A degradation approach to accelerate simulations to ... - Springer Link

( Springer-Verlag 1998

Climate Dynamics (1998) 14 : 101—116

O. Aumont · J. C. Orr · D. Jamous · P. Monfray O. Marti · G. Madec

A degradation approach to accelerate simulations to steady-state in a 3-D tracer transport model of the global ocean

Received: 18 March 1997 / Accepted: 27 July 1997

Abstract We have developed a new method to accelerate tracer simulations to steady-state in a 3-D global ocean model, run off-line. Using this technique, our simulations for natural 14C ran 17 times faster when compared to those made with the standard non-accelerated approach. For maximum acceleration we wish to initialize the model with tracer fields that are as close as possible to the final equilibrium solution. Our initial tracer fields were derived by judiciously constructing a much faster, lower-resolution (degraded), off-line model from advective and turbulent fields predicted from the parent on-line model, an ocean general circulation model (OGCM). No on-line version of the degraded model exists; it is based entirely on results from the parent OGCM. Degradation was made horizontally over sets of four adjacent grid-cell squares for each vertical layer of the parent model. However, final resolution did not suffer because as a second step, after allowing the degraded model to reach equilibrium, we used its tracer output to re-initialize the parent model (at the original resolution). After re-initialization, the parent model must then be integrated only to a few hundred years before reaching equilibrium. To validate our degradation-integration technique (DEGINT), we compared 14C results from runs with and without this approach. Differences are less than 10& throughout

O. Aumont ( ) · J.C. Orr · O. Marti Laboratoire de Mode´lisation du Climat et de l’Environnement, DSM, CE Saclay, CEA, L’Orme des Merisiers, Bt. 709, F-91191 Gif sur Yvette Cedex, France D. Jamous · P. Monfray Centre des Faibles Radioactivite´s, Laboratoire mixte CNRS-CEA, L’Orme des Merisiers, Bt. 709/LMCE, CE Saclay, F-91191 Gif sur Yvette Cedex, France G. Madec Laboratoire d’Oceánographie Dynamique et de Climatologie, (CNRS/ORSTOM/UPMC) Universite´ Paris VI, 4 place Jussieu, Paris, France

98.5% of the ocean volume. Predicted natural 14C appears reasonable over most of the ocean. In the Atlantic, modeled *14C indicates that as observed, the North Atlantic Deep Water (NADW) fills the deep North Atlantic, and Antartic Intermediate Water (AAIW) infiltrates northward; conversely, simulated Antarctic Bottom Water (AABW) does not penetrate northward beyond the equator as it should. In the Pacific, in surface eastern equatorial waters, the model produces a north—south assymetry similar to that observed; other global ocean models do not, because their resolution is inadequate to resolve equatorial dynamics properly, particularly the intense equatorial undercurrent. The model’s oldest water in the deep Pacific (at !239&) is close to that observed (!248&), but is too deep. Surface waters in the Southern Ocean are too rich in natural 14C due to inadequacies in the OGCM’s thermohaline forcing.

Notation

DEGINT D0 D1 D2 C v F *Z 1 j k E G A i *t i

Name of our acceleration method Original model OPA used in DEGINT First degraded model Second degraded model Tracer concentration Velocity vector Gas exchange flux of tracer at the sea surface Thickness of the first layer of the model Radioactive decay of the radiocarbon Rate of change of surface ocean tracer Gas exchange coefficient Total gain linked to the use of the acceleration method Specific acceleration ratio of Di compared to D0 Integration duration (in y) of Di

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º, », ¼ u, v, w g, f i K ,K X Y k ,k x y a, a* X, X*

Aumont et al.: A degradation approach to accelerate simulations to steady-state in a 3-D tracer transport model

Zonal, meridional and vertical velocities of the degraded model Zonal, meridional and vertical velocities of the parent model Horizontal meridional and zonal length of a parent model’s box diffusion tensor Horizontal zonal and meridional diffusion coefficients of the degraded model Horizontal zonal and meridional diffusion coefficients of the parent model Horizontal box surface area of the parent and degraded model, respectively Box volume of the degraded and parent model, respectively

1 Introduction Since Bryan’s (1969) pioneering efforts, 3-D ocean models have proved themselves essential for the study of ocean circulation and climate. Simulations of the ocean’s carbon cycle have also been made with such models (e.g., Maier-Reimer and Hasselmann 1987). However for the latter, global-scale runs must be made with coarser resolution, because multiple tracers must be integrated to steady-state, typically requiring several thousand model years. Coarse-resolution models have a certain number deficiencies. For instance, such models with explicitly prescribed vertical eddy diffusivities tend to produce a vertical thermocline structure which is too diffuse, and thus too warm (Bryan 1987). Near the equator, coarseresolution models upwell massive amounts of deep water, largely in the central Pacific in a distribution which is symmetric about the Equator; observations suggest the contrary, i.e., that upwelling in the Pacific does not derive directly from the deep ocean, is distributed asymmetrically occurring mostly just off of Peru, and that it is fed almost entirely by the equatorial undercurrent (Toggweiler et al. 1991; Toggweiler and Carson 1995). These problems becomes less severe when models use much higher resolution, but the added computational expense limits the integration time and the type of simulation. Techniques exist already to speed up 3-D ocean model simulations, i.e., to accelerate their convergence towards equilibrium. One approach, conceptually simple, would be to initialize the model with observations. If data were sufficient, without error, and synoptic, and if the model were perfect, steady-state would be attained virtually instantaneously. But in the real world, models are imperfect and data coverage is far from ideal, particularly as concerns the carbon cycle. Another approach, developed by Bryan (1984) for potential temperature H and salinity S, relies on the use

of a time step, that increases with each depth layer. Toggweiler et al. (1989) apply approach to Bryan’s (1984) a passive tracer, natural 14C. Unfortunately, this acceleration technique does not conserve the tracer’s mass when there is a source or sink term. For instance, for radioactive decay, accelerated (deeper) layers would represent a net loss to the system if it were not for additional compensatory input due to a resulting increase in the air—sea gas exchange. For natural 14C this does not represent a problem, for other tracers without such an outside source in the model (e.g., for phosphate), there would be a net loss with time. Another problem is that conceptually, Bryan’s (1984) approach is designed only for annual-mean models. For instance, the seasonal cycle is altered in models when the accelerated time step becomes the order of a season or more. For example, when using Bryan’s (1984) time-step acceleration method to simulate the equatorial undercurrent in the model developed at the Laboratoire d’Oceáno-graphie Dynamique et de Climatologie (LODyC), its seasonal cycle disappears completely. Other means to speed simulations cannot really be classified as techniques to accelerate convergence. Some models might be recoded for better vectorization or parallel processing. Others might be able to take advantage of improved theories to parameterize subgrid scale mixing, thereby effectively allowing larger grid spacing. And then of course the speed of computers seems always to be improving. In practice, modelers will employ as many existing techniques as are available (and practical to implement), in concert, to make more realistic simulations technically feasible. We present here a new method to accelerate convergence of a tracer transport model towards a steady state. The principle of this approach consists in obtaining, with the least expense, a 3-D initial state which is not far from the final equilibrium solution. By definition then, this initialization field must be compatible with fields of advection and turbulence from the on-line model. To obtain this initial state rapidly, we first degraded on-line model output by averaging it horizontally over sets of four gridbox squares for each vertical level. Subsequently, we made off-line simulations to equilibrium with this lower resolution model, whose output was then used to initialize the original model. This degradationinitialization-integration process (DEGINT) speeds simulations by a factor of 17 in the tracer-transport version of the general circulation model (GCM) developed at LODyC. With simulations of natural 14C, we show that the original model’s steady state is near that of our runs which employ DEGINT. Finally with these same results, we evaluate the deep circulation predicted by the model, i.e., with respect to observations of natural 14C.

Aumont et al.: A degradation apprach to accelerate simulations to steady-state in a 3-D tracer transport model

2 Model description For the simulations here, we used output from a seasonal ocean GCM from LODyC known as OPA (Oceán Parallelise´). The code was first described by Andrich (1988). Subsequently, it has been modified and used in both regional (Madec and Cre´pon 1991; Madec et al. 1991a, b; Blanke and Delecluse 1993) and global configurations (Marti 1992; Delecluse et al. 1993; Delecluse 1994; Madec and Imbard 1996; Guilyardi and Madec 1997). As with most other general circulation models, OPA solves the primitive equations of motion and uses the rigid lid and Boussinesq approximations. The global model contains 30 vertical levels, which vary in thickness from 10 m at the surface to 500 m at depth. There are ten layers in the first 100 m. The deepest level reaches 5000 m. Horizontal grid size varies between 0.5° and 2° with the higher meridional resolution at the Equator to take into account its enhanced dynamics (Fig. 1). The OPA model is still developing rapidly and has relatively high resolution for a global ocean model. Hence, dynamic (on-line) simulations have been made semi-diagnostically, i.e., over much of the ocean H and S were restored toward monthly averaged observations (Levitus 1982) throughout the water column (with a time constant that varies from 50 days at the surface to 1 y at depth). Toggweiler et al. (1989) use a similar approach for their ‘‘robust-diagnostic’’ version of the GCM from GFDL. However, they restore H and S throughout the water column over the entire ocean. Also their restoring time

constant at depth is 50 times longer (i.e., 50 y). In OPA, as detailed by Marti (1992), H and S restoration is relaxed in four regions: (1) in the mixed layer, (2) near the equator, to avoid the propagation of the error of the density on the velocity field due to the decrease of the Coriolis acceleration (Fujio and Imasato 1991), (3) near the boundaries of the ocean (where advection and turbulence are most rapid) and (4) in high latitudes where measurements are seasonally biased, most being made during summer. OPA uses a curvilinear grid (Madec and Imbard 1996). This grid is distorted so that the northern singularity is not over the ocean (i.e., directly over the North Pole as in other models). That is, the singularity is shifted over Asia to allow longer time steps while still respecting the CFL stability criterion (Courant et al. 1928). The grid is also orthogonal; it retains numerical accuracy to the second order (Marti et al. 1992). The GCM is coupled to a model that predicts the vertical mixing coefficient k over the entire water column. As z a consequence, OPA is able to describe, prognostically, turbulence in and below the mixed layer, as driven by heat fluxes and wind stress, according to equations describing turbulent kinetic energy (TKE) (Gaspar et al. 1990; as modified by Blanke and Delecluse 1993). Relative to other simpler approaches, where k ’s are z prescribed either explicitly or as a function of the Richardson number, the TKE parametrization improves (1) equatorial dynamics, particularly regarding the representation of sea-surface temperatures, (2) the equatorial undercurrent, and (3) the thermocline as shown in a high resolution model study by Blanke and Delecluse (1993). We have studied passive tracers with a tracer-transport (off-line) version of OPA (Marti 1992) driven by monthly averaged fields of advection and vertical turbulent diffusion from the dynamic (on-line) model. Horizontal turbulence occurs exactly as in the on-line model, i.e., a Laplacian operator with a uniform horizontal diffusion coefficient of 2000 m2 s~1, constant in time. Regarding advection, we employed the fluxcorrected-transport scheme from Smolarkiewicz (1982, 1983) and Smolarkiewicz and Clark (1986). This advection scheme is little diffusive and guarantees positive tracer concentrations. To model the natural component of 14C, we treated the 14C/12C ratio as a concentration and neglected effects due to fractionation; measurements reported as *14C are thus directly comparable to modeled results (Toggweiler et al. 1989). Within the ocean, the model transports radiocarbon as a passive tracer according to the conservation equation LC "!v ) +C#+ ) (i+C)!jC Lt

Fig. 1 Equidistant cylindrical projection of the numerical masked grid used in OPA global ocean model. One mesh parallel and meridian over three are shown

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(1)

which describes the rate of change of radiocarbon C as a function of advection, diffusion, and radioactive

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decay, respectively. Here v is the velocity vector, i is the diffusion tensor, j is the decay rate of 14C (from its half-life t "5730 y). 1@2 For the surface boundary condition following Toggweiler et al. (1989a), air—sea gas exchange F is the difference in 14C between atmosphere C and ocean C times a o the rate of change of the DIC (dissolved inorganic carbon) concentration in the upper layer (k), so that F"k(C !C ) (2) a o For natural 14C, C is held constant at 0& and C is a o that predicted in each surface grid box of the ocean model. The ocean was initialized everywhere to !150&, the global mean in natural 14C as determined from GEOSECS). For air—sea transfer, the normalized rate (in s~1) at which DIC is replenished is k" CO

E

(3) *z 2463& ) 1 where CO is surface DIC concentration, held con2463& stant at 2.0 mol m~3 (Toggweiler et al. 1989), *z is the 1 depth of the first layer. E is the gas transfer coefficient (in units in mol m~2 s~1) as given by Eq. (8) from Wanninkhof (1992), which is a quadratic function of wind speed that passes through the global mean value derived from bomb 14C (Broecker et al. 1986); E contains a chemical enhancement term for low wind speeds. For wind speed, we used climatological monthly mean values derived from satellite observations (Boutin and Etcheto, personal communication). The same boundary conditions are used as the standard for the Ocean Carbon-Cycle Model Intercomparison Project (OCMIP) of IGBP/GAIM (International Geosphere-Biosphere Program/Global Analysis, Intercomparison, and Modeling Task Force).

3 The acceleration method From the original (or parent) model, we first constructed the degraded model, which is coarser in resolution. The dynamics of the parent and degraded models must be as compatible as possible. Thus the degraded model must always be constructed from the original dynamic output of the parent model. We emphasize that our approach is based on output from only one dynamic model; our approach is not equivalent to constructing a lower resolution online model and using output from that to initialize a different on-line model at higher resolution. The latter approach would probably not work very well because ocean dynamics are nonlinear and would differ between two on-line models of different resolutions. The horizontal resolution of our degraded model has only one fourth the number of grid cells of the parent model. That is, fields of advection, turbulent diffusion, and tracers were essentially averaged onto ‘‘squares’’ of two boxes longitudinally by two boxes latitudinally. The vertical resolution is not degraded. The degradation procedure is designed to conserve both water fluxes and tracer fluxes at the boundaries of each degraded grid cell, with respect to the corresponding borders for each set of four boxes of the original model (Fig. 2).

Fig. 2 Comparison of the dynamic and tracer properties of the parent and degraded model. Bold capital letters correspond to the parent model, dashed lines and slanted characters denote the degraded model; a and A are the zonal water fluxes, b and B the water fluxes along direction j, u, v and U, V the horizontal velocities, t and T the tracer concentrations, e and e both horizontal lengths of the 1 2 parent grid box. The vertical velocity of the parent and degraded models (w and W, respectively), are located immediately above the tracer grid points but between two tracer boxes

As for degrading tracers, they are averaged from each of the four grid-boxes weighted by their respective ocean area. Tracer sources and sinks (i.e., gas exchange and radioactive decay for 14C) must also be degraded, as decribed in the Appendix. Because the degradation procedure doubles the size of grid boxes in the x and y directions, the time step can also be enlarged by a factor of two. Combining both spatial and temporal changes, the degraded model should run, theoretically, eight times faster. The second step of the method is the re-initialization of the parent model by the tracer fields from the degraded model. After the degraded model reaches equilibrium, its output must be transferred to the grid of the parent model. For such, we simply set the concentration of each set of four grid boxes of the parent model to the concentration of the corresponding one grid box of the degraded model. This approach, the reciprocal operation to that of the degradation of on-line output, is more appropriate than is interpolation. For seasonal models such as OPA, the parent model must be initialized with instantaneous output from the degraded model at the identical time of year. Monthly or annual averaged output are not appropriate. Degradation of model output can be repeated several times, making the degradation-initialization procedure a multi-step process. The number of degradations possible depends on the resolution and the bathymetry of the original model. Any model should be able to be degraded at least once. In the case of the current global version of OPA, two degradations are possible. A third degradation would open the strait of Panama, thus resulting in a radical change in ocean circulation. For later discussion, we denote the parent model as D0 (not degraded), the first degradation as D1, and the second degradation as D2. In practice, only the coarsest resolution model (D2) was integrated the equivalent of several thousand years to reach steady state. Subsequently, the intermediate resolution model (D1) was initialized by equilibrium output from D2, then run to equilibrium (requiring


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Fig. 3 Schematic description of DEGINT. Solid lines denote the integration of the models with the corresponding integration period below each line. Dashed lines symbolizes the re-initializations of D1 (re-initialization 1) and D0 (re-initialization 2) by output of D2 and D1, respectively

only several hundred years). Finally the original model (D0) was integrated (also for a much shorter period) after being initialized with equilibrium output from D1 (Fig. 3).

4 Results We first evaluated DEGINT by comparing natural 14C results from the first combination (D2—D1: equilibrium output from D2 used to initialize D1, followed by a short integration) to a standard non-accelerated run (D1 only, to equilibrium). Unfortunately, the same type of validation is not practical for the second transition from D1 to D0 simply because, without DEGINT, the cost to run D0 to equilibrium is exorbitant. However, we did the next best thing by evaluating the behavior of the parent model (D0) following re-initialization, i.e., we studied its drift in 14C and how that changed with time. 4.1 First re-initialization

For D2, we arbitrarily defined its ‘‘equilibrium’’ to have been reached after a 3000-y simulation. After that time, only 1% of the volume of the model ocean drifts by more than 0.01& y~1; in contrast, after only 2000 y, more than 20% of the model ocean drifts by the same amount (Fig. 4a). For D1, there is a greater need for an equilibrium solution relative to runs D2 and D2—D1 because output of both the latter are referenced to D1. We arbitrarily define that D1 has reached a ‘‘satisfactory’’ steady-state when 98% of the model ocean is within 1& of the true equilibrium. Since a true equilibrium solution would require an infinitely long integration, our criterion must be converted to a practical form using model drift. Assuming that the slowest ocean response time is of the order of 1000 y, our arbitrary steady-state criterion above is equivalent to the case when model drift at every grid point is below 0.001& y~1 over 98% of the ocean. This criterion was met in D1 after a 3000-yr integration (Fig. 4b).

Fig. 4a, b Drifts in natural 14C for a D2 after integrations of 2000 y (solid), 3000 y (dashed), and 4000 y (dotted), and b for D1 integrated 2000 y (solid), 3000 y (dashed), and 3500 y (dotted). Class limits are $0.1, 0.25, 0.5, 0.75, 1, and 2]10~3 &/y

Figure 5 shows the time evolution of the globally averaged concentrations of simulations made with D2 alone, D1 alone, and D1 subsequent to re-initialization with 2700-y output from D2 (denoted here as D2—D1). For the latter, D2 was integrated for 2700 y so that 300 y after using D2’s output to initialize D1, all runs could be compared at 3000 y. From zero to 300 y, both D1 and D2 became globally richer in 14C. This global enrichment corresponds to invasion of atmospheric 14C into the thermocline (see Fig. 5b and c up to about 500 y). Like the rest of the ocean, these waters are initialized to !150&, which is, on average, substantially less than their 14C content at steady-state. For the deep ocean, change occurs much more slowly, but in the opposite sense to waters above. That is, initial deep values at !150& are richer in 14C than those at steady-state. In the thermocline, this slower trend became perceptible only after more intense near-surface mixing had a chance to act out its part. Absolute values of differences between D2 (after transfer on the grid of D1) and D1, both run separately,

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Fig. 5a–d Evolution of the globally averaged 14C concentrations for D2 (solid), D1 (dashed), and D1 initialized by D2 (dotted). Panels are given for 14C concentrations averaged over a the whole ocean, b level 5 at 45 m, c level 22 at 1033 m, and d level 28 at 3780 m

average 3.0& after 2700 y of simulation. These differences arise because both models converge toward slightly different steady-states (see below). Re-initialization remedies this problem. Subsequent to re-initialization, the first combination (D2—D1) readily converges toward the steady-state of the complete run of D1. After the 300-y integration (D2—D1[300]), the difference between D2—D1 and D1 is reduced to 0.9&. Figure 6 shows differences in *14C between the model D2—D1 versus the run using only D1. Subsequent to re-initialization, differences decrease rapidly even after only 100 y of integration. After 300 y (D2—D1 [300]), most of the ocean (84%) differs by less than 1&. At $5& (roughly the precision of measurements during GEOSECS), 98.5% of the ocean falls within this range. But there still remain regions that differ by more than 10& between model versions D2—D1 [300] and D1 (alone). These regions comprise 1.2% of the ocean’s volume, and are located in two areas whose land-ocean boundaries differ dramatically between degraded model versions D1 and D2: the deep Gulf of Mexico and the deep Celebes Sea. In D1 (and D0) these zones are isolated and stagnant; in D2, both areas are wellventilated because the bathymetry defining these regions has been smoothed and partially removed as

a result of the degradation procedure. The largest residuals in these two problem areas (up to 150& between D2 and D1, when each run separately) decreased relatively little between 100 y and 300 y are re-initialization. More than 1200 y would have been necessary to completely eleminate the largest differences. But given that both basins in D1 (and D0) are stagnant, and thus do not affect surrounding waters significantly, we did not go to the trouble of making such expensive integrations. Currently, these two isolated regions hold relatively little interest when it comes to validating global ocean models. Few data are available in the Gulf of Mexico; none have been reported for the Celebes Sea. Although the parent model D0 has horizontal diffusion coefficients which are constant everywhere, such is not the case for degraded model versions D1 and D2 (see Appendix). That is, we found it necessary to degrade these coefficients which are used to parametrize horizontal turbulence. Without degradation of horizontal diffusion coefficients, the difference between D1 and D2—D1 increases from 0.9& to 2.2& on average at 300 y after re-initialization. In some areas located mainly in the deep ocean where the bathymetry is changed most, the difference reached 20&. Thus one should not use the same horizontal diffusion


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Fig. 6 *14C differences (&) between a D2 after an integration of 3000 y (solid), D1, 50 y after its initialization by D2 (dashed), and a 3000 y simulation of D1. b Same as a with D1, 100 y after its initialization (solid), and D1, 300 y after its initialization (dashed). Dotted vertical lines on both graphs denote the $5& precision of the measurements. Class limits are #1, 2, 5 and 10&

coefficients in the degraded model as in the parent model. We found little gain in integrating D1 more than 300 y after re-initialization with output from D2 (i.e., D2—D1 [300]). That is, 99% of the re-initialized D2—D1 [300] simulation was within 5& of D1 when run alone. Thus we went no further with degraded version D1. Instead, we used output from D2—D1 [300] to initialize the original model D0. 4.2 Second re-initialization

We validated the final step of DEGINT procedure (i.e., D1PD0) by studying the temporal drift of modeled levels of natural 14C in D0, which is the conventional method of determining how close model results are to equilibrium.

Fig. 7a–c Drift of the natural 14C levels (&y~1) for a D1 integrated 100 y with equilibrium output from D2 (solid) versus the same but integrated 300 y and b output from the full series DEGINT after integration of D0 for 100 y (solid) and 300 y (dashed). Dotted line on b represents the same as the before but with D0 directly initialized with equilibrium output from D2. D0 is integrated for 135 y so that the total computing cost is equivalent to a full series of DEGINT as presented on Fig. 2b. c Represents the temporal evolution of the quadratic mean of the drifts for D2-D1 (dashed) for D2-D1-D0 (solid)

Figure 7 shows the drift in D0 after it was re-initialized with output from D2—D1 [300] and followed by integrations for 100 and for 300 y (D2—D1—D0 [100] and D2—D1—D0 [300]. Figure 7 also shows comparable output from D2—D1 [100] and D2—D1 [300]. The original model D0 converges toward its steady state faster than the first degraded version D1 (Fig. 7c). At 100 y after re-initialization, only 16% of the ocean in

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D0 drifted by more than 0.01& y~1; after 100 y in D1, 37% of the modeled ocean drifted by the same amount. These different behaviors of D1 and D0 after their re-initialization come from the design of the degradation procedure. Model D1 has some walls of zero thickness (see Appendix) which result from the procedure used for the first degradation (D0PD1, see Appendix). These walls were not conserved in the second degradation (D1PD2). The opening of such walls explains the absence in D2 of some quasi-stagnant waters which were present in D1. Thus D1 is more like D0 than D2. Equilibrium output from D1 provided better initial conditions for D0 than did D2 equilibrium output for D1. But what is the point of the intermediate model? Would it not be easier and just as good to initialize the original model D0, directly with output from D2 (i.e., without using D1)? Figure 7b shows the drift of output from D0 after running it for 135 y, following initialization with output directly from D2. We chose that particular sequence, D2—D0 [135], because it is equivalent in computer time to our standard run D2—D1 [300]—D0[100]. The latter two-step approach is preferable because at the end, only 5% of the ocean volume drifts by more than 0.02& y~1; in the former, 20% of the ocean drifts by the same amount. Because D0 reaches its steady state in fewer model years than does D1 and because D0 is extremely computer intensive, we chose to perform only a 100 y integration after re-initialization in the standard DEGINT procedure. Our choice is justified for two reasons: (1) in D2—D1 [100], 98.8% of the ocean volume differs by less than 10& from D1, and (2) D0 converges toward its steady-state faster than does D1. Thus after a 100 y simulation in D0, results should be at least as good. 4.3 Modeled circulation

Figure 8 shows the simulated global mean profile of natural C-14 from OPA, the observed profile (the mean from GEOSECS observations), and simulated results from the Prognostic 3-D ocean model (version P) from GFDL (Toggweiler et al. 1989). Observations above 800 m are contaminated with the bomb 14C transient, and are thus not included. The difficulty for model P to match the observations in the thermocline (1000—2000 m) is a well-known problem, due to the manner in which the GFDL model’s eddy diffusion coefficients are defined a priori; this detail likewise produces a mean vertical temperature profile which is too diffuse (Toggweiler et al. 1989). In OPA, agreement is better because eddy diffusion coefficients are calculated prognostically with the TKE model (see Sect. 2); vertical structure is much more sharply defined as in the real ocean. However, natural 14C indicates that there still remains a problem in OPA, near the base of the thermocline. Deep waters in OPA and in model

Fig. 8 Globally averaged vertical profiles of *14C in & according to observations (GEOSECS), and model run D2-D1-D0[100], as well as the P experiment from Toggweiler et al. (1989)

version P from GFDL are too old. We show below that in OPA this problem is largely due to inadequate formation of AABW; the P simulation from GFDL underestimates deep-ocean 14C, mostly because its NADW (young; rich 14C) does not penetrate into the deep North Atlantic as it should. Figure 9 shows a surface map of the predicted natural *14C from OPA using output from D2[2700]— D1[300]—D1[100], which has now become our standard approach. In upwelling or convective regions, concentrations are low because of contact with old waters from the deep ocean which are depleted in radiocarbon. The oldest water (minimum of !60&) in the surface equatorial Pacific is found just off of Peru where relatively shallow waters supplied by the equatorial undercurrent upwell to the surface. Once at the surface, this water spreads to the west, mostly in the Southern Hemisphere in agreement with observations (Toggweiler et al. 1991). Maximum values in surface *14C are found in the sub-tropical gyres, whose waters have had a much longer time to equilibrate the atmosphere. But even there, waters do not fully equilibrate with atmospheric *14C (at 0&), due to transport by downwelling, subduction, and diffusion, as necessary to satisfy losses due to radioactive decay in the deep ocean. In the Southern Ocean, modeled levels of *14C are much too high (minimum at !90&); the observed minimum is at about !160& (see Toggweiler and Samuels 1993). Figure 10 shows the observed and modeled distributions for a north—south section in the west Atlantic as sampled during GEOSECS. The southward extent and structure of NADW at mid-depth is much like that observed. Deeper down, the modeled NADW


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Fig. 9 Surface distribution of pre-bomb *14C (&) simulated by the present model D0 after the use of DEGINT

fills the deep North Atlantic basin (unlike in model P from GFDL in which NADW does not penetrate below 2500 m). In the north, the 80& and 100& isolines do not extend quite deep enough nor far enough southward (the latter isoline falling short by about 10° of latitude). These relatively minor difficulties may be partially remedied when formulations for isopycnal mixing (e.g., Redi 1982) are incorporated in OPA. Still at depth, the 120& isoline in OPA is too far to the south indicating that along-bottom, northward transport of AABW is inadequate. On the other hand, the natural 14C signature of the Antarctic Intermediate Water (AAIW) seems to closely match the observations. Figure 11 shows north—south section taken in the west Pacific during GEOSECS. The simulated midwater minimum (!239&) is close to that observed (!248&) but it is pushed toward the bottom. Surrounding the minimum, the difference in the modeled versus observed structure seems to be caused by a combination of two problems: (1) too much penetration of 14C from overlying waters in the north and (2) insufficient northward extension of AABW, which limits supply from the south. The first problem is evident in northern waters nearer the surface: modeled isolines have a downward tilt when moving north, whereas observed isolines slope in the opposite direction. One result is that northward of 20 °N, the modeled 200& isoline dips below 2000 m; in the observations the same isoline never drops below 1200 m. These problems in the North Pacific seem due to excessive downwelling in the model. The second problem is discussed in Sect. 5.3.

5 Discussion 5.1 Computation time gain

Table 1 lists the necessary computer time and relative gain for simulations of natural 14C in the off-line version of OPA using (1) the conventional technique, i.e., no degradation D0[3000], (2) a one-step degradation D1[3000]—D0[100], and (3) our now standard twostep degradation D2[2700]—D1[300]—D0[100]. Our standard approach runs 17 times faster than the conventional non-degraded technique. That total acceleration factor results from the cumulative gain from both degradations. In theory each degraded model should run eight times faster than its parent (Sect. 3). However in practice, two additional factors enter in. The gain due to degradation may be either higher or lower. First, during the process of degradation, one smoothes fields of advection sufficiently to substantially decrease velocity maxima. Thus we increased the time step even further, beyond a simple doubling, as permitted by the CFL criterion. The time-steps in model D1 and D2 are 2.4 and 5.4 larger than in the reference model D0. Secondly, one must also consider the vector length of the supercomputer in use. The second factor was negligible for the first degradation, D1 runs 9.9 times faster than D0. However, the vector length is important for the second degradation, from D1 to D2. For D2, the vector length on the Cray C90 is reduced to below its optimum value, making simulations less efficient (see column 4 in Table 1). Thus even with a larger time step, the 30% loss in computer efficiency makes simulations in D2 run only 6.1 times as fast as those in its parent D1. To counteract the loss of efficiency due to vector

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length decrease, one could always recode the model converting the 2—D horizontal arrays into 1—D arrays. We avoid such complexities however, because the total gain would be relatively small. Our simulations ran 17 times faster when using DEGINT, but the total gain G varies as a function of the total integration time as well as the relative amount of time spent integrating D2, D1, and D0. *t@ 0 G" , (4) *t0#*t1#*t2 0 1 2 A A A where the A is the individual acceleration ratio, and the *t represents the integration time. Subscripts 0, 1 and 2 denote degraded models D0 (non-degraded), D1, and D2, respectively. For OPA, the acceleration ratios are A "1, A "9.9 and A "61.1, and the integration 0 1 2 times are *t "100 y, *t "300 y, and *t "2700 y, 0 1 2 for our standard technique with DEGINT. The term *t@ refers to a complete integration in D0 without 0 DEGINT (*t@ "3000 y). 0 Equation 4 shows that G increases with *t@ . For 0 example, if instead D2 were to have been integrated for 10000 y (while keeping the duration of other integrations as before), the total gain would be 31, nearly twice that of our standard run. The maximum gain cannot exceed the asymptotic value of A (61.1 in OPA), i.e., 2 when *t is so long that the computer time spent doing 2 other integrations of D1 and D0 become negligible. The maximum gain would be somewhat higher if the vector length in D2 were not limiting (see earlier).

5.2 Limits of the method

A degraded model does not reach its steady state at the same rate as does its parent model (Fig. 5). Thus technically the degradation approach cannot currently be used to study transient tracers, such as bomb 14C, anthropogenic CO , or CFCs. In practice, this concern 2 is larger between model versions D2 and D1 than between versions D1 and D0. The different transient behavior arises from the increase in volume that occurs because of degradation. For consistency, mass fluxes, and source and sink terms (i.e., all right hand terms in Eq. (1)) in the degraded model must be adjusted accordingly for the volume changes incurred during degradation (see Appendix). But, we have not, so far, bothered to apply corrections for the degraded volume changes to the left hand term LC/Lt in Eq. (1). Our primary interest has been to speed simulations to equilibrium, where we needed acceleration techniques the most. c Fig. 10a–c a The western Atlantic GEOSECS section along which natural 14C was estimated from b GEOSECS measurements (Broecker et al. 1995) and from c model DO. The vertical scale of the upper 1000 m is displayed with expanded scale

Aumont et al.: A degradation apprach to accelerate simulations to steady-state in a 3-D tracer transport model Fig. 11a–c The western Pacific GEOSECS section a, along which natural 14C was estimated from b GEOSECS measurements and c model DO. The upper 1000 m is expanded as in Fig. 10

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Table 1 Computation times, gain, and acceleration factors for the simulations with DEGINT. All results were obtained on CRAY C90 super computer

Models

Standard method

1-step degradation (D1—D0)

2-step degradation (D2—D1—D0)

MFLOPS

D0 D1 D2

740 h (3000 y) — —

24.5 h (100 y) 75 h (3000 y) —

24.5 h (100 y) 7.5 h (300 y) 10.9 h (2700 y)

480 460 324

Total

740 h

99.5

42.9

1

7.5

17.2

Acceleration factor

One could apply an analogous volume correction to LC/Lt term, as we have done to the source and sink terms (see Appendix). But when we divide that local change term by the necessary volume correction (whose value varies between less than 0.25 and 1), this corrected term can become quite large. Problems arise in regions where volume corrections are large and circulation is intense. In such regions, the LC/Lt terms is substantially increased by this volume correction. When LC/Lt is too large (numerically speaking), it is compensated on the next time step by also a large change but of the opposite sign. Thus tracer values oscillate between much too high and much too low values at the frequency of a time step, rendering the model numerically unstable. We did not pursue further our attempts to use the degraded model for the study of the transient tracers, because for short runs, there is less need for an accelerating method. If the transient is not reproduced, then one might also worry about the seasonal cycle. The seasonal response can be compared to the transitory state prior to a model’s equilibrium because of changes in the nearsurface ocean (i.e., despite zero change over a given year, instantaneous change LC/Lt is non-zero for H, S, and other tracers as well as for advection, turbulence). However, degradation alters least the surface ocean: area changes there are only 5% (see Appendix). We have found that the slight inconsistencies in the seasonal cycle between the degraded and parent models are quickly remedied in the short integration that follows re-initialization. DEGINT may not be usable if the scale of dynamical and geochemical processes are to be altered dramatically. For example, when studying a phytoplankton bloom in a frontal zone, an eddy resolving model would be appropriate. Degradation of output from such model would not preserve the fine-scale structure because the sharp gradients would be altered substatially. As another example, let us consider a model which must simultaneously describe both the coastal zone and the open ocean. If degradation would completely eliminate the boundary between these two zones, then DEGINT would not be usable. In short, we have used DEGINT with coarse-resolution models to describe geochemical processes occurring over largescale circulation features of the global ocean.

Use of DEGINT with an eddy-resolving model may be feasible, but requires further study. Figueroa and Olson (1994) offer some insight. From an eddy resolving model with a 20 km horizontal resolution, they constructed a 200-km coarse resolution model: its advective velocity field is the Eulerian mean velocity field calculated from the eddy-resolving model. Eddy mixing is parametrized by an eddy diffusivity distribution obtained from the eddy velocity field. Figueroa and Olson (1994) showed that such a coarse-resolution model cannot simultaneously reproduce the tracer distributions nor the meridional tracer transport as obtained in the eddy-resolving model. Nonetheless, some gain may come from initializing the eddy-resolving model with equilibrium output from the coarse-resolution model. More research is needed to determine if a modified degradation approach might be a key to unlocking our present incapacity to make multiple tracer simulations to equilibrium in global-scale eddy resolving models. 5.3 Natural 14C

The utility of natural 14C as a tracer is obvious from the pluses and minuses it reveals about OPA’s modeled circulation. Toggweiler et al. (1991) used coral records to show that natural 14C-impoverished water found in the southern part of the equatorial region seems to have its origin in the upwelling which occurs just off Peru. Their study indicates that the major supplier of this upwelled water is not the deep ocean, but instead the equatorial undercurrent which itself derives its low natural 14C signal through its connection with Subantarctic Mode Water. Toggweiler et al. (1991) go on to show that the coarse-resolution 3-D model from GFDL is unable to match the observed north—south asymmetry derived from coral records, because it upwells too much deep water to the surface in the center of the basin of the equatorial Pacific. Results from the 3-D Hamburg LSG ocean model seem to show the same tendency (MaierReimer 1993). Conversely, natural 14C results from OPA roughly reproduce the observed north—south assymetry (Fig. 9). OPA’s most depleted tropical *14C, !60&, is found in the Peru upwelling and spreads out mainly into the Southern Hemisphere. Much of the


improvement is due to OPA’s higher resolution at the equator, which requires smaller values of horizontal viscosity than in other global models of coarse resolution. Additional reduction of the horizontal viscosity is possible in the equatorial regions because the typical scale of eddies is larger than in the higher latitudes. The higher resolution and the lower values of the horizontal viscosity allow the model to reproduce the intense equatorial undercurrent, which is further invigorated due to OPA’s parametrization of turbulent kinetic energy (TKE) in the vertical (Blanke and Delecluse 1993). Maximum speed of the modeled equatorial undercurrent reaches 1.06 m s~1 in OPA as compared to 0.15—0.2 m s~1 in the GFDL model (Toggweiler et al. 1991). The observed speed lies close to 1 m s~1 (Wyrtki 1981). Despite reasonable model behavior at the Equator, the model’s Southern Ocean exhibits a serious flaw. Vertical mixing is too sluggish in the waters around Antarctica. South of 50 °S, surface levels of 14C are much too high. However, below 300 m, 14C drops to observed levels. Surface 14C in the Southern Ocean provides one half of a useful diagnostic. The 14C difference between surface waters in the Southern Ocean and deep waters in the North Pacific, where 14C is most depleted (at mid-depth from GEOSECS observations), provides an indicator of how long the latter have been isolated from the surface (Toggweiler and Samuels 1993). This difference in OPA is !183&, whereas observations indicate that the difference lies somewhere between !80 to !110&; the corresponding observed apparent age lies between 700 and 1000 y. Other 3-D models such as that from GFDL also tend to over predict the observed difference (Toggweiler and Samuels 1993) but not by nearly as much. For OPA, this diagnostic suggests that ventilation of the deep North Pacific from the Southern Ocean surface requires roughly twice as long as observed. Fig. 11 clarifies the OPA model’s problem. If instead the southern endmember is taken from below 300 m in the Antarctic, and not at the surface, this diagnostic for OPA falls within the observed range. In other words, surface concentrations are much too high, and the near surface vertical 14C gradient is much too intense. But what is the cause? Near-surface vertical mixing is too weak in the Southern Ocean. The cause appears due to the incompatibility between two climatologies used to force the model, that for sea ice cover (Reynolds 1982) and that for H and S (Levitus 1982). For technical reasons, the interaction of the model ocean with sea ice is not like that in the real ocean. When ice forms in the real ocean, the density of surrounding water increases due to salt rejection; when ice is formed in the model, just the opposite was imposed. That is, during ice formation in the model, extra salt was removed from adjacent waters so that they become more buoyant. This approach was found necessary for the on-line model runs in order

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that sea ice cover be maintained acording to the climatology. Specifically, when the seasonally varying sea ice climatology designated that a grid cell is covered with ice, the model’s surface temperature was then restored to the freezing temperature of sea-water (!1.86 at 35 psu), instead of using the temperature from the Levitus (1982) climatology. Thus the heat flux was fixed. But the surface salt flux under sea ice cannot be determined with the sea ice climatology, because the climatology contains no information about ice volume, just ice cover. When salt was rejected during sea ice formation in the model, as in the real ocean, through restoring to the observations as in most models, then the mixed layer deepened. Heat was brought to the surface by this deepening as well as by the corresponding thinning of the thermocline, which thereby increased the diffusive flux of heat from the deep ocean (Martinson 1990; Gordon and Huber 1990). This heat flux warmed the surface water enough so that the sea ice specified in the climatology could not be maintained. Although some form of this problem is known to those working with prognostic models which restore H and S to observations at the surface, particularly when southern salinities are raised artificially (England 1992), it is exacerbated in models with semi-diagnostic forcing. For the on-line model, the choice was made to maintain sea ice cover. By simply removing salt during sea ice formation, the model’s surface waters were better isolated from the deep ocean. Maintaining ice cover in the on-line ocean model was a high priority because it was first designed to be used in coupled model simulations studying interannual variability during the 1980s and 1990s. Problems related to slower deep-ocean circulation were less immediate, but are now beginning to be investigated, as demonstrated here. Our future work will entail making natural 14C simulations in the prognostic version of the same ocean model, which is now coupled to a prognostic model of sea ice that accounts for ice volume expicitly. Natural 14C runs suggest that artifacts caused by this pragmatic approach to maintaining modeled sea ice cover are largely confined to the top few hundred meters in ice-forming regions of the Southern Hemisphere. In this case, spreading of the adverse affects is limited, probably due to corrections made by the model’s semi-diagnostic forcing. These convective artifacts may be more significant for other tracers where transport by convection is important; ocean concentrations of some other tracers, such as O and CFCs are 2 much more sensitive to convection than is natural 14C (Joos et al. 1997). To infer the sensitivity of the 14C distribution to the mixing intensity in this region, we have performed a simple experiment with the off-line model. In this test, we set the mixed layer depth of the model to that of the estimates of Levitus (1982), south of 60 °S during winter. Results showed a substantial decrease of the mean

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Fig. 12 Annual mean meridional stream function associated with the total water mass transport from east to west for the Pacific Ocean

surface 14C concentration which drops from !60& to !80&; the low in the Weddell Sea drops from !97& to !140&. Still though, the deep ocean distribution remained close to that obtained in the standard run. The long equilibration time of 14C between the air and sea limits build up of 14C in the deep ocean. The West Pacific GEOSECS section shows clearly the structure of the deep circulation of the Pacific. The AABW flows along the ocean floor toward the Northern Hemisphere. This northward inflow is compensated by a return flow to the south between 2000 and 3500 m. The radiocarbon observations suggest a northward extension of this cell beyond 40 °N. The 14C concentrations in the return flow increase slowly toward the south by a slow vertical diffusive intrusion of radiocarbon from upper layers of the ocean. OPA presents an analogous structure. Nevertheless, the oldest water was found at the bottom of the ocean north of 30 °N. Figure 12 shows a zonally integrated meridional stream function for the combined Indian and Pacific Oceans. The deep cell remains confined south of 10 °S. This feature appears to be characteristic of the robust-diagnostic models (Toggweiler et al. 1989a; Semtner and Chervin 1988). The density structure of the deep Pacific which was used to constrain the model, does not allow an extension of the cell far into the Northern Hemisphere. A test performed with OPA shows an extension of this cell toward the north when there is no diagnostic restoring within the ocean.

6 Conclusions We have developed a new method, DEGINT, to accelerate steady-state simulations in an ocean tracertransport model. DEGINT efficiently determines initial conditions which are close to the final equilibrium solution. To obtain such initial conditions, we constructed a series of two coarser-resolution models, both based on output from one on-line OGCM at higher resolution. Then, by successive re-initialization and integration, DEGINT allowed a 17-fold reduction in computing time. Final radiocarbon fields obtained when using this method have been shown to be faithful to predictions extending from the parent model. DEGINT has been developed for steady-state simulations in coarse-resolution tracer transport models. We have shown that DEGINT does work for simulation of natural 14C. Furthermore, our preliminary simulations of the natural carbon-cycle with DEGINT, using OPA coupled to a simple biogeochemical model, closely match results from non-accelerated runs. Similar tracers should work as well. The gain in computation time offered by DEGINT depends on how many degradations can be made, which depends on the resolution and the bathymetry of the original model. In OPA, a third degradation is not feasible because crucial features of the bathymetry,


such as the Isthmus of Panama, would be lost. That would dramatically change the steady-state distribution of the tracer. Higher resolution models may well be able to withstand more than two degrations; however, changes in the spatial scale of the dynamic parametrized processes may limit the application. At present, DEGINT does not include sophisticated means to transform high resolution dynamics into subgrid-scale mixing in the degraded model, as would probably be necessary to use DEGINT with eddy resolving models. As for the predicted circulation, most of the largescale features of the radiocarbon distribution are reproduced (e.g., formation of NADW and AAIW, realistic equatorial dynamics). The 14C also reveals a major deficiency: incorrect thermohaline forcing related to sea—ice formation which leads to insufficient near-surface mixing in the Southern Ocean.

Appendix Description of the degradation procedure All equations presented here are written for a C-grid (Arakawa 1972) as used in OPA. DEGINT is applicable to other grids, but would require minor modifications. Degraded horizontal velocities are determined as #u u )g )g i`1,j`1 i`1,j`1 , º " i`1,j i`1,j I,J g #g i`1,j i`1,j`1 v )f #v )f i`1,j`1 i`1,j`1 , » " i,j1 ij`1 I,J f #f i,j`1 i`1,j`1

(A1)

where º and » are the velocities of the degraded model (with coordinates I, J), u and v are the velocities of the parent model (with coordinates i, j), f is the zonal grid space and g is the longitudinal grid space of the parent model (Fig. 2). These equations are directly deduced from the conservation of the water fluxes at the border of the domain composed by each set of four boxes of the parent model. The horizontal diffusion coefficients are constructed in a manner which is analogous to that of horizontal velocities: d )d )g )k K " i`1,j i`2,j i`1,j x i`1,j XI,J g #g i`1,j i`1,j`1 d )d )g )k # i`1,j`1 i`2,j`1 i`1,j`1 x i`1,j`1 (A2) g #g i`1,j i`1,j`1 d )k )f )d K " i,j`1 i,j`2 i,j`1 yi,j`1 YI,J f #f i`1,j i`1,j`1 d d ) )f )k # i`1,j`1 i`1,j`2 i`1,j`1 yi`1,j`1 , f #f i`1,j i`1,j`1 where k and k are the horizontal diffusion coefficients of the parent x y model along direction x and y, K and K are the corresponding X Yand d is the land mask degraded horizontal diffusion coefficients, (equal to zero if the box contains no water and one otherwise). The values of k and k are everywhere constant (k"2000 m~2 s~1) in x y D0; the degraded coefficients K and K are altered in the regions located near the coasts or at theX bottomY of the ocean.

115

Vertical advection in the degraded models ¼ is given by +4 w ) a (A3) ¼ " n/1 n n I,J +4 a n/1 n where w is the vertical velocity of the parent model, a is the horizontal area of the box, and n is the shorthand notation of the i, j boxes in x—y space (see Fig. 3). The definition of the degraded vertical mixing coefficients follows an analogous equation. The degradation procedure conserves one of the fundamental properties of the ocean: the non-divergence of the velocities. For non-divergence to be maintained, one must not neglect an important detail of DEGINT. Water fluxes must be conserved at land—sea borders. In practice, that means that for any grid box of the degraded grid, its water column depth must be taken as the deepest of the four constituent grid cells of the parent model. Globally, the volume of the degraded model is thus larger, 22% in our case. Deeper levels differ most. In the surface layer, the volume of the degraded model is but 5% larger, but in the deepest layer, the volume differs by 30%. Such differences, if not accounted for, can seriously affect the rate of the ventilation. Although some isthmuses and ridges disappear during degradation, corresponding fluxes remain null in the degraded model because the velocity and the diffusion coefficient normal to the boundary are zero on both its sides. Thus, the degradation procedure creates artificial walls of zero thickness across which no tracer is transported. Potential temperatures and salinities are averaged over the four boxes of the parent model according to their respective volume. The boundary condition at the air—sea interface (i.e., the gas exchange rate) is analogous to a vertical tracer flux due to vertical advection or diffusion. It must also be degraded. We do so by applying a equation analogous to Eq. (A3). The global air—sea flux is identical in both parent and degraded models. Without correction, the increase of ocean volume would augment sources and sinks within the ocean, so the resulting degraded tracer concentrations would not be representative of those in the original model. For simulations of 14C, the degraded model would thus become globally older: the rate of decay would be identical, but the mass of 14C lost in the degraded model would be greater. To avoid this, we apply a volume correction to the radioactive sink. In the parent model, the mass that disappear in each set of four grid boxes (which comprise one grid box of the degraded model) during one time step *t is 4 *m" + !j ) X ) d *t, (A4) n n n n/1 where *m is the variation of the mass, j the decay rate of the 14C, and X the box volume. In the degraded model, the decay rate must be corrected by j*"j

+4 X ) d +4 a ) d n/1 n n"j n/1 n n X* a*

(A5)

where j* is the decay rate of the 14C of the degraded model; X* is the box volume of the degraded volume; and a and a* are the horizonn tal box areas of the parent and of the degraded model, respectively. Thus the degraded model’s decay rate is not constant, unlike that for the parent model. This same type of volume correction may also be applied to other source and sink terms throughout the water column and at the surface as necessary when modeling other tracers (e.g., CO and O ). 2 2 Acknowledgements We are grateful to J. C. Dutay for helpful discussions concerning the OPA model. Many thanks to E. MaierReimer and an anonymous reviewer for constructive comments. Support for computations was provided by the CEA/DSM and French CNRS/IDRIS. This work was funded by the Environment and Climate Programme of the European Community (contract ENV4-CT95-0132). This is LMCE Contribution 411 and CFR contribution 1969.

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