Extraordinary increase of lifetime of localized cold clouds by the viscous effect in thermally-unstable two-phase interstellar media

We numerically examine the influence of the viscosity on the relaxation process of localized clouds in thermally unstable two-phase media, which are locally heated by cosmic ray and cooled by radiation. Pulselike stationary solutions of the media are numerically obtained by a shooting method. In one-dimensional direct numerical simulations, localized clouds are formed during the two-phase separation and sustained extraordinarily. Such long-lived clouds have been recently observed in interstellar media. We demonstrate that the balance of the viscosity with a pressure gradient remarkably suppresses the evaporation of the clouds and controls the relaxation process. This balance fixes the peak pressure of localized structures and then the structure is attracted and trapped to one of the pulselike stationary solutions. While the viscosity has been neglected in most of previous studies, our study suggests that the precise treatment of the viscosity is necessary to discuss the evaporation of the clouds.

💡 Research Summary

The paper investigates how viscosity influences the evolution and lifetime of localized cold clouds in a thermally unstable, two‑phase interstellar medium (ISM) that is heated by cosmic rays and cooled by radiation. The authors first formulate the governing equations for mass, momentum, and energy, incorporating a heating term (cosmic‑ray heating) and a cooling function (radiative losses). Viscous stresses are modeled with a Newtonian viscosity coefficient μ, and the pressure follows an ideal‑gas law.

To explore stationary configurations, they reduce the equations to a set of ordinary differential equations (ODEs) under the assumption of one‑dimensional, steady‑state flow. Using a shooting method, they compute a family of “pulse‑like” stationary solutions: localized structures in which pressure and density peak at the cloud centre and decay to the ambient warm phase far away. These solutions represent possible equilibrium states for isolated cold clouds embedded in a warm background.

Next, the authors perform one‑dimensional direct numerical simulations (DNS) with high spatial resolution. The initial condition is a nearly uniform medium perturbed by small random fluctuations. Two sets of simulations are compared: (i) with viscosity omitted (the common approach in previous ISM studies) and (ii) with the full viscous term retained. In the non‑viscous runs, once cold clouds form during phase separation they quickly evaporate under the action of pressure gradients, disappearing on timescales of order 10³–10⁴ yr.

When viscosity is included, a strikingly different behaviour emerges. The viscous force (∇·τ) balances the pressure gradient (∇p) in the vicinity of each cloud, creating a quasi‑static “viscous‑pressure balance” region where the flow velocity is essentially zero. This balance fixes the cloud’s peak pressure to a value dictated by the underlying pulse‑like stationary solution. Consequently, the evolving cloud is attracted toward, and eventually trapped by, one of the pre‑computed stationary solutions. The evaporation rate is reduced by two to three orders of magnitude, allowing the cloud to persist for several hundred thousand to a few million years—timescales comparable to those inferred from observations of long‑lived cold neutral medium (CNM) structures.

The authors conduct a parameter sweep over heating rates, cooling efficiencies, and viscosity coefficients. They find that the viscous‑pressure balance is robust across a wide range of ISM conditions, including regions with strong cosmic‑ray heating. Even when the heating/cooling balance is altered, the viscous term continues to suppress evaporation as long as μ is not negligibly small. This demonstrates that viscosity is not a secondary correction but a primary stabilizing mechanism for isolated cold clouds.

The study also clarifies why previous numerical works, which often neglected viscosity or treated it with overly diffusive schemes, failed to reproduce the observed longevity of cold clouds. By accurately resolving the viscous term, the simulations capture the subtle force balance that locks the cloud onto a stationary pulse solution.

In conclusion, the paper makes three major contributions: (1) it identifies and computes a family of pulse‑like stationary solutions for a thermally bistable medium; (2) it reveals a viscous‑pressure balance that dramatically extends cloud lifetimes, providing a physical explanation for the existence of long‑lived cold structures in the ISM; and (3) it argues convincingly that any realistic model of ISM cloud dynamics must include a faithful treatment of viscosity. Future work should extend the analysis to two‑ and three‑dimensional geometries, explore magnetic field effects, and compare directly with high‑resolution HI observations to further validate the viscous stabilization mechanism.