Multilayer shallow-water model with stratification and shear

• F. J. Beron-Vera ^[1]
  1. [1] Department of Atmospheric Sciences Rosenstiel School of Marine & Atmospheric ScienceUniversity of Miami, Miami, FL, USA
• Localización: Revista Mexicana de Física, ISSN-e 0035-001X, Vol. 67, Nº. 3, 2021, págs. 351-364
• Idioma: inglés
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• Resumen

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