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