The system that describes the radiative forcing from H2 emissions modeled by Warwick et al. (2022) and later used in Ocko and Hamburg (2022) is a representation of the following differential equations.

The change of H2 molar mass relative to an unperturbed background condition, as a function of the time horizon t in units of year, is represented by a source function fH2(t) and a decay term mH2τH2, where mH2 is the molar mass of H2 and τH2 is the perturbation lifetime of H2: 1dmH2dt=fH2(t)-mH2τH2.

The presence of additional H2 in the atmosphere changes the decay of atmospheric CH4 and also results in the production of ozone (O3) and stratospheric water vapor (H2O). Underlying equations for perturbations to atmospheric molar masses of CH4, O3, and stratospheric H2O induced from additional atmospheric H2 (denoted by superscripts) are 2admCH4H2dt=aCH4mH2-mCH4H2τCH4,2bdmO3H2dt=aO3mH2-mO3H2τO3,2cdmH2OH2dt=aH2OmH2-mH2OH2τH2O. Here, mCH4H2, mO3H2, and mH2OH2 are the respective molar masses of CH4, O3, and H2O resulting from additional atmospheric H2; aCH4, aO3, and aH2O are factors representing the impact of remaining H2 in the atmosphere on the atmospheric molar mass of these respective species; and τCH4, τO3, and τH2O are perturbation lifetimes of these respective species.

For the special case of a unit pulse perturbation of H2 into an unperturbed background condition at time zero, these equations can be solved analytically. The respective solutions to Eqs. (1) and (2) under conditions mH2(0)=1, fH2(t)=0, mCH4H2(0)=0, mO3H2(0)=0, and mH2OH2(0)=0 are 3mH2(t)=e-tτH2, and 4amCH4H2(t)=aCH41τH2-1τCH4e-tτCH4-e-tτH2,4bmO3H2(t)=aO31τH2-1τO3e-tτO3-e-tτH2,4cmH2OH2(t)=aH2O1τH2-1τH2Oe-tτH2O-e-tτH2. The corresponding radiative forcing is the product of the resulted molar mass and scaling factors ACH4, AO3, and AH2O that convert molar mass to watts per meter squared (Wm-2). The chemically adjusted radiative forcing, ACH4∗, for CH4 uses factors f1 and f2 (Myhre et al., 2013) to represent the effect on O3 and stratospheric H2O: 5ACH4∗=(1+f1+f2)ACH4.

The indirect radiative forcing from a unit pulse emission of H2, RH2, is thus the sum of radiative forcing from all three radiatively active perturbations: 6a6aRH2(t)=ACH4∗mCH4H2(t)+AO3mO3H2(t)+AH2OmH2OH2(t). Inserting Eq. (4), we have 6bRH2(t)=ACH4∗aCH41τH2-1τCH4e-tτCH4-e-tτH2+AO3aO31τH2-1τO3e-tτO3-e-tτH2+AH2OaH2O1τH2-1τH2Oe-tτH2O-e-tτH2.

For a 1 kg unit pulse emission case, the time-integrated radiative forcing to a specified time horizon, H, is defined to be the absolute global warming potential (AGWP) (Myhre et al., 2013). Thus, AGWP can be represented as 7aAGWPH2(H)=∫0HRH2(t)dt, which can be rewritten as 7b7bAGWPH2(H)=ACH4∗aCH4τH2τCH4τCH41-e-HτCH4-τH21-e-HτH2τCH4-τH2+AO3aO3τH2τO3τO31-e-HτO3-τH21-e-HτH2τO3-τH2+AH2OaH2OτH2τH2OτH2O1-e-HτH2O-τH21-e-HτH2τH2O-τH2.

Equation (7) is the response to a unit pulse emission of H2 taking into consideration radiative forcing adjustments to CH4 as in Ocko and Hamburg (2022). Because we are considering a linear system, we can use this impulse response function to derive the radiative forcing from an arbitrary H2 emission function fH2: 8RH2^(t)=∫0tfH2(τ)RH2(t-τ)dτ.

Considering a continuous unit emission scenario where 9fH2(t)=1, this leads to radiative forcing under a continuous emission scenario: 10RH2,cont(t)=ACH4∗aCH4τH2τCH4τH2e-tτH2-1-τCH4e-tτCH4-1τCH4-τH2+AO3aO3τH2τO3τH2e-tτH2-1-τO3e-tτO3-1τO3-τH2+AH2OaH2OτH2τH2OτH2e-tτH2-1-τH2Oe-tτH2O-1τH2O-τH2.

In a linear system, the time-integrated radiative forcing from a unit pulse emission to some time horizon t0 is mathematically equivalent to the radiative forcing at time t0 from continuous unit emissions: 11AGWPH2(t0)=RH2,cont(t0).

Ocko and Hamburg (2022) used a metric that is equal to the time-integrated radiative forcing of continuous emissions to time horizon H. Since the AGWP has been defined as the time-integrated radiative forcing from the instantaneous release of 1 kg of a trace substance (Myhre et al., 2013), here we define the time-integrated radiative forcing under a continuous emission scenario as continuous absolute global warming potential (CAGWP): 12aCAGWPH2(H)=∫0HRH2,cont(t)dt,12b=∫0H∫0tRH2(τ)dτdt,12c=∫0H∫tHRH2(τ)dtdτ,12d=∫0H∫tHdtRH2(τ)dτ,12e=∫0H(H-t)RH2(t)dt.

Comparing Eq. (12) with Eq. (7a), we can see that the CAGWP metric is equivalent to the AGWP metric, except that the radiative forcing at time 0 is weighted by H, and the radiative forcing at time H is weighted at 0, with a linear ramping of weights in between by the number of years to the end of the time horizon. Equation (12) illustrates that time-integrated metrics under continuous emissions put heavy weights on the short-term effect.

Expanding Eq. (12), we have 13CAGWPH2(H)=ACH4∗aCH4τH2τCH4τH221-e-HτH2-τCH421-e-HτCH4+HτCH4-τH2τCH4-τH2+AO3aO3τH2τO3τH221-e-HτH2-τO321-e-HτO3+HτO3-τH2τO3-τH2+AH2OaH2OτH2τH2OτH221-e-HτH2-τH2O21-e-HτH2O+HτH2O-τH2τH2O-τH2.

Equations (10) and (13) consider continuous emissions through the whole period. Equations considering a continuous emission to time tp are shown in Sect. S1 in the Supplement. Reproductions of the three components in Warwick et al. (2022) and Ocko and Hamburg (2022) are shown in Sect. S2. When used to estimate radiative forcing for identical cases, numerical differences between our equations and equations presented by Warwick et al. (2022) are small and are unlikely to make a material difference.

Here, we show radiative forcing and time-integrated radiative forcing functions for CO2 and CH4. Parameters are defined and values are given in Table S1. Radiative forcing for a unit pulse emission of CO2 and CH4 is represented as (Myhre et al., 2013) 14RCO2(t)=ACO2a0+∑i=13aie-tτi,15RCH4(t)=(1+f1+f2)ACH4e-tτCH4.

The AGWP for a unit pulse emission is 16AGWPCO2(H)=ACO2a0H+∑i=13aiτi1-e-Hτi,17AGWPCH4(H)=(1+f1+f2)ACH4τCH4(1-e-HτCH4).

Radiative forcing for continuous emissions of CO2 and CH4 can be represented as 18RCO2,cont(t)=ACO2a0t+∑i=13aiτi1-e-tτi,19RCH4,cont(t)=(1+f1+f2)ACH4τCH4(1-e-tτCH4).

And the corresponding CAGWP is 20CAGWPCO2(H)=ACO2(a0H22+∑i=13aiτiH+τie-Hτi-1),21CAGWPCH4(H)=(1+f1+f2)ACH4τCH4×H+τCH4e-HτCH4-1.

For a linear system, the absolute global temperature change potential (AGTP), defined as the change of global mean surface temperature realized at a given time horizon from a pulse or continuous emissions of any gas i, can be represented as a convolution function (Myhre et al., 2013; Gasser et al., 2017): 22AGTPi(H)=∫0HRi(t)T(H-t)dt.

In Eq. (22), Ri(t) is the radiative forcing for unit pulse or continuous emissions of gas i and T(t) indicates the temperature response to a unit forcing that can be represented as a sum of exponentials: 23T(t)=λ∑j=1Mcjdje-tdj, where λ is a constant that corresponds to the equilibrium climate sensitivity, ∑j=1Mcj=1, and dj is the response time. Two exponential terms (M=2) are normally used in previous studies, with the first term associated with the response of the ocean mixed layer and the higher order associated with the response of the deep ocean (Gasser et al., 2017). In our central cases, we focus on using the equation from Geoffroy et al. (2013): 24T(t)=0.8850.5874.1e-t4.1+0.413249e-t249.

Uncertainty in the temperature response function is shown in Sect. S3.

As in Ocko and Hamburg (2022), we focus on comparing the climate impact of replacing fossil fuel technologies with clean hydrogen alternatives. Climate impacts from hydrogen or fossil fuels are the summation of climate impacts of one or more components in a linear system (Sect. S4).

In this commentary, we analyze the climate impact per 1 kg consumption of green and blue H2 and corresponding impacts from the avoided CO2 emissions. We consider consistent assumptions as in Ocko and Hamburg (2022). For example, the kilogram amount of CH4 required to produce blue H2 is 3 times the kilogram amount of H2 used, 1 kg consumption of H2 would avoid 11 kg of CO2 emissions (additional cases, i.e., 5 or 15 kg of avoided CO2 emissions, are examined as well), and burning 1 kg of CH4 would emit 2.75 kg of CO2. Also, we take the same consistent leakage rate assumptions for CH4 and H2 to generate two central cases: a low-leakage case with 1 % H2 and 1 % CH4 leakage rates, and a high-leakage case with 10 % H2 and 3 % CH4 leakage rates (see detailed discussion underlying these assumptions in their paper).

We focus our discussions on three H2 consumption scenarios: a 1 kg pulse consumption, 0.01 kgyr-1 continuous consumption lasting for 100 years, and 0.01 kgyr-1 continuous consumption lasting for 500 years. Results for 20-, 100-, and 500-year horizons are summarized in Tables S2 to S5.

We first examine the time-evolving climate impact from emissions of carbon dioxide (CO2), methane (CH4), and hydrogen (H2), respectively. Similarly, we consider three emission scenarios: a 1 kg pulse emission, 0.01 kgyr-1 continuous emissions lasting for 100 years, and 0.01 kgyr-1 continuous emissions lasting for 500 years.

Figure 1 shows the climate impact of individual species under various emission scenarios. Results showing ratios of CH4 and H2 to CO2 are plotted in Fig. S1 in the Supplement. For the 1 kg pulse emission scenario, all species produce the largest climate impacts within the first few years and decay over time. Soon after emission, per kilogram emitted, CH4 and H2 show much larger impacts compared to CO2. The global warming potential is typically defined for a 1 kg pulse emission of gas (Myhre et al., 2013), which will lead to different immediate changes in their atmospheric concentration when viewed on a molar basis. Figure S2 shows that when considering a 1 ppb increase of these gases, CH4 still generates a much larger warming potential, whereas H2 and CO2 show the same order of magnitude impacts on radiative forcing and temperature in the first decade.

The climate impact of CH4 and H2 decays substantially faster than CO2 along with their concentrations (perturbation lifetime used is 11.8 years for CH4 and 1.9 years for H2). For example, the radiative forcing of CH4 and H2 for the 1 kg pulse emission scenario is smaller than that of CO2 after ∼ 65 and 50 years and approaches zero after 100 years. We do not consider conversions of the decayed CH4 to CO2, which will add more long-term impacts for CH4 emissions (Forster et al., 2021) as shown in Fig. S3. This conversion should not be considered in the case of CH4 perturbations brought about by H2 emissions, because there is no net addition of carbon to the atmosphere in this case. In contrast, the radiative forcing of CO2 is still 28 % of its maximum value at the 500-year time horizon. Temperature response behaves similarly to radiative forcing but at a slower rate due to the inertia of the climate system. Impacts of considering different H2 perturbation lifetimes (i.e., 1.4 and 2.5 years) are shown in Fig. S4.

For 0.01 kgyr-1 continuous emission cases, there is an accumulation of CO2 concentration in the atmosphere, leading to monotonic increases in radiative forcing and temperature. If emissions stop abruptly after 100 years, the climate impacts of CO2 slowly converge with those under the 1 kg emission case and stay roughly stable, because the effects of atmospheric concentration decrease are approximately offset by the effects of ocean warming. Due to the shorter perturbation lifetime of CH4 and H2, their atmospheric concentrations reach equilibrium under continuous emission scenarios with magnitudes depending on the emission rates, and radiative forcing reaches a stable level after a few decades. Global mean temperature continues to increase slowly due to the thermal inertia of the climate system. If emissions stop abruptly after 100 years, their concentrations would decrease rapidly and reach zero within decades. The longer perturbation lifetime of CO2 results in more prominent longer-term climate impacts under both pulse and continuous emission scenarios.

Under the low-leakage scenario (i.e., 1 % H2 and 1 % CH4 leakage rate), both green and blue H2 produce smaller radiative forcing and global mean temperature increases compared to the avoided CO2 emissions (Figs. 2 and 3), indicating net climate benefits of replacing fossil fuels with clean hydrogen alternatives. Compared to green H2, leakages of CH4 from blue H2 add substantial additional warming within the first few decades. For the 1 kg pulse consumption scenario, the climate impact of green and blue H2 decays rapidly to zero within the first few decades (conversion of decayed CH4 to CO2 not included), whereas the climate impact of avoided CO2 emissions becomes roughly stable with time. Continuous consumption of H2 would lead to stable radiative forcing and temperature change at longer timescales with magnitudes depending on consumption levels, and such impacts will adjust quickly if future leakage rates change. Meanwhile, continuous consumption of fossil fuels leads to accumulation of CO2 concentration and increasing climate responses. Even if CO2 consumption is ceased, its impacts would last for hundreds of years.

Under the high-leakage scenario, the additional leakage of H2 (i.e., 10 % vs. 1 % H2 leakage rate) reduces the short-term climate benefits of green H2, and the additional leakage of CH4 (i.e., 3 % vs. 1 % CH4 leakage rate) further leads to net disbenefits for blue H2 in the first few years, when compared to avoided CO2 emissions (Figs. 2 and 3). In both the low- and high-leakage cases, CH4 adds more warming than H2 does (Fig. S5). Because of the shorter lifetimes of H2 and CH4, net climate benefits for blue H2 are observed after ∼ 12 and 20 years under different emission scenarios for the high-leakage case. The climate impacts of H2 become orders of magnitude smaller than that of CO2 emissions as time evolves.

In our central cases, we do not include CH4 leakages when calculating climate impacts for the avoided CO2 emissions, which are included for blue H2. Under all emission scenarios, using the same CH4 leakage rates for CH4 combustion and H2 production (Figs. S6 and S7) substantially increases the warming potentials from the avoided CO2 emissions, especially for the short-term results, leading to net climate benefits for both clean hydrogen alternatives. Consideration of conversion of the decayed CH4 to CO2 further increases the long-term climate impacts for both blue H2 and the avoided CO2 emission cases that contain CH4 leakage (Figs. S6 and S7). As in Ocko and Hamburg (2022), considering that different amounts of avoided CO2 emission for per kilogram of H2 consumption (e.g., 5 or 15 kg CO2 avoided per kilogram of H2 consumption) can affect both short-term and long-term climate impacts (Fig. S8). In contrast, considering different H2 perturbation lifetimes or climate response functions has minor effects (Figs. S6, S7, and S9). Here, we do not cover all uncertainties but give some first-level impressions of how different parameters can affect results presented in this analysis.

Finally, Ocko and Hamburg (2022) quantified the net climate benefits of consuming H2 compared to the avoided CO2 emissions by comparing the time-integrated radiative forcing from continuous emissions of both gases. This metric overpredicts the amount of warming that would be produced by CH4 and H2 leakage relative to the warming that would have been caused by the avoided CO2 emissions over time (Fig. S10). This result is similar to that of Allen et al. (2016) showing that, for pulse emissions and any time horizon longer than a decade, the global mean relative temperature response metric (i.e., GTP) would be lower than values of the time-integrated relative radiative forcing (i.e., GWP).