The atmosphere and ocean are stratified fluids and as such support the propagation of disturbances through internal waves. These internal waves may deform nonlinearly and break by overturning, leading to the mixing of the ambient fluid. Both the atmosphere and ocean also display strong shear flows that may become unstable, producing rolls that can also lead to mixing and local homogenization of the density. The investigators study the issue of which of these two processes prevails in a given flow configuration. Based on preliminary work, the investigators conjecture that in the shallow water regime there is a sharp boundary below which the dynamics disallow shear instabilities, leaving only wave breaking as the possible mixing mechanism. In mathematical terms, they consider systems of partial differential equations of mixed type, where the hyperbolic domain corresponds to the internal waves and the elliptic domain to shear instability. The question of nonlinear stability of the flow can then be formulated in terms of whether the solutions themselves can make the system become elliptic. The investigators have proved that this cannot happen for a simple system and here extend the result to much more general scenarios. In addition to this stability result, they propose a closure that quantifies the mixing taking place when waves break.
Understanding and quantifying fluid mixing is a key ingredient in global weather and climate studies. The atmosphere and ocean are stratified fluids: fluids whose density varies (primarily) with height due to temperature, salinity and other effects. Stratified fluids allow for the propagation of internal waves, and these waves may eventually break and mix the fluid. Another possible source of mixing is due to shear instabilities: the formation of eddies at the interface between flows of different speeds. In this project the investigators study which of these two effects is more likely to prevail given the ambient conditions. Such a study has far-reaching implications: the atmospheric and ocean mixing layers control the coupling between the two, and hence exert a critical control on the evolution of the climate. The work advances the predictive capabilities of coupled atmosphere-ocean models, by improving their parameterization of fluid entrainment and mixing. It also trains undergraduate and graduate students in the use of applied mathematical tools for the advancement of the natural sciences.
