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Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows
1. | School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China |
2. | Department of Earth and Atmospheric Sciences, Indiana University, Bloomington, IN 47405, USA |
3. | College of Mathematics, Sichuan University, Chengdu, Sichuan, 610065, China |
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.
References:
[1] |
P. Berloff and S. Meacham,
On the stability of the wind-driven circulation, J. Mar. Res., 56 (1998), 937-993.
doi: 10.1357/002224098765173437. |
[2] |
P. Cessi and G. R. Ierley,
Symmetry-breaking multiple equilibria in quasi-geostrophic, wind-driven flows, J. Phys. Oceanogr., 25 (1995), 1196-1205.
doi: 10.1175/1520-0485(1995)025<1196:SBMEIQ>2.0.CO;2. |
[3] |
J. Charney and D. Straus,
Form-drag instability, multiple equilibria and propagating planetary waves in baroclinic, orographically forced, planetary wave systems, J. Atmos. Sci., 37 (1980), 1157-1176.
doi: 10.1175/1520-0469(1980)037<1157:FDIMEA>2.0.CO;2. |
[4] |
J. G. Charney and J. DeVore,
Multiple flow equilibria in the atmosphere and blocking, J. Atmos. Sci., 36 (1979), 1205-1216.
doi: 10.1175/1520-0469(1979)036<1205:mfeita>2.0.co;2. |
[5] |
J. G. Charney, J. Shukla and K. C. Mo,
Comparison of a barotropic blocking theory with observation, J. Atmos. Sci., 38 (1981), 762-779.
doi: 10.1175/1520-0469(1981)038<0762:COABBT>2.0.CO;2. |
[6] |
Z. Chen, M. Ghil, E. Simonnet and S. Wang,
Hopf bifurcation in quasi-geostrophic channel flow, SIAM J. Appl. Math., 64 (2003), 343-368.
doi: 10.1137/S0036139902406164. |
[7] |
Z. Chen and X. Xiong,
Equilibrium states of the charney-devore quasi-geostrophic equation in mid-latitude atmosphere, J. Math. Anal. Appl., 444 (2016), 1403-1416.
doi: 10.1016/j.jmaa.2016.07.021. |
[8] |
H. Dijkstra, T. Sengul, J. Shen and S. Wang,
Dynamic transitions of quasi-geostrophic channel flow, SIAM J. Appl. Math., 75 (2015), 2361-2378.
doi: 10.1137/15M1008166. |
[9] |
R. N. Ferreira and W. H. Schubert,
Barotropic aspects of itcz breakdown, J. Atmos. Sci., 54 (19997), 261-285.
doi: 10.1175/1520-0469(1997)054<0261:BAOIB>2.0.CO;2. |
[10] |
M. Ghil,
The wind-driven ocean circulation: applying dynamical systems theory to a climate problem, Discrete Contin. Dyn. Syst., 37 (2017), 189-228.
doi: 10.3934/dcds.2017008. |
[11] |
M. Ghil, M. D. Chekround and E. Simonnete,
Climate dynamics and fluid mechanics: Natural variability and related uncertainties, Physica D, 237 (2008), 2111-2126.
doi: 10.1016/j.physd.2008.03.036. |
[12] |
D. Han, M. Hernandez and Q. Wang,
Dynamic bifurcation and transition in the Rayleigh-Bénard enard convection with internal heating and varying gravity, Commun. Math. Sci., 17 (2019), 175-192.
doi: 10.4310/CMS.2019.v17.n1.a7. |
[13] |
D. Han, M. Hernandez and Q. Wang,
On the instabilities and transitions of the western boundary current, Commun. Computa. Phys., 26 (2019), 35-56.
doi: 10.4208/cicp.oa-2018-0066. |
[14] |
D. Han, M. Hernandez and Q. Wang,
Dynamic Transitions and Bifurcations for a Class of Axisymmetric Geophysical Fluid Flow, SIAM J. Appl. Dyn. Syst., 20 (2020), 38-64.
doi: 10.1137/20M1321139. |
[15] |
S. Jiang, F. F. Jin and M. Ghil,
Multiple equilibria, periodic, and aperiodic solutions in a wind-driven, double-gyre, shallow-water model, J. Phys. Oceanogr., 25 (1995), 764-786.
doi: 10.1175/1520-0485(1995)025<0764:MEPAAS>2.0.CO;2. |
[16] | E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, 2002. Google Scholar |
[17] |
C. Kieu, T. Sengul, Q. Wang and D. Yan,
On the Hopf (double Hopf) bifurcations and transitions of two-layer western boundary currents, Commun. Nonlinear Sci. Numer. Simul., 65 (2018), 196-215.
doi: 10.1016/j.cnsns.2018.05.010. |
[18] |
Y. A. Kuznetsov, Elements of applied bifurcation theory, Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-3978-7. |
[19] |
B. Legras and M. Ghil,
Persistent anomalies, blocking and variations in atmospheric predictability, J. Atmos. Sci., 42 (1985), 433-471.
doi: 10.1175/1520-0469(1985)042<0433:PABAVI>2.0.CO;2. |
[20] |
C. Lu, Y. Mao, Q. Wang and D. Yan,
Hopf bifurcation and transition of three-dimensional wind-driven ocean circulation problem, J. Differ. Equ., 267 (2019), 2560-2593.
doi: 10.1016/j.jde.2019.03.021. |
[21] |
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.
doi: 10.1016/j.physd.2019.132296. |
[22] |
T. Ma and A. Wang,
Rayleigh-Bénard convection: dynamics and structure in the physical space, Commun. Math. Sci., 5 (2007), 553-574.
|
[23] |
T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, New York, 2014.
doi: 10.1007/978-1-4614-8963-4. |
[24] |
S. P. Meacham,
Low-frequency variability in the wind-driven circulation, J. Phys. Oceanogr., 30 (2000), 269-293.
doi: 10.1175/1520-0485(2000)030<0269:LFVITW>2.0.CO;2. |
[25] |
B. T. Nadiga and B. P. Luce,
Global bifurcation of shilnikov type in a double-gyre ocean model, J. Phys. Oceanogr., 31 (2001), 2669-2690.
doi: 10.1175/1520-0485(2001)031<2669:GBOSTI>2.0.CO;2. |
[26] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987.
doi: 10.1007/978-1-4612-4650-3. |
[27] |
S. Rambaldi and K. Mo,
Forced stationary solutions in a barotropic channel: Multiple equilibria and theory of nonlinear resonance, J. Atmos. Sci., 41 (1984), 3135-3146.
doi: 10.1175/1520-0469(1984)041<3135:FSSIAB>2.0.CO;2. |
[28] |
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.
doi: 10.1006/jcph.1999.6344. |
[29] |
J. Shen, T. Tang and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer-Verlag, Berlin Heidelberg, 2011.
doi: 10.1007/978-3-540-71041-7. |
[30] |
V. A. Sheremet, G. R. Ierley and V. M. Kamenkovich,
Eigenanalysis of the two-dimensional wind-driven ocean circulation problem, J. Mar. Res., 55 (1997), 57-92.
doi: 10.1357/0022240973224463. |
[31] |
E. Simonnet, M. Ghi, K. Ide, R. Temam and S. Wang,
Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part I: Steady-state solution, J. Phys. Oceanogr., 33 (2003), 712-728.
doi: 10.1175/1520-0485(2003)33<712:LVISMO>2.0.CO;2. |
[32] |
E. Simonnet, M. Ghi, K. Ide, R. Temam and S. Wang,
Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part II: time-dependent solutions, J. Phys. Oceanogr., 33 (2003), 729-751.
doi: 10.1175/1520-0485(2003)33<729:LVISMO>2.0.CO;2. |
[33] |
E. Simonnet, M. Ghil and H. Dijkstra,
Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation, J. Mar. Res., 63 (2005), 931-956.
doi: 10.1357/002224005774464210. |
[34] |
T. Ma and S. Wang,
Stability and bifurcation of the taylor problem, Arch. Rational Mech. Anal., 181 (2006), 149-176.
doi: 10.1007/s00205-006-0415-8. |
[35] |
G. Veronis, Wind-driven ocean circulation: Part 1. linear theory and perturbation analysis, Deep-Sea Research, 13 (1966), 17–29.
doi: 10.1016/0011-7471(66)90003-9. |
[36] |
G. Veronis, Wind-driven ocean circulation: Part 2. numerical solutions of the non-linear problem, Deep-Sea Research, 13 (1966), 31–55.
doi: 10.1016/0011-7471(66)90004-0. |
[37] |
C. C. Wang and G. Magnusdottir,
The itcz in the central and eastern pacific on synoptic time scales, Mon. Wea. Rev., 134 (2006), 1405-1421.
doi: 10.1175/MWR3130.1. |
[38] |
Q. Wang, C. Kieu and T. A. Vu,
Large-scale dynamics of tropical cyclone formation associated with ITCZ breakdown, Atmos. Chem. Phys., 19 (2019), 8383-8397.
doi: 10.5194/acp-19-8383-2019. |
[39] |
G. Wolansky,
Existence, uniqueness, and stability of stationary barotropic flow with forcing and dissipation, Commun. Pure Appl. Math., 41 (1988), 19-46.
doi: 10.1002/cpa.3160410104. |
[40] |
G. Wolansky,
The barotropic vorticity equation under forcing and dissipation: Bifurcations of nonsymmetric responses and multiplicity of solutions, SIAM J. Appl. Math., 41 (1989), 1585-1607.
doi: 10.1137/0149096. |
show all references
References:
[1] |
P. Berloff and S. Meacham,
On the stability of the wind-driven circulation, J. Mar. Res., 56 (1998), 937-993.
doi: 10.1357/002224098765173437. |
[2] |
P. Cessi and G. R. Ierley,
Symmetry-breaking multiple equilibria in quasi-geostrophic, wind-driven flows, J. Phys. Oceanogr., 25 (1995), 1196-1205.
doi: 10.1175/1520-0485(1995)025<1196:SBMEIQ>2.0.CO;2. |
[3] |
J. Charney and D. Straus,
Form-drag instability, multiple equilibria and propagating planetary waves in baroclinic, orographically forced, planetary wave systems, J. Atmos. Sci., 37 (1980), 1157-1176.
doi: 10.1175/1520-0469(1980)037<1157:FDIMEA>2.0.CO;2. |
[4] |
J. G. Charney and J. DeVore,
Multiple flow equilibria in the atmosphere and blocking, J. Atmos. Sci., 36 (1979), 1205-1216.
doi: 10.1175/1520-0469(1979)036<1205:mfeita>2.0.co;2. |
[5] |
J. G. Charney, J. Shukla and K. C. Mo,
Comparison of a barotropic blocking theory with observation, J. Atmos. Sci., 38 (1981), 762-779.
doi: 10.1175/1520-0469(1981)038<0762:COABBT>2.0.CO;2. |
[6] |
Z. Chen, M. Ghil, E. Simonnet and S. Wang,
Hopf bifurcation in quasi-geostrophic channel flow, SIAM J. Appl. Math., 64 (2003), 343-368.
doi: 10.1137/S0036139902406164. |
[7] |
Z. Chen and X. Xiong,
Equilibrium states of the charney-devore quasi-geostrophic equation in mid-latitude atmosphere, J. Math. Anal. Appl., 444 (2016), 1403-1416.
doi: 10.1016/j.jmaa.2016.07.021. |
[8] |
H. Dijkstra, T. Sengul, J. Shen and S. Wang,
Dynamic transitions of quasi-geostrophic channel flow, SIAM J. Appl. Math., 75 (2015), 2361-2378.
doi: 10.1137/15M1008166. |
[9] |
R. N. Ferreira and W. H. Schubert,
Barotropic aspects of itcz breakdown, J. Atmos. Sci., 54 (19997), 261-285.
doi: 10.1175/1520-0469(1997)054<0261:BAOIB>2.0.CO;2. |
[10] |
M. Ghil,
The wind-driven ocean circulation: applying dynamical systems theory to a climate problem, Discrete Contin. Dyn. Syst., 37 (2017), 189-228.
doi: 10.3934/dcds.2017008. |
[11] |
M. Ghil, M. D. Chekround and E. Simonnete,
Climate dynamics and fluid mechanics: Natural variability and related uncertainties, Physica D, 237 (2008), 2111-2126.
doi: 10.1016/j.physd.2008.03.036. |
[12] |
D. Han, M. Hernandez and Q. Wang,
Dynamic bifurcation and transition in the Rayleigh-Bénard enard convection with internal heating and varying gravity, Commun. Math. Sci., 17 (2019), 175-192.
doi: 10.4310/CMS.2019.v17.n1.a7. |
[13] |
D. Han, M. Hernandez and Q. Wang,
On the instabilities and transitions of the western boundary current, Commun. Computa. Phys., 26 (2019), 35-56.
doi: 10.4208/cicp.oa-2018-0066. |
[14] |
D. Han, M. Hernandez and Q. Wang,
Dynamic Transitions and Bifurcations for a Class of Axisymmetric Geophysical Fluid Flow, SIAM J. Appl. Dyn. Syst., 20 (2020), 38-64.
doi: 10.1137/20M1321139. |
[15] |
S. Jiang, F. F. Jin and M. Ghil,
Multiple equilibria, periodic, and aperiodic solutions in a wind-driven, double-gyre, shallow-water model, J. Phys. Oceanogr., 25 (1995), 764-786.
doi: 10.1175/1520-0485(1995)025<0764:MEPAAS>2.0.CO;2. |
[16] | E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, 2002. Google Scholar |
[17] |
C. Kieu, T. Sengul, Q. Wang and D. Yan,
On the Hopf (double Hopf) bifurcations and transitions of two-layer western boundary currents, Commun. Nonlinear Sci. Numer. Simul., 65 (2018), 196-215.
doi: 10.1016/j.cnsns.2018.05.010. |
[18] |
Y. A. Kuznetsov, Elements of applied bifurcation theory, Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-3978-7. |
[19] |
B. Legras and M. Ghil,
Persistent anomalies, blocking and variations in atmospheric predictability, J. Atmos. Sci., 42 (1985), 433-471.
doi: 10.1175/1520-0469(1985)042<0433:PABAVI>2.0.CO;2. |
[20] |
C. Lu, Y. Mao, Q. Wang and D. Yan,
Hopf bifurcation and transition of three-dimensional wind-driven ocean circulation problem, J. Differ. Equ., 267 (2019), 2560-2593.
doi: 10.1016/j.jde.2019.03.021. |
[21] |
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.
doi: 10.1016/j.physd.2019.132296. |
[22] |
T. Ma and A. Wang,
Rayleigh-Bénard convection: dynamics and structure in the physical space, Commun. Math. Sci., 5 (2007), 553-574.
|
[23] |
T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, New York, 2014.
doi: 10.1007/978-1-4614-8963-4. |
[24] |
S. P. Meacham,
Low-frequency variability in the wind-driven circulation, J. Phys. Oceanogr., 30 (2000), 269-293.
doi: 10.1175/1520-0485(2000)030<0269:LFVITW>2.0.CO;2. |
[25] |
B. T. Nadiga and B. P. Luce,
Global bifurcation of shilnikov type in a double-gyre ocean model, J. Phys. Oceanogr., 31 (2001), 2669-2690.
doi: 10.1175/1520-0485(2001)031<2669:GBOSTI>2.0.CO;2. |
[26] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987.
doi: 10.1007/978-1-4612-4650-3. |
[27] |
S. Rambaldi and K. Mo,
Forced stationary solutions in a barotropic channel: Multiple equilibria and theory of nonlinear resonance, J. Atmos. Sci., 41 (1984), 3135-3146.
doi: 10.1175/1520-0469(1984)041<3135:FSSIAB>2.0.CO;2. |
[28] |
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.
doi: 10.1006/jcph.1999.6344. |
[29] |
J. Shen, T. Tang and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer-Verlag, Berlin Heidelberg, 2011.
doi: 10.1007/978-3-540-71041-7. |
[30] |
V. A. Sheremet, G. R. Ierley and V. M. Kamenkovich,
Eigenanalysis of the two-dimensional wind-driven ocean circulation problem, J. Mar. Res., 55 (1997), 57-92.
doi: 10.1357/0022240973224463. |
[31] |
E. Simonnet, M. Ghi, K. Ide, R. Temam and S. Wang,
Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part I: Steady-state solution, J. Phys. Oceanogr., 33 (2003), 712-728.
doi: 10.1175/1520-0485(2003)33<712:LVISMO>2.0.CO;2. |
[32] |
E. Simonnet, M. Ghi, K. Ide, R. Temam and S. Wang,
Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part II: time-dependent solutions, J. Phys. Oceanogr., 33 (2003), 729-751.
doi: 10.1175/1520-0485(2003)33<729:LVISMO>2.0.CO;2. |
[33] |
E. Simonnet, M. Ghil and H. Dijkstra,
Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation, J. Mar. Res., 63 (2005), 931-956.
doi: 10.1357/002224005774464210. |
[34] |
T. Ma and S. Wang,
Stability and bifurcation of the taylor problem, Arch. Rational Mech. Anal., 181 (2006), 149-176.
doi: 10.1007/s00205-006-0415-8. |
[35] |
G. Veronis, Wind-driven ocean circulation: Part 1. linear theory and perturbation analysis, Deep-Sea Research, 13 (1966), 17–29.
doi: 10.1016/0011-7471(66)90003-9. |
[36] |
G. Veronis, Wind-driven ocean circulation: Part 2. numerical solutions of the non-linear problem, Deep-Sea Research, 13 (1966), 31–55.
doi: 10.1016/0011-7471(66)90004-0. |
[37] |
C. C. Wang and G. Magnusdottir,
The itcz in the central and eastern pacific on synoptic time scales, Mon. Wea. Rev., 134 (2006), 1405-1421.
doi: 10.1175/MWR3130.1. |
[38] |
Q. Wang, C. Kieu and T. A. Vu,
Large-scale dynamics of tropical cyclone formation associated with ITCZ breakdown, Atmos. Chem. Phys., 19 (2019), 8383-8397.
doi: 10.5194/acp-19-8383-2019. |
[39] |
G. Wolansky,
Existence, uniqueness, and stability of stationary barotropic flow with forcing and dissipation, Commun. Pure Appl. Math., 41 (1988), 19-46.
doi: 10.1002/cpa.3160410104. |
[40] |
G. Wolansky,
The barotropic vorticity equation under forcing and dissipation: Bifurcations of nonsymmetric responses and multiplicity of solutions, SIAM J. Appl. Math., 41 (1989), 1585-1607.
doi: 10.1137/0149096. |










0.11934208 | 0.01 | 4(5) | 2.9824 | -0.5545 | -1.6917 | -1.8423 | -1.1788 |
0.11935935 | 0.03 | 4(5) | 3.4699 | -0.2779 | -0.8562 | -0.8955 | -0.5052 |
0.09768743 | 0.01 | 5(6) | 2.9518 | -0.6002 | -1.7218 | -1.8647 | -1.1083 |
0.09771521 | 0.03 | 5(6) | 3.4344 | -0.2983 | -0.8626 | -0.8986 | -0.4841 |
0.08267975 | 0.01 | 6(7) | 2.9345 | -0.6330 | -1.7385 | -1.8711 | -1.0615 |
0.08271008 | 0.03 | 6(7) | 3.4141 | -0.3127 | -0.8655 | -0.8982 | -0.4697 |
0.11934208 | 0.01 | 4(5) | 2.9824 | -0.5545 | -1.6917 | -1.8423 | -1.1788 |
0.11935935 | 0.03 | 4(5) | 3.4699 | -0.2779 | -0.8562 | -0.8955 | -0.5052 |
0.09768743 | 0.01 | 5(6) | 2.9518 | -0.6002 | -1.7218 | -1.8647 | -1.1083 |
0.09771521 | 0.03 | 5(6) | 3.4344 | -0.2983 | -0.8626 | -0.8986 | -0.4841 |
0.08267975 | 0.01 | 6(7) | 2.9345 | -0.6330 | -1.7385 | -1.8711 | -1.0615 |
0.08271008 | 0.03 | 6(7) | 3.4141 | -0.3127 | -0.8655 | -0.8982 | -0.4697 |
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