Inverse cascade based on a two-dimensional generalized nonlinear
Schrödinger equation

doi:10.16568/j.0254-6086.202303016

Abstract

Abstract:

For a fixed external potential field, the evolution of a highly localized finite-amplitude initial pulse is investigated by numerically solving the corresponding two-dimensional generalized nonlinear Schrödinger equation i∂_tE +p _▽²E+[V (x,y )+q|E|²] E =0 . The evolution depends crucially on the complex group dispersion coefficient p and the complex nonlinearity coefficient q of the equation. It is found that, in general, the wave field first suffers modulational instability, followed by abrupt collapse into a turbulent state containing the shortest wavelength modes allowed in the system. The latter is in turn followed by inverse cascade of the shortest wavelength modes back to the longer wavelength ones, until a statistically stationary turbulent state is reached. For p = 3.5 +0.5i and q = 8.0+ 0.9i , it is found that the energy is mainly concentrated in the wave vector region |k|≥100 of the energy spectrum, i.e., the inverse cascade is limited to the shorter wavelength modes with |k|≥100 . Furthermore, for the imaginary part p_i (which represents viscous damping) of the group dispersion coefficient in the range 0.1 < p_i <1.0, it is found that the region of inverse cascade gradually shrinks with increase of p _i( <1.1) . That is, in the range considered, the viscous damping coefficient p _i acts like a control switch, and it can regulate the degree of inverse cascade and the final results.

Key words: Nonlinear Schr?dinger equation, Inverse cascade, Turbulence, Viscous damping

摘要： 在外部势场和初始空间局部有限振幅脉冲给定的情况下，基于二维广义非线性薛定谔方程i∂_tE +p _▽²E+[V (x,y )+q|E|²] E =0 ，研究了波的不稳定性随时间的演化。波的不稳定性演化主要依赖于方程里的群色散系数 p = p_r+ip_i 和非线性系数 q= q_r+ iq_i ，数值结果发现系统会出现调制不稳定性、波坍缩、逆级联以及整个空间的湍流态。当 p = 3.5 +0.5i, q=q_r+iq_i 时，数值结果发现系统在出现部分逆级联之后波场的能量主要聚集在波矢量 k 空间中半径 |k| ≥100 的短波区域，同时形成了以区域中心为圆心，半径 |k|≈100 的圆形相对稳定区域。当粘性阻尼系数 p _i在[0.1,1.0] 时，数值结果发现产生逆级联的部分区域随着 p _i( <1.1) 的不断增大而逐渐缩小，相对稳定的圆域半径从零开始逐渐增大。因此，粘性系数p _i 相当于系统中的一个调节开关，调整它在一定范围 ( p _i≤1 )的取值可以控制逆级联的进程与最终结果。

关键词: 非线性薛定谔方程, 逆级联, 湍流, 粘性阻尼

CLC Number:

O534+ 2

CUI Shao-yan , YU Ming-yang. Inverse cascade based on a two-dimensional generalized nonlinear Schrödinger equation [J]. Nuclear Fusion and Plasma Physics, 2023, 43(3): 352-358.

崔少燕, 郁明阳. 基于二维广义非线性薛定谔方程的逆级联现象[J]. 核工业西南物理研究院, 2023, 43(3): 352-358.

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Inverse cascade based on a two-dimensional generalized nonlinear Schrödinger equation

基于二维广义非线性薛定谔方程的逆级联现象

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