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M. Oliver and S. Vasylkevych,
Generalized large-scale semigeostrophic approximations for the
f-plane primitive equations,
J. Phys. A: Math. Theor. 49 (2016), 184001.
Abstract:
We derive a family of balance models for rotating stratified flow in
the primitive equation setting. By construction, the models possess
conservation laws for energy and potential vorticity and are formally
of the same order of accuracy as Hoskins' semigeostrophic equations.
Our construction is based on choosing a new coordinate frame for the
primitive equation variational principle in such a way that the
consistently truncated Lagrangian degenerates. We show that the
balance relations so obtained are elliptic when the fluid is stably
stratified and certain smallness assumptions are satisfied. Moreover,
the potential temperature can be recovered from the potential
vorticity via inversion of a non-standard Monge-Ampère problem
which is subject to the same ellipticity condition. While the present
work is entirely formal, we conjecture, based on a careful rewriting
of the equations of motion and a straightforward derivative count,
that the Cauchy problem for the balance models is well posed subject
to conditions on the initial data. Our family of models includes, in
particular, the stratified analog of the L1 balance model of
R. Salmon.
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