Math/Oceanography Research Problems Juan M. Restrepo Mathematics Department Physics Department University of Arizona

BEFORE KATRINA

AFTER KATRINA

Themes Geophysical Fluid Dynamics Geophysical Fluid Dynamics Mathematics Mathematics Computational Computational Engineering Engineering Education Education

Geophysical Fluid Dynamics Sub-mesoscale transition between large-scale rotating stratified flows that accurately satisfy a diagnostic force balance but have great difficulty in forward cascade of energy to small-scale dissipation and flows (these are very efficient dissipation mechanisms). Sub-mesoscale transition between large-scale rotating stratified flows that accurately satisfy a diagnostic force balance but have great difficulty in forward cascade of energy to small-scale dissipation and flows (these are very efficient dissipation mechanisms). Interaction of surface waves, wind, and near- surface currents: wave boundary layers in ocean and atmosphere. Interaction of surface waves, wind, and near- surface currents: wave boundary layers in ocean and atmosphere.

Improvements in turbulence/mixing parameterizations (via measurements, large eddy simulations, theoretical inspiration). Improvements in turbulence/mixing parameterizations (via measurements, large eddy simulations, theoretical inspiration). Improvements in wave-breaking parameterization (via measurements, simulations, theoretical inspiration). Improvements in wave-breaking parameterization (via measurements, simulations, theoretical inspiration). Computation/visualization/measurement of the spatio-temporal statistics of real sea surfaces. Computation/visualization/measurement of the spatio-temporal statistics of real sea surfaces.

The thermohaline circulation at high Rayleigh numbers (mixing): how differential surface buoyancy forcing accomplishes meridional overturning circulation and lateral buoyancy flux in the absence of additional sources of turbulence to assist in the diapycnal mixing. The thermohaline circulation at high Rayleigh numbers (mixing): how differential surface buoyancy forcing accomplishes meridional overturning circulation and lateral buoyancy flux in the absence of additional sources of turbulence to assist in the diapycnal mixing. Wave/current interactions: dissipative mechanisms and wind forcing across the different spatio-temporal scales. Wave/current interactions: dissipative mechanisms and wind forcing across the different spatio-temporal scales.

Ocean/Atmosphere GCM coupling for small scale and for large scale simulations. Ocean/Atmosphere GCM coupling for small scale and for large scale simulations. Non-stationary (in the statistical sense) scattering (acoustic/RF) from randomly-rough ocean surface and bottom surfaces. Non-stationary (in the statistical sense) scattering (acoustic/RF) from randomly-rough ocean surface and bottom surfaces. River/Ocean Environments: plumes, sediment- laden flows, buoyancy/mixing effects, wave/currents, topographycal effects. River/Ocean Environments: plumes, sediment- laden flows, buoyancy/mixing effects, wave/currents, topographycal effects.

Sediment dynamics and the structure of large scale moving features of the bottom. Sediment dynamics and the structure of large scale moving features of the bottom. Near-shore erosion by waves, rip currents, long- shore currents. Some of these structures are: shore-connected ridges, ridge/runnel, shore- parallel bars. Near-shore erosion by waves, rip currents, long- shore currents. Some of these structures are: shore-connected ridges, ridge/runnel, shore- parallel bars. Rogue waves and the generation of tsunamis. Rogue waves and the generation of tsunamis. The modeling of oceanic and atmospheric CO 2. The modeling of oceanic and atmospheric CO 2.

Mathematical Estimation: nonlinear/non-Gaussian problems, especially in Lagrangian frame where these are more critical. nonlinear/non-Gaussian problems, especially in Lagrangian frame where these are more critical. Translating the statistics of measurements from Lagrangian to Eulerian frames. Translating the statistics of measurements from Lagrangian to Eulerian frames. Coarse graining and speedup techniques for Monte Carlo. Coarse graining and speedup techniques for Monte Carlo. Near-optimal measurement coverage in space/time from moving platforms and gliders (design of trajectories, schedules, etc). Near-optimal measurement coverage in space/time from moving platforms and gliders (design of trajectories, schedules, etc).

Multi-resolution in data assimilation to handle the large- data set case (more a problem in meteorology). Multi-resolution in data assimilation to handle the large- data set case (more a problem in meteorology). Quantification of uncertainties. Quantification of uncertainties. Stochastic Lagrangian models of fluids. Stochastic Lagrangian models of fluids. New vortex filament based models. New vortex filament based models. Development of an analytical theory of Kolmogorov equations for stochastic fluids. Development of an analytical theory of Kolmogorov equations for stochastic fluids. Applications of Wiener chaos to de-coupling of the Reynolds equation. Applications of Wiener chaos to de-coupling of the Reynolds equation.

Computational Numerical methods/algorithms for the computation of oceanic domains with changing boundaries (due to erosion, ice, evaporation, tides). Numerical methods/algorithms for the computation of oceanic domains with changing boundaries (due to erosion, ice, evaporation, tides). Wave run-up in 3-space dimensions. Wave run-up in 3-space dimensions. Large-scale filter/smoothers and data insertion. Large-scale filter/smoothers and data insertion. Sensitivity analysis packages. Sensitivity analysis packages. Arnoldi and QZD packages for stability calculations. Arnoldi and QZD packages for stability calculations.

Engineering Mesh generation for GCM. Mesh generation for GCM. Topographical tools (along the lines of GMT) that will input real topographies and shorelines into numerical models. Topographical tools (along the lines of GMT) that will input real topographies and shorelines into numerical models. Visualization and analysis tools. Visualization and analysis tools. Integration of tools such as clawpack and AMR Integration of tools such as clawpack and AMR Update isopycnic models (effort commensurate to POP). Update isopycnic models (effort commensurate to POP). GRID to couple models/data across institutions. GRID to couple models/data across institutions.

Educational (training scientists) Geophysical fluid dynamics (1 year). Geophysical fluid dynamics (1 year). Inverse methods and sensitivity analysis. (1/2 year). Inverse methods and sensitivity analysis. (1/2 year). Linear algebra, part of 1 year numerical analysis. Linear algebra, part of 1 year numerical analysis. Stochastic processes. (1 year, part of applied probability theory). Stochastic processes. (1 year, part of applied probability theory). Dynamical systems (1/2 year). Dynamical systems (1/2 year). Estimation theory (including map-making, data assimilation, fitting). (1/2 year). Estimation theory (including map-making, data assimilation, fitting). (1/2 year).