The temperature increases with depth in the ice, and becomes uniform at depths where Fq « 0, i.e. below the layer within which most of the solar radiation is absorbed. The deep ice temperature becomes larger as the solar flux penetrates deeper into the ice. To make this more clear, suppose that kt is constant and Fq = (1 — ag)Sg exp(z/HQ). Then, the deep ice temperature is
This temperature increases without bound as the penetration depth Hq increases. For sufficiently large Hq, the deep ice temperature increases to the melting point, which in that case essentially limits the ice thickness to the solar penetration depth. A precise calculation of the ice thickness is done in Problem ??. Note that much of what we have said about the effect of internal absorption of solar radiation applies equally well to other internal heat sources, notably tidal heating arising from flexing of the ice crust. This heat source may be particularly important in determining the ice crust thickness on Europa.
Exercise 6.9.2 Assume that Tg = 240K and (1 - a9)S9 = 100W/m2. If the decay of solar flux is exponential, how great does the penetration depth have to be in order to bring the deep ice temperature to the freezing point?
The interest in mechanisms for thinning tropical ice in Snowball conditions arises from two challenges facing the Snowball hypothesis. The first is the obvious need to find a way to exit from the globally glaciated state. Accumulation of CO2 can warm the planet, but it is not clear that this process is actually sufficient to deglaciate thick ice, or that the necessary levels of CO2 can be achieved. Thin ice can help make deglaciation easier, especially if solar radiation can penetrate the ice and warm the underlying ocean. The second challenge is that photosynthetic eukaryotes, who are rather fragile creatures in comparison to cyanobacteria, seem to have made it through the Neoproterozoic snowball without any evidence of a major crisis (such creatures were not yet around at the time of the putative Paleoproterozoic snowball, and so pose less of a problem then). Thin ice allows more fractures and leads, which can provide open water refugia. If the ice is thin and clear enough, enough solar radiation may even be able to penetrate the ice to support photosynthetic life beneath the ice layer.
The problem of solar absorption within ice is a radiative transfer problem nearly as challenging as that confronted for clouds. It depends on scattering off of air bubbles and brine pockets, and therefore requires some understanding of the distribution of these. The absorption and scattering is wavenumber dependent,so spectrally resolved radiative transfer should ideally be used, and is is the case for the atmosphere, the spectral absorption features generally lead to non-exponential attenuation of the solar beam. Theses issues are all at the frontier of climate research.
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