wave turbulence theory
Figure 1: (left) energy cascade from large to small vortices in flow turbulence; and (middle) from long to short waves in wave turbulence; (right) similar energy spectrum for flow or wave turbulence.
Figure 2: Typical evolution of spectrum with time predicted by the numerical simulation: Initial spectrum (dashed line); fully developed spectrum (solid line); Decayed spectra in quasi-stationary state due to dissipation (single dotted dash line and double dotted dash line). The slope corresponding to the analytical solution of -19/4 is indicated.
Developed by Zakharov, wave turbulence theory (WTT) describes the stationary field of an ensemble of waves in weakly nonlinear interactions. The equilibrium wave spectrum I(k) can be obtained from the theory as an analytical power-law solution over a range of wavenumber k, carrying an energy flux P from large to small scales. This solution can be understood in analogy with Kolmogorov’s description of the cascade process in the inertial range of turbulent flows (see figure 1 for an illustration).
Our study on WTT started from capillary waves (ocean surface waves at small scales), which play an important role in the upper ocean dynamics, and physically represents a class of dispersive waves allowing triad resonance on a two-dimensional surface. In theory, we reformulated the derivation, leading to I=CP1/2k-19/4, with the Komogorov constant C=2π×6.97, corrected from the previous value of 9.85 derived 15 years ago. Then we established a numerical simulation of the primitive Euler equations (see figure 2 for a typical simulation), which successfully confirmed the new analytical result. This resolves a long-term debate on the value of C due to previous inconsistent analytical, numerical and experimental attempts.
With the theoretical solution confirmed, we further studied the effect of assumptions (e.g. weak nonlinearity, infinite domain) involved in the derivation. This is important in understanding the realistic spectrum. We investigated particularly the effect of a finite domain (e.g. waves in a finite tank) on the capillary wave spectrum, starting from numerical simulations. The simulations revealed that the spectrum tends to have its power-law slope steepened and energy flux decreased as the domain size decreases. These variations were found to be caused by the wavenumber discreteness due to the finite domain, which to some extent suppresses the nonlinear interactions. Inspired by these findings, we developed a theoretical model named quasi-resonant kinetic equation which describes the property of stationary spectrum in a finite domain.
Our current study involves several aspects of wave turbulence, including turbulence closure for discrete wave system, wave turbulence of surface and internal gravity waves, MMT spectra, etc.
Publication:
Hrabski, A. and Pan, Y. 2020, The Effect of Discrete Resonant Manifold Structure on Discrete Wave Turbulence, to appear in Physical Review E. https://arxiv.org/abs/2003.10801
Pan, Y. and Yue, D.K.P. 2017, Understanding discrete capillary wave turbulence using quasi-resonant kinetic equation, Journal of Fluid Mechanics, 816, R1. 1-11. (PDF)
Pan, Y. and Yue, D.K.P. 2015, Decaying capillary wave turbulence under broad-scale dissipation, Journal of Fluid Mechanics, 780, R1. 1-11. (PDF)
Pan, Y. and Yue, D.K.P. 2014, Direct numerical investigation of turbulence of capillary waves, Physical Review Letters, 113, 094501.(PDF)