Lagrangian Averages, Averaged Lagrangians and Dimension Reduction in Modeling GFD Turbulence

Monday, February 11, 2002 - 2:00pm - 3:00pm
Keller 3-180
Darryl Holm (Los Alamos National Laboratory)
By using Averaged/Approximated Lagrangians one obtains a well known series of GFD model equations. Each of these preserves energetics and potential-vorticity/Kelvin-circulation dynamics at its own level of approximation. One is then faced with the additional task in these multiscale problems of reducing the number of degrees of freedom by finding an average description of the motion that incorporates the mean effects of the small scales on the large scales. This is the problem of subgrid-scale modeling.

Lagrangian averaging (LA) also preserves energetics and fluid transport properties in the process of averaging over fast times scales following Lagrangian fluid trajectories. This compatibility allows LA to be imposed at any level of this series of GFD model equations, in either order of approximation. The resulting nonlinear Lagrangian mean equations, however, are not closed.

We shall obtain closure by introducing a small amplitude expansion and Taylor's hypothesis of frozen-in turbulence into the Lagrangian at a given level of approximation, before averaging. This closure approximation also reduces the number of degrees of freedom by smoothing the solution.

Navier-Stokes models for incompressible turbulence and

GFD models (if time allows)