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Trevor J. McDougall

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 m2 s−1.

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Trevor J. McDougall

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/dθ2, rather than the usual form (Dθx)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 Dθxx (say 0.15 Dθxx). 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.

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Trevor J. McDougall

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 + ½ρ0 eΔh, where F Ħ is the double-diffusive heat flux across each interface, e the entrainment velocity across the interfaces (migration rate of the interfaces), Δh the difference in enthalpy between adjacent layers and ρ0 a reference density.

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Trevor J. McDougall

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.

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Trevor J. McDougall

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 γf 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.

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Trevor J. McDougall

Abstract

The potential vorticity equation has been vertically differentiated and used with the thermal wind equation to obtain the following expression for the absolute horizontal Eulerian velocity vector,
i1520-0485-25-5-705-eq1
where
i1520-0485-25-5-705-eq2
where ϕz 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 ∇ is the horizontal gradient of in situ density in the geopotential surface; εz, 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 zz, 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 qεz. 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.

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Trevor J. McDougall

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.

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Trevor J. McDougall

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 Rp 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.

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Andreas Klocker and Trevor J. McDougall

Abstract

In the absence of diapycnal mixing processes, fluid parcels move in directions along which they do not encounter buoyant forces. These directions define the local neutral tangent plane. Because of the nonlinear nature of the equation of state of seawater, these neutral tangent planes cannot be connected globally to form a well-defined surface in three-dimensional space; that is, continuous “neutral surfaces” do not exist. This inability to form well-defined neutral surfaces implies that neutral trajectories are helical. Consequently, even in the absence of diapycnal mixing processes, fluid trajectories penetrate through any “density” surface. This process amounts to an extra mechanism that achieves mean vertical advection through any continuous surface such as surfaces of constant potential density or neutral density. That is, the helical nature of neutral trajectories causes this additional diasurface velocity. A water-mass analysis performed with respect to continuous density surfaces will have part of its diapycnal advection due to this diasurface advection process. Hence, this additional diasurface advection should be accounted for when attributing observed water-mass changes to mixing processes. Here, the authors quantify this component of the total diasurface velocity and show that locally it can be the same order of magnitude as diasurface velocities produced by other mixing processes, particularly in the Southern Ocean. The magnitude of this diasurface advection is proportional to the ocean’s neutral helicity, which is observed to be quite small in today’s ocean. The authors also use a perturbation experiment to show that the ocean rapidly readjusts to its present state of small neutral helicity, even if perturbed significantly. Additionally, the authors show how seasonal (rather than spatial) changes in the ocean’s hydrography can generate a similar vertical advection process. This process is described here for the first time; although the vertical advection due to this process is small, it helps to understand water-mass transformation on density surfaces.

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Trevor J. McDougall and Raffaele Ferrari

Abstract

Ledwell, in a comment on McDougall and Ferrari, discusses the dianeutral upwelling and downwelling that occurs near isolated topographic features, by performing a buoyancy budget analysis that integrates the diffusive buoyancy fluxes only out to a set horizontal distance from the topography. The consequence of this choice of control volume is that the magnitude of the area-integrated diffusive buoyancy flux decreases to zero at the base of a topographic feature resulting in a net dianeutral upwelling of water. Based on this result, Ledwell argues that isolated topographic features are preferential locations for the upwelling of waters from the abyss. However the assumptions behind Ledwell’s analysis may or may not be typical of abyssal mixing in the ocean. McDougall and Ferrari developed general expressions for the balance between area-integrated dianeutral advection and diffusion, and then illustrated these general expressions using the very simple assumption that the magnitude of the buoyancy flux per unit area at the top of the turbulent boundary layer was constant. In these pedagogical illustrations, McDougall and Ferrari concentrated on the region near the top (rather than near the base) of isolated topographic features, and they found net sinking of abyssal waters. Here we show that McDougall and Ferrari’s conclusion that isolated topographic features cause dianeutral downwelling is in fact a result that applies for general geometries and for all forms of bottom-intensified mixing profiles at heights above the base of such topographic features.

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