a. Independent parameters

The overall goal of the model calculations is to find optimal values for a set of physical and biogeochemical parameters defined on the model grid:

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Included in the vector of independent parameters p* are zonal and meridional oceanic flows (u, υ), isopycnal and diapycnal mixing coefficients (K_h, K_υ), air–sea fluxes of heat and gases (Q), and biological production rates in the ocean surface layer (export production EP), as well as downward fluxes and remineralization rates of particulate matter (β). For reasons of computational efficiency and because of the sparseness of the available hydrographic and tracer datasets, which provide seasonal or interannual resolution at only a few time series stations worldwide, all physical and biogeochemical transports and rate constants in the model are considered to be time invariant, and temporal variability in these fields is neglected. The model results therefore represent climatological mean fields, which are valid for the time periods spanned by the datasets assimilated by the model (ca. 1950–2000).

The parameters in p* can be varied and adjusted independently by the model, and the ultimate objective is to find optimal values for the elements in p*. To be considered optimal, the parameters have to satisfy as closely as possible a number of formalized criteria that are all represented as real-valued penalty terms in the model’s cost function F. The most important cost function terms of the present model are constraints on the vertical velocity shear, linked to geostrophic shear estimates obtained from hydrographic data, and the requirement that the simulated distributions of temperature, salinity, nutrients, oxygen, carbon, radiocarbon, and CFC (hereinafter collectively called tracers) match the observations accurately. The velocity shear constraints in the cost function are the model’s way of implementing the geostrophic principle: the shape of the velocity profiles are required to be close to geostrophic profiles while constant offsets are allowed without penalty. Unlike in other inverse models, which enforce exact matches with estimated geostrophic shears, here, the model shears may deviate from geostrophic estimates within the uncertainties of the estimates. Details on additional dynamical constraints and smoothness terms in F can be found in Schlitzer (1993). The problem of finding optimal values for p* is equivalent to finding a minimum of F.

b. Tracer budgets

Model tracer values are defined at the centers of the grid boxes whereas flows are defined on the interfaces (Arakawa C grid). Mass and tracer budget equations are formulated for every model box, and for all tracers except CFC, long-term mean, steady-state tracer distributions are calculated as solutions of the steady-state budget equations formulated for the boxes of the model:

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In (2) summation is over all interfaces of a box, A is the area of an interface and has either a positive or negative sign depending on whether a positive flow u, υ, or w (eastward, northward, or upward) enters or leaves the box; c and ∇c are tracer concentration and surface-normal tracer gradients on the respective interfaces; and q includes air–sea fluxes and internal source or sink terms, such as radioactive decay, biogeochemical production, or particle remineralization. A weighted averaging scheme is used to express the interface concentrations c in terms of the upwind and downwind box concentrations c_u and c_d:

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Here, f_u = 0.7 is chosen, resulting in a hybrid tracer advection scheme that combines the numerical stability of the upwind scheme ( f_u = 1) with the low numerical diffusion of the centered scheme ( f_u = ½).

Setting c = 1, ∇c = 0, and q = 0 in (2) yields the mass conservation equations, which are used to calculate the vertical model flows w, based on the zonal and meridional flows u and υ. The w calculation is done column-wise, starting with the bottom-most box and assigning any net in- or outflow arising from the fluxes from the sides and from below to the transport across the top interface of a box.

Formulation of the budget Eq. (2) for a given tracer for all n_b boxes of the model results in a set of n_b coupled linear equations for the unknown box tracer concentrations c, which can be expressed in matrix–vector notation as

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where 𝗧 is the square advection–diffusion transport matrix and c is the vector of the tracer concentrations for all n_b boxes. The vector on the right-hand-side q contains the source–sink terms and boundary fluxes for each of the n_b boxes. The transport matrix 𝗧 depends on the circulation and mixing of the model, and is the same for all tracers, except for radiocarbon and CFC (see below). For the other tracers, the budget equations only differ in the boundary fluxes and source–sink terms contained in the right-hand-side q. Calculation of the different unknown tracer concentrations c therefore requires solving many systems of linear equations, which all have the same coefficient matrix 𝗧 and only differ in the right-hand-side q. Solutions are most efficiently obtained by applying a sparse 𝗟𝗨 factorization algorithm (Harwell 1995) on 𝗧, and then using the factorization and the different right-hand sides for computationally cheap solution steps. The availability of the 𝗟𝗨 factor matrices also allows efficient solving of the associated adjoint equations involving the transpose of the transport matrix 𝗧^T (see below). Surface boundary conditions for all tracers except radiocarbon and CFC are formulated as flux boundary conditions, and the respective air–sea tracer fluxes (if applicable) are contained in q.

c. Radiocarbon

For radiocarbon (and CFC; see below) surface concentration boundary conditions are used instead of flux conditions, because ¹⁴C and CFC concentrations are better known than fluxes. Consequently, the distributions of these two tracers are simulated only for model layers 2 and deeper (for a total of n_i interior boxes), while the values in layer 1 are specified. This leads to smaller tracer budget systems [Eq. (4)] as compared with the other hydrographic parameters. Another difference for radiocarbon is the contribution of the decay term with the diagonal elements of the radiocarbon transport matrix 𝗧. Internally, radiocarbon concentrations are represented as relative ¹⁴C/¹²C abundance ratios:

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where Δ¹⁴C is the ¹⁴C value in the usual Δ notation. The advantage of using ¹⁴C/¹²C ratios instead of absolute ¹⁴C concentrations is that biological effects can be largely ignored (Fiadeiro 1982), as they affect ¹⁴C and ¹²C in the same manner. The radiocarbon surface boundary concentrations are taken from Key et al. (2004), who provide estimates of natural ¹⁴C for the World Ocean. Natural Δ¹⁴C values in low- and midlatitude surface waters are typically between −45‰ and −65‰, while much lower values between −100‰ and −120‰ are found in the Southern Ocean. Solutions for the unknown model ¹⁴C values in the ocean interior are obtained by first calculating the 𝗟𝗨 factorization of the radiocarbon coefficient matrix and then using the factorization in the solution step. As for the other tracers, the factorization matrices are reused in solving the associated adjoint model (see below).

d. CFCs

CFC-11 and CFC-12 are the only time-dependent tracers in the present model. Their temporal evolution in the ocean interior is simulated by applying time-varying CFC surface boundary concentrations and an implicit forward time-stepping scheme. Analysis of observed CFC saturation levels shows that low- and midlatitude surface waters generally are in close equilibrium with the overlying atmosphere. In polar and subpolar waters, however, seasonal or permanent ice cover as well as vigorous vertical water mass exchange typically prevent surface waters from reaching full equilibrium, and saturation levels can be as low as 60%–80% (Orsi et al. 2002; Smethie and Jacobs 2005). Regional CFC saturation levels estimated from CFC observations are taken into account for the construction of the time-varying CFC surface boundary values of the model. It is assumed that in the surface water the degree of saturation remains constant in time, and time-dependent surface boundary values are then obtained by applying the observed atmospheric CFC trends (Walker et al. 2000).

Model CFC distributions are calculated for the period between 1950 and 2004 using an implicit time-stepping procedure and zero initial CFC concentrations at the start of the integration. Once the CFC concentrations c_k are known for a given time t_k, the concentrations c_k+1 at the next time t_k+1 = t_k + Δt are obtained as follows:

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In (6), 𝗧 is the advection–diffusion transport matrix for CFCs, c = (c_k+1 + c_k)/2 are the CFC concentrations evaluated at the middle of the time interval t = (t_k+1 + t_k)/2 that are actually transported by 𝗧, and b is the vector of CFC fluxes from the boundaries, also evaluated at midtime t. Substituting c and reordering the terms yields a matrix equation for the unknown concentrations c_k+1 at the new time:

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where and ^˜_b are the transport matrix 𝗧 and boundary flux vector b scaled with Δt/2 and Δt, respectively, and 𝗜 is the identity matrix. Finding the CFC concentrations for a new time step using an implicit numerical scheme thus involves solving a system of linear equations and in terms of algorithms and computational cost is equivalent to the task of calculating the steady-state distributions for any of the other tracers in the model.

It is important to note that the coefficient matrix of the system (𝗜 − ) remains the same for all time steps, and only the right-hand sides change from step to step. Again, a solver based on 𝗟𝗨 factorization of the matrix is most efficient, since the factorization step has to be performed only once, while the factor matrices can be used for all time steps in the relatively cheap solving phases. As an additional benefit, the 𝗟𝗨 factor matrices are also used to solve the associated CFC adjoint equations (see below).

The great advantage of implicit-in-time numerical schemes is their stability and the accommodation of relatively large time steps. In the present case the model was operated with a 2-yr step, which required only 27 time-step calculations and allowed for the retention of the full model CFC histories in the computer’s memory. This is essential for an efficient solving of the CFC adjoint as will be seen below. Sensitivity runs with a much smaller 0.2-yr time step showed negligible CFC concentration differences of less than 1‰. The 2-yr time step is much larger than the air–sea equilibration times for CFC; however, this does not imply CFC equilibrium at the sea surface since we apply realistic surface concentration boundary conditions (see above).

e. Dependent parameters

The vertical velocities w and the simulated distributions c of all tracers in the model (CFC fields for all time steps; steady-state fields for the other tracers) compose the set of dependent parameters of the model:

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Given values for the independent model parameters p*, all the dependent parameters p̃ can be calculated by applying the mass and tracer budget equations described above. Note that there is one dependent parameter for every budget equation considered. For mass and the steady-state tracers there are n_b budget equations each, for radiocarbon there are n_i equations, and for the CFCs there are n_i budget equations for every time step (n_i is the number of interior boxes). All tracer budget equations can be reordered and written in the following homogenous form:

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This form will be used in the adjoint formulation below.

f. Cost function

Once calculated, the simulated tracer concentrations c_i in the model are compared with values derived from measurements d_i, and misfit terms normalized by the errors of the data values σ_i are accumulated in the cost function F of the model:

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where the d_i are box averages calculated from the historical hydrographic and tracer datasets. For the steady-state tracers the box averages include all available data from all years, months, or seasons, whereas for the time-varying CFCs, box averages are calculated for 3-month periods. Comparisons are performed for those boxes (and time steps in the case of CFC) that contain data. Radiocarbon data for boxes with significant CFC concentrations, either in the upper ocean or in recently ventilated deeper water masses are believed to be contaminated with atomic bomb radiocarbon and are excluded from the ¹⁴C comparison. In addition to the misfit terms in (10), the cost function F contains many other terms, some of which also involve the independent parameters p*. For details see Schlitzer (1993, 1995, 2000, 2002, 2004).

When searching for the minimum of the cost function F, only the independent parameters p* may be varied freely, while the dependent parameters p̃ are uniquely determined by the n_e model budget equations E_i = 0. The problem of finding the smallest value of F therefore is a constrained minimization problem, which can be efficiently solved with the method of Lagrange multipliers, also denoted in meteorology and oceanography as the adjoint method (Le Dimet and Talagrand 1986; Schlitzer 2000; Thacker and Long 1988).

g. Adjoint equations

With the n_e model equations E_i = 0 and the cost function F, the Lagrangian L of the problem is given by

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where λ_j are n_e yet unknown Lagrange multipliers. Note that L is a function of model parameters p* and p̃ as well as of the Lagrange multipliers λ. In optimization theory it is shown by, for example, Hestenes (1975) that the minimum of F is a stationary point of L, at which all partial derivatives of L with respect to the Lagrange multipliers λ as well as the dependent and independent parameters p̃ and p* vanish:

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The derivatives in (11), (12), and (13) are calculated as though the λ_i, p̃_i, and p*_i were independent. Differentiation with respect to the Lagrange multipliers (12) recovers the model Eqs. (9), and differentiation with respect to the dependent and independent model parameters p̃ and p* yields the adjoint in Eqs. (13) and (14). Seeking the minimum of F is equivalent to finding a model solution that satisfies (12), (13), and (14). The Lagrange multipliers λ_j are calculated by solving (13) with ∂L/∂p̃_i set to zero. Introducing the λ_j into (14) then yields the gradient of F given by ∇F = ∂L/∂p*_i. This gradient is then passed to a descent algorithm (here a limited-memory, quasi-Newton conjugate gradient algorithm) to obtain a new, improved model state p*_new with a smaller value of the cost function F.

h. Adjoint implementation

The dimension n_e of the homogenous linear system (13),

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used to calculate the unknown Lagrange multipliers λ_j tends to be very large. For the present, relatively coarse resolution model there are more than 3.2 million equations, most of which arise from the CFC-11 and CFC-12 time-stepping equations in (7). It is important to note that this huge system can be separated into much smaller linear systems, one for each tracer in the model, and that the Lagrange multipliers associated with the different tracers can be calculated separately. This follows from the fact that the budget equations E_j = 0 of a given tracer do not contain the concentration values (dependent parameters p̃_i) of the other tracers, and the respective derivatives ∂E_j/∂p̃_i therefore vanish.

For a tracer with model equations 𝗧 · c − q = 0 [see (4) above] the partial derivative of the jth equation E_j with respect to the ith concentration p̃_i = c_i evaluates to ∂E_j/∂p̃_i = T_ji, and (15) can be rewritten as

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Thus, the Lagrange multipliers for a given tracer are the solution of a linear system closely related to the model equations of that tracer, and the transpose of the coefficient matrix in the model equations serves as a coefficient matrix of the Lagrange multiplier equations.

For the time-dependent CFCs the situation is somewhat more complicated. The concentration values c_k (dependent parameters p̃_i) for any given time step k, except the very last one n, occur in the budget equations of two steps—first in step k, during which c_k is calculated, and second in the following step k + 1:

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Nonzero partial derivatives of model equations with respect to c_k therefore occur for both steps and evaluate to ∂(E_k)_j/∂(c_k)_i = (𝗜 − )_ji (step k) and ∂(E_k+1)_j/ ∂(c_k)_i = −(𝗜 + )_ji (step k + 1). Summation in (13) over all model equations and reordering of terms yields

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For the last time step k = n, there is no contribution from a subsequent step k + 1, and one obtains

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Taken together, (20) and (19) provide a backward-stepping algorithm for the sequential calculation of all Lagrange multipliers associated with the CFC budget equations: one starts with (20) and solves for the Lagrange multipliers for the last time step λ_n, then λ_n is substituted into (19) and λ_n−1 is calculated. This procedure is repeated until the Lagrange multipliers for all time steps are calculated. For each step k of the backward calculation of Lagrange multipliers (k = n, . . . , 1), the CFC model–data differences have to be evaluated for the given time interval k, and the gradient of F with respect to the CFC concentrations c_k has to be calculated and substituted into the right-hand sides of (20) and (19). Therefore, the full time history of simulated CFC concentrations has to be accessible during backward Lagrange multiplier calculation. In the present model, the long time step of the implicit scheme and the small number of required time steps allows us to keep the full CFC model histories in the computer’s memory, thereby avoiding inefficient and time-consuming saving and retrieving to and from disk files.

Note that the calculation of the Lagrange multipliers in both the steady-state and time-dependent cases involves solving linear systems of equations that are very closely related to the original model equation systems. The coefficient matrices are given as a transpose of the original matrices, and the right-hand sides consist of the gradients of the cost function F with respect to the simulated tracer concentrations. Model tracer misfits as described in (10) contribute to the right-hand sides and therefore influence the magnitudes of the Lagrange multipliers, which in turn affect the gradients of the cost function F with respect to the independent parameters p* (14). Using a solver based on the 𝗟𝗨 factorization for the tracer simulations offers the advantage of calculating the Lagrange multipliers at very little additional computational cost, as the 𝗟𝗨 factor matrices can also be used for the solution of the transposed system.

Overall, the model calculations proceed according to the following steps:

1. initialize the independent parameters p*;

2. calculate the dependent parameters p̃ using steady-state and time-dependent tracer budget equations;

3. calculate the value of the cost function F;

4. calculate the gradient of F ∇F = ∂L/∂p*, by solving the adjoint equations in (13) and (14);

5. use ∇F in a descent algorithm to obtain a new, improved set of independent model parameters p*_new; and

6. return to step 2 unless a stopping criteria (i.e., indicating that the minimum has been reached) is met.

Step 2 is usually referred to as running the forward model. Its most computationally intensive part is the calculation of the model tracer fields, which involves solving large (but sparse) sets of linear equations. The calculation of the w is comparatively cheap. For each run of the forward model, one run of the adjoint model (step 4) is needed to complete one iteration. The availability of the 𝗟𝗨 factorization from the forward run allows for the efficient solving of the adjoint equations with little additional computational cost.

In the present model the number of boxes and the dimensions of the linear systems to be solved are on the order of 45 000. Because any given box is connected to only a very small number of neighboring boxes, the number of terms in any of the budget equations is quite small and the coefficient matrices are sparse. The sparse 𝗟𝗨 factorization method applied in the present case proved to be very efficient, and, in contrast to iterative solvers, has the advantage of providing exact solutions. Completing a single full model cycle consisting of forward simulations as well as cost function calculation and adjoint runs for ten steady-state tracers (potential temperature, salinity, oxygen, phosphate, nitrate, silicate, dissolved inorganic carbon, alkalinity, dissolved organic carbon, and natural radiocarbon) and two 60-yr-long simulations for CFC-11 and CFC-12 only takes about 40 s on a high-end personal computer. Being able to do single forward and adjoint runs quickly is an indispensable prerequisite for full model optimizations, which require thousands of these cycles.

i. Data

The present model assimilates a large number of classical hydrographic parameters and tracers (temperature, salinity, oxygen, phosphate, nitrate, silicate, dissolved inorganic carbon, alkalinity, radiocarbon, CFC-11, and CFC-12) and exploits existing large global data compilations for these variables. The recent Global Ocean Data Analysis Project (GLODAP) dataset of Key et al. (2004) is used for radiocarbon. This collection contains more than 20 000 ¹⁴C data points for almost 1200 stations from WOCE and pre-WOCE cruises (Fig. 1). The data coverage is good and all major water masses in the World Ocean are represented. The typical accuracy of individual Δ¹⁴C data values is between 2‰ and 5‰.

For the CFCs a separate data compilation is used, mainly consisting of WOCE measurements, but complemented with numerous pre-WOCE and two recent, post-WOCE cruises (Fig. 2). Overall, the CFC dataset covers the period from 1982 to 2003 and consists of more than 150 000 CFC-11 and CFC-12 measurements at more than 8000 stations. The spatial data coverage for the combined dataset is good; however, for shorter time periods or single years the respective CFC observations are sparse and distributed heterogeneously. Only very few areas such as the deep western boundary current (DWBC) region in the northwest and tropical Atlantic have repeated CFC observations over longer time periods. The estimated accuracy of individual CFC data values is 1% or 0.02 pmol kg⁻¹, whichever is larger.

a. Radiocarbon

Figure 3 shows the bottom water distribution of the natural radiocarbon for experiment B. In agreement with the data and consistent with the general concept of the global conveyor belt circulation (Broecker 1991), the model yields its highest Δ¹⁴C values in the North Atlantic (−70‰ to −80‰) and lowest radiocarbon concentrations in the northeast Pacific (−235‰). The Δ¹⁴C values in the Southern and Indian Oceans are intermediate and range between −150‰ and −165‰. There are clear signs of tongues with elevated Δ¹⁴C values along the major deep and bottom water spreading paths. This includes the southward flow of North Atlantic Deep Water (NADW) in the North Atlantic and the northward spreading of Antarctic Bottom Water (AABW) in the western South Atlantic as well as in the South Pacific and the Indian Oceans. The quantitative agreement between simulated and measured Δ¹⁴C values is excellent, and the model–data difference is only 1.3‰ on average (Table 1). The rms difference amounts to only 5.2‰ and is of the same order of magnitude as the uncertainties of the radiocarbon data. In the deep North Atlantic and parts of the Southern Ocean the measurements are systematically higher than the model simulations, owing to the contribution of bomb ¹⁴C in these waters (Key et al. 2004; Rubin and Key 2002). Note that bomb radiocarbon is not included in the model, and contaminated data values are excluded and not assimilated in the model. The very good agreement between the model and the data also holds at other depth levels not shown here. At 2000-m depth, for instance, where the lowest Δ¹⁴C values in the World Ocean are found in the northeast Pacific (−250‰), the mean and rms deviations are 0.3‰ and 5.6‰, respectively.

b. CFCs

The simulated CFC-11 bottom water concentrations for year 1998 are shown in Fig. 4 together with observations from the 1995–2000 period. As for radiocarbon, the simulated CFC-11 concentrations are in very good agreement with the data. High bottom water CFC-11 values are found in the northwest Atlantic and in the Southern Ocean at four locations around the Antarctic continent. In the North Atlantic the high-CFC bottom water concentrations are associated with deep-reaching convection in the Greenland, Norwegian, and Labrador Seas and the southward spreading of NADW in the DWBC. The highest bottom water CFC-11 concentrations in the Southern Ocean are found in the Weddell and Ross Seas as well as east of the Amery Basin and in the Australian–Antarctic Basin, indicating and confirming that deep convection and bottom water formation are occurring at multiple sites around Antarctica and are not confined to the Weddell Sea (Orsi et al. 2002). Tongues of elevated CFC concentrations spread from polar areas equatorward and reveal major bottom water flows. This is most clearly seen in the Atlantic, where the southward-flowing NADW and northward-flowing AABW produce marked CFC patches that have almost reached the equator in 1998. Bottom water flows are also reflected in the CFC-11 distribution in the South Pacific near 170°W and in the south Indian Ocean (30°E and 80°–120°E). All the bottom water spreading paths evident in the CFC distribution coincide and are consistent with paths derived from the bottom water radiocarbon distribution in Fig. 3. The major difference between the two tracers is that because of their recent origin, CFC-11 has not penetrated entire basins yet, leaving most of the deep Pacific, Indian, and eastern Atlantic Ocean still essentially free of CFCs.

Figure 5 shows the simulated CFC-11 distribution at 1700-m depth for year 1998, together with available CFC-11 data for the period from 1995 to 2000. There is some similarity with the bottom water CFC-11 distribution in Fig. 4 with regard to the relatively high concentrations in polar and subpolar regions and the small values at low latitudes. However, at 1700-m depth there is a marked north–south asymmetry, with CFC concentrations in the North Atlantic being much larger than those in the Southern Ocean. The CFC signal in the North Atlantic associated with the southward flow of the upper NADW (uNADW) has progressed farther south compared to the bottom water and has reached the equator by 1998. There is evidence for southward uNADW transport across the equator and eastward transport along the equator. As for the bottom water, the agreement between the model simulation and data is very good globally (see Table 1).

The zonal sections at 26°N in the Atlantic (Fig. 6) show that the model is also able to reproduce details of the CFC-11 distribution within the DWBC. This includes location, vertical and zonal extent, as well as the amplitude of the double CFC-11 concentration maxima associated with the upper and lower NADW cores.

c. Deep and bottom water circulation

Figure 7 shows the meridional model flow velocities at 26°N in the North Atlantic for the same section as the CFC-11 distributions in Fig. 6. Clearly visible are two velocity maxima in the DWBC located near 1750- and 4200-m depth associated with the CFC-11 maxima at nearly the same locations. Within the DWBC cores, the long-term mean model flow velocities are on the order of 1 cm s⁻¹. It should be noted that the general patterns of the meridional velocity field are quite similar in the two solutions A and B and that the two separate upper and lower cores in the model DWBC at 26°N are already present in experiment A, which did not assimilate CFC data. This indicates that information on the detailed structure of the boundary current is contained in the temperature, salinity, oxygen, and nutrient data. The structure and strength of the DWBC calculated from the classical tracers alone (experiment A) are in surprisingly good agreement with the CFC data, as shown by the relatively small CFC-11 misfits of −30% and +18% in the upper and lower NADW branches of experiment A (Table 1). Only small adjustments to the velocity field in the North Atlantic are required in experiment B to achieve full consistency with the CFC observations. These adjustments result in small changes of watermass transports (see below), but leave the flow patterns as shown in Fig. 7 largely unchanged.

Integrated water mass transports in the model are calculated for deep and bottom water layers using the interface definitions of Ganachaud and Wunsch (2000), which are based on the neutral density γⁿ. The deep water layer (27.72 < γⁿ < 28.11) is relatively thick and extends between circa 1200- and 4000-m depths in most of the low- and midlatitude ocean. In the Southern Ocean, this layer thins and rises to the surface, and south of 55°S it is typically located between 100- and 500-m depths. The entire water column below the deep water (28.11 < γⁿ) represents the bottom water layer. Its thickness varies between a few hundred meters in the northeast Atlantic and North Pacific Oceans to almost the entire water column in the polar Southern Ocean. At mid- and low latitudes the mean thickness of the bottom water layer is about 1100 m. In the North Atlantic the deep water layer mainly represents the upper NADW (uNADW), while the lower NADW (lNADW) as well as the AABW are included in the bottom water layer (see Fig. 7 for γⁿ isolines at 26°N). In the South Atlantic, the deep water layer encompasses uNADW and the largest part of lNADW, whereas the bottom water mainly represents the AABW. In the Southern Ocean the deep water density range corresponds to the upper Circumpolar Deep Water (CDW; Whitworth and Nowlin 1987), while the lower CDW and the AABW are part of the bottom water layer.

Schematic diagrams of the integrated deep and bottom water flows as well as transport estimates across selected cross sections and passages are shown in Fig. 8 for model experiment B. The deep water transport scheme (Fig. 8a) is dominated by the strong eastward flow of the deep ACC and a strong southward transport of the DWBC originating in the northern North Atlantic and continuing into the far South Atlantic. The transports of the DWBC in this layer are 14.7 Sv (Sv ≡ 10⁶ m³ s⁻¹) at 30°N and 8.6 Sv at 20°S. Near the equator a branch of the DWBC flows eastward and turns south along the African coast (2.2 Sv at 30°S). Another branch flows southeastward across the Brazil Basin, then turning eastward at about 20°S. In the model, there are three major counterclockwise recirculation gyres of the DWBC at 40°N, 20°N, and 40°S. The southern recirculation loop extends across the entire South Atlantic, while the recirculation at 40°N in the North Atlantic is confined to the western basin. In the Pacific and Indian Oceans the deep water flow pattern mainly consists of large, basin-scale gyres, and net meridional transports are apparently small.

The bottom water currents in Fig. 8b also exhibit a strong eastward flow associated with the ACC. Apart from this zonal component, the other main bottom water flows are mostly meridional, originating in polar regions and transporting bottom water equatorward. This includes the southward transport of lower NADW from the North Atlantic into the South Atlantic (5.2 Sv) and the northward transport of AABW from the Southern Ocean into the Pacific, Atlantic, and Indian Oceans. In the Pacific the dominant northward AABW transport is near 165°W. This flow passes through the Samoan Passage (10.3 Sv at 10°S) and continues into the northwestern Pacific. In the Atlantic the AABW flow is through the Vema and Hunter Channels (located at 28°S, 40°W and 33°S, 27°W, respectively; combined transport 4.2 Sv), passing through the Brazil Basin and continuing into the western North Atlantic (2.6 Sv at 20°N). The deep basins of the east Atlantic are renewed by bottom water flows through the Romanche and Vema Fracture Zones (equator–21°W, 11°N– 42°W), and the Cape Basin is flushed by a clockwise flow of AABW (1.5 Sv). In the Indian Ocean there are three main branches of northward AABW flows: near 35°E into the Mozambique Basin; near 55°E via the Crozet Basin into the Madagascar, Mascarene, and Somali Basins; and near 90°E into the South Australian, West Australian, and Central Indian Basins. Each of these bottom water flows carries between 2 and 3 Sv of AABW northward.

Zonally integrated layer transports for the deep and bottom water layers are listed in Table 2. In the Atlantic, experiment B shows combined southward NADW transports of 15.7, 17.9, and 15.6 Sv at 50°N, 30°N, and 30°S, and a relatively small net northward transport in the bottom layer at 30°S (1.5 Sv). The small net bottom layer transport arises because some of the southward transport of lNADW occurs in the bottom layer, which partially compensates for the northward AABW flux. There are only small differences among the zonally integrated transports in the Atlantic between experiments A and B. As noted above, A already achieves reasonable agreement with the CFC data in the North Atlantic (see Table 1), and only minor modifications are obviously required for full consistency.

CFC concentrations in South Atlantic bottom waters are highly sensitive to even small changes in the strength of the bottom water flows, as indicated by experiment A, which has only slightly higher (relative to experiment B) AABW transports through the Vema and Hunter Channels (5.2 Sv as compared with 4.2 Sv for B) and a slightly higher net bottom layer transport at 30°S (1.8 Sv as compared with 1.5 Sv for B). Yet these small transport differences are sufficient to produce CFC concentrations in the deep and bottom waters that exceed the observations by about 100% (Table 1). These significant CFC misfits in the South Atlantic are corrected in experiment B by a reduction of the northward AABW transport of about 1 Sv.

Both the Pacific and Indian Oceans show a meridional overturning consisting of northward bottom water flows and southward return flows in the deep water layer. In the Pacific, the net northward bottom water transports are 9.8 Sv at 30°S and 5.3 Sv at 30°N. The deep water return flows at these latitudes are somewhat smaller and amount to −6.4 and −3.2 Sv, respectively. At 30°S in the Indian Ocean, the net northward bottom water transport is 5.5 Sv, and the deep water return flow amounts to −4 Sv. There are drastic transport reductions between experiments A and B for all the integrated abyssal flows in the Pacific and Indian Oceans (Table 2). While the large northward AABW transports into the Pacific (19.3 Sv) and Indian Oceans (17.2 Sv) in experiment A are found to be compatible with the set of traditional tracers (T, S, oxygen, nutrients), they result in very large overestimations of the radiocarbon and CFC concentrations in the deep and bottom waters (Table 1) and produce CFC tongues that extend much too far into the Pacific and Indian Oceans. To reach consistency with the ¹⁴C and CFC data, bottom water transports into the Indian and Pacific Oceans have to be reduced in experiment B by factors of 3 and 2, respectively.

Corresponding to the large differences in bottom water transports in the Pacific and Indian Oceans, there are also large differences in the production and export of dense bottom water from the Southern Ocean. Experiment A exhibits very vigorous vertical overturning in the Southern Ocean and a net equatorward AABW export of about 40 Sv at 55°S (Table 2). This large bottom water export feeds the large AABW inflows into the Pacific and Indian Oceans described above and is responsible for the much too high radiocarbon and CFC deep and bottom water values over the entire Southern Ocean (Table 1). Again, radiocarbon and CFC data are seen to provide additional constraints on water mass formation and transport rates not provided by the classical hydrographic parameters. This holds especially in the Southern Ocean. To achieve consistency with these tracers a threefold reduction of the export rate of AABW at 55°S to about 16 Sv is required in experiment B (Table 2). The 16 Sv of newly formed dense bottom water from the Southern Ocean are of comparable size as the contribution of NADW from the North Atlantic.

The general patterns of deep and bottom water flows and most of the transport estimates in Fig. 8 and Table 2 are consistent with published work based either on direct observations or model reconstructions. In the subpolar North Atlantic there is broad agreement with the deep water circulation derived from current-meter observations and hydrographic sections (Schott et al. 2004), with the middepth circulation obtained from neutrally buoyant floats (Lavender et al. 2005), and with the spreading pathways of Labrador Sea Water (LSW) obtained from inverse model analysis of hydrographic lines (Paillet et al. 1998). In addition, the combined deep and bottom water transport of 16.8 Sv in the DWBC at the latitude of the Grand Banks agrees very well with the 16-Sv estimate of Schmitz and McCartney (1993) and the 17.2-Sv NADW formation rate of Smethie and Fine (2001). The mean spreading rate of circa 1 cm s⁻¹ in the model’s DWBC at 26°N is consistent with independent estimates based on CFC data (Smethie et al. 2000; Smethie 1993; Weiss et al. 1985). These studies also find evidence for strong recirculation gyres and exchanges of the DWBC with the basin interior, consideration of which helps to reconcile the apparent mismatch between current-meter velocity measurements and tracer-based spreading rates.

The deep and bottom water flows in the tropical Atlantic are consistent with the scheme of McCartney (1993) and the zonal deep water flows in the Brazil Basin are in agreement with the analysis of Treguier et al. (2003) using float data and dynamical model results. The deep water flow directions of the model in the South Atlantic are also consistent with the ³He distributions of Rüth et al. (2000), which indicate westerly flow over the Mid-Atlantic Ridge at 30°S, and little net zonal flows at 19° and 11°S. In the Argentine Basin the bottom water circulation resembles the patterns found by Peterson (1992) and Coles et al. (1996).

The northward model transport of 3.3 Sv of AABW through the Vema Channel is somewhat smaller than the direct current measurements and dynamic calculations of Hogg et al. (1982) and McDonagh et al. (2002) (4 ± 1.2 and 3.7 Sv, respectively). A major discrepancy is found in the Hunter Channel, where the model flow (0.9 Sv) is small relative to the transport estimate of 2.9 ± 1.2 Sv by Zenk et al. (1999). The combined northward AABW model transport of 4.2 Sv is significantly smaller than the 6.9 ± 1.2 Sv estimate of Hogg (2001). Given the large sensitivity of bottom water CFC concentrations in the southwest Atlantic to even slight increases of the northward bottom water transport (as indicated by experiment A; see above), an AABW transport of 6.9 Sv appears to be incompatible with CFC data and the model transport of 4.2 Sv is considered to be an upper limit for the long-term mean northward AABW transport across 30°S in the Atlantic. Consistent with the present study, Meredith et al. (2001) also report rather small northward AABW transports on the basis of a CFC data analysis in the Southern Ocean and South Atlantic, and McDonagh et al. (2002) using hydrographic sections find an even smaller net AABW flow of 1.7 ± 1.1 Sv through the Vema Channel.

In the Pacific and Indian Oceans the bottom water circulation pattern (Fig. 8b) is consistent with previous work (Fukasawa et al. 2004; Hogg 2001; Kawabe et al. 2003; Mantyla and Reid 1995; van Aken et al. 2004; Warren and Johnson 2002), and the deep water flow directions in the central and eastern Pacific as well as in the Indian Ocean (Fig. 8a) agree with the spreading directions of deep water ³He plumes originating from hydrothermal vents (Lupton 1998; Srinivasan et al. 2004). The northward model transport of AABW through the Samoan Passage (10.3 Sv) is virtually identical with the estimate of Roemmich et al. (1996) (10.6 ± 1.6 Sv) based on current-meter measurements combined with hydrographic sections.

The zonally integrated layer transports in Table 2 agree with the results of Ganachaud and Wunsch (2000) and Ganachaud et al. (2000) within their error bars. There are only two exceptions: the net 6 ± 1.6 Sv northward bottom water transport in the South Atlantic found by Ganachaud and Wunsch (2000) appears to be too large and incompatible with the CFC data (see discussion above) and the bottom water transport across 25°N in the North Pacific (1.5 ± 1 Sv) appears to be small relative to the 5.3 Sv in experiment B. It remains to be verified whether such small bottom water inflows are compatible with the radiocarbon balance of the deep North Pacific.

The net northward transport below 3200 m into the Indian Ocean of 17 Sv shown by Ferron and Marotzke (2003) is much larger than the 5.5 Sv in experiment B for the same cross section and appears to be inconsistent with the CFC observations. Experiment A, with about the same net northward transport into the Indian Ocean as in Ferron and Marotzke (2003), produces bottom water CFC-11 concentrations that are on average 0.11 pmol kg⁻¹ or 600% higher than the observations.

The finding in the present model that the production rates of deep and bottom waters in the North Atlantic and Southern Oceans are of about the same magnitude (about 16 Sv) is consistent with the study of Orsi et al. (2002), who exploited CFC inventories in the Southern Ocean for the estimation of watermass production rates, and is also in agreement with the postulations of Broecker et al. (1998) on the basis of the PO*₄ parameter. Many previous AABW production estimates (see Fahrbach et al. 2001; Foldvik et al. 2004 and references therein) are much smaller; however, values are often difficult to compare because of differences in the watermass definitions used and because most studies focus on specific regions and do not provide circum-Antarctic production rates. Garabato et al. (2002) find significant AABW production in the Weddell Sea and a northward transport into the Scotia Sea of 9.7 ± 3.7 Sv. This rate is compatible with the present study and the result of Orsi et al. (2002), when combined with contributions from other production sites in the Southern Ocean. In the model, there are four regions around the Antarctic continent where significant downwelling of dense and CFC-rich waters occurs: the Weddell and Ross Seas, the region east of the Amery Basin, and the Australian–Antarctic Basin (Fig. 8b). All of these areas show enhanced CFC concentrations at the bottom.

d. CFC forecast

It has been shown above that model solution B produces radiocarbon and CFC distributions that are in very good agreement with the available observations. Here, this solution is combined with best-guess estimates of atmospheric CFC trends for the next 70 yr (WMO 2003) to simulate the evolution of oceanic CFC distributions in the foreseeable future. This experiment implicitly assumes that the ocean circulation remains unchanged with time and that temporal CFC changes in the ocean are driven by atmospheric CFC changes only. The results of this run allow for the identification of the regions of largest future CFC changes in the ocean, provide guidance for future CFC observational programs, and can serve as benchmark distributions against which future CFC observations can be compared. Major differences between predictions and future data may be indications of changes in the ocean circulation.

According to the WMO (2003) prediction, the atmospheric concentrations of CFC-11 and CFC-12 will decrease over the next 70 yr. Because of differences in the emissions and lifetimes of the two substances, however, the timing and amplitude of the decreases are different. CFC-11 concentrations reached their maximum in the mid-1990s and have been decreasing since then. The future decrease is expected to be nearly exponential with a lifetime of 45 yr. By 2030 the anticipated atmospheric CFC-11 concentrations are expected to be only 60% of their present values. For CFC-12, the decrease has only begun in recent years, and its rate of change will be slower because of continued emissions and a larger estimated atmospheric lifetime (100 yr). In 2030 the predicted atmospheric CFC-12 concentrations will be about 80% of their present levels.

Figure 9 shows the model-simulated temporal evolution of the CFC-11 concentrations in the upper branch of the DWBC at 40°N in the North Atlantic together with available measurements from the region. The data and simulation are in very good agreement and show rapid concentration increases during the period from 1980 to 2010. Afterward, the rate of increase declines, and maximal CFC-11 concentrations in the uNADW at this location are expected for 2025. At locations farther south in the DWBC the general trends are similar to those in Fig. 9, but the concentration maxima appear later in time and the subsequent decreases are slower. In the tropical Atlantic, CFC-11 concentrations in uNADW and lNADW will continue to increase beyond 2060.

The temporal evolution of CFC-11 in the bottom waters of the global ocean are shown in Fig. 10. It is found that of all ocean basins the western Atlantic is the one that is most rapidly flushed with CFC-bearing bottom waters, and that this invasion is occurring at comparable rates from the north and south. By 2030 the bottom water CFC-11 concentrations in the entire western Atlantic are expected to exceed 0.1 pmol kg⁻¹. CFC-rich bottom waters also spread into the South Pacific and into the Indian Ocean along the bottom water pathways depicted in Fig. 8b; however, the spatial extent is limited, and by 2030 the bottom water CFC-11 concentrations in most of the Indian Ocean and North Pacific are expected to remain below 0.1 pmol kg⁻¹. CFC-11 bottom concentrations above the current detection limit of 0.02 pmol kg⁻¹ are expected for 2030 in the entire Atlantic, up to about 20°S in the Indian Ocean and as far north as 15°N in the central Pacific (180°). According to the present model, bottom water CFC-11 concentrations in 2030 will remain below the detection limit in the entire northeast Pacific and in the Indian Ocean north of 10°S.