Probability densities and preservation of randomness in wave turbulence

Abstract

Time evolution equation for the Probability Distribution Function (PDF) is derived for system of weakly interacting waves. It is shown that a steady state for such system may correspond to strong intermittency.Introduction -Wave Turbulence (WT) is a common name for the fields of dispersive waves which are engaged in stochastic weakly nonlinear interactions over a wide range of scales. Numerous examples of WT are found in oceans, atmospheres, plasmas and Bose-Einstein condensates [1][2][3][4][5][6][7][8][9]. For a long time, describing and predicting the energy spectra was the only concern in WT theory. More recently, some attention was given to the study of turbulence intermittency. WT intermittency, or "burstiness" of the turbulent signal, was observed experimentally and numerically and was attributed, as in most turbulent systems, to the presence of coherent structures. Examples include collapsing filaments in Bose-Einstein condensates with attractive potentials [9,10], condensate quasi-solitons in systems with repulsive potentials [9,11,12], white caps of sea waves at small scales [13], freak ocean waves at larger scales [14]. Often, such coherent structures are intense but quite sparse so that in most of the space waves remain weakly nonlinear and mostly unaffected by these structures.

Wave turbulence

“…Note that the wave turbulence approach permits the description of statistical objects which are more general than the spectra, in particular, the one-mode and multi-mode probability density functions (PDFs) [3,[74][75][76]. The evolution of the one-mode PDF is interesting when the initial statistics has random phases and amplitudes, but is not Gaussian.Also, it predicts solutions with fluxes in the probability space which describe intermittency -an anomalously high probability of strong waves [3,75].…”

Weak magnetohydrodynamic turbulence and intermittency

“…In this case, vorticity filaments are aligned with the rotation axis, which means, in particular, that the presence of structures is not necessarily indicative of intermittency. The classical WT theory has been extended to the case of random amplitudes (with phases and amplitudes statistically independent), and with the introduction of generating functions the time evolution equation for the PDF of the amplitudes has been derived for three-and four-wave processes (Choi, Lvov & Nazarenko 2004;Nazarenko 2011). When wave breaking is present, intermittency is predicted with the tail of the PDF arbitrarily far from the exponential distribution required by the Gaussian form.…”

Anisotropic Wave Turbulence for Reduced Hydrodynamics with Rotationally Constrained Slow Inertial Waves

“…Akin to classical quasigeostrophic theory, nonlinear vertical advection, W∂ Z , is an asymptotically subdominant process and does not appear. However, unlike quasigeostrophic theory the velocity field is isotropic in magnitude with |u ⊥ | ∼ |W|, hence the appearance of a prognostic equation for W. Physically, the R-RHD (5) and (6) state that unbalanced vertical pressure gradients drive vertical motions that are materially advected in the horizontal, in turn, vortical stretching due to vertical gradients in W produce vortical motions. The vertical velocity, W, also generates an ageostrophic velocity fleld, u ag ⊥ , such that incompressibility, ∇ ⊥ · u ag 1⊥ + ∂ Z W = 0, holds to O(Ro).…”

“…The theory of weak wave dynamics (i.e., the stochastic theory of nonlinear wave interactions) has been extensively studied since the seminal works of Kadomstev [1], Galeev et al [2], Zakharov and Filonenko [3] and more recently reviewed by Zakharov [4], Balk [5], Choi et al [6] and Nazarenko [7]. This theory remains one of the few areas where a mathematical framework exists with predictive capabilities for studying the energetics and dynamics associated with fluid turbulence.…”