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Melvin E. Stern

Abstract

Separation from the continental slope of stratified jets like the Gulf Stream involves the sliding of successive isopycnal layers from a nearly horizontal bottom to the adjacent offshore isopycnal in the deep ocean. One mechanism for producing such an effect is due to a downstream convergence of slope isobaths, as shown herein for a 1-layer density model. Upstream of the convergence, a geostrophically balanced jet is assumed with an inshore region of cyclonic vorticity resting on the continental slope and an offshore anticyclonic region resting on the isopycnal interface above heavier water. For O(1) Rossby number and cross-stream topographic variation, the steady transverse current displacements forced by slowly varying downstream topography are computed. For “supercritical” upstream flow (i.e., fast compared to free topographic waves) offslope displacements are produced by converging isobaths; extrapolation of the small amplitude result suggests that the mechanism is quantitatively important for the explanation of the full separation of the Gulf Stream from the bottom of the continental slope. The kinematics involved in this process should apply to a continuously stratified jet, as well as to other forcing mechanisms known to be of importance in continental boundary current separation.

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Melvin E. Stern

Abstract

When a low Rossby number barotropic flow accelerates in the laterally converging half of a strait, the local propagation speed of long topographic waves can be reduced to zero, thereby blocking or preventing the formation of a steady flow downstream from the strait. An inviscid longwave theory is presented for the new steady upstream and downstream states that evolve from the blocking wave. The enhanced inshore cyclonic vorticity extending far downstream suggests that topographic jetogenesis, rather than lateral eddy diffusion, in major ocean straits (e.g., Yucatan and Florida) may be important in generating or reforming boundary currents.

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Melvin E. Stern

Abstract

The evolution of large-amplitude disturbances at the outer edge of a quasi-geostrophic shear layer depends on the sign of the outward gradient of potential vorticity. Entrainment of ambient water can occur when the gradient of relative vorticity dominates in the potential vorticity, and detrainment from the current can occur when gradient of isopycnal thickness dominates. In the latter case long, thin filaments of finite area are “pinched off” into the surrounding water mass. This is verified using a quasi-geostrophic model having piecewise uniform potential vorticity. Contour dynamical calculations for many initial conditions allow us to define and tabulate an entrainment/detrainment velocity. This is used for an order of magnitude estimate of the flux of heat or salt on an isopycnal surface in a warm core ring.

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Melvin E. Stern

Abstract

We compute the evolution of disturbances on a circularly symmetric eddy having uniform vorticity in a central core, in a surrounding annulus, and in the irrotational exterior water mass. This vortex is known to be (Kelvin-Helmholtz) unstable when its annular width is less than the core radius. Our calculations for the nonlinear regime show that amplification of azimuthal wavenumber n = 2 causes the vortex to split into two dipoles, in agreement with previous numerical calculations for a smoothed version of our vorticity field. This paper concentrates on the evolution of large-amplitude disturbances on the outer edge of a stable and robust eddy. We show that lateral wave breaking of vorticity isopleths causes intrusions of the (irrotational) exterior water mass into the central core of the vortex, a physical process which is relevant to lateral diffusion and isopycnal mixing in baroclinic ocean eddies. Similar intrusive features occur for an n = 1 disturbance, which also causes a “self-propagation” of the entire eddy. The large-amplitude disturbances on the eddy can be initiated by the action of external eddies or currents. A simple model for this case exhibits filaments detraining from the eddy, as well as intrusive features.

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Melvin E. Stern

Abstract

The temporal evolution of large amplitude quasi-geostrophic disturbances in a piecewise uniform potential vorticity flow is elucidated by numerical solutions of the “contour dynamica” equations. Lateral wavebreaking occurs when the initial disturbance amplitude exceeds a certain value, and at later times tongues of the 1ower vorticity fluid are engulfed or entrained into the higher vorticity shear flow. The effect appears to be important for the evolution of “shingles” observed between the coastal water and the cyclonic side of the Gulf Stream. The effect may also be in important phase in initiating the mixing process at the perimeter of an eddy embedded in another water mass.

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Melvin E. Stern

Abstract

It is suggested that the inshore shear of continental boundary flows like the Florida Current can be accounted for by a countergradient vorticity flux, rather than by lateral diffusion to the shore. Two simple barotropic models with cross-stream and downstream topographic variations illustrate the point. In the first case, a broad jet with piecewise uniform vorticity accelerates through a slowly converging strait, eventually becoming locally critical (in the hydraulic sense) and then undergoing a transition to a different downstream state in which the maximum inshore vorticity is increased. The topographic conditions for this to occur are determined by a nonlinear long-wave theory. In the second model, the computed flow around an idealized cape illustrates the role of lee waves in generating mean downstream vorticity. For large-amplitude capes a nonlinear long-wave theory shows that a downstream transition (similar to the strait problem) can occur as well as upstream “blocking.”

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Melvin E. Stern

Abstract

The free discharge of a layer of bottom water in a wide strait (e.g., the Denmark Strait) differs from the classical control problem because of the strong geostrophic turbulence. As a consequence, the cross-stream (x) variation of the time-averaged downstream velocity υ(x) is “underdetermined” and depends on more parameters than available conditions. To resolve this “degeneracy” the classical control condition is generalized; the result requires the discharge Q to be extremized with respect to the degeneracy parameters and with respect to the constraints. One of these constraints is that a branch point or a local stationary wave can be supported at some section of the long channel. By either maximizing Q or by requiring a stationary wave, useful approximations are obtained. Future work should consider the joint variational problem. Specific calculations are made for nonuniform potential vorticity in a rectangular channel and also for variable cross-stream bottom topography. In the latter case, the current has a free streamline whose point of intersection with the bottom is computed. It is suggested that a necessary condition for hydraulic control at any section is that the flow does not separate from either side of the channel. The mean isopycnals in the Denmark Strait satisfy this condition, thereby suggesting that it does in fact (topographically) control the upstream state in the Greenland Sea.

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Timour Radko
and
Melvin E. Stern

Abstract

This study examines dynamics of finescale instabilities in thermohaline–shear flows. It is shown that the presence of the background diapycnal temperature and salinity fluxes due to double diffusion has a destabilizing effect on the basic current. Using linear stability analysis based on the Floquet theory for the sinusoidal basic velocity profile, the authors demonstrate that the well-known Richardson number criterion (Ri < ¼) cannot be directly applied to doubly diffusive fluids. Rigorous instabilities are predicted to occur for Richardson numbers as high as—or even exceeding—unity. The inferences from the linear theory are supported by the fully nonlinear numerical simulations. Since the Richardson number in the main thermocline rarely drops below ¼, whereas the observations of turbulent patches are common, the authors hypothesize that some turbulent mixing events can be attributed to the finescale instabilities associated with double-diffusive processes.

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Timour Radko
and
Melvin E. Stern

Abstract

Analytical solutions of the shallow-water system of equations in a 1½-layer f-plane model are obtained for stable quasi-monopolar vortices propagating due to a dipolar (azimuthal mode m = 1) component whose amplitude is small compared to the circularly symmetric (m = 0) part. The ability of the f-plane eddies with O(1) Rossby number to propagate distances greater than their size is demonstrated by finite amplitude numerical calculations. Numerical and analytical considerations also indicate that significant departures may occur between the “centroid of mass anomaly” (commonly used in theoretical estimates of the speed of oceanic vortices) and other measures of the eddy center.

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Melvin E. Stern
and
Timour Radko

Abstract

If an azimuthally symmetric barotropic eddy on the f plane is subject to a relatively small amplitude disturbance of unit azimuthal wavenumber (m = 1), it can propagate very many diameters away from its origin, as shown by a weak nonlinear theory for a piecewise uniform vorticity eddy, and also for one with continuous vorticity inside a finite area. In the former case an initial value contour dynamical calculation shows that the analytical solution is realizable over long distances; the same is true in the latter case, as shown by spectral calculations using the full two-dimensional vorticity equation (with small dissipation). The oceanographic significance of this effect lies in the ability of almost symmetric eddies to self-propagate over large distances and collide with other eddies, currents, and continents; this produces important mixing effects, as illustrated herein. It is also shown how the analysis and the effect is generalizeable to a 1½-layer density model on the β plane.

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