,- , ry.-j.'i , Yajnik and Lieber 



a=(V V -V V ),/3=V /2 = (V + V ) '2. (32) 



\x>: yy xy yx''*^ zz ^xx yy ' ^ ' 



Since the invariants M and N are given by 



M=V V +V V +V V -V V =a-4/32, 



XX yy yy zz zz xx xy yx '^ 



N=V (V V -V V) = 2a/3, 



zz ^ XX yy xy yx' '^ ' 



(33) 



the whirlicity can be expressed as 



4M^ + 27N2 = 4 (a - /3^) [(a - /3^) + 9/32] 2 . (34) 



Hence the whirlicity is positive if, and only if, the swirl a- /3^ is positive and 

 the theorem follows. 



Since the whirlicity is an invariant under Galilean transformations, an eddy 

 in one inertial frame is also an eddy in any other. 



Unlike vorticity, the whirlicity is nonlinear. Whereas the superposition of 

 two irrotational flows always leads to an irrotational flow, flows with zero 

 whirlicity behave differently. Two plane Couette flows in perpendicular direc- 

 tions (u = Ay, V = w = 0; V = - Ax, u = w = ), when superposed, lead to rigid 

 body rotation. It is to be expected then that an unstable flow with a wavelike 

 disturbance may have eddies, although neither the streamlines of the flow nor 

 those of the disturbance may display any whirling property. Many flows and 

 disturbances analyzed in the hydrodynamic stability theory have this property. 



It is known that the appearance of eddies transforms the flow in a marked 

 way and that the dynamic and thermodynamic consequences of the transition are 

 unmistakable, whereas the picture of concentrated vorticity does not give any 

 clue to the difference in kind between the flows with eddies and those without 

 them (the definition given above fills this lack). Let the z axis be chosen as in 

 the proof of Theorem in. Now if the whirlicity is zero or negative at a point, 

 there is a line element at the point in the xy plane with zero angular velocity, 

 and X the axis can be chosen in its direction. Then Eqs. (8) and (14) imply that 

 Bv/3x, Bw/Bx , and 3w/3y are zero, so that the convective part of acceleration 

 can be written in the canonical form: 



uBu/'Bx + vBu/By + w3u/3z 



vBv/3y -(- w3v/Bz 

 wBw/Bz. 



If, on the other hand, we similarly examine a particle in an eddy, the form 

 for the convective part of acceleration would be 



450 



