Restoring boundary conditions, wherein the temperature and salinity are restored to surface target fields of temperature and salinity, are traditionally used for studies of the ocean circulation in ocean general circulation models. The canonical problem with these boundary conditions is that, when the target fields are chosen as the observed fields, accurate simulation of the surface fields of temperature and salinity would imply that the surface fluxes and therefore the ocean heat transports approach zero, a clearly unrealistic situation. It is clear that the target fields cannot be chosen as the observed fields. A simple but effective method of modifying conventional restoring boundary conditions is introduced, designed to keep the calculated values of surface temperature and salinity as close to observations as possible. The technique involves calculating the optimal target fields in the restoring boundary conditions by an iterative procedure. The method accounts for oceanic processes, such as advection and eddy mixing in the derivation of the new boundary conditions. A reduced version of this method is introduced that produces comparable results but offers greater simplicity in implementation. The simplicity of the method is particularly attractive in idealized studies, which often employ restoring surface boundary conditions. The success of the new method is, however, limited by several factors that cannot be easily compensated by the adjustment of the target profiles. These factors include inaccurate model dynamics, errors in the observations, and the too-simplified form of restoring surface boundary conditions themselves. The application of the method in this study with a coarse-resolution model leads to considerable improvements of the simulation of sea surface temperature (SST) and sea surface salinity (SSS). Both amplitude and phase of the annual cycle in SST greatly improve. The resulting magnitudes of surface heat and freshwater fluxes increase on average, and the meridional heat transport gets stronger. However, the fluxes in some regions remain unrealistic, notably the too-strong freshwater forcing of the western boundary currents in the Northern Hemisphere. Southern Ocean cooling and freshening are also likely to be too strong. The subsurface values of temperature improve greatly, proving that a large part of errors in the subsurface temperature distribution in our model can be corrected by reducing errors at the surface. In contrast, the reduction of errors in surface salinity fails to improve uniformly the simulated subsurface salinity values.
The oceanic circulation is driven from the surface by the air–sea fluxes of heat, water, and momentum. In order to accurately simulate the ocean state in a general circulation model (GCM), it is therefore necessary to realistically represent these fluxes at the surface and use accurate measurements of key quantities. As originally demonstrated by Haney (1971), the surface heat fluxes over the ocean can be approximated by the relaxation of the ocean surface temperature T to a “target temperature” T*:
where T* and λT generally vary with location and time and can be computed from observed air temperature, humidity, and surface radiative and turbulent fluxes of heat. The freshwater fluxes can be approximated by a prescribed sum of precipitation, river runoff, evaporative loss of water, and freshwater exchanges under sea ice. However, neither the accuracy of the observation-based estimates of key quantities needed for calculating the heat and freshwater fluxes nor the current skill of the numerical modeling of the ocean is sufficient for realistic simulations of the temperature and salinity.
The ocean owes its stratification in large part to surface processes. Therefore, realistic simulation of sea surface temperature (SST) and sea surface salinity (SSS) is crucial for successful modeling of the ocean state, and a choice of surface boundary conditions is often dictated by the need to keep the values of simulated SST and SSS as close to observations as possible. For these reasons, and for the sake of simplicity, ocean modelers often resort to using simplified form of restoring and take T* as the observed SST and set λT to a constant within the range 30–70 W m−2 K−1 (Chu et al. 1998). In addition, restoring similar to (1a) has also been often used for freshwater fluxes:
where S* is a “target salinity” and S is the SSS. Although lacking in physical justification—the flux of freshwater from the atmosphere is not dependent on ocean values of SSS—the restoring of SSS to observations is designed to prevent surface values from deviating too much from observations. However, as is shown in this and in several previous studies cited below, choosing the target fields to be the observed fields is not the best way to keep the modeled surface SST and SSS near the observed values in ocean simulations.
This simple form of the restoring (“conventional restoring” hereinafter) inevitably leads to significant errors in both SST and SSS since, by the use of conventional restoring, the surface fluxes are directly proportional to the deviations in SST and SSS from their observed values. These deviations are particularly large wherever the surface fluxes themselves are large, which means that the values of SST and SSS are in error in all dynamically significant regions. Chu et al. (1998) criticized the conventional restoring and reported poor correlation between fluxes resulting from conventional restoring (1a) and those from the National Centers for Environmental Prediction (NCEP) reanalysis. Pierce (1996) showed that using the conventional restoring boundary conditions underestimates the amplitude of the annual cycle of SST. He introduces a method of modifying the target profile for temperature that improves simulation of surface variability. His method, however, explicitly neglects oceanic processes that redistribute heat at the surface, such as advection and mixing. This technique, therefore, cannot successfully work in a steady state, where the divergence of oceanic heat transfers balance the surface fluxes. For example, Killworth et al. (2000) stress the importance of the regions with strong advection. The authors demonstrate both theoretically and numerically that in the regions where the surface heat fluxes are balanced by advection, both annual mean and annual cycle of temperature are bound to be in error if the conventional restoring is employed. Clearly, if both the fluxes and SST and SSS are to be correct, the restoring cannot be to the observed values of these quantities.
Reducing errors in SST and SSS is not straightforward since the causes for these errors are not limited to inaccuracy of simplified conventional restoring [(1a), (1b)], but include, among others, coarse spatial resolution and crude parameterization of subgrid processes. However, it is tempting to explore whether the errors in SST and SSS can be significantly reduced if conditions (1a) and (1b) are modified but their simplicity is retained. Making the restoring coefficient λT large can, of course, reduce errors in SST but would change the stability characteristics of the thermohaline circulation, would worsen the representation of heat fluxes (Pierce 1996), and would decrease decay time of temperature anomalies, which is already unrealistically short even for “normal” values of λT. Increasing λS is even more problematic, since the mere existence of the feedback of SSS on freshwater fluxes is unphysical (e.g., Paiva and Chassignet 2001), and λS should be at a minimum kept as small as possible. An alternative method is therefore to find target profiles T* and S* whose use would reduce the errors in SST and SSS. The model parameters, T* and S*, could be successfully estimated by using the methods of adjoint modeling. However, such methods, while proven to be effective, are complex and time consuming. Another approach is to use a combination of prescribed surface fluxes estimated from observations plus relaxation terms (1a) and (1b) (Tziperman and Bryan 1993; Jiang et al. 1999; Paiva and Chassignet 2001). This method permits a model to correctly simulate both the surface fluxes and SST and SSS. In practice, however, inaccurate model dynamics can lead to errors in both the simulated surface heat (freshwater) fluxes and surface temperature (salinity). This approach also suffers from large errors in the observation-based estimates of surface fluxes of heat and freshwater.
In this study, we propose a new method of finding target profiles that minimizes surface errors in the simulated stationary seasonal cycle of SST and SSS and explicitly takes oceanic processes into account. Unlike the conventional restoring, this method allows one to simulate observed SST and SSS in a model with nonzero surface fluxes. Our objectives are the following. First, we want to develop a simple practical method for improving simulated values of surface temperature and salinity in ocean-only models. This approach can be viewed as a method that is more effective in assimilating data for SST and SSS in a model than is the conventional restoring. A second objective is to estimate to what degree the deviations of subsurface values of temperatures and salinities from observations are explained by inaccurate simulation of SST and SSS. Last, we will identify regions where errors in SST and SSS are caused by inaccurate ocean dynamics and cannot be corrected by any reasonable surface fluxes. We derive this new method together with its reduced versions in section 2. We then compare the results from simulations with the new method with those with conventional restoring in section 3. Section 4 presents a discussion and conclusions.
2. Formulation of the new restoring boundary conditions
We introduce here a new method for calculating the target profile T* for restoring boundary conditions for the temperature. We only consider temperature in this section since the derivation for S* is identical. The goal is to reduce the deviations in the resulting surface values from the observed climatology TL taken from Levitus and Boyer (1994). We do not account for the errors in the observations themselves, but rather aim to improve ability of a numerical model to reproduce a given distribution of surface values. This can then be used in ocean modeling to obtain a stationary climatological state of the ocean by using these monthly target temperatures T* to simulate the annual cycle in surface flux forcing.
The method is iterative: each iteration n + 1 involves updating the target profile T*n, which generally has an annual cycle repeated every year, from the previous iterative step n. After each iterative determination of T*n, the ocean model is integrated for a period of time titer, during which the upper ocean in the model would have sufficient time to come to equilibrium with the new surface boundary conditions. Assume that on an iterative step n at a given location the modeled SST is Tn. The time evolution of the heat stored in the upper ocean layer of depth d1 is caused by (i) the heat flux through the oceanic surface given by (1a), (ii) the horizontal convergence of the oceanic heat transport, and (iii) the vertical exchanges of heat with the ocean below:
where Fo(u, T) is the convergence of the horizontal advective and diffusive heat transports in the upper layer, as well as the vertical heat exchanges with the ocean below; Fo(u, T) is a function of the upper-layer velocities un and temperature Tn and temperature in the layer below the surface, all known in the model. The convection term is conv. We stress that at this stage all quantities in (2) are assumed to be continuous functions of time; Cp is the specific heat capacity, and ρ is the density.
We then find a new target profile T *n+1 that would balance the ocean transfers Fo as in (2) but with SST exactly equal to the observations TL. Since the velocities in such a state are unknown, we assume that u is the same as in (2). Because the upper-layer velocities are to a large degree controlled by surface winds, this assumption is reasonable. We also assume that the convection term, conv, is the same as in (2). It would be difficult to proceed without this assumption, since the convective heat flux is a complicated function of temperature and salinity. Note that Fo is a linear function of T. We can write
Since the Levitus values are available as monthly means only, we next consider the average of (4) over a time interval [0 ta], which can be either one month or an entire year. Using notation ΔTn for TL − Tn, we obtain
where the overbars denote time-averaged values. Note that we use an approximation for a time average of Fo: Fo(u, T) = Fo(u, T).
From (5), our best guess for the target profile is updated by (i) the errors in SST (the first term in the “SST correction“), (ii) the “ocean transfer of errors in SST” (the second term in the SST correction), (iii) the errors in the rate of change of SST (the “SST evolution correction”). As the ocean adjusts to a new surface heat flux, the ocean velocities in Fo change in response to changing SST, and the SST assumes a new value Tn+1; the procedure is then repeated. Convergence can take place in two distinct cases. In the first case, both the model-simulated values of SST and its time evolution are reproduced correctly, and all the terms on the right-hand side of (5) approach zero. In the second case, all the terms in the SST correction and SST evolution correction are nonzero but mutually compensating.
The convergence may not, however, be possible at all if the simulated oceanic heat transfer Fo is in error. In that case, it is not possible to balance such Fo by any reasonable surface flux and therefore to find a target profile T* that would eliminate the errors in SST. Identification of such regions in a specific model is one of the objectives of this study; these regions are model dependent. The convergence in such areas can never be achieved and the surface fluxes may be ever increasing in a futile attempt to force the model to reproduce observations. To prevent such a drift, we restrict deviations of the target temperature from the observations ΔT* = (T* − TL) from becoming too large. It is equivalent to restricting the magnitude of a surface flux necessary to keep SST very close to the observations since the surface heat flux λT(T* − T) is equal to λTΔT* if SST assumes the observed values. Increasing a maximum allowed ΔT* will reduce errors in SST at the expense of increasing surface fluxes, often making them unrealistically large. The choice of the maximum allowed ΔT* should be made in each case on an individual basis.
a. Reduced version of the method
Equilibration of the entire ocean, which typically takes thousands of years, on each step makes the iterative procedure time-consuming. For example, in our test run of the full method, we set titer to 2000 yr and performed 20 iterations; the required length of integration then becomes 40 000 yr. It would therefore be far more practical to choose a much shorter length of integration titer that follows each iterative change of target SST and SSS. To check the accuracy of such an approach, we consider here a reduced version of the method with a much shorter titer = 100 yr. The results of the test runs with the shorter iteration time are practically indistinguishable from those with titer = 2000 yr. The integration for titer = 100 yr after each iterative step n, therefore, gives the upper ocean sufficient time to come near to equilibrium and keeps the method practical and accurate. In the discussion in the following sections, we do not distinguish between the two cases.
For added simplicity of the method, one can also consider omitting two groups of terms in (5): the ocean flux terms in the SST correction and the SST evolution correction terms. Unlike the shortening of titer, this simplification is not designed to shorten the integration time, but rather to make the implementation of the method more straightforward. It is noteworthy that removing the ocean flux terms in (5) does not mean that the ocean processes themselves are neglected. Rather, we assume that the divergence of heat transport in the upper layer changes very little with each iterative change in T* and S*. In neglecting the SST evolution correction terms, we assume that the sensitivity of the seasonal evolution of the upper-layer heat (salt) storage to changes in T* (S*) is much smaller than the sensitivity of the surface fluxes themselves. The resulting equation takes a very simple and attractive form:
Unlike the full method (5), the convergence using (6) can take place only if SSTs approach their observed values. Using (6) generally means that while SST would be driven closer to observations, convergence will be more difficult to achieve. In the following sections we present results of the method (5). In addition, we also test the more simplified method (6) and conclude that the differences between the results of the methods (5) and (6) are sufficiently small to justify the use of (6) in more idealized studies.
3. Implementation and results
We first apply our new method to the case of the annual-mean forcing that is constant in time (section 3b). The SST evolution correction terms in (5) are exactly zero. We will evaluate the effectiveness of the new method in improving simulation of the SST, SSS, and, consequently, the subsurface values of temperature and salinity. We will also isolate regions in which the convergence is impossible because of inaccurate dynamics. We then consider the case with a stationary annual cycle where T* marches through its annual cycle and repeats itself every year (section 3c). This annual-cycle case will allow the success of our method in improving time evolution of SST to be assessed.
a. The numerical model
The model is based on the Geophysical Fluid Dynamics Laboratory (GFDL) Modular Ocean Model (MOM3) code (Pacanowski and Griffies 1999). The horizontal resolution is 4° in longitude and 3° in latitude. The geometry is global, but the Arctic Ocean is not included (see Fig. 1) and the northernmost ocean point is at 82°N. The model has 25 levels in the vertical direction with resolution increasing from 17 m at the surface layer to 510 m at the bottom. The bathymetry of the model is derived from the Scripps Topography. The maximum depth is 5 km; the minimum depth is 50 m. No-slip conditions for velocity are applied at all lateral walls; linear bottom drag is applied at the bottom. Boundary conditions for tracers are insulating at the solid lateral walls and bottom. Vertical diffusivity varies from 0.25 × 10−4 m2 s−1 at the surface to 1.0 × 10−4 m2 s at the bottom. This profile reflects the increase of the vertical mixing in the direction from the thermocline to the deep ocean (Bryan and Lewis 1979) and the intensification of mixing by rough bottom topography (Polzin et al. 1997). Horizontal diffusion with the coefficient of 103 m2 s−1 is used. We use asynchronous integration for tracer and momentum equations with time steps of 12 h and 30 min. This acceleration ratio is sufficient to accurately simulate monthly means of temperature and salinity in a numerical model (Kamenkovich et al. 2002).
The wind stress used to force the model is taken from the NCEP reanalysis, has both zonal and meridional components, and is fixed for all subsequent experiments. All surface conditions used for forcing are either fixed to annual-mean values (annual-mean forcing) or march through an annual cycle. The model is forced by the restoring boundary conditions in (1a) and (1b) with the coefficients λT and λS taken inversely proportional to the restoring time scale with a value of 60 days (for a 50-m mixed layer) for temperature and 180 days for salinity. Target salinity is also linearly increased to 35.0 psu near the Antarctic coast in the local winter months to simulate the effects of brine rejection during sea ice formation; see England (1993) and Goodman (1998). This adjustment is needed for a decent representation of Antarctic Bottom Water (AABW), which is crucial in maintaining deep densities in the simulated ocean. In order to preserve the properties of AABW, surface boundary conditions are kept unchanged south of 69°S throughout this study.
b. Annual-mean forcing
We begin by spinning up the model using the observed annual-mean (Levitus) values of SST and SSS as target fields in (1a) and (1b)—this is the conventional restoring (denoted experiment CONVR). The model is driven to equilibrium by 5000 years of integration; the drift in the globally averaged temperature at the end of integration is less than 0.002°C (100 yr)−1. The difference between the simulated surface values of temperature and salinity and observations is shown in Figs. 1a,b. The oceanic transfers Fo have to be balanced by surface fluxes, which in the case of conventional restoring are only possible if the values of SST and SSS deviate from observations. The regions of the western boundary currents, Kuroshio and Gulf Stream, North Atlantic, and most of the Southern Ocean show surface temperature warmer than observations—this supports cooling over these areas and balances heat loss by the ocean (Fig. 2a). The surface is too cold in the equatorial regions because of the heat uptake by the ocean (Fig. 2a). Mid- and low latitudes are fresher and high latitudes are more saline than observations in order to maintain corresponding freshwater fluxes (Fig. 1b). Since most of these errors are clearly caused by having to simulate nonzero fluxes by the restoring boundary conditions, it seems plausible that choosing appropriate target profiles will reduce most of these errors. It is noteworthy that the absence of the annual harmonic is an additional source of errors; these errors are small, however, as our analysis reveals.
We next apply our method to obtain a best possible estimate of the target profiles for this case (experiment NEWR). Note, that the errors in CONVR are largest in regions where the ocean is most active. In addition to the simplified choice of target profiles T* and S* in CONVR, the errors in these regions come from an inadequate simulation of the swift surface currents, such as the western boundary currents or of the Antarctic Circumpolar Current (ACC), as well as of convection and mixing. As discussed in section 2, it is unreasonable to attempt to fully compensate these sources of errors by altering T* and S*, and we restrict maximum allowed difference between target profiles and observations ΔT*max and ΔS*max. We choose values of ΔT*max = 5°C and ΔS*max = 0.75 psu; expressed in flux units (see section 2), they correspond to 200 W m−2 and 1.8 m yr−1. Two additional experiments, one with large maximum allowed deviations, ΔT*max = 10°C and ΔS*max = 1.5 psu, and another with smaller values, ΔT*max = 3°C and ΔS*max = 0.5 psu, were also carried out but will be only briefly discussed below wherever appropriate.
The model was integrated for 2000 years with titer = 100 yr integrations between iterations; a total number of 20 iterations were performed. As discussed in section 2a, increasing titer to 2000 yr does not noticeably change the results. The target profiles T* and S* were then fixed, and the integration was continued for another 3000 years, until the drift in the global mean temperature was less than 0.002°C (100 yr)−1. The difference of the resulting annual-mean values of SST and SSS with the observations is shown in Figs. 1c and 1d. As one can see, the errors are greatly reduced: The method is effective in bringing the SST and SSS values closer to the observations. However, errors persist in some areas, and, in particular, the Kuroshio and Gulf Stream regions remain warmer and saltier than the observations. The Indian sector of the Southern Ocean is too warm and too salty, although errors are reduced relative to CONVR. In all these regions, the magnitudes of ΔT* and ΔS* have reached the maximum allowed values ΔT*max and ΔS*max; additional iterations therefore would not reduce errors at these locations. Increasing ΔT*max to 10°C and ΔS*max to 1.5 psu was not enough to completely eliminate errors in these regions, but instead resulted in a more unrealistic simulation of the surface fluxes. The most probable cause for these errors is inaccurate simulation of the heat advection by the western boundary currents and ACC, and improving the restoring boundary conditions is evidently incapable of correcting this problem. Another plausible cause for the errors in the Southern Ocean is tied to convection, which is very active in coarse-resolution models with horizontal diffusion, but whose dependence on temperature we neglect in derivation of (5) and (6).
Surface fluxes of heat are generally larger in amplitude in NEWR than in CONVR (Figs. 2a and 2c). As a result, the heating in the tropical Pacific and Atlantic and cooling over the Gulf Stream become more realistic (Fig. 2). At the same time, the observed heat gain by the northern Indian Ocean is still lacking in the model simulation, and the cooling of the northern North Atlantic remains too strong. NEWR exhibits very strong cooling of the Indian sector of the Southern Ocean and of the area near Drake Passage. Although the observations over the Southern Ocean are sparse (World Ocean Atlas 2001; Conkright et al. 2001), and therefore the observational estimates of the heat fluxes are uncertain, these cooling patterns are most likely unrealistic. Increased magnitudes of surface heat fluxes lead to the larger meridional heat transport in NEWR as compared with CONVR (Fig. 3). The peak value in the Northern Hemisphere (at 24°N) increases from 0.5 to 0.7 PW, a clear improvement, whereas the Southern Hemisphere peak value (at 12°S) becomes more unrealistic and increases from 2.4 to 2.75 PW.
The magnitudes of the freshwater fluxes (Fig. 4) increase in NEWR from CONVR values, and there is some improvement in several regions. The magnitudes of freshwater loss in the midlatitudes of the Atlantic, Indian, and North Pacific Oceans become closer to estimates from observations (synthesized by Jiang et al. 1999). The freshwater gain in the equatorial Pacific intensifies but remains too weak in the western tropical Pacific. In other regions, the simulated fluxes deteriorate in NEWR in comparison with CONVR. The unrealistic water fluxes into the ocean in the western boundary regions of the Northern Hemisphere become even larger in NEWR. In contrast, observations demonstrate freshwater loss in the western boundary currents of both North Pacific and Atlantic. The freshwater fluxes into the regions of the western boundary currents of the Northern Hemisphere become even larger if ΔS*max is increased to 1.5 psu. As in the case of the heat fluxes above, it is difficult to determine how unrealistic is the very strong freshening of the Indian sector and Drake Passage region in NEWR because of the sparsity of the observations in the region.
The errors in the subsurface temperatures are reduced by more than 1°C in NEWR in comparison with CONVR (Fig. 5). Improvements in the subsurface values are linked to more realistically simulated SSTs in several parts of the domain. For example, the most substantial improvement in the subsurface temperatures takes place north of 60°N in the Atlantic; this is clearly visible in the global zonal means in Fig. 5 since the Atlantic is the only ocean present at these latitudes. This change is most likely caused by the SST in the northern North Atlantic in NEWR being cooler and more realistic than in CONVR (Fig. 1a). This, in turn, results in the colder North Atlantic Deep Water in NEWR. The thermohaline circulation in the North Atlantic also intensifies as a result of the denser North Atlantic Deep Water: the maximum mass overturning (not shown) increases from 11 Sv (Sv ≡ 106 m3 s−1) in CONVR to more than 13 Sv in NEWR. Noticeable improvement in the temperature in the upper 1000 m at around 45°N is attributed to more realistic temperatures in the Gulf Stream and Kuroshio regions. Subsurface temperature values are closer to observations near the equator where strong upwelling is important in the heat balance (Killworth et al. 2000). The water at intermediate depths at the southern edge of both the Pacific and Atlantic basins, where Antarctic Intermediate Water (AAIW) enters the basins, becomes colder and more realistic. The most likely cause for this change is the intensified cooling near the tip of South America in the Southern Ocean— an area crucial for the AAIW formation. The remaining errors are caused by a number of factors, including a highly diffusive thermocline, coarse resolution, and inaccurately simulated eddy fluxes of heat parameterized by diffusion in the model.
The salinities in the upper 1000 m are too high in CONVR and the deep ocean below 2000 m is too fresh (Figs. 6a,b). The ocean becomes saltier in NEWR, which brings the values below 2000 m closer to the observations but increases errors in the upper 1000 m. The increase in the deep part of the ocean is most likely explained by the higher salinities and more vigorous overturning of the NADW, which increases the effective salt flux to the deep ocean. The inflow of the AAIW also becomes saltier in NEWR, which is the main reason for the increase in the salinity at the intermediate depths (400–1000 m) (Fig. 6b). The increase in salinity in the upper 1000 m of the Southern Ocean is a curious feature of the experiment. The change cannot directly originate from the surface, because the SSS in the region decreases and becomes closer to the observations (Figs. 1b,d). Overall, the reduction of errors in SSS fails to improve simulation of the subsurface salinities; improvements in the model dynamics below the surface are required for better simulation.
c. Annual cycle
In simulating an annual cycle from monthly means of the Levitus data, the values at the beginning TL(0) and the end TL(ta) of the month are determined by linear interpolation between the two nearest monthly points1; the corresponding model values Tn(0) and Tn(ta) are obtained in the same way for consistency. We carry out two experiments, one with the conventional restoring (CONVR) and one with the new method (NEWR). As in the case of the annual mean forcing, we choose values of ΔT*max = 5°C and ΔS*max = 0.75 psu in NEWR.
The deviations from Levitus values in SST in December–February and in June–August are shown in Fig. 7 for both experiments. In CONVR, the winter hemispheres are consistently warmer than the observations, whereas the summer hemispheres are too cold. This results in reduced amplitude of the seasonal cycle (Pierce 1996) and in too small an interhemispheric SST gradient. NEWR exhibits noticeable improvement in both winter- and summertime SSTs, and the seasonal cycle becomes much more realistic. As in the case of the annual-mean forcing, the errors remain in the western boundary regions of the Northern Hemisphere and in the Southern Ocean. In addition, the interior of the subtropical gyre in the North Pacific is too warm. The increase in monthly ΔT* in all these regions was restricted by our method, which resulted in nonzero errors but prevented the surface fluxes from becoming unrealistic. Because the deviations of the winter- and summertime SSS from observations are both very similar to that of the annual mean values, we choose not to report them separately in this section.
The time series of SST at several locations are presented in Fig. 8. We choose four locations: three with remaining errors in NEWR—the Indian sector of the Southern Ocean (SO), the Gulf Stream (GS), and the North Pacific (NP), as well as the equatorial Pacific (EQ). The amplitude of the annual cycle in the Gulf Stream and in the North Pacific improves considerably and now compares well to the observations. However, the simulated annual cycle in NEWR remains delayed when compared with the observed one, although clearly improving from the one in CONVR. The lagging annual cycle is characteristic for restoring boundary conditions (Pierce 1996; Killworth et al. 2000) and manifests itself in errors in the seasonal means (Fig. 7). In the Southern Ocean, the amplitude of SST is simulated with errors, although it is closer to observations than in CONVR. The location EQ exhibits a good simulation of the annual cycle.
4. Summary and conclusions
We introduced a simple but effective method of reducing errors in the values of SST and SSS in a numerical ocean model forced by restoring surface boundary conditions. By using an iterative procedure, this method calculates target profiles for restoring boundary conditions that are designed to keep the values of SST and SSS close to observations while explicitly accounting for oceanic processes, such as advection and diffusion. This allows for improvement in simulated values of SST and SSS in situations in which such oceanic processes are important in balancing surface fluxes—in the annual-mean values and in the regions with strong heat divergence. The results show that the method is both effective and, unlike more sophisticated but complex methods of adjoint modeling, straightforward to use. The technique is not overly expensive computationally and typically takes a fraction of time required to drive a numerical model to equilibrium. In practice, therefore, the method can be used for coarse-resolution models.
We applied the new technique for coarse-resolution simulations of the annual-mean surface forcing and with the annual cycle in the target SST and SSS. Main results are the following.
The method greatly improves the distributions of SST and SSS in comparison with the conventional method of restoring to observations.
The surface fluxes change as a result of modified target fields; it is, therefore, plausible that some of these changes could be specific to this particular model configuration. The magnitude of the surface heat fluxes increases, which leads to stronger meridional heat transports. The freshwater fluxes, as expected, do not become more realistic. Adjusting the target profile for salinity, although effective in reducing surface errors in SSS, is unable to improve the simulation of the freshwater fluxes. More physically meaningful surface boundary conditions for salinity are obviously needed.
The subsurface values of temperature improve considerably, demonstrating a link between several surface regions and the subsurface ocean. These crucial regions include formation sites of the Antarctic Intermediate Water and the North Atlantic Deep Water, the western boundary currents, and the equatorial regions. In contrast, the subsurface salinity does not improve uniformly and the upper 1000 m becomes too salty. The results suggest that the main cause for errors in the subsurface salinities in the model is not in the errors in the simulated SSS, but rather in inaccurate model dynamics.
Both amplitude and phase of the simulated annual cycle of SST improve considerably over the entire domain. However, the simulated annual cycle still lags the observed one in several locations, in particular in the Gulf Stream region and the North Pacific, resulting in errors in seasonal values of SST.
A reduced version of the method works equally well and is recommended for situations in which greater simplicity is desired. Although errors in simulated surface values can increase and convergence could be compromised, the simplicity of the reduced method is attractive for practical purposes.
The success of our method is limited by several factors, most notably by the inaccurate model dynamics. Coarse-resolution models often fail to accurately simulate swift surface currents, such as western boundary currents and the ACC, whose structure is essential for obtaining correct temperature and salinity distributions. In addition, subgrid processes such as mixing and convection are also represented crudely in such models. The resulting erroneous oceanic fluxes that redistribute the heat and salt near the surface lead to deviations of the values of SST and SSS from reality. The modification of the restoring boundary conditions presented in this study cannot completely eliminate these errors, and the method often does not converge with any reasonable value of the surface heat and freshwater fluxes. Rather than allow unrealistically large surface fluxes, we choose to limit the deviations of target profiles from the observed values for SST and SSS, thus limiting a surface flux magnitude but also restricting an ability of the method to bring the surface values closer to the observations. Nevertheless, the errors are substantially reduced even in these regions when compared with conventional restoring.
Another source of errors, not possible to correct by the presented method, is in the simplified form of the surface boundary conditions in (1a) and (1b). Restoring conditions for heat (1a) with a constant coefficient λT is a crude approximation to the conditions proposed by Haney (1971), and the dependence of the freshwater fluxes on SSS is unphysical. The limitations of restoring boundary conditions, even with an optimal choice of target profiles in our study, are clear in the areas of the ACC and western boundary currents of the Northern Hemisphere. Our method, of course, cannot substitute for the need for more physically meaningful surface boundary conditions. Bulk formulas, for example, are shown to produce more realistic distribution of temperature and salinity by Large et al. (1997). However, because of large uncertainties in observational estimates of surface fluxes, restoring remains an attractive tool for numerical modelers, and the restoring for salinity is often used even in some state-of-the-art GCMs (Gent et al. 1998). In addition, dependence of convection on SST and SSS was neglected in derivation of our method, which could have further impeded success of the method in the Southern Ocean and the northern North Atlantic. Last, the accuracy of the observations themselves is not addressed in this study, but could be another factor limiting the success of our method.
The simplicity of our method is particularly attractive in idealized ocean-only studies, which often employ restoring surface boundary conditions. The method can also be applied in selected areas where improvement in surface values is needed, such as the tropical regions, the northern North Atlantic, and the ocean interior at midlatitudes.
We thank two anonymous reviewers for their helpful comments on the manuscript. This work was supported by a grant from the NOAA Office of Global Programs to the Joint Institute for the Study of the Atmosphere and Ocean (JISAO) Center for Science in the Earth System and by the National Science Foundation through the Office of Polar Programs under Grant 0126208. This publication is funded by JISAO under NOAA Cooperative Agreement NA17RJ11232.
Corresponding author address: I. Kamenkovich, JISAO—University of Washington, Box 354235, Seattle, WA 98195-4235. Email: firstname.lastname@example.org
Joint Institute for the Study of Atmosphere and Ocean Contribution Number 1007.
Although such interpolation does not preserve monthly means, the errors are negligible. More advanced methods of simulating an annual cycle are available (Gent et al. 1998).