The half-order energy balance equation – Part 1: The homogeneous HEBE and long memories

Abstract. The original Budyko–Sellers type of 1D energy balance models (EBMs) consider the Earth system averaged over long times and apply continuum mechanics heat equation. When these more phenomenological box are extended to include time-varying anomalies, they have a key weakness: neither model explicitly nor realistically treats conductive–radiative surface boundary condition that is necessary for correct treatment storage. In this first two-part series, I standard Laplace Fourier techniques equation, solving it with radiative–conductive conditions obtaining an equation directly temperature anomalies in terms anomalous forcing. Although classical, half-ordered not integer-ordered: half-order (HEBE). A quite general consequence although Newton's law cooling holds, flux across surfaces proportional (not first-ordered) time derivative temperature. This implies has memory, depends on entire previous history forcing, temperature–heat relationship no longer instantaneous. then case which periodically forced. classical diurnal forcing; extend annual forcing show thermal impedance complex valued quantity equal (complex) climate sensitivity. Using simple semi-empirical how HEBE can account phase lag between summer maximum response. Part 2, all results spatially inhomogeneous full horizontally problem varying specific heats, diffusivities, advection velocities, sensitivities. consequences macroweather (monthly, seasonal, interannual) forecasting projections.