Abstract
The two-time energy spectrum of weak magnetohydrodynamic turbulence is found by applying a wave-turbulence closure to the cumulant hierarchy constructed from the dynamical equations. Solutions are facilitated via asymptotic expansions in terms of the small parameter
$\varepsilon$
, describing the ratio of time scales corresponding to Alfvénic propagation and nonlinear interactions between counter-propagating Alfvén waves. The strength of nonlinearity at a given spatial scale is further quantified by an integration over all possible delta-correlated modes compliant in a given set of three-wave interactions that are associated with energy flux through the said scale. The wave-turbulence closure for the two-time spectrum uncovers a secularity occurring on a time scale of order
$\varepsilon ^{-2}$
, and the asymptotic expansion for the spectrum is reordered in a manner comparable to the one-time case. It is shown that for the regime of stationary turbulence, the two-time energy spectrum exponentially decays on a lagged time scale
$(\varepsilon ^2 \gamma _k^s)^{-1}$
in proportion to the strength of the associated three-wave interactions, characterized by nonlinear decorrelation frequency
$\gamma _k^s$
. The scaling of the form
$k_{\perp } v_0 \chi _0$
exhibited by this frequency is reminiscent of random sweeping by the outer scale with characteristic fluctuation velocity
$v_0$
that is modified due to competition with Alfvénic propagation (characterized by
$\chi _0$
) at the said scale. A brief calculation of frequency broadening of the power spectrum due to nonlinear interactions is also presented.
Funder
U.S. Department of Energy
Publisher
Cambridge University Press (CUP)