Dynamics of transport under random fluxes on the boundary

Abstract

The impact of boundary noise on the dynamical evolution of the scalar transport equation in shear flows is studied, taking off from earlier studies in shear-flow dispersion in internal waves, a mechanism for horizontal mixing in the ocean. In particular, we model a gravity current evolving under an assumed shear-flow. The transport equation is deterministic, with a noise term at the inlet boundary. This was motivated by observed seasonal fluctuations in some known sources of salty, dense water in the oceans, like the Red Sea overflow, as well as observed thermal and saline anomalies in the thermohaline circulation. The noises used were: Wiener white, Wiener colored, Lévy white, and Lévy colored noise. Lévy processes form a more general class of processes which are generally non-Gaussian in distribution, and may have infinitely many jumps in finite time. They have been used to model pollutant point-sources, the flight time of particles in vortices, and linear and nonlinear anomalous diffusion. The major finding was that white noises (Wiener and Lévy ) and colored Wiener noise all have the effect of impeding the diffusion process, by as much as 33%. However, colored Lévy noise (non-Gaussian, time-correlated) does not have this effect on diffusion. This would suggest that time-correlation is more important in distinguishing noises than the distribution of the process that produced the noise. This also explains why Lévy colored noise showed great sensitivity to the stability parameter α, while Lévy white noise is unaffected by its stability parameter.