The GHOSDT Simulations: II. Missing H$_2$ in Simulations of a Self-Regulated Interstellar Medium

Observations in the Galaxy and nearby spirals have established that the HI-to-H$_2$ transition at solar metallicity occurs at gas weight of $P_{\rm DE}/k_B\approx 10^4 \ \rm K \ cm ^{-3}$, similar to solar neighbourhood conditions. Even so, state-of-the-art models of a self-regulated interstellar medium underproduce the molecular fraction ($R_{\rm mol} \equiv M_{{\rm H}_2}/M_{HI}$) at solar neighbourhood conditions by a factor of $\approx2-4$. We use the GHOSDT suite of simulations at a mass resolution range of $100-0.25\ M_{\odot}$ (effective spatial resolution range of $\sim 20-0.05\ \rm pc$) run for 500 Myr to show how this problem is affected by modeling choices such as the inclusion of photoionizing radiation, assumed supernova energy, numerical resolution, inclusion of magnetic fields, and including a model for sub-grid clumping. We find that $R_{\rm mol}$ is not converged even at a resolution of 1 $M_{\odot}$, with $R_{\rm mol}$ increasing by a factor of 2 when resolution is improved from 10 to $1\ M_{\odot}$. Models excluding either photoionization or magnetic fields result in a factor 2 reduction in $R_{\rm mol}$. The only model that agrees with the observed value of $R_{\rm mol}$ includes our sub-grid clumping model, which enhances $R_{\rm mol}$ by a factor of $\sim3$ compared with our fiducial model. This increases the time-averaged $R_{\rm mol}$ to $0.25$, in agreement with the Solar circle value, and closer to the observed median value of $0.42$ in regions comparable to the solar neighbourhood in nearby spirals. Our findings show that small-scale clumping in the ISM plays a significant role in H$_2$ formation even in high-resolution numerical simulations.