MT11B Weather Systems Analysis
Eleanor Highwood, December 2003
Lecture W3
Vorticity
Reading: FWC pp223-226.
W3 1 - Circulation:
Winds in the atmosphere, at least in the midlatitudes, are dominated by circulating or
swirling motions. The winds blow around, rather than into, centres of high or low
pressure. On the larger scale, especially away from the Earth's surface, winds tend to
blow around the cold poles, in predominantly westerly flow. We could do with a measure
of the strength of these circulations.
A measure of this circulation is provided by taking a given region in the atmosphere,
dividing its boundary into straight line segments, and adding up the component of wind
parallel to each segment times the length of the segment, i.e.,
C   v s
n
We adopt the sign convention that anticlockwise circulation is positive.
Consider a rectangular circuit in the atmosphere with sides of length x and y . Then the
circulation around the rectangle is:
C  ux  (v  v) y  (u  u) x  vy .
Expanding the brackets, this expression simplifies to:
In this form, circulation is a rather unhelpful quantity, since it is specific to the particular
circuit we have chosen. We can make a more general measure of circulation by dividing
by the area of our circuit A  xy , and taking the limit of x, y  0 . Then:
C
v u


.
A
x y
MT11B Weather Systems Analysis
Eleanor Highwood, December 2003
The quantity  is called the relative vorticity. Relative vorticity has units of s-1. It may
be thought of as the circulation per unit area.
[Note, strictly, the vorticity, like spin or rotation rate, is a vector quantity and it is
necessary to specify the direction of the spin axis as well as the magnitude of the spin.
The calculation above in fact gives the component of vorticity about an axis perpendicular
to the area enclosed by the loop. Components about other axes can be calculated
similarly.]
In meteorology, the vertical component of vorticity is much more significant (though
numerically smaller) than the horizontal components of vorticity. For this reason, when
meteorologists speak of “the vorticity”, they often mean the “vertical component of
vorticity”.
W3 2 - Vorticity and spin:
Vorticity is closely related to the spin of individual fluid elements. Let’s consider the spin
of the rectangular air parcel discussed in the previous section. We mark the rectangle
with a cross. A short time later, the wind shears across the rectangular parcel mean that
the arms of the cross will have been rotated through a small angle.
Consider the arm AB to start with:
End B is moving in the y-direction v faster than the end A. So, after a time t , the arm
AB will therefore have rotated through an angle:
 
v t
.
x
That is, the arm AB will be rotating at an angular velocity:
 AB 
 v
.

t x
In exactly the same way, the angular velocity of arm CD can be calculated. The only
subtlety is to keep track of the signs - remember our sign convention for the direction of
positive circulation. The result is:
MT11B Weather Systems Analysis
Eleanor Highwood, December 2003
Combining these two results, we can derive an expression for the average angular
velocity or rate of spin of the cross. If the limit in which the length of the arms tend to
zero is taken, we have:

1
1   v  u 
 AB   CD   


 .
2
2   x  y 2
That is, the relative vorticity is simply twice the angular velocity of spinning
fluid elements.
This calculation has derived an expression for the spin of fluid elements relative to the
solid Earth. But that in turn is spinning. At latitude , the vertical component of the
Earth's rotation is sin() ; relative to an inertial frame of reference, the parcel of air will
have an extra contribution to its vorticity due to the Earth's rotation. The vorticity in an
inertial frame of reference is called the absolute vorticity. Absolute vorticity may be
written:
  2sin ()    f   .
W3 3 – Vortex stretching:
The vorticity of an air parcel can change as a result of several different processes. But on
the synoptic scale, one process is dominant. It is called “vortex stretching”. The diagram
illustrates the process.
Consider a column of air with zero relative vorticity, i.e., it is not spinning relative to the
solid Earth, and it is spinning with an angular velocity sin() relative to an inertial frame
of reference. Suppose some process stretches the column.
1. Its radius must decrease in order to conserve its volume.
2. To conserve angular momentum (ok if we assume no friction) it must spin more
rapidly.
3. Relative to the solid Earth, it gains positive (cyclonic) relative vorticity.
MT11B Weather Systems Analysis
Eleanor Highwood, December 2003
Q. What happens if the column is squashed instead of stretched?
This useful picture can be made quantitative. If the height of the air column is H and
friction is negligible, the quantity:
P
 f  
H
is conserved by the column as it moves around. So if H is increased (by stretching), 
must also increase in order that P should remain constant. P is called the potential
vorticity. It is a measure of the vorticity the column would have if it were stretched or
squashed to some standard length. Confusingly, P does not have the same units as .
Q. What units does P have?
W3 4 - Vorticity and thermal wind:
Imagine a cross section through a circular vortex. A particular pressure
surface is shown by the line AB on the diagram. If the vortex is cyclonic,
the pressure surface will be lower in the centre of the vortex than on its
periphery. Now suppose that the vortex is colder towards its centre than
at its periphery. The thickness in the centre of the vortex will be smaller
than further out, and so the vorticity increases with height.
The opposite arguments hold for a warm cored vortex. The thickness
will be increased in the centre and so the vorticity will decrease with
height. Similarly, the intensity of a cold cored anticyclone decreases
with height, whereas a warm cored anticyclone becomes more intense
with height.
MT11B Weather Systems Analysis
Eleanor Highwood, December 2003
These results are made more quantitative using the geostrophic vorticity. Since
g 
To get this we write ug and vg in
terms of height gradient using the
geostrophic wind relationships and
substitute into the vorticity equation.
g 2
 Z
f
it follows that, using the thickness (or hydrostatic) equation:
 g
R 2

 T.
p
pf
This is, in fact, an alternative statement of the thermal wind relationship.
Worked Example: Air blows around a circular depression at 60N with a speed of 20 m s-1 at 90 kPa at a
distance of 300 km from its centre. The wind speed drops linearly to zero at the centre of the depression.
Estimate the relative and absolute vorticity of the flow. If the centre of the depression is 5 K colder than its
periphery, estimate the vorticity at 50 kPa.
Step
1:
What
do
A  20 / 3  10  6.7  10
5
we
5
know
about
the
wind?
The
wind
is
given
by
U ( r )  Ar
where
s .
-1
Step 2: For circular flow, it is easier to use the so-called “kinematic” formulae for relative vorticity (see S10)
where  
U t U T

. Note that in this case,  U /  r  U / r so that   2 A , i.e., 1.3310-4s-1.
r
r
Step 3: The Coriolis parameter is 1.2610-4s-1.
Step 4: The absolute vorticity   f   is 2.5910-4s-1.
Step 5: By thermal wind balance, the wind will increase with height, so that
U
R T
.

p
pf  r
Replacing the derivative with simple finite differences we can find U50 and then the vorticity at 50kPa:
U 50  U 90 
R p T
.
pfr
Substituting numbers, U 50 is 41.6 m s-1, and  50 is 2.7710-4s-1.