Correcting Budyko-Sellers boundary conditions: The Half-order Energy Balance Equation (HEBE)
Bibliographic record
Abstract
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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.016 | 0.003 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".