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## Abstract

The effect of three different parameterizations of dissipation on the nonlinear dynamics of unstable baroclinic waves is studied. The model is the two-layer *f*-plane model and the dynamics is quasigeostrophic. The dissipation mechanisms are 1) dissipation due to Ekman layers at the horizontal boundary surfaces, 2) the addition of interfacial Ekman friction, or 3) dissipation proportional to the perturbation potential vorticity.

We find, as anticipated by weakly nonlinear theory, a strong effect on the nonlinear amplitude dynamics for supercriticalities as large as four times the threshold value for instability. The use of interfacial friction or potential vorticity damping expunges the vacillating behavior common to the system with type 1 dissipation.

At high supercriticality a barotropic vacillation involving the mean flow and harmonics of the fundamental is superimposed on the basic baroclinic wave dynamics. Examination of the critical transition for the emergence of the barotropic oscillation reveals that the enhanced linear instability of the higher harmonics is responsible for the self-sustained vacillation.

## Abstract

The effect of three different parameterizations of dissipation on the nonlinear dynamics of unstable baroclinic waves is studied. The model is the two-layer *f*-plane model and the dynamics is quasigeostrophic. The dissipation mechanisms are 1) dissipation due to Ekman layers at the horizontal boundary surfaces, 2) the addition of interfacial Ekman friction, or 3) dissipation proportional to the perturbation potential vorticity.

We find, as anticipated by weakly nonlinear theory, a strong effect on the nonlinear amplitude dynamics for supercriticalities as large as four times the threshold value for instability. The use of interfacial friction or potential vorticity damping expunges the vacillating behavior common to the system with type 1 dissipation.

At high supercriticality a barotropic vacillation involving the mean flow and harmonics of the fundamental is superimposed on the basic baroclinic wave dynamics. Examination of the critical transition for the emergence of the barotropic oscillation reveals that the enhanced linear instability of the higher harmonics is responsible for the self-sustained vacillation.

## Abstract

The nonlinear dynamics of a slightly unstable baroclinic wave is studied for a two-layer *f*-plane system in which the basic flow is strongly sheared in the horizontal direction. The basic flow is purely baroclinic, i.e., equal and opposite in each layer. In addition, the basic flow vanishes on the channel walls containing the flow. Weakly nonlinear theory predicts that for small supercriticality, the basic wave eigenfunction has the same horizontal structure as the basic flow although it is vertically barotropic. Moreover, weakly nonlinear theory predicts growth of the wave amplitudes, which is unrestrained by wave–mean flow interaction. This prediction is verified by direct numerical calculation. The numerical calculations further reveal the manner by which the wave eventually equilibrates. The strongly growing wave cascades energy to higher zonal harmonics. These harmonics alter the meridional structure of the fundamental that allows wave–mean flow interaction to operate, leading finally to equilibration. If the cascade to higher zonal wavenumbers is artificially blocked by truncating the numerical model to a single zonal wavenumber, equilibration artificially requires the annihilation of the basic shear.

## Abstract

The nonlinear dynamics of a slightly unstable baroclinic wave is studied for a two-layer *f*-plane system in which the basic flow is strongly sheared in the horizontal direction. The basic flow is purely baroclinic, i.e., equal and opposite in each layer. In addition, the basic flow vanishes on the channel walls containing the flow. Weakly nonlinear theory predicts that for small supercriticality, the basic wave eigenfunction has the same horizontal structure as the basic flow although it is vertically barotropic. Moreover, weakly nonlinear theory predicts growth of the wave amplitudes, which is unrestrained by wave–mean flow interaction. This prediction is verified by direct numerical calculation. The numerical calculations further reveal the manner by which the wave eventually equilibrates. The strongly growing wave cascades energy to higher zonal harmonics. These harmonics alter the meridional structure of the fundamental that allows wave–mean flow interaction to operate, leading finally to equilibration. If the cascade to higher zonal wavenumbers is artificially blocked by truncating the numerical model to a single zonal wavenumber, equilibration artificially requires the annihilation of the basic shear.

## Abstract

A two-layer thermocline model is modified by adding an essentially passive mixed layer above it. The surface temperature variation is simulated by a moving outcrop line. It is found that, in contrast to a surface wind stress, a surface temperature variation causes strong variability in the ventilated zone through subducted water, while it affects the shadow zone little.

Two types of buoyancy-forced solution are found. When the outcrop line moves slowly, the solutions are nonentrainment solutions. For these solutions, the surface beat flux is mainly balanced by the horizontal advection in the permanent thermocline. The mixed layer never entrains. The time-mean thermocline is close to the steady thermocline with the time-mean outcrop line.

When the outcrop line moves southward rapidly during the cooling season, the solutions become entrainment solutions. Now, deep vertical convection must occur because the horizontal advection in the permanent thermocline is no longer strong enough to balance the surface cooling. The time-mean thermocline resembles the steady thermocline with the early spring mixed layer, as suggested by Stommel. The local variability in the permanent thermocline is most efficiently produced by decadal forcings.

## Abstract

A two-layer thermocline model is modified by adding an essentially passive mixed layer above it. The surface temperature variation is simulated by a moving outcrop line. It is found that, in contrast to a surface wind stress, a surface temperature variation causes strong variability in the ventilated zone through subducted water, while it affects the shadow zone little.

Two types of buoyancy-forced solution are found. When the outcrop line moves slowly, the solutions are nonentrainment solutions. For these solutions, the surface beat flux is mainly balanced by the horizontal advection in the permanent thermocline. The mixed layer never entrains. The time-mean thermocline is close to the steady thermocline with the time-mean outcrop line.

When the outcrop line moves southward rapidly during the cooling season, the solutions become entrainment solutions. Now, deep vertical convection must occur because the horizontal advection in the permanent thermocline is no longer strong enough to balance the surface cooling. The time-mean thermocline resembles the steady thermocline with the early spring mixed layer, as suggested by Stommel. The local variability in the permanent thermocline is most efficiently produced by decadal forcings.

## Abstract

A theory that describes the ventilated part of the ocean thermocline in the presence of a continuous density distribution is developed. The theory is based on the Sverdrup relation, on the conservation of the potential vorticity, and it assumes that the thermocline is fully ventilated in order to have a simplified dynamics. A finite density step is allowed between the bottom of the thermocline and the underlying quiescent abyss. If the outcrop lines have constant latitude, the potential vorticity and Montgomery function are proved to be inversely proportional. Their product is a function of the fluid density only, and it can be determined numerically from an arbitrary density distribution at the sea surface. The dependence of the coefficient of proportionality on the sea surface density distribution and on the parameter that controls both the nonlinearity and the baroclinicity of the solution is investigated and an analytical expression is proposed. The theory results in an integral–differential equation, which allows the derivation of the vertical stratification in the thermocline from the sea surface density distribution. The equation is solved numerically for a typical midlatitude ocean gyre. The solution shows the presence of a region of low vorticity fluid at the bottom of the thermocline as a consequence of a fully inviscid model physics. This theory is the generalization of the Lionello and Pedlosky many-layer model to an infinite number of layers of infinitesimal thickness. It is therefore shown that the layer model of the thermocline can be considered the discrete approximation of the continuous system.

## Abstract

A theory that describes the ventilated part of the ocean thermocline in the presence of a continuous density distribution is developed. The theory is based on the Sverdrup relation, on the conservation of the potential vorticity, and it assumes that the thermocline is fully ventilated in order to have a simplified dynamics. A finite density step is allowed between the bottom of the thermocline and the underlying quiescent abyss. If the outcrop lines have constant latitude, the potential vorticity and Montgomery function are proved to be inversely proportional. Their product is a function of the fluid density only, and it can be determined numerically from an arbitrary density distribution at the sea surface. The dependence of the coefficient of proportionality on the sea surface density distribution and on the parameter that controls both the nonlinearity and the baroclinicity of the solution is investigated and an analytical expression is proposed. The theory results in an integral–differential equation, which allows the derivation of the vertical stratification in the thermocline from the sea surface density distribution. The equation is solved numerically for a typical midlatitude ocean gyre. The solution shows the presence of a region of low vorticity fluid at the bottom of the thermocline as a consequence of a fully inviscid model physics. This theory is the generalization of the Lionello and Pedlosky many-layer model to an infinite number of layers of infinitesimal thickness. It is therefore shown that the layer model of the thermocline can be considered the discrete approximation of the continuous system.

## Abstract

The ocean thermocline is resolved in a very large number of layers by means of a recursive relation that extends the LPS model of the ventilated flow from a small to an arbitrary number of layers. In order to have simplified dynamics, the basin is semi-infinite in the zonal direction, the thermocline is fully ventilated, and its thickness vanishes at the northern boundary. In this model, the potential vorticity of each layer is shown to be inversely proportional to the Bernoulli function. The high vertical resolution adopted for the thermocline allows the study of the dependence of its motion on the ratio between the density contrast at the sea surface and the density step separating the thermocline bottom from the underlying quiescent abyss. This ratio controls both the nonlinearity and the baroclinicity of the solution. The behavior of the solution as this ratio varies from zero (linear and barotropic case) to infinity (“fully nonlinear” and baroclinic case) is described. The singularity that is found in the fully nonlinear case is discussed.

## Abstract

The ocean thermocline is resolved in a very large number of layers by means of a recursive relation that extends the LPS model of the ventilated flow from a small to an arbitrary number of layers. In order to have simplified dynamics, the basin is semi-infinite in the zonal direction, the thermocline is fully ventilated, and its thickness vanishes at the northern boundary. In this model, the potential vorticity of each layer is shown to be inversely proportional to the Bernoulli function. The high vertical resolution adopted for the thermocline allows the study of the dependence of its motion on the ratio between the density contrast at the sea surface and the density step separating the thermocline bottom from the underlying quiescent abyss. This ratio controls both the nonlinearity and the baroclinicity of the solution. The behavior of the solution as this ratio varies from zero (linear and barotropic case) to infinity (“fully nonlinear” and baroclinic case) is described. The singularity that is found in the fully nonlinear case is discussed.

## Abstract

Low-frequency, large-scale baroclinic Rossby basin modes, resistant to scale-dependent dissipation, have been recently theoretically analyzed and discussed as possible efficient coupling agents with the atmosphere for interactions on decadal time scales. Such modes are also consistent with evidence of the westward phase propagation in satellite altimetry data. In both the theory and the observations, the scale of the waves is large in comparison with the Rossby radius of deformation and the orientation of fluid motion in the waves is predominantly meridional. These two facts suggest that the waves are vulnerable to baroclinic instability on the scale of the deformation radius. The key dynamical parameter is the ratio *Z* of the transit time of the long Rossby wave to the *e*-folding time of the instability. When this parameter is small the wave easily crosses the basin largely undisturbed by the instability; if *Z* is large the wave succumbs to the instability and is largely destroyed before making a complete transit of the basin. For small *Z,* the instability is shown to be a triad instability; for large *Z* the instability is fundamentally similar to the Eady instability mechanism. For all *Z,* the growth rate is on the order of the vertical shear of the basic wave divided by the deformation radius. If the parametric dependence of *Z* on latitude is examined, the condition of unit *Z* separates latitudes south of which the Rossby wave may successfully cross the basin while north of which the wave will break down into small-scale eddies with a barotropic component. The boundary between the two corresponds to the domain boundary found in satellite measurements. Furthermore, the resulting barotropic wave field is shown to propagate at speeds about 2 times as large as the baroclinic speed, and this is offered as a consistent explanation of the observed discrepancy between the satellite observations of Chelton and Schlax and simple linear wave theory. Here it is suggested that Rossby basin modes, if they exist, would be limited to tropical domains and that a considerable part of the observed midlatitude eddy field north of that boundary is due to the instability of wind-forced, long Rossby waves.

## Abstract

Low-frequency, large-scale baroclinic Rossby basin modes, resistant to scale-dependent dissipation, have been recently theoretically analyzed and discussed as possible efficient coupling agents with the atmosphere for interactions on decadal time scales. Such modes are also consistent with evidence of the westward phase propagation in satellite altimetry data. In both the theory and the observations, the scale of the waves is large in comparison with the Rossby radius of deformation and the orientation of fluid motion in the waves is predominantly meridional. These two facts suggest that the waves are vulnerable to baroclinic instability on the scale of the deformation radius. The key dynamical parameter is the ratio *Z* of the transit time of the long Rossby wave to the *e*-folding time of the instability. When this parameter is small the wave easily crosses the basin largely undisturbed by the instability; if *Z* is large the wave succumbs to the instability and is largely destroyed before making a complete transit of the basin. For small *Z,* the instability is shown to be a triad instability; for large *Z* the instability is fundamentally similar to the Eady instability mechanism. For all *Z,* the growth rate is on the order of the vertical shear of the basic wave divided by the deformation radius. If the parametric dependence of *Z* on latitude is examined, the condition of unit *Z* separates latitudes south of which the Rossby wave may successfully cross the basin while north of which the wave will break down into small-scale eddies with a barotropic component. The boundary between the two corresponds to the domain boundary found in satellite measurements. Furthermore, the resulting barotropic wave field is shown to propagate at speeds about 2 times as large as the baroclinic speed, and this is offered as a consistent explanation of the observed discrepancy between the satellite observations of Chelton and Schlax and simple linear wave theory. Here it is suggested that Rossby basin modes, if they exist, would be limited to tropical domains and that a considerable part of the observed midlatitude eddy field north of that boundary is due to the instability of wind-forced, long Rossby waves.

## Abstract

The question of convective (i.e., spatial) instability of baroclinic waves on an *f*-plane is studied in the context of the two-layer model. The viscous and inviscid marginal curves for linear convective instability are obtained. The finite-amplitude problem shows that when dissipation is O(1) it acts to stabilize the waves that are of Eady type. For very small dissipation the weakly nonlinear analysis reveals that at low frequencies, contrary to what is known to occur in the temporal problem, in addition to the baroclinic component a barotropic correction to the “mean” flow is generated by the nonlinearities, and spatial equilibration occurs provided the ratio of shear to mean flow does not exceed some critical value. In the same limit, the slightly dissipative nonlinear dynamics reveals the presence of large spatial vacillations immediately downstream of the source, even if asymptotically (i.e., very far away from the source) the amplitudes are found to reach steady values. No case of period doubling or aperiodic behavior was found. The results obtained seem to be qualitatively independent of the form chosen to model the dissipation.

## Abstract

The question of convective (i.e., spatial) instability of baroclinic waves on an *f*-plane is studied in the context of the two-layer model. The viscous and inviscid marginal curves for linear convective instability are obtained. The finite-amplitude problem shows that when dissipation is O(1) it acts to stabilize the waves that are of Eady type. For very small dissipation the weakly nonlinear analysis reveals that at low frequencies, contrary to what is known to occur in the temporal problem, in addition to the baroclinic component a barotropic correction to the “mean” flow is generated by the nonlinearities, and spatial equilibration occurs provided the ratio of shear to mean flow does not exceed some critical value. In the same limit, the slightly dissipative nonlinear dynamics reveals the presence of large spatial vacillations immediately downstream of the source, even if asymptotically (i.e., very far away from the source) the amplitudes are found to reach steady values. No case of period doubling or aperiodic behavior was found. The results obtained seem to be qualitatively independent of the form chosen to model the dissipation.

## Abstract

A model for the vertical structure of the oceanic circulation is presented that combines elements of the theory of the ventilated thermocline, given by Luyten, Pedlosky and Stommel, with the theory of Rhines and Young for the wind driven circulation of an unventilated ocean.

Our model consists of a ventilated thermocline region above an unventilated zone in which motion is limited to pools of constant potential vorticity. The model is nonlinear and hence the presence of ventilation affects the dynamics of the unventilated motion and vice-versa.

The planetary geostrophic equations are used and so the quasi-geostrophic assumption of Rhines and Young is relaxed, allowing large isopycnal excursions.

It is shown that the presence of ventilation generally shrinks and weakens the size and vigor of the subsurface pools of homogenized potential vorticity. At the same time, within those domains, the strength of circulation in the ventilated zone is somewhat diminished as the subsurface layers carry a portion of the Sverdrup transport.

We argue that the (mathematically) consistent circulation in the absence of sub-thermocline constant potential-vorticity pools is unstable.

The non-uniqueness of the nondissipative Sverdrup dynamics is demonstrated by the ambiguity in the specification of potential vorticity in the deeper, unventilated layers. The study emphasizes the subtle importance of dissipation in selecting a unique solution.

## Abstract

A model for the vertical structure of the oceanic circulation is presented that combines elements of the theory of the ventilated thermocline, given by Luyten, Pedlosky and Stommel, with the theory of Rhines and Young for the wind driven circulation of an unventilated ocean.

Our model consists of a ventilated thermocline region above an unventilated zone in which motion is limited to pools of constant potential vorticity. The model is nonlinear and hence the presence of ventilation affects the dynamics of the unventilated motion and vice-versa.

The planetary geostrophic equations are used and so the quasi-geostrophic assumption of Rhines and Young is relaxed, allowing large isopycnal excursions.

It is shown that the presence of ventilation generally shrinks and weakens the size and vigor of the subsurface pools of homogenized potential vorticity. At the same time, within those domains, the strength of circulation in the ventilated zone is somewhat diminished as the subsurface layers carry a portion of the Sverdrup transport.

We argue that the (mathematically) consistent circulation in the absence of sub-thermocline constant potential-vorticity pools is unstable.

The non-uniqueness of the nondissipative Sverdrup dynamics is demonstrated by the ambiguity in the specification of potential vorticity in the deeper, unventilated layers. The study emphasizes the subtle importance of dissipation in selecting a unique solution.

## Abstract

A recently developed nonlinear inviscid model of the equatorial undercurrent is coupled to wind-driven surface layer. Wind stress drives a poleward Ekman flow, causing equatorial divergence of surface layer transport. This divergence is balanced by upwelling of fluid supplied by zonal convergence of the undercurrent. In this manner, the imposed wind stress controls the zonal structure of the undercurrent transport. The meridional structure of the undercurrent is determined from the undercurrent transport, the thermocline structure outside the undercurrent, and conservation of potential vorticity and Bernoulli function by the inviscid undercurrent. Solutions are presented for two zonal profiles of zonal wind stress. For westward wind stress increasing linearly westward, eastward transport increases nearly linearly westward. For westward wind stress with a midbasin maximum, eastward transport has a maximum just west of the basin middle, and there is recirculation along the equator. Solutions are also presented for uncoupled models with several layers and with a deep constant potential vorticity layer.

## Abstract

A recently developed nonlinear inviscid model of the equatorial undercurrent is coupled to wind-driven surface layer. Wind stress drives a poleward Ekman flow, causing equatorial divergence of surface layer transport. This divergence is balanced by upwelling of fluid supplied by zonal convergence of the undercurrent. In this manner, the imposed wind stress controls the zonal structure of the undercurrent transport. The meridional structure of the undercurrent is determined from the undercurrent transport, the thermocline structure outside the undercurrent, and conservation of potential vorticity and Bernoulli function by the inviscid undercurrent. Solutions are presented for two zonal profiles of zonal wind stress. For westward wind stress increasing linearly westward, eastward transport increases nearly linearly westward. For westward wind stress with a midbasin maximum, eastward transport has a maximum just west of the basin middle, and there is recirculation along the equator. Solutions are also presented for uncoupled models with several layers and with a deep constant potential vorticity layer.

## Abstract

The properties of baroclinic, quasigeostrophic Rossby basin waves are examined. Full analytical solutions are derived to elucidate the response in irregular basins, specifically in a (horizontally) tilted rectangular basin and in a circular one. When the basin is much larger than the (internal) deformation radius, the basin mode properties depend profoundly on whether one allows the streamfunction to oscillate at the boundary or not, as has been shown previously. With boundary oscillations, modes occur that have low frequencies and, with scale-selective dissipation, decay at a rate less than or equal to that of the imposed dissipation. These modes approximately satisfy the long-wave equation in the interior. Using both unforced and forced solutions, the variation of the response with basin geometry and dissipation is documented. The long-wave modes obtain with scale-selective dissipation, but also with damping that acts equally at all scales. One finds evidence of them as well in the forced response, even when the dissipation is weak and the corresponding free modes are apparently absent.

## Abstract

The properties of baroclinic, quasigeostrophic Rossby basin waves are examined. Full analytical solutions are derived to elucidate the response in irregular basins, specifically in a (horizontally) tilted rectangular basin and in a circular one. When the basin is much larger than the (internal) deformation radius, the basin mode properties depend profoundly on whether one allows the streamfunction to oscillate at the boundary or not, as has been shown previously. With boundary oscillations, modes occur that have low frequencies and, with scale-selective dissipation, decay at a rate less than or equal to that of the imposed dissipation. These modes approximately satisfy the long-wave equation in the interior. Using both unforced and forced solutions, the variation of the response with basin geometry and dissipation is documented. The long-wave modes obtain with scale-selective dissipation, but also with damping that acts equally at all scales. One finds evidence of them as well in the forced response, even when the dissipation is weak and the corresponding free modes are apparently absent.