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.

 

 

                                                            (1)

 

 

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.

                 

                                                                                         (2)

 

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

 

 

                                              (3)

 

 

Use the chain rule to expand right hand term.

 

 

                                        (4)

 

 

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

 

 

                                          (5)             

                                                                                                  

 

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

 

                          (6a)

 

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

 

 

 

                                                     (6b)

 

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

 

 

                            (7)

 

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

 

 

 

     (10)

 

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.

 

 

 

                                                                              (11)

 

 

 

 

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

 

                                    (12)                                                                  

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.