Bob Wentworth Ph.D. (Applied Physics)
[Corrigendum 4/27: The analysis in this essay is flawed. For a corrected analysis, please see
After I offered “Deconstructing Wilde and Mulholland’s Analysis of Earth’s Energy Budget,” I realized I had focused on a paper by Stephen Wilde and Philip Mulholland that was less than ideal for addressing the heart of their work. So, today, I’d like to examine Mulholland & Wilde’s “Modelling the Climate of Noonworld: A New Look at Venus.”
I very much enjoyed the setup of the situation Mulholland & Wilde (M&W) chose to examine. They chose to examine the thermal behavior of a hypothetical tidally-locked (same side always faces the Sun) planet with an atmosphere transparent to all wavelengths of radiation.
Here is the figure M&W use to illustrate the energy flows on the planet they call “Noonworld.”
Sunlight warms the Lit hemisphere. The planetary surface radiates some of the absorbed energy flux into space, unhindered by the transparent atmosphere. The remainder of the absorbed insolation energy flux is conducted into the atmosphere. The warm air rises, then flows to the Dark hemisphere, where the air sinks to the cold surface, warms it, then circulates back to the Lit hemisphere. The dark hemisphere radiates its absorbed energy flux into space, again unhindered by the atmosphere.
The convective circulation is described as being much like a Hadley Cell on Earth.
Sounds good, so far, right?
Not so fast.
You see, natural convection is a process that occurs in a fluid in a gravitational field when there is a heat source, a heat sink, and a “thermal head.” What is a “thermal head”? It’s a pressure differential that is created because the heat sink is elevated above the heat source. It’s that pressure differential that drives the circulation process. No thermal head, no circulation.
On Noonworld, the heat sink (the planetary surface on the Dark side) is at the same elevation as the heat source (the surface on the Lit side). There is no thermal head. There will be no convection.
But surely Noonworld is just like Earth in that respect, isn’t it? On Earth, convective circulation cells form by hot air rising over a warm region and then sinking over a cold region, and the hot and cold regions are all at the same elevation, right?
Yes, but that description leaves out a key difference between Earth and Noonworld.
Earth has greenhouse gases which radiate and cool the atmosphere, providing an elevated heat sink. That is what provides the thermal head that drives convective cells on Earth.
Noonworld doesn’t have greenhouse gases to provide an elevated heat sink. There are not going to be any convective cells forming to circulate gases between the hemispheres.
Yet, it wouldn’t be very satisfying to end the story right there. So, to allow the story to continue, let’s stipulate that convective cells magically circulate atmosphere between the hemispheres despite the lack of any pressure differential to drive the process.
Corrigendum 4/21: The “Oops” needs to be for my error. Natural convection will, in fact, occur on Noonworld. Since writing this, I’ve realized that a “thermal head” is required for natural convection to occur in a closed loop of fluid, but it is not required in an open container of fluid, such as an atmosphere. The difference arises because in a closed loop of fluid the upper and lower circulatory flows are thermally isolated from one another, while in an open container the upper and lower circulatory flows are in thermal contact with one another. This leads to different dynamics.
* * *
M&W set out to examine how energy flows through the system and establishes equilibrium temperatures. To do this, they make an assumption that there is a fixed “Diabatic Energy Partition Ratio,” which I will denote 𝛾, between the surface and the atmosphere.
What this means is that each time an energy flux, P, arrives at the interface between the surface and the atmosphere, the energy flux is assumed to “partition” itself so that the atmosphere receives energy flux 𝛾P and the surface receives, and is assumed to radiate, energy flux (1-𝛾)P.
One oddity of this assumption is that where the energy flux ends up does not depend on where it starts. In other words, if 𝛾=0.2, then it is assumed that any flux present will partition into 20% being in the air and 80% going into the surface and being radiated. So, that means that if insolation is absorbed into the surface, 20% of that flux will be transferred to the air; yet if a flux starts out in the air, then 80% of it will apparently be transferred to the surface. I would think that if conduction between the surface and the air was weak, then energy flux would tend to stay where it started out. Yet, that’s not the way energy fluxes are assumed to behave. The assumption that energy fluxes have a preference for being in air or being in the surface (or dividing evenly) seems not at all justifiable.
That’s concerning, but let’s continue.
For Noonworld, M&W assume 𝛾=1/2 on both the Lit and Dark hemispheres. Later, when modeling Venus, they use a value for the Lit side 𝛾ₗ which is distinct from the value 𝛾ₒ used for the Dark side.
So, how does this play out?
The summary is that, starting from the solar irradiation flux, the energy fluxes get partitioned, circulate with the atmosphere, get partitioned again, and this repeats indefinitely. This creates infinite series of terms which can be added up to compute the radiant flux of thermal radiation on the Lit side and the Dark side, respectively. From these radiant fluxes, one can calculate the temperature of each side.
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Here are the mathematical details, in case you want them, but feel free to skip this segment.
Given a Solar constant, S (watts/m²), the average isolation energy flux on the Lit hemisphere is S/2.
This initial energy flux gets partitioned and circulates, again and again:
- The absorbed insolation S/2 is taken to lead to the Lit side radiating an energy flux Rₗ ₀ = (1- 𝛾ₗ) S/2, while the atmosphere receives an energy flux Aₗ ₀ = 𝛾ₗ S/2.
- The warmed air travels to the Dark side, where its energy flux is partitioned, leading to the surface radiating Rₒ ₀ = (1-𝛾ₒ) 𝛾ₗ S/2 and the atmosphere retains Aₒ ₀ = 𝛾ₒ 𝛾ₗ S/2.
- The now cold air travels to the Lit side, where its energy flux is partitioned, leading to the surface radiating an additional amount Rₗ ₁ = (1- 𝛾ₗ ) 𝛾ₒ 𝛾ₗ S/2 and the atmosphere retaining an additional amount Aₗ ₁ = 𝛾ₒ 𝛾ₗ² S/2.
As the energy fluxes circulate back and forth, they get partitioned again and again, adding additional terms to the amount radiated and in the atmosphere.
The incremental additions to these energy fluxes form geometric series, making it easy to add up the infinite series. The resulting thermal radiation fluxes are:
𝜀σTₗ⁴ = Rₗ = (1- 𝛾ₗ ) (S/2)/(1- 𝛾ₒ 𝛾ₗ)
𝜀σTₒ⁴ = Rₒ = (1-𝛾ₒ) 𝛾ₗ (S/2)/(1- 𝛾ₒ 𝛾ₗ)
where Tₗ and Tₒ are the temperatures of the Lit side and Dark side, respectively.
For Noonworld, one finds Rₗ = (2/3) S/2 and Rₒ = (1/3) S/2.
* * *
The end result is that M&W have a recipe for finding the temperatures of the Lit side and the Dark side as a function of the “Diabatic Energy Partition Ratio” 𝛾 on each side of the planet.
Given the temperatures of the two sides of the planet, one can solve for the two partition ratios. M&W use an iterative Inverse Modelling process to numerically solve for the partition ratios that correspond to temperatures on Venus.
Is it surprising that M&W are able to find parameters that fit the temperatures on Venus?
Not really. Their model led to a fairly general function mapping two partitioning parameters to two temperature parameters. Fitting this model to data is simply a curve fitting process involving two tunable parameters and two data points to be fit. When a fit is achieved this isn’t surprising and doesn’t have any inherent significance.
Yet, could M&W have captured some real physics, demonstrating that convection can account for planetary temperatures?
Unfortunately, the energy partitioning rule M&W used to calculate their results is completely non-physical. Heat transport can’t work that way.
M&W are treating convection as if it behaves somewhat like radiant flux, with the heat flux carried by convection being able to be split, almost as if by partially reflective mirrors.
Convection is a means of carrying a heat flux. Heat fluxes flow from hot to cold. Unlike radiation, they can’t ever recycle, building up power like radiation in a resonant cavity.
If one looks careful at step #3 in the process described above, it involves cold air circulating to the hot side of the planet and then giving some of its heat to the hot surface. It involves heat flowing from cold to hot, violating the Second Law of Thermodynamics.
* * *
Air can circulate around and around in a cycle, but a heat flux can’t.
Convecting air can carry a heat flux, but that heat flux is not constrained to stay with the air. The heat flux will only travel from hot to cold. If the air circulates back to a hotter place, the heat flux will not go with it.
So, on Noonworld, the Dark side partitioning ratio is always 𝛾ₒ = 0. The entire convective heat flux from the warm air always flows into the cold surface.
Any other outcome violates the Second Law of Thermodynamics.
Corrigendum 4/21: Another “Oops” on my part is needed here. While I remain convinced that M&W’s model is inconsistent with thermodynamic principles, my explanation here of the nature of the problem is wrong. One can analyze a problem from an “energy flow” perspective or a “heat flow” perspective. I incorrectly applied a “heat flow” perspective to an “energy flow” diagram. So, for the moment, the “trouble in Noonworld” is my own. Apologies to all.
As far as I can tell, this leaves M&W’s analytic approach nowhere to go. I don’t see how it could be salvaged.
While I’ve enjoyed learning and thinking about M&W’s energy partitioning convection model, it has nothing to do with the way actual convective heat transfer works.
Nor does it correspond to how temperatures on planets get determined.
* * *
I feel quite sad to relay this news to Philip Mulholland and Stephen Wilde, who I imagine have given enormous attention, thought, and heart to developing this model and sorting through its implications.
I don’t know how it is for them, but for me, my creations sometimes seem like a part of me. It would be a significant loss, if I learned that something that I had invested deeply in creating wasn’t what I had hoped it was.
I trust that Philip, Stephen, and I share a common desire to understand reality as it is. I hope this essay supports that.
Corrigendum 4/21: The preceding conclusions of mine are no longer justified on the basis of the contents of this essay.