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Variations on a beta-plane: derivation of non-traditional beta-plane
equations from Hamilton's principle on a sphere

P. J. Dellar (2010) Variations on a
beta-plane: derivation of non-traditional beta-plane equations from
Hamilton's principle on a sphere To appear in J. Fluid Mech.

Preprint available (BetaPlane.pdf
164K)

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Abstract

Starting from Hamilton's principle on a rotating sphere, we derive a
series of successively more accurate b-plane
approximations. These are Cartesian approximations to motion in
spherical geometry that capture the change with latitude of the angle
between the rotation vector and the local vertical. Being derived using
Hamilton's principle, the different b-plane
approximations each conserve energy, angular momentum, and potential
vorticity. They differ in their treatments of the locally horizontal
component of the rotation vector, the component that is usually
neglected under the traditional approximation. In particular, we derive
an extended set of b-plane
equations in which the locally vertical and locally horizontal
components of the rotation vector both vary linearly with latitude.
This was previously thought to violate conservation of angular momentum
and potential vorticity. We show that the difficulty in maintaining
these conservation laws arises from the need to express the rotation
vector as the curl of a vector potential while approximating the true
spherical metric by a flat Cartesian metric. Finally, we derive
depth-averaged equations on our extended b-plane with topography, and show
that they coincide with the extended non-traditional shallow water
equations previously derived in Cartesian geometry.

Related papers:

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

P. J. Dellar and R. Salmon
(2005) *Shallow
water equations with a complete Coriolis force and topography
*Phys. Fluids**
17**, 106601