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 is 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.a)

Alfvén wave collisions, the fundamental building block of plasma turbulence. IV. Laboratory experiment

“…Since the measured daughter wave signal satisfies the theoretical predictions, we conclude that we have measured the nonlinear interaction between two counterpropagating Alfvén waves. This evidence supports the use of such idealized models, as discussed in Paper I, 22 for weakly collisional plasmas which are relevant to various space and astrophysical plasma environments. It is important to note that this procedure, of subtracting the parent wave signals and bandpass filtering the data, is not required to see the nonlinear daughter wave signal since the three waves are in different locations in the k-plane and vector orientation of δB ⊥ in the plane is different.…”

“…(1) is the linear term representing the propagation of the Elsässer fields along the mean magnetic field at the Alfvén speed, the first term on the right-hand side is the nonlinear term representing the interaction between counterpropagating waves, and the second term on the right-hand side is a nonlinear term that ensures incompressibility. 22,26,27 As emphasized in Paper III, the mathematical properties of Eqs. (1) and (2) dictate that the fundamental building block of turbulence in an incompressible MHD plasma is the nonlinear interaction between perpendicularly polarized, counterpropagating Alfvén waves.…”

“…36 in Paper I. 22 Thus, if the amplitude of either antenna signal is reduced by half, the amplitude of the daughter wave should also be reduced by half. If the signal at 270 kHz was related to one of the primary waves, this would not occur.…”

Revisiting a Classic: The Parker–moffatt Problem

“…These properties not only provide motivation for the Alfvén wave packet collision problem, but also enter into some of its complexity as an elementary interaction that generates turbulence (Kraichnan 1965;Dobrowolny et al 1980b;Howes & Nielson 2013;Drake et al 2016).…”

Spectral moments for the analysis of frequency shift, broadening, and wavevector anisotropy in a turbulent flow

“…In addition, the frequency broadening occurs whenever (1) the flow has large-scale variations with the zero mean such that the root-mean-square of the frequency deviation (measured from the Doppler-shifted frequency) is linearly proportional to the root-mean-square of the flow velocity variation, so we associate the frequency broadening with the relation �δω 2 � = k s U swp , by regarding the sweeping velocity U swp as the root-mean-square of the flow velocity fluctuation U rms . Here the angular bracket denotes the operation of statistical averaging, or (2) wave-wave interactions produce waves with deviating frequencies from the Doppler shift such as linear modes, sideband waves, nonlinear modes (Howes and Nielson 2013). In the spectral moment method, we determine the shift velocity and the broadening velocity from the measurements, and compare with the Doppler shift and broadening.…”