Measurements of Dispersion and Damping for Kinetic and Intertial Shear Alfvén WavesD.J. Thuecks, C.A. Kletzing, F. Skiff, S.R. Bounds

Department of Physics and Astronomy, University of IowaS. Vincena

Department of Physics, U.C.L.A.

Abstract Experiments to test the dispersion relation of shear Alfvén waves as a function of perpendicular wave number were performed using the LArge Plasma Device (LAPD) at UCLA. The waves are launched using a 48-element antenna in which the user can “tune” the perpendicular structure of the Alfvén wave, allowing for good control over the perpendicular wave numbers of the wave. Amplified magnetic search coil probes placed along the length of the chamber are used to measure the wave across the center of the wave pattern. The measured signals are processed to separate the signal into perpendicular wave number components. Signal components at two different locations for the same wave number are cross-correlated and the propagation time with the probe separation leads to the parallel phase velocity for each wave number. Relative amplitudes of each wave number between two probes are also found and allow the damping to be determined. The parallel phase velocity and the damping rate are then compared to the theoretical dispersion relation and damping for both the kinetic and inertial cases. For the kinetic limit measurements, the theoretical dispersion relation and damping curves are taken from warm-plasma theory with the inclusion of charged-particle collisional effects via a Krook collision operator. For the inertial Alfvén wave, the theoretical dispersion and damping curves are determined using a two-fluid theory developed by Braginskii [1965] which includes effects such as ion viscosity, among others. An extension to this theory developed by Catto [1994] and Helander [1994] allow the inclusion of neutral particle effects. Additionally, charged-particle collisions are included through the use of Krook collision operators. We find that it is essential to include all of these effects in order to have good agreement between the measurements and theoretical curves.

ConclusionsShear Alfvén waves have been launched and measured at different points along the axis of the LAPD plasma column to determine wave phase velocity and damping. The UI antenna is shown to be tunable to drive a variety of desired wave numbers, but wave damping limits the amount of power that can be driven for higher values of . Dispersion and damping measurements have been compared to theoretical results with reasonable success. Experimental values of dispersion are in good agreement with theory for both kinetic and inertial cases. The inclusion of e-i collisional effects is essential in the inertial regime; the comparison between theory and data is quite poor without them. In the kinetic regime, the calculated e-i damping must be significantly reduced (by 95%) in order to match observed values of damping. This suggests that the usual e-i collision rate is not applicable for the parameters in the kinetic limit. It may be that the electrons thermalize sufficiently quickly in presence of the parallel electric field that they are effectively always Boltzmann distributed. Therefore the motions of the electrons with respect to the ions is less severe than in the inertial case (where the relatively cold electrons are treated ballistically), allowing the effects of e-i collisions to be reduced. Other methods of including collisions into warm-plasma theory are being investigated in order to describe e-i collisions more adequately in both regimes.

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Motivation

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Shear Alfvén waves are thought to be important for acceleration of auroral electrons. In this model, waves resonantly accelerate electrons through a process similar to a single instance of Fermi acceleration. These data to the left are from the HIBAR rocket flight, and show higher energy electrons arriving at the observation point earlier than lower energy electrons.

To the right is the result of a computer model. It assumes the same conditions and observation point as HIBAR had. The shape of the dispersion trace is comparable to the results from HIBAR, and the dispersion times are close. Many simulations like this assume the shear Alfvén wave dispersion equation that is the focus of this investigation.

Plots courtesy of Li-Jen Chen et al.,J. Geophys. Res., 110 (A9), 2005.

Large Plasma Device (LAPD) at UCLA

A Helium discharge plasma is created within the LAPD. The discharge occurs every second and has a duration of 7-12 ms. Dispersion experiments are performed during the discharge for kinetic Alfvén waves and 50-100 ms after the end of the discharge for inertial Alfvén waves. Typical plasma conditions are listed below for both cases.

Antenna Cathode and Anode

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-3Kinetic: B=400G n =1.1e12 cm T =6.00eV T ~1.25eV d=c/w =.51cm r=c /w =1.25cme e i e pe s s ci

f =1.12GHz f =152KHz f =9.41GHz f =110MHz V /V =2.47ce ci pe pi te A

-3Inertial: B=2300G n =7.5e11 cm T =2.0eV T ~1.25eV d=c/w =.61cm r=c /w =.13cme e i e pe s s ci

f =6.44GHz f =877KHz f =7.77GHz f =91MHz V /V =0.20ce ci pe pi te A

Launching and Measuring Shear Alfvén Waves

Current is driven on 48 copper mesh elements of an antenna, each separated by .25”. Each element has a limited-duration sine wave signal applied on top of a positive bias. By tuning the current amplitude supplied by each element, different perpendicular wave numbers can be driven.

A 3-axis powered b-dot probe is used to measure and amplify the time varying magnetic field associated with the shear Alfvén wave. By taking measurements in time at many different positions across the diameter of the plasma column, a map of the magnetic field in both position and time can be created. Above are two plots of the y-component of the magnetic field.

Antenna SketchAntenna Element Settings

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Analysis

Pictured are signal components from two different ports for the same perpendicular wave number. The red component is from port 36, while the black component is from port 31. The black component trails and is smaller than the red component as one would expect.

The reduced chi-squared (black) and correlation (red) values found when comparing the two signals can be minimized (for a given amplitude) by shifting the signal at port 31 with respect to the signal at port 36. The reduced chi-squared value can be further minimized by adjusting the amplitude of the port 31 signal.

The two signal components are shown after the correlation has been performed.Once the best time shift and amplitude shift have been found, the phase velocity and the damping rate can be determined for each k number.

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Shear Alfvén Wave Dispersion RelationsBraginskii TheoryWarm Plasma Theory

The equations above represent the two approaches for finding the theoretical phase velocity and damping rate. The left set is a warm-plasma approximation that includes collisions via a Krook operator. Damping effects include Landau damping and collisional damping. The right set of equations represent a two-fluid approach developed by Braginskii [1965] where the stress tensor introduces effects due to electron-electron and ion-ion collisions and the friction force term includes electron-ion collisions in the electron momentum equation. Neutral effects can be included by changing n to n +n in the ion Braginskii parameters (h) [Catto, 1994; Helander,1994] and including an i i n ss

additional isotropic e-n collision term in the friction force term of the electron momentum equation.

Results

Comparison of the experimental dispersion relation with warm-plasma theory (black) and Braginskii theory (red) including e-i collisions for the inertial Alfvén wave.

Comparison to Landau damping plus collisional damping using n (red) and 5% of ei n (black).ei

Comparison of experimental damping to theory for the inertial case.

Comparison of the experimental dispersion relation with warm plasma theory. The red curve includes . The solid black includes an electron collision frequency of 5% of The dotted black lines represent +- 20% in density and electron temperature.

n=n n.col ei ei

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250 Khz

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76 Khz

Kinetic regime:

Inertial regime: