Journal cover Journal topic
Earth System Dynamics An interactive open-access journal of the European Geosciences Union
Journal topic

Journal metrics

Journal metrics

  • IF value: 3.866 IF 3.866
  • IF 5-year value: 4.135 IF 5-year
  • CiteScore value: 7.0 CiteScore
  • SNIP value: 1.182 SNIP 1.182
  • IPP value: 3.86 IPP 3.86
  • SJR value: 1.883 SJR 1.883
  • Scimago H <br class='hide-on-tablet hide-on-mobile'>index value: 33 Scimago H
    index 33
  • h5-index value: 30 h5-index 30
© Author(s) 2020. This work is distributed under
the Creative Commons Attribution 4.0 License.
© Author(s) 2020. This work is distributed under
the Creative Commons Attribution 4.0 License.

  03 Apr 2020

03 Apr 2020

Review status
A revised version of this preprint is currently under review for the journal ESD.

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

Shaun Lovejoy Shaun Lovejoy
  • Physics dept., McGill University, Montreal, Que. H3A 2T8, Canada

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.

Shaun Lovejoy

Interactive discussion

Status: final response (author comments only)
Status: final response (author comments only)
AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment
[Login for Authors/Editors] [Subscribe to comment alert] Printer-friendly Version - Printer-friendly version Supplement - Supplement

Shaun Lovejoy

Shaun Lovejoy


Total article views: 407 (including HTML, PDF, and XML)
HTML PDF XML Total BibTeX EndNote
275 108 24 407 31 30
  • HTML: 275
  • PDF: 108
  • XML: 24
  • Total: 407
  • BibTeX: 31
  • EndNote: 30
Views and downloads (calculated since 03 Apr 2020)
Cumulative views and downloads (calculated since 03 Apr 2020)

Viewed (geographical distribution)

Total article views: 383 (including HTML, PDF, and XML) Thereof 383 with geography defined and 0 with unknown origin.
Country # Views %
  • 1



No saved metrics found.


No discussed metrics found.
Latest update: 04 Aug 2020
Publications Copernicus
Short summary
Monthly, seasonal and decadal scale modelling of the atmosphere is possible using the principle of energy balance. Yet the scope of classical approaches is limited because they do not adequately deal with energy storage in the Earth system. We show that the introduction of a vertical coordinate implies that the storage has a huge memory. This memory can be used for macroweather (long range) forecasts and climate projections.
Monthly, seasonal and decadal scale modelling of the atmosphere is possible using the principle...