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February  2014, 34(2): 421-436. doi: 10.3934/dcds.2014.34.421

On the existence and asymptotic stability of solutions for unsteady mixing-layer models

Tómas Chacón-Rebollo 1, , Macarena Gómez-Mármol 1, and  Samuele Rubino 1,

1. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Calle Tarfia, s/n, 41012, Sevilla, Spain, Spain, Spain

Received  November 2012 Revised  May 2013 Published  August 2013

We introduce in this paper some elements for the mathematical analysis of turbulence models for oceanic surface mixing layers. We consider Richardson-number based vertical eddy diffusion models. We prove the existence of unsteady solutions if the initial condition is close to an equilibrium, via the inverse function theorem in Banach spaces. We use this result to prove the non-linear asymptotic stability of equilibrium solutions.
Keywords: inverse function theorem., Oceanic turbulent mixing-layer models, gradient Richardson number.
Mathematics Subject Classification: Primary: 35Q86, 37N10; Secondary: 76F4.
Citation: Tómas Chacón-Rebollo, Macarena Gómez-Mármol, Samuele Rubino. On the existence and asymptotic stability of solutions for unsteady mixing-layer models. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 421-436. doi: 10.3934/dcds.2014.34.421
References:
[1]

A.-C. Bennis, T. Chacón-Rebollo, M. Gómez Mármol and R. Lewandowski, Numerical modelling of algebraic closure models of oceanic turbulent mixing layers, M2AN Math. Model. Numer. Anal., 44 (2010), 1255-1277. doi: 10.1051/m2an/2010025.  Google Scholar

[2]

A.-C. Bennis, T. Chacón-Rebollo, M. Gómez Mármol and R. Lewandowski, Stability of some turbulent vertical models for the ocean mixing boundary layer, Appl. Math. Lett., 21 (2008), 128-133. doi: 10.1016/j.aml.2007.02.016.  Google Scholar

[3]

H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations," Universitext, Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7.  Google Scholar

[4]

T. Chacón-Rebollo, M. Gómez Mármol and S. Rubino, Analysis of numerical stability of algebraic oceanic turbulent mixing layer models, submitted to Appl. Math. Model., (2013). Google Scholar

[5]

S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 251, Springer-Verlag, New York-Berlin, 1982. doi: 10.1007/978-1-4613-8159-4.  Google Scholar

[6]

A. Defant, Schichtung und zirkulation des atlantischen ozeans, (German) Wiss. Ergebn.: Deutsch. Atlant. Exp. Forsch., 6 (1936), 289-411. Google Scholar

[7]

L. C. Evans, "Partial Differential Equations," $2^{nd}$ edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.  Google Scholar

[8]

P. R. Gent, The heat budget of the TOGA-COARE domain in an ocean model, J. Geophys. Res., 96 (1991), 3323-3330. doi: 10.1029/90JC01677.  Google Scholar

[9]

H. Goosse, E. Deleersnijder, T. Fichefet and M. H. England, Sensitivity of a global coupled ocean-sea ice model to the parameterization of vertical mixing, J. Geophys. Res., 104 (1999), 13681-13695. doi: 10.1029/1999JC900099.  Google Scholar

[10]

Z. Kowalik and T. S. Murty, "Numerical Modeling of Ocean Dynamics," Advanced Series on Ocean Engineering, Vol. 5, World Scientific, Singapore, 1993. doi: 10.1142/1970.  Google Scholar

[11]

O. A. Ladyženskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI, 1968.  Google Scholar

[12]

M. Lesieur, "Turbulence in Fluids," $3^{rd}$ edition, Fluid Mechanics and its Applications, 40, Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-94-010-9018-6.  Google Scholar

[13]

R. Lewandowski, "Analyse Mathématique et Océanographie," (French) Masson, Paris, 1997. Google Scholar

[14]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," (French) Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[15]

R. C. Pacanowski and S. G. H. Philander, Parametrization of vertical mixing in numerical models of the tropical oceans, J. Phys. Oceanogr., 11 (1981), 1443-1451. Available from: http://journals.ametsoc.org/loi/phoc. Google Scholar

[16]

J. Pedloski, "Geophysical Fluid Dynamics," $2^{nd}$ edition, Springer-Verlag, New York-Berlin, 1987. Google Scholar

[17]

S. Rubino, Numerical modelling of oceanic turbulent mixing layers considering pressure gradient effects, in "Mascot10 Proceedings: IMACS Series in Comp. and Appl. Math." (eds. F. Pistella and R. M. Spitaleri), 16 (2011), 229-238. Google Scholar

[18]

R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations, Math. Comp., 62 (1994), 445-475. doi: 10.2307/2153518.  Google Scholar

[19]

J. Vialard and P. Delecluse, An ogcm study for the TOGA decade. Part I: Role of salinity in the physics of the western Pacific fresh pool, J. Phys. Oceanogr., 28 (1998), 1071-1088. Available from: http://journals.ametsoc.org/loi/phoc. Google Scholar

show all references

References:
[1]

A.-C. Bennis, T. Chacón-Rebollo, M. Gómez Mármol and R. Lewandowski, Numerical modelling of algebraic closure models of oceanic turbulent mixing layers, M2AN Math. Model. Numer. Anal., 44 (2010), 1255-1277. doi: 10.1051/m2an/2010025.  Google Scholar

[2]

A.-C. Bennis, T. Chacón-Rebollo, M. Gómez Mármol and R. Lewandowski, Stability of some turbulent vertical models for the ocean mixing boundary layer, Appl. Math. Lett., 21 (2008), 128-133. doi: 10.1016/j.aml.2007.02.016.  Google Scholar

[3]

H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations," Universitext, Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7.  Google Scholar

[4]

T. Chacón-Rebollo, M. Gómez Mármol and S. Rubino, Analysis of numerical stability of algebraic oceanic turbulent mixing layer models, submitted to Appl. Math. Model., (2013). Google Scholar

[5]

S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 251, Springer-Verlag, New York-Berlin, 1982. doi: 10.1007/978-1-4613-8159-4.  Google Scholar

[6]

A. Defant, Schichtung und zirkulation des atlantischen ozeans, (German) Wiss. Ergebn.: Deutsch. Atlant. Exp. Forsch., 6 (1936), 289-411. Google Scholar

[7]

L. C. Evans, "Partial Differential Equations," $2^{nd}$ edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.  Google Scholar

[8]

P. R. Gent, The heat budget of the TOGA-COARE domain in an ocean model, J. Geophys. Res., 96 (1991), 3323-3330. doi: 10.1029/90JC01677.  Google Scholar

[9]

H. Goosse, E. Deleersnijder, T. Fichefet and M. H. England, Sensitivity of a global coupled ocean-sea ice model to the parameterization of vertical mixing, J. Geophys. Res., 104 (1999), 13681-13695. doi: 10.1029/1999JC900099.  Google Scholar

[10]

Z. Kowalik and T. S. Murty, "Numerical Modeling of Ocean Dynamics," Advanced Series on Ocean Engineering, Vol. 5, World Scientific, Singapore, 1993. doi: 10.1142/1970.  Google Scholar

[11]

O. A. Ladyženskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI, 1968.  Google Scholar

[12]

M. Lesieur, "Turbulence in Fluids," $3^{rd}$ edition, Fluid Mechanics and its Applications, 40, Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-94-010-9018-6.  Google Scholar

[13]

R. Lewandowski, "Analyse Mathématique et Océanographie," (French) Masson, Paris, 1997. Google Scholar

[14]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," (French) Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[15]

R. C. Pacanowski and S. G. H. Philander, Parametrization of vertical mixing in numerical models of the tropical oceans, J. Phys. Oceanogr., 11 (1981), 1443-1451. Available from: http://journals.ametsoc.org/loi/phoc. Google Scholar

[16]

J. Pedloski, "Geophysical Fluid Dynamics," $2^{nd}$ edition, Springer-Verlag, New York-Berlin, 1987. Google Scholar

[17]

S. Rubino, Numerical modelling of oceanic turbulent mixing layers considering pressure gradient effects, in "Mascot10 Proceedings: IMACS Series in Comp. and Appl. Math." (eds. F. Pistella and R. M. Spitaleri), 16 (2011), 229-238. Google Scholar

[18]

R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations, Math. Comp., 62 (1994), 445-475. doi: 10.2307/2153518.  Google Scholar

[19]

J. Vialard and P. Delecluse, An ogcm study for the TOGA decade. Part I: Role of salinity in the physics of the western Pacific fresh pool, J. Phys. Oceanogr., 28 (1998), 1071-1088. Available from: http://journals.ametsoc.org/loi/phoc. Google Scholar

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