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

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MilewskiDMS-0604635TabakDMS-0604520 The atmosphere and ocean are stratified fluids and as suchsupport the propagation of disturbances through internal waves. These internal waves may deform nonlinearly and break byoverturning, leading to the mixing of the ambient fluid. Boththe atmosphere and ocean also display strong shear flows that maybecome unstable, producing rolls that can also lead to mixing andlocal homogenization of the density. The investigators study theissue of which of these two processes prevails in a given flowconfiguration. Based on preliminary work, the investigatorsconjecture that in the shallow water regime there is a sharpboundary below which the dynamics disallow shear instabilities,leaving only wave breaking as the possible mixing mechanism. Inmathematical terms, they consider systems of partial differentialequations of mixed type, where the hyperbolic domain correspondsto the internal waves and the elliptic domain to shearinstability. The question of nonlinear stability of the flow canthen be formulated in terms of whether the solutions themselvescan make the system become elliptic. The investigators haveproved that this cannot happen for a simple system and hereextend the result to much more general scenarios. In addition tothis stability result, they propose a closure that quantifies themixing taking place when waves break. Understanding and quantifying fluid mixing is a keyingredient in global weather and climate studies. The atmosphereand ocean are stratified fluids: fluids whose density varies(primarily) with height due to temperature, salinity and othereffects. Stratified fluids allow for the propagation of internalwaves, 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 ofdifferent speeds. In this project the investigators study whichof these two effects is more likely to prevail given the ambientconditions. Such a study has far-reaching implications: theatmospheric and ocean mixing layers control the coupling betweenthe two, and hence exert a critical control on the evolution ofthe climate. The work advances the predictive capabilities ofcoupled atmosphere-ocean models, by improving theirparameterization of fluid entrainment and mixing. It also trainsundergraduate and graduate students in the use of appliedmathematical tools for the advancement of the natural sciences.