The atmosphere

The earth’s atmosphere keeps the planet warmer than it would be otherwise. This is demonstrated by the much colder surface temperature of the moon at the same distance from the sun but without an atmosphere. This warming effect is called the greenhouse effect, but few people really understand what that actually means or how it works. There is so much contradictory information on the internet that I wanted to try and explain in words rather than equations how I see that the greenhouse effect works on earth.

Our atmosphere consists mainly of 78% nitrogen, 20% oxygen, 0.04% CO2 and roughly ~0.25% H2O (water vapour). H2O levels are concentrated in the lower atmosphere and vary on a daily basis, whereas the others components are nearly constant throughout the atmosphere. They are said to be “well mixed”. Recently  CO2 has been increasing due to human activity mainly since the industrial revolution. So far CO2 levels have risen from about 300ppm to 400ppm since 1750, and may reach 600ppm by 2100.

Density structure of the atmosphere
The earth’s gravity keeps the atmosphere trapped to the planet. Despite this there is still a tiny loss of molecules to space in the upper reaches of the atmosphere. For this reason all the light hydrogen molecules originally in the atmosphere have already escaped to space. Gravity creates an exponential pressure gradient with the highest pressures at sea level at 15 lb/sq inch (1013 hPa). As you move upwards, for example when driving up a mountain, so the air pressure decreases and your ears begin to pop. This  change in “hydrostatic pressure” and the exponential distribution of pressure with height is easy to derive.

Pressure changes with height as \frac{DP}{DZ} = -\rho.g where rho is density and g is gravity which simply says that the change in pressure (force per unit area) moving up by a height DZ is equal to the change in “weight” of the air per unit area by moving to  height Z+dz . This is -\rho.g .  So applying the gas laws we get.

P = P_0 e^\frac{-RTZ}{g} where T is the ‘temperature’ of the atmosphere

Temperature structure of the atmosphere
When a gas expands it cools. This is because temperature is just the average kinetic energy of all the molecules in one volume, so the gas loses internal energy by expanding into a larger volume. We know this intuitively because compressing air to pump up a bicycle tyre warms the inner tube. So as air rises in the atmosphere to lower pressure so it cools and likewise as it falls down to higher pressure so it warms up. Gravity maintains the pressure difference with height in the atmosphere but it is motion of air masses (convection) which sets up a temperature gradient so that it is warmer near the surface and cooler higher up. But as we shall see this does not happen by itself because you need the energy of the sun to drive that convection and you also need greenhouse gases. Simple thermodynamics shows that there is a perfect balance between convection and temperature gradient called the adiabatic lapse rate. When the atmosphere is stable the air is perfectly still and the temperature falls off exactly at the adiabatic lapse rate. If the sun rises in the morning and heats the surface so it induces a steeper temperature gradient thereby inducing convection. The sun’s energy drives a global heat engine moving heat from hot to cold according to the 2nd law of thermodynamics.

Figure 1: Temperature profile of the atmosphere

Figure 1: Temperature profile of the atmosphere

Remember that the atmosphere is stable at the adiabatic lapse rate. The dry adiabatic lapse rate is -g/Cp where Cp is the specific heat of air at constant pressure. Adiabatic means that that no external heat enters the gas so the work done in expanding a unit volume of air that rises upwards comes from the internal energy of the gas. Imagine a mass of atmosphere m that rises up a height DH against gravity then the work done is mgDH which is equal to the loss in internal energy(heat) or -mCpDT so DH/DT = -g/Cp. When the atmosphere is exactly at the lapse rate air can move up or down without external work. During the day the surface warms up fast increasing the lapse rate above this thereby inducing convection. If the environmental lapse rate is below the DALR then air will fall otherwise it rises until the DALR is restored. When evaporation is included we get weather such as storms, thunderstorms and precipitation.

Effect of CO2
The earth’s surface radiates heat as IR into the atmosphere just like a black body. CO2 absorbs photons of very specific wavelengths through quantum excitation of vibrational and rotational lines. These are concentrated in the 15micron band (see Fig 1).

Figure 1: The fine structure of transition lines that make up the 15 micron CO2 absorption band.

Figure 2: The fine structure of transition lines that make up the 15 micron CO2 absorption band.

This energy is quickly thermalised with surrounding air including other CO2 molecules. These quickly re-emit IR photons loosing vibrational energy according to the local thermalized temperature. As the temperature falls with increasing altitude so too does the relative emission of CO2 molecules. Multiple such re-emission steps “transfer” radiative energy in CO2 sensitive bands up through the atmosphere until the density falls sufficiently so that it escapes to space. This is because eventually there are so few CO2 molecules left higher up that the probability for absorption of a CO2 IR photon essentially falls to zero. During this process radiative energy is absorbed by the atmosphere tending to increase the temperature gradient because the density is highest near the surface. As this temperature gradient steepens away from the stable DALR so convection of air is induced to restore balance. IR radiation is the “energy source” that drives the convection heat engine that maintains the earth’s lapse rate.

Figure 3: Taken  from Richard Lindzen. Pure radiative equilibrium would be the temperature gradient if the atmosphere only relied on radiation to cool the surface (without any convection) and the surface temperature would be >20C warmer than today ! Thermodynamics drives the lapse rate towards the moist adiabatic lapse rate through convection and Latent heat of evaporation.

The greenhouse effect works as follows.

  •  Solar energy during the day warms the surface.
  • The surface radiates IR upwards into the atmosphere increasing the lapse rate by absorption of photons by CO2 ( and water) at different levels through radiative transfer.
  • This absorption then drives convection and evaporation (latent heat) to restore the lapse rate back towards adiabatic stability.
  • Most IR photons escape to space at higher and therefore colder altitudes.

The lapse rate and convection stops at the tropopause. Above that lies the stratosphere where temperatures first remain constant, and then increase because ozone absorbs sunlight to warm the upper atmosphere. What determines the height of the tropopause? The tropopause is the height where the net radiation loss to space exceeds the radiation absorbed from all lower levels. The greenhouse effect stops there and the atmosphere cools by radiation alone. Total radiation must balance incoming solar radiation so the average temperature is the “effective temperature” or about 225K. To look into more detail of how CO2 effects surface temperatures and how any increases will lead to global warming we need to study the wavelength dependence of the height where CO2 radiates to space. This then defines the effective emission height for CO2 in the atmosphere. The following is one way to calculate this height distribution.

The density of air with altitude is determined by barometric pressure. For a well-mixed gas the density of CO2 with height is determined by the overall concentration in ppm. Therefore at any altitude we know the number of CO2 molecules/m^3. The cross section for the absorption of IR photons of wavelength ? is given by the HITRAN database. Therefore we can calculate at which altitude a fraction = 0.5 of incident photons arriving from the TOA have been absorbed. Since absorption + transmission = 1 this height is exactly the same as the effective emission height for radiative transfer photons upwards of wavelength ? transmitted from all lower levels in the atmosphere and surface. The only criteria that determines the emission of IR photons of wavelength ? by CO2 molecules in all the optically thick layers below is the local thermalized temperature (Kirchoff’s law). This emission intensity is given by Boltzman’s distribution.

Now imagine a downward flux of IR photons originating from space. We assume a US standard atmosphere. Then for each wavelength using HITRAN we can calculate the height at which more than half of the incident photons are absorbed by CO2 molecules in the atmosphere. This is the same as the effective emission height for upwelling photons from lower atmospheric levels in local thermodynamic equilibrium. This is what you get

Fig 5b: The CO2 emission height profile for 300ppm and for 600ppm smoothed with a resolution of 20 lines.

Fig 3: The CO2 emission height profile for 300ppm and for 600ppm smoothed with a resolution of 20 lines.

Then for each wavelength we can also calculate the emitted radiance and the result agrees almost perfectly with Nimbus spectra.

Fig 7: Calculated IR spectra for 300ppm and 600ppm using Planck spectra. Also shown are the curves for 289K and 220K

Fig 4: Calculated IR spectra for 300ppm and 600ppm using Planck spectra. Also shown are the curves for 289K and 220K

Furthermore by varying the concentration of CO2 you can calculate the change in OLR at the TOA for each wavelength. Integrating over wavelength gives you the net CO2 forcing as a function of CO2.Furthermore we can also allows “derive” the formula RF = 5.3 ln(C/C0) which had previously been a mystery to me (actually I got 6.0!). You can also calculate the net temperature effects of different CO2 concentrations on earth -ignoring all other feedbacks. The result is shown below.

Fi 1: Dependence of radiative forcing on CO2 concentration in the atmosphere. The red curve is a logarithmic fit.

Fi 1: Dependence of radiative forcing on CO2 concentration in the atmosphere. The red curve is a logarithmic fit.

Summary
The greenhouse effect depends on there being a lapse rate. This is stabilized at the (moist) adiabatic lapse rate through convection and evaporation. Convection is driven by solar heating of the surface and IR radiation absorption by the atmosphere. The lapse rate stops at the tropopause because this is where the radiative losses to space exceed the radiative absorption from below. The height of the tropopause effectively determines the surface temperature because radiative energy balance fixes the tropopause temperature at about 225K.   CO2 just affects a narrow band of wavelengths centered on 15 microns. Increasing CO2 levels affect mostly the side lines leading to a weak logarithmic dependence of forcing because the central lines are already saturated way up to the stratosphere. This CO2 effect essentially increases the tropopause height somewhat leading to slightly larger surface temperatures at equilibrium.  This is often called radiative forcing at the “top of the atmosphere” but really it is because the rise in the tropopause causes an energy imbalance until surface IR increases at the higher temperature to rebalance once again outgoing IR with incoming solar radiation.

12/6 update: corrected according to comment of Eli below.
13/6 update: corrected atmospheric pressure at sea level – see Ro below.

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22 Responses to The atmosphere

  1. Eli Rabett says:

    There are a number of points where Eli would disagree. First of all, when you say

    “These same CO2 molecules quickly re-emit IR photons and loose vibrational energy according to their local thermalized temperature. ”

    The implication is that the energy stays in the CO2 molecule that absorbs it. This is not the case. The energy is thermalized in a few microseconds by collisions with nearby molecules mostly of N2 and O2. A fuller explanation of the process can be found at

    http://rabett.blogspot.co.uk/2013/04/this-is-where-eli-came-in.html

    It is this thermalization that controls the relative emission of CO2 molecules as a function of altitude and temperature as you conclude.

  2. A C Osborn says:

    Clive another point that you might like to clarify is the comparison to the Moon, the average temperature may be cooler on the Moon, but the daily high temperature is much hotter and the night time/shodow temperature is much colder. It is the Earth’s atmosphere and Oceans that drastically reduce the high/low swing of the temperatures on Earth.

    • Clive Best says:

      Yes that is correct. The surface of the moon does gets hotter than the earth during the lunar day and much colder during the lunar night. The moon radiates directly to space from the surface. The main climate stabilizing component of the earth’s atmosphere is water vapour and especially clouds. There are greater extremes of temperature in the desert because of low water vapour and clouds at night allowing more radiation form the surface to space. Clouds keep the surface warmer at night. You need clear skies to get frost.

  3. Euan Mearns says:

    Clive, I’ve been thinking of asking you to write something like this for a while. As a non-physicist I still find some of the concepts difficult to grasp and conceptualise. I really like you first chart. If the height of the tropopause were to change does this change the temperature gradient through the troposphere (temperature remains the same at surface and tropopause) or does the temperature gradient stay the same and surface temperatures go up or down?

    Houghton clearly states that convection is the main process for moving heat away from surface. A thunder storm may move heat from surface to tropopause. I have this simplistic view of the world. If surface temperature rises so will the amount of heat transfer by convection. Heat convected to tropopause can radiate away to space unhindered by water vapour. I imagine this is the main mechanism for maintaining atmosphere stability. Its a self regulating negative feedback. Do you know if GCMs include this is their functionality?

    Finally, returning to your first chart. Do you know how changes in spectral output from the Sun might affect where radiation is captured in the atmosphere and how this might impact the structure shown in your chart?

    Best,

    Euan

    • Clive Best says:

      Euan,

      For the Dry Adiabatic Lapse rate (DALR) the slope is linear and remains the same, so if the height of the tropopause increases then the surface temperature must rise as well linearly. This essentially is the same argument as given in Houghton’s book, but it is not quite right. The real atmosphere is far closer to the moist adiabatic lapse rate especially in the tropics which radiates the most. Th moist ALR is not linear and the temperature gradient decreases with surface temperature reducing the surface warming compared to DALR. This is essentially a negative feedback to CO2 forcing. Despite this however the surface temperature will still rise as shown correctly in the figure 2 diagram.

      The spectrum from the sun is almost exactly a block body spectrum so it is unlikely to change unless the sun’s temperature changed or else some dust passed between the earth and the sun. However the spectrum reaching the surface depends critically on clouds and water vapour. As we showed in our study clouds may well act as the thermostat for the earth’s climate.

  4. Roberto says:

    Hello,

    The ‘standard’ atmosphere has a pressure at sea level of 1013.25 hPa, not 985 as you’vee stated, Clive.
    Apart from that a very informative piece, thanks.

    R.

  5. drew says:

    It was my understanding that H2O was the most significant greenhouse gas. Is that understanding incorrect? I ask because you don’t seem to mention it much in this article.

    • Clive Best says:

      Yes that is correct. H2O is by the most important greenhouse gas on earth and contributes 80-90% of the overall average greenhouse effect (33C). CO2 contributes about only 5C of warming overall. However H2O effect is complicated by cloud formation and it’s involvement in convection/evaporation. So water also helps cool the surface, increase albedo while also increasing the greenhouse the effect. H2O concentration also varies on a daily basis at any location so overall the effect is very complicated. Therefore this article describes only CO2 and the relatively small effect it has on earth’s climate compared to H2O.

  6. Bryan says:

    An expanding gas only drops its temperature if it does work.
    If it expands into a vacuum there is no loss of internal energy as the speed of individual molecules is unchanged

    • Clive Best says:

      A gas expanding into the vacuum of space would cool to 3K – the temperature of the microwave background. It would lose all its internal energy through expansion.

      • Bryan says:

        You seem to confuse a vacuum with a vacuum in outer space.
        With a high pressure pump you can create a near vacuum here on earth.
        If (say) a vessel with two litres of gas at two atmospheres and 275K temperature is allowed to expand into an evacuated connected three litre vessel the temperature of the gas stays the same.
        It’s a Boyles Law problem

  7. Alan McIntire says:

    “When the atmosphere is stable the air is perfectly still and the temperature falls off exactly at the adiabatic lapse rate.” I suppose this is true in the daytime, as a limit as the earth heats up, but at night the ground can cool off a lot faster than the air- as happens on clear nights, especially in the desert. In that case you can get a temperature inversion, where temperatures INCREASE with altitude. Those temperature inversions are ALSO stable, in the sense that you don’t immediately get convection to bring the lapse rate back to the adiabetic lapse rate.

  8. Andyj says:

    Clive you said, “This energy is quickly thermalised with surrounding air including other CO2 molecules. These quickly re-emit IR photons loosing vibrational energy according to the local thermalized temperature.”

    I have an issue where an assumption of radiation exists in latent heat.
    Sure the excited CO2 molecule will equalise with the surrounding air but is it photonic?

    • clivebest says:

      I think it is simplest to treat all energy the same way so it doesn’t matter whether the atmosphere is heated by radiation, convection or latent heat. What matters is the temperature at each level which tends towards the moist adiabatic lapse rate. CO2 molecules and H2O molecules at any height emit radiation following black body radiation for their wavelength. Near the troposphere enough radiation escapes to space to stall convection because the primary energy source (radiative transfer from below plus latent heat) dissipates. Essentially the fuel runs out to drive the atmospheric heat engine and it stops !

      • R Graf says:

        “CO2 molecules and H2O molecules at any height emit radiation following black body radiation for their wavelength. ”

        I think you meant black body radiation for their temperature.

        Clive, BTW thanks for this site. I have learned a lot.

        Also Euan Mearns asked above: “Do you know how changes in spectral output from the Sun might affect where radiation is captured in the atmosphere?”

        I think he was curious if the temperature rising from the ozone layer when getting beamed by the sun has an effect warming the tropopause increasing emissivity since ozone will be radiating it’s own black body IR signature with the peak getting intercepted on the way down by CO2 as it’s first customer in line. On a similar question, since CO2 has absorption peak at 1.9 and 2.8 micron, well within the sun’s spectral tail, wouldn’t this help to warm the tropopause to some extent too? Or, would most of that radiation get spit out at 12 microns before it had the chance to thermalize, if you know or care to guess?

        • Clive Best says:

          UV absorption by Ozone is an important effect and warms the stratosphere. That is why the temperature gradient increases above 20km. Ozone also emits IR as a greenhouse gas through vibrational excitation. This can be seen from satellite spectra where the effective BB temperature is higher than CO2. So yes the Ozone layer emits Irrelevant downwards at a higher temperature than the tropopause. This effectively warms the troposphere.

          Likewise any absorption of solar radiation by CO2 would warm the troposphere.

          • R Graf says:

            Clive,

            Thanks for confirming my thought that ozone above the troposphere is causing a possible temperature inversion when sunlight gets strong enough. The stratosphere is still lower density so as the effective tropopause rises due to increased CO2 it is not necessarily going to need to cool. In fact it might just might warm in which case black body radiation efficiency rises. Or, does rad efficiency go down some too with lower air density?

            On a different thought, if CO2 has made the previous tropopause height more IR reflective does this mean it could shield the ozone heat from penetrating as deeply as before, preventing interference with upward convection?

            One last effect is the thermal concentration effect allowing the tropopause to rise to a higher temp just because there is more IR being thermalized in a smaller area. Could this be yet another effect to increases temp and BBR ?

  9. Susan Oliver says:

    Thank you Clive. You’ve explained this very well. I was wondering if you had any thoughts on this paper by Hermann Harde which finds a lower forcing for CO2 than the commonly expected value:
    http://www.hindawi.com/journals/ijas/2013/503727/

  10. gymnosperm says:

    The fine structure of the 667.4 CO2 band and its rotational sidekicks can also be plotted against absorption. The result is very similar to your figure 2 with the left axis showing near complete absorption in one meter for the 667.4 band at an atmosphere and 400ppm.

    The rotational bands to the left and right are called P and R. The P rotations are constructive and increase the energy of the molecule but the R bands are destructive and reduce the energy. The net effect of the rotational bands is slightly negative, and the rotational bands are generally 2-3 orders of magnitude weaker than the central Q band.

    Nevertheless, the whole lot plotted against 280ppm transmission to the tropopause virtually defines the zone of zero transmission.

    Furthermore, MODTRAN indicates that the atmosphere below 1km simply does not radiate in the CO2 bands.

    https://geosciencebigpicture.com/2016/08/21/modtran-up-and-down/

    What the satellites are seeing radiating in the tropopause is NOT surface energy. The CO2 effective radiative level in the tropopause is above the lapse rate. Raising it a bit has no effect anyway.

  11. Hammad says:

    Hello Clive i need a article of Richard Lindzen in which he describe figure.3(green house effect ) Lapase rate

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