improver.temperature.virtual_temperature module

Calculate the virtual temperature.

class VirtualTemperature

Bases: BasePlugin

Plugin class to handle virtual temperature calculations from humidity mixing ratio.

Virtual Temperature Calculations

Two virtual temperature calculations are currently provided:

1. VirtualTemperature Class Calculation

Input diagnostics:

1. temperature (T)

2. humidity_mixing_ratio (q)

This IMPROVER plugin uses a first order approximation for virtual temperature (\(T_v\)).

You can show:

\[T_v = T \left[ \frac{q}{\epsilon} + (1 - q) \right] = T \left[ 1 + q\left( \frac{1}{\epsilon} - 1 \right) \right]\]

Since:

\[\frac{1}{\epsilon} - 1 \approx 0.62\]

Gives:

\[T_v \approx T (1 + 0.62q)\]

where \(\epsilon\) is the ratio of the gas constants (or molecular weights) of dry air and water vapour:

\[\epsilon = \frac{R_d}{R_v}\]

where:

• \(R_d \approx 287.05 \,\mathrm{J\,kg^{-1}\,K^{-1}}\) is the gas constant for dry air.

• \(R_v \approx 461.5 \,\mathrm{J\,kg^{-1}\,K^{-1}}\) is the gas constant for water vapour.

So:

\[\epsilon \approx \frac{287.05}{461.5} \approx 0.622\]

An equivalent definition using molecular weights is:

\[\epsilon = \frac{M_v}{M_d}\]

where:

• \(M_v \approx 18.016 \,\mathrm{g\,mol^{-1}}\) (water vapour)

• \(M_d \approx 28.964 \,\mathrm{g\,mol^{-1}}\) (dry air)

giving the same value:

\[\epsilon \approx 0.622\]

2. VirtualTemperatureFromSpecificHumidityClass Calculation

Input diagnostics:

1. temperature (T)

2. humidity_mixing_ratio (q)

3. cloud_water_mixing_ratio

4. cloud_ice_mixing_ratio

This calculates virtual temperature using the specific humidity, which is desirable in the calculation of air density.

This virtual temperature (\(T_v\)) calculation also uses condensates, if provided.

Condensed water (rain, snow and liquid cloud droplets) add weight to a volume of air without contributing to the pressure. Their effect can be included in the calculation of virtual temperature.

When the data on the mixing ratios of liquid water (\(q_{cl}\)) and ice (\(q_{cf}\)) are available, a slightly improved estimate of \(T_v\), accounting for this weight, is:

\[T_v = T \left[ \left(\frac{q}{\epsilon}\right) + (1 - q - q_{cl} - q_{cf}) \right]\]

This is only significant lower down in the atmosphere, and where there is cloud, and in most cases a very small adjustment.

Given the densest clouds have condensate specific humidities typically no higher than 5 g kg^-1, the adjustment is unlikely to ever be more than 0.5%.

The virtual temperature calculation can be modified if condensate information is available (Equation 4), which it is from StaGE. This can include condensate information on liquid water, ice, and rain. Subtracting this condensate moisture from within the virtual temperature calculation will have the effect of reducing density slightly.

Where:

• \(T_v\) = virtual temperature (K)

• \(T\) = temperature (K)

• \(q\) = specific humidity (a measure of the actual amount of water vapour in the air, expressed as kg kg^-1)

• \(\epsilon\) = 0.62198, the ratio of gas constants of dry to moist air

  (See EARTH_REPSILON constant in IMPROVER.constants)

• \(q_{cl}\) = mixing ratio of liquid water (cloud_water_mixing_ratio_on_pressure_levels; kg kg^-1)

• \(q_{cf}\) = mixing ratio of ice (cloud_ice_mixing_ratio_on_pressure_levels; kg kg^-1)

Excluding condensates from the virtual temperature calculation will reduce the amount of moisture and will therefore reduce density. This is only significant lower down in the atmosphere and where there is cloud, and in most cases leads to a very small adjustment. Given the densest clouds have condensate specific humidities typically no higher than 5 g kg^-1, the adjustment is unlikely to ever be more than 0.5%.

_abc_impl = <_abc._abc_data object>

static get_virtual_temperature(temperature, humidity_mixing_ratio)

Calculate the virtual temperature from temperature and humidity mixing ratio.

Parameters:

• temperature (Cube) – Cube of temperature.

• humidity_mixing_ratio (Cube) – Cube of humidity mixing ratio.

Return type:

Cube

Returns:

Cube of virtual_temperature.

process(*cubes)

Main entry point for this class.

Parameters:

cubes (Union[Cube, CubeList]) –

air_temperature:

Cube of temperature.

humidity_mixing_ratio:

Cube of humidity mixing ratios.

Return type:

Cube

Returns:

Cube of virtual_temperature.

class VirtualTemperatureFromSpecificHumidity

Bases: BasePlugin

Plugin class to handle virtual temperature calculations from specific humidity.

This calculates virtual temperature using the specific humidity output from StaGE as an input, which is desirable in the calculation of Air Density.

This virtual temperature (Tv) calculation also uses condensates, if provided.

Condensed water (rain, snow and liquid cloud droplets) add weight to a volume of air without contributing to the pressure. Their effect can be included in the calculation of virtual temperature.

The data on the mixing ratios of liquid water (qcl) and ice (qcf) are available so a slightly improved estimate of Tv, accounting for this weight, is:

Tv = T [(q / epsilon) + (1 - q - qcl - qcf - qR) ].

This is only significant lower down in the atmosphere, and where there is cloud, and in most cases a very small adjustment. Given the densest clouds have condensate specific humidities typically no higher than 5 g kg-1, the adjustment is unlikely to ever be more than 0.5%.

Virtual Temperature Calculations

Two virtual temperature calculations are currently provided:

1. VirtualTemperature Class Calculation

Input diagnostics:

1. temperature (T)

2. humidity_mixing_ratio (q)

This IMPROVER plugin uses a first order approximation for virtual temperature (\(T_v\)).

You can show:

\[T_v = T \left[ \frac{q}{\epsilon} + (1 - q) \right] = T \left[ 1 + q\left( \frac{1}{\epsilon} - 1 \right) \right]\]

Since:

\[\frac{1}{\epsilon} - 1 \approx 0.62\]

Gives:

\[T_v \approx T (1 + 0.62q)\]

where \(\epsilon\) is the ratio of the gas constants (or molecular weights) of dry air and water vapour:

\[\epsilon = \frac{R_d}{R_v}\]

where:

• \(R_d \approx 287.05 \,\mathrm{J\,kg^{-1}\,K^{-1}}\) is the gas constant for dry air.

• \(R_v \approx 461.5 \,\mathrm{J\,kg^{-1}\,K^{-1}}\) is the gas constant for water vapour.

So:

\[\epsilon \approx \frac{287.05}{461.5} \approx 0.622\]

An equivalent definition using molecular weights is:

\[\epsilon = \frac{M_v}{M_d}\]

where:

• \(M_v \approx 18.016 \,\mathrm{g\,mol^{-1}}\) (water vapour)

• \(M_d \approx 28.964 \,\mathrm{g\,mol^{-1}}\) (dry air)

giving the same value:

\[\epsilon \approx 0.622\]

2. VirtualTemperatureFromSpecificHumidityClass Calculation

Input diagnostics:

1. temperature (T)

2. humidity_mixing_ratio (q)

3. cloud_water_mixing_ratio

4. cloud_ice_mixing_ratio

This calculates virtual temperature using the specific humidity, which is desirable in the calculation of air density.

This virtual temperature (\(T_v\)) calculation also uses condensates, if provided.

Condensed water (rain, snow and liquid cloud droplets) add weight to a volume of air without contributing to the pressure. Their effect can be included in the calculation of virtual temperature.

When the data on the mixing ratios of liquid water (\(q_{cl}\)) and ice (\(q_{cf}\)) are available, a slightly improved estimate of \(T_v\), accounting for this weight, is:

\[T_v = T \left[ \left(\frac{q}{\epsilon}\right) + (1 - q - q_{cl} - q_{cf}) \right]\]

This is only significant lower down in the atmosphere, and where there is cloud, and in most cases a very small adjustment.

Given the densest clouds have condensate specific humidities typically no higher than 5 g kg^-1, the adjustment is unlikely to ever be more than 0.5%.

The virtual temperature calculation can be modified if condensate information is available (Equation 4), which it is from StaGE. This can include condensate information on liquid water, ice, and rain. Subtracting this condensate moisture from within the virtual temperature calculation will have the effect of reducing density slightly.

Where:

• \(T_v\) = virtual temperature (K)

• \(T\) = temperature (K)

• \(q\) = specific humidity (a measure of the actual amount of water vapour in the air, expressed as kg kg^-1)

• \(\epsilon\) = 0.62198, the ratio of gas constants of dry to moist air

  (See EARTH_REPSILON constant in IMPROVER.constants)

• \(q_{cl}\) = mixing ratio of liquid water (cloud_water_mixing_ratio_on_pressure_levels; kg kg^-1)

• \(q_{cf}\) = mixing ratio of ice (cloud_ice_mixing_ratio_on_pressure_levels; kg kg^-1)

Excluding condensates from the virtual temperature calculation will reduce the amount of moisture and will therefore reduce density. This is only significant lower down in the atmosphere and where there is cloud, and in most cases leads to a very small adjustment. Given the densest clouds have condensate specific humidities typically no higher than 5 g kg^-1, the adjustment is unlikely to ever be more than 0.5%.

_abc_impl = <_abc._abc_data object>

static get_virtual_temperature_specific_humidity(temperature, specific_humidity, cloud_water_mixing_ratio=None, cloud_ice_mixing_ratio=None)

Calculate the virtual temperature from temperature, specific humidity and condensates, if provided.

Parameters:

• Required –

  air_temperature:

  Cube of temperature.

  specific_humidity:

  Cube of specific humidity on pressure levels.

• Optional –

  cloud_water_mixing_ratio_on_pressure_levels:

  Cube of cloud water mixing ratio on pressure levels.

  cloud_ice_mixing_ratio_on_pressure_levels:

  Cube of cloud ice mixing ratio on pressure levels.

Returns:

Cube of virtual_temperature (K).

process(*cubes)

Main entry point for this class. Use Specific Humidity and optional condensate cubes to calculate

the Virtual Temperature cube.

Parameters:

• Required –

  cube:

  air_temperature

  cube:

  specific_humidity

• Optional –

  cube:

  cloud_liquid_water_mixing_ratio

  cube:

  cloud_ice_mixing_ratio

Returns:

virtual_temperature (K).

Return type:

Cube