Numerical investigation of turbulence of surface gravity waves

Abstract

In this paper, we numerically study the wave turbulence of surface gravity waves in the framework of Euler equations of the free surface. The purpose is to understand the variation of the scaling of the spectra with wavenumber $k$ and energy flux $P$ at different nonlinearity levels under different forcing/free-decay conditions. For all conditions (free decay and narrow-band and broad-band forcing) that we consider, we find that the spectral forms approach the wave turbulence theory (WTT) solution $S_\eta \sim k^{-5/2}$ and $S_\eta \sim P^{1/3}$ at high nonlinearity levels. With a decrease of nonlinearity level, the spectra for all cases become steeper, with the narrow-band forcing case exhibiting the most rapid deviation from WTT. We investigate bound waves and the finite-size effect as possible mechanisms causing the spectral variations. Through a tri-coherence analysis, we find that the finite-size effect is present in all cases, which is responsible for the overall steepening of the spectra and the reduced capacity of energy flux at lower nonlinearity levels. The fraction of bound waves in the domain generally decreases with the decrease of nonlinearity level, except for the narrow-band case, which exhibits a transition at a critical nonlinearity level below which a rapid increase is observed. This increase serves as the main reason for the fastest deviation from WTT with the decrease of nonlinearity in the narrow-band forcing case.

Do surface gravity waves have a frozen turbulence state?

“…Such a transition implies a bifurcation of solutions depending on the initial condition of energy distribution in discrete turbulence of surface gravity waves. In addition, we remark that the frozen turbulence observed here for small r D is consistent with the simulation of primitive Euler equations at low nonlinearity level [13], which suggests a frozen turbulence state when the forcing is located in the range of k ∈ [1,19].…”

“…The importance of (1) in understanding the wave dynamics at low nonlinearity has been demonstrated in capillary wave turbulence [7][8][9], MMT turbulence [10] and surface gravity wave turbulence [11][12][13] (also see review in [14]). * yulinpan@umich.edu…”

“…For R → ∞, this is a very hard number theoretic problem (through private communication with number theorists) which is beyond our scope in this paper. We will instead consider an S R with finite R ∼ 10 3 that is compatible with (or beyond) most state-of-the-art simulations [13,[20][21][22][23][24][25][26][27] performed for wave turbulence.…”

A multiple time scale approach for anisotropic inertial wave turbulence

“…2022; Rodda et al. 2022; Zhang & Pan 2022; Dematteis & Lvov 2023; Galtier 2023 a ; Lanchon et al. 2023; Novkoski, Pham & Falcon 2023; Zhu et al.…”

Feynman rules for forced wave turbulence

“…A 3. This includes: elastic plate wave turbulence[8][9][10][11][12][13][14], turbulence in surface gravity waves[15][16][17][18][19], in gravitational waves[20][21][22][23], acoustic turbulence[24], planetary Rossby waves[25], waves on bubbles[26], emergent hydrodynamics[27], in a diatomic chain[28], in plasmas[29,30], optical waves[31][32][33], and rotating waves[34,35], and in FPUT chains[36].…”