Experimental study of Alfvén wave reflection from an Alfvén-speed gradient relevant to the solar coronal holes

Figure 2 shows the wave magnetic field time series recorded by B-dot probes at $z_{1}$ and $z_{2}$ for gradient U. Initially, the peaks and troughs of the wave signal at $z_{1}$ leads $z_{2}$, which is consistent with an incident wave propagating in the $+z$ direction. Some time after the incident wave in Fig. 2, we see a much smaller wave in the region between the dotted lines. The signal of this smaller wave is observed at $z_{2}$ before $z_{1}$, indicating the waveform is propagating in the $-z$ direction, towards the antenna.

Our analysis suggests that the smaller wave signal is reflected from the plasma source at the far end of LAPD. Furthermore, reflection introduces an additional $180^{\circ}$ phase difference between the reflected and incident wave; the smaller signal in Fig. 2 has three peaks and two troughs, while in the incident wave has two peaks and three troughs. We performed time-of-flight calculations using the logic that if the smaller wave at $z_{1}$ is reflected from the plasma source, then the time lag between the incident and the inverted smaller wave should be equal to twice the time of flight of the incident wave from $z_{1}$ to the plasma source. The time lag ($t_{\rm{lag}}^{\rm{U}}$) between the incident and the smaller waves obtained from their phase difference at $z_{1}$ is $95.3\pm 0.4\>~{}\rm{\mu s}$. The error bar represent the $1\sigma$ uncertainty obtained by calculating the standard deviation of the time lag measured at various radial locations. The time of flight ($t_{\rm{tof}}^{\rm{U}}$) of the incident wave from $z_{1}$ to the plasma source determined by dividing the distance between $z_{1}$ and the cathode, 14.31 m, by the velocity of the wave. The velocity of the wave is $3\pm 0.1\times 10^{5}~{}\rm{m~{}s^{-1}}$, as determined by dividing the distance between $z_{1}$ and $z_{2}$ by the time lag between the wave signals at those two locations. Therefore, $t_{\rm{tof}}^{\rm{U}}$ is $47.7\pm 1.6~{}\mu\rm{s}$. Hence, $t_{\rm{lag}}^{\rm{U}}$ agrees with $2t_{\rm{tof}}^{\rm{U}}$, indicating that the smaller wave has been reflected from the plasma source. Our findings agree with previous experiments where physical objects in the plasma, such as metal and insulator plates, were found to reflect Alfvén waves (Leneman, 2007).

We introduced a strong $v_{\rm{A}}$ gradient between the antenna and the far end of LAPD and detected a wave reflected from the $v_{\rm{A}}$ gradient. The minimum value of the $v_{\rm{A}}$ length scale within the gradient was $L_{\rm{A,min}}=0.74~{}\rm{m}$. The wave magnetic field data before the gradient is given in Fig. 3(a). Initially, the peaks and troughs of the wave signal at $z_{1}$ leads $z_{2}$, which is expected for an incident wave. However, in the region between the dotted lines, a small wave signal is observed at $z_{2}$ before $z_{1}$, suggesting the wave is propagating in the $-z$ direction. This is likely due to a partial superposition of waves reflected from gradient I and the plasma source, as we explain shortly. Here we remind the reader that since $v_{\rm A}\propto B_{0}$, the time window for the wave reflected by the plasma source at $z_{1}$ and $z_{2}$ for the strong $v_{\rm A}$ gradient and uniform $B_{0}$ profile will be different as the values of $B_{0}$ near the plasma source is different in the two cases.

In order to help disentangle wave reflection from the gradient from wave reflection by the plasma source, we also measured the wave magnetic field on the high field side at $z_{3}$ and $z_{4}$ (Fig. 3(b)). The first two peaks and troughs at $z_{3}$ leads $z_{4}$, which is consistent with wave propagation in the $+z$ direction. However, the third trough of $z_{3}$ lags $z_{4}$, suggesting the presence of a wave reflected from the far end in the measured signal. The amplitude of the third trough at $z_{4}$ is also more than at $z_{3}$, suggesting constructive interference between waves travelling in the $+z$ (incident) and $-z$ (reflected from far end) directions. The presence of a far-end reflected wave is also supported by the amplitude of the third peak at $z_{4}$, which is greater than $z_{3}$. This observation suggests that the far-end reflected wave may have reached $z_{4}$ around $t=68.3\pm 3.5~{}\mu\rm{s}$. The upper limit corresponds to the time of the second peak at $z_{4}$, where we observe a signature of a reflected wave from the far end. The lower limit refers to the time of the second trough at $z_{4}$ where we do not observe a sign of reflection from the far end.

Building on this, we put forward two sets of analyses that imply that the first cycle of the wave propagating in the $-z$ direction at $z_{2}$ is reflected from gradient I only and does not have any contribution due to far-end reflection. First, we estimate the time at which a wave reflected from the plasma source will reach $z_{2}$. The arrival time of the reflected wave from the source is equal to $68.3\pm 3.5~{}\mu s$ plus the time of flight of the wave from $z_{4}$ to $z_{2}$. Since the speed of Alfvén wave propagation in the $+z$ and $-z$ direction is the same, the time of flight of an Alfvén wave from $z_{4}$ to $z_{2}$ (i.e., the $-z$ direction) is determined from the phase difference of the incident wave (i.e., the $+z$ direction) between $z_{2}$ and $z_{4}$. The inferred time of flight is $21.4\pm 0.7~{}\mu\rm{s}$. Therefore, the part of the waveform at $z_{2}$ beyond $t=89.7\pm 3.6~{}\mu\rm{s}$ may contain a contribution from a wave reflected from the plasma source. Hence, the first cycle of the waveform in the inset graph of Fig 3(a) is not affected by reflection from the source.

Next, we show that the location of reflection of the first cycle of the wave propagating in the $-z$ direction at $z_{2}$ is within gradient I. Since gradient I is located between $z_{2}$ and $z_{3}$, for the first cycle to be reflected from the gradient I, the time lag between the incident and reflected wave at $z_{2}$, $t_{\rm lag}^{\rm I}$, must be less than the twice the time of flight of the wave from $z_{2}$ to $z_{3}$, $2t_{\rm tof}^{\rm I}$. We used the results of the simulations given in Section 4 to correlate the phases of incident and gradient-reflected wave. Simulations showed that reflection from a $v_{\rm{A}}$ gradient introduces a $180^{\circ}$ phase difference between the incident and the reflected wave. In addition, the simulations also suggest that the leading low amplitude trough of the incident Alfvén wave may not produce a sufficiently strong reflected wave to be experimentally detectable. The subsequent peaks and troughs of the incident Alfvén wave, though, are predicted to produce a detectable reflected wave. Therefore, in Fig. 3(a), the first peak of the incident wave should correspond to the first trough of the wave moving in the $-z$ direction if it is reflected from the $v_{\rm{A}}$ gradient. The time lag between the first peak of the incident wave and the first trough of the reflected wave is $t_{\rm{lag}}^{\rm{I}}=36.0\pm 0.2~{}\mu\rm{s}$. The time of flight of the wave from $z_{2}$ to $z_{3}$ obtained by comparing the phase difference between the $+z$ propagating part of the wave at $z_{2}$ and $z_{3}$ is $t_{\rm{tof}}^{\rm{I}}=20.4\pm 0.7~{}\mu\rm{s}$. Hence, $t_{\rm{lag}}^{\rm{I}}<2t_{\rm{tof}}^{\rm{I}}$ indicating that the source of reflection is located within gradient I. This experimental finding is supported by our simulations described in Section 4.

To further confirm experimentally that gradient I is indeed reflecting an Alfvén wave, we moved the gradient further away from the antenna and checked if the time lag between the incident and reflected wave increased accordingly. We refer to this shifted gradient as gradient II. A comparison of the wave signal at $z_{2}$ for gradients I and II is presented in Fig. 4.

We performed further analysis to quantitatively show that the phase difference between the reflected wave from gradients I and II agrees with the extra distance traversed by the wave to reach gradient II and return. The investigation was done using the first wave cycle propagating in $-z$ direction for both gradients. This is because an analysis for gradient II following the methodology in subsection 3.2.2 that was used for gradient I showed that the reflector for the first cycle is located within the gradient II region. The time lag between waves reflected from I and II is $t_{\rm{lag}}^{\rm{I,II}}\approx 2\Delta z/v_{\rm{ph},\parallel}$, where $\Delta z=1.72~{}\rm{m}$ is the distance between gradients I and II (Fig. 1), and $v_{\rm{ph},\parallel}$ is the phase velocity of the Alfvén wave parallel to $B_{0}$ in the low-field side. We find $v_{\rm{ph},\parallel}$ = $3.3\pm 0.1\times 10^{5}~{}\rm{m~{}s^{-1}}$, as determined by dividing the distance between $z_{1}$ and $z_{2}$ by the phase difference of the incident wave at $z_{1}$ and $z_{2}$. This gives, $t_{\rm{lag}}^{\rm{I,II}}\approx 10.4\pm 0.3~{}\mu\rm{s}$. The phase difference between the reflected wave from I and II is $\Delta t_{\rm{I,II}}^{\rm{ref}}\approx 11.0\pm 0.2~{}\mu\rm{s}$. In Fig. 4, we find a minor phase offset between the incident waves for cases I and II possibly due to small difference in density. This minor phase offset adds to the phase difference between the reflected waves from the two gradients. Correcting for the phase difference between incident waves, we find $\Delta t_{\rm{I,II}}^{\rm{ref}}\approx 10.0\pm 0.3~{}\mu\rm{s}$. Thus, $t_{\rm{lag}}^{\rm{I,II}}$ closely agrees with $\Delta t_{\rm{I,II}}^{\rm{ref}}$ further confirming that the gradient is reflecting Alfvén waves. This experimental finding is supported by our simulation discussed in Section 4.

As a further test to prove that the first cycle of the reflected wave in Fig. 3(a) is a gradient driven effect, we weakened the $v_{\rm{A}}$ gradient and repeated the experiment. The $L_{\rm{A,min}}$ within the gradient was $3.24~{}\rm{m}$. The wave magnetic field data at $z_{1}$ and $z_{2}$ for III is shown in Fig. 5. Initially, the troughs and peaks of the wave signal at $z_{1}$ leads $z_{2}$, which is consistent for a wave propagating in the $+z$ direction. However, some time after the incident wave in Fig. 5, we see a much smaller wave in the region between the dotted lines. This small wave is propagating in the $-z$ direction, as the peaks and troughs of the small wave at $z_{2}$ leads $z_{1}$.

Analysis shows that the small wave is reflected from the plasma source of LAPD and that there is no signature of the wave being reflected from the gradient. For the small wave to be reflected from the source, the time lag ($t_{\rm{lag}}^{\rm{III}}$) between the incident and small wave at $z_{1}$ should be equal to twice the time of flight ($t_{\rm{tof}}^{\rm{III}}$) of the wave from the $z_{1}$ to the plasma source. The $t_{\rm{lag}}^{\rm{III}}$ measured by cross-correlating the incident and small wave after accounting for $180^{\circ}$ phase shift caused by reflection is $67.2\pm 1.5~{}\mu\rm{s}$. The time of flight calculated using $t_{\rm{tof}}^{\rm{III}}=\int_{z=z_{1}}^{z=16.3}dz/\int_{z=z_{1}}^{z=16.3}v_{\rm{A}}\,dz$ is $35.7\pm 1.9\,\mu\rm{s}$. Here, the $z$ dependence of $B_{0}$ primarily causes an axial variation of $v_{\rm{A}}$ as $v_{\rm{A}}\propto B_{0}$, while a minor variation in $n$ doesn’t effect $v_{\rm{A}}$ much as $v_{\rm{A}}\propto 1/\sqrt{n}$. Therefore, $t_{\rm{lag}}^{\rm{III}}$ agrees with $2t_{\rm{tof}}^{\rm{III}}$ to within the measurement uncertainty, proving that the small wave is reflected from the plasma source. Since we do not observe any other wave between the incident wave and the small source-reflected wave in Fig. 5, we conclude that gradient III does not reflect Alfvén waves. This experimental finding is supported by our simulations, which shows that gradient III does not reflect (see Section 4).

The values of $\mathcal{R}$ calculated by taking the ratio of the reflected to incident wave power at $z_{2}$ is given by color filled symbols in Fig. 6, where the magenta and yellow colored data points represent measurements for gradients I and III, respectively. $\mathcal{R}$ is presented as a percentage by multiplying Eq. 4 by 100. For convenience we have referred to $\mathcal{R}$ measured at $z_{2}$ as $\mathcal{R}_{z_{2}}$.

$\mathcal{R}_{z_{2}}$ data points for gradient I shows that longer wavelength waves are reflected more strongly from a $v_{\rm{A}}$ gradient than shorter wavelengths. The wavelengths were changed by varying the wave frequency. For the longest wavelength, $\lambda_{\parallel}=5.6\pm 1.2~{}\rm{m}$ ($\lambda_{\parallel}/L_{\rm{A,min}}=7.5\pm 1.6$), and $\mathcal{R}_{z_{2}}$ is $\approx 9\%$. For the shortest wavelength, $\lambda_{\parallel}=3.2\pm 0.7~{}\rm{m}$ ($\lambda_{\parallel}/L_{\rm{A,min}}=4.3\pm 1~{}\rm{m}$), and $\mathcal{R}_{z_{2}}$ is $4.6\%$. Note, the errorbar of $\lambda_{\parallel}$ is due to uncertainty in frequency measurement which in turn is due to the finite temporal length of the incident wave. We estimated the frequency uncertainty using the full width at half maximum of the frequency peak in the Fourier spectrum of the incident wave.

A comparison of $\mathcal{R}_{z_{2}}$ data points for gradients I and III suggests that the length scale of the $v_{\rm{A}}$ gradient affects wave reflection. $\mathcal{R}_{z_{2}}$ is zero for gradient III, as the measurement of wave magnetic field along the $x$ axis spanning the width of the plasma column at $y=0$ did not reveal any signature of a reflected wave from the $v_{\rm{A}}$ gradient. However, $\mathcal{R}_{z_{2}}$ is non-zero for I. The key difference between I and III are their values of $L_{\rm{A,min}}$ because $\lambda_{\parallel}$ for both gradients lie in the same range. $L_{\rm{A,min}}$ for I and III are 0.74 m and 3.24 m, respectively. Note that $L_{\rm{A,min}}$ signifies the degree of inhomogeneity of a $v_{\rm{A}}$ gradient and a small $L_{\rm{A,min}}$ indicates a strong gradient.

The blue curve in Fig. 6 showing the variation of $\mathcal{R}_{z_{2}}$ vs. $\lambda_{\parallel}/L_{\rm{A,min}}$ as predicted by Gaussian process regression (GPR), a class of machine learning algorithms (Rasmussen, 2005). GPR is a Baysean non-parametric regression technique that predicts a probability distribution over possible functions that fit a set of discrete data points. The blue curve in Fig. 6 gives mean of the probability distribution i.e., most probable characterization of the data (Pedregosa et al., 2011; Bose et al., 2022).

The values of $\mathcal{R}_{z_{2}}$ in Fig. 6 are the lower limit of the coefficient of reflection. This is because the incident Alfvén wave damps as it propagates from $z_{2}$ to the location of the reflector in the gradient, causing the incident wave energy to be smaller at the location of the reflector than at $z_{2}$. Similarly, the reflected wave damps as it propagates from the site of the reflector in the gradient, causing the reflected wave energy to be smaller at $z_{2}$. As a result, the incident wave energy is overestimated while the reflected wave energy is underestimated at $z_{2}$ compared to the corresponding energies at the site of the reflector in the gradient.

We have used a simple model to estimate the incident and reflected wave energy at the site of the wave reflector to make a zeroth order correction to the directly measured values of $\mathcal{R}_{z_{2}}$. In the model, the incident wave energy at $z_{2}$ ($\mathcal{E}_{\rm{i}}$) is related to the incident wave energy at the site of wave reflector ($\mathcal{E}_{\rm{i,0}}$) by

where $\Delta z$ is the distance between $z_{2}$ and location of the reflector, and $d$ is the collisional damping length of Alfvén wave, i.e., the length over which the Alfvén wave amplitude decreases to $1/e$ of its initial value. Note, the factor 2 appears in the exponential term of Eq. (5) because $\mathcal{E}\propto b^{2}$ and the damped amplitude of $b\propto\exp{\left(-\Delta z/d\right)}$. Similarly, the reflected wave energy at the reflector ($\mathcal{E}_{\rm{r,0}}$) and at $z_{2}$ ($\mathcal{E}_{\rm{r}}$) are related by

Therefore, the coefficient of reflection at the site of the reflector ($\mathcal{R}_{0}$) is given by

The resulting value of $\mathcal{R}_{0}$ has a monotonic variation with respect to $\lambda_{\parallel}/L_{\rm{A,min}}$ as shown in Fig. 7. The data points are shown by symbols while the red curve is most probable characterization of the data points as per GPR. The $\mathcal{R}_{0}$ vs. $\lambda_{\parallel}/L_{\rm{A,min}}$ curve shows that as the inhomogeneity increases, so does the reflection of Alfvén waves.

The values of $\mathcal{R}_{0}$ are approximate quantities as we estimated $d$ of the Alfvén wave in the gradient using the dispersion relation for a uniform plasma. A comprehensive analysis to accurately determine $d$ in a strong non-WKB gradient relevant to our experiments is beyond the scope of this paper. However, past experiments of Bose et al. (2019) does provide an insight on the damping of Alfvén waves in a strong non-WKB gradient. They found the energy reduction of an Alfvén wave in a strong non-WKB gradient to be more than in an almost uniform plasma suggesting that the damping length is smaller (i.e. damping is more pronounced) in the gradient than in the uniform case. This implies that for the uniform plasma the values of $d$ used to calculate $\mathcal{R}_{0}$ overestimates the damping length in the gradient and thus underestimates $\mathcal{R}_{0}$. Therefore, $\mathcal{R}_{0}$ in Fig. 7 is likely to be a lower bound on the coefficient of reflection after correcting for collisional damping.

Fig. 8(a) shows the wave propagation along the cylindrical axis of LAPD in the $tz$-plane for gradient U. The color gives the $y$-component of the wave magnetic field, which in turn also indicates the phase of the wave. The slope of the band traced by the phase of the wave indicates the propagation direction of the wave; positive and negative slope corresponds to $+z$ and $-z$ direction, respectively. In Fig. 8(a), the slope of the band traced by the phase of the wave is positive, consistent for an incident wave propagating away from the antenna. Fig. 8(b) shows the time variation of the incident wave at $z=3.5~{}\rm{m}$. Unlike the experiment, we do not see a wave reflected from the far end in the time series data because the simulation was carried out up to $\approx 78\;\mu s$, and the wave reflected from the far end requires a longer time to reach $z=3.5~{}\rm{m}$.

Fig. 8(c) shows the wave propagation in the $tz$-plane for gradient I, where we observe experimentally wave reflection from the $v_{\rm{A}}$ gradient. Initially, from $t=0$ to $\lesssim 40~{}\rm{\mu s}$ and $z\lesssim 5~{}\rm{m}$, the wave propagates in $+z$ direction as expected for an incident wave. Upon reaching the gradient region, where $L_{\rm{A}}\approx L_{\rm{A,min}}$, a portion of the incident wave is reflected. The negatively sloped arrows trace the phases of the wave reflected from the $v_{\rm{A}}$ gradient.

We observe that reflection from a $v_{\rm{A}}$ gradient introduces a $180^{\circ}$ phase difference between the wave magnetic field of the incident and the reflected wave. In Fig. 8(c), the deep blue band of the incident wave, marked by the positively sloped white arrow, correlates to the negatively sloped red band (reflected wave), marked by the negatively sloped white arrow. Similarly, the positively sloped deep red band of the incident wave corresponds to the negatively sloped blue band of the reflected wave, as shown by $\rm{follow\;the\;black\;arrows\;in\;Fig.~{}\ref{fig:simulation}(c)}$. The change in the color between the corresponding phases of the incident and the reflected wave represents a $180^{\circ}$ phase difference. We performed a second test to check the phase difference. Simulations using an incident wave having only a strong negative $b_{y}$ (i.e. no strong positive $b_{y}$) gave rise to a reflected wave with only a positive $b_{y}$, further demonstrating a $180^{\circ}$ phase difference between incident and reflected wave due to reflection. See Appendix A for the additional details on this second test.

The time variation of $b_{y}$ before the gradient at $z=3.5~{}\rm{m}$ in Figs. 8 (c) and (d) show that the first low amplitude trough of the incident wave does not produce a strong detectable reflected wave. The reasons for this observation may be related to the process associated with Alfvén wave reflection at the boundary, such as establishing currents at the boundary to support a reflected Alfvén wave. A discussion on the interaction of the Alfvén wave with the $v_{\rm{A}}$ gradient boundary is beyond the scope of this paper and will be discussed elsewhere.

Fig. 8(e) shows the wave propagation in the $tz$-plane for case III, i.e., the $B_{0}$ profile with gradient III. We do not observe any wave reflection from the $v_{\rm{A}}$ gradient, but the transmitted wave is reflected from the far end of the simulation domain. The $b_{y}$ time series data in Fig. 8(f) only shows a well-formed incident wave and does not exhibit any signature of a reflected wave from the $v_{\rm{A}}$ gradient.