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

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.
Citation: Tómas Chacón-Rebollo, Macarena Gómez-Mármol, Samuele Rubino. On the existence and asymptotic stability of solutions for unsteady mixing-layer models. Discrete and 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.

[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.

[3]

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

[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).

[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.

[6]

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

[7]

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

[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.

[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.

[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.

[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.

[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.

[13]

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

[14]

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

[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.

[16]

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

[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.

[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.

[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.

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.

[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.

[3]

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

[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).

[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.

[6]

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

[7]

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

[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.

[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.

[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.

[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.

[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.

[13]

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

[14]

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

[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.

[16]

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

[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.

[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.

[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.

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