Some Problems Related to Fluid and Geophysical Fluid Dynamics

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The investigator studies three classes of problems relatedto fluid and geophysical fluid dynamics. The first class ofproblems involves connections between the Navier-Stokes equationsof incompressible fluids and conventional turbulence theory. Theinvestigator studies links between the estimates of energydissipation rate of three-dimensional boundary-driven flows basedon the Navier-Stokes equations and Kolmogorov's heuristicestimate of energy dissipation rate for turbulent flows, andconnections between the asymptotic degrees of freedom predictedby the Navier-Stokes equations (either as the fractal dimensionof the global or exponential attractor, or as the number ofdetermining modes, nodes or volume of the system) and those fromconventional turbulence theory in combination with the heuristicLandau-Lifschitz argument. The second class of problems isboundary layer behavior associated with incompressible viscousfluids whose dynamics are governed by the Navier-Stokesequations. Both the case with non-permeable boundary and the casewith permeable boundary are studied. Results are checked vianumerical simulation and laboratory data. Some relevant efficientnumerical algorithms are developed as well. The third clas ofproblems is related to geophysical flows, emphasizing emergenceand persistence of large scale coherent structures based on thebarotropic quasi-geostrophic model. Three prototype problems arestudied. The first is a selective decay problem for geophysicalflows that predicts the emergence of Rossby waves directly fromthe dynamic equations. The second is a spin-up problem withrandom forcing of small scale positive vortices. The third isthe barotropic quasi-geostrophic equations on the sphere. Fluids are indispensable to our life. They range from theair we breathe and water we drink to oceans and atmosphere thatcover the surface of our the earth. Study of the motion of thesefluids is crucial to us. At this stage full resolution of thecomplete dynamic equations lies beyond the capability of the mostpowerful computing system available today. Hence approximationmethods (both numerical and asymptotic) are needed to providequalitative and quantitative results on the behavior of thefluids for simplified but prototypical situations. The problemsproposed here are prototypical and understanding them willfacilitate our understanding of fundamental concepts related tofluid and geophysical fluid dynamics. Study of these typicalexamples will further our understanding of more realistic andsophisticated problems that are important in many engineeringapplications (such as wing design or turbulent combustion) aswell as in weather prediction, climatology and environmentalphenomena.