The Temperature Feedback Problem

Abstract

In the past 810 ka, inferred variability of absolute temperature has barely exceeded ±3 K, or ±1 % of the period mean. This thermostatic behavior seems inconsistent with a feedback sum f t = (σ i f t,i) ≫ 0 over any term of years t. In electronic circuits, as the loop gain g t exceeds 1, the Bode system-gain relation G t = (1 − g t)−1 models the abrupt transition of the output voltage from V t → + ∞ to − ∞ ← V t ( G t, V t being momentarily undefined at the singularity, where g t = 1). Yet in the climate, where g t might plausibly exceed unity owing to, say, the near-exponential Clausius-Clapeyron increase in the water-vapor carrying capacity of the atmospheric space as it warms, +Δ f t ⇒ +Δ T t ⇏ −Δ T t for all f t > 0. In this and several other material respects the Bode relation, which as clearly specifies a negative dynamical-system response to g t > 1 as a positive response to g t < 1, seems inapplicable to the climate. In general-circulation models, the equilibrium system gain G∞ = (1 − g∞)−1 = (1 − Λ0 f∞)−1, where Λ0 is the Planck parameter, doubles or triples a direct warming. But if Bode is inapplicable to the climate, what is the appropriate relation between loop gain g t and system gain G t, and thus between instantaneous (Δ T0), transient (Δ T t), and equilibrium (Δ T∞) climate sensitivity? What representation of system gain in the climate should replace the inapplicable Bode relation?