a unit area of the lower surface of the ice sheet. This heat budget 

 equation must satisfy Fourier's heat transfer equations! 



2 



L s £ Z—J. in the ice (o < X < £ ) (1) 



dt ' 





ond _Il = 0C2 ^~ y in the water (£< X ) , (2) 



dt dx d 



where s • 



•.-■-. T]_ = temperature in the ice, 



Tj = temperature in the water, 



ki 



OCi = — - — - is thermal diffusivity in ice, 



k ' ' 



^ „ _J1_&_ is thermal diffusivity in water, 



CL 2 ~ c 2 £ 2 



t = time 



ki - thermal conductivity in ice, 



k2 = thermal conductivity in water, 



C][ = specific heat of ice, 



C2 - specific heat of water, 

 p\~ density of ice, and 

 p2 - density of water e 



The temperature of the boundary surface of ice and water (at x - f ) 

 must always be 0°C (under this simplified formulation) and there will be 

 continual formation of new ice. If the thickness increased by d£in time 

 dt, there will be set free for each unit of area an amount of heat 



Q = L A<, (3) 



where L is the latent heat of fusion. This heat must escape upward by 

 conduction through the ice, and in addition heat must be carried away 

 from the water below, so that the total amount of heat that flows out- 

 ward through a unit area of the lower surface of the ice sheet is 



d«V. (4*4 :* f w 



of this amount the quantity 



X ~i 



0x 'x=£ 



