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## Abstract

The effectiveness of mixing near a sloping boundary is reduced not just by the ratio of the stratification there to that in the interior but by the square of this ratio. This result has previously been derived and interpreted by invoking the upgradient vertical buoyancy flux associated with secondary circulation as this opposes the downgradient vertical diffusive flux. Here it is shown that the result may be derived very simply by considering the diffusive flux across an isopycnal, rather than a horizontal, surface. The reduction in the buoyancy flux comes from the increased isopycnal spacing and the reduced distance to the boundary. Derivation of the secondary circulation in the isopycnal/diapycnal framework is more subtle, however, as it involves allowing for the curvilinear nature of the coordinates.

## Abstract

The effectiveness of mixing near a sloping boundary is reduced not just by the ratio of the stratification there to that in the interior but by the square of this ratio. This result has previously been derived and interpreted by invoking the upgradient vertical buoyancy flux associated with secondary circulation as this opposes the downgradient vertical diffusive flux. Here it is shown that the result may be derived very simply by considering the diffusive flux across an isopycnal, rather than a horizontal, surface. The reduction in the buoyancy flux comes from the increased isopycnal spacing and the reduced distance to the boundary. Derivation of the secondary circulation in the isopycnal/diapycnal framework is more subtle, however, as it involves allowing for the curvilinear nature of the coordinates.

## Abstract

The “near-inertial” part of the internal wave continuum is dominant and also different from the rest of the spectrum. A simple possible reason for the difference is that waves generated at the surface are not reflected or scattered from the seafloor until they have propagated equatorward to a latitude where their frequency exceeds the local inertial frequency. This excess is easily estimated and is of the order of 10% of *f* at midlatitudes. The estimate is in reasonable agreement with data on the depth dependence of the peak frequency over smooth topography and on the frequency band within which there is little upward propagating energy. Internal wave propagation and interactions with bottom topography may thus be just as important as wave–wave interactions in controlling the energetic parts of the internal wave spectrum and, hence, in determining mixing rates in the ocean.

## Abstract

The “near-inertial” part of the internal wave continuum is dominant and also different from the rest of the spectrum. A simple possible reason for the difference is that waves generated at the surface are not reflected or scattered from the seafloor until they have propagated equatorward to a latitude where their frequency exceeds the local inertial frequency. This excess is easily estimated and is of the order of 10% of *f* at midlatitudes. The estimate is in reasonable agreement with data on the depth dependence of the peak frequency over smooth topography and on the frequency band within which there is little upward propagating energy. Internal wave propagation and interactions with bottom topography may thus be just as important as wave–wave interactions in controlling the energetic parts of the internal wave spectrum and, hence, in determining mixing rates in the ocean.

## Abstract

Internal solitary waves (ISWs) are a common feature of the coastal zone and marginal seas, especially close to shelf breaks, and are observed to mix the water column at the depth of maximum density gradient. For a two-layer system separated by a thin interface with a finite density gradient, the Richardson number in the interface fails below 1/4 if, in the simplest case, the ISW amplitude exceeds 2(*hH*
_{1})^{1/2}, where *h* is the interface thickness and *H*
_{1} the thickness of the upper layer. Assuming that mixing then thickens the interface and that the potential energy for this comes from the ISW, we derive formulas for the damping rate of the ISW.

The model is generalized to allow for a stratified upper layer; a Richardson number of less than 1/4 now requires that the displacement of the base of the upper layer exceeds 0.82 times the thickness of the layer. The ISW damping rate is sensitive to the ratio of the mixing depths above and below the base of the upper layer but can be plausibly matched to field observations.

## Abstract

Internal solitary waves (ISWs) are a common feature of the coastal zone and marginal seas, especially close to shelf breaks, and are observed to mix the water column at the depth of maximum density gradient. For a two-layer system separated by a thin interface with a finite density gradient, the Richardson number in the interface fails below 1/4 if, in the simplest case, the ISW amplitude exceeds 2(*hH*
_{1})^{1/2}, where *h* is the interface thickness and *H*
_{1} the thickness of the upper layer. Assuming that mixing then thickens the interface and that the potential energy for this comes from the ISW, we derive formulas for the damping rate of the ISW.

The model is generalized to allow for a stratified upper layer; a Richardson number of less than 1/4 now requires that the displacement of the base of the upper layer exceeds 0.82 times the thickness of the layer. The ISW damping rate is sensitive to the ratio of the mixing depths above and below the base of the upper layer but can be plausibly matched to field observations.

## Abstract

The temperature dependence of the expansion coefficient of seawater can lead to a nonzero annual-average surface buoyancy flux even if the annual-average heat flux is zero. In the Mediterranean Sea, for example, this effect apparently gives the same buoyancy flux into the sea as a heat flux of 6 W m^{−2}. This does not, however, lead to an increase in the surface layer buoyancy over the course of a year; compensating cabbeling occurs mainly in the winter and spring when there is intensive mixing. The magnitude of the apparent buoyancy flux is proportional to the area inside the hysteresis loop of the seasonal cycle of the sea surface temperature versus the total heat content of the ocean. The output of a simple mixed layer model, however, shows only a weak sensitivity of mixed layer properties, such as depth, to inclusion of the nonlinearity in the equation of state.

## Abstract

The temperature dependence of the expansion coefficient of seawater can lead to a nonzero annual-average surface buoyancy flux even if the annual-average heat flux is zero. In the Mediterranean Sea, for example, this effect apparently gives the same buoyancy flux into the sea as a heat flux of 6 W m^{−2}. This does not, however, lead to an increase in the surface layer buoyancy over the course of a year; compensating cabbeling occurs mainly in the winter and spring when there is intensive mixing. The magnitude of the apparent buoyancy flux is proportional to the area inside the hysteresis loop of the seasonal cycle of the sea surface temperature versus the total heat content of the ocean. The output of a simple mixed layer model, however, shows only a weak sensitivity of mixed layer properties, such as depth, to inclusion of the nonlinearity in the equation of state.

## Abstract

Gull's theory for rotating hydraulics is applied, as a reduced gravity model, to the surface inflow through the Strait of Gibraltar into the Mediterranean as it is shown that rotation is mostly important east of Tarifa Narrows where only the upper layer is active. Details of the flow depend on the volume flux, *Q*, and on the potential vorticity of the inflow. We examine two extreme cases for this. In both examples the model predicts that supercritical inflow separates from the north share about half way between Tarifa and Gibraltar, so that at Gibraltar the flow is separated. The comparison of (i) conditions for criticality, (ii) separation widths and (iii) interface depths and velocities in midstrait, between this 2D model and the ID models of Farmer and Armi and of Bormans and Garrett suggests that the use of a simple 1D model is appropriate for the Strait of Gibraltar for both subcritical and separated supercritical flows. However, for the latter, the prediction of the point of separation and of the interface depth using the 1D model requires allowance for the effect of rotation.

## Abstract

Gull's theory for rotating hydraulics is applied, as a reduced gravity model, to the surface inflow through the Strait of Gibraltar into the Mediterranean as it is shown that rotation is mostly important east of Tarifa Narrows where only the upper layer is active. Details of the flow depend on the volume flux, *Q*, and on the potential vorticity of the inflow. We examine two extreme cases for this. In both examples the model predicts that supercritical inflow separates from the north share about half way between Tarifa and Gibraltar, so that at Gibraltar the flow is separated. The comparison of (i) conditions for criticality, (ii) separation widths and (iii) interface depths and velocities in midstrait, between this 2D model and the ID models of Farmer and Armi and of Bormans and Garrett suggests that the use of a simple 1D model is appropriate for the Strait of Gibraltar for both subcritical and separated supercritical flows. However, for the latter, the prediction of the point of separation and of the interface depth using the 1D model requires allowance for the effect of rotation.

## Abstract

The ratio of the buoyancy force driving thermal convection to the surface wave vortex-force driving Langmuir circulation in the Craik–Leibovich mechanism involves the Hoenikker number Ho. The critical value Ho_{c}, at which wave forcing and thermal convection contribute equally to the circulation, is found to increase with decreasing Langmuir number La and approaches 3 in the small La limit. For a typical wind speed and surface cooling, Ho is of order O(10^{−2}) to O(10^{−1}). Thus, wave forcing dominates over thermal convection in driving Langmuir circulation.

Stratification induced by strong surface heating suppresses the circulation generated by wave forcing and could completely inhibit the CL instability. In the physically plausible range of −0.1 < Ho < 0, however, this does not happen for small La and the dynamical effect of heating is very small.

For a given heat flux, the temperature difference between the regions of surface divergence and convergence in Langmuir circulation depends on Ho, Pr, and La and on the depth distribution of the heating, but is typically 0(10^{−2}) K.

## Abstract

The ratio of the buoyancy force driving thermal convection to the surface wave vortex-force driving Langmuir circulation in the Craik–Leibovich mechanism involves the Hoenikker number Ho. The critical value Ho_{c}, at which wave forcing and thermal convection contribute equally to the circulation, is found to increase with decreasing Langmuir number La and approaches 3 in the small La limit. For a typical wind speed and surface cooling, Ho is of order O(10^{−2}) to O(10^{−1}). Thus, wave forcing dominates over thermal convection in driving Langmuir circulation.

Stratification induced by strong surface heating suppresses the circulation generated by wave forcing and could completely inhibit the CL instability. In the physically plausible range of −0.1 < Ho < 0, however, this does not happen for small La and the dynamical effect of heating is very small.

For a given heat flux, the temperature difference between the regions of surface divergence and convergence in Langmuir circulation depends on Ho, Pr, and La and on the depth distribution of the heating, but is typically 0(10^{−2}) K.

## Abstract

The restratification in the surface mixed layer driven by a horizontal density gradient following a storm is examined. For a constant layer depth *H* and constant buoyancy gradient |*b _{x}*| =

*M*

^{2}, geostrophic adjustment leads to new stratification with

*N*

^{2}=

*M*

^{4}/

*f*

^{2}and Richardson number Ri = 1. With the inclusion of time dependence, inertial oscillations result and give Ri = ½. If the horizontal buoyancy gradient is confined, the minimum Ri for an initial distribution of buoyancy

*b*(

*x*) is given by 1 − ½

*H*|

*b*|

_{xx}_{max}/

*f*

^{2}. The resulting maximum restratification is

*N*

^{2}=

*M*

^{4}/ (

*f*

^{2}− ½

*H*|

*b*). This restratification can be significant in coastal oceans and possibly in some frontal areas of the open ocean.

_{xx}## Abstract

The restratification in the surface mixed layer driven by a horizontal density gradient following a storm is examined. For a constant layer depth *H* and constant buoyancy gradient |*b _{x}*| =

*M*

^{2}, geostrophic adjustment leads to new stratification with

*N*

^{2}=

*M*

^{4}/

*f*

^{2}and Richardson number Ri = 1. With the inclusion of time dependence, inertial oscillations result and give Ri = ½. If the horizontal buoyancy gradient is confined, the minimum Ri for an initial distribution of buoyancy

*b*(

*x*) is given by 1 − ½

*H*|

*b*|

_{xx}_{max}/

*f*

^{2}. The resulting maximum restratification is

*N*

^{2}=

*M*

^{4}/ (

*f*

^{2}− ½

*H*|

*b*). This restratification can be significant in coastal oceans and possibly in some frontal areas of the open ocean.

_{xx}## Abstract

The inviscid two-layer hydraulic model of Farmer and Armi has two types of solution in the eastern end of the Strait of Gibraltar for flow that is critical at Camarinal Sill. The solution corresponding to maximal exchange has a fast supercritical flow with a shallow interface whereas the solutions corresponding to submaximal exchange show a much slower subcritical flow with a rather deep interface. We extend their model to include physically important effects such as nonrectangular cross section, friction and barotropic fluctuations. Allowing for realistic cross sections reduces the exchange and raises the interface everywhere along the strait. Realistic interfacial friction has little influence on the exchange but lateral friction at the sloping sides of the cross sections has a significant effect on the flow and brings the two solutions (supercritical and subcritical) closer together. The effect of the earth's rotation is also summarized but dealt with in more detail in a companion paper by Bormans and Garrett.

Low-frequency barotropic fluctuations significantly alter the volume flux and the interface depth everywhere along the strait and so should be taken into account in any comparison between theory and observations. They also affect sea-level differences across and along the strait, with a ratio of these that is positive for maximal exchange and negative for submaximal exchange. This provides a useful diagnostic using sea level data alone.

It appears that the exchange through the Strait of Gibraltar alternates between periods of maximal and submaximal exchange, although we do not yet know the frequency of, or conditions for, each state.

## Abstract

The inviscid two-layer hydraulic model of Farmer and Armi has two types of solution in the eastern end of the Strait of Gibraltar for flow that is critical at Camarinal Sill. The solution corresponding to maximal exchange has a fast supercritical flow with a shallow interface whereas the solutions corresponding to submaximal exchange show a much slower subcritical flow with a rather deep interface. We extend their model to include physically important effects such as nonrectangular cross section, friction and barotropic fluctuations. Allowing for realistic cross sections reduces the exchange and raises the interface everywhere along the strait. Realistic interfacial friction has little influence on the exchange but lateral friction at the sloping sides of the cross sections has a significant effect on the flow and brings the two solutions (supercritical and subcritical) closer together. The effect of the earth's rotation is also summarized but dealt with in more detail in a companion paper by Bormans and Garrett.

Low-frequency barotropic fluctuations significantly alter the volume flux and the interface depth everywhere along the strait and so should be taken into account in any comparison between theory and observations. They also affect sea-level differences across and along the strait, with a ratio of these that is positive for maximal exchange and negative for submaximal exchange. This provides a useful diagnostic using sea level data alone.

It appears that the exchange through the Strait of Gibraltar alternates between periods of maximal and submaximal exchange, although we do not yet know the frequency of, or conditions for, each state.

## Abstract

The restratification of a mixed layer with horizontal density gradients above a stratified layer is considered. Solutions are obtained on the assumption that the width across this front is much larger than the local radius of deformation *b*h̄*f*|*b*, mean mixed layer depth *R _{d}*h̄, and the Coriolis parameter

*f*, where

*b*is defined as

*R*−

_{d}*g*(ρ−ρ

_{0})ρ

_{0}, but the fractional change in the mixed layer depth is not required to be small. For an initially quiescent mixed layer, created by homogenizing a fluid of constant stratification to a depth that varies horizontally, the isopycnals in the mixed layer tilt about their intersections with the top surface in the adjusted state, and the base of the mixed layer flattens slightly in the frontal region. Other cases considered include mixed layer fronts with initial momentum out of geostrophic balance, created by vertical mixing of a layer with horizontal gradients previously in thermal wind balance. For a wide front, the isopycnals pivot about the middepth for this case. In all cases, for a wide front, the new vertical buoyancy gradient is

*R*

_{d}*M*

^{4}/

*f*

^{2}h̄, where

*R*

_{d}*M*

^{2}=|

*b*|h̄ is the magnitude of the horizontal buoyancy gradient, and the Richardson number of the adjusted state is 1, as in an earlier constant depth case.

## Abstract

The restratification of a mixed layer with horizontal density gradients above a stratified layer is considered. Solutions are obtained on the assumption that the width across this front is much larger than the local radius of deformation *b*h̄*f*|*b*, mean mixed layer depth *R _{d}*h̄, and the Coriolis parameter

*f*, where

*b*is defined as

*R*−

_{d}*g*(ρ−ρ

_{0})ρ

_{0}, but the fractional change in the mixed layer depth is not required to be small. For an initially quiescent mixed layer, created by homogenizing a fluid of constant stratification to a depth that varies horizontally, the isopycnals in the mixed layer tilt about their intersections with the top surface in the adjusted state, and the base of the mixed layer flattens slightly in the frontal region. Other cases considered include mixed layer fronts with initial momentum out of geostrophic balance, created by vertical mixing of a layer with horizontal gradients previously in thermal wind balance. For a wide front, the isopycnals pivot about the middepth for this case. In all cases, for a wide front, the new vertical buoyancy gradient is

*R*

_{d}*M*

^{4}/

*f*

^{2}h̄, where

*R*

_{d}*M*

^{2}=|

*b*|h̄ is the magnitude of the horizontal buoyancy gradient, and the Richardson number of the adjusted state is 1, as in an earlier constant depth case.

## Abstract

A recent parameterization of mesoscale eddies by Gent and McWilliams (GMc) represents their effects as advective and diffusive fluxes along isopycnals. The form chosen for the added transport velocity due to eddies flattens isopycnals as in baroclinic instability but implicitly assumes purely viscous dissipation of the available potential energy released. If, however, the energy dissipation occurs in the ocean interior due to a process such as internal wave breaking, it is likely to cause diapycnal mixing. The implied diffusivity is large in a frontal situation, but the analysis of the spindown equation for a quasigeostrophic front shows that it causes only small changes in the frontal evolution. The spindown equation also permits analysis of the relative importance of various terms describing subgrid-scale fluxes of momentum and buoyancy, and may be interpreted in terms of Eliassen–Palm fluxes. Another possibility for the dissipation of the eddy energy that is generated from the mean available potential energy in the GMc mechanism involves air–sea interaction and subsequent water mass modification, but this is also clearly diabatic across mean isopycnals. The GMc parameterization does accomplish diabatic transfer across mean isopycnals near the surface due to the boundary conditions on the advective eddy flux, though it is not clear that this is the same as if the effect air–sea interaction on the eddies were treated explicitly. The cross-frontal volume flux must be compatible with the buoyancy budget. In the case of the Southern Ocean, this may require the net meridional circulation cell to be weak if the air–sea buoyancy flux is small.

## Abstract

A recent parameterization of mesoscale eddies by Gent and McWilliams (GMc) represents their effects as advective and diffusive fluxes along isopycnals. The form chosen for the added transport velocity due to eddies flattens isopycnals as in baroclinic instability but implicitly assumes purely viscous dissipation of the available potential energy released. If, however, the energy dissipation occurs in the ocean interior due to a process such as internal wave breaking, it is likely to cause diapycnal mixing. The implied diffusivity is large in a frontal situation, but the analysis of the spindown equation for a quasigeostrophic front shows that it causes only small changes in the frontal evolution. The spindown equation also permits analysis of the relative importance of various terms describing subgrid-scale fluxes of momentum and buoyancy, and may be interpreted in terms of Eliassen–Palm fluxes. Another possibility for the dissipation of the eddy energy that is generated from the mean available potential energy in the GMc mechanism involves air–sea interaction and subsequent water mass modification, but this is also clearly diabatic across mean isopycnals. The GMc parameterization does accomplish diabatic transfer across mean isopycnals near the surface due to the boundary conditions on the advective eddy flux, though it is not clear that this is the same as if the effect air–sea interaction on the eddies were treated explicitly. The cross-frontal volume flux must be compatible with the buoyancy budget. In the case of the Southern Ocean, this may require the net meridional circulation cell to be weak if the air–sea buoyancy flux is small.