1. 12: Greenhouses and the Earth System
ENPh257: Thermodynamics
12: Greenhouses and the Earth System
© Chris Waltham, UBC Physics & Astronomy, 2018

2. Direct heating: greenhouse
How hot can you get in sunlight? Try an extreme greenhouse:
Consider a black sphere, radius 𝑎, uniform temperature 𝑇 𝑖𝑛𝑡 inside a glass sphere of uniform temperature 𝑇 𝑒𝑥𝑡 , radial distance between the inner and outer surfaces 𝛿𝑎. The incident solar intensity is 𝐼= 1 kW/m2, ambient temperature 𝑇 𝑎𝑚𝑏 .
Assume we have “low-E” glass*, which has a thin-film coating on the outer surface to reduce the emissivity ε. Take the extreme case of ε = 0 and reflectance r = 0. This turns off the radiation terms except for direct sunlight. *
𝑇 𝑖𝑛𝑡
𝑇 𝑒𝑥𝑡
𝑇 𝑎𝑚𝑏
© Chris Waltham, UBC Physics & Astronomy, 2018

3. Direct heating: greenhouse
𝑃𝑖𝑛 = 𝐼𝜋𝑎2
𝑃𝑜𝑢𝑡 = 𝑘𝑐 4𝜋𝑎2 ( 𝑇 𝑒𝑥𝑡 − 𝑇 𝑎𝑚𝑏 )
𝑇 𝑒𝑥𝑡 − 𝑇 𝑎𝑚𝑏 = 𝐼 4 𝑘 𝑐
Using a typical value of 𝑘 𝑐 = 8 W/m2/K for free convection (i.e. no wind) around a sphere, this gives a temperature difference of about 30 K.
© Chris Waltham, UBC Physics & Astronomy, 2018

4. Direct heating: greenhouse
Now consider the conductive heat flow from the interior black ball through the glass, which in this simple model has to be equal to the flow of heat from the Sun into the black ball. Equate 𝑃𝑖𝑛 with power flowing through glass:
𝑃𝑔𝑙𝑎𝑠𝑠 = 𝑘4𝜋𝑎2 ( 𝑇 𝑖𝑛𝑡 − 𝑇 𝑒𝑥𝑡 )/𝛿𝑎
𝑇 𝑖𝑛𝑡 − 𝑇 𝑒𝑥𝑡 = 𝐼 4 𝛿𝑎 𝑘
Here 𝑘 is the effective conductivity of the air and glass interface between the ball and the outside world. We take 𝛿𝑎 to be very much smaller than the external radius of the sphere.
© Chris Waltham, UBC Physics & Astronomy, 2018

5. Direct heating: greenhouse
In building industry terms, 𝛿𝑎 𝑘 is just the “R” insulation factor or 1/U where U is the conduction factor. For a low-E glass panel, U can be as low as 2 W/m2/K (equivalent to an R value of about 3 in customary units).
In other words, 𝑇 𝑖𝑛𝑡 − 𝑇 𝑒𝑥𝑡 can in principle be as high as 125 K, i.e. 𝑇 𝑖𝑛𝑡 − 𝑇 𝑎𝑚𝑏 ≈ 155 K.
Even with single-glazing (like in a vehicle), U is about 6 W/m2/K so 𝑇 𝑖𝑛𝑡 − 𝑇 𝑒𝑥𝑡 would be 40 K, i.e. 𝑇 𝑖𝑛𝑡 − 𝑇 𝑎𝑚𝑏 ≈ 70 K.
Don’t leave your dog in the car on a sunny day.
© Chris Waltham, UBC Physics & Astronomy, 2018

6. Thermodynamics of the Earth
The intensity of sunlight at the top of our atmosphere is called the solar constant: S ≈ 1367 W/m2 The Earth does not absorb all this (we look bright blue and white from space). Our albedo A (what planetary scientists call the reflectance) is about 0.3 which means we absorb: 𝑃 𝑖𝑛 = 1−𝐴 𝑆𝜋 𝑟 𝐸 2 Over a total area of 4𝜋 𝑟 𝐸 2 . Hence the time averaged intensity is: 𝐼 𝑎𝑣 = 1−𝐴 𝑆 4 ≈240 W/ m 2
© Chris Waltham, UBC Physics & Astronomy, 2018

7. Earth radiation
If we were in steady-state (which we are not any more, apparently), the Earth would radiate 1−𝐴 𝑆/4 back into space as thermal radiation. This requires us to have a temperature, 𝑇 𝑎 , at the “top” of the atmosphere – a planet’s “radiation temperature” - whose emissivity we assume to be unity: 1−𝐴 𝑆/4=𝜎 𝑇 𝑎 4 In this case 𝑇 𝑎 = 255 K or -18 C, which is about right for the level of the atmosphere that last radiates infrared.
© Chris Waltham, UBC Physics & Astronomy, 2018

8. Map of temperature at 5500 m altitude
Thermal map
Mean “radiation temperature” -18 C
(Don’t forget its late spring on the other side)
© Chris Waltham, UBC Physics & Astronomy, 2018

9. Greenhouse effect
Why are we warmer than -18 C? Fourier speculated, Tyndall experimented, and Arrhenius finally established (in 1896) that the atmosphere is largely transparent to incoming solar radiation, but absorbs and reradiates terrestrial thermal infrared, and this keeps the Earth’s surface a lot warmer than the upper atmosphere. By analogy with the properties of glass greenhouses used for growing plants, this is called the greenhouse effect.
© Chris Waltham, UBC Physics & Astronomy, 2018
Simple one-layer model of a greenhouse atmosphere

10. reality
To maintain the balance, the Earth’s surface has to double it output, so 𝑇 𝑒 4 =2 𝑇 𝑎 4 i.e. 𝑇 𝑒 =303 K, which would be 15 C higher than the mean now, and a disaster for us (it would be OK in Yellowknife but would push the temperature in already hot, highly populated states way beyond the survivable).
The real situation is more complicated – the Earth’s atmosphere has “holes” in the greenhouse layer.
The graph shows the total outgoing flux measured at the top of Earth's atmosphere (blue curve). This is compared to the radiation of a perfect blackbody (red curve). The difference between the red and the blue curve is due to absorption. Most of the absorption is due to the presence of water vapour, ozone, and carbon dioxide.
© Chris Waltham, UBC Physics & Astronomy, 2018

11. Negative feedback
Our mean surface temperature is now (June 2018) about 288 K. The stability of this temperature is maintained by negative feedback. The primary negative feedback comes from Planck’s Law - the hotter we get, the more we radiate heat away: 𝐼 𝑟𝑎𝑑 =𝜎 𝑇 𝑎 4 𝑑 𝐼 𝑟𝑎𝑑 𝑑 𝑇 𝑎 =4𝜎 𝑇 𝑎 3 ≈ 6 W/m2/K This mechanism has kept us more-or-less stable, up to now.
© Chris Waltham, UBC Physics & Astronomy, 2018

12. Positive feedback Unfortunately, there are positive feedbacks, e.g.:
The more the permafrost melts, the more methane (a powerful GHG) pours into the atmosphere, filling in the holes in our IR emissions.
The more ice melts, the lower our albedo becomes.
If the sum of these positive feedbacks add more than 6 W/m2/K to our thermal budget, we are ...
© Chris Waltham, UBC Physics & Astronomy, 2018