Generation of residual energy by many interacting Alfvén waves*

Counter-propagating Alfvén wave interactions which transfer energy from large to small spacial scales lie at the heart of astrophysical plasma turbulence.  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 and shows that residual energy lies in nonlinearly-generated Alfvénic quasimodes with excess magnetic energy.  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 ε, the normalized initial wave amplitude. The analytic solution shows that residual energy lies in daughter modes with k_||VA<<ω produced by the non-linear interaction of perpendicularly polarized modes of similar k_⊥.  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 ε<<1 limit, secularly growing Alfvén waves with zero residual energy dominate the expansion and the condensate is difficult to discern.  As ε is increased, the strength of the non-linear terms also increases, and residual energy containing modes become more apparent.  Planned Large Plasma Device experiments to identify this residual energy resulting from counter-propagating interactions will also be discussed.