SECT. 1] 



THE FLOW OF HEAT THROUGH THE FLOOR OF THE OCEAN 



221 



purely radial. The temperature, 0, of the probe at a time t after entry may then 

 be shown to be (Bullard, 1954) 



d = di+doF{a,T), 

 where di and ^o are constants and 



4a r°° 



F{cc, r) = - ' 



exp ( —TX^) dx 



tt'^ Jo x[{xYq{x) -aYi{x)Y + {xJo{x) -aJi{x)Y] 

 Here r is a non-dimensional time given by 



(1) 



(2) 



0.3 



o 

 o 



oT 



o 



§ 0.2 



0) 



2 0.1 



E 







0.05 



0.10 



0.15 



Fig. 3. Correction for lack of equilibrium. The temperature difference between the top 

 and bottom of the probe is plotted against the theoretical function F of (1). The 

 intercept of the straight line fitted to the points gives the equilibrium temperature 

 that would be attained if the probe were left in the sediment for an indefinitely long 

 time. (After Bullard et al., 1956, fig. 3.) 



where k is the thermometric conductivity of the sediment and a is the radius 

 of the probe; r is about O.U if t is in minutes and a is 1.3 cm. a is given by 



where p and o- are the density and specific heat of the sediment and m is the 

 water equivalent of a 1-cm length of the probe (that is, the mass of water that 

 would require the same amount of heat to raise its temperature 1°C as a 1-cm 

 length of the probe does), a is usually about 2. The T's and J's in (2) are Bessel 

 functions. 



The function F{a, r) has been tabulated by Bullard (1954) and by Jaeger 

 (1956). It is 1 for t zero, decreases rapidly at first as t increases and for large t 

 approaches zero as mj4:rrkt. The observed temperatures may be fitted to (1) 

 by least squares. The fit is usually excellent as is shown in Fig. 3 ; the constants 

 of the best fitting line give estimates of di and ^o- (^i+ ^o) is the temperature 



