Numerical modelling of algebraic closure models of oceanic turbulent mixing layers
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 6, p. 1255-1277

We introduce in this paper some elements for the mathematical and numerical analysis of algebraic turbulence models for oceanic surface mixing layers. In these models the turbulent diffusions are parameterized by means of the gradient Richardson number, that measures the balance between stabilizing buoyancy forces and destabilizing shearing forces. We analyze the existence and linear exponential asymptotic stability of continuous and discrete equilibria states. We also analyze the well-posedness of a simplified model, by application of the linearization principle for non-linear parabolic equations. We finally present some numerical tests for realistic flows in tropical seas that reproduce the formation of mixing layers in time scales of the order of days, in agreement with the physics of the problem. We conclude that the typical mixing layers are transient effects due to the variability of equatorial winds. Also, that these states evolve to steady states in time scales of the order of years, under negative surface energy flux conditions.

DOI : https://doi.org/10.1051/m2an/2010025
Classification:  76D05,  35Q30,  76F65,  76D03
Keywords: turbulent mixing layers, Richardson number, first order closure models, conservative numerical solution, stability of steady states, tests for tropical seas
@article{M2AN_2010__44_6_1255_0,
     author = {Bennis, Anne-Claire and Chac\'on Rebollo, Tomas and G\'omez M\'armol, Macarena and Lewandowski, Roger},
     title = {Numerical modelling of algebraic closure models of oceanic turbulent mixing layers},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {6},
     year = {2010},
     pages = {1255-1277},
     doi = {10.1051/m2an/2010025},
     zbl = {pre05835021},
     mrnumber = {2769057},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2010__44_6_1255_0}
}
Bennis, Anne-Claire; Chacón Rebollo, Tomas; Gómez Mármol, Macarena; Lewandowski, Roger. Numerical modelling of algebraic closure models of oceanic turbulent mixing layers. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 6, pp. 1255-1277. doi : 10.1051/m2an/2010025. http://www.numdam.org/item/M2AN_2010__44_6_1255_0/

[1] B. Blanke and P. Delecluse, Variability of the tropical atlantic ocean simulated by a general circulation model with two different mixed-layer physics. J. Phys. Oceanogr. 23 (1993) 1363-1388.

[2] J.H. Bramble, J.E. Pasciak and O. Steinbach, On the stability of the l2 projection in h1. Math. Comp. 7 (2001) 147-156. | Zbl 0989.65122

[3] H. Burchard, Applied turbulence modelling in marine water. Ph.D. Thesis, University of Hambourg, Germany (2004). | Zbl 1068.86001

[4] P. Gaspar, Y. Gregoris and J.-M. Lefevre, A simple eddy kinetic energy model for simulations of the oceanic vertical mixing: test at Station Papa and long-term upper ocean study site. J. Geophys. Res. 16 (1990) 179-193.

[5] P.R. Gent, The heat budget of the toga-coare domain in an ocean model. J. Geophys. Res. 96 (1991) 3323-3330.

[6] H. Goosse, E. Deleersnijder, T. Fichefet and M.H. England, Sensitivity of a global coupled ocean-sea ice model to the parametrization of vertical mixing. J. Geophys. Res. 104 (1999) 13681-13695.

[7] J.H. Jones, Vertical mixing in the equatorial undercurrent. J. Phys. Oceanogr. 3 (1973) 286-296.

[8] Z. Kowalik and T.S. Murty, Numerical modeling of ocean dynamics. World Scientific (1993).

[9] W.G. Large, C. Mcwilliams and S.C. Doney, Oceanic vertical mixing: a review and a model with a nonlocal boundary layer parametrization. Rev. Geophys. 32 (1994) 363-403.

[10] G. Madec, P. Delecluse, M. Imbard and C. Levy, OPA version 8.0, Ocean General Circulation Model Reference Manual. LODYC, Int. Rep. 97/04 (1997).

[11] M. Mcphaden, The tropical atmosphere ocean (tao) array is completed. Bull. Am. Meteorol. Soc. 76 (1995) 739-741.

[12] G. Mellor and T. Yamada, Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys. Space Phys. 20 (1982) 851-875.

[13] R.C. Pacanowski and S.G.H. Philander, Parametrization of vertical mixing in numericals models of the tropical oceans. J. Phys. Oceanogr. 11 (1981) 1443-1451.

[14] J. Pedloski, Geophysical fluid dynamics. Springer (1987). | Zbl 0713.76005

[15] M. Potier-Ferry, The linearization principle for the stability of solutions of quasilinear parabolic equations. Arch. Ration. Mech. Anal. 77 (1981) 301-320. | Zbl 0497.35006

[16] A.R. Robinson, An investigation into the wind as the cause of the equatiorial undercurrent. J. Mar. Res. 24 (1966) 179-204.