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, +Δf _t ⇒ +ΔT _t ⇏ −ΔT _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 − Λ₀f_∞)⁻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 (ΔT₀), transient (ΔT _t ), and equilibrium (ΔT_∞) climate sensitivity? What representation of system gain in the climate should replace the inapplicable Bode relation?