Abstract
We present a set of equations describing the nonlinear dynamics of flows constrained by environmental rotation and stratification (Rossby numbers Ro∈[0.1,0.5] and Burger numbers of order unity). The fluid is assumed incompressible, adiabatic, inviscid and in hydrostatic balance. This set of equations is derived from the Navier Stokes equations (with the above properties), using a Rossby number expansion with second order truncation. The resulting model has the following properties: 1) it can represent motions with moderate Rossby numbers and a Burger number of order unity; 2) it filters inertia-gravity waves by assuming that the divergence of horizontal velocity remains small; 3) it is written in terms of a single function of space and time (pressure, generalized streamfunction or Bernoulli function); 4) it conserves total (Ertel) vorticity in a Lagrangian form, and its quadratic norm (potential enstrophy) at the model order in Rossby number; 5) it also conserves total energy at the same order if the work of pressure forces vanishes when integrated over the fluid domain. The layerwise version of the model is finally presented, written in terms of pressure. Integral properties (energy, enstrophy) are conserved by these layerwise equations. The model equations agree with the generalized geostrophy equations in the appropriate parameter regime. Application to vortex dynamics are mentioned.
Highlights
We present a set of equations describing the nonlinear dynamics of flows constrained by environmental rotation and stratification
The resulting model has the following properties: 1) it can represent motions with moderate Rossby numbers and a Burger number of order unity; 2) it filters inertia-gravity waves by assuming that the divergence of horizontal velocity remains small; 3) it is written in terms of a single function of space and time; 4) it conserves total (Ertel) vorticity in a Lagrangian form, and its quadratic norm at the model order in Rossby number; 5) it conserves total energy at the same order if the work of pressure forces vanishes when integrated over the fluid domain
( ) O Ro2 in vorticity-divergence or integral invariants; it appears under two forms, one in generalized streamfunction, and one in Bernoulli function
Summary
Strongly constrained in the horizontal plane by stratification and by background rotation, are often found in geophysical. The importance of quasi-2D jets and vortices in geophysical flows and laboratory experiments has motivated the search for asymptotic limits of the primitive equations on the slow quasimanifold (see for instance [4]) This regime is governed by the time evolution of potential vorticity ( called Ertel vorticity, a combination of vertical vorticity with vertical density gradient). The projection of these equations onto a set of homogeneous layers is shown to be in agreement with equations directly derived from the shallow-water equations Such a model is important for the study of mesoscale vortex dynamics in the ocean; for these vortices, Ro is not very small but often remains smaller than 1.
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