NOTE: Different vertical scales! The size of a scale height in each case compared with the height of the troposphere - the troposphere is about 2-3 scaleheights.
How do we QUANTIFY this greenhouse effect? How do we MODEL the absorption/radiation?
First start with a very, very simple model..... see Chapter 3 pages 52-61 of Goody & Walker
NOTE - Te = Teff is the EFFECTIVE temperature - that's the "temperature" that is derived by measuring all the (blackbody - or thermal radiation) IR flux from a planet
The OPAQUE SLAB MODEL - sometimes called "grey model" - does not depend on wavelength.
Starting at the top of the atmosphere and working down layer by layer, equating the flux into each layer with the flux out
Layer 0 - T1 = Teff
Layer 1 - sT24 = sT14 + sT14 ....so, T24 = 2T14
Layer 2 - sT34 + sT14 = sT24 + sT24 ....so, T34 = 2T24 - T14 = 4T14 - T14= 3T14
Layer N - TN4 = NT14
So, if you imagine you keep going down the layers until you reach the surface of the planet - where the temperature is Tg - then you can envisage an expression
Tg4 = (1+t) Teff4
where we call t = OPTICAL THICKNESS or OPTICAL DEPTH
(why (1+t) rather than t? Good question. I have no idea - probably due to some alternative derivation).
Earth: t = (Tg/Teff)4 -1 = (288K/255K)4 -1 = 0.6 - the Earth's atmosphere is thin - not even one totally absorbing layer.
Venus: t = (Tg/Teff)4 -1 = (750K/238K)4 -1 = 98 - Venus atmosphere is very thick
(G&W quotes a surface temperature of 700 K and gets t = 68, I've seen different values for Teff too).
The next step would be to add heating and cooling due to absorption and radiation... and make a much more complicated model.
Rates are in units of Kelvin per day
In reality.... it's further complicated....by the fact that there are horizontal variations - some places have clouds, some do not. And there are vatiations with latitude.... etc. But, when you average over the globe, for Earth, this is what you get for a NET RADIATION BUDGET - visible light in (sunlight) - and IR light out (thermal radiation). The units are Watts per square meter
But we just want a simple comparison of Earth, Venus and Mars -
Isn't there a simpler way of looking at this?
For a THIN atmosphere - e.g. Earth, Mars - we can use a simpler model. The book has this...
We can go simpler....
Think of the atmosphere as a single slab. We want to build a model similar to the colored radiation budget diagram above for the Earth but much simpler. So, starting at the top, we have the Flux of visible sunlight coming in
Fin = 1368/(4 a2) Watts m-2 where a is the distance from the Sun in AU.
(Whoa!! where did the 4 come from? Why does the color diagram above have 342 Watts m-2 coming into the atmosphere? This is the AVERAGE flux. The net flux is 1368 x p R2 which must be averaged over the total area of 4p R2 - making an average of 1368/4 = 342 Watts m-2 hitting the earth).
Of this a fraction of the is reflected - the fraction being the Albedo - A. So, (1-A) of Fin goes into the planet (either into the atmosphere or the ground - does not strictly matter for this simple model).
Now let's look at the infrared side - the Flux of energy going outwards.
From the ground a flux of s Tg4 Watts m-2 is emitted (where s is the Stefan-Boltzmann constant of 5.67 x 10-8 in SI units).
Some of this is absorbed by the atmosphere - we are going to calculate how much. So, look at the net amount that is radiated to space - it must be the same as the net amount that is absorbed - Fout = 1368/(4 a2) (1-A) Watts m-2 - but this must be the same as the flux of energy that is radiated by the atmosphere = s Teff 4
So - what happens in the middle? Let's define the fraction that gets absorbed by the atmosphere to be the OPACITY = e
Thus, the Flux that is absorbed is e s Tg4 Watts m-2 and the amount that is radiated to space is then (1-e) s Tg4 = s Teff 4 Watts m-2
This then means that we can define
OPACITY e = = 1 - (Teff/Tg)4
which is not to be confused with OPTICAL DEPTH t = (Tg/Te)4 -1
Opacity is useful for a THIN atmosphere where only a small fraction of the outgoing IR radiation is absorbed (e.g. Mars). Optical depth is useful for a THICK atmosphere where most of the outgoing IR radiation is absorbed (e.g. Venus).
|Distance from Sun, a (AU)||1||0.723||1.5|
|OPACITY e = 1 - (Teff/Tg)4||0.39||0.98||0.17|
OPTICAL DEPTH t = (Tg/Te)4 -1
So...40% of the energy is absorbed on its way from the ground out to space on Earth - at Venus this is 98%! but only 17% at Mars - seems a little but still half of the percentage of Earth even though Mars' atmospheric pressure is only 0.6% of a bar - shows how CO2 is an effective absorber.
You could now draw your own energy budget diagrams for Earth, Venus and Mars - with appropriate colors, objects on the surface, etc - showing the AVERAGE flow of energy through the system.
For all planets temperature decreases with altitude for the tropospheric layer, the first couple of scaleheights - WHY?
ADIABATIC LAPSE RATE - dT/dz
What does ADIABATIC mean?
dT/dz = -g/cp
g= gravitational acceleration cp = Specific Heat Capacity at Constant Pressure
cp = squiggle R / M
M = mean molecular mass =<amu> 0.001 - e.g. air = 29 x 0.001 kg / mol
R = gas constant = 8.31 J K-1 mol-1
squiggle = 5/2 for monoatomic gas, = 7/2 for diatomic gas, = 9/2 for triatomic gas
UNITS: cp - (kg m2 s-2 K-1 mole-1 )/(kg mole-1) = m2 s-2 K-1
dT/dz = g/cp - m s-2 /(m2 s-2 K-1) = K m-1
What's cp for E, V, M? What's dT/dz for Earth, Venus Mars?
m2 s-2 K-1
K / km
|V||CO2||44 x 0.001||850||8.87||10.4|
|E||O2 , N2||29 x 0.001||1003||9.81||9.8|
|M||CO2||44 x 0.001||850||3.71||4.4|
So, Earth and Venus about the same - down in the troposphere - but at Mars the adiabatic temperature profile drops off more slowly with height.
- - - - - = adiabatic lapse rate, White line = actual lapse rate
- if cools faster than adiabatic - stable (cool air sinks)
- if cools slower than than adiabatic - unstable (warm air rises)
..... which leads us to circulation patterns - next class.