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



disturbance has arisen from dynamical instability, or from disturbances transmitted 

 from the surface of the earth. The rate at which x-momentum leaves a layer of 

 thickness Sz is 



But U, is constant over the plane xy and since there is no resultant flow of fluid 

 across a horizontal plane pU z w'dxdy = U z \\ pw'dx'dy = 0. 



Hence, if we write I for the value of the expression in square brackets 

 I == j| jj^lL+O w'dxdy = p^u'w'dx dy 



The equation of continuity is 



<-\ / i~~ f 



fill fltii 



||+|^=0 (5) 



Since the motion is confined to two dimensions 



^ / T T /\ ^ / 



rt I I J 1 o/ I f' J/i 



-- = twice the vorticity of the fluid at the point (x, y, z). 



(J% (J JC 



And since every portion of the fluid retains its vorticity throughout the motion, this 

 must be equal to twice the vorticity which the fluid at the point (x, y, z) had before 

 the disturbance set in. This is equal to the value of (dUJdz) at the height, z u ,* of the 

 layer from which the fluid at the point (x, y, z) originated. If this value be expressed 

 by the symbol [dU,/dz~]. a we see that the dynamical equations of fluid motion lead to 

 the equation 



d\J, Su' {)?</ _ 



~ "I" ~^ ^ = 



dz 



^ f ^N / 



Substituting in (4) the values of ^- and ~ given by (5) and (6), we find 



The first term integrates and vanishes when a large area is considered ; but the 

 second term does not vanish. 



* is evidently a function of x and z when the motion is confined to two dimensions. 



