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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

Density-compensated temperature and salinity gradients are often observed in mixed layer fronts. A possible explanation Of this Observation is that there is a systematic relation between the “strength” of a front, defined as the buoyancy jump across the front, and the thickness of a front. If stronger fronts tend to be thicker, then in an ensemble of random fronts, in which the temperature and salinity jumps am independent random variables, the temperature and salinity gradients will he correlated. This correlation between the thermohaline gradients is such that heat and salt make antagonistic contributions to the buoyancy gradient–-that is, them is buoyancy compensation. The statistics of heat and salt fluxes across nearly compensated fronts are counterintuitive: strong heat fluxes can occur across a front with weak thermal gradients and strong salinity gradients, and vice versa.

As a specific model that relates the width of a front to the strength of a front, a pair of coupled nonlinear diffusion equations for heat and salt are used. The nonlinear diffusion coefficient, proportional to the square of the buoyancy gradient, arises from quasi-steady shear dispersion driven by thermohaline gradients. This nonlinear mixing prevents stirring by mesoscale advection from indefinitely filamenting mixed layer tracer distributions. The model predicts that the thickness of a front varies as the square root of the strength and inversely as the one-quarter power of the mesoscale strain.

## Abstract

Density-compensated temperature and salinity gradients are often observed in mixed layer fronts. A possible explanation Of this Observation is that there is a systematic relation between the “strength” of a front, defined as the buoyancy jump across the front, and the thickness of a front. If stronger fronts tend to be thicker, then in an ensemble of random fronts, in which the temperature and salinity jumps am independent random variables, the temperature and salinity gradients will he correlated. This correlation between the thermohaline gradients is such that heat and salt make antagonistic contributions to the buoyancy gradient–-that is, them is buoyancy compensation. The statistics of heat and salt fluxes across nearly compensated fronts are counterintuitive: strong heat fluxes can occur across a front with weak thermal gradients and strong salinity gradients, and vice versa.

As a specific model that relates the width of a front to the strength of a front, a pair of coupled nonlinear diffusion equations for heat and salt are used. The nonlinear diffusion coefficient, proportional to the square of the buoyancy gradient, arises from quasi-steady shear dispersion driven by thermohaline gradients. This nonlinear mixing prevents stirring by mesoscale advection from indefinitely filamenting mixed layer tracer distributions. The model predicts that the thickness of a front varies as the square root of the strength and inversely as the one-quarter power of the mesoscale strain.

## Abstract

The density of the mixed layer (ML) is approximately uniform in the vertical, but there are dynamically important horizontal gradients. The subinertial mixed layer (SML) approximation is a small Rossby number filtering of the primitive equation that isolates the low frequency (ω ≪ *f*) dynamics.

A linear stability analysis based on the SML approximation shows that the horizontal density gradients within the mixed layer (ML) support baroclinically unstable waves with inverse wavenumbers in the range 1 to 10 km. This conclusion follows from both a slab ML model, in which the horizontal velocity has no vertical shear, and a geostrophic ML model, in which the horizontal velocity is sheared according to the thermal wind relation. In the geostrophic case the instability is identical to the long wavelength limit of baroclinically unstable Eady waves.

An interesting difference between the slab and geostrophic ML is the dynamics of thermal and saline anomalies. In the slab case, thermohaline anomalies are advected without shear dispersion, and the initial *T*–*S* relation is preserved. In the geostrophic case, the shear dispersion associated with the thermal wind produces a flux of heat and salt orthogonal to the buoyancy gradient. This flux varies as the cube of the thermohaline gradients, and it acts so as to mix heat and salt while leaving buoyancy unchanged on fluid particles. The mechanism tighten an initially diffuse *T*–*S* relation so that a cloud of points in the *T*–*S* plane condenses onto a curve.

## Abstract

The density of the mixed layer (ML) is approximately uniform in the vertical, but there are dynamically important horizontal gradients. The subinertial mixed layer (SML) approximation is a small Rossby number filtering of the primitive equation that isolates the low frequency (ω ≪ *f*) dynamics.

A linear stability analysis based on the SML approximation shows that the horizontal density gradients within the mixed layer (ML) support baroclinically unstable waves with inverse wavenumbers in the range 1 to 10 km. This conclusion follows from both a slab ML model, in which the horizontal velocity has no vertical shear, and a geostrophic ML model, in which the horizontal velocity is sheared according to the thermal wind relation. In the geostrophic case the instability is identical to the long wavelength limit of baroclinically unstable Eady waves.

An interesting difference between the slab and geostrophic ML is the dynamics of thermal and saline anomalies. In the slab case, thermohaline anomalies are advected without shear dispersion, and the initial *T*–*S* relation is preserved. In the geostrophic case, the shear dispersion associated with the thermal wind produces a flux of heat and salt orthogonal to the buoyancy gradient. This flux varies as the cube of the thermohaline gradients, and it acts so as to mix heat and salt while leaving buoyancy unchanged on fluid particles. The mechanism tighten an initially diffuse *T*–*S* relation so that a cloud of points in the *T*–*S* plane condenses onto a curve.

## 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

Laboratory experiments show that ageostrophic instability can “break up” a parallel flow into a sequence of axisymmetric eddies. This is a plausible scenario for the generation of sub-mesoscale coherent Vortices (SCVs). Here we show that conservation of mass, energy and potential vorticity enables one to very simply calculate the radius of the axisymmetric eddy and the wavelength of the nonlinear instability. The latter agrees more closely with laboratory experiments than does the wavelength predicted by linearized stability theory. It energy is not conserved, say because of wave radiation into the lower layer, then the preceding calculation establishes an upper bound on the radius of the eddy. We offer this as an explanation of the observed small size of SCVs.

## Abstract

Laboratory experiments show that ageostrophic instability can “break up” a parallel flow into a sequence of axisymmetric eddies. This is a plausible scenario for the generation of sub-mesoscale coherent Vortices (SCVs). Here we show that conservation of mass, energy and potential vorticity enables one to very simply calculate the radius of the axisymmetric eddy and the wavelength of the nonlinear instability. The latter agrees more closely with laboratory experiments than does the wavelength predicted by linearized stability theory. It energy is not conserved, say because of wave radiation into the lower layer, then the preceding calculation establishes an upper bound on the radius of the eddy. We offer this as an explanation of the observed small size of SCVs.

## 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

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.