Piacsek 



disturbances were free to propagate. They have found that the value of the criti- 

 cal Rayleigh number in this case is 6.5 times smaller than the value obtained in 

 the usual case. Veronis (1963) has also studied penetrative convection in a case 

 where an unstable layer of liquid was bounded above by a stable region. He 

 found that a finite amplitude instability sets in at values of the Rayleigh number 

 below critical (given by the linear theory). He argued that any finite amplitude 

 motion which mixes liquid above and below the upper boundary of the unstable 

 layer would create a deeper layer that would be gravitationally more unstable 

 than if it were in a conduction state only. 



In view of the two-dimensional nature of the penetrating sheets, the numeri- 

 cal experiments were confined to two-dimensional flows only. The relevant 

 system of equations were left in dimensional form. They may be obtained from 

 Eqs. (7), (4), (9), and (5a) by neglecting curvature and putting fi = v = 0: 



:V2T 



(20) 



Bi/; 



b7 (21) 



(22) 



where we used Eqs. (22) to put the left-hand side of Eqs. (20) into conservative 

 form. The volume of the fluid is assumed to be contained between the surfaces 

 X = and L, and z = and D. The results presented here were designed for 

 the following cases: 



1. An initially homogeneous rectangular volume of fluid is subject to con- 

 stant heat loss at its top surface. The dimensions of the volume are so chosen 

 that a semi- infinite region is simulated: the lateral dimensions are such as to 

 allow two to four cells to develop, and the vertical dimensions are anywhere 

 from 10 to 50 times the boundary layer thickness at the cooling surface. All 

 the boundaries are assumed to be "frictionless lids" (i.e., at which only the 

 normal velocity vanishes), and all except the top surface are assumed to be 

 thermal insulators. Based on the results of laboratory experiments, cooling 

 rates are so chosen that the thickness of the thermal boundary layer at the top 

 is small compared to the total depth of the fluid. 



2. A situation similar to that described above, but with the bottom surface 

 cooled at the same rate as the top, to set up a stable layer near the bottom. 



The procedure was to solve the equation of thermal conduction until the 

 cooling effect penetrated to a depth judged to be sufficient to support convection. 

 Then the temperature field had a perturbation added to it of the form 



764 



