Arising From Cain et al. npj Climate and Atmospheric Science https://doi.org/10.1038/s4161201900864 (2019)
GWP* was recently proposed^{1} as a simple metric for calculating warmingequivalent emissions by equating a change in the rate of emission of a shortlived climate pollutant (SLCP) to a pulse emission of carbon dioxide. Other metrics aiming to account for the timedependent impact of SLCP emissions, such as CGWP, have also been proposed^{2}. In 2019 an improvement to GWP* was proposed by Cain et al.^{3}, hereafter CLA, combining both the rate and change in rate of SLCP emission, justified by the rate of forcing decline required to stabilise temperatures following a recent multidecade emissions increase. Here we provide a more direct justification of the coefficients used in this definition of GWP*, with a small revision to their absolute values, by equating CO_{2} and SLCP forcing directly, without reference to the temperature response. This provides a more direct link to the impulseresponse model used to calculate GWP values and improves consistency with CGWP values.
The formula for CO_{2}warmingequivalent emissions using GWP* in CLA is:
where E(t) are CO_{2}equivalent emissions defined using GWP with a timehorizon H, much longer than the SLCP lifetime, and s was a coefficient introduced by CLA and estimated by reproducing the response to a simple climate model to various emission scenarios. \({\Delta} E\left( t \right) = E\left( t \right)  E(t  {\Delta} t)\), the change in emissions over a recent time period Δt. Twenty years has been used in implementations of GWP* to date^{1,3} and appears to work well for methane (here we explain why this is the case).
Setting E*(t) to zero in Eq. (1) shows the ratio \(s/\left[ {H\left( {1  s} \right)} \right]\) defines the decay rate of SLCP emissions required to have the same warming impact as zero CO_{2} emissions. CLA justify a value of 0.33% per year, giving s = 0.25 for H = 100 years, as the decline rate required to give stable temperatures under typical values of the Equilibrium Climate Sensitivity (ECS) and Transient Climate Response (TCR). They further justify this formulation using the constraint that total CO_{2}warmingequivalent emissions over H years corresponding to a steady emission of an SLCP starting in year zero should be equal to total CO_{2}equivalent emissions over the same period, arguing that equal constant CO_{2}equivalent emissions give, by construction, the same forcing at the GWP timehorizon, and redistributing CO_{2} emissions over time has minimal impact on final warming. An advantage of the above formula is that it involves no new modeldependent coefficients other than s.
Although confirmed by fitting the warming response to methane emissions in an explicit climate model, this justification is not entirely satisfactory: if the aim is to produce a CO_{2} emissions series that generates the same forcing trajectory as that generated by the SLCP, there should be no need to invoke the warming response. The relationship between CO_{2}warmingequivalent emissions and radiative forcing should, by construction, replicate the relationship between CO_{2} emissions and radiative forcing.
We can focus on timescales of 30–200 years, on the grounds that on shorter timescales the temperature response is dominated by internal variability^{4}, so exact reproduction of forcing timeseries is irrelevant, while 200 years captures at least the initial cumulative impact of CO_{2} emissions. By restricting the timescale of interest, CO_{2} emissions and radiative forcing can be approximately related by the firstorder equation:
where ρ is the rate of decline of radiative forcing over these timescales under zero emissions, and α is a constant representing the forcing impact of ongoing CO_{2} emissions. In terms of the linear impulseresponse model used to provide GWP values in AR5^{5,6}, this formulation assumes the short adjustment timescales are fully equilibrated and neglects the very long cumulative timescale, in effect fitting an exponential to the midrange impulseresponse function. As we show below, this turns out to be a surprisingly good approximation.
We express α in familiar terms by noting that the forcing response after H years to steady CO_{2} emissions of 1 kg per year, starting in year 0, is by definition the Absolute Global Warming Potential of CO_{2}, or AGWP_{H} (this is identical to the standard definition^{5,6} because the calculation of AGWP_{H} values is based on a linear model). Hence, integrating equation (2) for E_{CO2} = 1
So \(\alpha = {\mathrm{AGWP}}_H\rho \left( {1  e^{  \rho H}} \right)^{  1}\), or 1.08 W m^{−2} per 1000 GtCO_{2} with ρ = 0.33% per year, H = 100 years and the AR5 value^{5} of AGWP_{100} of 91.7 Wyears m^{−2} per 1000 GtCO_{2}. With these coefficients, this expression (solid black line in Fig. 1) reproduces the forcing response to constant unit CO_{2} emissions computed using the full impulseresponse model used for GWP calculations in AR5 (solid red line) accurately over multidecade to century timescales. Decreasing ρ (dotted line) causes the fit to deteriorate on all timescales, since it fails to capture the curvature of the AGWP as a function of H, while increasing ρ (dashed line) causes the fit to deteriorate on greater than 100year timescales, by failing to capture the cumulative impact of CO_{2} emissions. Clearly there is an element of subjectivity inherent in all metric approximations as to what constitutes a ‘good enough’ approximation, but the above expression with ρ = 0.33% per year appears to capture the forcing response to constant CO_{2} emissions very well, and certainly well within the uncertainties of the climate and carbon cycle response^{6}. Defining an ‘optimal’ value of ρ depends on the choice of goodnessoffit statistic: we focus here on reproducing the absolute forcing per tonne of CO_{2} as plotted in Fig. 1. This is most relevant to expressing forcing changes in terms of cumulative CO_{2} emissions, and represents the timeintegral of the forcing impulseresponse function. Using a higher value of ρ gives better agreement on short timescales at the expense of downplaying the cumulative impact of CO_{2} emissions, and vice versa. The fact that the value of ρ implied by the timedependence of the AGWP coincides with the value implied by the ECS and TCR in CLA is the reason net zero CO_{2} emissions is expected to be consistent with no further CO_{2}induced warming, and a further reason to use a consistent value.
Using the substitution \(\rho = s/\left[ {H\left( {1  s} \right)} \right]\) we can reexpress Eq. (2) in a form similar to Eq. (1):
where
The function g(s) is approximately unity for small s, and is implicitly approximated to unity by CLA, but it actually has a value g = 1.13 for s = 0.25 and H = 100 years.
The radiative forcing due to a constant SLCP emission of 1 kg CO_{2}equivalent per year starting in year 0 can be expressed:
provided \(\tau \ll H\), so \({\mathrm{e}}^{  H/\tau } \ll 1,\)where AGWP_{H} is the AGWP of CO_{2} for the timehorizon used to evaluate CO_{2}equivalent emissions and τ is the SLCP lifetime.
Substituting this into Eq. (4) gives an expression for the CO_{2}warmingequivalent emissions corresponding to this constant SLCP emission:
Hence the CO_{2}warmingequivalent emissions corresponding to this CO_{2}equivalent SLCP emission are a constant gs kg per year plus an emission totalling of \(gH\left( {1  s} \right)\) kg almost all of which occurs in the first ~2τ years (using \({\int\nolimits_0^\infty} \left( {{\mathrm{e}}^{  t/\tau }/\tau } \right)dt = 1\). GWP* approximates this pulse as a constant additional emission spread over the first Δt years, and explains why Δt = 20 years works for an SLCP with a lifetime of order one decade. The initial adjustment time of the solid blue curve in Fig. 1 is of this order: hence using 20 years approximately matches the initial gradients of the blue solid and dashed lines, which correspond to the instantaneous radiative forcing impact of the release of one tonne of methane relative to that of CO_{2}.
Hence a more consistent definition of CO_{2}warmingequivalent emissions under GWP* is:
This is identical to that of CLA but scaled by g = 1.13 and now justified without reference to the temperature response. Including this scaling improves the consistency with simulated warming responses under ambitious mitigation scenarios, at the expense of consistency with warming responses under higher emissions, as shown in Fig. 2, which reproduces Fig. 1 of CLA but now including the scaling factor g. It appears that the reproduction of simulated warming under the higher emissions scenarios noted in CLA was coincidental: additional methane emissions have less warming impact per tonne if introduced into a higher background emission scenario, compensating for the use of g = 1 in the calculation of warmingequivalent emissions.
Given the approximations involved in greenhouse gas metrics in the first place, such as the choice of background emissions trajectory against which to linearise, it is debateable whether scaling factors of order 10% are worth any additional complexity. The parameter g, however, is an unambiguous function of s, not an additional tuneable parameter, so we propose that it should be included in the definition of GWP* for greater consistency with the linear models used for metric calculations. As these linear models are updated the forcing decay rate corresponding to zero CO_{2} emissions will change, potentially resulting in a change in s; however, given the weak dependence seen in Fig. 1, any changes are likely to be small. Including g means that the expression for CO_{2} warmingequivalent emissions of methane becomes \(E^ \ast \left( t \right) = 128 \times E_{{\mathrm{CH4}}}\left( t \right)  120 \times E_{{\mathrm{CH4}}}\left( {t  20} \right)\), where E_{CH4} is methane emissions in tCH_{4} per year, with AR5 GWP values. For a generic SLCP, \(E^ \ast \left( t \right) = 4.53 \times E_{100}\left( t \right)  4.25 \times E_{100}\left( {t  20} \right)\), where E_{100} are CO_{2}equivalent emissions calculated using GWP_{100}, with residual discrepancies due to rounding. A shorter than 20year period might be better suited to representing shorterlived climate pollutants, but given this choice has no impact on cumulative warmingequivalent emissions, we propose a consistent value is used for all SLCPs for simplicity.
Data availability
The code to produce Figs 1 and 2 is available at https://gitlab.ouce.ox.ac.uk/OMP_climate_pollutants/co2warmingequivalence.
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Acknowledgements
We would like to acknowledge helpful and timely comments from the anonymous reviewer and the editorial team. M.A.S., M.C. and M.R.A. acknowledge support from the Natural Environment Research Council award number NE/T004053/1–A practical tool and robust framework for evaluating greenhouse gas emissions from landbased activities, and the Oxford Martin Programme on Climate Pollutants.
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M.R.A. initiated the work with M.A.S. and M.C. developing the work to bring it to submission. M.R.A. produced Fig. 1. M.C. produced Fig. 2. All authors contributed to developing the scientific questions, discussion of the results, subsequent drafts of the paper and in editing the final version.
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M.C. and M.R.A. are both authors of “Improved calculation of warmingequivalent emissions for shortlived climate pollutants”.
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Smith, M.A., Cain, M. & Allen, M.R. Further improvement of warmingequivalent emissions calculation. npj Clim Atmos Sci 4, 19 (2021). https://doi.org/10.1038/s41612021001698
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