A comparison of global ocean general circulation model solutions … · 2004. 5. 18. · The Bering Strait and Hudson Bay are open, but the Gibraltar Strait and the Red Sea Outﬂow

Ocean Modelling 7 (2004) 323–341

www.elsevier.com/locate/ocemod

A comparison of global ocean general circulationmodel solutions obtained with synchronous and

accelerated integration methods

Gokhan Danabasoglu *

National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307, USA

Available online 13 November 2003

Abstract

A 10 000-year synchronous control integration using a comprehensive ocean general circulation model

(OGCM) subject to realistic, time-dependent forcing in a global domain is performed to quantitatively

determine how well its solution is reproduced by two accelerated, but otherwise identical, equilibrium

integrations. To our knowledge, this is the first long synchronous integration of such an OGCM. The two

accelerated cases use tracer time steps increasing with depth and unequal momentum and tracer time steps,respectively. After accelerated equilibration, these cases are integrated further synchronously to achieve

synchronous equilibration. The equilibration time is defined as the time when the annual- and global-mean

potential temperature trend is below 10�5 �Cyear�1. The synchronous control integration achieves equi-

librium in about 3500 years. The accelerated equilibration times are 685 and 3000 surface years for the two

cases, respectively. Their synchronous extensions require about 100 and 700 surface years. The accelerated

solutions do differ from each other and from those of the control experiment. However, for practical

purposes, many aspects of the accelerated and control solutions are more similar than otherwise. Because of

severe non-conservation issues associated with tracer time-step variations with depth, this technique is notrecommended. Instead, unequal momentum and tracer time steps should be used. Compared to the control

case, this method of acceleration provides a factor of 2.5 reduction in the computational cost (acceler-

ated + synchronous equilibration) with the possibility of further reductions. Any accelerated integration

must be followed by a synchronous extension to recover the correct seasonal cycle and to eliminate any

possible oscillatory behavior present in the accelerated phase. If the incremental gains with further syn-

chronous extensions are judged to be minimal or of minor significance, shorter (<700 years) integrations

may certainly suffice. Among others, the equilibration times are affected by the choice of surface forcing

method.� 2003 Elsevier Ltd. All rights reserved.

* Tel.: +1-303-497-1604; fax: +1-303-497-1700.

E-mail address: [email protected] (G. Danabasoglu).

1463-5003/$ - see front matter � 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ocemod.2003.10.001

mail to: [email protected]

324 G. Danabasoglu / Ocean Modelling 7 (2004) 323–341

Keywords: Synchronous equilibrium; Accelerated equilibrium; Global ocean general circulation model

1. Introduction

The abyssal ocean diffusive processes with time scales of many thousand years ultimatelycontrol the approach to equilibrium of oceanic general circulation models (OGCMs). Despite everincreasing computational capabilities, particularly overall speed, obtaining equilibrium solutionsis still practically unaffordable even for non-eddy-resolving models commonly used in climatesystem studies. Therefore, accelerated integration methods retain their wide usage in the ocean-ographic community to obtain equilibrium solutions (e.g. Large et al., 1997; Wood, 1998).Arguably, the most commonly applied technique is due to Bryan et al. (1975) in which the modeltracer time step, also followed by the model calendar, is much longer than the momentum timestep (order days vs. order hours/minutes). This approach was later extended by Bryan and Lewis(1979) to allow even longer tracer time steps with depth to exploit increased advective numericalstability limit due to diminishing current speeds in the deep ocean. Such a distorted physicstechnique has proven useful when an equilibrium baseline solution is sought without any interestin the transient behavior of the model. Detailed analyses of this method are presented in Bryan(1984) and Killworth et al. (1984), considering its effects on baroclinic instability, Rossby waves,and gravity wave dispersion and numerical stability, respectively.

Clearly, the governing equations for the original and accelerated systems are identical only atsteady state, and the primary assumption with time-independent forcing is the uniqueness of thefinal steady state solution. The usefulness of the distorted physics technique with unsteady forcingis explored by Danabasoglu et al. (1996). Without actually performing a companion synchronousintegration (in which all model time steps are equal), the method is shown to work provided thatthe accelerated integration is followed by a short synchronous period to recover the properseasonal cycle. This key issue is also recently addressed in Huang and Pedlosky (2002), where thelowest mode is shown to be no longer strictly barotropic when accelerated, leading to severelycontaminated seasonal cycles.

Using a series of OGCM integrations, including a companion synchronous one, Wang (2001)shows that although the choice of the acceleration parameters does affect the mean states of theaccelerated equilibrium, these mean states are not too far from either each other or the syn-chronous mean state for practical purposes. The study also restresses the importance of a syn-chronous period following an accelerated integration. Although this work is the first in which asynchronous equilibrium solution is obtained for an OGCM, the model configuration is verysimplistic, i.e. a box-type domain with simple sub-grid-scale parameterizations and withoutsalinity and wind forcing. Furthermore, temperature forcing is of restoring type.

In the present study, we revisit the accelerated integration technique one more time, using astate-of-the-art OGCM with the most recent parameterizations. A 10 000-year synchronousintegration is performed subject to realistic, time-dependent forcing in a global domain. To ourknowledge, this is the first, long synchronous integration with such a model. We compare twoaccelerated, but otherwise identical, integrations to this control case and to each other to de-termine if these acceleration methods do truly work. We also perform several multi-century

G. Danabasoglu / Ocean Modelling 7 (2004) 323–341 325

synchronous continuation integrations for the accelerated cases to assess their synchronousadjustments and to document any changes in their seasonal cycles and mean states. The equili-bration times and the number of model time steps to get there are computed for each accelerated-synchronous integration combination to measure computational gains.

A brief description of the model along with some forcing details are given in Section 2. Section3 describes our integration strategy and cases. The results are presented in Section 4, and someconclusions are given in Section 5.

2. Model description

The oceanic model is the ocean component of the second generation of the National Center forAtmospheric Research (NCAR) Community Climate System Model (CCSM2). It is a state-of-the-art, Bryan–Cox type (Bryan, 1969), level-coordinate model based on the Parallel OceanProgram (POP 1.4) of the Los Alamos National Laboratory (Smith et al., 1992). The model solvesthe primitive equations in general orthogonal coordinates in the horizontal (Smith et al., 1995)with the hydrostatic and Boussinesq approximations. An implicit free surface formulation is usedfor the barotropic equations (Dukowicz and Smith, 1994). Although the surface layer thickness isallowed to vary, we opt not to use the natural freshwater fluxes. Instead, these fluxes are convertedto virtual salt fluxes using a reference salinity, and the globally integrated ocean volume remainsconstant. Only a summary of the ocean model physics and parameters is given here and furtherdetails can be found in Smith and Gent (2002).

In order to complete this study in a reasonable time frame and to reduce the burden on ourcomputational resources, the coarser, ·3 resolution version of the model is used. The domain isglobal. The Bering Strait and Hudson Bay are open, but the Gibraltar Strait and the Red SeaOutflow are closed. The grid is in spherical coordinates in the Southern Hemisphere. In theNorthern Hemisphere, the North Pole is displaced into Greenland at 40�W and 77�N. The zonalresolution is constant at 3.6�. The meridional resolution is variable with 0.86� at the Equator,monotonically increasing to 1.85� at 31�S and staying constant further south. In the NorthernHemisphere high latitudes, the minimum and maximum meridional resolutions are about 1.3� and2.2�, occurring in the Atlantic and Pacific Oceans, respectively. This horizontal resolution resultsin a 100 (zonal)· 116 (meridional) grid. There are 25 vertical levels, monotonically increasingfrom 12 m near the surface to 450 m in the abyss. The minimum and maximum depths are 49 and5000 m, respectively.

The model tracer equations use the skew-flux form (Griffies, 1998) of the Gent and McWilliams(1990) isopycnal transport parameterization with a mixing coefficient of 800 m2 s�1. A generalform of the anisotropic horizontal viscosity formulation due to Smith and McWilliams (2003) isimplemented in the momentum equations. Its parameters are chosen following Large et al. (2001)with 1000 m2 s�1 as the minimum (subject to numerical diffusive stability) background horizontalviscosity. The vertical mixing coefficients are determined from the KPP scheme of Large et al.(1994). In the ocean interior, the background internal wave mixing diffusivity varies in the verticalfrom 0.2· 10�4 m2 s�1 near the surface to 1.0· 10�4 m2 s�1 in the deep ocean. The increase indiffusivity occurs around 1000-m depth, as a crude representation of the enhanced deep vertical


mixing observed over rough topography (Ledwell et al., 2000). The background vertical viscosityobeys the same form, but with a turbulent Prandtl number of 10.

Similar to other OGCMs, POP uses the Leap–Frog time-stepping method. The associated time-splitting error is eliminated via a time-averaging step in which the model calendar is advanced byone half the surface tracer time step.

The model is driven by the net fluxes of heat, salt (freshwater), and momentum computed usingthe bulk forcing scheme described in Large et al. (1997). The same forcing method was alsoadopted in a recent inter-annual variability study by Doney et al. (2003). In the present work, a 3-year cycle of atmospheric data sets covering the period 1991–1993 is used. Thus, the monthlymean International Satellite Cloud Climatology Project (ISCCP) solar radiation (Bishop andRossow, 1991; Bishop et al., 1997) and cloud cover (Rossow and Schiffer, 1991) data sets includethe most recent updates. To eliminate the low bias in the cloud cover data, it is further modified athigh latitudes following the Hahn et al. (1987) observations. The monthly mean precipitation datarepresent a blending of the Microwave Sounding Unit (MSU) (Spencer, 1993) and Xie and Arkin(1996) data sets (see Doney et al., 2003 for blending details). The remaining atmospheric fields(winds, temperature, humidity, and density) are based on the 6-hourly NCEP/NCAR reanalysisdata (Kalnay et al., 1996). A comparison of the NCEP/NCAR air temperatures to the availablestation data along the Antarctic coast shows an excessive cold bias in the NCEP/NCAR data(F. Bryan, personal communication). This bias is eliminated by limiting the minimum air tem-peratures south of 60�S to roughly match the station data. When observed SSTs are used with thesedata sets, the net heat flux shows a positive bias. This is largely eliminated by subtracting uniformly4 Wm�2 from the longwave-down heat flux component. The bulk forcing formulation uses theevolving model surface temperature. All the atmospheric data sets are on a nominal 2�-grid and areinterpolated to the ocean tracer points. The daily-means of the fluxes drive the OGCM.

As described in Large and Yeager (2003), a climatological river runoff distribution from thecontinents contributes to the surface salt fluxes. Following Large et al. (1997), a precipitationcorrection factor is computed for each year based on the change in the global-mean salinity duringthat year. When the tracer time step increases with depth, the normalizing time for the tendencycomputation does accordingly increase. This global factor is then used to multiply precipitationand runoff fluxes to partially balance the evaporation for the next year. Due to the lack of anyappreciable feedback between the model surface salinities and salt fluxes, any flux errors can leadto unbounded local salinity trends. Therefore, as a means for some local constraint, a local, weaksalinity restoring is applied globally using a blending of the monthly mean salinity climatologiesfrom Levitus et al. (1998) and Steele et al. (2001). The salinity enhancements along the Antarcticcoast due to Doney and Hecht (2002) are also included in this blended climatology. The coefficientfor this weak restoring is 11.5 mgm�2 s�1 psu�1, corresponding to a 1-year time scale over 12 m.Because the global-mean of this flux is subtracted at each time step, it does not contribute to theglobal salt budget. The surface salt fluxes over the land-locked, marginal seas are balanced overeach marginal sea individually to prevent any unbounded, local salinity trends in these regions.Here, any deficit/excess is taken from/given to a designated open ocean region. This can beinterpreted as an approximation to include some of the outflow effects of these seas which areotherwise closed.

At high latitudes, the daily sea-ice concentration distributions from the Special SensorMicrowave/Imager (SSM/I) data set (Comiso, 1999) are used to determine the ice extent. The


usual air-sea fluxes are computed for open water and the wind stress passes directly through thesea-ice. No restoring boundary conditions are used under ice covered regions. Instead, we applythe freshwater fluxes diagnosed monthly from a companion coupled ocean–ice integration. In afew regions, there is no observed ice, but the ocean needs to form sea-ice to keep its surfacetemperature from falling below freezing. In such instances, the surface water is heated to thefreezing point with a corresponding increase in salt. This heat flux is the only form of anyadditional heat flux applied under ice. This kind of ice is assumed to be 1-m thick for the purposeof computing a sea-ice concentration. It is not transported, and is melted locally before the surfacetemperature can rise above the freezing temperature of )1.8 �C.

3. Acceleration details and cases

Following Bryan (1984), the momentum and tracer equations for the distorted system are

Table

List o

Cas

SYN

AðzÞS_AS_AAC

S_A

S_A

DtT an

steps

values

aou

ot¼ F ; cðzÞ oðh; SÞ

ot¼ G:

Here, u is the horizontal velocity vector, h is potential temperature, S is salinity, and t is time. Fand G represent the rest of the terms in these equations. Finally, a and c, which can have avertical, z, variation, are the acceleration factors. When a ¼ c ¼ 1, the original, synchronousequations are recovered.

A list of the numerical experiments is given in Table 1. The 10 000-year synchronous controlintegration is designated as SYNC. In the first accelerated integration AðzÞ, a ¼ 1 but the tracertime step varies in z. Here, we use c ¼ 1 between 0- and 1099-m and c ¼ 0:02 between 2877- and5000-m depth. In between, c has a linear variation. As discussed in Danabasoglu et al. (1996),these changes in the tracer time step result in tracer non-conservation. Therefore, the changes arerestricted to intermediate and deep levels where the vertical fluxes are expected to be relativelysmall. In AC, the tracer time step is constant in z, but a ¼ 5. So, the momentum time step is fivetimes smaller than the tracer one. This particular choice of a for AC is artificially dictated by the

1

f numerical experiments

e Initial condition

(surface years)

Integration

length (surface/

deep years)

Equilibration

time (surface

years)

Equilibration

time-step

(·106)

DtT (surface/

deep in s)

DtU (s)

C Levitus/rest 10 000 3500 16.6 6912 6912

Levitus/rest 700/35 000 685 (450) 6.5 (4.3) 3456/172 800 3456

ðzÞ_450 Year 450 of AðzÞ 400 200 0.9 6912 6912

ðzÞ_700 Year 700 of AðzÞ 500 100 0.5 6912 6912

Levitus/rest 10 000 3000 3.3 34 560 6912

C_3 Year 3000 of AC 800 700 3.3 6912 6912

C_10 Year 10 000 of AC 800 750 3.5 6912 6912

d DtU are the tracer and momentum time steps, respectively. Equilibration time-step refers to the number of time

to reach the equilibration time using the surface DtT . In AðzÞ, the numbers in parentheses represent alternative

based on the global-mean time. See text for the definitions.


CCSM2 requirement of at least daily coupling. This is further compounded by some POP modeldetails, e.g. one averaging time step per coupling interval. Therefore, AC does not necessarily takefull advantage of longer tracer time steps. SYNC, AðzÞ, and AC are all initialized with the Jan-uary-mean Levitus et al. (1998) h and S and state of rest.

Four additional, multi-century synchronous integrations are performed to study the synchro-nous adjustment following the accelerated integrations. S_AðzÞ_450 and S_AðzÞ_700 start fromyear 450 and 700 of AðzÞ, respectively. Similarly, S_AC_3 and S_AC_10 are the synchronouscontinuations of AC, starting at year 3000 and 10 000, respectively.

4. Results

4.1. Equilibration times

The equilibration time (Table 1) is defined as the time when the annual- and global-mean hhtitrend is below 10�5 �C year�1, excluding any early transient behavior. This is a rather stringentcriterion on the equilibration time for a global OGCM with realistic forcing. Indeed, even after afairly monotonic decrease below the given thresh-hold, hhti trend may occasionally be muchhigher. Our estimates, therefore, are somewhat subjective, but conservative. The trend is deter-mined using

hhtiyearþ n � hhtiyear

nhti ;

where hti ¼ 1 year is the global-mean time for all estimates except one estimate for AðzÞ. Here, weuse hti ¼ 25:1 years to reflect the depth acceleration effects, i.e. 1 surface year equals 25.1 global-mean years.

Due to the 3-year forcing cycle, we use n ¼ 3 in all cases except AC. In AC, we set n ¼ 66,because a 66-year, regular oscillation with a hhti amplitude of 3· 10�3 �C develops after about 800years (e.g. see Figs. 2 and 3). The associated anomalies start in the northern North Atlantic withlocal h and S amplitudes in excess of 1 �C and 1 psu, respectively, in the upper ocean. Theanomalies propagate southward along the American continent, decaying significantly when theyreach the Southern Ocean. Therefore, the oscillations are more prominent in the Atlantic Oceanrelated fields. Unlike Wood (1998), there is no wave-like variability in the Southern Oceanassociated with baroclinic instability (Bryan, 1984). Unfortunately, any further analysis of thepresent solutions does not easily indicate the culprit among many possibilities including surfaceforcing and the value of the acceleration factor (see also Wood, 1998 for other causes).

We do not use S to determine the equilibration time. The surface forcing method described inSection 2 maintains the initial global-mean S value when the tracer time step has no z variation.Therefore, SYNC and all AC cases show no S trends (Fig. 1).

For our purposes, the equilibration time-step count, representing the actual cost of an inte-gration, is the most relevant information in Table 1. Here, because the model equations areintegrated as usual, the time-averaging time steps are counted as full time steps. The table showsthat the SYNC equilibrium is achieved in about 3500 years, taking 16.6· 106 time steps. Based on

Fig. 1. Time series of the annual-mean potential temperature and salinity. Global and 3268-5000-m means are plotted.

The lines are based on mean values for every third year. In all time series figures, the AðzÞ lines are plotted after

multiplying the integration times by the global-mean time, 25.1. The black lines represent the synchronous extensions of

the accelerated integrations.


surface years, AðzÞ and AC take 6.5 · 106 and 3.3 · 106 time steps, corresponding to 685 and 3000years, respectively, for equilibration. Their synchronous extensions, S_AðzÞ_700 and S_AC_3,need additional 0.5· 106 and 3.3 · 106 steps. Thus, the suite of AðzÞ and AC integration methodsrequire approximately 7.0· 106 and 6.6· 106 steps, respectively. These represent about a factor of2.5 reduction in the computational cost of an equilibrium solution compared to SYNC. Analternative count based on branching from the AðzÞ case earlier is computed as 5.2· 106 using AðzÞand S_AðzÞ_450. Finally, S_AC_10 is included to assess any effects of long, diffusive time scales.Even such, relatively long integrations are still much cheaper than a comparable length SYNCintegration (14.5 · 106 vs. 47.4· 106 time steps).

The synchronous equilibration times in Table 1 for S_AC cases appear to be independent of thebranching time. In contrast, S_AðzÞ cases require a shorter synchronous integration after a longeraccelerated integration.


4.2. Time series

We first consider the evolutions of h and S in Fig. 1. In addition to the global-mean time series,the volume-means for 3268–5000-m depth are included to represent the abyssal trends. Also, in alltime series plots, AðzÞ line is plotted after multiplying its time axis by 25.1. The figure shows thatthe global hhti values for all synchronous extensions following accelerated integrations are withinless than 0.01 �C of the SYNC mean of 4.13 �C. The AC global hhti time series (along with theabyssal hSti below) definitely show that particularly this type of accelerated integration, in which itis more than 0.12 �C warmer than in SYNC, must be followed by a synchronous integration. BothSYNC and AC reveal minor drifts in the deep ocean hhti. The AC deep hhti stays 0.05 �C warmerthan in SYNC even after the synchronous extensions. In contrast, the S_AðzÞ cases appear toapproach the SYNC deep-mean faster. However, neither S_AðzÞ nor S_AC deep hhtis are as closeto the SYNC deep-mean as their global-means to the SYNC one.

As noted earlier, if the accelerated integration technique is conservative, hSti changes very littledue to our surface forcing method. The extent of non-conservation in AðzÞ, particularly obvious inS, is quite detrimental, showing a difference of more than 0.08 psu compared to either SYNC orAC. Similar magnitude differences are also present in the abyssal time series. The deep 0.04-psufresh bias existing in AC vanishes upon switching to synchronous integrations.

Fig. 2 shows the time series for the maximum (in [30�–60�N, 500–1200 m]) and minimum (in[60�S–30�N, 2000–5000 m]) global meridional overturning circulation computed using the Eule-rian-mean velocity. The SYNC maximum and minimum transports are about 17.7 and 19.2 Sv,respectively. Both S_AðzÞ cases reproduce the same maximum transport as SYNC. The S_ACmaximum transports are only slightly (0.3 Sv) higher than in SYNC. For the minimum transport,all of the synchronous extensions are within 0.2 Sv of the SYNC one. Note that the AC oscillationamplitude is about 1.7 and 0.2 Sv, respectively, for the maximum and minimum transports.

Fig. 2. Time series of the annual-mean global maximum and minimum meridional overturning circulations (MOCs).

The maximum and minimum transports are searched in [30�–60�N, 500–1200 m] and [60�S–30�N, 2000–5000 m],

respectively. The black lines represent the synchronous extensions of the accelerated integrations. Also, 1 Sv ” 106 m3 s�1.

Fig. 3. Annual-mean transport time series for the Antarctic circumpolar current (ACC) at Drake Passage, Indonesian

Throughflow, Florida Strait, and Mozambique Channel. The black lines represent the synchronous extensions of the

accelerated integrations.


The vertically integrated (barotropic) mass transports are given in Fig. 3 for the Antarcticcircumpolar current (ACC) at Drake Passage, Indonesian Throughflow, Florida Strait, andMozambique Channel. All transports show that the synchronous integrations succeeding theaccelerated segments do produce the SYNC transports mostly within a few tenths of a Sv. Thebiggest difference is about 1 Sv for S_AðzÞ cases in the ACC. The amplitude of the AC oscillation isthe largest (but still <1 Sv) in the Florida Strait transport.

4.3. Equilibrium solutions

We first perform a quantitative assessment of how similar the solutions for synchronousequilibria following AðzÞ and AC are to those of SYNC. For this purpose, the global-mean root-mean-square (rms) differences for h and S (i.e. S_AðzÞ)SYNC and S_AC)SYNC) are computed


using 3-year means every 100 years (Fig. 4). The years 9997–9999 mean is used for SYNC. Therms h differences monotonically diminish with continued synchronous integrations. However,their trends become much smaller with time, suggesting that significantly longer integrations areneeded for similar marginal reductions. The year 0 values actually show the rms differences usingthe last 3 years of the corresponding accelerated integrations, displaying that the AðzÞ methodresults in h distributions much closer to the SYNC ones than the AC method. At their respectiveequilibria, all synchronous extentions show 0.06–0.07 �C rms differences. Note also that furtheraccelerated integration with AðzÞ reduces both the accelerated rms difference and the length of thesynchronous integration for comparable rms differences.

In contrast with the very favorable rms h differences for AðzÞ cases, the corresponding rms Sdifference plots, again, reveal how significant the associated non-conservation issues are for thismethod of acceleration. Both S_AðzÞ cases have a difference of about 0.09 psu, unaffected byextended integrations. This is of course a significant concern for water mass properties. In con-trast, the rms S differences diminish monotonically with time for S_AC cases with about 0.01-psudifference at equilibrium.

Fig. 4 shows that half of the approach (half-life) towards the SYNC solution occurs in less than200 years for all cases. This time scale is consistent with most of the time series given in Figs. 1–3,following an initial, fast transient adjustment. The exceptions are the global hhti and ACC timeseries for AC where the half-life is a few decades.

The zonal-mean h and S distributions for the entire globe and the Atlantic Ocean are presentedin Figs. 5 and 6 in comparison to the SYNC mean for years 9997–9999. For all other cases exceptAC, the 3-year means are used at their respective equilibration times. In synchronous distribu-tions, the largest global differences in h occur in the abyssal Arctic Ocean. Here, S_AðzÞ and S_AC

Fig. 4. Global-mean root-mean-square (rms) potential temperature and salinity difference time series for the syn-

chronous extensions of the accelerated integrations. The differences are from the years 9997–9999 mean of SYNC and

are computed every 100 years using 3-year means. Year 0 values correspond to the rms differences for the last 3 years of

the corresponding accelerated integrations.

Fig. 5. Time- and zonal-mean global and Atlantic potential temperature. The top two panels are for SYNC and the

contour interval is 2 �C. The other panels show difference distributions between SYNC and the accelerated equilibrium

and synchronous equilibrium of the accelerated integrations. The color key is for the difference plots with a contour

interval of 0.1 �C. The Atlantic distributions exclude the Hudson Bay, Labrador Sea, and the Arctic Ocean. A 66-year

mean (years 3001–3066) is used for AC.


have comparable magnitude differences; >0.4 �C colder and >0.3 �C warmer, respectively. Thecold bias is larger in S_AðzÞ_450 than in S_AðzÞ_700, suggesting that the equilibration time based

Fig. 6. Time- and zonal-mean global and Atlantic salinity. The top two panels are for SYNC and the contour interval is

0.4 psu. The other panels show difference distributions between SYNC and the accelerated equilibrium and synchro-

nous equilibrium of the accelerated integrations. The color key is for the difference plots with a contour interval of 0.02

psu. The Atlantic distributions exclude the Hudson Bay, Labrador Sea, and the Arctic Ocean. A 66-year mean (years

3001–3066) is used for AC.


on the global-mean time may not be very reliable, i.e. the accelerated integration needs to becontinued. This is also evident in AðzÞ–SYNC plots which show smaller differences than those of


S_AðzÞ_450)SYNC. In contrast, as indicated earlier, there is little gain in integrating AC beyond3000 years. In the North Atlantic, S_AðzÞ cases are more than 0.1 �C colder at intermediatedepths. The differences are somewhat higher locally for S_AC cases in the northern NorthAtlantic. The S_AC abyssal Atlantic distributions are uniformly warmer by more than 0.1 �C,associated with the changes in the meridional overturning circulation (see Fig. 7).

Fig. 7. Time-mean zonally integrated global and Atlantic meridional overturning circulation obtained with the Eule-

rian-mean velocity. The top two panels are for SYNC and the contour interval is 4 Sv. The other panels show difference

distributions between SYNC and the synchronous equilibrium of the accelerated integrations with a contour interval of

0.1 Sv. The Atlantic distributions include the Hudson Bay, Labrador Sea, and the Arctic Ocean.


In S (Fig. 6), the fresh bias of AðzÞ and S_AðzÞ cases is quite evident. Indeed, this bias is morethan 0.1 psu in the Arctic and North Atlantic. In contrast, S_AC cases reveal much reduced basinscale differences. Here, the largest differences are confined to the deep Arctic (�0.04 psu) and nearthe surface in the North Atlantic (�0.12 psu).

The AC accelerated equilibrium solutions (Figs. 5 and 6) show non-negligible differences mostlyconfined to the upper 1000 m in both h and S. This further demonstrates that an AC acceleratedequilibrium is simply not good enough and that a synchronous extension is absolutely necessary.

The global and Atlantic meridional overturning circulation distributions due to the Eulerian-mean velocity are given in Fig. 7. In S_AðzÞ, the biggest differences (�0.3 Sv) occur in the SouthernOcean. Elsewhere, the comparisons show minor (�0.1 Sv), relatively small scale differences. Incontrast, S_AC differences are a little larger both in magnitude and their spatial extents. TheNorthern Hemisphere overturning circulation, dominated by the thermohaline circulation pri-marily occurring in the North Atlantic, penetrates slightly deeper in S_AC than in SYNC, pro-ducing 0.6 Sv difference. In the abyssal ocean, the circulation associated with the AntarcticBottom Water appears to have somewhat higher transports into the Equatorial NorthernHemisphere. Note that there is little change in the maximum and minimum transports. Thesedifferences in the circulations are too small to result in any noticeable, significant changes in thenorthward heat transports given in Fig. 8. The peak differences are below 0.02 PW.

Fig. 8. Time-mean global and Atlantic northward heat transport due to the Eulerian-mean velocity. The top panels are

for SYNC. The bottom panels show differences between SYNC and the synchronous equilibrium of the accelerated

integrations. The Atlantic distributions include the Hudson Bay, Labrador Sea, and the Arctic Ocean.


4.4. Seasonal cycle

The monthly mean transport time series presented in Fig. 9 show that AðzÞ and AC cycles arenon-negligibly different from that of SYNC both in phase and amplitude. The phase errors dis-appear almost instantaneously upon switching to synchronous integrations. This is also true forthe amplitude for some transports, e.g. Indonesian Throughflow and Mozambique Channel. In

Fig. 9. Monthly mean transport time series for the Antarctic circumpolar current (ACC) at Drake Passage, Indonesian

Throughflow, Florida Strait, and Mozambique Channel for intra- and inter-annual cycle comparisons. The SYNC

equilibrium is given by the red line. The AðzÞ and AC cycles at about 700 and 3000 years are represented by the solid

blue and green lines, respectively. The remaining blue and green lines are for S_AðzÞ_700 and S_AC_3, respectively. The

time series are shown for 3-year periods, corresponding to forcing years 1991–1993.


contrast, the S_AðzÞ_700 and S_AC_3 ACC time series are still far from the SYNC one even after18 years of integration. This slow adjustment process is likely due to deep water formationprocesses off the Antarctic coast and associated diffusive time scales in the abyssal ocean. Gentet al. (2001) show the effects of this deep water formation on the ACC transport. After 18 years,the Florida Strait transport amplitude mismatches are also evident for S_AC_3. In general, earlyparts of the AðzÞ synchronous extensions appear to be closer to the SYNC cycle. These transporttime series suggest that longer synchronous integrations are needed to fully recover the SYNCcycle, as evidenced by the equilibrium time series, in contrast with Danabasoglu et al. (1996)where no synchronous equilibrium solutions were obtained.

In the ocean interior, away from the convective regions, there are no appreciable phase oramplitude differences between SYNC and all other synchronous or accelerated near-surface h timeseries (not shown) due to strong feedbacks between the surface h and surface heat fluxes. Incontrast, likely as a result of lack of any significant feedbacks between the surface S and fresh-water fluxes, the near-surface S time series (not shown) reveal some amplitude differences incomparison to the SYNC cycle. The only exception to this is the S_AC_3 equilibrium cycle whichmatches the SYNC one quite remarkably. The AC cycle also displays some phase differences.Further down in the deep ocean, the seasonal cycle amplitudes are very small. Nevertheless, thereare no major phase and amplitude differences for h and S cycles for all cases except for the offset ofthe mean distributions. This offset is of course an important concern especially for S in all AðzÞand S_AðzÞ cases. Finally, the AðzÞ seasonal cycle amplitudes are significantly larger than in othercases in the abyssal ocean, in agreement with Danabasoglu et al. (1996).

5. Discussion and conclusions

We have performed a 10 000-year synchronous integration using a comprehensive OGCMsubject to realistic, time-dependent forcing in a global domain. To our knowledge, this is the firstsuch integration of an OGCM. Two accelerated, but otherwise identical, equilibrium integrationsalong with their multi-century equilibrated synchronous extensions have also been obtained toquantitatively determine how well they reproduce the synchronous control solution. The twoaccelerated cases use tracer time steps increasing with depth and unequal momentum and tracertime steps with no depth variations, respectively (Bryan, 1984).

We define the equilibration time as the time when the annual- and global-mean potentialtemperature trend is below 10�5 �C year�1. The synchronous control integration achieves equi-librium in about 3500 years. The accelerated equilibration times are 685 and 3000 surface years forthe two cases, respectively. Their synchronous extensions require about 100 and 700 surface years.

The accelerated solutions do indeed differ from each other and from those of the controlexperiment, in agreement with Wang (2001). However, for practical purposes, many aspects of theaccelerated and control solutions are more similar to each other than they are different. Oneimportant exception here is the detrimental effects of non-conservation due to the depth accel-eration method, particularly obvious in salinity distributions. Depending on the surface forcingtype and details, the level of non-conservation may vary, and further synchronous integrationsmay not necessarily eliminate these errors. Therefore, we do not recommend this kind of accel-eration for any meaningful integration.


Obviously, the desired equilibrium solution can only be obtained via a synchronous integrationmethod. However, if one wishes to expedite this extended process, we do recommend using un-equal momentum and tracer time steps instead. Even in our present unfavorable set-up wherefurther increases in the tracer time step are limited by some technical issues, we get a factor of 2.5reduction in the computational cost (accelerated+ synchronous extension) compared to thesynchronous equilibration. We speculate that these savings could approach order 10 when ourtechnical limitations are eliminated.

Any accelerated integration must definitely be followed by a synchronous segment. Among thereasons are the adjustments of the intra- and inter-annual cycles and any possible oscillatorybehavior present in the accelerated phase. In our experience, these oscillations disappear promptlyupon switching to synchronous integration. With our present definition of equilibrium, the syn-chronous equilibration times reported here are much longer than those of Danabasoglu et al.(1996) where a synchronous control case was not obtained. Having this control integration athand now, one can decide how long to integrate synchronously after acceleration considering theroot-mean-square difference time series. If the incremental gains with further integration arejudged to be minimal or of minor significance, shorter integrations may certainly suffice.

We note that the equilibration times will undoubtedly depend on the model physics andparameter choices, particularly diffusion coefficients in the abyssal ocean giving the longest timescales. Nevertheless, because most non-eddy-resolving, coarse resolution climate models, includingthe ·1 version of the CCSM2, use similar physics and parameter choices as in our present study, wedo not anticipate significant deviations from our current equilibration times. However, the per-missible values of the acceleration factors are likely to be lower at higher resolutions (Killworthet al., 1984), thus somewhat limiting the degree of payoff. Usage of any acceleration methods is ofcourse not permissible in eddy-permitting or -resolving, high resolution models, because thepresence of unstable meso-scale eddies will make their solutions highly time-dependent.

Finally, we find that the equilibration times are strongly affected by the choice of surfaceboundary conditions. For example, replacing the present under-ice boundary conditions withrestoring type boundary conditions (with order 10-day time scales over 12 m) produces shorterequilibration times in two sensitivity experiments with synchronous and depth accelerated timesteps, respectively. We speculate that when surface properties are clamped as in restoring espe-cially in these deep water formation regions, occurrence of more regular convective events leads toreduced diffusive time scales (via smaller length scales) in the abyssal ocean.

Acknowledgements

The author would like to thank Drs. P.R. Gent, W.G. Large, and F.O. Bryan for helpfuldiscussions and suggestions. The National Center for Atmospheric Research is sponsored by theNational Science Foundation.

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