On the existence and asymptotic stability of solutions for unsteady mixing-layer models. (English) Zbl 1283.35145
This paper deals with unsteady turbulence models. The existence of regular solutions is shown, using inverse function theorem, under the assumption that the initial condition is close to the equilibrium. Finally, the nonlinear asymptotic stability of equilibrium solution is established.
Reviewer: V. D. Sharma (Mumbai)
MSC:
35Q86 | PDEs in connection with geophysics |
37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |
76F40 | Turbulent boundary layers |
35B65 | Smoothness and regularity of solutions to PDEs |
35B35 | Stability in context of PDEs |
Keywords:
oceanic turbulent mixing-layer models; gradient Richardson number; inverse function theoremReferences:
[1] | A.-C. Bennis, <em>Numerical modelling of algebraic closure models of oceanic turbulent mixing layers</em>,, M2AN Math. Model. Numer. Anal., 44, 1255 (2010) · Zbl 1427.76084 · doi:10.1051/m2an/2010025 |
[2] | A.-C. Bennis, <em>Stability of some turbulent vertical models for the ocean mixing boundary layer</em>,, Appl. Math. Lett., 21, 128 (2008) · Zbl 1139.86001 · doi:10.1016/j.aml.2007.02.016 |
[3] | H. Brezis, “Functional Analysis, Sobolev Spaces and Partial Differential Equations,”, Universitext (2011) · Zbl 1220.46002 · doi:10.1007/978-0-387-70914-7 |
[4] | T. Chacón-Rebollo, <em>Analysis of numerical stability of algebraic oceanic turbulent mixing layer models</em>,, submitted to Appl. Math. Model. (2013) |
[5] | S. N. Chow, “Methods of Bifurcation Theory,”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 251 (1982) · Zbl 0487.47039 · doi:10.1007/978-1-4613-8159-4 |
[6] | A. Defant, <em>Schichtung und zirkulation des atlantischen ozeans</em>,, (German) Wiss. Ergebn.: Deutsch. Atlant. Exp. Forsch., 6, 289 (1936) |
[7] | L. C. Evans, “Partial Differential Equations,”, \(2^{nd}\) edition, 19 (2010) · Zbl 1194.35001 |
[8] | P. R. Gent, <em>The heat budget of the TOGA-COARE domain in an ocean model</em>,, J. Geophys. Res., 96, 3323 (1991) · doi:10.1029/90JC01677 |
[9] | H. Goosse, <em>Sensitivity of a global coupled ocean-sea ice model to the parameterization of vertical mixing</em>,, J. Geophys. Res., 104, 13681 (1999) · doi:10.1029/1999JC900099 |
[10] | Z. Kowalik, “Numerical Modeling of Ocean Dynamics,”, Advanced Series on Ocean Engineering (1993) · doi:10.1142/1970 |
[11] | O. A. Ladyženskaya, “Linear and Quasi-Linear Equations of Parabolic Type,”, Translations of Mathematical Monographs (1968) · Zbl 0174.15403 |
[12] | M. Lesieur, “Turbulence in Fluids,”, \(3^{rd}\) edition, 40 (1997) · Zbl 0876.76002 · doi:10.1007/978-94-010-9018-6 |
[13] | R. Lewandowski, “Analyse Mathématique et Océanographie,”, (French) Masson (1997) · Zbl 1347.86001 |
[14] | J.-L. Lions, “Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,”, (French) Dunod; Gauthier-Villars (1969) · Zbl 0189.40603 |
[15] | R. C. Pacanowski, <em>Parametrization of vertical mixing in numerical models of the tropical oceans</em>,, J. Phys. Oceanogr., 11, 1443 (1981) |
[16] | J. Pedloski, “Geophysical Fluid Dynamics,”, \(2^{nd}\) edition (1987) · Zbl 0713.76005 |
[17] | S. Rubino, <em>Numerical modelling of oceanic turbulent mixing layers considering pressure gradient effects</em>,, in, 16, 229 (2011) |
[18] | R. Verfürth, <em>A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations</em>,, Math. Comp., 62, 445 (1994) · Zbl 0799.65112 · doi:10.2307/2153518 |
[19] | J. Vialard, <em>An ogcm study for the TOGA decade. Part I: Role of salinity in the physics of the western Pacific fresh pool</em>,, J. Phys. Oceanogr., 28, 1071 (1998) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.