As the turbulent cascade proceeds to smaller scales, the ideal MHD description of the Alfvén wave is no longer valid; instead the cascade is made up of kinetic shear Alfvén waves. The kinetic shear Alv\'en waves (KAW) differ from the shear Alfvén waves of ideal MHD both linearly (Landau damping) and nonlinearly (from secondary instabilities, also known as negative eddy viscosity). The KAW cascade will be simulated using the electromagnetic gyrokinetic code GS2. A preliminary nonlinear calculation (above) demonstrates that GS2 correctly describes shear Alfvén turbulence (i.e., Alfvén turbulence in the MHD regime, where kinetic effects are irrelevant). This is an important demonstration of the correct treatment of the nonlinear terms for the KSA problem. It also provides an independent test of the Goldreich-Sridhar theory, developed in the MHD regime. [Note that only the grid resolution changes as one moves from a simulation of MHD to the KSA regime with a gyrokinetic code.] Here, the plasma was stirred at the box scale with a forcing function whose amplitude was taken from a Langevin equation, which in turn was constructed to produce Alfvén waves with a decorrelation time comparable to the linear wave period. This approximates the flow of energy into the simulated scales from larger scales. Hyperviscosity and hyperresistivity were employed at the small scales, since in this case the scales at which kinetic damping would occur were not included in the simulation. The hyper-damping terms become important at the scale of the vertical dashed line. The expectation from the theory of Goldreich and Sridhar is that an inertial range should result, with the Kolmogorov spectrum E ~ k_perp^-5/3 appearing in the perpendicular cascade. This corresponds to a slope of Phi²(k_perp) ~ k_perp^-11/3 which agrees well with the GS2 simulation, as shown in the figure above.

The existing quasilinear theory of kinetic shear Alfvén waves indicates that much of the energy of the cascade can be drained by wave-particle interactions as kinetic effects cause the development of a finite parallel electric field. The associated wave damping for typical LAPD parameters is shown here. For these LAPD parameters, the damping is significant, and corresponds mainly to electron heating. CMPD supports a comprehensive theoretical and computational investigation of Alfvén wave cascades. We expect to understand whether the existing quasilinear estimates of the development of the turbulent cascade at small scales are qualitatively correct. Other physics may enter the cascade, such as secondary instabilities or coupling to other waves. However the cascade develops, one would like to know the relative heating of ions and electrons. The heating time scale is long compared to the frequency of small-scale fluctuations. It remains to be proven that the only interaction between these scales occurs via the cascade. Such proof may only come from multiscale computations. Each of these questions requires carefully designed nonlinear simulations and the development of nonlinear theory to interpret the results.

This project should establish that our kinetic simulations of electromagnetic turbulence (in the form of KAW's) correctly describe the experimental observations from LAPD. The results will feed directly into all other research tasks of the Center.