Schultz, D. M., P. N. Schumacher, and C. A. Doswell III, 2000: The
intricacies of instabilities. *Mon. Wea. Rev.,* **128,**
4143-4148. [AMS]
[PDF]
[HTML]

© Copyright 2000, American Meteorological Society

**The Intricacies of Instabilities**

Philip N. Schumacher

*NOAA/National Weather Service, Sioux Falls, South Dakota*

Charles A. Doswell III

*NOAA/National Severe Storms Laboratory, Norman, Oklahoma*

NOTES AND CORRESPONDENCE

*Monthly Weather Review*

Received April 19, 2000; Revised May 30, 2000

Published in the December 2000 issue: pp. 4143-4148

To be published with and after

Sherwood, S. C., 2000: On moist instability.
*Mon. Wea. Rev.*, **128,** 4139-4142.

*Corresponding author address:*

Dr. David M. Schultz

NOAA/National Severe Storms Laboratory

1313 Halley Circle

Norman, OK 73069

E-mail: `schultz@nssl.noaa.gov`

In response to Sherwood's comments and in an attempt to restore proper usage of terminology associated with moist instability, the early history of moist instability is reviewed. This review shows that many of Sherwood's concerns about the terminology were understood at the time of their origination. Definitions of conditional instability include both the lapse-rate definition (i.e., the environmental lapse rate lies between the dry- and the moist-adiabatic lapse rates) and the available-energy definition (i.e., a parcel possesses positive buoyant energy; also called latent instability), neither of which can be considered an instability in the classic sense. Furthermore, the lapse-rate definition is really a statement of uncertainty about instability. The uncertainty can be resolved by including the effects of moisture through a consideration of the available-energy definition (i.e., convective available potential energy, CAPE) or potential instability. It is shown that such misunderstandings about conditional instability were likely due to the simplifications resulting from the substitution of lapse rates for buoyancy in the vertical acceleration equation. Despite these valid concerns about the value of the lapse-rate definition of conditional instability, consideration of the lapse rate and moisture separately can be useful in some contexts (e.g., the ingredients-based methodology for forecasting deep, moist convection). It is argued that the release of potential (or convective) instability through layer lifting may occur in association with fronts, rather than with isolated convection, the terminology ``convective'' being an unfortunate modifier. The merits and demerits of slantwise convective available potential energy (SCAPE) are discussed, with the hope of improving diagnostic methodologies for assessing slantwise convection. Finally, it is argued that, when assessing precipitation events, undue emphasis may appear to be placed on instability, rather than the forcing for ascent, which should be of primary importance.

Sherwood (2000) raises some excellent points regarding usage of the
term conditional instability in atmospheric science. He discusses two
ways that conditional instability is commonly used: the *
lapse-rate definition* (i.e., the environmental lapse rate lies
between the dry- and the moist-adiabatic lapse rates) and the *
available-energy definition* (i.e., a parcel possesses positive
buoyant energy). As Sherwood argues, the lapse-rate definition does
not fit the strict definition of an instability and the
available-energy definition is a subcritical instability, arguments
with which we agree fully. Despite his valid concerns, we argue that
the lapse-rate definition is useful in some contexts.

Sherwood's comments lead to questions about how the terminology for moist instability arose. Why would the founding fathers of scientific meteorology use the terms conditional and potential (convective) instability for situations that are not even instabilities in the strictest sense? As will be demonstrated in this paper, they appear to have understood the subtleties of this terminology-more recently, some meteorologists appear to have forgotten them.

Sherwood's concerns also affect the analogy Schultz and Schumacher (1999) drew between moist gravitational instability and moist symmetric instability through the ingredients-based methodology for forecasting deep moist gravitational convection (e.g., McNulty 1978, 1995; Doswell 1987; Johns and Doswell 1992). Consequently, in our response, we wish to differentiate among our concerns about the application of the terminology associated with conditional instability (section 2), potential instability (section 3), the ingredients-based methodology (section 4), and moist symmetric instability (section 5). Concluding thoughts are found in section 6.

In section 2a, the origins of the terminology for conditional instability are explored. The fallacies noted by Sherwood were actually understood at the time this terminology was defined in the 1930's, but a popular mathematical derivation demonstrating the supposed equivalence between the lapse-rate and the available-energy definitions of conditional instability (reproduced in section 2b) appears to have reinforced these fallacies, or at least caused them to be more readily forgotten.

The mathematical basis for the vertical stability of moist atmospheres appears to have its roots in the 1860's (e.g., Kutzbach 1979, 53-58). Independently, Reye (1864) in Germany and Peslin (1868) in France determined the lapse-rate criteria for what is now commonly termed absolute instability (environmental lapse rate greater than the dry-adiabatic lapse rate), absolute stability (environmental lapse rate less than the moist-adiabatic lapse rate), and conditional instability. At one time, conditional instability was known as ``liability of environment for saturated air" (Shaw 1926) or ``Feucht-labilität" (Refsdal 1930) before Rossby (1932) apparently coined the term now used today for environmental lapse rates between the dry- and moist-adiabatic lapse rates.

Conditional instability was explored further by Normand (1938), who
recognized two of the problems identified by Sherwood. First, Normand
(1938) acknowledged that an unsaturated conditionally unstable
atmosphere is not a true instability in the sense that a linearized
stability analysis would show infinitesimal perturbations growing at
an exponential rate. In fact, for unsaturated conditionally unstable
environments, finite displacements of unsaturated air parcels are
needed in order to realize the instability (if it can be realized at
all). Normand (1938, p. 52) described this state as ``instability for
big upward displacements" or *latent instability* (Normand
1931a,b), analogous to the metastability or subcritical-instability
concepts from fluid dynamics (e.g., Sohoni and Paranjpe 1937; Emanuel
1997; Sherwood 2000). The term latent instability, however, appears
to have fallen into disuse since Normand's time.

Second, as Sherwood has illustrated in his Fig. 1, a state of
conditional instability is really a statement of *uncertainty*
with regard to stability. Normand (1938) understood that not all
conditionally unstable atmospheres lead to unsettled weather. Because
moisture is not accounted for in assessing conditional instability
(i.e., q_{es} or q_{e}^{*}, the saturated equivalent
potential temperature, is a function of temperature and pressure only,
not of humidity), some measure of the moisture profile is needed to
refine the classification of stability. Thus, the concept of
available energy was introduced. Normand (1938) subdivided
conditional instability into additional classifications based on what
we now term convective inhibition (CIN) and convective available
potential energy (CAPE).^{1}

Table 1 illustrates Normand's classification scheme. When a
conditionally unstable atmosphere is saturated, then a true state of
instability is present since any perturbation will grow. [For
example, Kain and Fritsch (1998) and Bryan and Fritsch (2000)
demonstrate that layers of saturated conditional instability exist in
advance of squall lines, features they call *moist-adiabatic
unstable layers*.] When a conditionally unstable atmosphere is
unsaturated, CAPE must be evaluated to determine the degree of
instability. If the conditionally unstable layer is surmounted by a
stable layer such that CAPE is zero, for example, then Normand (1938)
used the oxymoron *stable conditional instability*; that is, no
vertical displacement of parcels, however large, will produce any
positive buoyancy. A conditionally unstable atmosphere with positive
CAPE is viewed as possessing a subcritical or latent instability. If
CAPE exceeds CIN, then this state is termed termed *real (latent)
instability* (Sohoni and Paranjpe 1937). If CIN exceeds CAPE,
implying that more energy is put into lifting parcels than is obtained
once convection starts, then Sohoni and Paranjpe (1937) term this
situation *pseudoinstability* or *pseudolatent instability*.
Whereas Sohoni and Paranjpe (1937) and Normand (1938) compared the
relative sizes of CIN and CAPE to determine stability, modern
forecasters consider each separately. CIN is an important element in
the forecast process, but, in our opinion, not in classifying
stability [see also Petterssen (1956, 139-140)]. Thus, as Sherwood
has argued, the available-energy definition for assessing instability
appears to be much more useful than the lapse-rate definition, a point
we will return to in section 4. For the purposes of this
paper, ``conditional instability'' will hereafter refer to the
lapse-rate definition.

The previous section showed that not all soundings possessing layers
with conditionally unstable lapse rates can produce parcel
instability. The traditional association between conditional
instability and parcel instability, however, may owe its origins to
the following simple, but flawed, analysis using parcel theory.
Parcel theory employs observed or model-forecast soundings to assess
instability. In forecasting, these so-called *environmental
soundings* approximate the conditions that exist prior to convective
initiation, presumably because they are meant to represent the
``environment" that storms will form in. There are caveats with this
interpretation, however. First, defining a sounding representative
enough to be considered ``environmental" has a host of problems (e.g.,
Brooks et al. 1994). Second,
as mentioned in Schultz and Schumacher (1999, section 3e), parcel
theory neglects potentially important processes (e.g., mixing, ice
phase). Finally, processes that lift parcels in the atmosphere also
affect the ``environment" such that the sounding evolves during parcel
ascent (e.g., Normand 1938). Therefore, although the parcel method is
not without its problems, we are faced with using it to assess
instability, until a new paradigm for moist convection is developed.

Following Hess (1959, 97-98), the vertical acceleration of a parcel
([(d^{2}z)/( dt^{2})])
is proportional to the acceleration due to gravity (g)
times the difference between the parcel temperature (T_{p}) and the
environmental temperature (T_{e}):

Equation (3) will be a linear ordinary differential equation
when the coefficient of z, g(G_{p} - g)/T_{0}, is constant
with height. In that case, if G_{p} - g < 0,
perturbations will grow exponentially in time, resulting in release of
the instability. Thus, instability in the classic sense will be
manifest given two conditions: (a) G_{p} is constant over the
displacement z (i.e., the parcel remains saturated or unsaturated)
and (b) g is constant over the displacement z. Clearly, for
conditional instability, (a) would only be met if
the parcel was initially saturated. Thus, accounting for the process
by which finite displacements of unsaturated air parcels become
saturated is unaddressed by this formulation. In general, (b) is not
met either, as the lapse rate of the environmental sounding typically
varies with height.^{3} Thus, the problems with
this derivation are that the theory does not include the finite
displacement necessary to reach saturation and that the linearization
is often not valid, except in a very small neighborhood about the
original state. Thus, the standard mathematical exercise for equating
parcel/environment temperatures with parcel/environment lapse rates is
not appropriate for conditional instability in general.

This point is significant because situations occur in the atmosphere where parcel stability differs from lapse-rate stability. Consider a sounding with sufficient low-level moisture and warmth to yield substantial CAPE for some low-level parcel, from its level of free convection (LFC) to its equilibrium level (e.g., at 250 hPa). As that parcel ascends along its associated moist adiabat, it is quite possible that in some layer (e.g., 400-450 hPa), the environmental lapse rate could be absolutely stable, even to the extent of having a negative lapse rate. The presence of absolute stability in that layer, however, does not alter the fact that the ascending parcel is positively buoyant. In such cases, the stability of the layer (as defined by its lapse rate) can differ dramatically from the parcel stability (as defined by its buoyancy). Hence, it may be preferable in the future to use the term conditional instability only for the lapse-rate definition and resurrect the term latent instability for the energy-based definition.

As discussed in the previous section, the inability of the lapse-rate definition of conditional instability to ascertain unambiguously parcel stability led to the realization that the vertical profile of moisture needs to be accounted for. Thus, potential (or convective) instability was developed. The historical basis is described in section 3a. Although the layer-lifting mechanism originally perceived to release potential instability is generally not believed to occur in isolated convective storms, certain situations, discussed in section 3b, may be found where layer lifting of parcels produces deep moist convection.

The concept of *convective instability* was originally defined by
Rossby (1932) to be when one of the three following equivalent
conditions is met over a layer of the atmosphere: (1) the lapse rate
of wet-bulb temperature exceeds the moist-adiabatic lapse rate, (2)
the equivalent potential temperature q_{e} decreases with height,
or (3) the wet-bulb potential temperature q_{w} decreases with
height. As formulated by Rossby, convective instability is present
when a layer bounded by two pressure levels is lifted such that the
bottom of that layer reaches saturation before the top does.
Continued ascent steepens the lapse rate within the layer by the
difference between the dry- and moist-adiabatic lapse rates, until the
entire layer reaches saturation. A steepening of the lapse rate in
this fashion might produce CAPE in situations where none had
previously existed.

Although the terminology ``convective'' instability suggests a close
relationship to isolated convective cells,^{4} Rossby (1932, p. 26) suggested that layer lifting during
ascent over fronts was a possible means to realize convective
instability. ``Obviously,'' he stated, ``the nature of convection is
not such that a large body of air is lifted solidly to higher levels,
but it is equally plain that we can not hope to find a dependable
criterion for convection on the highly artificial assumption that one
small element at a time is lifted while the rest of the atmosphere
remains in equilibrium.'' To avoid misassociations with isolated
convection, Hewson (1937), a British meteorologist, coined the term
*potential instability* for the same concept. In Normand (1938,
p. 57) and the subsequent discussion (especially by Hewson on p. 66),
it appears that the argument over use of the terms convective or
potential instability initially was drawn down American versus British
lines. The terms now are used interchangeably.

This layer-lifting process, however, is *not* typically associated
with the development of isolated upright deep moist convection. If it
were, layer lifting initially would produce stable stratiform clouds,
which would then develop into deep moist convection. Although this
process does not appear to be acting in isolated convective storms, it
does appear to occur in other circumstances. For example, Schultz and
Schumacher (1999, section 5a), discuss examples of so-called *
downscale convective-symmetric instability* (Xu 1986) in which the
ascent occurring above warm fronts is punctuated with isolated buoyant
convective elements [e.g., the warm-frontal elevator/escalator concept
of Neiman et al. (1993)].
It is important to recognize that the paths of parcels in such
situations are likely to undergo slantwise displacements to their
lifting condensation levels (LCL's) and LFC's before releasing the
buoyant instability and becoming more upright; this process should be
distinguished from slantwise convection due to the release of CSI.

Potential instability is often diagnosed to be present in situations where deep, moist convection is likely, even if the process of convective initiation is not at all related to the layer-lifting process envisioned by Rossby (1932). Not all ``convective'' environments are characterized by the presence of potential (convective) instability, which is a potentially confusing aspect of the term, as noted by Hewson (Normand 1938, p. 66). Generally speaking, the terminology associated with moist instability in the context of gravitational convection is not particularly lucid. Sherwood's comments offer an opportunity to remind ourselves of this. Normand (1938) shows that the concepts associated with conditional instability and its associated metastable state in the presence of CAPE were understood 63 years ago. Regrettably, the clarity of Normand's exposition has not always been manifest in the literature on gravitational convection.

The ingredients-based methodology states that three ingredients (lift,
instability, and moisture) are required for deep, moist convection.
Sherwood argues that the ingredient of instability is unfortunately
labeled since the three ingredients are actually attempting to
diagnose the instability (of which the visible manifestation is deep,
moist convection). As we have seen in this paper already, there are
numerous problems associated with the meaning and use of the term
``instability.'' The use of the term instability among the
ingredients was intended to infer the lapse-rate definition of
conditional instability (e.g., Doswell 1987). The three ingredients
were chosen (a) to imply the presence of CAPE via conditional
instability and moisture, and (b) to realize that convective potential
via the ascent of parcels to their LFC's. Clearly, if a sounding has
no layer of conditional instability, deep, moist convection is
precluded (i.e., there can be no CAPE if lapse rates do not exceed
moist adiabatic somewhere in the environmental sounding). The three
ingredients are the *necessary* conditions for the initiation of
deep, moist convection and are not intended to diagnose an
instability, per se. If CAPE is present and the lift is enough to
attain an LFC, then these conditions also become *sufficient* for
deep, moist convection. The ingredients-based methodology does not
provide details regarding the distribution of conditionally unstable
lapse rates or moisture, because CAPE can be found in vertical
soundings under many different circumstances. Thus, we see the
success of the energy-based definition for evaluating instability: two
of the three ingredients are accounted for in a single combined
parameter: either CAPE or potential instability.

Although we have demonstrated problems with the lapse-rate definition, the advantage of the ingredients-based methodology is that the lapse rates and moisture are considered independently. Thus, forecasters anticipating the atmospheric changes allowing deep, moist convection to develop can more easily visualize the destabilizing influences of lapse rate and moisture separately, rather than trying to visualize the processes changing CAPE or potential instability. Therefore, we see no actual disagreement between ourselves and Sherwood in this matter, because the available-energy definition is accounted for by treating the lapse rate and moisture separately in the ingredients-based methodology. Sherwood's comments allow us the opportunity to clarify the intentions of the ingredients-based methodology.

Analogous to its gravitational counterpart, CSI can be diagnosed in
more than one way: the lapse-rate definition of CSI and the
available-energy definition of CSI. As discussed in Schultz and
Schumacher (1999), the lapse-rate definition of CSI is equivalent to
the M_{g}-q_{e}^{*} relationship or negative MPV_{g}^{*}, whereas
the available-energy definition of CSI is equivalent to SCAPE
(slantwise convective available potential energy). Sherwood argues
that SCAPE, not MPV_{g}^{*}, should be the favored form for diagnosis.
Unfortunately, SCAPE has not been found useful in what little
available literature has been published (Schultz and Schumacher 1999,
section 3d). Thus, Schultz and Schumacher (1999) argued that the most
logical and consistent way to diagnose CSI was to locate regions of
negative MPV_{g}^{*}. There are three possible explanations for this
apparent nonutility of SCAPE compared to MPV_{g}^{*}.

First, CAPE typically represents available energy through the
entire troposphere once the parcel reaches its LFC. The LCL and LFC
are usually fairly close to each other in gravitational
convection. With SCAPE, however, the vertical extent of the
instability is usually much more shallow (less than a couple hundred
mb) and the horizontal and vertical distances between the slantwise
LCL and slantwise LFC can be much greater. The location of CSI with
respect to the location of the lift may be critical because of the
typically shallow nature of the instability (and its location far
removed from the slantwise LCL). By using
MPV_{g}^{*}, the spatial relationship between the
instability and lift can be visualized. For example, knowledge of the
relative locations between the region of MPV_{g}^{*} <
0 and the lifting mechanism allows the forecaster to know how strong
the lift has to be to reach the instability. Operational experience
suggests that a separation of 100 or 200 mb between the region of
frontogenesis and the negative MPV_{g}^{*} layer
implies that a strong frontal circulation is required to access the
instability.

Second, the computation and display of SCAPE is problematic. Unlike CAPE, where a single vertically integrated value can be associated with each horizontal location and can be easily displayed on a horizontal map, SCAPE is an integrated quantity along a slantwise path. Thus, what is the best way to display such a quantity? At what horizontal location does it apply?

Third, observed values of SCAPE tend to be relatively small, whereas mesoscale numerical models tend to produce larger values. Perhaps an analogy can be drawn between gravitational and slantwise convection to explain this observation. Before convection starts, soundings in the environment would indicate positive available potential energy. Once convection occurs, soundings taken within storms would indicate little, if any, CAPE/SCAPE. Previous work on SCAPE has focused on the period when slantwise convection is already occurring (e.g., Emanuel 1988). This may explain the absence of SCAPE (i.e., the symmetrically neutral state) in such observational studies. In contrast, mesoscale models typically do not have the resolution or the parameterization to reduce SCAPE (e.g., Schultz and Schumacher 1999, section 7). Therefore, SCAPE may build up to values larger than those observed.

Although Schultz and Schumacher (1999) favored the lapse-rate
definition of CSI, further research on different approaches to
diagnose the potential for slantwise convection was encouraged. Based
on Sherwood's comments and to maintain consistency, SCAPE would appear
to be preferred over MPV_{g}^{*}. But, as we argued in
section 4 for gravitational convection, it may be preferable
to consider symmetric lapse rates (via MPV_{g}^{*}) separately from
moisture for some purposes. Also, research supporting SCAPE as an
effective diagnostic tool is lacking, prohibiting us from
enthusiastically supporting SCAPE. Previous work by Shutts (1990) and
more recent research by Dixon (2000) suggests some utility of SCAPE in
numerical models, particularly in anticipating the development of
cloud heads in oceanic cyclones. Gray and Thorpe (2000) also
reexamine the computation of SCAPE and suggest an alternative method
based on trajectories from a mesoscale model. Hopefully, this body of
recent research will encourage others to explore techniques for
diagnosis and evaluate the utility of SCAPE in a forecasting
environment.

Although we agree with much of what Sherwood says about the
terminology associated with moist gravitational and slantwise
convection, this dialog perhaps draws attention away from how to
assess the potential for slantwise convection. As stated in Schultz
and Schumacher (1999), we feel that if slantwise convection occurs in
the atmosphere, then it must be initiated by finite-amplitude
disturbances, rather than infinitesimal perturbations. Therefore, an
emphasis on instability may distract scientists and forecasters from
the real issue: what is the *lift mechanism* responsible for the
precipitation? This question is relevant whether the precipitation is
a result of moist gravitational convection, moist slantwise
convection, or even stable ascent of moist air. Thus, we wish to
emphasize that, in diagnosing precipitation processes, assessing the
mechanism for forcing ascent should be the primary concern. The
degree of instability (as measured by lapse rate or available energy)
merely modulates the *response* to the forcing.

Finally, the debate between lapse-rate-based and energy-based measures of moist symmetric instability should not detract from the message that scientific debate usually indicates the sign of a healthy discipline. The purpose of the present discussion is to improve the diagnostic methodology, not to invalidate it.

*Acknowledgments.*
We thank Steve Sherwood and Kerry Emanuel for engaging in this dialog
with us and providing their feedback.
Katherine Day (NOAA Library, Boulder) performed an extensive
literature survey on the early history of moist instability and its
terminology, initiated by references provided by Leo Donner
(NOAA/GFDL, Princeton).

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A. Lapse rates exceeding dry adiabatic: dry absolute instabilty

B. Lapse rates less than moist adiabatic: absolute stability

C. Lap¯se rates between moist and dry adiabatic: conditional instability

1. Saturated: moist absolute instability

2. Uns¯aturated: stability unknown

a. No CAPE: stablity to all vertical displacements

b. CAP¯E > 0: ins¯tability to some finite
vertical displacements

(metastability or latent
instability)

i. CAPE > CIN: real latent instability

ii. CIN > CAPE: pseudolatent instability

^{1} Although the term CAPE would not be
coined until Moncrieff and Miller (1976), the concept of available
energy had been discussed previously. In this paper, we define CIN as
the negative area/energy on a thermodynamic diagram and CAPE as the
positive area, in contrast to Emanuel (1994, p. 171) where CAPE is the
difference between the positive and negative areas.

^{2} Neglecting terms nonlinear in z requires that
gz/T_{0} << 1. If z = 3 km, this approximation will err by only about
10%.

^{3} It can be shown that, in order to neglect
the nonlinear terms in the Taylor expansion (2),
g >> ^{1}/_{2}z[(d g)/ dz] must occur. In other words, the
change in environmental lapse rate over z is small compared to the
environmental lapse rate. If g were constant with height, the
linearized Taylor expansion would be exact.

^{4} During the time
that Rossby defined convective instability (1932), the term
``convective'' was often used in the same way as ``advective'' is used
today. Such a tradition continues today in some branches of fluid
dynamics. It was likely that Rossby innocently used convective in
that sense (K. Emanuel 2000, personal communication). Of course, this
in no way changes the fact that the terminology can be very misleading
today.