Generalized Theory of Weak Magnetohydrodynamic Turbulence

The heliosphere is replete with Alfvenic fluctuations which are of solar origin. These are believed to be important in the context of the solar coronal heating and solar wind acceleration processes. The study of such low-frequency hydromagnetic turbulent phenomena is thus of compelling nature, especially in view of the contemporary space missions such as NASA's Parker Solar Probe and ESA's Solar Orbiter missions. Among the mathematical tools to study such turbulent phenomena is the weak turbulence theory. The weak magnetohydrodynamic (MHD) turbulence theory found in the literature often starts from the equations of incompressible MHD theory expressed via the Elsasser variables, and the derivation is carried out via a truncated solution at the second-order of iteration under the assumption of zero residual energy, which is the difference between the turbulent energy associated with the velocity fluctuation and the magnetic field fluctuation associated with the shear Alfven waves. The present paper generalizes the formulation of weak MHD turbulence theory by relaxing the assumption on the residual energy, and by retaining the iterative solution up to the third order. For this purpose, it is found that the pristine form of incompressible MHD equation, rather than that expressed in terms of Elsasser fields, offers a more straightforward theoretical platform. It is also found that the residual energy is generally finite, but in order to treat this proble properly, it is necessary to include the third-order nonlinear correction.