Partial and spectral-viscosity models for geophysical flows

Abstract

Two models based on the hydrostatic primitive equations are proposed. The first model is the primitive equations with partial viscosity only, and is oriented towards large-scale wave structures in the ocean and atmosphere. The second model is the viscous primitive equations with spectral eddy viscosity, and is oriented towards turbulent geophysical flows. For both models, the existence and uniqueness of global strong solutions are established. For the second model, the convergence of the solutions to the solutions of the classical primitive equations as eddy viscosity parameters tend to zero is also established.

Keywords

Primitive equations / Partial viscosity / Spectral-eddy viscosity

References

[1] Adams R. A.. Sobolev Spaces, 1975, New York, London: Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers]

[2] Avrin J., Xiao C.. Convergence of Galerkin solutions and continuous dependence on data in spectrallyhyperviscous models of 3D turbulent flow. J. Diff. Eqs., 2009, 247(10): 2778-2798

[3] Berselli L. C., Iliescu T., Layton W. J.. Mathematics of Large Eddy Simulation of Turbulent Flows, Scientific Computation, 2006, Berlin: Springer-Verlag

[4] Calhoun-Lopez M., Gunzburger M. D.. A finite element, multiresolution viscosity method for hyperbolic conservation laws (electronic). SIAM J. Numer. Anal., 2005, 43(5): 1988-2011

[5] Calhoun-Lopez M., Gunzburger M. D.. The efficient implementation of a finite element, multi-resolution viscosity method for hyperbolic conservation laws. J. Comput. Phys., 2007, 225(2): 1288-1313

[6] Cao C. S., Titi E. S.. Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model. Comm. Pure Appl. Math., 2003, 56(2): 198-233

[7] Cao C. S., Titi E. S.. Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. Ann. of Math. (2), 2007, 166(1): 245-267

[8] Chen Q., Laminie J., Rousseau A. A 2.5D model for the equations of the ocean and the atmosphere. Anal. Appl. (Singapore), 2007, 5(3): 199-229

[9] Constantin P., Foias C.. Navier-Stokes Equations, Chicago Lectures in Mathematics, 1988, Chicago, IL: University of Chicago Press

[10] Díez D. C., Gunzburger M., Kunoth A.. An adaptive wavelet viscosity method for hyperbolic conservation laws. Numer. Meth. Part. Diff. Eqs., 2008, 24(6): 1388-1404

[11] Frederiksen J. S., Dix M. R., Kepert S. M.. Systematic energy errors and the tendency toward canonical equilibrium in atmospheric circulation models. J. Atmosph. Sci., 1996, 53(6): 887-904

[12] Gill A. E.. Atmosphere-Ocean Dynamics, 1982, New York: Academic Press

[13] Guermond J.-L., Prudhomme S.. Mathematical analysis of a spectral hyperviscosity LES model for the simulation of turbulent flows. M2AN Math. Model. Numer. Anal., 2003, 37(6): 893-908

[14] Guillén-González F., Masmoudi N., Rodríguez-Bellido M. A.. Anisotropic estimates and strong solutions of the primitive equations. Diff. Int. Eqs., 2001, 14(11): 1381-1408

[15] Gunzburger, M., Lee, E. Saka, Y., et al, Analysis of nonlinear spectral eddy-viscosity models of turbulence, J. Sci. Comput., to appear. DOI: 10.1007/s10915-009-9335-8

[16] Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities, reprint of the 1952 edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. MR944909 (89d:26016)

[17] Hu C. B., Temam R., Ziane M.. The primitive equations on the large scale ocean under the small depth hypothesis. Discrete Contin. Dyn. Syst., 2003, 9(1): 97-131

[18] Ju N.. The global attractor for the solutions to the 3D viscous primitive equations. Discrete Contin. Dyn. Syst., 2007, 17(1): 159-179

[19] Karamanos G.-S., Karniadakis G. E.. A spectral vanishing viscosity method for large-eddy simulations. J. Comput. Phys., 2000, 163(1): 22-50

[20] Kobelkov G. M.. Existence of a solution “in the large” for ocean dynamics equations. J. Math. Fluid Mech., 2007, 9(4): 588-610

[21] Kukavica I., Ziane M.. On the regularity of the primitive equations of the ocean. Nonlinearity, 2007, 20(12): 2739-2753

[22] Lions J. L., Temam R., Wang S. H.. New formulations of the primitive equations of atmosphere and applications. Nonlinearity, 1992, 5(2): 237-288

[23] Lions J. L., Temam R., Wang S. H.. On the equations of the large-scale ocean. Nonlinearity, 1992, 5(5): 1007-1053

[24] Majda A. J., Wang X. M.. Non-linear Dynamics and Statistical Theories for Basic Geophysical Flows, 2006, Cambridge: Cambridge University Press

[25] Mcwilliams J. C.. The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mechanics Digital Archive, 1984, 146(1): 21-43

[26] Nessyahu H., Tadmor E.. The convergence rate of approximate solutions for nonlinear scalar conservation laws. SIAM J. Numer. Anal., 1992, 29(6): 1505-1519

[27] Pedlosky J.. Geophysical Fluid Dynamics, 1987 2nd ed. New York: Springer-Verlag

[28] Petcu M., Temam R., Ziane M.. Temam R., Ciarlet P. G., Tribbia J.. Mathematical problems for the primitive equations with viscosity, Handbook of Numerical Analysis, Special Issue on Some Mathematical Problems in Geophysical Fluid Dynamics. Handb. Numer. Anal., 2008, New York: Elsevier

[29] Rousseau A., Temam R., Tribbia J.. Chen G. Q., Gasper G., Jerome J. J.. Boundary conditions for an ocean related system with a small parameter, 2005, Providence, RI: Contemporary Mathematics, A. M. S. 231-263

[30] Rousseau A., Temam R., Tribbia J.. The 3D primitive equations in the absence of viscosity: boundary conditions and well-posedness in the linearized case. J. Math. Pures Appl. (9), 2008, 89(3): 297-319

[31] Smagorinsky J.. General circulation experiments with the primitive equations. I. the basic experiment. Monthly Weather Review, 1963, 91: 99-152

[32] Stolz S., Schlatter P., Kleiser L.. High-pass filtered eddy-viscosity models for large-eddy simulations of transitional and turbulent flow. Physics of Fluids, 2005, 17(6): 065103

[33] Temam R.. Navier-Stokes Equations, Theory and Numerical Analysis, 2001, Providence, RI: A. M. S.