http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=340134
Journal of Fluid Mechanics (1995), 303:169-201 Cambridge University Press
Copyright © 1995 Cambridge University Press
doi:10.1017/S0022112095004228
On improving a one-layer ocean model with thermodynamics
P. Ripa a1
a1
Centro de Investigación Científica y de Educación Superior de Ensenada, Km. 107 Carretera
Tijuana-Ensenada, (22800) Ensenada, B. C., México
Abstract
A popular method used to incorporate thermodynamic processes in a shallow water model (e.g. one used to
study the upper layer of the ocean) is to allow for density variations in time and horizontal position, but
keep all dynamical fields as depth independent. This is achieved by replacing the horizontal pressure
gradient by its vertical average. These models have limitations, for instance they cannot represent the
‘thermal wind’ balance (between the horizontal density gradient and the vertical shear of the velocity)
which dominates at low frequencies. A new model is now proposed which uses velocity and density fields
varying linearly with depth, with coefficients that are functions of horizontal position and time. This model
can explicitly represent the thermal wind balance, but its use is not restricted to low- frequency dynamics.
Volume, mass, buoyancy variance, energy and momentum are conserved in the new model. Furthermore,
these integrals of motion have the same dependence on the dynamical fields as the exact (continuously
stratified) case. The evolution of the three components of the absolute vorticity field are correctly
represented. Conservation of density–potential vorticity is not fulfilled, though, owing to artificial removal
of the vertical curvature of the velocity field.
The integrals of motion are used to construct a ‘free energy’ [script E]f, which is quadratic to the lowest
order in the deviation from a steady state with (at most) a uniform velocity field. [script E] f is positive
definite, and therefore the free evolution of the system cannot lead to an ‘explosion’ of the dynamical
fields. (This is not the case if the velocity shear and/or the density vertical gradient is excluded in the
model, which results in a non-negative definite free energy.)
In a model with one active layer, linear waves on top of a steady state with no currents are, to a very good
approximation, those of the first two vertical modes of the continuously stratified model. These are the
familiar geophysical gravity and vortical waves (e.g. Poincaré, Rossby, and coastal Kelvin waves at midlatitudes, equatorial waves, etc.).
Finally, baroclinic instability is well represented in the new model. For long perturbations (wavelengths of
the order of the deformation radius of the first mode) the agreement with more precise calculations is
excellent. On the other hand, the comparison with the eigenvalues of Eady's problem (which corresponds to
wavelengths of the order of the deformation radius of the second mode) shows differences of the order of
40%. Nevertheless, the new model does have a high-wavenumber cutoff, even though it is constrained to
linear profiles in depth and therefore cannot reproduce the exponential trapping of Eady's problem
eigensolutions.
In sum, the integrals of motion, vorticity dynamics, free waves and baroclinic instability results all give
confidence in the new model. Its main novelty, however, lies in the ability to incorporate thermodynamic
processes.
(Published Online April 26 2006)
(Received February 7 1995)
(Revised July 6 1995)
http://www.agu.org/pubs/crossref/1996/95JC02899.shtml
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 101, NO. C1, PAGES 1233–1245,
1996
Linear waves in a one-layer ocean model with thermodynamics
P. Ripa
Centro de Investigación Científica y de Educación Superior de Ensenada, Ensenada,
Baja, California, México
Abstract
The simplest member of a family of ocean models with inhomogeneous layers is used to
study the physics of an active mixed layer overlying a quiescent denser layer. The
dynamical variables of the system are the depth-averaged horizontal velocity u and
buoyancy ϑ fields and the layer thickness h. The potential vorticity q is not conserved in
these models, unless ϑ is constant. An integral of motion quadratic in the deviation from
a steady and symmetric basic state is derived and shown not to be positive definite; that
is, there is no formal stability theorem (except in the case of uniform flow). This is due to
a term proportional to the square of the distance between the isolines of q and ν, which is
a measure of the nonconservation of q. A reference state without mean currents must
have ΘH 2 = constant, where Θ and H are the buoyancy and layer thickness, respectively,
in that state. Linear waves superimposed on this basic state are Poincaré and Rossby
waves, analogous to those of a homogeneous layer problem but with a different
topography. In addition, there is a force-compensating mode (FCM), which corresponds
to changes in buoyancy and layer thickness that represent a vanishing contribution to the
depth-averaged pressure force. The phase speed of gravity waves (in the absence of
rotation effects) is c = Θ H . Rossby waves are driven by the gradient of ƒ/c 2 (where ƒ is
the Coriolis parameter) rather than by the gradient of ƒ/H, as they are in the homogeneous
layer case. The quadratic integrals of motion are used to classify the types of instability.
First, this is done in an a priori sense, looking for properties of an unstable basic flow that
constrain the structure of growing perturbations. Second, this is done in an a posteriori
sense, through the evaluation of the rate change of selected parts of those integrals of
motion, as a function of the perturbation fields. It is stressed that for this method to give
unambiguous results it is necessary that the integrals of motion be quadratic, to lowest
order, in the perturbation. Unlike the case of the classical shallow water equations, a
uniform flow in a channel may be unstable, if the buoyancy increases toward its right
(left) in the northern (southern) hemisphere. In the ƒ plane this necessary instability
condition corresponds to mean flow in the direction opposite to that of the propagation of
Rossby waves. A growing perturbation is very well represented by the combination of a
Rossby wave and a FCM (from the system without currents) which interact due to the
presence of the uniform flow.
Received 20 October 1994; accepted 19 September 1995.
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Citation: Ripa, P. (1996), Linear waves in a one-layer ocean model with thermodynamics,
J. Geophys. Res., 101(C1), 1233–1245.
Copyright 1996 by the American Geophysical Union.