2004 LOM Workshop Monday 11:30 - 11:50 a.m.
Toward the Implementation of Ripa's Inhomogeneous Layer Model
F. J. Beron-Vera, M. J. Olascoaga and J. Zavala-Garay
RSMAS/AMP, UMiami, 4600 Rickenbacker Cwy., Miami, FL 33149 USA
fberon@rsmas.miami.edu
ABSTRACT
Inhomogeneous layer models in which the velocity and buoyancy fields are allowed to vary only in the horizontal position and time have been very extensively exploited in ocean modeling. One example is the widely used Miami Isopycnic-Coordinate Model, whose upper layer is chosen as an inhomogeneous layer of this kind. These so-called "slab" models have the ability to partially incorporate thermodynamic processes, which are of fundamental importance in the ocean. For instance, in addition to momentum fluxes, these models can accommodate nonuniform heat and freshwater fluxes through the ocean surface. However, the slab models are known to have several limitations and deficiencies. In particular: (i) they cannot represent explicitly the thermal-wind balance which dominates at low frequencies; (ii) they have a zero-frequency mode not present in the exact fully three-dimensional model; and, in close relation to this, (iii) they cannot prevent spurious instabilities from developing. To cure the slab model limitations and deficiencies, Ripa proposed an improved closure to incorporate thermodynamic processes in a one-layer model. In addition to allowing arbitrary velocity and buoyancy variations in horizontal position and time, Ripa's model allows the velocity and buoyancy fields to vary \emph{linearly} with depth. Ripa's model enjoys a number of properties which make it very promising: (i) it represents explicitly the thermal-wind balance at low frequencies; (ii) the free waves supported by the model (Poincaré, Rossby, midlatitude coastal Kelvin, equatorial, etc.) are a very good approximation to the first and second vertical modes in the fully three-dimensional model; and, very importantly, (iii) in the absence of dissipation and external forcing, Ripa's model has a general invariant, quadratic in the departure from a state of rest (or at most with a uniform current), which is positive definite. This property, which is present in the fully three-dimensional model, prevents the system to explode by itself, unlike the slab models for which this integral of motion is nonnegative definite. In this work we generalize Ripa's model to an arbitrary number of layers, including the possibility of a free surface and irregular bottom topography or the (mathematically equivalent) case in which the stack of layers floats on top of a quiescent infinitely deep layer. As a test we consider the problem of ageostrophic upper-ocean baroclinic instability, by allowing for the bottom boundary in the classical Stone's model to move freely.
LOM Users' Workshop, February 9-11, 2004