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Glenn R. Ierley

Abstract

A nonlinear ordinary differential equation which models the western boundary layer of quasi-geostrophic barotropic models of the wind-driven circulation has been investigated by Il'in and Kamenkovich and more recently by Ierley and Ruehr. For mild nonlinearity the boundary layer model has two possible outflow solutions which match to a Sverdrup interior. For stronger nonlinearity, no solutions exist. In the weakly nonlinear case we use linear stability theory to resolve the problem of multiple solutions. A highly stretched regional model of the full quasi-geostrophic equations is used to investigate the disappearance of solutions of the boundary layer model. We find that a failure of the boundary layer model is coincident with the onset of recirculation in the solution of the partial differential equations at a sufficiently large ratio of basin scale to viscous boundary-layer scale. For less extreme ratios, the onset of recirculation is deferred, and hence its relation to a failure of the boundary layer model is obscured.

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Paola Cessi and Glenn R. Ierley

Abstract

The classical Munk problem of barotropic flow driven by an antisymmetric wind stress exhibits multiple steady solutions in the range of moderate to high forcing and moderate to low dissipation. Everywhere in the parameter space a perfectly antisymmetric solution exists in which the strength of the cyclonic gyre is equal and opposite to that of the anticyclonic gyre. This kind of solution has been well documented in the literature.

In a subset of the parameter a pair of nonsymmetric stationary solutions coexists with the antisymmetric solution. For one member of the pair the amplitude of the cyclonic circulation exceeds that of the anticyclonic flow. The other member of the pair is obtained from the quasigeostrophic symmetry y&minusy and ψ→−ψ. As a result, the point at which the western boundary current separates from the coast can be either south or north of the latitude at which the antisymmetric Ekman pumping changes sign. This is the first oceanogrphic example of spontaneous breaking of the quasigeostrophic symmetry.

Within the region of parameter space where three solutions are found, a second pair of nonsymmetric stationary solutions emerges, bringing the total number of stationary solutions to five. This last pair of nonsymmetric solutions is characterized by basin-filling gyres with amplitudes much above the Sverdrup prediction. Once again, the separation point is displaced from the latitude of vanishing wind stress curl.

The existence of nonsymmetric double gyres in an antisymmetrically forced basin shows that there can be no general rule for determining the point of separation of the boundary current in terms of the relative strength of the subtropical and subpolar forcings.

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Paola Cessi and Glenn R. Ierley

Abstract

Viscous shear instability is proposed as a primary mechanism for generating time-dependent eddies at western boundaries. Thus, the authors examine the stability of the Munk flow with constant transport flowing along a straight coast, tilted at an angle with respect to the north-south direction. Various properties of the marginally unstable wave are calculated as a function of the tilting angle, such as the critical Reynolds number and the phase and group velocities. The effects of weak nonlinearity are also examined, and the authors find that the instability is supercritical for the whole range of tilting angles examined. Thus, the marginally unstable mode can equilibrate at a small finite amplitude, and we derive the equation governing its slow evolution. The flow that results after the disturbance has equilibrated to finite amplitude is in agreement with the eddying boundary currents obtained in many wind-driven general circulation models.

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