Emergence of a non-linear global surface air temperature response to rapidly increasing effective radiative forcing

Abstract. A time series of global surface air temperature anomalies (GSAT) and effective radiative forcing (ERF) obtained from IPCC Sixth Assessment Report (AR6) for 1977 to 2025 provides evidence that GSAT is a non-linear, power-law function of ERF when ERF is rapidly increasing. With ERF > 0 W m⁻² (n = 48), the GSAT-ERF relationship is described by a power-law, GSAT = 0.41 + 0.11 × ERF^1.90 (R² = 0.896; likely range = ERF^1.74-2.24). A linear relationship was rejected (F(1,45) = 15.41, p < 0.0003). GSAT, ERF^1.90, and the Niño 3.4 sea surface temperature anomaly index (ONI) were used in an autoregressive distributed lag model [ARDL(1,0,0)] yielding GSAT_t = 0.32 + 0.23 GSAT_t-1 + 0.08 ERF^1.90 + 0.11 ONI. This model is equivalent to a two-layer energy-balance model with the deep-ocean temperature held constant. The ARDL model explains 96 % of GSAT variability from 1977 to 2025 (R² = 0.959). Residual diagnostics indicate no evidence of residual autocorrelation or heteroskedasticity and bounds testing (F = 40.51) supports cointegration. The model was validated by expanding-window recursive out-of-sample prediction from 1994 to 2025 (Theil’s U = 0.53). Earth’s climate system has entered a new regime in which the radiative response to the energy imbalance at the top of the atmosphere is progressively decreasing. ARDL projections for GSAT in 2041–2060 with ONI = 0 exceed the AR6 best estimate by 0.6 °C. If ERF continues to rise 0.7 W m⁻² decade⁻¹, GSAT by mid-century will likely be 2.8 ± 0.3 °C.