The commonly adopted biogeochemistry spin-up process in an Earth system model (ESM) is to run the model for hundreds to thousands of years subject to periodic atmospheric forcing to reach dynamic steady state of the carbon–nitrogen (CN) models. A variety of approaches have been proposed to reduce the computation time of the spin-up process. Significant improvement in computational efficiency has been made recently. However, a long simulation time is still required to reach the common convergence criteria of the coupled carbon–nitrogen model. A gradient projection method was proposed and used to further reduce the computation time after examining the trend of the dominant carbon pools. The Community Land Model version 4 (CLM4) with a carbon and nitrogen component was used in this study. From point-scale simulations, we found that the method can reduce the computation time by 20–69 % compared to one of the fastest approaches in the literature. We also found that the cyclic stability of total carbon for some cases differs from that of the periodic atmospheric forcing, and some cases even showed instability. Close examination showed that one case has a carbon periodicity much longer than that of the atmospheric forcing due to the annual fire disturbance that is longer than half a year. The rest was caused by the instability of water table calculation in the hydrology model of CLM4. The instability issue is resolved after we replaced the hydrology scheme in CLM4 with a flow model for variably saturated porous media.

The initial starting values of carbon–nitrogen (CN) models are not commonly available, especially for large-scale applications, but they have an important influence on the subsequent C–N states simulated by the models. Typically, Earth system model (ESM) simulations are initialized in the pre-industrial period to allow sufficient time for the coupled system to respond to the various forcings. Initialization of the CN model is usually achieved by a spin-up run of the CN model given an arbitrary initial condition until an approximate C equilibrium is reached. This time marching of the model requires several hundreds to thousands of years of model simulations before a dynamic steady state is reached. The length of the transient state to reach a dynamic steady state is dependent on the initial conditions of the system. It has long been recognized that the spin-up process of CN models is time consuming due to the slow turnover rates of the soil carbon pools, which significantly affect the computational efficiency for global modeling. A number of approaches have been proposed in the past to improve upon the explicit forward time integration of ordinary differential equations in their native form and rate parameters for CN models. These approaches include the initialization of soil organic matter carbon pools with observations (Zhang et al., 2002), the accelerated decomposition method using a higher decomposition rate for litter and soil carbon pools (Thornton and Rosenbloom, 2005), decelerated bulk denitrification and leaching method (Shi et al., 2013), and a semi-analytical steady-state solution for soil organic C and N pools (Xia et al., 2012). Except for the semi-analytical approach, the other approaches mentioned above have been summarized and compared in Shi et al. (2013). The semi-analytical model needs initial spin-up values of net primary productivity (NPP), which still requires a long simulation time for stabilization, because C and N are coupled in CN models. We had previously restructured the CN model in Community Land Model version 4 (CLM4-CN) (Lawrence et al., 2011) and developed a steady-state solution directly using annually averaged rate parameters (Fang et al., 2013, 2014). Using our approach, we were able to implement the semi-analytical method in Xia et al. (2012). Our numerical experiment showed that the semi-analytical method is not necessarily faster compared to the modified form of the “accelerated decomposition” approach in Koven et al. (2013).

Recently, Koven et al. (2013) used a modified form of the “accelerated decomposition” (hereafter referred to as the AD approach) by numerically increasing the decomposition rates for the two slowest soil carbon pools (named Soil3 C and Soil4 C) to a level so that their turnover rates are similar to the fast pools during the initialization. Numerical evaluation found that the approach significantly reduced the spin-up time (Koven et al., 2013). Figure 1 shows the structure of the soil C pool represented in CLM4-CN. Note that heterotrophic respiration fractions are not shown. The reason that the AD approach can accelerate the spin-up is because these two slowest pools are essentially decoupled from the rest of the ordinary differential equations, in that all other pools do not need input from them. The approach, however, cannot be applied to the coarse woody debris (CWD) pool even though its turnover rate is on the same order of Soil3 C, because it is an input to the litter pools. Changing the rate of CWD will give a different solution of other pools during each integration step using the same initial condition, which will lead to a state far from equilibrium if the state is used in a restart simulation.

In the AD approach, once the solution is obtained from the accelerated run,
the state of Soil3 C and Soil4 C can be analytically solved. From Fig. 1, the
flux of Soil3 C and Soil4 C pools can be described by the following equations:

Soil carbon pool structure of CLM4-CN. The arrows represent the
decomposition pathways, and

Even with this modified accelerator approach, long simulation times cannot be avoided in dry and cold places because the decomposition scaling factor is associated with soil water potential and temperature. Hence, new methods are needed to address the spin-up problem. In implicit time integration approaches, based on knowledge about the trajectory of the solution of the initial value problem, linear extrapolation from time integration was often used to find a good initial value for iterative multirate multidisciplinary processes (Birken et al., 2014, and references therein). A number of explicit Euler steps with small time steps followed by an explicit Euler step with a large time step when the change in components due to fast processes become negligible have been shown to efficiently solve stiff ordinary differential equations (Eriksson et al., 2003; Gear and Kevrekidis, 2003). We made use of these concepts, referred to as the gradient projection (GP) approach in this study, to further improve the computational efficiency of the biogeochemistry spin-up processes.

Community land model CLM4 is the land component of the Community Earth System Model (CESM) (Lawrence et al., 2011). Processes simulated in CLM4 include biogeophysics (solar and longwave radiation, momentum, heat transfer in soil and snow, hydrology of canopy, soil, and snow, and stomatal physiology and photosynthesis) and biogeochemistry (phenology, autotrophic respiration, heterotrophic respiration, carbon and nitrogen allocation, and nitrogen source/sink). The vegetation structures (leaf area index, stem area index and height) in CLM4-CN are represented through the predictive state variables of leaf and stem carbon, which are coupled to simulate fluxes of carbon and nitrogen state variables in vegetation, litter, and soil organic matter (Lawrence et al., 2011; Thornton and Zimmermann, 2007). The tree, shrub and grass plant functional types (PFTs) are divided into tropical, temperate and boreal climate groupings using the PFT physiology and climate rules of Nemani and Running (1996) and C3/C4 photosynthetic pathways in the case of grasses (Lawrence and Chase, 2007). For this study, we used CLM4-CN in offline mode, which is not coupled to an atmosphere model.

If

At the end of the accelerated run, a dynamic steady-state water table should
be reached in the soil column. Due to the slow turnover rates, the total soil
carbon gradually approaches steady state from one cycle to the next
(Fig. 2a). We can approximate C at a future time

Annual average total soil carbon change with respect to
time

We call the method shown in Eq. (7) the gradient projection (GP) method. This
method is analogous to that described in Eriksson et al. (2003), which uses a
large time step that satisfies the stability requirement for integrating the
slowest processes once the contributions from fast processes become
negligible. We allow

Location, PFT, soil type and number of years of atmospheric forcing for each site.

A total of 38 single point tower sites from the FLUXNET (Baldocchi et al., 2001) were selected to assess the gradient projection method. These sites include temperate, boreal, tropical, and subtropical climatic environments and four ecosystem types (tropical forests, temperate forests, boreal forests, grasslands, and Mediterranean-type ecosystems) (Table 1).

The meteorological forcing, site information such as soil texture, vegetation
cover, and satellite-derived phenology at each site are provided by the North
American Carbon Program (NACP) site synthesis team for the sites located in
North America and by the Large Scale Biosphere-Atmosphere Experiment in
Amazônia Model Intercomparison Project (LBA-MIP) for the sites located in
South America. The NACP site synthesis and LBA-MIP data sets are detailed in
Schwalm et al. (2010) and at

Table 2 shows the comparison of total simulation years till a certain
convergence criterion is met. Three convergence threshold values in

Comparison between the gradient projection method (in bold) and the accelerated spin-up method.

Note that the computation cost reduction for sites US-Me2, RJA, US-IB1 and US-SO2 is not shown in Fig. 3. Site US-Me2 met the convergence criteria before the GP method is applied. Sites RJA and US-IB1 show oscillation of the annual average total C from one full length (multiple years) of forcing cycle to the next, and site US-SO2 shows a carbon periodicity much longer (81 years) than that of the atmospheric forcing (9 years) (Fig. 4). The oscillation noted in the simulations at RJA and US-IB1 differs from the variability within the forcing cycle, which happens when soil C has a fast turnover rate such that soil C dynamics are primarily controlled by variability of the forcing (Luo et al., 2014). Due to the aforementioned reasons, the GP method failed at those three sites.

Stacked bar chart of percent reduction in computation cost for three convergence threshold values.

We first checked whether the oscillation and longer periodicity were caused by fire disturbance. However, this can only explain the oscillation at site US-SO2. The annual fire disturbance at site US-SO2 is longer than half a year, while it is less than a month at the other two sites. In the original CLM4, soil water is calculated first for the top ten soil layers (3.8 m below the ground surface) and one aquifer layer using a water-content-based formulation for water mass conservation and a groundwater table as the bottom boundary condition (Oleson et al., 2010); the Niu et al. (2005, 2007) parameterizations are then used to simulate groundwater–soil water interaction and update the water table depth. If the water table is below 3.8 m, groundwater does not contribute to the soil moisture in the overlaying soil layers. We found that, after 100 years, the water table calculation scheme in CLM4 has resulted in a significantly different evolution of water table depth from one cycle to the next when repeatedly forcing the model with atmospheric data at sites RJA and US-IB1. The issue has also been found previously and an effort has been made to eliminate the oscillations (Oleson et al., 2010), but such oscillations can still occur under specific conditions such as at RJA and US-IB1. When we turned off the groundwater component, i.e., applying a zero flux boundary condition at the bottom of the soil column, we did not see oscillations in SOC at RJA and US-IB1. In the Niu et al. (2007) groundwater model, after solving the mass conservation equations (Richards' equation) in the top ten layers, water is then moved around to account for recharge and subsurface runoff and in the meantime to satisfy two conditions for water content in each layer; i.e., the water content has to be greater than the minimum content and smaller than the effective porosity of the layer. By moving water mass around after the Richards' equation is solved, the Richards' equation at each node is no longer satisfied if its moisture deviates from its previous solution. We have confirmed the local mass conservation error of water in the original model of CLM4. The error is large when recharge or subsurface runoff is high. The water content formulation itself has been previously shown to cause solution instability for soils near saturation (Hills et al., 1989). Instead of solving the soil water and groundwater separately, we use a flow model for variably saturated porous media, STOMP (Subsurface Transport Over Multiple Phases) (White and Oostrom, 2000), to see if it can resolve the oscillation in the total soil C.

Annual average total soil C with respect to time at sites RJA, US-IB1 and US-SO2.

The STOMP simulator was developed to predict non-isothermal hydrological flow and reactive transport in variably saturated subsurface environments. In STOMP, the water mass conservation equation balances the time rate of change of water mass within a control volume with the flux of water mass crossing the control volume surface. The nonlinear equations describing mass conservation are discretized spatially on structured orthogonal grids using the integral finite difference approach of Patankar (1980), which is locally and globally mass conserving. The equations are discretized temporally using first-order backward Euler differencing or implicit time stepping that is suitable for the solution of the equations that are numerically unstable (LeVeque, 2007). Newton–Raphson iteration is used to resolve the nonlinearities from the constitutive equations that relate the primary and secondary variables.

Comparison of water table depth

Detailed information regarding STOMP, such as the user's guide, theory guide
and code availability, can be found at

Comparison of water table depth

Figure 7a and b shows the oscillations of water table depth resolved at both
RJA and US-IB1; i.e., the oscillations between forcing cycles noted in the
original hydrology scheme in CLM4 are caused by the local water mass balance
error. Each cycle of atmospheric forcing at both sites has a length of 3 years. The GP method is successful at those two
sites. The little jumps in Fig. 7c and d are where the GP method is applied.
Both sites show higher total soil C predicted compared to Fig. 4a and b
because of the new flow model. The issue of the original groundwater model in
CLM4 might explain why it took so long (

We described a gradient projection method to further speed up the spin-up process based on the slow nature of soil organic C decomposition. Comparison between our approach and the modified accelerator approach showed that 20–69 % of simulation years can be reduced with our approach. While the approach was specifically evaluated using CLM4-CN, it can also be readily applied to other CN models in Earth system models. No matter what modification is made to improve the spin-up efficiency, a final spin-up is always needed to reach a converged solution due to disequilibrium caused by the modification. Our approach is especially useful when a new model formulation is proposed and a high-quality solution (small convergence threshold) is needed for a fair comparison.

Comparison of water table depth simulated by the original soil
hydrology scheme in CLM4 (solid line) and STOMP (dashed line) at site
RJA

In addition, we also found that the original numerical hydrology scheme, especially the water table calculation in CLM4, creates numerical oscillations in simulated water tables, leading to a challenge in achieving the common convergence criteria for soil C. To resolve the issue, we replaced the hydrological model using a flow model for variably saturated porous media. The new flow model caused an increase of about 10 % in computation time, but gives more accurate results that corrected the oscillation behavior of the original hydrological model. Comparing the total C predicted by the old and new flow models, we also see more C being predicted using the new flow model. Whether the prediction of more C is realistic depends on other factors besides the hydrology, so we have not attempted to evaluate the simulated C using observations. Nevertheless, a correct implementation of numerical schemes is always desirable for reducing uncertainty in model prediction.

The source code of CLM4.0 and STOMP can be requested through

This research has been accomplished through funding support from Pacific Northwest National Laboratory's Laboratory Directed Research and Development Program. We thank the North American Carbon Program Site-Level Interim Synthesis team, the Large Scale Biosphere-Atmosphere Experiment in Amazônia Model Intercomparison Project team, and the site investigators for collecting, organizing, and distributing the data required for this analysis. We thank the anonymous referee and Yiqi Luo for comments that improved the manuscript. A portion of this research was performed using PNNL Institutional Computing at Pacific Northwest National Laboratory. PNNL is operated by Battelle for the US Department of Energy under contract DE-AC05-76RL01830. Edited by: S. Arndt