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. In Part 1, I considered the zero-dimensional heatequation, showing quite generally that conductive–radiative surfaceboundary conditions lead to half-ordered derivative relationships betweensurface heat fluxes and temperatures: the half-ordered energy balanceequation (HEBE). The real Earth, even when averaged in time over theweather scales (up to ≈ 10 d), is highly heterogeneous. Inthis 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 varyingdiffusivities, thermal capacities, climate sensitivities, and forcings. Forthis I use Babenko's operator method, which generalizes Laplace and Fouriermethods. By expanding the inhomogeneous space–time operator at both high andlow frequencies, I derive 2D energy balance equations that can be used formacroweather forecasting, climate projections, and studying the approachto new (equilibrium) climate states when the forcings are all increased andheld constant.

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