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Physical interpretation of spiralling-columnar convection in a rapidly rotating annulus with radial propagation properties of Rossby waves

Published online by Cambridge University Press:  16 October 2008

SHIN-ICHI TAKEHIRO
SHIN-ICHI TAKEHIRO*
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japantakepiro@gfd-dennou.org
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Abstract

To aid the physical understanding of spiralling-columnar convection emerging in rapidly rotating spheres and spherical shells, two-dimensional thermal convection in a rapidly rotating annulus is investigated through the radial propagation properties of topographic Rossby waves. Two kinds of the boundaries containing the fluid in the axial direction are considered: a convex type modelling a spherical geometry and a concave type for comparison. The linear stability of a basic state with no motion and uniformly unstable stratification is examined and spirally elongated structures of critical convection are obtained for small Prandtl numbers. An analysis of the energy budget shows that a part of the kinetic energy generated in the region with slightly inclined boundaries is dynamically transferred and dissipates through viscosity in the region with strongly inclined boundaries. This indicates that the Rossby waves propagate from the region with slightly inclined boundaries to the region with strongly inclined boundaries. It is presented that the appearance of a spiral structure corresponds to an increase of the local radial wavenumber of the Rossby waves propagating in the radial direction. The flow patterns obtained using the dispersion relation of the Rossby waves coincide with those of the tailing part of the spiral structure obtained numerically. As the Prandtl number increases, the Rossby waves barely propagate because of strong viscous dissipation, and the flow pattern is localized in the region with slightly inclined boundaries. For convex boundaries with unstable stratification concentrating near the outer boundary and concave boundaries with unstable stratification confined near the inner boundary, the flow patterns tilt in the direction inverse to the case of uniform unstable stratification. The tilting direction of the flow pattern is not determined by the curvature of the boundaries considered but instead by the radial propagation direction of the Rossby waves excited by thermal convection.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 614 , 10 November 2008 , pp. 67 - 86
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Aurnou, J. M. & Olson, P. L. 2001 Strong zonal winds from thermal convection in a rotating spherical shell. Geophys. Res. Lett. 28, 2557–2559.CrossRefGoogle Scholar
Baldwin, M. P., Rhines, P. B., Huang, H.-P. & McIntyre, M. E. 2007 The jet-stream conundrum. Science 315, 467–468.CrossRefGoogle ScholarPubMed
Busse, F. H. 1970 Thermal instabilities in rapidly rotating systems. J. Fluid Mech. 44, 441–460.CrossRefGoogle Scholar
Busse, F. H. 1983 a Convection-driven zonal flows in the major planets. PAGEOPH 121, 375–390.CrossRefGoogle Scholar
Busse, F. H. 1983 b A model of mean zonal flows in the major planets. Geophys. Astrophys. Fluid Dyn. 23, 153–174.CrossRefGoogle Scholar
Busse, F. H. 1986 Asymptotic theory of convection in a rotating, cylindrical annulus. J. Fluid Mech. 173, 545–556.CrossRefGoogle Scholar
Busse, F. H. 2002 Convective flows in rapidly rotating spheres and their dynamo action. Phys. Fluids 14, 1301–1314.CrossRefGoogle Scholar
Busse, F. H. & Hood, L. L. 1982 Differential rotation driven by convection in a rapidly rotating annulus. Geophys. Astrophys. Fluid Dyn. 21, 59–74CrossRefGoogle Scholar
Busse, F. H. & Or, A. C. 1986 Convection in a rotating cylindrical annulus: thermal Rossby waves. J. Fluid Mech. 166, 173–187.CrossRefGoogle Scholar
Christensen, U. R. 2001 Zonal flow driven by deep convection in the major planets. Geophys. Res. Lett. 28, 2553–2556.CrossRefGoogle Scholar
Dormy, E., Soward, A. M., Jones, C. A., Jault, D. & Cardin, P. 2003 The onset of thermal convection in rotating spherical shells. J. Fluid Mech. 501, 43–70.CrossRefGoogle Scholar
Jones, C. A., Soward, A. M. & Mussa, A. I. 2000 The onset of thermal convection in a rapidly rotating sphere. J. Fluid Mech. 405, 157–179.CrossRefGoogle Scholar
Lin, R.-Q. 1990 Double column instabilities in the barotropic annulus. Geophys. Astrophys. Fluid Dyn. 54, 161–188.CrossRefGoogle Scholar
Or, A. C. & Busse, F. H. 1987 Convection in a rotating cylindrical annulus. Part 2. Transitions to asymmetric and vacillating flow. J. Fluid Mech. 174, 313–326.CrossRefGoogle Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics. Springer.CrossRefGoogle Scholar
Plaut, E. & Busse, F. H. 2005 Multicellular convection in rotating annuli. J. Fluid Mech. 528, 119–133.CrossRefGoogle Scholar
Schnaubelt, M. & Busse, F. H. 1992 Convection in a rotating cylindrical annulus. Part 3. Vacillating and spatially modulated flows. J. Fluid Mech. 245, 155–173.CrossRefGoogle Scholar
Takehiro, S. & Hayashi, Y.-Y. 1999 Mean zonal flows excited by critical thermal convection in rotating spherical shells. Geophys. Astrophys. Fluid Dyn. 90, 43–77.CrossRefGoogle Scholar
Takehiro, S., Ishiwatari, M., Nakajima, K. & Hayashi, Y.-Y. 2002 Linear stability of thermal convection in rotating systems with fixed heat flux boundaries. Geophys. Astrophys. Fluid Dyn. 96, 439–459.CrossRefGoogle Scholar
Takehiro, S., Odaka, M., Ishioka, K., Ishiwatari, M., Hayashi, Y.-Y. & SPMODEL Development Group. 2006 A series of hierarchical spectral models for geophysical fluid dynamics. Nagare Multimedia 2006, !http://www.nagare.or.jp/mm/2006/spmodel/.Google Scholar
Vasavada, A. R. & Showman, A. P. 2005 Jovian atmospheric dynamics: an update after Galileo and Cassini. Rep. Prog. Phys. 68, 1935–1996.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley & Sons.Google Scholar
Yano, J.-I. 1992 Asymptotic theory of thermal convection in rapidly rotating systems. J. Fluid Mech. 243, 103–131.CrossRefGoogle Scholar
Yano, J.-I., Talagrand, O. & Drossart, P. 2005 Deep two-dimensional turbulence: an idealized model for atmospheric jets. Geophys. Astrophys. Fluid Dyn. 99, 137–150.CrossRefGoogle Scholar
Zhang, K. 1992 Spiralling columnar convection in rapidly rotating spherical fluid shells. J. Fluid Mech. 236, 535–556.CrossRefGoogle Scholar
Zhang, K.-K. & Busse, F. H. 1987 On the onset of convection in rotating spherical shells. Geophys. Astrophys. Fluid Dyn. 39, 119–147.CrossRefGoogle Scholar