The feedback framework
Start from the linearized energy balance near equilibrium temperature $T_0$. A radiative forcing $\Delta F$ perturbs the system; each feedback $i$ generates an additional flux $\lambda_i \Delta T$:
\[C\frac{d(\Delta T)}{dt} = \Delta F + \sum_i \lambda_i \Delta T\]At equilibrium ($d\Delta T/dt = 0$):
\[\Delta T = \frac{\Delta F}{\lambda_0 + \sum_i \lambda_i} = \frac{\Delta F}{\lambda_{eff}}\]| Defining the feedback factor $f_i = \lambda_i/ | \lambda_0 | $ (positive = amplifying, negative = damping): |
where $\Delta T_0 \approx 1.2$ K is the Planck-only (no-feedback) response to CO₂ doubling. The gain factor $g = 1/(1-\Sigma f_i)$; currently $g \approx 2.5$–3.5. If $\Sigma f_i \to 1$, sensitivity diverges — runaway warming.
| Feedback | $f_i$ | Physical mechanism |
|---|---|---|
| Planck | −1.00 (reference) | $\sigma T^4$ restoring force |
| Water vapor | +0.55 | More WV absorbs more LW |
| Lapse rate | −0.16 | Tropical upper-trop. warms more |
| Cloud (net) | +0.19 | Net positive in CMIP6 |
| Surface albedo | +0.11 | Ice/snow area decreases |
| Total | +0.69 |
Planck response: the restoring force
The Planck feedback is the fundamental stabilizing mechanism — without it the climate would be unstable. For a blackbody at temperature $T$, the outgoing longwave radiation (OLR) is $F = \sigma T^4$, so:
\[\lambda_0 = \frac{\partial OLR}{\partial T} = 4\sigma T^3 \approx 3.2 \text{ W m}^{-2}\text{K}^{-1}\](with negative sign convention: $\lambda_0 = -3.2$ W m⁻² K⁻¹). The e-folding relaxation timescale is:
\[\tau_0 = \frac{C}{|\lambda_0|} \approx \frac{2\times10^8 \text{ J m}^{-2}\text{K}^{-1}}{3.2 \text{ W m}^{-2}\text{K}^{-1}} \approx 2 \text{ years}\]Water vapor feedback
Warming increases atmospheric water vapor through the Clausius-Clapeyron relation:
\[\frac{d\ln e_s}{dT} = \frac{L_v}{R_v T^2} \approx 7\% \text{ K}^{-1}\]where $e_s$ is saturation vapor pressure, $L_v = 2.5\times10^6$ J kg⁻¹ is latent heat of vaporization, and $R_v = 461$ J kg⁻¹ K⁻¹. Assuming constant relative humidity (supported by observations and models), specific humidity rises ~7% per kelvin. This increases absorption of longwave radiation proportionally, amplifying warming.
The water vapor feedback is the strongest positive feedback (~+1.77 W m⁻² K⁻¹ in AR6), but its magnitude is robust across models precisely because it is thermodynamically constrained. The lapse rate and water vapor feedbacks are anticorrelated — they are typically combined as the WV+LR feedback of ~+1.27 W m⁻² K⁻¹.
Lapse rate feedback
The lapse rate feedback is negative in the tropics (stabilizing) and positive in the Arctic (amplifying). In the tropics, convection ties the temperature profile to the moist adiabat. Warming increases the moist adiabatic lapse rate more at upper levels (Clausius-Clapeyron effect on latent heat release), so the upper troposphere warms faster than the surface:
\[\frac{\partial T}{\partial z}\bigg|_{moist} = -\Gamma_m(T) \approx -\Gamma_d\frac{1 + (L_v q_s)/(R_d T)}{1 + (L_v^2 q_s)/(c_p R_v T^2)}\]Enhanced upper-tropospheric warming increases OLR more than if the surface alone warmed, providing a restoring force ($\lambda_{LR} < 0$).
In polar regions, temperature inversions trap warming near the surface (polar amplification), making the lapse rate feedback positive at high latitudes.
Cloud feedbacks
Cloud feedbacks remain the largest source of uncertainty in ECS. The key distinction is between low clouds (optically thick, strong shortwave cooling) and high clouds (thin, LW trapping dominates).
Low-cloud (marine boundary layer) feedback: As SST rises, the stability of the boundary layer decreases, reducing stratocumulus coverage. The SW cloud feedback:
\[\lambda_{SW}^{cloud} = \frac{S_0}{4}\frac{\partial \alpha_c f_c}{\partial T} \approx +0.4 \text{ W m}^{-2}\text{K}^{-1}\]High-cloud altitude feedback: The anvil-top temperature remains approximately fixed (Fixed Anvil Temperature hypothesis), so as the troposphere warms, high clouds rise. Higher clouds are colder and emit less LW, reducing OLR:
\[\lambda_{LW}^{cloud} \approx +0.2 \text{ W m}^{-2}\text{K}^{-1}\]Ice-albedo feedback
Sea ice and snow cover decrease with warming, exposing darker ocean/land surfaces. The feedback:
\[\lambda_{alb} = \frac{S_0}{4}\frac{d(1-\bar\alpha)}{dT} > 0\]Arctic sea ice albedo ~0.5–0.8 vs. open ocean ~0.06. Loss of sea ice has contributed to Arctic amplification — the Arctic warms 2–4× faster than the global mean:
\[\frac{\Delta T_{Arctic}}{\Delta T_{global}} \approx 2\text{–}4\]driven by albedo feedback, lapse rate feedback, and reduced poleward heat transport.
Tipping points and nonlinear feedbacks
When $\sum f_i$ approaches 1, or when feedbacks become state-dependent (nonlinear), the system can undergo tipping: an irreversible transition to a qualitatively different state. Examples include:
- West Antarctic Ice Sheet collapse (threshold ~1.5–2°C above pre-industrial)
- Amazon dieback (deforestation + drought feedback)
- AMOC collapse (freshwater hosing, hysteresis)
- Permafrost carbon release (soil carbon-climate feedback)
Mathematically, tipping near a fold bifurcation: the potential $V(T) = -\int R(T)\,dT$ develops a single minimum as the barrier shrinks. Critical slowing down (increased autocorrelation and variance before tipping) provides early warning signals that can in principle be detected in time-series data.
The full nonlinear sensitivity is:
\[\lambda_{eff}(T) = \lambda_0 + \sum_i \lambda_i(T)\]State-dependence of feedbacks means ECS computed from historical warming may differ from ECS under doubled CO₂ — this is the pattern effect and a central open question in climate sensitivity research.