måndag 14 juni 2010

Elementary Climate Mathematics

Let us collect some basic facts about global climate:
  • temperature of stratopause: 273 K
  • temperature of tropopause: 218  K 
  • temperature at Earth surface: 288 K
  • observed lapse rate in troposphere: 7 K drop per km 
  • total temperature drop in troposphere: 288 - 218 = 70 K = 7 x 10 km 
  • isentropic thermodynamic theoretical lapse rate: 10 K/km 
  • 180 Watts/m2 absorbed by Earth surface
  • 120 Watts/m2 returned by convection/latent heat
  • 60 Watts/m2 returned by radiation 
  • Earth-atmosphere effective blackbody temperature: 273 K = temperature at stratopause.
We observe:
  • Convective heat transport in the troposphere decreases the theoretical isentropic lapse rate by 3 K/km (from 10 K to 7 K), by evaporation at the Earth surface removing heat and condensation a higher altitudes adding heat.
  • Stratopause temperature of 273 K fixed by blackbody radiation at given insolation.
  • Tropopause temperature determined by temperature distribution in the stratosphere.
  • Surface temperature is determined by troposphere lapse rate and tropopause temperature.
  • It is natural to compute temperatures outside in, starting at the stratopause at 273 K and ending at the Earth surface at 288 K.
  • 60 Watts of radiation dQ is consistent with the relation dQ ~ 4 dT (differentiated SB Stefan-Boltzmann radiation law) with a temperature drop dT of 15 K (from Earth surface to stratopause).
  • Observations indicate a climate sensitivity of dQ ~ 6 dT (negative feedback from dQ =4 dT).
Main question:
  • Suppose the heat transport in the troposphere changes so that less heat is radiated and more heat is transported by convection from the Earth surface, at a constant total, for example to 124 Watts by convection and 56 Watts by radiation.  Will then the Earth surface temperature increase or decrease? Warming or cooling?
Tentative answer:
  • The temperature will drop as the intensity of evaporation/condensation increases and the troposphere lapse rate is further decreased, assuming that the tropopause temperature stays constant (assuming the stratosphere temperature does not change). The change in temperature could come from a change of lapse rate of 4/120 x 3 = 0.3 K/km resulting in a 3 K drop of surface temperature.
  • A shift from radiative to convective heat transfer in the troposphere can be expected by by increasing the effect of GHG greenhouse gases (mainly water vapour). 
  • This could correspond to a decrease of surface temperature under increased cloud cover from increased GHG.
  • IPCC predicts an increase of surface temperature of 1.5 - 4. 5 C, from 1 C basic greenhouse effect/doubled CO2 based on SB plus assumed ad hoc positive feedbacks.
  • The above argument indicates instead a decrease of surface temperature from increased greenhouse effect, under constant insolation.
  • An increase of insolation by 4 Watts can by SB by expected to give an overall increase of 1 C.
The above argument uses more physics and more data than a direct application of SB argued by IPCC. Both arguments are simplistic. Which one is more realistic? Or none?

It is remarkable that not even the sign of climate sensitivity (warming or cooling by adding greenhouse gases) can be convincingly predicted by some form of mathematical analysis of the thermodynamics of an atmosphere. Or maybe it can, by a correct analysis based on computing turbulent solutions of the Navier-Stokes equations...stay tuned...

Compare with the canonized description of the greenhouse effect:
  • The greenhouse effect is a process by which radiative energy leaving a planetary surface is absorbed by some atmospheric gases, called greenhouse gases. They transfer this energy to other components of the atmosphere, and it is re-radiated in all directions, including back down towards the surface. This transfers energy to the surface and lower atmosphere, so the temperature there is higher than it would be if direct heating by
  • solar radiation were the only warming mechanism.
  • The Earth receives energy from the sun in the form of visible light. This light is absorbed at the Earth's surface, and re-radiated as thermal radiation. Some of this thermal radiation is absorbed by the atmosphere, and re-radiated both upwards and downwards; that radiated downwards is absorbed by the Earth's surface. Thus the presence of the atmosphere results in the surface receiving more radiation than it would were the atmosphere absent; and it is thus warmer than it would otherwise be.
  • This highly simplified picture of the basic mechanism needs to be qualified in a number of ways, none of which affect the fundamental process.
  • This mechanism is fundamentally different from that of an actual greenhouse, which works by isolating warm air inside the structure so that heat is not lost by convection.
We see that the lapse rate with an elevated surface temperature is viewed to come from radiation only, more precisely from atmospheric "re-radiation in all directions". We also see that the term "greenhouse gas" is admitted to be a (deliberately) misleading misnomer.  Like the Democratic People's Republic of North Korea.

Nevertheless, this is the essence of the scientific basis of climate alarmism: Re-radiation in all directions without any thermodynamics. Convincing science? Convincing to you?
What is the physics of 
  • the fundamental process, fundamentally different from that of a greenhouse, which results in the surface receiving more radiation? 

What physics books describe this fundamental process? I would like to learn about this phenomenon!

Note that the "fundamental process" referred to (probably) is "photons emitted by the Earth surface" which are "being trappedb y greenhouse gases in the atmosphere" and then "re-emitted back to the Earth". 

But is this the real physics of radiation as an electromagnetic wave phenomenon?
I don't think so. The idea of photons being trapped like fish in a net, is too simple. I would rather think of the situation as "dominance of the stronger over the weaker" as a flow of heat energy from higher to lower temperatures, instead of flows in both directions (at different strengths).

10 kommentarer:

  1. Your fixation on the stratopause and stratosphere is increasingly bizarre.

    Your first collection of statements is fine except for the last one:
    "Earth-atmosphere blackbody temperature: 273 K = temperature at stratopause."

    "Earth-atmosphere" is not a black body! The stratopause is not a black body! Satellite observations show extreme deviations from the black body spectrum in the infrared-to-visible spectrum of radiation emitted and reflected by our planet (and both emission and reflection need to be included to come up with 273 K).

    And when you say "Tropopause temperature determined by heat transfer in the stratosphere." - what exactly is the mechanism of heat transfer in the stratosphere? Do you understand why it is called the stratosphere?

    In reality, it is the *tropopause* temperature that is determined by radiation.

    This is easily understood through simple thermodynamics: heat travels from hotter to colder places. The coldest place we have any contact with is the cosmic microwave background at about 2 K - "outer space". The coldest place in our atmosphere must be the part that is most directly in thermal contact with the 2 K background. The only way to be "directly in thermal contact" with outer space is through radiation. And that is what fixes the coldest part of our atmosphere - the tropopause.

  2. Arthur,

    The Greenhouse Effect supposedly keeps the tropopause much colder than it would otherwise be. Explain to me why

    "This is easily understood through simple thermodynamics: heat travels from hotter to colder places"

  3. The stratosphere is heated by absorbing light from the Sun.
    The tropopause at 218 K is too cold to radiate enough to empty space.

  4. Anders - the greenhouse effect is like adding insulation. It doesn't literally increase the energy received from the sun, it just keeps hot stuff hotter, and cold stuff colder. The main bulk of the GHG's lie in the troposphere, so the insulation is applied between the surface and the tropopause, across the width of the troposphere.

    Claes - when you say "the tropopause at 218 K is too cold to radiate enough to empty space" - that's wrong in a couple of ways. First, the tropopause is a zero-width region, so by itself it radiates nothing at all, no matter what temperature it is. But the air around it contains radiating molecules and they emit plenty at around 220 K - for example look closely at the "20 km looking down" graphic in this article:


    That's for the cloud-free Arctic atmosphere, so the effects are almost free of water-vapor and liquid water absorption and emission. So the outgoing radiation is primarily on the one hand from the surface (close to 270 K for this case) or from near the tropopause (close to 220 K). When you combine what goes straight out from the surface into space, what goes out from cloud tops, and what goes out from near the tropopause, that gives you the temperature distribution and heat flows of the troposphere and it all adds up to essentially the entire incoming energy from the sun. The role of the stratosphere in the radiative balance is very small.

  5. I don't know if the origin of radiation matters, as concerns the radiation
    passing e.g. the stratopause. Isn't it possible to view the Earth plus tropo/stratosphere as a black body emitting radiation from/through the stratopause at 273 K?

  6. Claes, judging from your last comment I think you are more and more approaching my view that the temperature is the same everywhere in equilibrium (by definition) and that the thermometers simply measure the wrong temperature almost everywhere. Then it is of course possible to define the place where they actually do measure the correct temperature (if such a concept is meaningful) as the "effective radiating shell" of the earth. You could go further up in the atmosphere until you no longer find any matter at all and you could still conclude from the radiation intensity that the temperature is 273 K.

  7. I agree that blackbody temperature (based on a fictitious vibrating medium) may be different from what you measure by a thermometer.
    But 273 C at the stratosphere coincides with the theoretical blackbody
    temperature (with absorbitivity = emissivity) of a blackbody one AU from the Sun. Coincidence or real?

  8. It may not be a coincidence.

    I can convince myself that stratospheric ozone cannot cause the stratosphere to warm above the radiative temperature of the planet below, which is nearly identical to the radiative temperature as seen from space because the atmosphere above the stratopause has negligible optical depth.

    Is this right? I have never thought about it before but it sounds right to me.

    I think Arthur is right that it is not relevant. If there were no stratospheric ozone, or insufficient ozone to reach the maximum temperature at the stratopause, the temperature profile of the troposphere would not be significantly changed.

  9. I agree that the temp in the stratosphere may not be relevant. Maybe temperature can have a different meaning from thermodynamic and radiation point of view.