Bifurcation Properties of a Stratospheric Vacillation Model

Shigeo Yoden Department of Atmospheric Sciences, University of Washington, Seattle, WA 98195

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Abstract

Nonlinear properties of a stratospheric vacillation model are investigated numerically in the light of bifurcation theory. The model is exactly the same as that used by Holton and Mass, which describes the wave-zonal flow interaction in a β-channel under a nonconservative constraint with zonal-flow forcing and wave dissipation. A set of 81 nonlinear ordinary differential equations with variables depending on time is obtained by a severe truncation and vertical differencing. All of the external parameters are fixed in time. The amplitude of the wave forcing or the intensity of zonal wind forcing at the bottom boundary is changed as a bifurcation parameter.

Three branches of the steady solutions are obtained by use of Powell's hybrid method and the pseudo-arclength continuation method. Linear stability of these solution branches is investigated by solving an eigenvalue problem in the linearized system. In some range of the bifurcation parameter, there exists a multiplicity of stable steady solutions with different vertical structures.

Periodic solutions a series of stratospheric vacillations originally found by Holton and Mass, are obtained by time-integrations. It is found that the periodic solutions branch off from a steady solution by a Hopf bifurcation. For a finite increment of the parameter from the bifurcation point, the time average of the periodic solution is significantly different from the unstable steady solution. The nonlinear transience causes the difference.

The multiplicity of stable solutions (steady and periodic) is a possible explanation for the interannual variability of the stratosphere circulation in the middle and high latitudes during winter.

Abstract

Nonlinear properties of a stratospheric vacillation model are investigated numerically in the light of bifurcation theory. The model is exactly the same as that used by Holton and Mass, which describes the wave-zonal flow interaction in a β-channel under a nonconservative constraint with zonal-flow forcing and wave dissipation. A set of 81 nonlinear ordinary differential equations with variables depending on time is obtained by a severe truncation and vertical differencing. All of the external parameters are fixed in time. The amplitude of the wave forcing or the intensity of zonal wind forcing at the bottom boundary is changed as a bifurcation parameter.

Three branches of the steady solutions are obtained by use of Powell's hybrid method and the pseudo-arclength continuation method. Linear stability of these solution branches is investigated by solving an eigenvalue problem in the linearized system. In some range of the bifurcation parameter, there exists a multiplicity of stable steady solutions with different vertical structures.

Periodic solutions a series of stratospheric vacillations originally found by Holton and Mass, are obtained by time-integrations. It is found that the periodic solutions branch off from a steady solution by a Hopf bifurcation. For a finite increment of the parameter from the bifurcation point, the time average of the periodic solution is significantly different from the unstable steady solution. The nonlinear transience causes the difference.

The multiplicity of stable solutions (steady and periodic) is a possible explanation for the interannual variability of the stratosphere circulation in the middle and high latitudes during winter.

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