Convection Effects

The oscillation equations presented in the preceding sections neglect the thermal and mechanical effects of convection. GYRE provides functionality for controlling how the thermal effects are suppressed, and how the mechanical effects can be included in a limited way.

Frozen Convection

In the derivation of the linearized equations, a term \(\delta (\rho^{-1} \nabla \cdot \vFcon)\) is dropped from the perturbed heat equation. This is known as a frozen convection approximation, and is grounded in the assumption that the energy transport by convection remains unaffected affected by the pulsation. There’s more than one way to freeze convection; Pesnell (1990) presents a systematic review of different approaches. GYRE currently implements a subset of these:

  • Pesnell’s case 1, neglecting \(\delta (\rho^{-1} \nabla \cdot \vFcon)\) in the perturbed heat equation.

  • Pesnell’s case 4, neglecting \(\delta \Lcon\) (the Lagrangian perturbation to the convective luminosity) in the perturbed heat equation.

For further details, see the conv_scheme parameter in the Oscillation Parameters section.

Turbulent Damping

The Reynolds number in stars is very large, and thus convection tends to be turbulent. Following the treatment by Savonije & Witte (2002), GYRE can partially incorporate the mechanical effects of this turbulence by adding a term

\[f_{r,{\rm visc}} = - \frac{1}{r^{2}} \pderiv{}{r} \left( \rho \nu r^{2} \pderiv{v'_{r}}{r} \right)\]

to the radial component of the linearized momentum equation (6), representing the viscous force arising from radial fluid motion. Because this term depends on \(v'_{r}\), it is phase-shifted by a quarter cycle relative to the other terms in the equation, and acts like a drag force that damps oscillations. The turbulent viscous coefficient \(\nu\) is evaluated as

\[\nu = \frac{(\alphatrb H_{P})^{2}}{\tconv} \left[ 1 + \tconv \frac{\sigma}{2\pi} \right]^{-1},\]

where \(H_{P}\) is the pressure scale height, \(\alphatrb\) is the turbulent mixing length (in units of \(H_{P}\)), and \(\tconv\) the convective turnover timescale. This expression is adapted from equation (18) of Savonije & Witte (2002), with an exponent \(s=1\).

In GYRE \(\alphatrb\) is implemented as a switch (see the Physics Switches section). A reasonable choice is to set this parameter equal to the MLT mixing length parameter \(\alpha_{\rm MLT}\) of the stellar model.