In this study we used Hector v 2.3.0 as the parent model, providing both of HIRM's primary and only inputs. We selected Hector because it is open source, well documented, fast-executing, and has a structure that makes it easy to obtain “clean” IRFs from model runs (Schwarber et al., 2019). As noted above, however, HIRM can be coupled with any parent model that can provide its inputs. Hector has been well documented (Hartin et al., 2015), but we provide a brief summary here.

Hector (Hartin et al., 2015; Link et al., 2019) is an open-source, process-based SCM carbon–climate model available at https://github.com/jgcri/hector (last access: 11 January 2021). The model is written in C++ and has an object-oriented structure, allowing for substitutions of different model components; it has both internal and external automated testing, e.g., enforced unit-checking, which provide robustness and quality assurance. Hector models carbon and energy flows between the ocean, atmosphere, and terrestrial biosphere, starting with a preindustrial steady-state system that is then perturbed by anthropogenic emissions provided as input files. The model runs on an annual time stamp, although the carbon cycle as an adaptive time step solver to ensure smooth numerical changes when fluxes (primarily ocean uptake) are large. The terrestrial carbon cycle is divided into biota, litter, and soil across multiple biomes; the ocean features surface, intermediate, and deep pools in different hemispheres, with heat uptake governed by an implementation of the DOECLIM (Kriegler, 2005; Urban et al., 2014) one-dimensional heat diffusion sub-model. Hector models the dynamics of 37 different radiative forcing agents. The total radiative forcing in turn affects global temperature change, with all of Hector's radiative forcing agents exhibiting the same temperature response to change in radiative forcing. In effect, Hector can be considered to interpret forcing assumptions as effective radiative forcing values, which are more closely related to surface temperature changes than the previously used values for stratospheric-adjusted radiative forcing (Richardson et al., 2019; Smith et al., 2018b). This has no impact on the model dynamics that are our focus here and only impacts how numerical values are selected as input settings. Note that Hector also assumes that the temporal shape of the response function is the same for all forcers, a simplifying assumption that has consequences for HIRM configuration but also the consequences of which we examine below.

HIRM's total atmospheric temperature response is calculated as the sum of the Green's function of a temperature response to a radiative forcing perturbation with radiative forcing time series, an approach taken by many SCMs (Joos et al., 1999, 2013; Van Vuuren 2011; Millar et al., 2015; Boas, 2006). By relying on a process-based climate model to compute response function (RF) values, HIRM is able to use a linear IRF in a simple impulse response model and capture the major nonlinear dynamics between the emissions to radiative forcing calculations by using radiative forcing time series as input data.

HIRM calculates the atmospheric temperature change from preindustrial temperature (T) as the sum of the temperature contribution from individual forcing agents Ti (Eq. 1): 1T(t)=∑i=1nTi(t). Here the individual temperature contribution is equal to the convolution of the radiative forcing time series RFi with the temperature response to a radiative forcing pulse IRFi for a single radiative forcing agent (Eq. 2). 2Ti(t)=∫t0tRFi(t′)IRFi(t-t′)dt′ The method we used to obtain RFi and IRFi for HIRM relies on output from a parent process-based model. The subsequent sections discuss how we obtained RFi and IRFi specifically from Hector. It is important to note that while HIRM can be set up with unique IRFs for each radiative forcing agent (as demonstrated below), this was not done in this application since Hector uses one IRF for all species.

HIRM is an open-source R package (https://github.com/jgcri/hirm, last access: 11 January 2021) with Doxygen-style comments, unit tests, and online documentation via pkgdown (Wickham and Hesselberth, 2020). The online documentation available at https://jgcri.github.io/HIRM/ (last access: 11 January 2021) documents all of the package functions and links with a vignette (example) that demonstrates how to set up and run HIRM. The package contains all of the IRFs and RF inputs used in this paper that can be used in a customizable configuration matrix to set up and run HIRM.

As previously mentioned, one of Hector's assumptions is that all of Hector's radiative forcing agents elicit the same temperature response to a change in radiative forcing. Even though HIRM can use a unique IRF for each radiative forcing agent, for the purposes of HIRM validation exercises in this study, HIRM's setup must be analogous to that of its parent model Hector. In this study we configured HIRM with a single IRF that characterizes Hector's temperature response to all of its 37 radiative forcing agents, derived from a reference run and a black carbon (BC) emissions perturbation run of Hector. In Hector, BC emissions are converted directly to radiative forcing, and therefore an emissions pulse of BC is analogous to a radiative forcing pulse. BC was chosen as the forcing agent since there are no gas cycle or forcing interactions with other species within Hector, making it straightforward to derive the IRF, but other forcing agents could have been selected for the perturbation run. During the reference model run Hector was driven with the Representative Concentration Pathway (RCP) 4.5 scenario, while for the perturbation model run BC emissions were doubled relative to RCP 4.5 BC emissions in a single year. RCP 4.5 CO2 concentrations were prescribed during these runs, suppressing Hector's normal carbon cycle–temperature feedbacks.

For the two validation experiments we did not include carbon cycle–climate feedbacks into the IRF as we wanted to examine the relative importance of nonlinearities in emission to forcing calculations, at least as represented in Hector, as compared to nonlinearities in Hector's forcing to temperature calculations (as represented within DOECLIM). For this reason the IRF should represent only the response of temperature to radiative forcing; otherwise, the temperature response from these feedback mechanisms would be incorporated into the IRF, which would then be doubled-counted as forcing time series are being used as inputs. For the replication experiments we also focus on reproducing Hector temperature without carbon–climate feedbacks. Other applications of HIRM may require IRFs that include the temperature response from carbon cycle feedbacks.

The temperature response (Tresponse) to the BC emissions perturbation is equal to the difference between the reference (Tref) and perturbation temperature Tp (Eq. 3), with the perturbation occurring at year t0: 3Tresponset-t0=Tpt-t0-Treft-t0. The temperature response to a radiative forcing perturbation was calculated by dividing the temperature response to the emissions perturbation by the size of the radiative forcing pulse (Eq. 4). The size of the radiative forcing pulse (Xy) was set equal to the difference in radiative forcing between the reference and emissions perturbation runs (described in the paragraph above) in the perturbation year: 4IRFi(t-t0)=Tresponset-t0/Xy. This IRF had a length of 300 years, in order to ensure the IRF was long enough to be convolved with the RF inputs; the end of the IRF was extrapolated with an exponential decay function to a length of 3000 years with a decay constant of 0.20. Extending the length of the IRF prevents the IRF from being padded with zeros and having to truncate the RF inputs.

The majority of Hector's temperature response to a radiative forcing pulse occurs within the first 50 years after the perturbation (Fig. 1). The strongest response occurs during the perturbation year itself, with a maximum value of 0.09 (∘C W-1 m-2); by year 35 the temperature response has decreased by 97 % and continues to approach zero for the remainder of the IRF. This IRF is used in both of the validation experiments and case studies except where noted.

Emulation is used to validate HIRM by illustrating that the HIRM framework reproduces the dynamics of a process-based SCM with a minimal loss of information. If HIRM can accurately reproduce or emulate the atmospheric temperature of a more complex, process-based model such as Hector, then we assume that HIRM is able to capture important nonlinear dynamics of the climate system using this setup, at least to the extent these are captured in the SCM. Conversely, if HIRM is unable to reproduce Hector's global temperature outputs, this would indicate that important processes are not being captured by the HIRM framework.

In the first validation experiment, HIRM was set up to reproduce Hector temperature for RCP 2.6, RCP 4.5, RCP 6.0, and RCP 8.5. HIRM was configured for each RCP scenario with a single IRF derived from Hector (Fig. 1) together with a complete set of time series from Hector's 37 radiative forcing agents. The radiative forcing time series for these validation experiments came from Hector output from RCP 2.6, 4.5, 6.0, and 8.5 with prescribed CO2 concentrations. The global mean temperature outputs from Hector driven with RCP 2.6, RCP 4.5, RCP 6.0, and RCP 8.5 were saved and used as validation data for HIRM.

HIRM was able to emulate Hector's temperature for the four RCPs with a minimal loss of information (Fig. 2a). The difference between HIRM and Hector total temperature, measured as the root-mean-squared error (RMSE), was 1.3 × 10-9 ∘C (Fig. 2a) for each RCP scenario. The cumulative percentage difference between HIRM and Hector temperature was 0 % (rounded from 1.0 × 10-5; other 0 % results are similar) for each RCP scenario.

The second validation experiment tested HIRM's ability to reproduce Hector's temperature response to an abrupt 4 times CO2 concentration step. The abrupt 4 times CO2 concentration step is a test commonly used by climate modelers to understand the climate system's response to CO2 (Taylor et al., 2012). In this experiment HIRM was set up with the Hector-derived IRF and a RF input from an abrupt 4 times CO2 concentration step. The radiative forcing time series was obtained from Hector runs following the CMIP5 protocol (Taylor et al., 2012). HIRM's radiative forcing time series input was the difference in Hector radiative forcing from Hector driven with a constant CO2 concentration of 278 ppm and Hector driven with a CO2 concentration of 278 ppm until the year 2010 when the CO2 concentration increased by a magnitude of 4 and remained constant for the rest of the run. The difference in Hector's global mean temperature anomaly between the constant reference run and the perturbed step run was then compared with HIRM's output.

HIRM reproduced Hector's abrupt 4 times CO2 concentration step temperature response with a high degree of accuracy (Fig. 2b). The RMSE between HIRM and Hector temperature output from the abrupt CO2 concentration step was 1.5 × 10-19 ∘C with a cumulative percent difference of 0 %. The abrupt CO2 concentration step is a standard diagnostic test used to examine climate model responses (Taylor et al., 2012; Eyring et al., 2016). Since HIRM was able to accurately emulate Hector's temperature response to a large step perturbation, we conclude that the majority of the nonlinearities within Hector are occurring during the emissions-to-radiative forcing portion of the emissions-to-temperature causal chain. While this is to be expected from the general principles of SCMs, it nonetheless provides a useful check that our understanding of the parent model's behavior is correct.

Uncertainties in the magnitude of historical and future radiative forcing effects continue to be a crucial challenge for climate science research, and this is particularly true for aerosol effects (Forest, 2018). In this first case study HIRM was used to explore a range of future temperature change when accounting for uncertainty in some aerosol radiative forcing effects, specifically black carbon (BC), organic carbon (OC), indirect SO2 effects (SO2i), and direct SO2 effects (SO2d). To do so, HIRM was again set up to recreate Hector's RCP 4.5 temperature. In this analysis, BC, OC, SO2i, and SO2d RF inputs were varied. Aerosol cloud indirect effects are represented in Hector as a function of SO2 emissions only, and thus we refer to that as SO2 indirect forcing. We present a simple demonstration of the model in this case study and note that we have not produced probabilistic results but an illustrative range of temperature pathways that result from aerosol uncertainties (e.g., Smith and Bond, 2014). A full probabilistic analysis would also involve varying additional parameters, such as climate sensitivity, ocean heat update, and carbon cycle parameters.

The aerosol uncertainty scalers were generated from the 2011 aerosol radiative forcing ranges reported in IPCC AR5 8.SM Table 5 (Myhre et al., 2013). The BC, OC, SO2i, and SO2d radiative forcing IPCC ranges were individually sampled at intervals of 0.04 W m-2 in 2011 (Table 1), resulting in a total of 29 000uncertainty scalar combinations. Default HIRM 2011 BC, OC, SO2i, and SO2d radiative forcing values were then divided by the values sampled from the respective IPCC ranges to obtain the uncertainty scalers.

HIRM was set up to run every possible combination of the scaled RF time series a total of 29 000 times. This created an ensemble of uncertainty runs, whose results were constrained (i.e., filtered) by historical radiative forcing and temperature. HIRM total radiative forcing was constrained to match IPCC historical estimates in radiative forcing and temperature change. The 2011 aerosol (SO2i, SO2d, BC, + OC) radiative forcing was constrained to pass through an uncertainty range [-1.66 to 0.14 W m-2] (similar to Myhre et al., 2013, but adjusted to account for nitrate and dust forcing and empirical constraints; see the discussion in Smith and Bond, 2014). HIRM temperature trend was calculated as the slope of a linear regression and then compared to the observed temperature trend range of [0.65 to 1.1] ∘C over 1880–2012 reported by Hartmann et al. (2013). Cases that did not meet these constraints were removed (see Fig. 3).

We found that the historical constraints had an unequal impact on the scaled radiative forcing impacts. The temperature at the end of the century for the unconstrained ensemble ranged over 2.5–3.1 ∘C; incorporating the historical constraints into the uncertainty analysis narrowed uncertainty in future temperature to 2.7–2.9 ∘C (Fig. 3). The historical constraints had different impacts on the sampled aerosol uncertainty scalers. All of the sampled OC scalers passed through the historical constraints (Fig. 4b), while the constraints had a modest effect on the OC, BC, and SO2d scalers (Fig. 4a, b, and c).

The historical constraints have the most noticeable effect on the SO2i uncertainty scalers. This is because of the large absolute magnitude of the uncertainty in aerosol indirect effects (Myhre et al., 2013), which results in a large role for assumptions about the strength of aerosol indirect cooling (Tomassini et al., 2007; Meinshausen et al., 2009). This shows that strong (negative) aerosol indirect forcing is consistent with only a few numerical combinations of forcing values from other species, at least for default Hector climate system parameters. The sample analysis using HIRM illustrates how this modeling framework can be utilized to calculate the range of past and future temperature changes under assumed uncertainty in aerosol radiative forcing.

Radiative forcing effects from aerosols are complex (Fan et al., 2016; Bond et al., 2013), and while the physics driving these complexities have been incorporated into ESMs, they are not considered in most SCMs. For example, consider black carbon (BC): unlike cooling effects from aerosols that scatter shortwave radiation back into space, BC heats within the atmosphere and also at the surface when deposited on snow or ice, potentially contributing to both cloud indirect cooling and heating effects (Bond et al., 2013). It can also increase cloud amounts, as BC atmospheric heating stabilizes the atmospheric thermal profile (Bond et al., 2013). Experiments conducted with ESMs have found large differences in the response to a step change in BC emissions compared to a step change in CO2 (Sand et al., 2015; Yang et al., 2019).

Incorporating these dynamics into Hector would be a nontrivial task, but HIRM can be used to estimate what effect they would have on the model's global temperature. For this case study, HIRM was set up to emulate Hector RCP 4.5 as before but with one difference: instead of pairing the BC RCP 4.5 RF time series with Hector's single IRF, the BC RCP 4.5 RF time series was paired with a BC-specific IRF. Since HIRM is set up with a BC-specific IRF, the results will no longer be equivalent to Hector's. Instead, the results illustrate what Hector's temperature could be if the BC dynamics were modified.

The BC-specific IRF was derived using output from a study that performed BC emission step tests with the ESM NorESM-1 (Sand et al., 2015). Mathematically, the derivative of a step response is equal to the impulse response function, and therefore we can derive an impulse response function from the step response results reported in the Sand et al. (2015) ESM experiment. The temperature response to a BC step in ESM experiments is well fitted by a single exponential approach to a constant response (see Yang et al., 2019, for details). We fit the Sand et al.(2015) abrupt BC step response as follows: 5Tt=A(1-e-tτ). The results of a nonlinear optimization of this function returned values of and τ that were 1.8 ∘C and 2.1 years, respectively. These optimal values were used in Eq. (6), the differentiated form of Eq. (5), to provide a numerical BC temperature impulse response function corresponding to the Sand et al. (2015) result: 6Rtt=Aτe-ytdt. The numerical result of Eq. (6) is converted to a BC impulse response per unit forcing by dividing by the forcing from a 133 Tg BC emissions change (used in Sand et al., 2015) using Hector's default forcing per unit BC emission assumptions. With this transformation we have replaced Hectors' default BC representation in HIRM with the Sand et al. (2015) temperature response in both magnitude and temporal behavior.

We found that the BC Sand et al. (2015) IRF has a weaker temperature response in the perturbation year and a more rapid decline in temperature response compared to Hector's global IRF (Fig. 5a). The maximum IRF response for the BC Sand et al. (2015) IRF is 0.06 (∘C W-1 m-2), which is 0.03 (∘C W-1 m-2) cooler than Hector's IRF. In addition, the BC Sand et al. (2015) IRF approaches 0 (∘C W-1 m-2) faster than Hector's IRF. These differences are expected since the BC Sand et al. IRF was derived from the NorESM-1 ESM, meaning that this IRF incorporates the complex cooling and warming effects of BC emissions, the net warming over land as compared to no net warming over oceans (Sand et al., 2015). When HIRM was configured with the BC Sand et al. (2015) IRF, the global temperature was lower by 0.2 ∘C from 1750 to 2100 under the RCP 4.5 scenario (Fig. 5b). Based on these results, if Hector were modified to emulate this BC response, we predict that the model's global temperature would be cooler by approximately 0.2 ∘C in 2100.

We note that the idea of different forcing agents has been around for quite some time. For example, this has been incorporated mechanistically for aerosols in the MAGICC model for around 30 years now (Wigley and Raper, 1992), and more recently inferred by Shindell (2014) from General Circulation Model results. Richardson et al. (2019) used separate response functions for CO2, CH4, solar insolation, and aerosols, although the differences in these response functions were not discussed. As further information on species-specific IRFs become available it will be important to quantify the consequences of these different IRFs using tools such as HIRM.