S10 Estimating vorticity
S10 1 - Application of kinematic formulae:
The definition of relative vorticity is:
UT
v u
 .
x y
This is not a very practical formula, for its application involves
decomposing the wind into its components, and then attempting to
estimate derivatives. A more practical formula, which is certainly
useful for determining the sign of the vorticity, is:

r
L
Notation used in
text.
U T U T

r
r
The first term on the RHS is called the “curvature term” and the second is the “shear
term”. In the case of flow in straight lines, the wind varying across the streamlines,
the curvature term is zero, and the vorticity is associated with the shear. In the case of
flow around a depression which does not depend on the distance from the centre of
the depression, the shear term is zero, and vorticity is entirely associated with the
curvature term.

An extreme example is provided by a hurricane. The wind blows around the eye of
the hurricane and is approximately proportional to 1/r. In this case, the shear and
curvature terms cancel out, so that the relative vorticity is zero.
r
UT
Estimating vorticity.
If the wind is blowing parallel to curved isobars, and the flow
is reasonably steady, the shear/curvature formula can be used
to estimate the vorticity. First, determine the radius of
r
curvature of the flow (see lecture 1). Then given the tangential
component of wind at this point, the curvature can be
calculated. To determine the shear term, estimate the
tangential component of the wind at two points, one closer to
the centre of curvature and one further away, both along the same radius. The shear
term is approximated by:
U T U T

r
r
The diagram illustrates the calculation. Notice the difficulty. UT is calculated by
taking the difference between two nearly equal wind speeds. Unless they are known to
great precision, the result will not be at all accurate. This problem can be avoided by
choosing r larger. But then our difference formula for the derivative becomes a poor
approximation.S10 2 - Geostrophic vorticity:
If the geostrophic approximation holds, a simpler method of estimating the
geostrophic vorticity can be devised. Recall the two components of the geostrophic
ug  
g Z
,
f y
vg 
g Z
.
f x
wind:
Substitute into the formula for relative vorticity:
g 
g  2Z 2Z  g 2

  Z

f  x 2 y 2  f
A vorticity cross.
Z2
x
Z3
Z0
Z1
x
Z4
In fact, a simple finite difference
formula exists which can be used to
estimate the Laplacian of the height
field. Imagine a square cross whose
arms are x long. Then:

Z  Z 2  Z 3  Z 4  4Z 0
2Z  1
x 2
and so the geostrophic vorticity can
be written:
g 
g  Z1  Z 2  Z 3  Z 4  4Z 0 


f 
x 2

Of course, there is still the problem of subtracting two large, comparable numbers,
which makes even this estimation of vorticity inaccurate. But the computational
labour is a lot less than using the kinematic formulae.