An Oceanic Ultra-Violet Catastrophe, Wave-Particle Duality and a Strongly Nonlinear Concept of Geophysical Turbulence

Abstract: 

Nonlinear interactions between high frequency internal waves interacting with larger vertical and horizontal scale waves having inertial frequency are investigated using ray tracing techniques, analytic approximations to kinetic equations, solutions for the moments of a diffusive approximation to the resonant kinetic equation and Taylor's identity for relative dispersion. Tracing high frequency waves in one and two inertial wave backgrounds demonstrates that the infinitesimal amplitude and finite amplitude limits are phenomenologically distinct: the finite amplitude state is characterized by the coalescing of the two small scale members of the triad and a transition to a bound wave phenomena. This coalescence marks the transition from the coupled oscillator paradigm to a particle (wave packet) in a potential well paradigm at extremely small amplitude. Tracing high frequency waves in stochastic inertial wave backgrounds does not reveal any such transition. Rather, the ray tracing results are phenomenologically consistent with the particle in a (stochastic) well paradigm, independent of amplitude.

Tracing high frequency waves in a stochastic background of inertial oscillations provides estimates of the temporal evolution for the ensemble mean and variance of vertical wavenumber of a test wave distribution. These estimates are compared to the evolution of the first and second moments of Fokker-Plank equations derived from the resonant kinetic equation and a slowly varying approximation to the Landau equation in which dispersion of wave packets in the spectral domain is equivalent to particle dispersion given by Taylor's identity for Lagrangian particles. Despite the disparate phenomenology noted above, the resonant closure manages to describe the evolution of the first two moments at energy levels an order of magnitude smaller than background oceanic values and predicts no transport of action to smaller scales. At realistic energy levels the growth of the second moment is inhibited relative to the first, implying a finite downscale action transport. The resulting flux, though, exceeds that anticipated for oceanic parameters by an order of magnitude. We argue that this is a defect of the one-dimensional model and demonstrate that the coupled oscillator paradigm is fundamentally unable to represent wave refraction in inhomogeneous flows having dimensions greater than one.