×

Hopf bifurcations and transitions of two-dimensional quasi-geostrophic flows. (English) Zbl 1470.37109

Summary: This study examines the Hopf (double Hopf) bifurcations and transitions of two dimensional quasi-geostrophic (QG) flows that model various large-scale oceanic and atmospheric circulations. Using the Kolmogorov function to represent an external forcing in the tropical region, it is shown that the equilibrium of the QG model loses its stability if the combination of the Rossby number, the Ekman number, and the eddy viscosity satisfies a specific condition. Further use of the center manifold technique reveals two different types of the dynamical transition from either a pair of simple complex eigenvalues or a double pair of complex conjugate eigenvalues. These dynamical transitions are confirmed in the numerical analyses of the QG dynamics at the equilibrium, which capture Hopf (double Hopf) bifurcations due to the existence of a nonzero imaginary part of the first eigenvalue. The transition from a pair of simple complex eigenvalues is of particular interest, because it indicates the existence of a stable periodic pattern that is similar to atmospheric easterly waves and related tropical cyclone formation in the tropical atmosphere.

MSC:

37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
37L15 Stability problems for infinite-dimensional dissipative dynamical systems
76E20 Stability and instability of geophysical and astrophysical flows
76U60 Geophysical flows
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] P. Berloff; S. Meacham, On the stability of the wind-driven circulation, J. Mar. Res., 56, 937-993 (1998)
[2] P. Cessi; G. R. Ierley, Symmetry-breaking multiple equilibria in quasi-geostrophic, wind-driven flows, J. Phys. Oceanogr., 25, 1196-1205 (1995)
[3] J. Charney; D. Straus, Form-drag instability, multiple equilibria and propagating planetary waves in baroclinic, orographically forced, planetary wave systems, J. Atmos. Sci., 37, 1157-1176 (1980)
[4] J. G. Charney; J. DeVore, Multiple flow equilibria in the atmosphere and blocking, J. Atmos. Sci., 36, 1205-1216 (1979)
[5] J. G. Charney; J. Shukla; K. C. Mo, Comparison of a barotropic blocking theory with observation, J. Atmos. Sci., 38, 762-779 (1981)
[6] Z. Chen; M. Ghil; E. Simonnet; S. Wang, Hopf bifurcation in quasi-geostrophic channel flow, SIAM J. Appl. Math., 64, 343-368 (2003) · Zbl 1126.76327
[7] Z. Chen; X. Xiong, Equilibrium states of the charney-devore quasi-geostrophic equation in mid-latitude atmosphere, J. Math. Anal. Appl., 444, 1403-1416 (2016) · Zbl 1351.86017
[8] H. Dijkstra; T. Sengul; J. Shen; S. Wang, Dynamic transitions of quasi-geostrophic channel flow, SIAM J. Appl. Math., 75, 2361-2378 (2015) · Zbl 1329.35232
[9] R. N. Ferreira; W. H. Schubert, Barotropic aspects of itcz breakdown, J. Atmos. Sci., 54, 261-285 (19997)
[10] M. Ghil, The wind-driven ocean circulation: applying dynamical systems theory to a climate problem, Discrete Contin. Dyn. Syst., 37, 189-228 (2017) · Zbl 1418.86002
[11] M. Ghil; M. D. Chekround; E. Simonnete, Climate dynamics and fluid mechanics: Natural variability and related uncertainties, Physica D, 237, 2111-2126 (2008) · Zbl 1143.76440
[12] D. Han; M. Hernandez; Q. Wang, Dynamic bifurcation and transition in the Rayleigh-Bénard enard convection with internal heating and varying gravity, Commun. Math. Sci., 17, 175-192 (2019) · Zbl 1415.37097
[13] D. Han; M. Hernandez; Q. Wang, On the instabilities and transitions of the western boundary current, Commun. Computa. Phys., 26, 35-56 (2019)
[14] D. Han; M. Hernandez; Q. Wang, Dynamic Transitions and Bifurcations for a Class of Axisymmetric Geophysical Fluid Flow, SIAM J. Appl. Dyn. Syst., 20, 38-64 (2020) · Zbl 1461.76172
[15] S. Jiang; F. F. Jin; M. Ghil, Multiple equilibria, periodic, and aperiodic solutions in a wind-driven, double-gyre, shallow-water model, J. Phys. Oceanogr., 25, 764-786 (1995)
[16] C. Kieu; T. Sengul; Q. Wang; D. Yan, On the Hopf (double Hopf) bifurcations and transitions of two-layer western boundary currents, Commun. Nonlinear Sci. Numer. Simul., 65, 196-215 (2018) · Zbl 1456.76055
[17] Y. A. Kuznetsov, Elements of applied bifurcation theory, Springer-Verlag, New York, 2004. · Zbl 1082.37002
[18] B. Legras; M. Ghil, Persistent anomalies, blocking and variations in atmospheric predictability, J. Atmos. Sci., 42, 433-471 (1985)
[19] C. Lu; Y. Mao; Q. Wang; D. Yan, Hopf bifurcation and transition of three-dimensional wind-driven ocean circulation problem, J. Differ. Equ., 267, 2560-2593 (2019) · Zbl 1415.35032
[20] C. Lu, Y. Mao, T. Sengul and Q. Wang, On the spectral instability and bifurcation of the 2d-quasi-geostrophic potential vorticity equation with a generalized kolmogorov forcing, Physica D, 43 (2020), 132296.
[21] T. Ma; A. Wang, Rayleigh-Bénard convection: dynamics and structure in the physical space, Commun. Math. Sci., 5, 553-574 (2007) · Zbl 1133.35426
[22] T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, New York, 2014. · Zbl 1285.82004
[23] S. P. Meacham, Low-frequency variability in the wind-driven circulation, J. Phys. Oceanogr., 30, 269-293 (2000)
[24] B. T. Nadiga; B. P. Luce, Global bifurcation of shilnikov type in a double-gyre ocean model, J. Phys. Oceanogr., 31, 2669-2690 (2001)
[25] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. · Zbl 0713.76005
[26] S. Rambaldi; K. Mo, Forced stationary solutions in a barotropic channel: Multiple equilibria and theory of nonlinear resonance, J. Atmos. Sci., 41, 3135-3146 (1984)
[27] J. Shen, T. T. Medjo and S. Wang, On a wind-driven, double-gyre, quasi-geostrophic ocean model: numerical simulations and structural analysis, J. Comput. Phys., 155, 387-409. · Zbl 0954.76070
[28] J. Shen, T. Tang and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer-Verlag, Berlin Heidelberg, 2011. · Zbl 1227.65117
[29] V. A. Sheremet; G. R. Ierley; V. M. Kamenkovich, Eigenanalysis of the two-dimensional wind-driven ocean circulation problem, J. Mar. Res., 55, 57-92 (1997)
[30] E. Simonnet; M. Ghi; K. Ide; R. Temam; S. Wang, Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part I: Steady-state solution, J. Phys. Oceanogr., 33, 712-728 (2003)
[31] E. Simonnet; M. Ghi; K. Ide; R. Temam; S. Wang, Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part II: time-dependent solutions, J. Phys. Oceanogr., 33, 729-751 (2003)
[32] E. Simonnet; M. Ghil; H. Dijkstra, Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation, J. Mar. Res., 63, 931-956 (2005)
[33] T. Ma; S. Wang, Stability and bifurcation of the taylor problem, Arch. Rational Mech. Anal., 181, 149-176 (2006) · Zbl 1089.76019
[34] G. Veronis, Wind-driven ocean circulation: Part 1. linear theory and perturbation analysis, Deep-Sea Research, 13 (1966), 17-29.
[35] G. Veronis, Wind-driven ocean circulation: Part 2. numerical solutions of the non-linear problem, Deep-Sea Research, 13 (1966), 31-55.
[36] C. C. Wang; G. Magnusdottir, The itcz in the central and eastern pacific on synoptic time scales, Mon. Wea. Rev., 134, 1405-1421 (2006)
[37] Q. Wang; C. Kieu; T. A. Vu, Large-scale dynamics of tropical cyclone formation associated with ITCZ breakdown, Atmos. Chem. Phys., 19, 8383-8397 (2019)
[38] G. Wolansky, Existence, uniqueness, and stability of stationary barotropic flow with forcing and dissipation, Commun. Pure Appl. Math., 41, 19-46 (1988) · Zbl 0651.76009
[39] G. Wolansky, The barotropic vorticity equation under forcing and dissipation: Bifurcations of nonsymmetric responses and multiplicity of solutions, SIAM J. Appl. Math., 41, 1585-1607 (1989) · Zbl 0682.76015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.