Applications of ‘IPV’ thinking for time-dependent dynamical

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					  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 synoptic-
  scale 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   N
    maxima and
    minima, correspond,
                              max   min   max   min
    respectively to
    troughs and ridges    N



  • instantaneous winds
Consider a PV reference state
     (as for Fig. 1.150):
• Consider the PV             larger PV
                                          PV+2dPV
  contours at right       N
  with increasing PV                      PV+dPV

  northward (owing
  primarily to increase                   PV

  of the Coriolis
                                           PV-dPV
  parameter)
Consider the introduction of alternating PV
      anomalies (as for Fig. 1.151):
  • The sense of the wind
    field that is induced by       larger PV
    the PV anomalies           N
  • There will be a
    propagation to the left
    or to the west (largest                     PV+2dPV

    effect for large                        L
    anomalies                      +    -   PV+dPV
                                                   +
  • This effect is opposed                        PV
    by the eastward
    advective effect                               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,
  877-946.

				
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posted:5/27/2010
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