Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T18:01:49.878Z Has data issue: false hasContentIssue false
  • English
  • Français

On the late-time behaviour of a bounded, inviscid two-dimensional flow

Published online by Cambridge University Press:  13 October 2015

David G. Dritschel ,
Wanming Qi  and
J. B. Marston
David G. Dritschel*
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK
Wanming Qi
Affiliation:
School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030, USA Department of Physics, Brown University, Providence, RI 02912-1843, USA
J. B. Marston
Affiliation:
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030, USA Department of Physics, Brown University, Providence, RI 02912-1843, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Using complementary numerical approaches at high resolution, we study the late-time behaviour of an inviscid incompressible two-dimensional flow on the surface of a sphere. Starting from a random initial vorticity field comprised of a small set of intermediate-wavenumber spherical harmonics, we find that, contrary to the predictions of equilibrium statistical mechanics, the flow does not evolve into a large-scale steady state. Instead, significant unsteadiness persists, characterised by a population of persistent small-scale vortices interacting with a large-scale oscillating quadrupolar vorticity field. Moreover, the vorticity develops a stepped, staircase distribution, consisting of nearly homogeneous regions separated by sharp gradients. The persistence of unsteadiness is explained by a simple point-vortex model characterising the interactions between the four main vortices which emerge.

JFM classification

Mixing: Turbulent mixing Vortex Flows: Vortex dynamics Vortex Flows: Vortex Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 783 , 25 November 2015 , pp. 1 - 22
Check for updates
Copyright
© 2015 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

Abramov, R. V. & Majda, A. J. 2003 Statistically relevant conserved quantities for truncated quasigeostrophic flow. Proc. Natl Acad. Sci. USA 100, 3841–3846.Google Scholar
Bouchet, F. & Corvellec, M. 2010 Invariant measures of the 2D Euler and Vlasov equations. J. Stat. Phys. 2010, P08021.Google Scholar
Bouchet, F. & Venaille, A. 2012 Statistical mechanics of two-dimensional and geophysical flows. Phys. Rep. 515, 227–295.Google Scholar
Brands, H., Stulemeyer, J., Pasmanter, R. A. & Schep, T. J. 1997 A mean field prediction of the asymptotic state of decaying 2D turbulence. Phys. Fluids 9, 2815–2817.Google Scholar
Bricmont, J., Kupiainen, A. & Lefevere, R. 2001 Ergodicity of the 2D Navier–Stokes equations with random forcing. Commun. Math. Phys. 224, 65–81.Google Scholar
Charney, J. 1949 On a physical basis for numerical prediction of large-scale motions in the atmosphere. J. Meteorol. 6, 371–385.Google Scholar
Chavanis, P. H. 2002 Dynamics and Thermodynamics of Systems with Long-Range Interactions, Lecture Notes in Physics, vol. 602, pp. 208–289. Springer.Google Scholar
Chavanis, P. H. 2009 Dynamical and thermodynamical stability of two-dimensional flows: variational principles and relaxation equations. Eur. Phys. J. B 70, 73–105.Google Scholar
Dritschel, D. G. 1988 Contour surgery: a topological reconnection scheme for extended integrations using contour dynamics. J. Comput. Phys. 77, 240–266.Google Scholar
Dritschel, D. G. 1990 The stability of elliptical vortices in an external straining flow. J. Fluid Mech. 210, 223–261.Google Scholar
Dritschel, D. G. & Ambaum, M. H. P. 1997 A contour-advective semi-lagrangian algorithm for the simulation of fine-scale conservative fields. Q. J. R. Meteorol. Soc. 123, 1097–1130.Google Scholar
Dritschel, D. G. & Fontane, J. 2010 The combined Lagrangian advection method. J. Comput. Phys. 229, 5408–5417.CrossRefGoogle Scholar
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.Google Scholar
Dritschel, D. G. & McIntyre, M. E. 2008 Multiple jets as PV staircases: the Phillips effect and the resilience of eddy-transport barriers. J. Atmos. Sci 65, 855–874.CrossRefGoogle Scholar
Dritschel, D. G. & Scott, R. K. 2009 On the simulation of nearly inviscid two-dimensional turbulence. J. Comput. Phys. 228, 2707–2711.Google Scholar
Dritschel, D. G., Scott, R. K., Macaskill, C., Gottwald, G. A. & Tran, C. V. 2009 Late time evolution of unforced inviscid two-dimensional turbulence. J. Fluid Mech. 640, 215–233.Google Scholar
Dritschel, D. G. & Tobias, S. M. 2012 Two-dimensional magnetohydrodynamic turbulence in the small Prandtl number limit. J. Fluid Mech. 703, 85–98.Google Scholar
Dritschel, D. G., de la Torre Juárez, M. & Ambaum, M. H. P. 1999 On the three-dimensional vortical nature of atmospheric and oceanic flows. Phys. Fluids 11, 1512–1520.CrossRefGoogle Scholar
Dritschel, D. G., Tran, C. V. & Scott, R. K. 2007 Revisiting Batchelor’s theory of two-dimensional turbulence. J. Fluid Mech. 591, 379–391.Google Scholar
Dubinkina, S. & Frank, J. 2010 Statistical relevance of vorticity conservation in the Hamiltonian particle-mesh method. J. Comput. Phys. 229, 2634–2648.Google Scholar
Eyink, G. L. & Sreenivasan, K. R. 2006 Onsager and the theory of hydrodynamic turbulence. Rev. Mod. Phys. 78, 87–135.Google Scholar
Fjørtoft, R. 1953 On the changes in the spectral distribution of kinetic energy for two dimensional, nondivergent flow. Tellus 5, 225–230.CrossRefGoogle Scholar
Fontane, J. & Dritschel, D. G. 2009 The HyperCASL Algorithm: a new approach to the numerical simulation of geophysical flows. J. Comput. Phys. 228, 6411–6425.CrossRefGoogle Scholar
Heikes, R. & Randall, D. A. 1995a Numerical integration of the shallow-water equations on a twisted icosahedral grid. Part I. Basic design and results of tests. Mon. Weath. Rev. 123, 1862–1880.2.0.CO;2>CrossRefGoogle Scholar
Heikes, R. & Randall, D. A. 1995b Numerical integration of the shallow-water equations on a twisted icosahedral grid. Part II. A detailed description of the grid and an analysis of numerical accuracy. Mon. Weath. Rev. 123, 1881–1887.Google Scholar
Herbert, C. 2013 Additional invariants and statistical equilibria for the 2D Euler equations on a spherical domain. J. Stat. Phys. 152, 1084–1114.Google Scholar
Jiménez, J. 1994 Hyperviscous vortices. J. Fluid Mech. 279, 169–176.Google Scholar
Majda, A. J. & Wang, X. 2006 Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows. Cambridge University Press.CrossRefGoogle Scholar
Marcus, P. S. 1993 Jupiter’s great red spot and other vortices. Annu. Rev. Astron. Astrophys. 31, 523–573.Google Scholar
Mariotti, A., Legras, B. & Dritschel, D. G. 1994 Vortex stripping and the erosion of coherent structures in two-dimensional flows. J. Fluid Mech. 6, 3954–3962.Google Scholar
Matthaeus, W. H., Stribling, W. T., Martinez, D., Oughton, S. & Montgomery, D. 1991a Decaying, two-dimensional, Navier–Stokes turbulence at very long times. Physica D 51, 531–538.Google Scholar
Matthaeus, W. H., Stribling, W. T., Martinez, D., Oughton, S. & Montgomery, D. 1991b Selective decay and coherent vortices in two-dimensional incompressible turbulence. Phys. Rev. Lett. 66, 2731–2734.Google Scholar
Miller, J. 1990 Statistical mechanics of Euler equations in two dimensions. Phys. Rev. Lett. 65, 2137–2140.Google Scholar
Miller, J., Weichman, P. B. & Cross, M. C. 1992 Statistical mechanics, Euler’s equation, and Jupiter’s Red Spot. Phys. Rev. A 45, 2328–2359.Google Scholar
Mohebalhojeh, A. R. & Dritschel, D. G. 2007 Assessing the numerical accuracy of complex spherical shallow water flows. Mon. Weath. Rev. 135, 3876–3894.Google Scholar
Montgomery, D., Matthaeus, W. H., Stribling, W. T., Martinez, D. & Oughton, S. 1992 Relaxation in two dimensions and the ‘Sinh–Poisson’ equation. Phys. Fluids A 4, 3–6.Google Scholar
Montgomery, D., Shan, X. & Matthaeus, W. H. 1993 Navier–Stokes relaxation to Sinh–Poisson states at finite Reynolds numbers. Phys. Fluids A 5, 2207–2216.CrossRefGoogle Scholar
Morita, H.2011 Collective oscillation in two-dimensional fluid, preprint, arXiv:1103.1140.Google Scholar
Newton, P. K. 2001 The $N$ -vortex problem. In Analytical Techniques, Applied Mathematical Sciences, vol. 145, p. 415. Springer.Google Scholar
Polvani, L. M. & Dritschel, D. G. 1993 Wave and vortex dynamics on the surface of the sphere: equilibria and their stability. J. Fluid Mech. 255, 35–64.Google Scholar
Qi, W. & Marston, J. B. 2014 Hyperviscosity and statistical equilibria of Euler turbulence on the torus and the sphere. J. Stat. Mech. 2014 (7), P07020.Google Scholar
Robert, R. 1991 A maximum-entropy principle for two-dimensional perfect fluid dynamics. J. Stat. Phys. 65, 531–553.Google Scholar
Robert, R. & Sommeria, J. 1991 Statistical equilibrium states for two-dimensional flows. J. Fluid Mech. 229, 291–310.Google Scholar
Scott, R. K. & Dritschel, D. G. 2012 The structure of zonal jets in geostrophic turbulence. J. Fluid Mech. 711, 576–598.Google Scholar
Segre, E. & Kida, S. 1998 Late states of incompressible 2d decaying vorticity fields. Fluid Dyn. Res. 23, 89–112.CrossRefGoogle Scholar
Whitaker, N. & Turkington, B. 1994 Maximum entropy states for rotating vortex patches. Phys. Fluids 6 (12), 3963.Google Scholar
Williams, P. D. 2009 A proposed modification to the Robert–Asselin time filter. Mon. Weath. Rev. 137, 2538–2546.Google Scholar
Yao, H. B., Dritschel, D. G. & Zabusky, N. J. 1995 High-gradient phenomena in two-dimensional vortex interactions. Phys. Fluids 7, 539–548.Google Scholar
Yin, Z., Montgomery, D. C. & Clercx, H. J. H. 2003 Alternative statistical-mechanical descriptions of decaying two-dimensional turbulence in terms of ‘patches’ and ‘points’. Phys. Fluids 15, 1937–1953.Google Scholar
Zermelo, E. 1902 Hydrodynamische Untersuchungen über die Wirbelbewegungen in einer Kugelfläche. Z. Math. Phys. 47, 201–237.Google Scholar