Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-14T07:01:50.478Z Has data issue: false hasContentIssue false
  • English
  • Français

The turbulent equilibration of an unstable baroclinic jet

Published online by Cambridge University Press:  06 March 2008

J. G. ESLER
J. G. ESLER*
Affiliation:
Department of Mathematics, University College London, 25 Gower Street, London WC1E 6BT, UKgavin@math.ucl.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

The evolution of an unstable baroclinic jet, subject to a small perturbation, is examined numerically in a quasi-geostrophic two-layer β-channel model. After a period of initial wave growth, wave breaking leads to turbulence within each layer, and to the eventual equilibration of the flow. The equilibrated flow must satisfy certain dynamical constraints: total momentum is conserved, the total energy is bounded and the flow must be realizable via some area-preserving (diffusive) rearrangement of the initial potential vorticity field in each layer. A theory is introduced that predicts the equilibrated flow in terms of the initial flow parameters. The idea is that the equilibrated state minimizes available potential energy, subject to the constraints on total momentum and total energy, and the further ‘kinematic’ constraint that the potential vorticity changes through a process of complete homogenization within well-delineated regions in each layer. Within a large region of parameter space, the theory accurately predicts the cross-channel structure and strength of the equilibrated jet, the regions where potential vorticity mixing takes place, and total eddy mass (temperature) fluxes. Results are compared with predictions from a maximum-entropy theory that allows for more general rearrangements of the initial potential vorticity field, subject to the known dynamical constraints. The maximum-entropy theory predicts that significantly more available potential energy is released than is observed in the simulations, and that an unphysical ‘exchange’ of bands of fluid will occur across the channel in the lower layer. The kinematic constraint of piecewise potential vorticity homogenization is therefore important in limiting the ‘efficiency’ of release of available potential energy in unstable baroclinic flows. For a typical initial flow, it is demonstrated that if the dynamical constraints alone are considered, then over twice as much potential energy is available for release compared to that actually released in the simulations.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 599 , 25 March 2008 , pp. 241 - 268
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Arbic, B. K. & Flierl, G. R. 2004 Baroclinically unstable geostrophic turbulence in the limits of strong and weak bottom ekman friction: Application to midocean eddies. J. Phys. Oceanogr. 64, 2257–2273.2.0.CO;2>CrossRefGoogle Scholar
Brands, H., Chavanis, P. H., Pasmanter, R. & Sommeria, J. 1999 Maximum entropy versus minimum enstrophy vortices. Phys. Fluids 11, 3465–3477.Google Scholar
Chavanis, P. H. & Sommeria, J. 1998 Classification of robust isolated vortices in two-dimensional hydrodynamics. J. Fluid Mech. 356, 259–296.CrossRefGoogle Scholar
Dritschel, D. G. & McIntyre, M. E. 2007 Multiple jets as PV staircases: the Phillips effect and the resilience of eddy-transport barriers. J. Atmos. Sci. 65, in press.Google Scholar
Esler, J. G. & Haynes, P. H. 1999 Mechanisms for wave packet formation and maintenance in a quasi-geostrophic two-layer model. J. Atmos. Sci. 56, 2457–2489.Google Scholar
Feldstein, S. B. & Held, I. M. 1989 Barotropic decay of baroclinic waves in a two-layer beta plane model. J. Atmos. Sci. 46, 3416–3430.2.0.CO;2>CrossRefGoogle Scholar
Gutowksi, W. J. 1985 Baroclinic adjustment and midlatitude temperature profiles. J. Atmos. Sci. 42, 1733–1745.Google Scholar
Held, I. M. 2005 The gap between simulation and understanding in climate modeling. Bull. Am. Met. Soc. 86, 1069.Google Scholar
Held, I. M. 2007 Progress and problems in large-scale atmospheric dynamics. In The Global Circulation of the Atmosphere: Phenomena, Theory, Challenges, chap. 1. Princeton University Press.Google Scholar
Held, I. M. & Larichev, V. 1996 A scaling theory for horizontally homogeneous, baroclinically unstable flow on a beta plane. J. Atmos. Sci. 53, 946–952.Google Scholar
James, I. N. 1987 Suppression of baroclinic instability in horizontally sheared flows. J. Atmos. Sci. 44, 3710–3720.2.0.CO;2>CrossRefGoogle Scholar
Lapeyre, G. & Held, I. M. 2003 Diffusivity, kinetic energy dissipation, and closure theories for the poleward eddy heat flux. J. Atmos. Sci. 60, 2907–2916.Google Scholar
Lee, S. & Held, I. M. 1993 Baroclinic wave packets in models and observations. J. Atmos. Sci. 50, 1413–1428.Google Scholar
Majda, A. J. & Wang, X. 2006 Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows. Cambridge University Press.CrossRefGoogle Scholar
Mak, M. 2000 Does an unstable baroclinic wave equilibrate/decay baroclinically or barotropically? J. Atmos. Sci 57, 453–463.Google Scholar
McIntyre, M. E. 1994 The quasi-biennial oscillation (QBO): some points about the terrestrial QBO and the possibility of related phenomena in the solar interior. In The Solar Engine and its Influence on the Terrestrial Atmosphere and Climate (Vol. 25 NATO ASI Subseries I, Global environmental change), chap. 2, pp. 293–320. Springer.Google Scholar
McIntyre, M. E. & Shepherd, T. G. 1987 An exact local conservation theorem for finite-amplitude disturbances to non-local shear flows, with remarks on Hamiltonian structure and on Arnol'd's stability theorems. J. Fluid Mech. 181, 527–565.Google Scholar
McWilliams, J. C. 2006 Fundamentals of Geophysical Fluid Dynamics. Cambridge University Press.Google Scholar
Miller, J. 1990 Statistical mechanics of Euler's equations in two dimensions. Phys. Rev. Lett. 65, 2137–2140.Google Scholar
Nakamura, N. 1993 Momentum flux, flow symmetry, and the nonlinear barotropic governor. J. Atmos. Sci. 50, 2159–2179.2.0.CO;2>CrossRefGoogle Scholar
Nakamura, N. 1999 Baroclinic-barotropic adjustments in a meridionally wide domain. J. Atmos. Sci. 56, 2246–2260.2.0.CO;2>CrossRefGoogle Scholar
Pavan, V. & Held, I. M. 1996 The diffusive approximation for eddy fluxes in baroclinically unstable jets. J. Atmos. Sci. 52, 1262–1272.2.0.CO;2>CrossRefGoogle Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics. Springer.CrossRefGoogle Scholar
Phillips, N. A. 1951 A simple three-dimensional model for the study of large-scale extratropical flow patterns. J. Met. 8, 381–394.Google Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. 1996 Numerical Recipes in Fortran 77, Second edn. Cambridge University Press.Google Scholar
Prieto, R. & Schubert, W. H. 2001 Analytical predictions for zonally symmetric equilibrium states of the stratospheric polar vortex. J. Atmos. Sci 58, 2709–2728.Google Scholar
Rhines, P. 1975 Waves and turbulence on the beta plane. J. Fluid. Mech. 69, 417–443.CrossRefGoogle Scholar
Rhines, P. 1994 Jets. Chaos 4, 313–339.CrossRefGoogle ScholarPubMed
Robert, R. 1991 A maximum-entropy principle for two-dimensional perfect fluid dynamics. J. Statist Phys. 65, 531–553.CrossRefGoogle Scholar
Robert, R. & Sommeria, J. 1991 Statistical equilibrium states for two dimensional flows. J. Fluid Mech. 229, 291–310.Google Scholar
Schneider, T. & Walker, C. C. 2006 Self-organisation of atmospheric macroturbulence into critical states of weak nonlinear eddy-eddy interactions. J. Atmos. Sci. 63, 1569–1585.Google Scholar
Shepherd, T. G. 1988 Nonlinear saturation of baroclinic instability. Part I: The two-layer model. J. Atmos. Sci. 45, 2014–2025.2.0.CO;2>CrossRefGoogle Scholar
Simmons, A. J. & Hoskins, B. J. 1978 The lifecycles of some nonlinear baroclinic waves. J. Atmos. Sci. 35, 414–432.Google Scholar
Sommeria, J., Staquet, C. & Robert, R. 1991 Final equilibrium state of a two-dimensional shear layer. J. Fluid Mech. 233, 661–689.Google Scholar
Starr, V. P. 1968 Physics of Negative Viscosity Phenomena. McGraw-Hill.Google Scholar
Stone, P. H. 1978 Baroclinic adjustment. J. Atmos. Sci. 35, 561–571.2.0.CO;2>CrossRefGoogle Scholar
Swanson, K. & Pierrehumbert, R. T. 1994 Nonlinear wave packet evolution on a baroclinically unstable jet. J. Atmos. Sci. 51, 384–396.Google Scholar
Thompson, A. F. & Young, W. R. 2007 Two-layer baroclinic eddy heat fluxes: zonal flows and energy balance. J. Atmos. Sci. 63, 3214–3231.Google Scholar
Thorncroft, C. D., Hoskins, B. J. & McIntyre, M. E. 1993 Two paradigms of baroclinic wave lifecycle behaviour. Q. J. R. Met. Soc. 119, 17–55.Google Scholar
Turkington, B. & Whitaker, N. 1996 Statistical equilibrium computations of coherent structures in turbulent shear layers. SIAM J. Sci. Comput. 17, 1414–1433.Google Scholar
Vallis, G. K. 1988 Numerical studies of eddy transport properties in eddy-resolving and parameterized models. Q. J. R. Met. Soc. 114, 183–208.Google Scholar
Wardle, R. & Marshall, J. 2000 Representation of eddies in primitive equation models by a PV flux. J. Phys. Oceanogr. 30, 2481–2503.2.0.CO;2>CrossRefGoogle Scholar
Warn, T. & Gauthier, P. 1989 Potential vorticity mixing by marginally unstable baroclinic disturbances. Tellus 41A, 115–131.CrossRefGoogle Scholar
Zurita-Gotor, P. 2007 The relation between baroclinic adjustment and turbulent diffusion in the two layer model. J. Atmos. Sci. 64, 1284–1300.Google Scholar
Zurita-Gotor, P. & Lindzen, R. S. 2007 Theories of baroclinic adjustment and eddy equilibration. In The Global Circulation of the Atmosphere: Phenomena, Theory, Challenges, chap. 2. Princeton University Press.Google Scholar