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## 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

A study is made of the thermocline and current structures of a subtropical/subpolar basin. Tis paper explores the shape of the interface, and the flow patterns when the lower layer is infinitely thick. Simple frictional parameterizations are used to obtain a full solution including the structure of the boundary layers. When the amount of water in the upper layer is less than (or the wind stress is larger than) a critical value, the lower layer outcrops near the middle of the western boundary of the subpolar gyre. A dynamically consistent picture includes a strong, isolated western boundary current (i.e., bounded on one side by the wall and on the other by a streamline along which the upper layer thickness vanishes) flowing southward and an internal boundary current (i.e., a current that flows in the interior of the ocean along the outcrop, separating the two layers) flowing northeastward across the zero-wind-curl line. This interior boundary current models the Gulf Stream and the North Atlantic Current. In addition, there will be a normal western boundary layer in the subtropical ocean. When the amount of upper water is less than a second critical value, the upper layer water separates from the eastern wall and becomes a warm water pool in the southwest corner.

Our model describes the thermocline structure for a two-gyre basin. The surface temperature is determined from the dynamical balance of the entire basin. The subtropical and subpolar gyres combine to a unified pattern, which is asymmetric with the zero-wind-curl line.

## Abstract

A study is made of the thermocline and current structures of a subtropical/subpolar basin. Tis paper explores the shape of the interface, and the flow patterns when the lower layer is infinitely thick. Simple frictional parameterizations are used to obtain a full solution including the structure of the boundary layers. When the amount of water in the upper layer is less than (or the wind stress is larger than) a critical value, the lower layer outcrops near the middle of the western boundary of the subpolar gyre. A dynamically consistent picture includes a strong, isolated western boundary current (i.e., bounded on one side by the wall and on the other by a streamline along which the upper layer thickness vanishes) flowing southward and an internal boundary current (i.e., a current that flows in the interior of the ocean along the outcrop, separating the two layers) flowing northeastward across the zero-wind-curl line. This interior boundary current models the Gulf Stream and the North Atlantic Current. In addition, there will be a normal western boundary layer in the subtropical ocean. When the amount of upper water is less than a second critical value, the upper layer water separates from the eastern wall and becomes a warm water pool in the southwest corner.

Our model describes the thermocline structure for a two-gyre basin. The surface temperature is determined from the dynamical balance of the entire basin. The subtropical and subpolar gyres combine to a unified pattern, which is asymmetric with the zero-wind-curl line.

## 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

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

In this article, the effect shelflike topography has on the stability of a jet that flows along the smooth shelf is addressed. The linear stability problem is solved to determine for which nondimensional parameters a shelf can either destabilize or stabilize a jet. These calculations reveal an intricate dependence of growth rate on topography. In particular, the authors determine that retrograde topography (with the shallow water on the left) always stabilizes the jet (in relation to the flat-bottom equivalent), whereas prograde topography (with the shallow water on the right) can either stabilize or destabilize the jet depending on the particular values of the Rossby number and topographic parameters. For Rossby numbers of order 1 and larger, prograde topography is strictly stabilizing. For small Rossby numbers, small-amplitude topography destabilizes whereas large topography stabilizes. The nonlinear evolution of these instabilities is explored to confirm the predictions from the linear theory and, also, to illustrate how stabilization is directly related to fluid transport across the shelf.

## Abstract

In this article, the effect shelflike topography has on the stability of a jet that flows along the smooth shelf is addressed. The linear stability problem is solved to determine for which nondimensional parameters a shelf can either destabilize or stabilize a jet. These calculations reveal an intricate dependence of growth rate on topography. In particular, the authors determine that retrograde topography (with the shallow water on the left) always stabilizes the jet (in relation to the flat-bottom equivalent), whereas prograde topography (with the shallow water on the right) can either stabilize or destabilize the jet depending on the particular values of the Rossby number and topographic parameters. For Rossby numbers of order 1 and larger, prograde topography is strictly stabilizing. For small Rossby numbers, small-amplitude topography destabilizes whereas large topography stabilizes. The nonlinear evolution of these instabilities is explored to confirm the predictions from the linear theory and, also, to illustrate how stabilization is directly related to fluid transport across the shelf.

## Abstract

The instability of jets in the context of the shallow-water model for both quasigeostrophic (QG) and non-QG parameters is studied. First, the linear stability problem is solved for a wide range of Rossby and Froude numbers to elucidate the functional dependency of growth rate on these two nondimensional parameters. Then the nonlinear evolution of the instability is investigated through the use of numerical experiments. The QG scenarios produce a vortex street with cyclones and anticyclones that are symmetric in size, strength, and shape, as predicted by QG. In the non-QG regime the instability yields cyclones and anticyclones that can be asymmetric in all three properties. The authors comment on some interesting connections this work has with the Denmark Strait overflow.

## Abstract

The instability of jets in the context of the shallow-water model for both quasigeostrophic (QG) and non-QG parameters is studied. First, the linear stability problem is solved for a wide range of Rossby and Froude numbers to elucidate the functional dependency of growth rate on these two nondimensional parameters. Then the nonlinear evolution of the instability is investigated through the use of numerical experiments. The QG scenarios produce a vortex street with cyclones and anticyclones that are symmetric in size, strength, and shape, as predicted by QG. In the non-QG regime the instability yields cyclones and anticyclones that can be asymmetric in all three properties. The authors comment on some interesting connections this work has with the Denmark Strait overflow.

## Abstract

We examine the nonlinear evolution of barotropic β-plane jets on a periodic domain with a pseudospectral. A calculation of the linear growth rate yields an infected U-shaped curve on the β versus *k*
_{0} plane which separates regions of stability and instability. This curve aids in clarifying the morphology of the nonlinear structures which evolve from monochromatic small-amplitude perturbations of wavenumber *k*
_{0}. At very small or zero β, we recover and further quantify previously obtained results, including formation of: dipolar vortex structures or bound pools of opposite-signed vortex regions at small *k*
_{0}; staggered streets isolated vortex pools at intermediate *k*
_{0}; and “cat-eyes” or staggered connected pools of vorticity at large but still unstable *k*
_{0}.

As β is increased, the jet exhibits quite different evolutionary patterns. At low *k*
_{0}, where the laminar jet may be stable, we find a multistage instability. First, neutrally stable long-wavelength modes of small amplitude interact nonlinearly to produce harmonics in the linear unstable band. These grow at an exponential rate until a near-steady wake appears. However, the wake is unstable to the initial long wavelength modes and a rapid merger (i.e., backward energy cascade) occurs.

At an intermediate *k*
_{0}, the presence of β causes a “reversal” of vortex pools in the meridional direction of the near-steady vortex street. That is for a west-to-east flowing jet the Positive pools of vorticity are south of the negative pools causing a decrease in the near-steady velocity of the jet. Retrograde Rossby radiation is observed and weak “shingles” or cast-away vortex pools are observed. The meander amplitude pulsates, pumping Rossby radiation into the far field. The merger and binding processes also occur on a jet excited by many harmonics, with an ensuing chaos that is very sensitive to initial conditions.

## Abstract

We examine the nonlinear evolution of barotropic β-plane jets on a periodic domain with a pseudospectral. A calculation of the linear growth rate yields an infected U-shaped curve on the β versus *k*
_{0} plane which separates regions of stability and instability. This curve aids in clarifying the morphology of the nonlinear structures which evolve from monochromatic small-amplitude perturbations of wavenumber *k*
_{0}. At very small or zero β, we recover and further quantify previously obtained results, including formation of: dipolar vortex structures or bound pools of opposite-signed vortex regions at small *k*
_{0}; staggered streets isolated vortex pools at intermediate *k*
_{0}; and “cat-eyes” or staggered connected pools of vorticity at large but still unstable *k*
_{0}.

As β is increased, the jet exhibits quite different evolutionary patterns. At low *k*
_{0}, where the laminar jet may be stable, we find a multistage instability. First, neutrally stable long-wavelength modes of small amplitude interact nonlinearly to produce harmonics in the linear unstable band. These grow at an exponential rate until a near-steady wake appears. However, the wake is unstable to the initial long wavelength modes and a rapid merger (i.e., backward energy cascade) occurs.

At an intermediate *k*
_{0}, the presence of β causes a “reversal” of vortex pools in the meridional direction of the near-steady vortex street. That is for a west-to-east flowing jet the Positive pools of vorticity are south of the negative pools causing a decrease in the near-steady velocity of the jet. Retrograde Rossby radiation is observed and weak “shingles” or cast-away vortex pools are observed. The meander amplitude pulsates, pumping Rossby radiation into the far field. The merger and binding processes also occur on a jet excited by many harmonics, with an ensuing chaos that is very sensitive to initial conditions.

## Abstract

Motivated by the ubiquity of time series in oceanic data, the relative lack of studies of geostrophic turbulence in the frequency domain, and the interest in quantifying the contributions of intrinsic nonlinearities to oceanic frequency spectra, this paper examines the spectra and spectral fluxes of surface oceanic geostrophic flows in the frequency domain. Spectra and spectral fluxes are computed from idealized two-layer quasigeostrophic (QG) turbulence models and realistic ocean general circulation models, as well as from gridded satellite altimeter data. The frequency spectra of the variance of streamfunction (akin to sea surface height) and of geostrophic velocity are qualitatively similar in all of these, with substantial variance extending out to low frequencies. The spectral flux Π(*ω*) of kinetic energy in the frequency *ω* domain for the QG model documents a tendency for nonlinearity to drive energy toward longer periods, in like manner to the inverse cascade toward larger length scales documented in calculations of the spectral flux Π(*k*) in the wavenumber *k* domain. Computations of Π(*ω*) in the realistic model also display an “inverse temporal cascade.” In satellite altimeter data, some regions are dominated by an inverse temporal cascade, whereas others exhibit a forward temporal cascade. However, calculations performed with temporally and/or spatially filtered output from the models demonstrate that Π(*ω*) values are highly susceptible to the smoothing inherent in the construction of gridded altimeter products. Therefore, at present it is difficult to say whether the forward temporal cascades seen in some regions in altimeter data represent physics that is missing in the models studied here or merely sampling artifacts.

## Abstract

Motivated by the ubiquity of time series in oceanic data, the relative lack of studies of geostrophic turbulence in the frequency domain, and the interest in quantifying the contributions of intrinsic nonlinearities to oceanic frequency spectra, this paper examines the spectra and spectral fluxes of surface oceanic geostrophic flows in the frequency domain. Spectra and spectral fluxes are computed from idealized two-layer quasigeostrophic (QG) turbulence models and realistic ocean general circulation models, as well as from gridded satellite altimeter data. The frequency spectra of the variance of streamfunction (akin to sea surface height) and of geostrophic velocity are qualitatively similar in all of these, with substantial variance extending out to low frequencies. The spectral flux Π(*ω*) of kinetic energy in the frequency *ω* domain for the QG model documents a tendency for nonlinearity to drive energy toward longer periods, in like manner to the inverse cascade toward larger length scales documented in calculations of the spectral flux Π(*k*) in the wavenumber *k* domain. Computations of Π(*ω*) in the realistic model also display an “inverse temporal cascade.” In satellite altimeter data, some regions are dominated by an inverse temporal cascade, whereas others exhibit a forward temporal cascade. However, calculations performed with temporally and/or spatially filtered output from the models demonstrate that Π(*ω*) values are highly susceptible to the smoothing inherent in the construction of gridded altimeter products. Therefore, at present it is difficult to say whether the forward temporal cascades seen in some regions in altimeter data represent physics that is missing in the models studied here or merely sampling artifacts.