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Benoit Cushman-Roisin

Abstract

It has been previously noted that, if two warm-core rings were to merge to form a single new eddy, there would be more energy in the final state than in the initial, premerging state, and that therefore, merging is energetically prohibited. However, field and laboratory observations reveal spontaneous merging occurrences and thus challenge this energy argument. The paradox is resolved here by noting that the merging need not be complete; specifically, that the final state in fact consists of a center eddy containing almost all the energy but only a fraction of the mass, surrounded by a pair of thin filaments holding the mass difference, a residual of energy and most of the angular momentum. Existing numerical and experimental results point to the likelihood of such a composite state as the outcome of the merging of two warm-core rings.

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Benoit Cushman-Roisin

Abstract

In the North Pacific and Atlantic Oceans, the Subtropical Countercurrent is an eastward flow across the Subtropical Gyre at a latitude where a classical wind-driven circulation theory would predict a westward flow. It is in geostrophic balance with the Subtropical Front, a zonal density front which runs across the ocean basin. From previous numerical models, it is argued that convergence of Ekman transports can hardly be the primary reason for the existence of such a phenomenon and that thermodynamic effects play a crucial role. To elucidate how them may be responsible for the frontal structure, a very simple analytical model is constructed where the dynamics yield motions consisting of a nondivergent wind-driven Sverdrup current and a geostrophic thermal flow that is divergent on a beta-plane. The surface temperature is governed by a nonlinear hyperbolic equation, for which the corresponding characteristics intersect, separating the basin into two distinct regions limited by a temperature discontinuity. We then show how the beta-plane convergence of the thermally driven flow is responsible for this frontal formation and how all consequent results such as location of front, temperature jump across it, flow pattern and vertical velocity compare favorably with observations and numerical models. Disagreement along the eastern boundary is recorded and explained by the absence of California Current dynamics in the present simple model. The conclusion is that, aside from convergence of Ekman transports the convergence of the thermally driven flow on a beta-plane may be the primary mechanism responsible for forming and maintaining the Subtropical Front-Subtropical Countercurrent system.

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Benoit Cushman-Roisin

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From the primitive equations simplified dynamics are derived that apply to frontal situations in which interface slopes are important. The formalism, which eliminates inertial motions, is not Unlike the derivation of the quasi-geostrophic equation. The difference is two-fold: while quasi-geostrophic dynamics apply for length scales on the order of the deformation radius with limitation to small interface variations, frontal geostrophic dynamics apply for finite interface variations but only at length scales large compared to the deformation radius (three or more times). When the length scale is on the order of the deformation radius and, simultaneously, the interface variations are finite, inertial oscillations cannot be filtered out, and the primitive equations ought to be retained.

In a reduced-gravity context, frontal geostrophic dynamics yield a single equation for the upper-layer depth. Although this equation is cubic in the depth variable, it is nonetheless considerably simpler than the primitive equations. It is suggested that the use of this equation can further advance the theoretical investigation of frontal dynamical processes.

Some particular solutions are presented as illustrations. A new breed of waves is discovered. These waves propagate downstream (with the front on their left in the Northern Hemisphere) and are dispersive. They are unlike either Kelvin, Rossby or edge waves. Finally, a solution that corresponds to a time-dependent elliptical warm-core ring is also presented.

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Benoit Cushman-Roisin

Abstract

In contrast with the traditional view of midlatitude circulation driven by winds in the ocean interior and regulated by friction along the western boundary, it is hypothesized that some control can be attributed to surface cooling acting primarily in a recirculation region off the western boundary current. Several arguments suggests that this mechanism and the generation of eddies are the two major reactions of the midlatitude ocean under the action of the surface winds.

The theory is best described by tracing the journey of a water parcel around the Subtropical Gyre. In the interior, the anticyclonic wind-stress pattern acts to decrease the parcel's potential vorticity (PV). In reaction, the parcel first migrates south, where the planetary vorticity is less, then veers westward and participates in a western boundary jet, where the relative vorticity is less. Instead of calling upon friction within the jet, it is not difficult to conceive that PV restoration can be achieved by diabatic processes beyond the jet. As water parcels await to rejoin the interior circulation, they accumulate and the PV diabatic processes beyond the jet. As water parcels await to rejoin the interior circulation, they accumulate and the PV balance is retained by increased thickness between density surfaces. This forms a high pressure and a recirculation. There, such storage can be imagined to have proceeded until a steady regime has been reached whereby surface cooling triggered by anomalous amounts of warm water is effective in restoring PV and allowing parcels to rejoin the interior circulation.

This scenario is attractive, for it sees the Gulf Stream as an inertial current, whose width is thus set by the deformation radius and not a friction length. It explains why the Gulf Stream transport, augmented by the recirculation, far exceeds the Sverdrup wind-driven transport of the interior. It further gives an explicit dynamical role to the recirculation as a storage area, and it concludes that the recirculation is anticyclonic, a region of minimum potential vorticity and one of intense cooling.

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Benoit Cushman-Roisin

Abstract

The wind stress on the ocean surface induces horizontal advection and vertical mixing, two mechanisms which are capable of changing the surface water density. Advection and mixing can enhance or partially destroy the effects of each other. Frontogenesis due to convergent Ekman transports forced by a wind-stress field is one example. A bulk model for the study of advection and mixing in such circumstances is constructed from a one-dimensional model. Continuity of mass requires that water masses either downwell (convergence) or escape laterally (confluence). This distinction leads to a study of two extreme cases of frontogenesis, each herein treated separately. The model reduces to two coupled highly nonlinear prognostic equations for the mixed-layer buoyancy and mixed-layer depth. Scaling of the equations leads to the definition of a mixing parameter, a nondimensional number which measures the relative importance of advection and mixing. For large-scale ocean frontogenesis, this parameter is of the order of unity, implying that mixing is as efficient as advection. If the region of denser water is referred to as the north, the numerical results are: 1) the front is never symmetric; 2) in the case of weak mixing, the density jump across the pycnocline is stronger in the south and the mixed layer is deeper at the north, 3) in the case of strong mixing, the front is limited by a southern edge with a weak horizontal gradient to the south and a strong decreasing gradient to the north; 4) strong mixing can induce frontolysis south of the front, 5) after about one month, the Ekman downwelling resulting from convergence, if any, strongly controls the rate of deepening., and 6) frontal density gradients are about three times larger in the case of confluence than in the case of convergence. Because the emphasis is on the interaction between wind advection and wind stirring, both dissipation and surface buoyancy flux are neglected. Hence, the model does not reach a steady state and does not provide a length-scale for the width of the front.

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Benoit Cushman-Roisin and Vlado Malačič

Abstract

This paper reconsiders the classic problem of bottom Ekman pumping below a steady geostrophic flow by relaxing the assumption of a constant eddy viscosity. It is assumed instead that the eddy viscosity depends on the magnitude of the bottom stress, which itself is a function of the geostrophic flow. Results show that the vertical Ekman pumping is no longer directly proportional to the relative vorticity of the geostrophic flow, but is a far more complicated function of the geostrophic flow. Specific examples are discussed, which show that the Ekman pumping rate may be 50% or 100% larger than that predicted by the traditional theory.

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Benoit Cushman-Roisin and Thomas Keffer

Abstract

No abstract available.

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Benyang Tang and Benoit Cushman-Roisin

Abstract

In Part I of this series, generalized geostrophic equations were formulated for the two-layer system on a beta plane and over a flat bottom. Here numerical experiments with these equations are carried out to study freely evolving geostrophic turbulence. In contrast with the classical quasigeostrophic analysis, the emphasis is placed on the finite amplitude of the vertical displacement (the frontal effect).

A previous study with a reduced-gravity, generalized geostrophic equation has shown that geostrophic turbulence of finite amplitude (frontal geostrophic turbulence) evolves toward a statistical equilibrium state dominated by large, coherent anticyclones. The present study reveals that, in the presence of baroclinicity, this statistical equilibrium state can only be reached if the finite-amplitude turbulent flow evolves from scales smaller or equal to the baroclinic deformation radius. Although the emerging anticyclones should be unstable according to the classical quasigeostrophic theory of baroclinic instability, they nonetheless appear to be stable within the present, generalized-geostrophic formalism. By their stability, they prevent potential energy from being released by baroclinic instability to the barotrophic flow.

Finally, similarities and differences between the evolution of two-layer geostrophic turbulence in the quasi-geostrophic regime, on one hand, and the frontal (finite-amplitude) geostrophic regime, on the other, are discussed and summarized.

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Benoit Cushman Roisin and Benyang Tang

Abstract

Geostrophic turbulence has traditionally been studied within the framework of the classical, quasi-geostrophic equation. This equation, valid only when vertical displacements are weak, possesses a symmetry between cyclonic and anticyclonic vortices that is not present in the primitive equations. Moreover, previous studies were restricted by length scales not in excess of the deformation radius. In an attempt to advance the study of unforced geostrophic turbulence, we address here the following questions: How is the energy cascade toward longer length scales affected beyond the deformation radius? And, what is the result of the cyclonic-anticyclonic asymmetry brought on by finite vertical displacements?

Some answers are provided by numerical experiments using a generalized geostrophic equation. The energy cascade is found to come to a halt beyond the deformation radius. There, a statistical equilibrium is reached at a length scale prescribed as a combination of the deformation radius, the beta effect and the energy level of the system. Also, over the long run, one witnesses the emergence of few, large eddies, which all are anticyclonic and drift in a weaker, shorter-scale, quasi-geostrophic background. A simple theory capturing the essence of this bimodal distribution correctly predicts the bulk characteristics of the statistical equilibrium. Finally, some arguments are outlined to explain the selection of anticyclonic eddies and its relation to the statistical equilibrium.

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Benoit Cushman-Roisin and Shabnam Merchant-Both

Abstract

The rodon solution for an elliptical vortex with outcropping interface is extended from the reduced-gravity limit to the two-layer model. Motions in the lower layer consist of a reaction to the presence of the upper-layer vortex, or of an independent shear flow, or both. For simplicity and clarity, the analysis is performed in a geostrophic limit, which restricts the attention to vortices somewhat larger than the baroclinic radius of deformation.

In a first part, geostrophic equations are established and discussed to identify clearly when the vortex motions induced in the lower layer and the ambient lower-layer shear must be retained or may be neglected.

In the second part, the equations are solved for various cases. It is found that lower-layer motions induced by the upper-layer vortex provide a feedback on the latter and that this reaction is a mere increase of the anticyclonic rotation rate of the rodon. Furthermore, this increased rate has a straightforward interpretation in terms of potential vorticity conservation. When an additional shear flow in the lower layer is important, the upper-layer rodon can exhibit one of three behaviors: it can execute complete rotations, albeit at a varying angular rate, it can oscillate within two extreme orientations, or it can be sheared away with time. Results also show that anticyclonic shear flows (same sign as the rodon) have relatively benign effects unless they are very strong, while moderate cyclonic shear flows (opposite sign to the rodon) have very significant effects.

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