[Paper Review] Nonlinear Development of Thermal Instability without External Forcing

Hiroshi Koyama, Shu‐ichiro Inutsuka|arXiv (Cornell University)|May 20, 2006

Astrophysics and Star Formation Studies2 references3 citations

TL;DR

This study investigates the self-sustained development of supersonic turbulence in a thermally unstable, two-phase interstellar medium without external forcing. Using 2D and 3D hydrodynamic simulations with radiative cooling, thermal conduction, and physical viscosity, the authors show that turbulent motions persist due to continuous energy supply from local inhomogeneous cooling, with saturation amplitude strongly dependent on domain size relative to the cooling length and Prandtl number, while thermal conduction plays a key role in suppressing viscous dissipation.

ABSTRACT

Supersonic turbulent motions are the remarkable properties of interstellar medium. Previous numerical simulations have demonstrated that the thermal instability in a shock-compressed layer produces the supersonic turbulent motion that does not decay. In this paper we focus on two- and three-dimensional numerical simulations of the non-linear development of simple thermal instability incorporating physical viscosity but without any external forcing, in order to isolate the effects of various processes responsible for the long-lasting turbulent motion. As the initial condition for our simulations, we set up spatially uniform gas with thermally unstable temperature in a box with periodic boundaries. After the linear growth stage of the thermal instability, two-phase medium forms where cold clumps are embedded in warm medium, and turbulent fluid flow clearly visible as translational motions of the cold clumps does not decay in a viscous dissipation timescale. The amplitude of the turbulent velocity increases when we reduce the Prandtl number that is the non-dimensional ratio of kinetic viscosity to thermal conduction: the saturation amplitude does not change when we increase the viscosity and thermal conduction coefficients simultaneously in order to keep the Prandtl number. This shows that the thermal conduction plays an important role in maintaining turbulent motions against viscous dissipation. The amplitude also increases when we increase the ratio of the computational domain length $L$ to the cooling length $λ_{ m c}$ that is defined by the product of the cooling time and the sound speed, as long as $L \la 100 λ_{ m c}$.

Motivation & Objective

• To isolate the mechanisms responsible for long-lasting turbulent motions in thermally unstable interstellar gas without external forcing.
• To determine the role of physical viscosity and thermal conduction in sustaining turbulence against dissipation.
• To investigate how system size (domain length) and the ratio of cooling length to field length affect turbulent saturation amplitude.
• To clarify the dependence of turbulence amplitude on the Prandtl number, particularly in relation to viscous and thermal dissipation.

Proposed method

• Numerical simulations solve the compressible Navier-Stokes equations with radiative cooling, heating, thermal conduction, and physical viscosity.
• Initial conditions consist of a spatially uniform, thermally unstable gas in a periodic box with no external forcing.
• The simulations track the nonlinear evolution of thermal instability from linear growth to saturation, focusing on two- and three-dimensional configurations.
• The cooling length $\lambda_{\rm c}$ is defined as the product of sound speed and cooling time, and its ratio to domain length $L$ is used as a key control parameter.
• The Prandtl number (ratio of kinematic viscosity to thermal diffusivity) is varied to isolate the effects of thermal conduction on turbulence maintenance.
• Resolution is tested by ensuring the Field length $\lambda_{\rm F}$ is resolved and $\lambda_{\rm F} < \lambda_{\rm c}$ to avoid suppression of the most unstable mode.

Experimental results

Research questions

• RQ1 What mechanisms sustain supersonic turbulence in a two-phase interstellar medium when no external forcing is applied?
• RQ2 How does the ratio of domain size $L$ to cooling length $\lambda_{\rm c}$ influence the saturation amplitude of turbulence?
• RQ3 What is the role of thermal conduction in counteracting viscous dissipation of turbulent motions?
• RQ4 How does the Prandtl number affect the amplitude of turbulence in the absence of external forcing?
• RQ5 What resolution requirements are necessary to accurately capture the nonlinear development of thermal instability?

Key findings

• Turbulent kinetic energy does not decay over 25 Myr—equivalent to 125 sound crossing times of the warm phase—despite a viscous dissipation timescale of ~6 Myr.
• The saturation amplitude of turbulent velocity increases with decreasing Prandtl number, scaling approximately as $v \propto \text{Pr}^{-0.15 \text{ to } -0.5}$, indicating thermal conduction is essential for turbulence maintenance.
• When viscosity and thermal conductivity are increased simultaneously to preserve a constant Prandtl number, the saturation amplitude remains unchanged, confirming thermal conduction's dominant role in energy balance.
• Turbulence saturates only when $L/\lambda_{\rm c} > 1$, with full saturation observed at $L/\lambda_{\rm c} \gtrsim 100$, where the maximum turbulent velocity reaches about half the sound speed of the cold phase.
• The maximum saturation velocity is approximately 0.5 times the sound speed of the cold neutral medium when the Prandtl number is 2/3.
• To avoid resolution-dependent results, simulations must resolve both the Field length $\lambda_{\rm F}$ and cover a dynamic range from $\lambda_{\rm F}$ to $100\lambda_{\rm c}$, requiring $\sim 10^4$ grid points per dimension for realistic parameters.