The subject of atmospheric heating rates is one which is worth spending time on.
What is a heating rate?
To see the usefulness of a heating rate let’s consider the per capita income of a country.
Per capita income compares the ratio of total $ to the total population. If we compare the total income of China to the total income of Laos we don’t have a useful comparison. If we compare the per capita income of China to the per capita income of Laos.. well, who knows whether we have a meaningful comparison – but at least we have something more useful. Something more relevant.
Energy absorbed in a layer of the atmosphere causes heating at a certain rate. Energy lost from a layer of the atmosphere causes cooling at a certain rate.
Heating rates tell us something different from total energy lost or gained. Suppose a 1m layer of the atmosphere gains 1,000 J/m², what will the temperature change be?
The specific heat capacity of the atmosphere at constant pressure is 1005 J/(K.kg) – which means it takes just over 1,000 J to lift the temperature of 1 kg of the atmosphere by 1K (=1°C).
However, the atmospheric density decreases with height:
At the surface, where pressure = 1000 mbar, the density = 1.2 kg/m³.
So 1,000 J/m² lifts the temperature of a 1m layer of the atmosphere at the surface by 0.83 K (calculated by ΔT=1,000/[1.2 x 1005]) .
At the top of the stratosphere, near 50km where the pressure = 1 mbar, the density = 0.0016 kg/m³.
Here, 1,000 J/m² lifts the temperature of a 1 m layer of the atmosphere by 620 K (calculated by ΔT=1,000/[0.0016 x 1005]).
So it’s a bit like sharing out the total income of China among the residents of Laos.
That’s why heating rates are useful – they relate the amount of energy with the amount of atmosphere.
More CO2 equals More Absorption but More CO2 equals More Emission
One of the confusing aspects in atmospheric radiation comes as people start to consider the fact that the atmosphere emits as well as absorbs.
So more radiatively-active gases (=”greenhouse” gases) causes more absorption? Or more emission? Or doesn’t one just balance out the other and so there is no change?
There are legitimate questions to ask.
The only way to answer these questions is to solve the Schwarzschild equation – see Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Six – The Equations.
What we will do is first of all review the heating/cooling rates vs altitude and try and understand some features qualititively.
Here is the right way to think about the problem:
Absorption at any given wavelength depends on:
- the quantity of gases that absorb at that wavelength
- the effectiveness of each gas at absorbing at that wavelength
- the “amount” of radiation travelling through that part of the atmosphere (note 1).
Emission at any wavelength depends on:
- the quantity of gases that absorb at that wavelength (and therefore emit at the same wavelength)
- the effectiveness of each gas at absorbing (and therefore emitting) at that wavelength
- the temperature of the gas
In the case of shortwave (=solar radiation) the atmosphere absorbs but does not emit. This is because the atmosphere is not hot enough to radiate significantly below 4 μm, see The Sun and Max Planck Agree – Part Two.
In the case of longwave (= terrestrial / atmospheric radiation) absorption is from radiation from above and below. But usually the radiation from below is a lot higher than from above. This is because the earth’s surface emits close to blackbody radiation (the surface has a very high emissivity over all wavelengths), and the atmosphere (which doesn’t emit as a blackbody) is hotter closer to the surface.
Doesn’t a Heating or Cooling Rate Mean that the Atmosphere is Heating up or Cooling Down?
The sun heats the atmosphere (a heating rate), but the atmosphere radiates to space (a cooling rate), and also convection moves heat through the troposphere.
We can still have a heating rate, a cooling rate and convective heat transfer while the atmosphere is in approximate energy balance (=not changing in temperature). If the temperature of one part of the atmosphere is not changing then these will sum to zero.
So heating rates vs height give us insight into the strength of these effects, and we can break the effects up between the responsible gases (water vapor, CO2, ozone, etc).
From the always excellent Grant Petty, A First Course in Atmospheric Radiation, the solar heating of the atmosphere, for a standardized tropical atmosphere:
Figure 2 – Solar heating
If we showed total energy absorbed at each height in the atmosphere, then the troposphere would overwhelm the stratosphere (upper atmosphere). But because we are showing energy absorbed in proportion to the density of the atmosphere, the upper atmosphere appears more important.
We see that ozone causes the highest heating rate in the stratosphere, whereas water vapor causes the highest heating rate in the troposphere, and CO2 has a very small effect.
The water vapor heating rate is – of course – concentrated in the bottom few km of the atmosphere because water vapor is concentrated here.
Most of the absorbed solar radiation is absorbed by the earth’s surface. The surface absorption is not shown in this graph. In turn, the surface heats the atmosphere primarily through convection. The convective heat transfer is also not shown.
Now let’s look at longwave heating (cooling) rates for a few different regions:
We see that the heating rates are mostly negative, meaning that these are really cooling rates. Most of the atmosphere is cooling via longwave radiation. However, one small part of the atmosphere experiences a heating rate due to longwave radiation – the tropical tropopause.
The tropopause is the coldest part of the atmosphere – the top of the troposphere and bottom of the stratosphere. And the coldest part of the atmosphere radiates less than it absorbs.
Let’s see the breakdown of cooling rates by individual gases:
We see that water vapor has a peak longwave cooling at around 3 km and another maximum at 10 km. The lower peak is caused by the “continuum” – also shown separately on the graph – which we will return to shortly.
Ozone shows a heating rate in the stratosphere below 30km. If we had the graph extend up to the top of the stratosphere, around 50km, we would see ozone with a cooling to space higher up.
We also see that CO2 has a very small cooling effect until we get into the stratosphere. Generally, each layer experiences a very small heating/cooling effect from CO2 because CO2 has a such a strong absorption that energy is absorbed from layers very close by – which are at very similar temperatures. The tropopause is the coldest part of the atmosphere so absorbs a little more radiation via CO2 than it emits – consequently a small heating effect.
As we get up into the stratosphere we see a progressively stronger cooling to space from CO2. In part, this is because of the reduction in pressure broadening at lower pressures = higher altitudes (see Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part Nine). This effect causes the absorptivity of CO2 to reduce at higher altitudes meaning that the radiation from the layer below can “get through” to space.
Water vapor has an unusual absorption profile. In the so-called “atmospheric window” of 8-12 μm where there are no strong absorption bands from any atmospheric molecule (apart from ozone at 9.6 μm), water vapor still absorbs. Here the absorption coefficient is proportional to the atmospheric pressure.
In general, the more we have of a particular gas, the more absorption. This is expressed in the Beer-Lambert law.
But for other gases, the absorption coefficient is a constant for a given wavelength – not proportional to pressure. In the Beer-Lambert law we multiply the mass of absorbing molecules in the path by the absorption coefficient to find the optical thickness (note 2).
For the water vapor continuum the absorption coefficient is proportional to pressure. And absorptivity is a function of the absorption coefficient and the total mass in the path (which is proportional to pressure). Therefore, total absorption in the continuum band is a very strong function of pressure.
Water vapor concentration is concentrated at lower levels in the atmosphere so the total absorption due to the water vapor continuum falls off very quickly with height.
This is why the lower peak cooling rate occurs. The absorption by water vapor due to the continuum above 3 km is very small – so around 3km the atmosphere (in these wavelengths) can very effectively cool to space. Other bands of water vapor absorb more strongly, so effective cooling to space doesn’t really begin until the concentration of water vapor drops to very low values. Hence the second peak at 10 km.
A long time we had a look at Stratospheric Cooling. This strange phenonemon is expected from more CO2 in the atmosphere. All other things being equal, the troposphere will warm and the stratosphere will cool.
Radiative-convective models predict this. Once you’ve got to grips with basic radiation in the atmosphere, it is easy to see why the troposphere will warm.
But why will the stratosphere cool?
Some will look at Figure 4 and say “ah ha“. More CO2 will move the CO2 line over the left and so that’s why the stratosphere will cool.
As a cautionary note, the heating rate at level z is equal to:
Where among other terms, the italicized “T” is the band-averaged transmittance between z and z’, and the integrals are (obviously to the mathematicians) for each “level” (note 3) between the surface and z, or between the top of atmosphere (∞) and z..
If we went through this equation we would find that there are competing terms – terms which represent absorption of energy from other parts of the atmosphere (heating), and terms which represent emission of energy from this layer (cooling). Increasing CO2 increases absorption in the stratosphere. Increasing CO2 increases emission from the stratosphere.
Given that radiative-convective models predict stratospheric cooling we can say confidently that more CO2 will move the cooling curve in figure 4 to the left in the stratosphere (note 4). So emission will be higher than absorption.
However, we haven’t developed an intuitive understanding of why. At least, I haven’t.
To develop an intuitive understanding I would need the solution of these equations for a variety of conditions, and after playing around with changed parameters and reviewing results it would all start to make sense. That’s what I would hope.
But that’s just me. Others can perhaps just see it all dance out of the equations in a flash (think – the crazy one in The Hangover in the casino). Or out of the fundamental physics.
Heating rates help give insight into how the atmosphere absorbs and emits radiation from different “greenhouse” gases at different levels.
Generally the peak cooling rates for each band occur when that band “thins out” enough in the layers above to allow significant radiation to space, rather than just to the level immediately above.
Convection is the most important mechanism for moving heat in the troposphere (but not the stratosphere).
This article hasn’t considered convection at all – which just demonstrates the ongoing plot to hide the importance of convection. Once people realize how important convection is, radiative heating and radiative cooling to space will be.. the same.
Other articles in the series:
Part One - a bit of a re-introduction to the subject.
Part Two - introducing a simple model, with molecules pH2O and pCO2 to demonstrate some basic effects in the atmosphere. This part – absorption only.
Part Three - the simple model extended to emission and absorption, showing what a difference an emitting atmosphere makes. Also very easy to see that the “IPCC logarithmic graph” is not at odds with the Beer-Lambert law.
Part Four - the effect of changing lapse rates (atmospheric temperature profile) and of overlapping the pH2O and pCO2 bands. Why surface radiation is not a mirror image of top of atmosphere radiation.
Part Five – a bit of a wrap up so far as well as an explanation of how the stratospheric temperature profile can affect “saturation”
Part Six – The Equations – the equations of radiative transfer including the plane parallel assumption and it’s nothing to do with blackbodies
Part Seven – changing the shape of the pCO2 band to see how it affects “saturation” – the wings of the band pick up the slack, in a manner of speaking
Part Eight - interesting actual absorption values of CO2 in the atmosphere from Grant Petty’s book
Part Nine - calculations of CO2 transmittance vs wavelength in the atmosphere using the 300,000 absorption lines from the HITRAN database
Part Ten - spectral measurements of radiation from the surface looking up, and from 20km up looking down, in a variety of locations, along with explanations of the characteristics
Part Eleven – Heating Rates - the heating and cooling effect of different “greenhouse” gases at different heights in the atmosphere
Part Twelve – The Curve of Growth - how absorptance increases as path length (or mass of molecules in the path) increases, and how much effect is from the “far wings” of the individual CO2 lines compared with the weaker CO2 lines
And Also -
Theory and Experiment – Atmospheric Radiation – real values of total flux and spectra compared with the theory.
Note 1 – Absorptivity is a different parameter from absorption. Absorptivity is the proportion of radiation absorbed and is dependent on the number of molecules of different radiatively-active gases. Absorption is the total amount of energy absorbed and so depends on the intensity of radiation passing through that part of the atmosphere and the absorptivity.
Note 2 - The Beer-Lambert law can be expressed in a number of different ways. Essentially the units for the amount of the gas (e.g. number of molecules, mass) in the radiation path has to match the units for the absorption coefficient. The same result is obtained.
Note 3 - There are no discrete “layers” in the atmosphere. This is a convenient term for explaining the physics in plainer English (as with many other inexact and non-formal explanations). All the properties of the atmosphere we are considering have continuous change with pressure and, therefore, with height.
Note 4 - The derivation of the equations for heating rates comes from the same equations which are used in radiative-convective models.