Budyko-Sellers 2.0: the classical and fractional heat equations, and the fractional energy balance equation

2021 ◽  
Author(s):  
Shaun Lovejoy
Keyword(s):  
Energy Balance ◽  
Heat Equation ◽  
Surface Temperature ◽  
Continuum Mechanics ◽  
Balance Equation ◽  
Energy Balance Equation ◽  
Forced Response ◽  
Climate Projections ◽  
Surface Boundary ◽  
Seasonal Forecasts

<p>The highly successful Budyko-Sellers energy balance models are based on the classical continuum mechanics heat equation in two spatial dimensions. When extended to the third dimension using the correct conductive-radiative surface boundary conditions, we show that surface temperature anomalies obey the (nonclassical) Half-order energy balance equation (HEBE, with exponent H = ½) implying heat is stored in the subsurface with long memory. </p><p> </p><p>Empirically, we find that both internal variability and the forced response to external variability are compatible with H ≈ 0.4.  Although already close to the HEBE and classical continuum mechanics, we argue that an even more realistic “effective media” macroweather model is a generalization: the fractional heat equation (FHE) for long-time (e.g. monthly scale anomalies).  This model retains standard diffusive and advective heat transport but generalize the (temporal) storage term.  A consequence of the FHE is that the surface temperature obeys the Fractional EBE (FEBE), generalizing the HEBE to 0< H ≤1.  We show how the resulting FEBE can be been used for monthly and seasonal forecasts as well as for multidecadal climate projections.  We argue that it can also be used for understanding and modelling past climates.</p>

scholarly journals The Half-order Energy Balance Equation, Part 1: The homogeneous HEBE and long memories

2020 ◽  
Author(s):  
Shaun Lovejoy
Keyword(s):  
Heat Flux ◽  
Energy Balance ◽  
Heat Equation ◽  
Surface Temperature ◽  
Continuum Mechanics ◽  
Balance Equation ◽  
Energy Balance Equation ◽  
Surface Boundary ◽  
The Earth ◽  
The Continuum

Abstract. The original Budyko–Sellers type 1-D energy balance models (EBMs) consider the Earth system averaged over long times and applies the continuum mechanics heat equation. When these and the more phenomenological zero (horizontal) – dimensional box models are extended to include time varying anomalies, they have a key weakness: neither model explicitly nor realistically treats the surface radiative – conductive surface boundary condition that is necessary for a correct treatment of energy storage. In this first of a two part series, we apply standard Laplace and Fourier techniques to the continuum mechanics heat equation, solving it with the correct radiative – conductive BC's obtaining an equation directly for the surface temperature anomalies in terms of the anomalous forcing. Although classical, this equation is half – not integer – ordered: the Half - ordered Energy Balance Equation (HEBE). A quite general consequence is that although Newton's law of cooling holds, that the heat flux across surfaces is proportional to a half (not first) ordered 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, the relationship is no longer instantaneous. We then consider the case where the Earth is periodically forced. The classical case is diurnal heat forcing; we 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, we show how this can account for the phase lag between the summer maximum forcing and maximum surface temperature Earth response. In part II, we extend all these results to spatially inhomogeneous forcing and to the full horizontally inhomogeneous problem with spatially varying specific heats, diffusivities, advection velocities, climate sensitivities. We consider the consequences for macroweather forecasting and climate projections.

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

2021 ◽  
Vol 12 (2) ◽  
pp. 469-487 ◽  
Author(s):  
Shaun Lovejoy
Keyword(s):  
Heat Flux ◽  
Energy Balance ◽  
Heat Equation ◽  
Surface Temperature ◽  
Continuum Mechanics ◽  
Balance Equation ◽  
Energy Balance Equation ◽  
The Earth ◽  
Order Energy ◽  
The Continuum

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.

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

2021 ◽  
Vol 12 (2) ◽  
pp. 489-511
Author(s):  
Shaun Lovejoy
Keyword(s):  
Energy Balance ◽  
Balance Equation ◽  
Operator Method ◽  
Energy Balance Equation ◽  
Space Time ◽  
Heat Fluxes ◽  
Climate Projections ◽  
Surface Boundary ◽  
Low Frequencies ◽  
Surface Heat Fluxes

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.

scholarly journals The Half-order Energy Balance Equation, Part 2:The inhomogeneous HEBE and 2D energy balance models

2020 ◽  
Author(s):  
Shaun Lovejoy
Keyword(s):  
Energy Balance ◽  
Balance Equation ◽  
Operator Method ◽  
Energy Balance Equation ◽  
Space Time ◽  
Heat Fluxes ◽  
Climate Projections ◽  
Surface Boundary ◽  
Low Frequencies ◽  
Surface Heat Fluxes

Abstract. In part I, we 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 days) – is highly heterogeneous, in this part II, we thus extend our treatment to the horizontal direction. We first consider a homogeneous Earth but with spatially varying forcing. Using Laplace and Fourier techniques, we derive the Generalized HEBE (the GHEBE) based on half-ordered space-time operators. We analytically solve the homogeneous GHEBE, and show how these operators can be given precise interpretations. We then consider the full inhomogeneous problem with horizontally varying diffusivities, thermal capacities, climate sensitivities and forcings. For this we use Babenko's operator method which generalizes Laplace and Fourier methods. By expanding the inhomogeneous space-time operator at both high and low frequencies, we derive 2-D energy balance equations that can be used for macroweather forecasting, climate projections and for studying the approach to new (thermodynamic equilibrium) climate states when the forcings are all increased and held constant.

Global and Regional Temperature Projections Using the Fractional Energy Balance Equation

2021 ◽  
Author(s):  
Roman Procyk ◽  
Shaun Lovejoy ◽  
Raphaël Hébert ◽  
Lenin Del Rio Amador
Keyword(s):  
Energy Storage ◽  
Energy Balance ◽  
White Noise ◽  
Climate Sensitivity ◽  
Balance Equation ◽  
Historical Data ◽  
Energy Balance Equation ◽  
Error Model ◽  
Climate Projections ◽  
Equilibrium Climate Sensitivity

<p>We present the Fractional Energy Balance Equation (FEBE): a generalization of the standard EBE.  The key FEBE novelty is the assumption of a hierarchy of energy storage mechanisms: scaling energy storage.  Mathematically the storage term is of fractional rather than integer order.  The special half-order case (HEBE) can be classically derived from the continuum mechanics heat equation used by Budyko and Sellers simply by introducing a vertical coordinate and using the correct conductive-radiative surface boundary conditions (the FEBE is a mild extension).</p><div> <p>We use the FEBE to determine the temperature response to both historical forcings and to future scenarios.  Using historical data, we estimate the 2 FEBE parameters: its scaling exponent (<em>H</em> = 0.38±0.05; <em>H</em> = 1 is the standard EBE) and relaxation time (4.7±2.3 years, comparable to box model relaxation times). We also introduce two forcing parameters: an aerosol re-calibration factor, to account for their large uncertainty, and a volcanic intermittency exponent so that the intermittency volcanic signal can be linearly related to the temperature. The high frequency FEBE regime not only allows for modelling responses to volcanic forcings but also the response to internal white noise forcings: a theoretically motivated error model (approximated by a fractional Gaussian noise). The low frequency part uses historical data and long memory for climate projections, constraining both equilibrium climate sensitivity and historical aerosol forcings. <span>Parameters are estimated in a Bayesian framework using 5 global observational temperature series, and an error model which is a theoretical consequence of the FEBE forced by a Gaussian white noise.</span></p> <p>Using the CMIP5 Representative Concentration Pathways (RCPs) and CMIP6 Shared Socioeconomic Pathways (SSPs) scenario, the FEBE projections to 2100 are shown alongside the CMIP5 MME. The Equilibrium Climate Sensitivity = 2.0±0.4 <sup>o</sup>C/CO<sub>2</sub> doubling implies slightly lower temperature increases.   However, the FEBE’s 90% confidence intervals are about half the CMIP5 size so that the new projections lie within the corresponding CMIP5 MME uncertainties so that both approaches fully agree.   The mutually agreement of qualitatively different approaches, gives strong support to both.  We also compare both generations of General Circulation Models (GCMs) outputs from CMIP5/6 alongside with the projections produced by the FEBE model which are entirely independent from GCMs, contributing to our understanding of forced climate variability in the past, present and future.</p> <p>Following the same methodology, we apply the FEBE to regional scales: estimating model and forcing parameters to produce climate projections at 2.5<sup>o</sup>x2.5<sup>o</sup> resolutions. We compare the spatial patterns of climate sensitivity and projected warming between the FEBE and CMIP5/6 GCMs. </p> </div>

Using the fractional energy balance equation for accurate temperature projections through 2100

2020 ◽  
Author(s):  
Roman Procyk ◽  
Shaun Lovejoy ◽  
Lenin Del Rio Amador
Keyword(s):  
Energy Storage ◽  
Energy Balance ◽  
Power Law ◽  
Balance Equation ◽  
General Circulation ◽  
Numerical Models ◽  
General Circulation Models ◽  
Energy Balance Equation ◽  
Forced Response ◽  
Linear Differential

<p>The conventional energy balance equation (EBE) is a first order linear differential equation driven by solar, volcanic and anthropogenic forcings.  The differential term accounts for energy storage usually modelled as one or two “boxes”.  Each box obeys Newton’s law of cooling, so that when perturbed, the Earth’s temperature relaxes exponentially to a thermodynamic equilibrium.</p><p>However, the spatial scaling obeyed by the atmosphere and its numerical models implies that the energy storage process is a scaling, power law process, a consequence largely of turbulent, hierarchically organized oceans currents but also hierarchies of land ice, soil moisture and other processes whose rates depend on size.</p><p>Scaling storage leads to power law relaxation and can be modelled via a seemingly trivial change - from integer to fractional order derivatives - the Fractional EBE (FEBE): with temperature derivatives order 0 < H  < 1 rather than the EBE value H = 1.  Mathematically the FEBE is a past value problem, not an initial value problem.    Recent support for the FEBE comes from [Lovejoy, 2019a] who shows that the special H = 1/2 case (close to observations), the “Half-order EBE” (HEBE), can be analytically obtained from classical Budyko-Sellers energy balance models by improving the boundary conditions.</p><p>The FEBE simultaneously models the deterministic forced response to external (e.g. anthropogenic) forcing as well as the stochastic response to internal forcing (variability) [Lovejoy, 2019b].  We directly exploit both aspects to make projections based on historical data estimating the parameters using Bayesian inference.  Using global instrumental temperature series, alongside CMIP5 and CMIP6 standard forcings, the basic FEBE parameters are H ≈ 0.4 with a relaxation time ≈ 4 years.  </p><p>This observation-based model also produces projections for the coming century with forcings prescribed by the CMIP5 Representative Concentration Pathways scenarios and the CMIP6 Shared Socioeconomic Pathways.</p><p>We compare both generations of General Circulation Models (GCMs) outputs from CMIP5/6 alongside with the projections produced by the FEBE model which are entirely independent from GCMs, contributing to our understanding of forced climate variability in the past, present and future.  When comparing to CMIP5 projections, we find that the mean projections are about 10- 15% lower while the uncertainties are roughly half as large.  Our global temperature projections are therefore within the  CMIP5 90% confidence limits and thus give them strong, independent support.</p><p> </p><p><strong>References</strong></p><p>Lovejoy, S., The half-order energy balance equation, J. Geophys. Res. (Atmos.), (submitted, Nov. 2019), 2019a.</p><p>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, 2019b.</p>

A Mathematical Model for Frost Formation on Ground Aircraft

2011 ◽  
Vol 141 ◽  
pp. 147-151
Author(s):  
Li Wen Wang ◽  
Dan Dan Xu ◽  
Zhi Wei Xing
Keyword(s):  
Mathematical Model ◽  
Growth Rate ◽  
Energy Balance ◽  
Relative Humidity ◽  
Surface Temperature ◽  
Balance Equation ◽  
Atmospheric Temperature ◽  
Energy Balance Equation ◽  
Frost Formation ◽  
The Mathematical Model

A mathematical model is developed to describe frost formation on ground aircraft. The mathematical model was based on frost formation physics together with the mass and energy balance equation developed by Mason. It can be used to forecast the frost formation on ground aircraft. Particular attention is paid to the study of the effects of the important factors, such as surface temperature, atmospheric temperature and relative humidity on the frost growth rate over ground aircraft.

scholarly journals The Fractional Energy Balance Equation for Climate projections through 2100

2020 ◽  
Author(s):  
Roman Procyk ◽  
Shaun Lovejoy ◽  
Raphael Hébert
Keyword(s):  
Energy Balance ◽  
Climate Sensitivity ◽  
Balance Equation ◽  
Ad Hoc ◽  
Internal Variability ◽  
Energy Balance Equation ◽  
Climate Projections ◽  
Scaling Exponent ◽  
Shape Parameters ◽  
Factor Α

Abstract. We produce climate projections through the 21st century using the fractional energy balance equation (FEBE) which is a generalization of the standard EBE. The FEBE can be derived either from Budyko–Sellers models or phenomenologically by applying the scaling symmetry to energy storage processes. It is easily implemented by changing the integer order of the storage (derivative) term in the EBE to a fractional value near 1/2. The FEBE has two shape parameters: a scaling exponent H and relaxation time τ; its amplitude parameter is the equilibrium climate sensitivity (ECS). Two additional parameters were needed for the forcing: an aerosol re-calibration factor α to account for the large aerosol uncertainty, and a volcanic intermittency correction exponent ν. A Bayesian framework based on historical temperatures and natural and anthropogenic forcing series was used for parameter estimation. Significantly, the error model was not ad hoc, but was predicted by the model itself: the internal variability response to white noise internal forcing. The 90 % Confidence Interval (CI) of the shape parameters were H = [0.33, 0.44] (median = 0.38), τ = [2.4, 7.0] (median = 4.7) years compared to the usual EBE H = 1, and literature values τ typically in the range 2–8 years. We found that aerosols were too strong by an average factor α = [0.2, 1.0] (median = 0.6) and the volcanic intermittency correction exponent was ν = [0.15, 0.41] (median = 0.28) compared to standard values α = ν = 1. The overpowered aerosols support a revision of the global modern (2005) aerosol forcing 90 % CI to a narrower range [−1.0, −0.2] W m−2 compared with the IPCC AR5 range [1.5, 4.5] K (median = 3.2 K). Similarly, we found the transient climate sensitivity (TCR) = [1.2, 1.8] K (median = 1.5 K) compared to the AR5 range TCR = [1.0, 2.5] K (median = 1.8 K). As commonly seen in other observational-based studies, the FEBE values are therefore somewhat lower but still consistent with those in IPCC AR5. Using these parameters we made projections to 2100 using both the Representative Carbon Pathways (RCP) and Shared Socioeconomic Pathways (SSP) scenarios and shown alongside the CMIP5/6 MME. The FEBE hindprojections (1880–2019) closely follow observations (notably during the hiatus, 1998–2015). Overall the FEBE were 10–15 % lower but due to their smaller uncertainties, their 90 % CIs lie completely within the GCM 90 % CIs. The FEBE thus complements and supports the GCMs.

SURFTEMP: Simulation of Soil Surface Temperature Using the Energy Balance Equation

1991 ◽  
Vol 20 (1) ◽  
pp. 11-15
Author(s):  
S.D. Wullschleger ◽  
J.E. Cahoon ◽  
J.A. Ferguson ◽  
D.M. Oosterhuis
Keyword(s):  
Energy Balance ◽  
Surface Temperature ◽  
Balance Equation ◽  
Soil Surface ◽  
Energy Balance Equation ◽  
Soil Surface Temperature
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