A semi-analytical solution to accelerate spin-up of a coupled carbon and nitrogen land model to steady state

A semi-analytical solution to accelerate spin-up of a coupled carbon and nitrogen land model to steady state
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  Geosci. Model Dev., 5, 1259–1271, doi:10.5194/gmd-5-1259-2012© Author(s) 2012. CC Attribution 3.0 License. GeoscientificModel Development A semi-analytical solution to accelerate spin-up of a coupled carbonand nitrogen land model to steady state J. Y. Xia 1 , Y. Q. Luo 1 , Y.-P. Wang 2 , E. S. Weng 3 , and O. Hararuk 11 Department of Microbiology and Plant Biology, University of Oklahoma, OK, USA 2 CSIRO Marine and Atmospheric Research, Centre for Australian Weather and Climate Research,Aspendale, Victoria, Australia 3 Department of Ecology and Evolutionary Biology, Princeton University, NJ, USA Correspondence to:  J. Y. Xia ( 14 March 2012 – Published in Geosci. Model Dev. Discuss.: 17 April 2012Revised: 15 August 2012 – Accepted: 5 September 2012 – Published: 11 October 2012 Abstract.  The spin-up of land models to steady state of cou-pled carbon–nitrogen processes is computationally so costlythat it becomes a bottleneck issue for global analysis. In thisstudy, we introduced a semi-analytical solution (SAS) forthe spin-up issue. SAS is fundamentally based on the ana-lytic solution to a set of equations that describe carbon trans-fers within ecosystems over time. SAS is implemented bythree steps: (1) having an initial spin-up with prior pool-sizevalues until net primary productivity (NPP) reaches stabi-lization, (2) calculating quasi-steady-state pool sizes by let-ting fluxes of the equations equal zero, and (3) having afinal spin-up to meet the criterion of steady state. Step 2is enabled by averaged time-varying variables over one pe-riod of repeated driving forcings. SAS was applied to bothsite-level and global scale spin-up of the Australian Com-munity Atmosphere Biosphere Land Exchange (CABLE)model. For the carbon-cycle-only simulations, SAS saved95.7% and 92.4% of computational time for site-level andglobal spin-up, respectively, in comparison with the tradi-tional method (a long-term iterative simulation to achieve thesteady states of variables). For the carbon–nitrogen coupledsimulations, SAS reduced computational cost by 84.5% and86.6% for site-level and global spin-up, respectively. The es-timated steady-state pool sizes represent the ecosystem car-bon storage capacity, which was 12.1kgCm − 2 with the cou-pled carbon–nitrogen global model, 14.6% lower than thatwith the carbon-only model. The nitrogen down-regulationin modeled carbon storage is partly due to the 4.6% decreaseincarboninflux(i.e.,netprimaryproductivity)andpartlydueto the 10.5% reduction in residence times. This steady-stateanalysis accelerated by the SAS method can facilitate com-parative studies of structural differences in determining theecosystem carbon storage capacity among biogeochemicalmodels. Overall, the computational efficiency of SAS poten-tially permits many global analyses that are impossible withthe traditional spin-up methods, such as ensemble analysis of land models against parameter variations. 1 Introduction Modeling ecosystem biogeochemical cycles is highly de-pendent on initial values because of long-term persistenceof ecosystem state properties. It requires setting up initialvalues of all state variables (e.g., carbon and nitrogen poolsizes) in any biogeochemical models before scientists canuse the models for any analysis. The initial values are ei-ther estimated from observations (D’Odorico et al., 2004;Luo and Reynolds, 1999) or assumed to be at steady state.The latter is usually achieved by traditional spin-up meth-ods that perform long model simulations until no trend of change in pool sizes over many periods of the repeated cli-mate forcing, even though the pool sizes vary seasonallyand inter-annually within one period of the repeated forc-ing (Johns et al., 1997; McGuire et al., 1997; Thornton et al.,2002; Yang et al., 1995). Spinning biogeochemical models tosteadystateiscomputationallyexpensive,especiallysowhensimulations are performed with global biogeochemical mod-els of coupled C and N cycles (Thornton and Rosenbloom,2005). In general, a fully coupled earth system model with Published by Copernicus Publications on behalf of the European Geosciences Union.  1260 J. Y. Xia et al.: Spin-up of biogeochemical models biogeochemical cycles needs to be spun up in several se-quential steps, and each step can take several thousands of simulation years (Doney et al., 2006). Spin-up of a fully cou-pled earth system model with a relatively coarse resolution( ≈ 3.75 ◦ ) atmospheric model is estimated to take hundredsof real-world days of computation using the present NationalCenter for Atmosphere Research (NCAR) supercomputers(Jochum et al., 2010). As a consequence, spin-up has becomea serious constraint on global modeling analysis of biogeo-chemical cycles.Thornton and Rosenbloom (2005) have explored a fewspin-up methods for achieving steady states of a coupledcarbon–nitrogen ecosystem model. The spin-up of theirmodel begins with initial values of no soil organic matter(SOM) and very small plant pools (Thornton et al., 2002).During the spin-up, the accumulation rate of SOM stronglydepends on the nitrogen addition rate in their model. Theyperiodically increase the mineral nitrogen supply during theearly stage of the spin-up to acclerate the spin-up. Each ni-trogen addition period is followed by a period with reduc-ing nitrogen input linearly to the normal level (Thornton andRosenbloom, 2005; Thornton et al., 2002). The efficiency of this punctuated nitrogen addition method is low without anoptimal combination of the nitrogen addition and reductionperiods. A prior analysis for searching such optimal com-binations is needed when this method is applied to a newmodel. An accelerated decomposition method is based onan assumption of linear scaling between decomposition ratesand litter/soil pool sizes. A linearity test between decom-position rates and pool sizes is needed before performingthis method for each new model (Thornton and Rosenbloom,2005). The accelerated decomposition method can distort in-teractions between carbon and nitrogen cycles in the a globalbiogeochemical model of terrestrial carbon, nitrogen, andphosphorus (CASACNP) (Y.-P. Wang, unpublished data) andocean physics for spinning up an ocean model (Bryan andLewis, 1979; Danabasoglu et al., 1996). Although the abovemethods have been used in some modeling studies (Rander-son et al., 2009), most models still use the traditional spin-upmethod with long-term iterative simulations to achieve thesteady states of variables.Carbon processes in terrestrial ecosystems can be repre-sented by first-order, linear differential equations (Bolker etal., 1998; Luo and Weng, 2011; Luo et al., 2012). This prop-erty renders a possibility to obtain an analytical solution of steadystatesforterrestrialcarboncyclemodels(Bolkeretal.,1998; Comins, 1994; Govind et al., 2011; King, 1995; Luoet al., 2001). Ludwig et al. (1978), for example, have useda two-step approximation. They first calculated the steady-state pool sizes of the fast variables by holding the slow vari-ables fixed, and then analyzed the slow variables with thefast variables held at corresponding steady-state pool sizes.Most of the analytical solutions (Comins, 1994; Govind etal., 2011; King, 1995; Luo et al., 2001) are obtained by usingconstant net primary productivity (NPP). However, temporalclimate fluctuations and seasonal plant growth make it dif-ficult to get an analytical solution for state variables in themodels. Some studies (Lardy et al., 2011; Martin et al., 2007)have attempted to obtain the analytical solution of steady-state pool sizes via managing the climate fluctuations. Thesemethods have used matrix-based analysis but still need tosolve several relatively complicated equations. For these rea-sons, no effective analytical method has been developed tosave the spin-up time for global land models.An analytical solution is still possible to obtain steady-state pool sizes if we can overcome two obstacles. First,we need to get time-averaged approximations of the time-varying variables, such as environmental scalars and NPP.Since most spin-up uses repeated climate forcing variablesto estimate steady states of pool sizes, we can estimate av-erages of those time-varying variables within one period of the repeated forcing variables. Second, for most carbon–nitrogen coupled models, nitrogen regulates carbon cycle viaits influences on photosynthesis and decomposition. Nitro-gen pool sizes are usually related with carbon cycle via car-bon/nitrogen (C/N) ratios (Gerber et al., 2010; Wang et al.,2010). Nitrogen influences on photosynthesis are fast pro-cesses and can be accounted for by short, initial spin-up toreachastabilizationofNPP.TheC/Nratiosintheendloopof initial spin-up for stable NPP can be used to estimate steady-state nitrogen pool sizes. Using the above approximations of those time-varying variables will generate errors for estimat-ing steady states of carbon and nitrogen pools. Thus, someadditional spin-up may be necessary to achieve steady statesof all pools.This study was intended to develop a semi-analytical solu-tion (SAS) to accelerate spin-up of global carbon–nitrogencoupled models. We first discussed biogeochemical prin-ciples underlying SAS. SAS becomes permissible becausea set of first-order ordinary differential equations can ad-equately describe carbon transfers within ecosystems overtime and be analytically solved to obtain steady-state poolsizes. We applied SAS to the Australian Community Atmo-sphere Biosphere Land Exchange (CABLE) model and de-veloped a general procedure of SAS for spin-up. There arethree key steps of SAS. The first step is a short spin-upto obtain steady-state NPP, averaged environmental factorswithin one period of repeated forcing variables, and C/N ra-tios. The second step is to analytically solve the differentialequation to calculate steady-state carbon and nitrogen poolsizes. The last step is to make an additional spin-up to meetthe steady-state criterion for all pools. We evaluated the com-putational efficiency of SAS for the carbon-only and the cou-pled carbon–nitrogen models against the traditional spin-up,applications of SAS to other biogeochemical models, andpossible model analyses enabled by SAS. Geosci. Model Dev., 5, 1259–1271, 2012   J. Y. Xia et al.: Spin-up of biogeochemical models 1261   Leaf (X 1 )Root(X 2 )Woody(X 3 )Metabolic Litter (X 4 )Structural Litter (X 5 )Fast SOM(X 7 )Slow SOM(X 8 )Passive SOM(X 9 ) CO 2 CO 2 CO 2 CO 2 CO 2 CO 2 Canopy Photosynthesis CWD(X 6 ) Fig. 1.  Diagram of the carbon processes of the CASACNP modelon which Eq. (1) is based. SOM stands for soil organic matter. 2 Methods2.1 Biogeochemical principles for the semi-analyticalsolution (SAS) The semi-analytical solution (SAS) we introduced in thisstudy is built upon principles of biogeochemical cycles interrestrial ecosystems. The biogeochemical cycle of carbonin an ecosystem is usually initiated with plant photosynthe-sis, which fixes CO 2  from the atmosphere into an ecosystem.The photosynthetic carbon is partitioned into leaf, root, andwoody biomass. Dead biomass becomes litter to metabolic,structural, and coarse woody debris (CWD) litter pools.The litter carbon is partially released to the atmosphere asrespired CO 2  and partially converted to soil organic mat-ter (SOM) in fast, slow, and passive pools (Fig. 1). The meancarbon residence time varies greatly among different pools,from several months in leaves and roots to hundreds or thou-sands of years in woody tissues and SOM (Torn et al., 1997).Thecarbonprocessesinaterrestrialecosystemcanbemathe-matically expressed by the following first-order ordinary dif-ferential matrix equation (Luo et al., 2003):d X(t) d t  = ξACX(t) + BU(t),  (1)where  X(t) = (X 1 (t) ,  X 2 (t),...,X 9 (t)) T  is a 9 × 1 vector de-scribing nine carbon pool sizes in leaf, wood, root, metaboliclitter, structural litter, CWD, fast SOM, slow SOM, and pas-sive SOM, respectively, in the Community Atmosphere Bio-sphere Land Exchange (CABLE) model (Wang et al., 2011). A  and  C  are 9 × 9 matrices given by A =  − 1 0 0 0 0 0 0 0 00  − 1 0 0 0 0 0 0 00 0  − 1 0 0 0 0 0 0 a 41  a 42  0  − 1 0 0 0 0 0 a 51  a 52  0 0  − 1 0 0 0 00 0 1 0 0  − 1 0 0 00 0 0  a 74  a 75  a 76  − 1 0 00 0 0 0  a 85  a 86  a 87  − 1 00 0 0 0 0 0  a 97  a 98  − 1  ,  (2) C = diag (c),  (3)where  A  denotes the carbon transfer matrix, in which  a ij  represents the fraction of carbon transfer from pool  j   to  i .The diag( c)  is a 9 × 9 diagonal matrix with diagonal en-tries given by vector  c = (c 1 ,  c 2 , ...,  c 9 ) T  ; components  c j  ( j   = 1 , 2 ,..., 9) quantify the fraction of carbon left frompool  X j   ( j   = 1 , 2 ,..., 9) after each time step.  ξ   is anenvironmental scalar accounting for effects of soil type,temperature and moisture on carbon decomposition.  B = (b 1 ,b 2 ,b 3 , 0 ,..., 0 ) T  represents the partitioning coefficientsof the photosynthetically fixed carbon into different pools.  U  is the input of fixed carbon via plant photosynthesis. In gen-eral, Eq. (1) can adequately summarize C cycle processes inmost land models (Cramer et al., 2001; Parton et al., 1987).Equation(1)cannotbedirectlysolvedtoobtainthesteady-state values of carbon pools because matrices  A  and  B , theenvironmental scalar  ξ  , and ecosystem carbon influx  U(t) vary with time and driving variables. Since carbon influx in-volves fast processes, its steady-state value  U  ss  can be ob-tained from short spin-up. Most model spin-ups use repeateddriving variables in long-term, iterative simulations. Thus itis possible to calculate averaged values of the environmentalscalar ( ¯ ξ) , the carbon transfer (  ¯ A)  and partitioning ( ¯ B)  coef-ficients within one period of repeated driving variables. With U  ss  and the mean values of the time-varying variables ( ¯ ξ,  ¯ A, and  ¯ B) , we can analytically calculate the steady-state carbonpool sizes  X ss  by letting Eq. (1) equal zero as X ss =− ( ¯ ξ   ¯ AC) − 1 ¯ BU  ss .  (4)We divided  X ss  by C/N ratio in each pool to obtain steady-state nitrogen pool sizes  N  ss . The C/N ratios were the tem-poral average values of the last loop of the initial spin-upfor  U  ss . Using temporal averages of those time-varying vari-ables ( ¯ ξ  ,  ¯ A ,  ¯ B , and mean C/N ratios) will yield approxima-tion errors to estimate the steady states of carbon and nitro-gen pools. Equations (1)–(4) assume the terrestrial carboncycle as a linear system, while biogeochemical models simu-late some carbon processes nonlinearly. For example, alloca-tions of NPP to new leaf and stem growths are determined bya dynamic function of NPP in the Community Land Model(CLM-CN) (Oleson et al., 2010). As a result, using the linearsystem will generate some additional approximation errors Geosci. Model Dev., 5, 1259–1271, 2012  1262 J. Y. Xia et al.: Spin-up of biogeochemical models to estimate the steady states of carbon and nitrogen pools.Thus, the steady-state carbon and nitrogen pool sizes that areanalytically calculated by Eq. (4) need to be further adjustedwith additional spin-up to meet the criterion of steady statesfor all carbon and nitrogen processes.Overall, our semi-analytic solution of spin-up consists of threesteps:(1)aninitialspin-uptoobtainsteady-statecarboninflux  U  ss , temporally averaged values of the time-varyingvariables in Eq. (3) ( ¯ ξ  ,  ¯ A , and  ¯ B) , and C/N values; (2) calcu-lation of the steady-state carbon pool sizes  X ss  using Eq. (3)and the steady-state N pool sizes  N  ss  from dividing  X ss  byC/N ratios; and (3) additional spin-up to meet the steady-state criteria for all carbon and nitrogen processes. 2.2 Model description We applied the semi-analytic solution of spin-up to the CA-BLE model, which is one of the land surface models forsimulationofbiophysicalandbiochemicalprocesses.Kowal-czyk (2006) and Wang et al. (2011) have described the CA-BLE model in detail. The model includes 5 submodels: radi-ation, canopy micrometeorology, surface flux, soil and snow,and biogeochemical cycles. The CABLE model calls the ra-diation submodel first to compute absorption and transmis-sion of both diffuse and direct beam radiation in the two-big-leaf canopy and at soil surface (see the details in Wang andLeuning,1998).Thecanopymicrometeorologysubmodeles-timates the canopy roughness length, zero-plane displace-mentheightandaerodynamictransferresistancebasedonthetheory developed by Raupach (Raupach, 1989a, b, 1994) andRaupachetal.(1997).Thesurfacefluxsubmodelusestheab-sorbed radiation to estimate the water extraction and groundheat flux, which are required in the soil and snow submodel.The biogeochemical-cycles submodel is called last to com-pute the respiration of non-leaf plant tissues, the soil respira-tion, and the net ecosystem CO 2  exchange.The biogeochemical submodel of CABLE evolves fromthe CASACNP model, which was developed by Wang etal. (2010). It adopted the model structure of carbon pro-cesses from the CASA’ model (Randerson et al., 1997) andcontains coupled carbon, nitrogen, and phosphorus cycles.In this study, phosphorus cycle and its coupling with car-bon and nitrogen cycles were not activated. The CASACNPmodel has 9 pools, which include three plant pools (leaf,wood, and root), three litter pools (metabolic litter, struc-tural litter, and coarse woody debris), and three soil pools(microbial biomass, slow and passive soil organic matter)(Fig. 1). There is an additional pool of inorganic nitrogen(NO − 3 + NH + 4  )  when nitrogen cycle is coupled with carboncycle. The equations that describe changes in pool size withtime have been presented by Wang et al. (2010) and can berepresented by Eq. (1). In Eq. (1), parameter  C  is set to be aconstant in the CABLE model, while matrices  A  and  B , theenvironmental scalar  ξ  , and ecosystem carbon influx  U(t) vary with time and driving variables. In the  A  matrix, thecarbontransfercoefficientsaredeterminedbylignin/nitrogenratio from plant to litter pools, lignin fraction from litter tosoil pools, and soil texture among soil pools. In the  B  matrix,the carbon partitioning coefficients of the photosyntheticallyfixed carbon into plant pools are determined by availabilitiesof light, water and nitrogen as the carbon allocation schemedescribed by Friedlingstein et al. (1999). The environmen-tal scalar ( ξ)  regulates the leaf turnover rates by cold anddrought stresses on leaf senescence rate, the turnover rates of litter carbon pools via limitations of soil temperature, mois-ture, and nitrogen availability, and SOM turnover rates bysoil temperature, moisture, and texture. The soil texture isspatially fixed in the CABLE model. The soil nitrogen willlimit litter decomposition if the gross mineralization is lessthan the immobilization (Wang et al., 2010). We spun up themodel for about one hundred of simulation years to obtainthe steady-state carbon influx  U  ss .In the CABLE model, the optimal carbon decay rates (the C  matrix in Eq. 1) are preset and vary with vegetation types.The vegetation types for each 1 ◦ × 1 ◦ grid cell in the modelwere derived from the 0.5 ◦ × 0 . 5 ◦ International Geosphere-Biosphere Programme (IGBP) classification (Loveland et al.,2000). During the spin-up of the coupled carbon–nitrogenmodel, the carbon influx ( U)  and litter decomposition rateare regulated by the soil nitrogen availability (Wang et al.,2010). The nitrogen regulation may periodically occur untilall the nitrogen processes of the model reach steady states.In the CABLE model, the nitrogen inputs of deposition,fertilizer application and fixation are explicitly estimated.The nitrogen deposition in 1990 was estimated from Den-tener (2006) and nitrogen fixation from Wang and Houl-ton (2009). The global fertilizer application of nitrogen istaken as 0.86GtNyr − 1 from Galloway et al. (2004) and isdistributed uniformly within the cropland biome (Wang etal., 2010).The forcing variables required for the CABLE model in-clude incoming short- and long-wave radiation, air temper-ature, specific humidity, air pressure, wind speed, precipita-tionandambientCO 2  concentration.TheCABLEmodelfirstgenerated daily meteorological forcing (surface air tempera-ture, soil temperature and moisture). Then the daily forcingswere used to integrate the full model with a time step of oneday. In this study, the meteorological forcings of 1990 wereused to run the global version of the CABLE model to steadystates. A site version of CABLE (v2.01), which has been cal-ibrated by datasets from Harvard Forest, was used for thesite-level analysis in this study. We used forcing data of Har-vard Forest from 1992 to 1999 with the time step of half anhour for the site-level simulation. A detailed description of the data sources was provided by Urbanski et al. (2007). 2.3 The procedure of semi-analytic solution to spin-up For modeling analyses of biogeochemical responses toglobal change, models often have to be first spun up to steady Geosci. Model Dev., 5, 1259–1271, 2012   J. Y. Xia et al.: Spin-up of biogeochemical models 1263   1.Develop a flow diagram as Fig. 1.6. Calculate the analytical solution of the steady-state pool sizes.2. Organize the linkage between fluxes and pools into matrix  A , C  , and vector  B .4. Re-code the model (see Text S1). • Set up NPP criterion for the initial spin-up • Create new variables to store mean values of the time-varying variables • Add equations to calculate the analytical solutions of pools • Set up a criterion for the slowest pool for the final spin-up 3. Figure out the determinants of each element of the time-varying variables (  A ,  B , and ξ  in equation 1) in the model.5. Initial spin-up. • Read in the initial parameters and spin-up NPP (or plant carbon pools) to steady state • Meanwhile, save all the values of the variables in the equations in step 4 7. Final spin-up. • Read in the analytical solved carbon and nitrogen pools, and spin up all pools to steady states Fig. 2.  The spin-up strategies of the spin-up method with the semi-analytical solution (SAS) used in this study. state for all pools and fluxes. Traditionally, the biogeochem-ical models first read in all meteorological input parametervalues and initial pool sizes. Then the models continuouslyrun with recycled meteorological forcing variables for thou-sands of simulation years until steady states are reached forall pools and fluxes.To implement SAS with the CABLE model, we did thefollowing things (Fig. 2):1. Developing a flow diagram to link carbon pools andfluxes within ecosystems as in Fig. 1.2. Organizing the linkage between carbon pools and fluxesinto carbon transfer matrices  A  and  C , and plant carbonpartitioningcoefficientsintovector B .Thepresetvaluesof the optimal carbon decay rates were organized intothe  C  matrix.3. Figuring out how each element of the time-varying vari-ables ( A ,  B  and  ξ)  in the Eq. (1) was determined in themodel.4. Recoding a section in the model (e.g., the biogeochem-ical cycle submodel in the CABLE). The recoding of the model includes 4 steps: (1) setting up a criterion forthe stable NPP for the initial spin-up; (2) creating newvariables to store the mean values of the time-varyingparameters; (3) creating equations to calculate the ana-lytical solutions of each pool according to the structuresof matrix  A ,  C  and vector  B ; and (4) setting up a crite-rion for the steady state of the slowest pool for the finalspin-up. More details about the recoding can be foundin Text S1.5. Making an initial spin-up by running the model usingrepeated meteorological forcing until NPP (or all plantpools) reached stabilization ( U  ss ) . In this study, we ranthe model until the mean change in NPP over each loop(8yr) of site simulation at Harvard Forest was smallerthan 10 − 4 gCm − 2 . For the global simulation, we ranthe model until the mean changes in plant carbon poolsover each loop (1yr) were smaller than 0.01% per yearcompared to the previous cycle. Meanwhile, the meanvalues of all the time-varying parameters in Eq. (3) werewritten to those newly created variables. Those param-eters are the stable NPP ( U  ss ) , the mean environmentalscalar ( ¯ ξ  ) and matrices of carbon transfer (  ¯ A)  and par-titioning ( ¯ B)  coefficients within one period of repeatedforcing variables, as well as C/N ratios at the end of theinitial spin-up.6. Calculating the analytical solution of the steady statesof carbon and nitrogen pools. The steady-state carbonpools were solved by letting carbon influx equal effluxfor each pool (Eq. 4). Nitrogen pools are obtained bydividing the steady-state carbon pools by the mean C/Nratios of the end loop of the initial spin-up.7. Making the final spin-up by using the analyticallysolved carbon and nitrogen pools as initial values untilthe steady-state criterion for the soil carbon pools wasmet. The steady-state criterion set in this study was thatthe change in any soil carbon pool (  C soil )  within eachsimulation cycle was smaller than 0.5gCm − 2 yr − 1 (asone criterion in Thornton and Rosenbloom, 2005). Ac-cording to the difference in turnover rate, a slower poolneedsalongertimetoreachsteadystateduringthespin-up. The final spin-up is determined by the dynamic of the slowest carbon pool when the criterion of steady-state soil carbon pools is small enough. 3 Results3.1 Performances of SAS at Harvard Forest For Harvard Forest, the traditional spin-up method took 9768and 6768yr (1220 and 846 loops, respectively) to get thesteadystatesofthecarbon-onlyandcoupledcarbon–nitrogensimulations ( C soil  <  0.5gCm − 2 yr − 1 ) , respectively. Thefirst step of the SAS was to spin up the model to reach stableNPP. It took 64 and 336yr (8 and 42 loops, respectively) forthe carbon-only and coupled carbon–nitrogen simulations,respectively (Fig. 3). After the semi-analytical solution of steady-state values was obtained for all carbon and nitrogenpools, it took another 45 and 89 loops of the carbon-onlyand coupled carbon–nitrogen simulations, respectively, forthe change in any soil carbon pool in each loop of simula-tion (  C soil )  less than 0.5gCm − 2 yr − 1 . In comparison withthe traditional spin-up method, the SAS method saved about95.7% and 84.5% of computational time for getting steadystates of the carbon-only and coupled carbon–nitrogen sim-ulations, respectively. The differences in steady-state carbon Geosci. Model Dev., 5, 1259–1271, 2012
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