THEORY OF CYCLONES VON BEZOLD 363 



other hand, the generalization of the problem offers very con- 

 siderable difficulties. 



Nearly circular movements are to be observed, for instance, in the 

 case of tornadoes and waterspouts. 



If now we apply the considerations just introduced, to these latter 

 cases, we find that the critical surfaces have extraordinarily large 

 inclination near the axes and hence there must be present enormous 

 [counteracting barometric] gradients if these circular movements 

 are not to turn into centrifugal movements. For instance, from 

 the approximate formula (10) for r = 10 and v = 30 (or for a 

 wind velocity of 30 meters per second at a distance of 10 meters from 

 the axis) we find a gradient of 90,000, i. e., a diminution of pressure 

 of 0.81 mm for 1 meter of approach to the axis. 



Under the given assumptions, the inclination of the critical sur- 

 face will be about 84 . 



If then such motions are maintained at the cost of the energy that 

 is gained at other points, then great rarefactions of the air must 

 take place in the neighborhood of the axis that can be computed 

 when definite assumptions are made as to the diminution of the 

 velocity with the distance from the axis. 



Such computations have already been made for the tornado by 

 William Ferrel 14 which he considered as a simple centered whirl and 

 for which he represented in a diagram the form of the isobaric sur- 

 faces which, under the assumption of the given velocities, are the 

 same as our critical surfaces. 



It would therefore scarcely be necessary here again to touch on 

 this point, but that it seems to me that in one respect different con- 

 clusions are to be drawn from these studies than those drawn by 

 that investigator. 



The enormous gradients that must prevail in a very thin mantle 

 enclosing the axis of a tornado, if there are to be no centrifugal 

 movements, make it very improbable that any air penetrates this 

 mantle from without and moves inward to the axis, or that any 

 centripetal movements occur. 



In order to bring about such movements the isobaric surfaces 

 actually existing must be more strongly inclined than the critical 

 surfaces, or, which comes to the same thing, the effective gradients 

 must be still greater than the critical gradients, which themselves 

 already have such extraordinary large values. 



But if there be no continuous flow of air inward toward the axis 



14 Sprung: Lehrbuch, p. 224. 



