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M. Oliver,
Variational asymptotics for rotating shallow
water near geostrophy: A
transformational approach,
J. Fluid Mech. 551 (2006), 197-234.
Abstract:
We introduce a unified variational framework in which the classical
balance models for nearly geostrophic shallow water as well as several
new models can be derived. Our approach is based on consistently
truncating an asymptotic expansion of a near identity transformation
of the rotating shallow water Lagrangian. Model reduction is achieved
by imposing either degeneracy (for models in a semigeostrophic
scaling) or incompressibility (for models in a quasigeostrophic
scaling) with respect to the new coordinates.
At first order, we recover the classical semigeostrophic and
quasigeostrophic equations, Salmon's L1 and
large-scale semigeostrophic equations, as well as a one-parameter
family of models that interpolate between the two. We identify one
member of this family, different from previously known models, that
promises better regularity - hence consistency with large-scale
vortical motion - than all other first order models. Moreover, we
explicitly derive second order models for all cases considered. While
these second order models involve nonlinear potential vorticity
inversion and do not obviously share the good properties or their
first order counterparts, we offer an explicit survey of second order
models and point out several avenues for exploration.
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