EGU2020-11557
https://doi.org/10.5194/egusphere-egu2020-11557
EGU General Assembly 2020
© Author(s) 2020. This work is distributed under
the Creative Commons Attribution 4.0 License.

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

Shaun Lovejoy, Lenin Del Rio Amador, and Roman Procyk
Shaun Lovejoy et al.
  • McGill University, Department of Physics, Montreal, Canada (lovejoy@physics.mcgill.ca)

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.  We retain the basic EBE but add a vertical coordinate so that the imbalances drive the system by imposing heat fluxes through the surface.   While this is theoretically correct, it leads to (apparently) difficult mixed boundary conditions.  However, using Babenko’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].  The HEBE anomaly equation only depends on the local climate sensitivities and relaxation times.  We analytically solve the HEBE and GHEBE for Ts(x,t) and provide evidence that the HEBE applies at scales >≈10km.  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.  Several papers have empirically estimated H ≈ 0.5, as well as t ≈ 4 years, whereas the classical zero-dimensional EBE has H = 1 and t ≈ 4 years.   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.  Future generalizations include albedo-temperature feedbacks and more realistic treatments of past and future climate states.

References

 

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.

How to cite: Lovejoy, S., Del Rio Amador, L., and Procyk, R.: Correcting Budyko-Sellers boundary conditions: The Half-order Energy Balance Equation (HEBE), EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-11557, https://doi.org/10.5194/egusphere-egu2020-11557, 2020

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Presentation version 1 – uploaded on 25 Apr 2020
  • AC1: Updated references, Shaun Lovejoy, 25 Apr 2020

    The references have been updated, the following include hyperlinks to texts:

    References

    Lovejoy, S. (2019): Fractional relaxation noises, motions and the fractional energy balance equation, Nonlin. Processes Geophys. Discuss., https://doi.org/10.5194/npg-2019-39.

    http://www.physics.mcgill.ca/~gang/eprints/eprintLovejoy/neweprint/npg-2019-39-manuscript-version3.pdf

    Lovejoy, S., Weather, Macroweather and Climate: our random yet predictable atmosphere, 334 pp., Oxford U. Press, 2019b.

    Lovejoy, S., 2020: The Half-order Energy Balance Equation, Part 1: 
The homogeneous HEBE and long memories, Earth Syst. Dyn. Disc., https://doi.org/10.5194/esd-2020-12

    https://www.earth-syst-dynam-discuss.net/esd-2020-12/esd-2020-12.pdf

    Lovejoy, S., 2020: The Half-order Energy Balance Equation, Part 2: The inhomogeneous HEBE and 2D energy balance models, Earth Syst. Dyn. Disc., https://doi.org/10.5194/esd-2020-13

    https://www.earth-syst-dynam-discuss.net/esd-2020-13/esd-2020-13.pdf

    Lovejoy, S. 2020: The Fractional Heat Equation and Earth’s energy balance, submitted to Nonlin. Proc. In Geophys. (April 7) http://www.physics.mcgill.ca/~gang/eprints/eprintLovejoy/esubmissions/FHE.NPG.7.4.20ter.pdf 

  • CC1: PDF Version?, Goodwin Gibbins, 04 May 2020

    Hi Professor Lovejoy,

    Would it be possible to upload a PDF version of your slides as well? The formatting on the equations is going a little haywire for me trying to view it in Chrome on Mac. It works okay if I download it, but just through I'd flag this.

    Just reading it through it now!

    Best,

    Goody

  • AC2: PDF version of the presentation, Shaun Lovejoy, 04 May 2020

    Sorry about difficuloites with the math!

    Here is a pdf version:

     

    http://www.physics.mcgill.ca/~gang/ftp.transfer/NP1.1.EGU.HEBE.Lovejoy.25.4.20.pdf