Equivalent potential temperature
Equivalent potential temperature, commonly referred to as theta-e , is a quantity that is conserved during changes to an air parcel's pressure (that is, during vertical motions in the
is the
Its use in estimating atmospheric stability
Stability of incompressible fluid
Like a ball balanced on top of a hill, denser fluid lying above less dense fluid would be dynamically unstable: overturning motions (convection) can lower the center of gravity, and thus will occur spontaneously, rapidly producing a stable stratification which is thus the observed condition almost all the time. The condition for stability of an incompressible fluid is that density decreases monotonically with height.
Stability of compressible air: potential temperature
If a fluid is
To understand this, consider dry convection in the atmosphere, where the vertical variation in pressure is substantial and adiabatic temperature change is important: As a parcel of air moves upward, the ambient pressure drops, causing the parcel to expand. Some of the internal energy 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, as long as the temperature decrease with height is less than the
Effects of water condensation: equivalent potential temperature
A rising parcel of air containing water vapor, if it rises far enough, reaches its
Formula
The definition of the equivalent potential temperature is:[1][2]
Where:
- is the temperature [K] of air at pressure ,
- is a reference pressure that is taken as 1000 hPa,
- is the pressure at the point,
- and are the specific gas constants of dry air and of water vapour, respectively,
- and are the specific heat capacities of dry air and of liquid water, respectively,
- and are the total water and water vapour mixing ratios, respectively,
- is the relative humidity,
- is the latent heat of vapourisation of water.
A number of approximate formulations are used for calculating equivalent potential temperature, since it is not easy to compute integrations along motion of the parcel. Bolton (1980) [3] gives review of such procedures with estimates of error. His best approximation formula is used when accuracy is needed:
Where:
- is (dry) potential temperature [K] at the lifted condensation level (LCL),
- is (approximated) temperature [K] at LCL,
- is dew point temperature at pressure ,
- is the water vapor pressure (to obtain for dry air),
- is the ratio of the specific gas constant to the specific heat of dry air at constant pressure (0.2854),
- is mixing ratio of water vapor mass per mass [kg/kg] (sometimes value is given in [g/kg][4] and that should be divided by 1000).
A little more theoretical formula is commonly used in literature like Holton (1972) [5] when theoretical explanation is important:
Where:
- is saturated mixing ratio of water at temperature , the temperature at the saturation level of the air,
- is latent heat of evaporation at temperature (2406 kJ/kg {at 40 °C} to 2501 kJ/kg {at 0 °C}), and
- is specific heat of dry air at constant pressure (1005.7 J/(kg·K)).
Further more simplified formula is used (in, for example, Stull 1988[6] §13.1 p. 546) for simplicity, if it is desirable to avoid computing :
Where:
- = equivalent temperature
- = specific gas constant for air (287.04 J/(kg·K))
Usage
This applies on the
In the mesoscale, equivalent potential temperature is also a useful measure of the static stability of the unsaturated atmosphere. Under normal, stably stratified conditions, the potential temperature increases with height,
and vertical motions are suppressed. If the equivalent potential temperature decreases with height,
the atmosphere is unstable to vertical motions, and convection is likely. Situations in which the equivalent potential temperature decreases with height, indicating instability in saturated air, are quite common.
See also
Bibliography
- M K Yau and R.R. Rogers, Short Course in Cloud Physics, Third Edition, published by Butterworth-Heinemann, January 1, 1989, 304 pages. ISBN 0-7506-3215-1
References
- ^ Emmanuel, Kerry (1994). Atmospheric Convection. Oxford University Press.
- ^ "Equivalent potential temperature". AMS Glossary of Meteorology. American Meteorological Society. Retrieved 2020-11-03.
- ^ D Bolton, 1980: The Computation of Equivalent Potential Temperature. Mon. Wea. Rev., Vol. 108, pp.1046-1053.
- World Meteorological Organisation. Retrieved 2009-08-02.
- ^ J R Holton, An Introduction to Dynamical Meteorology. Academic Press, 1972, 319 pages.
- ISBN 9027727694.
- ..