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William K. Dewar
and
Peter D. Killworth

Abstract

The Rossby adjustment of an initially circular column of water, the so-called collapse of a cylinder, continues to be a widely used method for forming lenslike eddies in the laboratory. Here, we consider the structure of an eddy so formed as well as some ramifications of that formation. We demonstrate that the calculation of the eddy structure can be reduced to the extraction of the roots of two nonlinear, coupled algebraic equations. Analytical solutions in the limit of the collapse of a needle are given and roots are obtained numerically otherwise. It is concluded that in the collapse of a cylinder initially spanning the entire column of water, the eddy always maintains contact with both surfaces. (This is not the case in the seemingly equivalent two-dimensional case with no variation in one Cartesian direction.) In the event the initial cold column is separated only slightly from the surface, the above solution acts as the lowest order solution in a regular perturbation sequence.

Next, these “collapse eddy” solutions, which possess motions in both layers and finite energies, are used to examine lens merger. Two collapse eddies of equal volume jointly possess less energy than one collapse eddy of twice the volume. However, we argue that two collapse eddies of equal volume can have more energy than the circularly symmetric end-state eddy formed from them if the two initial eddies “mix.” We also offer evidence that the energy budgets may be balanced exactly if the end-state eddy is slightly asymmetric. Comparisons with some previous laboratory experiments are made.

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Sergey V. Kravtsov
and
William K. Dewar

Abstract

A six-box model is employed as a prototype of the coupled Atlantic ocean–atmosphere system. Ice dynamics are excluded. Numerical integration of this system shows that different thermohaline circulation patterns are possible under the same forcing conditions. They consist of a global thermal mode with oceanic poleward surface flow, a global saline mode with equatorward surface flow, and two intermediate modes that are combinations of the two global modes. The stability of the modern-day-like intermediate mode to finite amplitude freshwater flux perturbations in the high latitude North Atlantic (meant as a model of glacial melting) is explored. It is found that freshwater fluxes of the proper inferred magnitude are close to critical and can induce a transition of the coupled system to a saline mode. However, paleoclimatic data argues the last deglaciation was subcritical. Further, working in a realistic subcritical parameter regime, the box model yields an unrealistic temperature record. This argues, in turn, that additional physics (e.g., sea ice effects) must be included to properly describe the fundamental mechanics of the last glacial retreat.

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Guohui Wang
and
William K. Dewar

Abstract

A quasigeostrophic point vortex numerical model is used to explore interactions of eddies and seamounts. The ultimate objective of this study is to assess the role of meddy–seamount interaction as an input to Mediterranean salt tongue maintenance. Secondary objectives are to clarify the dynamics of meddy–seamount interaction. The results suggest that meddies survive seamount collisions with 60%–70% of their initial cores remaining intact as coherent vortices. Given observed formation rates, it appears meddies, in their interactions with seamounts, inject between one-quarter and one-half of the salt anomaly necessary to sustain the Mediterranean salt tongue. Other considerations suggest the anomalous mass flux by meddies is comparable to that due to the mean flow. In summary, meddies are important to the maintenance of the salt tongue, although other mechanisms are needed. Coherent vortex transport, of which meddies are one example, is a mesoscale process not well described by the downgradient mixing algorithms normally employed in general circulation models. More sophisticated mesoscale models are thus suggested by this study. In particular, survival by meddies of collisions with seamounts emerges as a potentially important limiting effect on the Mediterranean salt tongue. This effect has climatically significant implications for ocean simulations.

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William K. Dewar
and
Rui X. Huang

Abstract

The time dependence of the ventilated thermocline is examined via analytical and numerical means. The original Henderschott model is modified such that the outcrops all occur on the same geopotential surface, rather than at staggered geopotential surfaces. This model has the advantage that the ocean interior can be ventilated directly by the Sverdrup flow, rather than by western boundary processes. The propagation of disturbances governed by linearized forms of the three-layer or four-layer modified Henderschott model, nonlinear solutions of the full modified Henderschott model, and numerical solutions of the planetary geostrophic equations are considered.

Low-frequency disturbances are predicted by the linear models to move on characteristics jointly set by advection and wave dynamics. It is shown that perturbations due to wind stress anomalies project strongly onto the first mode and propagate westward similarly to the classical first baroclinic Rossby mode. They do not experience much interaction with the mean flow (the so-called non-Doppler effect). On the other hand, perturbations generated by buoyancy anomalies have strong projections onto the second or third modes, and propagate along pathways very close to the mean circulation. Their speed is somewhat slower than the current speed, however. These properties appear in the linearized and simplified nonlinear models and their occurrence in planetary geostrophic results argues the relevance of the Henderschott model. Also, these properties are consistent with results from other studies.

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Rui Xin Huang
and
William K. Dewar

Abstract

A highly idealized model for the oceanic haline circulation is studied. Specifically, loops filled with salty water and subjected to either the natural boundary condition, the virtual salt flux condition, or salinity relaxation are considered. It is shown that the characteristics of the solutions, especially the transition between steady and unsteady convection, depend critically on the applied boundary conditions. It is found that the relaxation condition generally modifies the location of the Hopf bifurcation so highly that models based on it should always remain in the regime of steady convection. On the other hand, the location of the Hopf bifurcation for models based on flux conditions is much less extreme. Thus, in these models, limit cycles or chaotic behavior can easily be excited.

Further, the nature of the Hopf bifurcation depends sensitively on the boundary condition. For example, if the frictional parameter is gradually reduced, the model based on the natural boundary condition goes through a supercritical Hopf bifurcation, while the model based on virtual salt flux goes through a subcritical Hopf bifurcation. Similar dependencies are found when other parameters are varied. Beyond the Hopf bifurcations, windows of limit cycle solutions alternate with windows of chaos. In addition, for a given set of parameters, the system can have multiple solutions, such as a limit cycle and a chaotic solution, or limit cycles which have distinctively different structure.

These results comment on the types of behavior that more complicated three-dimensional models may exhibit.

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William K. Dewar
and
Peter D. Killworth

Abstract

Oceanic rings tend to have length scales larger than the deformation radius and also to he long-lived. This latter characteristic, in view of the former, is particularly curious as many quasigeostrophic and primitive equation simulations suggest such eddies are quite unstable. Large eddies eventually break into smaller deformation scale vortices, with the attendant production of considerable variability.

Here it is argued that the stability characteristics of oceanic eddies and rings are sensitive to the presence of deep flows. In particular, eddies in which the deep flow is counter to the sense of the shallow flows are often more unstable than eddies with no deep flow, while eddies with circulations in the same sense as the shallow circulation can experience an enhanced stability. For a given vertical shear, oceanic eddy stability can vary dramatically. (This is in contrast to quasigeostrophic theory, where stability properties are largely determined by vertical shear.) The onset of these mechanics is quite pronounced for Gaussian oceanic eddies. Linear “f”- plane stability calculations reveal a marked suppression of unstable growth rates for warm corotating eddies with relatively weak deep flows. Cold eddies also experience a suppression of instability in the corotating state, although relatively weak unstable modes have been found. Comparisons of f- and β-plane numerical primitive equation experiments support these results, as well as demonstrate some relevant limitations. Finally, studies of dipolar eddies and non-Gaussian circular eddies are used to examine the generality of the results. We suggest such stability considerations may be partially responsible for the observed long lives of oceanic rings.

An examination of the unstable normal modes from the f-plane model demonstrates an intimate coupling between the suppression of instability and the appearance of multiple critical layers. The normal-mode energetics are used to demonstrate the role of upgradient momentum fluxes at the points of stabilization, and a heuristic argument involving critical layers is given.

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William K. Dewar
and
Huan Meng

Abstract

A combined analytical and numerical examination of submesoscale coherent vortex (SCV) dynamics and propagation is conducted. This study is prompted by observations of the movement relative to their surroundings of one class of SCVs, that is, Meddies. An asymptotic analysis is performed to study the mechanics governing SCV propagation. It is found that the large-scale flow plays a dominant role in determining the trajectory of SCVs and that the β effect and form drag of neighboring layers are weaker effects. As a result, SCVs propagate at a speed that is a density-weighted average of the flow in the surrounding layers. Meddies may thus move relative to the surrounding water, which is in accordance with observations.

This theory extends previous studies on eddy propagation by considering more general situations. For example, a lenslike eddy embedded in a nonzonal, vertically and horizontally sheared flow is studied. A significant difference between this study and most previous related work is that the submesoscale nature of SCVs is exploited. It is this nature that leads to our conclusions about SCV drift.

The theory is tested both by solving the asymptotic equations and through experiments with a primitive equation model. Agreement is found between the results of our numerical experiments and the analytical predictions, thus suggesting that the asymptotic analysis has captured the leading order behavior of SCV propagation.

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Trevor J. McDougall
and
William K. Dewar

Abstract

A consistent scheme for vertical mixing in layered numerical models is derived in this paper. The fact that the vertical coordinate (density) depends on the properties being transported (namely salinity and potential temperature) renders the inclusion of vertical mixing in layered models a subtle problem. The approach the authors have taken is based upon the entrainment into a layer being proportional to the turbulent activity in that layer. Across each interface there are then two entrainment velocities, one upward velocity that is the entrainment of fluid into the layer above the interface, and one downward velocity, being the entrainment velocity into the layer below the interface. This double entrainment accounts for both the diffusive and the advective consequences of turbulent mixing. The proposed scheme works without approximation for a nonlinear equation of state and can readily handle the production of density caused by cabbeling. Several examples are given.

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William K. Dewar
and
Michele Y. Morris

Abstract

The propagation of long, first mode, baroclinic planetary waves in eddy-resolving quasigeostrophic general circulation models is studied. Recent TOPEX/Poseidon observations argue oceanic first-mode planetary waves move with speeds other than those predicted by simple theory. These data have prompted theoretical analyses of wave propagation in a mean flow, with the results suggesting mean shear can have a controlling effect on the planetary wave guide. Some of the predicted effects appear to be relevant to the observations, while others are less obvious. This, coupled with other explanations for the observations, motivates the calculations.

Based on these experiments, the authors suggest that the predicted effects of mean shear on wave propagation are consistent with those computed in a fully geostrophically turbulent ocean. These are that a two-layer model misses the dominant component of long-wave interaction with a mean flow, a three-layer model captures this interaction qualitatively, and the correction to wave propagation is in the direction opposite to the mean flow. Quantitative comparisons between the theory and the numerical experiments are good in the northern latitudes and questionable in the southern latitudes. Reasons for the southern discrepancy are offered.

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William K. Dewar
and
John M. Bane Jr.

Abstract

The energy budgets of the eddies and the mean flow in the Gulf Stream near a topographic feature known as the Charleston bump are computed. First, we consider these results in the context of the amplification hypothesis for the development of Gulf Stream meanders. According to this hypothesis, the finite amplitude Gulf Stream fluctuations observed offshore of Onslow Bay are the result of the destabilizing effect of the bump on the Stream. The present dataset was obtained both immediately upstream and downstream of the bump, and the results of our analysis suggest: 1) Immediately south of the Charleston bump, the eddies perform net work on the Gulf Stream at a rate of (1.02 ± .66) × 10−2 ergs cm−3 s−1 by transporting momentum offshore; 2) The net work performed by the eddies south of the bump is not used locally to accelerate the mean; rather, it is exported to the rest of the ocean at a rate of (1.58 ± 1.39) × 10−2 ergs cm−3 s−1; 3) In spite of the net work performed by the eddies south of the bump, eddy kinetic energy apparently does not decrease; 4) Immediately north of the Charleston bump, the flow appears to be both barotropically and baroclinically unstable. These results support the amplification hypothesis by demonstrating the destabilizing effect of the bump on the eddies (points 1 and 4) and that upstream perturbations may survive to encounter the bump topography (point 3). Other results of our analysis are that the mean of mean kinetic energy by the eddies constitutes the dominant form of energy conversion and that eddy pressure work may be an important factor in the fluctuation energy budget.

The second application of our calculations is to a characterization of the mean Gulf Stream in the South Atlantic Bight (SAB). The results of this analysis indicate the following: 1) The mean Gulf Stream kinetic energy flux increases downstream at a rate of (2.17 ± .98) × 10−2 ergs cm−3 s−1; 2) The eddies tend to decelerate the mean flow at a rate of (-0.57 ± 1.3) × 10−2 ergs cm−3 s−1; 3) In order that the mean energy equation be balanced, the Gulf Stream in the SAB must be releasing mean potential energy by flowing down a mean pressure gradient. Thus we have evidence suggesting the existence of a component of the pressure gradient associated with the Gulf Stream which is not geostrophically balanced. The downstream pressure gradient inferred at our array site is consistent with published estimates of mean alongshore pressure gradients in the SAB; however, the partitioning of the pressure force between mean acceleration and eddy Reynolds stress most likely holds only near the bump. We also estimate the net loss from the mean potential energy in the SAB using our measured conversion rate and demonstrate that it compares in magnitude but is opposite in sign to that thought to occur downstream of Cape Hatteras. Thus we argue that the Gulf Stream in the SAB is exhibiting some of the properties of the inflow regions of western boundary layers in inviscid inertial models of the general ocean circulation. Our measurements, however, also indicate the presence of vigorous eddies whose effects in the mean energy equation are potentially sizeable. Such eddies are, of course, not contained in strictly inviscid, inertial models of the western boundary layer.

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