Geomtry and Fluids - Langrangian averaged alpha models for fluid dynamics

An "Alpha Model" is a model whose principle modeling mechanism is through the time-reversible material derivative rather than diffusion. These models include many variants such as Leray, Voigt, LANS-alpha. They have different behavior compared to eddy diffusion. These models include many variants such as Leray, Voigt, LANS-alpha. They have different behavior compared to eddy diffusion. In fact, one of the areas where more understanding is required, is understanding that these types of models often do not dissipate the energy at the small scales; they dissipate the 'turning energy'. As an example of how the nonlinearity is modified, see the equation below, eddy kinetic energy

Though there is scientific conversation about physical interpretations of the model (for example, is it, or is not, related to Generalized Lagrangian Mean theory?) an alternative view is that these are mathematical regularizations of the equations of motion. One of the original such models was due to Leray, 1942 who was trying to analyze the Navier-Stokes Equations.  He split the advective and advecting velocities and related them by a Helmholtz operator. In the absence of viscosity this operation is time-reversible.

Then in 1998, Holm et al. were able to derive a set of equations using ideas from geometric mechanics which adds an additional nonlinear term to Leray’s equations allowing them to preserve Kelvin’s Circulation. The quantities that describe the total energy and enstrophy change.

With my colleagues Matthew Hecht, Mark Petersen, and Darryl Holm we have investigated models like these for ocean physics. In ocean modeling, one of the most important processes to model is the conversion of available potential energy to kinetic. In our paper Baroclinic Instability of the two-layer quasigeostrophic alpha model we compared the time-reversible mechanism of the alpha model to a standard dissipative model to show that the Alpha models allow the baroclinic instability to exist on coarser meshes, while dissipative mechanisms overwhelm it with damping. For one project where we looked at how well the model could account for the transfer of available potential energy to kinetic energy we get the following results:

Figure 1, above. This figure shows 4 different resolutions for the POP ocean model at .8, .4, .2, and .1 degree grid spacing. This is an idealised representation of the Antarctic Circumpolar Current (ACC) with walls on the north and south boundaries and periodic boundary conditions in longitude.

Figure 2, above, shows the kinetic energy for 3 different model resolutions. The alpha model gives statistics like doubling the resolution.

Figure 3, left This figure shows the eddy kinetic energy for 3 model resolutions. The eddy kinetic energy is difficult to reproduce at coarse resolutions because coarse models do not represent baroclinic instability well. Again, the effect of the model is to give results like doubling the number of grid points.

Figure 4, right shows the depth of the 6C isotherm. Again the alpha model is getting close to giving results like a doubling of resolution.

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