Cosmic rays, solar activity and the climate

The equilibrium heat balance of the Earth is maintained by the re-radiation of thermal energy, mainly in the infrared (IR) range, at the same rate as the energy falling on it from the Sun which is mainly in the visible part of the spectrum.

The principle components of the atmosphere are nitrogen and oxygen which are diatomic gases whose symmetric molecules have zero electric dipole moment (from the principle of time reversal invariance). In consequence vibrational modes are not excited and these gases do not absorb in the IR. In contrast, triatomic molecules with asymmetric degrees of freedom, such as carbon dioxide (CO₂) and water vapour (H₂O) allow vibrational modes to be excited by IR radiation. In consequence, these act as strong absorbers of the Earth's IR radiation so that the atmosphere appears almost as a black body to such radiation. To maintain equilibrium the Earth radiates as a black body at an equilibrium temperature of 255 K. With zero CO₂ in the atmosphere, the Earth would be a colder place and less hospitable to life. A small amount of CO₂ allows the atmosphere to act as a blanket warming the Earth to its more hospitable temperature.

The presence of CO₂ and water vapour makes the atmosphere a strong absorber of IR radiation. Direct sunlight (mainly visible wavelengths) penetrates the atmosphere with little absorption and warms the surface of the Earth, the re-radiation from which warms the atmosphere because of this absorption. The atmosphere itself then re-radiates both upwards and downwards from layer to layer. However, at high altitude the pressure becomes small enough that the thickness of atmosphere to outer space is less than one absorption mean free path for IR radiation. At this altitude the IR is radiated to outer space and the temperature (∼255 K) is that at which the energy radiated by the Earth equals the energy falling on it from the sun to maintain equilibrium. At such an altitude water vapour plays little part since it has frozen out at lower altitudes.

The CO₂ is well mixed in the atmosphere. Before industrialization the CO₂ concentration was 280 ppm by volume and has increased to 400 ppm today. The effects of this are illustrated by the simple heuristic model shown in figure 1. Adding more CO₂ increases the altitude at which there is of order one mean free path to outer space i.e. increases the altitude at which re-radiation to space takes place. The thermodynamics of the atmosphere imposes a linear variation of temperature with altitude with fixed slope, dT/dh, known as the lapse rate [1]. This is fixed by the gas laws. Hence simple geometry shows that increasing the altitude of radiation to space together with the fixed lapse rate will cause the surface temperature of the Earth to increase (see figure 1 right hand box).

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The vibrational modes of molecules such as water and CO₂ give rise to absorption of IR radiation in bands of closely spaced lines. Temperature and pressure broadening of the lines cause them to merge together at atmospheric temperatures and pressure. According to the random band model for such absorption [2] the fraction of the radiation transmitted (transmittance T) through a thickness u (at atmospheric pressure) of gas is

$$\begin{eqnarray} &&T=\exp -\frac{S u}{\delta }\Biggl (1+\frac{S u}{\pi \alpha }\Biggr )^{-1/2}=\exp -\tau \end{eqnarray} \tag{ 1 }$$

where S atm⁻¹ cm⁻¹ is a constant known as the line strength, u is the gas thickness at 1 atmospheric pressure (in atm cm), δ is the line spacing in cm⁻¹ and α is the line width in cm⁻¹. In the strong absorption case, Su/πα ≫ 1, so that $\tau \simeq \frac{1}{\delta }(S u \pi \alpha )^{1/2}$.

We now compare the absorptions in the two cases of an atmosphere with CO₂ concentration of a₁ = 280 ppm and one with a concentration of a₂ = 400 ppm. In each case, radiation to outer space from the atmosphere will commence at an altitude at which the air reaches a density such that the mean free path τ is of order unity. Let the air thicknesses at 1 atmosphere pressure for unit absorption mean free path (τ = 1) for these be u₁ and u₂ in each case, so that the absorber thicknesses are a₁u₁ and a₂u₂. It follows that

$$\begin{eqnarray} &&\tau =1=\frac{1}{\delta }(S{a}_{1}{u}_{1}\pi {\alpha }_{1})^{1/2}=\frac{1}{\delta }(S{a}_{2}{u}_{2}\pi {\alpha }_{2})^{1/2} \end{eqnarray} \tag{ 2 }$$

and hence

$$\begin{eqnarray} &&{a}_{1}{u}_{1}{\alpha }_{1}={a}_{2}{u}_{2}{\alpha }_{2}. \end{eqnarray} \tag{ 3 }$$

The mass of a column of air of thickness u at 1 atmosphere pressure above an area dS is ρ_Au dS where ρ_A is the density of the air at 1 atmosphere pressure. The real atmosphere has a variable density so the mass above altitude z is $\int \nolimits _{z}^{\infty }\rho (z)\hspace{0.167em} \mathrm{d}S \hspace{0.167em} \mathrm{d}z$. They have the same mass so for unit area dS = 1 m² (dS cancels)

$$\begin{eqnarray} &&{\rho }_{\mathrm{A}}u=\int \nolimits _{z}^{\infty }\rho (z)\hspace{0.167em} \mathrm{d}z. \end{eqnarray} \tag{ 4 }$$

Assume that the air is an ideal gas with PV = mRT so that density ρ = P/RT and $P={P}_{0}\exp -\frac{z}{H}$ where H is the scale height of the atmosphere. We take the scale height H to be approximately independent of altitude above the tropopause and assume that the atmosphere is at a constant absolute temperature. Such approximations are good to of order 20% accuracy. Hence equation (4) becomes

$$\begin{eqnarray} &&{\rho }_{\mathrm{A}}u=\int \nolimits _{z}^{\infty }\frac{{P}_{0}}{R T}\exp -\frac{z}{H}\hspace{0.167em} \mathrm{d}z \end{eqnarray} \tag{ 5 }$$

i.e. 

$$\begin{eqnarray} &&u=\frac{{P}_{0}H}{{\rho }_{\mathrm{A}}R T}\exp -\frac{z}{H}. \end{eqnarray} \tag{ 6 }$$

Substituting this into equation (3) and assuming that the line width α varies linearly with pressure i.e. $\alpha =k P=k{P}_{0}\exp -\frac{z}{H}$ where k is a constant, we see that

$$\begin{eqnarray} &&{a}_{2}\exp -\frac{2{z}_{2}}{H}={a}_{1}\exp -\frac{2{z}_{1}}{H} \end{eqnarray} \tag{ 7 }$$

where z₁(z₂) is the altitude for radiation to outer space at CO₂ concentration a₁(a₂). Taking logs gives

$$\begin{eqnarray} &&\log {a}_{2}-\frac{2{z}_{2}}{H}=\log {a}_{1}-\frac{2{z}_{1}}{H} \end{eqnarray} \tag{ 8 }$$

i.e. 

$$\begin{eqnarray} &&\frac{{z}_{2}-{z}_{1}}{H}=\frac{\Delta z}{H}=0.5\log \frac{{a}_{2}}{{a}_{1}}=\log \sqrt{\frac{{a}_{2}}{{a}_{1}}}. \end{eqnarray} \tag{ 9 }$$

The conclusions of this simple reductionist model is that the increase of the CO₂ from a₁ = 280 ppm to a₂ = 400 ppm gives Δz = 1.1 km for a scale height (above the tropopause) of 6.35 km. For a lapse rate of 6 K km⁻¹ this gives a temperature rise of 6.6°.

Such a temperature rise is much larger than the observed increase of 0.8° seen since industrialization. To explain this difference feedback mechanisms and all the other complications of the climate need to be invoked. Some of these complications serve to reduce the temperature increase and some to decrease it. Presumably the mechanisms which decrease the rise have a greater effect than those which increase it so that the actual rise in temperature is smaller than that predicted by the simple model.

Note that this is a grossly simplified, reductionist model of the global warming which has occurred since industrialization. In practise, the Earth's climate is much more complicated with many different effects occurring simultaneously. Such effects are modelled in general circulation models (GCM) of the climate which include all known processes (including the one described above). Figure 2 [4] shows the results of these models compared to the measured mean global surface temperature. There is a good agreement between the measurements and the model calculations. However, if the effects of the increase in green house gases are ignored, as in the upper left hand panel of figure 2, then the models do not agree with the data.

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For the picture to be wrong, the effects of the increased CO₂ in the models of the climate must be wrong and the models, if they were correct, ought to produce results similar to the grey curves in the upper left hand panel (a) of figure 2. In that case there would have to be a process which is not included in the models. Since the models should include all known phenomena the process is presumably unknown to climate science at present.

Furthermore, the process or processes must reproduce the warming seen since industrialization rather than cooling. Such a chain of conditions reduces the probability that this is the state of things even though any one of the three conditions is not so improbable.

The Intergovernmental Panel on Climate Change (IPCC) [3] say that 'it is extremely likely that human influence has been the dominant cause of the observed warming since the mid 20th century' due to the observed increase in anthropogenic green house gas concentrations. Here 'extremely likely' is defined as more than 95% probable. Hence it is less than 5% probable that most of the warming is due to natural causes.

Let us suppose that the standard picture is wrong. We can then ask if solar activity or cosmic rays is the new effect which would cause the majority of the observed warming.

Palaeontological effects hinting at a connection between cosmic rays and the Earth's climate in the distant past have been reported. One clear correlation between the cosmic ray rate as measured by the ¹⁴C data from ancient trees and monsoon rainfall from ¹⁸O data is reported by Neff et al [5] from 9500 to 5500 years ago. This correlation is very sensitive to the local wind strength and direction and is thought to indicate a connection between the North Atlantic oscillation and solar activity. This affected the local monsoon rainfall at the epoch but it is unclear if it affects the climate as whole.

Another correlation was noted by Shaviv and Veizer [6]. They took the computations of a model by Shaviv [7] of the changes in the cosmic ray flux as the Solar System moves from one spiral arm of the Galaxy into the inter-arm region. They compared these fluxes with prehistoric temperature variations deduced by Veizer from ¹⁸O data and noted a distinct correlation. This work was criticized by Overholt et al [8] on the grounds that the Sun crossed the Galactic Spiral arms at different and irregular times rather than at the times proposed by Shaviv which had a regular period. Furthermore, Shaviv's model predicts a ratio of cosmic ray intensities in the spiral arm to inter-arm regions of 3:1. This seems excessive given that the scale height of the arms is roughly similar to the inter-arm separation. Hence, using the conventional picture of the Galaxy this ratio would expected to be much smaller and of order 20% [9]. Here, the scale height is the standard deviation of the distribution of cosmic rays about the centre of the spiral arms.

In conclusion, the palaeontological evidence for a correlation between cosmic rays and the global temperature is weak and confused. In addition, in those ancient times many other things were different than today such as the orientation of the Earth's orbit, Earth's atmosphere etc. Hence it is difficult to use such evidence to verify a modern connection between cosmic rays and the climate.

A more modern correlation between the solar modulation of the cosmic ray rate, as measured from neutron monitors, and the low level cloud cover (measured from the ISCCP IR data) [10] during the period 1983–1995 (solar cycle 22) was noted by Svensmark et al [11, 12]. Here the neutron monitor data were used as a proxy for the ionization rate in the atmosphere. Marsh and Svensmark [12] observed a dip in the low level cloud cover between 1983 and 1995 which followed closely the decrease in the neutron monitor data due to the 11 year solar cycle.

They used this correlation to hypothesize that there was a strong connection between the production of clouds and ionization from cosmic rays. They went on to show that, if this were a global phenomenon, it would explain most of the global warming seen in the twentieth century [12, 13]. The cosmic ray rate has been shown to have decreased during the twentieth century [14] so fewer cosmic rays implies less low cloud i.e. more solar radiation penetrating to the surface of the Earth i.e. more warming. However, the correlation was examined by Voiculescu et al [15] who showed that it was not a global phenomenon but that it only existed in certain regions covering a fraction of the globe. Hence the effect can only be used to explain a fraction of the global warming seen in the twentieth century.