Editorial Type:
Article
Article Type:
Research Article

Balance and the Slow Quasimanifold: Some Explicit Results

Rupert Ford Centre for Atmospheric Science, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom

Search for other papers by Rupert Ford in
Current site
Google Scholar
PubMed Close
,
Michael E. McIntyre Centre for Atmospheric Science, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom

Search for other papers by Michael E. McIntyre in
Current site
Google Scholar
PubMed Close
, and
Warwick A. Norton Centre for Atmospheric Science, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom

Search for other papers by Warwick A. Norton in
Current site
Google Scholar
PubMed Close
Print Publication:
01 May 2000
DOI:
https://doi.org/10.1175/1520-0469(2000)057<1236:BATSQS>2.0.CO;2
Page(s):
1236–1254
Restricted access

Abstract

The ultimate limitations of the balance, slow-manifold, and potential vorticity inversion concepts are investigated. These limitations are associated with the weak but nonvanishing spontaneous-adjustment emission, or Lighthill radiation, of inertia–gravity waves by unsteady, two-dimensional or layerwise-two-dimensional vortical flow (the wave emission mechanism sometimes being called “geostrophic” adjustment even though it need not take the flow toward geostrophic balance). Spontaneous-adjustment emission is studied in detail for the case of unbounded f-plane shallow-water flow, in which the potential vorticity anomalies are confined to a finite-sized region, but whose distribution within the region is otherwise completely general. The approach assumes that the Froude number F and Rossby number R satisfy F ≪ 1 and R ≳ 1 (implying, incidentally, that any balance would have to include gradient wind and other ageostrophic contributions). The method of matched asymptotic expansions is used to obtain a general mathematical description of spontaneous-adjustment emission in this parameter regime. Expansions are carried out to O(F4), which is a high enough order to describe not only the weakly emitted waves but also, explicitly, the correspondingly weak radiation reaction upon the vortical flow, accounting for the loss of vortical energy. Exact evolution on a slow manifold, in its usual strict sense, would be incompatible with the arrow of time introduced by this radiation reaction and energy loss. The magnitude O(F4) of the radiation reaction may thus be taken to measure the degree of “fuzziness” of the entity that must exist in place of the strict slow manifold. That entity must, presumably, be not a simple invariant manifold, but rather an O(F4)-thin, multileaved, fractal “stochastic layer” like those known for analogous but low-order coupled oscillator systems. It could more appropriately be called the “slow quasimanifold.”

* Current affiliation: Department of Mathematics, Imperial College, London, United Kingdom.

Current affiliation: Atmospheric, Oceanic, and Planetary Physics, Oxford University, Oxford, United Kingdom.

The Centre for Atmospheric Science is a joint initiative of the Department of Chemistry and the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge.

Corresponding author address: Dr. Michael E. McIntyre, Dept. of Applied Mathmatics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, United Kingdom.

Email: mem@damtp.cam.ac.uk

Abstract

The ultimate limitations of the balance, slow-manifold, and potential vorticity inversion concepts are investigated. These limitations are associated with the weak but nonvanishing spontaneous-adjustment emission, or Lighthill radiation, of inertia–gravity waves by unsteady, two-dimensional or layerwise-two-dimensional vortical flow (the wave emission mechanism sometimes being called “geostrophic” adjustment even though it need not take the flow toward geostrophic balance). Spontaneous-adjustment emission is studied in detail for the case of unbounded f-plane shallow-water flow, in which the potential vorticity anomalies are confined to a finite-sized region, but whose distribution within the region is otherwise completely general. The approach assumes that the Froude number F and Rossby number R satisfy F ≪ 1 and R ≳ 1 (implying, incidentally, that any balance would have to include gradient wind and other ageostrophic contributions). The method of matched asymptotic expansions is used to obtain a general mathematical description of spontaneous-adjustment emission in this parameter regime. Expansions are carried out to O(F4), which is a high enough order to describe not only the weakly emitted waves but also, explicitly, the correspondingly weak radiation reaction upon the vortical flow, accounting for the loss of vortical energy. Exact evolution on a slow manifold, in its usual strict sense, would be incompatible with the arrow of time introduced by this radiation reaction and energy loss. The magnitude O(F4) of the radiation reaction may thus be taken to measure the degree of “fuzziness” of the entity that must exist in place of the strict slow manifold. That entity must, presumably, be not a simple invariant manifold, but rather an O(F4)-thin, multileaved, fractal “stochastic layer” like those known for analogous but low-order coupled oscillator systems. It could more appropriately be called the “slow quasimanifold.”

* Current affiliation: Department of Mathematics, Imperial College, London, United Kingdom.

Current affiliation: Atmospheric, Oceanic, and Planetary Physics, Oxford University, Oxford, United Kingdom.

The Centre for Atmospheric Science is a joint initiative of the Department of Chemistry and the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge.

Corresponding author address: Dr. Michael E. McIntyre, Dept. of Applied Mathmatics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, United Kingdom.

Email: mem@damtp.cam.ac.uk

Save
Cover Journal of the Atmospheric Sciences
Journal of the Atmospheric Sciences

  • Abramowitz, M., and I. A. Stegun, 1964: A Handbook of Mathematical Functions. Dover, 1046 pp.

  • Aref, H., and N. Pomphrey, 1982: Integrable and chaotic motions of four vortices: I. The case of identical vortices. Proc. Roy. Soc. London,380A, 359–387.

  • Armi, L., D. Hebert, N. Oakey, J. Price, P. L. Richardson, T. Rossby, and B. Ruddick, 1988: The history and decay of a Mediterranean salt lens. Nature,333, 649–651.

  • Batchelor, G. K., 1967: An Introduction to Fluid Dynamics. Cambridge University Press, 615 pp.

  • Berry, M. V., 1978: Regular and irregular motion. Proc. AIP Conf. on Topics in Nonlinear Dynamics, Vol. 46, La Jolla, CA, American Institute of Physics, 16–120.

  • Bishop, C. H., and A. J. Thorpe, 1994: Potential vorticity and the electrostatics analogy: Quasi-geostrophic theory. Quart. J. Roy. Meteor. Soc.,120, 713–731.

  • Bokhove, O., and T. G. Shepherd, 1996: On Hamiltonian balanced dynamics and the slowest invariant manifold. J. Atmos. Sci.,53, 276–297.

  • Chang, K. M., and I. Orlanski, 1993: On the dynamics of storm tracks. J. Atmos. Sci.,50, 999–1015.

  • Charney, J. G., 1948: On the scale of atmospheric motions. Geofys. Publ.,17 (2), 3–17.

  • Crighton, D. G., 1981: Acoustics as a branch of fluid mechanics. J. Fluid Mech.,106, 261–298.

  • Crow, S. C., 1970: Aerodynamic sound generation as a singular perturbation problem. Stud. Appl. Math.,49, 21–44.

  • Errico, R. M., 1981: An analysis of interactions between geostrophic and ageostrophic modes in a simple model. J. Atmos. Sci.,38, 544–553.

  • Ford, R., 1993: Wave generation by vortical motion. Ph.D. thesis, University of Cambridge, 269 pp. [Available from Superintendent of Manuscripts, University Library, West Rd., Cambridge CB3 9DR, United Kingdom.].

  • ——, 1994a: The instability of an axisymmetric vortex with monotonic potential vorticity in rotating shallow water. J. Fluid Mech.,280, 303–334.

  • ——, 1994b: The response of a rotating ellipse of uniform potential vorticity to gravity wave radiation. Phys. Fluids,6A, 3694–3704.

  • Gill, A. E., 1982: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.

  • Gough, D. O., and M. E. McIntyre, 1998: Inevitability of a magnetic field in the sun’s radiative interior. Nature,394, 755–757.

  • Hoskins, B. J., M. E. McIntyre, and A. W. Robertson, 1985: On the use and significance of isentropic potential-vorticity maps. Quart. J. Roy. Meteor. Soc.,111, 877–946.

  • Kambe, T., 1986: Acoustic emissions by vortex motions. J. Fluid Mech.,173, 643–666.

  • Kleinschmidt, E., 1950a: Über Aufbau und Entstehung von Zyklonen (1 Teil). Meteor. Rundsch.,3, 1–6.

  • ——, 1950b: Über Aufbau und Entstehung von Zyklonen (2 Teil). Meteor. Rundsch.,3, 54–61.

  • ——, 1951: Über Aufbau und Entstehung von Zyklonen (3 Teil). Meteor. Rundsch.,4, 89–96.

  • Lamb, H., 1932: Hydrodynamics. 6th ed. Cambridge University Press, 738 pp.

  • Leith, C. E., 1980: Nonlinear normal mode initialization and quasi-geostrophic theory. J. Atmos. Sci.,37, 958–968.

  • Lighthill, M. J., 1952: On sound generated aerodynamically, I. General theory. Proc. Roy. Soc. London,211A, 564–587.

  • ——, 1958: An Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press, 79 pp.

  • Lorenz, E. N., 1980: Attractor sets and quasi-geostrophic equilibrium. J. Atmos. Sci.,37, 1685–1699.

  • ——, 1992: The slow manifold—What is it? J. Atmos. Sci.,49, 2449–2451.

  • ——, and V. Krishnamurthy, 1987: On the nonexistence of a slow manifold. J. Atmos. Sci.,44, 2940–2950.

  • Manney, G. L., R. W. Zurek, A. O’Neill, and R. Swinbank, 1994: On the motion of air through the stratospheric polar vortex. J. Atmos. Sci.,51, 2973–2994.

  • McIntyre, M. E., 1993: Isentropic distributions of potential vorticity and their relevance to tropical cyclone dynamics. Proc. ICSU/WMO Int. Symp. on Tropical Cyclone Disasters, Beijing, China, ISCU/WMO, 143–156.

  • ——, and W. A. Norton, 2000: Potential vorticity inversion on a hemisphere. J. Atmos. Sci.,57, 1214–1235.

  • Norton, W. A., 1988: Balance and potential vorticity inversion in atmospheric dynamics. Ph.D. thesis, University of Cambridge, 167 pp. [Available from Superintendent of Manuscripts, University Library, West Rd., Cambridge CB3 9DR, United Kingdom.].

  • Obukhov, A. M., 1962: On the dynamics of a stratified liquid. Sov. Phys. Dokl.,7, 682–684. First published in Dokl. Akad. Nauk SSSR,145, 1239–1242.

  • Polvani, L. M., and R. A. Plumb, 1992: Rossby wave breaking, microbreaking, filamentation, and secondary vortex formation: The dynamics of a perturbed vortex. J. Atmos. Sci.,49, 462–476.

  • Simmons, A. J., and B. J. Hoskins, 1979: The downstream and upstream development of unstable baroclinic waves. J. Atmos. Sci.,36, 1239–1254.

  • Thorncroft, C. D., B. J. Hoskins, and M. E. McIntyre, 1993: Two paradigms of baroclinic-wave life-cycle behaviour. Quart. J. Roy. Meteor. Soc.,119, 17–55.

  • van Dyke, 1964: Perturbation Methods in Fluid Mechanics. Academic Press, 229 pp.

  • Vautard, R., and B. Legras, 1986: Invariant manifolds, quasi-geostrophy and initialization. J. Atmos. Sci.,43, 565–584.

  • Warn, T., 1997: Nonlinear balance and quasigeostrophic sets. Atmos.–Ocean,35, 135–145 (originally written in 1983).

  • ——, and R. Ménard, 1986: Nonlinear balance and gravity-inertial wave saturation in a simple atmosperic model. Tellus,38A, 285–294.

  • ——, O. Bokhove, T. G. Shepherd, and G. K. Vallis, 1995: Rossby number expansions, slaving principles, and balance dynamics. Quart. J. Roy. Meteor. Soc.,121, 723–739.

  • Webster, R. B., 1970: Jet noise simulation on shallow water. J. Fluid Mech.,40, 423–432.

  • Whitaker, J. S., 1993: A comparison of primitive and balance equation simulations of baroclinic waves. J. Atmos. Sci.,50, 1519–1530.

  • Zeitlin, V., 1988: Acoustic radiation from distributed vortex structures. Sov. Phys. Dokl.,34, 199–191.

  • ——, 1991: On the backreaction of acoustic radiation for two-dimensional vortex structures. Phys. Fluids,3A, 1677–1680.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 392 104 13
PDF Downloads 248 88 10