Journal of Mathematical Fluid Mechanics

, Volume 20, Issue 3, pp 1173–1193 | Cite as

Dynamic Transitions and Baroclinic Instability for 3D Continuously Stratified Boussinesq Flows

  • Taylan ŞengülEmail author
  • Shouhong Wang


The main objective of this article is to study the nonlinear stability and dynamic transitions of the basic (zonal) shear flows for the three-dimensional continuously stratified rotating Boussinesq model. The model equations are fundamental equations in geophysical fluid dynamics, and dynamics associated with their basic zonal shear flows play a crucial role in understanding many important geophysical fluid dynamical processes, such as the meridional overturning oceanic circulation and the geophysical baroclinic instability. In this paper, first we derive a threshold for the energy stability of the basic shear flow, and obtain a criterion for local nonlinear stability in terms of the critical horizontal wavenumbers and the system parameters such as the Froude number, the Rossby number, the Prandtl number and the strength of the shear flow. Next, we demonstrate that the system always undergoes a dynamic transition from the basic shear flow to either a spatiotemporal oscillatory pattern or circle of steady states, as the shear strength of the basic flow crosses a critical threshold. Also, we show that the dynamic transition can be either continuous or catastrophic, and is dictated by the sign of a transition number, fully characterizing the nonlinear interactions of different modes. Both the critical shear strength and the transition number are functions of the system parameters. A systematic numerical method is carried out to explore transition in different flow parameter regimes. In particular, our numerical investigations show the existence of a hypersurface which separates the parameter space into regions where the basic shear flow is stable and unstable. Numerical investigations also yield that the selection of horizontal wave indices is determined only by the aspect ratio of the box. We find that the system admits only critical eigenmodes with roll patterns aligned with the x-axis. Furthermore, numerically we encountered continuous transitions to multiple steady states, as well as continuous and catastrophic transitions to spatiotemporal oscillations.


Baroclinic instability Shear flow instability Continuously stratified Boussinesq flows Dynamic transition Center manifold reduction Continuous transition Catastrophic transition Random transition 


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Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Bjerknes, V.: Das Problem der Wettervorhersage: betrachtet vom Standpunkte der Mechanik und der Physik (1904)Google Scholar
  2. 2.
    Cai, M.: An analytic study of the baroclinic adjustment in a quasigeostrophic two-layer channel model. J. Atmos. Sci. 49, 1594–1605 (1992)ADSCrossRefGoogle Scholar
  3. 3.
    Cai, M., Mak, M.: On the multiplicity of equilibria of baroclinic waves. Tellus A 39, 116–137 (1987)ADSCrossRefGoogle Scholar
  4. 4.
    Charney, J.: On the scale of atmospheric motion. Geofys. Publ. 17(2), 1–17 (1948)MathSciNetGoogle Scholar
  5. 5.
    Dijkstra, H., Sengul, T., Shen, J., Wang, S.: Dynamic transitions of quasi-geostrophic channel flow. SIAM J. Appl. Math. 75, 2361–2378 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dijkstra, H.A.: Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Niño. Kluwer Academic Publishers, Dordrecht (2000)CrossRefGoogle Scholar
  7. 7.
    Dijkstra, H.A., Ghil, M.: Low-frequency variability of the large-scale ocean circulations: a dynamical systems approach. Rev. Geophys. 43, 1–38 (2005)CrossRefGoogle Scholar
  8. 8.
    Eady, E.T.: Long waves and cyclone waves. Tellus 1, 33–52 (1949)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ghil, M., Childress, S.: Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics. Springer-Verlag, New York (1987)CrossRefzbMATHGoogle Scholar
  10. 10.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations, vol. 840. Springer, Berlin (2006)Google Scholar
  11. 11.
    Lions, J.-L., Temam, R., Wang, S.: New formulations of the primitive equations of atmosphere and applications. Nonlinearity 5, 237–288 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)ADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Lorenz, E.N.: The mechanics of vacillation. J. Atmos. Sci. 20, 448–464 (1963)ADSCrossRefGoogle Scholar
  14. 14.
    Ma, T., Wang, S.: Bifurcation and stability of superconductivity. J. Math. Phys. 46, 095112 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ma, T., Wang, S.: Bifurcation Theory and Applications. World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, vol. 53. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2005)zbMATHGoogle Scholar
  16. 16.
    Ma, T., Wang, S.: Dynamic transition theory for thermohaline circulation. Phys. D 239, 167–189 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ma, T., Wang, S.: Phase Transition Dynamics. Springer, Berlin (2013)zbMATHGoogle Scholar
  18. 18.
    Mak, M.: Equilibration in nonlinear baroclinic instability. J. Atmos. Sci. 42, 2764–2782 (1985)ADSCrossRefGoogle Scholar
  19. 19.
    Pedlosky, J.: Finite-amplitude baroclinic waves. J. Atmos. Sci. 27, 15–30 (1970)ADSCrossRefGoogle Scholar
  20. 20.
    Phillips, N.A.: The general circulation of the atmosphere: a numerical experiment. Q. J. R. Meteorol Soc. 82, 123–164 (1956)ADSCrossRefGoogle Scholar
  21. 21.
    Rossby, C.: On the solution of problems of atmospheric motion by means of model experiment. Mon. Weather Rev. 54, 237–240 (1926)ADSCrossRefGoogle Scholar
  22. 22.
    Stommel, H.: Thermohaline convection with two stable regimes of flow. Tellus 13, 224–230 (1961)ADSCrossRefGoogle Scholar
  23. 23.
    Veronis, G.: An analysis of wind-driven ocean circulation with a limited Fourier components. J. Atmos. Sci. 20, 577–593 (1963)ADSCrossRefGoogle Scholar
  24. 24.
    Veronis, G.: Wind-driven ocean circulation, part ii: numerical solution of the nonlinear problem. Deep-Sea Res. 13, 31–55 (1966)Google Scholar
  25. 25.
    von Neumann, J.: Some remarks on the problem of forecasting climatic fluctuations. In: Pfeffer, R.L. (ed.) Dynamics of Climate, pp. 9–12. Pergamon Press, Oxford (1960)CrossRefGoogle Scholar
  26. 26.
    Özer, S., Şengül, T.: Stability and transitions of the second grade Poiseuille flow. Phys. D Nonlinear Phenom. 331, 71–80 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsMarmara UniversityIstanbulTurkey
  2. 2.Department of MathematicsIndiana UniversityBloomingtonUSA

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