doi:10.1016/j.aml.2007.02.016
Copyright © 2007 Elsevier Ltd All rights reserved.
Stability of some turbulent vertical models for the ocean mixing boundary layer
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A.-C. Bennisa, T. Chacón Rebollob, M. Gomez Marmolb and R. Lewandowskia, 
, 
aIRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France
bDepartamento de Ecuaciones Diferenciales y Análisis Numerico, Universidad de Sevilla. C/Tarfia, s/n. 41080, Sevilla, Spain
Received 8 December 2006;
revised 16 January 2007;
accepted 22 February 2007.
Available online 21 March 2007.
Abstract
We consider four turbulent models for simulating the boundary mixing layer of the ocean. We show the existence of solutions to these models in the steady state case and then we study the mathematical linear stability of these solutions.
Keywords: Oceanography; Turbulence models; Stability; Partial differential equations
Fig. 1a. Graphs of k(R) and h(R) for R–2–2–4 model, where k is the quotient of the square of f1 with f2 and h(R)=CR, C depending on Q (see precise definitions below ). The graphs intersect in only one point in both cases Q>0 and Q<0, showing the uniqueness of the solution claimed in Theorem 2.1, as explained in the following.
Fig. 1b. Graphs of k(R) and h(R) for R–2–2 model. We see that the graph intersect in one or three points, yielding a non uniqueness result of the solution for this model.
Fig. 2. Numerical stability. The results are summarized here. The dot zone represents a zone where the solution is physically not valid. This is the case for the R–2–3 and R–2–1–3 formulations. The rectangle zone is an instability zone. All formulations have an instability zone. Nevertheless, one observes that for each model, mathematical stability holds for non-negative R.
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