Inertial, Symmetric and Conditional Symmetric Instability (CSI)
The following is not meant to be a self-contained tutorial. It is
meant to accompany active discussion and demonstration in the
classroom. Most of the figures are drawn from Markowski and
Richardson (2010).
1. Magic Number for Inertial Instability
I am going to start with equation (9) from the last handout and
renumber it.
dv
= - f (M - M g )
dt
(1)
The student can easily figure out how the terms on the right of (1)
can collaborate to contribute to bring a parcel perturbed out of the
base state (in this case, jet stream) back to its initial latitude.
However, the concept still may be elusive.
The example we were discussing starts with no pressure (or height)
gradient along the x-axis and an invarying pressure (or height)
gradient along the y-axis. Thus the base state geostrophic flow is
westerly. Say an air parcel initially in geostrophic balance at some
initial latitude, say 40N, in this sort of environment is perturbed to
higher latitudes.
The air parcel will be moving into regions in which the Coriolis
parameter has a greater value than that at its initial latitude. As the
air parcel moves northward, it will move into an environment in
which the there is no longer a balance between pressure gradient
and Coriolis accelerations. The net acceleration will be southward,
bringing the air parcel back to its initial latitude. It will overshoot
this latitude and oscillate about it.1 This is the state of inertial
stability.
By the way, in Metr 500/800 we’ll discuss another manifestation
of this leading to Rossby Waves, called Conservation of Absolute
Vorticity
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But now, let’s think about how we can get the air parcel to continue
to be accelerated northward out of the current. To do that,
remember that equation (1) has to return a positive result.
dv
>0
dt
(2)
Putting (2) into (1) we see that this can only occur for situations in
which the term in parentheses in equation (1) returns a negative
result. This occurs when the air parcel perturbed northward out of
the flow, conserving its M, moves into regions in which the value
of the ambient geostrophic momentum is greater than M.
That can only occur if Mg increases northward, or
(M g f - M gi ) > 0
(3a)
or, to put it in a way that fits the example we are considering
¶M g
>0
¶y
(3b)
Recall that the definition of the geostrophic momentum, Mg, is
ug - fy = M g
(4)
Inserting (4) into (3b) and making the so-called Beta Plane
approximation (which is that the north-south variation of f is so
small across the middle latitudes that it can be approximated by a
constant value of f), the left side of (3b) becomes
¶M g ¶ug
=
-f
¶y
¶y
(5)
Now, recall the definition of geostrophic relative vorticity
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¶vg ¶ug
zg =
¶x ¶y
(6)
Since, in the example we are assuming that the base state flow is
geostrophic and purely westerly, then equation (6) becomes
zg = -
¶ug
¶y
(7)
The absolute vorticity of the geostrophic wind is
hg = z g + f
(8)
Put (7) into (8) and that result into (5) to get
¶M g
= -hg
¶y
(9)
Equation (9) states that the condition for inertial instability is that
the absolute vorticity is negative. In other words, the geostrophic
momentum increases northward in regions in which the absolute
geostrophic vorticity is negative.
This is something that can be easily diagnosed on a weather map.
A real case in which precipitation bands developed in areas of
negative absolute geostrophic vorticity is provided in Fig. 1.
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Figure 1. Rapid Update Cycle initialization of 600 mb heights
(dm). Negative absolute geostrophic voriticy is shown as dashed
red contours at intervals of -3 X 10-5 s-1 and orange areas show
regions of negative potential vorticity. Thus, the orange and red
areas denote regions of inertial instability. The red areas were
generally coextensive with bands of precipitation observed on this
day and time. (Source: Figure 3.6, Markowski and Richardson
2010)
2. More on Symmetric Instability
In the last handout, we discussed how a combination of static
instability and inertial stability could produce a situation in which a
slantwise deflection of the air parcel out of the base state would
produce a situation in which the parcel continues to be accelerated
(rather than returning to its initial position in an oscillation). Recall
that the static stability can be estimated by the slope and vertical
packing of the isentropes and the inertial stability can be estimated
by the packing and slope of the lines of constant geostrophic
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momentum. That illustration (Fig. 3.9) of the procecure in the last
handout gave us our first overview of symmetric instability.
Here’s a closer look at that same diagram but labeled Fig. 2 here.
Figure 2. Zoomed in view of Fig. 3.9 in last handout. (Source:
Figure 3.10, Markowski and Richardson 2010)
Note that for the case of intertial stability a parcel would be
laterally moved along the line segment ∆y and would find itself in
an environment of lesser geostrophic momentum. The parcel
would therefore oscillate back to its initial position.
But, if one considers the impact of static stability, one notes that
along that same transect, this air parcel enters a region in which its
potential temperature is greater than that of the environmental air.
Hence, the air parcel will also be accelerated along the path shown
by ∆z. The net path is slantwise, as shown by the line segment ∆s.
The orange area encompasses the region in which a combination of
paths/vectors yields a northward acceleration of the air parcel.
Hence, there is a range of values of potential temperature and
geostrophic momentum that would yield a symmetrically unstable
set of conditions for that air parcel. Thus, possible exponential
growth of the slantwise velocity would occur along the paths
extrapolated into that orange area. The “release” of symmetric
instability is termed slantwise convection.
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3. Conditional Symmetric Instability
The real atmosphere contains many situations in which upright
convection occurs in the presence of a Level of Free Convection
(LFC). As you know, such a situation is termed conditionally
unstable. Conditional (upright or gravitational or static) instability,
like absolute instabilty, can also be assessed from cross sections.
However, whereas cross-sections of potential temperature can be
used to assess absolute instability, cross sections of equivalent
potential temperature can be used to assess layers of conditional
instability. Equivalent potential temperature includes the effects of
the latent heat released, thus raising the potential temperature of
ascending air parcels. Areas on cross-sections in which equivalent
potential temperature decreases with height are associated with
layers that are conditionally unstable. In general, surfaces of
constant equivalent potential temperature have steeper slopes than
isentropes.
Isentropes of equivalent potential temperature can be used to assess
moist symmetric instability. This is known as “conditional
symmetric instability” (CSI) because the “condition” is that
somewhere along its path the air parcel encounters an LCL (and
becomes saturated) and, then, eventually encounters something
analagous to an LFC in a slantwise fashion.
The rest of the discussion uses this as a conceptual underpinning of
CSI. Be aware that some controversy exists whether this
explanation is best, rather than an approach that utilizes moist
adiabatic processes to develop an instability criterion analgous to
that for inertial instability, as given in Equation (9).
Those approaches use a variable called Moist Potential Geostrophic
Vorticity (MPVg) by evaluating CSI on the basis of the relationship
of the slope of MPVg contours to the slope of contours of equivalent
potential temperature. For us, this dispute is interesting but a
sidelight.
The instability criterion for conditional symmetric instability that
can be visualized easily on charts centers on the slopes of the
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equivalent potential temperature and geostrophic momentum
contours. The criterion is given in Equation (10).
æ Dz ö
æ Dz ö
>
çè Dy ÷ø
çè Dy ÷ø
qe
Mg
(10)
Equation (10) states that regions in which the slope of the contours
of equivalent potential temperature exceeds the slope of the
contours of geostrophic momentum are conditionally symmetrically
unstable.
Note that when contours of equivalent potential temperature slope
upwards, the air column bounded by them has a higher mean
equivalent potential temperature and, hence, larger values of
thickness. Thus, in visualizing precipitation bands, one should find
that the bands are tangent to thickness contours. Some of the
characteristics of the precipitation bands related to release
(slantwise convection) of CSI are summarized in Table 3.1.
(Source: Table 3.1, Markowski and Richardson, 2010).
Please note that one must be certain that the atmosphere is neither
absolutely unstable nor interially unstable in these regions, because
by definition static instability and/or intertial instability takes
“precedence” (is released) first. This simply means that if the
atmosphere is inertially unstable, the static stabilty is immaterial
and a horizontal wave will develop. Likewise, if the atmosphere is
absolutely unstable, the intertial instability is immaterial and
upright convection will dominate.
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A situation in which the instability critierion is met is illustrated in
Figure 3. This shows a real case of mesoscale bands, parallel to
thickness contours (seen on Fig. 4) in the warm advection area
northwest of a surface warm front. The cross-section given is along
the blue line stretching from DDC to MLC on Fig 3a.
Figure 3. (a) Vance Air Force Base (VNX) 88-D reflectivity at
1658 UTC 30 November 2006. (b) A vertical cross section tnormal
to hte bands at 1700 UTC from roughly Dodge City to Kansas City.
Grey stippled areas on (b) indicate regions in which the equivalent
potential temperature contours have steeper slopes than contours of
geostrophic momentum. Grey shaded region is aproximately
neutral or slightly unstable with respect to CSI. (Source: Figure
3.11, Markowski and Richardson 2010).
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Regions analagous to areas of CAPE can be approximated by the
area bounded by the lines of constant geostrophic momentum and
the contours of equivalent potential temperature (shaded in grey on
Fig. 3b). To visualize this, recall that parcels actually move along
equivalent potential temperature contours. Where those contours
are steeper than the geostrophic momentum contours, the area
bounded is analagous the areas meteorologists shade (in red) on
conventional sounding analyses to indicate positive areas on
soundings.
Figure 4. NCEP Reanalysis of 1000-500 mb Thickness (dm) at
1200 UTC 30 November 2006.
Also, the maximum updraft speed in the slanted direction can be
estimated for CSI areas using the slantwise convective available
potential energy (SCAPE), which is basically the integration of
buoyancy in the area in which the equivalent potential temperature
contours are steeper than geostrophic momentum contours.
Vertical velocities within the updrafts of CSI areas are on the order
of 1 m s-1, compared to a few cm s-1 on the synoptic scale and 10 m
s-1 in the buoyant updrafts of strong thunderstorms.
4. Reference
Markowski, P. And Y. Richardson, 2010: Mesoscale meteorology
in midlatitudes. Wiley-Blackwell, 2010. ISBN: 978-0470742136.
430 pp.
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