1. Introduction

The motivation for the present work dates back to the paper of McDougall (1984) in which the conservation equations of heat and salt were combined in a form that did not include the dianeutral advection, leaving a balance between lateral advection along a density surface and a specific combination of the effects of mixing of both heat and salt. This motivation will be described in this introduction for the advective form of the conservation equations. To make this approach workable as an inverse method in the ocean we need to implement this combination of conservation equations in the divergence form, and this is tackled in section 2.

The conservation equations for salinity S and conservative temperature Θ [which is proportional to potential enthalpy and represents the “heat content” per unit mass of seawater (McDougall 2003)] are

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and

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These equations have been written in the advective form and with respect to neutral density (γⁿ) surfaces (Jackett and McDougall 1997) so that e is the vertical velocity through neutral density surfaces (i.e., the dianeutral velocity), and V is the thickness-weighted horizontal velocity obtained by temporally averaging the horizontal velocity between closely spaced neutral density surfaces. Similarly, the salinity and conservative temperature are the thickness-weighted values obtained by averaging between closely spaced pairs of neutral density surfaces (McDougall and McIntosh 2001). In these equations h(x, y) is the thickness between two closely spaced neutral density surfaces. The mixing processes that appear on the right-hand sides are simply lateral mixing of passive tracers (with diffusivity K) along the density surfaces and vertical small-scale turbulent mixing (with diffusivity D). We have not included double-diffusive convection or double-diffusive interleaving. Note that the distinction between conservative temperature and potential temperature is not central to this paper, and we will frequently refer to Θ as simply temperature.

Following McDougall (1984) we note that the temporal and spatial gradient of S and Θ along neutral tangent planes are related through the relevant thermal expansion coefficient (α = −ρ⁻¹ρ_Θ|_S,p) and haline contraction coefficient (β = ρ⁻¹ρ_S|_Θ,p); that is,

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Strictly, these relations hold locally in neutral tangent planes, but here we will assume that they hold in neutral density surfaces since McDougall and Jackett (2005a, b) have shown that neutral density surfaces are everywhere almost tangent to neutral tangent planes.

The dianeutral advection term can be eliminated between (1) and (2) by multiplying the equations by Θ_z and S_z, respectively, and subtracting. On doing this and using (3) we find (similarly to McDougall 1984)

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where the buoyancy frequency N is defined by N ² = g(αΘ_z − βS_z), and R_ρ ≡ αΘ_z/βS_z is the stability ratio of the vertical stratification. For completeness we also write down another important linear combination of (1) and (2), namely the one formed by cross-multiplying these equations by α and β, giving what might be loosely called the “density” conservation equation, or the equation for the dianeutral velocity, namely,

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where the cabbeling coefficient and the thermobaric coefficient (McDougall 1987, 1991a) are

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and represent the nonlinear nature of the equation of state of seawater.

The motivation for the present study comes from the rather simple balance in (4) between the lateral advection of salinity along isopycnals (the left-hand side) and the terms on the right that depend, respectively, on the vertical and lateral diffusivity. We describe this balance as “simple” for three reasons. First, (4) does not contain any dependence on the dianeutral velocity e or, second, on the vertical derivative of the vertical diffusivity D_z. Third, the term involving D in (4) is proportional to the curvature of the S–Θ diagram, which is much less subject to numerical noise caused by vertical heave than the second derivatives Θ_zz and S_zz in (1), (2), and (5); recall that

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For these three reasons we expect that the diapycnal mixing term in (4) can be evaluated from oceanographic data with less uncertainty than the diapycnal mixing and advection terms in (1) and (2). Quantitative knowledge of the flow out of the Mediterranean Sea into the North Atlantic Ocean, as described by Baringer and Price (1997), seems a suitable way of estimating the left-hand side of (4), so the eastern North Atlantic may be a region of the ocean where we could exploit the rather simple balance between advection and mixing processes, as given by (4), perhaps being able to perform an inverse model without the need to consider unknown reference level velocities, which are the traditional unknowns in oceanographic inversions. This idea is the motivation for the next section, in which we form rather specific combinations of the divergence forms of the basic conservation equations in order to arrive at an equation that satisfies the overall flux constraints over a finite volume but has the above three essential features of the “simple” curvature form conservation equation (4).

2. A careful linear combination of conservation equations

The three basic conservation (Boussinesq) equations, namely continuity, conservation of salinity, and conservation of conservative temperature, can be written in divergence form with respect to neutral density coordinates as (Griffies 2004)

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Here the superscripts u and l refer to the upper and lower interfaces bounding each layer and h is the vertical distance (layer thickness) between these bounding density interfaces. In the above we have ignored double-diffusive convection in that the only diapycnal mixing is done with the same diffusivity for salinity as for conservative temperature. The last term in (9) [and in (10)] is the difference in the diapycnal flux of salt (and temperature) across the upper and lower density interfaces.

For each layer we will consider a series of control volumes bounded by the source of Mediterranean Water in the east and a series of contours of salinity and conservative temperature, S₀ and Θ₀. The S₀ and Θ₀ contours coincide because they are on a density surface. We multiply (8) by the constant values S₁ and Θ₁ (which we may well take to be different to S₀ and Θ₀, see below) and subtract these equations from (9) and (10), respectively obtaining

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Here we have moved the dianeutral advection terms to the right-hand side since they only occur as a result of mixing processes and we wish to place all the mixing influences on the right-hand side. This step of subtracting the constant reference values S₁ and Θ₁ from the salinity and conservative temperature makes the divergence form of the conservation equations behave more like the advective forms (1) and (2). The resulting conservation equations for the anomaly variables are less sensitive to errors in the continuity equation (8) (McDougall 1991b; McIntosh and Rintoul 1997; Ganachaud and Wunsch 2000; Sloyan and Rintoul 2000). Since the lateral and vertical diffusion terms in (11) and (12) appear as gradients of S and Θ, we can regard (11) and (12) as exactly the same form of conservation equation as (9) and (10) except for the anomaly variables [S − S₁] and [Θ − Θ₁].

Now we ignore the temporal derivative term and spatially integrate (11) and (12) over the area from the Gulf of Cadiz to the S₀ contour (which almost exactly coincides with a Θ contour, Θ₀, on a neutral density surface). The volume flux into this layer from the Gulf of Cadiz is Q, which we will take as known from the work of Baringer and Price (1997). We will take the flow out across the S₀ contour to be cQ where we expect c to be not too different to unity because we will be considering contours that are not too distant from the Mediterranean outflow. In what follows we will assume that 0.7 < c < 1.0 . Taking the salinity and temperature of the Mediterranean outflow into this layer to be S_M and Θ_M, the area integrals of the left-hand sides of (11) and (12) {i.e., the area integrals of ∇_γ · (hV[S − S₁]) and ∇_γ · (hV[Θ − Θ₁])} are

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The values of Q, S_M, and Θ_M represent the values of the Mediterranean outflow after it has reached its equilibrium depth and has begun to spread out neutrally into the interior of the North Atlantic. That is, the values will be chosen as representative of the flow after the intense dianeutral mixing and entrainment in the downslope flow has occurred. This is done so that our values of D and K represent interior mixing processes in the North Atlantic rather than representing intense mixing and dilution in the outflow plume. It is convenient to rearrange these equations as

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where

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The second approximate equality follows because the salinity and temperature contrasts here are all measured along a neutral density surface and so are related though the thermal expansion coefficient and the saline contraction coefficient, which do not change substantially in this part of the North Atlantic. The values of S₁ and Θ₁ that we will choose will mean that F is little different to unity in our region of interest.

Now we come to the area integration of the right-hand sides of (11) and (12). The lateral diffusion terms become the lateral diffusion across the boundary of the control volume along the S₀ contour; namely,

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where n is the outward unit vector in two dimensions, normal to the S₀ contour, and dl is the element of length along the S₀ contour.

Before area-integrating the diapycnal mixing and advection terms in (11) and (12), we rewrite these terms in (11) as

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The first term, 0.5(D^u + D^l)(S^u_z − S^l_z), corresponds to the term DS_zz in (1), while the last term in (18) corresponds to D_zS_z in (1). The first of the dianeutral advection terms in (18) does not have an analogous term in (1) while the other term, −0.5(e^u + e^l)(S^u − S^l), corresponds to eS_z in (1). The next step in section 1 (where we were concerned with differential equations at a point) was to cross multiply the S and Θ equations by the vertical gradients of temperature and salinity. This step in section 1 eliminated the terms in D_zS_z and eS_z, so simplifying the effects of diapycnal mixing into a single term that was proportional to the curvature of the vertical S–Θ cast. We follow a similar procedure here, with the aim of minimizing the influence of dianeutral advection and of the vertical variation of D on the final equation. Only the contribution from the first term in (18) will be significant in the final equation.

We form the area-averaged vertical difference variables

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and form ΔS times the area integral of (12) minus ΔΘ times the area integral of (11), giving [using (15), (17), (18), and an equation corresponding to (18) for Θ]

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In addition, there are three other terms that should appear on the right-hand side of (20), namely, (A1)–(A3) of the appendix. We have omitted these terms here because in the appendix we show them to be negligible. Here A is the area from the Gulf of Cadiz to the salinity contour S₀, and the angle brackets indicate an area average over area A. This equation is the one used in this paper to deduce magnitudes for both the diffusivities D and K, given knowledge of FQ and of the hydrography in the eastern North Atlantic. We have used the symbol D in (20) for the average of the diapycnal diffusivity at the lower and upper interfaces, 0.5(D^u + D^l). In what follows we will take the diapycnal diffusivity D to be constant along a layer. It can also be noted from (20) that the numerical magnitudes that we obtain for both D and K in each layer are both directly proportional to the effective lateral advection, FQ.

Equation (20) corresponds to (4) in the advective approach, and notice that, like (4), (20) does not contain dianeutral advection and that diapycnal mixing enters proportionally to the curvature of the S–Θ diagram of the vertical water columns, averaged over the area A [since (S^u_z − S^l_z) and ΔΘ correspond to S_zz and Θ_z, respectively]. Vertical heave contributes considerable noise when evaluating S_zz and Θ_zz but, through cancellation, does not contribute significantly to the combination of terms (Θ_zS_zz − S_zΘ_zz), especially when the atlas data has been isopycnally averaged [as is the case with the atlas data of Gouretski and Koltermann (2004), which we use].

Equation (20) is, in fact, the conservation equation for the variable

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The choice of Θ₁ and S₁ as the volume-averaged conservative temperature and salinity of the control volume ensures that the term (A1) is minimized, while the choice of the ratio ΔS/ΔΘ as the ratio of the volume-averaged vertical gradients of salinity and temperature ensures that the vertical gradient of (21) is, on average, zero. This property minimizes the contribution of both dianeutral advection and the vertical gradient of D [terms (A2) and (A3)] to (20).

Note that at the level of maximum Mediterranean Water influence, where ΔS = 0 (20) reduces to

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In comparing this equation with (the negative of) the area integral of the original salinity conservation equation (9), it can be seen that in this special case where ΔS = 0 the price that has been paid for eliminating the influence of dianeutral advection across both the upper and lower interfaces is simply to introduce the factor F.

3. Diffusivities from hydrographic atlas data

Hydrographic data from the region of the North Atlantic near the Strait of Gibraltar have been taken from the atlas of Gouretski and Koltermann (2004). We have used the atlas at a horizontal resolution of ½° of longitude and latitude. Eight layers were chosen, bounded by nine neutral density surfaces, and in each of these layers we consider five areas from the Strait of Gibraltar to different salinity contours. In Fig. 1 we show S–Θ curves from the three locations (38°N, 350°E), (38°N, 345°E), and (38°N, 340°E). The easternmost cast has a salinity maximum close to a neutral density of γⁿ = 27.70 kg m⁻³, but the salinity maximum rises to less dense surfaces as one moves along the tongue to the west. As explained by McDougall and Giles (1987), such a diapycnal migration of a salinity maximum should not automatically be interpreted as implying asymmetric vertical mixing processes since salinity itself is an asymmetric variable. Another way of looking at this issue is to form a salinity anomaly variable that represents the presence of Mediterranean Water with respect to a background North Atlantic water mass field that exists in the rest of the North Atlantic. This variable has isolines that rise about 15°C for every unit increase in salinity, and the maximum of this anomaly variable does not so obviously move across isopycnals.

Based on the curvature of the casts at (38°N, 350°E) and (38°N, 345°E) we see that the peak salinity occurs at a density of approximately γⁿ = 27.65 kg m⁻³ and take this to be the central neutral density of the Mediterranean outflow. We will henceforth consider the hydrography between neutral densities of 27.50 and 27.80 kg m⁻³ in order to capture the majority of the outflow, consistent with the work of Baringer and Price (1997).

Looking closely at the S–Θ cast at (38°N, 350°E) we notice a “dented” section between the neutral densities 27.6 and 27.7. This dent represents a discontinuity in the curvature and is consistent with the observation that the Mediterranean Water does tend to have a strong signature on two distinct density surfaces (sometimes called two “cores”) close to the outflow region [as reviewed by Baringer and Price (1997)]. To minimize the influence of this vertical variation of S–Θ curvature we choose S₀ contours well away from the Gulf of Cadiz; that is, we choose contours that extend west of 340°E.

On the density surface γⁿ = 27.70 kg m⁻³ we show (Fig. 2) the five S₀ contours that are used as bounds for five overlapping areas on this density surface. For the other density interfaces used in this study we chose S₀ contours at the locations of the arrow tips in Fig. 2, thus forming regions of roughly equivalent areas on each density surface. In this way we have chosen S₀ contours passing through longitudes 332°, 336.5°, 338°, 340°, and 341.5°E, each at 35.5°N, on neutral density surfaces of 27.5, 27.55, 27.6, 27.65, 27.7, 27.725, 27.75, 27.775, and 27.8. Thus we have five sets of areas on each of eight density layers, giving 40 equations of the form (20). Note that the density intervals were chosen to give approximately equal vertical spacing (dbar) between layers. We have limited the area of our calculations to being west of 351°E in order to exclude the region of complicated geometry and complex mixing processes close to the Strait of Gibraltar and north of the Iberian Peninsula. Factoring in these areas would marginally increase the surface area and contour length, but we would expect it to produce a negligible change in our results. Table 1 gives the areas and lengths along the S₀ and Θ₀ contours corresponding to the largest area on Fig. 2. The salinity and conservative temperature of the Mediterranean Water entering each layer, S_M and Θ_M, are taken to be the maximum values close to (36°N, 352°E), which is close to the easternmost point of our dataset and the position of maximum salinity and conservative temperature.

Baringer and Price (1997) describe the Mediterranean outflow as it descends from the Strait of Gibraltar. After the bulk of the mixing and entrainment has occurred, the volume flux of “Mediterranean Water” into the North Atlantic is estimated as 1.52 Sv (Sv ≡ 10⁶ m³ s⁻¹). Based on this paper we take this as the transport of the outflow once it has stopped actively entraining and that this outflow was distributed vertically over a range of neutral densities of 0.75 kg m⁻³ centered at approximately γⁿ = 27.65 kg m⁻³. Plots of salinity and temperature on our surfaces affirm this range of densities. Hence we approximate the distribution of the flow as a sine wave with zeros at neutral densities of 27.65 ± 0.375 kg m⁻³ and an integral between this region equalling 1.52 Sv. Figure 3 shows the volume flux that is assumed to flow into each density bin of width 0.05 kg m⁻³. We use only the central half of the density range shown in Fig. 3, and the actual volume flux into our densest three layers is half that shown, simply because those layers have a density “width” of only 0.025 kg m⁻³. Baringer and Price’s estimate has been gathered from mooring data, and, although it is indeed not exact, we take it as a known for the purposes of this study. The sensitivity of changes in Q of ±10% is explored later in this section, and it is emphasized that the solutions for K and D are proportional to the magnitude of Q.

In Table 2 we present the three relevant coefficients that appear in (20) for each of the 40 areas considered (5 contours times 8 layers). The upper set of 40 numbers are of ∫_S0h(ΔS∇_γΘ − ΔΘ∇_γS) · n dl, which multiplies K in (20); the middle set are of A〈(Θ^u_z − Θ^l_z)ΔS − (S^u_z − S^l_z)ΔΘ〉, which multiplies D in (20); and the third set of coefficients are of Q(ΔΘ[S_M − S₀] − ΔS[Θ_M − Θ₀]), which multiplies F in (20).

We argue in the appendix that F lies in the range 0.85 < F < 1.0, or F = 0.925 ± 0.075, and in what follows actually take F to be unity because the uncertainty in estimating Q is considered to be larger than that associated with estimating F. Hence we will consider the left-hand side of (20) to be known for each of our 40 equations. As a first step, one might assume that the diffusivities K and D do not vary in space, either laterally or vertically. Then we have 40 equations in just two unknowns and find that the overdetermined least squares solution is K = 481 ± 114 m² s⁻¹ and D = 2.2 × 10⁻⁵ ± 7.5 × 10⁻⁵ m² s⁻¹. The estimate of the standard error of K was found by substituting D = 2.2 × 10⁻⁵ m² s⁻¹ into each of the 40 equations, and, similarly, the estimate for the standard error of D was found by substituting K = 481 m² s⁻¹ into each of the 40 equations. The spatially constant diapycnal diffusivity, at D = 2.2 × 10⁻⁵ ± 7.5 × 10⁻⁵ m² s⁻¹ while being relatively small, is also rather poorly known in that even its sign is in doubt.

The rather uncertain value found for the spatially invariant diapycnal diffusivity raises the question of whether this is simply indicative of resolving power of the method and the hydrography or whether this result was indicative of a real variation of diffusivity in space. Noting that each of our 40 equations are simply linear relations between K and D, we plotted these 40 linear relations on the D–K diagram and noticed that the straight lines for the five contours from each density layer tended to intersect, while these points of (near) intersection are different from one layer to the next. This is illustrated in Fig. 4, which shows 10 such straight lines, 5 from each of two different layers. There is a strong tendency for the lines from an individual layer to nearly cross at a point: This strongly suggested that the method has sufficient resolving power to be able to estimate both D and K and that these diffusivities were not constant in the vertical.

Hence, we performed least squares fits, assuming that the diffusivities K and D were constant along each layer. In this way we performed eight such fits with five equations in each fit, and the resulting values of the diffusivities for the eight layers are shown in Fig. 5 and Table 3. The errors bars σ_D and σ_K for these diffusivities are obtained by taking the constant value of the other diffusivity (either K or D) in each layer and the five equations, (20), then give five values of the diffusivity in question (either D or K) from which the standard deviation is calculated.

The first thing to notice in Fig. 5 is that the diffusivities are all positive and they vary quite smoothly from one layer to the next rather than varying randomly in the vertical. The standard error of these diffusivities is quite small, especially for the dianeutral diffusivity D. The error bars here do not reflect any uncertainty in the estimate of the effective transport FQ but rather the resolving power of the signature of the tracer patterns in the averaged hydrography to be able to distinguish between dianeutral and epineutral diffusion for a given FQ. We find the rather small standard errors of these diffusivities remarkable and surprising.

The vertical structure of the diapycnal diffusivity in Fig. 5 exhibits a rather strong vertical variation from the core of the Mediterranean Water at γⁿ ≈ 27.65 kg m⁻³ where we find D ≈ 2 × 10⁻⁴ m² s⁻¹ toward deeper layers where the diapycnal diffusivities are seen to be about 0.5 × 10⁻⁴ m² s⁻¹. The epineutral diffusivities in this region of about 200 m² s⁻¹ are much less than values nearer the sea surface but may be appropriate for this depth and region of the ocean. The orderly progression of both D and K with density is probably due to our use of the rather smooth hydrographic atlas data. Nevertheless, the small error bars on these diffusivities give us some hope that the idea of carefully choosing the reference temperatures and salinities and the ratios ΔS/ΔΘ in order to achieve the three properties described in section 2 may add skill to the determination of diffusivities in a more general inverse problem. There have been few observational studies of diffusivity in this region. In the North Atlantic Tracer Release Experiment (NATRE) region (25°N, 332°E), which is farther to the south and west of the area that we are considering, Ferrari and Polzin (2005) found D ≈ 10⁻⁵ m² s⁻¹ and K ≈ 300 m² s⁻¹. Our diapycnal diffusivity is a factor of about 10 greater than this and our epineutral diffusivity is about a factor of 2 smaller.

The relative importance of vertical and epineutral diffusion in explaining the effective lateral advection of Mediterranean Water through this part of the North Atlantic is indicated in Fig. 6. For each density layer we have used the least-squares-determined values of D and K to evaluate the fraction of the right-hand sides of the five equations, (20), that are due to vertical and epineutral diffusion. The regular progression of the lines in this figure is probably indicative of the smoothed nature of the hydrographic atlas data and the fact that the areas that we consider are cumulative so that the properties of the area might be expected to vary slowly. The dashed curve with the smallest fractional influence is the curve for the largest horizontal area (the westernmost contour), and the successive dashed lines are the contours proceeding to the east in a monotonic fashion. The opposite occurs for the full lines where the curve for the smallest fractional influence is the easternmost contour with the smallest horizontal area. Perhaps this regular progression indicates that there may be more information that can be extracted using this technique, such as trying to determine a spatial variation of the diffusivities inside each isopycnal layer. We have not attempted to do so in this exploratory study but note that, if we took the diapycnal diffusivity to decrease and the lateral diffusivity to increase away from the Gulf of Cadiz, this would be consistent with the differences in D and K between the NATRE site and our study region, as well as tending to have the five solid lines and five dashed lines in Fig. 6 collapsing onto single full and dashed lines.

We have checked the robustness of the technique presented in this paper by varying the values of the vertical difference in salinity and conservative temperature, ΔS and ΔΘ, separately by ±10% and then redoing the analysis. These four separate changes typically resulted in changes to the diffusivities of less that 1%, confirming that the terms (22) and (23) that we neglected in arriving at our conservation equation (20) are indeed small. Another assumption built into our results is the vertical structure of the volume flux of Mediterranean Water entering the North Atlantic. Since we examined only the central range of densities between neutral densities of 27.50 and 27.80, we effectively chose the volume flux per unit density interval to be nearly constant (see Fig. 3). Nevertheless, this is an assumption. and our results in each density layer are directly proportional to the volume flume entering the layer.

Another key assumption that we have made is to use the same diapycnal turbulent diffusivity for heat as for salt. This is equivalent to assuming that the diapycnal mixing is caused by isotropic turbulent mixing. Another assumption would be to take the ratio of the diapycnal fluxes of heat and salt to be that in salt fingers, but we have not attempted this here.

4. Diffusivities from Hallberg Isopycnal Model data

Model output from the same region of the North Atlantic as in section 3 has been gained from the Hallberg Isopycnal Model (HIM; Hallberg 2000). There are three main advantages of using HIM as a means of testing the accuracy of the method already applied to the atlas data: First, HIM is one of the few global circulation models to impose a volume transport through the Strait of Gibraltar. If only temperature and salinity fluxes are allowed through the strait the overall dynamics of the region are changed (Griffies et al 2005). Second, by utilizing a density coordinate system HIM avoids the unwanted mixing effects caused by the Veronis effect in z-coordinate GCMs (Veronis 1975). Third, as HIM is already on density surfaces, we avoid any error implicit in interpolating to density surfaces from a Cartesian grid of Θ, S, and p values. The HIM output used is the time average of the final 20 years of a 100-yr coupled experiment The model uses a σ₂-coordinate system with a total of 49 layers in the vertical and a horizontal resolution of 1° of longitude and latitude. The model is not eddy resolving, has an isopycnal diffusivity (K_model) of 600 m² s⁻¹, and employs a Richardson number (Ri) dependent vertical diffusion scheme that results in high vertical diffusivities [D_model O(10⁻⁴)] in regions of high Ri and relaxes to a background profile [depth-dependent O(10⁻⁵)] in the deep ocean. For the region explored by this study, D_model is close to 1.8 × 10⁻⁵ (i.e., within 10%).

The model stores velocity, advective flux, and diffusive fluxes of heat and salt in the meridional, zonal, and vertical directions. From the advective fluxes close to the Strait of Gibraltar we are able to assign a specific Q value to each layer in the vertical. Layers 36.0, 36.1, 36.2, and 36.3 are chosen as they represent the majority of the volume of the salt tongue and are bounded by five surfaces equally spaced in density coordinates. Contours are chosen such that they pass through latitude 35.5°N and longitudes 336°, 338°, 340°, 342°, and 344°E. One downside of using model output is the unsmoothed nature of the data close to the continents. Along both the Portuguese and Moroccan coasts the pressures on density surfaces are quite variable, and this data is removed by defining an eastern boundary to the region that is 1°–2° from the coast. The inflow Q may now be approximated as the volume flux through that boundary rather than simply the flux out of the Gulf of Cadiz. The region in question is shown in Fig. 7 along with the position of all five contours on the 36.2 σ₂ layer.

Once again we have five sets of areas, now on each of four density layers, giving 20 equations of the form (20). Our initial overdetermined least squares solution is K = 586 ± 131 m² s⁻¹ and D = 3.42 × 10⁻⁵ ± 1.79 × 10⁻⁵ m² s⁻¹. It was found in the previous section that, when K and D were solved for each layer, much greater precision resulted, and it is thus pertinent to look to the intersection of lines in K–D space formed by each equation. A comparison between the lines formed by areas on both the 36.2 σ₂ and 36.3 σ₂ layers is shown in Fig. 8. While the lines formed by the 36.2 σ₂ layer compare well to those of the hydrographic atlas, each having varied slopes and intersecting within a small range, the lines formed on the 36.3 σ₂ layer are close to parallel. Clearly the ratio of coefficients in front of K and D in Eq. (20) varies little within each extra area on the 29.0 σ₂ layer, and thus not enough information is present to solve for K and D within that layer individually. Possible explanations for this may lie in the averaging problem mentioned earlier and the fact that there may be something implicit in HIM that maintains a constant ratio between the K and D coefficients in (20), which is not observed in the real Mediterranean outflow.

We chose to solve for K and D for each set of equivalently sized areas in the vertical, making the assumption that K and D are constant on each set. The values obtained along with the average values used in HIM for these regions are displayed in Table 4 and Figs. 9a and 9b (for F = 1), and these results show that the diagnosed K and D values vary little throughout this region. The model values also vary little in space but differ from those diagnosed by around a factor of 2 in the D case and 20%–50% in the K case.

In the above calculations the assumption that FQ = Q has been made, as was the case in section 3, where an estimate for Q was all that was known. Since time-averaged volume fluxes are known at every point in space of the HIM data, we examine whether the inclusion of a better approximation for F will improve our ability to estimate K and D. In section 2, it was shown that FQ can be thought of as the average of the flow into the layer from the Mediterranean and the flow out through the contour that bounds it. Making this modification to the left-hand side of (20), the overall least squares estimates become K = 625 ± 109 m² s⁻¹ and D = 2.52 × 10⁻⁵ ± 1.87 × 10⁻⁵ m² s⁻¹, and K and D for each set of areas in the vertical are shown in Figs. 9a and 9b. Here K adjusts slightly upward, while D is reduced by 30%–50%, bringing the model D value well within the standard deviation of that diagnosed by this inverse technique. With F = 1, the gain or loss of volume within each layer is not accounted for. In the above case, diapycnal fluxes create a reduction in the outflow volume of each layer, and constructing an inverse model without taking this into account gives an overestimate of the vertical diffusivities. When advective fluxes are incorporated into the inverse model through the variable F, a better estimate for D results, with little change to our estimate for the lateral diffusivity K.

5. Summary

By taking a carefully chosen linear combination of the conservation statements of volume, salinity, and conservative temperature, we have arrived at (20), which is effectively a conservation equation for the variable ΔS(Θ − Θ₁) − ΔΘ(S − S₁) where the offset values Θ₁ and S₁ are chosen to be the volume-averaged conservative temperature and salinity of the control volume, and the vertical differences ΔS and ΔΘ are the area-averaged differences between the values at the upper and lower interfaces bounding a density layer. Because of its careful construction, the vertical gradient of this variable is on average zero, and hence dianeutral advection makes a negligible contribution to (20), as does the vertical gradient of the dianeutral diffusivity. Rather, the process of dianeutral mixing enters (20) only as the average of the dianeutral diffusivity at the upper and lower interfaces. In addition, this dianeutral diffusion term is proportional to the vertical curvature of the S–Θ diagram, which is much less subject to numerical noise than is traditionally the case when estimating second derivatives such as S_zz and Θ_zz. These advantages were the motivation for forming this rather careful linear combination of the conservation equations.

Needler and Heath (1975) analyzed the Mediterranean Water signature in the North Atlantic using a salinity anomaly measured at fixed values of potential temperature with respect to a linear S–θ relation. Diapycnal advection of this salinity anomaly was ignored in their study. This can be justified at the central density of the Mediterranean Water because there this type of salinity anomaly shares the same important property as our (21), namely that its vertical gradient is zero, so the effect of diapycnal advection can be neglected. Above and below the central density, the vertical gradient of the salinity anomaly is nonzero, and our technique, which has a different ratio ΔΘ/ΔS for each density layer, is preferred.

Before using this technique in a more general inverse model of ocean hydrography we have applied it to the hydrography in the eastern North Atlantic, which has two very important advantages for our purposes. First, the volume flux entering the North Atlantic from the Mediterranean Sea can be taken as known and regarded as steady and, second, the hydrographic data has a very obvious Mediterranean Water signature. The extra terms (A1)–(A3) due to dianeutral advection and the vertical variation of the dianeutral diffusivity that should appear in (20) have been shown to have a negligible influence on the results of our study.

The method seems able to distinguish between the effects of epineutral and dianeutral diffusion on the dilution of the Mediterranean Water signature in the eastern North Atlantic. Also, the overdetermined least squares “inversion” gives dianeutral and epineutral diffusivities that vary smoothly in the vertical. The data examined here is from an ocean atlas (Gouretski and Koltermann 2004), and it remains to be seen how the method performs on unaveraged ocean hydrography. In this region of the ocean meddies are known to transport a significant amount of heat and salt laterally (Armi and Zenk 1984), and it is clear that this mechanism will not be well parameterized as diffusion. The analysis of the HIM data provided a further test of the ideas. It is not clear why the coefficients of the two diffusivities were so closely proportional along one of the model layers but, in any case, the technique recognized this and attached large uncertainties to the diffusion coefficients. We are encouraged by the results of this technique in this rather special region and plan to apply it as a more general inverse technique in the ocean, where one does not have an a priori estimate of the volume flux entering one side of the box.