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A scenario for finite-time singularity in the quasigeostrophic model

Published online by Cambridge University Press:  14 October 2011

Richard K. Scott
Richard K. Scott*
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, Scotland, UK
*
†Email address for correspondence: rks@mcs.st-and.ac.uk
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Abstract

A possible route to finite-time singularity in the quasigeostrophic system, via a cascade of filament instabilities of geometrically decreasing spatial and temporal scales, is investigated numerically using a high-resolution hybrid contour dynamical algorithm. A number of initial temperature distributions are considered, of varying degrees of continuity. In all cases, primary, secondary, and tertiary instabilities are apparent before the algorithm loses accuracy due to limitations of finite resolution. Filament instability is also shown to be potentially important in the closing saddle scenario investigated in many previous studies. The results do not provide a rigorous demonstration of finite-time singularity, but suggest avenues for further investigation.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Instability: Instability

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Papers
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Journal of Fluid Mechanics , Volume 687 , 25 November 2011 , pp. 492 - 502
Copyright
Copyright © Cambridge University Press 2011

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References

1. Blumen, W. 1978 Uniform potential vorticity flow. Part I. Theory of wave interactions and two-dimensional turbulence. J. Atmos. Sci. 35, 774–783.2.0.CO;2>CrossRefGoogle Scholar
2. Constantin, P. 1995 Nonlinear inviscid incompressible dynamics. Physica D 86, 212–219.CrossRefGoogle Scholar
3. Constantin, P., Majda, A. J. & Tabak, E. 1994 Formation of strong fronts in the 2D quasigeostrophic thermal active scalar. Nonlinearity 7, 1495–1533.CrossRefGoogle Scholar
4. Constantin, P., Nie, Q. & Schorghofer, N. 1998 Nonsingular surface quasi-geostrophic flow. Phys. Lett. A 241, 168–172.CrossRefGoogle Scholar
5. Cordoba, D. 1998 Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann. Math. 148, 1135–1152.CrossRefGoogle Scholar
6. Cordoba, D., Fontelos, M. A., Mancho, A. M. & Rodrigo, J. L. 2005 Evidence of singularities for a family of contour dynamics equations. Proc. Natl Acad. Sci. 102, 5949–5952.CrossRefGoogle ScholarPubMed
7. Dritschel, D. G. 1989 Contour dynamics and contour surgery: numerical algorithms for extended, high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows. Comput. Phys. Rep. 10, 78–146.CrossRefGoogle Scholar
8. Dritschel, D. G. 2011 An exact steadily rotating surface quasi-geostrophic elliptical vortex. Geophys. Astrophys. Fluid Dyn. 105, 368–376.CrossRefGoogle Scholar
9. Dritschel, D. G. & Ambaum, M. H. P. 1997 A contour-advective semi-Lagrangian numerical algorithm for simulating fine-scale conservative dynamical fields. Q. J. R. Meteorol. Soc. 123, 1097–1130.Google Scholar
10. Dritschel, D. G., Haynes, P. H., Juckes, M. N. & Shepherd, T. G. 1991 The stability of a two-dimensional vorticity filament under uniform strain. J. Fluid Mech. 230, 647–665.CrossRefGoogle Scholar
11. Dritschel, D. G. & Scott, R. K. 2009 On the simulation of nearly inviscid two-dimensional turbulence. J. Comput. Phys. 228, 2707–2711.CrossRefGoogle Scholar
12. Held, I. M., Pierrehumbert, R. T., Garner, S. T. & Swanson, K. L. 1995 Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20.CrossRefGoogle Scholar
13. Hoskins, B. J. & Bretherton, F. P. 1972 Atmospheric frontogenesis models: mathematical formulation and solution. J. Atmos. Sci. 29, 11–37.2.0.CO;2>CrossRefGoogle Scholar
14. Hoyer, J.-M. & Sadourny, R. 1982 Closure modelling of fully developed baroclinic instability. J. Atmos. Sci. 39, 707–721.2.0.CO;2>CrossRefGoogle Scholar
15. Juckes, M. N. 1994 Quasigeostrophic dynamics of the tropopause. J. Atmos. Sci. 51, 2756–2779.2.0.CO;2>CrossRefGoogle Scholar
16. Juckes, M. N. 1995 Instability of surface and upper-tropospheric shear lines. J. Atmos. Sci. 52, 3247–3262.2.0.CO;2>CrossRefGoogle Scholar
17. Ohkitani, K. & Yamada, M. 1997 Inviscid and inviscid-limit behaviour of surface quasigeostrophic flow. Phys. Fluids 9, 876–882.CrossRefGoogle Scholar
18. Pierrehumbert, R. T., Held, I. M. & Swanson, K. L. 1994 Spectra of local and nonlocal two-dimensional turbulence. Chaos, Solitons Fractals 4, 1111–1116.CrossRefGoogle Scholar
19. Tran, C. V., Dritschel, D. G. & Scott, R. K. 2010 Effective degrees of nonlinearity in a family of generalized models of two-dimensional turbulence. Phys. Rev. E 81, 01630.CrossRefGoogle Scholar