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 Æx and Æy.

(2)

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

(3)

Use the chain rule to expand right hand term.

(4)

but Æx/Æt = u and Æy/Æt = v.  Substituted into equation (4), this gives

(5)

To obtain DIVh, divide both sides of (5) by the area, Æx Æy

(6a)

and take the limit as Æt->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.

Æs Æn = A

Equations (8a and 8b)

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

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

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

(11)

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

(12)

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

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