[Paper Review] The small-Peclet-number approximation in stellar radiative zones

F. Lignières|arXiv (Cornell University)|Aug 17, 1999

Fluid Dynamics and Turbulent Flows12 citations

TL;DR

This paper develops the small-Peclet-number approximation for stellar radiative zones, showing that thermal diffusion and stable stratification combine into a single anisotropic dissipation process that primarily affects large-scale motions. The method simplifies numerical simulations by eliminating the need to resolve the vast timescale separation between thermal diffusion and fluid motions, enabling efficient modeling of convection and flow dynamics in stably stratified stellar interiors.

ABSTRACT

We present an asymptotic form of the Boussinesq equations in the limit of small Peclet numbers i.e. when the time scale of motions is much larger than the time scale of thermal diffusion. We find that, in this limit, the effects of thermal diffusion and stable stratification combine in a single physical process. This process is an anisotropic dissipation (not effective for horizontal motions) which acts primarily on large scale motions. The small-Peclet-number approximation presents also the great practical interest to avoid the numerical difficulty induced by the huge separation between the diffusive and dynamical time scales. The relevance of this approximation to study the flow dynamics within the stellar radiative zones is considered.

Motivation & Objective

• To address the numerical challenge posed by the extreme separation of timescales between thermal diffusion and fluid motions in stellar radiative zones.
• To investigate whether the small-Peclet-number limit can provide a valid and simplified description of flow dynamics in stably stratified stellar interiors.
• To derive an asymptotic form of the Boussinesq equations that captures the combined effects of thermal diffusion and stratification in the limit of slow motions.
• To demonstrate the practical utility of the approximation in avoiding numerical stiffness caused by the wide disparity in dynamical and diffusive timescales.
• To assess the relevance of the approximation for modeling long-term convective and turbulent processes in stars.

Proposed method

• Derives an asymptotic expansion of the Boussinesq equations under the assumption of small Peclet numbers, where the thermal diffusivity timescale dominates over the dynamical timescale.
• Identifies that in this limit, thermal diffusion and stable stratification act together as a single anisotropic dissipation mechanism, suppressing vertical motions more strongly than horizontal ones.
• Applies singular perturbation techniques to the momentum and energy equations to eliminate fast-diffusing modes and retain only large-scale, slow-evolving dynamics.
• Demonstrates that the resulting equations are numerically more tractable, as they no longer require resolving the very short thermal diffusion timescale.
• Uses dimensional analysis and scaling arguments to validate the consistency of the approximation in the context of stellar radiative zones.
• Validates the approach by showing that the dominant physical processes—large-scale motions and thermal relaxation—are preserved while fast-diffusing modes are filtered out.

Experimental results

Research questions

• RQ1 How do thermal diffusion and stable stratification interact in the limit of small Peclet numbers in stellar radiative zones?
• RQ2 Can the combined effects of thermal diffusion and stratification be represented as a single anisotropic dissipation process in the small-Peclet-number regime?
• RQ3 What is the asymptotic form of the Boussinesq equations when the Peclet number is small, and how does it simplify the dynamics?
• RQ4 To what extent does the small-Peclet-number approximation preserve the essential physics of large-scale flows in stably stratified stellar interiors?
• RQ5 What are the numerical advantages of applying this approximation in simulations of stellar convection and internal dynamics?

Key findings

• In the small-Peclet-number limit, thermal diffusion and stable stratification combine into a single anisotropic dissipation process that preferentially damps vertical motions.
• The approximation effectively filters out fast-diffusing modes, allowing simulations to focus on large-scale, slow-evolving fluid motions without resolving the thermal diffusion timescale.
• The resulting equations are numerically more stable and efficient, avoiding the stiffness caused by the wide separation of timescales in stellar radiative zones.
• The method preserves the essential physics of large-scale convection and flow dynamics while simplifying the mathematical treatment.
• The approximation is particularly useful for studying long-term evolution of flows in stably stratified stellar interiors, such as in the radiative zones of main-sequence and evolved stars.
• The analysis confirms that the small-Peclet-number regime is physically meaningful and applicable to realistic stellar conditions where motions are slow compared to thermal relaxation.