Collaborative Research: Stability, Wave Breaking, and Mixing in Stratified Flows

Milewski

DMS-0604635

Tabak

DMS-0604520

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.