Cosmic-ray propagation in the bi-stable interstellar medium. I.
  Conditions for cosmic-ray trapping
  Cosmic rays (CR) propagate through the galactic scales down to the smaller
scales at which stars form. CRs are close to energy equipartition with the
other components of the interstellar medium and can provide a support against
gravity if pressure gradients develop. We study CRs propagation within a
turbulent and magnetised bi-stable interstellar gas and identify the conditions
necessary for CR trapping. We present a numerical study using 3D simulations of
the evolution of a mixture of interstellar gas and CRs, in which turbulence is
driven by stochastic forcing within a box of 40 pc. We explore a large
parameter space (CR diffusion coefficient, magnetisation, driving scale and
amplitude of the turbulence forcing, initial CR energy). We identify a clear
transition in the interstellar dynamics for CR diffusion coefficient below a
critical value, which depends on the characteristic length scale L as $D_{\rm
crit}\simeq 3.1~10^{23}~{\rm~cm^2/s}~(L/{\rm 1~pc})^{q+1}$, where the exponent
q relates the turbulent velocity dispersion to the length scale as $v\simeq
L^q$. In our simulations this transition occurs around $D_{\rm crit}\simeq
10^{24-25}$ cm^2/s. The transition is recovered in all cases of our parameter
study and is in very good agreement with our simple analytical estimate. In the
trapped CR regime, the induced CR pressure gradients can modify the gas flow
and provide a support against the thermal instability development. We discuss
possible mechanisms that can reduce significantly the CR diffusion coefficients
within the interstellar medium. CR pressure gradients can develop and modify
the evolution of thermally bi-stable gas for diffusion coefficients $D\leq
10^{25}$ cm^2/s or in regions where the CR pressure exceeds the thermal one by
factor>10. This study provides the basis of further works including more
realistic CR diffusion coefficients, as well as local CR sources.
