Temperature and specific humidity
Published:
This is mainly talking about the relation between temperature, relative humidity and specific humidity.
We first clarify some notations:
(a). $q$ specific humidity or the mass mixing ratio of water vapor to total air (dimensionless)
(b). $ w$ mass mixing ratio of water vapor to dry air (dimensionless)
(c). $ e_s(t)$ saturation vapor pressure (hPa)
(d). $ e_{s0}=6.111$ saturation vapor pressure at $ T_0$ (hPa)
(e). $ R_d=287$ specific gas constant for dry air $ (J\cdot kg^{-1}\cdot K^{-1})$
(f). $ R_v=461$ specific gas constant for water vapor $ (J \cdot kg^{-1}\cdot K^{-1})$
(g). $ p$ pressure (hPa)
(h). $ L_v(T)=2.26\cdot 10^6$ specific enthalpy of vaporization $ (J \cdot kg^{-1})$
(i). $ T$ temperature (K)
(j). $ T_0$ reference temperature (K)
From definition of RH, one knows
\[\displaystyle RH = \dfrac{e}{e_s}. \ \ \ \ \ (1)\]From Clausius-Clapeyron, using T with unit K,
\[\displaystyle e_s(T) = e_{s0}\exp\left[\left(\dfrac{L_v(T)}{R_v}\right)\left(\dfrac{1}{T_0} -\dfrac{1}{T} \right)\right] \approx 6.11\exp\left(\dfrac{17.67(T-T_0)}{T-29.65}\right). \ \ \ \ \ (2)\]As if one uses T with unit C,
\[\displaystyle e_s(T) \approx 6.11\exp\left(\dfrac{17.67T}{T-237.3}\right). \ \ \ \ \ (2a)\]If one applies (2a) to (1),
\[\displaystyle RH=\frac{\exp\left(\dfrac{17.67Td}{Td-237.3}\right) }{\exp\left(\dfrac{17.67T}{T-237.3}\right) }.\]where $ Td$ is dewpoint. Given $ RH$ and $ T$, one can also use above formula to get $ Td$.
Applying
\[\displaystyle P=\rho RT, \ \ \ \ \ (3)\]one gets
\[\displaystyle w = \dfrac{m_v}{m_d}=\dfrac{\rho_v}{\rho_d}=\dfrac{e\ R_d}{R_v(p-e)}. \ \ \ \ \ (4)\]By (a), (b) , specific humidity is defined as.
\[\displaystyle q = \dfrac{w}{w+1}. \ \ \ \ \ (5)\]Substitute (4) to (5) , one gets
\[\displaystyle q=\frac{\frac{R_d}{R_v}e}{\frac{R_d}{R_v}e+(p-e)}\approx \frac{0.622e}{p-(1-0.622)e}.\]Given $ RH$, $ T$ and $ P$, one can apply (1) and (2) to get $ e$. Finally, put the calculated $ e$ to above formula, one gets $ q$.