Multilayer shallow-water model with stratification and shear

Abstract

The purpose of this paper is to present a shallow-water-type model with multiple inhomogeneous layers featuring variable linear velocity vertical shear and startificaion in horizontal space and time. This is achieved by writing the layer velocity and buoyancy fields as linear functions of depth, with coefficients that depend arbitrarily on horizontal position and time. The model is a generalization of Ripa's (1995) single-layer model to an arbitrary number of layers. Unlike models with homogeneous layers the present model is able to represent thermodynamics processes driven by heat and freshwater fluxes through the surface or mixing processes resulting from fluid exchanges across contiguous layers. By contrast with inhomogeneous-layer models with depth-independent velocity and buoyancy, the model derived here can sustain explicitly at low frequency a current in thermal wind balance (between the vertical vertical shear and the horizontal density gradient) within each layer. In the absence of external forcing and dissipation, energy, volume, mass, and buoyancy variance constrain the dynamics; conservation of total zonal momentum requires in addition the usual zonal symmetry of the topography and horizontal domain. The inviscid, unforced model admits a formulation suggestive of a generalized Hamiltonian structure, which enables the classical connection between symmetries and conservation laws via Noether's theorem. A steady solution to a system involving one Ripa-like layer and otherwise homogeneous layers can be proved formally (or Arnold) stable using the above invariants. A model configuration with only one layer has been shown previously to provide: a very good representation of the exact vertical normal modes up to the first internal mode; an exact representation of long-perturbation (free boundary) baroclinic instability; and a very reasonable representation of short-perturbation (classical Eady) baroclinic instability. Here it is shown that substantially more accurate overall results with respect to single-layer calculations can be achieved by considering a stack of only a few layers. A similar behavior is found in ageostrophic (classical Stone) baroclinic instability by describing accurately the dependence of the solutions on the Richardson number with only two layers.

Nonlinear saturation of thermal instabilities

“…While β can have a stabilizing effect, the lack of a high-wavenumber cutoff of instability when Ū Uσ > 0 can be consequential for the nonlinear evolution of system (1), which however tends to show Kelvin-Helmoltzlike circulations that saturate at subdeformation scales rather than blowing up indefinitely 15 . Condition (7) was shown in Ripa 39 to be an a-priori condition for the formal stability of basic state (5), i.e., stability under small-amplitude perturbations of arbitrary structure. This result followed from the application of Arnold 2 method.…”

“…In both cases, the integral represents a norm that constrains the growth of perturbations. For the zonally symmetric basic state (5), such an integral of motion is given by 8) is positive-definite when (7) holds. (That the circulation perturbations δγ 0,W do not enter in (8) should not be taken as implying positive-semidefinitness of (8) and hence the possibility of unarrested growth along their directions in phase space: once initially specified, δγ 0,W remain the same at all times.…”

WAVETRISK-2.1: an adaptive dynamical core for ocean modelling

“…The dynamical equations of WAVETRISK-OCEAN are a multi-layer rotating shallow water model with inhomogeneous density layers, but with no vertical variation of velocity and buoyancy within each layer. This is an n-IL 0 model in the terminology of Beron-Vera (2021). In such a model, to be consistent with the piecewise constant representation of buoyancy in the vertical, a vertical average of the horizontal pressure gradient term in each layer is used to compute horizontal velocity.…”

“…n-IL 0 preserves important mimetic properties of the continuously stratified system (Kelvin's circulation theorem, advection of potential vorticity, conservation of Casimir invariants) and ensures a good approximation of the horizontal pressure gradient (similar to state-of-the-art models using terrain following coordinates). Beron-Vera (2021) improves n-IL 0 to n-IL 1 by allowing linear vertical variation within each layer.…”

Geometry of shallow-water dynamics with thermodynamics

“…1). This renewed interest motivated investigating the geometric properties of the system further, 3,2 following pioneering work by Ripa. 20,21,22,24 We review those properties here and also establish an explicit connection, so far overlooked, with seminal work by Morrison and Greene 16 on generalized Hamiltonians.…”