Evaluating the sounding instability with the Lifted Parcel Theory

www.elsevier.com/locate/atmos

Atmospheric Research 67–68 (2003) 455–473

Evaluating the sounding instability with the

Lifted Parcel Theory

Agostino Manzatoa,*, Griffith Morgan Jr.b

aAgenzia Regionale per la Protezione dell’Ambiente (ARPA), Osservatorio Meteorologico Regionale (OSMER),

c/o Villa Chiozza Via carso n3, Cervignano UD 33052, Italyb204 28th St., Boulder, CO 80305, USA

Accepted 28 March 2003

Abstract

The Lifted Parcel Theory is a simple method for evaluating the instability of a sounding and

computing a number of instability indices, for forecasting convective events. Very different results are

obtained, depending on the choice of the initial parcel to lift and on the method used for evaluating the

buoyancy (e.g., using the virtual or cloud-virtual temperature).

A new method, which retains the condensed water inside the rising parcel and freezes it

progressively and continuously, is investigated. For each of the three methods implemented, the

coherent adiabatic processes are applied, and finally, the different results are compared.

D 2003 Elsevier B.V. All rights reserved.

Keywords: Lifted Parcel Theory; Instability; Convective; Sounding

1. Introduction

This is part of a study performed at the Regional Meteorological Observatory of Friuli

Venezia Giulia (OSMER, Italy) to forecast convective events using sounding analysis-

derived indices. In this first part of the work, a Python program, SOUND_ANALYSIS.PY, was

developed to compute a large number of indices (thermodynamic and wind-derived). In this

program, three different schemes for evaluating the buoyancy and performing the adiabatic

processes have been implemented, and they are described in this study.

0169-8095/03/$ - see front matter D 2003 Elsevier B.V. All rights reserved.

doi:10.1016/S0169-8095(03)00059-0

* Corresponding author. Tel.: +39-0431382448.

E-mail addresses: [email protected] (A. Manzato), [email protected] (G. Morgan).

A. Manzato, G. Morgan Jr. / Atmospheric Research 67–68 (2003) 455–473456

In the second part of the work, sounding analysis using that software program has been

carried out on 5050 soundings made by the Italian Aereonautica Militare at the Udine

station (WMO code 16044, using a Vaisala RS 80 radio-sonde) from April to September

1995–2001. For each index, a ‘‘climatology’’ has been determined and an objective

method to relate the index values to the presence of convective events has been

developed. Results are presented in the report ‘‘A climatology of instability indices

derived from Friuli Venezia Giulia soundings, using three different methods’’ (hereafter,

ACII) also printed in this same issue.

2. Lifted Parcel Theory

The idea of adiabatically lifting a parcel in an environment described by a radio-

sounding and considering this moist parcel as a rising convective cloud has been

around for a long time (Bjerknes, 1938). This simple idea is based on the following

assumptions:

� the rising parcel does not mix with the environment;� the parcel pressure is always equal to the environment pressure at the same height;� the parcel rises along completely dry adiabat until it becomes saturated, and afterwards

it rises along a wet adiabat;� condensed water falls out of the parcel, so there is no freezing of the condensed

water.

There are some limits to this theory. For example, the lifted parcel unlikely to be totally

independent from the environment, or the sounding may be contaminated by convective

clouds and not really representative of the environment, or the pressure field inside the

cloud is changed by dynamic effects, etc.

In spite of this, the Lifted Parcel Theory is often useful for classifying the potential

instability of soundings and for nowcasting of deep convective events. Moreover, airplane

measurements have shown that the cumulo-nimbus core is often quite adiabatic, in terms of

both temperature and water content (see Heymsfield et al., 1978).

The ‘‘classical’’ Lifted Parcel Theory scheme is illustrated by the following steps:

1. We choose an initial low-level parcel, which represents the moist and warm air, that will

create the cloud. Let us assume that this parcel has ( p0, T0,Td0) thermodynamic initial

conditions.

2. We lift the parcel to a higher level p, along a dry adiabat, i.e. such that the new values

of ( p, T, Td) will conserve the potential temperature H0 and the initial mixing ratio q0,

until the parcel becomes saturated. This level is called Lifting Condensation Level

(LCL).

3. After that, we lift the parcel higher along a wet adiabat, i.e. such that the new values

of ( p, T= Td) will conserve only the equivalent potential temperature He0, while

q= qsat(T, p). If, during this process, the lifting parcel becomes lighter than the

environment, that level is called Level of Free Convection (LFC), and the sounding is

A. Manzato, G. Morgan Jr. / Atmospheric Research 67–68 (2003) 455–473 457

said to be potentially unstable. Afterwards, we can lift the parcel until the Equilibrium

Level (EL) is reached, i.e. until the parcel density becomes again equal to that of the

environment.

3. Which parcel to lift?

The choice of the initial parcel is very important, because it will define the entire process.

In particular, the ( p0,T0,Td0) point defines He0, i.e. the wet adiabat, because He is conserved

also during the dry adiabatic ascent, and it also defines the LCL because TLCL(T0,e0) (see

Eq. (3)) is the temperature that the parcel lifted along the dry adiabatic will reach at the

LCL.

To choose the initial ( p0, T0, Td0), the Most Unstable Parcel (MUP) in the sounding is

frequently used. Which parcel is that?

A simple algorithm is to take the mean values of ( p,H,q) in the first 100 hPa of the

sounding. This is probably less than the ‘‘ideal most unstable’’, because 100 hPa is a very

deep layer. Another choice is to take just the sounding level with the maximum of He in the

lowest 300 hPa. This chooses an unrealistically high He, as air for the parcel probably

comes from several layers. Moreover, this algorithm would be too subject to ‘‘spikes’’ in the

data, if the sounding has a high vertical resolution (as do our Udine soundings).

The choice made in SOUND_ANALYSIS.PY (suggested by Doswell and Brooks in 1998,

personal communication) is to define a layer 30 hPa deep, and to raise it stepwise through

the lowest 250 hPa (about 2400 m over the Udine sounding location), computing for each

‘‘central pressure’’ ( pc) of the layer the mean values of all the air levels, sampled from the

sounding, that fall inside that layer ( pc� 15 < p < pc + 15). It takes the layer at the pc level,

which has the ‘‘most unstable features’’, i.e. the largest ‘‘pressure-weighted’’ equivalent

potential temperature He0. Finally, it takes as pressure p0 the sounding level just below the

mean pressure of that layer (due to the high vertical resolution, this difference is less than 2

hPa), as dew point temperature Td0 the pressure-weighted mean of all the sounding levels

reported within that layer and as temperature T0 the best fit to obtain the right

He( p0,T,Td0) =He0.

As can be seen, the choice of the definition of the MUP is arbitrary, and it can greatly

change the final results. To see that, we compare three different sounding indices, which are

essentially the same thing, except for the choice of the initial parcel.

Many years ago, Showalter (1953) proposed an instability index (SI), which is the

difference of temperature between the sounding temperature and the lifted parcel at 500

hPa. As MUP, he decided to use the sounding air at the mandatory level of 850 hPa. A few

years later, Galway (1956) introduced a variant of the SI, the Lifted Index (LI), which

takes as MUP the boundary layer parcel, defined as the mean air of the lowest 500 m. The

program SOUND_ANALYSIS.PY computes the SI, LI and also the difference of temperature

between the lifted parcel and the air sounding at 500 hPa, using the above defined MUP,

calling the result DT@500 index. In the report ACII, we compared the performance

obtained by SI, LI and the DT@500 indices, and we found that the last has much better

capacity for forecasting the presence of a thunderstorm. This result attests to the aptness of

our choice to use MUP as the initial parcel.


4. Classical T scheme

If we perform an adiabatic process, the total heat exchanged will be null. We can write

the first law of the thermodynamic in this form:

dQ

dt¼ dU

dtþ p

dV

dt¼ dH

dt� V

dp

dt¼ 0 ð1Þ

where Q is the heat, U is the internal energy, V is the volume and H is the parcel enthalpy.

Using the gas law and considering the parcel just as dry air, neglecting the water vapour

(qp = qd and p =RdqdT, with Rd = 287.0 K� 1 J/kg) when we compute the enthalpy term, we

obtain that the thermodynamic equation that describes a dry adiabatic process has as

solution the Poisson law:

cpddT

dt¼ RdT

p

dp

dtZ T2 ¼ T1

p2

p1

� � Rdcpd

iT1p2

p1

� �27

ð2Þ

where cpd = 7/2Rdi1004.5 K� 1 J/kg is the specific heat at constant pressure for dry air,

which is approximated as a diatomic ideal gas. If we choose as second level the standard

level p2 = 1000 hPa, we obtain as T2 the potential temperature H( p,T ). This means that the

potential temperature Q is—by construction—an invariant of the dry adiabatic process.

In a similar way, it is possible to find the thermodynamic equation that describes a wet

adiabat, i.e. a process that involves an adiabatic lifting of a parcel with saturated air (dry

air and condensing water vapour). In such a parcel, the mixing ratio is always equal to the

saturated value for that temperature and pressure, q = qsat ( p,T ). Bolton (1980) showed that

the pseudo-adiabatic process that describes a saturated process, where the condensed water

is removed from the parcel (irreversible process), has as invariant the following equivalent

potential temperature He( p,T,q):

TLCLðT ; eÞi2840

3:5lnðTÞ � lnðeÞ � 4:805þ 55 ð3Þ

Heðp; T ; qÞiT1000

p

� �0:2854ð1�0:28qÞeqð1þ0:81qÞ 3376

TLCL�2:54

� �ð4Þ

where T is in Kelvin, p in hPa and q in kg/kg. Since the condensed water is supposed to

fall out of the parcel immediately, there is no latent heat of solidification released by the

freezing water and there is no heat retained by the liquid water (or by the ice).

The first parameter that one wants to compute for the rising parcel is the buoyancy

energy. Starting from the inviscid vertical equation of motion, the vertical acceleration (av)

of the parcel at pressure p (assumed to be equal to that of the environment at that level) is

given by:

av ¼dw

dt¼ 1

qp

dp

dz� g ð5Þ

where w indicates the parcel vertical velocity.


If we make the assumption that the environment is in hydrostatic equilibrium, we know

that its acceleration is null, i.e. dp/dz =� gqe and, if we write dw/dt as w(dw/dz), we finallyobtain:

wdw

dz¼ g

qe � qp

qp

¼ B ð6Þ

where B is nothing but the Archimedes buoyancy force per unit of mass.

From this general equation, we know that the rising parcel will continue to rise again at

colder temperatures, releasing latent heat of condensation and warming itself if its density

qp is less than that of the environment qe.

In the classical T scheme, we make the approximation of considering the parcel air

completely dry, so we can write qpiqd = p/(RdTp), so the equation to compute the force of

buoyancy in this scheme is simply:

B ¼ g

p

RdTe� p

RdTpp

RdTp

¼ gTp � Te

Teð7Þ

because we have assumed that p is the same for both the parcel and the environment, and

also we used the same Rd, assuming that both are dry. In this approximation, the parcel is

buoyant if it is warmer than the environment.

Looking at the left term of Eq. (6), we see that, if we integrate that equation from the

surface up to z, we obtain 1/2w2(z), i.e., a—specific—kinetic energy, expressed in [J/kg]=

[m2/s2]. The work done by the buoyancy force B in lifting our parcel from the level z1 up

to the level z2 (increasing the kinetic energy of the parcel) is the buoyacy energy.

At the beginning of the lifting, when the parcel rises along a dry adiabat, probably the

buoyancy energy will be negative and we must think that an external forcing agent (such

as an orographic lift or a low-level wind convergence) will supply to the parcel this kinetic

energy. The level when the parcel becomes ‘‘less dense’’ than the environment is the Level

of Free Convection.

If the LFC exists, following Colby (1984), we define the Convective Inhibition (CIN) as

the negative energy needed to raise the parcel from its initial level z0 up to zLFC, where the

parcel becomes buoyant: CIN ¼ mzLFCz0Bdz.

In the same way, Moncrieff and Green (1972) defined the Convective Available

Potential Energy (CAPE) as the work that the buoyancy force will do on the parcel,

when it will rise from LFC up to the Equilibrium Level, above which it will become

heavier than the environment: CAPE ¼ mzELzLFCBdz.

In the classical ‘‘T scheme’’, for doing the adiabatic lifting we use Eq. (7), with simple

temperature, to compute the CIN and CAPE values.

Because the LFC may be not unique (if the buoyancy changes sign in following layers),

SOUND_ANALYSIS.PY defines as LFC the first point above the LCL where the density of the

rising parcel becomes equal to that of the environment (i.e., the parcel is buoyant). Also,

the EL (equilibrium level) may be not unique: if, above the LFC, the parcel becomes

unbuoyant the energy becomes negative and the integrated CAPE decreases, until the


parcel becomes buoyant again. In this case, SOUND_ANALYSIS.PY chooses as EL that which

provides the larger value of CAPE.

5. Instability and Theta-Plot diagram

Bolton (1980) gave a formulation to compute the invariant of a pseudo-adiabatic

process, the equivalent potential temperature: He =He( p,T,q). Physically, the He is the

temperature that we obtain doing a pseudo-adiabatic ascent from the LCL up to where the

mixing ratio is negligible (so that all the condensation heat of the water is given to the dry

air) and then a—dry—descent down to the standard level of 1000 hPa, as shown in Fig. 1.

Note that, if we descend along a pseudo-adiabat from the LCL down to the parcel level,

we would find the wet bulb temperature Tw.

If we were to begin the pseudo-adiabatic ascent at the point ( p,T ), that would mean

assuming the air to be already saturated, i.e. increasing the air mixing ratio from q to

qsat( p,T ). So, if we compute Eq. (4) using the saturated mixing ratio of the parcel, we obtain

the saturated equivalent potential temperature: Hes =He ( p,T,qsat), which is a function of

only p and T.

Instead, if we were to begin the pseudo-adiabatic ascent at the point ( p,Td) that would

mean that the air is already saturated at its dew point temperature, i.e. changing its

Fig. 1. In this figure, it is possible to see a point ( p= 900 hPa, T= 15 jC, q= 8 g/kg) of non-saturated air and some

related variables, obtained with pseudo-adiabatic ascents and dry descents. Temperatures at the level of the air

parcel: dew point (Td), wet bulb (Tw), and equivalent (Te). Temperatures at the standard level of 1000 hPa: potential

(H), equivalent potential (He), saturated equivalent potential (Hes), and dew point equivalent potential (Hed).


temperature from T to Td and its mixing ratio from q to qsat( p,Td), which is always the orig-

inal q of the parcel for definition of Td. So, computing Eq. (4) using the dew point tem-

perature Td, we obtain the ‘‘dew point’’ equivalent potential temperature:Hed =He( p,Td,q).

These three quantities are used to draw the thermodynamic diagram called Theta-Plot

(Morgan, 1992), as can be seen in Fig. 2, made through the ZEBRA software (Corbet et al.,

1994): the left curve is Hed, the middle one is He, while the curve on the right is Hes.

Fig. 2. The Theta-Plot diagram of Udine 28 June 1998 at 12:00 UTC: the black curve is the sounding He, the right

curve is Hes (where you can read the temperature using the thin isotherms) and the left curve is Hed (where you

can read the dew point temperature, or the mixing ratio using the dotted isohygrometrics). The low-level segments

close respectively to the Hes and Hed represent the temperature and mixing ratio along the dry adiabatic lifting of

the Most Unstable Parcel ( p= 988.6 hPa, He = 337.2 K and q= 12.4 g/kg), while the vertical line over the LCL is

the first part of the wet pseudo-adiabat, and the marked LFC is calculated using normal T scheme. The horizontal

segment at 575 hPa represents the Maximum Buoyancy.


The thin solid isotherms (from 28 to � 28 jC) and the thin dotted isohygrometric lines

(from 25 to 1 g/kg) intersect on the Hes, the temperature T and the saturated mixing ratio qsat( p,T ) on the He, the wet-bulb temperature Tw and on the Hed, the dew point temperature Tdand the mixing ratio q.

The great advantage of using the Theta-Plot diagram is that the wet pseudo-adiabatic

process is a simple vertical line (differently from the skew-T diagram), and that simplifies

the drawing of the lifted parcel thermodynamic process. We can see in the same figure the

lifting of the MUP: the thick line from MUP to LCL is the dry adiabatic process, while the

thick line over the LCL tracks the wet pseudo-adiabat. Drawing the lift of the parcel is

simple, because the LCL is determined by the intersection of the dry adiabatic process with

the wet adiabat (the vertical line starting from the initial He, which is conserved); while the

LFC is determined by the intersection of the wet pseudo-adiabat with the sounding Hes line.

Thus, it is very easy to see if a sounding is potentially unstable for the Lifted Parcel

Theory: in fact, if the low level MUP He is greater than the high level Hes it means that,

during the lifting of the MUP, the parcel will become warmer than the environment (i.e., the

LFC exists). This kind of instability is considered as ‘‘potential’’ because it needs an

external agent, which lifts the MUP up to the LFC. Above that level, the parcel is buoyant

and the—positive—energy is the CAPE. Unfortunately, the area between the wet pseudo-

adiabat and the sounding Hes is not exactly proportional to this energy, as it is in the skew-T

diagram.

Morgan and Tuttle (1984) suggested the use of a new sounding-derived index, called

Maximum Buoyancy, which is exactly the difference between the maximum He in the lower

layer and the minimum Hes in the middle layer aloft.

MaxBuo ¼ maxðHejlowÞ �minðHesjmidÞ ð8Þ

As low level maximum He, the SOUND_ANALYSIS.PY program takes the 30-hPa layer with

the maximum mean He in the ‘‘first 250 hPa’’ (from about 1000 to 750 hPa, over

Udine), i.e. the MUP He. As middle-level minimum Hes, it takes the 30-hPa deep layer

with minimum mean value of Hes in the ‘‘second 250 hPa’’ of sounding data (about

from 750 to 500 hPa).

For the sounding shown in Fig. 2, the min(Hes | mid) is equal to 326.85 K at 575 hPa,

while the max(He | low) is 337.18 K at 989 hPa, leading to a MaxBuo of 10.3 K.

The low level He represents the unstable air which is lifted (and keeps constant its He,

equal to its Hes above the LCL). The middle level Hes at a fixed pressure level depends

only on the sounding temperature, so we can think that MaxBuo is related to the maximum

difference of temperature (buoyancy) achieved by the parcel during the adiabatic lifting. If

the low level maximum He is always less than the high level minimum Hes, this means

that the lifting parcel will never become lighter than the environment (no intersection

means no LFC), so the sounding is not potentially unstable.

With the use of such variable, the condition for potential instability is simply

MaxBuo>0, which is different from the classical convective instability condition dHe/

dz< 0. In fact, in the last case, it is necessary only that there be an upper level with

minimum He lower than the maximum He in low levels. The condition using MaxBuo

to define the potential instability is stronger than that using only the He lapse rate, and


we show in the ACII study that MaxBuo is an instability index that is more useful than

dHe/dz.

6. Tv scheme

In the simple T scheme, we consider the parcel to consist of only dry air, neglecting the

water vapour. If we also consider the non-saturated vapour during the ‘‘dry’’ lifting, we

obtain a ‘‘moist’’ adiabatic process, for which the mixing ratio is constant: q(T,p) = q0.

Considering qp = qd + qv, we find that the thermodynamic equation that represents the moist

adiabat is:

ðcpd þ q0cpvÞdlnT

dt¼ ðRd þ q0RvÞ

dlnp

dtð9Þ

where cpv = 4Rvi1845.6 K� 1 J/kg is the specific heat at constant pressure for the water

vapour, which is a triatomic molecule. This equation has, again, an exact solution, which is

only slightly different from its Poisson counterpart (see, for example, Paluch, 1982,

Appendix A.2):

T2 ¼ T1p2

p1

� � RdþRvq0cpdþcpvq0

iT1p2

p1

� �27

1þRvRd

q0

1þ87RvRd

q0 ð10Þ

where q0 is the constant mixing ratio of the rising parcel. Note that, if we take p2 = 1000

hPa, we define the invariant of this process, Hm( p,T,q0), which also depends on q0: it is not

possible to draw the isolines of Hm ‘‘a priori’’ in a classical thermodynamic diagram.

Fortunately, the difference in temperature using this moist equation instead of the dry

one is very small, since 8/7i1.

Starting from Eq. (1) and considering the water vapour density, it is possible to find that

the thermodynamic equation that represents the wet pseudo-adiabatic process is:

ðcpd þ qsatcpvÞdlnT

dtþ LcðTÞ

T

dqsat

dt¼ ðRd þ RvqsatÞ

dlnp

dtð11Þ

where Lc(T ) is the latent heat of condensation released by water vapour, approximately

given, in J/kg, by Lc(T ) = 2,500,800� 2.3� 103T, with T expressed in jC. This equationdoes not give an exact solution, since qsat is a transcendental function of T, but we may

integrate it numerically, making some small approximation between every couple of near

levels.

The dashed line of Fig. 5 shows the Bolton He along the adiabatic process obtained

solving numerically Eq. (11) for the sounding of 28 June 1998 at 12:00 UTC—it is almost a

vertical line.

In this scheme, the buoyancy force is computed as suggested by Doswell and Rasmussen

(1994), i.e. considering also the water vapour density, so q = qd + qv = qd(1 + q). In such a

case, it is possible to show how we can always use the dry air constant Rd for a mixture of


dry air and water vapour if we use the virtual temperature Tv, instead of the normal

temperature T:

q ¼ p

RdTvwhere Tv ¼ T

�1þ Rv

Rd

q

�

1þ qiTð1þ 0:6qÞ ð12Þ

Bv ¼ gqe � qp

qp

¼ gTvp � Tve

Tveð13Þ

This form of the buoyancy force is used to compute CIN and CAPE. Since Tv is always

greater than ‘‘normal’’ T and at high levels the parcel has more vapour than the environment

(which is usually not saturated as the parcel), the buoyancy Bv becomes greater than B, so

the CIN decreases and the CAPE increases, with respect to the classical T scheme.

The results obtained using that ‘‘virtual correction’’ and solving numerically Eq. (11),

instead of taking the Bolton He as constant, are not large, but they are statistically

significant, as shown in the ACII study.

7. Tvc scheme

The idea of this scheme is to retain the condensed water above the LCL inside the parcel.

That increases the total density of the parcel (qp = qd + qv + qcond), but the freezing of the

liquid water will release the latent heat of solidification, given approximately by

Ls(T ) = 338,200 + 2.3� 103T + 3.6(T + 35)2, where T is in jC and Ls is in J/kg.

This idea was already tested and was considered the best way to evaluate the sounding

instability in the tropics (e.g., Emanuel, 1989), and sounding analysis freezing all the

condensed water (e.g., at � 10 jC by Williams and Renno, 1993) was already performed.

Also, in cloud model simulations, the freezing of condensed water (as a function of cloud

temperature) was applied (e.g., Levi and Saluzzi, 1995), leading to interesting conclusions

for the weakly unstable atmosphere.

In our program, the transition between liquid water and ice is parameterized by the

following continuous function of the temperature:

f ðTÞ ¼ qice

qcond¼ NiceðTÞðT � TLCLÞ2

NiceðTÞðT � TLCLÞ2 þ 3000ð14Þ

with

qiceðTÞ ¼ q1ðTÞNiceðTÞðT � TLCLÞ2

3000ð15Þ

where T is in jC and Nice = e� 2.8� 0.262T is the concentration of ice nuclei suggested by

Meyers et al. (1992). As usual, the ice mixing ratio is qice = qice/qd, the liquid water mixing

ratio is ql = ql/qd and qcond = q0� qsat = ql + qice.

In that parametrization, there is also a weak dependence from the cloud-base temperature

TLCL, since the farther from the cloud-base we were, the more time for coalescence of cloud

drops there was, and consequently the larger the mass of droplets which freeze. The

Fig. 3. The transition function f (T,TLCL) imposes for each value of temperature the fraction of frozen water with

respect to the total condensed water. The three lines are referred to three values of TLCL= 20, 10 and 0 jC.


function f (T ) is shown in Fig. 3: above 0 jC, all water is liquid ( f = 0), below � 40 jC all

water is frozen ( f = 1), while the 50% rate occurs around � 16 jC (for a typical cloud-base

temperature of 10 jC).When we have liquid water and ice at the same time, the water vapour is not in

equilibrium, because its saturation pressure over ice esatice (T ) is different from that over

liquid water. The program SOUND_ANALYSIS.PY assumes an artificial equilibrium of water

vapour with respect to an assumed saturation pressure over mixture esatmix (T ), which is

taken as a transition from esatwat (T ) to esat

ice (T ), governed by the same function f (T ):

emixsat ðTÞ ¼ ewatsat ðTÞ þ f ðTÞ½eicesatðTÞ � ewatsat ðTÞ ð16Þ

So, below 0 jC, the saturated mixing ratio is computed as qsat=(Rd/Rv)esatmix/(p� esat

mix).

Using that parametrization to compute the quantity of ice and liquid water phase from

the condensing vapour, it is possible to show how the thermodynamic equation that

represents the reversible wet adiabatic process, in differential forms, is given by:

ðcpd þ qsatcpv þ qlcpl þ qicecpiÞdT

dtþ Lc

dqsat

dt� Ls

dqice

dt¼ ðRd þ RvqsatÞ

dlnp

dt

ð17Þ

where cpl(T ) i4186[0.9979 + 3.1�10� 6(T� 35)2 + 3.8� 10� 9(T� 35)4] K� 1 J/kg and

cpi(T )i4186[0.503 + 0.00175T ] K� 1 J/kg are the specific heat at constant pressure for the


liquid water and the ice (see Pruppacher and Klett, 1997, Eqs. (3-12) and (3-15)). Note that

the term with Ls has the minus because dqice > 0, meanwhile dqsat < 0, i.e. both terms

represent released latent heat. With small approximations between every couple of

following levels, we may compute iteratively the temperature that a parcel will have along

this wet adiabatic process.

In Fig. 4, it is possible to see the mixing ratio of the three water load phases along the

reversible adiabatic lifting for the Udine sounding of 28 June 1998, keeping in mind that

qice = f (T )( q0� qsat) and ql = [1� f (T )]( q0� qsat).

In Fig. 5, we may see how much different this wet reversible adiabat (dotted line) is with

respect to the wet pseudo-adiabat (dashed line); in fact, the Bolton He increases

considerably during the reversible process.

In this scheme, the buoyancy force B is computed considering not only the water

vapour density but also the condensed water load, so qp = qd + qv + ql + qice = qd(1 + q0). It

is possible to show that we can use the gas constant of dry air Rd for that mixture if we

use, instead of the temperature, the cloud-virtual temperature Tvc, which is defined as:

Tvc ¼ T

1þ Rv

Rd

q

� �

1þ q0iT

ð1þ 1:608qÞ1þ q0

ð18Þ

The smaller q becomes (far above the LCL), the smaller the Tvc gets with respect to Tv.

Fig. 4. How the different water phase contributions along a reversible wet adiabatic process change with pressure,

for the Udine sounding of 28 June 1998 at 12:00 UTC. Solid line is q, dashed line is ql while dotted line is qice.

Fig. 5. How the Bolton equivalent potential temperature changes along a moist adiabat following a wet pseudo-

adiabat (dashed line), or following a reversible wet adiabat (dotted line), or following a He const adiabat (thin

line, which is not exactly straight due to little computational approximations), for the Udine sounding of 28 June

1998 at 12:00 UTC.


Finally, the buoyancy force is computed using the virtual temperature for the

environment (it has no condensed load) and the cloud-virtual temperature for the parcel:

Bvc ¼ gqe � qp

qp

¼ gTvcp � Tve

Tveð19Þ

The use of this buoyancy produces, in general, lower CAPE values than using Bv, but if the

initial mixing ratio is great and if the top of the cloud is high, the CAPE can also be

considerable, due to the latent heat of freezing.

8. Conclusions

In Fig. 6, we show, for the Udine sounding of 28 June 1998 at 12:00 UTC (same as in the

previous Theta-Plot), the three buoyancy forces B(z) = g(qe� qp)/qp computed by using the

different density approximations defined in the T (solid line), Tv (dashed line) and Tvc(dotted line) methods and the corresponding adiabatic processes. Note the big difference

between the Tvc method and the rest, especially when the altitude exceeds 6000 m.

In each of the methods, the cloud starts its formation at the same cloud-base height,

LCL, and temperature, TLCL, because the moist adiabat is close to the classical dry adiabat.

Fig. 6. The different buoyancy force B computed by the three methods for the Udine sounding of 28 June 1998 at

12:00 UTC. The moist adiabat leads to warmer LCL than the dry adiabat used in the T method. Afterwards the

difference between T and Tv decreases along the wet adiabat, when the parcel mixing ratio decreases. In the Tvcmethod, it is possible to see before lower buoyancy and, during the ice phase, the strongest buoyancy force.

Above the LFC, the area between the curve and the 0 buoyancy (neutral level) is the CAPE.


Nevertheless, at the LCL, the buoyancy computed in the Tv and Tvc methods is greater than

in the T method, because in the former, the virtual correction has been used.

The LFC height is quite a bit lower for the Tv (greater low level buoyancy), while in the

Tvc method the weight of the condensed water balances the minor density due to the use of

the virtual correction (i.e., considering the lighter vapour density). In Tvc, at high levels,

freezing increases the buoyancy, which stops at the highest cloud-top, i.e. EL. In reality, the

cloud may rise somewhat above the EL by inertia.

To exemplify the differences between these methods, we report in Table 1 the com-

putations for some parameters of the 28 June sounding, made with the three methods

implemented in the program SOUND_ANALYSIS.PY.

Table 1

Variable T Tv Tvc

CAPE [J/kg] 1128 1239 1214

CIN [J/kg] � 46 � 15 � 26

Cloud-base [m] 1470 1468 1468

Level of Free Convection [m] 2932 2080 2608

Melting level [m] 4402 4587 4312

Cloud-top [m] 11,506 11,464 11,857

Fig. 7. (a) Difference in CAPE computed by the method Tv less that computed by the T scheme, calculated over

1292 soundings associated with thunderstorm presence in the 6-h period after their launch, in the plain of Friuli

Venezia Giulia region. (b) Same as (a) but with the CAPE computed by Tvc method less that by the T method. In

that case, the difference is much more spread out.


Fig. 8. (a) Difference in CIN computed by the method Tv less that computed by the T scheme, calculated over

1292 soundings associated with thunderstorm presence in the 6-h period after their launch, in the plain of Friuli

Venezia Giulia region. (b) Same as (a) but with the CIN computed by Tvc method less that by T method. In that

case the difference is much more spread out.



Note that the values of CAPE are around 1200 J/kg, which, based on the literature, may

not seem such a very high value. In reality, during that day we had severe weather events,

the records show 194 occurrences of C2G lightning and 11 damaged hailpads in our target

area (about 5000 km2) in just 2 h (from 15:00 to 17:00 UTC). This means that the thresholds

found for the instability indices in the literature may require some tuning for specific

regions.

Not only the absolute value of some Lifted Parcel Theory parameters changes with the

method used, but also the concept of an unstable sounding itself may change. If a sounding

is ‘‘weakly unstable’’ for the classical method T, i.e. it has MaxBuoi0, it can become more

unstable in the Tv method (greater buoyancy) or not unstable at all in the Tvc scheme,

because the added weight of condensed water can miss the LFC condition.

In Fig. 7a,b, we compare the difference between the CAPE computed in Tv and T

schemes, with that between the CAPE in Tvc and T schemes. Each point represents a

sounding associated with the presence of a thunderstorm (at least one lightning strike) in the

plain of our region during the 6-h period starting at the launch time of the sounding. There

were 1292 of such soundings during the period from April to September 1995–2001.

While the CAPE Tv is almost always greater than the CAPE T, and they are very well

correlated (i.e., the Tv method looks like a ‘‘systematic’’ correction of its T counterpart), the

CAPE Tvc is usually lower than CAPE T for low values and usually greater than CAPE T for

values over 1000 J/kg. In the last case, the result depends very much on the initial mixing

ratio.

The same is done in Fig. 8a,b for the CIN variable. In this case also, the difference

between CIN T and CIN Tv seems to be very low (at least for CIN < � 300 J/kg) if

compared with that between CIN Tvc and CIN T, which is much more ‘‘spread’’ out. In the

last case, the result depends on the added weight of the condensate, which can considerably

increase the LFC height, or miss it in the case of the Tvc method. In fact, out of the 1292

soundings with lightning activity, only 86% had the LFC in the Tvc scheme, while 92% had

the LFC in the Tv method.

Theoretically, the best method should be Tv or Tvc, depending on whether or not there is

a lot of precipitation (condensed water that falls out of the parcel), where the wet pseudo-

adiabat is a good approximation, or just cloud-drops (drops with diameter less than 0.3 mm)

as in ‘‘non-precipitating’’ cumulus, where the reversible wet adiabat may be more

appropriate. Practically, in the ACII study, we see how the parameters computed in the

simple T scheme often give the best statistical performance!

9. Summary

The SOUND_ANALYSIS.PY software is able to apply the Lifted Parcel Theory in three

different schemes: (1) using the classical temperature difference for evaluating the buoy-

ancy and the classical adiabatic process, (2) including the water vapour density, using the

virtual correction and a better approximation of the adiabatic processes, or (3) also

considering the liquid water density and the ice phase process (reversible adiabat).

The difference obtained in the three ways can be very large, in particular with the Tvcmethod, where we found very low values of buoyancy (or not buoyancy at all) when the


sounding was not greatly unstable, and the EL was lower than the region where the liquid

water froze. Running the SOUND ANALYSIS.PY software over 5050 different soundings (1292

with at least one thunderstorm in the target area of about 5000 km2 during the 6-h period),

we were able to test the sensitivity of the three schemes and to determine which method is

better related to the observed convective activity. Statistical results are reported in the ACII

study.

Acknowledgements

We are very grateful to the personnel of Campoformido Aereonautica Militare, who

collected the WMO 16044 radio-soundings with professional care.

Dr. Charles A. Doswell III and Dr. Harold Brooks, of NSSL (Norman OK) helped with

the choice of the Most Unstable Parcel.

We thank Dr. James E. Dye of NCAR (Boulder, CO) for his suggestions about the ice

phase, given to Manzato during the NCAR Summer Colloquium on ‘‘Ice formation’’

(Boulder, 1999).

Discussions with personnel of OSMER helped in developing this work, and in

particular those with Dr. Fulvio Stel.

We thank the reviewers for their help in making the text clearer. Last but not least, we

want to thank the ‘‘GNU generation’’ for providing such good—and free!—tools as Python,

R, LATEX, Emacs, as well as Linux itself.

References

Bjerknes, J., 1938. Saturated-adiabatic ascent of air through dry-adiabatically descending environment. Q. J. R.

Meteorol. Soc. 64, 325–330.

Bolton, D., 1980. The computation of equivalent potential temperature. Mon. Weather Rev. 108, 1046–1053.

Colby Jr., F.P., 1984. Convective inhibition as a predictor of convection during AVE-SESAME II. Mon. Weather

Rev. 112, 2239–2252.

Corbet, J.M., Mueller, C., Burghart, C., Gould, K., Granger, G., 1994. ZEB: software for integration, display and

management of diverse environmental datasets. Bull. Am. Meteorol. Soc. 75, 783–792.

Doswell III, C.A., Rasmussen, E., 1994. The effect of neglecting the virtual temperature correction on CAPE

calculations. Weather Forecast. 9, 625–629.

Emanuel, K.A., 1989. Dynamical theories of tropical convection. Aust. Meteorol. Mag. 37, 3–10.

Galway, J.G., 1956. The lifted index as a predictor of latent instability. Bull. Am. Meteorol. Soc. 37, 528–529.

Heymsfield, A.J., Johnson, P.N., Dye, J.E., 1978. Observations of moist adiabatic ascent in northeast Colorado

cumulus congestus clouds. J. Atmos. Sci. 35, 1689–1703.

Levi, L., Saluzzi, M.E., 1995. Effects of ice formation on convective cloud development. J. Appl. Metereol. 35,

1587–1595.

Meyers, M.P., DeMott, P.J., Cotton, W.R., 1992. New primary ice-nucleation parameterizations in an explicit

cloud model. J. Appl. Meteorol. 31, 708–721.

Moncrieff, M.W., Green, J.S.A., 1972. The propagation of steady convective overturning in shear. Q. J. R.

Meteorol. Soc. 98, 336–352.

Morgan Jr., G.M., 1992. ThetaPlot, an equivalent potential temperature diagram. Meteorol. Atmos. Phys. 47,

259–265.

Morgan Jr., G.M., Tutle, J., 1984. Some experimental techniques for the study of the evolution of atmospheric

thermodynamic instability. Paper presented at the Second International Conference on Hailstorms and Hail

Suppression (Proceedings). Printing Office of the Hydrometeorological Service, Sofia, pp. 192–196.


Paluch, I.R., 1979. Entrainment mechanism in Colorado cumuli. J. Atmos. Sci. 36, 2467–2478.

Pruppacher, H.R., Klett, J.D., 1997. Microphysics of Clouds and Precipitation. Kluwer Academic Publishers,

Dordrecht (The Netherlands).

Showalter, A.K., 1953. A stability index for thunderstorm forecasting. Bull. Am. Meteorol. Soc. 34, 250–252.

Williams, E., Renno, N., 1993. An analysis of the conditional instability of the tropical atmosphere. Mon.

Weather Rev. 121, 21–36.

Documents

Evaluating the sounding instability with the Lifted Parcel Theory