Transcript Potential vorticity and the invertibility principle (pp

Potential vorticity and the
invertibility principle (pp. 187-195)
To a first approximation, the
atmospheric structure may be
regarded as a superposition of
positive and negative anomalies.
Further, the observed wind field is
aproximately equal to the sum of
the winds associated with anomaly
separately.
The actual PV field is the sum
of the reference state and
each of the anomalies
• The reference state is that of constant f,
and zero relative vorticity
• PV in the reference atmosphere is a
function only of potential temperature
We impose a balance
condition on the wind field
Geostrophic, gradient, and hydrostatic assumptions are
examples of balance
Problem:
To demonstrate that it is
possible to determine,
uniquely, the distribution of
both vorticity and static
stability associated with a PV
field given the appropriate
boundary conditions
Given the balance condition,
assumption of a reference state,
and boundary conditions:
• Only one set of vorticity and static
stability values fit the global distribution
of PV
• The balance condition used must be
compatible with the space and time
scales of the observed air motion
Given gradient wind and
hydrostatic balance:
• The second-order PDE for the wind is
shown in eq. 1.9.22.
• It is highly non-linear when relative
vorticity, static stability, and f are
functions of either potential temperature
(vertically) or r (horizontally).
The invertibility principle:
• Winds are induced by the PV-anomaly
field
• Eq. 1.9.22 is similar in structure to that
of the quasi-geostrophic omega
equation.
The effect of scale:
• The vertical scale of a PV anomaly is
proportional to the horizontal scale, and
inversely proportional to the square root of
the static stability parameter (eq. 1.9.33)
• Typical vertical scale is about 60K (see
sample soundings), which is as deep as the
typical troposphere
• Therefore, a synoptic-scale PV anomaly at
upper-levels can induce a wind field all the
way down to the ground
Relationship of scale to wind
strength (eq. 1.9.38)
• Small-scale PV anomalies of a given strength
induce weak wind fields, whose vertical
influence is only in a shallow layer
• Large-scale PV anomalies of the same
strength induce strong wind fields, whose
vertical influence is deep.
• The response of the atmosphere to PV
anomalies is dependent on scale
Isentropic coordinates
(potential temperature is the
vertical coordinate)
• Air parcels will conserve potential
temperature for isentropic processes
• Vertical motions can be visualized
• moisture transports can be better
visualized than on pressure surfaces
• Isentropic surfaces can be used to
diagnose potential vorticity
Consider the comparison of the
cross sections we have been
viewing:
temperature cross section
potential temperature cross
section:
isentropes slope up to cold air
and downward to warm air
high/low pressure on a theta
surface corresponds to warm/
cold temperature on a pressure
surface
700 hPa heights (m; solid) and
Temperature (K; dashed)
292 K Montgomery stream function
((m2 s-2 /100) solid) and pressure
(hPa; dashed)
Potential vorticity structures
•
•
•
•
surface cyclone
surface anticyclone
upper-tropospheric trough
upper-tropospheric ridge
Surface cyclone (warm ‘anomaly’)
PV = g(-q/p)zaq
• warm air is associated
with isentropes
becoming packed near
the ground (more PV)
• surface cyclone is
associated with a warm
core with no
disturbance aloft (zT=
zgu- zgl=0-zgl<0
200
Pressure
(hPa)
cold
0
warm
cold
more stable
distance (km)
1000
4000
Surface anticyclone (cold ‘anomaly’)
PV = g(-q/p)zaq
• cold air is associated
with isentropes
becoming less packed
near the ground (less
PV and smaller static
stability)
• surface anticyclone is
associated with a cold
core with no
disturbance aloft (zT=
zgu- zgl=0-zgl>0
200
Pressure
(hPa)
warm
0
cold
warm
less stable
distance (km)
1000
4000
Upper-tropospheric trough (positive PV
‘anomaly’)
PV = g(-q/p)zaq
• cold tropospheric air is
associated with
isentropes becoming
more packed near the
tropopause (more PV
and greater static
stability)
• upper tropospheric trough is
associated with a cold core
cyclone with no disturbance
below (zT= zgu- zgl= zgu-0>0
200
cold
warm
0
warm
more
stable
cold Pressure
cold
less stable
distance (km)
(hPa)
warm
1000
4000
Upper-tropospheric ridge (negative PV
‘anomaly’)
PV = g(-q/p)zaq
• warm tropospheric air is
associated with
isentropes becoming
less packed near the
tropopause (less PV
and smaller static
stability)
• upper tropospheric ridge is
associated with a warm core
anticyclone with no
disturbance below (zT= zguzgl= zgu-0<0
200
warm
cold
cold
less
stable
warm
warm
Pressure
(hPa)
cold
more
stable
0
distance (km)
1000
4000
Comparison of potential vorticity
analyses with traditional quasigeostrophic analyses
• Focus is on the PV perspective of QG
vertical motions and the movement of
high and low pressure systems
OK, but what about PV????
Consider a positive PV anomaly (PV maximum) aloft in a
westerly shear flow:
z
+ PV anomaly
0
x
Now, consider a reference frame of the PV
anomaly in which the anomaly is fixed:
Consider the quasi-geostrophic
Vorticity equation in the reference
Frame of the positive PV anomaly
z
0= -vg(zg
+ PV anomaly
>0
CVA; >0
0
<0
AVA; <0
x
+ f)-f0
Now, consider the same PV anomaly in which
the anomaly is fixed from the perspective of the
thermodynamic equation:
z
+ PV anomaly
cool
0
x
+ PV anomaly
z
>0
0
0 = -vg  T + s(p/R)
cool
CA
WA
x
<0
Consider vertical motions in the vicinity of a warm
surface potential temperature anomaly (surrogate PV
anomaly) from the vorticity equation:
0= -vg(zg
z
CVA
>0
AVA
<0
>0
0
<0
x
+ PV
+q
+ f)-f0
Consider vertical motions in the vicinity of a warm
surface potential temperature anomaly (surrogate PV
anomaly) from the thermodynamic equation:
0 = -vg  T + s(p/R)
z
y
cold
<0
>0
WA
CA
warm
+ PV
+q
Movement of surface cyclones and anticyclones
on level terrain:
Consider a reference state of potential temperature:
North
q-q
q
q+q
Consider that air parcels are displaced alternately
poleward and equatorward within the east-west
channel. Potential temperature is conserved for
isentropic processes
Since =0 at the surface, potential temperature changes
Occur due to advection only
q-q
North
-
+
L/4
q
L/4
q + q
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.
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
z
q+q
q
q-q
x
Note that cyclones and anticyclones move with
higher terrain to their right, in the absence of
any other effects.
q+q
q-q
q
N
+
Mountain
Range
References
• Bluestein, H. B., 1993: Synoptic-dynamic meteorology in midlatitudes.
Volume II: Observations and theory of weather systems. Oxford
University Press. 594 pp.
• Dickinson, M. J., and coauthors, 1997: The Marcch 1993 superstorm
cyclogenesis: Incipient phase synoptic- and convective-scale flow
interaction and model performance. Mon. Wea. Rev., 125, 3041-3072.
• 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.
• Morgan, M. C., and J. W. Nielsen-Gammon, 1998: Using tropopause
maps to diagnose midlatitude weather systems. Mon. Wea. Rev., 126,
2555-2579.