Definitions: ECS, TCR, and TCRE

Equilibrium Climate Sensitivity (ECS): the equilibrium global mean surface temperature change following a sustained doubling of atmospheric CO₂:

\[ECS = \Delta T_{2\times CO_2}^{equil} = -\frac{\Delta F_{2\times}}{\lambda_{eff}}\]

where $\Delta F_{2\times} \approx 3.93$ W m⁻² (AR6) and $\lambda_{eff}$ is the net feedback parameter (W m⁻² K⁻¹, negative for a stable climate).

Transient Climate Response (TCR): warming at CO₂ doubling under 1% yr⁻¹ increase (70-year ramp). TCR $<$ ECS because ocean heat uptake delays equilibration:

\[TCR = ECS \cdot \frac{1}{1 + \kappa/\lambda_{eff}}\]

where $\kappa \approx 0.6$ W m⁻² K⁻¹ is the ocean heat uptake efficiency.

Transient Climate Response to Cumulative Emissions (TCRE): approximately constant at ~0.45°C per 1000 PgC (AR6), providing the basis for carbon budgets.

Metric	AR6 likely range	AR6 best estimate
ECS	2.5–4.0°C	3.0°C
TCR	1.2–2.4°C	1.8°C
TCRE	0.27–0.63°C/1000 PgC	0.45°C/1000 PgC

Feedback analysis

The net feedback parameter decomposes into additive contributions:

\[\lambda_{eff} = \lambda_0 + \sum_i \lambda_i\]

where $\lambda_0 = -3.2$ W m⁻² K⁻¹ is the Planck response (blackbody cooling). Individual feedback parameters (AR6 best estimates, W m⁻² K⁻¹):

Feedback	$\lambda_i$	Sign	Mechanism
Planck (reference)	−3.20	Stabilizing	Increased OLR with $T$
Water vapor	+1.77	Amplifying	More WV in warmer atm.
Lapse rate	−0.50	Stabilizing	Tropical upper-trop. warms more
Cloud (SW)	+0.42	Amplifying	Low-cloud reduction
Cloud (LW)	+0.20	Amplifying	High-cloud rise
Surface albedo	+0.35	Amplifying	Ice/snow retreat
Net	−0.96

The feedback factor $f$ measures the amplification of the no-feedback response:

\[f_i = \frac{\lambda_i}{\lambda_0}, \qquad \Delta T = \frac{\Delta T_{Planck}}{1-\sum f_i}\]

With the values above, $\sum f_i \approx 0.70$, amplifying the Planck-only response by $1/(1-0.70) \approx 3.3$.

Cloud feedbacks and the Charney range

Cloud feedbacks are the dominant source of inter-model spread. The AR6 likely range 2.5–4.0°C is largely determined by uncertainty in low-cloud (boundary layer) response.

Low-cloud feedback mechanism: Marine stratocumulus cover decreases with warming because (a) the lower troposphere becomes less stably stratified and (b) subsidence weakens. The shortwave cloud feedback:

\[\lambda_{SW,cloud} = \frac{S_0}{4}\frac{\partial(1-\alpha)}{\partial T} = -\frac{S_0}{4}\frac{\partial\alpha_{cloud} \cdot f_{cloud}}{\partial T}\]

CMIP6 models span $\lambda_{SW,cloud}$ from $-0.1$ to $+1.2$ W m⁻² K⁻¹ — a 1.3 W m⁻² K⁻¹ spread translating to ~1.5°C in ECS.

Emergent constraints

Cross-model relationships between observable present-day quantities and ECS provide emergent constraints. Key examples:

• Sherwood et al. (2014): Lower-tropospheric mixing (D index) correlates with ECS across CMIP5 models; observations constrain ECS $>$ 3°C.
• Klein & Hall (2015): Seasonal cycle of low-cloud cover correlates with ECS.
• Zelinka et al. (2020): CMIP6 high ECS models ($>$ 5°C) have unrealistically large cloud feedbacks compared to CERES observations.

In a Bayesian framework, let $x$ be the observable and $y = ECS$:

\[p(y | x_{obs}) \propto \int p(x_{obs} | x_{model}) p(y | x_{model}) p(x_{model})\,dx_{model}\]

With a Gaussian likelihood and a uniform prior, the constrained estimate is:

\[\hat{y} = \bar{y} + \frac{\rho\sigma_y}{\sigma_x}(x_{obs} - \bar{x}), \quad \sigma_{y|x}^2 = \sigma_y^2(1-\rho^2)\]

where $\rho$ is the cross-model correlation. This yields a narrower posterior distribution.

Historical energy balance constraint

The energy budget method uses instrumental records directly. Over a period with forcing $\Delta F$, warming $\Delta T$, and ocean heat uptake $N$:

\[ECS = \frac{\Delta F_{2\times}}{\Delta F - N}\Delta T\]

Using AR6 estimates ($\Delta T = 1.03$ K, $\Delta F = 2.20$ W m⁻², $N = 0.79$ W m⁻²):

\[ECS \approx \frac{3.93}{2.20 - 0.79} \times 1.03 \approx \frac{3.93}{1.41} \times 1.03 \approx 2.87\text{ K}\]

The pattern effect complicates this: the geographic pattern of SST warming affects the global radiative response. Historical warming has been relatively concentrated in regions of low cloud feedback (Pacific); future warming will spread to high-feedback regions, implying $\lambda_{eff}^{historical} > \lambda_{eff}^{future}$, and so the energy-budget method underestimates ECS by ~0.5°C.

Paleoclimate evidence

Last Glacial Maximum (LGM, ~21 ka): Global cooling of ~5–7°C with ice-sheet and CO₂ forcings of ~$-8$ W m⁻². The paleo-ECS from LGM:

\[ECS_{paleo} = \frac{\Delta T_{LGM}}{\Delta F_{LGM}}\times \Delta F_{2\times} \approx \frac{6}{8} \times 3.93 \approx 2.9\text{ K}\]

Eocene (~50 Ma): CO₂ ~1000 ppm, temperatures ~10–14°C warmer. Paleoclimate proxies (Mg/Ca, $\delta^{18}$O, TEX86) suggest ECS consistent with 2.5–4.5°C when ice-albedo feedback differences are accounted for. The combination of instrumental, model, and paleo evidence underpins the AR6 assessment that ECS is very likely 2.0–5.0°C and likely 2.5–4.0°C, with a best estimate of 3.0°C.