Relative Humidity and the Dewpoint Temperature

Key words: Relative Humidity (rh), Dewpoint Temperature (dewpoint), Water Vapor Mixing Ratio (qv)

• Bolton, D. (1980). The computation of equivalent potential temperature. Monthly weather review, 108(7), 1046-1053.
• Lawrence, M. G. (2005). The relationship between relative humidity and the dewpoint temperature in moist air: A simple conversion and applications. Bulletin of the American Meteorological Society, 86(2), 225-234.
  • How are the dewpoint temperature and relative humidity related, and is there an easy and sufficiently accurate way to convert between them without using a calculator?

Water vapor

Water vapor is the gaseous state of water (\(\text{H}_2\text{O}\)) existing below its boiling point, formed through vaporization or sublimation. It remains invisible unless water droplets are suspended within it (e.g., steam, fog). Its chemical formula is \(\text{H}_2\text{O}_{(g)}\), with molecules existing as dimers or trimers in the vapor phase.

Key Humidity Metrics

1. Water Vapor Mixing Ratio
- Definition: Mass of water vapor (\(m_v\)) per unit mass of dry air (\(m_d\)):
\[ w = \frac{m_v}{m_d} \quad (\text{units: g/kg or kg/kg}) \]
- Differs from specific humidity by <3% at typical atmospheric levels.
- Used in meteorological diagrams (e.g., skew-T) to analyze air parcels.

2. Absolute Humidity
- Definition: Mass of water vapor per unit volume of air:
\[ \rho_v = \frac{m_v}{V} \quad (\text{units: g/m}^3\text{)} \]
- Varies with temperature and pressure, making it less reliable for dynamic conditions.
- Maximum values near 30 g/m³ in saturated tropical air.

3. Specific Humidity
- Definition: Mass of water vapor (\(m_v\)) per total mass of moist air (\(m_v + m_d\)):
\[ q = \frac{m_v}{m_v + m_d} \quad (\text{units: g/kg}) \]
- Remains constant during temperature/pressure changes unless moisture is added/removed.
- Key for tracking air masses due to its stability.

4. Relative Humidity (RH)
- Definition: Ratio of actual vapor pressure (\(e\)) to saturation vapor pressure (\(e_s\)) at air temperature:
\[ \text{RH} = \frac{e}{e_s} \times 100\% \]
- Temperature-dependent: Warm air can hold more moisture, lowering RH at constant \(e\).
- At 100% RH, dew forms via condensation.

Saturation vapor pressure (SVP)

Saturation vapor pressure (SVP) is the equilibrium pressure exerted by water vapor in air that is fully saturated (unable to hold more moisture) at a given temperature. It is a critical parameter for understanding humidity, cloud formation, and weather processes. Below are the key concepts and formulas:

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Key Equations

1. Clausius-Clapeyron Equation (Theoretical Basis)

Describes the temperature dependence of SVP:
\[ \frac{d e_s}{d T} \approx \frac{L_v \cdot e_s}{\Re_v \cdot T^2} \]

• \(e_s\): Saturation vapor pressure
  
• \(L_v\): Latent heat of vaporization (\(2.5 \times 10^3 \, \text{J/g}\) for water, \(2.824 \times 10^3 \, \text{J/g}\) for ice)
  
• \(\Re_v\): Gas constant for water vapor (\(461.5 \, \text{J/kg·K}\))
  
• \(T\): Temperature (K)

This differential equation shows SVP increases exponentially with temperature.

2. Arden Buck Equation (Empirical)

Widely used in meteorology for calculating SVP over liquid water and ice:

Over liquid water (\(T > 0^\circ \text{C}\)):
\[ e_s(T) = 6.1121 \exp\left(\left(18.678 - \frac{T}{234.5}\right) \cdot \frac{T}{257.14 + T}\right) \, \text{(hPa)} \]

Over ice (\(T < 0^\circ \text{C}\)):
\[ e_s(T) = 6.1115 \exp\left(\left(23.036 - \frac{T}{333.7}\right) \cdot \frac{T}{279.82 + T}\right) \, \text{(hPa)} \]
- Valid for \(-80^\circ \text{C} < T < 50^\circ \text{C}\) with errors < 0.5%.

3. Simplified Magnus Formula

For quick approximations over water (\(0^\circ \text{C} < T < 100^\circ \text{C}\)):
\[ e_s(T) = 6.112 \exp\left(\frac{17.67 \cdot T}{T + 243.5}\right) \, \text{(hPa)} \]
- \(T\) in \(^\circ \text{C}\).

Temperature Dependence

SVP doubles approximately every \(10^\circ \text{C}\) (or \(20^\circ \text{F}\)) temperature increase. For example:
- At \(0^\circ \text{C}\): \(6.11 \, \text{hPa}\)
- At \(20^\circ \text{C}\): \(23.4 \, \text{hPa}\)
- At \(30^\circ \text{C}\): \(42.4 \, \text{hPa}\).

Applications

1. Relative Humidity (RH):
   \[ \text{RH} = \frac{\text{Actual Vapor Pressure}}{e_s(T)} \times 100\% \]
   Indicates air’s moisture content relative to its capacity.

2. Dewpoint: Temperature at which air becomes saturated (\(RH = 100\%\)). Calculated iteratively using \(e = e_s(T_d)\).

3. Weather Prediction: Used in thermodynamic diagrams (e.g., skew-T) to assess cloud formation and storm potential.

WRF

• relative humidity | 2020
  • The Relative Humidity (RH) is simply the mixing ratio divided by the saturation mixing ratio. wrfout files include mixing ratio and T, from which you can easily obtain RH.
  • Here is a small piece of Fortran to calculate relative humidity from mixing ratio:

    real, parameter :: svp1=611.2
    real, parameter :: svp2=17.67
    real, parameter :: svp3=29.65
    real, parameter :: svpt0=273.15
    real, parameter :: eps = 0.622
    rh = 1.E2 * (p*q/(q*(1.-eps) + eps))/(svp1*exp(svp2*(t-svpt0)/(T-svp3)))

  • Note that t (temperature), p (pressure) and q (mixing ratio) are output variables in wrfout.
• How to derive Specific Humidity from WRF output ? | 2024
  • You're right, that specific humidity is not a variable that is available to output with wrf. It looks like there is one (called QLEV_URB3D) if you happen to be using an urban parameterization scheme. Otherwise, you may have to calculate it during post-processing. I would recommend using either NCL or python to try to get the output you want.