G. I. TAYLOR ON EDDY MOTION IN THE ATMOSPHERE. 13 



To find the value of the second term expand [d\Jjdz], a in powers of z z. 



z (v-z) 3 #u, 



Hence (7) becomes 



So far nothing has been said about the magnitude of the disturbance ; (8) is true 

 even if the disturbance be large. Let us now suppose that the height, z z , through 

 which any portion of the fluid has moved from its undisturbed position is of such a 

 magnitude that the change in dUjdz in that height is small compared with dUjdz 

 itself. In that case (8) becomes 



z , / \jj 

 I = p -7-7^ w' (ZQ-Z) dx dy. 



tlz\ J J 



The rate at which ie-momentum leaves a layer of thickness Sz is therefore 



The effect of the disturbance is therefore to reduce the .(--momentum of a horizontal 



asTT 

 layer of thickness Sz at rate p .," Sz x [average value of ic' (z z)~\ per unit area. 



cz" 



The same effect would be produced on a layer of thickness Sz by a viscosity equal 

 to p x average value of w' (z z ) if the motion had not been disturbed. If, some time 

 after the disturbance has set in, all the air at any level mixes, no change will take 

 place in the average momentum of the layer. Deviations from the mean velocity of 

 the layer will disappear, and the velocity will be horizontal once more and uniform 

 over any horizontal layer. When, therefore, we wish to consider the disturbed motion 

 of layers of air, we can take account of the eddies by introducing a coefficient of eddy 

 viscosity equal to />x average value of IP' (z z a ), and supposing that the motion is 

 steady, z z is the height through which air has moved since the last mixture took 

 place. 



As before in the case of the eddy conduction of heat, we can express the average 

 value of w' (z z a ) in the form jt(wd), where d is the average height through which an 

 eddy moves before mixing with its surroundings, and w roughly represents the average 

 vertical velocity in places where w' is positive. It will be noticed that the value we 

 have obtained for eddy-viscosity is the same as that which we would have obtained 

 if we had neglected variations in pressure over a horizontal plane, and had assumed 

 that air in disturbed motion conserves the momentum of the layer from which it 

 originated till it mixes with its new surroundings, just as it conserves its potential 



