where the subscript ‘w’ denotes the physical quantities in the WNM. Equation (\[saturate\]) shows the balance between cooling and heating inside the transition layer. The pressure satisfying equation (\[saturate\]) is called saturation pressure.

In steady solutions with non-zero mass flux $j\ne0$, integrating equation (\[integ eq0\]) from $x=-\infty$ to $x=\infty$, one obtains $$j = - \frac{q}{c_p \left( T_\mathrm{w} - T_\mathrm{c}\right)},\;\;\mathrm{where}\;\;
        q = \int_{-\infty}^{\infty} \rho{\cal L}(\rho,T)dx, 
\label{j}$$ and we neglect the second term of equation (\[integ eq0\]) since the flow is subsonic and hence almost isobaric. In the saturation pressure, $q=0$ since $j=0$. For $P>P_\mathrm{sat}$, $q$ is positive because cooling dominates heating inside the front. Thus, $j<0$ from equation (\[j\]), i.e. the front corresponds to the condensation. In contrast, for $P<P_\mathrm{sat}$, the front describes the evaporation. In the ZP solutions, the mass flux $j$ is very small. Thus, the energy is mainly transferred by heat conduction. From equation (\[integ eq0\]), one can see that the thickness of the front is characterized by the Field length [@BM90] defined by $$l_\mathrm{F}\equiv \sqrt{\frac{\kappa T}{|\rho {\cal L}|}}.$$ The typical values of $l_\mathrm{F}$ is as large as $10^{-3}$ pc in the CNM, and $10^{-1}$ pc in the WNM. From equation (\[j\]), the flow speed across the transition 