Virtual temperature

Last updated

In atmospheric thermodynamics, the virtual temperature () of a moist air parcel is the temperature at which a theoretical dry air parcel would have a total pressure and density equal to the moist parcel of air. [1] The virtual temperature of unsaturated moist air is always greater than the absolute air temperature, however, as the existence of suspended cloud droplets reduces the virtual temperature.

Contents

The virtual temperature effect is also known as the vapor buoyancy effect. [2] It has been described to increase Earth's thermal emission by warming the tropical atmosphere. [3] [4]

Introduction

Description

In atmospheric thermodynamic processes, it is often useful to assume air parcels behave approximately adiabatically, and approximately ideally. The specific gas constant for the standardized mass of one kilogram of a particular gas is variable, and described mathematically as

where is the molar gas constant, and is the apparent molar mass of gas in kilograms per mole. The apparent molar mass of a theoretical moist parcel in Earth's atmosphere can be defined in components of water vapor and dry air as

with being partial pressure of water, dry air pressure, and and representing the molar masses of water vapor and dry air respectively. The total pressure is described by Dalton's law of partial pressures:

Purpose

Rather than carry out these calculations, it is convenient to scale another quantity within the ideal gas law to equate the pressure and density of a dry parcel to a moist parcel. The only variable quantity of the ideal gas law independent of density and pressure is temperature. This scaled quantity is known as virtual temperature, and it allows for the use of the dry-air equation of state for moist air. [5] Temperature has an inverse proportionality to density. Thus, analytically, a higher vapor pressure would yield a lower density, which should yield a higher virtual temperature in turn.

Derivation

Consider a moist air parcel containing masses and of dry air and water vapor in a given volume . The density is given by

where and are the densities the dry air and water vapor would respectively have when occupying the volume of the air parcel. Rearranging the standard ideal gas equation with these variables gives

and

Solving for the densities in each equation and combining with the law of partial pressures yields

Then, solving for and using is approximately 0.622 in Earth's atmosphere:

where the virtual temperature is

We now have a non-linear scalar for temperature dependent purely on the unitless value , allowing for varying amounts of water vapor in an air parcel. This virtual temperature in units of kelvin can be used seamlessly in any thermodynamic equation necessitating it.

Variations

Often the more easily accessible atmospheric parameter is the mixing ratio . Through expansion upon the definition of vapor pressure in the law of partial pressures as presented above and the definition of mixing ratio:

which allows

Algebraic expansion of that equation, ignoring higher orders of due to its typical order in Earth's atmosphere of , and substituting with its constant value yields the linear approximation


With the mixing ratio expressed in g/g. [6]

An approximate conversion using in degrees Celsius and mixing ratio in g/kg is [7]

Virtual potential temperature

Virtual potential temperature is similar to potential temperature in that it removes the temperature variation caused by changes in pressure. Virtual potential temperature is useful as a surrogate for density in buoyancy calculations and in turbulence transport which includes vertical air movement.

Density temperature

A moist air parcel may also contain liquid droplets and ice crystals in addition to water vapor. A net mixing ratio can be defined as the sum of the mixing ratios of water vapor , liquid , and ice present in the parcel. Assuming that and are typically much smaller than , a density temperature of a parcel can be defined, representing the temperature at which a theoretical dry air parcel would have the a pressure and density equal to a moist parcel of air while accounting for condensates: [8] :113

Uses

Virtual temperature is used in adjusting CAPE soundings for assessing available convective potential energy from skew-T log-P diagrams. The errors associated with ignoring virtual temperature correction for smaller CAPE values can be quite significant. [9] Thus, in the early stages of convective storm formation, a virtual temperature correction is significant in identifying the potential intensity in tropical cyclogenesis. [10]

Further reading

Related Research Articles

<span class="mw-page-title-main">Relative density</span> Ratio of two densities

Relative density, sometimes called specific gravity, is a dimensionless quantity defined as the ratio of the density of a substance to the density of a given reference material. Specific gravity for liquids is nearly always measured with respect to water at its densest ; for gases, the reference is air at room temperature. The term "relative density" is often preferred in scientific usage, whereas the term "specific gravity" is deprecated.

<span class="mw-page-title-main">Troposphere</span> Lowest layer of Earths atmosphere

The troposphere is the lowest layer of the atmosphere of Earth. It contains 75% of the total mass of the planetary atmosphere and 99% of the total mass of water vapor and aerosols, and is where most weather phenomena occur. From the planetary surface of the Earth, the average height of the troposphere is 18 km in the tropics; 17 km in the middle latitudes; and 6 km in the high latitudes of the polar regions in winter; thus the average height of the troposphere is 13 km.

<span class="mw-page-title-main">Bernoulli's principle</span> Principle relating to fluid dynamics

Bernoulli's principle is a key concept in fluid dynamics that relates pressure, speed and height. Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or the fluid's potential energy. The principle is named after the Swiss mathematician and physicist Daniel Bernoulli, who published it in his book Hydrodynamica in 1738. Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler in 1752 who derived Bernoulli's equation in its usual form.

<span class="mw-page-title-main">Lapse rate</span> Vertical rate of change of temperature in atmosphere

The lapse rate is the rate at which an atmospheric variable, normally temperature in Earth's atmosphere, falls with altitude. Lapse rate arises from the word lapse, in the sense of a gradual fall. In dry air, the adiabatic lapse rate is 9.8 °C/km. The saturated air lapse rate (SALR), or moist adiabatic lapse rate (MALR), is the decrease in temperature of a parcel of water-saturated air that rises in the atmosphere. It varies with the temperature and pressure of the parcel and is often in the range 3.6 to 9.2 °C/km, as obtained from the International Civil Aviation Organization (ICAO). The environmental lapse rate is the decrease in temperature of air with altitude for a specific time and place. It can be highly variable between circumstances.

In the calculus of variations, a field of mathematical analysis, the functional derivative relates a change in a functional to a change in a function on which the functional depends.

Equivalent potential temperature, commonly referred to as theta-e, is a quantity that is conserved during changes to an air parcel's pressure, even if water vapor condenses during that pressure change. It is therefore more conserved than the ordinary potential temperature, which remains constant only for unsaturated vertical motions.

<span class="mw-page-title-main">Rankine–Hugoniot conditions</span> Concept in physics

The Rankine–Hugoniot conditions, also referred to as Rankine–Hugoniot jump conditions or Rankine–Hugoniot relations, describe the relationship between the states on both sides of a shock wave or a combustion wave in a one-dimensional flow in fluids or a one-dimensional deformation in solids. They are named in recognition of the work carried out by Scottish engineer and physicist William John Macquorn Rankine and French engineer Pierre Henri Hugoniot.

The density of air or atmospheric density, denoted ρ, is the mass per unit volume of Earth's atmosphere. Air density, like air pressure, decreases with increasing altitude. It also changes with variations in atmospheric pressure, temperature and humidity. At 101.325 kPa (abs) and 20 °C, air has a density of approximately 1.204 kg/m3 (0.0752 lb/cu ft), according to the International Standard Atmosphere (ISA). At 101.325 kPa (abs) and 15 °C (59 °F), air has a density of approximately 1.225 kg/m3 (0.0765 lb/cu ft), which is about 1800 that of water, according to the International Standard Atmosphere (ISA). Pure liquid water is 1,000 kg/m3 (62 lb/cu ft).

<span class="mw-page-title-main">Stellar structure</span> Structure of stars

Stellar structure models describe the internal structure of a star in detail and make predictions about the luminosity, the color and the future evolution of the star. Different classes and ages of stars have different internal structures, reflecting their elemental makeup and energy transport mechanisms.

<span class="mw-page-title-main">Scale height</span>

In atmospheric, earth, and planetary sciences, a scale height, usually denoted by the capital letter H, is a distance over which a physical quantity decreases by a factor of e.

In atmospheric dynamics, oceanography, asteroseismology and geophysics, the Brunt–Väisälä frequency, or buoyancy frequency, is a measure of the stability of a fluid to vertical displacements such as those caused by convection. More precisely it is the frequency at which a vertically displaced parcel will oscillate within a statically stable environment. It is named after David Brunt and Vilho Väisälä. It can be used as a measure of atmospheric stratification.

The Ostwald–Freundlich equation governs boundaries between two phases; specifically, it relates the surface tension of the boundary to its curvature, the ambient temperature, and the vapor pressure or chemical potential in the two phases.

The Ergun equation, derived by the Turkish chemical engineer Sabri Ergun in 1952, expresses the friction factor in a packed column as a function of the modified Reynolds number.

The Kelvin equation describes the change in vapour pressure due to a curved liquid–vapor interface, such as the surface of a droplet. The vapor pressure at a convex curved surface is higher than that at a flat surface. The Kelvin equation is dependent upon thermodynamic principles and does not allude to special properties of materials. It is also used for determination of pore size distribution of a porous medium using adsorption porosimetry. The equation is named in honor of William Thomson, also known as Lord Kelvin.

<span class="mw-page-title-main">Volume (thermodynamics)</span> Extensive parameter used to describe a thermodynamic systems state

In thermodynamics, the volume of a system is an important extensive parameter for describing its thermodynamic state. The specific volume, an intensive property, is the system's volume per unit mass. Volume is a function of state and is interdependent with other thermodynamic properties such as pressure and temperature. For example, volume is related to the pressure and temperature of an ideal gas by the ideal gas law. The physical region covered by a system may or may not coincide with a control volume used to analyze the system.

Slip ratio in gas–liquid (two-phase) flow, is defined as the ratio of the velocity of the gas phase to the velocity of the liquid phase.

Isentropic nozzle flow describes the movement of a gas or fluid through a narrowing opening without an increase or decrease in entropy.

<span class="mw-page-title-main">Emden–Chandrasekhar equation</span>

In astrophysics, the Emden–Chandrasekhar equation is a dimensionless form of the Poisson equation for the density distribution of a spherically symmetric isothermal gas sphere subjected to its own gravitational force, named after Robert Emden and Subrahmanyan Chandrasekhar. The equation was first introduced by Robert Emden in 1907. The equation reads

Conservative temperature is a thermodynamic property of seawater. It is derived from the potential enthalpy and is recommended under the TEOS-10 standard as a replacement for potential temperature as it more accurately represents the heat content in the ocean.

The mass-flux fraction is the ratio of mass-flux of a particular chemical species to the total mass flux of a gaseous mixture. It includes both the convectional mass flux and the diffusional mass flux. It was introduced by Joseph O. Hirschfelder and Charles F. Curtiss in 1948 and later by Theodore von Kármán and Sol Penner in 1954. The mass-flux fraction of a species i is defined as

References

  1. Bailey, Desmond T. (February 2000) [June 1987]. "Upper-air Monitoring" (PDF). Meteorological Monitoring Guidance for Regulatory Modeling Applications. John Irwin. Research Triangle Park, NC: United States Environmental Protection Agency. pp. 9–14. EPA-454/R-99-005.
  2. "Cold air rises—what that means for Earth's climate". phys.org. Retrieved 2020-07-10.
  3. Yang, Da; Seidel, Seth D. (2020-04-01). "The Incredible Lightness of Water Vapor". Journal of Climate. 33 (7): 2841–2851. Bibcode:2020JCli...33.2841Y. doi: 10.1175/JCLI-D-19-0260.1 . ISSN   0894-8755.
  4. Seidel, Seth D.; Yang, Da (2020-05-01). "The lightness of water vapor helps to stabilize tropical climate". Science Advances. 6 (19): eaba1951. Bibcode:2020SciA....6.1951S. doi: 10.1126/sciadv.aba1951 . ISSN   2375-2548. PMC   7202867 . PMID   32494724.
  5. "AMS Glossary". American Meteorological Society. Retrieved 2014-06-30.
  6. Doswell, Charles A.; Rasmussen, Erik N. (1 December 1994). "The Effect of Neglecting the Virtual Temperature Correction on CAPE Calculations". Weather and Forecasting. 9 (4): 625–629. Bibcode:1994WtFor...9..625D. doi: 10.1175/1520-0434(1994)009<0625:TEONTV>2.0.CO;2 .
  7. U.S. Air Force (1990). The Use of the Skew-T Log p Diagram in Analysis and Forecasting. United States Air Force. pp. 4–9. AWS-TR79/006.
  8. Emanuel, Kerry A. (1994). "Moist Thermodynamic Processes". Atmospheric Convection. Oxford University Press. ISBN   0-19-506630-8 . Retrieved 18 October 2023 via Google Books.
  9. Doswell, Charles A.; Rasmussen, Erik N. (1994). "The Effect of Neglecting the Virtual Temperature Correction on CAPE Calculations". Weather and Forecasting. 9 (4): 625–629. Bibcode:1994WtFor...9..625D. doi: 10.1175/1520-0434(1994)009<0625:TEONTV>2.0.CO;2 .
  10. Camargo, Suzana J.; Sobel, Adam H.; Barnston, Anthony G.; Emanuel, Kerry A. (2007). "Tropical cyclone genesis potential index in climate models". Tellus A. 59 (4): 428–443. Bibcode:2007TellA..59..428C. doi: 10.1111/j.1600-0870.2007.00238.x .