1. Introduction

Traditionally, the effects of nonlinearities of the equation of state on the transport of buoyancy have often been assumed to be negligible. For example Tziperman (1986) investigates the density budget for a fixed volume with a linear equation of state, and Munk and Wunch (1998) investigate the oceanic energy budget using the same equation of state. However, there are exceptions. Davis (1994) looked into density production by the nonlinearities of the equation of state and found that the rate of change of density at several sites in the Atlantic, Pacific, and Southern Ocean were often more influenced by these nonlinear effects than by the divergence of the diffusive fluxes. More recent estimates of the effects of these nonlinearities also suggest that they may indeed by substantial. For example, Gnanadesikan et al. (2005) estimate the mechanical energy input necessary to balance the potential energy lost through cabbeling in the Modular Ocean Model (MOM). They find that 0.4 TW is needed, which is a very substantial power consumption given that their estimate of the global energy demand for balancing convection is 0.15–0.2 TW. Klocker and McDougall (2010) estimate the dianeutral advection and diffusion due to the nonlinearities of the equation of state and present estimates of dense water production due to these nonlinearities. They find that 6–10 Sv (1 Sv ≡ 10⁶ m³ s⁻¹) of dense water is produced in this way, where the lower estimate is from an ocean model and the higher one from using the World Ocean Circulation Experiment (WOCE) climatology (Gouretski and Koltermann 2004). This is a considerable amount given that the total dense water production is estimated to be 30 Sv (Munk and Wunch 1998).

In this article we will look at the vertical buoyancy transport in detail. The integral over the entire ocean volume of the buoyancy sink term, which is due to the nonlinearities of the equation of state, can in a steady state be determined from the heat fluxes at the ocean boundary. Those heat fluxes, which in our case are derived from bulk formulas and based on either in situ measurements or reanalysis data, are accurate enough to provide a first estimate of the oceanic buoyancy sink. Furthermore, this approach has the advantage that the analysis can be done directly from data without the use of an ocean circulation model.

However, when the buoyancy transport equation is integrated over the whole ocean volume the global buoyancy sink is a single quantity. Hence, it does not shed any light on the spatial locations where the buoyancy sink is more effective. This is a significant problem, since whether the buoyancy sink is dynamically important depends on whether it is large compared to the divergence of the local buoyancy fluxes. To overcome this difficulty we will also present some modeling result for the buoyancy fluxes and sink from the Nucleus for European Modeling of the Ocean (NEMO) (Madec 2008), where we have applied online diagnostics to extract the buoyancy fluxes from a near steady state run.

2. The oceanic buoyancy budget

In a steady state the surface integral of the heat fluxes over the ocean boundaries must be zero. This means that the geothermal heating at the ocean floor is accompanied by a net heat loss at the ocean atmosphere interface. We express this balance of heat fluxes in terms of fluxes of conservative temperature (McDougall 2003) as

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where η(x, y) is the vertical coordinate of the ocean atmosphere interface and −H(x, y) that of the ocean floor, F_Θ is the flux of conservative temperature at the boundary due to geothermal heating, latent heating, sensible heating, solar radiation, and heat loss due to longwave radiation, A(η) is the area of the ocean atmosphere interface, A(−H) is the area of the ocean floor, and n is an outward unit normal to the surface on which the integration takes place.

We may thus calculate the steady state heat loss at the sea surface from the geothermal heating. In our calculations we will use geothermal heat fluxes based on the age of the sea floor following Stein and Stein (1992).

Given a general nonlinear equation of state, the transport equation for buoyancy is different from that for conservative temperature or absolute salinity. This is because nonlinear processes can introduce a sink or source of buoyancy, which means that the steady state integral of boundary buoyancy fluxes does not vanish by necessity, as the steady state integral of boundary heat fluxes does. This was pointed out already by McDougall and Garrett (1992). As we will show, the magnitude of the buoyancy source or sink can be determined from the heat fluxes at the ocean boundary.

We start the derivation by defining the buoyancy flux as

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Here is buoyancy and ρ₀ = 1027 kg m⁻³ is a constant reference density, F_Θ is the net flux of conservative temperature from diffusion, advection, and penetrative shortwave radiation, and is the net absolute salinity flux from diffusion and advection. For conservative temperature and absolute salinity, which are conserved quantities, the transport equations are

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and

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By taking the divergence of (2) and substituting ∇ · F_Θ and from (3) and (4), respectively, we obtain the buoyancy transport equation

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where the source term Q for buoyancy can be written

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Assuming a steady state and integrating (5) over the entire volume we get

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Here V is the ocean volume, and A is the area of the ocean boundary. Using the Thermodynamic Equation of Seawater 2010 (TEOS 10) (Intergovernmental Oceanographic Commission 2010), we found that for seawater at surface pressure with absolute salinities between 34 and 37 g kg⁻¹ and conservative temperatures between 0° and 30°C, ∂b/∂S_A varies about 5% from its mean value, while ∂b/∂Θ varies almost 80% from its mean value. Moreover, we found that the area integrated buoyancy flux at the ocean atmosphere interface owing to heat fluxes was two orders of magnitude larger than that owing to freshwater fluxes, using heat and salt fluxes from the Global Ocean Data Assimilation System (GODAS) reanalysis (National Oceanic and Atmospheric Administration 2011) and the TEOS 10 equation of state. It is therefore a good approximation to assume a constant ∂b/∂S_A, which in a steady state leads to

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and hence we may write (7) as

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Consequently, the internal sink of buoyancy owing to the nonlinearities of the equation of state can be quantified from the boundary heat fluxes. Furthermore, by using Gauss theorem and the fact that ∇ · F_Θ = 0 in a steady state, (8) can be rewritten as

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it is thus the spatially varying ∂b/∂Θ that is the main agent behind this buoyancy sink.

We can investigate the buoyancy sink further by using an analytical expression for the equation of state. Here we use a simple nonlinear equation of state proposed by de Szoeke (2004) and used by, for example, Vallis (2006) and Nycander (2011), given as

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where c is the speed of sound taken as constant, β_T is the thermal expansion coefficient, β_S is the haline contraction coefficient, γ is the thermobaric constant that describes thermobaricity (the thermal expansion coefficient’s dependency on pressure), and is the second thermal expansion coefficient, describing the fact that thermal expansion increases with increased temperature, which gives rise to cabbeling. Using (10), ∂b/∂Θ can be written

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Substitution of (11) into (9) yields

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where z = (0, 0, 1). In a steady state the integral over a surface of constant z of F_Θ · z is simply equal to the geothermal heating below this surface. In most applications this is negligible. Therefore we may write

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This means that, with this simplified equation of state, cabbeling is consuming the buoyancy fluxes from the boundaries, and that thermobaricity does not contribute to the buoyancy budget. Since this result is derived using the simplified equation of state, it may not hold using a more accurate equation of state. In our data analysis we use TEOS 10, and to get some idea of the thermobaric contribution to the buoyancy sink using that equation of state we have looked at variations in the thermobaric parameter at constant pressure. The thermobaric parameter is defined as

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where α = ρ⁻¹∂ρ/∂Θ, β = ρ⁻¹∂ρ/∂S_A, and p is pressure. We found that absolute salinity variations do little to change in oceanic conditions. However, variations in conservative temperature can be important. We found that at 1000 dbar with conservative temperatures between 0° and 10°C and salinity of 35 g kg⁻¹ varies by almost 15% from its mean value. However, since conservative temperature does not change dramatically on surfaces of constant z apart from in relatively shallow areas it is likely cabbeling that dominates the buoyancy sink.

Assuming that the thermobaricity is negligible and using (8) and (11) we may write

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which shows that there has to be a correlation between Θ and F_Θ for the volume integral of Q to be nonzero.

McDougall and Garrett (1992) relate the buoyancy sink term to the dissipation of thermal variance. Interestingly enough, the correlation between surface values of Θ and F_Θ is also responsible for the generation of thermal variance in the ocean. This can be seen by multiplying (3) with Θ and integrating over the ocean volume, which gives

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where is the nonadvective part of the conservative temperature flux, and ∇ · u = 0 has been used. In a steady state we get

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where is the turbulent diffusive flux vector, and is the penetrative shortwave radiation flux vector. Both terms on the right are zero in the mixed layer. Below the mixed layer the downward shortwave flux gives a negative contribution to (17), since the vertical temperature gradient is mainly positive. The diffusive term is always negative because the diffusive flux is down the tracer gradient. Both processes can thus be said to dissipate thermal variance, however, the diffusive part is much larger because only a small fraction of reaches deeper than the mixed layer.

3. Data

We have seen that to evaluate the buoyancy sink from the nonlinear equation of state we need to evaluate the buoyancy fluxes at the boundaries. To do so, we have used climatological heat fluxes and calculated ∂b/∂Θ using TEOS 10 (Intergovernmental Oceanographic Commission 2010). The geothermal heat fluxes are based on Stein and Stein (1992), and we use salinity and temperature from the WOCE climatology, all interpolated onto an ORCA1 grid to calculate the buoyancy flux at the ocean floor. For the surface fluxes we use the National Oceanography Centre Southhampton (NOCS) v2.0 flux dataset (Berry and Kent 2009) and the GODAS reanalysis (National Oceanic and Atmospheric Administration 2011).

The NOCS v2.0 dataset comes with a spatial resolution of one degree in both latitude and longitude. It is based on ship-based measurements from which heat fluxes are derived through bulk formulas. The dataset contains heat fluxes for the years 1973 to 2006 with a monthly resolution. It also contains SST measurements that we use together with salinity from the WOCE climatology (Gouretski and Koltermann 2004) to calculate ∂b/∂Θ. The time-averaged heat flux, SST, and the calculated ∂b/∂Θ from the NOCS v2.0 dataset are shown in Fig. 1.

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The temporal average of some variables calculated using the NOCS v2.0 climatology. (a) The surface heat flux (W m⁻²), (b) the surface temperature (°C), and (c) ∂b/∂Θ (m s⁻² K⁻¹).

Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-063.1

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The GODAS dataset is an ocean reanalysis forced with the National Centers for Environmental Prediction (NCEP) atmospheric reanalysis 2 (Kanamitsu et al. 2002). This dataset comes with a spatial resolution of one degree in longitude and one-third degree in latitude. We use the monthly mean surface heat fluxes and the salinities and temperatures at 5-m depth from year 1980 to 2011. The time-averaged heat flux, 5-m temperature, and the calculated ∂b/∂Θ from the GODAS reanalysis are shown in Fig. 2.

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As in Fig. 1, but using the surfaces fluxes, SST, and ∂b/∂Θ from the GODAS reanalysis.

Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-063.1

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The NOCS v2.0 flux dataset does as its predecessor v1.0 contains a rather large and unphysical imbalance of the surface heat fluxes (the average flux is 25 W m⁻² and directed into the ocean.). However, since we are considering a steady state case we have subtracted a constant from our surface heat fluxes such that (1) holds. We have also tried two other correction strategies, namely, to lower only the ingoing or raising only the outgoing heat fluxes, in each case with a constant value, such that the resulting surface heat flux balances the geothermal heating at the bottom. The resulting buoyancy flux from the two different approaches differs by roughly a factor of 3. The imbalance in the GODAS dataset is much smaller roughly, 1 W m⁻², and the same corrections are applied to that dataset, however, in this case with only minor implications for the resulting buoyancy flux.

4. Results

Figures 3 and 4 show the horizontally integrated buoyancy fluxes at the ocean boundaries and the surface heat fluxes (solid curve). In Fig. 3 the surface heat fluxes are taken from the NOCS v2.0 climatology and in Fig. 4 the surface heat fluxes are from the GODAS reanalysis. The buoyancy flux at the bottom (dash-dotted curve) is always small in comparison. As we can see there is a strong seasonality in the surface buoyancy fluxes (dashed curve). This is due to the differences in surface area of the ocean on the northern and Southern Hemisphere. The ocean surface area on the Southern Hemisphere is larger than that on the Northern Hemisphere. Therefore, the ocean on the Southern Hemisphere gains more heat in the austral summer then the ocean on the Northern Hemisphere does in boreal summer. The ocean as a whole is thus more buoyant in the austral summer.

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Horizontally integrated buoyancy and heat fluxes at the boundaries. The surface buoyancy fluxes (dashed curve) are calculated using the NOCS v2.0 (Berry and Kent 2009) climatology and the fluxes at the ocean floor (dash-dotted curve) using the Stein and Stein (1992) parameterization. The solid curve shows the surface heat fluxes in NOCS v2.0. A positive value means that the flux is upward (i.e., out of the ocean for the surface fluxes).

Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-063.1

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As in Fig. 3, but using the surfaces fluxes from the GODAS reanalysis.

Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-063.1

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However, we are mainly interested in the long-term mean buoyancy flux, which is 9 · 10⁵ m⁴ s⁻³ using the NOCS v2.0 climatology and 5.3 · 10⁵ m⁴ s⁻³ using the GODAS air–sea fluxes. The buoyancy flux is directed into the ocean in both cases. The geothermal heating gives rise to a buoyancy flux of 1.2 · 10⁴ m⁴ s⁻³, which is also directed into the ocean. The surface flux calculated when lowering only the ingoing fluxes was 5.7 · 10⁵ m⁴ s⁻³ using the NOCS fluxes and 5.1 · 10⁵ m⁴ s⁻³ using the GODAS fluxes. When raising only the outgoing fluxes we got 1.54 · 10⁶ m⁴ s⁻³ for the NOCS fluxes and 5.5 · 10⁵ m⁴ s⁻³ for the GODAS. Thus, there is about a factor of 3 difference between the results from these rather extreme correction strategies in the NOCS case and only minor differences in the GODAS case, which gives some indication of the uncertainties involved.

The generation rate of thermal variance, defined as minus the left side of (17), amounts to 1.1 · 10¹⁰ K² m³ s⁻¹ using the NOCS fluxes, and 6.1 · 10⁹ K² m³ s⁻¹ using the GODAS fluxes. When raising only the outgoing fluxes we got 1.8 · 10¹⁰ K² m³ s⁻¹ in the NOCS case, and 6.3 · 10⁹ K² m³ s⁻¹ in the GODAS case. Lowering only the ingoing fluxes resulted in a generation rate of 6.9 · 10¹⁰ K² m³ s⁻¹ in the NOCS case, and 5.9 · 10⁹ K² m³ s⁻¹ in the GODAS case.

In a steady state the ocean must lose heat to the atmosphere at the same rate as it gains heat through geothermal heating. Nevertheless, we have seen that the buoyancy fluxes are directed into the ocean at both boundaries and are much greater at the ocean atmosphere interface. This could be due to both spatial and temporal correlations between ∂b/∂Θ and F_Θ (the spatial distribution of the surface heat flux, the SST, and ∂b/∂Θ are shown in Figs. 1 and 2). A way of investigating the relative importance of these spatial and temporal correlations is to decompose our variables into constant and fluctuating parts. A general variable k(x, y, t) can be decomposed as k(x, y, t) = [k](t) + k*(x, y, t), where the square brackets denote the horizontal average over the entire space domain and the asterisk the departure from that average. Similarly, k can also be decomposed as , where the overbar denotes the temporal average over the entire time domain and the prime the departure from that average. Combining these decompositions, the horizontal and temporal average of the buoyancy flux can be written, in a way where no further decomposition is possible, as

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Here, the first term on the right-hand side is the product of the average surface heat loss and the average ∂b/∂Θ. In a steady state with no geothermal heating this would be zero. The second term is due to temporal correlations between the spatial means, and the third term is due to spatial correlations between the temporal means. The fourth term contains a mixture of both temporal and spatial correlations.

The terms in (18) can be rearranged into two more familiar decompositions

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where the third and fourth term of (18) have been combined into the third term of (19a), and the second and fourth term of (18) have been combined into the third term of (19b).

In Table 1 we can see the relative importance of the terms in our decompositions. All the values have been multiplied by the ocean area so that the terms in either decomposition sums to the integrated surface buoyancy flux of 9 · 10⁵ m⁴ s⁻³ in the NOCS case and 5.3 · 10⁵ m⁴ s⁻³ in the GODAS case. A positive value indicates a flux into the ocean. The first term in Table 1, which is due to the net surface heat loss at the average ∂b/∂Θ, is slightly larger than the integrated buoyancy flux at the ocean floor in both cases, which was 1.2 · 10⁴ m⁴ s⁻³. This is because the heat loss at the ocean atmosphere interface occurs at a higher temperature than the heat gain at the ocean floor. The difference between the NOCS and GODAS fluxes in this respect is probably largely due to the fact that we use the temperature at five meters in the GODAS case and the appropriate surface temperature in the NOCS case. The second term, which is due to the temporal correlation of the spatial means, is small and negative in the NOCS case and small and positive in GODAS case. This smallness of this term can be understood if we assume that [F_Θ] ∝ [∂Θ/∂t], that [∂b/∂Θ] ∝ [Θ] and that [Θ(t)] is roughly sinusoidal because then the temporal average, , vanishes. The third term, the spatial correlation of the temporal means, is the biggest term in the NOCS case and second biggest in the GODAS case. This means that a very important buoyancy gain occurs because the ocean loses heat at high latitudes with low surface temperatures and gains heat in the low latitude at high surface temperatures. The fourth term is also large in both datasets. However, its mixed form makes it difficult to interpret in terms of correlations in time and space. For this application, the use of decomposition (19a) appears to be the most convenient choice because the third term on the right-hand side contains nearly all of the temporal and spatial mean buoyancy flux. This means, that it is enough to understand the physical interpretation of the third term on the right of (19a) to understand the important correlations between ∂b/∂Θ and F_Θ.

Table 1.

The magnitude of the terms in the decompositions (18), (19a), and (19b) multiplied by the ocean area from the NOCS v2.0 climatology and the GODAS reanalysis. A positive value indicates a flux into the ocean.

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5. An equivalent heat flux

To put these results into perspective, let us calculate an equivalent heat flux to the calculated buoyancy flux. First we define F as the heat flux vector

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where c_p = 4000 J (kg K)⁻¹ is the specific heat of seawater. Using the definition of the buoyancy flux (2), letting where α = ρ⁻¹∂ρ/∂Θ and using ρ ≈ ρ_o, g = 9.81 m s⁻², and α = 1.67 · 10⁻⁴ K⁻¹, we look at the horizontal average of F_b · n, that is, [F_b]. If we assume a constant ∂b/∂S_A we then get [F_b] = gα[F_Θ]. Combining this with (20) gives

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The horizontally averaged equivalent heat flux is [F] ≈ 6 W m⁻² in the NOCS case and [F] ≈ 4 W m⁻² in the GODAS case. Both are very large heat fluxes by oceanic standards, compared with, for example, the average geothermal heat flux, which is thought to be 86.4 mW m⁻², or with the vertical heat fluxes in NEMO which are in excess of 6 W m⁻² only in the upper 100 m. This is an indication that unless almost all the excess buoyancy is consumed by a buoyancy sink near the surface, the buoyancy sink must be an important part of the buoyancy budget of the global ocean. Hence, regarding buoyancy as a linear combination of conservative temperature and salinity cannot be justified a priori. Even though it may be a fair approximation in some ocean regions, it could lead to large errors in others.

6. Buoyancy fluxes in NEMO

In this section we present the buoyancy fluxes in the ocean model NEMO, using the ORCA1 configuration, which has a spatially varying horizontal grid resolution with a base resolution of one degree. We use 46 vertical levels and the ocean ice model LIM2 (Fichefet and Maqueda 1997). The run is forced with the Drakkar Forcing Set v4.3 (DFS4.3) (Brodeau et al. 2009), and the forcing years 1958–83 have been repeated so that roughly 3000 model years have been integrated. Because of the very long time integrations needed to get the deep ocean into a steady state there is still some drift in the deeper part of the model domain. However, even though the buoyancy budget in (22) is calculated by integration from the bottom and up, the budget is dominated by large contributions in the upper parts. Therefore the model drift has a negligible effect on the buoyancy budget except in the deepest regions. The surface heat fluxes in NEMO are derived from bulk formulas using the modeled sea surface temperature and the DFS4.3 forcing, which is based on the 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40). This is an approach that constrains the ocean model to observations and gives rather balanced surface fluxes (i.e., the integrated surfaces fluxes nearly balance those from the geothermal heating).

The buoyancy fluxes in NEMO are calculated according to (2), where and , using the potential temperature and practical salinity (S_P) based Jackett and McDougall (1995) equation of state. The thermal expansion coefficient α and the haline contraction coefficient β are calculated at every time step by the model and all flux calculations are done online. This results in closed budgets for potential temperature and practical salinity. However, for buoyancy this is not the case because of the sink. Nevertheless, having closed budgets of practical salinity and potential temperature makes it possible to accurately evaluate the buoyancy sink. Our approach is to take a time average of the volume integral of (5)

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where A(z) is the horizontal area at a given model level z, V(z) is the ocean volume beneath that level, t₀ is the start time of the analysis, and T is the time period of averaging (26 years). Here we use the model fluxes and sinks where, F_b is the net upward buoyancy flux, composed of fluxes from dianeutral and isoneutral diffusion, advection, penetrative shortwave radiation, and convection (here modeled as enhanced diffusion), and Q_b is the net buoyancy source from a bottom boundary layer parameterization and geothermal heating. Having closed budgets for potential temperature and practical salinity means that, if we look at the analog of (22) for potential temperature or practical salinity then the sum of the three terms on the left-hand side is very close to zero, in particular it is much smaller than any of the individual sources or flux terms. Therefore, and because ∂b/∂θ and ∂b/∂S_P are calculated at each time step and at each grid point, we can accurately calculate the second and third term on the left side of (22). The integrated buoyancy sink due to the nonlinear processes beneath a chosen model level [the right-hand side of (22)] is then evaluated by calculating the left-hand side of (22).

To find out what processes are more important on different depths we take ∂/∂z of (22), yielding

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where the first term on the left-hand side is just a time average of the horizontally integrated buoyancy trend at the vertical level z. The third term on the left-hand side and the first term on the right-hand side are the time average at the vertical level z of the horizontally integrated net buoyancy source Q_b and the sink Q, respectively. The second term on the left-hand side is the time averaged horizontally integrated divergence at the vertical level z of the buoyancy fluxes, that is, the time average of the horizontally integrated buoyancy trends due to the buoyancy fluxes F_b. Because of difficulties with the free surface we start our calculation at the first interior model level, which is situated at 6 m depth.

Figure 5 shows a 26-yr average of the buoyancy budget (22) and its vertical derivative with fluxes and sinks from different physical processes shown separately. The sign convention used is such that we plot minus the source terms and fluxes are positive if they are upward, which ensures that the sum of all the terms in Fig. 5 is zero at all depths. As an example, the term is calculated according to

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and the term is calculated according to

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The integrated nonlinear buoyancy sink is shown as squares in Figs. 5a and 5c. The first thing to stress is its close agreement with the surface buoyancy fluxes calculated using the climatological heat fluxes. The GODAS surface buoyancy flux is marked with a red plus sign, and the NOCS with a green one, in Fig. 5. The buoyancy sink integrated below the depth 6 m in NEMO was 8.9 · 10⁵ m⁴ s⁻³, which compares very well to the value 9 · 10⁵ m⁴ s⁻³ from the NOCS v2.0 climatology, and is a bit larger than the value 5.3 · 10⁵ m⁴ s⁻³ that we got from the GODAS fluxes.

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26-yr average of the buoyancy budget [(22)] for NEMO ORCA1 with the different fluxes and sources displayed individually is shown in (a) above 700 m and (c) below 700 m. Its vertical derivatives are shown in (b) above 700 m and (d) below 700 m. A positive value means an upward flux in the left figures and a negative trend in the right figures. is the advective buoyancy flux which incorporates a parameterized eddy advection (Gent and McWilliams 1990), is the buoyancy sink from the bottom boundary layer parameterization, is the dianeutral diffusive buoyancy flux which here incorporates convection, is the buoyancy source from geothermal heating, is the buoyancy flux from isoneutral diffusion, Q is the buoyancy sink due to the nonlinear equation of state, is the buoyancy flux from penetrative shortwave radiation, and B_t is the model drift [the first term on the left-hand side of (22) and b_t its vertical derivative]. (top) The red plus is the surface buoyancy flux calculated using the GODAS dataset and the green plus the surface buoyancy flux calculated using the NOCS v2.0 climatology.

Citation: Journal of Physical Oceanography 43, 1; 10.1175/JPO-D-12-063.1

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With the exception of the very top ocean, where the buoyancy budget is dominated by the penetrative shortwave radiation (blue) and convection (cyan), the nonlinear buoyancy sink is indeed large. In fact, below 200 m only dianeutral diffusion is larger, and between 400 and 2000 m it is the largest term in the buoyancy budget. At these depths both diffusion (cyan) and advection (pink) act to flux buoyancy downward, and this buoyancy is then mainly consumed by the nonlinear buoyancy sink. In the heat budget in the same depth region (not shown) the downward heat flux by dianeutral diffusion and advection is balanced by a large upward heat flux from isoneutral diffusion. (By definition, isoneutral diffusion should have no effect on the buoyancy budget. However, this is not exactly true in the numerical model, since the local neutral slopes are filtered in the isoneutral diffusion scheme and are therefore not strictly neutral. Nevertheless, the isoneutral diffusion has only a minor effect on the buoyancy budget, as seen in Fig. 5.)

However, it is not the magnitude of the flux that determines the evolution of a tracer, it is the divergence of the flux. Therefore we show the vertical derivative of (22) in Figs 5b and 5d. Here we can see that the impact on the buoyancy trend from the sink because of the nonlinear equation of state is especially large between 600- and 2500-m depth. Above that, advection, dianeutral diffusion, and shortwave penetration are more important. Davis (1994) used a fixed ratio of diapycnal to isopycnal diffusivity and Levitus data (Levitus 1982) from several sites in the Atlantic, Pacific, and Southern Ocean to estimate the importance of the nonlinear buoyancy sink relative to the other terms that determine the local buoyancy evolution. He found that the nonlinear buoyancy sink was often larger than the buoyancy trend owing to the divergence of the diffusive fluxes, and we find the same to be true of the global averages in our ocean model. He also separated the effect of thermobaricity from that of cabbeling and the dianeutral diffusive fluxes from the isoneutral ones, and found cabbeling owing to the isoneutral diffusive fluxes to usually dominate the budget. This is in agreement with our conclusion, based on the simplified equation of state, that thermobaricity is of less importance than cabbeling for the buoyancy sink. Although we have not separated isoneutral from dianeutral diffusion in our source term, it is likely that isoneutral fluxes dominate the cabbeling term in our case as well. The isoneutral diffusive fluxes in NEMO are treated in some detail in Hieroymus and Nycander (2012, manuscript submitted to Ocean Modell.).

In the very deep parts (below 2500 m) advection is the most important part; however, geothermal heating, the bottom boundary layer parameterization, the nonlinear buoyancy sink, dianeutral diffusion, and isoneutral diffusion are all important. Even though the run is not in a steady state the results appear to be very robust and have not changed much during hundreds of years of simulation. Since the model drift is explicitly taken into account in (22), model drift does not introduce errors in the calculation of the buoyancy sink; however, we cannot totally disregard the possibility that the steady state buoyancy budget could differ from that of the transient states leading there.

7. Conclusions

Because of the nonlinear equation of state there is a buoyancy sink in the interior of the ocean, which must be compensated by a downward buoyancy flux at the sea surface. Thus, the surface buoyancy flux can be used to measure the interior buoyancy sink. We have here quantified the buoyancy sink associated with the nonlinear equation of state using both empirical surface flux climatologies and an ocean model simulation, with similar results. The volume integrated buoyancy sink was close to 9 · 10⁵ m⁴ s⁻³ both in NEMO and when calculated using the NOCS v2.0 climatology, and somewhat smaller, 5.3 · 10⁵ m⁴ s⁻³, when calculated using the GODAS fluxes. These results are comparable to an equivalent surface heat flux of roughly 4–6 W m⁻² over the global ocean. This large net buoyancy flux is mostly due to spatial correlation between ∂b/∂Θ and F_Θ, that is, the ocean loses heat at high latitudes with low sea surface temperatures and gains it at low latitudes with high sea surface temperatures. After the submission of this manuscript we became aware of recent work by Griffies and Greatbatch (2012). In their work they develop an analysis framework to study global mean sea level and find the same correlations between the surface heat flux and the thermal expansion coefficient to be a dominant term affecting the sea level.

From our model run we have also studied the vertical distribution of the buoyancy sink. Those results (Figs. 5a and 5c) show that the buoyancy sink is indeed a first-order term in the buoyancy transport equation, perhaps with the exception of the top 50 m. In fact, in the buoyancy budget [(22)] of NEMO it is the largest term between 400- and 2000-m depth. When looking at the tracer trends at a specific depth (Figs. 5b and 5d), we find that the buoyancy sink owing to the nonlinearities of the equation of state has a large impact on the buoyancy evolution especially between 600- and 2500-m depth, but it is important even in the deeper region. It is only in the very top, where shortwave penetration and convection totally dominate the buoyancy evolution, where it is negligible.

Our findings suggest that regarding the density of seawater as a linear combination of salinity and conservative temperature can generally not be justified. We further found that thermobaricity hardly contributes to the buoyancy sink in a steady state with the simplified nonlinear equation of state in (10) because then the thermobaricity is constant on surfaces of constant z and the integral of heat fluxes through such surfaces is small. However, this is only true when we consider the total heat flux from all processes. Heat fluxes from specific physical processes, such as, for example, isoneutral diffusion or advection, may still give large thermobaric contributions to the buoyancy sink. Griffies and Greatbatch (2012) found that the thermobaric contribution to the sea level evolution from the isoneutral diffusive flux was about half of the cabbeling contribution from the same flux and hence hardly negligible. In the deep ocean where geothermal heating is important or if the ocean is far from in a steady state, we may still have important contributions to the buoyancy sink from thermobaricity because under those conditions the integral of heat fluxes through surfaces of constant z may not be small.

There are of course some rather large uncertainties in our investigation. The largest is likely to be the large imbalance in the NOCS v2.0 flux climatology of about 25 W m⁻². This imbalance has been corrected for by subtracting a constant value such that the heat gain at the ocean floor equals the heat loss at the surface. We also tried two other correction strategies. In the first case we lowered all the fluxes into the ocean with a constant value, and in the second we raised all the outgoing heat fluxes by a constant value, in both cases so that the heat loss at the surface balances the heat gain at the bottom. There was about a factor of 3 difference in the results using these rather drastic approaches. The imbalance in the GODAS fluxes was much smaller, about 1 W m⁻², and correcting those had only minor effects. The equivalent heat fluxes as calculated by (21) are then 4–11 W m⁻², so the contribution to the buoyancy budget is large in all cases. The fact that we got similar results from NEMO and the climatologies is quite encouraging and suggests that the imbalanced heat fluxes have not introduced too dramatic errors.