NWP | Specific Humidity

• Relative Humidity and the Dewpoint Temperature
• NWP | Why is RH preferred over mixing ratio for initial moisture in NWP models
• Tephigram | 溫熵圖

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To convert relative humidity (RH) to specific humidity (q), you need the following parameters: - Relative humidity (%) - Air temperature (in °C or K) - Atmospheric pressure (in hPa or Pa)

The formula for specific humidity is:

\[q = \frac{0.622 \cdot e}{P - (0.378 \cdot e)} \]

Where:

• \(e\) is the actual vapor pressure
• \(P\) is the atmospheric pressure
• 0.622 and 0.378 are constants related to the gas constants for dry air and water vapor

To calculate \(e\), you first need the saturation vapor pressure (\(e_s\)) and then use relative humidity:

\[e = \frac{\text{RH} \cdot e_s}{100} \]

The saturation vapor pressure (\(e_s\)) can be approximated using the Tetens formula (for temperatures over water):

\[e_s = 6.1078 \cdot 10^{\frac{7.5T}{T + 237.3}} \text{ (in hPa, if T is in °C)} \]

Step-by-Step Process

1. Input Parameters:
   • Relative humidity (RH, in %)
   • Temperature (T, in °C or K)
   • Pressure (P, in hPa or Pa)
2. Calculate Saturation Vapor Pressure (\(e_s\)):
   • If T is in °C, use the Tetens formula: \(e_s = 6.1078 \cdot 10^{\frac{7.5T}{T + 237.3}}\)
   • If pressure is in Pa, convert \(e_s\)to Pa by multiplying by 100.
3. Calculate Actual Vapor Pressure (\(e\)):
   • \(e = \frac{\text{RH} \cdot e_s}{100}\)
4. Calculate Specific Humidity (\(q\)):
   • Ensure pressure (\(P\)) is in the same units as \(e\)(hPa or Pa).
   • Use: \(q = \frac{0.622 \cdot e}{P - (0.378 \cdot e)}\)
   • The result \(q\)is in kg/kg (dimensionless, mass of water vapor per mass of air).

Example Calculation

Suppose:

• RH = 60%
• T = 25°C
• P = 1013.25 hPa

1. Saturation Vapor Pressure: \[e_s = 6.1078 \cdot 10^{\frac{7.5 \cdot 25}{25 + 237.3}} \] \[e_s = 6.1078 \cdot 10^{\frac{187.5}{262.3}} \approx 6.1078 \cdot 10^{0.7148} \approx 6.1078 \cdot 5.189 \approx 31.67 \, \text{hPa} \]

2. Actual Vapor Pressure: \[e = \frac{60 \cdot 31.67}{100} \approx 19.00 \, \text{hPa} \]

3. Specific Humidity: \[q = \frac{0.622 \cdot 19.00}{1013.25 - (0.378 \cdot 19.00)} \] \[q = \frac{11.818}{1013.25 - 7.182} \approx \frac{11.818}{1006.068} \approx 0.01175 \, \text{kg/kg} \]

So, the specific humidity is approximately 11.75 g/kg (or 0.01175 kg/kg).

Notes

• Ensure consistent units (e.g., convert pressure to Pa if needed: 1 hPa = 100 Pa).

References

1. 3.1 Ways to Specify Water Vapor | METEO 300 - Dutton Institute