Observation of an Alfvén Wave Parametric Instability in a Laboratory Plasma

A shear Alfvén wave parametric instability is observed for the first time in the laboratory. When a single finite $\omega /{\mathrm{\Omega }}_i$ kinetic Alfvén wave (KAW) is launched in the Large Plasma Device above a threshold amplitude, three daughter modes are produced. These daughter modes have frequencies and parallel wave numbers that are consistent with copropagating KAW sidebands and a low frequency nonresonant mode. The observed process is parametric in nature, with the frequency of the daughter modes varying as a function of pump wave amplitude. The daughter modes are spatially localized on a gradient of the pump wave magnetic field amplitude in the plane perpendicular to the background field, suggesting that perpendicular nonlinear forces (and therefore $k_{\perp }$ of the pump wave) play an important role in the instability process. Despite this, modulational instability theory with $k_{\perp }=0$ has several features in common with the observed nonresonant mode and Alfvén wave sidebands.

Figure 1

Experimental setup in LAPD. Top: An Alfvén wave antenna on the right end of the device launches the pump wave. Magnetic and Langmuir probes used to diagnose the interaction are shown. Bottom: Spatial pattern of the pump wave in the $xy$ plane measured by a magnetic probe at $z=2.6\mathrm{m}$ for the strap antenna (left, $B_0=1135\mathrm{G}$) and RMF antenna (right, $B_0=993\mathrm{G}$).

Figure 2

Observed kinetic Alfvén wave (KAW) parametric instability showing threshold behavior and parametric dependence. RMF antenna, RHCP mode, $B_0=993\mathrm{G}$. (a) Frequency spectrum from a magnetic probe at $x=0$, $y=-6\mathrm{cm}$, $z=2.6\mathrm{m}$ for three pump mode amplitudes. When the pump amplitude is above threshold for instability, three daughter modes are seen. (b) Parametric dependence of the daughter mode frequency as a function of pump amplitude $\delta B_{0\perp }/B_0$. The pump amplitude is 0 on the ${\log }_{10}$ color scale. White vertical dashed lines represent values of pump amplitude from (a).

Figure 3

Parallel wave number measurement showing daughter modes copropagating with the pump. The pump, $M-$, and $M+$ are identified as KAWs while $M1$ is a nonresonant mode. Strap antenna, $B_0=1140\mathrm{G}$, $\delta B_{0\perp }/B_0=1.9\times {10}^{-3}$. Magnetic probes at $z=5.11\mathrm{m}$, 5.75 m, and 6.39 m. The fluid dispersion relation for a KAW with the pump $k_{\perp 0}{\rho }_s=0.11$ and a line with slope $\omega /k_{||}=0.29V_A$ are plotted for comparison.

Figure 4

Spatial profile of $M-$ for the strap antenna suggesting the nonlinearity is perpendicular in nature. A cut of $\delta B_x$ is shown on the right. Strap antenna pump from Fig. 1, $B_0=1135\mathrm{G}$. Color represents fluctuating magnetic field amplitude $\delta B_{-\perp }$; white arrows show relative magnitude and direction. The peak in $M-$ amplitude occurs on a gradient of the pump mode magnetic field near the current channel center.

Figure 5

Dependence of the observed frequency spectrum on the polarization of the RMF antenna. Magnetic probe $x=0$, $y=-6\mathrm{cm}$, $z=2.6\mathrm{m}$. Inset: Polarization of the RMF pump mode from Fig. 1 along a cut at $x=0$. $B_0=993\mathrm{G}$.

Figure 6

Comparison between LAPD data and $k_{\perp }=0$ dispersion relation derived by Wong and Goldstein [6] and Hollweg [7]. (a) Solutions to the dispersion relation of Wong and Goldstein [6] and Hollweg [7] for experimental parameters of Fig. 3. Labeled: $s$: sound mode, $-b$: backward propagating lower Alfvénic sideband, $-f$: forward propagating lower Alfvénic sideband, $+f$: forward propagating upper Alfvénic sideband. Black curves represent stable modes; orange curves representing unstable modes are labeled with the appropriate instability. (b) Mode frequency of the modulational instability as a function of pump amplitude for experimental parameters in Fig. 2 (blue circles), theoretical predictions (red stars), and strap antenna results with similar parameters (yellow squares).