Variational Formulation of Budyko-Sellers Climate Models

On the Budyko-sellers Energy Balance Climate Model with Ice Line Coupling

Over 40 years ago, M. Budyko and W. Sellers independently introduced low-order climate models that continue to play an important role in the mathematical modeling of climate. Each model has one spatial variable, and each was introduced to investigate the role ice-albedo feedback plays in influencing surface temperature. This paper serves in part as a tutorial on the Budyko-Sellers model, with p...

A variational formulation of vertical slice models

A variational framework is defined for vertical slice models with three-dimensional velocity depending only on x and z. The models that result from this framework are Hamiltonian, and have a Kelvin– Noether circulation theorem that results in a conserved potential vorticity in the slice geometry. These results are demonstrated for the incompressible Euler– Boussinesq equations with a constant t...

Variational Formulation of Problems and Variational Methods

A variational formulation for PDEs

Definition 1. A classic solution (sometimes called strong solution) to the problem is a function u ∈ C2([a, b]) such that it vanishes at the endpoints and pointwise it satisfies the equation −u′′(x)+ u(x) = f(x). Let’s change approach. Suppose that we formally multiply the differential equation by an arbitrary function φ ∈ C1([a, b]) with φ(a) = φ(b) = 0 and we integrate over the interval [a, b...

Understanding the Budyko Equation

The Budyko equation has achieved iconic status in hydrology for its concise and accurate representation of the relationship between annual evapotranspiration and long-term-average water and energy balance at catchment scales. Accelerating anthropogenic land-use and climate change have sparked a renewed interest in predictive applications of the Budyko equation to analyze future scenarios import...