Bob Wentworth Ph.D. (Applied Physics)

In their 2020 paper, An Analysis of the Earth’s Energy Budget, Stephen Wilde and Philip Mulholland (W&M) examine the energy budget of the Earth. They infer the presence of energy fluxes associated with energy recycling, add the measured and inferred fluxes together to achieve a total energy flux which matches the flux needed to explain the Earth’s mean surface temperature. Based on this analysis, they offer various conclusions.

In this post, I’ll deconstruct W&M’s analysis. I expect this will be of interest primarily to those interested in their particular analytic methods and conclusions.

My prior post Atmospheric Energy Recycling is essential back-ground reading for understanding what I will be offering here. So, please read that if you want the rest of this post to make sense.

Regrettably, W&M’s analysis in this paper contains multiple serious errors which render the conclusions meaningless.

Let me outline key aspects of their analysis process, as I understand it:

- W&M (correctly) embrace the idea that there exists an “energy recycling” process in which energy circulates back and forth between the planetary surface and the atmosphere, increasing net energy flux in this region.
- W&M (incorrectly) assume that the recycling fraction associated with this recycling process is β = ½.
- W&M (correctly, within the limitations of the assumption that β=½) sum the net result of recycling back and forth (with infinite but geometrically declining power) to conclude that any initial energy flux introduced to the recycling process will result in the generation of an equal downwelling radiation flux towards the surface, and an another equal upward energy flux towards space.
- W&M assert (p. 57) “the atmosphere retains and stores an energy flux equal to that of the total intercepted flux” and (p. 60) talk about the “role of the atmosphere as an energy recycling reservoir.” They say the energy recycling process is the means by which the “Back Radiation of the canonical model is created and stored in the atmospheric reservoir.”

- For each energy flux absorbed by the atmosphere, whether that flux originates from the surface, or from direct absorption of solar radiation, W&M tally up an equal inferred “stored” flux due to energy recycling. (These fluxes are listed in tables under the label “Infinite Recycling Limit”).

- W&M compute (p. 58) a “total energy budget for the Earth” which includes: solar radiation absorbed by the surface, solar radiation absorbed by the atmosphere, net energy fluxes from the surface absorbed by the atmosphere, fluxes “stored” as a result of recycling of fluxes absorbed by the atmosphere. This yields a “total planetary energy budget of 558 W/m² [which] converts to a thermodynamic temperature for the Earth’s surface of 315 Kelvin (42° Celsius)” using the Stephan-Boltzmann law.
- W&M reason (p. 58) that energy fluxes away from the surface cool the surface and so must be subtracted from the previously calculated “total planetary energy budget.” They subtract net all net heat fluxes away from the surface, including radiative heat flux (net of radiation from surface subtracting back-radiation), and fluxes associated with thermals and evaporation. This produces what is described a “Surface Radiation flux” of 390 W/m². Applying the Stephan-Boltzmann law, they conclude “the average temperature of the Earth’s atmosphere for a net atmospheric power intensity flux of 390 W/m2 is shown to be 288 Kelvin (15° Celsius).”

Reviewing this, I notice some seeming oddity and vagueness in the way that locations are discussed.

- W&M (c.f., item #4) apparently think of recycled energy fluxes as being “stored” in the atmosphere. Yet, a flux is a flow (energy per unit time per unit area) at a particular location.
Calculations of “back-radiation” to the surface are well-defined and meaningful if one is talking about fluxes at the interface between the surface and the atmosphere. While fluxes exist at other points in the atmosphere, there is unlikely to be a simple relationship between those fluxes within the atmosphere and the fluxes at the interface with the Earth’s surface.

I am skeptical that it can be meaningfully be said that fluxes are “stored” anywhere. I wonder how this surprising way of thinking affects W&M’s reasoning.

- Solar irradiance absorbed anywhere in the atmosphere is assumed (cf., item #5) to be recycled to the surface in the same way, and to the same degree, as energy fluxes originating at the Earth’s surface. In any realistic model of the atmosphere, it seems unlikely that this would be the case.
- W&M (cf., item #7) vary between talking about their total energy flux as a “Surface Radiation flux” or an “atmospheric power intensity flux.” Similarly, they vary between talking about a “thermodynamic temperature for the Earth’s surface” and “the average temperature of the Earth’s atmosphere.” The average temperature of the surface and the average temperature of the atmosphere are not the same. It’s a bit worrisome that W&M shift their terminology in this way.

After the adjustments (cf., item #7), what fluxes are included in W&M’s “total energy budget”?

In subtracting all the net heat fluxes leaving the Earth’s surface, W&M removed an amount equal to the solar irradiance absorbed by the surface. Thus, their total energy budget includes (a) fluxes associated with energy recycling, and (b) energy fluxes absorbed by the atmosphere, whether originating from the surface or from solar radiation absorbed by the atmosphere.

This is where vagueness about location becomes problematic.

As I understand it, fluxes associated with energy recycling (back-radiation to the surface) are meaningfully defined only at the interface between the surface and the atmosphere. Yet, the fluxes of solar irradiance absorbed by the atmosphere relate to locations distributed throughout the atmosphere. So, some fluxes relate to energy arriving at the surface and other fluxes relate to energy leaving the surface (and entering the atmosphere), yet these are added together and not subtracted. What does it mean to add together fluxes that are going different directions or are associated with different locations?

And where is the location for which we are calculating the temperature?

- Are W&M calculating the temperature of the surface? To calculate the temperature of the surface one must do an energy balance calculation at the surface. Such a calculation would need to include solar irradiance absorbed by the surface, yet that has been excluded. And one would not include energy fluxes present only in the atmosphere, which have been included. The calculation of temperature clearly cannot relate to the temperature of the surface.

- Are W&M calculating temperature of the atmosphere? At first this seems likely, given that all energy fluxes absorbed by the atmosphere are included. Yet the total also includes fluxes associated with the result of energy recycling. Rigorously speaking, these are fluxes associated with energy leaving the atmosphere and being absorbed by the surface. Why would these be added to energy arriving in the atmosphere, rather than being subtracted? And, even if we had a correct figure for total energy absorbed by the atmosphere, it would not be correct to use an emissivity value of 1 to calculate temperature, as W&M have apparently done in their calculation. So, the calculation of temperature cannot relate to the temperature of the atmosphere.

In summary, energy fluxes with different destinations have been jumbled together. The “total energy budget” does not constitute an appropriate list of fluxes for any one location. The resulting temperature calculation for radiative balance is not meaningful for any location.

* * *

Perhaps if W&M were to correct their reasoning about which fluxes to include or exclude, their calculation could be fixed?

Unfortunately, even if that were done, the faulty assumption that the energy recycling fraction must be β = ½ would likely doom their efforts.

As discussed in my post on Atmospheric Energy Recycling. , the assumption that the energy recycling fraction β is ½ would be appropriate if the atmosphere consisted of a single layer which was opaque to longwave radiation yet thin enough to have a uniform temperature. However, for a thick atmosphere with a temperature that varies with altitude, this assumption is not valid. The energy recycling fraction β could, in principle, be anywhere in the range 0 < β < 1 (subject to additional limitations associated with the radiative properties of the particular gases involved).

Unfortunately, I expect that it is impossible to say, a priori, what the energy recycling fraction β of an atmosphere will be. I believe it is an emergent property of the system as a whole. It can only be computed after the fact, after one has accurately modeled or measured the thermal behavior of the atmosphere.

So, it is not clear to me how W&M could salvage their analytic approach. I hope they succeed in finding a way.

**Conclusions**

The analysis in An Analysis of the Earth’s Energy Budget has some serious flaws. There is no coherent model of for what location in the system W&M are computing a “planetary energy budget.” The energy fluxes included in the proposed energy budget do not correspond to the fluxes needed to calculate a valid temperature anywhere. In addition, the analysis relies on an invalid assumption that the energy recycling fraction of an atmosphere is ½.

Unfortunately, this means that none of the inferences and conclusions which W&M offer as a result of their analysis are justified.

I appreciate the level of original thought that it appears Wilde and Mulholland have given to this work. It’s a shame that they’ve built their work on flawed foundations.

**Appendix: The Role of Convection**

I address this topic as an appendix because it involves no quantifiable claims that I can address, just some comments that seem to hint at important implications.

An idea in W&M’s paper which invites attention relates to their perspective on “atmospheric mass movement.”

W&M (p. 60) seem to find it important to point out “the implicit role of atmospheric mass movement in the process of energy recycling that also heats the surface of our planet. In the presence of a gravity field that binds the atmosphere to the surface of a planet, what goes up must come down.”

They note that their “total atmospheric energy budget” includes terms associated with sensible heat flow (surface thermals) and latent heat flow (evaporation), as well as corresponding recycled energy (back-radiation) terms. These terms are both associated with “mass movement.”

W&M (p. 60-61) reason:

“if the proportion of flux carried by mass motion increases due to an increase in moist convection overturning, then the proportion of energy transmitted by radiant processes must decrease (or vice versa). A given energy flux cannot do two things at once, a balance is always maintained between these two distinct processes if the Bond albedo remains constant.”

W&M do not seem to unpack the perceived implications of this logic within this paper. Perhaps this idea is offered as a foundation for reasoning in other papers?

I feel a bit worried by the generalization to the idea of “atmospheric mass movement” when all that W&M have purported to show importance for is convection. Perhaps, elsewhere, W&M argue that additional types of “atmospheric mass movement” are important?

I’m not in a position right now to comment on inferences that aren’t unpacked in this particular paper.

But, I can try to examine these assertions as they’ve been presented.

* * *

First, we need to address the fact that W&M have been calculating their “total atmospheric energy budget” in a way that lacks any physical meaning.

Let’s address that deficiency by writing an energy balance equation for the surface:

S + B = 𝜀σT⁴ + V [Equation 1]

Here, S is the shortwave solar irradiance absorbed by the surface, B is longwave back-radiation absorbed by the surface, 𝜀σT⁴ is emitted longwave thermal radiation, and V is heat transport away from the surface related to latent or sensible heat flux (typically associated with moist convection). This equation balances the energy flux going into the surface with the energy flux leaving.

This equation suggests that convection tends to cool the surface.

Let’s try to incorporate the concept of energy recycling into the energy balance equation.

From my post on Atmospheric Energy Recycling, the flux of back-radiation could be written as B = S⋅β/(1-β) where β is the energy recycling fraction. So, we could write the energy balance equation as:

S/(1-β) = 𝜀σT⁴ + V [Equation 2]

Alternatively, to align with how W&M are analyzing things, we could write S = U + V where U is the net radiative flux away from the surface, U = 𝜀σT⁴ − B. That allows us to split the back-radiation into a component associated with radiant flux and a component associated with convection. However, energy leaving the surface via thermal radiation or by moist convection will in general be deposited in different portions of the atmosphere. Consequently, there is no reason to believe that the energy recycling fraction will be the same for both type of energy flux. So, we could write back-radiation as:

B = U⋅βᵤ/(1-βᵤ) + V⋅ βᵥ/(1-βᵥ)]

(Perhaps one should also include a term related to W, the amount of solar irradiance absorbed by the atmosphere. It might look like W⋅βᵥᵥ/(1-βᵥᵥ). For simplicity, I will forgo this added complication.)

Substituting our formula for B into our energy-balance equation yields:

𝜀σT⁴ = S + U⋅βᵤ/(1-βᵤ) + V⋅[(2βᵥ – 1)/(1-βᵥ)] [Equation 3]

Equation 3 is the form of the energy balance equation that would seem most aligned with W&M’s formalism.

We can see from this that if W&M were correct in thinking that the energy recycling fraction is βᵥ = ½, then changes to convective heat transfer (i.e., the “mass transfer” contribution to heat flow) would have zero impact on temperature!

(I don’t know what βᵥ is in practice, and I’m not likely to trust any informal estimate of this.)

Does Equation 3 support W&M’s contention that convection is important in determining planetary temperature?

To the extent that βᵥ might be close to ½, Equation 3 would seem to undermine the hypothesis that convection rates are important in determining surface temperature. However, the answer seems unknowable in the absence of a value for βᵥ.

It seems worth noting that Equation 3 is a very odd equation. It computes back-radiation in terms of the net surface radiant flux, U, which can only be calculated if one already knows the amount of back-radiation! (This was happening in W&M’s analysis all along.)

I might make more sense to use something like Equation 2 as a basis for reasoning. Though, there is no guarantee at all that β would really act like a constant, independent of other variables.

* * *

Overall, I have significant doubts about this whole approach to trying to make sense of planetary temperature.

I do wish W&M good fortune in their efforts.

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