The evolution of concentrations of atmospheric long-lived non-CO₂ species (and sometimes CO₂) is typically modelled by a mass balance model, alternatively known as a single-reservoir box model: 1dCxdt=Πx-Λx=Exβx-Cxτx. In this representation, the change in concentration of a given species x (Cx) is governed by its production (Πx) and loss rates (Λx). The production rate is simply the species emissions (Ex), converted to concentration units with a conversion factor βx (this is sometimes omitted in model descriptions). Emissions can be purely anthropogenic in origin, be caused by natural processes or a combination of both, depending on the species, the considered scenario, and the internal structure of the model. The loss rate is typically assumed to be an exponential decay, characterised by a global decay time τx. This assumption makes the scheme a particular case of the wider family of impulse-response models, as described in Sect. . The characteristic lifetime of the species may not directly correspond to any single physical process, but combine the effects of multiple sinks removing that species from the atmosphere at similar timescales. In general, the lifetime can be state-dependent, altering its value based on an number of climate properties. This is, for instance, what the FaIR model does (Sect. ) for CO₂ and CH₄, including carbon and temperature feedbacks for those species lifetimes. However, this is unusual in model representations, with most SCMs adopting a constant lifetime. This is particularly the case for minor GHGs (all GHGs except CO₂, CH₄ and N₂O). A more typical approach to increase the flexibility of the lifetime scheme involves the inclusion of additional sinks acting at different timescales. This is typically incorporated into models through the use of a total lifetime defined as: 21τtotal=1τ1+1τ2+…+1τn where n is the number of sinks considered. This is particularly relevant for tropospheric methane, represented in most models with a mass balance equation and a total lifetime with contributions from multiple sinks, typically including its reaction with hydroxyl radicals (OH) (∼11.2 years), stratospheric loss (∼120 years) and soil uptake (∼150 years) . Using Eq. () this results in a total lifetime of ∼9.6 years.

This mass balance formulation focuses on the amount of species x that remains airborne after some time t, without providing any additional information about the fate or potential impacts after its removal from the atmosphere. Often, this level of detail is enough to satisfy SCM objectives, particularly for chemical species lacking a natural cycle. Consequently, SCMs generally disregard the dynamics of these species after they cease to be relevant for the calculation of radiative forcing. However, this is not the case for CO₂, because this species possesses a strong natural cycle with large fluxes across four reservoirs in the Earth's system: atmosphere, ocean, land and biosphere. Understanding the response of this cycle to anthropogenic disturbances, as well as any potential tipping points, is therefore necessary for accurate predictions of future atmospheric concentrations and ultimately, to estimate compatible emission scenarios with climate stabilisation targets . Indeed, this is currently an area of intensive research, often discussed in terms of the Zero Emissions Commitment (ZEC), which quantifies the global temperature change in a post-net-zero world . SCM development teams, recognising the need for details on the carbon cycle, have generally either started or transitioned towards CO₂ representations more complex than this mass balance model. By far, the most popular representation for SCM carbon cycles, particularly for the land component, is box-based carbon cycle models, which simulate the flow of carbon through the different Earth's reservoirs.

Box-based carbon models consist of a number of conceptual boxes, also known as pools or reservoirs, that abstract away the carbon content of certain parts of the carbon cycle (e.g., vegetation, soil, ocean (layers), atmosphere). These boxes exchange carbon through fluxes which can be modulated by climate properties. For example, the carbon transport from vegetation to the soil can be simulated through a litterfall flux, which can be magnified by higher temperatures. The carbon inventories of the model boxes are determined by the initial stocks and the carbon fluxes. SCMs typically aim to represent all the main fluxes in the Earth's carbon cycle, such as net primary production (NPP), litterfall, decomposition and respiration. See, for instance, the depiction of the carbon cycles of Hector (Fig. a) and MAGICC (Fig. ). More details about the nature and magnitude of these fluxes, as well as the wider Earth's carbon cycle, can be found in .

SCMs can simulate these fluxes in different ways. Some fluxes may be modelled by relatively simple analytical expressions, accounting for climate feedbacks. This is particularly common for the land component. Other fluxes may be modelled by more complex schemes, such as carbonate schemes to estimate the dissolution of atmospheric carbon into the ocean mixed layer (OML), the uppermost layer of the ocean exhibiting relatively uniform properties due to turbulence homogenisation effects. A common approach, particularly for the ocean component, is the use of IRMs, which are discussed in Sect. .

This seemingly simple representation of carbon reservoirs and fluxes provides modelling teams with remarkable flexibility. It is easy to extend, with the inclusion of new components merely requiring the definition of the carbon content of the new pool and interacting fluxes. It also allows the inclusion of multiple fluxes, each considering different climate feedbacks and being calibrated independently. Furthermore, it is a computationally efficient approach, with the number of parameters and calculations required remaining relatively low. Crucially, box models offer an internally-consistent framework that can keep track of the flux of carbon in a closed cycle. These qualities are further amplified by the conceptual simplicity of these models, which aids interpretability and visualisation, facilitating their adoption by both users and developers.

The resolution of box models is typically global, with boxes representing the entire Earth's carbon content for that category (e.g., soil, vegetation). However, some models like Hector and OSCAR further partition these boxes into smaller units representing regions or even regional biomes.

Once the total radiative forcing has been calculated, SCMs need a method to translate that quantity into a global surface temperature anomaly. This is usually achieved by an EBM . This type of model relies on a series of assumptions and approximations that will be briefly described in this section. A more comprehensive review of the derivation and validity of these assumptions can be found in Appendix A of .

In an equilibrium state, incoming solar energy absorbed by the Earth system (Fin) must be equal to the outgoing radiated infrared energy (Fout). However, the atmosphere absorbs a significant amount of the outgoing radiation leaving the Earth's surface and emits it back down, increasing surface temperature. This is known as the natural greenhouse effect (G). Approximating the Earth as a black body, one can estimate the outgoing surface radiation using the Stefan–Boltzmann law: Fout=σTS4 where TS is the average global surface temperature and σ=5.67×10-8 W m⁻² K⁻⁴ is the Stefan-Boltzmann constant. In equilibrium, the outgoing energy flux must, therefore, equal the incoming radiation plus the energy absorbed by the atmosphere: 3Fout=σTS4=Fin+G. If that energy balance is forced with a small perturbation to the amount of incoming or absorbed energy (ΔE) the system will respond with a heat flux at the surface, changing the surface temperature until the system is back at equilibrium. By conducting a Taylor expansion of the outgoing energy around the original equilibrium surface temperature (T0) and retaining only the first order term, one can approximate the temperature response induced by this perturbation as: 4Fout(T)=σTS4≈σT04+4σT03ΔT⇒5ΔFout=Fout(T)-F0out≈σT04+4σT03ΔT-σT04=4σT03ΔT where ΔT is the difference between the new and original temperature, or temperature anomaly. This linear approximation is valid only for small temperature differences of a few degrees, with non-linear terms becoming relevant for larger anomalies . Consequently, the heat flux (dH/dt) induced by this perturbation in the energy balance during the transient state will be equal to the perturbation (ΔE) minus the surface temperature response: 6dHdt=ΔE-4σT03ΔT where the perturbation can take place either in the incoming radiation or the greenhouse effect: ΔE=ΔFin+ΔG. ΔG corresponds to the anthropogenic greenhouse effect (assuming the perturbation is human in origin). Typically with energy balance models this perturbation is assumed to be separable into two terms:

A radiative forcing term, F. In SCMs this is usually taken to be the stratospherically-adjusted radiative forcing or the effective radiative forcing (see Sect. ).

Under this assumption Eq. () can be rewritten as: 7dHdt=ΔE-4σT03ΔT=F+k⋅4σT03ΔT-4σT03ΔT=F-λΔT where λ=4σT03(1-k) is known as the climate feedback parameter and combines the effect of increased blackbody radiation (4σT03) and temperature feedbacks caused by the perturbation in the energy flux (4σT03k). While seemingly simple, Eq. () is arguably the single most important parametrisation in most SCMs, as it controls the model's temperature response for a given forcing. The k parameter (or, indirectly, the λ parameter) abstracts away the great complexity of the climate system and its numerous feedbacks effects (e.g., albedo change, aerosol interactions, etc.). In most instances, this is a constant model parameter that can be tuned to emulate results from other, more complex ESMs. However, more complex formulations are possible, like MAGICC's (Sect. ) time-varying, and WASP's (Sect. ) forcing-agent-specific time-varying λ parameters.

Typically, Eq. () is modified to assume that the heat flux results in an increase in surface temperature, mediated by a global heat capacity C, thereby converting it into the following first order differential equation: 8CdTdt=F-λT. Note that the surface temperature anomaly, T, is presented without the delta notation for the sake of simplicity, although it continues to represent the same anomaly.

The formalism outlined thus far is common to most simple climate models. Where they diverge is in their treatment of the heat flux distribution and effects across the Earth system. While most models include a representation of the heat flux towards the ocean, the largest heat reservoir in the Earth system, how that heat is distributed across the ocean varies. Heat diffusion to the deep ocean is often included, with the occasional addition of heat transport by ocean currents. Representations of the land heat sink are less frequently included.

Equation () provides an approximate representation of the temperature response to the energy imbalance induced by radiative forcing. Assuming a constant climate feedback parameter (λ), the resulting relationship provides an estimate of the global temperature necessary for the climate system to reach equilibrium with the new radiative forcing.

However, this equilibrium is not achieved instantaneously. The climate system exhibits a large thermal inertia (relative to human timescales), primarily due to the stabilising influence of the deep ocean, which acts as a large heat reservoir. The ocean is estimated to have absorbed around 89 % of the additional heat coming into the Earth system as a result of the historical energy imbalance . Consequently, the transient temperature anomaly induced by a certain amount of positive radiative forcing is smaller than the eventual temperature once the system reaches a new equilibrium.

A commonly-used method to incorporate this deep-ocean thermal inertia to SCMs is the two-layer or two-box model of temperature response. This model, referred to as the “Held et. al two-layer model” in , is based on the work of . Fundamentally, this representation adds an additional box with a large heat capacity to the EBM described by Eq. (). As a result, the model possesses the original “rapid” box simulating the fast temperature response of the atmosphere, land and ocean boundary layer to the changes in radiative forcing, and an additional second “slow” box coupled to the first that emulates the slow response of the deep ocean. Mathematically, this can be expressed as follows: 9CFdTdt=F-λT-HoCodTodt=Ho where CF and Co are the heat capacities for the fast and slow boxes, respectively, with Co≫CF. Ho is the exchanged heat between the two layers, which is assumed to be proportional to the difference in temperature anomaly between the two boxes, characterised by a heat exchange coefficient κ, Ho≡κ(T-To).

However, an important limitation to this model is its inability to resolve evolving spatial warming patterns that occur as the system approaches equilibrium, as discussed in and . These evolving patterns modulate the outgoing energy flux to space, impacting the global temperature response. Since this phenomenon can be related to the ocean heat uptake, introduced an efficacy parameter, ϵ, to account for this effect: 10CFdTdt=F-λT-ϵκT-ToCodTodt=κT-To. This new formulation is equivalent to Eq. () but with heat exchange and deep-ocean exchange coefficients scaled by ϵ, defining κ′≡ϵκ and Co′≡ϵCo. reported values for this efficacy parameter greater than one for nearly all models under 1pctCO2 experiments. provided analytical solutions to this system of differential equations, and compared them with other EBMs, specifically a two-region model – which they demonstrated to be mathematically equivalent to the two-layer model in its temperature response – a one-layer model with a temperature-dependent feedback, and a hybrid of the two.

The two-layer model can be easily generalised to an n-layer model by adding new ocean layers beneath the extra layer introduced in Eq. (), defining a system of n differential equations : 11C1dT1dt=F-κ1T1-κ2T1-T2C2dT2dt=κ2T1-T2-κ3T2-T3⋮Cn-1dTn-1dt=κn-1Tn-2-Tn-1-ϵκnTn-1-TnCndTndt=κnTn-1-Tn. In this generalised formulation, the top layer corresponds to the “fast” layer in Eq. () (C1, T1, κ1, κ2 and T2 are CF, T, λ, κ and To, respectively). Note that the efficiency parameter, ϵ, is only present in the penultimate layer, as the deepest layer is still the largest reservoir of heat (i.e., it possesses the largest heat capacity) dominating the deep ocean uptake.

A higher number of layers increases the number of distinct timescales in the system (see τi in Eqs.  and ), and therefore increases the complexity the model is able to display in its transient response. The equilibrium state, however, remains independent from the number of layers, as Eq. () reduces to T1=F/κ1 when T1=T2=…=Tn. Typically, the number of layers is limited to two or three in SCM implementations. A two-layer model, calibrated to Coupled Model Intercomparison Project Phase 5 (CMIP5) model output , has been employed as the EBM module in several SCMs, including OSCAR, AR5-IR, and FaIR until v2.0. FaIR v2.0 increased the number of layers to three by default based on evidence from and that suggested three layers are usually sufficient to accurately capture the temperature response of ESMs. Notwithstanding this finding, to further enhance the model's flexibility, FaIR v2.1 allows an arbitrary large number of n layers (as long as it is larger than one), although the number of tuning parameters quickly increases as n becomes larger. It is important to note that n-layer EBMs might not be immediately recognizable as such, as they are sometimes presented in an alternative formulation related to a broader family of models, known as IRMs.

As previously discussed, the primary purpose of SCMs is to emulate the behaviour of more complex climate models efficiently. In the preceding sections, methods were explored that aimed to achieve this emulation through the development of computationally efficient approximations of the Earth system maintaining physical intuition. IRMs take a different route; they are not derived from fundamental physical principles but rather, offer a mathematical framework able to approximate the dynamics of any non-linear system through empirical parameter tuning.

In particular, IRMs tune empirical Impulse-Response Functions (IRFs) to simulate the behaviour (response) of a variable of interest to a perturbation (impulse) in a related property. This tuning is typically based on the output of more complex models, such as AOGCMs or ESMs. IRFs are also known as “Green's functions” in other fields of physics.

IRFs can fully characterise the dynamical response of a linear system, providing only an approximation for non-linear systems. The quality of the approximation depends on the extent of the system's deviation from linearity. As previously discussed in the derivation of Eq. () for the standard EBM, a quasi-linear assumption of the climate system is often employed when working with SCMs, justifying the use of IRFs to approximate climate properties.

For climate-related quantities, IRFs typically take the form of a sum of n decaying exponentials with characteristic timescales τi and magnitudes Ai. Thus, the value of a property of interest x(t) at time t can be approximated by evaluating the convolution of the magnitude of the impulse F(t′) up to time t with the tuned response to that impulse (i.e., the sum of exponentials): 12x(t)=∫0tF(t′)⋅IRF(t-t′)dt′=∫0tF(t′)∑iAiexp⁡-t-t′τidt′. Alternatively, taking the time derivative of Eq. () yields the equivalent differential equation which is also often employed: 13dx(t)dt=∑iAiF(t)-xi(t)τi. As an example, the temperature anomaly T(t) after a unit of radiative forcing at t=0 (represented by a dirac delta function δ centered at time t′=0) can be approximated using an IRM by : 14T(t)=∫0tδ(t′=0)∑iAiexp⁡-t-t′τidt′=∑iAiτiexp⁡-tτi. Through the tuning of the τi and Ai parameters, Eq. () can be calibrated to emulate the temperature response of more complex climate models. Alternatively, IRFs can model the increase in atmospheric CO₂ concentration following an emissions pulse at t=0, as seen in the AR5-IR and FaIR models reviewed below and discussed in Sect. , or the transfer of carbon from the ocean mixed layer to the deep ocean . This last IRM has been a particularly influential scheme in the SCM field, being used to simulate the ocean carbon cycle in three prominent SCMs: CICERO-SCM, OSCAR and MAGICC. More details about this scheme are presented in Sect. .

The family of n-time-constant temperature IRMs described by Eqs. () and () is particularly interesting because it is mathematically equivalent to the family of n-layer temperature models described by Eq. (). In its general IRM form, Eq. () can be expressed as: 15dTidt=AiF-Tiτi;T=∑inTi which is a diagonalised form of the equation one would obtain if Eq. () was expressed in matrix form . Table 1 in provides a list of conversions between the constants in the two formulations for the case n=2. Note that in this formulation, the temperature anomaly at the surface, T, is the addition of the contributions from the different temperature components.

While the n-time-constant temperature IRMs and the n-layer temperature models are mathematically equivalent, and could therefore be fundamentally considered the same model describing the same dynamics, the IRM formulation has the advantage of a simple relationship between the parameters in Eq. () and two of the most critical and widely discussed quantities in climate science, the equilibrium climate sensitivity (ECS) and the TCR: 16ECS=F2×CO2∑i=1nAi17TCR=F2×CO2∑i=1nAi1-τiD1-exp⁡-Dτi where D=ln⁡(2)/ln⁡(1.01)≈69.7 years, is the time required to double the atmospheric CO₂ concentration under a 1 % yearly increase scenario. These relations are highly advantageous because, for the case n=2, and given τ1 and τ2, they can be inverted to determine A1 and A2 as functions of ECS and TCR. This allows for a straightforward definition of an EBM consistent with any combination of ECS and TCR values. Even for n>2, Eqd. () and () remain relevant, as they define a hyperplane where any combination of ECS and TCR values can be easily obtained. Characteristic timescales, τi, are typically taken following previous studies of n-layer models. FaIR v1.0, for instance, based its 2-time IRM on the characteristic timescales of the multi-model mean of a 2-layer model tuned to CMIP5 AOGCMs by .

The core of the model is defined by the following relationship between monthly temperature anomaly (T) and sources of radiative forcing: 18T=1+γλPFiGHG+FiAER+FiLUC-Qiocean+C0+C1⋅SAODi-6+C2⋅TSIi-1+C3⋅ENSOi-2+C4⋅AMOCi+C5⋅PDOi+C6⋅IODi. The terms between squared brackets include anthropogenic sources of radiative forcing – elevated GHG concentrations (FiGHG), aerosols (FiAER), LULCC changes in albedo (FiLUC) – as well as the ocean heat sink (Qiocean). These terms are multiplied by a (1+γ)/λP factor to account for climate feedbacks, where λP=3.2 W m⁻² °C⁻¹ is the Planck feedback parameter, determining the temperature response in the absence of feedbacks, and γ is a calibrated factor determining the strength of those feedbacks. The GHG term comprises forcings from CO₂, CH₄ (including 15 % increase from stratospheric water vapour), N₂O, and 31 halogenated compounds. In early versions of the model, the equations from were used to estimate the forcing from these species concentration time series, along with radiative efficiencies from . Forcing related to tropospheric ozone was also added by to this term directly as an additional forcing time series. Similarly, the aerosol and LULCC terms are included in the model directly as forcing time series data. The former was taken from a combination of the SSP and RCP databases, while the latter was based on Table AII.1.2 in . In the latest version of the model , GHG and aerosol forcings were updated to follow the expressions and magnitudes from chap. 7 and annex III of the AR6 report .

The ocean heat sink is a prognostic quantity in the model which, since , is calculated by computing the difference between temperature anomaly in the atmosphere and in the OML (the deep ocean is not considered in this model). This difference is further modulated by an ocean heat uptake efficiency that varies according to the accumulated ocean heat content and past radiative forcing. The ocean heat content is a key metric of the model, as it is one of the two quantities, along with global temperature anomaly, that is used to calibrate it.

The terms outside the brackets represent natural sources of variability: volcanos through stratospheric aerosol optical depth (SAOD_i−6), total solar irradiance (TSI_i−1), El Niño–Southern Oscillation (ENSO_i−2), the Atlantic Meridional Overturning Circulation (AMOC_i), the Pacific Decadal Oscillation (PDO_i), and the Indian Ocean Dipole (IOD_i). These are time series required as input by the model. Detailed explanations of the indices used to characterise each of these processes can be found in the studies cited above. It is worth noting that these sources of natural variability are typically turned off when the model is used to make future climate predictions due to the difficulty in deriving a timeseries that can reasonably predict the behaviour of these natural processes. C0–6 are calibration coefficients determined by minimising a cost function defined as the difference between model predictions and historical observations for temperature and ocean heat content. The subscripts i reference the monthly resolution of the model, and the time lags between the sources of radiative forcing and their effects (e.g., the model assumes that the volcano contribution, SAOD_i−6, takes six months to take effect).

The only GHG species included in AR5-IR is carbon dioxide. Its gas cycle is simulated by a 4-time-constant IRM based on the work of , which argued that four time components are enough to emulate the evolution of atmospheric carbon concentrations from ESMs following a 100 Gt C pulse. This approach can be interpreted as a distribution of the atmospheric carbon content into four different reservoirs (Ri), each governed by a mass balance equation as described in Sect.  (compare to Eq. ): 19dRidt=aiE-Riτi,fori=1,…,4 where ai is the proportion of the total anthropogenic emissions (E, in ppm per year) allocated to each reservoir. These reservoirs do not correspond to any single physical entity, but rather combine various atmospheric sinks operating at similar timescales.

Values for a1–4 and τ1–4 are provided in , borrowed from . Broadly speaking, these pools account for:

Indefinite airborne fraction (a0=0.2173 and τ0 = infinite years – usually implemented as a large number to allow incorporation into an exponential-sum framework, e.g., 10⁶ years in FaIR v1.0 and 10⁹ years in FaIR v2.0).

Once the different Ri are calculated, the total atmospheric concentration of CO₂ is simply the sum of pre-industrial concentrations (C0) and all considered reservoirs: C(t)=C0+∑iRi.

To compute the resulting radiative forcing from the previously-simulated carbon concentration, AR5-IR multiplies the atmospheric burden (the sum of all Ri pools) by the carbon radiative efficiency (A). This factor represents the radiative forcing per additional unit of carbon mass, and is approximated by taking the limit of carbon concentration anomaly as it approaches 0 in the common logarithmic relationship of . This limit results in a radiative efficiency of carbon of A=1.7517×10-15 W m⁻² kg⁻¹. The applicability of such scheme is limited to small perturbations, which is why SCMs typically use more complex forcing schemes with wider applicability.

The last step in the emissions-climate change chain is to calculate the increase in surface temperature resulting from this radiative forcing. AR5-IR follows and employs a two-time-constant IRM to produce temperature anomaly estimation, taking n=2 in Eq. (). Similarly to the equivalent 2-layer model, this temperature response can be interpreted as the addition of two contributions: a fast contribution, including effects from atmosphere, land and the OML, and a slow contribution accounting for deep-ocean heat uptake.

Initially, FaIR v1.0 extended the IRM used to simulate the carbon cycle in the AR5-IR model (Sect. ) with an additional equation to allow climate- and carbon-carbon feedbacks. This scheme was later applied to all other included GHGs (CH₄, N₂O, and 40 other halogenated gases, as well as aerosols) in FaIR v2.0, albeit in a simplified manner. Specifically, FaIR v1.0 modified the IRM described by Eq. () in the AR5-IR model to include a state-dependent gas lifetime through the addition of a scale factor α: 20dRidt=aiE-Riατi,fori=1,…,N and 21C(t)=C0+∑i=1nRi(t) which effectively alters the sink strength for that gas species. Similarly to AR5-IR, the number of carbon pools was set to four (N=4), although this is user-definable, while the number of sinks for all other species is kept to one by default. Note that since v2.0, the state-dependent factor α is also applied to all other species, but other than CO₂ and CH₄ the default α=1 parameter is not modified.

To determine the appropriate value of this parameter for carbon, FaIR v1.0 used the 100-year integrated impulse-response function (iIRF₁₀₀) derived by . This function multiplies the estimated average airborne fraction by the integration time over a 100-year time span, thereby capturing temporal variations in the remaining airborne carbon. By equating this to a linear function dependent on temperature (T) and land-ocean carbon stock anomalies (Gu), the value of α can be determined at each each time step, incorporating temperature and carbon feedbacks into the gas cycle. However, solving this equation is computationally expensive, so from v2.0 onwards a simplified exponential solution was adopted, which they present as a reasonable approximation for a “wide range of values”: 22α(t)=g0exp⁡r0+ruGu(t)+rtT(t)+raGa(t)g1 with Ga(t)=∑i=1nRi(t)Gu(t)=∑s=t0tE(s)-Ga(t)g0=exp⁡-∑i=1naiτi1-e-100/τig1g1=∑i=1naiτi1-1+100/τie-100/τi where g0 and g1 are new parameters controlling the magnitude and gradient of α. The raGa(t) term represents the sensitivity of the gas species to its own atmospheric burden. This has a small effect on CO₂ atmospheric lifetime, but it is an important factor in methane lifetime. In v2.1 this formulation was further refined for methane, with a new atmospheric lifetime modulated by the burden of an arbitrarily large number of species: 23ln⁡αCH4=ln⁡1+rtT(t)+∑iln⁡1+riGi(t) where ri denotes the sensitivity to the abundance of species i, Gi. This Gi represents either atmospheric concentrations for GHGs or emission rates for short-lived climate forcers, since their rapid decay prevents any significant accumulation in the atmosphere.

No lifetime sensitivities are assumed for nitrous oxide and halogen gases (ru, rT, ra=0). Lifetime estimates for these species are user-definable. The latest available calibration employed the AR6 values . Before V2.1, aerosols were converted from emissions to concentrations by setting τ=1 and taking a conversion factor between emissions and concentrations of 1. Since V2.1, however, to account for their short lifetimes, the concentration step is bypassed and the forcing is computed using emissions directly.

Finally, while FaIR does not natively simulate permafrost thaw, extended v1.6 by coupling it with a simplified permafrost carbon response model.

To calculate the ERF, FaIR v2.0 generalised the expressions offered in to a single equation, approximating the concentration-forcing relationships for all well-mixed greenhouse gases (WMGHGs) (or emissions-forcing in the case of aerosols due to their short atmospheric lifetimes) by: 24Fdet(t)=∑xforcingagentsf1x⋅ln⁡Cx(t)C0x+f2x⋅Cx(t)-C0x+f3x⋅Cx(t)-C0x+Fext(t). Each term is multiplied by an fix factor, allowing the model to account for indirect effects by modulating the direct forcing effects from GHG concentration (i.e., generating ERF estimations). This factor also enables the model to completely switch off a term for a given species. For instance, CO₂ forcing is approximated by a logarithmic and squared root term , so f2CO2=0. Methane and nitrous oxide contributions are approximated by the square-root term exclusively, f1,2CH4,N2O=0. Equally, setting f1,3x=0 reduces Eq. () to the common linear expression often used for minor GHGs, with f2x becoming the radiative efficiency. This is used by FaIR to approximate the contribution from halogenated gases and the direct effects of aerosols (scaling with sulfate, organic carbon, and black carbon emissions). Fext covers any exogenous forcing, such as natural forcings (volcanic activity and solar cycles) and albedo effects. These are included in the model directly as forcing time series.

Beyond Eq. (), FaIR also includes other sources of radiative forcing which may depend on one or multiple species. It parametrises both tropospheric and stratospheric ozone contributions following as a linear function of methane; nitrous oxide and ozone-depleting substances (ODSs) concentrations; as well as nitrate aerosol, carbon monoxide, and volatile organic compounds (VOCs) emissions. Contributions from stratospheric water vapour, black carbon on snow, and aviation contrails are scaled linearly with, respectively, tropospheric methane concentrations, black carbon emissions, and aviation sector NO_x emissions. Finally, indirect aerosol forcing effects due to cloud interactions are approximated as the addition of a logarithmic term from sulfate aerosol emissions and a linear term from organic carbon and black carbon emissions.

FaIR v2.1 increased the model's flexibility by implementing the other three main approaches to radiative forcing: , and . Users can choose which scheme to use, with the model defaulting to as the most accurate among the four. The expression computing the forcing from aerosol-cloud interaction was also updated following , generalising it to potentially include the effects from more species.

FaIR calculates the temperature response using the IRM formulation of an n-layer EBM, similar to AR5-IR (see Sect. ). However, successive versions of FaIR have increased the complexity of the EBM. V2.0 increased the number of temperature components (or layers) from two to three, following the findings of and , which suggest three layers are better suited to emulate impulse-like forcing scenarios. Subsequently, version 2.1 adopted the model of , incorporating stochastic terms in the temperature and radiative forcing responses, as well as allowing for an arbitrarily large number of ocean layers greater than two. In the equivalent n-layer formulation, the three-layer case can be expressed as: 25C1dT1(t)dt=Ftot(t)-k1T1(t)-k2T1(t)-T2(t)+ξ(t)C2dT2(t)dt=k2T1(t)-T2(t)-ϵk3T2(t)-T3(t)C3dT3(t)dt=k3T2(t)-T3(t) where C1–3, k1–3 and T1–3 denote the heat capacities, heat transfer coefficients with the layer above (k1 being the climate feedback parameter) and temperature of the three layers respectively. ϵ is the deep ocean efficacy parameter as discussed in Sect. . The only difference with the standard 3-layer EBM, as described in Eq. (), is the addition of two stochastic disturbances emulating climate's internal variability: one directly affecting the temperature response (ξ), and another affecting the total radiative forcing term (Ftot), which results from the combination of the deterministic ERF determined earlier (Fdet) and a red-noise component (ζ) simulating time-correlated variations from the mean (ζ): 26Ftot=Fdet+ζ27dζdt=-γζ+η where γ is a parameter controlling the degree of temporal auto-correlation and η represents a white noise addition.

The new carbon cycle module was introduced in the latest version of the model, v1.2 , where further details can be found. Fundamentally, this module consists of a four-box model for the ocean-atmosphere system and an impulse response scheme for the land component. Section  and  provide more details on these types of models.

Starting with the ocean-atmosphere component, this is implemented as a four-box scheme (analogous to Eq.  with n=4): 28dc0dt=-η1hscs+η1h1c1+E-fdc1dt=η1hscs-η1+η2h1c1+η2h2c2dc2dt=η2h1c1-η2+η3h2c2+η3h3c3dc3dt=η3h2c2-η3h3c3 where ck is the excess carbon in layer k, hk is the depth of the layer, ηk is the exchange coefficient between layers k-1 and k, E represents anthropogenic emissions, and f denotes the carbon uptake by the land component. A peculiarity of this model is that the top layer (c0) represents both the atmospheric and ocean mixed layer carbon pools. Consequently, c0 is partitioned into atmospheric (ca) and oceanic (cs) excess carbon, with the distribution between them being calculated through a complex chemical equilibrium scheme that includes temperature feedbacks. Parameters hk and ηk have been calibrated such that the evolution of c0 tracks an equivalent four-constant IRM for airborne fraction in . The characteristic timescales are set to 1.271, 12.17, 59.52 and 236.5 years, calibrated on a three-dimensional ocean carbon cycle model .

Terrestrial carbon uptake is governed by an IRM with four characteristic timescales (τi), corresponding to four categories of land carbon: vegetation, wood, detritus and soil organic carbon. Carbon input to the terrestrial system originates from an NPP flux, which is modulated by a sigmoid function dependent on atmospheric CO₂ concentration to account for fertilisation effects (βf([CO2])). Thus, the land carbon uptake is expressed as: 29f(t)=∑i=04dcidt=∑i=04βf(t)NPP0Aĩτi-ciτi where NPP₀ is the pre-industrial net primary production, ci is the carbon anomaly of the ith land carbon category and Aĩ is the amplitude of the ith IRF. Notice that the coefficient Ai in Eq. () corresponds to the Aĩτi product in Eq. (). Values for τi and Aĩ are set to 2.9, 20, 2.2 and 100 years and 0.70211, 0.013414, -0.71846, and 0.0029323 yr⁻¹, respectively. These values were borrowed from , where the IRM was originally presented.

The model's representation of non-CO₂ sources of radiative forcing remains limited. It does not include non-CO₂ gas cycles, requiring the use of prescribed concentration time series for CH₄, N₂O and halogenated gases to calculate the resulting radiative forcing. This is set to change in a future V1.3 version, where a simple gas cycle model will predict non-CO₂ concentrations from emissions (private communication). Similarly, MCE includes radiative impacts of tropospheric and stratospheric ozone; stratospheric water vapour; aerosols, including volcanic aerosols; solar irradiance; contrails; and LULCC albedo as prescribed forcing time series.

The radiative forcing from CO₂ is calculated using the standard logarithmic expression for concentrations up to twice the pre-industrial: 30FCO2(x)=αln⁡CCO2(t)C0CO2 where α is a scaling parameter and x is the ratio of CO₂ concentration relative to the pre-industrial level. For higher concentrations, up to four times the pre-industrial level, MCE uses: 31F̃2<x<4CO2(x)=(β-1)FCO2(x)-2FCO2(2)⋅2FCO2(x)FCO2(2)-1+βFCO2(x) where β=F̃CO2(4)/FCO2(4) represents a model-dependent scaling factor for the transition from the first to the second doubling of CO₂. Both α and β have been calibrated to replicate results from CMIP models . For x>4, the quadratic term in Eq. () is omitted, and Eq. () is reused with an adjustment such that the forcing is continuous at x=4.

If non-CO₂ concentration time series are provided, MCE uses the expressions from to calculate the forcing from methane and nitrous oxide, and from for halogenated gases.

To translate the total radiative forcing (F) into a temperature anomaly (T), the model's default is a three-time IRM (a two-time IRM is also available) as described in Eq. (): 32T(t)=∫0tF(t′)λ∑i3Aiτiexp⁡-t-t′τidt′, where τi and Ai are the characteristic times and amplitudes of the ith decaying exponential used in the IRM, and λ is the climate feedback parameter. The three characteristic times τi are approximately 1, 10 and >100 years. Notably, similar to Eq. (), the amplitudes from Eq. () have been slightly redefined in Eq. (), so Ai in Eq. () correspond to Ai/λτi in Eq. ().

The only species whose atmospheric concentration is computed prognostically in WASP is carbon dioxide. All other species are included indirectly, either through radiative forcing data in V1 and V2, or through prescribed concentration data in V3. WASP's carbon cycle is an eight-box scheme, with five of those boxes dedicated to the ocean. For all boxes, carbon stocks are tracked using the carbon anomaly with respect to pre-industrial stocks.

The key mechanism determining the distribution of anthropogenic carbon across the system in this SCM is ocean carbon undersaturation . This measure indicates the amount of carbon the ocean needs to absorb to reach equilibrium with the atmosphere, and is calculated using a relatively complex scheme depending on atmospheric carbon content, cumulative airborne emissions, an equivalent carbon emissions term accounting for the ocean temperature-CO₂ solubility feedback and the buffered carbon inventory . The further the ocean is from equilibrium, the more carbon it absorbs, with the evolution of ocean layers towards an equilibrium with the OML and the atmosphere modelled via decaying exponentials. How rapidly each layer approaches equilibrium is dictated by the volume of each box and the characteristic restoring e-folding timescale of each ocean layer. This model also uses the amount of carbon in the atmosphere and the carbon undersaturation in the OML to simulate ocean acidification, using the carbonate chemistry solver by to estimate the pH change.

Comparatively, the land carbon cycle is simpler than its ocean counterpart. NPP transfers carbon from the atmosphere to a vegetation box, which eventually decays to a soil box through a litterfall flux and returns to the atmosphere via a heterotrophic respiration flux. The magnitude of NPP is influenced by temperature (linear dependence) and CO₂ fertilisation effects (log dependence). Similarly, the effects of surface warming on heterotrohic respiration are incorporated through a linear influence on soil residence time.

Originally, WASP included three sources of radiative forcing. First, atmospheric carbon dioxide, whose radiative forcing is estimated via the logarithm of the increase since pre-industrial concentrations. Second, Kyoto-protocol agents (WMGHGs and CFCs), taking a forcing timeseries from RCP scenarios as input. Third, non-Kyoto-protocol agents (mainly aerosols), scaled proportionally to Kyoto-protocol agents. This division is still employed if the model is used to run any RCP scenario. However, since its update to SSP scenarios in , WASP's representation of radiative forcing takes non-CO₂ species concentrations as input and distinguishes between the following agents: CO₂, calculated following the logarithmic expression in ; CH₄ and N₂O, following ; 27 halogenated species, using radiative efficiencies from ; and aerosols, including both direct and indirect contributions from black carbon, organic carbon, sulphates, nitrous oxides, ammonia and VOCs, borrowing the scheme from FaIR V1.3 . Additionally, since , the model includes the effects of solar and volcanic radiative forcings via forcing time series.

To represent the internal variability in Earth's energy imbalance, included a noise term in the model's radiative forcing with parameters tuned to emulate the monthly and annual root-mean-square energy imbalance in .

WASP follows the common energy balance model approach (see Eq. ), with the increase in surface temperature being proportional to the total radiative forcing minus the heat absorbed by the Earth system, which in this model is limited to the ocean. To determine this heat uptake, a similar approach to the carbon cycle is employed: total radiative forcing pushes the heat content of the atmosphere away from equilibrium with the OML, inducing a heat flux into this mixed layer and the four deeper ocean layers. This disequilibrium is quantified by computing the eventual heat content of the OML required to balance the instantaneous total radiative forcing through a linear equivalence relationship. The ocean heat uptake at each time step is then determined based on the difference between the current heat content of the OML and its equilibrium heat content. Similar to the carbon cycle, the flux of heat from the OML to deeper ocean layers is modelled by decaying exponentials that push the system towards a new equilibrium. The characteristic timescales for each ocean box are the same for both the heat and carbon schemes.

further refined the model's EBM by incorporating i forcing-agent-specific climate feedbacks that can vary independently on a set of j feedback processes (λi,j), instead of a single climate feedback applicable to all forcing agents (λ). In practice, this translated to a reformulation of Eq. () into: 33ΔT=1-N(t)F(t)∑iFiλplanck+∑jλi,j(t) where N is the heat flux towards the surface (dH/dt in Eq. ), λplanck is the Planck climate sensitivity , and F and Fi are the total and agent-specific radiative forcings. Values for the new parameters were constrained by constructing a large ensemble of simulations using ranges of climate feedbacks from CMIP5 models and applying observational constraints.

The EBM described so far produces a deterministic temperature response. However, since , this response has been further modified by a stochastic source of variability. Specifically, a noise term was introduced for both surface air temperature and sea surface temperature, which was calibrated to reproduce the magnitude and auto-correlation properties observed during the historical period.

Lacking a representation of any gas cycles beyond water vapour dynamics (see hydrological cycle below), GREB estimates global temperature anomaly through a 3-layer EBM (atmosphere, OML, subsurface ocean). This EBM possesses two main peculiarities that set it apart from other SCMs in this review. First, this EBM focuses on the surface energy balance, as opposed to other SCMs that focus on the top-of-atmosphere energy balance. This allows GREB to include explicitly turbulent heat fluxes like the sensible and latent heat fluxes. Second, its resolution is unusually high for an SCM, operating at 12h timesteps in a 3.75°×3.75° cell grid. For each cell, the surface temperature anomaly is governed by: 34CdTsurfdt=Fsolar+Fthermal+Flatent+Fsense+Focean+Fice+Fcorrect. These are all the forcing agents included in the model, which also determine the remaining prognostic variables: atmosphere temperature anomaly, subsurface temperature anomaly, humidity of surface layer, and thickness of ice cover since GREB-ISM . Processes associated with these agents are described below, except Fcorrect, which is an empirical heat flux to correct for model error. The cell heat capacity, C, varies depending on the nature of the cell (ice-free ocean, frozen ocean or land). Note this is a similar equation to that describing the typical SCM EBM (see Eq. ) with the only difference being that the temperature response (λT) is part of the forcing terms in Eq. (). Table 1 of offers a list of all prognostic and diagnostic GREB's variables, as well as the required boundary conditions.

These forcing terms are determined by several parameterised processes, which are illustrated in Fig. 2 from . These are:

Solar radiation: the absorbed incoming solar radiation (Fsolar) is determined by the 24 h average of radiation reaching the surface, modulated by the day of the year and surface as well as cloud albedo effects.

Hector includes a representation of the gas cycle for CO₂, N₂O, CH₄ and 27 halocarbons. Atmospheric concentrations for all these species except CO₂ are governed by the mass balance equation detailed in Sect. , incorporating anthropogenic emissions and sinks. Carbon concentrations are determined by the box carbon cycle model described below. Additionally, Hector estimates concentrations of tropospheric ozone based on methane concentrations and emissions of nitrogen oxides, carbon monoxide, and non-methane volatile organic compounds (NMVOCs). For full details, see Sect. 4 in .

The carbon cycle in Hector has remained largely unchanged since V1. There are four categories of carbon reservoirs: a well-mixed atmosphere, land, ocean and Earth. The latter represents long-term carbon storage, including fossil fuels and carbon capture, which, since V3, can be specified independently. A diagram of the model is provided in Fig. a. An interesting peculiarity of Hector is that the model performs a spin up stage before each run to ensure that its carbon cycle is in equilibrium.

The land component consists of five categories of carbon: vegetation, soil, detritus, permafrost and thawed soil. By default, the model operates with global carbon pools, but users can specify an arbitrarily large number of “biomes”, each with its carbon pools and parameter values. These parameters include a warming factor to convert the global temperature anomaly into a biome-specific temperature anomaly. If multiple biomes are configured, the processes described below apply independently for each biome.

Net primary production transfers carbon from the atmosphere to the land component, distributing it across three carbon pools: vegetation, detritus and soil. This is modulated by a logarithmic CO₂ fertilisation factor and a land-use change (LUC) factor, which accounts for vegetation loss and gain (added in V3). A fraction of the carbon in the vegetation pool flows to the detritus pool as litter, and a fraction of both vegetation and detritus flows to the soil pool. Both detritus and soil pools lose carbon to the atmosphere through heterotrophic respiration fluxes, modelled by first-order decay equations modulated by temperature and carbon pool size. Carbon residence times for detritus and soil pools are four and fifty years, respectively. As of V3, Hector allows for two independent gross LUC flows (rather than one net flow), one for carbon loss and one for carbon uptake, impacting all three carbon pools. Additionally, a permafrost module was added in V3, which is controlled by land temperature, and releases CO₂ and CH₄ into the atmosphere through an intermediate “thawed soil” pool.

The ocean component of Hector's carbon cycle is based on the work of and . As depicted in Fig. a, the ocean carbon cycle is divided into four boxes: two surface boxes for low and high latitudes, an intermediate box and a deep ocean box. The exchange of carbon between the atmosphere and the ocean surface boxes is governed by a linear function of the differential in carbon partial pressures between the two reservoirs, further influenced by temperature and salinity. This carbonate scheme resolves the following prognostic variables: carbon partial pressure (pCO₂), pH, concentrations of HCO3- and CO32-, and saturation states of aragonite (ΩAR) and calcite (Ωca). Typically, the higher-latitude box, representing subpolar gyres, acts as a carbon sink, while the lower-latitude box outgasses carbon. Once dissolved into the surface boxes, the carbon flows to the intermediate and deep ocean (if not outgassed back to the atmosphere) through advection and mass exchange, simulating a simple thermohaline circulation.

At each time step, Hector computes the total radiative forcing relative to the year 1750, including 39 sources (see Table S1 in ): concentrations of carbon dioxide, methane, nitrous oxide, stratospheric water vapour, tropospheric ozone, and 27 halogenated compounds; emissions of black carbon, ammonia, organic carbon, sulphur dioxide; as well as simulating aerosol cloud interactions and taking externally defined time series for forcing related to LULCC albedo, volcanic activity and miscellaneous sources. The last contribution is zero by default, but allows the user to specify an additional forcing time series. Since V3, Hector uses the forcing equations from AR6 to estimate the forcing for most species (see Supplement in ), except for tropospheric O₃ and stratospheric H₂O, which still follow the formulation in using radiative efficiency factors.

Since V2, Hector has used the Diffusion Ocean Energy balance CLIMate (DOECLIM) model to estimate temperature anomaly. Originally formulated by , DOECLIM combines a zero-dimensional EBM with a one-dimensional ocean heat diffusion scheme, as depicted in Fig. b. Near-surface air temperature anomaly is calculated via a linear relationship with the total radiative forcing as described in Sect. , using a scaling climate feedback parameter λ. The DOECLIM scheme distinguishes between forcing over land (Fland) and ocean (Focean), although Hector assumes both these forcings are equal to the global forcing. Heat transfer to the Earth's system is then simulated through a standard two-box scheme coupled to a 1D diffusion ocean with uniform diffusivity across the water column. The two boxes correspond to the combination of the upper land layer and the atmosphere over land, and the ocean mixed layer and the atmosphere over ocean. These two boxes are also allowed to exchange heat between them based on their temperature gradient. The model's transient behaviour is primarily determined by the heat exchange with the deep ocean, due to its much larger heat capacity. The model's estimation of global temperature is the area-weighted average of these land and ocean box temperatures.

Atmospheric CO₂ concentrations in CICERO-SCM are determined using a carbon cycle module following IRM principles (see Sect. ). The atmosphere-ocean carbon exchange is simulated using the mixed-layer IRM described by , while terrestrial carbon uptake is modelled through an IRM that reduces the “effective” emissions seen by the ocean component.

The scheme by simulates ocean carbon uptake in two stages. First, it calculates the difference in carbon partial pressures between atmosphere and the OML to estimate the carbon uptake by the OML (fo(t)), where the ocean partial pressure is approximated via a parametrisation modulated by mean global concentration of Dissolved Organic Carbon (DOC (δm(t)). Second, it calculates this global concentration of DOC in the OML using an IRM to approximate the transport to the deep ocean. The IRM is represented by the convolution of the historical carbon uptake by the OML with an IRF representing deep ocean uptake (R(t)), akin to Eq. (): 35δm(t)=cd∫t0tfo(u)R(t-u)du where c is a coefficient for unit conversion and d represents the depth of the mixed layer. The original calibration of this scheme by was conducted using data from two models: the HILDA model and the Princeton 3D model . While CICERO-SCM employs the HILDA calibration, MAGICC – another SCM utilising this scheme, reviewed in Sect.  – employs the Princeton calibration.

For the terrestrial component, CICERO-SCM also adopts an IRM approach, distinguishing itself from most other models in this review. Specifically, land carbon uptake is modelled through an “effective” NPP flux, which modifies the flux from anthropogenic emissions (E): 36Eeff(t)=E(t)-NPPeff(t). The effective NPP term incorporates a CO₂-dependent term to account for carbon fertilisation effects, along with a convolution term to represent carbon returning to the atmosphere through overturning of terrestrial carbon : 37NPPeff(t)=NPP(t)-∫-∞tNPP(t′)rb(t-t′)dt′38NPP(t)=NPP0⋅β⋅ln⁡CO2(t)278ppm. It is this effective emissions term, Eeff, that the ocean component uses to calculate the atmosphere-ocean carbon exchange, thus accounting for the terrestrial component of the carbon cycle. A full description of this scheme can be found in .

Concentrations for all remaining non-CO₂ species (CH₄, N₂O, and 27 halogenated species) in CICERO-SCM are governed by mass balance equations as described in Sect. . The model includes a time-varying characteristic lifetime (τCH4) for methane, considering three contributions: OH chemistry, stratospheric sink and soil sink, as per Eq. ().

In terms of radiative forcing, CICERO-SCM uses the expressions found in to estimate the radiative forcing induced by elevated atmospheric concentrations of CO₂, CH₄ and N₂O, accounting for stratospheric adjustments and the overlap between absorption bands. Additionally, efficiency factors for each species can be used to account for tropospheric adjustments, producing an estimation for effective radiative forcings. Forcing from 27 other WMGHGs (SF₆, CFCs, HFCs, HCFCs) is calculated through the usual linear efficiency approach, scaling with the concentration anomaly since pre-industrial. Other prognostic sources of forcing included in the model are tropospheric and stratospheric ozone, stratospheric water vapour and aerosols. The contribution to total forcing of tropospheric ozone scales based on its concentrations, which are estimated based on methane concentrations and emissions of NO_x, CO, and NMVOCs. In the case of stratospheric ozone concentration, it decreases based on the concentration of chlorine- and bromine-containing species three years prior to the evaluation step, to account for atmospheric transport. Forcing related to stratospheric water vapour scales linearly with methane concentrations, while aerosol forcing scales linearly with emissions of sulfates, fossil fuels, biofuels, black carbon, organic carbon and biomass burning. Finally, albedo changes and natural forcing agents (volcanic aerosols and solar irradiance) can be added through prescribed forcing timeseries.

As described below, CICERO-SCM possesses a hemispheric EBM, which requires a hemispheric partition of radiative forcing. This is only relevant for three forcing agents whose forcings are not split equally: tropospheric O₃, albedo changes and aerosols. The first is weighted by 1.45 for the Northern Hemisphere and 0.55 for the Southern Hemisphere, following , while the other two are split following the results from .

To convert the total radiative forcing into a temperature response, the CICERO-SCM implements a semi-hemispheric UD-EBM following . As illustrated in Fig. , this UD-EBM consists of four main parts for each hemisphere: atmosphere, the OML (default depth 107 m), deep-ocean layers (39 layers), and a sinking column of water negligible in area compared to the ocean. The purpose of this water column is to represent thermohaline circulation, simulating polar deep water formation which later upwells through the ocean layers. Using the standard EBM relationships described in Sect. , total hemispheric radiative forcing is translated into temperature anomaly and additional heat content in the atmosphere. This heat flows into the OML and eventually into the deep ocean through thermal diffusion across ocean layers, as well as via the sinking water column mentioned earlier. Water in this column starts at the OML, flowing downwards to the ocean bottom, then upwelling through the ocean layers back to the OML, restarting the cycle. The upwelling velocity in CICERO-SCM depends on the global temperature anomaly, linearly decreasing as the temperature anomaly increases, following . While the land is not integrated as a heat-exchanging component, the model does incorporate the difference in ocean extension between hemispheres. Atmospheric interhemispheric heat exchange is included in the model, but it is rarely used. A full mathematical description of the UD-EBM in CICERO-SCM can be found in .

This is a similar scheme to the EBM used in MAGICC, as described in , with two main differences: entrainment (or alternatively, depth-dependent area profile, as discussed in Sect. ) was never added to the CICERO-SCM, and heat exchange with the land is disregarded. An advantage of UD-EBM schemes like these is the improved ability to estimate both ocean temperature anomaly and ocean heat content, as a representation of the different ocean layers and their temperatures exists, and a more flexible inclusion of forcing with possible hemispheric variations.

MAGICC offers a comprehensive treatment of the various agents currently believed to influence the climate. MAGICC6 calculates concentrations for 30 species (expanded to 43 in MAGICC7) by simulating the gas cycles for CO₂, CH₄, N₂O, and other 28 halogenated gases (40 in MAGICC7). Generally, the standard approach of combining emissions with sinks and using decaying exponentials with characteristic lifetimes is employed in this model to simulate gas cycles (see Sect. ). However, MAGICC's treatment of gas cycles has accrued modifications over the years, making it relatively complex. Generally, these modifications involve additional parameterisations enabling the modification of species lifetimes over time, to account for phenomena such as interaction with OH radicals, increased Brewer-Dobson circulation , and interactions between species. These representations were further refined in MAGICC7. Another peculiarity of this model is that it resolves hemispheric emissions and concentrations for non-well-mixed GHGs, a consequence of the hemispheric EBM it possesses. Due to the complexity and number of enhancements a full description is outside the scope of this review and the reader is directed to for full details.

The carbon cycle module consists of an ocean component following the implementation of calibrated to the Princeton 3D model results (see Sect. ) and a three-box terrestrial component as outlined in and described in . These boxes represent global vegetation, litter and soil carbon reservoirs, each exchanging carbon with the atmosphere. A diagram illustrating the various carbon pools and associated fluxes is provided in Fig. . These fluxes are:

NPP flux: this flux transfers carbon from the atmosphere to the vegetation (35 %), litter (60 %), and soil (5 %) pools. This distribution of NPP across different pools, along with the similar distribution of the litter flux below, is atypical in SCM carbon cycles which usually implement this flux taking carbon exclusively to the vegetation pool. The objective is to account for the long time steps that SCM are usually run with (usually one year). By including effects not only in the target pool (vegetation), but also in subsequent pools in the carbon cycle, MAGICC aims to create a carbon cycle model that is more sensitive to NPP and litter flux changes.

The litter, deforestation and respiration flows in MAGICC are proportional to the carbon content in their respective source pools and are governed by associated turnover times. These turnover times are constant, which implies that following a land-use change event, the carbon in the various pools will asymptotically return to their original levels, assuming no further changes occur. This behaviour effectively simulates a regrowth process after deforestation. To better approximate real-world dynamics where land-use changes usually result in persistent alterations to carbon stocks, the MAGICC carbon cycle module uses a regrowth fraction parameter to adjust the turnover time and thereby allow for partial regrowth . The scheme was further modified by , where a direct impact on NPP by the extent of the deforestation was implemented, thereby impacting long-term equilibrium in carbon stocks.

The return to original carbon content in the carbon pools is contingent on stable climatic conditions, as the model incorporates carbon- and climate-carbon feedbacks. Specifically, the model resolves two key processes: CO₂ fertilisation of NPP and temperature-induced changes in NPP, respiration and decomposition fluxes. CO₂ fertilisation can be simulated using the usual logarithmic formulation, a hyperbolic formulation, a sigmodial formulation since , or a linear combination of the three. Temperature effects are modelled as an exponential modulation in the aforementioned flows in response to the temperature anomaly, with the added possibility of a sigmodial modulation since . For a more detailed explanation, refer to the Appendix A1.1 in and .

The terrestrial carbon cycle described here was further enhanced by to include the limiting effects of nitrogen that have been observed in more complex models . MAGICC is, therefore, the first and only SCM in this review to include a nitrogen cycle and emulate its impact on the carbon cycle. This is a relatively complex scheme, and only a brief summary is offered here. It mirrors the structure of the MAGICC's carbon cycle, comprising four global nitrogen pools: vegetation, litter, soil, and mineral. The flow of nitrogen through the different pools is depicted in Fig.  and behaves as follows: similarly to the NPP flux in the carbon cycle, two fluxes transport nitrogen from the inorganic pools into the organic pools (vegetation, litter and soil), plant uptake from the mineral pool and biological nitrogen fixation from the atmosphere. Then, vegetation loses nitrogen to the litter and soil pools through a litter production flux. Litter loses nitrogen to the soil and mineral pools through litter decomposition. Soil loses carbon to the mineral pool through soil respiration. Additionally, the organic pools can lose carbon to the atmosphere through an anthropogenic land use emission flux. Finally, the mineral pool can gain nitrogen through fertiliser application and through atmospheric deposition (from the atmosphere pool), and lose it through a mineral loss flux. Similarly to the carbon cycle, most of these fluxes are governed by first order decay functions with characteristic turnover times.

This nitrogen cycle is coupled to the carbon cycle mainly through the NPP flux, which is modulated by the possible plant uptake of nitrogen. This uptake is computed as a function of the size of NPP (to account for declining carbon : nitrogen ratios), nitrogen availability and temperature. The nitrogen availability, in turn, is approximated using the fluxes in the nitrogen cycle described above.

Since MAGICC7, , the carbon cycle also possesses a permafrost module based on the work of . This module divides the permafrost stocks into a number of zonal (latitudinal) bands – 50 by default – each with different carbon content and thawing thresholds. When the local temperature excedes the thawing threshold, CO₂ (and potentially CH₄) is released into the atmosphere. The quantity of gases released depends on the type of soil (mineral or peatland) and the temperature. This is a relatively complex module, and readers are referred to for further details.

For WMGHGs, MAGICC uses standard methods from the literature. MAGICC6 employed the usual logarithmic relation to calculate CO₂ forcing , and followed for the combined CH₄ and N₂O forcing. These forcings were further refined in MAGICC7, adopting the scheme from to compute forcing for these three species. For halogenated gases a radiative efficiency approach is taken, multiplying this factor by concentrations. MAGICC accounts for the direct and indirect contributions of tropospheric aerosols on radiative forcing directly from their emissions, given their short atmospheric lifetimes. The direct contribution is estimated as a linear relationship between concentrations and forcing, while the indirect effects are modelled by using optical thickness timeseries for the relevant species: sulfates, nitrates, black carbon and organic carbon. In addition, the model includes the contributions of both tropospheric and stratospheric ozone, which are simulated through simple relationships based on ozone concentrations. Similarly, forcing from stratospheric water vapour due to methane-induced enhancement is estimated linearly (default 15 %) with the pure methane forcing (without absorption band overlaps). Natural forcings (volcanic aerosols and solar irradiance), as well as LULCC albedo effects on forcing, can be added as prescribed time series. Full details can be found in with the MAGICC7 modifications described in . Finally, MAGICC also includes a contrail scheme that estimates forcing effects from aviation emissions supplied by the user. However, this is seldom used and mentioned in the literature.

Each individual forcing contribution is assigned an efficacy value to account for indirect effects, leading to an estimation of ERF, which is subsequently used to estimate temperature anomalies using the model's EBM. These efficacies are allowed to vary over time and space, with potential different values for each of the hemispheric ocean and land boxes. The hemispheric partition is a consequence of the EBM employed by the model, although only three species have hemispheric differences in their forcing contributions: tropospheric ozone, halogenated gases, and aerosols. Hemispheric differences of tropospheric ozone and aerosols are a consequence of the different hemispheric emissions and concentration for these species, while the difference for halogenated gases is dependent on the species lifetime, following .

MAGICC's EBM traces back to the origins of the model, with a hemispheric UD-EBM initially formulated by , based on the work of . This hemispheric separation can be useful for spatially inhomogeneous forcings associated with human activities such as aerosols and tropospheric ozone. A similar scheme would be adopted by the CICERO-SCM years later, as described in Sect.  where a summary on the basic principles on how it works can be found. There are some differences however. Unlike CICERO-SCM, MAGICC considers heat exchange between land and ocean. In fact, since MAGICC6, varying heat-exchange coefficients between land and ocean have been employed as a mechanism to alter the temporal profile of the model's effective climate sensitivity. Additionally, the model allows for time-dependent feedback parameters to modify its climate sensitivity over time.

Originally, most of the parameters associated with this scheme were fixed, but they were gradually relaxed overtime. For instance, upwelling and downwelling rates were allowed to evolve in time . Starting with MAGICC6, a warming-dependent gradient of thermal diffusivity was introduced to account for the increased stratification of the ocean as temperatures rise. This version also saw the inclusion of the “entrainment” component of the module. This entrainment mechanism became necessary with the introduction of a depth-dependent area profile in the ocean column, as illustrated in Fig. b. To satisfy conservation of mass with a vertically constant upwelling rate, water had to be added, leading to the incorporation of water entrainment from the sinking column into each of the ocean layers (the default being 50 layers).

The carbon cycle in SCM4OPT integrates various components that have been discussed elsewhere in this review. The land component, for instance, employs the same three-box model used in MAGICC6 , which includes carbon pools for the vegetation, detritus and soil, as described in Sect. . The key distinction in SCM4OPT compared to MAGICC lies in its representation of forest regrowth, which is modelled as a variable with a linear dependence on the relaxation times of each carbon pool, rather than through a parameterisation of permanent deforestation.

Similarly, the ocean carbon cycle in SCM4OPT initially followed the approach of , implementing the OML-IRM of . However, in SCM4OPT version 2, this scheme was replaced with a four-box ocean carbon cycle model, as implemented in Hector v1.1 , and described in Sect. , where more details can be found. A full description of both the land and ocean carbon cycles in SCM4OPT is provided in the Supplement of .

Over time, the number of species whose concentrations are determined from emissions in this model has expanded. The latest Version 3.3 included a representation of CO₂, CH₄, N₂O, 39 halogenated gases, tropospheric and stratospheric ozone, and aerosols (comprising SO₄, black carbon, nitrates, and primary and secondary organic aerosols). SCM4OPT adopts the conventional approach of defining mass balance models with emissions, sinks, and lifetimes (see Sect. ) for most species – CH₄, N₂O and halogenated gases – with some ad-hoc expressions for aerosols and ozone. The aerosol concentration scheme is based on a different SCM, OSCAR .

Similarly to the representation of WMGHGs, the number of radiative forcing agents included in SCM4OPT has increased over time. Version 3.3 included radiative forcing effects from CO₂, CH₄, N₂O, 39 halogenated gases, aerosols (direct) including mineral dust, clouds, stratospheric and tropospheric ozone, stratospheric water vapour, land albedo, black carbon on snow, as well as natural forcings (solar and volcanic). The forcing emulation generally follows expressions from the literature. Contributions from CO₂, CH₄ and N₂O are calculated following the formulation, although was used in V3.3 alongside this formulation to estimate the CH₄ contribution in a Monte Carlo simulation; direct effects of aerosols, clouds, tropospheric ozone and land albedo effects calculation follows OSCAR's ; the contribution from stratospheric ozone is based on the equivalent effective stratospheric chlorine concentration, following ; impacts from halogenated gases are estimated through a radiative efficiency approach to species concentrations, after MAGICC6; and the contribution from black carbon on snow is linearly scaled with black carbon emissions. Natural forcings can be included through prescribed forcing time series. The latest comprehensive reference for the forcing calculation in SCM4OPT is provided in the Supplement of , with a subsequent modification to account for the effects of overlapping absorption bands of CO₂, CH₄ and N₂O .

The conversion of total radiative forcing into a temperature response was initially performed using a two-box EBM as in the original DICE model. However, in SCM4OPT v2.0, this component was replaced by the Diffusion Ocean Energy balance CLIMate (DOECLIM) model . As discussed previously in Sect.  on Hector, where more details can be found, DOECLIM couples a 0D energy balance model with a 1D ocean heat diffusion scheme. In V3.3, an observationally-constrained statistical model was introduced to correct for temperature biases stemming from ocean variability using an ENSO index. This is similar to EM-GC's use of ocean indices to account for natural variability; however SCM4OPT is restricted to ENSO alone. As a result, it faces the same limitation: it can only be applied in historical simulations, where the ENSO index is available, since there is currently no reliable method for projecting the index into the future.

The carbon cycle in ACC2 consists of four-box representations for both the terrestrial and oceanic components, which are coupled together through the OML-atmosphere box. Details can be found in and , both offering the same model description.

For the ocean component, the model simulates progressive carbon uptake using a combined OML-atmosphere layer and three deep ocean layers. The combination of atmosphere and mixed-ocean layer is justified by the short time required for these carbon reservoirs to reach equilibrium, which is significantly shorter than the one-year timestep used in the model. Carbon emissions from fossil fuels and land-use change are added to this layer. A relatively complex carbonate chemistry scheme then determines the amount of dissolved inorganic carbon (DIC) in the OML, as well as its pH. The model also accounts for the effects of rising OML temperatures on carbonate uptake, with a recommended model emulation range of up to four times the pre-industrial level of atmospheric concentrations. Carbon is subsequently exported to the deeper ocean layers through diffusion. The model parameters were calibrated using the IRF for ocean carbon uptake from (R01 experiment), which emulates the response of the HAmburg Model of the Ocean Carbon Cycle version 3i (HAMOCC 3i) to a small increase in atmospheric carbon.

Similarly, the terrestrial carbon cycle is represented by four boxes that roughly correspond to vegetation, detritus, wood and soil organic carbon. These boxes interact exclusively with the OML-atmosphere layer rather than with each other. This atypical structure is a consequence of the IRM used in the model for heterotrophic respiration, which is a diagonalisation of the Bern-CC model . The diagonalisation decouples the carbon reservoirs from each other, allowing a single interaction with the OML-atmosphere layer. The downside of this diagonalisation is that the resulting boxes are no longer a representation of physical biospheric reservoirs. This is similar to the relationship between n-time-constant temperature IRMs and n-layer temperature models discussed in Sect. . Each box has two processes: (i) a carbon gain term associated with NPP, modulated by the standard logarithmic expression to account for CO₂ fertilisation effects; and (ii) a carbon loss term representing the turnover of carbon within the box, characterised by a decay time that simulates the effects of respiration. The carbon loss term also incorporates temperature effects through an exponential factor based on the land surface temperature anomaly.

For all other GHGs, ACC2 employs the standard mass balance treatment to simulate their life cycle (see Sect. ). This includes CH₄ (with a characteristic lifetime accounting for different sink times through Eq. ), N₂O, and 30 halogenated species. ACC2 also includes a representation of OH and tropospheric ozone concentrations, which follow the OxComp Workshop results , and affect GHG concentrations. Table 2.1 in presents a useful summary of GHG concentration calculations in this model.

ACC2 includes a comprehensive list of radiative forcing agents: CO₂, CH₄, N₂O, 30 species of halogenated gases (29 halocarbons and SF₆), tropospheric and stratospheric ozone, stratospheric water vapour and aerosols (sulfate and carbonaceous, plus indirect effects). The model generally adheres to standard treatments for these species, primarily based on and . For instance, it follows to calculate the contribution from CH₄ and N₂O, accounting for overlapping effects, as well as using its logarithmic expression for the CO₂ forcing. Contributions from halogenated species are scaled based on their radiative efficiencies . Aerosol forcing is parametrised as a function of SO₂, organic carbon and black carbon, based on the work of and . Natural forcings, volcanic aerosols and solar irradiance, are included as prescribed forcing time series, using estimations from and , respectively. Since the model uses DOECLIM as its EBM, the radiative forcing must be separated between its land and ocean components. All forcing contributions are roughly split in half, except the carbonaceous aerosol and the tropospheric ozone contributions, which are divided following . A full description of non-CO₂ and forcing estimation is provided in Sect. 2.2 of , with a useful summary in Table 2.1 of the same publication.

ACC2 supports two different EBMs to compute the temperature response to a given total radiative forcing. One option is a box-based formulation of the forcing describing the temperature response to forcing from , which is recommended only for inverse calculations aiming to estimate the climate sensitivity. The default and recommended EBM for general model use, particularly when a computationally lighter model is necessary, is the DOECLIM scheme presented in and discussed in Sect. . This consists of a zero-dimensional EBM with a one-dimensional diffusion scheme to represent the heat exchange with the deep ocean. For more details about this scheme see, in increasingly level of detail, Sect.  in this text, Sect. 2.3 in , and the original publication of .

Generally speaking, this model operates with anomalies with respect to the pre-industrial state, rather than absolute quantities. This is the case for OSCAR's carbon cycle, which was broadly established in its current form by V2.2 , with V3.1 adding a few minor fluxes and recalibrating its pre-industrial steady state to output from the GCB 2018 . Its primary strength resides in its land component, which includes a complex representation of LULCC disturbances across model boxes that represent different regions and “biomes” within regions. The ocean component is based on the work of with several modifications.

In OSCAR, land is divided into multiple boxes representing a number of regions, each containing several biomes. These boxes provide an average characterisation for each biome within each region. Users can choose the number of biomes and, since V3, of regions; however, typical configurations include around ten regions and five biomes. For instance, used ten regions following , and five biomes: forests, other natural lands (grasslands, shrublands, bare soil), croplands, pastures and urban lands. This can vary, however. For instance, the latest GCBs include 210 regions. Each biome in the model contains three carbon pools: vegetation, litter and soil. The flow of carbon through these boxes is governed by the following fluxes:

NPP: carbon flux from atmosphere to vegetation, modulated by temperature (linear relationship), precipitation (linear) and CO₂ concentration (logarithmic or hyperbolic).

Through an elaborate book-keeping approach, OSCAR tracks carbon across each region-biome pair, and emulates the movement of carbon associated with LULCC disturbances. These disturbances are currently limited to anthropogenic activities (no dynamic vegetation). For each biome and region, three wood product pools are defined that receive carbon after a LULCC disturbance. These pools are characterised by distinct turnover times that determine the rate at which carbon is subsequently released into the atmosphere. Broadly speaking, these pools represent fuel wood (∼1 year), pulp-based products (∼ a few years) and hardwood-based products (∼ dozens of years). The allocation of carbon to each wood pool, as well as the disturbance-induced movement of carbon between vegetation, litter and soil carbon pools across different biomes, depend on the specific nature of the LULCC disturbance. The model includes three types of disturbances: land-cover change, wood harvest and shifting cultivation. Given the large number of potential combinations (five biomes, three carbon pools and three wood pools) the detailed mechanics of these transitions can be complex, and interested readers are referred to the Appendix of for further information.

OSCAR's ocean component of the carbon cycle builds on the mixed-layer pulse response function proposed by , discussed in Sect. . However, several modifications have been implemented to improve upon the original Joos model:

Following , OSCAR replaced the convolution of the atmosphere-ocean flux in (Eq. ), with an equivalent box model. These boxes represent different turnover times of carbon mixing between the mixed layer and the deep ocean, rather than different ocean basins.

Since V2.3 , OSCAR also includes a representation of permafrost carbon stocks and potential thawing. This addition introduced a frozen carbon pool for two regions (Eurasia and North America). Thawing is driven by local temperature, following an empirical S-shaped function. Similarly to the LULCC module, thawed carbon is not immediately emitted to the atmosphere, but is instead distributed among thawed carbon pools (default number is three), each with its characteristic respiration turnover time. Respiration from these thawed pools is influenced by local temperatures, following the same relationships as standard soil respiration functions, but with different parameter values.

Beyond CO₂, OSCAR includes a comprehensive list of GHGs: methane, nitrous oxide and 37 halogenated compounds. It uses the usual one-box approach with emissions and sinks (see Sect. ), along with Eq. () to combine different sink contributions. Additionally, OSCAR implements time-dependent lifetimes with relatively complex dependencies on temperature and other atmospheric species. Due to this complexity, only a brief summary is offered here, with full details available in . A notable peculiarity of OSCAR is its implementation of stratospheric concentrations for methane, nitrous oxide and halogenated compounds, which follows a time-lagged linearisation of the corresponding tropospheric concentration based on .

The methane representation accounts for four sinks: OH tropospheric oxidation, stratospheric loss, soil uptake and OML uptake. These oxidation processes are modelled through a complex scheme depending on several quantities: temperature, atmospheric methane concentrations, stratospheric ozone concentrations (related to OH production), and emission of three ozone precursors (NO_x, CO, NMVOCs), although not all quantities are relevant for all processes. It also includes an estimate of wetland methane emissions, calculating both changes in wetland area and in emissions per unit of area. Wetland area depends on atmospheric CO₂, local temperature, and local precipitation, while emissions per unit of area scale based on total heterotrophic respiration since v3.1 .

Similar schemes are used to emulate the concentrations of N₂O and halogenated species. For N₂O, a single atmospheric sink – the stratospheric sink – is considered. In contrast, halogenated species are subject to three atmospheric sinks: tropospheric OH oxidation, stratospheric oxidation, and surface oxidation comprising both land and ocean contributions. The lifetimes associated with these sinks for both N₂O and halogens can vary based on several climate factors, such as the stratospheric concentrations of the corresponding species, the equivalent effective stratospheric chlorine and global temperature, with the specific dependencies varying based on the particular sink. For species where some of these processes are negligible or irrelevant, an infinite lifetime is defined.

OSCAR includes a comprehensive list of forcing agents: carbon dioxide, methane, nitrous oxide, 37 halogenated species, tropospheric and stratospheric ozone, aerosols (direct and indirect), stratospheric water vapour, albedo change, aviation contrails, volcanic emissions and solar irradiance. The last three are included through prescribed forcing time series, while the rest are computed prognostically by the model.

The calculation of radiative forcing in OSCAR is, generally, simpler than its treatments of gas cycles. In the latest available version, V3.3, OSCAR was updated to use the AR6 expressions to estimate the ERF of long-lived GHGs. However, since this version has not yet been described in the literature (aside from a brief mention in the model's CHANGELOG), and earlier versions remain widely used, we provide a brief overview of the pre-V3.3 forcing calculations, with the caveat that post-V3.3 versions will incorporate AR6 parameterisations.

In earlier versions, CO₂ forcing is estimated via the standard logarithmic formula , whereas CH₄ and N₂O forcings are derived from a modified square root expression accounting for absorption band overlaps, following . This square root expression (without overlap effects) is also used to estimate forcing related to water vapour in the stratosphere, based on lagged stratospheric methane concentrations (changed to a simpler linear scaling with methane concentrations in V3.3).

Contributions from halogenated compounds, tropospheric ozone and stratospheric ozone are assumed to scale linearly with the relevant atmospheric species. Ozone contributions are calculated directly from its atmospheric burden, which is based on methane concentrations, ozone precursors (NO_x, CO, VOC), stratospheric concentrations of chlorine, bromine, and nitrous oxide, as well as global temperature. Ozone contributions, like aerosols', are regionalised with region-specific weights, although these are logically-different regions from those in the biospheric module. Instead, OSCAR uses values from the four Hemispheric Transport of Air Pollution (HTAP) regions to derive the values for these weights. In terms of aerosol forcing, OSCAR considers both direct and indirect contributions. The direct contribution follows the linear radiative efficiency approach and considers five sources of anthropogenic aerosols: sulfate aerosols, primary organic aerosols, black carbon, nitrate aerosols, and secondary organic aerosols. Each of these contributions depends on two precursors and global temperature. The indirect effects are scaled linearly on black carbon emissions and logarithmically with the five sources mentioned earlier. Finally, the land surface albedo is based on the LULCC module, which estimates regional changes of land cover. OSCAR takes this estimations, as well as estimations of black carbon deposition on snow and averaged yearly albedo, and calculates the forcing contribution via the associated radiative efficiency. Three forcing agents, aviation contrails, volcanic aerosols, and solar irradiance, are taken directly as forcing time series from . For the last two, OSCAR uses forcing efficacies to compute the associated ERF.

OSCAR employs a two-box formulation of an IRM as EBM (see Sect. ). The two constituent boxes are the global surface and the deep ocean. An exchange coefficient governs the heat exchange between the two, while the climate sensitivity parameter translates the total radiative forcing into a heat flux. This formulation also includes two inertia factors to modulate the time lag in the temperature response of the two layers. Thus, an estimation for the global temperature anomaly is produced, which is then scaled linearly to the different regions in the model. Additionally, OSCAR also computes an estimation for the ocean heat content, based on the total forcing, the ocean temperature, and the climate sensitivity.

ESMICON possesses a variable called “concentration of greenhouse gases in the atmosphere” which is made up of two contributions: carbon and methane. Furthermore, methane is only contemplated as a product of permafrost thawing, and it is expressed in CO₂ equivalent units, like the GHG concentration variable. Therefore, ESMICON only resolves the ecosytem cycle of CO₂. Its system dynamics framework is similar to a box-based model (see Sect. ), being based on inventories and fluxes.

The carbon cycle in ESMICON consists of seven pools: fossil reserves, atmosphere, biomass, permafrost, surface ocean, deep ocean and sediments. Figure  shows a diagram of these pools in its right side. As a module following system dynamics principles, the carbon cycle in ESMICON is governed by a series of feedback processes :

Anthropogenic emissions: carbon is emitted into the atmosphere as a result of fossil fuel combustion. This is required input by the model.

While not using explicitly an EBM framework to determine temperature anomalies, ESMICON uses instead an analogous system dynamics framework to determine those anomalies based on the distribution of heat across the system. In particular, it uses the heat inventories from the atmosphere and surface, along with the associated heat capacities, to estimate GMST. The key distinction between the more common EBM and ESMICON's temperature module is the absence of an explicit radiative forcing concept and the parametrisation of feedbacks through a λ parameter, as in Eq. (). Instead, state variables in the system, such as CO₂ concentrations and ice volume, impact directly various feedback loops driving the model's dynamics, and, ultimately, the heat distribution. The eight primary feedback loops, as identified in Fig. , are:

Higher GMST leads to increased outgoing radiation to space (negative feedback).

Figure  provides a diagram of these eight primary feedbacks loops, which explain most of the dynamics in the ESMICON SCM. More processes are included in the model (e.g., low and high cloud reflection and radiation, convection, evaporation, volcanic aerosols, sunspots, desertification), with a comprehensive list available in the original publication. Notably, these feedback loops are often non-linear, with many processes in ESMICON exhibiting time delays and saturation or depletion effects.