Abstract
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.
Highlights
In Part 1, I showed that when the surface of a body exchanges heat both conductively and radiatively, its flux depends on the half-order derivative of the surface temperature (Lovejoy, 2021)
The result directly followed by assuming that the continuum mechanics heat equation was obeyed and the depth of the media was of the order of a few diffusion depths; for the Earth, this is perhaps several hundred meters
The homogeneous heat equation in a semi-infinite domain is a classical problem, and conductive–radiative surface boundary conditions naturally lead to fractional-order operators: the half-order energy balance equation (HEBE) and generalized half-order energy balance equation (GHEBE)
Summary
In Part 1, I showed that when the surface of a body exchanges heat both conductively and radiatively, its flux depends on the half-order derivative of the surface temperature (Lovejoy, 2021). This implies that energy stored in the subsurface effectively has a huge power-law memory. S. Lovejoy: The half-order energy balance equation ment of the surface boundary condition), the discussion was restricted to zero horizontal dimensions. Lovejoy: The half-order energy balance equation ment of the surface boundary condition), the discussion was restricted to zero horizontal dimensions I include several appendices focused on empirical parameter estimates (Appendix A), the implications for two-point and space– time temperature statistics (when the system is stochastically forced with internal variability; Appendix B), and (Appendix C) the changes needed to account for the Earth’s spherical geometry, including the definition of fractional operators on the sphere
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