__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]