Modern Applied Mathematics for the Atmospheric and Oceanic Sciences

Week 3: Stochastics

July 28 - August 2, 2003

Stochastic modeling represents a relatively new and exciting approach to old problems in geophysics. While statistical theories of geophysical flows and the study of stochastic differential equations are problems of long standing, geophysical problems are taking an increasingly stochastic outlook. This is evident in several lines of investigation: Ensemble forecasting is becoming more routine; stochastic models of climate and seasonal variability are becoming more common place ^{e.g., the Chorin-Germany group, see also 12}; stochastic representations of small scale processes, including atmospheric convection ^{4, 9} are also beginning to emerge. Despite the proliferation of stochastics in the geosciences, typical curricula in geophysics fail to keep up with these ideas, leading to students who are wholly unfamiliar with these ways of thinking. In addition, interesting problems in geophysics which appear amenable to stochastic treatments are often overlooked. For this reason a unit focusing on stochastics seems timely.

One particular avenue which we wish to explore in this respect is the use of particle models for geophysical problems. Such models have been used to good effect to describe complex processes in material science ^{e.g., 6, 7}, but are only beginning to be applied to problems in the earth sciences ^{e.g., 10}. In the stochastics week we would like to focus on the use of particle models to study problems in geophysics, particularly moist atmospheric convection. Additional topics include the description of tracer transport by ocean eddies ^{2, 3}, this being a particular point of resonance in the recent IPAM short program; as was the formulation of stochastic climate models through mode elimination ¹³.

Specific topics to be covered in the stochastic week will include:

• Atmospheric thermodynamics: Moisture variables, moist-adiabatic processes, virtual temperature, stability, phase changes.
• Moist Convection: Phenomenology, stability conditions, including meta-stability. Convective available potential energy (CAPE), convective inhibition, down-draft CAPE, radiative convective equilibrium, dry and moist-convective scaling.
• Kinetic theory: Newtonian dynamics, probabilistic formulation and Liouville equation, derivation of kinetic equations (Boltzmann, Vlassov, etc), conservation laws, H-theorem, relative entropy, asymptotic limits, moment closures and parameterization, radiation.
• Interacting particle systems: Discrete Markov processes, random walks, simple examples of interacting particle systems (simple exclusion and zero-range processes), Ising-type systems and dynamics, particle systems and relaxation, coarse-graining procedures, hydrodynamic limits to PDE and stochastic PDE, hierarchies of parameterizations in atmospheric convection.
• Stochastic Modeling: Introduction to Stochastic calculus and SDE, invariant measures, stochastic Langevin equations, large deviations and WKB expansions, multiple scales and averaging limits, stochastic climate modeling.