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The effect of convection on pulsational stability

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.


Introduction
Stars with relatively low surface temperatures show distinctive envelope convection zones which affect mode stability. Among the first problems of this nature was the modelling of the red edge of the classical instability strip (IS) in the Hertzsprung-Russell (H-R) diagram. The first pulsation calculations of classical pulsators without any pulsation-convection modelling predicted red edges which were much too cool and which were at best only neutrally stable. What follows were several attempts to bring the theoretically predicted location of the red edge in better agreement with the observed location by using timedependent convection models in the pulsation analyses (Dupree 1977;Baker & Gough 1979;Gonzi 1982;Stellingwerf 1984). More recently several authors, e.g. Bono et al. (1995Bono et al. ( , 1999, Houdek (1997Houdek ( , 2000, Xiong & Deng (2001, 2007, Dupret et al. (2005) were successful to model the red edge of the classical IS. These authors report, however, that different physical mechanisms are responsible for the return to stability. For example, Bono et al. (1995) and  The effect of convection on pulsational stability
Nearly all of the time-dependent convection models assume the Boussinesq approximation to the equations of motion. The Boussinesq approximation relies on the fact that the height of the fluid layer is small compared with the density scale height. It is based on a careful scaling argument and an expansion in small parameters (Spiegel & Veronis 1960;Gough 1969). The fluctuating convection equations for an inviscid Boussinesq fluid in a static plane-parallel atmosphere are supplemented by the continuity equation for an incompressible gas, ∂ j u j = 0, where u = (u, v, w) is the turbulent velocity field, ρ is density, p is gas pressure, g is the acceleration due to gravity, T is temperature, c p is the specific heat at constant pressure, α = −(∂ ln ρ/∂ ln T ) p / T , F i is the radiative heat flux, β is the superadiabatic temperature gradient and δ ij is the Kronecker delta. Primes ( ′ ) indicate Eulerian fluctuations and overbars horizontal averages. These are the starting equations for the two physical pictures describing the motion of an overturning convective eddy, illustrated in Fig. 1.
In the first physical picture, adopted by Unno (1967), the turbulent element, with a characteristic vertical length ℓ, evolves out of some chaotic state and achieves steady motion very quickly. The fluid element maintains exact balance between buoyancy force and turbulent drag by continuous exchange of momentum with other elements and its surroundings. Thus the acceleration terms ∂ t u i and ∂ t T ′ are neglected and the nonlinear advection terms provide dissipation (of kinetic energy) that balances the driving terms. The nonlinear advection terms are approximated by This leads to two nonlinear equations which need to be solved numerically together with the mean equations of the stellar structure. The second physical picture, which was generalized by Gough (1965Gough ( , 1977a to the time-dependent case, interprets the turbulent flow by indirect analogy with kinetic gas theory. The motion is not steady and one imagines the convective element to accelerate from rest followed by an instantaneous breakup after the element's lifetime. Thus the nonlinear advection terms are neglected in the convective fluctuation equations (1)-(2) but are taken to be responsible for the creation and destruction of the convective eddies (Gough 1977a,b). By retaining only the acceleration terms the equations become linear with analytical solutions w ∝ exp(σt) and T ′ ∝ exp(σt) subject to proper periodic spatial boundary conditions, where t is time and ℜ(σ) is the linear convective growth rate. The mixing length ℓ enters in the calculation of the eddy's survival probability, which is proportional to the eddy's internal shear (rms vorticity), for determining the convective heat and momentum fluxes. Although the two physical pictures give the same result in a static envelope, the results for the fluctuating turbulent fluxes in a pulsating star are very different (Gough 1977a). The main differences between Unno's and Gough's convection model are summarized in Table 1.
Application on mode stability in δ Scuti stars The results were computed with the time-dependent, nonlocal convection model by Gough (1977a,b). As demonstrated in the right panel of Fig. 2, the dominating damping term to the work integral W for a star located near the red edge is the contribution from the turbulent pressure fluctuations W t . Gabriel (1996) and more recently Grigahcène et al. (2005) generalized Unno's time-dependent convection model for stability computations of nonradial oscillation modes. They included in their mean thermal energy equation the viscous dissipation of turbulent kinetic energy, ǫ, as an additional heat source. The dissipation of turbulent kinetic energy is introduced in the conservation equation for the turbulent kinetic energy K := u i u i /2 (e.g. Tennekes & Lumley 1972, §3.4;Canuto 1992;Houdek & Gough 1999): where D t is the material derivative, U i is the average (oscillation) velocity, i.e. the total velocityũ i = U i + u i , and ν is the constant kinematic viscosity (in the limit of high Reynolds numbers the molecular transport term can be neglected). The first and second term on the right of Eq. (3) are the shear and buoyant productions of turbulent kinetic energy, whereas the last term ǫ = ν(∂ j u i + ∂ i u j ) 2 /2 is the viscous dissipation of turbulent kinetic energy into heat. This term is also present in the mean thermal energy equation, but with opposite sign. The linearized perturbed mean thermal energy equation for a star pulsating radially with complex angular frequency ω = ω r + iω i can then be written, in the absence of nuclear reactions, as ('δ' denotes a Lagrangian fluctuation and I omit overbars in the mean quantities): where m is the radial mass co-ordinate, ∇ ad = (∂ ln T /∂ ln p) s and L is the total (radiative and convective) luminosity. Grigahcène et al. (2005) evaluated ǫ from a turbulent kinetic energy equation which was derived without the assumption of the Boussinesq approximation. Furthermore it is not obvious whether 6 The effect of convection on pulsational stability the dominant buoyancy production term, g αu j T ′ (see Eq. 3), was included in their turbulent kinetic energy equation and so in their expression for ǫ. Dupret et al. (2005) applied the convection model of Grigahcène et al. (2005) to Delta Scuti and γ Doradus stars and reported well defined red edges. The results of their stability analysis for Delta Scuti stars are depicted in Fig. 3. The left panel compares the location of the red edge with results reported by Houdek (2000, see also Fig. 2) and Xiong & Deng (2001). The right panel of Fig. 3 displays the individual contributions to the accumulated work integral W for a star located near the red edge of the n = 3 mode (indicated by the 'star' symbol in the left panel). It demonstrates the near cancellation effect between the contributions of the turbulent kinetic energy dissipation , W ǫ , and turbulent pressure, W t , making the contribution from the fluctuating convective heat flux, W c , the dominating damping term. The near cancellation effect between W ǫ and W t was demonstrated first by Ledoux & Walraven (1958, §65) (see also Gabriel 1996) by writing the sum of both work integrals as: where M is the stellar mass, m b is the enclosed mass at the bottom of the envelope and γ 3 ≡ 1+(∂ ln T /∂ ln ρ) s (s is specific entropy) is the third adiabatic exponent. Except in ionization zones γ 3 ≃ 5/3 and consequently W ǫ + W t ≃ 0.
The convection model by Xiong (1977Xiong ( , 1989 uses transport equations for the second-order moments of the convective fluctuations. In the transport equation for the turbulent kinetic energy Xiong adopts the approximation by Hinze (1975) for the turbulent dissipation rate, i.e. ǫ = 2χk(u i u i ρ 2 /3ρ 2 ) 3/2 , where χ = 0.45 is the Heisenberg eddy coupling coefficient and k ∝ ℓ −1 is the wavenumber of the energy-containing eddies. However, Xiong does not provide a work integral for ǫ (neither does Unno et al. 1989, §26,30) but includes the viscous damping effect of the small-scale turbulence in his model. The convection models considered here describe only the largest, most energycontaining eddies and ignore the dynamics of the small-scale eddies lying further down the turbulent cascade. Small-scale turbulence does, however, contribute directly to the turbulent fluxes and, under the assumption that they evolve isotropically, they generate an effective viscosity ν t which is felt by a particular pulsation mode as an additional damping effect. The turbulent viscosity can be estimated as (e.g. Gough 1977b;Unno et al. 1989, §20) ν t ≃ λ(ww) 1/2 ℓ, where λ is a parameter of order unity. The associated work integral W ν can be written in Cartesian co-ordinates as (Ledoux & Walraven 1958, §63)  where e ij = (∂ j ξ i + ∂ i ξ j )/2 and ξ is the displacement eigenfunction. Xiong & Deng (2001, 2007 modelled successfully the IS of Delta Scuti and red giant stars and found the dominating damping effect to be the turbulent viscosity (Eq. 6). This is illustrated in Fig. 4 for two Delta Scuti stars: one is located inside the IS (left panel), the other outside the cool edge of the IS (right panel). The contribution from the small-scale turbulence was also the dominant damping effect in the stability calculations by Xiong et al. (2000) of radial p modes in the Sun, although the authors still found unstable modes with orders between 11 ≤ n ≤ 23. The importance of the turbulent damping was reported first by Goldreich & Keeley (1977) and later by Goldreich & Kumar (1991), who found all solar modes to be stable only if turbulent damping was included in their stability computations. In contrast, Balmforth (1992), who adopted the convection model of Gough (1977a,b), found all solar p modes to be stable due mainly to the damping of the turbulent pressure perturbations, W t , and reported that viscous damping, W ν , is about one order of magnitude smaller than the contribution of W t . Turbulent viscosity (Eq. 6) leads always to mode damping, where as the perturbation of the turbulent kinetic energy dissipation, δǫ (see Eq. 4), can contribute to both damping and driving of the pulsations (Gabriel 1996). The driving effect of δǫ was shown by Dupret et al. (2005) for a γ Doradus star.

8
The effect of convection on pulsational stability

Summary
We discussed three different mode stability calculations of Delta Scuti stars which successfully reproduced the red edge of the IS. Each of these computations adopted a different time-dependent convection description. The results were discussed by comparing work integrals. All convection descriptions include, although in different ways, the perturbations of the turbulent fluxes. Gough (1977a), Xiong (1977Xiong ( , 1989, and Unno et al. (1989) did not include the contribution W ǫ to the work integral because in the Boussinesq approximation (Spiegel & Veronis 1960) the viscous dissipation is neglected in the thermal energy equation. In practise, however, this term may be important. Grigahcène et al. (2005) included W ǫ but ignored the damping contribution of the smallscale turbulence W ν , which was found by Xiong & Deng (2001, 2007 to be the dominating damping term. The small-scale damping effect was also ignored in the calculations by Houdek (2000). A more detailed comparison of the convection descriptions has not yet been made but Houdek & Dupret have begun to address this problem.