Virtual potential temperature (changes)

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The **virtual potential temperature** $\theta_V$ is an important concept in atmospheric boundary meteorology. It can serve as a **stability criterion for an atmosphere with a moisture gradient**.

When $\theta_V$ is constant, the atmosphere is statically neutral. When it decreases with elevation, the atmosphere is statically unstable. When it increases with elevation, the atmosphere is statically stable.

The virtual potential temperature is given by:

$\theta_V = (1+ 0.61 q) T (p_0/p)^\kappa ,$

with $T$ the temperature of moist air, $p$ its pressure and $p_0$ a reference pressure (usually 1000 hPa). $q$ is the specific humidity of the air, which is the mass of the water vapour per unit mass of moist air, i.e. $q=\rho_v/(\rho_v + \rho_d)$ with $\rho_v$ the density of the water vapour and $\rho_d$ the density of dry air. And, for moist air,

$\kappa = R_d (1- 0.23 q)/c_pd .$

$R_d$ is the specific gas constant for dry air. For dry air, $q=0$.

In order to understand the virtual potential temperature, it is important to introduce the concepts **virtual temperature** and **potential temperature** first.

Moist air is lighter than dry air at the same pressure and temperature. In terms of virtual temperature, wet air has a higher virtual temperature than dry air.

The virtual temperature (of moist air) is the temperature dry air should have to have the same density as moist air with a given $q$, $p$ and $T$. It is defined by

$T_V = (1+ (M_{air}/M_{\mathrm{H}_2\mathrm{O}} -1) q) T .$

$M_{\mathrm{H}_2\mathrm{O}}$ is the molar mass of water, $M_{air}$ the molar mass of dry air, and $M_{air}/M_{\mathrm{H}_2\mathrm{O}}-1\approx0,61$.

The definition is obtained from the equation of state of moist air, i.e.

$p = \rho R_d T (1 + (M_{air}/M_{\mathrm{H}_2\mathrm{O}}-1) q)$

which can be derived by making use of Dalton’s law.

The potential temperature is the temperature which would result if the air were brought adiabatically to a standard pressure level $p_0=1000 hPa$. Combining the first law of thermodynamics with the equation of state of air gives

$\theta = T (p_0/p)^\kappa$

The potential temperature can serve as a stability criterion for an atmosphere which has a uniform moisture content $q$; when $\theta$ decreases with elevation, the atmosphere is unstable, and vice versa.

The potential temperature is conserved during an adiabatic displacement.

The virtual potential temperature is the virtual temperature a substance would have if changed adiabatically from its actual state to the standard pressure $p_0$.

$\theta_V = (1+ (M_{air}/M_{\mathrm{H}_2\mathrm{O}} -1) q) T (p_0/p)^\kappa$

The potential virtual temperature (of moist air) is the potential temperature of dry air at the same pressure and density. It is given by the same formula as the virtual potential temperature, but with $\kappa=R/c_{pd}$. For many purposes, it can be used instead of the virtual potential temperature.

category: methodology

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