Equations for methane radiative forcing in atmospheric chemistry models

I recently wrote a review and analysis of the methane atmospheric chemistry feedback (Holmes, 2018 JAMES). After it was published, a colleague asked me for some details about calculating the CH4 radiative forcing using output from the GEOS-Chem chemical transport model. Although the relevant equation can be derived from the JAMES paper, it doesn’t explicitly appear there.

The direct radiative forcing (RF) resulting from a change in methane burden (δm) in the atmosphere is
RFCH4= ERF,CH4 δm,     (1)
where ERF,CH4 is the radiative forcing efficiency of CH4. This efficiency can be calculated from an IPCC (Ramaswamy et al., 2001) formula originally developed by Myhre et al. (1998). For present-day levels of CH4 and N2O, ERF,CH4 = 270 mW m-2 ppm(CH4)-1.

Many atmospheric chemistry models prescribe CH4 concentrations so, by design, they can’t calculate δm. However, these models can and do simulate chemical loss of CH4 by reactions with tropospheric OH and other oxidants. As Eq. 10 in the JAMES paper shows, the mass change resulting from a steady state change in CH4 emissions (δE) or CH4 lifetime (δτ) is
δm/m0 = f δE/E0 = f δτ/τ0.     (2)
The variables with subscript zero represent the present-day values and f≈ 1.37 is the feedback factor.
From Eq. 2, we can derive
δm = f m0 δE/E0 (3)
δm = f m0 δτ/τ0. (4)

As a result, for a change in CH4 emissions, the resulting steady-state CH4 RF is
RFCH4 = ERF,CH4 f τ0 δE/E0. (5)
For a change in CH4 lifetime (e.g. due to changing NOx emissions), the steady-state CH4 RF is
RFCH4 = ERF,CH4 f m0 δτ/τ0. (6)

Atmospheric chemistry models that prescribe CH4 concentrations will also lack the tropospheric O3 change that is induced by CH4 changes. The CH4-induced O3 provides an additional RF of
RFO3-from-CH4 = ERF,O3 f m0 δτ/τ0 d(O3)/d(CH4). (7)
Reasonable values are ERF,O3 = 36 mW m-2 DU(O3)-1 and d(O3)/d(CH4) = 3.5 DU(O3) ppm(CH4)-1 (Holmes et al., 2011, PNAS). If the changes are driven by CH4 emission changes then Eq. 7 still applies with δE/E0 taking the place of δτ/τ0. Parameter values in the equations can be updated based on recent literature as needed.