Preprints
https://doi.org/10.5194/egusphere-2023-1994
https://doi.org/10.5194/egusphere-2023-1994
12 Sep 2023
 | 12 Sep 2023

Variational Techniques for a One-Dimensional Energy Balance Model

Gianmarco Del Sarto, Jochen Bröcker, Franco Flandoli, and Tobias Kuna

Abstract. A one-dimensional climate energy balance model (1D-EBM) is a simplified climate model that describes the evolution of Earth's temperature based on the planet's energy budget. In this study, we examine a 1D-EBM that incorporates a bifurcation parameter representing the impact of carbon dioxide on the energy balance. Firstly, independent of the value of the additive parameter, we demonstrate the existence of a steady-state solution by solving the associated variational problem, which involves minimizing a potential functional. Again using variational techniques, we can give sufficient conditions to prove the existence of at least three-steady state solutions. Secondly, we establish the uniqueness of the solution for the variational problem by examining the differentiability of the value function, which represents the minimum value of the potential functional across all temperature profiles. Lastly, we explore how this characterization provides valuable insights into the structure of the bifurcation diagram. Specifically, we demonstrate a one-to-one correspondence between the derivative of the value function and the mean value of the minimizer for the variational problem. Furthermore, we show the applicability of our findings to more general reaction-diffusion spatially heterogeneous models.

Gianmarco Del Sarto, Jochen Bröcker, Franco Flandoli, and Tobias Kuna

Status: closed

Comment types: AC – author | RC – referee | CC – community | EC – editor | CEC – chief editor | : Report abuse
  • RC1: 'Comment on egusphere-2023-1994', Anonymous Referee #1, 06 Nov 2023
    • AC1: 'Reply on RC1', Gianmarco Del Sarto, 21 Jan 2024
  • RC2: 'Comment on egusphere-2023-1994', Anonymous Referee #2, 27 Nov 2023
    • AC2: 'Reply on RC2', Gianmarco Del Sarto, 21 Jan 2024

Status: closed

Comment types: AC – author | RC – referee | CC – community | EC – editor | CEC – chief editor | : Report abuse
  • RC1: 'Comment on egusphere-2023-1994', Anonymous Referee #1, 06 Nov 2023
    • AC1: 'Reply on RC1', Gianmarco Del Sarto, 21 Jan 2024
  • RC2: 'Comment on egusphere-2023-1994', Anonymous Referee #2, 27 Nov 2023
    • AC2: 'Reply on RC2', Gianmarco Del Sarto, 21 Jan 2024
Gianmarco Del Sarto, Jochen Bröcker, Franco Flandoli, and Tobias Kuna
Gianmarco Del Sarto, Jochen Bröcker, Franco Flandoli, and Tobias Kuna

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Short summary
We investigate a 1D-energy balance model incorporating a bifurcation parameter to represent the impact of carbon dioxide on Earth's energy budget. We provide conditions for the existence of at least three steady-state solutions, studying the minimum points of a potential functional. Additionally, we characterize the uniqueness of the global minimum point and explore its implications on the bifurcation diagram, with possible applications to reaction-diffusion spatially heterogeneous models.