Vorticity
We have
previously discussed the angular velocity as a measure of rotation of a
body. This is a suitable quantity for a
body that retains its shape but a fluid can distort and we must consider two
components to rotation: shear and curvature.
We introduce the term vorticity as a measure of rotation within a fluid.
Consider a marker
in the fluid such as the dashed line crossing the streamlines. Initially, the
marker might be normal to the streamlines, as at time t1, but later will
be rotated to the position at t2. The marker line has undergone an
anticlockwise rotation (cyclonic in the N-hemisphere). We evaluate the
contribution to the vorticity as 
If, at radial
distance r, the velocity is V, the curvative component of vorticity is
evaluated as V/r in the same way we define angular velocity.
Convention: both components
are assessed the same way using the convention that cyclonic (anticlockwise in
the N-hemisphere) rotation is positive, and anticyclonic rotation is negative.
The vorticity of
a volume of atmosphere, when evaluated in this way from the velocity of the air
relative to the earth’s surface is called relative vorticity 
(Greek zeta)
Relative vorticity 
If the atmosphere rotated like a solid body, the two components
would be equal and thus the vorticity is equivalent to twice the angular
velocity of a solid object.
To apply basic
laws of motion (such as the conservation of angular momentum), we need to
consider rotation in an absolute frame of reference, and must add in the
rotation due to that of the earth itself. Twice the angular velocity of the
earth about a local vertical is, of course, equal to the Coriolis parameter 
, so we define absolute vorticity as
On certain upper
air charts (see, for example, the NGM 500 mb forecast maps), absolute vorticity
is plotted in units of 10-5 s-1. Thus, in mid latitudes,
where 
, a region of zero relative vorticity would have an absolute
vorticity of 10 units. Centers of maximum vorticity are useful in identifying
trough locations, which are sometimes difficult to identify, otherwise,
especially since the shear component is difficult to assess.
As we’ll see
later the relationship between the flow field and the vorticity pattern yields
an important forecasting tool.
Constant absolute vorticity trajectories
It has been shown
that, to a reasonable approximation, the atmosphere moves in such a manner as
to conserve its absolute vorticity. That is,
Atmospheric
trajectories with constant absolute vorticity execute sinusoidal-like paths
around the hemisphere. Remember that the Coriolis parameter f (twice the
angular velocity of the earth about a local vertical) increases towards the
pole (). If f increases as a mass of air moves northward, then must decrease, and
vice versa. The result is a wavy trajectory around the hemisphere as shown
below:
To understand
this, remember that positive is cyclonic
(anticlockwise) rotation, while negative is anticyclonic
(clockwise) motion, and cycles between
positive and negative values as the air executes a sine wave.
Rossby waves
These large
wave-like perturbations observed in the mid-latitude westerly flow are called
Rossby waves after the Swedish meteorologist (who founded the first meteorology
department in the
Analysis of
Rossby waves results in the following expression for the speed at which the
waves propagate towards the east:
Rossby wave speed
where
V is the wind speed at mid-tropospheric levels (specifically at the level of
non-divergence LND),
L
is the wavelength of the Rossby wave
where
Since the last
term in the equation 
is always positive,
the wave speed C must be smaller than the wind speed. Thus, the air passes
through the troughs and ridges at a speed greater than the pattern itself
propagates.
Notice that the
shorter the wavelength L, the larger the wave speed C. The longer the
wavelength, the smaller the wave speed. Long waves seen on upper air maps thus
move slowly while shorter waves ripple through them. At any one time, the
global circulation pattern is a summation of waves of different length in
different stages of being in or out of phase.
From the
equation, one can see that a sufficiently large wave could have a negative
speed of propagation (C < 0 if 
) and the wave will retrograde (move towards the west).
Example
calculations of wave speed C:
Suppose V = 30 m/s, 
for a wavelength
L = 4000 km
C = 30 – (0.405 x 10-12)(4
x 106)2
= 30 – 6.4 = 23.6 m/s
for a wavelength
L = 8000 km
C = 30 – (0.405 x 10-12)(8
x 106)2
= 30 – 25.9 = 4.1 m/s
Thus, the longer
wave propagates towards the east at a much lower rate than the short wave.
Divergence and convergence
In our numerous
discussions of current and prognostic weather patterns, we have spoken of
divergence and convergence in the horizontal fields of motion at different
levels within the troposphere. Analysis of wave-like disturbances (Rossby
waves) shows that the rate of change of vorticity relates to fields of
divergence and convergence, as shown in the following diagram:
The curve labeled
V represents the profile of wind speed through the troposphere, reaching a
maximum at the tropopause (the jet stream). When 
as in the upper portion of the troposphere, convergence
occurs in the horizontal wind field upwind of the trough line (west of the
trough) and divergence occurs down wind of the trough. In the lower part of the
troposphere (
), the opposite is the case. Compensatory vertical motions
occur with ascent ahead of the trough (downwind) and subsidence behind the
trough, together with the creation of areas of low and high pressure at the
surface, as shown.
On weather maps
which show upper air flow patterns, a way to identify divergence and
convergence is to examine the relationship between height contours (say of 500
mb) and calculated vorticity values (such as plotted on the NGM forecast 500 mb
maps)
As the air flows
through the trough and moves downstream, it loses vorticity and is forced to
diverge horizontally (recall that this is the same relationship as dictated by
the conservation of angular momentum). Where the “boxes” formed by the crossing
of the contours and the vorticity isopleths are smallest, divergence (or
convergence on the upwind side of the trough) is greatest. Alternatively, as
air moves downstream ahead of the trough, we can say that positive values of
vorticity are advected by the air parcels carrying the positive vorticity
acquired at the trough line. The downstream area is said to be one of positive
vorticity advection (PVA), while the upstream region is one of negative
vorticity advection (NVA). PVA is associated with upper air divergence, and NVA
is associated with upper air convergence.