Termisk elektromagnetisk strålning i atmosfären – ett ofta missförstått koncept

Karl Schwarzschild (1873–1916, bild från Wikipedia)

Detta tema (speciellt den del som handlar om ”återstrålning”) har ju ofta varit uppe här på KU under åren i ganska animerade diskussioner. Strax efter det att jag började intressera mig för ”global uppvärmning” ca 2003 fick jag via Yahoo-diskussionsgruppen ”ClimateSceptics” kontakt med en person vid namn Hartwig Volz. Han hörde till kärnan bland de tidiga ”skeptikerna”och jag tror han var disputerad geofysiker. Man hittar många referenser till Volz, liksom inlägg och kommentarer på den år 2004 tyvärr tidigt bortgångne John Daly’s blogg.

Volz sände mig för många år sedan nedanstående email, som innehåller en pedagogisk redogörelse för de sammanhang som så ofta missförstås, och som är svåra att läsa ut ur standarverken inom området, exempelvis Goody och Young’s bok.

Han har senare också bidragit med detta intressanta konferensbidrag, rörande havsytans högst varierande emissivitet vid olika vindförhållanden – tillräckligt för att ha en mycket väsentlig inverkan på strålningsbalansen. Jag vet inte om det senare blev publicerat i någon tidskrift, men det borde absolut uppmärksammas av dem som arbetar inom fältet.

Jag har inte kunnat nå Volz på email i dagarna, men är säker på att han inte har något emot att jag här återger vad han tidigare skrev. Jag hade tänkt göra det tidigare under åren, men vår bloggplattform har vissa begränsningar för ekvationer och annat. Möjligen har jag för flera år sedan kommenterat något med en länk till en PDF jag hade.

Nu hittade jag en väg runt ekvationshanteringens begränsningar. Bättre sent än aldrig. Jag har också lagt till lite länkar, för den som är intresserad av att läsa mer.

Subject: Forcing equation

The forcing by greenhouse gases is not calculated with aid of the Lambert Beer law, but
with the radiative transfer equation (which contains Lambert Beer’s law, but more than
this). This is a text I wrote some years ago as a starter for those not familiar with these
physics.

This is a the discussion of the radiative transfer equation (Schwarzschild equation) and its
consequences and hereafter the verification of the discussion by comparison with some
experimental results.

The Schwarzschild equation was written in 1906, i.e. in a time when quantum physics was
already well developed. Actually, Karl Schwarzschild was one of the leading quantum
physicists of his time.

In a simple one-dimensional form the equation can be written as follows:

dI = −I ∗ k ∗ ρ ∗ dz + B ∗ k ∗ ρ ∗ dz        (1)

with I = radiation entering the gaseous phase, k = mass absorption coefficient, ρ = density
of the absorbing/emitting gas, dz = differential thickness of the gaseous phase, dI =
differential change in radiation, B = Planck function.

The first term of the right-hand side is called sink function, the second term is called
source function.

When you integrate and consider the simplest of all cases, the one-dimensional
radiation in only one direction, e.g. from the surface of a flat earth in the direction of
space, you receive, with transmittance instead of absorption for the first term, the
following:

I = I_o ∗ exp(−k ∗ ρ ∗ z) + B ∗ [1 − exp(−k ∗ ρ ∗ z)]         (2)

This is an equation intuitively easy to understand and very practical to work with.

z = thickness of the gaseous layer. Greenhouse physics, including e.g. the quantification of
forcings or calculation of anthropogenic temperature rises, are based on this
equation.

Now some discussion of the physical meaning and some applications.

1. You will recognize that the sink function of equation 1 has transformed into Lambert-Beers law.

2. Both the Lambert-Beer sink function and the Planck source function contain a kind of weighting factor which are interconnected.

3. Naturally k of a gas is a strong function of the wave number and a weak function of
temperature. B itself depends strongly on wave number and temperature.

4. At the wave numbers where absorption is very high, I_o will rapidly run down to zero .
On the other hand, the Planck function is here fully developed (I = I_o ∗ 0 + B ∗ 1).
As a consequence the gas will radiate according to its temperature at this wave
number.

5. When a black body radiator generating I_o and the gas have equal temperature, I_o equals
B. I will equal I_o, independent of the size of the weighting factor (absorption
coefficient). Example: when 90 percent of I_o is absorbed, B is developed to 90
percent (I = I_o ∗ 0.1 + B ∗ 0.9). That is why in radiation transfer a term like
absorption is saturated or something similar does not make sense. This also explains
why one can measure the temperature of a flame with the aid of a black body
radiator and at appropriate temperature the absorption or emission lines disappear
completely.

6. Note that B is not directly interrelated with absorption, but is a function of gas
temperature (and wave number, naturally). A gas (the atmosphere) has many different
heating mechanisms, e.g. convection, condensation of water, absorption of UV radiation
etc. Absorption of IR radiation as heating mechanism is of importance, but of
comparatively minor importance.

7. The terms absorption and re-emission of IR radiation, frequently used by
meteorologists, do not make physical sense according to equation 2.

8. Also the idea of some kind of photon hopping in the cascade model does not make
physical sense.

9. Naturally equation 2 does not imply that a gas emits like a black body radiator. The
weighting factor is of utmost importance. No absorption at a specific wave length means
also no emission. (I = I_o ∗ 1 + B ∗ (1 − 1)). This, by the way, is also required from
Kirchhoffs law.

10. Equation 2 does not require thermal equilibrium in the sense that the energy of
incoming radiation (I_o) and outgoing radiation (B) are of the same size. In contrast
equation 2 can be and is used, after integration over all wave numbers, to calculate the
net energy flow from bodies of different temperature. What is required is local
thermodynamic equilibrium, which mainly means a steady temperature and mass
gradient in the gaseous phase.

This is always fulfilled in the earth/atmosphere system. (It is, for example, not fulfilled in an agglomeration of stars like an astronomic nebula).

Let me consider and comment various specific cases now in using equation 2.

1. Black body. Absorption is complete at all wave numbers, transmittance zero. According
to equation 2 a black body consequently emits like a Planck radiator. Prediction correct.

2. Grey body. Absorption is of the same size at any wave number and <1, e.g.
0.8. The source function then becomes B*0.8 at all wave numbers. Prediction
correct.

3. Coloured body. I call a gaseous phase, which absorbs and emits at specific lines or
bands, a coloured body. Absorption and emission are not constant over the entire range of
wave numbers, like with black or grey bodies. Because of the weighting factor
[1 − exp(−k ∗ ρ ∗ z)], quantifying absorption and emission, equation 2 describes the
situation correctly.

Experimental example

An interesting and easy to understand example is the zenith measurement in the
atmosphere from the surface of earth. Because the temperature of space is close to zero,
during night I_o becomes zero and equation 2 simplifies to the second term only,
i.e.

I = B ∗ [1 − exp(−k ∗ ρ ∗ z)]                                     (3)

This Figure shows the experimental results

As one would expect, within the limits of experimental error the radiation around 15μm is
fully developed, with the intensity described by the Planck function. With a high value of
the total water vapor optical path (1g/cm²) also the water radiation is well developed
between 17 and 22 μm. This is no longer so at a lower value (0.1g/cm²) of the water optical
path.

I also attach a graph which shows a measurement of Earth’s emission spectrum as measured from a satellite and the corresponding calculated spectrum, calculated by equation 2 already in 1971 and taking all greenhouse gases except ozone into account.

I wonder whether this helps? Again, this is the mathematical basis how radiative transfer and forcings are calculated and anything else is simply wrong physics.

Dr. Hartwig Volz

Fotnot: Någon kanske undrar över Svante Arrhenius beräkningar från 1896. Det är oklart för mig hur han egentligen räknade. Han kom ändå fram till ”det logaritmiska sambandet”, men det bör påpekas att de IR-spektra för både koldioxid och vatten han byggde beräkningarna på saknade all likhet med verkligheten. Så den ursprungliga siffran för ”klimatkänsligheten” från 1896 är inte mycket värd.

Givetvis har den ändå lyfts fram i nutid, eftersom den är ganska nära den IPCC och motsvarande hävdar som trolig. Symptomatiskt knystar man normalt aldrig om att Arrhenius 1906 reviderade sin egen beräkning, och närmast halverade sin uppskattning av nämnda storhet.  Det hela finns bl.a. sammanfattat i sista stycket av avsnittet ”Greenhouse effect” i Wikipedia-artikeln om Arrhenius.

0

Peter S

Professor emeritus i Fysikalisk Kemi vid KTH. Klimatdebattör sedan 2003.