142 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



The interpretation of the chart will be easiest in the case where the given lower 

 limiting surface of the sheet is a surface of flow. The transport in each tube being 

 constant, we conclude by the solenoidal condition that the upper limiting surface 

 will also be a surface of flow. 



We have thus obtained a method of constructing the topography of one surface 

 of flow relatively to another, and thus of arriving at those representations of vertical 

 motions which are illustrated by the figures 43 a and b and 45 A and b. 



If the lower limiting surface of the sheet is not a surface of flow, the upper 

 surface (the topography of which we have determined) will not be one either. But 

 still it will characterize that part of the vertical motion which arises as a consequence 

 of the horizontal motion within the sheet. 



(C) Vertical component of specific momentum. If we wish to find the vertical 

 component of specific momentum, we have simply to use the solenoidal condition 

 in its differential form. By equation (/) of section 1 7 1 , we have 



(0 f'=-*v,v 



or, when we multiply by dz, 



(g) dV 2 = (-div 2 V)dz 



By this equation we can draw a chart of the increase d V z of vertical specific momen- 

 tum within a sheet of any thickness dz within which we know the horizontal specific 

 momentum V. 



As in the preceding cases, it will be convenient to begin with the case of a sheet 

 of unit thickness dz = 1 . The corresponding increase of vertical specific momentum 

 will be 

 (A) dV liZ = -div 2 V 



From the given chart which represents the field of the horizontal vector V we derive 

 the field of the divergence div 2 V, using the method developed in section 170. This 

 field of divergence will, after change of sign, represent the increase dV 1>z of vertical 

 specific momentum from bottom to top in a sheet of unit thickness. 



In order to get the increase d V z for a sheet of any thickness we have to perform 

 the multiplication by the thickness of dz. If dz is constant, this will simply be a 

 change of the interval between the curves for constant values of dV liZ . In the 

 general case where dz is variable, and is represented by a chart which gives the 

 topography of the upper limiting surface of the sheet relatively to the lower, we 

 have to perform the graphical multiplication of the fields of dV iiZ and of dz. 



187. Change of Variables. The horizontal mass-transport was given by the 

 formula 



T = Vdndz 



It is the dz appearing here which brings in the vertical dimension in the formulae of 

 the preceding section and allows us to describe the motion in reference to this 

 dimension. 



