Tegegn Tesfalem Abate
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria 0204, South Africa.
Entropy (Basel). 2022 Jun 16;24(6):833. doi: 10.3390/e24060833.
The spectral slope of magnetohydrodynamic (MHD) turbulence varies depending on the spectral theory considered; -3/2 is the spectral slope in Kraichnan-Iroshnikov-Dobrowolny (KID) theory, -5/3 in Marsch-Matthaeus-Zhou and Goldreich-Sridhar theories, also called Kolmogorov-like (K-41-like) MHD theory, the combination of the -5/3 and -3/2 scales in Biskamp, and so on. A rigorous mathematical proof to any of these spectral theories is of great scientific interest. Motivated by the 2012 work of A. Biryuk and W. Craig (Physica D 241(2012) 426-438), we establish inertial range bounds for K-41-like phenomenon in MHD turbulent flow through a mathematical rigor; a range of wave numbers in which the spectral slope of MHD turbulence is proportional to -5/3 is established and the upper and lower bounds of this range are explicitly formulated. We also have shown that the Leray weak solution of the standard MHD model is bonded in the Fourier space, the spectral energy of the system is bounded and its average over time decreases in time.
磁流体动力学（MHD）湍流的谱斜率会根据所考虑的谱理论而变化；在克莱奇南 - 伊罗什尼科夫 - 多布罗沃利（KID）理论中谱斜率为 -3/2，在马尔施 - 马泰厄斯 - 周理论以及戈德赖希 - 斯里达尔理论（也称为类柯尔莫哥洛夫（K - 41 类）MHD 理论）中为 -5/3，在比斯坎普理论中是 -5/3 和 -3/2 尺度的组合，等等。对这些谱理论中的任何一个进行严格的数学证明都具有极大的科学意义。受 A. 比柳克和 W. 克雷格 2012 年工作（《物理 D》241(2012) 426 - 438）的启发，我们通过数学严谨性为 MHD 湍流中的 K - 41 类现象建立了惯性范围界限；确定了 MHD 湍流谱斜率与 -5/3 成比例的波数范围，并明确给出了该范围的上下界。我们还表明标准 MHD 模型的勒雷弱解在傅里叶空间中是有界的，系统的谱能量是有界的，并且其时间平均值随时间减小。
