Information Theory, Statistical Mechanics, and Predicting Jupiter's Red Spot

Information Theory, Statistical Mechanics, and Predicting Jupiter's Red Spot.

Feb 12, 2008
3:00pm-4:00pm, CORNETT B111
Galileo Journey to Jupiter This lecture blends ideas from probability theory, PDE’s, numerical analysis and physical reasoning in the style of modern applied mathematics. First elementary concepts of information theory involving the lack of information in one probability measure relative to another are introduced. These ideas are then applied to give a unified explanation of the competing equilibrium statistical theories for coherent structures in fluids. The potential application of these ideas to predict coherent structures in flows with random forcing and dissipation is then discussed briefly including the first theorem (Xiaoming Wang and Majda, CPAM 2006) justifying the approach. All of this is utilized to develop a successful prediction for the exact location and structure of Jupiter’s Red Spot in agreement with the Voyager Mission of the 1970’s by utilizing a physically based prior encoding the key features of small scale observations from the Galileo mission of the 1990’s. An apriori independent justification of this overall strategy is presented through novel numerical methods for detecting statistically relevant conserved quantities in fluid flow (Abramov and Majda, 2003, 2004.) The lecturer’s recent book with X. Wang, Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flow, Cambridge Univ. Press (2006) discusses in detail most of the material presented here.

Applied & Theoretical Challenges for Multi-Scale Hyperbolic PDE's in the Tropics:

Feb 14, 2008
3:30pm-4:30pm, ECS 125
Geophysical flows are a rich source of novel problems for applied mathematics and the contemporary theory of partial differential equations. The reason for this is that many physically important geophysical flows invlolve complex nonlinear interaction over multi-scales in both time and space so developing simplified reduced models which are simpler yet capture key physical phenomena is of central importance. In mid-latitudes, the fact that the rotational Coriolis terms are bounded away from zero leads to a strict temporal frequency scale seperation between slow potential potential vorticity dynamics and vast gravity waves: this physical fact leads to new theorems justifying the quasi-geostrophic midlatitude dynamics even with general unbalanced initial data for both rapidly rotating shallow water equations and completely stratified flows. At the equator, the tangential projection of the Coriolis force from rotation vanishes identically so that there is no longer a time scale seperation between potential vortical flows and gravity waves. This has profound consequences physically that allow the tropics to behave as a waveguide with extremely warm surface temperatures. The resulting behavior profoundly influences longer term mid-latitude weather prediction and climate change through hurricanes, monsoons, El Nino, and global teleconnections with the midlatitude atmosphere. How this happens through detailed physical mechanisms is one of the most important contemporary problems in the atmosphere-ocean science community with a central role played by nonlinear interactive heating involving the interaction of clouds, moisture, and convection. The variable coefficient degeneracy of the Coriolis term at the equator alluded to earlier leads to both important new physical effects as well as fascinating new mathematical phenomena and PDE's. In this equatorial context, new multi-scale reduced dynamical PDE models are even relatively recent in origin.