Derive a model for the formation of sea ice in terms of ice thickness, h, and the air-sea temperature difference ∆T.
Consider the formation of sea ice at quasi-equilibrium with a winter atmosphere. If one assumes that the formation of sea ice at the bottom of an ice pack is given by
W = ρi Li dh/dt
where ρi is the density of sea ice, Li is the latent heat (see below) and dh/dt is the rate of ice formation in terms of floe thickness, h. Now if ice is being steadily formed in balance with the conduction of heat through the ice above, the balance of heat released by the freezing process is matched by the flux of heat upward
through the ice,
Fi = - ki dT/dz.
This is just Fick’s law where ki is the thermal conductivity of sea ice as given by the formulas below. Then if this flux balances the latent heat release
Fi – W = 0, or after substitution
- ki dT/dz = ρi Li dh/dt.
Now if the temperature gradient through the ice floe is approximated by dT/dz = (To – Tf)/h,
where To (=To(t)) is the atmospheric temperature and Tf is the freezing point of sea water one can write an equation for the change in ice thickness. Note here the assumption is that there is a constant heat flux through the ice for a given thickness. What happens as to the heat flux as ice accumulates?
a) Write down the equation and solve it in terms of To(t) the winter air temperature. Note: This will involve and integral over time.
b) Plot this for the case of constant atmospheric temperature and discuss what is happening at longer times
As given in the question that
Part (b): The (To - Tf) is negative, therefore the left hand side will become positive. Thus the derived equation is equation of parabola. So at longer times the height will become constant as shown in figure.
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