There are many circumstances in which one would like to know the thickness attained by a layer of ice floating on an ocean once it comes into a state of thermal equilibrium. This problem is relevant to the state of the sea ice cover which forms in the polar regions on Earth today and in other icy climates. It is also relevant to the thick-ice regimes prevailing on a globally glaciated Snowball Earth. Another application is the determination of the ice-crust thickness for icy moons such as Europa, which consist of a crust of water ice floating on a deep brine ocean. In this section, we'll lay out some elementary models which shed light on the determination of ice thickness. While the physics set forth here would apply equally well to the freezing of any liquid whose solid form has lower density than the liquid form, in practice, the condition that the ice floats for the most part restricts the applicability to water. An important exception to this is the determination of the thickness of the crust of rocky planets, which can be viewed a a form of "rock-ice" supported by a more fluid interior.
In order to proceed we need to know a bit about the flux of heat through an immobile solid. Heat is conducted through such a substance through collisions of molecules with one another, which propagate information about changes in temperature in one part of the body to the remainder of the body. Both experiment and theory show that in most circumstances, the flux of heat is proportional to the temperature gradient, and the flux is in the direction opposite to the gradient. In other words, the heat flows in such a way a to try to wipe out temperature gradients and make the body isothermal. The constant of proportionality is called the thermal conductivity, and we shall call it kt . The thermal conductivity is determined by molecular properties of the solid, the specific heat, and the density. Low density substances generally have lower thermal conductivity, since there is less mass available to transmit heat. The thermal conductivity of various substances will be discussed at greater length in Chapter 7. For now, it will suffice to know that for water ice kt « 2.24W/m • K, and that of new fluffy snow is kt « .08W/m • K.
If there are no internal sources or sinks of heat within the ice layer, then the heat flux must be constant once the system reaches equilibrium. The value of the flux is set by the heat flux delivered to the bottom of the ice. In some cases, for example the heat flux through a quiescent ocean for a globally glaciated Snowball Earth, the flux would be just the geothermal heat flux escaping the interior of the planet. In other cases, for example in the case of sea ice or shelf ice abutting an open ocean, the flux would be the much larger value delivered by dynamic ocean heat transport - the delivery of heat in the form of relatively warm water by ocean currents. Regardless of the source, we will call this flux Fi. In this section, we'll denote the temperature profile within the ice as T(z), with T(0) being the temperature of the ice or snow surface (also called Tg) and T(—h) being the temperature at the base of the ice, where h is the ice thickness. Then, the constant flux requirement yields the following differential equation for the temperature profile:
Note that this equation remains valid even if the thermal conductivity varies with depth. For example, snow has much smaller conductivity than ice, owing to it's low density ahd the immobilization of air in the pore space. Hence,if a layer of ice were blanketed with snow, the temperature gradient within the snow layer would be much steeper than the temperature gradient within the rest of the ice. Suppose that the ice layer has not frozen all the way to the rock, so that the ice is floating on a layer of the same liquid (generally water) which freezes to make the ice. Where the ice is in contact with the liquid, the temperature of the ice must equal the freezing point of the liquid, which we will call Tf. Thus, the bottom boundary condition on temperature is T(—h) = Tf. Note that the freezing point of sea water or any other brine is lower than the freezing point of pure water.
The energy balance at the ice surface impose another condition on the temperature profile. This energy balance is identical to the surface energy budget given in Eq. 6.33, except that one must add in the contribution from the heat flux through the ice. Thus, at z = 0 we require
The internal flux is usually small, and makes a negligible contribution to the surface balance. In most cases, we can drop the term and compute the ice surface temperature as if it were not there. This equation determines Tg (which is now the ice surface temperature, called T(0)) as before, and the inclusion of F-i would only make the ice surface temperature ever so slightly warmer than it otherwise would have been without heat diffusion through the ice.
Sublimation takes away mass as well as heat, and in general this mass loss must be taken into account when formulating the conditions for equilibrium thickness. Let's assume first that there is no net mass loss or gain at the surface. This could be because the temperature is so low that sublimation is negligible, or it could be because all the sublimated mass precipitates back out onto the surface locally. In this case, since there is no mass loss at the surface, there is no freezing at the base of the ice once equilibrium is attained, and hence no latent heat release there, and the only heat flux that needs to escape through the ice is Fj. If we divide Eq. 6.39 by kT and integrate over the ice layer, we find
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