1. Introduction
Each grid cell in coupled climate–terrestrial vegetation models is bound to include major spatial heterogeneities that span various orders of magnitude and influence the processes represented (Giorgi and Avissar 1997). The basic approaches accounting for subgrid cell heterogeneity in these models can be divided into two main categories. The composite (also named aggregated) approach computes land–atmosphere exchanges as a function of a single “representative” state of the grid cell (Giorgi and Avissar 1997). The approach allows for the coexistence of different plant functional types (PFT), but these all experience the same environmental conditions. The second category groups the mosaic, mixture, and statistical–dynamical approaches, which all account explicitly for the contribution of different portions of the grid cell. The mosaic approach subdivides the grid cell into tiles that respond independently to the same climate input variables so that soil temperature, for example, can differ among tiles (Avissar and Pielke 1989). Different criteria can serve to subdivide the grid cell into different tiles, and the approach is often applied in a nested framework: for example, first to separate vegetated from nonvegetated areas (e.g., lakes) and then to subdivide the vegetated area into different PFT tiles (Bonan et al. 2002). Given that each tile is homogeneous, the mosaic approach implicitly uses the composite approach, albeit at a finer scale (i.e., within a PFT tile instead of across the entire grid cell). The mixture approach is a variation of the mosaic approach, in which the latent and sensible heat fluxes from the different tiles of the same grid cell interact together (Koster and Suarez 1992). The statistical–dynamical approach aims to represent the full range of heterogeneity across the entire grid cell through continuous probability density functions (Avissar 1992). The resulting equations can be integrated analytically for very simplified representations of land–atmosphere interactions, but applying the approach to state-of-the-art Earth system climate models seems hardly feasible because of the practical (discretization of the probability density functions) and conceptual (modification of each probability density function in response to climate change) issues. In large-scale climate models, which currently do not resolve mesoscale subgrid circulation (Pielke et al. 2007), none of the previous approaches account for the specific location of the heterogeneous elements within the grid cell.
The previous approaches have been developed to represent static subgrid cell heterogeneity (Giorgi and Avissar 1997). Research on how to better account for the impacts caused by static heterogeneity is still ongoing, particularly in the context of topographic variations (Ke et al. 2013; Newman et al. 2014; Rotach et al. 2014; Subin et al. 2014). Yet the spatial heterogeneity of the land surface is dynamic, among others due to changes in vegetation cover resulting from human or natural causes. Over the last centuries, agriculture has led to substantial and permanent (or long term) changes in land cover (Ramankutty and Foley 1999). These deforestation events can adequately be simulated in climate models by permanently decreasing the grid cell fraction occupied by the forested tile(s) and increasing the agricultural tile(s) (e.g., Matthews et al. 2004). (Afforestation is more difficult to represent because the regrowing forest differs considerably from the rest of the forest for many years.) Other anthropogenic and natural disturbances, however, lead to more complex changes in spatial heterogeneity through the temporary removal of tree cover over extensive areas, followed by forest regrowth. Fire and logging are important examples of such events, which we define as stand-clearing disturbances. At the global scale, fire burns about 350 Mha yr−1 (Giglio et al. 2013) and emits 1.5–3 PgC yr−1 (Mieville et al. 2010; van der Werf et al. 2010), while logging impacts about 30 Mha yr−1 and removes between 0.2 and 0.9 PgC yr−1 from forests (Hurtt et al. 2006; Pan et al. 2011). Although the previous numbers include some treeless ecosystems and permanent changes (e.g., deforestation fires), fire and logging events that are followed by tree regrowth affect extensive areas of forest each year. These events have been shown to be consequential for climate through their impacts on carbon cycling and energy exchanges, including the possibly large effect of aerosols in the case of fire (Jacobson 2004; Jones et al. 2007; Unger et al. 2010; Lawrence et al. 2012; Ward et al. 2012; Jacobson 2014; Landry et al. 2015). Furthermore, stand-clearing disturbances play a major role in ecological succession and can trigger vegetation shifts (Turner 2010). For all these reasons, representing the transient impacts from stand-clearing disturbances would improve the simulation of vegetation dynamics, land surface properties, and exchanges of carbon, energy, and water with the atmosphere.
Stand-clearing disturbances introduce dynamic (i.e., year after year) changes in subgrid cell heterogeneity, even if the area of the forested portion of the grid cell stays constant. Representing these recurring events is thus challenging, particularly for the first generation of Dynamic Global Vegetation Models (DGVM). These DGVM resort to large-scale parameterizations that, although usually distinguishing various PFT, do not explicitly treat subgrid heterogeneity for the same PFT (Prentice et al. 2007; Quillet et al. 2010). Examples of first-generation DGVM include IBIS (Foley et al. 1996; Kucharik et al. 2000), TRIFFID (Cox 2001), LPJ-DGVM (Sitch et al. 2003), CTEM (Arora and Boer 2005), ORCHIDEE (Krinner et al. 2005), JSBACH (Raddatz et al. 2007; Brovkin et al. 2009), and CLM4 Oleson et al. 2010) (expansions of acronyms are available at http://www.ametsoc.org/PubsAcronymList). Second-generation DGVM are more amenable to the representation of subgrid heterogeneity because they simulate small-scale interactions among many individual plants (including for the same PFT) during the establishment, growth, and mortality phases (Fisher et al. 2010). Most of these second-generation DGVM consist of stochastic gap models and include Hybrid (Friend et al. 1997), LPJ-GUESS (Smith et al. 2001), and SEIB-DGVM (Sato et al. 2007). The other class of second-generation DGVM are composed of the ecosystem demography (ED) family, which simulates the mean behavior of an ensemble of stochastic individual-based models over a large spatial scale through a deterministic “scaling up” approximation (Moorcroft et al. 2001; Medvigy et al. 2009). Second-generation DGVM are, however, considerably less computationally efficient than large-scale parameterizations (Scherstjanoi et al. 2013), which become problematic for large-scale and long-term climate simulations. Consequently, the vast majority of Earth system climate models currently use first-generation DGVM.
Various efforts have recently been made to “scale down” first-generation DGVM, usually aiming to better represent demographic processes as well as the subgrid vegetation heterogeneity caused by land-use changes (Shevliakova et al. 2009), forest self-thinning, human-caused thinning, and stand-clearing harvest (Bellassen et al. 2010), generic stand-clearing disturbances (Scherstjanoi et al. 2013), or fire (Haverd et al. 2013; Yue et al. 2013; Haverd et al. 2014). These studies focused on improving the simulation of the carbon cycle directly, or indirectly through more realistic forest structural attributes (e.g., allometry), but did not consider the biogeophysical fluxes that also influence the climate. Reduced evapotranspiration and surface roughness from deforestation increases the local temperature almost everywhere on land, but the cooling impact from the increased albedo has a stronger effect globally and over most extratropical regions (Davin and de Noblet-Ducoudré 2010; Lee et al. 2011). In boreal regions, the albedo-induced cooling (warming) caused by permanent forest cover reduction (increase) has generally been found to be greater than the associated warming (cooling) from reduced (increased) terrestrial carbon storage (Betts 2000; Claussen et al. 2001; Bala et al. 2007; Bathiany et al. 2010; Bernier et al. 2011) or to at least have the same magnitude (Arora and Montenegro 2011; Pongratz et al. 2011). In the case of stand-clearing fire in boreal forests, studies have also concluded that the albedo cooling was stronger than the carbon warming (Randerson et al. 2006; O’Halloran et al. 2012).
The purpose of our study was to quantify how different representations of the subgrid cell dynamic heterogeneity caused by fire or logging stand-clearing disturbances affect the modeling of albedo in boreal forests. In section 2, we present the Heterogeneous Landscape Model (HLM) we developed for this purpose as well as the three generic approaches we assessed. Using the simpler HLM instead of state-of-the-art models facilitated the interpretation of results (section 3) and allowed us to obtain more generalizable outcomes. In section 4, we discuss the importance of accounting for the impacts of subgrid dynamic heterogeneity in climate models as well as the implications of the HLM results on the representation of this heterogeneity in existing models.
2. Methodology
2.1. Heterogeneous Landscape Model
2.1.1. Overview
The HLM reproduced the computation of solar radiation exchanges used by many climate models (i.e., as the mean value of fluxes from the vegetated and “canopy gap” portions of the grid cell), but the HLM used empirical data for the different variables required instead of estimating them through prognostic simulations. We modeled a grid cell representative of the black spruce [Picea mariana (Mill.) B.S.P.] boreal forest in Canada, with albedo values that were the same from one year to another but did vary daily. We compared the grid-level mean annual reflected solar radiation (
Spatial heterogeneity in the grid cell came from stand-clearing disturbances only, which were either logging or fire. Each year, a stand-clearing disturbance event affected a constant proportion of the grid cell 
2.1.2. Main equations
2.1.3. Approaches assessed
The three approaches could all, conceptually at least, be implemented into real climate models, so we first present them in general terms before providing more details regarding their implementation in the HLM. The first approach consisted of adding one specific tile for each new patch created by a stand-clearing disturbance event and keeping track of all patches indefinitely. This Full approach would provide the most accurate results regarding subgrid dynamic heterogeneity because all land–atmosphere exchanges would be computed for each patch individually but would eventually become prohibitive in computing time due to the ever-increasing number of tiles. The second approach consisted of immediately merging any newly created patch with the rest of the grid cell, thereby leading to a single tile. This Average approach would have the smallest computing burden but would entirely neglect subgrid cell dynamic heterogeneity.
The third approach consisted of keeping track indefinitely of all patches created by stand-clearing disturbance events, but grouping these patches into a much smaller number of tiles for the evaluation of land–atmosphere exchanges. In this Landscape Fractions approach, the patches having a similar 
Conceptual representation of the Landscape Fractions approach for 10 patches (nine ellipses created by fire events and one patch grouping the yet undisturbed part of the grid cell; the specific location of each patch in the grid cell is not accounted for). (left) Each of the 10 patches has its own 
Citation: Earth Interactions 20, 5; 10.1175/EI-D-15-0022.1
The Landscape Fractions would be intermediate between the Full and Average approaches, both in precision (as illustrated in section 3 with the HLM) and in computing time, because land–atmosphere exchanges would be computed independently for 
In the context of albedo computation with the HLM, the three approaches differed only in their evaluation of 
2.1.4. Input data
Computing RSOL in the HLM [Equation (1)] required different sets of input data on 
For the turnover-related parameter f, we used values of 0.980, 0.985, and 0.990. 
Time-varying HLM input data. (a) Daily incoming solar radiation at the ground level. (b) Daily albedo for treeless areas. (c) Daily albedo for areas with tree cover. (d) Net primary productivity as a function of patch age for 
Citation: Earth Interactions 20, 5; 10.1175/EI-D-15-0022.1
The lower, representative, and upper values we used for 
Citation: Earth Interactions 20, 5; 10.1175/EI-D-15-0022.1
2.2. List of simulations
For each disturbance type (logging or fire), the total number of combinations of the different sets of input data is 
Possible values and level for the HLM input variables.
The three level-1 variables (
For each disturbance type, we performed the 51 simulations (27 for level-1 variables and 24 for level-2 variables) for 1) the Full approach, 2) the Average approach, and 3) the Landscape Fractions approach using 
3. Results
3.1. HLM results
For the same level of disturbance 
Henceforth, 
Figure 4 shows 
Boxplot of the difference in 
Citation: Earth Interactions 20, 5; 10.1175/EI-D-15-0022.1
The previous results illustrate the potential of the Landscape Fractions approach to substantially improve the RSOL results compared with the Average approach, even for a small 
Difference in 
Citation: Earth Interactions 20, 5; 10.1175/EI-D-15-0022.1
The robustness of the improvement brought by the Landscape Fractions approach was valid not only for the most representative combination of input data but for all 51 simulations. Furthermore, the domains (in terms of 
3.2. Limitations of the HLM
Our computation of 
Our model was dynamic at the scale of individual patches, but we performed the computations for a grid cell in equilibrium with constant disturbance regimes. Fire, however, does fluctuate markedly over long (Bergeron et al. 2010) and short (Stocks et al. 2002) time scales. The strict enforcement of even-aged forest management over various decades also seems highly unlikely. Nonconstant disturbance regimes should thus prevent any grid cell from ever reaching a perfect equilibrium. Nevertheless, the out-of-equilibrium grid cell can be conceived of as being in a transition between conditions similar to the ones we modeled. We therefore consider that the main HLM outcomes likely apply to out-of-equilibrium conditions, although the specific numerical results would certainly change.
Except for the spatial heterogeneity caused by logging or fire, we assumed that all factors were uniform in space as well as constant or periodic over time. The main reasons for this simplification were to keep results intelligible and be able to solve analytically the carbon dynamics of the HLM. Yet even in the state-of-the-art climate models for which our study is relevant, many abiotic (e.g., soil properties) and biotic (e.g., traits variation among and within tree species) terrestrial heterogeneities remain underrepresented (Moorcroft 2006).
Instead of simply using the most representative values, we performed a range of simulations to show that the main HLM outcomes were robust to changes in the specific values of input variables. Nonetheless, the choice of these values still involved some uncertainty. In the supplemental material, we further discuss the uncertainty related to the 
3.3. Summary
The HLM results presented previously led to the following outcomes. 1) On average (both mean and median values), the difference between fully accounting for the subgrid dynamic heterogeneity and performing computations based on a single average value was ~3 W m−2, reaching values higher than 5 W m−2 in 8 out of the 51 comparisons we performed. 2) The intermediate Landscape Fractions approach considerably improved the results compared with the Average approach, with an accuracy that increased quickly with the user-selected 
4. Discussion
4.1. Consequences of subgrid cell dynamic heterogeneity
4.1.1. Net radiative forcing from stand-clearing disturbances
The HLM results showed that the approach used to account for subgrid dynamic heterogeneity can have a substantial impact on the estimation of 
To estimate the magnitude of this potential error, we compared the radiative forcing of carbon emission and albedo change resulting from fire simulated in the HLM (see supplemental material 3.1 for more details). We assumed that the entire region composed of boreal ecozones in Canada was either completely free from any disturbance (the Old Growth scenario) or in equilibrium with a spatially random fire disturbance regime having 
Radiative forcing from fire simulated in the HLM (W m−2).
Simulating boreal forests stand-clearing disturbances in climate models through the Average approach could thus bring a bias toward a warming impact. Although the estimation of radiative forcing with the HLM was relatively crude, we consider that the results obtained would likely have the same order of magnitude with state-of-the-art methods. Indeed, the results were similar to the ones of Randerson et al. (2006) when applying the HLM to the specific fire event they studied (see supplemental material 3.2).
4.1.2. Additional impacts
The albedo estimated for snow-covered conditions varies considerably among climate models, which strongly affects the simulated strength of the snow-albedo feedback for current conditions and future projections (Qu and Hall 2014). Despite having consequential impacts on future warming in the Northern Hemisphere, this source of uncertainty has apparently not been reduced over the last decade (Qu and Hall 2014). Essery (2013) concluded that the main cause of this uncertainty does not reside in the mathematical form of the vegetation–snow albedo algorithms but rather in the simulated distribution of vegetation cover and/or the specific values of albedo parameters. The inadequate simulation of the south-to-north decline in tree cover is an example of such shortcomings in current climate models (Loranty et al. 2014). Our results suggest that even if the vegetation distribution and the albedo parameters were appropriate, climate models would need to properly account for the subgrid heterogeneity created by stand-clearing disturbances in order to precisely estimate albedo, at least over disturbance-prone portions of boreal forests.
We suggest that the carbon cycle could also be affected by the approach taken to implement stand-clearing disturbances in climate models. Using the CTEM DGVM coupled to the CLASS land surface scheme (Arora and Boer 2005), Li and Arora (2012) examined the differences in simulated energy and carbon exchanges between the mosaic and composite approaches under the static coexistence of two PFT; for the simulations performed in boreal or temperate locations, the grid-level differences in the major energy fluxes between the two approaches were all below 10%, yet the differences in the equilibrium values of carbon-related variables (
4.2. Implications for current climate models
4.2.1. Representing subgrid cell dynamic heterogeneity
Many models referred to as DGVM actually perform only a subset of all the computations required for full integration within climate models (Prentice et al. 2007; Quillet et al. 2010). For the sake of clarity, we henceforth designate as Dynamic Vegetation–Land Surface Models (DVLSM) the process-based models that compute the land–atmosphere exchanges of energy, carbon, water, and momentum required by climate models, while allowing vegetation to change dynamically (both the spatial distribution and the seasonal cycle) as a function of the climatic conditions. In other words, DVLSM correspond to the union of first- to third-generation land surface schemes, as defined by Pitman (2003), with dynamic vegetation. To the best of our knowledge, only one Earth system climate model currently uses a DVLSM (i.e., SEIB–DGVM) that includes a second-generation treatment of vegetation interactions by allowing many individuals of the same PFT in a single grid cell. We therefore focus on DVLSM using a first-generation representation of vegetation before coming back to this exception below.
Few studies in Earth system climate models have directly addressed the impacts of stand-clearing disturbances. Lawrence et al. (2012) evaluated the climatic impacts of logging (along with long-term changes in land cover) at the global scale, from 1850 to 2100. They represented subgrid dynamic heterogeneity through the Average approach within a PFT-tiling nested framework, resorting to MODIS and potential vegetation data to derive transient PFT coverage datasets. Landry et al. (2015) estimated the effect of different fire scenarios on global carbon stocks and the global atmospheric surface temperature over the 2015–2300 period. They once again represented subgrid dynamic heterogeneity through the Average approach, within a PFT-tiling framework based on simulated competition. According to the HLM results, these two studies may thus have underestimated the albedo-induced cooling caused by stand-clearing disturbances, at least over boreal forests. As far as we can tell, all first-generation DVLSM representing fire and/or logging in Earth system climate models (e.g., ORCHIDEE, JSBACH, and CLM4) resort to the Average approach, except LM3V. Unfortunately, the Average approach misrepresents the transient biogeophysical impacts resulting from temporary loss of forest cover as we showed for 
As mentioned above, SEIB–DGVM is, to our knowledge, the only second-generation DVLSM that has been included in an Earth system climate model (Watanabe et al. 2011). As many stochastic gap models, SEIB–DGVM simulates the coexistence of various PFT in a single tile of ~0.1 ha (Sato et al. 2007). Accurately representing within such a small area undisturbed and various burned or logged patches does not seem possible (e.g., a few surviving trees could shield most of the tile’s ground). Due to the stochastic nature of modeled processes, gap models need to simulate various replicates, typically 10 or more, for each grid cell. Some of these replicates could be entirely burned or logged in order to accurately represent the impacts from stand-clearing disturbances. This would however require increasing the number of replicates following each disturbance event, thereby increasing the computing time that is already an issue with gap models (Scherstjanoi et al. 2013). Consequently, accounting for the whole age-class distribution of patches created by stand-clearing disturbances seems challenging in SEIB–DGVM and similar DVLSM, especially in slow-growing boreal forests where decades can elapse before canopy closure. Incidentally, we note that the SEIB–DGVM built-in fire module (Sato et al. 2007) was not activated for the CMIP5 simulations (Arora et al. 2013). We also note that the ED model, which now computes for each patch the land–atmosphere fluxes required from DVLSM (Medvigy et al. 2009), basically implements the Full approach by default, except that patches are merged together to minimize computing time when they become sufficiently similar (Fisher et al. 2010). However, the ED model has apparently not been integrated yet into an Earth system climate model or even used for global simulations as a stand-alone DVLSM, likely reflecting its computing demand.
Finally, increasing the spatial resolution of DVLSM would generally be insufficient to resolve subgrid dynamic heterogeneity in global simulations; for example, individual fire events of less than 500 km2, each covering less than 7% of a 1° × 1° grid cell (centered on 50°N), are responsible for 55% of the mean area burned in the boreal ecozones of Canada (Stocks et al. 2002).
4.2.2. Implementing the Landscape Fractions approach
Given the potential biases of the Average approach and the high computing time of the Full approach, our HLM results suggest that approaches similar to the Landscape Fractions could constitute an interesting middle point for the representation of stand-clearing disturbances in first-generation DVLSM. Assigning patches to specific LF and updating their status each month or year should involve a small computing burden so that the additional time required by the Landscape Fractions approach should increase roughly linearly with 
5. Conclusions
Stand-clearing disturbances lead to the temporary removal of tree cover over extensive areas each year but are then followed by forest regrowth. Given that these events are only starting to be the subject of dedicated studies in Earth system climate models, the timing seems appropriate for a thorough consideration of the consequences from different approaches that could be implemented to account for the resulting subgrid cell dynamic heterogeneity.
In this study, we specifically assessed three such approaches. The Full approach, which consisted of creating an individual tile for each new patch created by a disturbance event, would result in the most accurate representation of subgrid dynamic heterogeneity but would quickly become prohibitive for long simulations. The Average approach, which consisted of immediately merging any new patch with the rest of the (forested part of the) grid cell, would be a simple and time-efficient strategy but would actually neglect subgrid dynamic heterogeneity. The third approach that we assessed aimed to be intermediate both in accuracy and computing time. The basic idea behind this Landscape Fractions approach was to temporarily group the patches having a similar proportion of tree cover 
We illustrated the consequences of the three approaches in the Heterogeneous Landscape Model (HLM) we developed for this study by computing the annual reflection of solar radiation (
We do not believe that precisely accounting for subgrid cell dynamic heterogeneity should be a priority under all circumstances. When the objective is to compare widely diverging scenarios of anthropogenic emissions, the biases resulting from the Average approach are likely acceptable because they probably cancel each other to a substantial extent among scenarios and are anyway much smaller than other sources of uncertainty, for example, the simulation of clouds and their interactions with aerosols. However, when the objective is to directly assess stand-clearing disturbances, neglecting the resulting subgrid dynamic heterogeneity might be misleading, particularly when studying the net climatic impact from fire or logging in boreal forests.
Acknowledgments
We thank David Price for helpful discussions about the development of the HLM and for providing us with the monthly mean incoming solar radiation data. We also thank Günther Grill, Gabriela Jamett, and Bano Mehdi for their comments on a previous version of the manuscript as well as Damon Matthews and Julia Pongratz for discussions about the Landscape Fractions approach. Thoughtful comments from the two reviewers have further helped us improve the manuscript. J.-S. L. was supported by an NSERC postgraduate scholarship (CGSD) and an FRQNT doctoral scholarship (B2).
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