Prof. Brad Marston
Professor of Physics - Brown University
Direct Statistical Simulation of Flows by Expansions in Cumulants
Thursday, March 13, 2014, 4:00pm to 5:00pm | Room 5-134
Low-order statistics of model geophysical and astrophysical fluids may be directly accessed by solving the equations of motion for the statistics themselves as proposed by Lorenz in 1967. I implement such Direct Statistical Simulation by systematic expansion in equal-time cumulants. Live simulations are performed using a barotropic model on the sphere to illustrate the approach. The first cumulant is the zonally averaged vorticity as a function of latitude, and the second and higher cumulants encode information about nonlocal teleconnections. No assumptions of homogeneity or isotropy are imposed. Closure of the equations of motion at second order (CE2) is realizable and retains the eddy -- mean-flow interactions, but neglects eddy-eddy interactions. Eddy-eddy interactions appear at third (CE3) order, but care must be taken to maintain realizability with a non-negative probability distribution function.
The cumulant expansion is conservative, order-by-order, in the absence of forcing and dissipation: CE2 conserves the total angular momentum, total energy, and enstrophy; CE3 further conserves the cubic vorticity Casimir, and CE-N conserves N Casimirs. An intermediate approximation, CE2.5, is related to the Eddy-Damped Quasi-Normal Markovian (EDQNM) approximation and maintains realizability at the expense of the introduction of a phenomenological timescale for eddy damping. First and second cumulants accumulated by numerical simulation are compared with those obtained at fixed points found at CE2, CE2.5, and CE3 levels of approximation. CE2 reproduces qualitative features of the zonal jets. CE2.5 and CE3 improve quantitative agreement in both the zonal means, and in the teleconnections. I conclude by discussing two-layer primitive equations, including a comparison with statistics obtained from reanalysis of the mid-level atmosphere of the Earth.
Brad Marston joined the Brown Physics Department in 1991. A graduate of Caltech, he received his Ph.D. from Princeton University in 1989. He did postdoctoral work at Cornell University and was a visiting scientist at the Institute for Theoretical Physics at UC Santa Barbara, a visiting professor at MIT, and a visiting associate at Caltech.