### Multilayer shallow water equations with complete Coriolis force.
Part I: Derivation on a non-traditional beta-plane

A. L. Stewart & P. J. Dellar (2010) Multilayer shallow water equations with
complete Coriolis force. Part I: Derivation on a non-traditional
beta-plane J. Fluid Mech. 651 387-413

Preprint available ( MultilayerSW1.pdf
386K)

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Abstract

We derive equations to describe the flow of multiple superposed layers
of inviscid, incompressible fluids with constant densities over
prescribed topography in a rotating frame. Motivated by geophysical
applications, these equations incorporate the complete Coriolis
force. We do not make the widely used "traditional approximation"
that omits the contribution to the Coriolis force from the locally
horizontal part of the rotation vector. Our derivation is performed by
averaging the governing Euler equations over each layer, and from two
different forms of Hamilton's variational principle that differ in
their treatment of the coupling between layers. The coupling may be
included implicitly through the map from Lagrangian particle labels to
particle coordinates, or explicitly by adding terms representing the
work done on each layer by the pressure exerted by the layers
above. The latter approach requires additional terms in the
Lagrangian, but extends more easily to many layers. We show that our
equations obey the expected conservation laws for energy, momentum, and
potential vorticity. The conserved momentum and potential vorticity are
modified by non-traditional effects. The vertical component of the
rotation vector that appears in the potential vorticity for each layer
under the traditional approximation is replaced by the component
perpendicular to the layer's midsurface. The momentum includes an
additional contribution that reflects changes in angular momentum
caused by changes in a fluid element's distance from the rotation axis
as it is displaced vertically. Again, this effect is absent in the
traditional approximation.

@article{StewartDellar10a,

author = {Stewart, A. L. and Dellar, P. J.},

title = {{Multilayer shallow water equations with
complete Coriolis

force. Part I. Derivation on a non-traditional beta-plane}},

journal = {J. Fluid Mech.},

year = {2010},

volume = {651},

pages = {387--413},

DOI = {doi:10.1017/S0022112009993922}

}