Fundamental Definition of Divergence


Horizontal Divergence is defined as the fractional rate of change of cross-sectional area of an air column (ratio of the area change in a unit time to the original area).  This is known as the fundamental definition of divergence.






Horizontal Divergence in Rectangular Coordinates


Consider an air column with a rectangular-shaped cross-sectional area that experiences a change in time (Fig. 1).



The dimensions of the original air column are x and y.




The finite difference version of   dA/dt,  after substituting (2),  is






Use the chain rule to expand right hand term.






but x/t = u and y/t = v.  Substituted into equation (4), this gives






To obtain DIVh, divide both sides of (5) by the area, x y




and take the limit as t->0, and substitute (5) into (1)






This result is the same, as expected, as the operation





Horizontal Divergence in Natural Coordinates


What is the horizontal cross-sectional area of the air column in the natural coordinate system? Hence the fundamental definition of Divergence can be expanded in the same manner.


s n = A


Equations (8a and 8b)



Expand the right hand side of (8b) and substitute the definition of velocity in natural coordinates




                  V =  s / t                                                                (9)



into (8b) and simplify expression






The growth of ABCD to A'B'C'D' is proportional to the original area plus the area of the pie-wedged shape labeled with the arrow.  This pie-shaped area has dimensions in the s and n directions as well as shown by the inset. 


Since S can always be related to the speed V by the relation V = S/t, the dimensions of the side S is as shown in the inset, where V is constant during the time interval considered here.



The distance N is an arclength that can be obtained as the product of the angle subtended and the distance S.









Substitute (11) into (10) and the take the limit as t  etc -->0



which is the fundamental definition of divergence in natural coordinates.  The first right hand term is called diffluence and the second is called speed divergence[1]

[1] Negative diffluence is called confluence and negative speed divergence is called confluence.