2011© by Kelly Boyd

Atmospheric Thermodynamic Stability Indices

Presented to Dr. Jill Coleman

Ball State University

Department of Geography

Presented By: Kelly Boyd

11/28/2011

GEOG 547 Lit Review of Thermodynamic Indices 2011 © by Kelly Boyd

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Atmospheric thermodynamics is an area of atmospheric science which helps to explain

severe thunderstorm forecasting quantitatively. Specifically, the thermodynamic concepts of

atmospheric convection and stability are necessary to understand in order to forecast correctly.

Atmospheric convection is the transport of some property (usually energy) by fluid movement,

and this concept is most often associated with heat transport through the troposphere (Doswell

2001). Convective stability indices have been proven to forecast these events throughout many

years. Typically, these indices described by thermodynamic variables such as temperature,

pressure and dew point, can be used to forecast deep moist convection (DMC), an important

factor for thunderstorm development (Ibid). DMC is an evolutionary process that exhibits a

response to atmospheric disequilibrium of energy (Ibid). For this unbalance of energy, the

atmosphere responds with a release of energy to realign the atmosphere back into

thermodynamic equilibrium. Thermodynamic stability indexes measure this disequilibrium and

realignment of energy. With that in mind, it is important to note that no particular

thermodynamic index is better than another when forecasting instability, but an integration or

combination of indices can help forecast thunderstorms and severe weather better when DMC

takes place. This essay will first examine the different types of static stability concepts in

thunderstorm forecasting and then highlight select indices to help explain their derivations, their

theories and their uses to forecast thunderstorms.

Under the realm of parcel theory, static stability is defined as the steadiness of an

atmosphere in hydrostatic equilibrium with respect to vertical displacements (Peppler 1988).

Parcel theory is based on displacements of air introduced to a steady state atmosphere under the

assumption that only the parcel is moved and the environment remains unchanged (Ibid). The

displaced parcel is assumed to undergo adiabatic temperature changes, meaning that it is

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thermally insulated from its environment disallows energy transfers between the parcel medium

as the parcel lifts through the atmosphere (Ibid). If the parcel is unstable with abundant low-level

moisture and has a triggering mechanism such as a front or mid-latitude cyclone to lift,

convective weather and rainfall can develop. Therefore, stability is one of the main

measurements of the atmosphere within which thunderstorms and severe weather develop.

Thermodynamic indices can help forecast these events.

There are several types of static stability scenarios. The four major types of static stability

which help to describe the thermodynamic indices mentioned in this essay are conditional

instability, absolute instability, latent instability and convective instability. Conditional

instability can described the stability of the atmosphere considering unsaturated and saturated air

parcels. It can be formally defined as “the state of a column of air in the atmosphere when its

lapse rate of temperature is less than the dry-adiabatic lapse rate but greater than the saturation-

adiabatic lapse rate” (Ibid). It is then said that the vertical displacement of a parcel of air is to be

unstable when it rises as saturated but is stable if it sinks and is unsaturated. Next, absolute

instability refers to an environmental lapse rate (ELR) which is greater than the dry adiabatic

lapse rate so that all displaced parcels of air around the parcel, including both saturated and

unsaturated, will be unstable (Ibid).

Latent instability is the state of a portion of a conditionally unstable air column lying

above the level of free convection (LFC) which is obtained only if an initial impulse on a parcel

gives it sufficient kinetic energy to carry it through the boundary layer below the LFC (Ibid).

Latent instability is caused by latent heat given off during the process of water condensation

above the LFC. This process adds energy to the atmosphere. Finally, the concept of convective

instability is defined as the state of an unsaturated layer or column of air in the atmosphere

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whose wet-bulb potential temperature or equivalent potential temperature decreases with

elevation where the body becomes completely saturated as it rises further in the atmosphere

(Ibid). Convective instability usually takes place when the lower portion of the atmosphere is

warm and moist and the remainder of the atmosphere above the boundary layer rapidly dries out.

Stability indices can be arranged according to these definitions of stability above and by their

originally-intended or traditional purpose (Ibid). The seven indices examined in this essay are the

CAPE Index (including sub-indexes associated with CAPE), the Lifted Index, the K-Index, the

Total-Totals Index, the Deep Convective Index and the Severe Weather Threat Index.

These seven indices are the most widely used indices used in forecasting today and have been

used for many decades since the early 1950s (Ibid).

The first and most widely used thermodynamic index used for severe thunderstorm

forecasting is the measurement of Convective Available Potential Energy (CAPE). This index

was developed by M. W. Moncrieff and J. S. A. Green in 1972 followed by further analysis by

many other atmospheric scientists during the mid to late 1970’s (Doswell 2001). This concept of

energy is based upon parcel theory and is a measure of the buoyant energy available to accelerate

a parcel of air into the upper troposphere, where buoyancy is defined as

, where g is

defined by the acceleration due to gravity (9.80665 m s-2

), TP is the temperature of a parcel of air

in Kelvin, and TE is the temperature of the environment surrounding the air parcel also in Kelvin

(Ibid). There are four major assumptions of the CAPE theory: (i) a rising parcel exhibits no

environmental entrainment, (ii) the parcel rises moist adiabatically, (iii) all precipitation falls out

of the parcel and (iv) the parcel pressure is equal to the environmental pressure at each level

(NWS Louisville 2011). CAPE can be defined as an area on a thermodynamic diagram between

the area to the right of the dry air temperature ascent line and to the left of the first moist adiabat

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line were this area can be integrated from the LFC to the equilibrium level (EL) (Blanchard

1998). The mathematical definition of this theory is:

∫ (

)

,

where TVp is the virtual potential temperature of the parcel and TVe is the virtual potential

temperature of the environment and g is the constant of gravity (Moncrieff and Green 1972).

CAPE is measured in joules per kilogram (J kg-2

) and has several thresholds of stability (NWS

Louisville 2011). Typically, the greater the difference of temperature between two parcels of air,

the larger the opportunity is for the warmer parcel to rise, hence the larger the CAPE value and

the greater chance for severe weather to take place (Ibid). CAPE values of 0 J kg-2

to 1000 J kg-2

are generally stable, values of 1000 J kg-2

to 2500 J kg-2

are generally considered moderately

unstable, while CAPE values of 2500 J kg-2

to 4000 J kg-2

are extremely unstable (Ibid).

There are several different categories of CAPE used for various purposes in forecasting.

Three predominate categories of CAPE are Most Unstable Convective Available Potential

Energy (MUCAPE), Surface Based Convective Available Potential Energy (SBCAPE) and

Mixed Layer Convective Available Potential Energy (MLCAPE) (Evans 2003). MUCAPE

calculation represents the total amount of potential energy available to the most unstable parcel

of air found within the lowest 300-millibar of the atmosphere while being lifted to its LFC (Ibid).

This type of CAPE is useful in non-surface based instability forecasting and can show how high

in the atmosphere instability is reaching (Ibid). The SBCAPE values denote the total amount of

potential energy available to a parcel of air originating at the surface and then theoretically lifted

to its level of free convection (Ibid). This CAPE value can give an estimate of energy available at

the surface that will eventually rise, giving way to thunderstorm development (Ibid). This type of

CAPE stability must first overcome the convective inhibition parameter (CIN) in order for

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thunderstorms to develop (Ibid). CIN is the amount of negative buoyant energy available to

inhibit / suppress upward vertical acceleration or the amount of additional work the environment

must do on a parcel to raise the parcel to its LFC (NWS Louisville 2011). Basically, CIN is

negative CAPE which pushes air parcels down instead of the environment allowing it to rise.

This is sometimes called a “lid” (Blanchard 1998). CIN is computed in a manner similar to

CAPE and is defined as:

∫ (

)

where, LFCZ is the height of the LFC, SFCZ is the height of the surface, TVP is the virtual

potential temperature of the parcel, and TVe is the virtual potential temperature of the

environment and g is gravity (Ibid). CIN values of less than or equal to 10 J kg-1

are usually

small, whereas values greater than or equal to 50 J kg-1

are usually large which allow for CAPE

to build until the “lid” is broken (Markowski and Richardson 2010). In typical daytime

temperature and moisture profiles, CIN values are decreased when the effects of moisture are

included (Ibid). Finally, MLCAPE is a measurement of energy in the lowest 100 mb of the

troposphere where the parcel of air is lifted using the mean temperature and moisture (Evans

2003). This calculation of CAPE is usually used during the latter part of the day when the

boundary layer is well mixed with moisture and convection (Ibid). This measurement of CAPE is

often the best indicator of severe weather over a large area (Ibid).

In concluding the index of CAPE, there are two widely used derived indices calculated

using the CAPE formula, these are the Energy-Helicity Index (EHI) and the Bulk Richardson

Number (BRN). The EHI is a straight-forward equation that combines helicity and instability

(CAPE) into one number for estimating and assessing the potential for supercell thunderstorms

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and tornadoes (Davies 1993). Helicity is simply a measure of the amount of rotation found in a

storm’s updraft measured in m2s

-2 (The University of Illinois at Urbana-Champaign 2011).

Helicity primarily takes into account vertical wind shear in the upper levels of the atmosphere

(Ibid). The EHI equation is ( ) , where the output of EHI is

a dimensionless number indicating whether tornadic initiation is possible (Davies 1993). The

constant of 160,000 is used to reduce the scale of the EHI to single digits (Ibid). The EHI scale

can range from zero, no possible chance of tornadoes or supercell thunderstorms, to five, a great

threat of tornadoes and supercell thunderstorms (Ibid). In practice, anything over two and one

half is worthy of possible tornado development (Ibid).

Lastly, the second widely used CAPE derived index is the BRN. This derived indice

from the thermodynamic concept of CAPE also integrates the dynamic concept of vertical wind

shear. However in this equation, the wind velocity is inputted directly into the algorithm instead

of calculating helicity before integrating into the equation. The equation for the BRN is

[ ( )] ,where U is the mean wind shear between the surface and 500 m above the surface

(Stensrud, Cortinas and Brooks 1997). Similar to EHI, this formula produces a dimensionless

number (units cancel) that specifies the convective storm type within a given environment (Ibid).

A BRN of nine or below signals strong vertical wind shear and weak CAPE where supercell

thunderstorms may develop (depending on forcing), while a BRN number of ten to fifty are

associated with supercell development because of high cape and moderate shear (Ibid). Anything

over a BRN of fifty usually identifies strong CAPE and weak shear which indicates linear or

multicellular thunderstorm development (Ibid).

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Another widely used thermodynamic index is the Lifted Index (LI). The LI is another

measure of convective available potential energy that uses the difference of an air parcel’s

temperature in Celsius at the surface to the 500 mb level of the parcel to the temperature of the

surrounding environment at the 500 mb level (Galway 1956). The mathematical formula for the

LI is

The lifted index was developed by Joseph Galway to forecast severe thunderstorms and

tornadoes in the Midwest (Peppler 1988). It is predominately calculated using morning

soundings and has been used to forecast afternoon convective activity (Ibid). This index is

different from CAPE because it assumes a constant mean mixing ratio based on the lowest 3000

feet of the atmosphere (moisture), and the potential temperature corresponds to the dry-adiabat

passing through a predicted afternoon maximum temperature (lapse rate in a static atmosphere)

(Ibid). These assumptions were later turned into a formula based upon observing the

environmental temperature at the 500 mb level and then subtracting the temperature at the 500

mb level inside the hypothetical parcel of air. The difference between these two numbers can

determine the stability of the atmosphere with zero being the most stable and negative nine

indicating extreme instability (NWS Louisville 2011).

There are many different ways to calculate the LI. Two major methods include the

Surface-Based Lift Index (SBLI, shown above) and the Showalter Index (SI). The SI was also

developed by Joseph Galway and his colleague Albert Showalter, for whom it was named after

(Showalter 1953). The SI is extensively used in the desert southwest portion of the United States

given the lack of low level moisture throughout a majority of the year (Ibid). It is computed as a

parcel of air is lifted to the 850 mb level by the dry adiabat to saturation and then pseudo-

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adiabatically to the 500mb level. Both temperatures are recorded with a difference of a negative

number representing instability and a positive number indicating stability (Galway 1956). The SI

may be better than the LI in showing instability aloft given the possibility of a shallow low-level

cool air-mass north of a frontal boundary. However, the SI is an unrepresentative index and

inferior to the LI in showing instability if the low-level moisture does not extend up to the 850

mb level (Peppler 1988). The SBLI is more widely used of the two indices.

A fourth broadly used stability index is the K-Index (KI). This index is a measure of

thunderstorm potential built upon the vertical temperature lapse rate plus the vertical range of

low-level moisture in the lower atmosphere (NWS Louisville 2011). The KI was developed by

an airline pilot named Joseph George (George 1960). The mathematical formula he developed is:

,

where temperatures are measured in degrees Celsius, Td is the dew point at 850mb and DD is the

dew-point depression at 700mb (NWS Louisville 2011). A KI value of below thirty usually

indicates a slight chance of thunderstorms with heavy rain, while a KI of over thirty indicates a

good chance of thunderstorms with heavy rain (Ibid). KI are also used to determine the potential

of airmass thunderstorms and flooding. When the K-index is quite high (above thirty five), it

means there is a likelihood to see numerous thunderstorms develop that train over an area

causing heavy rainfall and flooding (Ibid).

A fifth commonly used index in severe weather forecasting is the Total Totals-Index

(TT). This index is usually used if the lift index indicates a strong chance of severe weather

(Miller 1972). In order to calculate this index, the vertical totals and cross totals must first be

calculated. The Vertical Total (VT) measures vertical stability without regard for moisture and is

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found by subtracting the 500 mb temperature from the 850 mb temperature to get a difference

(Ibid). The Cross Total (CT) is a measure of stability that includes moisture and is found by

subtracting the 500 mb temperature from the 850 mb dew point temperature (Ibid). The TT index

is then found by adding the two measures of stability:

[ ],

where T (temperature) and Td (dew point) is measured in Celsius (NWS Louisville 2011). If the

TT is below forty, thunderstorms are unlikely since the VT will be close to the dry adiabatic

lapse rate, however if a CT value is greater than eighteen convection is possible (Ibid). Anything

over forty is considered a chance for thunderstorms.

The sixth thermodynamic index is the Deep Convective Index (DCI). This index is

similar to the LI by attempting to combine the properties of equivalent potential temperature ( )

at 850 mb with instability (Barlow 1993). Theta E is the temperature a parcel of air would reach

if all the water vapor in the parcel were to condense releasing latent heat and the parcel was

brought adiabatically to a standard U.S. atmospheric pressure (Peppler 1988). The formula for

the DCI is:

( ) ,

where, T and Td are measured in Celsius at the values and the Lift Index (LI) is the measure of

stability from the surface to the 500 mb level (Barlow 1993). Values of thirty or higher indicate

the potential for strong thunderstorms whereas values below indicate non-convective

environments (Ibid).

The seventh and last stability index is the Severe Weather Threat Index (SWEAT)

developed by the U. S. Air Force in 1972 (Miller 1972). This index combines all severe weather

dynamic and thermodynamic parameters into one equation. These factors include low-level

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moisture (the 850mb dew point) , instability (TT), lower and middle-level wind speeds (at

850mb and 500mb levels), and warm air advection (NWS Louisville 2011). The equation for

SWEAT is:

( ) ( )

where the first term is set to zero if the 850 mb dew point temperature in degrees Celsius is

negative, TT is the Total Totals Index (if the TT is less than forty-nine, the second term is set to

zero), are the wind speed at those pressure levels in knots and where “s” is the

sine of the difference in wind directions at 500 mb and 800mb (Peppler 1988). It is also

important to note that the shear term, 125(s + 0.2), is set to zero if any of the following

conditions are not met: (i) 850 mb wind direction is in the range of 130 to 250 degrees, (ii) the

500 mb wind direction is in the range of 210 to 310 degrees, (iii) the 500 mb wind direction to

the 850 mb wind direction is greater than zero and (iv) both the 850 mb and 500 mb wind speeds

are greater than or equal to fifteen knots (Ibid). SWEAT values from zero to three hundred

indicate the potential for thunderstorms, where SWEAT values over three hundred indicate the

possibility of severe thunderstorms and tornadoes (NWS Louisville 2011). It is also important to

note that SWEAT values can change dramatically throughout the day, so low valued data can

give a false impression of severe weather (Ibid). It is important to compare this index to others.

In closing, these atmospheric thermodynamic stability indices are useful to help predict

severe weather and thunderstorms. There are many more stability parameters used by a myriad

of professionals. However, there is no correct indice to use over another in any certain situation

but from these indexes meteorologists and climatologists can be better informed on how to

answer questions such as when and where thunderstorms will develop.

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BIBLOGRAPHY

Blanchard, David O., 1998: Assessing the Vertical Distribution of Convective Available

Potential Energy. Weather Forecasting, 13, 870–877.

Colby, Frank P., 1984: Convective Inhibition as a Predictor of Convection during AVE-

SESAME II. Monthly Weather Review, 112, 2239–2252.

Davies, J.M., 1993: Hourly helicity, instability, and EHI in forecasting supercell tornadoes. The

17th Conference on Severe Local Storms. St. Louis, MO, American Meteorological

Society, 107-111.

Doswell, C.A., 2001: Severe Convective Storms—an overview. Severe Convective Storms

Monograph, No. 50, American Meteorology Society, 1-26.

Evans, J. S., 2003: Severe Thunderstorm Forecasting-- An Overview. Severe Weather

Conference Presentation at North Carolina State University. Raleigh, NC, presented on

18th, Sept, 2003.

Galway, J.G., 1956: The lifted index as a predictor of latent instability. Bulletin of the American

Meteorological Society, 37, 528-529.

George, J.J., 1960: Weather Forecasting for Aeronautics. Academic Press, 673 pp.

Markowski, P. M., and Y. P. Richardson, 2010: Mesoscale Meteorology in Midlatitudes--

Advancing Weather and Climate Science. John Wiley and Sons, 33-34.

Miller, R.C., 1972: Notes on analysis and severe storm forecasting procedures of the Air Force

Global Weather Central. Tech. Rept. 200(R), Headquarters, Air Weather Service, USAF,

190 pp.

Moncrieff, M.W., and J.S.A. Green, 1972: The propagation of steady convective overturning in

shear. Quarterly Journal of the Royal Meteorological. Society, 98, 336-352.

Peppler, Randy, 1988: A Review of Static Stability Indices and Related Thermodynamic

Parameters. The Illinois State Water Survey Division, Department of Energy and Natural

Resources. 2-6.

Showalter, A.K., 1953: A stability index for thunderstorm forecasting. Bulletin of the American

Meteorological Society, 34, 250-252.

Stensrud, David J., John V. Cortinas, Harold E. Brooks, 1997: Discriminating between Tornadic

and Non-tornadic Thunderstorms Using Mesoscale Model Output. Journal of Weather

Forecasting, 12, 613–632. doi: http://dx.doi.org/10.1175/1520-

0434(1997)012<0613:DBTANT>2.0.CO;2

The Louisville NWS, cited 2011: National Weather Service Louisville-Convective Parameters

and Indices. [Available online at http://www.crh.noaa.gov/lmk/soo/docu/indices.php.]

The University of Illinois at Urbana-Champaign cited 2011: Helicity. [Available online at

http://ww2010.atmos.uiuc.edu/%28Gh%29/guides/mtr/svr/modl/fcst/params/hel.rxml.]