Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-29T11:34:46.987Z Has data issue: false hasContentIssue false
  • English
  • Français

Bifurcation structure of two-dimensional viscous zonal flows on a rotating sphere

Published online by Cambridge University Press:  05 June 2015

Eiichi Sasaki ,
Shin-ichi Takehiro  and
Michio Yamada
Eiichi Sasaki
Affiliation:
Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
Shin-ichi Takehiro
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
Michio Yamada
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the bifurcation structure of zonal flows on a rotating sphere. The setting of our problem is similar to the Kolmogorov problem on a flat torus, where the vorticity forcing is given by a single eigenfunction of the Laplacian. First we prove the global stability of two-jet zonal flow for arbitrary Reynolds number and the rotation rate of the sphere. Then we study the bifurcation structure of steady solutions arising from three-jet zonal flow. In the non-rotating case, we find that two steady travelling-wave solutions bifurcate from a three-jet zonal flow via Hopf bifurcation. As the Reynolds number increases, steady-travelling solutions arise via pitchfork bifurcation from the steady-travelling solutions. On the other hand, in the rotating case, we find saddle-node bifurcations and closed-loop branches. We carry out time integration to study the properties of unsteady solutions at high Reynolds numbers. In the non-rotating case, the unsteady solution is chaotic and it wanders around the steady-travelling solutions bifurcating from three-jet zonal flow. We show that a linear combination of the steady and steady-travelling solutions gives a good approximation of the zonal-mean zonal flow of the unsteady solution, suggesting that the chaotic solution at high Reynolds numbers exists mostly within a relatively low-dimensional space spanned by the steady and steady-travelling solutions, which become unstable at low Reynolds numbers.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Nonlinear Dynamical Systems: Chaos Geophysical and Geological Flows: Rotating flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 774 , 10 July 2015 , pp. 224 - 244
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

Baines, P. G. 1976 The stability of planetary waves on a sphere. J. Fluid Mech. 73 (2), 193–213.CrossRefGoogle Scholar
Inubushi, M., Kobayashi, M. U., Takehiro, S. & Yamada, M. 2012 Covariant Lyapunov analysis of chaotic Kolmogorov flows. Phys. Rev. E 85, 016331.CrossRefGoogle ScholarPubMed
Ishioka, K.2011 ISPACK-0.96, http://www.gfd-dennou.org/arch/ispack/, GFD Dennou Club.Google Scholar
Iudovisch, V. I. 1965 Example of the generation of a secondary stationary or periodic flow when there is loss of stability of the laminar flow of a viscous incompressible fluid. Z. Angew. Math. Mech. 29, 527–544.CrossRefGoogle Scholar
Kartashova, E. & L’vov, V. S. 2008 Cluster dynamics of planetary waves. Europhys. Lett. 83, 50012.CrossRefGoogle Scholar
Kim, L. & Moehlis, J. 2008 Characterizing the edge of chaos for a shear flow model. Phys. Rev. E 78, 036215.CrossRefGoogle ScholarPubMed
Kim, S. C. & Okamoto, H. 2010 Stationary vortices of large scale appearing in 2D Navier–Stokes equations at large Reynolds numbers. Japan J. Ind. Appl. Maths 27, 47–71.CrossRefGoogle Scholar
Kuo, H. 1949 Dynamic instability of two-dimensional nondivergent flow in a barotropic atmosphere. J. Meteorol. 6, 105–122.2.0.CO;2>CrossRefGoogle Scholar
Lucas, D. & Kerswell, R. R. 2015 Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow. Phys. Fluids 27, 045106.CrossRefGoogle Scholar
Meshalkin, L. D. & Sinai, Y. G. 1962 Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous liquid. Z. Angew. Math. Mech. 25, 1700–1705.CrossRefGoogle Scholar
Obukhov, A. M. 1983 Kolmogorov flow and laboratory simulation of it. Russian Math. Surv. 38 (4), 113–126.CrossRefGoogle Scholar
Obuse, K., Takehiro, S. & Yamada, M. 2010 Long-time asymptotic states of forced two-dimensional barotropic incompressible flows on a rotating sphere. Phys. Fluids 22, 056601.CrossRefGoogle Scholar
Okamoto, H. & Shōji, M. 1993 Bifurcation diagrams in Kolmogorov’s problem of viscous incompressible fluid on 2-D flat tori. Japan J. Ind. Appl. Maths 10, 191–218.CrossRefGoogle Scholar
Platt, N., Sirovich, L. & Fitzmaurice, N. 1991 An investigation of chaotic Kolmogorov flows. Phys. Fluids 3, 681–696.CrossRefGoogle Scholar
Sasaki, E., Takehiro, S. & Yamada, M. 2012 A note on the stability of inviscid zonal jet flows on a rotating sphere. J. Fluid Mech. 710, 154–165.CrossRefGoogle Scholar
Sasaki, E., Takehiro, S. & Yamada, M. 2013 Linear stability of viscous zonal jet flows on a rotating sphere. J. Phys. Soc. Japan 82, 094402.CrossRefGoogle Scholar
Silberman, I. 1953 Planetary waves in the atmosphere. J. Meteorol. 11, 27–34.2.0.CO;2>CrossRefGoogle Scholar
Smith, T. S., Moehlis, J. & Holmes, P. 2005 Low-dimensional models for turbulent plane Couette flow in a minimal flow unit. J. Fluid Mech. 538, 71–110.CrossRefGoogle Scholar
Takehiro, S., Sasaki, Y., Morikawa, Y., Ishioka, K., Odaka, M., Takahashi, Y. O., Nishizawa, S., Nakajima, K., Ishiwatari, M. & Hayashi, Y.-Y.2011 SPMODEL Development Group, 2011 Hierarchical Spectral Models for GFD (SPMODEL), http://www.gfd-dennou.org/library/spmodel/, GFD Dennou Club.Google Scholar