G. I. TAYLOE ON EDDY MOTION IN THE ATMOSPHERE. 25 



has removed it, increases with time. This evidently includes the case of exponentially 

 unstable simple harmonic waves. 



In unstable motion therefore =- 1 1 (z a -zf dx dy must be positive. 



Hence the rate at which z-momentum enters the slab A is positive or negative 

 according as d 2 \J/dz 2 is positive or negative. In an unstable disturbance of a fluid for 

 which d 21 U/dz 2 is everywhere positive the momentum of every layer must increase. But 

 if there is perfect slipping at the boundaries no momentum can be communicated by 

 them. Hence, as there is no other possible source from which the momentum can be 

 derived, instability cannot possibly occur. The argument applies equally well if 

 d 2 U/dz 2 is everywhere negative. Lord RAYLEIGH'S result is therefore proved for a 

 generalised disturbance. In a case where d 2 TJ/dz 2 changes sign at some point in the 

 fluid any disturbance reduces the x-momentum in a layer where d 2 U/dz 2 is negative, 

 while it increases the .r-momentxim in layers in which d 2 U/dz 2 is positive. A type of 

 disturbance which removes ^-momentum from places where d 2 \J/dz 2 is negative and 

 replaces it in regions where d-U/dz 2 is positive, so that there is no necessity for the 

 boundaries to contribute, may be unstable. 



Now consider what modifications must be made in the conditions in order that 

 instability may be possible in the case where d 2 \J/dz~ is of the same sign throughout 

 (say negative). Suppose that instability is set up so that ./--momentum flows outwards 

 from the central regions as the disturbance increases. The amount of ^-momentum 

 crossing outwards towards the walls through a plane perpendicular to the axis of z, 

 increases as the walls are approached. In order that instability may be set up this 

 momentum must be absorbed by the walls. There seems to be no particular reason 

 why an infinitesimal amount of viscosity should not cause a finite amount of momentum 

 to be absorbed by the walls. 



In connection with this two points should be noticed. Firstly, the momentum is 

 only communicated to the walls while the disturbance is being produced. The time 

 necessary to produce" a given .disturbance may increase as the viscosity diminishes. 

 Experimental evidence, however, does not suggest that this is the case. 



The second point is suggested by the conclusion arrived at on pp. 11-22, that a 

 very large amount of momentum is communicated by means of eddies from the 

 atmosphere to the ground. This momentum must ultimately pass from the eddies 

 to the ground by means of the almost infinitesimal viscosity of the air. The actual 

 value of the viscosity of the air does not affect the rate at which momentum is 

 communicated to the ground, although it is the agent by means of which the 

 transference is effected. 



In any case it is obvious that there is a finite difference, in regard to slipping at 

 the walls, between a perfectly inviscid fluid and one which has an infinitestimal 

 viscosity. The distribution of velocity acquired by a viscous fluid flowing between 



VOL. CCXV. A. E 



