\[fig:sol\_diag\_lowpre\], one can see that C-c solutions exist only for $P_\mathrm{c}>2P_\mathrm{min}$. This is because if $P_\mathrm{c}<2P_\mathrm{min}$, the solutions meet the equilibrium state before arriving at the critical point (see equation (\[P crit\])).

### Structure of Transition Layers {#sec:jcrit}

We investigate how the structure of transition layers depends on the mass flux. As mentioned in section \[sec:ZP\], in the ZP solutions, the thickness of the front is characterized by the Field length because the mass flux is very small. As $|j|$ increases, the effect of advection (the first term of equation (\[integ eq0\])) becomes important compared with the heat conduction (the third term of equation (\[integ eq0\])) in the energy transfer. We can define a critical mass flux $j_\mathrm{crit}$ where the effect of advection becomes comparable to the heat conduction as follows: $$- c_pj_\mathrm{crit}\frac{T_\mathrm{m}}{l_\mathrm{F,m}} = \kappa \frac{T_\mathrm{m}}{l_\mathrm{F,m}^2}
        \Rightarrow j_\mathrm{crit}=-\frac{l_\mathrm{F,m}}{\gamma l_\mathrm{c,m}} \rho_\mathrm{m}c_\mathrm{m},
        \label{jcrit}$$ where the subscript ‘m’ denotes a reference state in the solutions, $l_\mathrm{c,m}$ is the cooling length defined by $$l_\mathrm{c,m} = c_m \frac{P_\mathrm{m}}{(\gamma-1)\rho_\mathrm{m}{\cal L}_\mathrm{m}},$$ and $c_\mathrm{m}=\sqrt{\gamma P_\mathrm{m}/\rho_\mathrm{m}}$ is the sound speed. As the reference state, we consider the state where $|\cal L|$ has the maximum value for $T>T_\mathrm{c}$ under the constant pressure $P=P_\mathrm{c}$. This state roughly gives minimum Field length 