In Earth's middle atmosphere, above the ionization peak due to cosmic rays, the atmospheric conductivity nearly exponentially increases with altitude [Holzworth, 1995]. The main reason for this phenomenon is the presence of the collision frequency in the denominator of the expression for conductivity given above. Therefore, the mean free path of the ions increases as the neutral number density drops exponentially with altitude, and the collision frequency decreases, which more than makes up for the drop off in ionization rate with altitude.
As a simple example, consider an isothermal atmosphere in hydrostatic equilibrium. Typically, the change in ion number density, n, with time is given by
where S is the ion pair production rate, a is the recombination coefficient, ß is the attachment coefficient, and NA is the number density of aerosols. Ignoring the effect of aerosols and solving for a steady state solution,
Since S decreases with atmospheric number density above the ionization peak,
and the ion number density
Now the collision frequency must be proportional to the ion number density and the neutral number density, so
Finally,
So this result states that the atmospheric conductivity should exponentially increase with altitude with the same scale height, H, with which the atmospheric number density decreases. However, Gringel et al. [1986] showed that the mobility, k, is inversely proportional to the neutral number density, nn. Therefore,
and conductivity increases more slowly. Regardless of its exact form, both methods result in an exponentially increasing atmospheric conductivity.
These estimates are in actuality upper limits on conductivity. In this simple treatment, we have neglected the effects of aerosols. Aerosols essentially decrease conductivity by vastly increasing the mass of the charge carriers. Typically aerosols such as dust are found in greater quantities near the surface, and, therefore, lower the atmospheric conductivity near the surface.
Briefly, we can estimate the total resistance of the Martian atmosphere if we make some simplifying assumptions. First, assume that the atmospheric conductivity at the surface is 10-11 S/m as given by Grard [1995] and exponentially increases with altitude only due to ion pair production by cosmic rays. Secondly, assume that the scale height of the atmospheric conductivity is the same as the neutral atmosphere scale height, that is, 10 km. Finally, assume that the conducting ionosphere occurs at an altitude of 100 km. Then
columnar resistance = P = § p dz = Hõ[exp(-zo/H) - exp(-z/H)] ~ 1015 Ohms m2
Therefore, the global resistance of the Martian atmosphere appears to be on the order of 10 Ohms. Aerosols may increase the specific resistance of the lower atmosphere, and , thus, increase this value. This "back of the envelope" estimate is probably within an order of magnitude or so of the true value.
So far we have shown that the Martian atmosphere should have a finitely conducting atmosphere as a result of galactic cosmic rays, radioactive crustal material, and ionizing solar ultraviolet radiation which reaches the surface. The total atmospheric conductivity can vary on daily (due to solar ultraviolet radiation), yearly (due to global atmospheric pressure variations), and 11 year (due to the solar cycle) cycles. Also, the atmosphere is located between the highly conducting surface and ionosphere. Finally, the atmospheric conductivity increases exponentially with altitude.
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Introduction and Sources of Atmospheric Conductivity
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