Authors: Seth Dorfman (Space Science Institute), Christopher H K Chen (Queen Mary University of London), Stas Boldyrev (University of Wisconsin Madison), Mel Abler (Space Science Institute)

Counter-propagating Alfvén wave interactions which transfer energy from large to small spacial scales lie at the heart of magnetohydrodynamic turbulence in the solar wind. An unexpected feature of the turbulence is the generation of residual energy – excess energy in the magnetic fluctuations compared to the velocity fluctuations. By contrast, an MHD Alfvén wave has equal amounts of energy in fluctuations of each type. Howes, et. al. 2013 showed that purely magnetic fluctuations develop in non-linear interactions and suggested that this may explain residual energy generation.

The current work examines a solution to the reduced MHD equations in the presence of multiple non-linear interactions. We first consider the interaction of two sinusoidal Alfvén modes with arbitrary frequencies and wavenumbers and use the approach of Howes, et. al. 2013 to solve for generalized interaction terms. This analytic result is then used to iteratively solve the interaction of counter-propagating Alfvén waves to 80th order in the nonlinearity parameter χ₀, defined as the normalized initial wave amplitude times the initial anisotropy.

The analytic solution for a non-linear interaction between two arbitrary Alfvén waves contains both a particular solution at the frequency of the nonlinear drive and a homogeneous solution at the frequency of the associated normal mode. The particular solution may contain either positive or negative residual energy, depending on the relationship between the phase speed of the driven child mode and the phase speed of an Alfvén normal mode. The net negative residual energy of the full solution is due to the chosen initial conditions, which lead to resonant modes growing in time, not in space. We show that subsequent interactions involving these secularly growing modes preferentially produce negative residual energy. This result shows up in the iterative solution as the condensed region of residual energy near k_||=0 first derived by Wang, et. al. 2012. In the χ₀<<1 limit, secularly growing Alfvén waves with zero residual energy dominate the expansion and the condensate is difficult to discern. As χ₀ is increased, the strength of the non-linear terms also increases, and residual energy containing modes become more apparent. Large Plasma Device experiments which have successfully verified residual energy in a driven child mode will be presented in a companion poster.

Supported by NASA grant 80NSSC18K1235 and DOE Grant DE-SC0021291.