[Paper Review] The effect of convection on pulsational stability

G. Houdek|ArXiv.org|Oct 29, 2008

Geotechnical and Geomechanical Engineering4 references3 citations

TL;DR

This paper reviews time-dependent convection models in classical pulsators, comparing Gough's (1977) and Unno's (1967) approaches to assess their impact on pulsational stability. It finds that turbulent viscosity, momentum flux (turbulent pressure), and convective heat flux each play dominant stabilizing roles in different models, with Xiong & Deng (2001) identifying turbulent viscosity as the primary damping mechanism for Delta Scuti stars, successfully reproducing the red edge of the instability strip.

ABSTRACT

A review on the current state of mode physics in classical pulsators is presented. Two, currently in use, time-dependent convection models are compared and their applications on mode stability are discussed with particular emphasis on the location of the Delta Scuti instability strip.

Motivation & Objective

• To evaluate the role of different time-dependent convection models in determining pulsational stability in classical pulsators.
• To resolve the discrepancy between theoretical and observed locations of the red edge of the Delta Scuti instability strip.
• To compare the relative contributions of turbulent pressure, convective heat flux, and turbulent viscosity in stabilizing pulsation modes.
• To assess the validity and limitations of current convection models in pulsation calculations, particularly regarding small-scale turbulence and energy dissipation.

Proposed method

• Uses the Boussinesq approximation to model turbulent convection in a static, plane-parallel stellar atmosphere, solving coupled equations for velocity, temperature, and pressure fluctuations.
• Applies two distinct time-dependent convection models: Gough's (1977a,b) model emphasizing turbulent pressure and momentum flux, and Unno's (1967) model focusing on buoyancy-drag balance and nonlinear advection.
• Computes work integrals for mode damping and driving, including contributions from turbulent pressure ($W_{ m t}$), turbulent viscosity ($W_{ u}$), and turbulent kinetic energy dissipation ($W_{ m u}$).
• Compares results from different models—Gough, Unno, Xiong, and Dupret—using stability calculations on 1.7 M⊙ Delta Scuti stars, with emphasis on the red edge of the instability strip.
• Evaluates the role of small-scale turbulence via the turbulent viscosity estimate $\nu_{\rm t} \simeq \lambda (\overline{ww})^{1/2} \ell$, incorporated into the work integral $W_{\nu}$.
• Analyzes the impact of perturbations in turbulent fluxes, including the inclusion or omission of $W_{\epsilon}$ (dissipation work integral), in different model formulations.

Experimental results

Research questions

• RQ1 What is the relative importance of turbulent pressure, convective heat flux, and turbulent viscosity in stabilizing pulsation modes at the red edge of the Delta Scuti instability strip?
• RQ2 How do different time-dependent convection models—Gough (1977a,b), Unno (1967), and Xiong (1977, 1989)—affect the predicted location of the instability strip's red edge?
• RQ3 Why do some models predict stable modes at cooler temperatures while others fail, and what physical mechanisms resolve this discrepancy?
• RQ4 To what extent does the omission of small-scale turbulence damping ($W_{\nu}$) in certain models affect the accuracy of pulsational stability calculations?
• RQ5 Can the inclusion of the turbulent dissipation work integral $W_{\epsilon}$ improve the agreement between theoretical and observed instability strip boundaries?

Key findings

• Xiong & Deng (2001, 2007) found that turbulent viscosity ($W_{\nu}$) is the dominant damping mechanism in Delta Scuti stars, successfully reproducing the observed red edge of the instability strip.
• In contrast, Balmforth (1992) using Gough's (1977a,b) model found turbulent pressure perturbations ($W_{\rm t}$) to be the primary stabilizing factor, with viscous damping being an order of magnitude weaker.
• Dupret et al. (2005) demonstrated that perturbations in turbulent kinetic energy dissipation ($\delta\epsilon$) can contribute to both driving and damping, with a net driving effect observed in $\gamma$ Doradus stars.
• The model by Gough (1977a,b) neglects the turbulent dissipation work integral $W_{\epsilon}$, which may be important in practice despite being omitted in the Boussinesq approximation.
• Grigahcène et al. (2005) included $W_{\epsilon}$ but neglected the viscous damping contribution $W_{\nu}$, which Xiong & Deng (2001) showed to be the dominant damping term.
• Houdek (2000) and others omitted the small-scale turbulence damping ($W_{\nu}$), leading to potential inaccuracies in stability predictions, highlighting the need for a more complete model comparison.