Chapter 11
Imbalance and Vertical Motion
(1) Wind-Parallel Accelerations
In order for wind to flow in a cyclonically
curved manner, (an acceleration) it has to be
sub-geostrophic.
In order for wind to flow in an anticyclonically
curved manner (an acceleration) it must be
super-geostrophic.
We see in the real world that the wind also
speeds up and slows down (also
accelerations).
If the wind is
blowing parallel to
height contours or
isobars in the
absence of friction,
how can the wind
speed up or slow
down since the
Pressure
Gradient Force and the Coriolis Force are
acting at right angles to the wind direction
(they can only cause a change in direction)?
If the wind is
accelerating
(changing
direction - even if
parallel to
contours or
isobars,
speeding up or
slowing down), it
cannot be in
Above the friction layer, air will
exact
Geostrophic speed up if it has a component
towards lower heights or lower
Balance.
pressures. Air will slow down if it is
drifting toward higher heights or
higher pressures.
Ageostrophic Wind Vector - a vector
that represents the difference between
what the wind is actually doing and what
it would be doing if it were in perfect
geostrophic balance.
va  v  v g

Consider air flow about an anticyclone.
The actual winds are stronger (super-geostrophic)
than the geostrophic winds.
The Coriolis
force is stronger
than the PGF.
Thus, the
Ageostrophic
wind points the
same way the
wind is blowing.
Consider air flow about a cyclone.
The actual winds are weaker (sub-geostrophic) than
the geostrophic winds.
The Coriolis
force is weaker
than the PGF.
Thus, the
Ageostrophic
wind points the
opposite
direction from
which the wind is
blowing.
Consider air that is speeding up.
The actual wind must be oriented toward lower pressures or
lower contours, thus, the ageostrophic wind is pointed
toward lower pressures; at right angles to the geostrophic
wind.
Consider air that is slowing down.
The wind must be partly blowing toward
higher pressures and would be oriented to
the right of the geostrophic wind.
The ageostrophic wind would be oriented
at right angles, to the right of the
geostrophic wind.
Putting it all together (for northern hemisphere).
(opposite for S. H.)
Anticyclonic curvature - air accelerating (turning)
to the right. Wind faster than geostrophic and
ageostrophic wind pointing in forward direction.
Cyclonic curvature - air accelerating (turning) to
the left. Wind slower than geostrophic and
ageostrophic wind pointing in the backward
direction.
Air speeding up - air accelerating forward and
ageostrophic wind pointed to the left.
Air slowing down - air accelerating backward and
the ageostrophic wind pointed to the right.
In all four cases, the acceleration is 90o to the right of
the ageostrophic wind.
Consider the vector wind equation from
chapter 10 and the geostrophic equation.
Dv h
 f k  v h   go  h Z  Fh
Dt
f k  v g  g o  h Z


Subtracting the equations gives:



Dv h
  f k  v h  v g   Fh

Rearranging
Dt at substituting
gives:

Dv h
  f k  v a   Fh
Dt

va  vh  v g
Dv h
  f k  v a   Fh
Dt
If friction ( F)his
negligible, the

negative
cross
product of the unit
 k and the
vector
ageostrophic wind,
shows that the
acceleration will be
90o to the right of
the ageostrophic
wind in the northern
hemisphere.
Correction
(2) Wind Motion and System Motion
Consider a jet streak - a localized region of
very fast winds within a jet stream.
Air entering the jet streak is speeding up
(accelerating).
Air leaving the jet streak is slowing down
(accelerating).
Where it is speeding up, there must be crosscontour flow toward lower heights (to the left of
wind flow).
Where it is slowing down, there must be crosscontour flow toward higher heights (to the right of
the wind flow).
At entrance:
Ageostrophic wind toward left, acceleration to right
of it (speeding up).
At exit:
Ageostrophic wind toward right, acceleration to
right of it (slowing down).
What happens if the jet streak itself is
moving faster than the wind within it?
At exit:
The streak is catching up to the air parcels and
they would thus move faster. Should be crosscontour flow toward lower heights.
At entrance:
Jet streak is moving away faster than air parcels
are moving so parcels are essentially exiting the
streak and slowing down.
They should move toward higher contour heights.
Which is happening?
Compare successive maps to see if jet streak
is moving faster than the geostrophic wind
speed.
Check wind direction with contour analysis to
see how winds are crossing contours. (Need
accurate analysis.)
General rule: Air moves faster than weather
systems (e.g., jet streak) above 600 mb and
slower than weather systems below 700 mb.
Consider an upper-level trough.
The trough axis typically moves with a
component toward the east. (E, NE, SE)
Behind the trough air moves from the
northwest toward the southeast.
Thus, by the time the northwest air reaches
the point it should recurve toward the
northeast, the trough axis has moved so it
continues moving from the northwest.
Even more so if the trough is deepening.
Therefore, in this situation for the air
parcels moving from the northwest:
Their acceleration is weaker (not changing
direction toward the northeast).
The ageostrophic wind is less (not pointed
as strongly into the geostrophic wind which makes the acceleration [which is
always to the right of the ageostrophic
wind] not as strong toward lower height
contours - thus, it is not diverging from its
path).
(3) Convergence and Ageostrophic Wind
Consider the definition of the
ageostrophic wind.
va  vh  v g
Taking the divergence of both sides
gives:  va   vh   v g
However,
the divergence of the

geostrophic wind is almost exactly

nondivergent,
so this becomes:
  va    vh
Therefore, any divergence or convergence of
the horizontal wind is almost entirely
accounted for by the divergence or
convergence of the ageostrophic wind.
If you can infer the ageostrophic wind and,
using your understanding of the accelerations
that are occurring, you can make a good
guess about the patterns of divergence and
convergence in the upper air and from this,
the patterns of vertical motion that are
occurring.
Consider the trough again.
Air moves from northwest, through the trough axis,
and then toward the northeast.
The greatest curvature is in the trough axis.
Winds should depart most from geostrophic balance
because that is where the greatest acceleration
(changing direction) is occurring.
Thus, the ageostrophic wind should be greatest in
the trough axis and weaker on either side.
The wind in the trough axis is subgeostrophic (as
about a low).
The ageostrophic wind is pointed into the horizontal
wind (downstream toward upstream).
Acceleration is to the right (toward lower contour
heights).
Therefore:
Upstream of the trough axis (where ageostrophic
wind pointed into the horizontal wind is greatest)
there should be convergence.
Downstream of the trough axis (where
ageostrophic wind is weaker) there should be
divergence.
Also, upstream of a ridge axis there should be
divergence.
Downstream of a ridge axis there should be
convergence
Since vertical motion is restricted by the
ground and stratosphere,
Regions of convergence are associated with
downward motion in the interior of the
troposphere.
Regions of divergence are associated with upward
motion in the interior of the troposphere.
Consider again the jet streak.
The entrance region would have air
flowing across contours toward lower
heights (ageostrophic wind directed from
high heights toward low heights) - (and air
is speeding up).
The exit region would have air flowing
across contours higher heights
(ageostrophic wind directed toward higher
heights - (acceleration opposite to wind
flow - slowing down).
Entrance Side:
Thus, there should be convergence on the low
height side of the entrance to the jet streak.
There should be divergence on the high height
side of the entrance to the jet streak.
Exit Side:
There should be convergence on the high height
side of the exit to the jet streak.
There should be divergence on the low height
side of the exit to the set streak.
(4) Isallobaric Wind
An isallobar is a line of equal change in
atmospheric pressure.
When pressure changes, the initial response of air molecules
is to move toward the region of lower pressure.
Then, the Coriolis Force for this movement begins to act to
bring the forces into balance,
But, there is continual changes in pressure, so there is a
continual attempt to arrive at balance.
The motion of the air at balance is considered the steadystate response to a pressure change.
The instantaneous response is the to the pressure change.
The Isallobaric wind is the instantaneous response.
Mathematically, it is a component of the ageostrophic wind.
Will not be tested on.
(5) Convergence and the Chain Reaction
of Forces
Read for own interest.
Will not be tested on.
Homework:
Do 3, 4
(4) Isallobaric Wind
Consider the horizontal equations of motion
given in chapter 10, where v is the actual
horizontal wind.
Du
Z
 fv  go
 Fx
Dt
x
Dv
Z
 fu  go
 Fy
Dt
y
If we ignore friction, we have:


Du
Z
 fv  go
Dt
x
We can write:
Dv
Z
 fu  go
Dt
y
p 
p  z 
      
x z
z x x p
The minus sign is used since when δz>0,
δp<0.


Du
Z
Dv
Z
 fv  go
 fu  go
Dt
x
Dt
y
p 
p  z 


 
   
x z
z x x p
Then since from the hydrostatic
p
  g o
equation,

z
1 p z

p

z

we have:
 g o and:
go x x
x
x
Then we
 have:
 1  p
Du
 fv  g o 

 g o  x
Dt
Which becomes:

1  p
Du
 fv   
Dt
  x
 1 p
Dv

  fu  g o 
Dt

g
 o y
1 p
Dv
 fu   
Dt
 y
All we did was change the equation from one
typically expressed for upper-air maps
(change in height) to one typically for surface
maps (change in pressure).
1 p  z 
We can see that:
g
 
 x 
1  p
Du
 fv   
Dt
  x


 
x 
1 p
Dv
 fu   
Dt
 y
The term on the right (using any of these
horizontal wind equations) can be expressed
using the geostrophic wind.
1  p
Du
 fv   
Dt
  x

1 p
Dv
 fu   
Dt
 y
The geostrophic wind components are
given by:
fv g 



1 p
 x
fu g  
Du
 fv   fv g
Dt

Du
 f v - vg
Dt

1 p
 y
Dv
 fu  fu g
Dt

Dv
  f u  ug
Dt


Du
 f v - vg
Dt



Dv
  f u  ug
Dt

This can be written using the
ageostrophic wind definition as:
Du
 f v a 
Dt

Dv
  f ua 
Dt