34 



DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



Instead of vector-tubes we shall in the two-dimensional field get vector-bands 

 bordered by vector-lines. Transport must be referred to lines instead of to surfaces. 

 A n being the component of the vector normal to the curve s, we get 



(a) transport through curve s = J Ads 



Instead of surfaces we get curves of equal transport. The solenoidal condition is 

 expressed by 



(b) JAJs = o 



the integral being extended over a closed curve. When condition (b) is fulfilled, 

 the curves of equal transport can be left out, and the field be represented by bands 

 of equal transport, most conveniently of unit transport. If unit bands be used, the 

 numerical value of the vector is given by the reciprocal of the number expressing 

 the breadth of the band. 



If a unit band gets infinitely narrow, the solenoidal vector will be infinite. 

 Excluding infinite values, we get this important result: 



In the two-dimensional solenoidal field the lines of flow can not touch each other. 



> 



> > 



> 



> 



> 



> 



> 



> > 



> 



* * 



> 



> > 



> 



A > 



Fig. 35 Translation-field. 



Fig. 36. Vector-components of a plane deformation-field. 



120. Examples of Two-Dimensional Solenoidal Fields. We shall consider first 

 the case that the two-dimensional field is solenoidal. Let the surface containing 

 the field be plane. The simplest field will be that of a vector having the same 

 direction and the same intensity at all points of the plane. If the vector is velocity 

 the field will represent simple motion of translation . The field evidently fulfils the 

 solenoidal condition. It can be represented geometrically by a set of parallel and 

 equidistant vector- lines (fig. 35). 



Let us next consider a field where the component A x parallel to the axis x is 

 proportional to x, and the component A y parallel to the axis y is proportional to y: 



(a) A x = ax A y = by 



