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## Abstract

Foster and Carmack (1976) and Middleton and Foster (1980) have observed a series of “diffusive” double-diffusive interfaces in the deep, central Weddell Sea which are very close to the conditions under which cabbeling is thought to be possible. Prompted by these observations, McDougall (1981b) has studied the fluxes and the entrainment across a single horizontal double-diffusive interface when the equation of state is significantly nonlinear. This work has shown how the nonlinearity of the equation of state affects the double-diffusive convection process by causing entrainment across the interface. For values of the density anomaly ratio *R*
_{ρ} < 2, it was concluded that this modified double-diffusive convection is much more important than the cabbeling mechanism. In this note we use these recent results to predict what signature these processes would have on a series of such interfaces in the Weddell Sea. We show that regular CTD profiles over a 48 h period would shed considerable light on the magnitude of the fluxes caused by entrainment across the interfaces as compared to the fluxes of ordinary (symmetrical) double-diffusive convection. By considering a regular series of migrating double-diffusive interfaces which are steady in the mean we show that the total flux of heat through a fixed depth is *F*
_{ħ} + ½ρ_{0}
*e*Δ*h*, where *F*
_{Ħ} is the double-diffusive heat flux across each interface, *e* the entrainment velocity across the interfaces (migration rate of the interfaces), Δ*h* the difference in enthalpy between adjacent layers and ρ_{0} a reference density.

## Abstract

Foster and Carmack (1976) and Middleton and Foster (1980) have observed a series of “diffusive” double-diffusive interfaces in the deep, central Weddell Sea which are very close to the conditions under which cabbeling is thought to be possible. Prompted by these observations, McDougall (1981b) has studied the fluxes and the entrainment across a single horizontal double-diffusive interface when the equation of state is significantly nonlinear. This work has shown how the nonlinearity of the equation of state affects the double-diffusive convection process by causing entrainment across the interface. For values of the density anomaly ratio *R*
_{ρ} < 2, it was concluded that this modified double-diffusive convection is much more important than the cabbeling mechanism. In this note we use these recent results to predict what signature these processes would have on a series of such interfaces in the Weddell Sea. We show that regular CTD profiles over a 48 h period would shed considerable light on the magnitude of the fluxes caused by entrainment across the interfaces as compared to the fluxes of ordinary (symmetrical) double-diffusive convection. By considering a regular series of migrating double-diffusive interfaces which are steady in the mean we show that the total flux of heat through a fixed depth is *F*
_{ħ} + ½ρ_{0}
*e*Δ*h*, where *F*
_{Ħ} is the double-diffusive heat flux across each interface, *e* the entrainment velocity across the interfaces (migration rate of the interfaces), Δ*h* the difference in enthalpy between adjacent layers and ρ_{0} a reference density.

## Abstract

The exponentially growing solutions of the linear stability analysis of double-diffusive interleaving are allowed to grow to finite amplitude. At this stage the double-diffusive fluxes across the intrusion boundaries must be parameterized differently to those in the growing solution because every second finger interface becomes a diffusive interface in the steady state solution. Using the wavenumbers that are set by the initial, growing intrusions from the linear stability problem, together with the changed parameterization of the double-diffusive fluxes, it is shown that a steady state is reached. The ratio of the gradients of potential temperature and salinity along any particular intrusion is the same as in the linear stability analysis. When expressed in terms of density changes, this ratio is close to 0.9 rather than the commonly assumed value of 0.5 (being the buoyancy-flux ratio of a single salt finger interface). After solving for the velocity field, the isopycnal and diapycnal fluxes of potential temperature, salinity and density are formed. The diapycnal fluxes of potential temperature, salinity and density are all found to be up-gradient, implying negative diapycnal diffusivities for all three quantities. This is an unexpected and potentially important result. Following the approach of Garrett, a basin-average diapycnal diffusivity is estimated and is found to be of comparable size, but of opposite sign, to the modern day metric of 10^{−5} m^{2} s^{−1}.

## Abstract

The exponentially growing solutions of the linear stability analysis of double-diffusive interleaving are allowed to grow to finite amplitude. At this stage the double-diffusive fluxes across the intrusion boundaries must be parameterized differently to those in the growing solution because every second finger interface becomes a diffusive interface in the steady state solution. Using the wavenumbers that are set by the initial, growing intrusions from the linear stability problem, together with the changed parameterization of the double-diffusive fluxes, it is shown that a steady state is reached. The ratio of the gradients of potential temperature and salinity along any particular intrusion is the same as in the linear stability analysis. When expressed in terms of density changes, this ratio is close to 0.9 rather than the commonly assumed value of 0.5 (being the buoyancy-flux ratio of a single salt finger interface). After solving for the velocity field, the isopycnal and diapycnal fluxes of potential temperature, salinity and density are formed. The diapycnal fluxes of potential temperature, salinity and density are all found to be up-gradient, implying negative diapycnal diffusivities for all three quantities. This is an unexpected and potentially important result. Following the approach of Garrett, a basin-average diapycnal diffusivity is estimated and is found to be of comparable size, but of opposite sign, to the modern day metric of 10^{−5} m^{2} s^{−1}.

## Abstract

Scalar properties in the ocean are stirred (and subsequently mixed) rather efficiently by mesoscale eddies and two-dimensional turbulence along “neutral surfaces”, defined such that when water parcels are moved small distances in the neutral surface, they experience no buoyant restoring forces. By contrast, work would have to be done on a moving fluid parcel in order to keep it on a potential density surface. The differences between neutral surfaces and potential density surfaces are due to the variation of α/β with pressure (where α is the thermal expansion coefficient and β is the saline contraction coefficient).

By regarding the equation of state of seawater as a function of salinity, potential temperature, and pressure, rather than in terms of salinity, temperature, and pressure, it is possible to quantify the differences between neutral surfaces and potential density surfaces. In particular, the spatial gradients of scalar properties (e.g., *S*, θ, tritium or potential vorticity) on a neutral surface can be quite different to the corresponding gradients in a potential density surface. For example, at a potential temperature of 4°C and a pressure of 1000 db, the lateral gradient of potential temperature in a potential density surface (referenced to sea level) is too large by between 50% and 350% (depending on the stability ratio *R _{p}* of the water column) compared with the physically relevant gradient of potential temperature on the neutral surface. Three-examples of neutral surfaces are presented, based on the Levitus atlas of the North Atlantic.

## Abstract

Scalar properties in the ocean are stirred (and subsequently mixed) rather efficiently by mesoscale eddies and two-dimensional turbulence along “neutral surfaces”, defined such that when water parcels are moved small distances in the neutral surface, they experience no buoyant restoring forces. By contrast, work would have to be done on a moving fluid parcel in order to keep it on a potential density surface. The differences between neutral surfaces and potential density surfaces are due to the variation of α/β with pressure (where α is the thermal expansion coefficient and β is the saline contraction coefficient).

By regarding the equation of state of seawater as a function of salinity, potential temperature, and pressure, rather than in terms of salinity, temperature, and pressure, it is possible to quantify the differences between neutral surfaces and potential density surfaces. In particular, the spatial gradients of scalar properties (e.g., *S*, θ, tritium or potential vorticity) on a neutral surface can be quite different to the corresponding gradients in a potential density surface. For example, at a potential temperature of 4°C and a pressure of 1000 db, the lateral gradient of potential temperature in a potential density surface (referenced to sea level) is too large by between 50% and 350% (depending on the stability ratio *R _{p}* of the water column) compared with the physically relevant gradient of potential temperature on the neutral surface. Three-examples of neutral surfaces are presented, based on the Levitus atlas of the North Atlantic.

## Abstract

Potential temperature is used in oceanography as though it is a conservative variable like salinity; however, turbulent mixing processes conserve enthalpy and usually destroy potential temperature. This negative production of potential temperature is similar in magnitude to the well-known production of entropy that always occurs during mixing processes. Here it is shown that potential enthalpy—the enthalpy that a water parcel would have if raised adiabatically and without exchange of salt to the sea surface—is more conservative than potential temperature by two orders of magnitude. Furthermore, it is shown that a flux of potential enthalpy can be called “the heat flux” even though potential enthalpy is undefined up to a linear function of salinity. The exchange of heat across the sea surface is identically the flux of potential enthalpy. This same flux is not proportional to the flux of potential temperature because of variations in heat capacity of up to 5%. The geothermal heat flux across the ocean floor is also approximately the flux of potential enthalpy with an error of no more that 0.15%. These results prove that potential enthalpy is the quantity whose advection and diffusion is equivalent to advection and diffusion of “heat” in the ocean. That is, it is proven that to very high accuracy, the first law of thermodynamics in the ocean is the conservation equation of potential enthalpy. It is shown that potential enthalpy is to be preferred over the Bernoulli function. A new temperature variable called “conservative temperature” is advanced that is simply proportional to potential enthalpy. It is shown that present ocean models contain typical errors of 0.1°C and maximum errors of 1.4°C in their temperature because of the neglect of the nonconservative production of potential temperature. The meridional flux of heat through oceanic sections found using this conservative approach is different by up to 0.4% from that calculated by the approach used in present ocean models in which the nonconservative nature of potential temperature is ignored and the specific heat at the sea surface is assumed to be constant. An alternative approach that has been recommended and is often used with observed section data, namely, calculating the meridional heat flux using the specific heat (at zero pressure) and potential temperature, rests on an incorrect theoretical foundation, and this estimate of heat flux is actually less accurate than simply using the flux of potential temperature with a constant heat capacity.

## Abstract

Potential temperature is used in oceanography as though it is a conservative variable like salinity; however, turbulent mixing processes conserve enthalpy and usually destroy potential temperature. This negative production of potential temperature is similar in magnitude to the well-known production of entropy that always occurs during mixing processes. Here it is shown that potential enthalpy—the enthalpy that a water parcel would have if raised adiabatically and without exchange of salt to the sea surface—is more conservative than potential temperature by two orders of magnitude. Furthermore, it is shown that a flux of potential enthalpy can be called “the heat flux” even though potential enthalpy is undefined up to a linear function of salinity. The exchange of heat across the sea surface is identically the flux of potential enthalpy. This same flux is not proportional to the flux of potential temperature because of variations in heat capacity of up to 5%. The geothermal heat flux across the ocean floor is also approximately the flux of potential enthalpy with an error of no more that 0.15%. These results prove that potential enthalpy is the quantity whose advection and diffusion is equivalent to advection and diffusion of “heat” in the ocean. That is, it is proven that to very high accuracy, the first law of thermodynamics in the ocean is the conservation equation of potential enthalpy. It is shown that potential enthalpy is to be preferred over the Bernoulli function. A new temperature variable called “conservative temperature” is advanced that is simply proportional to potential enthalpy. It is shown that present ocean models contain typical errors of 0.1°C and maximum errors of 1.4°C in their temperature because of the neglect of the nonconservative production of potential temperature. The meridional flux of heat through oceanic sections found using this conservative approach is different by up to 0.4% from that calculated by the approach used in present ocean models in which the nonconservative nature of potential temperature is ignored and the specific heat at the sea surface is assumed to be constant. An alternative approach that has been recommended and is often used with observed section data, namely, calculating the meridional heat flux using the specific heat (at zero pressure) and potential temperature, rests on an incorrect theoretical foundation, and this estimate of heat flux is actually less accurate than simply using the flux of potential temperature with a constant heat capacity.

## Abstract

_{z}is the rate of turning with height of the epineutral

*q*contours, ∇

_{nq}is the gradient of potential vorticity in the neutral surface,

**k**is the unit vector antiparallel to gravity, and ∇

_{2ρ}is the horizontal gradient of in situ density in the geopotential surface; ε

_{z}, is the vertical derivative of the dianeutral velocity plus the effective vortex stretching due to unsteadiness and the lateral mixing of potential vorticity and

*V*

^{⊥}is the horizontal velocity component normal to the epineutral contours of potential vorticity. A new finding is that the velocity component that mixing induces

*along*the epineutral

*q*contours,

*V*

^{⊥}

_{z}/ϕ

_{z}, is equal to the rate of change of the

*V*

^{⊥}component up the cast per unit change in ϕ (measured in radians).

The equation for the absolute velocity vector **V** is used to explore the conditions required for **V** = **0**. In the absence of mixing processes, **V** = **0** occurs only where the epineutral gradient of potential vorticity is zero; however, with mixing processes, the lateral velocity can be arrested by mixing without requiring this condition on the gradient of potential vorticity. Conversely, in the presence of mixing, the flow is nonzero at points where ∇_{nq} = **0**, and it is shown that at thew points the lateral velocity is determined by the epineutral gradient of *q*ε_{z}. A zero of the lateral Eulerian velocity will occur only at isolated points on any particular surface. Similarly, points of zero three-dimensional velocity will be so rare that one may never expect to encounter such a point on any particular surface.

Mesoscale eddy activity is assumed to transport potential vorticity *q* in a downgradient fashion along neutral surfaces, and this leads directly to a parameterization of this contribution to the lateral Stokes drift. This extra lateral velocity is typically 1 mm s^{−1} in the direction toward greater |*q*| along neutral surfaces, that is, broadly speaking, in the direction away from the equator and toward the poles. This Stokes drift will often make a large contribution to the tracer conservation equations.

## Abstract

_{z}is the rate of turning with height of the epineutral

*q*contours, ∇

_{nq}is the gradient of potential vorticity in the neutral surface,

**k**is the unit vector antiparallel to gravity, and ∇

_{2ρ}is the horizontal gradient of in situ density in the geopotential surface; ε

_{z}, is the vertical derivative of the dianeutral velocity plus the effective vortex stretching due to unsteadiness and the lateral mixing of potential vorticity and

*V*

^{⊥}is the horizontal velocity component normal to the epineutral contours of potential vorticity. A new finding is that the velocity component that mixing induces

*along*the epineutral

*q*contours,

*V*

^{⊥}

_{z}/ϕ

_{z}, is equal to the rate of change of the

*V*

^{⊥}component up the cast per unit change in ϕ (measured in radians).

The equation for the absolute velocity vector **V** is used to explore the conditions required for **V** = **0**. In the absence of mixing processes, **V** = **0** occurs only where the epineutral gradient of potential vorticity is zero; however, with mixing processes, the lateral velocity can be arrested by mixing without requiring this condition on the gradient of potential vorticity. Conversely, in the presence of mixing, the flow is nonzero at points where ∇_{nq} = **0**, and it is shown that at thew points the lateral velocity is determined by the epineutral gradient of *q*ε_{z}. A zero of the lateral Eulerian velocity will occur only at isolated points on any particular surface. Similarly, points of zero three-dimensional velocity will be so rare that one may never expect to encounter such a point on any particular surface.

Mesoscale eddy activity is assumed to transport potential vorticity *q* in a downgradient fashion along neutral surfaces, and this leads directly to a parameterization of this contribution to the lateral Stokes drift. This extra lateral velocity is typically 1 mm s^{−1} in the direction toward greater |*q*| along neutral surfaces, that is, broadly speaking, in the direction away from the equator and toward the poles. This Stokes drift will often make a large contribution to the tracer conservation equations.

## Abstract

Quasi-horizontal interleaving between water masses is frequently observed in the frontal regions between different water masses where there are significant compensating isopycnal gradients of temperature and salinity. It is believed that these intrusions are driven by double-diffusive convection which transports salt, heat and density across sharp, quasi-horizontal interfaces (i.e., across regions of large vertical property gradients). Stern and other investigators have parameterized the vertical flux of salt in these intrusions by an eddy diffusivity multiplied by the vertical gradient of salinity, whereas laboratory experiments show that sharp interfaces tend to occur by double-diffusive convection and that the vertical fluxes of properties depend on the property contrasts across the interfaces. In this paper it is assumed that the vertical fluxes of double-diffusive convection are *proportional* to the salinity contrast across the sharp interfaces that occur between the quasi-horizontal intrusions. The previous approaches had the fluxes proportional to this salinity difference divided by the height of the intrusions. It is shown that the results of the linear stability analysis are the same as those found by Toole and Georgi if their nondimensional vertical wavenumber is simply reinterpreted.

In addition to finding the three wavenumbers of the fastest growing linear instability, the ratios of several of the growing perturbation quantities are also derived. It is shown that the fastest growing intrusions move directly across the front with zero velocity component in the alongfront direction. The only effect of rotation is to introduce an alongfront tilt to he instrusions: the wavenumbers, velocity components and the growth rate are all shown to be independent of the rotation rate. The instantaneous (synoptic) ratio of the changes of potential temperature and salinity along any particular intrusion is also derived and is found to be much nearer to 1 (with a typical value of 0.9) than to the buoyancy flux ratio γ_{f} of salt fingers. The ratios of the exponentially growing quantities provide further quantities to be compared with oceanographic and laboratory observations and so test the model.

## Abstract

Quasi-horizontal interleaving between water masses is frequently observed in the frontal regions between different water masses where there are significant compensating isopycnal gradients of temperature and salinity. It is believed that these intrusions are driven by double-diffusive convection which transports salt, heat and density across sharp, quasi-horizontal interfaces (i.e., across regions of large vertical property gradients). Stern and other investigators have parameterized the vertical flux of salt in these intrusions by an eddy diffusivity multiplied by the vertical gradient of salinity, whereas laboratory experiments show that sharp interfaces tend to occur by double-diffusive convection and that the vertical fluxes of properties depend on the property contrasts across the interfaces. In this paper it is assumed that the vertical fluxes of double-diffusive convection are *proportional* to the salinity contrast across the sharp interfaces that occur between the quasi-horizontal intrusions. The previous approaches had the fluxes proportional to this salinity difference divided by the height of the intrusions. It is shown that the results of the linear stability analysis are the same as those found by Toole and Georgi if their nondimensional vertical wavenumber is simply reinterpreted.

In addition to finding the three wavenumbers of the fastest growing linear instability, the ratios of several of the growing perturbation quantities are also derived. It is shown that the fastest growing intrusions move directly across the front with zero velocity component in the alongfront direction. The only effect of rotation is to introduce an alongfront tilt to he instrusions: the wavenumbers, velocity components and the growth rate are all shown to be independent of the rotation rate. The instantaneous (synoptic) ratio of the changes of potential temperature and salinity along any particular intrusion is also derived and is found to be much nearer to 1 (with a typical value of 0.9) than to the buoyancy flux ratio γ_{f} of salt fingers. The ratios of the exponentially growing quantities provide further quantities to be compared with oceanographic and laboratory observations and so test the model.

## Abstract

Fluid motion in the sea is known to occur predominantly along quasi-horizontal neutral surfaces but the very small diapycnal (i.e., across isopycnal) velocities often make a significant contribution to the conversation equations of heat, salt and tracer. By eliminating the diapycnal advection term between the conservation equations for (i) heat and (ii) salt, an equation is derived for the rate of change (Lagrangian derivative) of potential temperature θ on a neutral surface which has terms caused by (a) turbulent mixing along isopycnal surfaces (i.e., isopycnal mixing), (b) diapycnal turbulent mixing and (c) double-diffusive convection. Bemuse of the nature of the isopycnal reference frame, the diapycnal mixing terms do not take their expected forms. For example, the diapycnal turbulent mixing term is proportional to the diapycnal eddy diffusivity *D* multiplied by the curvature of the θ-*S* curve, *d*
^{2}
*S*/*d*θ^{2}, rather than the usual form (*D*θ_{x})_{x}. If the θ−*S* curve is locally straight, small-scale turbulent mixing can have no effect on the temperature (or salinity) measured on an isopycnal surface. For values of the θ−*S* curvature appropriate to the Central Waters of the World's Oceans, the rate at which diapycnal turbulent mixing changes potential temperature on isopycnals is a fraction of *D*θ_{xx} (say 0.15 *D*θ_{xx}). This surprising result is due to the ability of the isopycnal surface to migrate quasi-vertically through the water column (or equivalently, for water to move diapycnally through the isopycnal surface) in response to the divergence (∇·) of the fluxes of both heat and salt. In the interpretation of oceanographic data sets, it is not yet possible to estimate the diapycnal advection velocity and so it is customarily omitted from the conservation equations. It is the main aim of this paper to show that by so neglecting the diapycnal advective terms, the diapycnal mixing processes enter the conservation equations in greatly altered forms. The conservation equations for scalers (both active and passive) on a neutral surface which we develop are then the appropriate equations to be used in future studies of subsurface water mass conversion.

## Abstract

Fluid motion in the sea is known to occur predominantly along quasi-horizontal neutral surfaces but the very small diapycnal (i.e., across isopycnal) velocities often make a significant contribution to the conversation equations of heat, salt and tracer. By eliminating the diapycnal advection term between the conservation equations for (i) heat and (ii) salt, an equation is derived for the rate of change (Lagrangian derivative) of potential temperature θ on a neutral surface which has terms caused by (a) turbulent mixing along isopycnal surfaces (i.e., isopycnal mixing), (b) diapycnal turbulent mixing and (c) double-diffusive convection. Bemuse of the nature of the isopycnal reference frame, the diapycnal mixing terms do not take their expected forms. For example, the diapycnal turbulent mixing term is proportional to the diapycnal eddy diffusivity *D* multiplied by the curvature of the θ-*S* curve, *d*
^{2}
*S*/*d*θ^{2}, rather than the usual form (*D*θ_{x})_{x}. If the θ−*S* curve is locally straight, small-scale turbulent mixing can have no effect on the temperature (or salinity) measured on an isopycnal surface. For values of the θ−*S* curvature appropriate to the Central Waters of the World's Oceans, the rate at which diapycnal turbulent mixing changes potential temperature on isopycnals is a fraction of *D*θ_{xx} (say 0.15 *D*θ_{xx}). This surprising result is due to the ability of the isopycnal surface to migrate quasi-vertically through the water column (or equivalently, for water to move diapycnally through the isopycnal surface) in response to the divergence (∇·) of the fluxes of both heat and salt. In the interpretation of oceanographic data sets, it is not yet possible to estimate the diapycnal advection velocity and so it is customarily omitted from the conservation equations. It is the main aim of this paper to show that by so neglecting the diapycnal advective terms, the diapycnal mixing processes enter the conservation equations in greatly altered forms. The conservation equations for scalers (both active and passive) on a neutral surface which we develop are then the appropriate equations to be used in future studies of subsurface water mass conversion.

## Abstract

Submesoscale coherent vortices (SCVs, “meddies” or “bullets”) are shown not to move along either potential density surfaces or neutral surfaces, due to the compressibility of sea water being a function of potential temperature and salinity. While it has been recognized that SCVs may make an important contribution to *lateral* property fluxes in the ocean, it is shown in this paper that they can also cause significant *vertical* fluxes across neutral surfaces. Since they represent the slow vertical translation of macroscopic water masses, these fluxes are not measurable with microstructure measurements of the dissipation of kinetic energy, but are nonetheless real. These vertical fluxes of heat and salt can be either down-gradient or up-gradient, corresponding to either a positive or a negative vertical diffusivity. The magnitude of the effective diffusivity for salt is different from that for heat. Formulae are derived for estimating the mean vertical velocity of SCVs through neutral surfaces.

## Abstract

Submesoscale coherent vortices (SCVs, “meddies” or “bullets”) are shown not to move along either potential density surfaces or neutral surfaces, due to the compressibility of sea water being a function of potential temperature and salinity. While it has been recognized that SCVs may make an important contribution to *lateral* property fluxes in the ocean, it is shown in this paper that they can also cause significant *vertical* fluxes across neutral surfaces. Since they represent the slow vertical translation of macroscopic water masses, these fluxes are not measurable with microstructure measurements of the dissipation of kinetic energy, but are nonetheless real. These vertical fluxes of heat and salt can be either down-gradient or up-gradient, corresponding to either a positive or a negative vertical diffusivity. The magnitude of the effective diffusivity for salt is different from that for heat. Formulae are derived for estimating the mean vertical velocity of SCVs through neutral surfaces.

## Abstract

Double-diffusive interleaving is examined as it progresses from a linear instability toward finite amplitude. When the basic stratification is in the “finger” sense, the initial series of finger interfaces is unstable and one grows in strength at the expense of the others. At an intermediate stage of its development, the interleaving motions pass through a stage when every second interface in the vertical is stable to double diffusion. At a later time this interface turns into a “diffusive” double-diffusive interface. This study takes the fluxes of heat and salt across both the finger and diffusive interfaces to be given by the laboratory flux laws, and the authors ask whether a steady state is possible. It is found that the fluxes across the diffusive interfaces must be many times stronger relative to the corresponding fluxes across the finger interfaces than is indicated from existing flux expressions derived from laboratory experiments. The total effect of the interleaving motion on the vertical fluxes of heat and of salt are calculated for the steady-state solutions. It is found that both the fluxes of heat and salt are upgradient, corresponding to a negative vertical diffusion coefficient for all heat, salt, and density. For moderate to large Prandtl numbers, these negative effective diapycnal diffusivities of heat and salt are approximately equal so that the interleaving process acts to counteract some of the usual turbulent diapycnal diffusivity due to breaking internal waves.

## Abstract

Double-diffusive interleaving is examined as it progresses from a linear instability toward finite amplitude. When the basic stratification is in the “finger” sense, the initial series of finger interfaces is unstable and one grows in strength at the expense of the others. At an intermediate stage of its development, the interleaving motions pass through a stage when every second interface in the vertical is stable to double diffusion. At a later time this interface turns into a “diffusive” double-diffusive interface. This study takes the fluxes of heat and salt across both the finger and diffusive interfaces to be given by the laboratory flux laws, and the authors ask whether a steady state is possible. It is found that the fluxes across the diffusive interfaces must be many times stronger relative to the corresponding fluxes across the finger interfaces than is indicated from existing flux expressions derived from laboratory experiments. The total effect of the interleaving motion on the vertical fluxes of heat and of salt are calculated for the steady-state solutions. It is found that both the fluxes of heat and salt are upgradient, corresponding to a negative vertical diffusion coefficient for all heat, salt, and density. For moderate to large Prandtl numbers, these negative effective diapycnal diffusivities of heat and salt are approximately equal so that the interleaving process acts to counteract some of the usual turbulent diapycnal diffusivity due to breaking internal waves.

## Abstract

Recent work on the global overturning circulation and its energetics assumes that processes caused by nonlinearities of the equation of state of seawater are negligible. Nonlinear processes such as cabbeling and thermobaricity cause diapycnal motion as a consequence of isopycnal mixing. The nonlinear equation of state also causes the helical nature of neutral trajectories; as a consequence of this helical nature, it is not possible to define a continuous “density” surface that aligns with neutral tangent planes. The result is an additional diapycnal advection, which needs to be accounted for in water-mass analysis. In this paper, the authors take advantage of new techniques in constructing very accurate continuous density surfaces to more precisely estimate isopycnal and diapycnal processes caused by the nonlinear equation of state. They then quantify the diapycnal advection due to each of these nonlinear processes and show that they lead in total to a significant downward diapycnal advection, particularly in the Southern Ocean. The nonlinear processes are therefore another source of dense water formation in addition to high-latitude convection. To maintain the abyssal stratification in the global ocean, these dense water masses have to be brought back toward surface layers, and this can occur by either diabatic or adiabatic processes. Including these nonlinear processes into the advection–diffusion balance, the authors show that observed diapycnal diffusivities are unlikely to be able to support the amount of dense water produced in the global ocean, thus placing more importance on the adiabatic way of bringing the deep waters back to the surface.

## Abstract

Recent work on the global overturning circulation and its energetics assumes that processes caused by nonlinearities of the equation of state of seawater are negligible. Nonlinear processes such as cabbeling and thermobaricity cause diapycnal motion as a consequence of isopycnal mixing. The nonlinear equation of state also causes the helical nature of neutral trajectories; as a consequence of this helical nature, it is not possible to define a continuous “density” surface that aligns with neutral tangent planes. The result is an additional diapycnal advection, which needs to be accounted for in water-mass analysis. In this paper, the authors take advantage of new techniques in constructing very accurate continuous density surfaces to more precisely estimate isopycnal and diapycnal processes caused by the nonlinear equation of state. They then quantify the diapycnal advection due to each of these nonlinear processes and show that they lead in total to a significant downward diapycnal advection, particularly in the Southern Ocean. The nonlinear processes are therefore another source of dense water formation in addition to high-latitude convection. To maintain the abyssal stratification in the global ocean, these dense water masses have to be brought back toward surface layers, and this can occur by either diabatic or adiabatic processes. Including these nonlinear processes into the advection–diffusion balance, the authors show that observed diapycnal diffusivities are unlikely to be able to support the amount of dense water produced in the global ocean, thus placing more importance on the adiabatic way of bringing the deep waters back to the surface.