Energy Balance of the Lake and Waterrenewal Time

In the absence of a lake, the geothermal heat would be transmitted to the glacier by simple conduction. The lake collects the geothermal heat that it finally restores to the glacier. Neglecting the energy exchanges linked to ice advection and export and the other possible losses within the lake, we can state that the geothermal heat will be totally used for melting the glacier ice and then fully transmitted to the glacier as accretion ice is formed. Mass balance will be in steady state when the volume of glacier ice supplying the lake is compensated by the export of an equivalent volume of accreted ice out of the lake.

The latent heat of accretion ice is partly released into the glacier by the freezing of the interstitial water in the accretion area and partly released into the lake water as the latent heat of frazil formation. As the lake is isothermal, and if we assume the accretion phenomenon requires a proportion of frazil that remains constant with time, the latent heat from frazil formation must be diffused out of the lake. From the interface properties (see Sect. 7.2.2 and Fig. 7.10) this latent heat can only be diffused through the glacier in the melting area.

The total energy available from geothermal heat is given by the product of the heat flux (G) by the lake surface area (S). The total ice volume (Vg) that melts or accretes is:

Using a lake surface area S of ~ 14 000 km2 and a geothermal heat flux G of 46mW/m2, a likely value for the Vostok area (Siegert and Dowdeswell, 1996) and deduced hereafter (see Table 7.2), and Lf, the latent heat for water or ice and corresponding to a heat flux of 9.78mW/m2 per mm of ice, Vg is ~ 70 x 10+6 m3/a. This corresponds to a fresh-water production of about ~ 2.2m3/s, implying a flow of about 2200m3/s for the intermediate water moving into the melting area.

The renewal time of the lake water (Tr) can now be deduced. If "H" is the mean water depth, we obtain:

Giving ~ 80000 years for Lake Vostok (H = 400m).

The mean melting (Vmmi) and mean freezing rate (Vmfr) over the entire lake are:

With a geothermal heat flux of 46mW/m2, about 4.75mm of glacier ice could be melted annually. Conversely the formation of 4.75mm of accretion ice per year releases an equivalent heat flux of 46 mW/m2 into the glacier. Accretion and melting area of the lake are physically separated: one corresponds to the deep part of the glacier lake interface, the other to the upper level of the lake. The effective melting rate (Vml) and accretion rate (Vfr) should depend on their respective area. Adopting "m" as the lake part for melting, or more precisely the area without accretion, and "(1 — m)" as the part for accretion, the melting and freezing rates can be written:

When the melting area and freezing area are equivalent (m = 0.5), both the accretion rate and melting rate are 9.5mm/a. This value appears consistent with the rate we deduced for accretion ice II (see Sect. 7.1.3).

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