Bifurcations and Chaotic Solutions of Two-dimensional Zonal Jet Flow on a Rotating Sphere
We study bifurcation structure of the zonal jet flow the
streamfunction of which is expressed by a single spherical harmonics
on a rotating sphere. In the non-rotating case, we find that a steady
traveling wave solution arises from the zonal jet flow through Hopf
bifurcation. As the Reynolds number increases, several traveling
solutions arise only through the pitchfork bifurcations and at high
Reynolds number the bifurcating solutions become Hopf unstable. In
the rotating case, on the other hand, under the stabilizing effect of
rotation, as the absolute value of rotation rate increases, the number
of the bifurcating solutions arising from the zonal jet flow decreases
monotonically. We also carry out time integration to study unsteady
solutions at high Reynolds number and find that in the non-rotating
case the unsteady solutions are chaotic, while not in the rotating cases
calculated. This result reflects the general tendency that the rotation
stabilizes nonlinear solutions of Navier-Stokes equations.
[1] P. G. Williams, Planetary circulations: 1. Barotropic representation of
Jovian and Terrestrial turbulence J. Atmos. Sci., 35, 1399-1426, 1978.
[2] S. Yoden, S. and M. Yamada, A numerical experiment on twodimensional
Decaying turbulence on a rotating sphere. J. Atmos. Sci.,
50, 631-643, 1993.
[3] S. Takehiro, S., M. Yamada and Y-Y. Hayashi, Circumpolar jets emerging
in two-dimensional non-divergent decaying turbulence on a rapidly
rotating sphere. Fluid Dyn. Res., 39, 209-220, 2007.
[4] T. Nozawa and S. Yoden, Formation of zonal band structure in forced
two-dimensional turbulence on a rotating sphere, Phys. Fluid , 9, 2081-
2093, 1997.
[5] K. Obuse, S. Takehiro and M. Yamada, Long-time asymptotic states of
forced two-dimensional barotropic incompressible flows on a rotating
sphere. Phys. Fluid, 22, 056601, 2010.
[6] BAINES, P. G., 1976 The stability of planetary waves on a sphere.
J. Fluid Mech. 73-2, 193-213.
[7] E. Sasaki, S. Takehiro, and M, Yamada, A note on the stability of inviscid
zonal jet flows on a rotating sphere, J. Fluid Mech., 710, 154-165, 2012.
[8] E. Sasaki, S. Takehiro and M. Yamada, Stability of two-dimensional
viscous zonal jet flows on a rotating sphere, in preparation.
[9] I. V. Iudovisch, 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. J. Appl. Math. Mech., 29, 527-544, 1965.
[10] D. L. Meshalkin and Y. G. Sinai, 1962 Investigation of the stability of a
stationary solution of a system of equations for the plane movement of
an incompressible viscous liquid, J. Appl. Math. and Mech., 25, 1700-
1705, 1962
[11] H. Okamoto and M. Sh¯oji, Bifurcation Diagrams in Kolmogorov-s
Problem of Viscous Incompressible Fluid on 2-D Flat Tori. Japan J.
Indus Appl. Math., 10, 191-218, 1993.
[12] C. S. Kim, S. C. and H. Okamoto, Stationary vortices of large scale
appearing in 2D Navier-Stokes equations at large Reynolds numbers.
Japan J. Indust. Appl. Math., 27, 47-71, 2010.
[13] N. Platt, L. Sirovich and N. Fitzmaurice, An investigation of chaotic
Kolmogorov flows. Phys. Fluid , 3, 681-696, 1991.
[14] M. Inubushi, M. U. Kobayashi, S. Takehiro and M. Yamada, Covariant
Lyapunov analysis of chaotic Kolmogorov flows. Phys. Rev. E 85,
016331, 2012.
[15] I. Silberman, Planetary waves in the atmosphere, J. Meteo., 11, 27-34,
1953.
[16] E. Sasaki, S. Takehiro and M. Yamada, Bifurcation structure of twodimensional
viscous zonal jet flows on a rotating sphere, in preparation.
[17] K Ishioka, ispack-0.96, http://www.gfd-dennou.org/arch/ispack/, GFD
Dennou Club, 2011.
[18] S. Takehiro, Y. SASAKI, Y. Morikawa, K. Ishioka, M. Odaka,
O. Y. Takahashi, S. Nishizawa, K. Nakajima, M. Ishiwatari, and
Y.-Y, Hayashi, SPMODEL Development Group, Hierarchical Spectral
Models for GFD (SPMODEL), http://www.gfd-dennou.org/library/
spmodel/, GFD Dennou Club, 2011.
[1] P. G. Williams, Planetary circulations: 1. Barotropic representation of
Jovian and Terrestrial turbulence J. Atmos. Sci., 35, 1399-1426, 1978.
[2] S. Yoden, S. and M. Yamada, A numerical experiment on twodimensional
Decaying turbulence on a rotating sphere. J. Atmos. Sci.,
50, 631-643, 1993.
[3] S. Takehiro, S., M. Yamada and Y-Y. Hayashi, Circumpolar jets emerging
in two-dimensional non-divergent decaying turbulence on a rapidly
rotating sphere. Fluid Dyn. Res., 39, 209-220, 2007.
[4] T. Nozawa and S. Yoden, Formation of zonal band structure in forced
two-dimensional turbulence on a rotating sphere, Phys. Fluid , 9, 2081-
2093, 1997.
[5] K. Obuse, S. Takehiro and M. Yamada, Long-time asymptotic states of
forced two-dimensional barotropic incompressible flows on a rotating
sphere. Phys. Fluid, 22, 056601, 2010.
[6] BAINES, P. G., 1976 The stability of planetary waves on a sphere.
J. Fluid Mech. 73-2, 193-213.
[7] E. Sasaki, S. Takehiro, and M, Yamada, A note on the stability of inviscid
zonal jet flows on a rotating sphere, J. Fluid Mech., 710, 154-165, 2012.
[8] E. Sasaki, S. Takehiro and M. Yamada, Stability of two-dimensional
viscous zonal jet flows on a rotating sphere, in preparation.
[9] I. V. Iudovisch, 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. J. Appl. Math. Mech., 29, 527-544, 1965.
[10] D. L. Meshalkin and Y. G. Sinai, 1962 Investigation of the stability of a
stationary solution of a system of equations for the plane movement of
an incompressible viscous liquid, J. Appl. Math. and Mech., 25, 1700-
1705, 1962
[11] H. Okamoto and M. Sh¯oji, Bifurcation Diagrams in Kolmogorov-s
Problem of Viscous Incompressible Fluid on 2-D Flat Tori. Japan J.
Indus Appl. Math., 10, 191-218, 1993.
[12] C. S. Kim, S. C. and H. Okamoto, Stationary vortices of large scale
appearing in 2D Navier-Stokes equations at large Reynolds numbers.
Japan J. Indust. Appl. Math., 27, 47-71, 2010.
[13] N. Platt, L. Sirovich and N. Fitzmaurice, An investigation of chaotic
Kolmogorov flows. Phys. Fluid , 3, 681-696, 1991.
[14] M. Inubushi, M. U. Kobayashi, S. Takehiro and M. Yamada, Covariant
Lyapunov analysis of chaotic Kolmogorov flows. Phys. Rev. E 85,
016331, 2012.
[15] I. Silberman, Planetary waves in the atmosphere, J. Meteo., 11, 27-34,
1953.
[16] E. Sasaki, S. Takehiro and M. Yamada, Bifurcation structure of twodimensional
viscous zonal jet flows on a rotating sphere, in preparation.
[17] K Ishioka, ispack-0.96, http://www.gfd-dennou.org/arch/ispack/, GFD
Dennou Club, 2011.
[18] S. Takehiro, Y. SASAKI, Y. Morikawa, K. Ishioka, M. Odaka,
O. Y. Takahashi, S. Nishizawa, K. Nakajima, M. Ishiwatari, and
Y.-Y, Hayashi, SPMODEL Development Group, Hierarchical Spectral
Models for GFD (SPMODEL), http://www.gfd-dennou.org/library/
spmodel/, GFD Dennou Club, 2011.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:59623", author = "Eiichi Sasaki and Shin-ichi Takehiro and Michio Yamada", title = "Bifurcations and Chaotic Solutions of Two-dimensional Zonal Jet Flow on a Rotating Sphere", abstract = "We study bifurcation structure of the zonal jet flow the
streamfunction of which is expressed by a single spherical harmonics
on a rotating sphere. In the non-rotating case, we find that a steady
traveling wave solution arises from the zonal jet flow through Hopf
bifurcation. As the Reynolds number increases, several traveling
solutions arise only through the pitchfork bifurcations and at high
Reynolds number the bifurcating solutions become Hopf unstable. In
the rotating case, on the other hand, under the stabilizing effect of
rotation, as the absolute value of rotation rate increases, the number
of the bifurcating solutions arising from the zonal jet flow decreases
monotonically. We also carry out time integration to study unsteady
solutions at high Reynolds number and find that in the non-rotating
case the unsteady solutions are chaotic, while not in the rotating cases
calculated. This result reflects the general tendency that the rotation
stabilizes nonlinear solutions of Navier-Stokes equations.", keywords = "rotating sphere, two-dimensional flow, bifurcationstructure", volume = "7", number = "2", pages = "242-7", }
