Study of Thermoelastic behaviour, Efficiency and Effectiveness of Rectangular Fin with Fractional Order Energy Balance Equation

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.

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Correcting Budyko-Sellers boundary conditions: The Half-order Energy Balance Equation (HEBE)

10.5194/egusphere-egu2020-11557 ◽

2020 ◽

Author(s):

Shaun Lovejoy ◽

Lenin Del Rio Amador ◽

Roman Procyk

Keyword(s):

Energy Balance ◽

Boundary Conditions ◽

Balance Equation ◽

Mixed Boundary ◽

Energy Balance Equation ◽

Empirical Orthogonal Functions ◽

Heat Fluxes ◽

Climate Projections ◽

Vertical Coordinate ◽

Order Energy

The conventional 1-D energy balance equation (EBE) has no vertical coordinate so that radiative imbalances between the earth and outer space are redirected in the horizontal in an ad hoc manner.&#160; We retain the basic EBE but add a vertical coordinate so that the imbalances drive the system by imposing heat fluxes through the surface.&#160;&#160; While this is theoretically correct, it leads to (apparently) difficult mixed boundary conditions.&#160; However, using Babenko&#8217;s method, we directly obtain simple analytic equations for (2D) surface temperature anomalies Ts(x,t): the Half-order Energy Balance Equation (HEBE) and the Generalized HEBE, (GHEBE) [Lovejoy, 2019a].&#160; The HEBE anomaly equation only depends on the local climate sensitivities and relaxation times.&#160; We analytically solve the HEBE and GHEBE for Ts(x,t) and provide evidence that the HEBE applies at scales >&#8776;10km.&#160; We also calculate very long time diffusive transport dominated climate states as well as space-time statistics including the cross-correlation matrix needed for empirical orthogonal functions.The HEBE is the special H = 1/2 case of the Fractional EBE (FEBE) [Lovejoy, 2019b], [Lovejoy, 2019c] and has a long (power law) memory up to its relaxation time t.&#160; Several papers have empirically estimated H &#8776; 0.5, as well as t &#8776; 4 years, whereas the classical zero-dimensional EBE has H = 1 and t &#8776; 4 years.&#160;&#160; The former values permit accurate macroweather forecasts and low uncertainty climate projections; this suggests that the HEBE could apply to time scales as short as a month.&#160; Future generalizations include albedo-temperature feedbacks and more realistic treatments of past and future climate states.References&#160;Lovejoy, S., The half-order energy balance equation, J. Geophys. Res. (Atmos.), (submitted, Nov. 2019), 2019a.Lovejoy, S., Weather, Macroweather and Climate: our random yet predictable atmosphere, 334 pp., Oxford U. Press, 2019b.Lovejoy, S., Fractional Relaxation noises, motions and the stochastic fractional relxation equation Nonlinear Proc. in Geophys. Disc., https://doi.org/10.5194/npg-2019-39, 2019c.

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