Carlson identifies 8 "Steps".
 
Except for "Step 0", view these as "parts" of the
           
frontogenesis; the "Steps" organize the presentation, and
           
step "n" does not need to finish before step "n+1" can commence.
   
0.   Initial state isentropes might be oriented with
           
a steeper slope (i.e. closer to vertical) than in later
           
diagrams.
- Confluent flow frontogenesis. Confluence causes the
isentropes in the warm and cold sides of the frontal zone to
approach each other. The flow can be geostrophic. Similar to
confluence term in eqn (13.2)
- Cross-isobar view of ageostrophic flow.
- Having a
horizontal gradient:
/
y' <0 implies that
there must be a thickness gradient
h/
y' <0
in the y' direction as well.
- frontogenesis means both gradients are increasing
- Consider first the upper level.
- Falling h in the cold air and
rising h in the warm air
- the height gradient between the warm and cold sides
shall be increased (the change is in the same fashion as
that between the upper solid line
becoming the upper dot-dashed line in fig. 14.6)
- Where there is a horizontal pressure gradient, we might expect
ageostrophic wind:
- ageostrophic wind normally blows from higher to lower pressure.
- Hence an ageostrophic wind is formed at the top which drives
some mass from the warm side into the cold side.
- Now consider the lower level.
- The addition of mass at upper levels raises the surface
pressure on the right side. (note change of lower solid
line becoming lower dot-dashed line: the curve rises).
- The loss of mass at upper levels lowers the surface
pressure on the left side.
- Note the h changes are still as we expect: smaller h
on the cold (right) side, larger h on the warm (left) side.
- A pressure gradient at low levels implies an
ageostrophic wind at low levels as well
- Vag directed to warm air side at low levels;
opposite to upper levels
- Understanding where these Vag winds comes
from is key to the rest of what follows.
- Unbalanced change
creates Vag
- This step and the next are a bit weird because Carlson
is trying to split apart his discussion of a simultaneous
process. So, in these two steps a different thing is held
constant.
- hold u constant
- y' momentum equation states that: dv/dt = -y
- fu.
Since the gradient in
y direction must be increasing (step 2) then the v motion must
be accelerating, since we are holding u constant.
- if Vag is initially 0, then the result that dv/dt >0 implies
Vag >0 must develop.
- this is Carlson's way to get the upper Vag. It is
just another way to reach the same conclusion we stated in
step 2.
- Change in u creates Vag.
- now hold constant.
- The flow is confluent, and is crossing isotachs
(as depicted in fig. 14.5). Therefore, from the standpoint
of an air parcel, the air parcels must be accelerating.
i.e. du/dt >0
- x' momentum equation states that: du/dt = -x
+ fv.
Since du/dt >0, using the x' momentum eqn Carlson
then concludes that v must be positive since
gradient has been
held constant. i.e. Vag >0 as before.
- My suggestion: don't place too much emphasis on this chain
of events to bring formulas back into balance.
- Instead: View the discussion here as showing how the ageostrophic
circulation set up maintains the momentum balances.
- Frontogenesis from convergence due to Vag
- Carlson finally looks at those ageostrophic motions: Vag
- towards larger y' at upper levels; opposite at lower levels. That
means:
towards cold air at upper levels; towards warmer air at lower levels.
- Since height gradient largest in middle, expect largest
Vag there.
-
Vag/
y'
implies areas of divergence and convergence
- Vag decellerates in upper right part of diagram,
so
Vag/
y' <0 there which means
convergence there. Convergence also takes place
in the lower left part of the chart.
- Divergence in the upper left and lower right.
- Intuitively, where there is convergence you might expect the
isentropes to be moving closer together. The isentropes would
spread apart where there is divergence.
- Hence, convergence fosters frontogenesis, divergence inhibits
it.
- The Vag field creates an asymmetric, tilted frontal
zone (see fig. 14.6) with isentropes concentrated in a diagonal
band from lower left to upper right on the figure.
- Note 1: divergence & convergence imply vertical motions.
- Here that means sinking on the cold side, rising on
the warm side.
- Carlson does not comment on this, but the tilting term indicates
that this works against frontogenesis
- vertical motions are trying to tilt the isentropes in a
more horizontal orientation, thus weakening the horizontal
gradient of
- to put it another way: sinking means adiabatic warming on the cold side;
rising means adiabatic cooling on the warm side. This reduces the
horizontal gradient of
- Note 2: the pattern of Vag, divergence, and U
here are all consistent with what we saw in the jet streak (entrance
region) discussion. Again, that is consistent with fig. 14.5 as well.
- Changes in u maintain thermal wind balance
- here we are working the analysis in step 4 in the opposite
order. Changes in Vag cause accelerations in u.
- Note how Vag reverses sign between upper and lower
elevations.
- From the x' momentum equation, this means that du/dt >0
in upper levels and du/dt <0 in lower levels. Hence the vertical
shear must be increasing.
- Increasing vertical shear is consistent with thermal wind
balance, since the horizontal
gradient is increasing.
- Three important conclusions
- Convergence patterns augment the confluence and are asymmetric.
Hence, the axis of dilatation rotates: toward warmer air at low
levels; toward colder air at upper levels. The frontal zone
becomes more tilted.
- The jet tends to migrate towards the warm air side of the
frontal zone. Consider what happens at the upper levels:
- Assme u is initially constant in y' direction (no jet)
- Consult vorticity eqn. d/dt
= - fD.
- Assume vorticity changes are from shear vorticity, not curvature vorticity:
- Thus: convergence increases vorticity which increases
-u/
y'.
- Opposite for areas of divergence.
- Next remember that a straight jet (having no curvature vorticity)
would have its maximum speed where the vorticity is zero. There
would be positive vorticity on the left side (looking downwind) and
negative vorticity on the right side.
- Thus, the convergence/divergence fields cause the jet
to migrate to a location in between the upper
level divergence and convergence. Since the front develops a
tilt back over the cold air, then the jet has migrated
to the warm air side of the frontal zone. (see for example, fig. 14.6;
look for the circled J)
- Since the confluence term depends upon the
y' gradient of ,
then the process accelerates. The rate of intensification
is proportional to the intensity of the front. As the front gets
stronger its rate of intensification keeps getting faster.