Greenhouse effect on planetary equilibrium temperature help

[AotrsCommander]

I am trying to back-calculate the greenhouse effect on planetary equilibrium temperature for Earth (in the hope I can then attempt applying it to other calculations.)

I know that the actual equilibrium temperature of Earth is 287.89K, +33K above the calculated equilibrium temperature:

T_eq = (Q x (1- albedo)/ 4 x Stefan-Boltzmann constant)^0.25

When Q = 1400 W/m², (1 - albedo) = 0.7.

From http://www.lpl.arizona.edu/~showman/greenhouse.html, if I am understanding it aright, the greenhouse effect is (in a simplified form enough for my purposes) basically a multiplier of Q, respresenting the amount of energy reflected back off the atmosphere to the planet.

I put the T_eq as a spread sheet calculation, with an additional factor (let's call it Z) which multiplies Q that I could alter manually; experimentally it looks like for Earth it's Z is about 1.6ish.

T_eq = (Q x Z x (1- albedo)/ 4 x Stefan-Boltzmann constant)^0.25

T_eq = 254.89K for Earth

What I would like to do is see if I can work back, using the known values from Earth, and see if I can determine Z, which would at least give me something more to work with. (At the moment, my estimations for greenhouse effects are "add about 33ºK")

I tried rearranging the formula

287.89K = 33K + 254.89K = (Q × Z × (1- albedo)/ 4 × 5.67 × 10^-8)0.25287.89⁴= Q × Z × (1- albedo)/ 4 × 5.67 × 10^-8287.89⁴ × 4 × 5.67 × 10^-8 = Q × Z × (1- albedo)287.89⁴ × 4 × 5.67 × 10^-8/ Q × (1- albedo) = Z

However, this came out as 3.97, which is clearly wrong. I've obviosuy messed up somewhere, but for the life of me I'm not sure where.

 

[bigfooted]

To compute the greenhouse effect, you model the Earth as well as the atmospheric layer around it. The energy balance for the atmosphere is like this: The incoming radiation from the sun is in balance with the outgoing radiation from the atmosphere plus the outgoing radiation from the earth:
$\frac{Q}{4} (1-\alpha) = \epsilon \sigma T_a^4 + (1-\epsilon)\sigma T_s^4$
The atmosphere is radiating into space with an emissivity $\epsilon$ at temperature $T_a$. That means that $1-\epsilon$ of energy radiated from the Earth (with surface temperature $T_s$) gets transmitted through the atmosphere into space.

The second equation that you need to determine the atmosphere and surface temperature is the energy balance at the surface of the earth:
The incoming radiation from the sun and the atmosphere balance the outgoing radiation at the surface:
$\frac{Q}{4} (1-\alpha) + \epsilon \sigma T_a^4 = \sigma T_s^4$

Now the surface temperature can be computed:
$T_s^4 = \frac{Q(1-\alpha)}{2\sigma(2-\epsilon)}$

If emissivity is 0.75, then the surface temperature is around 288K.

 

[AotrsCommander]

Right. I put the numbers through, and after a little trouble with the calculator (I was trying to work out T_s by taking the forth root and not taking it to the power of 0.25!) and got it all worked through!

Thanks!

That ought to be a much more accurate stab at it, and I can now do a bit more reading up and seeing if I can find about some more about atmospheric emissivity and the variance thereof.

(And if not, I can always have a bit of a play at it; the main thing with an emissivity factor like this I can afford to play with it up and down a bit for a Earth-close atmosphere. I don't play too far out of Earth-standard anyway, on the basis of keeping the variables fairly managable anyway.)

Much obliged!

 

[Andrew1955]

The term planetary equilibrium temperature seems wrong to me. There cannot be an equilibrium if the sun is heating a planet that is sending radiation to the far reaches of the universe. It is just a steady state.

 

[fizzy]

I'm really not sure that this kind of hand-waving estimation process is worth much.

Where does the ballpark 'average' albedo come from ? Poles are much more reflective to solar than tropical seas. Yet the latter are about 25 kevin hotter. Reflectivity for solar wavelengths has little to with emissivity at IR wavelengths.

I think the 0.3 albedo has been picked to give about the right answer to start with so your calculations are just back-trackng the assumptions made in deducing it. They do not really give you the "known" figure you are looking for to apply elsewhere.

Trying to summarise a very complex situation in one "effective" albedo that is a total frig factor anyway is not likely to be much use elsewhere, IMO.

Does 0.3 mean 0.3+/-0.1 ? How much does the 33K change if you take those limits?

 

[sandersr]

Poles are much more reflective to solar than tropical seas. Yet the latter are about 25 kevin hotter.

That's what you would expect though - that areas with lower albedo (so lower α) would have a larger left hand side of the equation, and consequentially have a higher temperature on the right hand side. In general terms, the more incoming radiation is reflected back to space, the less warming you get. This is not the main driver of cooler polar regions, of course, but there is no way a higher albedo would be at odds with lower temperatures.

I think the 0.3 albedo has been picked to give about the right answer to start with so your calculations are just back-trackng the assumptions made in deducing it.

0.3 (which is actually 0.297 +/- 0.005) is the known average albedo from satellite measurements, and confirmed by Earth'shine measurements of the moon - read more here: http://onlinelibrary.wiley.com/doi/10.1029/2000GL012580/epdf

It is not a randomly picked factor

 

[fizzy]

Thanks, I was forgetting the Earthshine experiment. However, you should note that this is "optical" reflectivity which has nothing to do with IR emissivity. Both water and ice behave very differently at IR wavelengths. Don't confuse the two or assume they are "roughly" the same.

There is also glazing incidence at the poles which will increase surface reflection over open water and melt pools which will radiate a lot of IR especially in the Arctic, without even discussing changes to evaporation and what that implies. Be wary of one size fits all explanations.

 

[sandersr]

Thanks, I was forgetting the Earthshine experiment. However, you should note that this is "optical" reflectivity which has nothing to do with IR emissivity. Both water and ice behave very differently at IR wavelengths. Don't confuse the two or assume they are "roughly" the same.

There is also glazing incidence at the poles which will increase surface reflection over open water and melt pools which will radiate a lot of IR especially in the Arctic, without even discussing changes to evaporation and what that implies. Be wary of one size fits all explanations.

The albedo is measured with satellites, though... As I said, it is not a randomly picked number.

 

[fizzy]

"I am trying to back-calculate the greenhouse effect on planetary equilibrium temperature for Earth"

TSI is about 1633 W/m2 across the disk ( circle ) on one side . Earth radiates over a full sphere in the INFRA-RED . Plus non isotropic reflectivity ( albedo).

You are not going to get anywhere until you grasp the differennce between reflectivity at solar wavelengths and emission in IR bands. Also most IR emission happens at high altitude due absorption and re-emission. IR only makes it something like 2m at ground level without being re-emitted.

That means the the S-B temperature is probably more like -60 deg. C of the stratosphere , not your circa 14C or whatever,

If you think you will get a one line equation to even give you a ballpark figure, you will be disappointed.

 

[whomever]

I would like to know why the climate models do not take into account the direct effect of the human population.

Simple calculations based on a population of 7 billion yields some rather startling results.

Global human expended heat (excretia) 1.13e+11 MW-hours/year.
that is 0.35 kg fecal matter + 1 liter urine per person at an average temperature of 37C to 10.2C (global average ocean film temperature = 0.5*(surface + base)). Eventually all of that crap ends up in the ocean.
That is roughly 41 times the yearly North American Electrical energy generation from hydrocarbon sources (circa 2013).

 

[fizzy]

In terms of energy balance, you need to consider where this heat energy comes from, it is not magically produced by the human body. It comes from chemical energy that the body breaks down and converts to heat ( amongst other things ).

So where does the chemical energy come from?
From plants and meat from animals that eat plants.

And where do plants get their energy from?
From the sun.

The heat in your pile of crap is basically somewhat delayed solar energy and is already accounted for in the energy budget calculations.

But don't worry, there plenty of crap in climate models, mostly of human origin ;)

 

[whomever]

Since as you put it all energy derives from the sun, is not all of the hydrocarbon energy already accounted for?
Burning hydrocarbons would simply return the Earth to its original equilibrium condition.
Human population does distort the global temperature distribution.
Look at the temperature profile in the North Pacific Ocean as compared with the temperature profile of the South Pacific. Higher temperatures in the North are due to the higher proportion of the worlds population living in the northern hemisphere.

 

[fizzy]

The hydrocarbon energy is also, utlimately, solar. However, since it has been out of action fro millions of years, it is not accounted for in current energy balance calculations.

Your assertions about the pacific are spurious. I suppose you think that poles are cold because no one lives there. Correlation is not proof of causation.

SH is more stable ( slower to change ) than NH because it has a greater proportion of ocean.

 

[whomever]

Correlation may not be causation, but it does indicate that something should be looked at.

Human waste yearly amounts to 9.67e+19 calories. Heat generation by Earth's core =2.28e+20 calories/year.
This is 42% of what the core generates!

 

[fizzy]

So we should urgently investigate whether the polar regions are so cold because the lack of human excreta, is that what are proposing?

What you still don't seem to get is that human heat is not an additional input to the energy balance of the troposphere ( lower part of the climate system).

Geothermal heat is an input from below, solar is an input from above. There is an equal and opposite heat loss to space by thermal radiation. That is the eneryg budget that matters. How humans, termites or microbes process that energy between the time it comes into the biospohere and leaves on it's way to space is IRRELEVANT to overall budget.

That is why it is not being counted in such calculations. We are not *making" that energy, it is just an internal transformation which is part of the process we call life on Earth.
Burning fossil fuels ( and Uranium ) are unlocking stored chemical energy which constitute an additional input to the energy equation. Massive though it is, it is small beer in relation to the main energy fluxes if you calculate W/m^2 over the entire surface of the Earth.

The big question is what effect the extra CO2 ( and other GHG ) have on the energy budget. While we know the physics of E-M radiation and transmission to a good degree of accuracy we do not understand the key processes of evaporation , condensation and precipitation ( which determine how the climate works ) well enough to model them mathematically.

IPCC AR5 WG1 Full Report Jan 2014 : Chapter 7 Clouds and Aerosols:
7.3.4.2

====
The responses of other cloud types, such as those associated with deep convection, are not well determined.

7.4.4.2

Satellite remote sensing suggests that aerosol-related invigoration of deep convective clouds may generate more extensive anvils that radiate at cooler temperatures, are optically thinner, and generate a positive contribution to ERFaci (Koren et al., 2010b). The global influence on ERFaci is
unclear.
====

Most of the key physical mechanisms get approximated as educated guesses called "parametrisations"

Another key problem is that the principal metric for assessing how much warming is happening : global mean surface temperature is an average of an intensive quantity which can not therefore be added ( nor averaged ) in a meaningful way. In short you can't average temperatures of different physical media.

The principal metric for assessing global warming, which we are aiming to control by adjusting radiative feedbacks from GHG, does not respect the conservation of energy. That , sadly, is the level of "science" this whole messy subject is based upon.

https://climategrog.wordpress.com/2016/02/09/are-land-sea-averages-meaningful-2/

 

[Bored Wombat]

Since as you put it all energy derives from the sun, is not all of the hydrocarbon energy already accounted for?
Burning hydrocarbons would simply return the Earth to its original equilibrium condition.

The effect of the resulting CO2 in the atmosphere on the rate at which the Earth's thermal emissions escape into space is why burning hydrocarbons is affecting the global mean temperature.

Not the thermal energy of the combustion.

 

[D H]

Thread closed; this is going nowhere. To make matters worse, every post in this thread is in severe violation of this site's rules on discussions of global warming / climate change.