The half-order energy balance equation – Part 2: The inhomogeneous HEBE and 2D energy balance models

Abstract

In Part 1, I considered the zero-dimensional heat equation, showing quite generally that conductive–radiative surface boundary conditions lead to half-ordered derivative relationships between surface heat fluxes and temperatures: the half-ordered energy balance equation (HEBE). The real Earth, even when averaged in time over the weather scales (up to ≈ 10 d), is highly heterogeneous. In this Part 2, the treatment is extended to the horizontal direction. I first consider a homogeneous Earth but with spatially varying forcing on both a plane and on the sphere: the new equations are compared with the canonical 1D Budyko–Sellers equations. Using Laplace and Fourier techniques, I derive the generalized HEBE (the GHEBE) based on half-ordered space–time operators. I analytically solve the homogeneous GHEBE and show how these operators can be given precise interpretations. I then consider the full inhomogeneous problem with horizontally varying diffusivities, thermal capacities, climate sensitivities, and forcings. For this I use Babenko's operator method, which generalizes Laplace and Fourier methods. By expanding the inhomogeneous space–time operator at both high and low frequencies, I derive 2D energy balance equations that can be used for macroweather forecasting, climate projections, and studying the approach to new (equilibrium) climate states when the forcings are all increased and held constant.