Some Review From Metr 430
http://tornado.sfsu.edu/Geosciences/classes/m430/Vorticity/Circulation.htm
Absolute Circulation
The general expression for circulation, a macroscopic measure of swirl, for an air
column with circular cross-secitonal area is:
C = 2pwr 2
where the lower case omega is angular velocity and r is the radius of the air
column.
The relationship between circulation and vorticity is simply
C
C
= 2w = z =
2
pr
A
or vorticity per unit area. Thus, for a given air column, vorticity and circulation
are directly proportional.
In the absence of torques, absolute circulation is conserved, or
æ dC ö d (z a A)
=0
ç ÷ =
è dt øa
dt
where the absolute vorticity is the relative vorticity plus earth’s vorticity, the
Coriolis Parameter.
Expanding the derivative on the right, in Metr 430 we got
d (z a A)
dt
æ dA ö
æ dz a ö
= 0 = za ç ÷ + Aç
÷
è dt ø
è dt ø
or
dz a
z a æ dA ö
= - ç ÷ = -z a DIV
dt
A è dt ø
which states when there are no real torques (related to, for example, solenoids),
that the absolute vorticity of air parcels only changes because of synoptic scale
divergence (or convergence).
Conservation of Absolute Vorticity
For a non-divergent atmosphere, this then reduces to
dz a
=0
dt
or
(z + f ) 0 = (z + f )1
Substitution of the relation between tangential and angular velocities, V = wr , into
the definition above for circulation results in this expression for conservation of
absolute circulation, say for an air column that changes its latitude.
dCa
=0
dt
or
(Ce + C ) 0 = (Ce + C )1
or
2
2
f
p
r
+
2
p
Vr
=
f
p
r
(
) ( + 2p Vr)
0
1
To understand Rossby Waves, consider a situation in which there is NO relative
circulation at the initial latitude, and, thus, no tangential velocity at the initial
latitude. To determine the tangential velocity at the final latitude
r
V1 = ( f0 - f1 )
2
or
V1 = k ( f0 - f1 )
This says that a tangential velocity will develop entirely due to the variation of the
Coriolis Parameter with latitude, which is also known as Beta.
V1 » bDy
For example, consider a long wave with a half wavelength of about 5000 km (say,
with a ridge axis on the West Coast and a trough axis on the East Coast) formed
when an air column initially with no circulation relative to the earth is taken due
northward along the West Coast about 15 degrees of latitude. Such an air column
would have a diameter of around 2500 km, or a radius of about 1250 km.
Substitution into the equations just above yields a velocity at the periphery of this
air column when it reaches the new latitude of about 11.5 m s-1, or about 25 mph in
an anticyclonic sense. This appears to be a reasonable estimate of the actual flow at
jet stream levels, although perhaps a bit low to be perfectly realistic. Yet it is
important to note that, in this situation, the starting conditions included no velocity
relative to the earth, yet simply moving the air column northward resulted in
velocities relative to the earth.
As you will see, once this relative circulation develops, the application of the same
principles (conservation of absolute circulation and conservation of absolute
vorticity) will result in the development of cyclonic relative vorticity in the
southward moving air streams on the east side of the anticyclone thus formed, and
anticyclonic relative vorticity in the northward moving air streams on the west side
of the air column. Thus, anticyclonic circulation increases on the west side and
cyclonic circulation increases on the east side of the air column, manifested as a
retrogression of the long wave ridge.