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

Lovejoy, S.

doi:10.5194/esd-12-489-2021

Cited by 7 publications

(8 citation statements)

References 37 publications

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“…Hence, the box and HEBE models are special cases of the fractional-order energy balance equation (FEBE; Lovejoy, 2019a, b) derived phenomenologically in Lovejoy et al (2021). Whereas the box model changes its heat content instantaneously with its current temperature (T (t)), at any moment, the energy stored in the HEBE model depends on the past temperatures, and since their weights fall off slowly -there is a long memory -it potentially depends on the temperature and hence energy stored in the distant past.…”

Section: The Hebe Zero-dimensional and Box Models Andmentioning

confidence: 99%

“…where h is for "horizontal inhomogeneity" as in Lovejoy et al, 2021; there is an analogous calculation for the FEBE with h = 1/2. In the North et al (1983) 1D model, the top-of-theatmosphere forcing is exactly a cosine variation, i.e., with a single wavenumber k = 1 cycle around the Earth.…”

Section: Empirical Estimates Of Complex Climate Sensitivitiesmentioning

confidence: 99%

“…longwave emissions. EBMs avoid explicit treatment of this critical surface boundary condition, treating it phenomenologically in ways that are flawed; in this two-part paper, I show how they can easily be improved with significant benefits: first, the (idealized) homogeneous case (Part 1) and then the general horizontally inhomogeneous (2D) case (Part 2; Lovejoy, 2021).…”

Section: Introductionmentioning

confidence: 99%

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

Lovejoy

2021

Earth Syst. Dynam.

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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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Section: The Hebe Zero-dimensional and Box Models Andmentioning

confidence: 99%

Section: Empirical Estimates Of Complex Climate Sensitivitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Lovejoy

2021

Earth Syst. Dynam.

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“…This temperature calculation is based on the earth's outgoing energy and incoming energy from the Sun. The zero-dimensional heat equation was considered in Lovejoy (2021) [27] to show that generally, conductive, radiative surface boundary condition leads to a half-order derivative relationship. The relationship is based on temperature and surface heat fluxes.…”

Section: Introductionmentioning

confidence: 99%

“…One of the simple and effective climate system models is energy balance models (EBMs). These models were proposed about 50 years ago by Lohman (2020) [26] and Lovejoy (2021) [27]. These EBMs aimed to predict temperatures and how the climate system is sensitive to external forces.…”

Section: Introductionmentioning

confidence: 99%

A Computational Approach to a Mathematical Model of Climate Change Using Heat Sources and Diffusion

2022

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The present work aims to extend the climate change energy balance models using a heat source. An ordinary differential equations (ODEs) model is extended to a partial differential equations (PDEs) model using the effects of diffusion over the spatial variable. In addition, numerical schemes are presented using the Taylor series expansions. For the climate change model in the form of ODEs, a comparison of the presented scheme is made with the existing Trapezoidal method. It is found that the presented scheme converges faster than the existing scheme. Also, the proposed scheme provides fewer errors than the existing scheme. The PDEs model is also solved with the presented scheme, and the results are displayed in the form of different graphs. The impact of the climate feedback parameter, the heat uptake parameter of the deep ocean, and the heat source parameter on global mean surface temperature and deep ocean temperature is also portrayed. In addition, these recently developed techniques exhibit a high level of predictability. Doi: 10.28991/CEJ-2022-08-07-04 Full Text: PDF

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The Future of Climate Modelling: Weather Details, Macroweather Stochastics—Or Both?

Lovejoy

2022

Meteorology

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Since the first climate models in the 1970s, algorithms and computer speeds have increased by a factor of ≈1017 allowing the simulation of more and more processes at finer and finer resolutions. Yet, the spread of the members of the multi-model ensemble (MME) of the Climate Model Intercomparison Project (CMIP) used in last year’s 6th IPCC Assessment Report was larger than ever: model uncertainty, in the sense of MME uncertainty, has increased. Even if the holy grail is still kilometric scale models, bigger may not be better. Why model structures that live for ≈15 min only to average them over factors of several hundred thousand in order to produce decadal climate projections? In this commentary, I argue that alongside the development of “seamless” (unique) weather-climate models that chase ever smaller—and mostly irrelevant—details, the community should seriously invest in the development of stochastic macroweather models. Such models exploit the statistical laws that are obeyed at scales longer than the lifetimes of planetary scale structures, beyond the deterministic prediction limit (≈10 days). I argue that the conventional General Circulation Models and these new macroweather models are complementary in the same way that statistical mechanics and continuum mechanics are equally valid with the method of choice determined by the application. Candidates for stochastic macroweather models are now emerging, those based on the Fractional Energy Balance Equation (FEBE) are particularly promising. The FEBE is an update and generalization of the classical Budyko–Sellers energy balance models, it respects the symmetries of scaling and energy conservation and it already allows for both state-of-the-art monthly and seasonal, interannual temperature forecasts and multidecadal projections. I demonstrate this with 21st century FEBE climate projections for global mean temperatures. Overall, the projections agree with the CMIP5 and CMIP6 multi-model ensembles and the FEBE parametric uncertainty is about half of the MME structural uncertainty. Without the FEBE, uncertainties are so large that climate policies (mitigation) are largely decoupled from climate consequences (warming) allowing policy makers too much “wiggle room”. The lower FEBE uncertainties will help overcome the current “uncertainty crisis”. Both model types are complementary, a fact demonstrated by showing that CMIP global mean temperatures can be accurately projected using such stochastic macroweather models (validating both approaches). Unsurprisingly, they can therefore be combined to produce an optimum hybrid model in which the two model types are used as copredictors: when combined, the various uncertainties are reduced even further.

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

Cited by 7 publications

References 37 publications

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

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

A Computational Approach to a Mathematical Model of Climate Change Using Heat Sources and Diffusion

The Future of Climate Modelling: Weather Details, Macroweather Stochastics—Or Both?

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