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Eric Simonnet, Michael Ghil, Kayo Ide, Roger Temam, and Shouhong Wang

Abstract

Successive bifurcations—from steady states through periodic to aperiodic solutions—are studied in a shallow-water, reduced-gravity, 2½-layer model of the midlatitude ocean circulation subject to time-independent wind stress. The bifurcation sequence is studied in detail for a rectangular basin with an idealized spatial pattern of wind stress. The aperiodic behavior is studied also in a North Atlantic–shaped basin with realistic continental contours. The bifurcation sequence in the rectangular basin is studied in Part I, the present article. It follows essentially the one reported for single-layer quasigeostrophic and 1½-layer shallow-water models. As the intensity of the north–south-symmetric, zonal wind stress is increased, the nearly symmetric double-gyre circulation is destabilized through a perturbed pitchfork bifurcation. The low-stress steady solution, with its nearly equal subtropical and subpolar gyres, is replaced by an approximately mirror-symmetric pair of stable equilibria. The two solution branches so obtained are named after the inertial recirculation cell that is stronger, subtropical or subpolar, respectively. This perturbed pitchfork bifurcation and the associated Hopf bifurcations are robust to changes in the interface friction between the two active layers and the thickness H 2 of the lower active layer. They persist in the presence of asymmetries in the wind stress and of changes in the model's spatial resolution and finite-difference scheme. Time-dependent model behavior in the rectangular basin, as well as in the more realistic, North Atlantic–shaped one, is studied in Part II.

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Eric Simonnet, Michael Ghil, Kayo Ide, Roger Temam, and Shouhong Wang

Abstract

The time-dependent wind-driven ocean circulation is investigated for both a rectangular and a North Atlantic–shaped basin. Multiple steady states in a 2½-layer shallow-water model and their dependence on various parameters and other model properties were studied in Part I for the rectangular basin. As the wind stress on the rectangular basin is increased, each steady-state branch is destabilized by a Hopf bifurcation. The periodic solutions that arise off the subpolar branch have a robust subannual periodicity of 4–5 months. For the subtropical branch, the period varies between sub- and interannual, depending on the inverse Froude number F 2 defined with respect to the lower active layer's thickness H 2. As F 2 is lowered, the perturbed-symmetric branch is destabilized baroclinically, before the perturbed pitchfork bifurcation examined in detail in Part I occurs. Transition to aperiodic behavior arises at first by a homoclinic explosion off the isolated branch that exists only for sufficiently high wind stress. Subsequent global and local bifurcations all involve the subpolar branch, which alone exists in the limit of vanishing wind stress. Purely subpolar solutions vary on an interannual scale, whereas combined subpolar and subtropical solutions exhibit complex transitions affected by a second, subpolar homoclinic orbit. In the latter case, the timescale of the variability is interdecadal. The role of the global bifurcations in the interdecadal variability is investigated. Numerical simulations were carried out for the North Atlantic with earth topography-5 minute (ETOPO-5) coastline geometry in the presence of realistic, as well as idealized, wind stress forcing. The simulations exhibit a realistic Gulf Stream at 20-km resolution and with realistic wind stress. The variability at 12-km resolution exhibits spectral peaks at 6 months, 16 months, and 6–7 years. The subannual mode is strongest in the subtropical gyre; the interannual modes are both strongest in the subpolar gyre.

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Kyung-Il Chang, Michael Ghil, Kayo Ide, and Chung-Chieng Aaron Lai

Abstract

Multiple equilibria as well as periodic and aperiodic solution regimes are obtained in a barotropic model of the midlatitude ocean’s double-gyre circulation. The model circulation is driven by a steady zonal wind profile that is symmetric with respect to the square basin’s zonal axis of north–south symmetry, and dissipated by lateral friction.

As the intensity of the wind forcing increases, an antisymmetric double-gyre flow evolves through a pitchfork bifurcation into a pair of steady mirror-symmetric solutions in which either the subtropical or the subpolar gyre dominates. In either one of the two asymmetric solutions, a pair of intense recirculation vortices forms close to and on either side of the point where the two western boundary currents merge to form the eastward jet. To the east of this dipole, a spatially damped stationary wave arises, and an increase in the steady forcing amplifies the meander immediately to the east of the recirculating vortices. During this process, the transport of the weaker gyre remains nearly constant while the transport of the stronger gyre increases.

For even stronger forcing, the two steady solution branches undergo Hopf bifurcation, and each asymmetric solution gives rise to an oscillatory mode, whose subannual period is of 3.5–6 months. These two modes are also mirror-symmetric in space. The time-average difference in transport between the stronger and the weaker gyre is reduced as the forcing increases further, while the weaker gyre tends to oscillate with larger amplitude than the stronger gyre. Once the average strength of the weaker gyre on each branch equals the stronger gyre’s, the solution becomes aperiodic. The transition of aperiodic flow occurs through a global bifurcation that involves a homoclinic orbit. The subannual oscillations persist and stay fairly regular in the aperiodic solution regime, but they alternate now with a new and highly energetic, interannual oscillation. The physical causes of these two oscillations—as well as of a third, 19-day oscillation—are discussed. During episodes of the high-amplitude, interannual oscillation, the solution exhibits phases of either the subtropical or subpolar gyre being dominant. Even lower-frequency, interdecadal variability arises due to an irregular alternation between subannual and interannual modes of oscillation.

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