Spacetime structure of weak magnetohydrodynamic turbulence

The two time energy spectrum of weak magnetohydrodynamic turbulence is found  via application of a weak turbulence closure to the cumulant hierarchy constructed from the dynamical equation. Solutions are facilitated via asymptotic expansions in terms of the small parameter $\epsilon$, describing the ratio of timescales corresponding to Alfvénic propagation and nonlinear interactions between counter-propagating Alfvén waves. The strength of non-linearity at a given spatial scale is further quantified by an integration over all possible delta function correlated modes compliant in a given set of 3-wave interactions that are associated with energy flux through said scale. The weak turbulence closure for the two time spectrum uncovers a secularity occurring on timescale $\epsilon^{-2}$ and the asymptotic expansion for the spectrum is reordered in a manner comparable to the one-time case. It is shown that the two-time energy spectrum will oscillate in accordance with Alfvénic propagation on the linear timescale and exponentially decay on timescale $\epsilon^{-2}$ in proportion to the strength of the associated nonlinear interactions.