray}&&{{\boldsymbol{U}}}_{\perp }={{\boldsymbol{U}}}_{E}+{{\boldsymbol{U}}}_{D}+{{\boldsymbol{U}}}_{{\rm{A}}}+{{\boldsymbol{U}}}_{p}+{{\boldsymbol{U}}}_{\pi }+\delta {{\boldsymbol{U}}}_{\pi },\end{eqnarray} \tag{ 14 }$$

where

$$\begin{eqnarray}&&\begin{array}{l}{{\boldsymbol{U}}}_{E}=-\displaystyle \frac{{\boldsymbol{b}}}{B}\times {\boldsymbol{E}},\quad {{\boldsymbol{U}}}_{D}=\displaystyle \frac{{\boldsymbol{b}}}{{qnB}}\times {{\rm{\nabla }}}_{\perp }{p}_{\perp },\\ {{\boldsymbol{U}}}_{{\rm{A}}}=\left({p}_{\parallel }-{p}_{\perp }\right)\displaystyle \frac{{\boldsymbol{b}}}{{qnB}}\times \left({\boldsymbol{b}}\cdot {\rm{\nabla }}\right){\boldsymbol{b}},\quad {{\boldsymbol{U}}}_{B}=\displaystyle \frac{{p}_{\perp }{\boldsymbol{b}}}{{{qnB}}^{2}}\times {{\rm{\nabla }}}_{\perp }B\end{array}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{U}}}_{p} & = & \displaystyle \frac{{\boldsymbol{b}}}{{\rm{\Omega }}}\times \left(\displaystyle \frac{\partial }{\partial t}+{\boldsymbol{U}}\cdot {\rm{\nabla }}\right){\boldsymbol{U}},\quad {{\boldsymbol{U}}}_{\pi }=\displaystyle \frac{{\boldsymbol{b}}}{{qnB}}\times {{\boldsymbol{e}}}_{\beta }\displaystyle \frac{\partial {\pi }_{\alpha ,\beta }}{\partial {x}_{\alpha }},\\ \delta {{\boldsymbol{U}}}_{\pi } & = & \displaystyle \frac{{\boldsymbol{b}}}{{qnB}}\times {\left({\rm{\nabla }}\cdot {\boldsymbol{\pi }}\right)}_{\mathrm{curv}}.\end{array}\end{eqnarray} \tag{ 16 }$$

For completeness, in the above list we have also added the grad-B drift velocity ${{\boldsymbol{U}}}_{B}$, although it does not appear explicitly in expression (14), but it will turn up later in the continuity equation, by virtue of the term ${\rm{\nabla }}\cdot {\boldsymbol{U}}$.

Using Equation (13), the stress-related drift and the leading contribution of the stress to the parallel momentum (Equation (9)) take the form

$$\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{U}}}_{\pi } & = & -\displaystyle \frac{{T}_{\perp }}{2m{{\rm{\Omega }}}^{2}}{{\rm{\nabla }}}_{\perp }^{2}{{\boldsymbol{U}}}_{\perp }\\ & & -\,\displaystyle \frac{1}{{qnB}}\left\{\left[\left({\boldsymbol{b}}\times {{\rm{\nabla }}}_{\perp }\displaystyle \frac{{p}_{\perp }}{2{\rm{\Omega }}}\right)\cdot {{\rm{\nabla }}}_{\perp }\right]{\boldsymbol{b}}\times {{\boldsymbol{U}}}_{\perp }+\left({{\rm{\nabla }}}_{\perp }\displaystyle \frac{{p}_{\perp }}{2{\rm{\Omega }}}\cdot {{\rm{\nabla }}}_{\perp }\right){{\boldsymbol{U}}}_{\perp }\right.\\ & & \left.+\,\left({\boldsymbol{b}}\cdot {\rm{\nabla }}\right)\left[\displaystyle \frac{{p}_{\perp }}{{\rm{\Omega }}}{{\rm{\nabla }}}_{\perp }{U}_{\parallel }+\displaystyle \frac{2{p}_{\parallel }-{p}_{\perp }}{{\rm{\Omega }}}\left({\boldsymbol{b}}\cdot {\rm{\nabla }}\right){{\boldsymbol{U}}}_{\perp }\right]\right\},\quad \quad \quad \end{array}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\pi }_{\alpha ,b}}{\partial {x}_{\alpha }}={\boldsymbol{b}}\cdot \left\{{{\rm{\nabla }}}_{\perp }{U}_{\parallel }\times {{\rm{\nabla }}}_{\perp }\displaystyle \frac{{p}_{\perp }}{{\rm{\Omega }}}-{{\rm{\nabla }}}_{\perp }\times \left[\displaystyle \frac{2{p}_{\parallel }-{p}_{\perp }}{{\rm{\Omega }}}\left({\boldsymbol{b}}\cdot {\rm{\nabla }}\right){{\boldsymbol{U}}}_{\perp }\right]\right\}.\end{eqnarray} \tag{ 18 }$$

In the regime of slow variations (compared to Ω), weak dependence along the magnetic field lines, and small perturbations of the magnetic field in all spatial directions, which is described by the scalings given in Equations (11) and (12), we can ignore the last term in Equation (17) that contains the second parallel derivative ${(\partial /\partial {x}_{b})}^{2}$. Noting also that the contribution of the magnetic curvature to the divergence of the stress tensor is a small quantity of the order ${\epsilon }^{2}{\rho }_{L}^{2}{{\rm{\nabla }}}^{2}$, viz.

$$\begin{eqnarray}&&{\left({\rm{\nabla }}\cdot {\boldsymbol{\pi }}\right)}_{\mathrm{curv}}={\pi }_{\alpha ,b}\,\left(\partial {\boldsymbol{b}}/\partial {x}_{\alpha }\right)-{\boldsymbol{b}}\,{\pi }_{\alpha ,\beta }\left[{{\boldsymbol{e}}}_{\beta }\cdot \left(\partial {\boldsymbol{b}}/\partial {x}_{\alpha }\right)\right]+{ \mathcal O }\left({\epsilon }^{3}\right),\end{eqnarray} \tag{ 19 }$$

and using Equations (13) and the approximations in Equation (30), we obtain the following leading-order expressions:

$$\begin{eqnarray}&&\begin{array}{l}\delta {{\boldsymbol{U}}}_{\pi }={{\boldsymbol{e}}}_{{\alpha }_{\perp }}\,{\rho }_{L}^{2}\left(\displaystyle \frac{\partial {\boldsymbol{b}}}{\partial {x}_{{\alpha }_{\perp }}}\cdot {\rm{\nabla }}{U}_{\parallel }\right),\\ {\boldsymbol{b}}\cdot {\left({\rm{\nabla }}\cdot {\boldsymbol{\pi }}\right)}_{\mathrm{curv}}={{qn}}_{0}{\rho }_{L}^{2}\,\displaystyle \frac{\partial {{\boldsymbol{U}}}_{\perp }}{\partial {x}_{{\alpha }_{\perp }}}\cdot {{\rm{\nabla }}}_{\perp }\displaystyle \frac{\partial {A}_{z}}{\partial {x}_{{\alpha }_{\perp }}},\end{array}\end{eqnarray} \tag{ 20 }$$

where ${\rho }_{L}=\sqrt{{T}_{\perp }/m{{\rm{\Omega }}}_{0}^{2}}$ is the (unperturbed) Larmor radius and ${\alpha }_{\perp }=x,y$.

The leading-order expression (in ) for the stress-related drift ${{\boldsymbol{U}}}_{\pi }+\delta {{\boldsymbol{U}}}_{\pi }$ is given by the first term on the right-hand side of Equation (17). Then, using the fact that the polarization drift ${{\boldsymbol{U}}}_{p}$ and the curvature-related drifts ${{\boldsymbol{U}}}_{{\rm{A}}}$ and $\delta {{\boldsymbol{U}}}_{\pi }$ are small compared to ${\boldsymbol{E}}\times {\boldsymbol{B}}$ and the diamagnetic drifts and noting that ${\rm{\nabla }}\cdot ({{\boldsymbol{U}}}_{E}+{{\boldsymbol{U}}}_{D})={ \mathcal O }\,\left({\epsilon }^{2}\right)$ (see Equation (23)), with accuracy to leading order in , we can write ${{\boldsymbol{U}}}_{\perp }\approx {{\boldsymbol{U}}}_{E}+{{\boldsymbol{U}}}_{D}-{\rho }_{L}^{2}{{\rm{\nabla }}}_{\perp }^{2}{{\boldsymbol{U}}}_{\perp }/2$, which is formally solved as ${{\boldsymbol{U}}}_{\perp }\approx {{\boldsymbol{U}}}_{\perp }^{\mathrm{apr}}$, where

$$\begin{eqnarray}&&{{\boldsymbol{U}}}_{\perp }^{\mathrm{apr}}\equiv {\left(1+{\rho }_{L}^{2}{{\rm{\nabla }}}_{\perp }^{2}/2\right)}^{-1}\left({{\boldsymbol{U}}}_{E}+{{\boldsymbol{U}}}_{D}\right).\end{eqnarray} \tag{ 21 }$$

Within the adopted accuracy, this expression for ${{\boldsymbol{U}}}_{\perp }$ can be used on the right-hand sides of Equations (16)–(18) and (22). Likewise, in the convective derivatives, we use ${\boldsymbol{U}}\cdot {\rm{\nabla }}\,\approx {{\boldsymbol{U}}}_{\perp }\cdot {{\rm{\nabla }}}_{\perp }\approx {{\boldsymbol{U}}}_{\perp }^{\mathrm{apr}}\cdot {{\rm{\nabla }}}_{\perp }$. Under the scaling (Equations (11) and (12)), we can also ignore the right-hand side of Equation (9). Conversely, on the right-hand side of Equation (10), we ignore the second-order terms coming from the polarization, charge separation, displacement current, and the curvature of the magnetic field. Only the leading part of the last term remains, and the equation simplifies to

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\nabla }}}_{\perp }\left(\displaystyle \frac{{c}^{2}{\epsilon }_{0}{B}^{2}}{2}+{p}_{{e}_{\perp }}+{p}_{{i}_{\perp }}\right)\\ \,=\,-{\boldsymbol{b}}\times \left(\displaystyle \frac{{p}_{{i}_{\perp }}}{2{{\rm{\Omega }}}_{i}}\,{{\rm{\nabla }}}_{\perp }^{2}{{\boldsymbol{U}}}_{{i}_{\perp }}+\displaystyle \frac{{p}_{{e}_{\perp }}}{2{{\rm{\Omega }}}_{e}}\,{{\rm{\nabla }}}_{\perp }^{2}{{\boldsymbol{U}}}_{{e}_{\perp }}\right),\end{array}\end{eqnarray} \tag{ 22 }$$

which reduces to the equation of pressure balance when the Larmor radius corrections can be ignored. Now, using

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\nabla }}\cdot \left({{\boldsymbol{U}}}_{E}+{{\boldsymbol{U}}}_{D}\right) & \approx & -\left[\displaystyle \frac{\partial }{\partial t}+\left({{\boldsymbol{U}}}_{E}+{{\boldsymbol{U}}}_{D}\right)\cdot {{\rm{\nabla }}}_{\perp }\right]\mathrm{log}B\\ & & -\left({{\boldsymbol{U}}}_{E}+{{\boldsymbol{U}}}_{D}\right)\cdot \left({\boldsymbol{b}}\cdot {\rm{\nabla }}\right){\boldsymbol{b}}-\displaystyle \frac{1}{{{qn}}^{2}B}\left({\boldsymbol{b}}\times {\rm{\nabla }}{p}_{\perp }\right)\cdot {\rm{\nabla }}n,\end{array}\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{\boldsymbol{b}}{\rm{\nabla }}\cdot {{\boldsymbol{U}}}_{{\rm{A}}}\approx \left({\boldsymbol{b}}\cdot {\rm{\nabla }}\right)\left[\displaystyle \frac{{p}_{{\parallel }_{0}}-{p}_{{\perp }_{0}}}{{{qn}}_{0}{B}_{0}}\,\,{\rm{\nabla }}\cdot \left({{\boldsymbol{e}}}_{z}\times {\boldsymbol{b}}\right)\right],\end{eqnarray} \tag{ 24 }$$

and the expressions (17) and (19) for ${\rm{\nabla }}\cdot {{\boldsymbol{U}}}_{\pi }$, we evaluate continuity and parallel momentum equations to leading order in , setting ${\boldsymbol{U}}\cdot {\rm{\nabla }}\approx {{\boldsymbol{U}}}_{\perp }^{\mathrm{apr}}\cdot {{\rm{\nabla }}}_{\perp }$ and ${{\boldsymbol{U}}}_{\perp }\approx {{\boldsymbol{U}}}_{\perp }^{\mathrm{apr}}$, and dropping the last term on the right-hand side of Equation (23) that vanishes if ${p}_{\perp }$ is an arbitrary function of n, described by Equation (6) when ${\gamma }_{\perp }={\gamma }_{\perp }(n)$. We obtain

$$\begin{eqnarray}&&\begin{array}{l}\left(\displaystyle \frac{\partial }{\partial t}+{{\boldsymbol{U}}}_{\perp }^{\mathrm{apr}}\cdot {{\rm{\nabla }}}_{\perp }\right)\left(\mathrm{log}n-\mathrm{log}B\right)+\left({\boldsymbol{b}}\cdot {\rm{\nabla }}\right)\left[{U}_{\parallel }+\displaystyle \frac{{p}_{{\parallel }_{0}}-{p}_{{\perp }_{0}}}{{{qn}}_{0}{B}_{0}}\,\,{\rm{\nabla }}\cdot \left({{\boldsymbol{e}}}_{z}\times {\boldsymbol{b}}\right)\right]\\ \ +\,\displaystyle \frac{1}{{{\rm{\Omega }}}_{0}}{{\rm{\nabla }}}_{\perp }\cdot \left\{\left[\displaystyle \frac{\partial }{\partial t}+\left({{\boldsymbol{U}}}_{\perp }^{\mathrm{apr}}+{{\boldsymbol{U}}}_{B}-{{\boldsymbol{U}}}_{D}\right)\cdot {{\rm{\nabla }}}_{\perp }\right]{{\boldsymbol{e}}}_{z}\times {{\boldsymbol{U}}}_{\perp }^{\mathrm{apr}}\right\}+{\rho }_{L}^{2}\left({{\rm{\nabla }}}_{\perp }^{2}{\boldsymbol{b}}\cdot {\rm{\nabla }}{U}_{\parallel }-{\boldsymbol{b}}\cdot {\rm{\nabla }}{{\rm{\nabla }}}_{\perp }^{2}{U}_{\parallel }\right)=0,\end{array}\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&\begin{array}{l}\left[\displaystyle \frac{\partial }{\partial t}+\left({{\boldsymbol{U}}}_{\perp }^{\mathrm{apr}}+{{\boldsymbol{U}}}_{B}-{{\boldsymbol{U}}}_{D}\right)\cdot {\rm{\nabla }}\right]{U}_{\parallel }=\displaystyle \frac{q}{m}\left({\boldsymbol{b}}\cdot {\boldsymbol{E}}-{\rho }_{L}^{2}\,\displaystyle \frac{\partial {{\boldsymbol{U}}}_{\perp }^{\mathrm{apr}}}{\partial {x}_{{\alpha }_{\perp }}}\cdot {{\rm{\nabla }}}_{\perp }\displaystyle \frac{\partial {A}_{z}}{\partial {x}_{{\alpha }_{\perp }}}\right)\\ \,-\displaystyle \frac{1}{{n}_{0}m}\left({\boldsymbol{b}}\cdot {\rm{\nabla }}\right)\left[{p}_{\parallel }-\left({p}_{{\parallel }_{0}}-{p}_{{\perp }_{0}}\right)\displaystyle \frac{B}{{B}_{0}}+\displaystyle \frac{2{p}_{{\parallel }_{0}}-{p}_{{\perp }_{0}}}{{{\rm{\Omega }}}_{0}}\,{{\rm{\nabla }}}_{\perp }\cdot \left({\boldsymbol{b}}\times {{\boldsymbol{U}}}_{\perp }^{\mathrm{apr}}\right)\right].\end{array}\end{eqnarray} \tag{ 26 }$$

It is convenient to subtract the electron and ion continuity equations, i.e., to use the continuity equation for electric charge, which, after ignoring the electrons' polarization and FLR effects, yields

$$\begin{eqnarray}&&\begin{array}{l}\left({{\boldsymbol{U}}}_{{i}_{\perp }}^{\mathrm{apr}}-{{\boldsymbol{U}}}_{{e}_{\perp }}^{\mathrm{apr}}\right)\cdot {{\rm{\nabla }}}_{\perp }\left(\mathrm{log}n-\mathrm{log}B\right)+\left({\boldsymbol{b}}\cdot {\rm{\nabla }}\right)\left[{U}_{{i}_{\parallel }}-{U}_{{e}_{\parallel }}+\displaystyle \frac{{p}_{{i}_{\parallel 0}}-{p}_{{i}_{\perp 0}}+{p}_{{e}_{\parallel 0}}-{p}_{{e}_{\perp 0}}}{{{en}}_{0}{B}_{0}}\,\,{\rm{\nabla }}\cdot \left({{\boldsymbol{e}}}_{z}\times {\boldsymbol{b}}\right)\right]\\ \ +\,\displaystyle \frac{1}{{{\rm{\Omega }}}_{{i}_{0}}}{{\rm{\nabla }}}_{\perp }\cdot \left\{\left[\displaystyle \frac{\partial }{\partial t}+\left({{\boldsymbol{U}}}_{{i}_{\perp }}^{\mathrm{apr}}+{{\boldsymbol{U}}}_{{i}_{B}}-{{\boldsymbol{U}}}_{{i}_{D}}\right)\cdot {{\rm{\nabla }}}_{\perp }\right]{{\boldsymbol{e}}}_{z}\times {{\boldsymbol{U}}}_{{i}_{\perp }}^{\mathrm{apr}}\right\}+{\rho }_{\mathrm{Li}}^{2}\left({{\rm{\nabla }}}_{\perp }^{2}{\boldsymbol{b}}\cdot {\rm{\nabla }}{U}_{{i}_{\parallel }}-{\boldsymbol{b}}\cdot {\rm{\nabla }}{{\rm{\nabla }}}_{\perp }^{2}{U}_{{i}_{\parallel }}\right)=0.\end{array}\end{eqnarray} \tag{ 27 }$$

Similarly, multiplying the electron and ion momentum equations respectively by m_e and m_i and adding, we obtain the momentum equation for the plasma fluid, which in the limit of massless electrons m_e→0 has the form

$$\begin{eqnarray}&&\begin{array}{l}\left({{\boldsymbol{U}}}_{{i}_{\perp }}^{\mathrm{apr}}-{{\boldsymbol{U}}}_{{e}_{\perp }}^{\mathrm{apr}}\right)\cdot {{\rm{\nabla }}}_{\perp }\left(\mathrm{log}n-\mathrm{log}B\right)+\left({\boldsymbol{b}}\cdot {\rm{\nabla }}\right)\left[{U}_{{i}_{\parallel }}-{U}_{{e}_{\parallel }}+\displaystyle \frac{{p}_{{i}_{\parallel 0}}-{p}_{{i}_{\perp 0}}+{p}_{{e}_{\parallel 0}}-{p}_{{e}_{\perp 0}}}{{{en}}_{0}{B}_{0}}\,\,{\rm{\nabla }}\cdot \left({{\boldsymbol{e}}}_{z}\times {\boldsymbol{b}}\right)\right]\\ \ +\,\displaystyle \frac{1}{{{\rm{\Omega }}}_{{i}_{0}}}{{\rm{\nabla }}}_{\perp }\cdot \left\{\left[\displaystyle \frac{\partial }{\partial t}+\left({{\boldsymbol{U}}}_{{i}_{\perp }}^{\mathrm{apr}}+{{\boldsymbol{U}}}_{{i}_{B}}-{{\boldsymbol{U}}}_{{i}_{D}}\right)\cdot {{\rm{\nabla }}}_{\perp }\right]{{\boldsymbol{e}}}_{z}\times {{\boldsymbol{U}}}_{{i}_{\perp }}^{\mathrm{apr}}\right\}+{\rho }_{\mathrm{Li}}^{2}\left({{\rm{\nabla }}}_{\perp }^{2}{\boldsymbol{b}}\cdot {\rm{\nabla }}{U}_{{i}_{\parallel }}-{\boldsymbol{b}}\cdot {\rm{\nabla }}{{\rm{\nabla }}}_{\perp }^{2}{U}_{{i}_{\parallel }}\right)=0.\end{array}\end{eqnarray} \tag{ 28 }$$

Within the adopted drift and small but finite FLR scalings, Equations (11) and (12), and taking the compressional perturbation of the magnetic field to be of the same order as, or smaller than, the torsional component, the electromagnetic field can be expressed in terms of the electrostatic potential and of the z components of the vector potential and magnetic field, ϕ, A_z, and δB_z, viz.

$$\begin{eqnarray}&&\begin{array}{l}{\boldsymbol{E}}=-{\rm{\nabla }}\phi -\displaystyle \frac{\partial }{\partial t}\left({{\boldsymbol{e}}}_{z}{A}_{z}+{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\,{{\rm{\nabla }}}_{\perp }^{-2}\delta {B}_{z}\right),\\ {\boldsymbol{B}}={{\boldsymbol{e}}}_{z}\left({B}_{0}+\delta {B}_{z}\right)-{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }{A}_{z},\end{array}\end{eqnarray} \tag{ 29 }$$

which yields the following expressions, accurate to first order:

$$\begin{eqnarray}&&\begin{array}{l}{\boldsymbol{b}}={{\boldsymbol{e}}}_{z}-\displaystyle \frac{1}{{B}_{0}}\,{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }{A}_{z},\\ {{\boldsymbol{e}}}_{l}={{\boldsymbol{e}}}_{x}-\displaystyle \frac{{\boldsymbol{b}}}{{B}_{0}}\displaystyle \frac{\partial {A}_{z}}{\partial y},\\ {{\boldsymbol{e}}}_{m}={{\boldsymbol{e}}}_{y}+\displaystyle \frac{{\boldsymbol{b}}}{{B}_{0}}\displaystyle \frac{\partial {A}_{z}}{\partial x},\end{array}\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&\begin{array}{l}B=| {\boldsymbol{B}}| ={B}_{0}+\delta {B}_{z},\\ {\boldsymbol{b}}\cdot {\boldsymbol{E}}=-{\boldsymbol{b}}\cdot {\rm{\nabla }}\phi -\displaystyle \frac{\partial {A}_{z}}{\partial t},\\ {\boldsymbol{b}}\times {\boldsymbol{E}}=-{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\phi ,\end{array}\end{eqnarray} \tag{ 31 }$$

We conveniently rewrite our basic equations in dimensionless form using the following scaled variables and parameters:

$$\begin{eqnarray}&&\begin{array}{l}t^{\prime} =\omega t,x^{\prime} =kx,y^{\prime} =k\,y,\quad z^{\prime} =(\omega {/c}_{{\rm{A}}})z,\\ \phi ^{\prime} =\displaystyle \frac{{k}^{2}\phi }{{B}_{0}\omega },\quad {A}_{z}^{{\prime} }=\displaystyle \frac{{k}^{2}{c}_{{\rm{A}}}{A}_{z}}{{B}_{0}\omega },\quad {B}_{z}^{{\prime} }=\displaystyle \frac{{{\rm{\Omega }}}_{{i}_{0}}}{\omega }\displaystyle \frac{\delta {B}_{z}}{{B}_{0}},\\ n^{\prime} =\displaystyle \frac{{{\rm{\Omega }}}_{{i}_{0}}}{\omega }\displaystyle \frac{\delta n}{{n}_{0}},{U}_{\parallel }^{{\prime} }=\displaystyle \frac{{\omega }_{\mathrm{pi}}}{\omega }\displaystyle \frac{{U}_{\parallel }}{c},\quad p^{\prime} =\displaystyle \frac{{k}^{2}}{{{\rm{\Omega }}}_{{i}_{0}}\omega }\displaystyle \frac{\delta p}{{n}_{0}{m}_{i}},\quad \beta =\displaystyle \frac{2{p}_{0}}{{c}^{2}{\epsilon }_{0}{B}_{0}^{2}},\\ {d}_{e}^{{\prime} }=\displaystyle \frac{c\,k}{{\omega }_{\mathrm{pe}}},\quad {d}_{i}^{{\prime} }=\displaystyle \frac{c\,k}{{\omega }_{\mathrm{pi}}},\quad {\rho }_{i}^{{\prime} }=k\,{\rho }_{\mathrm{Li}}={d}_{i}^{{\prime} }\,\sqrt{\displaystyle \frac{{\beta }_{{i}_{\perp }}}{2}},\end{array}\end{eqnarray} \tag{ 32 }$$

where k and ω are some characteristic wavenumber and characteristic frequency, i.e., the inverse characteristic spatial and temporal scales, such as the width of the structure r₀ and the transit time ${r}_{0}/{u}_{\perp }$, ${u}_{\perp }$ being the speed of its propagation in the plasma frame transversely to the magnetic field. The normalization through k, ω is used for convenience and does not indicate any presence of wave phenomena. Other notations are standard, ${\omega }_{\mathrm{pi}}=\sqrt{{n}_{0}{e}^{2}/{m}_{i}{\epsilon }_{0}}$ is the ion plasma frequency and ${c}_{{\rm{A}}}=c\,{{\rm{\Omega }}}_{{i}_{0}}/{\omega }_{\mathrm{pi}}$ is the Alfvén speed. Note that the dimensionless parallel velocity ${U}_{\parallel }^{{\prime} }$, the pressure $p^{\prime}$, and the parameter β involve either the electrons or the ions, and the two latter quantities also involve the perpendicular or the parallel components, which are denoted below by the appropriate combination of subscripts $e,i,\perp$, and $\parallel$. The dimensionless versions of the charge continuity, Equation (27), the electron continuity and parallel momentum equations, Equations (25) and (26), and of the parallel and perpendicular fluid momentum, Equations (28) and (22), can be written as follows (for simplicity, here and in the rest of the paper, we omit the primes):

$$\begin{eqnarray}&&\begin{array}{l}\left[{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\left(\displaystyle \frac{{p}_{{i}_{\perp }}-{\rho }_{i}^{2}{{\rm{\nabla }}}_{\perp }^{2}\phi /2}{1+{\rho }_{i}^{2}{{\rm{\nabla }}}_{\perp }^{2}/2}+{p}_{{e}_{\perp }}\right)\right]\cdot {{\rm{\nabla }}}_{\perp }\left(n-{B}_{z}\right)\\ \quad -\,\left[\displaystyle \frac{\partial }{\partial z}-\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }{A}_{z}\right)\cdot {{\rm{\nabla }}}_{\perp }\right]\left(1-\displaystyle \frac{{\beta }_{{i}_{\parallel }}-{\beta }_{{i}_{\perp }}+{\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}}{2}\right){{\rm{\nabla }}}_{\perp }^{2}{A}_{z}\\ \quad -\,{{\rm{\nabla }}}_{\perp }\cdot \left(\left\{\displaystyle \frac{\partial }{\partial t}+\left[{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\left(\displaystyle \frac{\phi -{\rho }_{i}^{2}{{\rm{\nabla }}}_{\perp }^{2}{p}_{{i}_{\perp }}/2}{1+{\rho }_{i}^{2}{{\rm{\nabla }}}_{\perp }^{2}/2}\right.\right.\right.\right.\\ \quad \left.\left.\left.\left.+\,\displaystyle \frac{{\beta }_{{i}_{\perp }}}{2}\,{d}_{i}^{2}{B}_{z}\right)\right]\cdot {{\rm{\nabla }}}_{\perp }\right\}\cdot {{\rm{\nabla }}}_{\perp }\,\displaystyle \frac{\phi +{p}_{{i}_{\perp }}}{1+{\rho }_{i}^{2}{{\rm{\nabla }}}_{\perp }^{2}/2}\right)\\ \quad -\,{\rho }_{i}^{2}\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }{{\rm{\nabla }}}_{\perp }^{2}{A}_{z}\right)\cdot {{\rm{\nabla }}}_{\perp }{U}_{{i}_{\parallel }}\\ \quad -\,{\rho }_{i}^{2}\left[\displaystyle \frac{\partial }{\partial z}-\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }{A}_{z}\right)\cdot {{\rm{\nabla }}}_{\perp }\right]{{\rm{\nabla }}}_{\perp }^{2}{U}_{{i}_{\parallel }}=0,\end{array}\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}&&\begin{array}{l}\left\{\displaystyle \frac{\partial }{\partial t}+\left[{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\left(\phi -{p}_{{e}_{\perp }}\right)\right]\cdot {{\rm{\nabla }}}_{\perp }\right\}\left(n-{B}_{z}\right)\\ +\,\left[\displaystyle \frac{\partial }{\partial z}-\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }{A}_{z}\right)\cdot {{\rm{\nabla }}}_{\perp }\right]\left({U}_{{e}_{\parallel }}-\displaystyle \frac{{\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}}{2}{{\rm{\nabla }}}_{\perp }^{2}{A}_{z}\right)=0,\end{array}\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&\begin{array}{l}\left[\displaystyle \frac{\partial }{\partial z}-\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }{A}_{z}\right)\cdot {{\rm{\nabla }}}_{\perp }\right]\left(\phi -{p}_{{e}_{\parallel }}+\displaystyle \frac{{\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}}{2}\,{d}_{i}^{2}{B}_{z}\right)\\ +\,\displaystyle \frac{\partial {A}_{z}}{\partial t}=0,\end{array}\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&\begin{array}{l}\left\{\displaystyle \frac{\partial }{\partial t}+\left[{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\left(\displaystyle \frac{\phi -{\rho }_{i}^{2}{{\rm{\nabla }}}_{\perp }^{2}{p}_{{i}_{\perp }}/2}{1+{\rho }_{i}^{2}{{\rm{\nabla }}}_{\perp }^{2}/2}+\displaystyle \frac{{\beta }_{{i}_{\perp }}}{2}\,{d}_{i}^{2}{B}_{z}\right)\right]\cdot {{\rm{\nabla }}}_{\perp }\right\}\,{d}_{i}^{2}{U}_{{i}_{\parallel }}\\ \quad =\,-\,\left[\displaystyle \frac{\partial }{\partial z}-\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }{A}_{z}\right)\cdot {{\rm{\nabla }}}_{\perp }\right]\left[{p}_{{i}_{\parallel }}+{p}_{{e}_{\parallel }}\right.\\ \left(-\,\displaystyle \frac{{\beta }_{{i}_{\parallel }}-{\beta }_{{i}_{\perp }}+{\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}}{2}\,{d}_{i}^{2}{B}_{z}-\left(\displaystyle \frac{2{\beta }_{{i}_{\parallel }}}{{\beta }_{{i}_{\perp }}}-1\right)\displaystyle \frac{{\rho }_{i}^{2}{{\rm{\nabla }}}_{\perp }^{2}\left(\phi +{p}_{{i}_{\perp }}\right)}{1+{\rho }_{i}^{2}{{\rm{\nabla }}}_{\perp }^{2}/2}\right]\\ \quad -\,{\rho }_{i}^{2}\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\,\displaystyle \frac{\partial }{\partial {x}_{{\alpha }_{\perp }}}\displaystyle \frac{\phi +{p}_{{i}_{\perp }}}{1+{\rho }_{i}^{2}{{\rm{\nabla }}}_{\perp }^{2}/2}\right)\cdot {\rm{\nabla }}\displaystyle \frac{\partial {A}_{z}}{\partial {x}_{{\alpha }_{\perp }}},\end{array}\end{eqnarray} \tag{ 36 }$$

$$\begin{eqnarray}&&{d}_{i}^{2}{B}_{z}+{p}_{{i}_{\perp }}+{p}_{{e}_{\perp }}-\displaystyle \frac{{\rho }_{i}^{2}{{\rm{\nabla }}}_{\perp }^{2}\left(\phi +{p}_{{i}_{\perp }}\right)/2}{1+{\rho }_{i}^{2}{{\rm{\nabla }}}_{\perp }^{2}/2}=0,\end{eqnarray} \tag{ 37 }$$

where we ignored the charge separation (n_e = n_i = n) and the displacement current (${U}_{{e}_{\parallel }}^{{\prime} }-{U}_{{i}_{\parallel }}^{{\prime} }={{{\rm{\nabla }}}_{\perp }^{{\prime} }}^{2}{A}_{z}^{{\prime} }$), and we considered the electrons to be massless (i.e., we took m_e → 0 and accordingly, $\omega /{{\rm{\Omega }}}_{e}={\rho }_{e}^{{\prime} }={d}_{e}^{{\prime} }=0$).

Pressure perturbations are expressed in terms of density via dimensionless versions of the equations of state (6), viz.

$$\begin{eqnarray}&&{p}_{j\,\zeta }^{{\prime} }={\gamma }_{j\zeta }\left({\beta }_{j\zeta }/2\right)\,{{d}_{i}^{{\prime} }}^{2}n^{\prime} ,\quad \mathrm{where}\quad j=e,i\quad \mathrm{and}\quad \zeta =\parallel ,\perp .\end{eqnarray} \tag{ 38 }$$

It is worth noting that the acoustic perturbations, which are associated with the parallel plasma motion described by the momentum Equation (36), are coupled with the rest of the system only through the last terms in Equation (33), the first of which has come from the divergence of the curvature correction to the FLR drift, ${\rm{\nabla }}\cdot \delta {{\boldsymbol{U}}}_{i\pi }$. It will be shown below that in the regimes of our interest, these terms are small and the acoustic perturbations are fully decoupled.

Equations (33)–(35) and (37) constitute our basic set of equations for the functions $\phi ,{A}_{z},{B}_{z}$, and n. In a plasma with arbitrary values of ${\beta }_{j\zeta }$, these equations describe variations that are slowly varying both in time and along the magnetic field lines, but have an arbitrary spatial scale perpendicular to the magnetic field.

We readily note that all nonlinear terms in our Equations (33)–(36) have the form of mixed products that vanish when the solution is 1D, either in Cartesian or in cylindrical coordinates. As a consequence, there exist trivial solutions that are independent of z and stationary in time, $\partial /\partial z=\partial /\partial t=0$, such as current sheets (planar discontinuities) and localized cylindrically symmetric solutions that may have arbitrary radial dependencies. The latter have the form of Petviashvili–Pokhotelov monopoles and are constrained only by the equations of the pressure balance, Equation (37), and state, Equation (38). The monopoles come in two distinct varieties, as the electrostatic and magnetostatic convective cells (the latter are often referred to as the current filaments or force-free currents; Sugai et al. 1983, 1984). Force-free currents, in the form of filaments parallel to the magnetic field lines (Sturrock 1994; Scott 2015), are often encountered in space plasmas—in the solar corona, planetary magnetospheres, solar wind, etc.—and they are thought to have a sufficiently long lifetime to be carried by the solar wind for several astronomical units. The evolution of electrostatic cells has been extensively studied both numerically and experimentally (see, e.g., Lynov et al. 1980; Pecseli et al. 1984). The experiments in quiescent plasma, as well as the water-tank experiments (Beckers et al. 2003), in which the perturbations are described by 2D model equations that are similar to those in the plasma but also include a small but finite viscosity, revealed that stationary monopoles either disperse or slowly transform into propagating dipolar or rotating tripolar vortices, depending on the amount of shielding included in the radial profile of the initial state. Finding a general solution of the above equations is a formidable task because of their complexity, but suitable particular solutions have been found in a number of relevant cases. Thus, in our earlier papers, we studied in detail the electron-scale nonlinear structures in plasmas with cold electrons and very cold ions, ${\beta }_{{e}_{\perp }}\ll 1$ and ${\beta }_{{i}_{\perp }}\ll {m}_{e}/{m}_{i}$ on the MHD temporal scale (Jovanovic et al. 1987; Jovanovic & Horton 1994). For various traveling solutions in plasma regimes in which all diamagnetic drift and FLR effects are negligible, see also Petviashvili & Pokhotelov (1992) and Alexandrova et al. (2006). An extensive study of drift-Alfvén vortices in the regime β ≪ 1 was presented by Kuvshinov et al. (1999). In the warm plasma, mostly electron-scale structures were considered, such as whistler-frequency perturbations in the regime ${\beta }_{{e}_{\perp }}\gtrsim 1$ with immobile ions (Jovanović et al. 2015).

In the present paper, first we present a general solution that is slightly bigger than the ion scale (i.e., we keep the leading FLR terms) in a plasma with ${\beta }_{{i}_{\perp }}\sim {\beta }_{{e}_{\perp }}\sim 1$, described by the Kadomtsev–Pogutse–Strauss equations of reduced MHD (Kadomtsev & Pogutse 1973, 1974; Strauss 1976, 1977) and identified as a Chaplygin vortex (Chaplygin 1903). A class of stationary, magnetic-field-aligned monopoles can be derived from such traveling solutions in the limit of zero inclination and zero propagation speed. Due to their smooth transition to the linear response at large distances, these can be expected to be more stable than the monopoles (magnetostatic cells) with arbitrary radial dependencies. Additionally, in a plasma with ${\beta }_{{i}_{\perp }}\sim {\beta }_{{e}_{\perp }}\lesssim 1$, we find two kinds of dipolar Larichev & Reznik-type (Larichev & Reznik 1976) ion-scale vortices, which can be identified as the high-β shear Alfvén and kinetic slow nonlinear modes, respectively. The corresponding quasi-monopoles contain thin (electron-scale) current sheets and/or charge layers at the core edge and are probably less stable, particularly those in the kinetic slow-mode branch. Like all Larichev & Reznik-type solutions, inside the core, the vortices are fundamentally nonlinear structures, while outside of the core, they are described by the corresponding linear equations and thus follow (at least asymptotically) the linear dispersion relation in cylindrical coordinates of shear Alfén and kinetic slow magnetosonic modes. Thus, they can also be regarded as the generalization and the regularization of the singular point vortex model that is broadly used in the study of 2D turbulence in incompressible fluids and plasmas (Kirchhof 1876; Kuvshinov & Schep 2016); see also the collection of theoretical and numerical works (Caflisch 1989). Such nonlinear regularization will also broaden the envisaged spectrum of an agglomerate of point vortices.

3.1. Solution Sufficiently Bigger than the Ion Larmor Radius

One can easily verify that our model Equations (33)–(35) reduce to the Kadomtsev–Pogutse–Strauss equations of reduced MHD when all terms arising from the diamagnetic drift and FLR effects can be ignored. Using our notation introduced in Equation (32), this condition is expressed as ${\rho }_{i}^{{\prime} }={d}_{i}^{{\prime} }\,\sqrt{{\beta }_{{i}_{\perp }}/2}\ll 1$, which is obviously satisfied when either the plasma β is sufficiently small or the characteristic scale largely exceeds the ion inertial length $c/{\omega }_{\mathrm{pi}}$, i.e., for ${d}_{i}^{{\prime} }\ll 1$. However, it is worth noting that the dimensionless ion Larmor radius ${\rho }_{i}^{{\prime} }$ can be sufficiently small also in plasmas with a modest value of β, even if the size of the solution is not much bigger than the ion inertial length, i.e., ${d}_{i}^{{\prime} }$ is not too small (e.g., when ${d}_{i}^{{\prime} }\lesssim 0.4$). For example, in the magnetosheath region with ${\beta }_{{i}_{\perp }}=0.5$, the Cluster mission detected structures whose spatial scale was r₀ ≃ 500 km (Alexandrova et al. 2004), more than 10 times the ion inertial depth c/ω_pi ≃ 40 km. Under such conditions, we have ${d}_{i}^{{\prime} }\lesssim 0.1$ and a very small value of ${\rho }_{i}^{{\prime} }$, viz. ${{\rho }_{i}^{{\prime} }}^{2}=({\beta }_{{i}_{\perp }}/2)\,{{d}_{i}^{{\prime} }}^{2}\approx .0025\ll 1$. Recently, a quasi-monopolar Alfvén vortex with the transverse radius of ∼10 proton gyroscales was identified and studied in detail (Wang et al. 2019) using the data collected by the MMS in Earth's turbulent magnetosheath.

Obviously, for perturbations that comply with the small ion Larmor radius scaling,

$$\begin{eqnarray}&&{\rho }_{i}^{{\prime} }=\left(k/{{\rm{\Omega }}}_{i}\right)\sqrt{{T}_{{i}_{\perp }}/{m}_{i}}\sim \epsilon ,\end{eqnarray} \tag{ 39 }$$

where is the small parameter introduced in Equations (11) and (12); we can set ${{\rho }_{i}^{{\prime} }}^{2}\to 0$. Then, Equation (38) yields ${p}_{j\zeta }^{{\prime} }\to 0$ even if the dimensionless density perturbation is finite, $n^{\prime} \sim 1$. The latter also implies ${{\boldsymbol{E}}}_{\parallel }=0$, which excludes both kinetic-Alfvén and magnetosonic waves from our description; see Equation (41). Under these conditions, within the drift and weak z-dependence scalings, Equations (11) and (12), and in the massless electron limit (i.e., for perturbations whose spatial scale is much bigger than the electron inertial length, viz. ${d}_{e}^{\prime} \to 0$), our Equations (33) and (35) decouple from the rest of the system. They possess the form of the standard Kadomtsev–Pogutse–Strauss reduced MHD system for the potentials ϕ and A_z, including the effects of anisotropic temperatures (henceforth, we drop the primes for simplicity),

$$\begin{eqnarray}&&\begin{array}{l}\left[\displaystyle \frac{\partial }{\partial t}+\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\phi \right)\cdot {{\rm{\nabla }}}_{\perp }\right]{{\rm{\nabla }}}_{\perp }^{2}\phi \\ =\,-\left[\displaystyle \frac{\partial }{\partial z}-\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }{A}_{z}\right)\cdot {{\rm{\nabla }}}_{\perp }\right]\left(1-\displaystyle \frac{{\beta }_{{i}_{\parallel }}-{\beta }_{{i}_{\perp }}+{\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}}{2}\right){{\rm{\nabla }}}_{\perp }^{2}{A}_{z},\end{array}\end{eqnarray} \tag{ 40 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {A}_{z}}{\partial t}+\left[\displaystyle \frac{\partial }{\partial z}-\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }{A}_{z}\right)\cdot {{\rm{\nabla }}}_{\perp }\right]\phi =0,\end{eqnarray} \tag{ 41 }$$

while the parallel velocity, the density, and the compressional magnetic field are subsequently determined from

$$\begin{eqnarray}&&\begin{array}{l}\left[\displaystyle \frac{\partial }{\partial t}+\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\phi \right)\cdot {{\rm{\nabla }}}_{\perp }\right]\,{U}_{{i}_{\parallel }}+\displaystyle \frac{{\beta }_{{i}_{\perp }}}{2}\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\,\displaystyle \frac{\partial \phi }{\partial {x}_{{\alpha }_{\perp }}}\right)\cdot {\rm{\nabla }}\displaystyle \frac{\partial {A}_{z}}{\partial {x}_{{\alpha }_{\perp }}}\\ \quad =\,-\,\left[\displaystyle \frac{\partial }{\partial z}-\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }{A}_{z}\right)\cdot {{\rm{\nabla }}}_{\perp }\right]\left(\displaystyle \frac{{\beta }_{{i}_{\parallel }}+{\beta }_{{e}_{\parallel }}}{2}\,n\right.\\ \left.-\,\displaystyle \frac{{\beta }_{{i}_{\parallel }}-{\beta }_{{i}_{\perp }}+{\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}}{2}\,{B}_{z}-\displaystyle \frac{2{\beta }_{{i}_{\parallel }}-{\beta }_{{i}_{\perp }}}{2}\,{{\rm{\nabla }}}_{\perp }^{2}\phi \right),\end{array}\end{eqnarray} \tag{ 42 }$$

$$\begin{eqnarray}&&\begin{array}{l}\left[\displaystyle \frac{\partial }{\partial t}+\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\phi \right)\cdot {{\rm{\nabla }}}_{\perp }\right]\left(n-{B}_{z}\right)\\ =\,-\left[\displaystyle \frac{\partial }{\partial z}-\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }{A}_{z}\right)\cdot {{\rm{\nabla }}}_{\perp }\right]\left[{U}_{{i}_{\parallel }}+\left(1-\displaystyle \frac{{\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}}{2}\right){{\rm{\nabla }}}_{\perp }^{2}{A}_{z}\right],\end{array}\end{eqnarray} \tag{ 43 }$$

$$\begin{eqnarray}&&{B}_{z}+\displaystyle \frac{{\beta }_{{i}_{\perp }}+{\beta }_{{e}_{\perp }}}{2}\,n-\displaystyle \frac{{\beta }_{{i}_{\perp }}}{4}\,{{\rm{\nabla }}}_{\perp }^{2}\phi =0.\end{eqnarray} \tag{ 44 }$$

Here, in the regime ${\rho }_{i}^{2}\to 0$, we have used ${\gamma }_{{i}_{\perp }}={\gamma }_{{e}_{\perp }}=1$ as discussed above; see Equation (6).

Note that Equations (40)–(44) have no spatial scales, i.e., the characteristic wavenumber and frequency, k and ω, introduced in Equation (32) (i.e., the width of the structure, r₀, and its speed in the plasma frame, ${u}_{\perp }$) are arbitrary. Multiplying Equations (40) and (41) by ϕ and $[1-(1/2)({\beta }_{{i}_{\parallel }}\,-{\beta }_{{i}_{\perp }}+{\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }})]\,{{\rm{\nabla }}}_{\perp }^{2}{A}_{z}$, respectively, using the identities $({{\boldsymbol{e}}}_{z}\times {\rm{\nabla }}f)\cdot {\rm{\nabla }}g={\rm{\nabla }}\cdot (g\,{{\boldsymbol{e}}}_{z}\times {\rm{\nabla }}f)$ and ${\rm{\Phi }}\,{{\rm{\nabla }}}_{\perp }^{2}\partial {\rm{\Phi }}/\partial t={\rm{\nabla }}\cdot {({\rm{\Phi }}{{\rm{\nabla }}}_{\perp }\partial {\rm{\Phi }}/\partial t)-(\partial /\partial t)({{\rm{\nabla }}}_{\perp }{\rm{\Phi }})}^{2}/2$, and integrating over the entire space (provided ϕ = A_z = 0 at infinity), we readily obtain that the energy W is conserved, $\partial W/\partial t=0$, where

$$\begin{eqnarray}&&\begin{array}{l}W=\left(1/2\right){\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{dx}\,{dy}\,{dz}\left\{{\left({{\rm{\nabla }}}_{\perp }\phi \right)}^{2}\right.\\ \left\{+\,\left[1-\left(1/2\right)\left({\beta }_{{i}_{\parallel }}-{\beta }_{{i}_{\perp }}+{\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}\right)\right]{\left({{\rm{\nabla }}}_{\perp }{A}_{z}\right)}^{2}\right\}.\end{array}\end{eqnarray} \tag{ 45 }$$

Our Equations (40) and (41) are identical (apart from constant factors that come from the temperature anisotropy) to the Kadomtsev–Pogutse–Strauss system that has been studied in detail in the literature (see Petviashvili & Pokhotelov 1992 and references therein). Following the standard procedure of Larichev & Reznik (1976) and of more recent works (Kuvshinov et al. 1999), we seek its solution that is traveling with the (nonscaled) velocity ${{\boldsymbol{u}}}_{\perp }={{\boldsymbol{e}}}_{y}\,k/\omega$ and is tilted to the z-axis by a small angle $\theta =\omega /{{ku}}_{z}$, implying that the solution depends only on the dimensionless variables $x^{\prime}$ and $y^{\prime} +({c}_{{\rm{A}}}/{u}_{z})z^{\prime} -t^{\prime}$. Then, using $\partial /\partial t^{\prime} =-\partial /\partial y^{\prime}$ and $\partial /\partial z^{\prime} =({c}_{{\rm{A}}}/{u}_{z})\,\partial /\partial y^{\prime}$, the parallel electron momentum Equation (41) is readily solved as

$$\begin{eqnarray}&&\phi =\left({u}_{z}/{c}_{{\rm{A}}}\right)\,{A}_{z},\end{eqnarray} \tag{ 46 }$$

indicating that the parallel electric field is equal to zero, ${E}_{\parallel }=0$, short-circuited by the massless electrons. As a consequence, $\delta {{\boldsymbol{B}}}_{\perp }$ is aligned with ${{\boldsymbol{U}}}_{\perp }$. Substituting the above into Equation (40) yields a simple 2D Euler equation,

$$\begin{eqnarray}&&\left[{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\left(\phi -x\right)\right]\cdot {{\rm{\nabla }}}_{\perp }{{\rm{\nabla }}}_{\perp }^{2}\phi =0\quad \Rightarrow \quad {{\rm{\nabla }}}_{\perp }^{2}\phi =G\left(\phi -x\right).\end{eqnarray} \tag{ 47 }$$

Here, G is an arbitrary function of its argument that is adopted here to be part-by-part linear (Larichev & Reznik 1976; Kuvshinov et al. 1999), viz.

$$\begin{eqnarray}&&G(\xi )=(\xi -{\xi }_{0}){G}_{1},\end{eqnarray} \tag{ 48 }$$

where the parameter ξ₀ and the slope G₁ take different constant values ${\xi }_{0}^{\mathrm{in}},{G}_{1}^{\mathrm{in}}$ and ${\xi }_{0}^{\mathrm{out}},{G}_{1}^{\mathrm{out}}$ inside and outside, respectively, of a moving circle in the x, y plane whose radius is r₀ (usually referred to as the vortex core). For a spatially localized solution, we have ${G}_{1}^{\mathrm{out}}=0$, which implies that both the vorticity ${{\rm{\nabla }}}_{\perp }^{2}\phi$ and the parallel current $-{{\rm{\nabla }}}_{\perp }^{2}{A}_{z}$ are localized inside the vortex core, while ${G}_{1}^{\mathrm{in}}$ will be determined from the smoothness of the potentials ϕ and A_z (i.e., from the absence of the surface charges and surface currents along ${{\boldsymbol{B}}}_{0}$) at the edge of the vortex core.

Equation (47) separates variables in cylindrical coordinates. The solution is easily written in terms of the Bessel functions,

$$\begin{eqnarray}&&\phi (r,\varphi )=\sum _{k}[{\alpha }_{k}{J}_{k}(r)+{\beta }_{k}{Y}_{k}(r)]\exp ({ik}\varphi ),\end{eqnarray} \tag{ 49 }$$

where $r=\{{x}^{2}+{[y+({c}_{{\rm{A}}}/{u}_{z})z-t]}^{2}\}{}^{\tfrac{1}{2}}$, $\varphi =\arctan \{[y\,+({c}_{{\rm{A}}}/{u}_{z})z-t]/x\}$, and J_k and Y_k are the Bessel functions of the kth order and of the first and second kinds, respectively. The constants of integration α_k and β_k are determined from the finiteness of the solution at r = 0 and $r\to \infty$, and from the physical conditions of continuity and smoothness at the core edge r = r₀ of the potential ϕ. The ensuing solution takes the form of a Chaplygin (1903) vortex, constructed more than a century ago as the traveling solution of a 2D Euler equation for the incompressible flow in ordinary fluids. It consists of a circularly symmetric "rider" that is appropriately superimposed on a Lamb dipole that also provides its propagation, viz.

$$\begin{eqnarray}\begin{array}{l}\phi \left(r,\varphi \right)=\left({u}_{z}/{c}_{{\rm{A}}}\right)\,{A}_{z}\left(r,\varphi \right)\,=\,\left\{\begin{array}{ll}\cos \varphi \,\left({r}_{0}^{2}/r\right), & r\geqslant {r}_{0},\\ \cos \varphi \,\left\{r-(2{r}_{0}/{j}_{1})\,[{J}_{1}\left({j}_{1}\,r/{r}_{0}\right)/{J}_{1}^{{\prime} }\left({j}_{1}\right)]\right\}+{\psi }_{0}\left[{J}_{0}\left({j}_{1}\,r/{r}_{0}\right)-{J}_{0}\left({j}_{1}\right)\right], & r\lt {r}_{0},\end{array}\right.,\end{array}\end{eqnarray} \tag{ 50 }$$

where j_k is one of the zeros of the Bessel function J₁, viz. ${J}_{1}({j}_{k})=0$, and the amplitude ψ₀ of the monopole component is arbitrary. The solution (50) does not possess a characteristic spatial scale, and the radius r₀ is arbitrary. An identical Alfvén vortex was presented by Petviashvili & Pokhotelov (1992), albeit in Cartesian coordinates.

The corresponding density, compressional magnetic field, and parallel ion velocity are found from Equations (42)–(44) and using Equation (46), viz.

$$\begin{eqnarray}\begin{array}{rcl}n\equiv { \mathcal N }\,{{\rm{\nabla }}}_{\perp }^{2}\phi & = & {\left[1+\displaystyle \frac{{\beta }_{{e}_{\perp }}+{\beta }_{{i}_{\perp }}}{2}+\displaystyle \frac{{c}_{{\rm{A}}}^{2}}{{u}_{z}^{2}}\left(\displaystyle \frac{{\beta }_{{e}_{\perp }}+{\beta }_{{i}_{\perp }}}{2}\displaystyle \frac{{\beta }_{{e}_{\perp }}-{\beta }_{{e}_{\parallel }}+{\beta }_{{i}_{\perp }}-{\beta }_{{i}_{\parallel }}}{2}-\displaystyle \frac{{\beta }_{{e}_{\parallel }}+{\beta }_{{i}_{\parallel }}}{2}\right)\right]}^{-1}\\ & & \times \,\left\{\displaystyle \frac{{\beta }_{{i}_{\perp }}}{4}+\displaystyle \frac{{c}_{{\rm{A}}}^{2}}{{u}_{z}^{2}}\left[1-\displaystyle \frac{{\beta }_{{i}_{\parallel }}}{2}+\displaystyle \frac{{\beta }_{{e}_{\perp }}-{\beta }_{{e}_{\parallel }}+{\beta }_{{i}_{\perp }}-{\beta }_{{i}_{\parallel }}}{2}\left(1+\displaystyle \frac{{\beta }_{{i}_{\perp }}}{4}\right)\right]\right\}\,{{\rm{\nabla }}}_{\perp }^{2}\phi ,\end{array}\end{eqnarray} \tag{ 51 }$$

$$\begin{eqnarray}&&{B}_{z}\equiv { \mathcal B }\,{{\rm{\nabla }}}_{\perp }^{2}\phi =\left[-\displaystyle \frac{{\beta }_{{i}_{\perp }}+{\beta }_{{e}_{\perp }}}{2}\,{ \mathcal N }+\displaystyle \frac{{\beta }_{{i}_{\perp }}}{4}\right]\,{{\rm{\nabla }}}_{\perp }^{2}\phi .\end{eqnarray} \tag{ 52 }$$

$$\begin{eqnarray}&&{U}_{{i}_{\parallel }}\equiv {{ \mathcal U }}_{i}\,{{\rm{\nabla }}}_{\perp }^{2}\phi =\displaystyle \frac{{u}_{z}}{{c}_{{\rm{A}}}}\left[{ \mathcal N }-{ \mathcal B }-\displaystyle \frac{{c}_{{\rm{A}}}^{2}}{{u}_{z}^{2}}\left(1-\displaystyle \frac{{\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}}{2}\right)\right]{{\rm{\nabla }}}_{\perp }^{2}\phi .\end{eqnarray} \tag{ 53 }$$

The above density and magnetic field perturbations are clearly proportional, viz. B_z/n = constant. Depending on the plasma parameters and the characteristic velocity u_z, they can be either correlated or anticorrelated, B_z/n > 0 or <0, respectively. In a general case, the dependence of sign(B_z/n) on plasma parameters is rather complicated, and we present only the results in isothermal plasma, with ${\beta }_{{i}_{\perp }}={\beta }_{{i}_{\parallel }}={\beta }_{{e}_{\perp }}={\beta }_{{e}_{\parallel }}\equiv \beta$, in which the density and magnetic field perturbations are correlated, B_z/n > 0, either for sufficiently large u_z, viz. ${u}_{z}^{2}/{c}_{{\rm{A}}}^{2}\,\gt \max (4-\beta ,2\mbox{--}4/\beta )\,\geqslant 3-\sqrt{5}$ or for small parallel phase speeds, ${u}_{z}^{2}/{c}_{{\rm{A}}}^{2}\,\lt \min (4-\beta ,2\mbox{--}4/\beta )\leqslant 3-\sqrt{5}$ (the latter case can be realized only for large thermal pressures, 2 < β < 4, and it also involves acoustic perturbations). Otherwise, B and n are anticorrelated.

The large-scale vortex presented in Equations (50)–(53) is displayed in Figure 1. Note that the vorticity ${{\rm{\nabla }}}_{\perp }^{2}\phi$ associated with the monopolar component of the solution, Equation (50), has a finite jump at r = r₀ and that the function G is discontinuous for the corresponding value of its argument. Such discontinuity is permitted, because it does not yield a singularity of the vector product in Equation (47). At the core edge, the latter comprises the product of the derivatives in the directions parallel and perpendicular to the isoline r = r₀. As the derivative of a function along its isoline is always zero, the corresponding product remains finite even if the perpendicular derivative of ${{\rm{\nabla }}}_{\perp }^{2}\phi$ is infinite. In such a case, the vector product acquires an isolated point and remains a continuous function elsewhere.

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A stationary, nonpropagating monopole aligned with the background magnetic field is readily obtained in the limit ${u}_{\perp }\to 0$ from the moving, tilted, quasi-monopole, Equation (50), whose monopolar component has an arbitrary amplitude, and the amplitude of the dipole is proportional to the propagation velocity ${u}_{\perp }=\omega /k$. Here, 1/ω and 1/k are the characteristic temporal and spatial scales introduced in Equation (32).

The stability of propagating dipoles and quasi-monopoles, and of stationary monopoles, was studied in the experiments in water tanks (Beckers et al. 2003), in which the perturbations are also described by the 2D Euler Equation (47). However, they lack any 3D properties but include an additional small but finite viscosity of the fluid, not involved in our analysis. In these experiments, a dipole (solution with ψ₀ = 0 in Equation (50)) appears to be remarkably stable, and it can easily survive collisions with other dipoles (van Heijst & Flor 1989). Propagating quasi-monopoles, described by Equation (50) when ${\psi }_{0}\ne 0$, are found to travel over a distance that is an order of magnitude bigger than its diameter, which has also been suggested by the weak nonlinear theory and numerical simulations (Stern & Radko 1998), as well as by experiments in nonrotating water tanks with a rectangular shape (Voropayev et al. 1999). In these works, the moving quasi-monopoles were much more stable than the stationary monopoles, which were found to either disperse or slowly transform into dipolar or tripolar vortices depending on their initial profile, i.e., on the amount of shielding in the initial state.

The properties of our theoretical solution, Equations (50)–(53) (spatial and temporal scales, magnetic and electric field structures, temperature variations and parallel flows, compressibility, etc.), closely mimic those of the Alfvén vortex (Wang et al. 2019) identified in the high-resolution data collected by the MMS mission in Earth's turbulent magnetosheath.

3.2. Approaching Ion Scales

When the plasma β is close to, or or bigger than, unity, the dimensionless ion Larmor radius becomes comparable to and even bigger than the dimensionless ion inertial length. Then, the anisotropic reduced MHD Equations (40) and (41), derived in the regime ${{\rho }_{i}^{{\prime} }}^{2}\to 0$, do not provide an accurate description at ion scales, i.e., when ${d}_{i}^{{\prime} }\lesssim 1$ or >1. Below, we construct a localized, stationary, 2D (in cylindrical geometry) solution of the full system of model Equations (33)–(37), assuming that its spatial extent is comparable with ion scales (i.e., with the ion collisionless skin depth, viz. ${d}_{i}^{{\prime} }\sim 1$, and with the ion Larmor radius, viz. ${\rho }_{i}^{{\prime} }\lesssim 1$), but much larger than the electron skin depth, ${d}_{e}^{\prime} \to 0$. As in the preceding subsection, we seek a solution that is traveling with the (nonscaled) velocity ${{\boldsymbol{u}}}_{\perp }={{\boldsymbol{e}}}_{y}\,k/\omega$ and is tilted to the z-axis by a small angle $\theta =\omega /{{ku}}_{z}$, which depends only on the dimensionless variables $x^{\prime}$ and $y^{\prime} +({c}_{{\rm{A}}}/{u}_{z})z^{\prime} -t^{\prime}$. Then, the equations of parallel electron momentum (35), electron continuity (34), parallel fluid momentum (36), charge continuity (33), and pressure balance (37) can be conveniently cast in the following form:

$$\begin{eqnarray}&&\begin{array}{l}\left[{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\left(\psi -x\right)\right]\cdot {{\rm{\nabla }}}_{\perp }\left[\phi -{p}_{{e}_{\parallel }}-x\right.\\ \,\left.+\,\left(1/2\right)\left({\beta }_{{e}_{\parallel }}-{\beta }_{e\perp }\right)\,{d}_{i}^{2}{B}_{z}\right]=0,\end{array}\end{eqnarray} \tag{ 54 }$$

$$\begin{eqnarray}&&\begin{array}{l}\left[{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\left(\phi -{p}_{{e}_{\perp }}-x\right)\right]\cdot {{\rm{\nabla }}}_{\perp }\left(n-{B}_{z}\right)\\ \,-\,\left[{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\left(\psi -x\right)\right]\cdot {{\rm{\nabla }}}_{\perp }{V}_{\parallel }+\left({c}_{{\rm{A}}}^{2}/{u}_{z}^{2}\right)1-\left(1/2\right)\,\\ \,\times \,({\beta }_{{e}_{\parallel }}\left.\left.\left.\,-\,{\beta }_{{e}_{\perp }}\right)\right]{{\rm{\nabla }}}_{\perp }^{2}\psi \right\}=0,\end{array}\end{eqnarray} \tag{ 55 }$$

$$\begin{eqnarray}&&\begin{array}{l}\left[{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\left({\rm{\Phi }}-{p}_{{i}_{\perp }}+{\rho }_{i}^{2}{B}_{z}-x\right)\right]\cdot {{\rm{\nabla }}}_{\perp }{d}_{i}^{2}{V}_{\parallel }\\ \,-\,\left({c}_{{\rm{A}}}^{2}/{u}_{z}^{2}\right)\left[{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\left(\psi -x\right)\right]\cdot {{\rm{\nabla }}}_{\perp }\left[{p}_{{e}_{\parallel }}+{p}_{{i}_{\parallel }}\right.\\ \,-\,\left({\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}+{\beta }_{{i}_{\parallel }}-{\beta }_{{i}_{\perp }}\right)\left({\rho }_{i}^{2}/{\beta }_{{i}_{\perp }}\right){B}_{z}-\left(2{\beta }_{{i}_{\parallel }}/{\beta }_{{i}_{\perp }}-1\right){\rho }_{i}^{2}{{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}\\ \,=\,-{\rho }_{i}^{2}\left({c}_{{\rm{A}}}^{2}/{u}_{z}^{2}\right)\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\,\partial {\rm{\Phi }}/\partial {x}_{i}\right)\cdot {{\rm{\nabla }}}_{\perp }\partial \psi /\partial {x}_{i},\end{array}\end{eqnarray} \tag{ 56 }$$

$$\begin{eqnarray}&&\begin{array}{l}\,\left[{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\left(-{\rho }_{i}^{2}{{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}/2+{p}_{{i}_{\perp }}+{p}_{{e}_{\perp }}\right)\right]\cdot {{\rm{\nabla }}}_{\perp }\left(n-{B}_{z}\right)\\ \,+\,\left({c}_{{\rm{A}}}^{2}/{u}_{z}^{2}\right)\left[1-\left(1/2\right)\left({\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}+{\beta }_{{i}_{\parallel }}-{\beta }_{i\perp }\right)\right]\left[{{\boldsymbol{e}}}_{z}\right.\\ \,\times \,{{\rm{\nabla }}}_{\perp }\left(\psi -x\right)\,\cdot {{\rm{\nabla }}}_{\perp }{{\rm{\nabla }}}_{\perp }^{2}\psi +{\rho }_{i}^{2}\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }{V}_{\parallel }\right)\cdot {{\rm{\nabla }}}_{\perp }{{\rm{\nabla }}}_{\perp }^{2}\psi \\ \,-\,\left[{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\left({\rm{\Phi }}-{p}_{{i}_{\perp }}+{\rho }_{i}^{2}{B}_{z}-x\right)\right]\\ \,\cdot \,{{\rm{\nabla }}}_{\perp }{{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}+{\rho }_{i}^{2}\left[{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\left(\psi -x\right)\right]\cdot {{\rm{\nabla }}}_{\perp }{{\rm{\nabla }}}_{\perp }^{2}{V}_{\parallel }\\ \,=\,\left[{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\left(\partial /\partial {x}_{i}\right)\left({\rho }_{i}^{2}{B}_{z}-{p}_{{i}_{\perp }}\right)\right]\cdot {{\rm{\nabla }}}_{\perp }\,\partial {\rm{\Phi }}/\partial {x}_{i},\end{array}\end{eqnarray} \tag{ 57 }$$

$$\begin{eqnarray}&&{d}_{i}^{2}{B}_{z}+{p}_{{i}_{\perp }}+{p}_{{e}_{\perp }}-{\rho }_{i}^{2}{{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}/2=0,\end{eqnarray} \tag{ 58 }$$

where, as before, we have omitted the primes and have used the following notations:

$$\begin{eqnarray}&&\begin{array}{c}\begin{array}{l}{\rm{\Phi }}={\left(1+{\rho }_{i}^{2}{{\rm{\nabla }}}_{\perp }^{2}/2\right)}^{-1}\left(\phi +{p}_{{i}_{\perp }}\right),\,\psi =\left({u}_{z}/{c}_{{\rm{A}}}\right)\,{A}_{z}\\ {\rm{and}}\,\,\,\,{V}_{\parallel }=\left({c}_{{\rm{A}}}/{u}_{z}\right)\,{U}_{{i}_{\parallel }}.\end{array}\end{array}\end{eqnarray} \tag{ 59 }$$

Electron and ion pressures are related to the density by the equation of state (38), ${p}_{j\zeta }={\gamma }_{j\zeta }\,({\beta }_{j\zeta }/2)\,{d}_{i}^{2}n$, where j = e, i and $\zeta =\parallel ,\perp$. It is worth noting that the linearized version of our basic Equations (54)–(58) in the regime of small but finite ion Larmor radius corrections, ${\rho }_{i}^{2}{{\rm{\nabla }}}_{\perp }^{2}\sim \sqrt{\epsilon }\lt 1$, ${\rho }_{i}^{4}{{\rm{\nabla }}}_{\perp }^{4}\sim \epsilon \to 0$, and in the absence of acoustic perturbations that occurs when ${u}_{z}\gt \max ({v}_{{\mathrm{Ti}}_{\perp }},\,\sqrt{{\gamma }_{{\mathrm{Ti}}_{\parallel }}}\,{v}_{{\mathrm{Ti}}_{\parallel }})$, reduces to

$$\begin{eqnarray}&&\left({{\rm{\nabla }}}_{\perp }^{2}-{\kappa }^{2}\right){{\rm{\nabla }}}_{\perp }^{2}\psi =0,\end{eqnarray} \tag{ 60 }$$

where

$$\begin{eqnarray}\begin{array}{rcl}{\kappa }^{2} & = & \displaystyle \frac{2}{{\rho }_{i}^{2}}\left(1-\displaystyle \frac{2\,{u}_{z}^{2}/{c}_{{\rm{A}}}^{2}}{2-{\beta }_{{e}_{\parallel }}+{\beta }_{{e}_{\perp }}-{\beta }_{{i}_{\parallel }}+{\beta }_{{i}_{\perp }}}\right)\\ & & \times \,\left\{1-\displaystyle \frac{{\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}}{2}+\left[1+\displaystyle \frac{\left(2-{\beta }_{{e}_{\parallel }}+{\beta }_{{e}_{\perp }}\right)\left(4/{\beta }_{{i}_{\perp }}\right)}{2-{\beta }_{{e}_{\parallel }}+{\beta }_{{e}_{\perp }}-{\beta }_{{i}_{\parallel }}+{\beta }_{{i}_{\perp }}}\right]\right.\\ & & \times \,{\left.\left[\displaystyle \frac{{\beta }_{{e}_{\perp }}+{\beta }_{{e}_{\parallel }}\left({\gamma }_{{e}_{\parallel }}-1\right)+{\beta }_{{i}_{\perp }}{\gamma }_{{i}_{\perp }}}{2+{\beta }_{{e}_{\perp }}{\gamma }_{{e}_{\perp }}+{\beta }_{{i}_{\perp }}{\gamma }_{{i}_{\perp }}}+\displaystyle \frac{{\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}}{2}\right]\right\}}^{-1}.\end{array}\end{eqnarray} \tag{ 61 }$$

The linear response has the form of waves when κ² < 0. The response consists of two modes, whose wavenumbers are equal to zero and to iκ. These are identified as the large-β versions of the shear Alfvén wave (actually, the latter acquires a finite perpendicular wavenumber on the electron scale; for a simple example in the case of a very-low-β plasma, see Jovanovic et al. 1987; Jovanovic & Horton 1994) and of the kinetic slow mode, respectively. We expect that in the nonlinear regime there may exist two nonlinear vortex modes analogous to these.

A simple analysis shows that the last term in Equation (61) is strictly positive when the ion temperature is anisotropic with ${\beta }_{{i}_{\parallel }}\leqslant {\beta }_{{i}_{\perp }}$, if the parallel electron temperature is not exceptionally high, ${\beta }_{{e}_{\parallel }}\leqslant 2+{\beta }_{{e}_{\perp }}$. This indicates that the kinetic slow wave is evanescent (κ² > 0) only if its parallel phase velocity is sufficiently small,

$$\begin{eqnarray}&&{u}_{z}^{2}{/c}_{{\rm{A}}}^{2}\lt 1-\left({\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}+{\beta }_{{i}_{\parallel }}-{\beta }_{{i}_{\perp }}\right)\,/2\,\sim { \mathcal O }\,\left(1\right),\end{eqnarray} \tag{ 62 }$$

while for larger u_z, we have a propagating wave. Temperature anisotropies are common in space plasmas; see Appendices A and B for examples in the solar wind and Earth's magnetosheath.

Seeking traveling/tilted solutions, it is possible to integrate also the parallel electron momentum Equation (54), viz.

$$\begin{eqnarray}&&\phi -x={p}_{{e}_{\parallel }}-\left(1/2\right)\left({\beta }_{{e}_{\parallel }}-{\beta }_{e\perp }\right)\,{d}_{i}^{2}{B}_{z}+F\left(\psi -x\right),\end{eqnarray} \tag{ 63 }$$

where F is an arbitrary function of the nonlinear characteristic $\psi -x$.

It is difficult to proceed further because solving Equation (55) in a general case is a formidable task. Likewise, Equations (56) and (57) contain higher derivatives of unknown functions on their right-hand sides and are practically impossible to integrate. In particular, the coupling with acoustic perturbation increases the complexity of our equations and makes an analytic solution virtually impossible. For this reason, we will restrict our study to the perturbations whose parallel phase velocity satisfies the conditions $({c}_{a}^{2}/{u}_{z}^{2})({\beta }_{i\perp }/2)\lesssim ({c}_{a}^{2}{/u}_{z}^{2})({\gamma }_{i\parallel }{\beta }_{i\parallel }/2)\sim \epsilon \to 0$ or equivalently, ${u}_{z}\gt \max \,({v}_{{\mathrm{Ti}}_{\perp }},\,\sqrt{{\gamma }_{{i}_{\parallel }}}\,{v}_{{\mathrm{Ti}}_{\parallel }})$ when, according to Equation (56), we can ignore the parallel fluid velocity and set ${V}_{\parallel }\to 0$.

3.2.1. Large-β Shear Alfvén Solution with $\delta n/{n}_{0}=\delta {B}_{z}/{B}_{0}$ and ${E}_{\parallel }=0$

First, we exclude acoustic waves from our analysis, adopting ${u}_{z}\gt \max ({v}_{{\mathrm{Ti}}_{\perp }},\sqrt{{\gamma }_{{i}_{\parallel }}}\,{v}_{{\mathrm{Ti}}_{\parallel }})\,\Rightarrow \,{V}_{\parallel }\to 0$; see Equation (56). In such regime, we have ${\gamma }_{{i}_{\perp }}=2$ and ${\gamma }_{{e}_{\perp }}=1$; see Equation (6) and the subsequent discussion. An analytic solution can be constructed following the standard Larichev & Reznik procedure only in the special case when the relative perturbations of density and compressional magnetic field are equal (i.e., n = B_z in dimensionless units) when, after making use of Equation (58), both continuity Equations (55) and (57) are drastically simplified to 2D Euler equations, viz.

$$\begin{eqnarray}&&\begin{array}{l}\left[{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\left(\psi -x\right)\right]\cdot {{\rm{\nabla }}}_{\perp }{{\rm{\nabla }}}_{\perp }^{2}\psi =0,\\ \left[{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\left({\rm{\Phi }}-x\right)\right]\cdot {{\rm{\nabla }}}_{\perp }{{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}=0,\end{array}\end{eqnarray} \tag{ 64 }$$

which can be readily integrated as

$$\begin{eqnarray}&&{{\rm{\nabla }}}_{\perp }^{2}\psi =G\left(\psi -x\right),\quad \quad {{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}=H\left({\rm{\Phi }}-x\right).\end{eqnarray} \tag{ 65 }$$

Same as before, G and H are arbitrary functions of their argument, which we take to be part-by-part linear. Thus, G(ξ) is adopted as $G(\xi )=(\xi -{\xi }_{0}){G}_{1},$, with the constants ξ₀ and G₁ taking different values ${\xi }_{0}^{\mathrm{in}},{G}_{1}^{\mathrm{in}}$ and ${\xi }_{0}^{\mathrm{out}},{G}_{1}^{\mathrm{out}}$ inside and outside, respectively, of a (moving) vortex core defined by ${x}^{2}+{[y+({c}_{{\rm{A}}}/{u}_{z})z-t]}^{2}={r}_{0}^{2}$. Obviously, for a spatially localized solution we must have ${G}_{1}^{\mathrm{out}}=0$, while ${G}_{1}^{\mathrm{in}}$ will be determined from the smoothness of the vector potential ψ (i.e., from the absence of the z component of the surface current) at the edge of the vortex core. The function $G(\psi -x)$, and consequently the parallel current $-{{\rm{\nabla }}}_{\perp }^{2}\psi$, can have a finite jump for some value of $\psi -x$; see discussion following Equation (48) in the preceding subsection. Now we can readily write the solution of the 2D Euler Equation (65) as a Chaplygin vortex, identical to that obtained in the preceding subsection, Equation (50), viz.

$$\begin{eqnarray}\begin{array}{l}\psi \left(r,\varphi \right)=\left\{\begin{array}{ll}\cos \varphi \,\left({r}_{0}^{2}/r\right), & r\geqslant {r}_{0},\\ \cos \varphi \,\left\{r-(2{r}_{0}/{j}_{1})\,[{J}_{1}\left({j}_{1}\,r/{r}_{0}\right)/{J}_{1}^{{\prime} }\left({j}_{1}\right)]\right\}+{\psi }_{0}\left[{J}_{0}\left({j}_{1}\,r/{r}_{0}\right)-{J}_{0}\left({j}_{1}\right)\right], & r\lt {r}_{0}\end{array}\right..\end{array}\end{eqnarray} \tag{ 66 }$$

The density n and the compressional magnetic field B_z are calculated from the generalized pressure balance, Equation (58), viz.

$$\begin{eqnarray}&&n={B}_{z}={\left(1+{\beta }_{{i}_{\perp }}+\displaystyle \frac{{\beta }_{{e}_{\perp }}}{2}\right)}^{-1}\displaystyle \frac{{\beta }_{{i}_{\perp }}}{4}\,{{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }},\end{eqnarray} \tag{ 67 }$$

which, after substitution into the parallel electron momentum Equation (54) and using Equation (65), yields ${\rm{\Phi }}\,-x\,=F\left(\psi -x\right)$, where ψ is given by Equation (66). Obviously, for r > r₀, the slope of the function F is given by F₁ = 1, which yields n = 0 outside of the vortex core. The definition of the stream function Φ further implies that outside of the vortex core, we also have ϕ = ψ. As these potentials satisfy the same continuity conditions at the core edge, we must have ϕ = ψ on the entire x, y plane, which corresponds to ${E}_{\parallel }=0$, i.e., $\phi ={u}_{z}{A}_{z}$ in dimensional units. One easily sees that the dipolar components of the potentials ϕ and ψ (that are $\propto \cos \varphi$) are continuous and smooth functions and that the corresponding density n and compressional magnetic field B_z are continuous at r = r₀. Conversely, the monopolar component of the compressional magnetic field has a finite jump ${\rm{\Delta }}{B}_{z}^{(0)}$ at the edge of the vortex core, where

$$\begin{eqnarray}&&\begin{array}{c}\begin{array}{l}{\rm{\Delta }}{B}_{z}^{(0)}={\psi }_{0}\,{\beta }_{{i}_{\perp }}\left({j}_{1}^{2}/4{r}_{0}^{2}\right)\\ \,\times \,{\left[1+{\beta }_{{i}_{\perp }}+{\beta }_{{e}_{\perp }}/2+\left(1+{\beta }_{{e}_{\perp }}/2\right)\left({\rho }_{i}^{2}{j}_{1}^{2}/2{r}_{0}^{2}\right)\right]}^{-1}.\end{array}\end{array}\end{eqnarray} \tag{ 68 }$$

Such discontinuity corresponds to a surface current (with zero thickness and infinite density) located at r = r₀, flowing in the poloidal direction. The latter is regarded as nonphysical, and it probably gives rise to an instability of Chaplygin's monopolar component. In other words, the monopolar Chaplygin component may exist only when the compressional magnetic field is negligible; that is usually the case when the plasma β is small. The ion-scale shear Alfvén vortex, Equations (66)–(67), is very similar to the RMHD-scale shear Alfvén vortex found in the preceding subsection, and they are both displayed in Figure 1. Both have a vanishing parallel electric field, corresponding to $\phi =({u}_{z}/{c}_{{\rm{A}}}){A}_{z}$, and outside of the vortex core, they have ${{\rm{\nabla }}}_{\perp }^{2}\phi ={{\rm{\nabla }}}_{\perp }^{2}{A}_{z}=n={B}_{z}=0$. Inside the core, large-scale structures feature $n/{B}_{z}=\mathrm{constant}\ne 1$, while on the ion scale, we have been able to find analytically only vortices with n/B_z = 1.

In the regime when both the parallel electric field and the ion sound are absent, ${E}_{\parallel }=0$ and ${u}_{z}\gt {v}_{{\mathrm{Ti}}_{\perp }},\,\sqrt{{\gamma }_{{i}_{\parallel }}}\,{v}_{{\mathrm{Ti}}_{\parallel }}$, the ions are thermalized in the perpendicular direction, $d/{dt}\ll {v}_{{\mathrm{Ti}}_{\parallel }}\,| {{\rm{\nabla }}}_{\perp }|$, and the density and the parallel magnetic field perturbations are fully correlated, $\delta n/{n}_{0}=\delta {B}_{z}/{B}_{0}$, the energy conservation can be readily obtained from the charge continuity equation for arbitrary dependence on t and z that has the simple form (see Equation (64))

$$\begin{eqnarray}&&\left[\partial /\partial t+\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }{\rm{\Phi }}\right)\cdot {{\rm{\nabla }}}_{\perp }\right]{{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}=0.\end{eqnarray} \tag{ 69 }$$

Multiplying by Φ and integrating for the entire space (provided Φ = 0 at the infinity), we obtain $\partial W/\partial t=0$, where

$$\begin{eqnarray}&&W=\left(1/2\right){\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{dx}\,{dy}\,{dz}{\left({{\rm{\nabla }}}_{\perp }{\rm{\Phi }}\right)}^{2}.\end{eqnarray} \tag{ 70 }$$

3.2.2. Large-β Kinetic Slow Magnetosonic Solution, with $\delta n/{n}_{0}\ne \delta {B}_{z}/{B}_{0}$, u_z < c_A, and ${{\boldsymbol{B}}}_{\perp }=0$

In the regime with a negligible contribution of acoustic perturbation defined above, when ${V}_{\parallel }\to 0$, ${\gamma }_{{i}_{\perp }}=2$, ${\gamma }_{{e}_{\perp }}=1$, and ${u}_{z}\gt \max \,({v}_{{\mathrm{Ti}}_{\perp }},\sqrt{{\gamma }_{{i}_{\parallel }}}{v}_{{\mathrm{Ti}}_{\parallel }})$, we seek a traveling solution whose perturbations of density and compressional magnetic field are not fully correlated, i.e., with $n\ne {B}_{z}$. As outside of the vortex core the localized nonlinear solution is essentially a linear evanescent response to the nonlinearities located within the core, we ascertain from Equation (61) that in the plasma regimes of interest, featuring ion temperature anisotropy, ${\beta }_{{i}_{\parallel }}\leqslant {\beta }_{{i}_{\perp }}$, and a moderate parallel electron temperature, ${\beta }_{{e}_{\parallel }}\leqslant 2+{\beta }_{{e}_{\perp }}$, the parallel phase velocity of kinetic slow-mode vortices cannot be much bigger than the Alfvén speed; see Equation (62). We adopt a somewhat more rigorous restriction for u_z, viz. ${c}_{{\rm{A}}}\gt {u}_{z}\gt \max \,({v}_{{\mathrm{Ti}}_{\perp }},\sqrt{{\gamma }_{{i}_{\parallel }}}{v}_{{\mathrm{Ti}}_{\parallel }})$, that permits us to simultaneously set ${V}_{\parallel }\ll 1$ and $\psi \sim {u}_{z}^{2}/{c}_{{\rm{A}}}^{2}\ll 1$. The corresponding kinetic slow wave is localized, κ² > 0 (see Equation (62)), and it can be realized when the perpendicular ion temperature is of the order ${\beta }_{{i}_{\perp }}\lesssim 1$ and the parallel ion temperature is sufficiently small, ${v}_{{{Ti}}_{\parallel }}^{2}/{c}_{{\rm{A}}}^{2}={\beta }_{{i}_{\parallel }}/2\ll 1$. Such ordering is easy to achieve in Earth's magnetosheath downstream of a quasi-perpendicular bow shock, possibly also in the fast solar wind, but more difficult in the slow solar wind where the separation between c_A and ${v}_{{\mathrm{Ti}}_{\parallel }}$ is smaller; see Appendices A and B. Besides, if the parallel electron temperature is not extremely small, ${T}_{{e}_{\parallel }}/{T}_{{i}_{\parallel }}\geqslant {m}_{e}/{m}_{i}$, the electrons are isothermal along the magnetic field, too. In such regime, we have ${\gamma }_{{e}_{\parallel }}={\gamma }_{{e}_{\perp }}=1$, and making use of Equation (63), we can rewrite the electron continuity Equation (55) as

$$\begin{eqnarray}&&\begin{array}{l}\left[{{\boldsymbol{e}}}_{z}\times {\rm{\nabla }}F\left(\psi -x\right)\right]\cdot {\rm{\nabla }}\left(n-{B}_{z}\right)\\ \quad -\,\displaystyle \frac{{c}_{{\rm{A}}}^{2}}{{u}_{z}^{2}}\left(1-\displaystyle \frac{{\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}}{2}\right)\left[{{\boldsymbol{e}}}_{z}\times {\rm{\nabla }}\left(\psi -x\right)\right]\cdot {\rm{\nabla }}{{\rm{\nabla }}}_{\perp }^{2}\psi \\ \quad =\,-\left[{{\boldsymbol{e}}}_{z}\times {\rm{\nabla }}\left({p}_{{e}_{\parallel }}-{p}_{{e}_{\perp }}-\displaystyle \frac{{\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}}{2}\,{d}_{i}^{2}{B}_{z}\right)\right]\cdot {\rm{\nabla }}\left(n-{B}_{z}\right).\end{array}\end{eqnarray} \tag{ 71 }$$

We readily note that the right-hand side of this equation vanishes for isothermal electrons. Namely, the electrons have a negligibly small Larmor radius and thus the isothermal equation of state (38), we have ${p}_{{e}_{\parallel }}-{p}_{{e}_{\perp }}=(1/2)({\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}){d}_{i}^{2}\,n$, and the right-hand side of Equation (71) reduces to zero as a mixed product of two colinear vectors. This enables the equation to be integrated as

$$\begin{eqnarray}&&n-{B}_{z}=\displaystyle \frac{{c}_{{\rm{A}}}^{2}}{{u}_{z}^{2}}\left(1-\displaystyle \frac{{\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}}{2}\right)\displaystyle \frac{{{\rm{\nabla }}}_{\perp }^{2}\psi }{F^{\prime} \left(\psi -x\right)}+H\left(\psi -x\right),\end{eqnarray} \tag{ 72 }$$

where H is an arbitrary function. It can be shown that solutions with arbitrary u_z/c_A can meet all physical continuity conditions at the core's edge only if they contain both the shear and the kinetic slow-mode components described in Equation (61). However, in such case, the nonlinear term on the right-hand side of Equation (57) is finite on the entire x−y plane, which presents a formidable obstacle for an analytic treatment and requires extensive numerical calculations that are outside the scope of the present paper. To proceed, we restrict ourselves to the phase velocities that are much smaller than the Alfvén speed, ${c}_{{\rm{A}}}\gg {u}_{z}\gt {v}_{{\mathrm{Ti}}_{\parallel }}$, when a solution involving only one Alfvén mode becomes possible (in the solar wind event recorded by Cluster (Perrone et al. 2016) such a strong constraint might not be fulfilled–see Appendix A–and those structures are likely to be coupled either with the ion sound or with the torsional magnetic field). In such regime, the electron continuity Equation (55) gives $\psi \sim ({u}_{z}^{2}/{c}_{{\rm{A}}}^{2})\,\phi \to 0$, which in turns yields that the arguments of the functions F and G reduce to $\xi \equiv \psi -x\to -x$. As before, these functions are adopted to be part-by-part linear, in the form of Equation (48), where the slopes F₁ and H₁ take different constant values ${F}_{1}^{\mathrm{in}},{H}_{1}^{\mathrm{in}}$, and ${F}_{1}^{\mathrm{out}},{H}_{1}^{\mathrm{out}}$ inside and outside of the vortex core determined by $\xi ({r}_{0})={\xi }_{0}$. We note that the separatrix r = r₀ is not an isoline of the functions F and H, whose argument is given by $\xi =\psi -x\to -x=-r\,\cos \varphi$, which obviously is not constant at the separatrix r = r₀. This prohibits the slopes to jump at the circle r = r₀ and implies that ${F}_{1}^{\mathrm{in}}={F}_{1}^{\mathrm{out}}=1$ and ${H}_{1}^{\mathrm{in}}={H}_{1}^{\mathrm{out}}=0$. As a consequence, Equation (72) is decoupled from the rest of the system, while from Equation (54) we readily obtain

$$\begin{eqnarray}&&\phi -{p}_{{e}_{\parallel }}+\left(1/2\right)\left({\beta }_{{e}_{\parallel }}-{\beta }_{e\perp }\right)\,{d}_{i}^{2}{B}_{z}=0.\end{eqnarray} \tag{ 73 }$$

Setting ${\gamma }_{{i}_{\perp }}=2$ and ${\gamma }_{{e}_{\parallel }}$, the quantities n, B_z, and ϕ can now be expressed from Equations (58), (59), and (63) as follows (for easier reading, here and below we use the mathcal font to denote true constants, such as ${ \mathcal N },{ \mathcal B },{ \mathcal F },{ \mathcal Q },{ \mathcal A }$, and ${ \mathcal U }$, which depend only on the plasma parameters and not on the slopes ${G}_{1}^{\mathrm{in}}$ and ${G}_{1}^{\mathrm{out}}$):

$$\begin{eqnarray}&&\,n={{ \mathcal N }}_{0}{\rm{\Phi }}+{{ \mathcal N }}_{2}{{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}\equiv \displaystyle \frac{1}{{\rho }_{i}^{2}{ \mathcal Q }}\,{\rm{\Phi }}\,+\,\displaystyle \frac{2+{\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}}{4{ \mathcal Q }}\,{{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }},\end{eqnarray} \tag{ 74 }$$

$$\begin{eqnarray}&&\,{B}_{z}={{ \mathcal B }}_{0}{\rm{\Phi }}+{{ \mathcal B }}_{2}{{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}\equiv -\displaystyle \frac{2{\beta }_{{i}_{\perp }}+{\beta }_{{e}_{\perp }}}{2{\rho }_{i}^{2}{ \mathcal Q }}\,{\rm{\Phi }}\,+\displaystyle \frac{{\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}}{4{ \mathcal Q }}\,{{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }},\end{eqnarray} \tag{ 75 }$$

$$\begin{eqnarray}\begin{array}{lcl}\phi & = & {{ \mathcal F }}_{0}{\rm{\Phi }}+{{ \mathcal F }}_{2}{{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}\\ & \equiv & \displaystyle \frac{{ \mathcal Q }-2}{{ \mathcal Q }}{\rm{\Phi }}+\,\displaystyle \frac{{ \mathcal Q }-2-{\beta }_{{e}_{\parallel }}+{\beta }_{{e}_{\perp }}}{2{ \mathcal Q }}\,{\rho }_{i}^{2}\,{{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }},\,\end{array}\end{eqnarray} \tag{ 76 }$$

$$\begin{eqnarray}\begin{array}{rcl} & & { \mathcal Q }=\displaystyle \frac{{\beta }_{{e}_{\parallel }}+2{\beta }_{{i}_{\perp }}}{{\beta }_{{i}_{\perp }}}+\displaystyle \frac{2{\beta }_{{i}_{\perp }}+{\beta }_{{e}_{\perp }}}{2}\,\displaystyle \frac{{\beta }_{{e}_{\parallel }}-{\beta }_{{e}_{\perp }}}{{\beta }_{{i}_{\perp }}}\quad \quad \mathrm{and}\\ & & \quad { \mathcal U }=1-\displaystyle \frac{{\beta }_{{i}_{\parallel }}-{\beta }_{i\perp }}{2-{\beta }_{{e}_{\parallel }}+{\beta }_{{e}_{\perp }}}.\end{array}\,\end{eqnarray} \tag{ 77 }$$

It is worth noting that due to the ions' FLR effects, Equation (75) implies that the stream function Φ is not proportional to the magnetic field B_z, which essentially decouples the velocity and magnetic fields. Using Equations (63) and (72) and some simple manipulations, the above expressions permit us to rewrite the charge continuity Equation (57) as follows:

$$\begin{eqnarray}\begin{array}{rcl} & & \left[{{\boldsymbol{e}}}_{z}\times {\rm{\nabla }}\left(n-{B}_{z}\right)\right]\cdot {\rm{\nabla }}\left[{\rm{\Phi }}-{ \mathcal U }\,x-{{ \mathcal A }}_{1}\,{\rho }_{i}^{2}\left(n-{B}_{z}\right)\right]\\ & & \quad +\left\{{{\boldsymbol{e}}}_{z}\times {\rm{\nabla }}\left[{\rm{\Phi }}-x-{\rho }_{i}^{2}\left(n-{B}_{z}\right)+{{ \mathcal A }}_{2}\,{\rho }_{i}^{2}\,{{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}\right]\right\}\cdot {\rm{\nabla }}{{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}\\ & & \quad =\,\left[{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }\left(\partial /\partial {x}_{i}\right){\rho }_{i}^{2}\left(n-{B}_{z}\right)\right]\cdot {\rm{\nabla }}\left(\partial {\rm{\Phi }}/\partial {x}_{i}\right),\end{array}\end{eqnarray} \tag{ 78 }$$

where ${{ \mathcal A }}_{1}$ and ${{ \mathcal A }}_{2}$ are arbitrary constants introduced for algebraic convenience. Adopting these in the following way,

$$\begin{eqnarray}\begin{array}{rcl} & & {{ \mathcal A }}_{1}={ \mathcal U }-\displaystyle \frac{{ \mathcal U }-{ \mathcal U }/{ \mathcal Q }-1}{{\rho }_{i}^{2}\left({{ \mathcal N }}_{0}-{{ \mathcal B }}_{0}\right)},\\ & & {{ \mathcal A }}_{2}=\displaystyle \frac{{ \mathcal U }-{ \mathcal U }/{ \mathcal Q }-1}{{ \mathcal U }}\,\displaystyle \frac{{{ \mathcal N }}_{2}-{{ \mathcal B }}_{2}}{{\rho }_{i}^{2}\left({{ \mathcal N }}_{0}-{{ \mathcal B }}_{0}\right)},\end{array}\end{eqnarray} \tag{ 79 }$$

using Equations (74) and (75), and after some algebra, we can cast Equation (78) in a simple form, viz.

$$\begin{eqnarray}\begin{array}{rcl} & & \left[{{\boldsymbol{e}}}_{z}\times {\rm{\nabla }}\left({\rm{\Phi }}+{ \mathcal V }\,x\right)\right]\cdot {\rm{\nabla }}\left({{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}+{\kappa }^{2}{ \mathcal V }\,x\right)\\ & & \,=2\,{ \mathcal C }\left({{\boldsymbol{e}}}_{z}\times {\rm{\nabla }}\,\partial {\rm{\Phi }}/\partial {x}_{i}\right)\cdot {\rm{\nabla }}{{\rm{\nabla }}}_{\perp }^{2}\partial {\rm{\Phi }}/\partial {x}_{i},\end{array}\end{eqnarray} \tag{ 80 }$$

where

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{\kappa }^{2} & = & \displaystyle \frac{{ \mathcal U }\left({{ \mathcal N }}_{0}-{{ \mathcal B }}_{0}\right)}{1-{ \mathcal U }\left({{ \mathcal N }}_{2}-{{ \mathcal B }}_{2}\right)},\\ { \mathcal V } & = & -\displaystyle \frac{1-{ \mathcal U }\left({{ \mathcal N }}_{2}-{{ \mathcal B }}_{2}\right)}{1-1/{ \mathcal Q }-{\rho }_{i}^{2}\left({{ \mathcal N }}_{0}-{{ \mathcal B }}_{0}\right)-\left({{ \mathcal N }}_{2}-{{ \mathcal B }}_{2}\right)},\\ { \mathcal C } & = & \displaystyle \frac{{\rho }_{i}^{2}{\kappa }^{2}}{2}\,\displaystyle \frac{{ \mathcal V }}{{ \mathcal U }}\,\,\displaystyle \frac{2{{ \mathcal N }}_{2}-{{ \mathcal B }}_{2}}{{{ \mathcal N }}_{0}-{{ \mathcal B }}_{0}}.\end{array}\end{array}\end{eqnarray} \tag{ 81 }$$

In the regime of small but finite FLR corrections ${ \mathcal C }{{\rm{\nabla }}}_{\perp }^{2}\ll 1$, the right-hand side of Equation (80) comprises a small correction and, iteratively, it can be approximated by using the leading-order solution of Equation (80) ${{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}\approx -{\kappa }^{2}{ \mathcal V }\,x\,+G({\rm{\Phi }}+{ \mathcal V }\,x)$, where G is an arbitrary function of its argument. Then, using the identity

$$\begin{eqnarray}\begin{array}{rcl} & & 2\,{{\rm{\nabla }}}_{\perp }\cdot \left\{\left[\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }f\right)\cdot {{\rm{\nabla }}}_{\perp }\right]{{\rm{\nabla }}}_{\perp }G\left(f\right)\right\}\\ & & \,=\left({{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }f\right)\cdot {{\rm{\nabla }}}_{\perp }{{\rm{\nabla }}}_{\perp }^{2}G\left(f\right)-\left[{{\boldsymbol{e}}}_{z}\times {{\rm{\nabla }}}_{\perp }{{\rm{\nabla }}}_{\perp }^{2}f\right]\cdot {{\rm{\nabla }}}_{\perp }G\left(f\right),\end{array}\end{eqnarray} \tag{ 82 }$$

we can rewrite Equation (80) in the following form

$$\begin{eqnarray}\begin{array}{rcl} & & \left[{{\boldsymbol{e}}}_{z}\times {\rm{\nabla }}\left({\rm{\Phi }}+{ \mathcal V }\,x\right)\right]\cdot {\rm{\nabla }}\left[{{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}+{\kappa }^{2}{ \mathcal V }\,x\right]\\ & & +\left[{ \mathcal C }{{\rm{\nabla }}}_{\perp }^{2}G\left({\rm{\Phi }}+{ \mathcal V }\,x\right)\right]=\left[{{\boldsymbol{e}}}_{z}\times {\rm{\nabla }}G\left({\rm{\Phi }}+{ \mathcal V }\,x\right)\right]\cdot {\rm{\nabla }}\,{ \mathcal C }{{\rm{\nabla }}}_{\perp }^{2}\left({\rm{\Phi }}+{ \mathcal V }\,x\right),\end{array}\end{eqnarray} \tag{ 83 }$$

that is, with the accuracy to the first order in the small quantity ${ \mathcal C }{{\rm{\nabla }}}_{\perp }^{2}$, equivalent to

$$\begin{eqnarray}\begin{array}{rcl} & & \left[{{\boldsymbol{e}}}_{z}\times {\rm{\nabla }}\left(1+{ \mathcal C }{{\rm{\nabla }}}_{\perp }^{2}\right)\left({\rm{\Phi }}+{ \mathcal V }\,x\right)\right]\\ & & \,\cdot {\rm{\nabla }}\left[\left(1+{ \mathcal C }{{\rm{\nabla }}}_{\perp }^{2}\right)\left({{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}+{\kappa }^{2}{ \mathcal V }\,x\right)\right]=0,\end{array}\end{eqnarray} \tag{ 84 }$$

and is readily integrated one time, viz.

$$\begin{eqnarray}&&\left(1+{ \mathcal C }\,{{\rm{\nabla }}}_{\perp }^{2}\right)\left({{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}+{\kappa }^{2}{ \mathcal V }\,x\right)=G\,\left[\left(1+{ \mathcal C }\,{{\rm{\nabla }}}_{\perp }^{2}\right)\left({\rm{\Phi }}+{ \mathcal V }\,x\right)\right].\end{eqnarray} \tag{ 85 }$$

We adopt G(ξ) in the form of a continuous part-by-part linear function, G(ξ) = G₁ξ, whose constant slope G₁ takes different values ${G}_{1}^{\mathrm{in}}$ and ${G}_{1}^{\mathrm{out}}$ inside and outside of the circle r = r₀, respectively. Remarkably, with such choice of G(ξ), the parameter ${ \mathcal C }$ coming from the nonlinear term on the right-hand side of Equation (80) and Equation (57) cancels out in Equation (85) and does not affect its solution. As the function G(ξ) must be continuous (see discussion in the paragraph following Equation (90) at the end of this section), a jump is permitted only if the argument vanishes at such a circle, $\xi ({r}_{0},\varphi )=0$. Noting that for a localized solution we must have ${G}_{1}^{\mathrm{out}}={\kappa }^{2}$ and setting ${G}_{1}^{\mathrm{in}}=-{\lambda }^{2}$, we obtain the following equations for the stream function Φ outside and inside the circle r = r₀:

$$\begin{eqnarray}&&\begin{array}{l}\left({{\rm{\nabla }}}_{\perp }^{2}-{\kappa }^{2}\right){{\rm{\Phi }}}^{\mathrm{out}}=0,\qquad r\gt {r}_{0},\\ \left({{\rm{\nabla }}}_{\perp }^{2}+{\lambda }^{2}\right)\left[{{\rm{\Phi }}}^{\mathrm{in}}+\left(1+\displaystyle \frac{{\kappa }^{2}}{{\lambda }^{2}}\right){ \mathcal V }\,x\right]=0,\ r\lt {r}_{0}.\end{array}\end{eqnarray} \tag{ 86 }$$

These equations separate variables in cylindrical coordinates, ${\rm{\Phi }}={\sum }_{k}{{\rm{\Phi }}}_{k}\exp ({ik}\varphi );$ the amplitude of the kth harmonic is given by

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{k}^{\mathrm{out}}={c}_{k}^{\mathrm{out}}\,{K}_{k}\left(\kappa r\right),\quad \quad {{\rm{\Phi }}}_{k}^{\mathrm{in}}={c}_{k}^{\mathrm{in}}\,{J}_{k}\left(\lambda r\right),\end{eqnarray} \tag{ 87 }$$

where ${c}_{k}^{{\rm{in}}}$ and ${c}_{k}^{{\rm{out}}}$ are arbitrary constants. It can be argued that the stream function Φ must be a dipole, i.e., it may contain only the dipole component k = 1 (for a discussion, see the paragraph at the end of this section). Then, the continuity of the function G readily yields

$$\begin{eqnarray}\begin{array}{rcl} & & {\left({{\rm{\Phi }}}^{\mathrm{out}}+{ \mathcal V }x\right)}_{r={r}_{0}}={\left({{\rm{\Phi }}}^{\mathrm{in}}+{ \mathcal V }x\right)}_{r={r}_{0}}={\left({{\rm{\nabla }}}_{\perp }^{2}{{\rm{\Phi }}}^{\mathrm{out}}+{\kappa }^{2}{ \mathcal V }x\right)}_{r={r}_{0}}\\ & & ={\left({{\rm{\nabla }}}_{\perp }^{2}{{\rm{\Phi }}}^{\mathrm{in}}+{\kappa }^{2}{ \mathcal V }x\right)}_{r={r}_{0}}=0,\end{array}\end{eqnarray} \tag{ 88 }$$

which from Equations (74)–(76) also provides the continuity of the functions n, B_z, and ϕ. Finally, matching the above "in" and "out" solutions at r = r₀ we obtain a standard Larichev& Reznik-type dipole (Larichev & Reznik 1976):

$$\begin{eqnarray}{\rm{\Phi }}\left(r,\varphi \right)={ \mathcal V }\,{r}_{0}\,\cos \varphi \times \left\{\begin{array}{ll}-\tfrac{{K}_{1}\left(\kappa r\right)}{{K}_{1}\left(\kappa {r}_{0}\right)}, & r\gt {r}_{0}\\ -\left(1+\tfrac{{\kappa }^{2}}{{\lambda }^{2}}\right)\displaystyle \frac{r}{{r}_{0}}+\displaystyle \frac{{\kappa }^{2}}{{\lambda }^{2}}\displaystyle \frac{{J}_{1}\left(\lambda r\right)}{{J}_{1}\left(\lambda {r}_{0}\right)}, & r\lt {r}_{0}\end{array}\right.,\end{eqnarray} \tag{ 89 }$$

while the plasma density, compressional magnetic field, and the electrostatic potential are expressed in Equations (74)–(76). This solution is localized in space only if the "out" e-folding length κ defined in Equation (81) or, equivalently obtained from Equation (61) in the limit ${u}_{z}\ll {c}_{{\rm{A}}}$ and ${\gamma }_{{e}_{\parallel }}=1$, is a real quantity, i.e., for κ² > 0, which yields the condition for the existence of kinetic slow-mode vortices with a complicated dependence on the values of the plasma ${\beta }_{{j}_{\zeta }}$, with j = e, i and $\zeta =\parallel ,\perp$. In contrast to its shear Alfvén counterparts, Equations (50) and (66), which are essentially MHD nonlinear modes and do not have a spatial scale, the kinetic slow-mode vortex Equation (89) has a distinct scale comparable to the ion Larmor radius 1/κ ∼ ρ_i; see Equations (61) and (81). Also, linear equations indicate that outside of the vortex core, perturbations of the compressional magnetic field and density of a kinetic slow-mode vortex are anticorrelated, n/B < 0.

The remaining free parameter λ is determined from the condition that the radial electric field is continuous at the edge of the core, ${(\partial {\phi }^{\mathrm{in}}/\partial r)}_{r={r}_{0}}={(\partial {\phi }^{\mathrm{out}}/\partial r)}_{r={r}_{0}}$, i.e., of the absence at r = r₀ of any surface charges. This gives rise to the following nonlinear dispersion relation:

$$\begin{eqnarray}\begin{array}{rcl} & & \left({{ \mathcal F }}_{0}+{{ \mathcal F }}_{2}{\kappa }^{2}\right)\displaystyle \frac{\kappa {r}_{0}\,{K}_{1}^{{\prime} }\left(\kappa {r}_{0}\right)}{{K}_{1}\left(\kappa {r}_{0}\right)}={{ \mathcal F }}_{0}\left(1+\displaystyle \frac{{\kappa }^{2}}{{\lambda }^{2}}\right)\\ & & \,-\left({{ \mathcal F }}_{0}-{{ \mathcal F }}_{2}{\lambda }^{2}\right)\displaystyle \frac{{\kappa }^{2}}{{\lambda }^{2}}\displaystyle \frac{\lambda {r}_{0}\,{J}_{1}^{{\prime} }\left(\lambda {r}_{0}\right)}{{J}_{1}\left(\lambda {r}_{0}\right)}.\end{array}\end{eqnarray} \tag{ 90 }$$

We have shown in Section 3.1 that Chaplygin's monopole component of the solution can exist only when the function $G({\rm{\Phi }}+{ \mathcal V }x)$ features a finite jump at the edge of the vortex core, which on the spatial scale sufficiently bigger than the ion Larmor radius considered there, produces a jump in the vorticity ${{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}$. However, approaching the ion scales and including FLR terms, in the kinetic slow-mode branch, we obtain Equation (85), which also contains the Laplacian of vorticity, ${{\rm{\nabla }}}_{\perp }^{4}{\rm{\Phi }}$. The latter obviously becomes singular when ${{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}$ has a jump. Such singularity, which implies also a singularity in the charge continuity Equation (57) or (78), is clearly prohibited for physical reasons. This indicates that (quasi)monopolar Chaplygin structures in the form of Equation (66) cannot develop on the ion scale, in the kinetic slow mode. As the singularity of ${{\rm{\nabla }}}_{\perp }^{4}{\rm{\Phi }}$ arises in Equation (84) due to the expansion in the powers of small but finite FLR inherent in the stress tensor Equation (13), one may expect that it vanishes in a plasma description that involves all orders in the FLR, such as the gyrofluids. It can be expected that a gyrofluid kinetic slow magnetosonic Chaplygin vortex features a sharp peak of charge density and/or a thin current layer with a large current density that may affect its stability. Numerical study of such structures requires extensive calculations and is beyond the scope of this paper.

In a kinetic slow-mode regime without ion sound and torsional magnetic field, $d/{dt}\gg {v}_{{\mathrm{Ti}}_{\parallel }}\,\partial /\partial {x}_{b}$ and ${{\boldsymbol{B}}}_{\perp }=0$ $(\,\Rightarrow d/{dt}\ll {c}_{{\rm{A}}}\,\partial /\partial {x}_{b})$, the charge continuity equation with arbitrary $(t,{{\boldsymbol{r}}}_{\perp },z)$ dependence has the form (see Equation (80))

$$\begin{eqnarray}\begin{array}{rcl} & & \left[{ \mathcal V }\,\partial /\partial t+\left({{\boldsymbol{e}}}_{z}\times {\rm{\nabla }}{\rm{\Phi }}\right)\cdot {\rm{\nabla }}\right]\left({{\rm{\nabla }}}_{\perp }^{2}-{\kappa }^{2}\right){\rm{\Phi }}\\ & & \,=2\,{ \mathcal C }\left({{\boldsymbol{e}}}_{z}\times {\rm{\nabla }}\,\partial {\rm{\Phi }}/\partial {x}_{i}\right)\cdot {\rm{\nabla }}{{\rm{\nabla }}}_{\perp }^{2}\partial {\rm{\Phi }}/\partial {x}_{i}.\end{array}\end{eqnarray} \tag{ 91 }$$

where κ, ${ \mathcal V }$, and ${ \mathcal C }$ are given in Equation (81). Multiplying Equation (91) by Φ and after some algebra, we can cast it in the form

$$\begin{eqnarray}\begin{array}{rcl} & & \left[{ \mathcal V }\,{\rm{\Phi }}\,\displaystyle \frac{\partial }{\partial t}+\left({{\boldsymbol{e}}}_{z}\times {\rm{\nabla }}\displaystyle \frac{{{\rm{\Phi }}}^{2}}{2}\right)\cdot {\rm{\nabla }}\right]\left({{\rm{\nabla }}}_{\perp }^{2}-{\kappa }^{2}\right){\rm{\Phi }}\\ & & ={ \mathcal C }\left\{\left({{\boldsymbol{e}}}_{z}\times {\rm{\nabla }}{{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}\right)\cdot {\rm{\nabla }}{\left({{\rm{\nabla }}}_{\perp }{\rm{\Phi }}\right)}^{2}\right.\\ & & \,-\,\left.\displaystyle \frac{\partial }{\partial {x}_{i}}\left[2\,{\rm{\Phi }}\left({{\boldsymbol{e}}}_{z}\times {\rm{\nabla }}\,{{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}\right)\cdot {\rm{\nabla }}\displaystyle \frac{\partial {\rm{\Phi }}}{\partial {x}_{i}}\right]\right\}.\end{array}\end{eqnarray} \tag{ 92 }$$

The term on the right-hand side is a divergence and vanishes in the integration for the entire space, provided the effective potential Φ vanishes at infinity. Thus, we obtain the expression for the energy conservation $\partial W/\partial t=0$, where

$$\begin{eqnarray}&&W=\left(1/2\right){\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{dx}\,{dy}\,{dz}\left[{\left({{\rm{\nabla }}}_{\perp }{\rm{\Phi }}\right)}^{2}+{\kappa }^{2}{{\rm{\Phi }}}^{2}\right].\end{eqnarray} \tag{ 93 }$$

We have studied fluid plasma vortices in a high-β plasma, on the spatial scale comparable to the ion inertial length and approaching the ion Larmor radius, including the effects of magnetic field compression and of the finite ion Larmor radius, in the regime where the acoustic perturbations are small. The vortices have the form of infinitely long filaments, slightly tilted toward the magnetic field. Our basic Equations (54)–(58) also possess a trivial stationary solution that is fully aligned with the z-axis, ∂/∂t = ∂/∂z = 0, and circularly symmetric ∂/∂φ = 0, i.e., strictly monopolar. However, water-tank experiments (Beckers et al. 2003), in which perturbations evolved according to the 2D Euler Equation (64) but also involved a small but finite viscosity of the fluid (nonexistent in our plasma regime), revealed that such stationary monopoles either disperse or slowly transform into dipolar or tripolar vortices, depending on the amount of shielding in the initial state. This may also be related with the jumps in the vorticity, ${{\rm{\nabla }}}_{\perp }^{2}{\rm{\Phi }}$, at the edge of a monopole. Conversely, the propagating Lamb dipole, corresponding to a shear Alfvén vortex with ψ₀ = 0 in Equations (50) and (66), was remarkably stable and could easily survive collisions with other dipoles (van Heijst & Flor 1989). A propagating quasi-monopolar vortex, i.e., a Chaplygin structure with a relatively small dipolar component, described by Equation (50) when ${\psi }_{0}\ne 0$, is much more stable than the stationary monopoles. In an ordinary fluid, a Chaplygin quasi-monopole may propagate over a distance that is an order of magnitude bigger than its diameter, as suggested by the weak nonlinear theory and numerical simulations (Stern & Radko 1998), as well as by experiments in nonrotating water tanks with a rectangular shape (Voropayev et al. 1999). More recent water-tank experiments (Cariteau & Flór 2006) have demonstrated that it still has a finite lifetime, because the secondary component of such strongly asymmetric vortex pair starts to wrap around the principal monopole, creating a strain that eventually gives rise to an elliptic instability due to the parametric resonance between the oscillation of inertial waves and the ambient strain field; for details, see Cariteau & Flór (2006). It should be noted that the behavior of Chaplygin vortices in a fully 3D geometry is still an open question, because no reliable 3D simulations and experiments have been reported in the literature and we are unable to predict whether the dynamical 3D turbulence of the solar wind and of Earth's magnetosheath is dominated by stable dipolar (Larichev & Reznik 1976; Petviashvili & Pokhotelov 1992; Kuvshinov et al. 1999) vortices, or by long-lived, mostly monopolar Chaplygin structures. Their monopolar components feature a jump of the plasma density and of the compressional magnetic field B_z at the edge. In the presence of a compressional magnetic field, such jumps are associated with a current shaped as a thin hollow cylinder at r = r₀, which probably reduces its stability in high-β turbulent plasmas.

In a high-β plasma, we have found two distinct types of coherent vortices propagating in the perpendicular direction. The first is identified as a generalized shear Alfvén structure that possesses both the torsional and the compressional components of the magnetic field perturbation. It has a zero parallel electric field and, being homogeneous along its axis that is inclined toward the ambient magnetic field, it sweeps along the z-axis with a velocity u_z that is in the Alfvén speed range; the transverse phase velocity is equal to ${u}_{z}\tan \theta$, where θ is the (small) pitch angle between the structure and the background magnetic field. While in a sufficiently incompressible plasma $\delta n/{n}_{0}\to 0,\delta {B}_{z}/{B}_{0}\to 0$, it has the structure of a moving Chaplygin vortex with a monopole superimposed on a dipole; in plasmas with β ∼ 1, its monopolar component is likely to be unstable and short-lived due to the emergence of a thin, (electron-scale) current layer and/or a sharp peak of charge density at its edge. The compressible magnetic field associated with such a vortex is restricted to the interior of the vortex core, while the transverse perturbation "leaks" from the core to larger distances.

The second type of propagating structure obtainable analytically possesses a finite compressional magnetic field and parallel electric field, as well as perpendicular fluid velocity and density perturbation, but vanishing parallel ion fluid velocity and transverse perturbations of the magnetic field. It is identified as a nonlinear kinetic slow magnetosonic structure. Its parallel phase velocity is much smaller than the Alfvén speed, which also yields a thermalized electron distribution. The transverse fluid velocity of the kinetic slow-mode vortex is better localized than that of its shear Alfvén counterpart, while its compressional magnetic field extends outside the core. A kinetic slow-mode structure that possesses only a compressional magnetic field perturbation exists only when ${v}_{{{Ti}}_{\parallel }}^{2}/{c}_{{\rm{A}}}^{2}={\beta }_{{i}_{\parallel }}/2\ll 1$; otherwise, it also involves the ion sound or the torsional magnetic field.

Our analytical study does not exclude the possibility of mixed shear Alfvén/kinetic slow magnetosonic vortices whose parallel phase velocity approaches Alfvén speed, u_z ≲ c_A, but their construction requires extensive numerical calculations that are out of the scope of the present paper. Conversely, we have demonstrated that fluid-type (quasi)monopolar Chaplygin filaments are not likely to emerge in the kinetic slow magnetoacoustic domain. The only viable propagating kinetic magnetoacoustic monopoles, possibly in the form of cigars (i.e., of filaments with a finite length), may emerge in situations with nonvanishing ${E}_{\parallel }$ and with the parallel phase velocities in the thermal range, not studied here. They involve particles trapped both in the electrostatic potential wells and in magnetic depressions that provide an additional nonlinearity capable to produce the spatial localization of a vortex. Coherent vortex structures in a high-β plasma, with $\delta n/{n}_{0}\ne \delta {B}_{z}/{B}_{0}$ and with a finite parallel electric field, $\phi \ne ({u}_{z}/{c}_{{\rm{A}}}){A}_{z}$, that include kinetic phenomena such as particle trapping has been studied elsewhere (Jovanović et al. 2017) via a high-β gyrokinetic theory. Our results can explain observations of the solar wind and the magnetosheath turbulence in a plasma with β ∼ 1, in particular the Alfvén vortices and the compressible magnetic filaments. The structures at large scales (${ \mathcal L }\sim 30{\rho }_{i}$) and at the ion scales (${ \mathcal L }\gtrsim {\rho }_{i}\sim c/{\omega }_{\mathrm{pi}}$), described in Sections 3.1 and 3.2, can be an important ingredient of the kinetic turbulent cascade. The latter produces power-law spectra of δB fluctuations ∼k^−5/3 and ∼k^−2.8 at MHD and ion scales, respectively (Alexandrova et al. 2009; Perrone et al. 2016, 2017), while at the dissipative (electron) scale the dependence is exponential $\sim \exp (-k{\rho }_{\mathrm{Le}})$ (Alexandrova et al. 2012). The compressible component of the inertial range solar wind turbulence at 1 au has been shown (Howes et al. 2012) to belong almost entirely to the kinetic slow mode, which determines the nature of the density fluctuation spectrum and of the the cascade of kinetic turbulence to short wavelengths.

This work was supported in part (D.J. and M.B.) by the MPNTR 171006 and NPRP 11S-1126-170033 grants. D.J. and O.A. acknowledge financial support from the CIAS, from The French National Centre for Space Studies (CNES), and of the CNRS, and the hospitality of the LESIA laboratory in Meudon.

Plasma parameters in the region of the slow solar wind where vortex structures were observed (Perrone et al. 2016) are listed below. Such plasma can be regarded as weakly magnetized, because $| {{\rm{\Omega }}}_{e}| /{\omega }_{\mathrm{pe}}\sim 0.0056$, ${{\rm{\Omega }}}_{i}/{\omega }_{\mathrm{pi}}\sim 0.00013$.

Plasma density $n\sim (25\mbox{--}30)\,{\mathrm{cm}}^{-3}$	average magnetic field $\langle B\rangle \sim 9\,\mathrm{nT}$,
Alfvén speed ${c}_{{\rm{A}}}=c\,{{\rm{\Omega }}}_{i}/{\omega }_{\mathrm{pi}}\sim 36\,\mathrm{km}\,{{\rm{s}}}^{-1}$,	acoustic velocity ${c}_{S}\sim 50\,\mathrm{km}\,{{\rm{s}}}^{-1}$,
Ion gyrofrequency Ωi ∼ 0.8 1/s,	electron gyrofrequency ${{\rm{\Omega }}}_{e}\sim 1.46\times {10}^{3}\,1\,{\rm{s}}$
Ion plasma frequency ωpi ∼ 6.6 × 103 1/s,	electron plasma frequency ωpe ∼ 280 × 103 1/s
Ion plasma length ${d}_{i}=c/{\omega }_{\mathrm{pi}}\sim 46\,\mathrm{km}$,	electron plasma length de = c/ωpe ∼ 1 km
Ion Larmor radius ${\rho }_{{Li}}={v}_{{\mathrm{Ti}}_{\perp }}/{{\rm{\Omega }}}_{i}\sim (40\mbox{--}110)\,\mathrm{km}$,	electron Larmor radius ${\rho }_{\mathrm{Le}}={v}_{{\mathrm{Te}}_{\perp }}/{{\rm{\Omega }}}_{e}\sim (1\mbox{--}2.5)\,\mathrm{km}$
Ion beta ${\beta }_{{i}_{\perp }}=2{p}_{{i}_{\perp }}/{c}^{2}{\epsilon }_{0}{B}^{2}\sim 0.5\mbox{--}2.5$,	electron beta ${\beta }_{{e}_{\perp }}=2{p}_{{e}_{\perp }}/{c}^{2}{\epsilon }_{0}{B}^{2}\sim 1\mbox{--}2$
Ion temperature anisotropy ${A}_{i}={T}_{{i}_{\perp }}/{T}_{{i}_{\parallel }}\sim 1.6$,	electron temperature anisotropy ${A}_{e}={T}_{{e}_{\perp }}/{T}_{{e}_{\parallel }}\sim 0.9$
Ion thermal velocity vTi ∼ 40 km s−1,	electron thermal velocity vTe ∼ 1500 km s−1
Velocity of the slow solar wind vsw ∼ 360 km s−1,	angle between $\langle {\boldsymbol{B}}\rangle$ and the solar wind ${\theta }_{\mathrm{BV}}\sim {55}^{{\rm{o}}}\mbox{--}{125}^{{\rm{o}}}$

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characteristic diameter of the structure transverse to the magnetic field ${L}_{\perp }\sim (5\mbox{--}25)\,c/{\omega }_{\mathrm{pi}}\sim (6\mbox{--}30)\,{\rho }_{{Li}}$, velocity (in the plasma frame) of the structure perpendicular to the magnetic field ${u}_{\perp }=(0.5\mbox{--}4)\,{c}_{{\rm{A}}}\pm (1\mbox{--}4)\,{c}_{{\rm{A}}}$.

Relevant plasma parameters in the fast solar wind ($| {{\rm{\Omega }}}_{e}| /{\omega }_{\mathrm{pe}}\sim 0.013$, ${{\rm{\Omega }}}_{i}/{\omega }_{\mathrm{pi}}\sim 0.00030$) are (Perrone et al. 2017):

Plasma density n ∼ 4 cm−3	average magnetic field $\langle B\rangle \sim 8.3\,\mathrm{nT}$,
Ion temperatures ${T}_{{i}_{\perp }}=41\,\mathrm{eV}$, ${T}_{{i}_{\parallel }}=30\,\mathrm{eV}$	electron temperatures ${T}_{{e}_{\perp }}=18\,\mathrm{eV}$, ${T}_{{e}_{\parallel }}=14\,\mathrm{eV}$,
Parallel ion thermal velocity ${v}_{{\mathrm{Ti}}_{\parallel }}\sim 53\,\mathrm{km}\,{{\rm{s}}}^{-1}$,	parallel electron thermal velocity ${v}_{{\mathrm{Te}}_{\parallel }}\sim 1600\,\mathrm{km}\,{{\rm{s}}}^{-1}$,
Alfvén speed ${c}_{{\rm{A}}}=c{{\rm{\Omega }}}_{i}/{\omega }_{\mathrm{pi}}\sim 100\,\mathrm{km}\,{{\rm{s}}}^{-1}\sim 1.9{v}_{{\mathrm{Ti}}_{\parallel }}$,	ion beta ${\beta }_{{i}_{\perp }}=0.95$
Ion plasma length $c/{\omega }_{\mathrm{pi}}\sim 115\,\mathrm{km}$,	electron plasma length $c/{\omega }_{\mathrm{pe}}\sim 2.7\,\mathrm{km}$
Ion Larmor radius ${\rho }_{{Li}}={v}_{{\mathrm{Ti}}_{\perp }}/{{\rm{\Omega }}}_{i}\sim 110\,\mathrm{km}$,	electron Larmor radius ${\rho }_{\mathrm{Le}}={v}_{{\mathrm{Te}}_{\perp }}/{{\rm{\Omega }}}_{e}\sim 1.5\,\mathrm{km}$
Velocity of the fast solar wind ${v}_{\mathrm{sw}}\sim 600\,\mathrm{km}\,{{\rm{s}}}^{-1}$,	angle between $\langle {\boldsymbol{B}}\rangle$ and the solar wind ${\theta }_{\mathrm{BV}}\sim {50}^{{\rm{o}}}\mbox{--}{90}^{{\rm{o}}}$.

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Parameters in the magnetosheath region downstream of quasi-perpendicular bow shock ($| {{\rm{\Omega }}}_{e}| /{\omega }_{\mathrm{pe}}\sim 0.056$, ${{\rm{\Omega }}}_{i}/{\omega }_{\mathrm{pi}}\,\sim 0.0013$), and the properties of observed magnetic structures (Alexandrova et al. 2004, 2006; Alexandrova 2008):

Plasma density n ∼ 30 cm−3	average magnetic field $\langle B\rangle \sim 90\,\mathrm{nT}$,
Ion temperatures ${T}_{{i}_{\perp }}=360\,\mathrm{eV}$, ${T}_{{i}_{\parallel }}=170\,\mathrm{eV}$	electron temperatures ${T}_{{e}_{\perp }}=95\,\mathrm{eV}$, ${T}_{{e}_{\parallel }}=85\,\mathrm{eV}$,
Parallel ion thermal velocity ${v}_{{\mathrm{Ti}}_{\parallel }}\sim 130\,\mathrm{km}\,{{\rm{s}}}^{-1}$,	parallel electron thermal velocity ${v}_{{\mathrm{Te}}_{\parallel }}\sim 4000\,\mathrm{km}\,{{\rm{s}}}^{-1}$,
Alfvén speed ${c}_{{\rm{A}}}=c{{\rm{\Omega }}}_{i}/{\omega }_{\mathrm{pi}}\sim 390\,\mathrm{km}\,{{\rm{s}}}^{-1}\sim 3{{\rm{v}}}_{{\mathrm{Ti}}_{\parallel }}$,	ion beta ${\beta }_{{i}_{\perp }}=0.5$
Ion plasma length $c/{\omega }_{\mathrm{pi}}\sim 45\,\mathrm{km}$,	electron plasma length $c/{\omega }_{\mathrm{pe}}\sim 1\,\mathrm{km}$
Ion Larmor radius ${\rho }_{\mathrm{Li}}={v}_{{\mathrm{Ti}}_{\perp }}/{{\rm{\Omega }}}_{i}\sim 20\,\mathrm{km}$,	electron Larmor radius ${\rho }_{\mathrm{Le}}\,={({m}_{e}{T}_{e\perp }/{m}_{i}{T}_{i\perp })}^{\tfrac{1}{2}}\,{\rho }_{{Li}}\sim 0.25\,\mathrm{km}$,
Radius of the structure transverse to magnetic field ${R}_{\perp }\sim (400\mbox{--}500)\,\mathrm{km}\sim 10\,c/{\omega }_{\mathrm{pi}}\sim 20\,{\rho }_{{Li}}$,
Size of the structure along the magnetic field ${L}_{\parallel }\gt 1000\,\mathrm{km}$,
Velocities of the structure $\perp$ and $\parallel$ to the magnetic field ${u}_{\perp }=(35\mbox{--}100)\,\mathrm{km}\,{{\rm{s}}}^{-1}$, ${u}_{\parallel }\sim (70\mbox{--}200)\,\mathrm{km}\,{{\rm{s}}}^{-1}\sim {v}_{{\mathrm{Ti}}_{\parallel }}$
Bulk velocity of the plasma ${v}_{{p}_{0}}\sim 250\,\mathrm{km}\,{{\rm{s}}}^{-1}$.

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Here we demonstrate that in the regime studied in this paper, the process may be approximately regarded as polytropic. Our generic equation of state (6) has the form

$$\begin{eqnarray}&&{{dp}}_{\perp }/{p}_{\perp }={\gamma }_{\perp }\,{dn}/n,\quad \quad {{dp}}_{\parallel }/{p}_{\parallel }={\gamma }_{\parallel }\,{dn}/n,\end{eqnarray} \tag{ C1 }$$

in which the multipliers ${\gamma }_{\parallel }$ and ${\gamma }_{\perp }$ are some functionals of the plasma density n. We consider only regimes in which the perturbations of the density and of the magnetic field are sufficiently small—see Equation (12)—and the functionals ${\gamma }_{\parallel }$ and ${\gamma }_{\perp }$ can be estimated from the linearized Vlasov equation, viz.

$$\begin{eqnarray}&&\begin{array}{l}\left[\displaystyle \frac{\partial }{\partial t}+{\boldsymbol{v}}\cdot {\rm{\nabla }}+\displaystyle \frac{q}{m}\left({\boldsymbol{v}}\times {{\boldsymbol{e}}}_{z}{B}_{0}\right)\cdot \displaystyle \frac{\partial }{\partial {\boldsymbol{v}}}\right]\delta f=-\displaystyle \frac{q}{m}\left({\boldsymbol{E}}+{\boldsymbol{v}}\times {\boldsymbol{B}}\right)\cdot \displaystyle \frac{\partial {f}_{0}}{\partial {\boldsymbol{v}}},\\ {f}_{0}=\displaystyle \frac{{n}_{0}}{{(2\pi )}^{\tfrac{3}{2}}{v}_{{T}_{\parallel }}{v}_{{T}_{\perp }}^{2}}\,\exp \left(-\displaystyle \frac{{v}_{x}^{2}+{v}_{y}^{2}}{{v}_{{T}_{\perp }}^{2}}-\displaystyle \frac{{v}_{\parallel }^{2}}{{v}_{{T}_{\parallel }}^{2}}\right).\end{array}\end{eqnarray} \tag{ C2 }$$

The unperturbed distribution f₀ is Maxwellian with anisotropic temperature, ${T}_{\perp }\ne {T}_{\parallel }$. Using cylindrical coordinates in velocity space, ${v}_{\perp }=\sqrt{{v}_{x}^{2}+{v}_{y}^{2}}$, $\theta ={\tan }^{-1}({v}_{y}/{v}_{x})$, the linearized Vlasov equation takes the form

$$\begin{eqnarray}&&\begin{array}{c}\begin{array}{l}\left(\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}t}+{\boldsymbol{v}}\cdot {\rm{\nabla }}-{\rm{\Omega }}\,\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}\theta }\right)\delta f\\ =\,\displaystyle \frac{q}{m}\left({\rm{\nabla }}\phi -{\rm{\nabla }}\times {{\rm{\nabla }}}^{-2}{\boldsymbol{ \mathcal B }}-{\boldsymbol{v}}\times {\boldsymbol{ \mathcal B }}\right)\cdot \displaystyle \frac{{\rm{\partial }}{f}_{0}}{{\rm{\partial }}{\boldsymbol{v}}},\,\\ {\rm{where}}\,\,\,{\rm{\Omega }}=\displaystyle \frac{{{qB}}_{0}}{m}.\end{array}\end{array}\end{eqnarray} \tag{ C3 }$$

Next, we apply the Fourier transformation in time, $\partial /\partial t\to -i\omega$, and in space, ${\rm{\nabla }}\to i{\boldsymbol{k}}$, with ${\boldsymbol{k}}={k}_{y}{{\boldsymbol{e}}}_{y}+{k}_{z}{{\boldsymbol{e}}}_{z}$, yielding

$$\begin{eqnarray}\begin{array}{rcl} & & -i\left(\omega -{k}_{z}{v}_{z}-i{\rm{\Omega }}\displaystyle \frac{\partial }{\partial \theta }\right)\left[\delta f\,\exp \left(-i\,\displaystyle \frac{{k}_{y}{v}_{\perp }}{{\rm{\Omega }}}\,\sin \theta \right)\right]\\ & & =\displaystyle \frac{q}{m}\,{f}_{0}\,\left[{v}_{\perp }\left({u}_{x}\sin \theta +{u}_{y}\cos \theta \right)+{u}_{z}{v}_{z}\right]\,\exp \left(-i\,\displaystyle \frac{{k}_{y}{v}_{\perp }}{{\rm{\Omega }}}\,\sin \theta \right),\end{array}\end{eqnarray} \tag{ C4 }$$

where

$$\begin{eqnarray}&&\,{u}_{x}\left(\omega ,{k}_{y},{k}_{z},{v}_{z}\right)=\displaystyle \frac{{B}_{z}}{{k}_{y}}\left[\displaystyle \frac{\omega }{{v}_{{T}_{\perp }}^{2}}+{v}_{z}{k}_{z}\left(\displaystyle \frac{1}{{v}_{{T}_{\parallel }}^{2}}-\displaystyle \frac{1}{{v}_{{T}_{\perp }}^{2}}\right)\right],\end{eqnarray} \tag{ C5 }$$

$$\begin{eqnarray}\begin{array}{rcl}{u}_{y}\left(\omega ,{k}_{y},{k}_{z},{v}_{z}\right) & = & \left({{ik}}_{y}\phi +\displaystyle \frac{\omega {k}_{z}{B}_{x}}{{k}_{y}^{2}+{k}_{z}^{2}}\right)\displaystyle \frac{1}{{v}_{{T}_{\perp }}^{2}}\\ & & +{v}_{z}{B}_{x}\left(\displaystyle \frac{1}{{v}_{{T}_{\parallel }}^{2}}-\displaystyle \frac{1}{{v}_{{T}_{\perp }}^{2}}\right),\,\end{array}\end{eqnarray} \tag{ C6 }$$

$$\begin{eqnarray}&&{u}_{z}\left(\omega ,{k}_{y},{k}_{z}\right)=\left({{ik}}_{z}\phi -\displaystyle \frac{\omega {k}_{y}{B}_{x}}{{k}_{y}^{2}+{k}_{z}^{2}}\right)\displaystyle \frac{1}{{v}_{{T}_{\parallel }}^{2}}.\,\end{eqnarray} \tag{ C7 }$$

Noting that the above exponential function is the generating function of Bessel functions, $\exp (i\zeta \sin \theta )={\sum }_{l=-\infty }^{\infty }{J}_{l}(\zeta )\,{e}^{{il}\theta }$ and expanding the left-hand side of Equation (C3) into cylindrical harmonics, we obtain

$$\begin{eqnarray}\begin{array}{rcl}\delta f & = & i\,\displaystyle \frac{q}{m}\,{f}_{0}\,\displaystyle \sum _{s-\infty }^{\infty }{J}_{s}\left(\displaystyle \frac{{k}_{y}{v}_{\perp }}{{\rm{\Omega }}}\right){e}^{{is}\theta }\\ & & \times \,\displaystyle \sum _{l=-\infty }^{\infty }\displaystyle \frac{{e}^{-{il}\theta }}{\omega -{k}_{z}{v}_{z}-l\theta }\,\\ & & \times \,\left(i{\rm{\Omega }}{u}_{x}\,\displaystyle \frac{d}{{{dk}}_{y}}+\displaystyle \frac{l{\rm{\Omega }}{u}_{y}}{{k}_{y}}+{u}_{z}{v}_{z}\right)\,{J}_{l}\left(\displaystyle \frac{{k}_{y}{v}_{\perp }}{{\rm{\Omega }}}\right),\end{array}\end{eqnarray} \tag{ C8 }$$

where we used the recurrence relations for Bessel functions. As the unperturbed distribution function f₀ does not depend on the angle θ, the above δf is easily integrated in θ, yielding the selection rule ${\int }_{0}^{2\pi }d\theta \,{e}^{i(s-l)\theta }=2\pi {\delta }_{s,l}$.

We also note that the functions u_x, u_y, u_z do not depend on ${v}_{\perp }$, which permits an easy integration of the distribution function in ${v}_{\perp }$. In the calculation of the density perturbation, we will encounter the following integral:

$$\begin{eqnarray}&&\begin{array}{c}{{ \mathcal J }}_{l}\left({k}_{y},{v}_{{T}_{\perp }}\right)\equiv {\int }_{0}^{\infty }{v}_{\perp }{{dv}}_{\perp }\exp \left(-\displaystyle \frac{{v}_{\perp }^{2}}{2{v}_{{T}_{\perp }}^{2}}\right)\,{J}_{l}^{2}\left(\displaystyle \frac{{k}_{y}{v}_{\perp }}{{\rm{\Omega }}}\right)\\ \,=\,{v}_{{T}_{\perp }}^{2}\,\exp \left(-\displaystyle \frac{{k}_{y}^{2}{v}_{{T}_{\perp }}^{2}}{{{\rm{\Omega }}}^{2}}\right)\,{I}_{l}\left(\displaystyle \frac{{k}_{y}^{2}{v}_{{T}_{\perp }}^{2}}{{{\rm{\Omega }}}^{2}}\right),\end{array}\end{eqnarray} \tag{ C9 }$$

and its partial derivative $\partial {{ \mathcal J }}_{l}({k}_{y},{v}_{{T}_{\perp }})/\partial {k}_{y}$. Likewise, in the calculation of the perturbation of the perpendicular pressure ${p}_{\perp }$, we will encounter the partial derivative $\partial {{ \mathcal J }}_{l}({k}_{y},{v}_{{T}_{\perp }})/\partial ({v}_{{T}_{\perp }}^{-2})$.

We perform the calculations with accuracy to first order in the small but finite Larmor radius. Then, in the infinite sum, we keep only the terms l = 0 and l = ±1, and we expand the Bessel functions to first order in ${k}_{y}^{2}{v}_{{T}_{\perp }}^{2}/{{\rm{\Omega }}}^{2}$. The integration in v_z is performed by expanding the functions ${(\omega -{k}_{z}{v}_{z})}^{-1}$ and ${[{(\omega -{k}_{z}{v}_{z})}^{2}-{{\rm{\Omega }}}^{2}]}^{-1}$ in two limits, $\omega \gg {k}_{z}{v}_{z}$ and $\omega \ll {k}_{z}{v}_{z}$, and keeping the terms up to ${k}_{z}^{2}{v}_{z}^{2}/{\omega }^{2}$ and ${\omega }^{2}/{k}_{z}^{2}{v}_{z}^{2}$. Calculations are further simplified under drift scaling, Equation (11), and for a weak dependence along magnetic field lines, Equation (12). Under the above conditions and after some lengthy calculations, we find that within the adopted accuracy, the parallel functional ${\gamma }_{\parallel }$ reduces to a constant, viz. ${\gamma }_{\parallel }=3$, when the characteristic parallel velocity of propagation u_z is bigger than the parallel thermal velocity ${v}_{{T}_{\parallel }}$, and the process can be considered adiabatic, and to ${\gamma }_{\parallel }=1$ when ${u}_{z}\ll {v}_{{T}_{\parallel }}$, i.e., the process is isothermal. Likewise, the perpendicular functional ${\gamma }_{\perp }$ reduces to ${\gamma }_{\perp }=1$ for arbitrary ratios ${u}_{z}/{v}_{{T}_{\parallel }}$ if the characteristic perpendicular size of the solution is much bigger than the Larmor radius. Conversely, for solutions whose transverse scale approaches ion scales, ${\gamma }_{\perp }$ can be approximated by a constant only in a limited number of cases, for which vortex solutions are found in Section 3. These are the large-β shear Alfvén solution with ${u}_{z}\gg {v}_{{\mathrm{Ti}}_{\parallel }}$ whose parallel electric field is zero ${E}_{\parallel }=0$, and the large-β kinetic slow-mode solution in the regime ${c}_{{\rm{A}}}\gg {u}_{z}\gg {v}_{{\mathrm{Ti}}_{\parallel }}$, when the coupling with acoustic perturbation and the torsion of the magnetic field are negligible. In both cases, we have ${\gamma }_{{i}_{\perp }}=2$.