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

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

2. BEFORE KATRINA

3. AFTER KATRINA

4. Themes • Geophysical Fluid Dynamics • Mathematics • Computational • Engineering • Education

5. 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). • Interaction of surface waves, wind, and near-surface currents: wave boundary layers in ocean and atmosphere.

6. Improvements in turbulence/mixing parameterizations (via measurements, large eddy 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.

7. 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.

8. 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. • River/Ocean Environments: plumes, sediment-laden flows, buoyancy/mixing effects, wave/currents, topographycal effects.

9. 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. • Rogue waves and the generation of tsunamis. • The modeling of oceanic and atmospheric CO2 .

10. Mathematical Estimation: • nonlinear/non-Gaussian problems, especially in Lagrangian frame where these are more critical. • Translating the statistics of measurements from Lagrangian to Eulerian frames. • 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).

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

12. Computational • 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. • Large-scale filter/smoothers and data insertion. • Sensitivity analysis packages. • Arnoldi and QZD packages for stability calculations.

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

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