Nonautonomous modelling in Energy Balance Models of climate. Limitations of averaging and climate sensitivity

TL;DR

This work develops a nonautonomous, zero-dimensional Budyko-Sellers-Ghil energy balance framework to study how fast, time-varying forcing from solar irradiance and cloud cover shapes Earth’s mean temperature. By employing skew-product dynamics, averaging theory, and nonautonomous response concepts, the authors show the existence of three nonautonomous equilibria, with two attracting solutions and one repeller, and they quantify when autonomous (averaged) reductions are valid. They reveal fundamental limitations of averaging under non-UKBM or chaotic forcing, while also showing that hyperbolic attractors confer robustness and allow extending error bounds to long times and even under small stochastic perturbations. The analysis of CO$_2$ forcing through SSP scenarios demonstrates a quasi-linear climate sensitivity and provides a framework to compare nonautonomous forcing with its averaged counterpart, highlighting the practical impact of time-dependent forcings on climate projections.

Abstract

Starting from a classical Budyko-Sellers-Ghil energy balance model for the average surface temperature of the Earth, a nonautonomous version is designed by allowing the solar irradiance and the cloud cover coefficients to vary with time in a fast timescale, and to exhibit chaos in a precise sense. The dynamics of this model is described in terms of three existing nonautonomous equilibria, the upper one being attracting and representing the present temperature profile. The theory of averaging is used to compare the nonautonomous model and its time-averaged version. We analyse the influence of the qualitative properties of the time-dependent coefficients and develop physically significant error estimates close to the upper attracting solution. Furthermore, previous concepts of two-point response and sensitivity functions are adapted to the nonautonomous context and used to value the increase in temperature when a forcing caused by CO2 and other emissions intervenes.

Figures (11)

• Figure 1: Qualitative shape of the total solar irradiance (TSI) map $I(t)$ for the quasi-periodic model (upper panel) and the almost periodic model (lower panel). For a comparison with an empirical model of the total solar irradiance extrapolated from real data, see Kopp and Lean kopp2011new.
• Figure 2: Modelling of thin clouds in the upper layer of the atmosphere responsible for the reflection of most of long-wave radiation from the surface. On the left-hand side a quasi-periodic model is shown, whereas on the right-hand side the quasi-periodic forcing is perturbed by a chaotic dynamical system.
• Figure 3: Graphs of $g_*(T)$ (blue solid line) and $g^*(T)$ (red solid line) with respect to the variable $T$. The coloured areas correspond to values of $T$ where the sign of the second derivative of $g$ with respect to $T$ is positive (blue shaded region) or negative (red shaded regions) for all values of $t$. It is possible to appreciate that the vector field $g$ remains negative for all values of $T\in[176,273.5]$, thus preventing the existence of an attractor-repeller pair in the convex region.
• Figure 4: Attractors (red curves), repeller (blue curve), sub- and super-equilibria and zones of concavity (red shaded regions) and convexity (blue shaded regions) for the quasi-periodic model.
• Figure 5: The hyperbolic solutions in \ref{['eq: atractor a_1']} of the quasi-periodic EBM: $a_1(t)$ above, $r(t)$ in the middle and $a_2(t)$ below. The maps $a_1(t)$ and $a_2(t)$ are depicted in red solid lines (attractors) and $r(t)$ in a blue solid line (repeller). Their averages between 1900 and 2024 are depicted in dashed lines. The hyperbolic equilibria of the averaged model are depicted in black solid lines. Note that the three panels do not have the same scale on the vertical axis.