Applications of ‘IPV’ thinking for
time-dependent dynamical
processes (p. 202, Bluestein, 1993)
The purpose of this discussion is to utilize ‘IPV’
thinking to explain the motions and development
of synoptic-scale weather systems
The basic concepts to be
discussed include:
• The atmospheric structure consists of a
superposition of upper-level positive and
negative IPV anomalies, positive and
negative surface potential-temperature
anomalies, along with a basic flow. The more
conventional interpretation is the atmospheric
structure consists of upper-level troughs and
ridges, along with surface cyclones and
anticyclones.
Basic ideas of time-dependent
dynamical processes (Continued)
• Gradient-wind balance holds to a first-order
approximation. We assume that the magnitudes of
the anomalies (perturbations) are weak enough, so
that quasi-geostrophic theory is valid: The diagnostic
equation relating PV to the wind field (eq. 1.9.29) has
a linear operator. Additionally, the atmosphere is
statically stable so that the equation (1.9.29) is elliptic.
Therefore, the total wind field that is induced by all of
the PV anomalies is the sum of the wind fields induced
by each anomaly separately. For typical synopticscale anomalies and for typical static stabilities, the
induced wind fields extend throughout the depth of the
troposphere.
Basic ideas of time-dependent
dynamical processes (Continued)
• Both diabatic heating and friction are
ignored, so that potential vorticity is
conserved. Therefore, potential vorticity
anomalies are advected on isentropic
surfaces and account for local changes
in the potential vorticity.
Basic ideas of time-dependent
dynamical processes (Continued)
• Each of the potential vorticity anomaly’s
induced wind field will therefore change
the distribution of PV.
• The consequent new distribution of PV
is associated with new induced wind
fields, which will change the distribution
of PV, etc.
The motion of upper-level troughs
and ridges in the baroclinic
westerlies
• Consider, as in the following figure, a
series of alternating positive and
negative upper-level PV anomalies in
the east-west direction, and inserted in
a uniform westerly flow:
Potential vorticity inversion may be used to
understand the motions of troughs and
ridges (as for Fig. 1.149:
• Potential vorticity
maxima and
minima, correspond,
respectively to
troughs and ridges
• instantaneous winds
N
max
N
min
max
min
Consider a PV reference state
(as for Fig. 1.150):
• Consider the PV
contours at right
with increasing PV
northward (owing
primarily to increase
of the Coriolis
parameter)
larger PV
N
PV+2dPV
PV+dPV
PV
PV-dPV
Consider the introduction of alternating PV
anomalies (as for Fig. 1.151):
• The sense of the wind
field that is induced by
the PV anomalies
• There will be a
propagation to the left
or to the west (largest
effect for large
anomalies
• This effect is opposed
by the eastward
advective effect
larger PV
N
L
+
-
PV+2dPV
PV+dPV
+
PV
East
The previous figure shows the
following:
1. The locations of the maximum southerly
component of the induced wind are L/4 to the
west of the most poleward parcel
displacements (whose locations are the sites
of the negative PV anomalies, or ridges).
2. The locations of the maximum northerly
component of the induced wind are L/4 to the
west of the most equatorward parcel
displacements (whose locations are the sites
of the positive PV anomalies, or troughs).
Therefore:
• The induced wind field advects lower
PV northward just to the east of the PV
maxima, and high PV southward just to
the west of PV maxima.
• Consequently, the wave pattern in the
PV field, as well as its induced velocity
field, propagates to the west.
Propagation effects as a
function of scale:
• Large-scale PV anomalies induce
relatively strong wind fields.
• Small-scale PV anomalies induce
relatively weak wind fields.
• Consequently, the westward
propagation effect is greatest for long
waves, and the smallest for short waves
Consider the effect of adding a
basic westerly advecting wind:
• This basic current acts to advect the entire
wave pattern to the east (eastward).
• Consequently, the effect of eastward
advection in dominant in short waves.
• Whereas, the effect of westward propagation
is dominant in long waves.
• Long waves tend to retrogade to the west,
while short waves travel to the east.
Movement of surface cyclones and anticyclones
on level terrain (as in Fig. 1.152):
Consider a reference state of potential temperature:
North
-

+
Consider that air parcels are displaced alternately
poleward and equatorward within the east-west
channel. Potential temperature is conserved for
isentropic processes (as in Fig. 1.153)
Since =0 at the surface, potential temperature changes
Occur due to advection only
-
North
-
+
L/4

L/4
+
The previous slide shows the maximum cold advection
occurs one quarter of a wavelength east of cold
potential temperature anomalies, with maximum warm
advection occurring one-quarter of a wavelength east of
the warm potential temperature anomalies. The entire
wave travels (propagates), with the cyclones and
anticyclones propagates eastward.
Just as with traditional quasi-geostrophic theory, surface cyclones
Travel from regions of cold advection to regions of warm advection.
Surface anticyclones travel from regions of warm advection to
regions of cold advection. Note that we did not need to consider
explicitly the effects of vertical motion, as we did when we used
isobaric, quasi-geostrophic reasoning.
Orographic effects on the motions of
surface cyclones and anticyclones
Consider a statically stable reference state in the vicinity of
mountains as shown below, with no relative vorticity on a potential
Temperature surface (as in Fig. 1.154)
z
+

-
x
Note that cyclones and anticyclones move with
higher terrain to their right, in the absence of
any other effects (as in Fig. 1.155).
+
-

N
+
Mountain
Range
The formation of upper-level systems;
baroclinic instability (pp. 208-211)
• Consider a two-layer atmosphere (Fig.
156.a), in which in each layer, we have
an alternating train of positive and
negative PV anomalies
(From Bluestein, 1993)
Top layer:
• PV increases to the north mostly because of increase
in the Coriolis parameter to the North.
• Additionally, the static stability increases to the North
• Also, the temperatures decrease to the north with the
horizontal temperature gradient being concentrated in
the center of the channel (with accompanying strong
thermal wind). Therefore, there is cyclonic shear to
the North, and anticyclonic shear to the South. This
relative vorticity gradient is much stronger near the
tropopause, than is found in the lower troposphere.
Bottom layer:
• The PV gradient is oriented towards the South
in the lower troposphere
• The justification for this opposite sense of the
gradient is the existence of warm, low-level air
to the south, with increasing cyclonic shear, and
higher static stability (with isentropes becoming
more packed together near the ground in a
warm anomaly).
At the interface, assume
there is no basic current:
• The basic current is easterly in the
lower layer
• The basic current is westerly in the
upper layer
Because of this two layer structure:
• Upper-level disturbances will propagate
to the west
• Lower-level disturbances will propagate
to the east
• Upper-level disturbances will advect to
the east
• Lower-level disturbances will advect to
the west
If the disturbances are
relatively small:
• The effects of advection overwhelm
those effects of propagation
• Therefore, disturbances in the lower
layer will travel to the west
• And disturbances in the upper layer will
travel to the east
• The disturbances in each layer will
travel in opposite directions.
However:
• The upper-level PV anomalies induce
vortices in the lower layer, affecting the
distribution of PV in the lower layer
• The lower-level PV anomalies induce
vortices in the upper layer, affecting the
distribution of PV in the upper layer
With the slight westward shift with
elevation of the anomalies:
• The wind fields in the top layer induced
by PV anomalies in the top layer and in
the bottom layer result in a greater
northward component of motion just
west of the PV minima - and a
greater southward component of motion
west of the PV maxima
+ than
would occur in the absence of the wind
field induced by the lower layer.
Therefore, the rate of westward
propagation of upper-level PV
anomalies is increased, and the net
rate of eastward motion is reduced
• Similarly, the sum of the wind fields in the
bottom layer induced by the PV anomalies in
the bottom and top layers results in a greater
northward component of induced wind east of
the PV maxima + and a greater southward
component of motion east of the PV minima than would occur in the absence of the wind
field induced by the upper layer alone
Therefore, the rate of eastward
propagation is increased below, and
the net rate of westward motion of
the lower wavetrain is reduced.
Therefore, the wavetrains try to
‘lock’ onto one another: Each
prevents the other from racing off in
the opposite direction
Let us assume that the wavetrains were shifted
more downstream, so that there is less tilt in the
vertical, so that the wavetrains were more in
phase with each other:
• The effects of wind fields induced by lower
wavetrain on upper wavetrain, plus the effects
of wind induced by upper on lower wavetrains
would act to increase the individual
propagation speeds.
• Therefore, the propagation effects would
increase in each layer, so that the wavetrains
would move into a configuration in which they
were again tilted more westward with height.
Conversely, if the wavetrains were
shifted upstream so that more
westward tilt was shown, the
propagation effects would
decrease, and advection by the
basic current would restore the
wavetrains to their original phase.
Therefore, there is an optimal
phase difference for which the two
wavetrains may lock onto one
another
• For very short wavelengths, however,
propagation could never be significant, if the
basic current were strong, and the wavetrains
could not lock onto one another
• For very long wavelengths, propagation
would always overwhelm the effects of
advection, and the wavetrains would still not
lock onto one another
• Therefore, for a given vertical shear, the
two wavetrains can lock onto one
another only for a certain range of
wavelengths.
• If L is within range for which the
wavetrains can lock onto one another,
then total induced velocity pattern is L/4
out of phase with the displacement
pattern
• Therefore, the locations at which the PV
contours are displaced farthest to the north
are subjected to more northward
displacements, while locations at which PV
contours are displaced farthest to the south
are subjected to more southward
displacement.
• Therefore, the waves grow in ampitude
• Therefore, for a certain range of wavelengths,
depending on the vertical shear, troughs and
ridges will grow in amplitude if they lean
westward with height
• Additionally, using PV thinking, if the
wavetrains lean eastward with
increasing height, then for a certain
range of wavelengths, the two
wavetrains can lock onto one another,
and decay in amplitude
Effect of static stability on
baroclinic instability:
• For a given wavelength, the depth of the layer
affected by a PV anomaly increases as the
static stability decreases
• Therefore, the effect of propagation is
enhanced at low static stabilities, because
the wind field induced by a wavetrain at one
level on the other level is enhanced.
• Therefore, while the induced wind field
is weak for typical static stabilities and
short wavelengths, it is relatively strong
if the static stability is low enough
• Thus, it may be possible for short wave
wavetrains (which could not lock onto
one another at typical static stabilities)
to lock onto one another.
• Furthermore, for long waves, induced winds
are also stronger for lower static stabilities.
• The induced winds may become so strong,
that long wave wavetrains that could lock
onto each other at typical static stabilities
cannot do so at lower static stabilities,
because the propagation effects are too
strong.
• Therefore, the effect of lower static stability is
to reduce the scales at which baroclinic
instability occurs.
• We would expect to find shorter wavelengths
growing in an environment of weak static
stability, such as is the case over relatively
warm oceans during the winter, in which
small, intense cyclogenesis occurs.
References:
• Bluestein, H. B., 1993: Synoptic-dynamic
meteorology in midlatitudes. Volume II:
Observations and theory of weather systems. Oxford
University Press. 594 pp.
• Hoskins, B. J., M. McIntyre, and A. Robertson, 1985:
On the use and significance of isentropic potential
vorticity maps. Quart. J. Roy. Meteor. Soc., 111, 877946.