Abstract
On hydrostatic flows in isentropic coordinates
by Onno Bokhove
The hydrostatic primitive equations of motion which have been used
in large-scale weather prediction over the last decades are analysed
with variational and Hamiltonian methods in a Lagrangian and
isentropic Eulerian framework. Firstly, a Lagrangian Hamilton's principle,
expressed in terms of fluid parcel positions,
is shown to be singular and Dirac's approach is used to derive a constrained
Hamiltonian formulation. The use of material isentropic coordinates for
the Eulerian hydrostatic equations is known to have distinct
conceptual advantages since fluid motion is, under inviscid and statically
stable circumstances,
confined to take place on quasi-horizontal isentropic surfaces.
Secondly, an Eulerian isentropic Hamilton's
principle, expressed in terms of fluid parcel variables,
is derived by transformation of the Lagrangian
Hamilton's principle to an Eulerian one. This Eulerian principle
explicitly describes the boundary dynamics of the time-dependent domain in terms
of advection of boundary isentropes s_{B}; these are the values
the isentropes have at their intersection with
the (lower) boundary.
A partial Legendre transform
for only the interior variables yields an Eulerian ``action'' principle.
Thirdly, Noether's theorem is used to derive energy and potential vorticity
conservation from the Lagrangian but most explicitly from the
Eulerian Hamilton's principle.
Fourthly, these conservation laws are used to derive a wave-activity invariant
which is second-order in terms of small-amplitude disturbances relative to
a resting or moving basic state. Linear stability criteria are derived
but only for resting basic states.
Fifthly, gravity modes are derived from
the linearised and constantly rotating hydrostatic equations in a domain
confined between two horizontal plates and relative to an isothermal basic state.
In mid-latitudes a time-scale separation between gravity and vortical
modes occurs. Finally, this time-scale separation suggests that
conservative geostrophic and ageostrophic approximations can be made to the
Eulerian action principle for hydrostatic flows. The outlined derivations
of approximate
models from the parent hydrostatic formulation extend
previous derivations of approximate conservative models
from the parent shallow-water equations. An explicit variational derivation is
given of an isentropic version of Hoskins and Bretherton's
model for atmospheric fronts.