Radiation Balance and Transparent Media – 3

Written by C. (Kees) le Pair on December 3, 2018. Posted in Current News

In a radiation field a transparent medium behaves unlike an opaque. That is important for the earth’s climate. 70{154653b9ea5f83bbbf00f55de12e21cba2da5b4b158a426ee0e27ae0c1b44117} of the earth’s surface is ocean. Just like the atmosphere, this is a transparent medium. So transparency is worth considering. In addition the heat content of the ocean is significantly greater than that of the atmosphere.

Countless treatises on climate implicitly assume an opaque earth. Whether that is justified thus needs consideration. In both previous articles in this series and here that assumption seemed to be doubtful.

Earlier, another important assumption in the aforementioned treatises proved inadequate. The atmospheric greenhouse would bridge the 30 °C difference between the measured temperature and that of an atmosphere-free earth.

That is incorrect. The difference is greater. So if the greenhouse produces it, it must be stronger than one assumes. Then it cannot be calculated correctly. If these calculations were correct, then there is another mechanism that helps to keep the earth livable.

 Roy W. Spencer, a renowned climatologist, indicated that the 30 °C – he writes 33 °C, the difference is irrelevant here – is simply the difference between the supposed radiation equilibrium and the measured temperature. He criticizes the too low estimate of the gap.

According to him it should be 60 °C. However he suggests a lower equilibrium for a different reason than non-uniformity of the surface temperature.
More about this in Note (1).

A transparent bar

In a thought experiment we examine an isolated transparent bar irradiated by a parallel beam perpendicular to the head. The absorption coefficient k is assumed to be equal for all wavelengths of the beam. In the stationary state the temperature remains unchanged at every cross-section. The energy received there must be discharged in its entirety.

Assuming that the medium is opaque to long-wave (red) light – we keep the ocean in mind – this is only possible through material flow and conduction. Conduction is described with coefficient λ. We do, as if the transport can be described by flow in the same way, and use the ‘energy transport coefficient’ λ’.

The outgoing energy flux is equal to the incoming, I_o. It is made up of conduction, surface radiation and possibly evaporation. The division between the three is not important if we are only interested in the temperature gradient in the longitudinal direction ΔT_x, where x is the distance to head surface. The reference temperature T_o may have any value.

We assume the parameters are temperature independent. Think of the infinitesimal layer just below the head surface, where the exchange with the environment takes place.

Parameters and algorithms;

• I_x = I_o.e^-kx Wm^-2 (absorption law), where I_o is the radiation influx and I_x what is left after x m.
• k (absorption coefficient) = 9.21034.10^-3m^-1. So chosen that I₁₀₀₀ / I_o = 10^-4. (Think about the ocean, 1000 m deep, a negligible 1/10000^th left of the radiation.)
• A layer at x m absorbs I_x.kdx Wm^-2
• λ’.dT/dx Wm^-2 (heat transport by temperature gradient).
• λ’ = 12 W.m^-1.K^-1. The conduction coefficient of water is 0,6; that of copper 390. In liquid, the transport mainly goes through mass flow. We are doing a guess, but λ’ could also be 500 (!).
• I_o = 1000 Wm^-2, about the maximum sun irradiation after the deduction of an average albedo.

In stationary state all radiation passing through layer x ends up as heat on the way back in layer x. Here it passes on through conduction/convection to the next layer. In comparison, the incoming radiation energy absorbed in the layer is negligible. The calculation of ΔT_x

With the indicated parameters this provides the temperature profile in table 1.

A transparent medium in balance with incoming radiation can show surprisingly high temperatures internally. The table profile does not resemble the measured profile, the thermocline, in the ocean.

That is not surprising because the model is completely different from the oceanic reality(2). Note furthermore that at large depths in the bar model the energy transports are so small that a heat-proof, swimming fish or sinking algae destroys the profile.

At 1000 m the radiation yields I_x.k = 0.0009 Wm^-3 only. (The relaxation time for setting a balance is therefore long: ~ 1500 years to heat up a cubic meter 1 °C.).
The conclusion of this thought experiment is:

Radiation causes higher temperatures within a weakly absorbing medium than it does on the surface according to Stefan-Boltzman.

This is important for climate considerations. Both the atmosphere and the ocean are transparent. In addition, the heat content of the part of the ocean, that is involved in the climate, is much greater than that of the air. Perhaps the greenhouse is not even needed or or only partly?

More insight into the behavior of transparent media would be illuminating certain climatic issues. We therefore need model calculations e.g.

• A divided bar with different k and λ’ (think of air and water).
• A rotating bar (imitation of day and night).
• A bar of material that is not opaque for infra red. A transparent body does not radiate from the surface alone, but also from the inside out.
• Variation in the distribution of the outgoing energy between conduction, evaporation and radiation. See Hughes for the influence on temperature.
• A sphere instead of a bar?

Notes 

1. With thanks to Hans W. Erren and C.A. (Kees) de Lange. The latter pointed to publications by
   ∘ Murry L. Salby “Fundamentals of atmospheric physics” (Acad.Press 1995),
   ∘ and a ‘Twitter’ by Ed Nikolov. Nikolev mentions a terrestrial radiation equilibrium temperature of 197 K.
   ∘ Erren drew attention to the webpage of Roy Spencer, mentioned already in the text, who like Salby and
   ∘ Egbert Boeker and Van Grondelle in “Environmental Physics” (Wiley 1999) treat the greenhouse effect as bridging the gap between measured temperature and the calculated radiation equilibrium.
   Boeker states explicitly that the difference from 255 K is of atmospheric origine, CO₂, O₃, N₂O & CH₄ in particular. Spencer suggests a 60 °C lower calculated temperature, due to ‘flow’, while Nikolev’s twitter attributes heating to the adiabatic lapse rate.
   The effect of the only partial temperature equalization on earth and the mathematical property <T⁴>≠<T>⁴ is not very well known in climate studies.

2. Convection is able to transport heat more effective than conduction. If we would have chosen λ’ = 500, all ΔT_x in table 1 would have been ~40x lower.

3. Ad Huijser corrected the formula for ΔT_x by including my remark about the irrelevance of the value of T_o and insert it in the formula. The correction does not change the results listed in the table.

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