Equivalent potential temperature
Encyclopedia
Equivalent potential temperature, commonly referred to as theta-e 
, is a quantity related to the stability
of a column of air in the atmosphere
.
is the temperature
a parcel of air would reach if all the water vapor in the parcel were to condense, releasing its latent heat
, and the parcel was brought adiabatically to a standard reference pressure, usually 1000 hPa (1000 mbar) which is roughly equal to atmospheric pressure
at sea level
. In stable conditions,
increases with altitude. If 
decreases with height, convection
can occur. The comparison of the equivalent potential temperature of parcels of air at different pressures thus provides a measure of the instability of the column of air.
than warm air (see gas laws
). Like a ball balanced on top of a hill, denser fluid
lying above less dense fluid is dynamically unstable: if cool air is positioned above warm air, the former will sink and the latter will rise, the two volumes of air passing around and through each other, and perhaps mixing to some extent, until a stable condition (with denser fluid below and lighter fluid above) is achieved. The temperature near the ceiling of a room is consistently warmer than that near the floor.
If a hydrostatic fluid is compressible, the criterion for dynamic stability is not simply that denser fluid must lie below light fluid, but that small perturbations must tend to correct themselves. When lower fluid is raised up into upper fluid, (during which process the density of the lower fluid decreases due to the drop in pressure), stability requires that it remain denser than the upper fluid, so that gravity pulls it back toward its original position. The fluid is unstable if small perturbations tend to amplify themselves, i.e. if dense lower fluid, when displaced upward, expands enough to become lighter than the surrounding upper fluid, and therefore continues to move upward.
of the parcel is used up in doing the work required to expand against the atmospheric pressure, so the temperature of the parcel drops, even though it has not lost any heat
. Conversely, a sinking parcel is compressed and becomes warmer even though no heat is added.
Air at the top of a mountain is usually colder than the air in the valley below, but the arrangement is not unstable: if a parcel of air from the valley were somehow lifted up to the top of the mountain, when it arrived it would be even colder than the air already there, due to adiabatic cooling; it would be heavier than the ambient air, and would sink back toward its original position. Similarly, if a parcel of cold mountain-top air were to make the trip down to the valley, it would arrive warmer and lighter than the valley air, and would float back up the mountain.
So cool air lying on top of warm air can be stable after all (as long as the temperature decrease with height is less than the adiabatic lapse rate); the dynamically important quantity is not the temperature, but the potential temperature—the temperature the air would have if it were brought adiabatically to a reference pressure. The air around the mountain is stable because the air at the top, due to its lower pressure, has a higher potential temperature than the warmer air below.
: it becomes saturated with water vapor. This occurs because the vapor pressure
of water decreases at lower temperatures (see Clausius-Clapeyron relation
). If the parcel of air continues to rise, water vapor begins to condense into liquid droplets. The condensing water releases its latent heat to the surrounding air, partially offsetting the adiabatic cooling. A saturated parcel of air therefore cools less than a dry one would as it rises (its temperature changes with height at the moist adiabatic lapse rate, which is smaller than the dry adiabatic lapse rate).
Saturated air can be unstable even though its potential temperature increases with height: if the warming due to condensation is enough that a parcel of saturated air which is displaced upward (and would otherwise cool to below the ambient temperature) ends up warmer (lighter) than the surrounding air, it will continue to rise. This is the reason for defining the equivalent potential temperature, in analogy with the potential temperature: potential temperature is a temperature adjusted for potential warming due to adiabatic compression; equivalent potential temperature factors in potential warming due to condensation as well. For saturated air, or for air which is likely to be lifted high enough to reach saturation, it is the equivalent potential temperature which must increase with height in order to ensure stability.
is the following:
Where:
, is a quantity related to the stabilityInstability
In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds...
of a column of air in the atmosphere
Earth's atmosphere
The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...
.
is the temperatureTemperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
a parcel of air would reach if all the water vapor in the parcel were to condense, releasing its latent heat
Latent heat
Latent heat is the heat released or absorbed by a chemical substance or a thermodynamic system during a process that occurs without a change in temperature. A typical example is a change of state of matter, meaning a phase transition such as the melting of ice or the boiling of water. The term was...
, and the parcel was brought adiabatically to a standard reference pressure, usually 1000 hPa (1000 mbar) which is roughly equal to atmospheric pressure
Atmospheric pressure
Atmospheric pressure is the force per unit area exerted into a surface by the weight of air above that surface in the atmosphere of Earth . In most circumstances atmospheric pressure is closely approximated by the hydrostatic pressure caused by the weight of air above the measurement point...
at sea level
Sea level
Mean sea level is a measure of the average height of the ocean's surface ; used as a standard in reckoning land elevation...
. In stable conditions,
increases with altitude. If decreases with height, convectionConvection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....
can occur. The comparison of the equivalent potential temperature of parcels of air at different pressures thus provides a measure of the instability of the column of air.
Stability
Cool air is denserDensity
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
than warm air (see gas laws
Gas laws
The early gas laws were developed at the end of the 18th century, when scientists began to realize that relationships between the pressure, volume and temperature of a sample of gas could be obtained which would hold for all gases...
). Like a ball balanced on top of a hill, denser fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
lying above less dense fluid is dynamically unstable: if cool air is positioned above warm air, the former will sink and the latter will rise, the two volumes of air passing around and through each other, and perhaps mixing to some extent, until a stable condition (with denser fluid below and lighter fluid above) is achieved. The temperature near the ceiling of a room is consistently warmer than that near the floor.
If a hydrostatic fluid is compressible, the criterion for dynamic stability is not simply that denser fluid must lie below light fluid, but that small perturbations must tend to correct themselves. When lower fluid is raised up into upper fluid, (during which process the density of the lower fluid decreases due to the drop in pressure), stability requires that it remain denser than the upper fluid, so that gravity pulls it back toward its original position. The fluid is unstable if small perturbations tend to amplify themselves, i.e. if dense lower fluid, when displaced upward, expands enough to become lighter than the surrounding upper fluid, and therefore continues to move upward.
Potential temperature
In the atmosphere, where vertical variation in pressure is much larger than in a room, the situation is complicated by adiabatic temperature change: as a parcel of air moves upward, the ambient pressure drops, causing the parcel to expand. Some of the internal energyInternal energy
In thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...
of the parcel is used up in doing the work required to expand against the atmospheric pressure, so the temperature of the parcel drops, even though it has not lost any heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
. Conversely, a sinking parcel is compressed and becomes warmer even though no heat is added.
Air at the top of a mountain is usually colder than the air in the valley below, but the arrangement is not unstable: if a parcel of air from the valley were somehow lifted up to the top of the mountain, when it arrived it would be even colder than the air already there, due to adiabatic cooling; it would be heavier than the ambient air, and would sink back toward its original position. Similarly, if a parcel of cold mountain-top air were to make the trip down to the valley, it would arrive warmer and lighter than the valley air, and would float back up the mountain.
So cool air lying on top of warm air can be stable after all (as long as the temperature decrease with height is less than the adiabatic lapse rate); the dynamically important quantity is not the temperature, but the potential temperature—the temperature the air would have if it were brought adiabatically to a reference pressure. The air around the mountain is stable because the air at the top, due to its lower pressure, has a higher potential temperature than the warmer air below.
Water vapor
A parcel of air containing water vapor, if it rises far enough, cools to its dew pointDew point
The dew point is the temperature to which a given parcel of humid air must be cooled, at constant barometric pressure, for water vapor to condense into liquid water. The condensed water is called dew when it forms on a solid surface. The dew point is a saturation temperature.The dew point is...
: it becomes saturated with water vapor. This occurs because the vapor pressure
Vapor pressure
Vapor pressure or equilibrium vapor pressure is the pressure of a vapor in thermodynamic equilibrium with its condensed phases in a closed system. All liquids have a tendency to evaporate, and some solids can sublimate into a gaseous form...
of water decreases at lower temperatures (see Clausius-Clapeyron relation
Clausius-Clapeyron relation
The Clausius–Clapeyron relation, named after Rudolf Clausius and Benoît Paul Émile Clapeyron, who defined it sometime after 1834, is a way of characterizing a discontinuous phase transition between two phases of matter. On a pressure–temperature diagram, the line separating the two phases is known...
). If the parcel of air continues to rise, water vapor begins to condense into liquid droplets. The condensing water releases its latent heat to the surrounding air, partially offsetting the adiabatic cooling. A saturated parcel of air therefore cools less than a dry one would as it rises (its temperature changes with height at the moist adiabatic lapse rate, which is smaller than the dry adiabatic lapse rate).
Saturated air can be unstable even though its potential temperature increases with height: if the warming due to condensation is enough that a parcel of saturated air which is displaced upward (and would otherwise cool to below the ambient temperature) ends up warmer (lighter) than the surrounding air, it will continue to rise. This is the reason for defining the equivalent potential temperature, in analogy with the potential temperature: potential temperature is a temperature adjusted for potential warming due to adiabatic compression; equivalent potential temperature factors in potential warming due to condensation as well. For saturated air, or for air which is likely to be lifted high enough to reach saturation, it is the equivalent potential temperature which must increase with height in order to ensure stability.
Formula
The formula for is the following:Where:
-
= equivalent temperatureEquivalent temperatureEquivalent temperature is the temperature of an air parcel from which all the water vapor has been extracted by an adiabatic process.Air contains water vapor that has been evaporated into it from liquid sources . The energy needed to do that has been taken from the air... -
= Temperature of air at pressure 
-
= pressure at the point (in same units as 
) -
= standard reference pressure (1000 mbar or 100 kPa) -
= specific gas constant for air (287 J/(kg·K)) -
= specific heat of dry air at constant pressure (1004 J/(kg·K)) -
= latent heatLatent heatLatent heat is the heat released or absorbed by a chemical substance or a thermodynamic system during a process that occurs without a change in temperature. A typical example is a change of state of matter, meaning a phase transition such as the melting of ice or the boiling of water. The term was...
of evaporation (2400 kJ/kg {at 25 °C} to 2600 kJ/kg {at −40 °C}) -
= mixing ratioMixing ratioIn chemistry and physics, the dimensionless mixing ratio is defined as the abundance of one component of a mixture relative to that of all other components...
of water vapor mass per mass of dry air (gGramThe gram is a metric system unit of mass....
/Kg or in
Kg/Kg.)
See also
- MeteorologyMeteorologyMeteorology is the interdisciplinary scientific study of the atmosphere. Studies in the field stretch back millennia, though significant progress in meteorology did not occur until the 18th century. The 19th century saw breakthroughs occur after observing networks developed across several countries...
- Moist static energyMoist static energyThe moist static energy is a thermodynamic variable that describes the state of an air parcel, and is similar to the equivalent potential temperature. The moist static energy is a combination of a parcels kinetic energy due to an air parcel's temperature, its potential energy due to its height...
- Potential temperature
- Weather forecastingWeather forecastingWeather forecasting is the application of science and technology to predict the state of the atmosphere for a given location. Human beings have attempted to predict the weather informally for millennia, and formally since the nineteenth century...