Cai D, Majda A J, McLaughlin D W, Tabak E G
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA.
Proc Natl Acad Sci U S A. 1999 Dec 7;96(25):14216-21. doi: 10.1073/pnas.96.25.14216.
Dispersive wave turbulence is studied numerically for a class of one-dimensional nonlinear wave equations. Both deterministic and random (white noise in time) forcings are studied. Four distinct stable spectra are observed-the direct and inverse cascades of weak turbulence (WT) theory, thermal equilibrium, and a fourth spectrum (MMT; Majda, McLaughlin, Tabak). Each spectrum can describe long-time behavior, and each can be only metastable (with quite diverse lifetimes)-depending on details of nonlinearity, forcing, and dissipation. Cases of a long-live MMT transient state dcaying to a state with WT spectra, and vice-versa, are displayed. In the case of freely decaying turbulence, without forcing, both cascades of weak turbulence are observed. These WT states constitute the clearest and most striking numerical observations of WT spectra to date-over four decades of energy, and three decades of spatial, scales. Numerical experiments that study details of the composition, coexistence, and transition between spectra are then discussed, including: (i) for deterministic forcing, sharp distinctions between focusing and defocusing nonlinearities, including the role of long wavelength instabilities, localized coherent structures, and chaotic behavior; (ii) the role of energy growth in time to monitor the selection of MMT or WT spectra; (iii) a second manifestation of the MMT spectrum as it describes a self-similar evolution of the wave, without temporal averaging; (iv) coherent structures and the evolution of the direct and inverse cascades; and (v) nonlocality (in k-space) in the transferral process.
针对一类一维非线性波动方程，对色散波湍流进行了数值研究。研究了确定性强迫和随机（时间上的白噪声）强迫。观察到四种不同的稳定谱——弱湍流（WT）理论的正向和反向级联、热平衡以及第四种谱（MMT；马伊达、麦克劳克林、塔巴克）。每种谱都可以描述长时间行为，并且每种谱都可能只是亚稳的（具有相当不同的寿命）——这取决于非线性、强迫和耗散的细节。展示了长寿命的MMT瞬态衰减到具有WT谱的状态的情况，反之亦然。在自由衰减湍流的情况下，即没有强迫时，观察到了弱湍流的两种级联。这些WT状态构成了迄今为止对WT谱最清晰、最显著的数值观测——跨越了四个数量级的能量和三个数量级的空间尺度。然后讨论了研究谱之间的组成、共存和转变细节的数值实验，包括：（i）对于确定性强迫，聚焦和散焦非线性之间的明显区别，包括长波长不稳定性、局域相干结构和混沌行为的作用；（ii）能量随时间增长在监测MMT或WT谱选择中的作用；（iii）MMT谱的第二种表现形式，即它描述了波的自相似演化，无需时间平均；（iv）相干结构以及正向和反向级联的演化；以及（v）转移过程中的非局域性（在k空间中）。
