Class 14 - Earth, Venus, Mars - 4

Reading:

1. Atmospheres - Goody & Walker Chapter 3 pp52-70
2. New Solar System - Ch 13.

• Radiative Transfer - quantifying the greenhouse effect - KISS
• Convection & dT/dz

Earth, Venus, Mars Atmospheres

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.

Radiative Transfer - absorption/radiation

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 = T_eff 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.

Assumptions:

1. Each layer totally absorbs all light coming into it and radiates at its own temperature
2. There are no additional heat sources (no heating from the inside of the planet, no additional absorption in one of the layers)

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 = T_eff

Layer 1 - sT₂⁴ = sT₁⁴ + sT₁⁴ ....so, T₂⁴ = 2T₁⁴

Layer 2 - sT₃⁴ + sT₁⁴ = sT₂⁴ + sT₂⁴ ....so, T₃⁴ = 2T₂⁴ - T₁⁴ = 4T₁⁴ - T₁⁴= 3T₁⁴

.....

Layer N - T_N⁴ = 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) T_eff⁴

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).

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.

Convection & dT/dz = Lapse Rate

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?

So, Earth and Venus about the same - down in the troposphere - but at Mars the adiabatic temperature profile drops off more slowly with height.

SO WHAT?

- - - - - = 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.