  We present a prognostic, one-equation model for eddy-mean flow interactions
to parameterize the divergence of the Eliassen-Palm flux tensor (EPFT) that
arises from thickness-weighted averaging (TWA) the hydrostatic Boussinesq
equations. The TWA system of equations does not invoke approximations beyond
those for which the hydrostatic Boussinesq equations are valid, constituting a
mathematically consistent framework with clear physical interpretations. This
model is intended for the adiabatic interior of zonally symmetric flows, in the
absence of topographic features, where terms corresponding to eddy interfacial
form drag in the EPFT dominate forces. We model eddy interfacial form drag
terms for vertical flux of horizontal momentum using the gradient hypothesis,
as the product of an eddy viscosity and the vertical gradient of horizontal
momentum. We use mixing length theory to relate viscosity to an eddy length
scale and an eddy velocity, which is proportional to the eddy energy in the TWA
system. The eddy length scale is modeled as the first Rossby radius of
deformation, which we calculate as a function of the mean flow. We use a
prognostic equation for vertically integrated eddy energy at each horizontal
location, which we derive from the TWA framework, and then simplify to the
flows of interest by ignoring transport, redistribution and diabatic terms. The
prognostic vertically integrated eddy energy is projected onto the water column
using the eigenvalue of the first baroclinic mode to obtain the eddy energy at
each vertical position. The eddy viscosity has horizontal as well as vertical
structure. We diagnosed the model equations in an eddy resolving numerical
simulation of a zonally re-entrant channel representative of the Southern
Ocean. We have implemented the model parameterization in an ocean model and
tested it to simulate a parameterized simulation of this flow.
