MT11B Weather Systems Analysis
Eleanor Highwood, December 2003
Lecture D5
The ageostrophic wind
D5 1 - Ageostrophic wind and vertical motion:
In the last lecture we discussed how the vertical velocity depends on the horizontal
divergence of the flow. However, if we assume the flow is geostrophic we run into a
problem because the horizontal divergence of the geostrophic wind is identically zero.
Since ug   g / f  Z /  y and v g  g / f  Z /  x , the horizontal divergence of the geostrophic
wind is identically zero:
 ug
g  2 Z  vg g  2 Z

,

.
x
f xy y
f xy
Hence
 ug  v g

 0.
x y
The vector difference between the actual wind and the geostrophic wind is called the
"ageostrophic wind", and it is denoted v a  (ua , va ) :
va  v  vg .
The ageostrophic wind in the midlatitudes is usually an order of magnitude smaller than
the geostrophic wind.
It follows that:
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MT11B Weather Systems Analysis
Eleanor Highwood, December 2003
D
 u  v  ua  va

.




x y x y
p
The vertical velocity therefore depends upon the small ageostrophic part of the wind field.
Also, by considering the tendency equation derived in the last lecture, changes in surface
pressure also depend directly on the ageostrophic wind.
D5 2- Acceleration of air parcels:
Consider the acceleration of an air parcel. Away from the ground, we may neglect friction.
The acceleration is then simply the sum of the Coriolis and pressure gradient forces. For
the x-component of the wind:
Du
Z
.
 fv  g
Dt
x
But the definition of the geostrophic wind is:
fv g  g
Z
.
x
Substituting into the equation of motion:
Du
Dv
 fva , similarly
  fua .
Dt
Dt
These equations show that any acceleration of the fluid parcel is a direct consequence of
the ageostrophic wind. If the wind were exactly geostrophic, no acceleration, and hence
no evolution of weather systems could take place. Small ageostrophic wind components
are of crucial interest to meteorologists and can be useful in working out where there is
likely to be vertical motion. Let’s have a closer look at two examples of this.
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MT11B Weather Systems Analysis
Eleanor Highwood, December 2003
D5 3 - An upper jet exit:
Consider a jet exit with the jet axis being oriented in the zonal direction in the upper
troposphere.
1. As the parcel moves from A towards B, it is deccelerating.
2. The momentum equation tells us that the ageostrophic wind must therefore be directed
to the south, as shown.
Away from the jet exit, the ageostrophic wind will be small: thus there will be
convergence of the ageostrophic wind to the south of the jet exit and divergence of the
ageostrophic wind to the north of the jet exit. By continuity, at lower levels in the
troposphere, we expect to see descent to the south of the jet exit and ascent to the north
of the jet exit.
Exactly the opposite pattern of ascent and descent will occur in a jet entrance region.
These regions of ascent favour the development or deepening of low pressure centres,
while regions of descent favour the intensification of anticyclones. These “development
regions” were used by forecasters before the availability of numerical weather predictions
as means of predicting the intensification or decay of midlatitude weather systems.
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MT11B Weather Systems Analysis
Eleanor Highwood, December 2003
D 5.4 Thermal advection and ageostrophic wind:
z
Imagine a situation where the atmosphere is initially motionless, with level pressure
surfaces, shown in the diagram.
p2
p1
x
Now suppose warm thermal advection takes place, bringing warm air from the south
locally. The lower pressure surface is supposed to remain flat, while the upper one must
be elevated over the region of warming.
z
Looking at the slope of the p2 surface, a southerly geostrophic wind is needed to the west
of the thermal advection, and a northerly geostrophic wind to the east of the warm
advection region. How can these be achieved?
p2
Thermal
advection
p1
From the equations for the acceleration given in the last section, there must be an
easterly ageostrophic wind to the west of the heating and a westerly ageostrophic wind
on its eastern side. There is upper level divergence of the ageostrophic wind in the centre
of the region of warm thermal advection. By continuity, this implies ascent at lower
levels.
x
z
Ascent is associated with warm thermal advection.
ua
ua
.
Descent is associated with cold thermal advection.
p2
Ascent
p1
4
x
MT11B Weather Systems Analysis
Eleanor Highwood, December 2003
Worked Example:
An upper tropospheric jet extends across the Atlantic at 45N. The jet core is at 30 kPa
and the maximum westerly wind is 50 m s-1 above the coast of North America, declining
to 15 m s-1 3000 km further east. Calculate the ageostrophic wind in the jet exit at
30 kPa, and hence estimate the vertical velocity in the jet exit region. You may assume
the jet width is 1000 km.
Step 1: Jet is oriented W-E so no acceleration/deceleration in y direction. Therefore
Dv/Dt=0 and ua = 0
Step 2: Find va arising from deceleration in zonal direction. Remember from basic
mechanics that accel=v2-u2/2s such that Du/Dt=-3.79x10-4 ms-2.
Step 3: At 45N, f=1.03x10-4s-1, and Du/Dt=fva so calculate va=-3.68 ms-1. I.e. the
ageostrophic wind is northerly, magnitude 3.7ms1.
Step 4: Away from jet there is no ageostrophic wind. Therefore va= zero at 500km from
jet core. Divergence= -(va at core)/500km = 7.4x10-6 s-1.
p

Step 5: pressure vertical velocity  ( p)   Ddp which can be adapted to
0
=-Dp=-7.4x10 x30x10 = -0.222 Pa s .
-6
3
-1
Step 6:    gw so w=0.02 ms-2 (assuming g=10 and =1).
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