## Mass balance and melting

When the surface consists of a solid ice which can undergo melting, the surface balance works in a somewhat different way once the ground was warmed to the melting temperature. We can no longer solve Eq. 6.33 for Tg, since Tg cannot rise above the melting temperature so long as there is any ice left to melt. Instead, we compute the surface residual at the melting temperature Tf, i.e.

Fnet (Tsa, Tf ) = Frad(Tsa,Tf ) - Fsens(Tsa,Tf ) - FL(Tsa,Tf ) (6.36)

If Fnet(Tsa,Tf) > 0, then the energy flux Fnet is available for melting. In that case, the mass melted per unit time per unit area is given by Fnet/Lf, where Lf is the latent heat of fusion. Often this is converted to liquid equivalent depth per unit time by dividing by the density of the liquid phase. Expressed that way, the rate corresponds to the rate of growth of liquid layer that would be caused by the melting, if the liquid did not run off to some other place. Melting is a very powerful means of ablation of ice, be it mountain glacier, ice sheet or sea ice. The latent heat of fusion is much smaller than the latent heat of sublimation, so a given amount of energy can ice into rapidly movable form much more rapidly by melting than by sublimation. For example, the ratio of latent heats is 0.118, so a given amount of energy can get rid of 8.5 times as much ice by melting as it can by sublimation. Sublimation always carries the vapor away from the ice surface, but for melt to actually become realized ablation, the meltwater needs to go away somewhere. It may flow to the base of a glacier through an abyss called a moulin, or it may run away to a melt pond and form a temporary lake. It may also percolate into the snow and re-freeze, releasing latent heat int the process. In that case the melting actually constitutes an energy transport mechanism rather than a true ablation.

4000 3500 3000 2500 2000 1500

m ig

 :---200 W/m**2, neutral : --•--300 W/m**2, neutral :^100 W/m**2, stable : -^-200 W/m**2, stable : -»-300 W/m**2, stable / / / / / L / _____- r 1 m 1 KT 1 1 aiJri 1 i mJfT\ 1 1 1 1 1 1 1 1 1 1 1 1 -

260 265 270 275 280 285 Air Temperature (K)

Figure 6.4: Melting rate in liquid water equivalent as a function of air temperature. Results are given for three different values of absorbed solar radiation. The calculations were performed with U = 5m/s, z* = .00033m, and hsa = .8.

Figure 6.4 shows the melting rate as a function of air temperature for three different values of surface absorbed solar radiation (see the caption for the rest of the conditions). To illustrate the importance of stable surface layer physics, each calculation is done in two ways: using Monin-Obukhov theory and using neutral surface layer theory. Note that melting can begin even when the air temperature itself is below freezing; this is simply because the energy balance allows the ground to be warmer than the air, if there is a sufficient supply of solar absorption. Note also that even for a fixed air temperature, the melting increases with solar absorption. This shows that the ablation of a glacier can be affected by the solar radiation, even if air temperature does not change. Various processes can change the absorbed solar radiation, among them clouds, changes in the Earth's orbital parameters (see Section 7), and the fact that fresh snow is more reflective than old ice. For any given amount of absorbed solar radiation, the melting rate increases dramatically as air temperature is increased, because this increases the delivery of heat to the surface by sensible heat flux and infrared heat flux. The development of a stable surface layer when the air temperature gets large sharply limits the increase of melting. In this regime, increasing the wind speed very strongly increases melting, since higher winds favor lower Richardson numbers and higher drag coefficients.

If some geological indicator of past glacial behavior tells us that a mountain glacier was more extensive at some particular time in the past, should we conclude that it was colder then or that it simply snowed more at that time? Similarly, if we see mountain glaciers retreating worldwide at present, should we take that as an indication of a warming climate, or of a reduction in snowfall? The sensitivity of melting rate to temperature has a great bearing on this question. Consider the case with 200 W/m2 of absorbed solar radiation in Figure 6.4. Increasing air temperature from 270K to 274K increases the melting rate from zero to 650mm/mo. To offset such an increase in melting, the snow accumulation rate would thus have to increase by 650mm/mo. This is a very substantial increase. To put it in perspective, the required increase in precipitation is over three times the maximum monthly precipitation rate observed in the past few years in central Iceland - which is a very snowy place. Recall, too, that the melting rate we have stated is in liquid water equivalent depth. Snow has low density, and the actual thickness of the corresponding snow accumulation would be 6.5 meters per month or more. It is not impossible for glacier extent to be affected by precipitation, but in situations where melting occurs, it takes a truly enormous change in precipitation to have the same impact as a rather modest change in temperature.

Melting is a very nonlinear process, which acts as a kind of rectifier capable of turning a fluctuating temperature signal into a secular growth or decay of a mountain glacier or continental ice sheet. Melting turns on when the air temperature approaches freezing, and greatly accelerates as temperature becomes warmer. On the other hand, the melting turns off sharply when the air temperature falls much below freezing. In consequence, ablation of ice cares little about just how cold it gets during the depth of winter, but a great deal about the length and warmth of the summertime melt season. This makes the growth and decay of midlatitude and polar mountain glaciers or continental ice sheets very sensitive to what is going on during the melt season.