Gough’s (1977a) concept of averaging over convective eddies is based on Spiegel’s (1963) finding that the linearized equations of motions for determining the convective growth rate σ of a convective eddy, i.e. the rate with which the convective fluctuations grow with time, satisfy a variational equation for σ, if the locally defined superadiabatic temperature gradient β(r0) is replaced by the averaged, i.e. non-local, value (Gough 1977a,b)
(2)
\begin{eqnarray*}
{\cal B}(r)=\frac{2}{\ell }\int \beta (r_0)\cos ^2\left[\pi (r_0-r)/\ell \right]\, {\rm d}r_0\, .
\end{eqnarray*}
In expression (2), the averaging is obtained by taking account of contributions from convective eddies centred at r0 and the range of integration is the vertical extent ℓ, which scales with the local pressure scale height. Based on Spiegel’s finding on the convective growth rate, Gough (1977a) suggested similar expressions for the averaged, non-local, convective heat, ${\cal {F_{\mathrm{c}}}}$, and momentum, $\mathcal {P}_{\rm t}$, fluxes, with β(r0) in equation (2) being replaced by the locally computed convective fluxes Fc(r0) and pt(r0), respectively. This would, however, lead to a system of integro-differential equations for the stellar structure, which would be difficult to solve numerically. Gough (1977a) suggested therefore to approximate the kernel ${\cal K}:=2\cos ^2[\pi (r_0-r)/\ell ]$ in equation (2) by the expression ${\cal K}\simeq a\exp (-a|r_0-r|/\ell)/2$, and setting the integration limits formally to ±∞, which leads to an integral expression that represents the solution to the second-order differential equation
(3)
\begin{eqnarray*}
\frac{{\rm d}^2{\cal {F_{\mathrm{c}}}}}{{\rm d}\ln p^2}=\frac{a^2}{\alpha ^2}\left({\cal {F_{\mathrm{c}}}}-{F_{\mathrm{c}}}\right)\,
\end{eqnarray*}
(Gough 1977a; Balmforth 1992a; Houdek & Dupret 2015) for the non-local convective heat flux, ${\cal {F_{\mathrm{c}}}}$, for example, where ln p (p is total pressure) is now the new independent depth variable (as implemented in the calculations), α ≔ −ℓdln p/dr is the mixing-length parameter, and a is another dimensionless, non-local, parameter. Similar expressions are obtained for the averaged, non-local, superadiabatic temperature gradient, $\cal B$, and turbulent pressure, $\cal P_{\mathrm{t}}$, introducing the remaining non-local (dimensionless) parameters b and c, respectively. These non-local parameters control the spatial coherence of the ensemble of eddies contributing to the total heat (a) and momentum (c) fluxes, and the degree to which the turbulent fluxes are coupled to the local stratification (b). Roughly speaking, the parameters a, b, and c control the degree of ‘non-locality’ of convection; low values imply highly non-local solutions, and in the limit a, b, c → ∞ the system of equations formally reduces to the local formulation (except near the boundaries of convection zones, where the local equations are singular). Theoretical values for the dimensionless, non-local, parameters a, b, and c can be obtained by demanding that the terms in a Taylor expansion about r of the exact, 2cos 2[π(r0 − r)/ℓ], and approximate, aexp (− a|r0 − r|/ℓ)/2, kernels differ only at fourth order, resulting in a theoretical estimate for a ≃ 7.8 (Gough 1977a). The standard mixing-length assumption of assigning a unique scale to turbulent eddies at any given location can, however, cause too much smoothing, leading to larger values for a, b, or c (Houdek & Gough 2002). Therefore, values for a, b, and c, which typically differ, need to be determined from calibration in a similar way as it is common for the mixing-length parameter, α, in stellar evolutionary theory.1It should be mentioned that our adopted non-local convection formulation does not treat the overshoot regions correctly, where the non-local enthalpy flux, $\cal {F_{\mathrm{c}}}$, remains positive although it should be negative. The overshoot regions may therefore not be suitable for calibrating the non-local parameters to 3D simulation results (see also discussion in Section 5.2). The effect of such positive $\cal {F_{\mathrm{c}}}$ values in the overshoot regions on the damping rates and eigenfrequencies is, however, negligible.
Citation: Houdek & Dupret 2015
Citation function: Uses; Uses