What Happens If You Block Out The Sun For A Month? (Revised)
I have been reading far too many back issues of the Bulletin of the American Meteorological Society recently. There's a good reason. I'm doing research for my intellectual property course, but I have to say that it is entirely too much fun to be doing for a class.
One question asked in passing in the last part of the first issue ever of the journal is to ask what would happen if the Sun were blocked from the Earth by means of a screen. It's a good question, and since this blog supposedly treats of weather topics on occasion, I thought I would propose a very simplified solution, even though I know something about the techniques whereby a more accurate solution could be obtained.
First, one may ask, is this at all realistic? Well, no, but my friend Ayla likes to blame all of the odd geological features of the terrestrial planets on aliens. I, of course, act like a good disciple of Lyell and argue against aliens based on the principle of uniformitarianism. However, aliens are as good as any cause of placing a screen between the Sun and the Earth.
The first assumption we will make is that the Earth radiates as a blackbody with initial surface temperature T0=288 K. The next assumption we will make is that there are no greenhouse gases, so the heat leaves the ground and is not reabsorbed on the way up. So we need to set up a simple differential equation.
dT/dt= -(sigT^4)/Cp, where sig is Stefan-Boltzmann's Constant and Cp is the heat capacity.
implying that:
dT/T^4=-(sig/Cp)dt
(T^-3/3) + C=(sig/Cp)t
At time=0, C=-(T0^3/3)
implying that:
(1) T(t)=(((3sig*t/Cp)+T0^-3)^-1)^(1/3))
But the heat capacity here is in terms of J kg^-1 K^-1, not J m^-2 K^-1 and so does not match the other units. To resolve this, we will calculate the mass of air over a square meter of the Earth's surface, which happens to be the pressure at the surface multiplied by one square meter divided by the acceleration due to gravity or 10326.2 kg. Hence, Cp=1.0367304*10^7 J m^-2 K^-1.
After a month (30 days actually), therefore, the surface temperature of the Earth should be 228 K (or about 45 degrees Celsius below freezing). What larger consequences this would have for the weather and climate, I leave you to ponder, but please don't think that the greenhouse gases neglected in this model would have more than a modest effect.
However, note that greenhouse gases have not quite been neglected. If there were no greenhouse gases (or technically, if there was no infrared absorption by the atmosphere), the surface temperature of the Earth would be 255 K (Teff), its effective radiating temperature. However, it is possible to give a slightly fuller analysis of this situation without neglecting the first order effects of greenhouse gases either before or after the insertion of the screen (the beginning of the interruption in incident solar radiation, whatever). Hence, using the slab model of radiative transfer, we define an effective optical depth at the ground km s.t.
T0=Teff(1+km)^(1/4), so km = 0.627.
Then we can insert the effective radiating temperature into (1) above, so that the effective radiating temperature (the temperature with which the Earth would radiate without the effects of greenhouse gases or the temperature the Earth has above the layer of effective IR atmospheric absorption.) We're assuming this model that the effective radiating temperature is achieved at the surface and then some slab of absorbing material is slapped over the surface very close to the ground to mimic greenhouse gases. It is quite a rough model, but it should illustrate the point. After 30 days, the effective radiating temperature decreases to 212 K, implying that that the surface temperature of the Earth is reduced to 239 K. Hence, the difference made by fully accounting for greenhouse gases is 11 K. Admittedly, this makes a great deal of difference in climate regimes when water is liquid and its equilibrium concentration of vapor is significant (it varies exponentially with temperature), but it makes little difference when the average temperature of the Earth is well below freezing.
ESA(20040216.1a)
I have been reading far too many back issues of the Bulletin of the American Meteorological Society recently. There's a good reason. I'm doing research for my intellectual property course, but I have to say that it is entirely too much fun to be doing for a class.
One question asked in passing in the last part of the first issue ever of the journal is to ask what would happen if the Sun were blocked from the Earth by means of a screen. It's a good question, and since this blog supposedly treats of weather topics on occasion, I thought I would propose a very simplified solution, even though I know something about the techniques whereby a more accurate solution could be obtained.
First, one may ask, is this at all realistic? Well, no, but my friend Ayla likes to blame all of the odd geological features of the terrestrial planets on aliens. I, of course, act like a good disciple of Lyell and argue against aliens based on the principle of uniformitarianism. However, aliens are as good as any cause of placing a screen between the Sun and the Earth.
The first assumption we will make is that the Earth radiates as a blackbody with initial surface temperature T0=288 K. The next assumption we will make is that there are no greenhouse gases, so the heat leaves the ground and is not reabsorbed on the way up. So we need to set up a simple differential equation.
dT/dt= -(sigT^4)/Cp, where sig is Stefan-Boltzmann's Constant and Cp is the heat capacity.
implying that:
dT/T^4=-(sig/Cp)dt
(T^-3/3) + C=(sig/Cp)t
At time=0, C=-(T0^3/3)
implying that:
(1) T(t)=(((3sig*t/Cp)+T0^-3)^-1)^(1/3))
But the heat capacity here is in terms of J kg^-1 K^-1, not J m^-2 K^-1 and so does not match the other units. To resolve this, we will calculate the mass of air over a square meter of the Earth's surface, which happens to be the pressure at the surface multiplied by one square meter divided by the acceleration due to gravity or 10326.2 kg. Hence, Cp=1.0367304*10^7 J m^-2 K^-1.
After a month (30 days actually), therefore, the surface temperature of the Earth should be 228 K (or about 45 degrees Celsius below freezing). What larger consequences this would have for the weather and climate, I leave you to ponder, but please don't think that the greenhouse gases neglected in this model would have more than a modest effect.
However, note that greenhouse gases have not quite been neglected. If there were no greenhouse gases (or technically, if there was no infrared absorption by the atmosphere), the surface temperature of the Earth would be 255 K (Teff), its effective radiating temperature. However, it is possible to give a slightly fuller analysis of this situation without neglecting the first order effects of greenhouse gases either before or after the insertion of the screen (the beginning of the interruption in incident solar radiation, whatever). Hence, using the slab model of radiative transfer, we define an effective optical depth at the ground km s.t.
T0=Teff(1+km)^(1/4), so km = 0.627.
Then we can insert the effective radiating temperature into (1) above, so that the effective radiating temperature (the temperature with which the Earth would radiate without the effects of greenhouse gases or the temperature the Earth has above the layer of effective IR atmospheric absorption.) We're assuming this model that the effective radiating temperature is achieved at the surface and then some slab of absorbing material is slapped over the surface very close to the ground to mimic greenhouse gases. It is quite a rough model, but it should illustrate the point. After 30 days, the effective radiating temperature decreases to 212 K, implying that that the surface temperature of the Earth is reduced to 239 K. Hence, the difference made by fully accounting for greenhouse gases is 11 K. Admittedly, this makes a great deal of difference in climate regimes when water is liquid and its equilibrium concentration of vapor is significant (it varies exponentially with temperature), but it makes little difference when the average temperature of the Earth is well below freezing.
ESA(20040216.1a)
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