Definition and types of radiative forcing
Instantaneous RF is the change in net irradiance at the tropopause immediately after imposing an agent, before any temperature or circulation adjustments:
\[\Delta F_{inst} = F_{perturbed} - F_{control} \quad [\text{W m}^{-2}]\]Stratospherically adjusted RF (conventional RF): the stratosphere is allowed to re-equilibrate (typically ~2 months) while tropospheric temperatures are fixed. This removes the large but fast stratospheric response and gives a forcing more predictive of surface temperature change.
Effective RF (ERF): tropospheric adjustments (cloud cover, water vapor, land albedo) are also allowed to respond — only surface temperature is fixed. ERF is now preferred in IPCC AR6 because it better predicts transient warming:
\[ERF = \Delta F_{inst} + \text{rapid adjustments}\]The forcing-response relationship is approximately linear:
\[\Delta T = \lambda \cdot ERF\]where $\lambda \approx 0.8$ K (W m⁻²)⁻¹ is the equilibrium climate sensitivity parameter.
CO₂ logarithmic forcing
The radiative forcing from CO₂ follows a logarithmic relationship because the main absorption bands are already saturated; additional CO₂ absorbs in the band wings:
\[\Delta F_{CO_2} = \alpha\ln\left(\frac{C}{C_0}\right)\]with $\alpha \approx 5.35$ W m⁻² (Myhre et al. 1998, IPCC AR5). At doubled CO₂ ($C = 2C_0$):
\[\Delta F_{2\times CO_2} = 5.35\ln 2 \approx 3.71 \text{ W m}^{-2}\]The logarithmic dependence arises from the Voigt line shape of CO₂ absorption features. At very high concentrations the forcing transitions toward a square-root dependence as continuum absorption dominates.
| Agent | RF (W m⁻²) | Uncertainty |
|---|---|---|
| CO₂ (1750–2019) | +2.16 | ±10% |
| CH₄ | +0.54 | ±20% |
| N₂O | +0.21 | ±10% |
| Halocarbons | +0.41 | ±10% |
| Total aerosol | −1.1 | ±50% |
| Solar | +0.01 | ±50% |
Aerosol forcing: direct and indirect effects
Aerosols scatter and absorb solar radiation (direct effect) and modify cloud properties (indirect/Twomey effect).
Direct RF depends on aerosol optical depth $\tau$, single-scatter albedo $\omega_0$, and asymmetry factor $g$:
\[\Delta F_{direct} = -\frac{S_0}{4}T_{atm}^2(1-A_c)(1-\omega_0 f)\left[2\omega_0\beta\tau\right]\]where $f = g^2$, $\beta$ is the backscatter fraction, and $A_c$ is cloud fraction.
Twomey (first indirect) effect: More aerosol $\Rightarrow$ more cloud condensation nuclei $\Rightarrow$ smaller droplets $\Rightarrow$ higher cloud albedo $\alpha_c$. For fixed liquid water path:
\[\frac{d\alpha_c}{d\ln N} \approx \frac{\alpha_c(1-\alpha_c)}{3}\]The second indirect (lifetime) effect: smaller droplets suppress precipitation, increasing cloud lifetime and coverage. Total aerosol ERF is estimated at $-1.1 \pm 0.4$ W m⁻² (AR6), the largest uncertainty in the forcing budget.
Solar forcing and stratospheric ozone
Solar forcing varies with the 11-year sunspot cycle ($\Delta F \approx \pm 0.1$ W m⁻²) and longer-term changes in total solar irradiance (TSI). Since 1750, the net solar ERF is only $+0.01$ W m⁻² — negligible compared to greenhouse gas forcing.
Stratospheric ozone depletion (CFCs) exerts a negative RF through two mechanisms:
- Reduced ozone absorbs less solar UV $\Rightarrow$ less warming of stratosphere $\Rightarrow$ more OLR to space
- Less stratospheric ozone also reduces downward IR emission
The net ozone RF was approximately $-0.05$ W m⁻² over 1750–2019.
Transient vs equilibrium response and forcing efficacies
The transient climate response (TCR) is the warming at the time of CO₂ doubling under a 1% per year increase:
\[TCR = \lambda_{eff} \cdot \Delta F_{2\times} \approx 0.55 \times ECS\]The ratio TCR/ECS $< 1$ because the ocean absorbs heat: the net forcing driving surface warming is $N = ERF - \lambda\Delta T$ where $N$ is ocean heat uptake rate.
Forcing efficacy $e$ accounts for different geographic/vertical patterns having different temperature impacts per unit forcing. Defined as:
\[e = \frac{ECS_i / \Delta F_{2\times CO_2}}{ECS_{CO_2} / \Delta F_{2\times CO_2}} = \frac{\lambda_{CO_2}}{\lambda_i}\]Solar forcing has efficacy ~1.0; volcanic forcing has ~0.6 (short-lived aerosols at low latitudes); land-use albedo ~1.0; black carbon on ice ~3 (very high, as it affects the ice-albedo feedback strongly).
Attribution of observed warming
The energy budget attribution links observed warming $\Delta T_{obs} \approx 1.1$ K (1850–2019) to forcing components via:
\[\Delta T = \sum_i \lambda_i \cdot ERF_i\]In a regression framework, observed temperature $\mathbf{y}$ is decomposed:
\[\mathbf{y} = \sum_k \beta_k \mathbf{x}_k + \boldsymbol{\varepsilon}\]where $\mathbf{x}_k$ are GCM-simulated fingerprints (greenhouse gas, aerosol, natural) and $\beta_k$ are scaling factors estimated by optimal fingerprinting:
\[\hat{\boldsymbol{\beta}} = (X^T C^{-1} X)^{-1} X^T C^{-1} \mathbf{y}\]Here $C$ is the internal variability covariance estimated from unforced control runs. IPCC AR6 concludes: it is unequivocal that human influence has warmed the climate. The greenhouse gas contribution is $+1.0$ to $+2.0$ K, partially offset by aerosol cooling of $−0.5$ to $−0.2$ K.