Hostname: page-component-6d856f89d9-fb4gq Total loading time: 0 Render date: 2024-07-16T03:36:42.892Z Has data issue: false hasContentIssue false
  • English
  • Français

Super- and sub-rotating equatorial jets in shallow water models of Jovian atmospheres: Newtonian cooling versus Rayleigh friction

Published online by Cambridge University Press:  07 June 2017

Emma S. Warneford  and
Paul J. Dellar Open the ORCID record for Paul J. Dellar [Opens in a new window]
Emma S. Warneford
Affiliation:
OCIAM, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
Paul J. Dellar*
Affiliation:
OCIAM, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Numerical simulations of the shallow water equations on rotating spheres produce mixtures of robust vortices and alternating zonal jets, as seen in the atmospheres of the gas giant planets. However, simulations that include Rayleigh friction invariably produce a sub-rotating (retrograde) equatorial jet for Jovian parameter regimes, whilst observations of Jupiter show a super-rotating (prograde) equatorial jet that has persisted over several decades. Super-rotating equatorial jets have recently been obtained in shallow water simulations that include a Newtonian relaxation of perturbations to the layer thickness to model radiative cooling to space, and in simulations of the thermal shallow water equations that include a similar relaxation term in their temperature equation. Simulations of global quasigeostrophic forms of these different models produce equatorial jets in the same directions as the parent models, suggesting that the mechanism responsible for setting the direction lies within quasigeostrophic theory. We provide such a mechanism by calculating the effective force acting on the thickness-weighted zonal mean flow due to the decay of an equatorially trapped Rossby wave. Decay due to Newtonian cooling creates an eastward zonal mean flow at the equator, consistent with the formation of a super-rotating equatorial jet, while decay due to Rayleigh friction leads to a westward zonal mean flow at the equator, consistent with the formation of a sub-rotating equatorial jet. In both cases the meridionally integrated zonal mean of the absolute zonal momentum is westward, consistent with the standard result that Rossby waves carry westward pseudomomentum, but this does not preclude the zonal mean flow being eastward on and close to the equator.

JFM classification

Geophysical and Geological Flows: Geostrophic turbulence Geophysical and Geological Flows: Shallow water flows Geophysical and Geological Flows: Waves in rotating fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 822 , 10 July 2017 , pp. 484 - 511
Check for updates
Copyright
© 2017 Cambridge University Press 

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

Andrews, D. & McIntyre, M. 1976a Planetary waves in horizontal and vertical shear: the generalized Eliassen–Palm relation and the mean zonal acceleration. J. Atmos. Sci. 33, 2031–2048.2.0.CO;2>CrossRefGoogle Scholar
Andrews, D. & McIntyre, M. 1976b Planetary waves in horizontal and vertical shear: asymptotic theory for equatorial waves in weak shear. J. Atmos. Sci. 33, 2049–2053.2.0.CO;2>CrossRefGoogle Scholar
Andrews, D. G. & McIntyre, M. E. 1978 Generalized Eliassen–Palm and Charney–Drazin theorems for waves on axisymmetric mean flows in compressible atmospheres. J. Atmos. Sci. 35, 175–185.Google Scholar
Beebe, R. 1994 Characteristic zonal winds and long-lived vortices in the atmospheres of the outer planets. Chaos 4, 113–122.CrossRefGoogle ScholarPubMed
Bühler, O. 2000 On the vorticity transport due to dissipating or breaking waves in shallow-water flow. J. Fluid Mech. 407, 235–263.Google Scholar
Bühler, O. 2014 Waves and Mean Flows, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Charney, J. G. 1949 On a physical basis for numerical prediction of large-scale motions in the atmosphere. J. Atmos. Sci. 6, 372–385.Google Scholar
Cho, J. Y.-K. & Polvani, L. M. 1996a The emergence of jets and vortices in freely evolving, shallow-water turbulence on a sphere. Phys. Fluids 8, 1531–1552.Google Scholar
Cho, J. Y.-K. & Polvani, L. M. 1996b The morphogenesis of bands and zonal winds in the atmospheres on the giant outer planets. Science 273, 335–337.Google Scholar
Daley, R. 1983 Linear non-divergent mass-wind laws on the sphere. Tellus A 35A, 17–27.Google Scholar
Dickinson, R. E. 1969 Theory of planetary wave-zonal flow interaction. J. Atmos. Sci. 26, 73–81.Google Scholar
Dowling, T. E. 1995 Estimate of Jupiter’s deep zonal-wind profile from Shoemaker–Levy 9 data and Arnol’d’s second stability criterion. Icarus 117, 439–442.Google Scholar
Dowling, T. E. & Ingersoll, A. P. 1989 Jupiter’s great red spot as a shallow water system. J. Atmos. Sci. 46, 3256–3278.Google Scholar
Gill, A. E. 1982 Atmosphere Ocean Dynamics. Academic.Google Scholar
Green, J. S. A. 1970 Transfer properties of the large-scale eddies and the general circulation of the atmosphere. Q. J. R. Meteorol. Soc. 96, 157–185.CrossRefGoogle Scholar
Haynes, P. H. & McIntyre, M. E. 1990 On the conservation and impermeability theorems for potential vorticity. J. Atmos. Sci. 47, 2021–2031.Google Scholar
Held, I. M.2000 The general circulation of the atmosphere: superrotation. Lectures presented at the 2000 Geophysical Fluid Dynamics Summer Program, Woods Hole Oceanographic Institution, Woods Hole, MA, available from http://www.whoi.edu/page.do?pid=13076.Google Scholar
Hirst, A. C. 1986 Unstable and damped equatorial modes in simple coupled ocean-atmosphere models. J. Atmos. Sci. 43, 606–632.Google Scholar
Hou, T. Y. & Li, R. 2007 Computing nearly singular solutions using pseudo-spectral methods. J. Comput. Phys. 226, 379–397.Google Scholar
Hupca, I. O., Falcou, J., Grigori, L. & Stompor, R. 2012 Spherical harmonic transform with GPUs. Lecture Notes in Computer Science 7155, 355–366.Google Scholar
Iacono, R., Struglia, M. V. & Ronchi, C. 1999a Spontaneous formation of equatorial jets in freely decaying shallow water turbulence. Phys. Fluids 11, 1272–1274.Google Scholar
Iacono, R., Struglia, M. V., Ronchi, C. & Nicastro, S. 1999b High-resolution simulations of freely decaying shallow-water turbulence on a rotating sphere. Il Nuovo Cimento C 22, 813–821.Google Scholar
Ingersoll, A. P. 1990 Atmospheric dynamics of the outer planets. Science 248, 308–315.Google Scholar
Ingersoll, A. P., Dowling, T. E., Gierasch, P. J., Orton, G. S., Read, P. L., Sánchez-Lavega, A., Showman, A. P., Simon-Miller, A. A. & Vasavada, A. R. 2007 Dynamics of Jupiter’s atmosphere. In Jupiter: The Planet, Satellites and Magnetosphere (ed. Bagenal, F., Dowling, T. E. & McKinnon, W. B.), pp. 105–128. Cambridge University Press.Google Scholar
Juckes, M. 1989 A shallow water model of the winter stratosphere. J. Atmos. Sci. 46, 2934–2956.Google Scholar
Khouider, B., Majda, A. J. & Stechmann, S. N. 2013 Climate science in the tropics: waves, vortices and PDEs. Nonlinearity 26, R1–R68.Google Scholar
Lavoie, R. L. 1972 A mesoscale numerical model of lake-effect storms. J. Atmos. Sci. 29, 1025–1040.Google Scholar
Limaye, S. S. 1986 Jupiter: new estimates of the mean zonal flow at the cloud level. Icarus 65, 335–352.Google Scholar
Liu, J. & Schneider, T. 2010 Mechanisms of jet formation on the giant planets. J. Atmos. Sci. 67, 3652–3672.Google Scholar
Majda, A. J. & Klein, R. 2003 Systematic multiscale models for the tropics. J. Atmos. Sci. 60, 393–408.Google Scholar
Marcus, P. S. 1988 Numerical simulation of Jupiter’s great red spot. Nature 331, 693–696.Google Scholar
Matsuno, T. 1966 Quasi-geostrophic motions in the equatorial area. J. Met. Soc. Japan 44, 25–43.Google Scholar
Matsuno, T. 1970 Vertical propagation of stationary planetary waves in the winter northern hemisphere. J. Atmos. Sci. 27, 871–883.Google Scholar
Matsuno, T. 1971 A dynamical model of the stratospheric sudden warming. J. Atmos. Sci. 28, 1479–1494.Google Scholar
McCreary, J. P. 1985 Modeling equatorial ocean circulation. Annu. Rev. Fluid Mech. 17, 359–409.Google Scholar
McCreary, J. P., Fukamachi, Y. & Kundu, P. K. 1991 A numerical investigation of jets and eddies near an eastern ocean boundary. J. Geophys. Res. 96, 2515–2534.Google Scholar
McCreary, J. P. & Kundu, P. K. 1988 A numerical investigation of the Somali current during the Southwest monsoon. J. Mar. Res. 46, 25–58.Google Scholar
McCreary, J. P. & Yu, Z. 1992 Equatorial dynamics in a 2(1/2)-layer model. Prog. Oceanogr. 29, 61–132.Google Scholar
McIntyre, M. E. 1981 On the ‘wave momentum’ myth. J. Fluid Mech. 106, 331–347.Google Scholar
McIntyre, M. E. & Norton, W. A. 1990 Dissipative wave-mean interactions and the transport of vorticity or potential vorticity. J. Fluid Mech. 212, 403–435.Google Scholar
Mofjeld, H. O. 1981 An analytic theory on how friction affects free internal waves in the equatorial waveguide. J. Phys. Oceanogr. 11, 1585–1590.Google Scholar
Obukhov, A. M. 1949 On the problem of the geostrophic wind. Izv. Akad. Nauk SSSR Geogr. Geofiz 13, 281–306.Google Scholar
Philander, S. G. H., Yamagata, T. & Pacanowski, R. C. 1984 Unstable air–sea interactions in the tropics. J. Atmos. Sci. 41, 604–613.Google Scholar
Polvani, L. M., Waugh, D. W. & Plumb, R. A. 1995 On the subtropical edge of the stratospheric surf zone. J. Atmos. Sci. 52, 1288–1309.Google Scholar
Porco, C. C., West, R. A., McEwen, A., Del Genio, A. D., Ingersoll, A. P., Thomas, P., Squyres, S., Dones, L., Murray, C. D., Johnson, T. V. et al. 2003 Cassini imaging of Jupiter’s atmosphere, satellites, and rings. Science 299, 1541–1547.CrossRefGoogle ScholarPubMed
Rhines, P. B. 1975 Waves and turbulence on a beta-plane. J. Fluid Mech. 69, 417–443.Google Scholar
Richards, A.2015 University of Oxford advanced research computing. Tech. Note, doi:10.5281/zenodo.22558.Google Scholar
Ripa, P. 1993 Conservation laws for primitive equations models with inhomogeneous layers. Geophys. Astrophys. Fluid Dyn. 70, 85–111.Google Scholar
Ripa, P. 1995 On improving a one-layer ocean model with thermodynamics. J. Fluid Mech. 303, 169–201.Google Scholar
Ripa, P. 1996a Low frequency approximation of a vertically averaged ocean model with thermodynamics. Rev. Mex. Fís. 41, 117–135.Google Scholar
Ripa, P. 1996b Linear waves in a one-layer ocean model with thermodynamics. J. Geophys. Res. 101, 1233–1245.Google Scholar
Røed, L. P. 1997 Energy diagnostics in a 1(1/2)-layer, nonisopycnic model. J. Phys. Oceanogr. 27, 1472–1476.Google Scholar
Røed, L. P. & Shi, X. B. 1999 A numerical study of the dynamics and energetics of cool filaments, jets, and eddies off the Iberian peninsula. J. Geophys. Res. 104, 29817–29841.Google Scholar
Saito, I. & Ishioka, K. 2015 Mechanism for the formation of equatorial superrotation in forced shallow-water turbulence with Newtonian cooling. J. Atmos. Sci. 72, 1466–1483.Google Scholar
Schneider, T. & Liu, J. 2009 Formation of jets and equatorial superrotation on Jupiter. J. Atmos. Sci. 66, 579–601.Google Scholar
Schopf, P. S. & Cane, M. A. 1983 On equatorial dynamics, mixed layer physics and sea surface temperature. J. Phys. Oceanogr. 13, 917–935.Google Scholar
Schubert, W. H., Taft, R. K. & Silvers, L. G. 2009 Shallow water quasi-geostrophic theory on the sphere. J. Adv. Model. Earth Syst. 1, 2.Google Scholar
Scott, R. K. & Polvani, L. M. 2007 Forced-dissipative shallow-water turbulence on the sphere and the atmospheric circulation of the giant planets. J. Atmos. Sci. 64, 3158–3176.Google Scholar
Scott, R. K. & Polvani, L. M. 2008 Equatorial superrotation in shallow atmospheres. Geophys. Res. Lett. 35, L24202.Google Scholar
Shepherd, T. G. 1993 A unified theory of available potential energy. Atmos.-Ocean 31, 1–26.Google Scholar
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, 3132–3157.Google Scholar
Srinivasan, K. & Young, W. R. 2012 Zonostrophic instability. J. Atmos. Sci. 69, 1633–1656.Google Scholar
Srinivasan, K. & Young, W. R. 2014 Reynolds stress and eddy diffusivity of 𝛽-plane shear flows. J. Atmos. Sci. 71, 2169–2185.CrossRefGoogle Scholar
Thompson, R. O. R. Y. 1971 Why there is an intense Eastward current in the North Atlantic but not in the South Atlantic. J. Phys. Oceanogr. 1, 235–237.Google Scholar
Thuburn, J. & Lagneau, V. 1999 Eulerian mean, contour integral, and finite-amplitude wave activity diagnostics applied to a single-layer model of the winter stratosphere. J. Atmos. Sci. 56, 689–710.Google Scholar
Vallis, G. K. 2006 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.Google Scholar
Vasavada, A. R. & Showman, A. P. 2005 Jovian atmospheric dynamics: an update after Galileo and Cassini. Rep. Prog. Phys. 68, 1935–1996.Google Scholar
Verkley, W. T. M. 2009 A balanced approximation of the one-layer shallow-water equations on a sphere. J. Atmos. Sci. 66, 1735–1748.Google Scholar
Walterscheid, R. L., Brinkman, D. G. & Schubert, G. 2000 Wave disturbances from the comet SL–9 impacts into Jupiter’s atmosphere. Icarus 145, 140–146.Google Scholar
Warneford, E. S.2014 The thermal shallow water equations, their quasi-geostrophic limit, and equatorial super-rotation in Jovian atmospheres. DPhil thesis, University of Oxford, http://ora.ox.ac.uk/objects/uuid:6604fcac-afe6-4abe-8a6f-6a09de4f933f.Google Scholar
Warneford, E. S. & Dellar, P. J. 2013 The quasi-geostrophic theory of the thermal shallow water equations. J. Fluid Mech. 723, 374–403.Google Scholar
Warneford, E. S. & Dellar, P. J. 2014 Thermal shallow water models of geostrophic turbulence in Jovian atmospheres. Phys. Fluids 26, 016603.Google Scholar
Williams, G. P. 1978 Planetary circulations: 1. Barotropic representation of Jovian and terrestrial turbulence. J. Atmos. Sci. 35, 1399–1426.2.0.CO;2>CrossRefGoogle Scholar
Williams, G. P. & Yamagata, T. 1984 Geostrophic regimes, intermediate solitary vortices and Jovian eddies. J. Atmos. Sci. 41, 453–478.Google Scholar
Yamagata, T. & Philander, S. 1985 The role of damped equatorial waves in the oceanic response to winds. J. Oceanogr. Soc. Japan 41, 345–357.Google Scholar