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Earth System Dynamics An interactive open-access journal of the European Geosciences Union
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© 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 2:The inhomogeneous HEBE and 2D energy balance models

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

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.

Shaun Lovejoy

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Shaun Lovejoy

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Latest update: 01 Dec 2020
Publications Copernicus
Short summary
Radiant energy is exchanged between the Earth's surface and outer space. Some of the local imbalances are stored in the subsurface and some are transported horizontally. In part I we showed how – in a horizontally homogeneous Earth – these classical approaches imply long memory storage useful for seasonal forecasting and multidecadal projections. In this part II, we show how to apply these results to the heterogeneous, real Earth.
Radiant energy is exchanged between the Earth's surface and outer space. Some of the local...