The (Limited) Effect of Viscosity in Multiphase Turbulent Mixing

Problem Statement
Multiphase gas is ubiquitous in astrophysical environments such as the interstellar medium (ISM), circumgalactic medium (CGM), and intracluster medium (ICM). The dynamics of these systems are governed by the interplay between turbulence, radiative cooling, and transport processes. While the role of magnetic fields in suppressing the Kelvin-Helmholtz instability (KHI) and altering mixing in turbulent radiative mixing layers (TRMLs) has been studied, the specific impact of viscosity remains under debate. Fully ionized plasmas are expected to be highly viscous (Spitzer 1962), yet the net effective viscosity in astrophysical systems is uncertain. A critical question is whether viscosity suppresses the turbulence and mixing necessary for the "universal" scaling relations between turbulent velocity and cooling rate (Fielding et al. 2020; Tan et al. 2021) to hold, particularly in regimes where the Karlovitz number ($Ka < 1$) suggests laminar flow.

Methodology
The authors employed idealized numerical simulations using the Athena code to isolate the effects of viscosity. The study proceeded in two main stages:

1. Adiabatic 2D Simulations: To determine the critical viscosity ($\eta_{Crit}$) required to suppress the KHI, the authors simulated shear flows with varying overdensities ($\chi = \rho_{cold}/\rho_{hot}$) and Mach numbers ($M$). They utilized a planar slab setup to test overdensity dependence and a planar sheet setup to test Mach number dependence, avoiding boundary artifacts. They derived an analytical estimate for $\eta_{Crit}$ based on the comparison of the viscous timescale and the KHI growth timescale, utilizing a harmonic mean for kinematic viscosity to account for momentum differences in multi-density fluids.
2. 3D Turbulent Radiative Mixing Layers (TRMLs): Using the critical viscosity derived for an overdensity of $\chi=100$, the authors conducted 3D simulations of TRMLs with explicit viscosity. They varied the viscosity fraction relative to $\eta_{Crit}$ (from 0% to 50%) and the Damköhler number ($Da = t_{turb}/t_{cool}$) by adjusting the perturbation wavelength. This allowed them to explore both weak cooling ($Da < 1$) and strong cooling ($Da > 1$) regimes. The simulations included radiative cooling (Townsend algorithm) but no explicit thermal conduction, ensuring $Ka < 1$.

Key Contributions and Results

• Critical Viscosity Scaling: The study confirms that the critical viscosity required to suppress the KHI scales linearly with overdensity ($\eta_{Crit} \propto \chi$) in the high-overdensity limit. This is because the cold, dense medium carries the majority of the momentum, requiring stronger friction to suppress the instability. The authors also verified that at high Mach numbers, compressibility effects stabilize the KHI even without viscosity, defining a critical Mach number ($M_{Crit}$) beyond which the instability is suppressed.
• Weak Cooling Regime ($Da < 1$): In this regime, cooling timescales are longer than turbulent mixing times. The authors found that viscosity strongly suppresses turbulence, transitioning the flow from turbulent to laminar. Consequently, the universal relation between turbulent velocity ($u'$) and surface brightness ($Q$) derived for inviscid fluids breaks down. However, counterintuitively, the total bolometric luminosity remains largely unaffected. The authors attribute this to a "bottleneck effect" where the long cooling time limits the overall energy loss regardless of the mixing mechanism (diffusion vs. convection).
• Strong Cooling Regime ($Da > 1$): In this regime, cooling timescales are shorter than viscous timescales. The authors discovered that viscosity has a limited effect on the system's dynamics. Despite the presence of viscosity, the cooling process sharpens the temperature and velocity gradients at the interface, effectively counteracting the smoothing effect of viscosity. This allows the KHI to develop and maintain turbulence levels similar to inviscid cases. As a result, the scaling relations between $u'$ and $Q$ remain intact, and the system behaves effectively as non-viscous.
• Viscosity vs. Cooling Competition: The authors derived an analytical condition ($Da > \zeta$, where $\zeta$ is the fraction of critical viscosity) for when cooling dominates over viscosity. Their numerical results align with this prediction. In the strong cooling regime, the increased amount of intermediate-temperature gas in viscous runs (due to a slightly broader front) leads to a modest increase in surface brightness (up to a factor of $\sim 2$) compared to inviscid runs, but the fundamental scaling laws hold.

Significance and Implications
The paper claims that the role of viscosity in multiphase turbulent mixing is more limited than naively expected, particularly in the strong cooling regime which characterizes many astrophysical environments (e.g., galactic outflows, CGM, ICM).

• Observational Diagnostics: The findings suggest that established theoretical predictions relating line widths (turbulence) to emissivity (cooling) can still be applied to observational data, even in systems where viscosity might be significant, provided the system is in the strong cooling regime.
• Subgrid Models: The results provide a physical justification for current multiphase subgrid models used in large-scale cosmological simulations. These models often rely on inviscid, small-scale simulations to inform mass transfer rates and cooling prescriptions. The authors demonstrate that these prescriptions remain valid even when viscosity is present, as the cooling-dominated regime effectively negates the viscous suppression of turbulence and mixing.
• Astrophysical Context: The study highlights that while viscosity may influence large-scale morphology (e.g., in ram-pressure stripped tails), it does not fundamentally alter the energy dissipation or mass transfer rates in the fast-cooling limit. This offers confidence in interpreting the evolution of multiphase gas in galaxy halos and clusters.

The authors note caveats regarding their idealized setup, specifically the use of constant viscosity rather than temperature-dependent Spitzer viscosity, the absence of magnetic fields (which induce anisotropic viscosity), and the lack of explicit thermal conduction. They suggest these as avenues for future work to further refine the understanding of transport processes in astrophysical plasmas.