Measured Reduction in Alfvén Wave Energy Propagating through Longitudinal Gradients Scaled to Match Solar Coronal Holes

The value of $\bar{\beta }$ is ≫1 in the low-field region, where the shear Alfvén waves are excited, and decreases to 1 in the high-field region (see Table 2). As a result, the excited shear Alfvén waves do not strictly match the definition of KAWs at all points in space. Hence, we use the more general term shear Alfvén waves to refer to the waves excited by the antenna.

The measured y component of the wave magnetic field is shown in Figure 5 before (a) and after (b) the v_A gradient. The data are shown for x = y = 0, f = 57.6 kHz, and ${B}_{\mathrm{hi}}=1600$ G. Figures 5(c) and (d) show the structure of the shear Alfvén wave on each side of the magnetic field gradient. Two well formed current channels are observed, with the separation between the current channels being smaller on the high-field side of the gradient, as is expected for shear Alfvén waves propagating along converging magnetic field lines.

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In order to confirm that the measured property of the shear Alfvén wave is in agreement with the expected theoretical value, we have tried to measure the parallel component of the wave magnetic field, b_∥, on the axis of LAPD before and after the gradient. According to theory (Hollweg 1999), before the gradient for the dominant ${k}_{\perp }{\rho }_{{\rm{s}}}$ of ≈0.21 the predicted value of b_∥ is ≈0.22 mG, while after the gradient for the dominant ${k}_{\perp }{\rho }_{{\rm{s}}}$ of ≈0.08 the predicted value of b_∥ is ≈0.01 mG. These values of b_∥ are below our measurement threshold of 0.5 mG. Therefore, the lack of detection of b_∥ is consistent with the theoretical prediction.

The energy of the shear Alfvén wave is obtained using the Poynting vector, S, crossing a plane perpendicular to the ambient magnetic field. This is given by (Karavaev et al. 2011)

$$\begin{eqnarray}&&S=\displaystyle \frac{1}{{\mu }_{0}}{b}^{2}{v}_{{\rm{g}},\parallel }=\displaystyle \frac{1}{{\mu }_{0}}{b}^{2}{v}_{\mathrm{ph},\parallel }.\end{eqnarray} \tag{ 6 }$$

S is the energy flux. In Equation (6), ${v}_{{\rm{g}},\parallel }$ is considered to be equal to ${v}_{\mathrm{ph},\parallel }$ because the experiments are limited to $\omega \leqslant 0.5{\omega }_{\mathrm{ci}},$ where the dispersion relation for shear Alfvén waves is nearly linear. Hence, the total wave energy, ${ \mathcal E }$, passing through the cross section of LAPD perpendicular to the ambient magnetic field can be expressed as

$$\begin{eqnarray}&&{ \mathcal E }=\int \left(\iint S\,{dx}\,{dy}\right){dt}=\displaystyle \frac{{v}_{\mathrm{ph},\parallel }}{4\pi }\int \left(\iint {b}^{2}\,{dx}\,{dy}\right){dt}.\end{eqnarray} \tag{ 7 }$$

The spatial integration is carried out over the cross section of LAPD and the integration in time is carried out over the duration of the wave train, examples of which are shown in Figure 5. The wave power, ${\Gamma }$, is related to the total wave energy by the relation

$$\begin{eqnarray}&&{\Gamma }=\displaystyle \frac{{E}}{{t}_{{\rm{d}}{\rm{u}}{\rm{r}}}},\end{eqnarray} \tag{ 8 }$$

where ${t}_{\mathrm{dur}}$ is the duration of the wave train. The value of ${v}_{\mathrm{ph},\parallel }$ used in Equation (7) was determined by simultaneously measuring the wave magnetic field using two axially separated B-dot probes. The probes were carefully aligned to ensure that both intersected the same axial magnetic field line. To calculate ${v}_{\mathrm{ph},\parallel }$, the axial distance between the probes was divided by the time lag between the phases of the wave magnetic field. The time lag was determined by a cross-correlation analysis of the time variation of the data acquired by the two probes. The probes located at z₁ and z₂ were used to measure ${v}_{\mathrm{ph},\parallel }$ on the low-field side, while the probes located at z₃ and z₄ were used to measure ${v}_{\mathrm{ph},\parallel }$ on the high-field side.

The reduction in power of the wave propagating through the gradient was measured using the ratio of the transmitted wave power ${{\Gamma }}_{3}$ at z₃ to the incident wave power ${{\Gamma }}_{2}$ at z₂. Since ${\Gamma }$ is related to ${ \mathcal E }$ by a constant factor, the reduction in wave power is equal to the decrease in wave energy, i.e., ${{\Gamma }}_{3}/{{\Gamma }}_{2}={{E}}_{3}/{{E}}_{2}$. The dependence of ${{\Gamma }}_{3}/{{\Gamma }}_{2}$ on λ/L_A was studied by varying λ/L_A in two ways. In the first set of experiments, L_A was changed while holding λ constant. In the second set, λ was varied and L_A was kept constant.

For the first set of experiments, the values of L_A were varied by increasing B_hi from 500 G to 800, 1000, 1200, and 1600 G. The increase in the value of B_hi enhances the steepness of the gradient in v_A. This, in turn, decreases L_A.

The variation of ${{\Gamma }}_{3}/{{\Gamma }}_{2}$ with λ/L_A while varying L_A is shown in Figure 6. Waves of frequency f = 57.6 kHz were excited to keep λ fixed. For the nearly homogeneous case of $\lambda /{L}_{{\rm{A}}}\approx 0.49$, we find ${{\Gamma }}_{3}/{{\Gamma }}_{2}\approx 0.52$. For a large non-uniformity of $\lambda /{L}_{{\rm{A}}}\approx 6.3$, we find ${{\Gamma }}_{3}/{{\Gamma }}_{2}\approx 0.08$. These results show that the wave power propagating through the gradient decreases as the steepness of the gradient increases.

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In the second set of measurements, λ was varied by changing f from 57.6 to 96 kHz in steps of 9.6 kHz. We confined f to this range so that the shear Alfvén waves followed the linear dispersion relation to a good approximation. L_A was kept fixed by setting B_hi = 1000 G. Figure 7 shows that ${{\Gamma }}_{3}/{{\Gamma }}_{2}$ decreases with increasing λ/L_A while varying λ. For example, ${{\Gamma }}_{3}/{{\Gamma }}_{2}\approx 0.40$ for $\lambda /{L}_{{\rm{A}}}\approx 2.38$, while ${{\Gamma }}_{3}/{{\Gamma }}_{2}\approx 0.26$ for $\lambda /{L}_{{\rm{A}}}\approx 4.17$. This shows that a wave with a longer wavelength loses more energy than one with a shorter wavelength while propagating through a constant gradient.

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The decrease in ${{\Gamma }}_{3}/{{\Gamma }}_{2}$ with increasing λ/L_A presented in Figure 7 shows the same quantitative behavior as seen in Figure 6. This confirms that it is neither λ nor L_A but rather λ/L_A that is the independent parameter describing the effect of inhomogeneity on the shear Alfvén waves.

The variation of ${{\Gamma }}_{3}/{{\Gamma }}_{2}$ for all measured values of λ/L_A is shown in Figure 8 and given in Table 3. Here, λ/L_A was varied by changing f from 57.6 to 96 kHz for each of the five values of B_hi given in Figure 2. The data in Figure 8 include all of the data plotted in Figures 6 and 7, as well as the additional values listed in Table 3 but not included in Figures 6 and 7.

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The ${B}_{\mathrm{hi}}=500$ G data points in Figure 8 correspond to the case of a flat magnetic field, where the inhomogeneity in v_A is small with $\lambda /{L}_{{\rm{A}}}\lt 0.5$. ${{\Gamma }}_{3}/{{\Gamma }}_{2}\approx 0.54$ for the three lower frequency measurements and ≈0.41 for the two higher frequencies. The vertical error bars for all five frequencies nearly overlap. We attribute these minor differences to damping mechanisms that are most readily observable in a uniform plasma. No such similar differences versus frequency were seen in our gradient-driven results, which show an almost monotonic reduction in ${{\Gamma }}_{3}/{{\Gamma }}_{2}$ with increasing λ/L_A. For the gradient cases, the observed energy reduction relative to the non-gradient cases is substantial, with a decrease by a factor of ≈5. Moreover, the monotonic nature of the decrease strongly suggests that the energy reduction is due to a gradient-driven effect.

The two-dimensional structure of the wave magnetic field in Figure 5 suggests that the ring antenna was in effect was driving two counterpropagating current channels along the axial magnetic field lines of LAPD. Field-aligned time-varying currents with frequencies below ${f}_{\mathrm{ci}}$ are known to radiate shear Alfvén waves (Morales et al. 1994; Gekelman et al. 1994; Morales & Maggs 1997). Hence, we have modeled the antenna as two current sources driving field-aligned currents that are 180° out of phase with one another.

More specifically, the ring antenna located at x = y = z = 0 is modeled as two disks separated by a distance equal to the diameter of the ring, which is ≈9 cm. The current density across the surface of each disk is assumed to have a Gaussian profile, ${j}_{0}\exp [-{r}^{2}/{a}^{2}]$, where j₀ is the amplitude of the surface current density, r is radial the distance from the center of the disk, and a is a measure of the width of the current source. The wave magnetic field due to each current source lies in the azimuthal plane. This azimuthal wave magnetic field, b_ϕ, due to each disk is given by (Morales & Maggs 1997; Vincena 1999)

$$\begin{eqnarray}&&{b}_{\phi }=\displaystyle \frac{2{j}_{0}\pi {a}^{2}}{{\rm{c}}}{\int }_{0}^{\infty }\exp \left[-\displaystyle \frac{{a}^{2}{k}_{\perp }^{2}}{4}\right]{J}_{1}\left({k}_{\perp }r\right)\exp \left[{{ik}}_{\parallel }\left({k}_{\perp }\right)z\right]{{dk}}_{\perp }.\end{eqnarray} \tag{ 9 }$$

Here, z is the axial distance from the antenna at which the wave magnetic field is calculated, $i=\sqrt{-1}$, J₁ is the Bessel function of the first kind of order one, and ${k}_{\parallel }={k}_{{\rm{r}},\parallel }+{{ik}}_{{\rm{i}},\parallel }$ is a complex quantity where the real and imaginary parts are inversely proportional to the wavelength and damping length, respectively. Note that ${k}_{\parallel }$ is a complex quantity only in the formulas mentioned here in Section 5.1. In all other sections and subsections in this paper, k_∥ is a real quantity as defined in Section 2.

In order to simplify the calculation, we have normalized b_ϕ by the constant factor $2{j}_{0}\pi {a}^{2}/c$ to obtain

$$\begin{eqnarray}&&{b}_{\phi ,n}=\displaystyle \frac{{b}_{\phi }c}{2{j}_{0}\pi {a}^{2}}.\end{eqnarray} \tag{ 10 }$$

The sole purpose of developing this model is to determine the damping length of the wave energy, which depends on the relative decrease in wave energy versus the distance from the antenna. Our results for damping length are not affected by this normalization.

Landau damping is described by the warm plasma collisionless dispersion relation of shear Alfvén waves derived from the linearized Vlasov equation and Maxwell's equations (Swanson 1989). This relation is given by (Stasiewicz et al. 2000; Lysak 2008; Thuecks et al. 2009)

$$\begin{eqnarray}&&Z^{\prime} \left(\xi \right)\left[\displaystyle \frac{{v}_{{\rm{A}}}^{2}}{2{v}_{\mathrm{te}}^{2}}\displaystyle \frac{\left(1-{\bar{\omega }}^{2}\right){\mu }_{{\rm{i}}}}{1-{{\rm{\Gamma }}}_{0}({\mu }_{{\rm{i}}})}-{\xi }^{2}\right]={k}_{\perp }^{2}{\delta }^{2},\end{eqnarray} \tag{ 11 }$$

where $\xi =\omega /\left(\sqrt{2}{k}_{\parallel }{v}_{\mathrm{te}}\right)$, ${Z}^{{\prime} }\left(\xi \right)=-2\{1+\xi Z\left(\xi \right)\}$ is the derivative of the plasma dispersion function, Z, (Fried & Conte 1961) with respect to ξ, ${\mu }_{{\rm{i}}}={k}_{\perp }^{2}{\rho }_{{\rm{i}}}^{2}$, ${\rho }_{{\rm{i}}}={m}_{{\rm{i}}}{v}_{\mathrm{ti}}/{{qB}}_{0}$ is the ion gyroradius, ${{\rm{\Gamma }}}_{0}\left({\mu }_{{\rm{i}}}\right)={e}^{-{\mu }_{{\rm{i}}}}{I}_{0}({\mu }_{{\rm{i}}})$, and I₀ is the modified Bessel function of order zero.

Collisional damping is modeled in the wave dispersion by including the Krook collision operator in the linearized Vlasov equation (Gross 1951; Swanson 1989). The resulting dispersion relation is given by (Gekelman et al. 1997; Thuecks et al. 2009)

$$\begin{eqnarray}&&Z^{\prime} \left(\eta \right)\left(1+i\displaystyle \frac{{\nu }_{{\rm{e}}}}{\omega }\right)\left[\displaystyle \frac{{v}_{{\rm{A}}}^{2}}{2{v}_{\mathrm{te}}^{2}}\displaystyle \frac{\left(1-{\bar{\omega }}^{2}\right){\mu }_{{\rm{i}}}}{1-{{\rm{\Gamma }}}_{0}({\mu }_{{\rm{i}}})}-{\xi }^{2}\right]={k}_{\perp }^{2}{\delta }^{2},\end{eqnarray} \tag{ 12 }$$

where $\eta =\xi \left(1+i{\nu }_{{\rm{e}}}/\omega \right)$ and ${\nu }_{{\rm{e}}}$ is the collision frequency for electrons. The collision frequency is ${\nu }_{{\rm{e}}}={\nu }_{\mathrm{ei}}+{\nu }_{\mathrm{en}}$, where ν_ei and ${\nu }_{\mathrm{en}}$ are the electron–ion and electron–neutral collision frequencies, respectively. From this dispersion relation we determine k_∥ as a function of k_⊥, which we then substitute into Equation (10) to include the effect of Landau and collisional damping in the model.

The electron–ion collision frequency is calculated in units of hertz using the expression (Braginskii 1965)

$$\begin{eqnarray}&&{\nu }_{\mathrm{ei}}=2.9\times {10}^{-6}\displaystyle \frac{{Z}_{\mathrm{ch}}n\mathrm{ln}\ {\rm{\Lambda }}}{{T}_{{\rm{e}}}^{3/2}},\end{eqnarray} \tag{ 13 }$$

where n is in cm⁻³ and T_e is in eV. The electron–neutral collision frequency is determined using the formula given by Baille et al. (1981). For our experimental parameters of $n\,=2.6\times {10}^{12}\,{\mathrm{cm}}^{-3}$, neutral pressure of 10⁻⁴ Torr, and ${T}_{{\rm{e}}}=5\,\mathrm{eV}$ in a He⁺ plasma, ${\nu }_{{\rm{ei}}}\approx 7.5\times {10}^{6}$ Hz and ${\nu }_{{\rm{en}}}\,\approx 4\times {10}^{5}$ Hz.

Equation (12) gives the dispersion relation of shear Alfvén waves in the presence of Landau and collisional damping. In order to determine the damping due to collisions only, we have used the collisional dispersion relation of shear Alfvén waves derived using the two-fluid theory in the $\bar{\beta }\gg 1$ limit (Vranjes et al. 2006; Gigliotti et al. 2009),

$$\begin{eqnarray}&&{\omega }^{2}-{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2}\left(1-{\bar{\omega }}^{2}+{k}_{\perp }^{2}{\rho }_{{\rm{s}}}^{2}\right)+i\omega {k}_{\perp }^{2}{\delta }^{2}{\nu }_{{\rm{e}}}=0.\end{eqnarray} \tag{ 14 }$$

This two-fluid dispersion relation can be solved algebraically, where ${k}_{{\rm{r}},\parallel }$ and ${k}_{{\rm{i}},\parallel }$ are given by

$$\begin{eqnarray}&&{k}_{{\rm{r}},\parallel }=\displaystyle \frac{1}{\sqrt{2}}\displaystyle \frac{\omega }{{v}_{{\rm{A}}}}{\left[\displaystyle \frac{\sqrt{1+{k}_{\perp }^{4}{\delta }^{4}{\left({\nu }_{{\rm{e}}}/\omega \right)}^{2}}+1}{1-{\bar{\omega }}^{2}+{k}_{\perp }^{2}{\rho }_{{\rm{s}}}^{2}}\right]}^{1/2},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{k}_{{\rm{i}},\parallel }=\displaystyle \frac{1}{\sqrt{2}}\displaystyle \frac{\omega }{{v}_{{\rm{A}}}}{\left[\displaystyle \frac{\sqrt{1+{k}_{\perp }^{4}{\delta }^{4}{\left({\nu }_{{\rm{e}}}/\omega \right)}^{2}}-1}{1-{\bar{\omega }}^{2}+{k}_{\perp }^{2}{\rho }_{{\rm{s}}}^{2}}\right]}^{1/2}.\end{eqnarray} \tag{ 16 }$$

Using Equations (15) and (16), we have determined k_∥ as a function of k_⊥, and substituted ${k}_{\parallel }({k}_{\perp })$ in Equation (10) to include the effect of only collisional damping in the model.

In order to compare the structure of the experimentally measured wave magnetic field with that predicted by the model we first calculated b_ϕ in a cylindrical coordinate system due to each current source using Equation (10). The value of b_ϕ for each source is then converted from the cylindrical coordinates to Cartesian coordinates as ${b}_{\phi }={b}_{x}\hat{x}+{b}_{y}\hat{y}$. The total wave magnetic field produced by the ring antenna can then be modeled by a linear superposition of ${b}_{x}\hat{x}+{b}_{y}\hat{y}$ produced by the two disk sources. We refer to this total wave magnetic field as b_⊥.

The experimentally measured b_⊥ is compared in Figure 9 to the field calculated at z₂ for ${B}_{0}=500\,{\rm{G}}$ and f = 57.6 kHz. The measured and calculated b_⊥, normalized by their maximum values, are denoted as ${b}_{\perp ,{\rm{n}}}$. In the antenna model we set a = 0.25 cm, which is the thickness of the ring antenna. The average values of n and T_e given in Table 2 were used for the numerical calculation. Figure 9 shows the wave structure predicted by the model using Equation (12). This structure is in excellent agreement with the measured wave magnetic field. The difference in the total wave energy, ${\Gamma }$, in the xy plane calculated from the experimental and numerical data is typically ∼4%.

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The damping of wave energy in the model was determined by calculating Equation (10) at multiple z locations along the LAPD axis. Figure 10 presents the calculated damping of the shear Alfvén wave shown in Figure 9. The energy decay follows an exponential curve to a good approximation. The damping length, d, is obtained by fitting a function of the form $A\exp [-z/d]$ to the numerically calculated data, where A is a constant.

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The reason that the energy damping curve of the shear Alfvén wave approximately follows an exponential curve may be understood as follows. The wave magnetic field is obtained by integrating over a number of k_⊥ as discussed by Morales & Maggs (1997). For a shear Alfvén wave of a given frequency, different values of k_⊥ have different damping lengths as shown by Gekelman et al. (1997), Kletzing et al. (2003), and Lysak (2008). The cumulative effect of these multiple k_⊥ results in the energy decay being approximately exponential.

The measured and modeled results for ${{\Gamma }}_{3}/{{\Gamma }}_{2}$ are shown in Figure 11. The errors in the numerically calculated data were determined using a Monte Carlo method, by considering the uncertainties in n and T_e given in Table 2, and are of the order of the size of the plotted symbols.

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Comparing our observed damping for the flat 500 G case to our model results, we find that the reduction in wave energy predicted by the model by considering both Landau and collisional damping is in good agreement with the experiment. The comparison shown in Figure 11 of the modeled results due to Landau and collisional damping, and to collisional damping alone, shows that collisional damping is dominant. Landau damping, while present, is very weak.

Light and other electromagnetic waves undergo reflection while propagating across a change in refractive index, corresponding to a change in the propagation velocity of the wave. Similarly, according to both theoretical studies and numerical simulations, shear Alfvén waves propagating through a strong longitudinal v_A gradient are predicted to undergo reflection (Moore et al. 1991a; Musielak et al. 1992; Perez & Chandran 2013).

According to the theory of Musielak et al. (1992), a shear Alfvén wave incident on a longitudinal v_A gradient is expected to undergo strong reflection when the frequency of the wave is less than the critical frequency f_cr given by

$$\begin{eqnarray}&&{f}_{\mathrm{cr}}=\displaystyle \frac{1}{2}\sqrt{{\left({v}_{{\rm{A}}}^{{\prime} }\right)}^{2}+| 2{v}_{{\rm{A}}}{v}_{{\rm{A}}}^{{\prime\prime} }| }.\end{eqnarray} \tag{ 17 }$$

Here, the double prime indicates the second spatial derivative. This expression was deduced for gradients in n and B₀ in one dimension.

In the wave experiments described in Section 4.2, the shear Alfvén waves passes through two gradients in v_A, labeled as I and II in Figure 2. The difference between these gradients is that v_A increases with distance in gradient I, while it decreases with distance in gradient II. In order to constrain the role of reflected waves in the observed reduction in the transmitted wave energy versus λ/L_A, we performed several measurements to measure the magnitude of any reflected waves.

The first two sets of wave-reflection experiments were carried out to search for reflection from gradient I. B-dot probes were positioned at ${z}_{1}$ and z₂ as indicated in Figure 12. Shear Alfvén waves were excited by applying a sinusoidal wave train of two cycles to the antenna. This reduced the temporal length of the wave train compared to the previously excited ten-cycle wave train. The gradient was also moved to the far end of the machine, as shown in Figure 12. This ensured that the time required for a wave to traverse the distance from the B-dot probe at z₁ to gradient I and return to z₁ was greater than twice the time period of the lowest wave frequency investigated. As a result, the incident wave and any reflected wave would be separated in time in the B-dot probe data. Lastly, the range of frequencies was selected to satisfy the criteria: (a) the wave was predicted to be strongly reflected by theory and (b) there was an overlap in the values of $\lambda /{L}_{{\rm{A}}}$ with the inhomogeneity observed in coronal holes of $\lambda /{L}_{{\rm{A}}}\gtrsim 4.5$.

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The magnetic field profile used in the first set of wave-reflection experiments is shown by the solid curve in Figure 12. This curve was obtained by increasing B_hi to ≈1382 G. The value of f_cr varied axially with the magnetic field variation and reached a maximum value of ≈724 kHz at z = 12.14 m within gradient I. In order to satisfy the theoretical criteria for strong reflection we excited wave frequencies below 724 kHz.

Figure 13 shows the y component of the wave magnetic field detected for f = 65 kHz (i.e., $f/{f}_{\mathrm{cr}}=0.09$). A well formed two-cycle incident wave was detected by probes at z₁ and z₂ between ≈112 and 148 $\mu {\rm{s}}$. For this wave $\lambda /{L}_{{\rm{A}}}\approx 5.6$. If the shear Alfvén wave were strongly reflected by gradient I, then the reflected wave would reach z₁ at 39 μs after the incident wave has passed it. Hence, a reflected wave is predicted to be observed in Figure 13 between ≈151 and 187 μs. However, we do not observe any reflected wave in this time window. The wave signal at z₁ always leads the signal at z₂, implying that the B-dot probes did not detect any waves reflected by gradient I. We also find no detectable reflected wave in the b_x and b_z directions

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There are some small-amplitude fluctuations trailing the applied two-cycle wave train, but these features have a frequency twice that of the applied waveform. We believe that these are excited by the second harmonic present in the antenna signal and are unrelated to reflection from gradient I.

We have also considered the possibility that waves did not reflect exactly along the axis, but found no evidence for reflected waves at any location in the LAPD cross section. The B-dot probes were scanned through a cross section in LAPD and the results were always similar to those shown in Figure 13.

We repeated all of these measurements while increasing f to 190 kHz in steps of 5 kHz. This caused $f/{f}_{\mathrm{cr}}$ to increase to 0.3. As before, we did not observe a detectable reflected wave at any of the measured frequencies.

We then carried out a second set of experiments to study the possible effects of tunneling through the high v_A region. For the above wave-reflection studies, λ was greater than the width, w, of the high-field region. For example, $\lambda /w\approx 2.1$ for $f/{f}_{\mathrm{cr}}=0.1$. To rule out the possibility that the shear Alfvén wave could be tunneling through the high v_A region, instead of undergoing reflection, we moved gradient I closer to the antenna, as represented by the dashed line in Figure 12. This resulted in $\lambda /w\approx 1.1$. We then repeated the wave-reflection experiments described above and again did not observe any detectable reflected waves.

The lack of an observable reflected wave from gradient I may imply that the amplitude of the reflected wave is too weak to be detected. For example, using the measured initial wave amplitude at z₁ and taking into account Landau and collisional damping as the wave propagates from z₁ to gradient I and back, we estimate that if there were 100% reflection at gradient I then we would measure a reflected wave signal at z₁ with an amplitude of 11 mG. This is much larger than the ≈1.5 mG fluctuations in Figure 13 trailing the applied two-cycle wave train and should be readily observable. The lack of an observed reflected signal indicates that the efficiency of any reflection by the gradient is much less than 100%. Taking 3 mG as a reasonable detectable level over the 1.5 mG fluctuations trailing the applied wave train, and taking into account Landau and collisional damping between z₁ and gradient I and back to z₁, we can put an upper limit on the reflected wave energy of ≈7.4% for λ/L_A ≈ 5.6 and f/f_cr = 0.09. This reflectance is too small to account for the observed loss of wave energy.

Finally, in the third set of wave-reflection experiments, we investigated the effects of gradient II. For this we set ${B}_{\mathrm{lo}}={B}_{\mathrm{hi}}=500$ G, the flat-field case shown in Figure 2. Here, f_cr had a maximum value of ≈364 kHz within gradient II at z = 13 m. As before, shear Alfvén waves were excited by applying a sinusoidal wave train comprising two cycles to the antenna located at z = 0. The corresponding variation of ${b}_{y}$ at z₁ and z₂ for f = 65 kHz is shown in Figure 14. Similar to the results from the first two sets of wave-reflection measurements, the phase of the wave signal at z₁ always leads the wave signal at z₂, indicating that the B-dot probes did not detect any reflection from the gradient.

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The energy of a shear Alfvén wave traveling through a v_A gradient may decrease if a part of the wave energy is converted into another mode. Inhomogeneity in the magnetic field can enable the propagation of compressible surface magneto-acoustic waves and incompressible surface Alfvén waves (Roberts 1981). A gradient in the magnetic field may convert some of the shear Alfvén wave energy to a slow wave (Southwood & Saunders 1985). A fast wave may get excited. All of these modes induce a parallel perturbation, which we have tried to detect using B-dot probes. The ratio ${b}_{\parallel }/{b}_{\perp }$ was measured before, within, and after the gradient. Mode conversion into these modes would produce an amplification of b_∥, but we did not detect any b_∥ above the noise level of ≈0.5 mG. This implies that mode conversion is unlikely.

In order to further confirm that the only mode propagating in the gradient is a shear Alfvén wave, we have measured the wave magnetic fields before, within, and after the gradient. Figures 15(a)–(c) show the structure of the wave before, within, and after the magnetic field gradient, respectively. Two well formed current channels are observed, with the separation between the current channels decreasing in the gradient and on the high-field side, as is expected for shear Alfvén waves propagating along converging magnetic field lines. Considering that we have not detected any wave other than the shear Alfvén wave in the gradient, and that the structure of the shear Alfvén wave within the gradient is consistent with that before and after the gradient, energy reduction due to mode conversion is unlikely.

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Large-amplitude shear Alfvén waves can lose energy due to nonlinear effects. The experiments reported in this paper were carried out using very low-amplitude waves of $b/{B}_{0}\,\lesssim 8\times {10}^{-5}$. Nonlinear effects associated with shear Alfvén waves, such as parametric instability, have been found to occur only for relatively large-amplitude waves. For example, Dorfman & Carter (2016) reported the threshold for observation of parametric instability of shear Alfvén waves to be $b/{B}_{0}\geqslant 2\times {10}^{-3}$. This is over 10 times greater than our wave amplitude. Low-amplitude shear Alfvén waves also exhibit nonlinear effects in a narrow band of frequencies around the ion cyclotron frequency due to ion cyclotron resonance. As the range of wave frequencies excited here is $\leqslant {f}_{\mathrm{ci}}/2$, nonlinear effects due to ion cyclotron resonance are expected to be absent.

A shear Alfvén wave may exert a mirror force on the electrons and ions, and contribute to additional damping of the wave. This damping mechanism is called transit-time damping. For uniform plasmas with ${v}_{\mathrm{ti}}^{2}\ll {v}_{{\rm{A}}}^{2}$ and ${T}_{{\rm{i}}}\lt {T}_{{\rm{e}}}$, the mirror force experienced by an electron is greater than that for an ion, and is given by $| {F}_{\mathrm{Me}}| \sim \left({m}_{{\rm{e}}}{v}_{\mathrm{te}}^{2}/2{B}_{0}\right){k}_{\parallel }{b}_{\parallel }$ (Hollweg 1999). In order to estimate the relative importance of transit-time damping with respect to collisional damping, we have compared $| {F}_{\mathrm{Me}}|$ with the frictional force experienced by an electron, $| {F}_{\mathrm{fric},{\rm{e}}}| \sim {m}_{{\rm{e}}}{v}_{\mathrm{te}}{\nu }_{{\rm{e}}}$ (Swanson 1989). For our experimental parameters of $n=2.8\times {10}^{12}\,{\mathrm{cm}}^{-3}$, neutral pressure of 10⁻⁴ Torr, ${T}_{{\rm{e}}}=4.9\,\mathrm{eV}$, maximum value for b_∥ of $0.5\,\mathrm{mG}$, B₀ = 500 G, and ${k}_{\parallel }=0.014$ m, the estimated $| {F}_{{\rm{M}}{\rm{e}}}| /| {F}_{{\rm{f}}{\rm{r}}{\rm{i}}{\rm{c}},{\rm{e}}}|$ is $\sim 8\times {10}^{-8}$. This is extremely small, mainly because ${b}_{\parallel }/{B}_{0}\lt {10}^{-6}$ in the gradient. Hence, the transit-time damping due to the mirror force is inconsequential when compared to collisional damping and cannot account for the reduction in wave energy.