Transcript Math/Oceanography Research Problems

Math/Oceanography Research Problems
Juan M. Restrepo
Mathematics Department
Physics Department
University of Arizona
BEFORE KATRINA
AFTER KATRINA
Themes
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Geophysical Fluid Dynamics
Mathematics
Computational
Engineering
Education
Geophysical Fluid Dynamics
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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 nearsurface currents: wave boundary layers in ocean
and atmosphere.
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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.
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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.
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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, sedimentladen flows, buoyancy/mixing effects,
wave/currents, topographycal effects.
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Sediment dynamics and the structure of large
scale moving features of the bottom.
Near-shore erosion by waves, rip currents, longshore currents. Some of these structures are:
shore-connected ridges, ridge/runnel, shoreparallel bars.
Rogue waves and the generation of tsunamis.
The modeling of oceanic and atmospheric CO2
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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).
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Multi-resolution in data assimilation to handle the largedata 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.
Computational
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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.
Engineering
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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.
Educational (training scientists)
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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).