Weak Turbulence Closures in Incompressible Magnetohydrodynamics
Azelis, Augustus Antanas
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In this work we apply the weak turbulence closure to the governing equation of incompressible homogeneous magnetohydrodynamics (MHD). It was originally thought that the method lacked applicability to linearly dispersive waves however MHD presents a unique exception to this notion. A perturbative scheme is used in combination with multiple scale analysis to find a dynamic equation for the wavenumber energy spectrum corresponding to weakly non-linear shear Alfv´enic turbulence. We begin by deriving an equation describing the time evolution of the Fourier transformed Els¨asser fields associated with linear propagation and non-linear interactions between counter-propagating Alfv´en waves. A cumulant hierarchy is then constructed from products of field amplitudes and the well known problem of closure is encountered. Closure of the energy spectrum is then sought for timescales of order ϵ −2 via a series of perturbative expansions where ϵ is a small parameter representing the strength of nonlinear effects. Terms are encountered corresponding to resonant interactions between Alfv´en waves that are unbounded in time and result in an expansion that is not well ordered. Multiple scale analysis is then used to modify the slow time behavior of the cumulants to reorder the expansion, yielding an equation for the evolution of the energy spectrum. The formal mathematical framework of weak turbulence is well discussed in the literature however the present applications to shear Alfv´en waves are lacking in rigor. This work forgoes the aforementioned brevity favored by the literature to demonstrate with clarity the closure for MHD. Explicit detail of the formalism results in heightened understanding of the underlying physics as well as the ability to generalize the theory for two time closures.