Experiments on fundamental nonlinear interactions between Alfvén waves are being performed in the Large Plasma Device (LAPD) at UCLA. Interactions between counterpropagating shear waves are fundamental to the Alfvén cascade. The focus of our studies is on: (1) individual wave-wave collisions between counter-propagating waves and (2) the cumulative effects of many wave-wave collisions, realized through ``stirring'' with a large amplitude Alfvén wave in a cavity (reflecting boundary conditions). We are studying these interactions for: anisotropy (preference for cascade in the direction perpendicular to the field); dependence on launch wave amplitude, frequency, and polarization; and influence of non-ideal effects (coupling to compressible modes (e.g., slow magnetosonic modes), kinetic and inertial shear wave cascades). These experimental results will be used to test both theories and simulations of the Alfvén cascade. In particular, experiments in the LAPD are well suited to simulation by the GS2 electromagnetic code. We intend to provide a important experimental benchmark for this code through detailed measurements of this nonlinear process in LAPD.

Using LAPD typical parameters and assuming a perpendicular wavelength of 30 centimeters for our outer scale wave, we find a constraint on the largest launch frequency for the stirring wave in order to avoid cyclotron damping before reaching Landau damping scales. Written as a constraint on the parallel wavelength of the stirring wave: λ_parallel > 32 meters. This is less than the wavelength of the largest possible wave in LAPD, making a cascade over two orders of magnitude in perpendicular scale possible in LAPD. The constraint on the launch frequency is relaxed if the initial perpendicular wavelength is decreased (e.g., if we consider cascades over shorter distances in k space).

There are also requirements on the launch wave amplitude in order to easily observe this nonlinear process in LAPD. If we want the cascade to be strong, this requires that the outer scale wave amplitude satisfy δB/B ~ k_par/k_perp ~ 1%. This wave amplitude has already been achieved in early experiments with high-amplitude shear wave generation in LAPD. In the strong cascade, the nonlinear eddy turnover time (time for the nonlinear perturbation to grow to unity) is comparable to the period of the interacting Alfvén waves. In this regime, nonlinear perturbations arising from single wavepacket collisions will likely be observable and can be studied. These interactions will be studied for anisotropy (preference for structure formation in the perpendicular direction) and dependence on launch wave parameters (frequency, polarization). The strong cascade will be studied through establishing reflecting boundary conditions in LAPD using the conducting anode/cathode as one boundary and a metal plate as the second, allowing multiple interactions between reflected counter-propagating Alfvén waves. We will study the wavenmuber spectrum established by the cascade (again, looking for anisotropy) and dissipation and heating at skin depth or ion gyroradius scales. We will also look for evidence of a continuation of the cascade in Whistler waves beyond these scales (on the order of 0.5 centimeters in LAPD).

The development of large amplitude shear Alfvén wave drivers for LAPD is central to our research program. Two techniques have now been successfully used to launch shear waves with δB/B ~ 1%. The first makes use of recent observation of an Alfvén wave ``maser'' in LAPD. Here an instability driven by inverse Landau damping in the anode-cathode region has been observed to generate large amplitude shear Alfvén waves in the source cavity. This instability excites the fundamental Alfvénic mode of the resonant cavity defined by the Nickel cathode and the partially reflecting Molybdenum mesh anode, 0.5 meters away. The figure above shows example traces of discharge current, signal from a magnetic pick-up loop, and pickup-loop wavelet power spectrum during a discharge where the instability is excited.

	This resonant cavity can also be externally excited to generate Alfvén waves. An initial test of this idea has been performed, using a broadband power push-pull driving circuit which modulates the discharge current in the anode-cathode region. This figure shows the result of such an external excitation, including a measurement of the Q of the anode-cathode cavity.

The second approach uses the same novel, broadband driver circuit to drive a simple loop antenna (here the loop normal points perpendicular to the field and the loop is rectangular, distended in the magnetic field direction). It has recently been found that large amplitude shear Alfvén waves (which are column eigenmodes) can be launched using modest currents (100 A). Using this type of antenna, a broader range of frequencies can be launched (i.e., not a resonant process). The current antenna size (15 centimeters across the field by 31 centimeters along) is limited by available port size on the LAPD vacuum vessel. Efficient coupling to lower frequency waves is critical in order to easily create stirring waves in the strong cascade regime and in order to maximize the potential range in wavenumber space of a cascade in LAPD. Development of new anode-cathode drivers (perhaps driving between a cathode on one end of the machine and an anode at the other) or larger loop antenna will be pursued to achieve creation of large amplitude, low frequency Alfvén waves.

Initial studies of ``stirring'' with a large scale wave in reflecting boundary conditions have been performed using the resonant anode-cathode cavity as a wave source. Reflecting boundary conditions are established by closing a conducting door internal to the LAPD device, shortening the length of the machine to 10 meters. In these initial studies, a strong nonlinear interaction between the launch shear wave and a low-frequency density fluctuation is observed. This is likely brought about by the presence of two frequencies in the shear wave spectrum emitted by the anode-cathode cavity (usually this is transient, and is thought to be due to evolution of the plasma profiles allowing two eigenmodes of the anode-cathode cavity to exist simultaneously).

Spectrum and bicoherence (floating potential) during experiments with reflecting boundary conditions

This figure shows how the spectrum and bicoherence of potential fluctuations changes with increasing Alfvén wave amplitude (increasing discharge current). The experiments have clearly already entered a nonlinearly relevant regime. As the amplitude is increased (the amplitude of the MASER wave increases with discharge current), sidebands develop around the original Alfvén wave and a low frequency fluctuation develops at the sideband separation frequency and harmonics. The bicoherence clearly indicates three-wave interactions between the Alfvén waves and the low frequency fluctuations. The low-frequency fluctuation is a propagating wave in the plasma and is possibly an ion-acoustic or slow magnetosonic wave created nonlinearly by the interaction of shear waves differing in frequency by the ion acoustic frequency. This possibility is currently being explored in new experimental campaigns.

These experiments will allow detailed measurements of the nonlinear physics of Alfvén wave interactions in the LAPD plasma. These measurements can be used to provide stringent tests of theoretical models and simulations in the LAPD parameter regime, providing more confidence in the application of these to other problems (e.g., tokamak or astrophysical plasmas). A more detailed plan of work for this project follows