Particle dispersion by random waves in the rotating Boussinesq system

We present a theoretical and numerical study of horizontal particle dispersion due to random waves in the three-dimensional rotating and stratified Boussinesq system, which serves as a simple model to study the dispersion of tracers in the ocean by the internal wave field. Specifically, the effective one-particle diffusivity in the sense of Taylor (Proc. Lond. Math. Soc., vol. 20, 1921, p. 196) is computed for a small-amplitude internal gravity wave field modelled as a stationary homogeneous and horizontally isotropic Gaussian random field whose frequency spectrum is bounded away from zero. Dispersion in this system does not arise simply because of a Stokes drift effect, as in the case of surface gravity waves, but in addition it is driven by the nonlinear, second-order corrections to the linear velocity field, which can be computed using the methods of wave-mean interaction theory. A formula for the one-particle diffusivity as a function of the spectrum of the random wave field is presented. It is shown that this diffusivity is much smaller than might be expected from heuristic arguments based on the magnitude of the Stokes drift or the pseudomomentum. This appears to stem from certain incompressibility constraints for the Stokes drift and the second-order velocity field. Finally, the theory is applied to oceanic conditions described by a typical model wave spectrum, the Garrett-Munk spectrum, and also by detailed field observations from the North Atlantic tracer release experiment.