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Elongation and Contraction of the Western Boundary Current Extension in a Shallow-Water Model: A Bifurcation Analysis

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  • 1 Department of Earth System Science, University of California, Irvine, Irvine, California
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Abstract

The double-gyre circulation, formulated in terms of the quasigeostrophic equations, has a symmetry about the basin midlatitude (y → −y, ψ → −ψ), which is absent in a formulation based on the shallow-water equations. As a result, the shallow-water model does not have the pitchfork bifurcation structures that, in the case of the quasigeostrophic model, connect together multiple solution branches with elongated and contracted recirculation gyres. For the shallow-water model, solution branches with elongated recirculation gyres are disconnected, and a one-parameter bifurcation analysis is unable to detect their existence. The deeply penetrating jet solution branches do, however, continue to exist, and can be found using a bifurcation analysis couched in terms of two parameters. An effective pair of parameters is the viscosity and a parameter controlling the symmetry of the wind stress profile. A bifurcation analysis with these parameters reveals the existence of new solution branches that were not found in previous bifurcation analyses of the shallow-water model. The new solutions have a jet extension that penetrates farther eastward and that is more stable than the jet-up and jet-down solutions found in previous studies. Furthermore, the origin of the low-frequency variability at low viscosities is associated with a sequence of bifurcations originating from one of the new steady-state solution branches. In particular, the eigenmode analysis of the new branch reveals that a so-called gyre mode is at the origin of the model’s low-frequency variability at decadal time scales.

Corresponding author address: François Primeau, Department of Earth System Science, University of California, Irvine, 3216 Croul Hall, Irvine, CA 92697-3100. Email: fprimeau@uci.edu

Abstract

The double-gyre circulation, formulated in terms of the quasigeostrophic equations, has a symmetry about the basin midlatitude (y → −y, ψ → −ψ), which is absent in a formulation based on the shallow-water equations. As a result, the shallow-water model does not have the pitchfork bifurcation structures that, in the case of the quasigeostrophic model, connect together multiple solution branches with elongated and contracted recirculation gyres. For the shallow-water model, solution branches with elongated recirculation gyres are disconnected, and a one-parameter bifurcation analysis is unable to detect their existence. The deeply penetrating jet solution branches do, however, continue to exist, and can be found using a bifurcation analysis couched in terms of two parameters. An effective pair of parameters is the viscosity and a parameter controlling the symmetry of the wind stress profile. A bifurcation analysis with these parameters reveals the existence of new solution branches that were not found in previous bifurcation analyses of the shallow-water model. The new solutions have a jet extension that penetrates farther eastward and that is more stable than the jet-up and jet-down solutions found in previous studies. Furthermore, the origin of the low-frequency variability at low viscosities is associated with a sequence of bifurcations originating from one of the new steady-state solution branches. In particular, the eigenmode analysis of the new branch reveals that a so-called gyre mode is at the origin of the model’s low-frequency variability at decadal time scales.

Corresponding author address: François Primeau, Department of Earth System Science, University of California, Irvine, 3216 Croul Hall, Irvine, CA 92697-3100. Email: fprimeau@uci.edu

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