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 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. The current work examines an analytic solution to the reduced MHD equations in the presence of multiple nonlinear interactions. We 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. The result contains both a particular solution at the frequency of the nonlinear drive and a homogeneous solution at the frequency of the associated normal mode. At the resonance where the two frequencies match, secularly growing Alfvén normal modes are produced. Due to the chosen initial conditions, these modes grow in time (not in space), and subsequent interactions involving the secularly growing modes preferentially produce negative residual energy. This result shows up as the condensed region of residual energy near k_||=0 first derived by Wang, et. al. 2012. Large Plasma Device experiments which have successfully verified residual energy in a driven child mode will be presented in a companion abstract.