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.

中文翻译：

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弱磁流体动力湍流的时空结构

通过将波湍流闭包应用于由动力学方程构建的累积量层次，找到了弱磁流体动力湍流的二次能谱。通过小参数的渐近展开来促进解决方案 $\瓦瑞普西隆$ ，描述与阿尔芬传播和反向传播阿尔芬波之间的非线性相互作用相对应的时间尺度之比。给定空间尺度上的非线性强度可以通过对所有可能的δ相关模式进行积分来进一步量化，这些模式符合给定的一组三波相互作用，这些相互作用与通过所述尺度的能量通量相关。二次谱的波湍流闭合揭示了在有序时间尺度上发生的世俗性 $\varepsilon ^{-2}$ ，并且谱的渐近展开以与一次性情况相当的方式重新排序。结果表明，对于稳态湍流状态，二次能谱在滞后时间尺度上呈指数衰减 $(\varepsilon ^2 \gamma _k^s)^{-1}$ 与相关三波相互作用的强度成比例，以非线性去相关频率为特征 $\伽玛_k^s$ 。表单的缩放 $k_{\perp } v_0 \chi _0$ 该频率所表现出的让人想起具有特征波动速度的外部尺度的随机扫描 $v_0$ 由于与 Alfvénic 传播的竞争而被修改（其特征为 $\chi_0$ ）在上述规模。还介绍了由于非线性相互作用而导致的功率谱频率展宽的简要计算。