Vorticity
r 
Relative vorticity
U
R

U
n
where U/R is the curvature term and -∂U/∂n
the shear term
Absolute vorticity  a   r  f
where f is the Coriolis parameter. ξ written
without a subscript normally refers to ξa.
Potential vorticity: ratio of absolute vorticity
to depth of an air column
PV 
a
p
or
a
z
More formally we can derive Ertel’s
formulation of PV:
PV 
1

a

p
 - ga

p
Vorticity equation
We can derive the barotropic
vorticity equation:
d   f 
dt
   f


p
Which relates stretching to
changes in vorticity
Ω
pt
Ω/h
conserved
h
r
pb
Dines compensation
In the stratosphere vertical
motion is inhibited - so ω
reaches a maximum in the
mid-troposphere
Divergence at jet stream level
=> Convergence at the surface
This is how the jet stream drives
the weather
Tropopause
D
C
D
C
D
C
ground
Rossby waves
A
Divergence
U
ξr < 0
Convergence
B
C
ξr > 0
Minimum in ξain ridge
Maximum in ξa in trough
So dξa/dt >0 – ξa increases as air goes from ridge to
trough
By conservation of PV, air columns must stretch.
Stratosphere resists vertical motion so the columns
stretch downwards – descending motion between A and
B
By Dines, this results in divergence at the surface –
anticyclone tends to form.
Example
Upper tropospheric air flows at a speed of
30 ms-1 through a sinusoidal trough-ridge
pattern at 50oN, of peak-to-peak
amplitude 500 km and wavelength 3000
km.
Calculate the change in absolute vorticity
between ridge and trough, and derive the
fractional change in the depth of an air
column as it traverses the pattern.
Draw a diagram to mark areas of upward
and downward motion in the flow, and
hence describe the effect of the pattern
on the surface weather.
(The radius of curvature of y = A sin(kx) is
(Ak2)-1 at the crests).
Example
Upper tropospheric air flows at a speed of
30 ms-1 through a sinusoidal trough-ridge
pattern at 50oN, of peak-to-peak
amplitude 500 km and wavelength 3000
km.
Calculate the change in absolute vorticity
between ridge and trough, and derive the
fractional change in the depth of an air
column as it traverses the pattern.
Draw a diagram to mark areas of upward
and downward motion in the flow, and
hence describe the effect of the pattern
on the surface weather.
(The radius of curvature of y = A sin(kx) is
(Ak2)-1 at the crests).
First, calculate Rmax from (Ak2)-1
A = 250 km, λ=3x106 m so k = 2.09x10-6 m-1
Ak2 = 1.09 x 10-6 m-1 so Rmax =916 km
U/Rmax = 3.28 x 10-5 s-1 (+ve in trough, -ve in ridge)
U U
1. Absolute vorticity ξ a  f  R n
2. Potential vorticity
ξa
p
 const
use 1, with no shear vorticity term (uniform speed).
500 km corr. to 4.5° lat = ±2.25° (1° = 111 km)
ftrough (47.75) = 1.0810-4 s-1 so ξa =1.4110-4 s-1
fridge (52.25) = 1.1510-4 s-1 so ξa = 0.8210-4 s-1
Fractional change in depth of air column is
calculated using 2. In the straight part of the flow, ξa
= f = 1.1210-4 s-1.
Since
 p trough
 p ri dge
ξa
p
 p s tra i ght
 p s tra i ght
Ω
is conserved:

ξ a , trough

ξ a , ri dge
.
ξ a , s tra i ght
pt
h
r
ξ a , s tra i ght
pb
Ω/h
conse
rved
 1.409
 0.821
1.12
1.12
 1.26
 0.73
Divergence aloft causes
pressure drop
Convergence at surface
causes cyclone to spin-up
Shear Vorticity
U
n
 0
on left side of jet
n
A
Jet
B
U
n
Isotachs (const. wind
speed)
 0
on right side of jet
Convergence
Divergence
Divergence
Jet
Convergence
Example
A zonal jet streak develops in a
uniform zonal flow of 30 m s-1 at
60°N. The jet has a maximum
speed of 80 m s-1. The cyclonic
side is 200 km wide and the
anticyclonic side 600 km wide. If
the initial depth of a column of air
which enters the jet is 100 mb, use
the barotropic vorticity equation to
estimate its depth at maximum
velocity if it is positioned:
(i) poleward of the jet core
(ii) equatorward of the jet core
(iii) directly upstream of the jet core.
What limits the accuracy of these
estimates?
Curved jet streams
C, D : convergence/divergence due to trough or ridge
C,D : convergence/divergence due to jet entrance/exit
Diffluent
trough
Confluent
trough
DD
DC
DD
DC
Black arrow denotes jet axis (location of wind maximum)
CD
CC
CC
Diffluent
ridge
CD
Confluent
ridge