`This discussion is
"conventional" in the way it describes the topic ... I am now of the
opinion that this approach needs to be discarded, in favor of an
approach that uses proper analogs with upright convection theory, and
which uses proper terminology and diagnostic approaches. I think that
the primary value of this "conventional" discussion now is in
illustrating the conceptual tools used to explore instability, in
general, rather than as a guide for application of the specific ideas
embodied in "conventional" CSI. I recommend consideration of the
revised concepts developed by Schultz et al. (2000), available here.`

Most everyone knows about ordinary buoyant instability, sometimes
incorrectly referred to as *convective*
instability.^{[1]} Basically, if a parcel
is displaced vertically, when the atmosphere is unstable in this
sense, the acceleration on the parcel is in the same direction as the
displacement. From simple parcel theory, the relevant equation is the
inviscid vertical equation of motion:

With appropriate manipulations, this can be transformed to:

where B is the buoyancy. Again, with some manipulations, this can be rewritten as:

where G_{d} is the dry
adiabatic lapse rate and *g* is the
actual lapse rate. This differential equation has the solution:

so that when the lapse rate exceeds the dry adiabatic, parcels are
considered unstable. __Note__: this simple parcel theory has some serious issues, as discussed in Doswell and Markowski (2004), available here.

The existence of this instability means that in the real atmosphere, lapse rates exceeding the dry adiabatic are pretty rare, except very near the surface. Whenever superadiabatic lapse rates exist, it's pretty likely that convective overturning will occur and remove the instability, such that the lapse rate will be driven back toward the dry adiabatic.

In the presence of moisture in the form of water vapor, however,
another option is present, the so-called *conditional*
instability. It's called conditional because the instability depends
on whether or not a parcel is saturated. A lapse rate *g* is considered conditionally unstable when
it is between the dry (G_{d}) and
the moist adiabatic (G_{m}< G_{d}) lapse rates.

In the real atmosphere, parcels become saturated during some sort
of forced ascent, during which the parcels are normally negatively
buoyant, so that energy must be supplied to continue their ascent. At
some point (the saturation point) during their ascent, the parcels
become saturated ... this is also known as the lifting condensation
level (LCL). It is likely that the parcel is still negatively buoyant
at the LCL, so the continued ascent typically still depends on the
presence of forced lifting, but now the ascent is no longer along a
dry adiabat. Rather, saturated ascent is along a moist adiabat. At
some point above the LCL, the parcel ascent curve crosses the
environmental curve; the crossing point is known as the level of free
convection (LFC), above which positive buoyancy commences. This
should all be familiar territory to convection afficianados and can
be illustrated readily on a thermodynamic diagram. As it is typically
applied, parcel theory involves finite displacements. However, the
mathematical theory I've just given is associated with a differential
equation and the notion of stability is defined in this theoretical
context simply by the sign of (G-*g*) ... where G is
either G_{d} or G_{m} - depending on whether or not
the parcel is saturated. The implied displacements are infinitesmal.
The sign of (G-*g*) is negative for saturated parcels in a
conditionally unstable lapse rate situation, the square root produces
a pure imaginary number and the time change of z becomes exponential
growth ... bingo! ... it's unstable according to the simple theory.

Conditional instability means that for a saturated parcel, we
**replace** G_{d} with G_{m} in our equation for z(t). Thus,
whereas parcels ascending unsaturated are stable, parcels ascending
saturated (hence, cooling at the moist adiabatic lapse rate, G_{m} ) are unstable, because G_{m} *< g
* < G_{d}.

Consider what true *convective* instability means (see
footnote #1, above). In a case where q_{e} decreases with height, it can be
shown that a parcel at the bottom of a finite layer (e.g., bounded by
two pressure surfaces, p_{1} and
p_{2}<p_{1}) will reach saturation sooner during
layer lifting than will a parcel at the top of that layer. Continued
layer lifting beyond that point where the lowest layer reaches
saturation means that the top will be cooling at G_{d}, whereas the bottom is cooling at
G_{m}. This differential cooling
rate could increase the layer's lapse rate rather quickly. Continued
lifting, however, would eventually bring the entire layer to
saturation, so this destabilization associated with environmental
conditions of convective instability does not continue indefinitely
during forced ascent.

Is this what happens in cases of convection? This process is not likely to be an important one in real convection, where parcels from low levels are simply forced upward to their LCLs and eventually to their LFCs. The existence of true convective instability in situations involving real convection is typically associated with significant buoyant instability for lifted parcels, and the hypothetical process of lapse rate increases due to layer lifting in such cases is really not very physically relevant. In some relatively infrequent cases involving ascent of a layer (perhaps in an extratropical cyclone), this differential saturation rate of parcels in convectively unstable layers might be instrumental in an overall steepening of the lapse rate, but this simply makes ordinary, upright convection more likely.

If we consider a saturated displacement of a parcel, then if it
ascends *reversibly* (i.e., the condensing moisture doesn't fall
out but it retained in the parcel), it can descend along the same
moist adiabat (i.e., the moist adiabat associated with the parcel's
value of q_{e}).
^{[2]} For a parcel rising in a
conditionally unstable atmosphere, it finds itself positively buoyant
for upward displacements and negatively buoyant for downward
displacements, so it is easy to see that it is unstable. Clearly,
when q_{e} *increases* with
height (the normal situation), the accelerations are opposed to the
displacement, as illustrated in Fig. 1

Figure 1. Illustration of convective instability. Parcel displacements are shown with the purple arrows, the acceleration for vertical displacements is shown in turquoise, the acceleration for a slantwise displacement is shown in red, and the vertical component of the slantwise displacement is shown as hatched lavender. The q

_{e}contours are colored such that warmer colors correspond to warmer q_{e}values, but they also are labeled in deg K. In green, the parcel's state relative to its environment is noted: For an upward vertical displacement, for example, it is cooler than its environment, so the acceleration is downward.

Also noted on Fig. 1 is a

Speaking physically, the force associated with convective stability is that due to gravity, so this might be called a conditional gravitational stability problem.

Now let us turn to inertial stability, a topic not necessarily as familiar to operational forecasters. Consider first the two inviscid horizontal equations of motion:

where *f* is the Coriolis parameter, (u,v) are the components
of the horizontal wind, and (u_{g}, v_{g}) are the
components of the geostrophic wind. Combining these into a single
horizontal vector wind (**V**) equation gives

which means that the acceleration (d**V**/dt) is directed to
the right of the ageostrophic wind. If the wind is supergeostrophic,
the ageostrophic wind has a component in the same direction as the
geostrophic wind, so the associated acceleration is directed toward
the right (towards high pressure), whereas if the wind is
subgeostrophic, the ageostrophic wind has a component against the
geostrophic wind, so the associated acceleration is directed toward
the left (towards low pressure). The effect of the acceleration is to
return the flow to a state of geostrophic balance; supergeostrophic
winds are reduced by being accelerated towards high pressue,
subgeostrophic winds are increased by being accelerated towards low
pressure.

In mid-latitudes, the wind through most of the troposphere is
predominantly westerly (i.e., the u-component). There may be
accelerations in either the u or the v component, but for simplicity,
we can consider the flow to be defined by a basic state of pure
westerly flow with perturbations in either u or v. We can define a
quantity, then, known as the geostrophic momentum, m_{g}:

where y is simply the local meridional coordinate. There is no
corresponding y-component of the geostrophic momentum in the base
state. In this base state, then, m_{g} decreases northward (y
increases) and upward (u_{g} generally increases with
height). Note that *f* increases northward.

Given this as a description of the basic state, consider Fig. 2. This illustrates the stability of horizontal displacements in a base state where geostrophic momentum increases upward and southward.

Figure 2. Illustration of inertial instability. Parcel displacements are shown with the purple arrows, the acceleration for horizontal displacements is shown in turquoise, the acceleration for a slantwise displacement is shown in red, and the horizontal component of the slantwise displacement is shown as hatched lavender. The m

_{g}contours are colored such that warmer colors correspond to larger m_{g}values, but they also are labeled in m s^{-1}. As in Fig. 1, the parcel's state relative to its environment is in green. For a northward displacement, it is faster than its environment, which means an acceleration southward.

Figure 2 reveals that for horizontal displacements, the parcel is inertially stable; that is, the accelerations associated with horizontal displacements tend to restore the parcel to its original position. When the displacement is southward, the parcel's zonal speed is subgeostrophic, so its acceleration is to the left (toward lower pressure), with the inverse happening for northward displacements.

Figure 3. Illustration of the condition of conditional symmetric stability, with the q

_{e}surfaces as hatched lines and the m_{g}surfaces as solid lines.

As was the case for convective instability, it can be seen that for a

Again taking a physical viewpoint, the restoring force in this case is the Coriolis force, and this situation is a conditional dynamic instability.

Figure 4. Illustration of the condition of conditional symmetric instability, with the q

_{e}surfaces as solid lines and the m_{g}surfaces as hatched lines.

CSI - A combination of the preceding

Finally, let us consider these situations combined. That is, we
have an atmosphere that is stable to **both** vertical and
horizontal displacements at the same time. This is illustrated in
Figs. 3 and 4. Note, in passing, that the presence of
*horizontal* thermal gradients makes this an instability
associated with baroclinity. In a barotropic atmosphere, the
isentropic surfaces (if any) are horizontal. Neither buoyancy nor
inertial instability require horizontal thermal gradients, so
symmetric instability is a type of *baroclinic* instability.

Basically, the question of CSI depends on how q_{e} varies along a surface of constant
m_{g}. For saturated parcels (recall, the "condition"
associated with CSI is that the parcel be saturated) moving along an
m_{g} surface, in the case of stability, they will find
themselves in a region where their acceleration is opposed by
buoyancy. As seen on the figure, parcels rising along the m_{g
}= 50 m s^{-1} surface find themselves negatively
buoyant (in terms of q_{e }) and
parcels sinking along that surface find themselves positively
buoyant. This amounts to a restoring force and constitutes
conditional symmetric *stability *. In this situation, q_{e} increases with height along an
m_{g} surface.

On the other hand, as seen in Fig. 4, the condition of
*instability* is associated with q_{e} decreasing with height along
m_{g} surfaces. The situation now is reversed, so that
parcels rising along m_{g} = 40 m s^{-1} now find
they have a higher q_{e} than
their environment, with the inverse true for descending parcels.

Unfortunately, CSI is (at least for the moment) a "fashionable"
topic. Thus, it has a tendency to appear in forecast discussions in
inappropriate ways. When the stratification is convectively unstable
(the q_{e} contours have a
negative slope), the situation might well also exhibit moist CSI.
However, in such cases, it is quite clear from theory and observation
that *upright convection would dominate any slantwise
convection* ... the growth rate of upright convection is
considerably larger than that of CSI, so any CSI is overwhelmed.
Strictly speaking, it is inappropriate to speak of CSI when the
atmospheric is either convectively or inertially unstable. Only in
cases where it is *both* convectively and inertially stable is
it appropriate to speak of the possibility of CSI.

Further, not all cases of "slantwise convection" involve CSI. When parcels that are stable to purely vertical displacements are carried along slantwise paths, they may reach a point where they become positively buoyant. From that point, the convection is identical to ordinary, upright convection. Typically, this upright convection is "elevated" in the sense that the parcels participating in the convection do not originate in the boundary layer directly beneath the convection. Rather, they originate elsewhere and have followed a slantwise path to reach the point where they are positively buoyant. CSI is only one of many mechanisms that can carry parcels along a slantwise path; extratropical cyclones often do this poleward of warm fronts and frontogenetically-induced motions also produce slantwise ascent. The presence of weak symmetric stability is not sufficient to validate the occurrence of CSI, since convective instability is often associated with weak symmetric stability.

Thus, it is not true that CSI and "slantwise convection" are
identical concepts! I believe the *terminology* here needs some
careful thought and revision, to make it clear what is going on
physically and to distinguish among the physical processes operating
in any given situation.