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## Abstract

This contribution seeks to understand the vertical structure of linearized quasigeostrophic baroclinic modes when they are modified by the presence of a baroclinic mean flow and associated potential vorticity gradients. It is found that even modest, *O*(0.05 m s^{−1}), mean flows can give rise to very substantial changes in modal structures, often in the sense of increased surface intensification. The extent to which stable modes are modified depends strongly on the direction of Rossby wave propagation. Further, baroclinically unstable solutions can appear, and a meaningful inviscid critical-layer solution can occur at the transition to instability when the horizontal gradient of potential vorticity changes sign at some depth within the water column. In addition, the gravest, *n* = 0, vertical stable mode is no longer strictly barotropic, but rather it can carry density variability at frequencies much higher than those possible for baroclinic (higher) Rossby wave modes. This finding appears to be consistent with oceanic current-meter observations that suggest temperature variability propagation even when the frequency is too high for traditional baroclinic Rossby waves to exist.

## Abstract

This contribution seeks to understand the vertical structure of linearized quasigeostrophic baroclinic modes when they are modified by the presence of a baroclinic mean flow and associated potential vorticity gradients. It is found that even modest, *O*(0.05 m s^{−1}), mean flows can give rise to very substantial changes in modal structures, often in the sense of increased surface intensification. The extent to which stable modes are modified depends strongly on the direction of Rossby wave propagation. Further, baroclinically unstable solutions can appear, and a meaningful inviscid critical-layer solution can occur at the transition to instability when the horizontal gradient of potential vorticity changes sign at some depth within the water column. In addition, the gravest, *n* = 0, vertical stable mode is no longer strictly barotropic, but rather it can carry density variability at frequencies much higher than those possible for baroclinic (higher) Rossby wave modes. This finding appears to be consistent with oceanic current-meter observations that suggest temperature variability propagation even when the frequency is too high for traditional baroclinic Rossby waves to exist.

## Abstract

Published observations of subinertial ocean current variability show that the vertical structure is often well described by a vertical mode that has a node of horizontal velocity at the bottom rather than the traditional node of vertical velocity. The theory of forced and free linear Rossby waves in a continuously stratified ocean with a sloping bottom and bottom friction is treated here to see if frictional effects can plausibly contribute to this phenomenon. For parameter values representative of the mesoscale, bottom dissipation by itself appears to be too weak to be an explanation, although caution is required because the present approach uses a linear model to address a nonlinear phenomenon. One novel outcome is the emergence of a short-wave, bottom-trapped, strongly damped mode that is present even with a flat bottom.

## Abstract

Published observations of subinertial ocean current variability show that the vertical structure is often well described by a vertical mode that has a node of horizontal velocity at the bottom rather than the traditional node of vertical velocity. The theory of forced and free linear Rossby waves in a continuously stratified ocean with a sloping bottom and bottom friction is treated here to see if frictional effects can plausibly contribute to this phenomenon. For parameter values representative of the mesoscale, bottom dissipation by itself appears to be too weak to be an explanation, although caution is required because the present approach uses a linear model to address a nonlinear phenomenon. One novel outcome is the emergence of a short-wave, bottom-trapped, strongly damped mode that is present even with a flat bottom.

## Abstract

The nonlinear dynamics of baroclinically unstable waves in a time-dependent zonal shear flow is considered in the framework of the two-layer Phillips model on the beta plane. In most cases considered in this study the amplitude of the shear is well below the critical value of the steady shear version of the model. Nevertheless, the time-dependent problem in which the shear oscillates periodically is unstable, and the unstable waves grow to substantial amplitudes, in some cases with strongly nonlinear and turbulent characteristics. For very small values of the shear amplitude in the presence of dissipation an analytical, asymptotic theory predicts a self-sustained wave whose amplitude undergoes a nonlinear oscillation whose period is amplitude dependent. There is a sensitive amplitude dependence of the wave on the frequency of the oscillating shear when the shear amplitude is small. This behavior is also found in a truncated model of the dynamics, and that model is used to examine larger shear amplitudes. When there is a mean value of the shear in addition to the oscillating component, but such that the total shear is still subcritical, the resulting nonlinear states exhibit a rectified horizontal buoyancy flux with a nonzero time average as a result of the instability of the oscillating shear. For higher, still subcritical, values of the shear, a symmetry breaking is detected in which a second cross-stream mode is generated through an instability of the unstable wave although this second mode would by itself be stable on the basic time-dependent current. For shear values that are substantially subcritical but of order of the critical shear, calculations with a full quasigeostrophic numerical model reveal a turbulent flow generated by the instability. If the beta effect is disregarded, the inviscid, linear problem is formally stable. However, calculations show that a small degree of nonlinearity is enough to destabilize the flow, leading to large amplitude vacillations and turbulence. When the most unstable wave is not the longest wave in the system, a cascade up scale to longer waves is observed. Indeed, this classically subcritical flow shows most of the qualitative character of a strongly supercritical flow. This result supports previous suggestions of the important role of background time dependence in maintaining the atmospheric and oceanic synoptic eddy field.

## Abstract

The nonlinear dynamics of baroclinically unstable waves in a time-dependent zonal shear flow is considered in the framework of the two-layer Phillips model on the beta plane. In most cases considered in this study the amplitude of the shear is well below the critical value of the steady shear version of the model. Nevertheless, the time-dependent problem in which the shear oscillates periodically is unstable, and the unstable waves grow to substantial amplitudes, in some cases with strongly nonlinear and turbulent characteristics. For very small values of the shear amplitude in the presence of dissipation an analytical, asymptotic theory predicts a self-sustained wave whose amplitude undergoes a nonlinear oscillation whose period is amplitude dependent. There is a sensitive amplitude dependence of the wave on the frequency of the oscillating shear when the shear amplitude is small. This behavior is also found in a truncated model of the dynamics, and that model is used to examine larger shear amplitudes. When there is a mean value of the shear in addition to the oscillating component, but such that the total shear is still subcritical, the resulting nonlinear states exhibit a rectified horizontal buoyancy flux with a nonzero time average as a result of the instability of the oscillating shear. For higher, still subcritical, values of the shear, a symmetry breaking is detected in which a second cross-stream mode is generated through an instability of the unstable wave although this second mode would by itself be stable on the basic time-dependent current. For shear values that are substantially subcritical but of order of the critical shear, calculations with a full quasigeostrophic numerical model reveal a turbulent flow generated by the instability. If the beta effect is disregarded, the inviscid, linear problem is formally stable. However, calculations show that a small degree of nonlinearity is enough to destabilize the flow, leading to large amplitude vacillations and turbulence. When the most unstable wave is not the longest wave in the system, a cascade up scale to longer waves is observed. Indeed, this classically subcritical flow shows most of the qualitative character of a strongly supercritical flow. This result supports previous suggestions of the important role of background time dependence in maintaining the atmospheric and oceanic synoptic eddy field.

## Abstract

Multiple alternating zonal jets observed in the ocean are studied with an idealized quasigeostrophic zonal-channel model, with the supercritical, zonal background flow imposed. Both eastward and westward background flows with vertical shear are considered. The underlying nonlinear dynamics is illuminated with analysis of the vertical-mode interactions and time-mean eddy fluxes.

Interactions between the vertical modes are systematically studied. The barotropic component of the jets is maintained by both barotropic–barotropic and baroclinic–baroclinic time-mean interactions; thus, the barotropic component of the jets cannot be accurately simulated with a randomly forced barotropic model. The roles of the vertical-mode interactions in driving the baroclinic component of the jets are also characterized. Not only the first but also the second baroclinic mode is found to be important for maintaining the baroclinic component of the jets, whereas the barotropic component of the jets is maintained mostly by the barotropic and first baroclinic modes.

The properties of the eddy forcing were systematically studied. It is shown that the baroclinic component of the jets is maintained by Reynolds stress forcing and resisted by form stress forcing only in the eastward background flow. In the westward background flow, the jets are maintained by form stress forcing and resisted by Reynolds stress forcing.

The meridional scaling and kinematical properties of the jets are studied as well as the roles of meridional boundaries. The Rhines scaling for meridional spacing of the jets is not generally confirmed, and it is also shown that there are multiple stable equilibria with different numbers of the time-mean jets. It is also found that the jets are associated with alternating weak barriers to the meridional material transport, but the locations of these barriers are not unique and depend on the direction of the background flow and depth. Finally, if the channel is closed with meridional walls, then the jets become more latent but the eddy forcing properties do not change qualitatively.

## Abstract

Multiple alternating zonal jets observed in the ocean are studied with an idealized quasigeostrophic zonal-channel model, with the supercritical, zonal background flow imposed. Both eastward and westward background flows with vertical shear are considered. The underlying nonlinear dynamics is illuminated with analysis of the vertical-mode interactions and time-mean eddy fluxes.

Interactions between the vertical modes are systematically studied. The barotropic component of the jets is maintained by both barotropic–barotropic and baroclinic–baroclinic time-mean interactions; thus, the barotropic component of the jets cannot be accurately simulated with a randomly forced barotropic model. The roles of the vertical-mode interactions in driving the baroclinic component of the jets are also characterized. Not only the first but also the second baroclinic mode is found to be important for maintaining the baroclinic component of the jets, whereas the barotropic component of the jets is maintained mostly by the barotropic and first baroclinic modes.

The properties of the eddy forcing were systematically studied. It is shown that the baroclinic component of the jets is maintained by Reynolds stress forcing and resisted by form stress forcing only in the eastward background flow. In the westward background flow, the jets are maintained by form stress forcing and resisted by Reynolds stress forcing.

The meridional scaling and kinematical properties of the jets are studied as well as the roles of meridional boundaries. The Rhines scaling for meridional spacing of the jets is not generally confirmed, and it is also shown that there are multiple stable equilibria with different numbers of the time-mean jets. It is also found that the jets are associated with alternating weak barriers to the meridional material transport, but the locations of these barriers are not unique and depend on the direction of the background flow and depth. Finally, if the channel is closed with meridional walls, then the jets become more latent but the eddy forcing properties do not change qualitatively.

## Abstract

A simple theoretical model for the oceanic thermocline and the associated field of current is presented. The model consists of a finite but arbitarily large number of inviscid, homogeneous fluid layers each with a different density. The dynamical balances everywhere are Sverdrupian. IN regions where the Ekman pumping is negative (downward) the surface density is specified, i.e., the position of the outcrop of density interfaces is specified. This outcropping of density layers allows deep motion to be excited by the ventilation provided by Ekman pumping even in latitudes far south of the outcrop where the layer is shielded from direct influence of the wind. Analytical solutions are presented in the case where the density-outcrop lines are coincident with latitude circles. The solutions are not self-similar and important sub-domains of the solution are defined by critical potential vorticity trajectories which separate the ventilated from the unventilated regions in the lower thermocline. These critical trajectories also separate regions of strong variations in potential vorticity from regions of fairly weak variation in potential vorticity although the small variations in potential vorticity in the latter are crucial to the dynamics.

Comparison is made between the predictions of the model and data from the Atlantic with encouraging results.

## Abstract

A simple theoretical model for the oceanic thermocline and the associated field of current is presented. The model consists of a finite but arbitarily large number of inviscid, homogeneous fluid layers each with a different density. The dynamical balances everywhere are Sverdrupian. IN regions where the Ekman pumping is negative (downward) the surface density is specified, i.e., the position of the outcrop of density interfaces is specified. This outcropping of density layers allows deep motion to be excited by the ventilation provided by Ekman pumping even in latitudes far south of the outcrop where the layer is shielded from direct influence of the wind. Analytical solutions are presented in the case where the density-outcrop lines are coincident with latitude circles. The solutions are not self-similar and important sub-domains of the solution are defined by critical potential vorticity trajectories which separate the ventilated from the unventilated regions in the lower thermocline. These critical trajectories also separate regions of strong variations in potential vorticity from regions of fairly weak variation in potential vorticity although the small variations in potential vorticity in the latter are crucial to the dynamics.

Comparison is made between the predictions of the model and data from the Atlantic with encouraging results.

## Abstract

A simple model of the oceanic mixed layer is coupled to a model of the ventilated thermocline. The model allows a combination of advection and surface heating to determine the position of the outcrop lines of the isopycnals. The resulting isopycnal outcrops determine the circulation in the ventilated thermocline as in the 1983 study by Luyten, Pedlosky and Stommel (LPS). The isopycnal outcrop line is affected by both Ekman wind drift and the surface geostrophic flow. Hence, the outcrop position and the thermocline circulation am coupled.

The mixed layer and the thermocline models are extremely simple. Each is modeled by layers of constant density. The mixed layer, in which the isopycnals are vertical, is distinguished by the ability of fluid to cross the interfaces between adjacent layers under the influence of atmospheric heating. The heating is parameterized in terms of the departure of the isopycnal line from the position it would have if the ocean were heated, but at rest.

Although in most major respects the thermocline circulation is qualitatively similar to the model of LPS, the effect of the variation of the outcrop latitude with longitude introduces the possibility of potential-vorticity minima along latitude circles.

The model also predicts cooling of the most southern portion of the subtropical gyre under the influence of northward Ekman wind drift.

## Abstract

A simple model of the oceanic mixed layer is coupled to a model of the ventilated thermocline. The model allows a combination of advection and surface heating to determine the position of the outcrop lines of the isopycnals. The resulting isopycnal outcrops determine the circulation in the ventilated thermocline as in the 1983 study by Luyten, Pedlosky and Stommel (LPS). The isopycnal outcrop line is affected by both Ekman wind drift and the surface geostrophic flow. Hence, the outcrop position and the thermocline circulation am coupled.

The mixed layer and the thermocline models are extremely simple. Each is modeled by layers of constant density. The mixed layer, in which the isopycnals are vertical, is distinguished by the ability of fluid to cross the interfaces between adjacent layers under the influence of atmospheric heating. The heating is parameterized in terms of the departure of the isopycnal line from the position it would have if the ocean were heated, but at rest.

Although in most major respects the thermocline circulation is qualitatively similar to the model of LPS, the effect of the variation of the outcrop latitude with longitude introduces the possibility of potential-vorticity minima along latitude circles.

The model also predicts cooling of the most southern portion of the subtropical gyre under the influence of northward Ekman wind drift.

## Abstract

The stability of baroclinic Rossby waves in large ocean basins is examined, and the quasigeostrophic (QG) results of LaCasce and Pedlosky are generalized. First, stability equations are derived for perturbations on large-scale waves, using the two-layer shallow-water system. These equations resemble the QG stability equations, except that they retain the variation of the internal deformation radius with latitude. The equations are solved numerically for different initial conditions through eigenmode calculations and time stepping. The fastest-growing eigenmodes are intensified at high latitudes, and the slower-growing modes are intensified at lower latitudes. All of the modes have meridional scales and growth times that are comparable to the deformation radius in the latitude range where the eigenmode is intensified. This is what one would expect if one had applied QG theory in latitude bands. The evolution of large-scale waves was then simulated using the Regional Ocean Modeling System primitive equation model. The results are consistent with the theoretical predictions, with deformation-scale perturbations growing at rates inversely proportional to the local deformation radius. The waves succumb to the perturbations at the mid- to high latitudes, but are able to cross the basin at low latitudes before doing so. Also, the barotropic waves produced by the instability propagate faster than the baroclinic long-wave speed, which may explain the discrepancy in speeds noted by Chelton and Schlax.

## Abstract

The stability of baroclinic Rossby waves in large ocean basins is examined, and the quasigeostrophic (QG) results of LaCasce and Pedlosky are generalized. First, stability equations are derived for perturbations on large-scale waves, using the two-layer shallow-water system. These equations resemble the QG stability equations, except that they retain the variation of the internal deformation radius with latitude. The equations are solved numerically for different initial conditions through eigenmode calculations and time stepping. The fastest-growing eigenmodes are intensified at high latitudes, and the slower-growing modes are intensified at lower latitudes. All of the modes have meridional scales and growth times that are comparable to the deformation radius in the latitude range where the eigenmode is intensified. This is what one would expect if one had applied QG theory in latitude bands. The evolution of large-scale waves was then simulated using the Regional Ocean Modeling System primitive equation model. The results are consistent with the theoretical predictions, with deformation-scale perturbations growing at rates inversely proportional to the local deformation radius. The waves succumb to the perturbations at the mid- to high latitudes, but are able to cross the basin at low latitudes before doing so. Also, the barotropic waves produced by the instability propagate faster than the baroclinic long-wave speed, which may explain the discrepancy in speeds noted by Chelton and Schlax.

## Abstract

Motivated by the fact that time-dependent currents are ubiquitous in the ocean, this work studies the two-layer Phillips model on the beta plane with baroclinic shear flows that are steady, periodic, or aperiodic in time to understand their nonlinear evolution better. When a linearly unstable basic state is slightly perturbed, the primary wave grows exponentially until nonlinear advection adjusts the growth. Even though for long time scales these nearly two-dimensional motions predominantly cascade energy to large scales, for relatively short times the wave–mean flow and wave–wave interactions cascade energy to smaller horizontal length scales. The authors demonstrate that the manner through which these mechanisms excite the harmonics depends significantly on the characteristics of the basic state. Time-dependent basic states can excite harmonics very rapidly in comparison to steady basic states. Moreover, in all the simulations of aperiodic baroclinic shear flows, the barotropic component of the primary wave continues to grow after the adjustment by the nonlinearities. Furthermore, the authors find that the correction to the zonal mean flow can be much larger when the basic state is aperiodic compared to the periodic or steady limits. Finally, even though time-dependent baroclinic shear on an *f* plane is linearly stable, the authors show that perturbations can grow algebraically in the linear regime because of the erratic variations in the aperiodic flow. Subsequently, baroclinicity adjusts the growing wave and creates a final state that is more energetic than the nonlinear adjustment of any of the unstable steady baroclinic shears that are considered.

## Abstract

Motivated by the fact that time-dependent currents are ubiquitous in the ocean, this work studies the two-layer Phillips model on the beta plane with baroclinic shear flows that are steady, periodic, or aperiodic in time to understand their nonlinear evolution better. When a linearly unstable basic state is slightly perturbed, the primary wave grows exponentially until nonlinear advection adjusts the growth. Even though for long time scales these nearly two-dimensional motions predominantly cascade energy to large scales, for relatively short times the wave–mean flow and wave–wave interactions cascade energy to smaller horizontal length scales. The authors demonstrate that the manner through which these mechanisms excite the harmonics depends significantly on the characteristics of the basic state. Time-dependent basic states can excite harmonics very rapidly in comparison to steady basic states. Moreover, in all the simulations of aperiodic baroclinic shear flows, the barotropic component of the primary wave continues to grow after the adjustment by the nonlinearities. Furthermore, the authors find that the correction to the zonal mean flow can be much larger when the basic state is aperiodic compared to the periodic or steady limits. Finally, even though time-dependent baroclinic shear on an *f* plane is linearly stable, the authors show that perturbations can grow algebraically in the linear regime because of the erratic variations in the aperiodic flow. Subsequently, baroclinicity adjusts the growing wave and creates a final state that is more energetic than the nonlinear adjustment of any of the unstable steady baroclinic shears that are considered.