• Andrews, D. G., 1981: A note on potential energy density in a stratified compressible fluid. J. Fluid Mech., 107 , 227236.

  • Arnol’d, V. I., 1965: Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid. Dokl. Akad. Nauk SSSR, 162 , 975978.

    • Search Google Scholar
    • Export Citation
  • Bretherton, F. P., 1966: Baroclinic instability and the short wavelength cut-off in terms of potential vorticity. Quart. J. Roy. Meteor. Soc., 92 , 335345.

    • Search Google Scholar
    • Export Citation
  • Bühler, O., 2009: Waves and Mean Flows. Cambridge University Press, 341 pp.

  • Charney, J. G., , and M. E. Stern, 1962: On the stability of internal baroclinic jets in a rotating atmosphere. J. Atmos. Sci., 19 , 159172.

    • Search Google Scholar
    • Export Citation
  • Dickinson, R. E., 1969: Theory of planetary wave-zonal flow interaction. J. Atmos. Sci., 26 , 7381.

  • Dritschel, D. G., , and M. E. McIntyre, 2008: Multiple jets as PV staircases: The Phillips effect and the resilience of eddy-transport barriers. J. Atmos. Sci., 65 , 855874.

    • Search Google Scholar
    • Export Citation
  • Dunkerton, T. J., , and R. K. Scott, 2008: A barotropic model of the angular momentum–conserving potential vorticity staircase in spherical geometry. J. Atmos. Sci., 65 , 11051136.

    • Search Google Scholar
    • Export Citation
  • Esler, J. G., 2008a: Robust and leaky transport barriers in unstable baroclinic flows. Phys. Fluids, 20 , 116602. doi:10.1063/1.3013631.

  • Esler, J. G., 2008b: The turbulent equilibration of an unstable baroclinic jet. J. Fluid Mech., 599 , 241268.

  • Fiedler, B. H., 1984: An integral closure model for the vertical turbulent flux of a scalar in a mixed layer. J. Atmos. Sci., 41 , 674680.

    • Search Google Scholar
    • Export Citation
  • Green, J. S. A., 1970: Transfer properties of the large-scale eddies and the general circulation of the atmosphere. Quart. J. Roy. Meteor. Soc., 96 , 157185.

    • Search Google Scholar
    • Export Citation
  • Haynes, P. H., , and M. E. McIntyre, 1990: On the conservation and impermeability theorems for potential vorticity. J. Atmos. Sci., 47 , 20212031.

    • Search Google Scholar
    • Export Citation
  • Held, I. M., 2001: The general circulation of the atmosphere. The General Circulation of the Atmosphere: 2000 Program in Geophysical Fluid Dynamics, Woods Hole Oceanographic Institution Tech. Rep. WHOI-2001-03, R. Salmon et al., Eds., WHOI, 1–54.

    • Search Google Scholar
    • Export Citation
  • Holliday, D., , and M. E. McIntyre, 1981: On potential energy density in an incompressible, stratified fluid. J. Fluid Mech., 107 , 221225.

    • Search Google Scholar
    • Export Citation
  • Hughes, C. W., 1996: The Antarctic Circumpolar Current as a waveguide for Rossby waves. J. Phys. Oceanogr., 26 , 13751392.

  • Killworth, P. D., , and M. E. McIntyre, 1985: Do Rossby-wave critical layers absorb, reflect, or over-reflect? J. Fluid Mech., 161 , 449492.

    • Search Google Scholar
    • Export Citation
  • Lahoz, W. A., , A. J. Geer, , and A. O’Neill, 2006: Dynamical evolution of the 2003 southern hemisphere stratospheric winter using Envisat trace-gas observations. Quart. J. Roy. Meteor. Soc., 132 , 19852008.

    • Search Google Scholar
    • Export Citation
  • McIntyre, M. E., 2008: Potential-vorticity inversion and the wave-turbulence jigsaw: Some recent clarifications. Adv. Geosci., 15 , 4756.

    • Search Google Scholar
    • Export Citation
  • Molemaker, M. J., , and J. C. McWilliams, 2010: Local balance and cross-scale flux of available potential energy. J. Fluid Mech., in press.

    • Search Google Scholar
    • Export Citation
  • Pasquill, F., , and F. B. Smith, 1983: Atmospheric Diffusion. E. Horwood, 437 pp.

  • Phillips, N. A., 1954: Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level quasi-geostrophic model. Tellus, 6 , 273286.

    • Search Google Scholar
    • Export Citation
  • Plumb, R. A., , and D. D. McConalogue, 1988: On the meridional structure of long-lived tropospheric constituents. J. Geophys. Res., 93 , 1589715913.

    • Search Google Scholar
    • Export Citation
  • Roullet, G., , and P. Klein, 2009: Available potential energy diagnosis in a direct numerical simulation of rotating stratified turbulence. J. Fluid Mech., 624 , 4555.

    • Search Google Scholar
    • Export Citation
  • Shepherd, T. G., 1993: A unified theory of available potential energy. Atmos.–Ocean, 31 , 126.

  • Shnirelman, A. I., 1993: Lattice theory and flows of ideal incompressible fluids. Russ. J. Math. Phys., 1 , 105114.

  • Showman, A. P., 2007: Numerical simulations of forced shallow-water turbulence: Effects of moist convection on the large-scale circulation of Jupiter and Saturn. J. Atmos. Sci., 64 , 31323157.

    • Search Google Scholar
    • Export Citation
  • Stull, R. B., 1984: Transilient turbulence theory, I: The concept of eddy mixing across finite distances. J. Atmos. Sci., 41 , 33513367.

    • Search Google Scholar
    • Export Citation
  • Taylor, G. I., 1915: Eddy motion in the atmosphere. Philos. Trans. Roy. Soc. London, A215 , 123.

  • Taylor, G. I., 1932: The transport of vorticity and heat through fluids in turbulent motion. Proc. Roy. Soc. London, 135 , 685705.

  • Thuburn, J., , and M. E. McIntyre, 1997: Numerical advection schemes, cross-isentropic random walks, and correlations between chemical species. J. Geophys. Res., 102 , 67756797.

    • Search Google Scholar
    • Export Citation
  • Welander, P., 1973: Lateral friction in the oceans as an effect of potential vorticity mixing. Geophys. Fluid Dyn., 5 , 173189.

  • Young, W. R., 2010: Dynamic enthalpy, conservative temperature, and the seawater Boussinesq approximation. J. Phys. Oceanogr., 40 , 394400.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 54 54 17
PDF Downloads 25 25 5

A General Theorem on Angular-Momentum Changes due to Potential Vorticity Mixing and on Potential-Energy Changes due to Buoyancy Mixing

View More View Less
  • 1 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom
© Get Permissions
Restricted access

Abstract

An initial zonally symmetric quasigeostrophic potential vorticity (PV) distribution qi(y) is subjected to complete or partial mixing within some finite zone |y| < L, where y is latitude. The change in M, the total absolute angular momentum, between the initial and any later time is considered. For standard quasigeostrophic shallow-water beta-channel dynamics it is proved that, for any qi(y) such that dqi/dy > 0 throughout |y| < L, the change in M is always negative. This theorem holds even when “mixing” is understood in the most general possible sense. Arbitrary stirring or advective rearrangement is included, combined to an arbitrary extent with spatially inhomogeneous diffusion. The theorem holds whether or not the PV distribution is zonally symmetric at the later time. The same theorem governs Boussinesq potential-energy changes due to buoyancy mixing in the vertical. For the standard quasigeostrophic beta-channel dynamics to be valid the Rossby deformation length LDϵL where ϵ is the Rossby number; when LD = ∞ the theorem applies not only to the beta channel but also to a single barotropic layer on the full sphere, as considered in the recent work of Dunkerton and Scott on “PV staircases.” It follows that the M-conserving PV reconfigurations studied by those authors must involve processes describable as PV unmixing, or antidiffusion, in the sense of time-reversed diffusion. Ordinary jet self-sharpening and jet-core acceleration do not, by contrast, require unmixing, as is shown here by detailed analysis. Mixing in the jet flanks suffices. The theorem extends to multiple layers and continuous stratification. A least upper bound and greatest lower bound for the change in M is obtained for cases in which qi is neither monotonic nor zonally symmetric. A corollary is a new nonlinear stability theorem for shear flows.

Corresponding author address: Richard B. Wood, University of Cambridge, Department of Applied Mathematics and Theoretical Physics, Wilberforce Road, Cambridge CB3 0WA, United Kingdom. Email: r.b.wood@damtp.cam.ac.uk

Abstract

An initial zonally symmetric quasigeostrophic potential vorticity (PV) distribution qi(y) is subjected to complete or partial mixing within some finite zone |y| < L, where y is latitude. The change in M, the total absolute angular momentum, between the initial and any later time is considered. For standard quasigeostrophic shallow-water beta-channel dynamics it is proved that, for any qi(y) such that dqi/dy > 0 throughout |y| < L, the change in M is always negative. This theorem holds even when “mixing” is understood in the most general possible sense. Arbitrary stirring or advective rearrangement is included, combined to an arbitrary extent with spatially inhomogeneous diffusion. The theorem holds whether or not the PV distribution is zonally symmetric at the later time. The same theorem governs Boussinesq potential-energy changes due to buoyancy mixing in the vertical. For the standard quasigeostrophic beta-channel dynamics to be valid the Rossby deformation length LDϵL where ϵ is the Rossby number; when LD = ∞ the theorem applies not only to the beta channel but also to a single barotropic layer on the full sphere, as considered in the recent work of Dunkerton and Scott on “PV staircases.” It follows that the M-conserving PV reconfigurations studied by those authors must involve processes describable as PV unmixing, or antidiffusion, in the sense of time-reversed diffusion. Ordinary jet self-sharpening and jet-core acceleration do not, by contrast, require unmixing, as is shown here by detailed analysis. Mixing in the jet flanks suffices. The theorem extends to multiple layers and continuous stratification. A least upper bound and greatest lower bound for the change in M is obtained for cases in which qi is neither monotonic nor zonally symmetric. A corollary is a new nonlinear stability theorem for shear flows.

Corresponding author address: Richard B. Wood, University of Cambridge, Department of Applied Mathematics and Theoretical Physics, Wilberforce Road, Cambridge CB3 0WA, United Kingdom. Email: r.b.wood@damtp.cam.ac.uk

Save