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On improving a one-layer ocean model with thermodynamics

Published online by Cambridge University Press:  26 April 2006

P. Ripa
P. Ripa
Affiliation:
Centro de Investigación Científica y de Educación Superior de Ensenada, Km. 107 Carretera Tijuana-Ensenada, (22800) Ensenada, B. C., México
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Abstract

A popular method used to incorporate thermodynamic processes in a shallow water model (e.g. one used to study the upper layer of the ocean) is to allow for density variations in time and horizontal position, but keep all dynamical fields as depth independent. This is achieved by replacing the horizontal pressure gradient by its vertical average. These models have limitations, for instance they cannot represent the ‘thermal wind’ balance (between the horizontal density gradient and the vertical shear of the velocity) which dominates at low frequencies. A new model is now proposed which uses velocity and density fields varying linearly with depth, with coefficients that are functions of horizontal position and time. This model can explicitly represent the thermal wind balance, but its use is not restricted to low- frequency dynamics.

Volume, mass, buoyancy variance, energy and momentum are conserved in the new model. Furthermore, these integrals of motion have the same dependence on the dynamical fields as the exact (continuously stratified) case. The evolution of the three components of the absolute vorticity field are correctly represented. Conservation of density–potential vorticity is not fulfilled, though, owing to artificial removal of the vertical curvature of the velocity field.

The integrals of motion are used to construct a ‘free energy’ [Escr ]f, which is quadratic to the lowest order in the deviation from a steady state with (at most) a uniform velocity field. [Escr ]f is positive definite, and therefore the free evolution of the system cannot lead to an ‘explosion’ of the dynamical fields. (This is not the case if the velocity shear and/or the density vertical gradient is excluded in the model, which results in a non-negative definite free energy.)

In a model with one active layer, linear waves on top of a steady state with no currents are, to a very good approximation, those of the first two vertical modes of the continuously stratified model. These are the familiar geophysical gravity and vortical waves (e.g. Poincaré, Rossby, and coastal Kelvin waves at mid-latitudes, equatorial waves, etc.).

Finally, baroclinic instability is well represented in the new model. For long perturbations (wavelengths of the order of the deformation radius of the first mode) the agreement with more precise calculations is excellent. On the other hand, the comparison with the eigenvalues of Eady's problem (which corresponds to wavelengths of the order of the deformation radius of the second mode) shows differences of the order of 40%. Nevertheless, the new model does have a high-wavenumber cutoff, even though it is constrained to linear profiles in depth and therefore cannot reproduce the exponential trapping of Eady's problem eigensolutions.

In sum, the integrals of motion, vorticity dynamics, free waves and baroclinic instability results all give confidence in the new model. Its main novelty, however, lies in the ability to incorporate thermodynamic processes.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 303 , 25 November 1995 , pp. 169 - 201
Copyright
© 1995 Cambridge University Press

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References

Anderson, D. 1984 An advective mixed-layer model with applications to the diurnal cycle of the low-level East African jet. Tellus 36, 278–291.Google Scholar
Anderosn, D. & Mccreary, J. 1985 Slowly propagating disturbances in a compled ocean- atmosphere model. J. Atmos. Sci. 42, 615–629.Google Scholar
Arnold, V. 1965 Condition for nonlinear stationary plane curvilinear flows of an ideal fluid. Dokl. Akad. Nauk. USSR 162, (English transl: Soviet Maths 6, 773–777, 1965).Google Scholar
Arnold, V. 1966 On an apriori estimate in the theory of hydrodynamical stability. Izu. Vyssh. Uchebn. Zaved. Matematika 54, 3–5 (English transl: Am. Math. Soc. Transl. Series 6, 2, Am. Math. Soc. Transl. Series 79, 267–259, 1969).Google Scholar
Balmaseda, M. A., Anderson, D. & Davey, M. 1994 ENSO Prediction using a dynamical ocean model coupled to statistical atmospheres. Tellus 46A, 497–511.Google Scholar
Cherniwsky, J. & Holloway, G. 1991 An upper-ocean general circulation model of the North Pacific: preliminary experiments. Atmos.-Ocean 29, 737–784.Google Scholar
Cherniawsky, J. Y., Yuen, C. W., Lin, C. A. & Mysak, L. A. 1990 Numerical experiments with a wind- and buoyancy-driven two-and-a-half-layer upper ocean model. J. Geophys. Res. 95, 16149–16167.Google Scholar
Darby, M. & Willmott, A. 1993 Vacillating ocean gyres: an instability mechanism in a thermody-namic reduced gravity ocean model. Tellus, submitted.
Dempsey, P. & Rotunno, R. 1988 Topographic generation of mesoscale vortices in mixed-layer models. J. Atmos Sci. 45, 2961–2978.Google Scholar
Fukamachi, Y., Mccreary, J. & Proehl, J. 1995 Instability of density fronts in layer & continuously stratified models. J. Geophys. Res. 100, 2559–2577.Google Scholar
Gill, A. 1982 Atmosphere-Ocean Dynamics. Academic Press.
Kueny, C. S. & Morrison, P. J. 1994 Nonlinear instability and chaos in plasma wave-wave interactions. I. Introduction. Tech. Rep. IFSR 682. The University of Texas.
Lavoie, R. L. 1972 A mesoscale numerical model of lake-effect storms. J. Atmos. Sci. 29, 1025–1040.Google Scholar
Mccreary, J. P., Fukamachi, Y. & Kundu, P. 1991 A numerical investigation of jets and eddies near an eastern ocean boundary. J. Geophys. Res. 96, 2515–2534.Google Scholar
Mccreary, J. & Kundu, P. 1988 A numerical investigation of the Somali current during the southwest monsoon. J. Mar. Res. 46, 25–58.Google Scholar
Mccreary, J. P., Lee, H. & Enfield, D. 1989 The response of the coastal ocean to strong offshore winds: with application to circulations in the gulfs of Tehuantepec and Papagayo. J. Mar. Res. 47, 81–109.Google Scholar
Mccreary, J. & Lu, P. 1994 Interaction between the subtropical and equatorial ocean circulations: The subtropical cell. J. Phys. Oceanogr. 24, 466–497.Google Scholar
Mccreary, J. & Yu, Z. 1992 Equatorial dynamics in a 2-layer model. Prog. Oceanogr. 29, 61–132.Google Scholar
Mcintyre, M. & Shepherd, T. 1987 An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnold's stability theorems. J. Fluid Mech. 181, 527–565.Google Scholar
Morrison, P. J. & Kotschenreuther, M. 1990 The free energy principle, negative energy modes, and stability. In Nonlinear World: IV lntl Workshop on Nonlinear and Turbulent Processes in Physics, Kiev, USSR, 1989 (ed. V. G. Bar'yakhtar, V. M. Chernousenko, N. S. Erokhin, A. G. Sitenko & V. E. Zakharov), pp. 910–932.
F'edlosky, J. 1979 Geophysical Fluid Dynamics. Springer.
Ripa, P. 1986 Evaluation of vertical structure functions for the analysis of oceanic data. J. Phys. Oceanogr. 16, 223–232.Google Scholar
Ripa, P. 1990 Positive, negative and zero wave energy and the flow stability problem, in the eulerian and lagragian-eulerian descriptions. Pure Appl. Geophys. 133, 713–732.Google Scholar
Ripa, P. 1991 General stability conditions for a multi-layer model. J. Fluid Mech. 222, 119–137.Google Scholar
Ripa, P. 1992a Wave energy-momentum and pseudoenergy-momentum conservation for the layered quasi-geostrophic instability problem. J. Fluid Mech. 235, 379–398.Google Scholar
Ripa, P. 1992b Instability of a solid-body rotating vortex in a two layer model. J. Fluid Mech. 242, 395–417.Google Scholar
Ripa, P. 1993a Conservation laws for primitive equations models with inhomogeneous layers. Geophys. Astrophys. Fluid Dyn. 70, 85–111.Google Scholar
Ripa, P. 1993b Hamiltonian GFD. In Geometrical Methods in Fluid Mechanics (ed. R. Salmon), pp. 332–336. WHOI.
Ripa, P. 1993c Integrals of motion and stability-instability properties of ocean models. In Trends in Oceanography (ed. J. Menon), pp. 141–151. Research Trends (India).
Ripa, P. 1995a Linear waves in a one-layer ocean model with thermodynamics. J. Geophys. Res. ([A-Z]), in press.Google Scholar
Ripa, P. 1995b Low frequency approximation of a one-layer ocean model with thermodynamics Rev. Mex. Fis. submitted.Google Scholar
Ripa, P. 1995c A low-frequency one-layer model with variable velocity shear and stratification, J. Phys. Oceeanogr. 41, in perss.
Schopf, P. & Cane, M. 1983 On ecuatorial dynamics, mixed layer physics and sea surface temper- ature. J. Phys. Oceanogr. 13, 917–935.Google Scholar
Shepherd, T. 1988 Rigorous bounds on the nonlinear saturation of instabilities to parallel shear flows. J. Fluid Mech. 196, 291–322.Google Scholar
Shepherd, T. 1990 Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. Adv. Geophys. 32, 287–338.Google Scholar
Szoeke, R. & Richman, De J. 1984 On wind-driven mixed layers with strong horizontal gradients A theory with applications to coastal upwelling. J. Phys. Oceanogr. 14, 364–377.Google Scholar
Tandon, A. & Garrett, C. 1994 Mixed layer restratification due to a horizontal density gradient. J. Phys. Oceanogr. 24, 1419–1424.Google Scholar
Vanneste, J. 1995 Explosive resonant interaction of Rossby waves and stability of multiplayer quasi-geostrophic flow. J. Fluid Mech., in press.
Young, W. 1994 The subinertial mixed layer approximation. J. Phys. Oceanogr. 24, 1812–1826.Google Scholar
Young, W. & Chen, L. 1995 Baroclinic instability and thermoaline gradient alignment in the mixed layer. J. Phys. Oceanogr., submitted.