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## Abstract

If the momentum, energy and circulation of a fluid in a periodic, quasi-geostrophic, *β*-plane channel are specified, then there is a minimum enstrophy implied. This minimum enstrophy flow is obtained using the calculus of variations and is found to be also a solution of the quasi-geostrophic equations. It is either a parallel flow or a finite-amplitude Rossby wave, depending on the aspect ratio of the channel and the amount of energy and momentum within it. The most geophysically relevant case is a channel whose zonal length is substantially greater than its meridional breadth. In this instance the form of the minimum enstrophy solution is decided by the ratio of the energy to the squared momentum. When this parameter is below a Critical value one has a parallel flow, while if this value is exceeded, the minimum enstrophy, solution is a Rossby wave.

Heuristic arguments based on the enstrophy cascade in two-dimensional turbulence suggest a “selective decay hypothesis”. This is that scale-selective dissipation will decrease the enstrophy more rapidly than the energy, momentum and circulation. If this is the case, then the system should approach the minimum enstrophy solution.

## Abstract

If the momentum, energy and circulation of a fluid in a periodic, quasi-geostrophic, *β*-plane channel are specified, then there is a minimum enstrophy implied. This minimum enstrophy flow is obtained using the calculus of variations and is found to be also a solution of the quasi-geostrophic equations. It is either a parallel flow or a finite-amplitude Rossby wave, depending on the aspect ratio of the channel and the amount of energy and momentum within it. The most geophysically relevant case is a channel whose zonal length is substantially greater than its meridional breadth. In this instance the form of the minimum enstrophy solution is decided by the ratio of the energy to the squared momentum. When this parameter is below a Critical value one has a parallel flow, while if this value is exceeded, the minimum enstrophy, solution is a Rossby wave.

Heuristic arguments based on the enstrophy cascade in two-dimensional turbulence suggest a “selective decay hypothesis”. This is that scale-selective dissipation will decrease the enstrophy more rapidly than the energy, momentum and circulation. If this is the case, then the system should approach the minimum enstrophy solution.

## Abstract

The rectification of oscillatory tidal currents on the sloping sides of a *low* submarine bank is discussed using the moment method. This method has been previously used in shear dispersion studies where it is used to analyze the advection-diffusion equation. In the present problem it is applied to the barotropic potential vorticity equation linearized about an oscillatory, spatially uniform tidal velocity. To apply the method it is necessary to assume that the topography produces only a small change in depth. The method economically provides the most important qualitative properties (*e.g.*, transport, location and width) of the time averaged current.

These results are obtained without making an harmonic truncation. They can then be used to assess the accuracy of the harmonic truncation approximation used by other authors. It is shown that harmonic truncation correctly predicts the transport and location of the rectified current when the bank is low. However if the width of the bank is much less than a tidal excursion distance, harmonic truncation may give a very mistaken impression of the width of the rectified current.

Finally, lateral vorticity diffusion is included in the moment calculation. It is shown that this dissipative process does not change the transport or location of the rectified current. It does however increase its width.

## Abstract

The rectification of oscillatory tidal currents on the sloping sides of a *low* submarine bank is discussed using the moment method. This method has been previously used in shear dispersion studies where it is used to analyze the advection-diffusion equation. In the present problem it is applied to the barotropic potential vorticity equation linearized about an oscillatory, spatially uniform tidal velocity. To apply the method it is necessary to assume that the topography produces only a small change in depth. The method economically provides the most important qualitative properties (*e.g.*, transport, location and width) of the time averaged current.

These results are obtained without making an harmonic truncation. They can then be used to assess the accuracy of the harmonic truncation approximation used by other authors. It is shown that harmonic truncation correctly predicts the transport and location of the rectified current when the bank is low. However if the width of the bank is much less than a tidal excursion distance, harmonic truncation may give a very mistaken impression of the width of the rectified current.

Finally, lateral vorticity diffusion is included in the moment calculation. It is shown that this dissipative process does not change the transport or location of the rectified current. It does however increase its width.

## Abstract

The density of the mixed layer is approximately uniform in the vertical but has dynamically important horizontal gradients. These nonuniformities in density result in a vertically sheared horizontal pressure gradient. Subinertial motions balance this pressure gradient with a vertically sheared velocity. Systematic incorporation of shear into a three-dimensional mixed layer model is both the goal of the present study and its majority novelty.

The sheared flow is partitioned between a geostrophic response and a frictional, ageostrophic response. The relative weighting of them two components is determined by a nondimensional parameters μ≡1/*f*τ_{
U
}, where τ_{
U
} is the timescale for vertical mixing of momentum and *f*
^{−1} is the inertial timescale.

If μ is of order unity, then the velocity has vertical shear at leading order. Differential advection by this shear flow will tilt over vertical isosurfaces of heat and salt so as to “unmix” or “restratify” the mixed layer. The unmixing process is balanced by intermittent mixing events, which drive the mixed layer back to a state of vertical homogeneity.

All of these processes are captured by a new set of reduced or filtered dynamics called the subinertial mixed layer (SML) approximation. The SML approximation is obtained by expanding the equations of motion in both Rossby number and a second small parameter that is the ratio of the vertical mixing timescale to the dynamic time scale. The subinertial dynamics of slab mixed layer models is captured as a special case of the SML approximation by taking the limit μ → ∞.

## Abstract

The density of the mixed layer is approximately uniform in the vertical but has dynamically important horizontal gradients. These nonuniformities in density result in a vertically sheared horizontal pressure gradient. Subinertial motions balance this pressure gradient with a vertically sheared velocity. Systematic incorporation of shear into a three-dimensional mixed layer model is both the goal of the present study and its majority novelty.

The sheared flow is partitioned between a geostrophic response and a frictional, ageostrophic response. The relative weighting of them two components is determined by a nondimensional parameters μ≡1/*f*τ_{
U
}, where τ_{
U
} is the timescale for vertical mixing of momentum and *f*
^{−1} is the inertial timescale.

If μ is of order unity, then the velocity has vertical shear at leading order. Differential advection by this shear flow will tilt over vertical isosurfaces of heat and salt so as to “unmix” or “restratify” the mixed layer. The unmixing process is balanced by intermittent mixing events, which drive the mixed layer back to a state of vertical homogeneity.

All of these processes are captured by a new set of reduced or filtered dynamics called the subinertial mixed layer (SML) approximation. The SML approximation is obtained by expanding the equations of motion in both Rossby number and a second small parameter that is the ratio of the vertical mixing timescale to the dynamic time scale. The subinertial dynamics of slab mixed layer models is captured as a special case of the SML approximation by taking the limit μ → ∞.

## Abstract

A variety of two-dimensional advection-diffusion models are investigated analytically with the goal of understanding the role of boundary layer in gyre-scale ocean mixing.

It is assumed throughout that the Péclet number of the flow in the Sverdrup interior, Pe = *UL*/*k* is large. (Here, *L*) is the length of the gyre, *U* is the velocity scale in the interior and *k* is the explicit diffusivity.) There are then two limits depending on the size of (*I*/*L*) where *l* is the width of the western boundary layer.

First, if (*I*/*L*)*P* is large, the diffusion is weak everywhere in the gyre and the tracer makes repeated passages through the boundary layer before mixing. In this case, the time taken to mix is (*Ll*/*k*). Second, if (*l*/*L*)*P* is small, the diffusion is strong in the boundary layer and the time taken to mix is the circulation time *L*/*U*. In any case, the mixing time is substantially less than the diffusion time based on the length scale of the gyre.

## Abstract

A variety of two-dimensional advection-diffusion models are investigated analytically with the goal of understanding the role of boundary layer in gyre-scale ocean mixing.

It is assumed throughout that the Péclet number of the flow in the Sverdrup interior, Pe = *UL*/*k* is large. (Here, *L*) is the length of the gyre, *U* is the velocity scale in the interior and *k* is the explicit diffusivity.) There are then two limits depending on the size of (*I*/*L*) where *l* is the width of the western boundary layer.

First, if (*I*/*L*)*P* is large, the diffusion is weak everywhere in the gyre and the tracer makes repeated passages through the boundary layer before mixing. In this case, the time taken to mix is (*Ll*/*k*). Second, if (*l*/*L*)*P* is small, the diffusion is strong in the boundary layer and the time taken to mix is the circulation time *L*/*U*. In any case, the mixing time is substantially less than the diffusion time based on the length scale of the gyre.

## Abstract

It is argued that the ideal fluid thermocline equations have “weak” (i.e., nondifferentiable) solutions that satisfy no mass-flux boundary conditions at the East. This conclusion is based on a local analysis of the eastern “corner” of a subtropical gyre. Specifically we suppose that the surface density is uniform while the density on the eastern boundary is either uniform (but different from that of the surface) or else is linearly stratified. The surface density is injected into the interior by specified Ekman pumping. In the absence f dissipation the resulting solution would have a discontinuity is “smoothed” and becomes an internal boundary layer which separates the light fluid originating at the surface from the denser fluid which abuts the eastern boundary.

This solution, which is of the similarity type, illustrates the applicability of solutions of the ideal fluid thermocline problem with discontinuities. It is these discontinuities which enable ideal fluid solutions to satisfy eastern boundary conditions. Thus, contrary to statements in the literature, there is no a priori need for an eastern boundary layer which exchanges mass with an ideal interior.

This concept of a weak solution is implicit in recent theories of the large-scale oceanic circulation. For example, in the continuously stratified, quasigeostrophic model developed by Rhines and Young, the solution is singular at the boundary between the moving pool of homogenized potential vorticity and the motionless shadow region. Analogous surfaces of discontinuity enable the models discussed in previous studies to satisfy eastern boundary conditions. The present study makes this assumption more explicit and shows how one particular dissipative mechanism (vertical density diffusion) heals the singularity.

## Abstract

It is argued that the ideal fluid thermocline equations have “weak” (i.e., nondifferentiable) solutions that satisfy no mass-flux boundary conditions at the East. This conclusion is based on a local analysis of the eastern “corner” of a subtropical gyre. Specifically we suppose that the surface density is uniform while the density on the eastern boundary is either uniform (but different from that of the surface) or else is linearly stratified. The surface density is injected into the interior by specified Ekman pumping. In the absence f dissipation the resulting solution would have a discontinuity is “smoothed” and becomes an internal boundary layer which separates the light fluid originating at the surface from the denser fluid which abuts the eastern boundary.

This solution, which is of the similarity type, illustrates the applicability of solutions of the ideal fluid thermocline problem with discontinuities. It is these discontinuities which enable ideal fluid solutions to satisfy eastern boundary conditions. Thus, contrary to statements in the literature, there is no a priori need for an eastern boundary layer which exchanges mass with an ideal interior.

This concept of a weak solution is implicit in recent theories of the large-scale oceanic circulation. For example, in the continuously stratified, quasigeostrophic model developed by Rhines and Young, the solution is singular at the boundary between the moving pool of homogenized potential vorticity and the motionless shadow region. Analogous surfaces of discontinuity enable the models discussed in previous studies to satisfy eastern boundary conditions. The present study makes this assumption more explicit and shows how one particular dissipative mechanism (vertical density diffusion) heals the singularity.

## Abstract

The question posed in the title of this paper is answered in the affirmative by investigating a two-layer, quasi-geostrophic model of the wind-driven circulation. The two layers model the thermocline rather than the whole depth of the ocean. The wind stress is balanced by interfacial and bottom drag. This is perhaps the simplest baroclinic extension of Stommel's (1948) barotropic circulation model. It differs from an earlier model of Welander (1966) in that the vortex stretching nonlinearity is of primary importance.

In this model the dynamics of the frictional western boundary layer determine the vertical structure of the wind-driven flow in the Sverdrup interior. Thus, in a sense, the boundary layer is “active” and cannot be appended to an arbitrary interior flow; rather it partially determines the interior circulation by setting the functional relationship between the streamfunction and the potential vorticity in the lower layer.

In previous studies (Rhines and Young 1982b) this functional relationship has been calculated using a generalized Prandtl-Batchelor theorem. This result does not apply to the present calculation because every lower layer streamline passes through a frictional boundary layer.

## Abstract

The question posed in the title of this paper is answered in the affirmative by investigating a two-layer, quasi-geostrophic model of the wind-driven circulation. The two layers model the thermocline rather than the whole depth of the ocean. The wind stress is balanced by interfacial and bottom drag. This is perhaps the simplest baroclinic extension of Stommel's (1948) barotropic circulation model. It differs from an earlier model of Welander (1966) in that the vortex stretching nonlinearity is of primary importance.

In this model the dynamics of the frictional western boundary layer determine the vertical structure of the wind-driven flow in the Sverdrup interior. Thus, in a sense, the boundary layer is “active” and cannot be appended to an arbitrary interior flow; rather it partially determines the interior circulation by setting the functional relationship between the streamfunction and the potential vorticity in the lower layer.

In previous studies (Rhines and Young 1982b) this functional relationship has been calculated using a generalized Prandtl-Batchelor theorem. This result does not apply to the present calculation because every lower layer streamline passes through a frictional boundary layer.

## Abstract

A sub-basin scale recirculation can be driven by imposing low values of potential vorticity in the northwest corner of a β-Plane box. Mesoscale eddies parametrized by lateral potential vorticity diffusion, carry this anomaly into the interior and establish the mean flow. While the structure of the flow is not sensitive to details of the boundary forcing or to the size of the diffusion coefficient, κ, the amplitude and length scale are. For instance, as, κ is reduced, the maximum transport scales as κ^{½} and the Reynolds number as κ^{−½}.

## Abstract

A sub-basin scale recirculation can be driven by imposing low values of potential vorticity in the northwest corner of a β-Plane box. Mesoscale eddies parametrized by lateral potential vorticity diffusion, carry this anomaly into the interior and establish the mean flow. While the structure of the flow is not sensitive to details of the boundary forcing or to the size of the diffusion coefficient, κ, the amplitude and length scale are. For instance, as, κ is reduced, the maximum transport scales as κ^{½} and the Reynolds number as κ^{−½}.

## Abstract

The stability of the western boundary layer is studied by idealizing it as a parallel flow and solving the Orr–Sommerfeld equation, generalized to include the gradient of planetary vorticity. The critical Reynolds number, at which the idealized flow first becomes unstable, is found to be between 20 and 100 depending on the details of the profile. The modes themselves are trapped within the boundary jet because their phase speeds exceed that of the fastest free Rossby wave with the same meridional wavenumber. However, in the important case of a jet with a broad exponential decay, corresponding to a highly inertial flow, we find that the phase speed of the critical mode exceeds that of a free Rossby wave by a very small amount. Consequently, the trapped mode has a very slowly decaying oscillatory tail and so is much wider than the basic state that supports it. The Reynolds stresses in the tail region induce a mean Eulerian flow opposite in direction to the basic state jet. However, Stokes drift is substantial so that the mean Lagrangian flow is an order of magnitude smaller than the mean Eulerian.

## Abstract

The stability of the western boundary layer is studied by idealizing it as a parallel flow and solving the Orr–Sommerfeld equation, generalized to include the gradient of planetary vorticity. The critical Reynolds number, at which the idealized flow first becomes unstable, is found to be between 20 and 100 depending on the details of the profile. The modes themselves are trapped within the boundary jet because their phase speeds exceed that of the fastest free Rossby wave with the same meridional wavenumber. However, in the important case of a jet with a broad exponential decay, corresponding to a highly inertial flow, we find that the phase speed of the critical mode exceeds that of a free Rossby wave by a very small amount. Consequently, the trapped mode has a very slowly decaying oscillatory tail and so is much wider than the basic state that supports it. The Reynolds stresses in the tail region induce a mean Eulerian flow opposite in direction to the basic state jet. However, Stokes drift is substantial so that the mean Lagrangian flow is an order of magnitude smaller than the mean Eulerian.

## Abstract

The Reynolds stress induced by anisotropically forcing an unbounded Couette flow, with uniform shear *γ*, on a *β* plane, is calculated in conjunction with the eddy diffusivity of a coevolving passive tracer. The flow is damped by linear drag on a time scale *μ*
^{−1}. The stochastic forcing is white noise in time and its spatial anisotropy is controlled by a parameter *α* that characterizes whether eddies are elongated along the zonal direction (*α* < 0), are elongated along the meridional direction (*α* > 0), or are isotropic (*α* = 0). The Reynolds stress varies linearly with *α* and nonlinearly and nonmonotonically with *γ*, but the Reynolds stress is independent of *β*. For positive values of *α*, the Reynolds stress displays an “antifrictional” effect (energy is transferred from the eddies to the mean flow); for negative values of *α*, it displays a frictional effect. When *γ*/*μ* ≪ 1, these transfers can be identified as negative and positive eddy viscosities, respectively. With *γ* = *β* = 0, the meridional tracer eddy diffusivity is *υ*′ is the meridional eddy velocity. In general, nonzero *β* and *γ* suppress the eddy diffusivity below *γ* varies as *γ*
^{−1} while the suppression due to *β* varies between *β*
^{−1} and *β*
^{−2} depending on whether the shear is strong or weak, respectively.

## Abstract

The Reynolds stress induced by anisotropically forcing an unbounded Couette flow, with uniform shear *γ*, on a *β* plane, is calculated in conjunction with the eddy diffusivity of a coevolving passive tracer. The flow is damped by linear drag on a time scale *μ*
^{−1}. The stochastic forcing is white noise in time and its spatial anisotropy is controlled by a parameter *α* that characterizes whether eddies are elongated along the zonal direction (*α* < 0), are elongated along the meridional direction (*α* > 0), or are isotropic (*α* = 0). The Reynolds stress varies linearly with *α* and nonlinearly and nonmonotonically with *γ*, but the Reynolds stress is independent of *β*. For positive values of *α*, the Reynolds stress displays an “antifrictional” effect (energy is transferred from the eddies to the mean flow); for negative values of *α*, it displays a frictional effect. When *γ*/*μ* ≪ 1, these transfers can be identified as negative and positive eddy viscosities, respectively. With *γ* = *β* = 0, the meridional tracer eddy diffusivity is *υ*′ is the meridional eddy velocity. In general, nonzero *β* and *γ* suppress the eddy diffusivity below *γ* varies as *γ*
^{−1} while the suppression due to *β* varies between *β*
^{−1} and *β*
^{−2} depending on whether the shear is strong or weak, respectively.

## Abstract

Zonostrophic instability leads to the spontaneous emergence of zonal jets on a *β* plane from a jetless basic-state flow that is damped by bottom drag and driven by a random body force. Decomposing the barotropic vorticity equation into the zonal mean and eddy equations, and neglecting the eddy–eddy interactions, defines the quasilinear (QL) system. Numerical solution of the QL system shows zonal jets with length scales comparable to jets obtained by solving the nonlinear (NL) system.

Starting with the QL system, one can construct a deterministic equation for the evolution of the two-point single-time correlation function of the vorticity, from which one can obtain the Reynolds stress that drives the zonal mean flow. This deterministic system has an exact nonlinear solution, which is an isotropic and homogenous eddy field with no jets. The authors characterize the linear stability of this jetless solution by calculating the critical stability curve in the parameter space and successfully comparing this analytic result with numerical solutions of the QL system. But the critical drag required for the onset of NL zonostrophic instability is sometimes a factor of 6 smaller than that for QL zonostrophic instability.

Near the critical stability curve, the jet scale predicted by linear stability theory agrees with that obtained via QL numerics. But on reducing the drag, the emerging QL jets agree with the linear stability prediction at only short times. Subsequently jets merge with their neighbors until the flow matures into a state with jets that are significantly broader than the linear prediction but have spacing similar to NL jets.

## Abstract

Zonostrophic instability leads to the spontaneous emergence of zonal jets on a *β* plane from a jetless basic-state flow that is damped by bottom drag and driven by a random body force. Decomposing the barotropic vorticity equation into the zonal mean and eddy equations, and neglecting the eddy–eddy interactions, defines the quasilinear (QL) system. Numerical solution of the QL system shows zonal jets with length scales comparable to jets obtained by solving the nonlinear (NL) system.

Starting with the QL system, one can construct a deterministic equation for the evolution of the two-point single-time correlation function of the vorticity, from which one can obtain the Reynolds stress that drives the zonal mean flow. This deterministic system has an exact nonlinear solution, which is an isotropic and homogenous eddy field with no jets. The authors characterize the linear stability of this jetless solution by calculating the critical stability curve in the parameter space and successfully comparing this analytic result with numerical solutions of the QL system. But the critical drag required for the onset of NL zonostrophic instability is sometimes a factor of 6 smaller than that for QL zonostrophic instability.

Near the critical stability curve, the jet scale predicted by linear stability theory agrees with that obtained via QL numerics. But on reducing the drag, the emerging QL jets agree with the linear stability prediction at only short times. Subsequently jets merge with their neighbors until the flow matures into a state with jets that are significantly broader than the linear prediction but have spacing similar to NL jets.