Alfvén wave collisions, the fundamental building block of plasma turbulence. I. Asymptotic solution

Abstract

The nonlinear interaction between counterpropagating Alfvén waves is the physical mechanism underlying the cascade of energy to small scales in astrophysical plasma turbulence. Beginning with the equations for incompressible MHD, an asymptotic analytical solution for the nonlinear evolution of these Alfvén wave collisions is derived in the weakly nonlinear limit. The resulting qualitative picture of nonlinear energy transfer due to this mechanism involves two steps: first, the primary counterpropagating Alfvén waves interact to generate an inherently nonlinear, purely magnetic secondary fluctuation with no parallel variation; second, the two primary waves each interact with this secondary fluctuation to transfer energy secularly to two tertiary Alfvén waves. These tertiary modes are linear Alfvén waves with the same parallel wavenumber as the primary waves, indicating the lack of a parallel cascade. The amplitude of these tertiary modes increases linearly with time due to the coherent nature of the resonant four-wave interaction responsible for the nonlinear energy transfer. The implications of this analytical solution for turbulence in astrophysical plasmas are discussed. The solution presented here provides valuable intuition about the nonlinear interactions underlying magnetized plasma turbulence, in support of an experimental program to verify in the laboratory the nature of this fundamental building block of astrophysical plasma turbulence. © 2013 AIP Publishing LLC.