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 longevity of cold, dense clouds that form in a thermally unstable, two‑phase interstellar medium (ISM). The authors begin by formulating the governing equations for a neutral, optically thin gas subject to uniform cosmic‑ray heating and radiative cooling, while neglecting self‑gravity, magnetic fields, and other body forces. The fluid dynamics are described by the continuity, momentum, and energy equations together with an ideal‑gas equation of state. Viscosity (µ) and thermal conductivity (K) are retained, and the equations are non‑dimensionalized using characteristic scales: the Field length (the thermal front width) and a sound‑speed based velocity. Two dimensionless numbers appear: the Prandtl number Pr = γ/(γ‑1)·k_B µ/(m_H K) and the energy‑to‑dynamics ratio E_p, which controls the relative importance of heating/cooling versus dynamical response.

For the heating‑cooling function the authors adopt a simple Ginzburg‑Landau‑type form
F(T,P)=a(T−1)−b(T−1)^3−c ln P,
with parameters a=1, b=2, c=1, and also test a more realistic astrophysical prescription (Γ and Λ(T) terms). Both produce the same qualitative S‑shaped radiative‑equilibrium curve in the (T,P) plane, featuring two stable branches (cold neutral medium, warm neutral medium) and an intermediate unstable branch that triggers thermal instability.

The core of the study is the construction of spatially localized, stationary “pulse‑like” solutions (homo‑clinic orbits) of the reduced steady equations. Assuming a plane‑parallel geometry (1‑D) and uniform pressure (∂_x P=0), the temperature satisfies
F(T,P)+∂_x^2 T=0,
with Neumann boundary conditions (∂_x T=0 at the centre and at infinity). By a shooting method the authors obtain families of solutions for various pressures P̃. For pressures slightly above the saturation pressure (P̃>1) the solutions have a cold, dense core (peak density ρ_0) surrounded by warm gas, while for P̃<1 they obtain “warm” pulses. The peak density moves away from the radiative‑equilibrium curve as P̃ increases, and in the limit P̃→1⁺ the solution converges to the classic Zeldovich‑Pikel’ner (ZP) front solution. Analogous calculations are performed for axisymmetric (2‑D) and spherically symmetric (3‑D) geometries, where curvature terms (d−1)/r·∂_r T appear. The curvature reduces the peak density for a given pressure, and a critical radius emerges below which stationary pulses cannot exist.

Direct numerical simulations (DNS) are then carried out in one dimension with periodic boundaries. The initial condition is a uniform medium perturbed by small random noise, which triggers phase separation. In simulations without viscosity, cold clouds form but quickly evaporate or merge, leading to a uniform state. When viscosity is included (with realistic Prandtl numbers), a strikingly different behavior emerges: each cold cloud develops a region where the viscous stress µ∇^2 u balances the pressure gradient ∇P. This “viscous‑pressure balance” suppresses the flow that would otherwise drive evaporation. Consequently the clouds become quasi‑static, their internal pressure settles at a value dictated by the stationary pulse solution, and they are attracted toward the corresponding homo‑clinic orbit. The evaporation time τ_evap is extended by more than an order of magnitude compared with the inviscid case, matching the lifetimes of tiny, long‑lived interstellar clouds observed in recent surveys.

The authors provide a theoretical interpretation of this effect. In the steady momentum equation, the viscous term and pressure gradient have the same dimensionality; when they cancel, the velocity field vanishes, and the energy equation reduces to the static balance used to compute the pulse solutions. In multi‑dimensional geometries the curvature term competes with viscosity, establishing a minimum cloud radius for which the balance can be maintained. Thus, the longevity of cold clouds is not merely a consequence of external confinement or artificial boundary conditions, but a natural outcome of the intrinsic viscous dynamics of the two‑phase ISM.

In the discussion, the paper emphasizes that previous studies of thermal instability have largely neglected viscosity because its magnitude is small on typical ISM scales. However, the present results demonstrate that even a modest viscosity can dominate the slow relaxation phase, dramatically altering cloud lifetimes. The authors suggest that future work should incorporate magnetic fields, self‑gravity, and external driving (e.g., supernova shocks) to assess how robust the viscous‑pressure balance is under more realistic astrophysical conditions. Overall, the study provides a compelling argument that precise treatment of viscosity is essential for any quantitative model of cloud formation, survival, and the multiphase structure of the interstellar medium.