Abstract

One of the serious flaws in the standard quasi-geostrophic equations, commonly used for understanding the evolution of mesoscale eddies, is the requirement that the change in thickness between density surfaces must be small compared to the mean thickness. In the case of warm core rings, the thickness of the thermostadt layer may range from 500 m at the center to zero at the edge. Yet the prediction of the evolution of such features should be vastly simplified by noting that there is a dominant equilibrium balance of forces in the fluid, with the beta effect and the time derivatives being relatively small.

In this paper, the author presents the nonquasi-geostrophic model for the evolution of a warm core ring using a two-layer fluid in which the upper layer has a finite volume so that the interface surfaces on a basically circular boundary. The lowest order flow in the warm pool is much faster than the Rossby wave speeds βL² and is not geostrophic but rather is assumed to be in a state of cyclostrophic balance. The time changes then occur on a time scale set by (βL)⁻¹ and can be calculated by balancing the net Coriolis forces due to the translation of the warm pool with the southward forces caused by the β effect and the form drag of the lower fluid caused by Rossby wave generation. The lower layer is assumed to be deep so that the lower layer dynamics are quasi-geostrophic, being forced by the motion of the warm pool.

For very deep lower layers, the generated waves can be calculated explicitly and the form drags can be shown to induce a southward motion of the upper pool and decay of its energy. This wave drag vanishes for very special choices of the size of the upper pool and the lower layer motions are then nonzero only in the region under the warm pool and have a net counterclockwise circulation with angular momentum equal and opposite to that of the upper layer. This is not expected to occur for most oceanic cases.

The calculation of the first-order structure in the warm pool shows a dipole character resembling a modon pair. The self-induced tendency of the pair is for westward motion. Although dynamically analogous to a modon with a strong rider, there are important differences due to the surfacing of the interface, the order-one Rossby number and the loss of energy owing to wave generation in the lower layer.

Abstract

We have used linear quasigeostrophic dynamics to calculate the motion, decay and dispersion rates of radially symmetric initial disturbances with scales appropriate to Gulf Stream Rings. Baroclinic rings translate westward at 1–3 km day⁻¹; asymmetries in the initial shape can give northward or southward components. The pressure maximum decays to half its value in about 6 months. Both short-time and long-time estimates of the translational speed and strength of the central pressure minimum are obtained. The dispersed waves can spread over much of the northwestern Atlantic (for linear dynamics). Using estimates of the number of Gulf Stream Rings present at any time, their ages, their average strength and their distribution, we have calculated the distribution of fluctuation kinetic energy from the Rings and the associated dispersed wave field. This order-of-magnitude estimate suggests that Rings and the near-neighbor dispersed waves from the Rings may be responsible for fluctuation energies comparable to those observed in the ocean.

Abstract

A thin-jet model predicts the location of the axis of a strong current such as the Gulf Stream by using the vertical and cross-stream integrated vorticity balance, under the assumption that the meandering scales are large compared to the width of the jet. We demonstrate that such an integral provides a matching condition upon the barotropic component of the wave or eddy fields which, on either side of the jet, have north–south scales on the order of the meander wavelength. For steady meanders, these exterior fields do not influence the path and our model reproduces the dynamics of Robinson and Niiler, but for the transient case the determination of the jet axis motion and of the external field is a coupled problem.

When the disturbances in the axis position are time-dependent but are very small, the exterior wave problem can be linearized and the matching conditions can be applied at the mean position of the jet. We can therefore derive a dispersion relation for the meandering motion, allowing us to compute the phase speed and growth rates for the meanders in terms of the wavenumber and two integral properties of the stream: the mass and momentum transports. This dispersion relation predicts instability for wave shorter than a critical scale.

We also derive via standard four-dimensional instability theory a long wave approximation to the dispersion relation for perturbations of a quasi-geostrophic jet with both horizontal and vertical shears. The result is identical to that from the thin-jet theory for an interesting class of perturbations which we therefore identity as meandering modes. Thus thin-jet theory has been calibrated by reduction to both finite amplitude steady meandering and infinitesimal instability cases. For the understanding of large amplitude, time-dependent motions of the Gulf Stream and their role in the general circulation, the thin jet theory offers a semi-analytical approach for process studies.

Abstract

Estimates of the repeatability and accuracy of XBT measurements were made using XBT data taken during MODE. The XBT observations of the depth of isotherms had systematic errors of up to 15 decibars (by comparison to simultaneous CTD drops) as well as random errors on the order of 8 declare. Maps of these observations show small-scale thermal structure which would imply sizeable increases in geostrophic shears.

Abstract

Low-frequency oceanic motions have banded structures termed “striations.” Since these striations embedded in large-scale gyre flows can have large amplitudes, the authors investigated the effect of mean flow on their directions as well as their contribution to energetics and mixing using a β-plane, barotropic, quasigeostrophic ocean model. In spite of the model simplicity, striations are always found to exist regardless of the imposed barotropic mean flow. However, their properties are sensitive to the mean flow. Rhines jets move with the mean flow and are not necessarily striations. If the meridional component of the mean flow is large, Rhines jets become high-frequency motions; low-frequency striations still exist, but they are nonzonal, have small magnitudes, and contribute little to energetics and mixing. Otherwise, striations are zonal, dominated by Rhines jets, and contribute significantly to energetics and mixing. This study extends the theory of β-plane, barotropic turbulence, driven by white noise forcing at small scales, to include the effect of a constant mean flow. Theories developed in this study, based upon the Galilean invariance property, illustrate that the barotropic mean flow has no effect on total mixing rates, but does affect the energy cascades in the frequency domain. Diagnostic frameworks developed here can be useful to quantify the striations’ contribution to energetics and mixing in the ocean and more realistic models. A novel diagnostic formula is applied to estimating eddy diffusivities.

Abstract

In this paper it is proposed that baroclinic instability of even a weak shear may play an important role in the generation and stability of the strong zonal jets observed in the atmospheres of the giant planets. The atmosphere is modeled as a two-layer structure, where the upper layer is a standard quasigeostrophic layer on a β plane and the lower layer is parameterized to represent a deep interior convective columnar structure using a negative β plane as in Ingersoll and Pollard. Linear stability theory predicts that the high wavenumber perturbations will be the dominant unstable modes for a small vertical wind shear like that inferred from observations. Here a nonlinear analytical model is developed that is truncated to one growing mode that exhibits a multiple jet meridional structure, driven by the nonlinear interaction between the eddies. In the weakly supercritical limit, this model agrees with previous weakly nonlinear theory, but it can be explored beyond this limit allowing the multiple jet–induced zonal flow to be stronger than the eddy field. Calculations with a fully nonlinear pseudospectral model produce stable meridional multijet structures when beginning from a random potential vorticity perturbation field. The instability removes energy from the background weak baroclinic shear and generates turbulent eddies that undergo an inverse energy cascade and form multijet zonal winds. The jets are the dominant feature in the instantaneous upper-layer flow, with the eddies being relatively weak. The jets scale with the Rhines length, but are strong enough to violate the barotropic stability criterion. It is shown that the basic physical mechanism for the generation and stability of the jets in the full numerical model is similar to that of the truncated model.

Abstract

An analytical theory is presented for the self-induced translation of an intense vortex relative to a uniform background flow on the β plane. The equivalent barotropic approximation is used to formulate the initial value problem within a polar coordinate frame translating with the vortex center. A contour dynamical model of the vortex is melded with the regular beta-plane model of the residual flow. Evolution of vortex asymmetries for azimuthal mode number one, the so-called beta gyres, which are responsible for the relative vortex motion, is considered for a period of time while the Rossby wave radiation is not important.

It is shown for an initially axisymmetric vortex that the beta gyres and corresponding vortex translational velocity consist of two parts. The first one is generated by advection of the background potential vorticity gradient and rotates differentially because of the symmetric vortex circulation. The second part arises due to distortion in the vortex shape represented by displacements of the piecewise constant potential vorticity contours relative to the vortex center. The distortion of the vortex shape is described by the sum of normal modes generated by the first part. Explicit solutions for both parts are obtained, and approximate expressions for different stages of the vortex motion are presented.

For a vortex with a uniform potential vorticity core (single contour), the beta gyres are found to consist only of the first part so that the vortex translation depends on the ratio of the core size to the radius of deformation. A small core corresponds to the geostrophic point vortex limit with initially predominantly meridional motion. Asymptotically, after a large number of fluid revolutions at a radial distance on the order of the radius of deformation, the westward translation dominates: the meridional velocity and the deviation of zonal velocity from the maximum linear Rossby wave speed decay linearly with time. This tendency is explained to be a result of effective symmetrization of the potential vorticity due to differential rotation of fluid around the vortex. The period of initial predominantly meridional motion is negligible when the core size is on the order of the deformation radius.

For the vortex with two steps in the potential vorticity, the normal mode rotates faster than the fluid if the potential vorticities in the core and at the periphery have different signs. The effect of the distortion in the vortex shape on the vortex translation increases with increasing deformation radius relative to the vortex size. In a stationary beta gyre, for a finite vortex, the relative contour shift contributes to the westward translation just up to the long Rossby wave speed.

In the nondivergent limit a universal approximate trajectory has been found for large outer contour radius. The center of a finite vortex moves northwestward with permanent meridional acceleration due to degeneracy of a zero-frequency normal mode. The zonal translational velocity approaches a limit proportional to the vortex area. The effect of the distortion in the vortex shape in this nondivergent limit results in decreasing the westward translation and increasing the meridional one.

Applications of the theory to hurricanes in the atmosphere and rings in the ocean are discussed.

Abstract

Some simple air-sea momentum transfer models, which include sea surface velocity and temperature, are considered for their effects on Gulf Stream rings. Perturbing the stress calculation with sea surface velocity results in a “top drag”, which causes interior motions to decay. Numerical experiments with equivalent barotropic quasi-geostrophic dynamics and reasonable estimates for the top drag suggest that this mechanism can amount for a significant fraction of observed isotherm subsidence rates in rings. Perturbing the stress calculation with a temperature sensitive drag coefficient produces a dipolar Ekman pumping field over a ring. For an eastward directed wind, the result is a tendency for the ring to self-propagate to the south. Integral constraints can be used to estimate the meridional propagation rate precisely, and for reasonable stress and thermal anomaly values, the estimate compares well with observations.

Abstract

This paper examines the plausibility of mesoscale eddy generation through local baroclinic instability of weak midocean gyre flows. The main tool is a statistically steady, two-layer quasigeostrophic turbulence model driven by an imposed, horizontally homogeneous, vertically sheared mean flow and dissipated by bottom Ekman friction. A wide range of friction strengths is investigated. In the weakly damped limit, flow is nearly barotropic, and the horizontal length scale of barotropic energy increases with decreasing friction, consistent with previous studies. The strongly damped limit, explored here for the first time, is equivalent barotropic (lower-layer velocities are nearly zero) and features an increase in the horizontal scale of potential energy with increasing friction. Current-meter data suggest that midocean eddies lie between the barotropic and equivalent barotropic limits. In accord with this suggestion, the moderately damped regime of the model compares well to observations of eddy amplitude, vertical structure, and horizontal scale, especially when stratification is surface intensified. A review of pertinent observations suggests that mesoscale eddies may indeed lie in the moderately damped limit. These arguments are first developed in f-plane simulations. Previous studies of beta-plane turbulence have had eastward mean flows, and in this case eddy energy has little sensitivity to friction. However, midocean gyre flows are generally nonzonal, and this nonzonality appears to be a significant factor in the production of energetic eddies. Beta-plane turbulence driven by nonzonal mean flows is sensitive to bottom friction, such that moderate damping is required for model eddies to compare well to observations, as on the f plane. A heuristic argument is presented in support of this similarity.

Abstract

The effects of mean flow direction on statistically steady, baroclinically unstable, beta-plane quasigeostrophic (QG) turbulence are examined in a two-layer numerical model. The turbulence is forced by an imposed, horizontally homogeneous, vertically sheared mean flow and dissipated by bottom Ekman friction. The model is meant to be an idealization of the midocean eddy field, which generally has kinetic energies larger than the mean and is isotropic. Energetic eddies can be generated even when planetary beta (β) dominates gradients of mean potential vorticity (PV; also, q), as long as the mean shear has a substantial meridional component. However, eddies are isotropic only when the angle between layer mean PV gradients exceeds approximately 90°. This occurs when planetary and shear-induced gradients are comparable. Maps of PV indicate that these gradients may indeed be comparable over much of the midocean. Coherent jets form when the mean flow has a substantial meridional component and β is large. When β is nonzero, but small enough to permit isotropy, and the zonal component of the mean flow is not strongly eastward, lattices of like-signed coherent vortices develop. Like-signed vortex formation from initial and forcing conditions without a vorticity preference has not been observed before in QG systems. The vortex arrays are sensitive to the details of small-scale dissipation. Both cyclonic and anticyclonic fields arise in the simulations, depending on initial conditions, but they have different energies, consistent with broken symmetries in the governing equations.