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arXiv 1002.1822physics.flu-dyn
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基于最小耗散模式的低$Rm$ MHD turbulence的直接数值模拟

Direct Numerical Simulations of Low-$Rm$ MHD turbulence based on the least dissipative modes

Alban Pothérat, Vitali Dymkou

备注：
24 pages. Article accepted for publication in the Journal of Fluid Mechanics
发表信息：
J. Fluid Mech., 655, 174-197 (2010)
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摘要：

本文提出了一种新的谱方法用于低磁雷诺数下磁流体动力学湍流的直接数值模拟。我们的方法的独创性是，而不是使用传统的基地的功能，它依赖于本征模的基础上的耗散运营商，这代表粘性和焦耳耗散。我们应用这个想法的简单情况下，一个周期域在空间的三个方向，在$\mathbf e_z$方向的均匀磁场。然后基仍然是傅立叶空间的子集，但是通过增长的线性衰减率$|λ|$（根据\emph{least dissipative modes}的\emph{i.e}）来排序。我们表明，因为在傅立叶空间中的恒定能量线倾向于遵循恒定$|λ|$的线，所以在强迫流中最小尺度$|λ^{\rm max}|$的标度可以使用这个单一参数表示，作为雷诺数的函数为$\sqrt{|λ^{\rm max}|}/(2πk_f)\simeq 0.5\Rey^{1/2}$，其中$k_f$是强迫波长，或者作为Grashof数$\Gr_f$的函数，其给出强迫的无量纲度量，作为$|λ^{\rm max}|^{1/2}/(2πk_f)\simeq 0.47\Gr_f^{0.20}$。还发现这种缩放与启发式缩放一致，并且我们能够将其数值量化为$k_\perp^{\rm max}/k_f\simeq 0.5 \Rey^{1/2}$和$k_z^{\rm max}/k_f\simeq 0.8k_f \Rey/Ha$。最后，我们表明，一组最少的耗散模式给出了一个相关的预测的规模的第一个三维结构出现在一个强制的，最初的二维湍流。这完成了我们的数值演示，最小耗散模式可以用来模拟二维和三维低Rm MHD流。

英文摘要：

We present a new spectral method for the Direct Numerical Simulation of Magnetohydrodynamic turbulence at low Magnetic Reynolds number. The originality of our approach is that instead of using traditional bases of functions, it relies on the basis of eigenmodes of the dissipation operator, which represents viscous and Joule dissipation. We apply this idea to the simple case of a periodic domain in the three directions of space, with an homogeneous magnetic field in the $\mathbf e_z$ direction. The basis is then still as subset of the Fourier space, but ordered by growing linear decay rate $|λ|$ (\emph{i.e} according to the \emph{least dissipative modes}). We show that because the lines of constant energy tend to follow those of constant $|λ|$ in the Fourier space, the scaling for the the smallest scales $|λ^{\rm max}|$ in a forced flow can be expressed using this single parameter, as a function of the Reynolds number as $\sqrt{|λ^{\rm max}|}/(2πk_f)\simeq 0.5\Rey^{1/2}$, where $k_f$ is the forcing wavelength, or as a function of the Grashof number $\Gr_f$, which gives a non-dimensional measure of the forcing, as $|λ^{\rm max}|^{1/2}/(2πk_f)\simeq 0.47\Gr_f^{0.20}$. This scaling is also found consistent with heuristic scalings, and which we are able to numerically quantify as $k_\perp^{\rm max}/k_f\simeq 0.5 \Rey^{1/2}$ and $k_z^{\rm max}/k_f\simeq 0.8k_f \Rey/Ha$. Finally, we show that the set of least dissipative modes gives a relevant prediction for the scale of the first three-dimensional structure to appear in a forced, initially two-dimensional turbulent flow. This completes our numerical demonstration that the least dissipative modes can be used to simulate both two- and three-dimensional low-Rm MHD flows.

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