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Introduction
Land-use change has received a lot of attention as the second most important human-caused perturbation of the global carbon cycle, recently estimated to release an amount of 0.9 ± 0.5 GtC yr −1 of CO 2 to the atmosphere (Le Quéré et al., 2013).Most land-use change is due to human-caused tropical deforestation.A better quantification of impacted biomass carbon stocks (Baccini et al., 2012), as well as forest loss area (Hansen et al., 2010;Harris et al., 2012), helps to reduce uncertainties in the land-use change CO 2 flux.Differences in land-use flux estimates between studies (Houghton et al., 2012) are also due to different system boundaries (e.g., the inclusion of soil carbon dynamics after a change in landuse, or the account for regrowth after deforestation) and to different definitions of the human perturbation of ecosystems (e.g., the inclusion of shifting agriculture and forest degradation in the land-use change CO 2 flux) (Houghton, 2010).Yet, in this paper we show that a more insidious source of discrepancy in estimates lies in the definition of "emissions from land-use changes" as a component of the net land-toatmosphere CO 2 flux.In order to investigate and quantify this definition-related uncertainty, we need to go back to the T. Gasser and P. Ciais: ELUC definition equations of the global carbon budget, and to discuss the partitioning of the net carbon dioxide flux from the terrestrial biosphere to the atmosphere.
It has become "usual" to define the instantaneous change in atmospheric CO 2 concentration (noted [CO 2 ]) as being the sum of four fluxes (Canadell et al., 2007;Denman et al., 2007).In this approach, two of those fluxes are emissions: one caused by fossil fuel burning and other secondary industrial processes (EFF), and another by land use, land-use change and forestry (ELUC).The two other fluxes are natural responses of the carbon cycle.These responses have been generally negative since the beginning of the industrial era, i.e., they have been removing carbon dioxide from the atmosphere, acting as two sinks of CO 2 : the oceanic sink (OSNK) and the land sink (LSNK).Thence, the instantaneous global carbon budget follows the equation: In Eq. ( 1), EFF and OSNK describe CO 2 exchanges between the atmosphere and two different reservoirs (geological and oceanic reservoirs), while ELUC and LSNK are two terms used to describe exchanges with only one reservoir: the terrestrial carbon reservoir.
The partition of the net land-to-atmosphere CO 2 flux (noted NetFlux hereafter) between ELUC and LSNK aims to separate the direct anthropogenic effect of land-use activities (ELUC; mainly emissions through tropical deforestation) and the indirect effect of all anthropogenic activities (LSNK; the natural response of the terrestrial biosphere, expected to be a sink driven by the combined effect of regional climate change, N-fertilization and global CO 2 -fertilization).However, exact definitions of ELUC and consequently of LSNK vary among studies.For instance, in studies on the global carbon budget, based on observations (e.g.Denman et al., 2007;Khatiwala et al., 2009;Le Quéré et al., 2009), the natural response of the terrestrial biosphere, LSNK, is calculated as the residue in Eq. (1), knowing d[CO 2 ]/dt, EFF, ELUC and OSNK.In global vegetation modeling studies, the problem is opposite: most models cannot make the partition between ELUC and LSNK.When they do, they may use definitions of ELUC that are inappropriate for intercomparison.
The goal of this study is to provide a rigorous mathematical framework suitable for defining different terms of the net land-to-atmosphere CO 2 flux, so as to be able to compare estimates from different modeling approaches as well as from observations.First, we break down the net land-toatmosphere CO 2 flux into four components, thanks to three idealized experiments illustrated in Fig. 1.Second, we combine those four components to propose three definitions of "emissions from land-use changes" (ELUC) and provide examples of published studies falling into each definition.The aim is to clarify the definitions of ELUC encountered in the literature, and to present a framework for designing model simulations so that one can compare simulated estimates of ELUC from different studies without the bias due to the choice of different and incompatible definitions.The mathematical aspect of this study ensures that the results are exact and applicable for any approach used to calculate that kind of emissions.For the purpose of illustration, however, we give numerical applications so as to roughly quantify definitionrelated differences in ELUC, using the OSCAR v2 carbon cycle model (Gasser et al., 2013, see also Appendix B).This model has previously been shown to perform satisfactorily in reproducing recent estimates and trends in the global carbon budget, as simulated fluxes for global land and ocean are within uncertainty ranges calculated by Denman et al. (2007) and Le Quéré et al. (2009).In our first (thought) experiment, we consider a historical simulation without any land-use activity.In this experiment, the terrestrial biosphere is disturbed only by three indirect effects of human activities: (i) the increase in atmospheric CO 2 ; (ii) the increase in nitrogen deposition; and (iii) the change in climate resulting from radiative forcing of greenhouse gases and aerosols produced by diverse human activities.The first two perturbations have a fertilization effect on the productivity of the biosphere, enhancing the CO 2 removal, while the third one leads to regional responses of various signs of CO 2 removal (Denman et al., 2007).We call this indirect perturbation of the carbon balance of the terrestrial biosphere the "CCN" perturbation, for "carbon, climate and nitrogen", noting that it conceptually includes other perturbed processes affecting the terrestrial carbon cycle such as the effect of elevated O 3 (Sitch et al., 2007), altered P cycling (Goll et al., 2012) or SO 4 aerosols deposition on wetland plants (Gauci et al., 2004).It should be noted that here we consider the CCN perturbation to be exogenous to the simulation, whereas the CCN perturbation actually impacts atmospheric CO 2 and then climate (and ultimately the CCN perturbation itself through a feedback loop).However, taking an exogenous or endogenous CCN perturbation does not change the mathematical demonstrations that follow.In this simulation, at each time t, the net land-toatmosphere CO 2 flux over a geographical and biological point (g, b) can be expressed as where F * (g, b) is the extensive net flux over an area S(g, b) typically expressed in gC yr −1 , and f * (g, b) is the intensive (areal) net flux typically expressed in gC m −2 yr −1 .In this section, since we consider only the CCN perturbation, all fluxes are written with the superscript * that is used to describe an equilibrium state relative to the LUC  and f * (g, b) are positive if they correspond to an emission of CO 2 to the atmosphere.Depending on the model used, for instance a box model, an earth system model of intermediate complexity (EMIC) or an earth system model (ESM), g can be a grid cell, a country or even the whole globe in very simple models; while b can be a plant functional type (PFT), a specific biome or even the global "mean" vegetation in the simplest case.Each variable can be broken down into a preindustrial value (subscript 0) and a perturbation at t since preindustrial times (prefix ).Hence: Note that the area S has no perturbation term since we made the hypothesis of no land-use change in this section.Under the hypothesis of a preindustrial equilibrium of the carbon cycle, the net carbon flux is equal to zero and the carbon stock of each couple (g, b) remains unchanged; thus, the preindustrial terms F * 0 (g, b) and f * 0 (g, b) are equal to zero.Consequently, the global net land-to-atmosphere CO 2 flux at time t is One could write f = + ρ − η, where is the areal emissions due to sporadic natural disturbances such as insect outbreaks and wildfires, ρ is the heterotrophic respiration and η is the net primary productivity (NPP).Under present-day conditions, it is generally admitted that net primary productivity is higher than during preindustrial times because of fertilization effects of N deposition and increased atmospheric CO 2 (i.e., η * > 0); that heterotrophic respiration is a delayed response to increased NPP (i.e., ρ * (t) = η * (t − τ d ) < η * (t), where τ d > 0 is the delay); and that there is no significant change in sporadic activities since preindustrial times (i.e., CO 2 .Yet, this example ignores two processes: (i) the natural variability of climate; and (ii) the natural long-term migration of vegetation induced by climate change and CO 2 (e.g.Cramer et al., 2001).Section 4 includes a discussion on how these two effects can be incorporated in our definition framework, but doing so in the following demonstrations would unnecessarily complicate the notations.

Simulation with land use at preindustrial times (Exp. 2: LUC perturbation)
There are two types of land-use activities.The first type regroups activities that do not affect land-cover (i.e., no change in S(g, b)) while the second type corresponds to activities that come with land-cover change (i.e., an area conversion δS from (g, b 1 ) to (g, b 2 ) typically expressed in m 2 yr −1 ).
The first land-use type encompasses what IPCC calls "land use" and "forestry" while the second formally corresponds to "land-use change" in the "LULUCF" terminology (Watson et al., 2000).Houghton (2010) provides a detailed discussion of land use and land-use changes, and on the anthropogenic activities that are generally included in the definition.In the following, land-use activities occurring with landcover change, as well as activities occurring without landcover change, are accounted for; the later being represented by a land conversion from one biome to itself (i.e., a conversion δS from (g, b 1 ) to (g, b 1 )).Thence, all land-use activities (be it with or without land-cover change) induce a local perturbation of the terrestrial carbon cycle that leads to a net emission or absorption of CO 2 over time.A long time after the perturbation, we suppose that a new equilibrium is reach where the net land-use-induced CO 2 flux has returned to zero.We call this perturbation the "LUC" perturbation.
In this second experiment, we consider that the LUC perturbation is occurring under preindustrial conditions, i.e., with a CCN perturbation equal to zero.As previously, we suppose that this CCN perturbation is exogenous, remaining equal to zero at all times, despite the actual effect of the LUC perturbation over the CCN perturbation (through CO 2 emissions and then changes in atmospheric CO 2 and climate).Under the hypothesis of no CCN perturbation, all perturbation variables (the ones with the -prefix, in our notation) are equal to zero, except for the area S affected by land-use changes and that we break down into where S − is the cumulative destroyed area of primary ecosystems, and S + is the cumulative created area of secondary ecosystems, of b over g since preindustrial.Both quantities are positive but not necessarily equal, since they do not come from the same land conversion (the first is due to a conversion from (g, b) to (g, b n ), while the second is due to a conversion from (g, b m ) to (g, b); hence, they are equal only for land-use activities that do not induce land-cover change).Moreover, the created areas are not at equilibrium for their CO 2 net flux and thus we will monitor their status as a cohort of disturbed (i.e., transitioning) ecosystems since the time of their "creation".This is called the "book-keeping" approach.
To do so we introduce the vector notation for cohorts of transitioning ecosystems of different age classes τ : where δS +,τ (g, b) is the area of transitioning b over the geographic point g that was created τ years before t and δS + (g, b) is the vector that describes all the values δS +,τ (g, b) along the τ axis.We note that here all δS +,τ (g, b) for τ > t are equal to zero, as no transition occurred before that date.As a consequence, we can further express the total created area of secondary ecosystems S + (g, b) at time t: The notation for cohorts is extended to all variables associated with the transitioning ecosystems, so that the cohort of net areal land-to-atmosphere CO 2 fluxes that corresponds to As a perturbation becomes old (i.e., as τ increases), a disturbed secondary ecosystem tends to become fully transitioned to a new state of the undisturbed equivalent ecosystem (g, b).Note that some ecosystems (like croplands), because of continual anthropogenic perturbations, may never really reach this "undisturbed" state, but we can still define an hypothetical -idealized -undisturbed state.Thus, mathematically: where f * is the value of f at equilibrium.Note that there is no reason for any f τ to equal f * before the termination of the transition.Following the illustration of f * given in the previous section, we can write that f = + ρ − η + w, where , ρ and η are the same fluxes as in Sect.2.1, and w is the CO 2 flux of decaying products (usually wood) formed at the time of the land-use change activity (with w > 0 and w * = 0).Despite not being part of local CO 2 fluxes, we keep accounting for w into the net flux f (g, b) because it is tied to the initial land-use perturbation at the point (g, b).As a consequence, our formalism does not consider the geographic location of harvested wood, or food, products (e.g., displacement and/or trade).
The local net land-to-atmosphere CO 2 flux in this second experiment is expressed by 1 9 0 0 1 9 1 5 1 9 3 0 1 9 4 5 1 9 6 0 1 9 7 5 1 9 9 0 2 0 0    and right panel shows the simulated fluxes when the LUC perturbation is stopped in 2005 but the CCN perturbation follows the RCP 8.5 scenario.ELUC 0 is the term driven only by the LUC perturbation (plain black line) and ELUC is the term due to the effect of the CCN perturbation over the LUC perturbation (dashed black line).Conversely, LSNK 0 is the term driven only by the CCN perturbation (plain grey line) and LSNK is the term due to the effect of the LUC perturbation over the CCN perturbation (dashed grey line).The net land-to-atmosphere flux is the sum of the four components (red line).Note that the scale for LSNK 0 and NetFlux is one-fourth of the scale for the three other fluxes.
disturbed lands (10) where the operation • is the multiplication term-by-term (a scalar product between two orthogonal vectors) of the cohort of transitioning areas and the cohort of net areal CO 2 fluxes.The first term of Eq. ( 10) is the net CO 2 flux over undisturbed lands, while the second term is the net flux over disturbed (i.e., transitioning) lands.Because we made the hypothesis of a preindustrial equilibrium in the first experiment, f * 0 is also equal to zero, and thus the global land-to-atmosphere flux is given by Cohorts are mathematical representations of the physical temporality of the land-use perturbation.Since an ecosystem disturbed by land-use change needs a few decades to meet its new equilibrium, it is necessary to keep track of fluxes and stocks legated by previous perturbations to make accurate estimations of CO 2 fluxes.The so-called "legacy" of emissions from land-use change (Jones et al., 2010;Houghton, 2010) is the concrete illustration of this physical property.ELUC at one time t are partially due to the perturbation at t, but also due to all perturbations before that time t.This has been illustrated, e.g. by Pongratz et al. (2009), and can be visualized on the right-hand panel of Fig. 2, where land-use activities are stopped after the year 2005 in the OSCAR v2 model (whereas legated emissions do not go to zero immediately after this date).

Historical simulation with land-use (Exp. 3: CCN + LUC perturbations)
Our reasoning applies at the level of each couple (g, b), but we will drop the (g, b) notation for clarity in the following, bringing it back only when necessary.
In this third experiment, we consider an historical simulation with both CCN and LUC perturbations.Note that it is the only experiment that is realistic, as the two previous ones ignored one of the two perturbations.The local net land-toatmosphere flux is deduced from Eqs. ( 3) and (10) as We make the same assumption of preindustrial equilibrium as in previous sections (i.e., f * 0 = 0), and we integrate the flux F over all couples (g, b): In Eq. ( 13), one can identify the first two terms, corresponding to Eq. (4) in Sect.2.1 and to Eq. ( 11) in Sect.2.2 (i.e., the fluxes due to the separate CCN and LUC perturbations) plus a term (noted NetFlux CCN×LUC ) representing the interactions between the CCN and LUC perturbations.This term is zero in the absence of at least one of the two perturbations.
T. Gasser and P. Ciais: ELUC definition 2.4 The four components of the net land-to-atmosphere flux

Equations
Following Eq. ( 10), the partition between undisturbed and disturbed lands in the local flux F (g, b) of Eq. ( 12) can be expressed as In this equation there is no separation of the CCN and LUC perturbations over disturbed lands.For old "almost transitioned" ecosystems where the LUC perturbation is becoming negligible, the simulated net flux over disturbed lands is dominated by the CCN perturbation.To separate the two effects, we isolate in Eq. ( 14) the term representing the CCN perturbation that would occur in hypothetical fully transitioned ecosystems of the same area as the cohort: . Subtracting this term from the "disturbed lands" part of Eq. ( 14) and adding it to the "undisturbed lands" one leads to Now, the first term (left) of Eq. ( 15) is mainly driven by the CCN perturbation and the second term (right) is mainly driven by the LUC perturbation.In Eq. ( 15), the two main fluxes due to the separate CCN and LUC perturbations are present, but this time we conceptually split the coupling term NetFlux CCN×LUC of Eq. ( 13) into two sub-terms.This provides the four generic components of the net land-to-atmosphere CO 2 flux when both CCN and LUC perturbations occur.Figure 1 shows a conceptual diagram of the three experiments we used to break down the net land-to-atmosphere flux into these four components.To follow the formalism developed in previous sections, we propose the following notations and formulations: ELUC 0 is the flux due to the LUC perturbation only (experiment 2) and LSNK 0 is the flux due to the CCN perturbation only (experiment 1).The two terms with the delta symbols are the two sub-fluxes due to the coupling of CCN and LUC perturbations.ELUC (resp.LSNK) is named on the basis of its formulation that is analogous to the one of ELUC 0 (resp.LSNK 0 ).
Conceptually, ELUC 0 are the emissions from land-use change that would have been observed if land-use change activities occurred under preindustrial climate, CO 2 and nitrogen conditions.ELUC are the extra-emissions from landuse change due to the CCN perturbation that has been affecting transitioning ecosystems (e.g., CO 2 -and N-fertilizations have made carbon stocks larger, and global warming has changed the rate of heterotrophic respiration).LSNK 0 is the global land sink that would have been observed under preindustrial land-cover (i.e., without LUC perturbation).LSNK is the altered land sink due to land-cover change, i.e., due to changes in areas of the different ecosystems when compared to the preindustrial ones.This last term was called "amplification effect" by Gitz and Ciais (2003) and "loss of sink capacity" by Pongratz et al. (2009); it is equal to zero for landuse activities that are not associated with land-cover change (see Sect.A4).

Simulation with OSCAR v2
Now we illustrate the magnitude of the four components of NetFlux CCN+LUC using a numerical model of the global carbon cycle, OSCAR v2.The simulation by OSCAR v2 of the four fluxes is shown in Fig. 2. The left-hand panel shows the results of the historical simulation and the right-hand panel the results when the CCN perturbation follows the RCP 8.5 scenario (Riahi et al., 2011) without LUC perturbation after the year 2005 (i.e., no new land conversion nor biomass harvest).The two main fluxes ELUC 0 and LSNK 0 , due to the LUC and CCN perturbations separately, behave as expected over the historical period.ELUC 0 is positive because of deforestation being more important than afforestation or reforestation, although it has declined since the beginning of the 1990s (Friedlingstein et al., 2010).LSNK 0 is a sink driven in OSCAR v2 mainly by CO 2 -fertilization but also affected by climate variability.The legacy of ELUC 0 is negative a few years after 2005 and it (slowly) tends toward zero when all transitioned ecosystems have recovered.In our model, the negative sign of these "committed emissions" is due to biomass regrowth (see also Houghton, 2010) induced by significant agricultural and pastoral abandonment in the 1990s (i.e., conversions from croplands or pastures to forests).Contrarily, LSNK 0 keeps on increasing (in absolute value) under the RCP 8.5 CCN perturbation as CO 2 atmospheric concentration is also increasing.The stagnation of the sink after 2080 is due to the carbon-climate feedback on the terrestrial biosphere in OSCAR v2 (Gasser et al., 2013) with the negative effect of warming climate countering the positive effect of CO 2 -fertilization, and thus reducing the land sink.
ELUC is the term of the net land-to-atmosphere flux that quantifies the impact of the CCN perturbation over the LUC perturbation.It is roughly proportional to ELUC 0 , with a proportionality factor equal to the ratio of change in carbon areal density to preindustrial carbon areal density (i.e., c * /c * 0 , see Sects.A1 and A2).In our simulation, the estimated value of ELUC is about +10 % that of ELUC 0 over the 1980-2000 period.However, the behavior of this term is not exactly similar to ELUC 0 .Contrary to ELUC 0 , the short-term legacy of ELUC after 2005 is positive.This result may be model dependent, and it is explained in OSCAR v2 by two effects.First, the change in biomass carbon density is faster than the change in soil carbon density (i.e., c * /c * 0 is greater in biomass than in soils), which implies that the relative role of dead biomass in the committed emissions is greater in ELUC than in ELUC 0 .Second, global warming induces an increase in heterotrophic respiration rate, which in turn leads to faster carbon soil emissions than it would have been under preindustrial CCN conditions (see Sect.A2 for detailed equations).
The last flux illustrated in Fig. 2 is the "amplification effect"/"altered sink capacity", LSNK.We can see that it is positive, mainly because deforestation causes a loss of sink capacity compared to leaving in place pristine forests.
LSNK can be seen as the portion of LSNK 0 that is "not realized" because it is affected by land-cover change, thus the two fluxes are roughly proportional (with a proportionality factor equal to S/S 0 , see Sect.A4).LSNK has a temporal profile similar to LSNK 0 .Indeed, LSNK becomes significant after 1950 (i.e., when atmospheric CO 2 starts to increase significantly), and it is strongly affected by climate variability.When land-use activities are stopped (after 2005) it does not tend toward zero.LSNK is about −15 % of LSNK 0 in the 1990s and later.Over the period 2005-2100, LSNK increases as CO 2 -fertilization strengthens the potential sink, and consequently the loss of potential sink.The causes of the stabilization after 2080 are exactly the same as for LSNK 0 , and are ultimately dependent on the model's sensitivity to CO 2 , climate change, and other environmental changes (the default setup of OSCAR v2 having a relatively high sensitivity to CO 2 increase, Gasser et al., 2013).

Three possible definitions of ELUC and LSNK
In this section, we consider both CCN and LUC perturbations.Irrespective of the chosen definition of ELUC and LSNK, mass conservation implies that the sum of the two fluxes must be equal to the net land-to-atmosphere flux: Net-Flux = ELUC + LSNK.NetFlux is defined globally by the difference between fossil fuel emissions and ocean uptake of anthropogenic CO 2 plus the atmospheric CO 2 growth rate (e.g.Canadell et al., 2007;Le Quéré et al., 2013).Thus, using one definition for one of the two fluxes implies a nonambiguous definition for the other, and users must take care not to use two inconsistent definitions for ELUC and LSNK.

Definition 1: simulations with/without land use
A first choice (called definition 1, and noted def 1) underpinning the calculation of ELUC and LSNK with terrestrial ecosystem models is to compare the simulated land-toatmosphere flux of two model experiments: one done with LUC and exogenous CCN conditions, and another done without LUC and the same CCN conditions.Then, ELUC is the difference between the first simulation and the second one, and LSNK is necessarily equal to the result of the second simulation (because of the mass conservation constraint).The local flux calculated by the first model experiment is given by Eq. ( 13) while the one calculated by the second experiment is given by Eq. ( 4).Hence and The choice of definition 1 is usually (although implicitly) made with ecosystem models that do not include explicit land-use cohorts (e.g.McGuire et al., 2001;Piao et al., 2009;Pongratz et al., 2009).With def 1, the flux due to the crossinteractions between CCN and LUC is fully accounted for as part of "emissions from land-use change" (i.e., in the ELUC term).

General definition 2
A second definition for ELUC and LSNK (definition 2, def 2) is suggested by Eq. ( 10), and subsequently by Eq. ( 14).One can consider that ELUC is the net land-to-atmosphere flux over disturbed lands, and that consequently LSNK is the net flux over undisturbed lands.The resulting definitions are and The few vegetation models that have an explicit treatment of cohorts usually use this definition (e.g.Shevliakova et al., 2009) one that corresponds to what is observable with direct measurements.Assuming that we know if a land is primary or secondary -which is feasible thanks to satellite land-cover observations and land-use historical data -the local measurements with, for example, flux towers will provide data consistent with this definition.However, this raises the issue of choosing a reference land-cover S 0 that should depend on the scope of the study.For instance, considering European forests of the 18th century as primary forests, and thus neglecting previous land-use activities, seems to be a reasonable approximation if the study focuses on the industrial era, i.e., a period when land use is mostly driven by the Americas (North and then South).Contrarily, studies on preindustrial land use (e.g.Kaplan et al., 2010) may prefer to define S 0 with only natural biomes (e.g., potential vegetation) in order to seize all human-induced impacts on the terrestrial biosphere.

Truncated definition 2
The main drawback of implementing definition 2 in a spatially explicit ecosystem model is that it requires to keep track of very old age classes of the cohort (which are almost transitioned) in each grid point, making it demanding in computing time for almost no improvement in the precision of the simulation.To avoid this, one solution is to arbitrarily define an age class τ lim after which the cohorts are considered "transitioned" and are then reallocated to the "undisturbed" group of ecosystems.In IPCC guidelines (Paustian et al., 2006) the default value of τ lim is 20 yr.Using this truncated approach, we give a variation of definition 2 (noted def 2, τ lim ) of ELUC and LSNK, with a variable parameter that defines the last disturbed age class τ lim considered in the accouting of the ELUC term, which gives The general definition 2 given in Eqs. ( 19) and ( 20) is verified in Eqs. ( 21) and ( 22) for τ lim = ∞ (i.e., ELUC def 2 = ELUC def 2,τ lim =∞ ).It is clear that the setting of τ lim is crucial in the truncated definition 2. If τ lim is too small, disturbed lands will be considered transitioned too early (the ELUC flux will be "underestimated").However, there are no rigorous mathematical conclusions regarding the consequences of a choice of τ lim , since the behavior of the cohorts is modeldependent.We illustrate the effect of choosing different values of τ lim in Fig. 4 and discuss it in Sect.3.4.2,using the OSCAR v2 model.

Definition 3: LUC/CCN perturbations
The third definition we propose (definition 3, def 3) is based on Eq. ( 15) which separates the LUC and CCN perturbations: and The separation of the two perturbations implied by def 3 is conceptual.Book-keeping models usually use this definition because they are developed to look at the "difference to the equilibrium" for every kind of land-use activity.However, a model such as the one developed by Houghton et al. (1983) even updated for recent evaluations of ELUC (Friedlingstein et al., 2010;Le Quéré et al., 2013) is not fully coupled with the CCN perturbation.Indeed, if the parameters of such models are calibrated on observed stocks and fluxes in, for example, the 1970s, then the simulated ELUC will always be nudged to the CCN perturbation of the 1970s (e.g., increased C stocks when compared to preindustrial) even for emissions calculated at other dates like 1850 or 2050.A solution would be to use a time-dependent calibration of the book-keeping model (e.g., updated every decade) in order to update the parameters that are changing because of the CCN perturbation.
The only fully coupled book-keeping model we know of this far, which uses def 3 for calculating emissions from land-use change, is the one developed by Gasser et al. (2013).

Equations
To compare the three definitions introduced above, we use the formal names for fluxes given in Sect.2.4.Thus, based on Eqs. ( 17) to ( 24), and using the notation of Eq. ( 16), we can write the three definitions as We previously explained that, for an historical simulation and at global scale, ELUC 0 , ELUC and LSNK are all positive (emissions of CO 2 to the atmosphere) while LSNK 0 is negative (sink of atmospheric CO 2 ).In absolute values, LSNK must be inferior to LSNK 0 because the loss of sink capacity cannot be superior to the total sink capacity.Thus, LSNK 0 + LSNK is necessarily of the sign of LSNK 0 (i.e., negative in present days).Finally, using Eq. ( 25) we can order the three definitions of ELUC as follows: This inequality is valid only when comparing the results of different simulations of one model, with all parameters being the same.Consequently, comparison between ELUC calculated by different models should be done only if a single definition of land-use change emissions has been agreed upon, to avoid definition-related biases when trying to assess and understand differences between models.With this formalism, the truncated definition 2 (with τ lim finite) is It is impossible to draw general conclusion so as to include this definition in the comparison of Eq. ( 26) because of the opposite mathematical relation (in an historical simulation where LSNK 0 is negative) between the two terms: Let us now consider an idealized simulation where historical land-use activities are stopped at a given time t 0 while other anthropogenic forcings (such as fossil fuel emissions) are not.We then look at the value of ELUC with our different definitions a long time after the LUC perturbation stopped.The values of ELUC 0 and ELUC must both tend toward zero (Eqs.9 and 16) as time since the last perturbation increases (i.e., the age of the younger non-zero element of the cohort increases), which gives That result is interesting because it shows that only the third definition allows ELUC to be zero a long time after the end of the LUC perturbation.Oppositely, when using definitions 1 or 2, if there has been a LUC perturbation at one time, there will always be emissions from land-use change calculated by the model.In the case of the truncated definition 2 (def 2, τ lim ), ELUC also tends toward zero for this simulation but with a discontinuity at t = τ lim + t 0 , when ELUC drops from the value of the (small) net flux of the τ lim -th element of the cohort to a value of exactly zero (and because all elements of the cohort younger than τ lim years are equal to zero as landuse activities have stopped).

Simulation with OSCAR v2
The OSCAR v2 model is used, forced by prescribed land cover changes and forestry since preindustrial for the LUC perturbation, and climate and CO 2 effects (but no nitrogen) for the CCN perturbation.See Appendix B for references of data.The model code was written to be tractable with the calculation of ELUC and LSNK fluxes under the three different definitions.Figure 3 displays ELUC calculated using def 1, def 2 and def 3, as well as two examples of the truncated definition 2 (def 2, τ lim ) with different τ lim values being set to 20 and 40 yr.The left-hand panel displays the simulated value from 1900 to 2005, for an historical simulation that starts in 1700.First, we observe that the simulation results shown in Fig. 3 fulfill the established inequality (Eq.26), and that the difference between def 3 and def 2, or between def 1 and def 3, can be up to about 20 % during the 1980s and the 1990s.Despite being clearly model-dependent, this result highlights the importance of the choice of the definition to quantify land-use-related emissions and compare different model estimates.In the previous section, we explained why the value of ELUC calculated under def 2, τ lim is variable when compared to the values simulated under the other definitions.Up to 1950, both ELUC def 2,τ lim curves (τ lim equal to 20 and 40 yr) are below the curves generated with the other three definitions; but after that date, ELUC def 2,τ lim can be either above or below the ELUC def 2 curve.Another interesting result here is that def 3 is the least affected by climate variability.ELUC def 2 is more variable during the represented period than ELUC def 1 , which itself varies more than ELUC def 3 .Equation ( 16) brings insights on the causes of this behavior: definitions 1 and 2 are functions of the LSNK 0 and LSNK terms that are mainly driven by the CCN perturbation, as explained in Sect.2.3, and are consequently affected by climate variability.By contrast, under definition 3, the only term affected by the CCN perturbation is ELUC (and even then, the formulation f − f * in Eq. ( 23) is expected to act as a "buffer" of the variability because the perturbation f and the transitioned state f * are both affected by the variability in a similar way).The right-hand panel of Fig. 3 shows the simulated ELUC from the different definitions between 2005 and 2100, in the idealized case where land-use activities would stop after 2005 but atmospheric CO 2 and subsequent climate change follow the RCP 8.5 scenario.This part of the simulation illustrates the consequences of adopting different ELUC definitions.First, about the "legacy" of land-use change, the stop of land-use activity after 2005 does not imply that ELUC become immediately equal to zero, as explained in Sect.2.3.Second, the very different behaviors of the three definitions during the period 2005-2100 are good illustrations of Eq. ( 29).While ELUC from def 3 tends slowly toward zero as theory predicts, emissions following def 1 and def 2 clearly diverge from zero when t 2005.This is due to the continuing CCN perturbation in the RCP 8.5 CO 2 and climate change scenario.Here, ELUC from def 1 remain positive and increase with time after 2005 because this definition includes the loss of potential sink due to past deforestation ( LSNK), and this lost potential sink is also increasing (due to CO 2 -fertilization, despite its attenuation 1 9 0 0 1 9 1 5 1 9 3 0 1 9 4 5 1 9 6 0 1 9 7 5 1 9 9 0 2 0 0 5 2 0 2 0 2 0 4 0 2 0 6 0 2 0 8 0 2 1 0 0 legacy under RCP8.5 by carbon-climate feedbacks).Oppositely, ELUC from def 2 are negative for t > 2005 (and increase in absolute value with time) because def 2 includes the net effect of the CCN perturbation (LSNK 0 + LSNK) over lands that have been disturbed at any previous time.In other words, def 2 takes into account the "land sink" that occurs above previously disturbed lands that have almost "recovered" from the LUC perturbation.Finally, the emissions defined by def 2, τ lim behave as explained in the previous section: they drop to zero at the year t = 2005 + τ lim .
For a better discussion on the implications of using a truncated definition 2, Fig. 4 provides a comparison of def 2, τ lim for different values of τ lim , at different times of the simulation: in 1850, 1990, 2005 and 2025.On the four subplots, ELUC def 2,τ lim tends toward ELUC def 2 , by construction (see Eqs. 19 to 22).ELUC def 2,τ lim is not a monotonic function of τ lim .For example, in 1990 and 2005, it decreases as τ lim increases for τ lim > 100 yr, but it increases with τ lim in the range 50 to 100 yr.In 1850, however, ELUC def 2,τ lim is generally increasing with τ lim .Emissions calculated with def 2, τ lim appears to be close to that of def 2, but not always.In 2005, the value of ELUC def 2,τ lim is even greater than that of ELUC def 3 for small values of τ lim (< 20 yr).Therefore, since it seems that the behavior of the truncated second definition is highly model-dependent, we cannot recommend any "best" value of τ lim and great care must be taken when comparing ELUC estimates from models that use this definition.

Discussion
We see two limitations to this theoretical framework.First, natural climate variability affects the hypothesis of a preindustrial equilibrium.For clarity, we decided to write down equations without accounting for this variability.However, we could break down the flux f into a mean and a variable terms which average value is equal to zero: f = < f > + ḟ with ḟ = 0.In this case, the mathematical demonstrations of Sect. 2 are still valid for the mean term < f >.Practically, so as to avoid biases due to climate variability affecting ecosystem fluxes in models, the four component fluxes of the CCN and LUC perturbations should be estimated either on average over a long enough time period (e.g., 10 yr) or as cumulative fluxes.Note that the biases will only appear for the experiment with LUC perturbation at preindustrial times (Exp.2; where a reference preindustrial CCN perturbation has to be defined, but cannot because of climate variability).
The second limitation is the migration of vegetation induced by CO 2 and climate changes.This phenomenon can be seen as a natural (yet indirectly human-induced) land-use change.A first option to include it in the framework is to consider three perturbations: CCN, direct anthropogenic LUC and indirect anthropogenic LUC (i.e., migration).However, adding a third perturbation would require to run more experiments so as to separate more component fluxes of the net land-to-atmosphere flux.Another option, which avoids supplementary simulations, is to include the migration as part of the CCN perturbation.To do so, at each time step and over each grid cell, all natural biomes b would have to be aggregated into one "mean" biome before applying our When this aggregation is used with all equations of Sect.2, the migration of natural biomes, which can be seen as a land conversion from one natural biome to another, could be finally accounted for in the net areal flux f (g, b) (i.e., in the CCN perturbation).The effect of direct anthropogenic landuse change, which is limited to conversions from natural to anthropogenic biomes (and conversely) and to conversions between anthropogenic biomes, would still appear in the area change δS (i.e., in the LUC perturbation).Finally, the work by Strassmann et al. (2008) must be mentioned, as they also separated different components of the land-to-atmosphere CO 2 flux, but in a fundamentally different manner as we did here.The starting point of their analysis was that part of the CCN perturbation is caused by the LUC perturbation itself, as stated in Sect.2.2.Hence, if we stop looking at the CCN perturbation as an exogenous perturbation (a forcing) and start seeing it as being endogenous, the disturbed areal fluxes f could be written as f = f LUC + f noLUC + f non-lin .The superscript "LUC" refers to the part of the CCN perturbation that is attributable to the LUC perturbation, the superscript "noLUC" to the part induced by everything else (e.g., fossil fuels, methane, aerosols), and the superscript "non-lin" accounts for the non-linearity of the system, which was forgotten (Strassmann et al., 2008).Going through the same demonstrations as in Sect.2, but with f broken-down as above, leads to the breakdown of LSNK 0 and LSNK (and ELUC) into three subcomponents.Thence, we could regroup these components and identify the different terms defined by Strassmann et al. (2008) and Stocker et al. (2011): LSNK LUC 0 + LSNK LUC is their "land-use feedback", LSNK noLUC is their "replaced source/sink", and LSNK noLUC 0 is their "(effective) potential sink".ELUC 0 corresponds to what they call "the book-keeping flux", and all the other terms (i.e., LSNK non-lin 0 + LSNK non-lin + ELUC) correspond to their "interaction term".However, the initial breakdown of f into three sub-fluxes raises the issue of (i) what is the system boundary (forced or coupled land carbon cycle)?(ii) what is the cause-effect chain within this boundary?and (iii) how should the non-linearity be dealt with?The three questions are beyond the scope of our paper, but have been partly addressed in studies about the "regional attribution of climate change" (also known as the "Brazilian Proposal" (see e.g.Gasser et al., 2013)).

Conclusions
By looking at the mathematical structure and properties of the net land-to-atmosphere CO 2 flux this study provides a theoretical framework so as to distinguish its different constitutive components.Rather than defining two component fluxes (as one would expect since the net flux is the result of two perturbations: CCN and LUC), we show that considering four components of the net flux is mathematically exact.Using those four components, we demonstrate that T. Gasser and P. Ciais: ELUC definition different modeling definitions of emissions from land-use change (ELUC) can be chosen, mainly depending on the way a model is built.We can draw three conclusions from this work: -Choosing a definition for ELUC (or having a definition imposed by a model's structure) implies a complementary definition for the land sink (LSNK).This is critical for studies that look at the global carbon budget since choosing two inconsistent definitions may lead to missing -or accounting for multiple times -some terms of the net land-to-atmosphere flux that are due to the coupled interaction between CCN and LUC perturbations.We suggest that might explain a part of the "residual" flux of the global carbon budget estimated by Le Quéré et al. (2009) where they use estimates of ELUC through book-keeping (def 3) and estimates of LSNK through modeling without explicit representation of land use (def 1).
-There is only one modeling definition that is comparable to what can be directly measured: the second definition (def 2) based on the undisturbed/disturbed status of lands.However, since calculating the ELUC flux based on this definition requires important computing memory and time, one might approximate it with the truncated definition 2 (def 2, τ lim ) based on a deliberately limited size of the cohort of transitioning ecosystems.
Here, we suggest that the parameter τ lim should be as high as possible, or at least carefully evaluated for each land conversion type and/or ecosystem so as to keep a maximum of information about the cohorts.
-The different possibilities of definition increase the discrepancy between ELUC estimates made through modeling.In the OSCAR v2 model used for illustration here, the difference between two definitions can be about 20 %.Since this adds to all other kinds of uncertainty (related to data: on area changes, carbon areal densities, emission dynamics; or to the structural difference between models), we highly recommend to compare modeling results in which the definition of ELUC is the same.For model intercomparison, it is even better to assess the values of the four component fluxes of the net land-to-atmosphere CO 2 flux, which is feasible thanks to the three simulations describe in Sect. 2 of this paper: one with carbon-climate-nitrogen perturbation only, one with land-use change perturbation only, and one with both perturbations.The mass conservation constraint gives the fourth and last flux.Now, we consider a normalized land conversion from (g, b 1 ) to (g, b 2 ) (i.e., δS = 1) that happens at time t 0 with no CCN perturbation (like in Sect.2.2).The successive values of the τ -th element of the cohort (i.e., f τ 0 ) taken at t = t 0 + τ can be written as being the total carbon stock per area unit ( ĉ, expressed in gC m −2 ) that is to be emitted during the whole transition (i.e., through all the years of the transition) multiplied by a time-dependent rate of emission that represent the dynamics of this emission (r, expressed in yr −1 ).We can write f τ 0 (t 0 + τ ) = ĉ0 r 0 (τ ) (A1) with the following condition on f τ 0 and thus on the function r 0 :

Fig. 1 .
Fig. 1.Conceptual diagram of the three experiments described in Sect. 2. See text for notations and mathematical development of the framework used to break down the net land-to-atmosphere CO 2 flux.

Fig. 2 .
Fig. 2. The four component fluxes of the net land-to-atmosphere CO 2 flux simulated by OSCAR v2.Left panel shows results of the simulation over the last century and right panel shows the simulated fluxes when the LUC perturbation is stopped in 2005 but the CCN perturbation follows the RCP 8.5 scenario.ELUC 0 is the term driven only by the LUC perturbation (plain black line) and ELUC is the term due to the effect of the CCN perturbation over the LUC perturbation (dashed black line).Conversely, LSNK 0 is the term driven only by the CCN perturbation (plain grey line) and LSNK is the term due to the effect of the LUC perturbation over the CCN perturbation (dashed grey line).The net land-to-atmosphere flux is the sum of the four components (red line).Note that the scale for LSNK 0 and NetFlux is one-fourth of the scale for the three other fluxes.

Fig. 3 .
Fig. 3. Illustration with the OSCAR v2 model of the three proposed definitions of ELUC.The three plain lines correspond to the three definitions: first definition is based on the difference between a simulation with land use and another without land use (def 1, red line); second definition is based on the distinction between disturbed and undisturbed lands (def 2, green line); third definition is based on the distinction between LUC and CCN perturbations (def 3, blue line).Two examples of the truncated definition 2 (see text) are given, with τ lim being 20 yr (dashed line) and 40 yr (dotted line).Left panel shows the results for an historical simulation, while right panel shows a simulation where land-use activities cease after the year 2005 but atmospheric CO 2 and global warming follow the RCP 8.5 scenario.

Fig. 4 .
Fig. 4.Value of ELUC defined following the truncated second definition (def 2, τ lim ) as a function of the last element of the cohort considered to be disturbed (τ lim ), at four different years of the simulation with OSCAR v2.The value of this definition (black line) is compared to the three main definitions (dashed horizontal lines of the same color as in Fig.3).
, for a normalized area change, the exact value of ĉ0 is the difference in carbon density between the primary ecosystem and the secondary "transitioned" ecosystem (i.e., ĉ0 = c * 0 (g, b 1 ) − c * 0 (g, b 2 )).Indeed, the total CO 2 flux integrated over time induced by a transition from (g, b 1 ) to (g, b 2 ) only depends on the local carbon densities of b 1 and b 2 at the point g, because of mass conservation constraint.That flux is positive, causing net CO 2 emission (resp.negative, causing a net sink of atmospheric CO 2 ), if the primary ecosystem holds more (resp.less) carbon per area unit than the secondary ecosystem.By way of consequence, land-use activities like forestry, modeled as transitions from (g, b 1 ) to itself, are carbon neutral when integrated over a long enough period (i.e., ĉ0 = c * 0 (g, b 1 ) − c * 0 (g, b 1 ) = 0).The function r 0 is a normalized impulse response function (IRF) for the normalized transition from b 1 to b 2 over g.Thus, if we know the impulse response function at each point g for each transition b 1 → b 2 noted r 0 (t ; g, b 1 , b 2 ), if we know the local carbon densities c * 0 (g, b) and the history of land-use change conversions δS(t; g, b 1 , b 2 ) > 0, the emissions from land-use change ELUC 0 (under preindustrial conditions) can be expressed at all times t by the following convolution: ELUC 0 (t) = t t =0 g,b 1 ,b 2 c * 0 (g, b 1 ) − c * 0 (g, b 2 ) r 0 t − t ; g, b 1 , b 2 δS t ; g, b 1 , b 2 .(A3)