Bob Wentworth Ph.D. (Applied Physics)
After offering to “deconstruct Wilde and Mulholland’s analysis of the Earth’s energy budget,I realized that I had focused on a paper by Stephen Wilde and Philip Mulholland that was not ideal to speak to the heart of their work. So today I would like to examine Mulholland & Wilde’s “Modeling the Midday World Climate: A New Look at Venus”.
I really enjoyed the situation that Mulholland & Wilde (M&W) investigated. They studied the thermal behavior of a hypothetical tide-locked planet (the same side always facing the sun) with an atmosphere that is transparent to all wavelengths of radiation.
Here is the illustration M&W is using to illustrate the energy flows on the planet they call “Noonworld”.
Sunlight warms the illuminated hemisphere. The planet’s surface radiates part of the absorbed energy flow unhindered from the transparent atmosphere into space. The rest of the energy flow from the absorbed solar radiation is directed into the atmosphere. The warm air rises and then flows to the dark hemisphere, where the air sinks onto the cold surface, warms it, and then circulates back to the illuminated hemisphere. The dark hemisphere reflects its absorbed flow of energy back into space, again unhindered by the atmosphere.
The convective circulation is similar to a Hadley cell on Earth.
Sounds good so far, doesn’t it?
Not so fast.
You see, natural convection is a process that takes place in a liquid in a gravitational field when there is a heat source, a heat sink and a “heat head”. What is a “thermal head”? It is a pressure difference that arises because the heat sink is above the heat source. It is this pressure difference that drives the circulation process. No thermal head, no circulation.
In Noonworld, the heat sink (the planet surface on the dark side) is at the same level as the heat source (the surface on the illuminated side). There is no thermal head. There will be no convection.
But surely Noonworld is just like Earth in this regard, isn’t it? On earth, convective circulation cells are formed by hot air rising over a warm region and then falling over a cold region, and the hot and cold regions are all at the same elevation, right?
Yes, but this description leaves out a crucial difference between the earth and the midday world.
The earth has greenhouse gases that radiate and cool the atmosphere, creating an elevated heat sink. This is what provides the thermal head that powers convective cells on earth.
Noonworld has no greenhouse gases to provide an elevated heat sink. Convective cells will not form to circulate gases between the hemispheres.
It wouldn’t be very satisfying to end the story right there, however. For the story to continue, we should state that convective cells magically circulate the atmosphere between the hemispheres, even though there is no pressure differential to propel the process.
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M & W wanted to study how energy flows through the system and establish equilibrium temperatures. In addition, they assume that there is a fixed “Diabatic Energy Partition Ratio” between the surface and the atmosphere, which I will call that.
This means that every time an energy flow P arrives at the interface between the surface and the atmosphere, it is assumed that the energy flow “divides” itself so that the atmosphere receives the energy flow 𝛾P and the surface is assumed to radiate, energy flow (1-𝛾) P.
One curiosity of this assumption is that the place where the flow of energy ends does not depend on where it begins. In other words, when 𝛾 = 0.2, it is assumed that any existing flow is divided into 20% in the air and 80% in the surface and is radiated. This means that 20% of this flux will be transferred into the air when the solar radiation is absorbed by the surface. However, when a flux starts in the air, it appears that 80% of it will transfer to the surface. I would think if the conduction between the surface and the air were weak, the flow of energy would tend to stay where it started. However, it is not assumed that energy flows behave in this way. The assumption that energy flows prefer to be in the air or on the surface (or to divide equally) does not seem at all justified.
This is worrying, but let’s move on.
For Noonworld, M & W assumes 𝛾 = 1/2 in both the lit and the dark hemisphere. Later, when they model Venus, they use a value for the lighted side (which is different from the value used for the dark side).
How does that go?
The summary is that starting from the solar radiation flux, the energy flows are distributed, circulate with the atmosphere, are divided again and this is repeated indefinitely. This creates an infinite number of terms that can be added to calculate the radiant flux of thermal radiation on the illuminated side or the dark side. The temperature of each side can be calculated from these radiation fluxes.
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Here are the math details in case you want them, but you can skip this segment as well.
With a solar constant S (watt / m²) the average energy flow of the insulation on the lit hemisphere is S / 2.
This initial flow of energy is split up and circulates over and over again:
- The absorbed solar radiation S / 2 leads to the fact that the Lit side emits an energy flow Rₗ ₀ = (1- 𝛾ₗ) S / 2, while the atmosphere receives an energy flow Aₗ ₀ = 𝛾ₗ S / 2.
- The heated air moves to the dark side, where its energy flow is distributed, which means that the surface radiates Rₒ ₀ = (1-𝛾ₒ) 𝛾ₗ S / 2 and the atmosphere retains Aₒ ₀ = 𝛾ₒ 𝛾ₗ S / 2.
- The now cold air moves to the illuminated side, where its energy flow is distributed, which means that the surface emits an additional amount Rₗ ₁ = (1- 𝛾ₗ) 𝛾ₒ / S / 2 and the atmosphere an additional amount Aₗ ₁ = 𝛾ₒ 𝛾ₗ² keeps S / 2.
As the flows of energy circulate back and forth, they are split up again and again, adding additional terms to the amount emitted and the atmosphere.
The incremental additions to these energy flows form geometric series, making it easy to add the infinite series. The resulting heat radiation flows 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 illuminated side and the dark side, respectively.
For Noonworld one finds Rₗ = (2/3) S / 2 and Rₒ = (1/3) S / 2.
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The end result is that M&W has a recipe for determining the light side and dark side temperatures as a function of the Diabatic Energy Partition Ratio 𝛾 on each side of the planet.
Given the temperatures on both sides of the planet, one can search for the two distribution ratios. M&W uses an iterative inverse modeling process to numerically determine the distribution ratios that correspond to the temperatures on Venus.
Is it surprising that M&W can find parameters that match the temperatures on Venus?
Not really. Their model resulted in a fairly general function that mapped two partitioning parameters to two temperature parameters. Fitting this model to data is simply a curve fitting process that involves fitting two adjustable parameters and two data points. When an adjustment is achieved, it is not surprising and has no inherent meaning.
However, could M&W have captured some real physics showing that convection can be responsible for planet temperatures?
Unfortunately, the energy distribution rule M&W used to calculate its results is completely non-physical. Heat transport cannot work like this.
M&W treats convection as if it behaves like a radiant flux, whereby the heat flux transmitted by convection can be split, almost like by partially reflecting mirrors.
Convection is a means of transporting a flow of heat. Heat flows from hot to cold. Unlike radiation, they can never recycle and, like radiation, build up energy in a resonant cavity.
If you look closely at step 3 of the above process, cold air is circulating to the hot side of the planet and then releasing some of its heat to the hot surface. Heat flows from cold to hot and violates the second law of thermodynamics.
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Air can circulate in a circuit, but heat flow cannot.
Convection air can carry a flow of heat, but that flow of heat is not limited to staying in the air. The heat flow only moves from hot to cold. If the air circulates back to a hotter place, the heat flow does not go along with it.
In Noonworld the distribution ratio of the dark side is always 𝛾ₒ = 0. The entire convective heat flow of the warm air always flows into the cold surface.
Any other result violates the second law of thermodynamics.
As far as I can tell, M & W’s analytical approach leaves no way out. I don’t see how it could be saved.
While I enjoyed learning and thinking about M & W’s convection model of energy distribution, it has nothing to do with how actual convective heat transfer works.
It also does not correspond to the determination of the temperatures on planets.
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I am quite sad to pass this message on to Philip Mulholland and Stephen Wilde who, in my opinion, have devoted tremendous attention, thought and passion to developing this model and sorting out its implications.
I don’t know how it is for her, but to me my creations seem to be a part of me at times. It would be a significant loss to learn that something I had invested deeply in was not what I had hoped for.
I trust that Philip, Stephen and I share a common desire to understand reality for what it is. I hope this essay supports that.