大尺度准地转磁流体动力学
Large Scale Quasi-geostrophic Magnetohydrodynamics
摘要：
本文研究了快速旋转行星或恒星表面流体层的理想磁流体动力学。假设存在背景环形磁场，并使用“浅水”β-平面近似。我们推导出一个慢的大尺度动力学方程。该方程的适用范围与地球外核顶部较轻流体的MHD相吻合。该方程的形式类似于准地转（Q-G）方程（适用于通常的海洋或大气），但参数有本质的不同。我们的方程也意味着逆级联，但与通常的Q-G情况相反，能量级联到较小的长度尺度，而涡度拟能级联到较大的尺度。我们发现逆级联的Kolmogorov型谱。频谱表明能量在更大尺度上的积累。除了能量和涡度拟能外，所得到的方程还具有一个额外的不变量。它的存在被证明意味着能量积累在$30^\circ$部门周围的纬向流。在特定的能量输入下，附加不变量会导致纬向流中能量的积累，当附加不变量的输入很小，而能量输入很大时，就会出现这种情况。
英文摘要：
We consider the ideal magnetohydrodynamics (MHD) of a shallow fluid layer on a rapidly rotating planet or star. The presence of a background toroidal magnetic field is assumed, and the "shallow water" beta-plane approximation is used. We derive a single equation for the slow large length scale dynamics. The range of validity of this equation fits the MHD of the lighter fluid at the top of Earth's outer core. The form of this equation is similar to the quasi-geostrophic (Q-G) equation (for usual ocean or atmosphere), but the parameters are essentially different. Our equation also implies the inverse cascade; but contrary to the usual Q-G situation, the energy cascades to smaller length scales, while the enstrophy cascades to the larger scales. We find the Kolmogorov-type spectrum for the inverse cascade. The spectrum indicates the energy accumulation in larger scales. In addition to the energy and enstrophy, the obtained equation possesses an extra invariant. Its presence is shown to imply energy accumulation in the $30^\circ$ sector around zonal flow. With some special energy input, the extra invariant can lead to the accumulation of energy in zonal flow; this happens if the input of the extra invariant is small, while the energy input is considerable.
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