Lyutikov Maxim
Department of Physics, Purdue University, 525 Northwestern Avenue, West Lafayette, Indiana 47907-2036, USA and The Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street Toronto, Ontario, Canada M5S 3H8.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Nov;88(5):053103. doi: 10.1103/PhysRevE.88.053103. Epub 2013 Nov 20.
We consider dynamics and turbulent interaction of whistler modes within the framework of inertialess electron magnetohydrodynamics (EMHD). We argue that there is no energy principle in EMHD: any stationary closed configuration is neutrally stable. On the other hand, the relaxation principle, the long term evolution of a weakly dissipative system towards Taylor-Beltrami state, remains valid in EMHD. We consider the turbulent cascade of whistler modes. We show that (i) harmonic whistlers are exact nonlinear solutions; (ii) collinear whistlers do not interact (including counterpropagating); (iii) waves with the same value of the wave vector k(1)=k(2) do not interact; (iv) whistler modes have a dispersion that allows a three-wave decay, including into a zero frequency mode; (v) the three-wave interaction effectively couples modes with highly different wave numbers and propagation angles. In addition, linear interaction of a whistler with a single zero mode can lead to spatially divergent structures via parametric instability. All these properties are drastically different from MHD, so that the qualitative properties of the Alfvén turbulence can not be transferred to the EMHD turbulence. We derive the Hamiltonian formulation of EMHD, and using Bogoliubov transformation reduce it to the canonical form; we calculate the matrix elements for the three-wave interaction of whistlers. We solve numerically the kinetic equation and show that, generally, the EMHD cascade develops within a broad range of angles, while transiently it may show anisotropic, nearly two-dimensional structures. Development of a cascade depends on the forcing (nonuniversal) and often fails to reach a steady state. Analytical estimates predict the spectrum of magnetic fluctuations for the quasi-isotropic cascade [proportionality]k(-2). The cascade remains weak (not critically balanced). The cascade is UV local, while the infrared locality is weakly (logarithmically) violated.
我们在无惯性电子磁流体动力学（EMHD）的框架内考虑哨声波模的动力学和湍流相互作用。我们认为EMHD中不存在能量原理：任何静止的封闭构型都是中性稳定的。另一方面，弛豫原理，即弱耗散系统向泰勒 - 贝尔特拉米状态的长期演化，在EMHD中仍然有效。我们考虑哨声波模的湍流级联。我们表明：（i）谐波哨声波是精确的非线性解；（ii）共线哨声波不相互作用（包括反向传播的情况）；（iii）具有相同波矢值k(1)=k(2)的波不相互作用；（iv）哨声波模具有允许三波衰变的色散，包括衰变成零频率模；（v）三波相互作用有效地耦合了具有高度不同波数和传播角度的模。此外，哨声波与单个零模的线性相互作用可通过参量不稳定性导致空间发散结构。所有这些性质都与磁流体动力学（MHD）有很大不同，因此阿尔文湍流的定性性质不能转移到EMHD湍流中。我们推导了EMHD的哈密顿表述，并使用博戈留波夫变换将其简化为规范形式；我们计算了哨声波三波相互作用的矩阵元。我们对动力学方程进行了数值求解，结果表明，一般来说，EMHD级联在很宽的角度范围内发展，而在瞬态情况下它可能呈现各向异性的、近乎二维的结构。级联的发展取决于强迫（非普适的），并且常常无法达到稳态。解析估计预测了准各向同性级联的磁涨落谱[与]k(-2)[成正比]。级联仍然较弱（不是临界平衡的）。级联是紫外局部的，而红外局部性受到微弱（对数形式）的破坏。
