A one-dimensional climate energy balance model (1D EBM) is a simplified climate model for the zonally averaged global temperature profile, based on the Earth's energy budget. We examine a class of 1D EBMs which emerges as the parabolic equation corresponding to the Euler–Lagrange equations of an associated variational problem, covering spatially inhomogeneous models such as with latitude-dependent albedo. Sufficient conditions are provided for the existence of at least three steady-state solutions in the form of two local minima and one saddle, that is, of coexisting “cold”, “warm” and unstable “intermediate” climates. We also give an interpretation of minimizers as “typical” or “likely” solutions of time-dependent and stochastic 1D EBMs.

We then examine connections between the

Specifically, the global mean temperature varies continuously as long as there is a unique minimizing temperature profile, but coexisting minimizers must have different global mean temperatures. Furthermore, global mean temperature is non-decreasing with respect to greenhouse gas concentration, and its jumps must necessarily be upward.

Applicability of our findings to more general spatially heterogeneous reaction–diffusion models is also discussed, as are physical interpretations of our results.

Energy balance models are a fundamental tool used to understand the Earth's climate system and its energy dynamics. They represent the energy budget within the Earth's atmosphere, land, oceans and ice by quantifying the balance between incoming solar radiation and outgoing solar radiation. Although highly simplified compared to general circulation models, energy balance models (EBMs) are appreciated for their interpretability, mathematical tractability and ability to capture the essential dynamics of the Earth system

A zero-dimensional (0D) EBM is the simplest version of an EBM describing the evolution in time for the annual averaged global mean temperature

A one-dimensional (1D) EBM is given by a parabolic partial differential equation where the space variable is one-dimensional

This paper focuses on the study of the properties and the interpretation of the steady-state solutions of a 1D EBM depending on a bifurcation parameter. Motivated by 0D EBMs, there is a wide consensus in the literature, supported mainly by numerical simulations, regarding the existence of either one or three “interesting” steady-state solutions for 1D EBMs. Firstly, in Theorem 1 we prove the existence of a steady-state solution for the 1D EBM by solving the associated variational problem

the viscosity

the space-averaged global radiation balance

In Theorem 3, we show that

Furthermore, the value function fails to be differentiable if and only if there are two or more co–existing global minimizers for

In Corollary 4, we demonstrate how the derivative of

Our results have a number of interesting physical interpretations. The elliptic 1D EBM not only describes stationary solutions of the time-dependent 1D EBM but moreover characterizes “likely” climates around which the solutions of the stochastic 1D EBM fluctuate. Global minimizers carry special importance as they are exponentially more likely than just local minimizers. Coexistence of global minimizers is just of special interest as these represent equally likely climate scenarios, and intuitively it seems plausible that rapid transitions between those climates are a dominant feature of the dynamics, although this point is not investigated further here. Furthermore coexistence of global minimizers implies a discontinuous change of global mean temperature which will jump upwards with increasing greenhouse gas concentration.

We expect that additional interesting physical conclusions can be drawn through identifying

This paper is organized as follows. In Sect.

The fundamental mechanism of 1D EBMs is that the temperature

Firstly, the absorbed radiation is assumed to have the following form:

Second, the emitted radiation is modelled using the Stefan–Boltzmann law, in other words assuming that the Earth radiates as a black body. Under this assumption, the energy radiated is proportional to the fourth power of its temperature and it is given by

The third component of the model is the term

For the parabolic problem (Eq.

We recall the formulation of stochastic EBMs using the theory of SPDEs

As mentioned in the introduction, this measure is concentrated on minimum points of the functional

The stationary problem associated with the 1D EBM is given by the elliptic equation for

In the following paragraph, we describe the properties of the solutions of Eq. (

A stability analysis can be conducted to determine the stability of the steady-state solutions. The results show that

Parameters and constants appearing in the Seller EBM (

In this section, we (i) provide an intuitive motivation for why the invariant measure for the stochastic EBM concentrates on minimum points of the functional

Firstly, consider the stochastic EBM (

Next, we discuss the properties of the functional

A rigorous proof of the previous result can be found in Sect. S3. The proof relies on standard arguments from the direct method of calculus of variation, exploiting the fact that the outgoing radiation in the EBM model prevents the temperature from being too high.

Concerning the existence of two local minimum points, let us describe a sufficient condition. Consider the potential function

If

Note how the previous result can be also interpreted as giving sufficient conditions for the convergence of the stable solutions of a space-inhomogeneous EBM to the stable solution of the corresponding space-averaged model, as the diffusion becomes large.

The key element of this section is the value function, which is given by

Potential functional

Summarizing, the numerical evaluations of

We also see numerically that

An additional property of the value function can be observed when comparing the bifurcation diagram (Fig.

Comparison between the value function graph (left) and bifurcation diagram (right) for the 1D EBM. The magenta-shaded area highlights the parts of the plots which are in one-to-one correspondence.

It is worth pointing out that by combining Theorem 3 and Corollary 4, a valuable property emerges, i.e. the global mean temperature of the functional minimizer is non-decreasing with respect to

In the second part of this section, we demonstrate the applicability of Corollary 4 to other reaction–diffusion equations. We use as an example a spatially heterogeneous Allen–Cahn equation (ACE), already considered in

Comparison between the value function and the bifurcation diagram for the non-homogeneous ACE. The magenta-shaded area highlights the parts of the plots which are in one-to-one correspondence.

In this paper, we have considered a one-dimensional energy balance model depending on a bifurcation parameter

We then introduced the value function

The diffusion function

Regarding the impact of our work on current climate change, we have characterized climate as an invariant measure within a stochastic equation that describes temperature. The emission of

Concerning future development of this work, one interesting aspect we are working on is to understand how the invariant measure for the stochastic EBM changes close to bifurcation points. This points in the direction of using statistical indicators to detect the approach of tipping points, which in our model correspond to points of discontinuity of the global mean temperature with respect to the parameter

In this section, we describe the numerical method adopted to approximate the solutions of the elliptic problem (

This work does not include any externally supplied code, data, or other material. All material in the text and figures was produced by the authors using standard mathematical and numerical analysis by the authors. The code is available at Zenodo (

The supplement related to this article is available online at:

GDS conceptualized the paper, performed the numerical simulations, and took the lead role in writing and revising the paper. JB, FF and TK conceptualized the paper and supervised the writing. All authors provided critical feedback and helped shape the research.

The contact author has declared that none of the authors has any competing interests.

Views and opinions expressed are those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

We are very grateful to the referee for carefully reading the paper and for their valuable comments. We acknowledge fruitful discussions with Valerio Lucarini and Robbin Bastiaansen. Gianmarco Del Sarto would like to thank the Department of Mathematics and Statistics, University of Reading, for its hospitality. Tobias Kuna and Jochen Bröcker would like to thank the Scuola Normale Superiore for its hospitality. Gianmarco Del Sarto is supported by the Italian national interuniversity PhD course in sustainable development and climate change.

Franco Flandoli's research is funded by the European Union (ERC, NoisyFluid, no. 101053472).

This paper was edited by Natale Alberto Carrassi and reviewed by two anonymous referees.