I am actively continuing my research program, as described below. My ongoing investigations are partially funded by the Simons Foundation under their program, Collaboration Grants for Mathematicians; project title: Nonequilibrium Statistical Models of Turbulent Dynamics.
For over twenty years my reseach program has focused on the interface between nonlinear partial differential equations and statistical mechanics. In general terms, I have been interested in adapting and applying the ideas and techniques of statistical mechanics to complex systems that are "turbulent" in the broad sense that they exhibit flucuations over a range of scales.
Together with a number of colleagues (R.S. Ellis, N. Whitaker, A. Majda, M. DiBattista) and graduate students (J. Heisler, C. Boucher, K. Haven) I have studied the modern statistical equilibrium theories of coherent structures in two-dimensional and quasi-geostrophic turbulence. These theories were first proposed in their modern form by R. Robert and J. Miller (independently), who devised a way to incorporate all the dynamical invariants of vorticity transport into a mean-field model of large-scale structures. This approach has now evolved in a formulation that is more directly relevant to physical applications, in which a prior distribution on small-scale vorticity flucuations is given and constraints on the large-scale dynamical invariants (energy, circulations, impulse) are imposed. The theory then determines the link between the statistical properties of the unresolved, small-scale fluctuations of vorticity and the mean-field equations that governs the coherent, large-scale flow.
A vivid real application of this approach is to the prediction of the persistent flows in the weather layer of Jupiter -- namely, the reversing zonal shear flow and embedded vortices such as the Great Red Spot. My colleagues and I showed that, in a statistical model based on one-and-a-half layer quasi-geostrophic dynamics in a zonal band, the Great Red Spot emerges at 23 degrees south when the small-scale potential vorticity fluctuations have an anticyclonic skewness, but that in the comparable northern hemisphere band strong zonal shears persist. The skewness of the fluctuations models the effects of convective forcing of the weather layer, as confirmed by Galileo data. The unseen zonal mean flow that underlies the weather layer is inferred from Voyager data, using a method of T. Dowling. Other investigators (Bouchet, Chavanis and Robert) have used a similar theory to obtain the shape and internal structure of the GRS. All these theoretical predictions agree remarkably well with the real observed phenomena, even though they are based on a relatively simple model.
This application of statistical mechanics is novel in the sense that the systems have long-range interactions, and hence local mean-field theory is exact in the appropriate continuum limit. Moreover, the most interesting equilibrium states occur when the absolute temperature is negative. For these reasons, we have made a rigorous analysis of these probabilistic models, studying their thermodynamics properties, the equivalence and nonequivalence of their defining (microcanonical and canonical) ensembles, and the nonlinear stability of their equilibrium states.
The same sort of approach may be applied to other dynamical systems which excite fluctuations on small scales but generate coherent structures on large scales. My colleagues (R. Jordan, R.S. Ellis), graduate students (P. Otto, A. Eisner) and I have studied nonintegrable, nonlinear Schroedinger equations as other examples of this behavior, analyzing their Gibbs distributions and comparing the predictions of the equilibrium statistics with long-time numerical simulations. The basic result is that such a dispersive wave system organizes into a ground state solitary wave on the large scales together with a Gaussian field of wave radiation on the small scales. A very similar behavior occurs for MHD turbulence, owing to the form of its dynamical invariants.
In my recent work I am focusing on nonequilibrium behavior of such systems, and I am developing a theoretical framework that connects microscopic dynamics to macroscopic predictions, as in the equilibrium context. I combine ideas from nonequilibrium statistical mechanics with information theory and optimization and control theory in a new approach to formulating reduced models of complex systems. This approach is intended to yield computationally tractable statistical closures for turbulent systems. I have published a general framework for deriving optimal closures of essentially arbitrary Hamiltonian systems in terms of a selected set of resolved variables. Using this approach my collaborators (R. Kleeman, S. Talabard, Q.Y. Chen) and I have implemented and validated closed reduced models of the spectrally-truncated Burgers equation, a shell model of turbulence, and the two-dimensional Euler equation. These results rely on the assumption that the unresolved scales of motion remain in statistical equilibrium, and in that sense they can be viewed as being near-equilibrium reductions. Currently we are therefore pursuing research that will allow us to compute predictive reduced models in regimes in which these conditions are not met, as in turbulent cascades. At the moment our focus remains on deriving closed, dissipative macrodynamics from underlying, reversible microdynamics by best-fitting a statistical-dynamical model. In far-from-equilibrium regimes, though, our procedure results in reduced dynamics that have the form of optimal control problems --- the mean, resolved variables play the role of state variables, while the irreversible fluxes are analogous to control (or input) variables. The predictions of the reduced model are then obtained numerically by using optimal control algorithms.