a. Walin’s relations for formation rate

Walin (1982) derived elegant relations between water mass formation and diffusive and radiative (nonadvective) heat fluxes, combining heat and volume budgets for an isothermal layer. Here, we repeat the derivation of Walin’s relations in terms of potential density and illustrate their use with idealized examples. Throughout the paper, when we refer to “density,” it should be understood as potential density.

1) Water mass formation and diapycnal volume fluxes

Consider the volume sandwiched between the isopycnal surfaces with potential densities ρ and ρ + Δρ. We consider a limited area of the ocean, such as the North Atlantic, with an open boundary (Fig. 2a).

We first define water mass formation using a volume budget of a density layer. We write

ΔV the volume of fluid with density between ρ and ρ + Δρ

ΔΨ the volume flux of fluid with density between ρ and ρ + Δρ out of the domain

G(ρ), G(ρ + Δρ) the diapycnal volume flux of fluid crossing the ρ and ρ + Δρ isopycnals respectively;note that Speer and Tziperman (1992) refer to this term as transformation and denote it by A. The sign convention is (see Fig. 2a) that G is positive if directed toward increasing ρ.

We assume incompressibility. Then the volume budget of the control volume bounded by the ρ and ρ + Δρ isopycnals, the ocean surface, and the open boundary is

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Summing the volume inflation and the outflow terms gives the water mass formation

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where the water mass formation per unit of density, M, is given by the convergence of the diapycnal volume flux

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2) Diapycnal volume fluxes and density sources

We now consider the density budget for the same layer (see Fig. 2b). We write D as the density flux per unit area and D as the area-integrated density flux across an isopycnal, defined to be positive in the direction of increasing density. We then have

D_diff(ρ), D_diff(ρ + Δρ) the integrated diapycnal density flux across the ρ and ρ + Δρ isopycnals, and

∫_outcrop D_in dA the total density flux into the ocean where the surface density lies between ρ and ρ + Δρ; here D_in is the density influx per unit area.

The density budget of the control volume between the ρ and ρ + Δρ isopycnals is a balance between advective and diffusive density fluxes:

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or more concisely

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where we have introduced F, the surface density influx per unit of density (Fig. 2b):

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This F is Speer and Tziperman’s (1992) transformation driven by air–sea fluxes.

But by volume conservation (4) M = −∂G/∂ρ, so the density content gain implicit in volume inflation and lateral outflux may be eliminated in (5), leaving

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Thus, a cross-isopycnal volume flux directed from light to dense, G(ρ) > 0, requires a density supply either from a convergence of diapycnal density fluxes, −∂D_diff/∂ρ > 0, or from the surface, F > 0. There is, in fact, also an extra density source in (5) arising from cabbeling—densification through mixing—for example, McDougall (1984). This term is generally small and is neglected here for simplicity.

Combining (7) and (4) gives (1)

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Note that this relation for water mass formation holds for both steady and time-varying cases.

3) The “extended isopycnal” and the “net” diffusive density flux

It is useful to generalize the diapycnal density flux D so as to include the surface density flux. We thus define the “surface extension” of the isopycnal to lie along the ocean surface where the isopycnal is too light to exist—this “surface extension” is denoted by the dashed line in the schematic Fig. 3a. The total flux across this surface extension

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(here ρ_m is the mixed layer density) is equal (see Fig. 3b) to the density influx into the outcrops of all the water denser than ρ,

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Hence

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and combining (7) and (10) gives

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where

D_netD_surfD_diff(12)

is the total nonadvective diapycnal density flux across the extended isopycnal made up of the isopycnal and its surface extension, that is, the flux into the waters denser than ρ.

Equation (11) then states that the diapycnal volume flux is given by the convergence of the net diapycnal density flux, −∂D_net/∂ρ, into the control volume made up of the region between the two isopycnals and their surface extensions (Fig. 3c).

Accordingly, the water mass formation, M, is related to the net diapycnal density flux across the extended isopycnal by

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b. Idealized example 1: Water mass formation by surface fluxes alone

As an illustration of Walin’s formulation, consider an idealized example where a basin gains density everywhere at the surface (F > 0) and there is no mixing; that is, D_net = D_surf. Suppose ρ_dense is the density of the densest isopycnal that outcrops and that the lightest water that exists anywhere in the basin has density ρ_min (see Fig. 4a). The surface density gain leads to D_surf being positive over all the isopycnals that outcrop (ρ_dense > ρ > ρ_min); but it is zero for denser isopycnals with ρ > ρ_dense (shaded in Fig. 4a) that are unventilated within the domain. A hypothetical form for D_surf in this cooling case is shown in Fig. 4b. Note that D_surf(ρ_min) is the same as the total density gain over the basin since the area ρ_m(x) > ρ_min in (8) covers the whole surface of the basin.

In this example, since there is density gain everywhere, ∂D_surf/∂ρ = −F is (Fig. 4c) always negative [see (8) and (10)], and the maximum value of D_surf is attained (Fig. 4b) for the lightest water at ρ_min.

The surface density gain (11) drives a diapycnal volume flux (Speer and Tziperman 1992):

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Since there is surface density influx, F > 0 (or alternatively a convergence in D_surf), the diapycnal volume flux G(ρ) is directed (Fig. 4c) from light to dense water.

The resulting water mass formation M(ρ) is a consequence of the convergence of the diapycnal volume flux,

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As the surface forcing induces G(ρ) > 0 within the domain and by definition G(ρ) = 0 for isopycnals outside the physical domain, then there is a convergence of G (hence positive M) at high ρ and a divergence of G (hence negative M) at low ρ (Figs. 4c, d). Therefore, the surface forcing drives a formation of dense water and a compensating destruction of light water.

This formation of dense water either leads to the basin filling up with denser water or the export of dense water from the basin through an open boundary. Thus, in this example, a steady-state solution over a closed domain cannot be achieved.

c. Idealized example 2: The steady state in a closed basin

In a steady state in a closed basin, there is no water mass formation at any density, so M ≡ G ≡ D_net ≡ 0. The total density flux D_net(ρ) = D_surf(ρ) + D_diff(ρ) must be zero down into the region (shaded in Fig. 5a) below any extended isopycnal; that is, the region occupied by water denser than ρ. Otherwise the density content of this region would change. In the artificial limit of no diffusion, the surface forcing would have to adjust such that D_surf(ρ) ≡ 0. However, diffusion always mixes fluid, so giving −ve D_diff (arrow going from dense water to light in Fig. 5a; the dashed line in Fig. 5b). Hence with diffusion a steady state with D_net ≡ 0 implies that the surface densities (or fluxes) must adjust so that D_surf > 0 (dotted line in Fig. 5b) for all ρ: the surface fluxes must act so as to increase the density contrast.

Note that the diffusive flux must disappear (D_diff = 0) across the isopycnal surfaces for the lightest ρ = ρ_min and heaviest ρ = ρ_max waters, as the area of these isopycnal surfaces falls to zero. Here D_diff reaches a maximum magnitude for some intermediate density with a large isopycnal area. The gradient of D_diff is negative for low ρ and positive for high ρ, so it makes (Fig. 5c) low ρ waters denser (G_diff(ρ) > 0) and high ρ waters lighter (G_diff(ρ) < 0). Conversely the surface fluxes F attempt to make light waters lighter and dense waters denser (Figs. 5a, c).

a. Lateral mixing within the mixed layer

Lateral mixing within the mixed layer is driven by eddies in baroclinically unstable regions. The horizontal density flux per unit area D is often parameterized by

Dκ_h∇ρ_m

with κ_h the horizontal (eddy) diffusivity. The total lateral flux across the ρ isopycnal is then given (here h is the depth of the mixed layer) by

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assuming that the isopycnals are aligned roughly east–west and writing L_EW as the zonal width of the basin. The overbar denotes an average taken along the outcrop. This mixing is most likely to be important where the mixed layer is both deep and has strong lateral gradients, as in the winter North Atlantic near the Gulf Stream and Labrador Current (see, e.g., Fig. 13 and the model diagnosis of Marshall et al. 1998).

In such regions the annual-average (denoted by the angle brackets) lateral flux 〈D_lat〉 will be about half its winter value, as the winter mixed layer depth h_winter > 200 m is much greater than the summer depth h_summer ∼ 20 m:

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Moreover the mean density gradient |∇ρ_m| will be dominated by its value in regions where it becomes large, such as western boundary currents (WBCs). Suppose that the isopycnal runs along a WBC for ∼10³ km, the diffusivity κ_h ∼ 1000 m² s⁻¹ (see, e.g., Haine and Marshall 1998), the density gradient is 10⁻⁵ kg m⁻⁴ (1 kg m⁻³ in ∼100 km) and the winter mixed layer depth reaches 400 m. Then 〈D_lat〉 ∼ 2 × 10⁶ kg s⁻¹.

b. Mixing at the base of the mixed layer

There is enhanced mixing near the base of the surface mixed layer through the action of the wind, penetrative convection, and internal wave breaking, which is parameterized in mixed layer models (see Large et al. 1994 for a review). In particular, mixing occurs whenever dense thermocline water is entrained into the lighter mixed layer.

1) The entrainment flux per unit area, D_ent

We now evaluate the entrainment flux across isopycnals lying at the base of the mixed layer. The rate of entrainment of fluid into the mixed layer

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where w_b and u_b are the vertical and horizontal velocities at the base of the mixed layer. Entrainment can result from either (i) sustained upwelling in equatorial and eastern boundary regions (w_b > 0), (ii) the seasonal deepening of the mixed layer in autumn and winter (∂h/∂t > 0), or (iii) net annual heat loss, giving a winter mixed layer deepening from year to year (u_b · ∇h > 0), as happens as fluid moves northward in the western and northern North Atlantic. However, the density differences at the base of the winter mixed layer are too small for the entrainment driven by winter-on-winter deepening (iii) to be significant in mixing density.

As thermocline water is entrained into the mixed layer, the water crosses the strong density gradient at the base of the mixed layer (Fig. 7). There is a density jump Δρ = ρ₋ − ρ_m, from its thermocline value ρ₋(x) to its value in the mixed layer ρ_m(x).

Over a time Δt, a thickness w_eΔt of thermocline water has its density decreased by Δρ from ρ₋ to ρ_m. Hence density content w_eΔtΔρ per unit area (the hashed region in Fig. 7) is removed from the entrained water and mixed up into the rest of the mixed layer. In other words, there is a density flux per unit area w_eΔρ = w_e(ρ₋ − ρ_m) up into the mixed layer and across the ρ_m isopycnal.

However, we wish to find the density flux across a general ρ surface lying within the density jump, ρ_m < ρ < ρ₋. As water is entrained up into the mixed layer, the density content of the isopycnals denser than ρ decreases by w_eΔt[ρ₋ (x) − ρ] per unit area (shaded in Fig. 7). Hence the instantaneous entrainment density flux per unit area D_ent(ρ, x) down across the ρ isopycnal is

D_entρ,xw_exρ₋xρ(19)

In reality, of course, the density jump is smoothed into a transition zone of strong but finite density gradient. An advection/diffusion transition layer of thickness κ_υ/w_e forms, where κ_υ is the vertical diffusivity. However, as long as the scale of variation of w is greater than the transition layer thickness so that fluid passes through all the isopycnals at a rate w_e, (19) still holds.

Values of diffusivity immediately below the mixed layer vary widely. The strong shear in equatorial upwelling zones gives comparatively large diffusivities, and hence thick transition layers, ∼50 m deep (Peters et al. 1988), a scale similar to that over which the vertical velocity w varies (Bryden and Brady 1985). Here, therefore, (19) will not hold, so the mixing driven by equatorial upwelling is best considered as a thermocline diffusion. In midlatitudes the shears and hence the diffusivity are weaker, and the density gradient is much sharper (see, e.g., Brainerd and Gregg 1996). The transition layer is thus typically much thinner than the scale over which w varies, so the mixing driven by the seasonal mixed layer cycle at midlatitudes can be thought of as an entrainment flux.

We thus focus on the process of seasonal entrainment and now estimate its basin-integrated importance.

2) The basin-integrated seasonal entrainment flux

Integrating (19) over the whole basin gives the basin-integrated entrainment flux

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Here A_base = {x : ρ_m(x) < ρ < ρ₋(x)} is the area where the ρ isopycnal lies within the base of the mixed layer (Fig. 8).

Along the northern boundary of the region A_base, the isopycnal outcrops (Fig. 8) into the mixed layer, ρ = ρ_m, and ρ₋ − ρ = Δρ (which will, in general, vary along this outcrop). Moving south through A_base, ρ₋ − ρ declines until at its southern end the isopycnal passes into the thermocline and ρ₋ − ρ falls to zero. Hence the basin-integrated entrainment flux across the ρ isopycnal might be

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where the overline denotes the along-outcrop mean of Δρ.

We now need to estimate the area A_base(ρ). The zonal-mean meridional extent of A_base(ρ) is

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where ∂ρ₋/∂y is the gradient of the “top of thermocline” density ρ₋. Like the entrainment flux, L_base is proportional to the density jump Δρ. The area A_base(ρ) is then ∼ L_baseL_EW, where L_EW is the zonal width of the basin.

Hence the basin-integrated contribution to the entrainment flux

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where again the overbar denotes the outcrop-mean.

Note that, since both the area-averaged entrainment flux, (21), and the area over which it exists (22) are proportional to the density jump Δρ, the integrated flux D_ent is proportional to (Δρ)².¹

In spring and early summer the mixed layer retreats and there is no density jump at its base, so Δρ ≈ 0; while from late summer to mid winter, Δρ ≈ 1.0–0.4 kg m⁻³. See the temperature profiles at OWS Papa (Fig. 1), where ΔT ranges from 4° to 2°C from July to November. Hence, the entrainment flux is only significant from late summer to early winter—“autumn.” Where the mixed layer becomes deeper than 80–100 m, the density jump generally becomes insignificant. We therefore introduce the notation

hh_{early winter}h_summer(24)

for the change (∼50–100 m) in mixed layer depth over the autumn period while there is still a substantial density jump at its base.

Using (24) we can integrate (23) over the year, assuming that the entrainment rate w_e ≈ ∂h/∂t, which gives an estimate for the annual-mean entrainment mixing driven by the seasonal cycling as

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where (Δρ)² is the entrainment-weighted mean over the autumn period and τ_year the length of a year.

A plausible density gradient in the midlatitudes is ∼6 × 10⁻⁷ kg m⁻⁴ (1.2 kg m⁻³ in 20° of latitude). From the OWS Papa data, a typical autumn value of the density jump Δρ might be ≈0.8 kg m⁻³, and the autumn mixed layer change [h] might be ∼80 m. Choosing a basin width appropriate to the Atlantic, L_EW ∼4 × 10⁶ m then gives 〈D_ent(ρ)〉 ∼5 × 10⁶ kg s⁻¹. Assuming c_pρ = 4 × 10⁶, dρ/dT = 0.25 kg m⁻³ °C⁻¹, this density flux is equivalent to a heat flux of ∼0.1 PW across the isopycnal.

Now the diapycnal volume flux across the isopycnal driven by this entrainment mixing G_ent = −∂D_ent/∂ρ. Assume that the annual average 〈D_ent〉 of 6 × 10⁶ kg s⁻¹ is achieved in the North Atlantic at σ ≈ 26.0 and that the outcrops become very small in area by σ ≈ 28.0 so that 〈D_ent〉 disappears at σ ≈ 28.0. Then a mean value of G_ent over these densities is Δ〈D_ent〉/Δρ ≈ 3 Sv (Sv ≡ 10⁶ m³ s⁻¹), consistent with the estimate of Garrett and Tandon (1997).

c. Diapycnal mixing in the thermocline

The basin-integrated interior diffusive density flux D_thermo(ρ) is defined as

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where ρ_bot (x) is bottom density and ∂ρ/∂z is the local vertical gradient of ρ. This flux, however, is difficult to scale, given the variability in vertical diffusivity (see Caldwell and Moum 1995 for a review) and stratification. Recent tracer release experiments have emphasized that the vertical diffusivity in the interior of the ocean is only of the order 10⁻⁵ m² s⁻¹ (Ledwell et al. 1993). In contrast, the mixing may be greatly enhanced near the boundaries, such as near sidewalls (Armi 1978; Toole et al. 1994) and near the seafloor (Polzin et al. 1995).

d. Comparison of mixing processes

In the steady state over a closed basin, as in the example discussed in section 2c, the total density flux down into waters deeper than ρ,

D_netD_surfD_entD_latD_thermo

Since it is difficult to scale D_thermo(ρ), we shall instead scale D_lat and the seasonal component of D_ent against D_surf, the flux across the ocean surface into waters denser than ρ. Now the annual mean

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where D_in is the annual-mean area-mean density gain north of the outcrop and, again, L_EW and L_NS are estimates of the zonal and meridional scale of the region poleward of the ρ outcrop.

Assuming D_in ∼ 2 × 10⁻⁶ kg m⁻² s⁻¹ (equivalent to a heat loss of ∼40 W m⁻²) over this region north of the isopycnal outcrop, together with L_EW ∼ 4 × 10⁶ m, L_NS ∼ 3 × 10⁶ m implies a 〈D_surf(ρ)〉 ∼ 24 × 10⁶ kg s⁻¹. This estimate is consistent with climatological values for the North Atlantic [see, e.g., that derived from the climatology of Esbensen and Kushnir (1981), the dotted line in Fig. 11] and is clearly considerably larger than the values (2–5 × 10⁶ kg s⁻¹) estimated in sections 3a and 3b for the lateral mixing and seasonal entrainment terms. In the steady state, this surface flux must largely be balanced by thermocline diffusion over a closed basin or possibly result in water mass transformation over an open domain.

The surface flux dominates, for example, the seasonal entrainment flux for two main reasons: (i) the magnitude of the annual-mean surface flux north of the outcrop D_in, is greater than the entrainment flux D_ent and (ii) the north–south extent of the region poleward of the outcrop, L_NS, is much greater then the north–south extent of the entrainment region, L_base.

We now estimate the relative importance of the two mixed layer contributions: the lateral diffusive flux and the seasonal entrainment flux. From (25) and (17)

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where the overbars denote the average along the outcrop and we have assumed that h_winter and [h] vary relatively little along the outcrop.

Hence, 〈D_lat(ρ)〉 will be relatively more important than 〈D_ent(ρ)〉 for (a) h_winter ≫ [h], that is, winter mixed layers much deeper than ∼100 m; (b) large diffusivity κ_h; (c) large |∇ρ_m|; (d) small Δρ in autumn; and (e) large (∂ρ/∂y)⁻¹, giving a small entrainment region width

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Hence for denser isopycnals, outcropping in the subpolar gyre where winter mixed layers are deep, lateral diffusion is more important, while for midlatitude isopycnals outcropping into shallower mixed layers seasonal entrainment is more significant, especially where there is a large density jump at the base of the mixed layer.

It is important to note that a large outcrop-averaged mixed layer density gradient can be consistent with a relatively wide entrainment region. The outcrop-averaged mixed layer density gradient is much greater than the reciprocal of the outcrop-averaged mean isopycnal spacing |∇ρ_m| ≫ [(∂ρ/∂y)⁻¹]⁻¹ if the isopycnal separation varies strongly across the basin, as it will do if the isopycnals lie close together within a western boundary current and then spread out in the ocean interior.

In the following section, we assess the importance of each of these diffusive terms in a general circulation model of the North Atlantic.

a. The ocean model

In the present study, we examine the formation of water masses in a coupled mixed layer and isopycnic model of the North Atlantic. We employ a 30-yr run of the Miami Isopycnic Model (MICOM), developed by Bleck and his coworkers (Bleck et al. 1989; Bleck et al. 1992). The forcing and model configuration are as discussed in New et al. (1995).

A Kraus–Turner mixed layer overlies 19 isopycnic layers representing the ocean interior with potential density anomaly σ ≡ ρ_pot − 1000 kg m⁻³ ranging from 25.65 to 28.15. Here ρ_pot is potential density referenced to atmospheric pressure; in the following it is simply written ρ. The model is run over a (closed) domain (approximately from 15°S to 82°N over the North Atlantic). The model grid is based on a rotated Mercator projection, with a horizontal resolution of around 1°.

b. Surface and diffusive integrated density fluxes

1) Surface forcing

We are interested in the net-annual water mass production. Hence we consider annual-averaged fluxes, evaluated by “online” integration over model year 31.

The annual-averaged surface density flux applied in the model is shown in Fig. 9a. Assuming an expansion coefficient α_E = −ρ⁻¹dρ/dT = 2.5 × 10⁻⁴ K⁻¹, a density influx of 10⁻⁶ kg s⁻¹ m⁻² is equivalent to a heat outflux of ∼17 W m⁻². As expected, there is a net gain of density generally north of a zero line running from 10°N, 60°W to 45°N, 10°W and a loss south of it. This surface flux is the sum of the climatological fluxes and the relaxation (29). In comparison, the climatological density flux shown in Fig. 9b shows a greater density loss in the Tropics and has a zero line that lies farther north.

In applying Walin’s relations, however, we need to calculate annual averages following isopycnals as they move from season to season. The annual-averaged diapycnal surface flux 〈D_surf(ρ)〉 across the surface extension of the ρ isopycnal is then given, writing τ_year as the length of the year, by

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The finite-difference forms for 〈D_surf(ρ)〉 and other diagnosed fluxes are shown in the appendix. For clarity, in the following, we shall drop the explicit notation 〈 〉 for the annual mean, and D, G, and M should be read as annual means.

The surface flux D_surf (the dotted line in Fig. 10a) is positive over all the outcropping layers in the domain σ_min ≈ 20.0 < ρ < σ_dense ≈ 28.15 and is zero for any denser, unventilated layers σ > σ_dense (like those shaded in the schematic Fig. 4a). Here D_surf is positive for the lightest water σ = σ_min ≈ 20.0 since there is a slight basin-integrated density gain by the model (average value ∼6 × 10⁻⁸ kg s⁻¹ m⁻² ∼ a heat loss of 1.3 W m⁻²). Note that waters lighter than σ = 25.65 (denoted by the vertical gray line in Fig. 10 and succeeding figures) only exist within the mixed layer.

Given that D_surf(ρ) is the surface density input into all the isopycnals with density greater than ρ (9),

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it follows that the negative of the slope of D_surf gives F, the surface density input into an isopycnal range per unit of density (the dotted line in Fig. 10b). Denser isopycnals (σ > 25.4) outcrop into the mixed layer to the north and so receive annual-averaged density gain, while the lighter isopycnals σ < 25.4 outcrop in the Tropics and so lose density. Hence D_surf peaks at σ = 25.4, and the slope of D_surf changes from positive for σ < 25.4 to negative for σ > 25.4. This lightening of light waters and making dense of dense waters means that the surface forcing acts to increase the density contrast across the basin, as in the idealized example of the steady state in section 2c.

The density flux D_surf(ρ) (30) is the sum of the climatological forcing and a Haney-style relaxation (29). The climatological component is denoted by the dotted line in Fig. 11. It is large and negative for σ = σ_min ≈ 20.0 because the Esbensen and Kushnir climatology gives a basin-integrated density loss over the North Atlantic. This density loss is concentrated over the Tropics (Fig. 9b); hence the steep positive slope of the dashed line in Fig. 11 over the lighter densities. The implied net lightening is contrary to the normal view of the North Atlantic as exporting cold, dense water and importing warm, light water. The Haney forcing (the dashed line in Fig. 11) opposes this excessive density loss over the Tropics, giving a much more realistic net basin-integrated density gain (solid line).

The climatological forcing also has a larger maximum at σ = 25.4 than the total surface flux; here the Haney relaxation is acting to weaken the “unmixing” driven by the surface forcing.

Different climatologies, of course, will give different F and D_surf. Speer and Tziperman (1992) used surface fluxes from the climatology of Isemer and Hasse (1987) and found F reaching a maximum of 30 Sv at σ = 26.3 (their Fig. 6b). However, the climatology of Esbensen and Kushnir (1981) used in our model gives (not shown) a maximum diapycnal volume flux of only 19 Sv, at σ = 26.5, which is reduced further (dotted line in Fig. 10b) to 12 Sv (both in the full domain and in that part of the domain north of 24°N) by including the Haney relaxation term.

c. The relative importance of mixing processes

These components of D_diff are shown as functions of ρ in Fig. 12. The largest components of D_diff are from the diapycnal mixing D_thermo and equatorial upwelling D_equat. These terms represents a key part of the oceanic density balance: the return path of the thermohaline circulation. Dense water formed by surface cooling at high latitudes is warmed and lightened as it upwells in the Tropics. Most of the mixing due to equatorial upwelling of the light waters σ < 25.65 that are not resolved by the model is accomplished by entrainment. Lateral diffusion and seasonal entrainment are comparatively unimportant over the whole model domain.

d. The spatial variation of diffusive fluxes

We now investigate where the density fluxes across isopycnals occur in our model. We consider two of the lighter isopycnals in the model thermocline, σ = 26.55 and σ = 25.8. We choose σ = 26.55 in particular to show where seasonal mixing is important and σ = 25.8 to show equatorial entrainment.

e. Extratropical balances

We now consider whether there are geographical regions of the model in which Speer and Tziperman’s (1992) technique of diagnosing diapycnal volume fluxes directly from the surface forcing might give reasonable results.

Since mixing is evidently strongest over the equatorial regions (see, e.g., Fig. 14), we now diagnose the model north of a nominal line of 24°N; the actual line follows a coordinate line in the rotated model grid. The model ought also to be more realistic over this subdomain than over the whole domain, as (i) the effect of the closed southern boundary should be less and (ii) the unresolved waters with σ < 25.65 are less important north of 24°N.

In Fig. 15a we plot D_surf, D_diff, D_net, and D_true diagnosed north of 24°N, similarly to the globally diagnosed values in Fig. 10a. In Fig. 15b we plot similarly F, G_diff, G_true, and the residual G_true − F.

Over this more restricted region, there is a much improved correspondence between F and G_true. Thus, surface forcing does provide a reasonable first-order estimate of the diapycnal volume flux outside of the Tropics. Consistent with this, mixing is much weaker than it was over the whole domain, with a peak magnitude of D_diff ∼ 4 × 10⁶ kg s⁻¹ compared with the peak of ∼20 × 10⁶ kg s⁻¹ over the whole domain. Density is being input into all isopycnals (see the plot of F, the dotted line in Fig. 15b). The domain-integrated density input, D_surf(σ_min), is 35 × 10⁶ kg s⁻¹, approximately equivalent to a heat loss over this domain of 0.6 PW. Over this domain, the model is not far from the idealized limit in section 2b with no mixing and density input everywhere.

However, while the broad structure is captured, the surface forcing F predicts (Fig. 15b) a diapycnal volume flux that is too large for waters denser than σ = 25.5 and too small for lighter waters. For example, F = 3 Sv at σ = 25 and 8 Sv at σ = 26.5, compared with the true diapycnal volume fluxes of 6 and 5 Sv, respectively. Thus, mixing still plays a significant role.

We again separate the mixing flux (Fig. 16a) into explicit diffusion D_thermo, upwelling entrainment D_equat, seasonal entrainment D_seas, and lateral mixing in the mixed layer D_lat. Over this domain, which excludes the tropical upwelling regions, D_equat almost disappears, and the thermocline mixing D_thermo is much weaker than over the whole domain, with a peak of 2 × 10⁶ kg s⁻¹ rather than the peak of 10 × 10⁶ kg s⁻¹ seen in Fig. 12. Seasonal entrainment is now of comparable importance, with a peak magnitude of ∼2.5 × 10⁶ kg s⁻¹, in reasonable agreement with the estimate made in section 3d.

The effects of these mixing processes on the diapycnal volume flux are shown in Fig. 16b. Seasonal entrainment drives a volume flux of ∼2 Sv from dense to light for the thermocline and intermediate waters, while lateral mixing in the mixed layer drives a flux of ∼6 Sv from dense to light for heavy waters σ ≈ 27.9. These processes perhaps “fine tune” the properties of the water masses, already approximately set by the surface fluxes.

f. Within localized regions

Speer and Tziperman’s (1992) technique of diagnosing diapycnal volume fluxes directly from the surface forcing might be expected to give the best results of all when applied to the formation regions. We thus consider the diagnosed formation rates in the Sargasso and Labrador Seas, where the model forms versions of 18° subtropical mode water and Labrador Sea Water, respectively; the Sargasso Sea is defined here by the region roughly from 24° to 35°N, 70° to 50°W and the Labrador Sea by 53° to 64°N, 55° to 28°W. Over the Sargasso Sea, the surface forcing leads to a formation rate of up to 15 Sv/(kg m⁻³) at σ = 26.55, which is close to the model truth revealed by the close correspondence between the dotted and solid lines in Fig. 17a. Over this region, mixing makes a relatively minor contribution, compared with the strength of the surface forcing.

Over the Labrador and Irminger Seas, the surface forcing and model formation rates closely agree, with a formation rate of 35 Sv /(kg m⁻³) at σ = 27.75 being compensated by destruction of 30 Sv /(kg m⁻³) at σ = 27.6 (Fig. 17b). Again mixing only plays a relatively minor role locally.

g. Implied vertical diffusivity

The underlying appeal of Walin’s formulation is not only that rates of water mass formation might be predictable from surface forcing, but also that the integrated mixing might be inferred by comparing air–sea fluxes with known formation rates. This can be either done globally or, if the in- and outflows are known, over restricted domains. Applications using real data have so far assumed that the (annual mean) ocean is in steady state.

Walin (1982) estimated an implied globally averaged vertical diffusivity in the ocean to be 2 × 10⁻⁴ m² s⁻¹ using surface fluxes calculated from COADS data by Andersson et el. (1982) and assuming a background uniform ∂T/∂z = 4 × 10⁻³ K m⁻¹. This diffusivity estimate is an order of magnitude larger than the canonical value for the thermocline of ∼2 × 10⁻⁵ m² s⁻¹ found by Ledwell et al. (1993). This discrepancy between the global diffusivity inferred from integrated tracer diagnostics and local measurements is undoubtedly the result of mixing being strongly enhanced in some regions. Speer (1997) estimated the effective diffusivity as a function of density over the North Atlantic by comparing the volume flux along density layers across a hydrographic section at 11°S with the formation rates implied from climatological forcing north of there. The implied diffusivity ranges from 10⁻⁴ m² s⁻¹ at σ = 23 to lower values varying around 2 × 10⁻⁵ m² s⁻¹ for denser surfaces.

We now apply this diagnostic approach to our model. We diagnose an effective diffusivity based on the traditional assumption that mixing only occurs within the thermocline, even though mixing is really also occurring within the mixed layer (through lateral mixing) and at its base (through entrainment). This effective diffusivity κ_eff(ρ) is then the total diffusive flux across the isopycnal D_diff, divided by the density gradient, integrated over the area A_thermo of the isopycnal where it lies within the thermocline:

View Expanded

The brackets are explicitly included to emphasize that annual means of these quantities are taken.²

We calculate effective diffusivities both over the whole model domain and over the extratropical domain north of 24°N.

Over the full domain, the variation of effective diffusivity with density over the model density range that is resolved in the thermocline, 25.65 < σ < 28.15, is shown as the thick full line in Fig. 18a. Values range from ∼0.1 × 10⁻⁴ m² s⁻¹ at σ < 26.0 to ∼0.9 × 10⁻⁴ m² s⁻¹ at σ = 27.85, about three times the explicit diapycnal diffusivity, itself fairly large at this density (28) because of the low N in these weakly stratified subpolar mode waters. This enhancement at high densities results from lateral mixing within the deep winter mixed layers. Seasonal entrainment begins to dominate over lateral mixing for σ < 27, as is seen better in Fig. 12, but at these lighter densities the explicit diapycnal diffusion dominates anyway (again see Fig. 12). However, the lateral and seasonal mixing do increase the diffusivity by about a half over the explicit diffusion for the mid to deep thermocline waters, 26.6 < σ < 27.4.

North of 24°N (Fig. 18b), the effective diffusivity shows a similar pattern of variation with ρ. Values of κ_eff are however larger, with the peak at σ = 27.85 now reaching ∼1.3 × 10⁻⁴ m² s⁻¹. Excluding the tropical mixing, which is driven by the strong density gradients under the tropical mixed layer, and hence is associated with weak diffusivity (28), gives a larger effective diffusivity for the remainder of the mixing. Moreover, the seasonal entrainment and lateral mixing are then relatively more important (see again the diffusive fluxes in Fig. 16a) and so increase κ_eff more.