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

Abstract

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

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

“…A1, we then calculated spatial zonal and meridional fluctuations loga( x) and loga( y), and from these their root mean square (rms) values were calculated. From the figure, we see that to a good approximation, log a ( x) ≈ x L EW The fluctuations are Haar fluctuations, but because H x ≈ H y > 0, they are nearly equal to difference fluctuations (Lovejoy and Schertzer, 2012). We see that the zonal and meridional lines are roughly parallel, with a "trivial" horizontal anisotropy factor ≈ 5 (typical north-south fluctuations are 5 times larger than typical east-west ones).…”

“…This two-part paper re-examined the classical heat equation with classical semi-infinite geometry. In the horizontally homogeneous case (Part 1, Lovejoy, 2021), the fundamental novelty is the treatment of the conductive-radiative boundary conditions; here (Part 2), it is the use of Babenko's method to extend this to the more realistic horizontally inhomogeneous problem. In both cases, the semi-infinite subsurface geometry is only important over a shallow layer of the order of the diffusion depth where most of the storage occurs (roughly estimated as ≈ 100 m in the ocean and ≈ < 10 m over land; see Table 1 and Appendix…”

“…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.…”

Multiscale fractal dimension analysis of a reduced order model of coupled ocean–atmosphere dynamics

“…Since this cannot be assured, adaptive methods (as the MEMD) could be more suitable for reducing some mathematical assumptions and a priori constraints (Huang et al, 1998;Huang and Wu, 2008;Rehman and Mandic, 2010). Moreover, geophysical data are usually also characterized by scale-invariant features over a wide range of scales with different complexity and show a scale-dependent behavior due to several factors like forcings, coupling, intrinsic variability, and so on (e.g., Lovejoy and Schertzer, 2013;Franzke et al, 2020). For the above reasons, in this work we put forward a novel approach based on combining two different data analysis methods for investigating the multiscale fractal behavior of the coupled oceanatmosphere system: multivariate empirical mode decomposition (MEMD; Rehman and Mandic, 2010) and generalized fractal dimensions (Hentschel and Procaccia, 1983).…”

“…Recently, by means of a 36-variable model displaying marked LFV Vannitsem et al (2015) demonstrated that the LFV in the atmosphere could be a natural outcome of the ocean-atmosphere coupling. Other sources could be invoked to explain and to contribute to the development of LFV in the atmosphere, such as the long-range system memory as a consequence of the heat storage mechanism of the land-ocean-atmosphere system (e.g., Lovejoy, 2021;Lovejoy et al, 2021), the internal dynamics of the atmosphere itself (e.g., Legras and Ghil, 1985), or even the interaction between the tropical and extratropical regions (e.g., Alexander et al, 2002;Vannitsem et al, 2021), just to quote a few.…”