Implementation of the thickness-weighted averaged residual-mean flow equations in an ocean general circulation model

It has recently been shown (Young 2012, Maddison and Marshall 2013) that in the thickness-weighted averaged (TWA) Boussinesq equations for the prognostic residual-mean flow first introduced by DeSzoeke and Bennet (1993), eddy-mean flow interactions appear in the horizontal momentum equations as the divergence of the Eliassen-Palm Flux tensor (EPFT). It has been proposed that, given its physical interpretation and exact mathematical tractability, the TWA is an appropriate framework for modeling ocean circulation with parameterized eddies. We test the feasibility of this proposition by implementing the TWA framework in the Model for Prediction Across Scales - Ocean (MPAS-O). We investigate the connections between the TWA framework and the conventional framework used in models, where Eulerian-mean flow prognostic variables are solved for.

We parameterize eddies in the TWA framework by representing eddy form drag as vertical diffusion of horizontal momentum and ignoring Reynolds stresses in the EPFT. This results in the parameterization proposed by Greatbatch and Lamb (1990) (GL). After assuming geostrophic balance, this reduces to the Gent and McWilliams (1990) (GM) representation of eddies through the bolus velocity used to advect tracers in the Eulerian-mean flow equations. With the exact TWA framework as a starting point, we show that using GM to represent eddies in the Eulerian-mean flow equations results from ignoring the time tendency and inertial terms associated with the bolus velocity in the TWA residual-mean flow equations. Using Ertel Potential vorticity, we show that the GL parameterization in the TWA framework and the GM parameterization in the Eulerian-mean flow framework impose the same eddy form drag on the flow. After implementing the TWA framework in MPAS-O, we verify our analysis by comparing the flows in an idealized Southern Ocean configuration simulated using the TWA and Eulerian-mean flow frameworks.

References:

De Szoeke, R. A., and A. F. Bennett, 1993: Microstructure fluxes across density surfaces. Journal of Physical Oceanography, 23 (10), 2254–2264.

Gent, P. R., and J. C. McWilliams, 1990: Isopycnal mixing in ocean circulation models. Journal of Physical Oceanography, 20 (1), 150–155.

Greatbatch, R. J., and K. G. Lamb, 1990: On parameterizing vertical mixing of momentum in non-eddy resolving ocean models. Journal of Physical Oceanography, 20 (10), 1634–1637.

Maddison, J. R., and D. P. Marshall, 2013: The eliassen–palm flux tensor. Journal of Fluid Mechanics, 729, 69–102.

Young, W. R., 2012: An exact thickness-weighted average formulation of the boussinesq equations. Journal of Physical Oceanography, 42 (5), 692–707.