Interaction Between Convection and Pulsation
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Abstract
This article reviews our current understanding of modelling convection dynamics in stars. Several semianalytical timedependent convection models have been proposed for pulsating onedimensional stellar structures with different formulations for how the convective turbulent velocity field couples with the global stellar oscillations. In this review we put emphasis on two, widely used, timedependent convection formulations for estimating pulsation properties in onedimensional stellar models. Applications to pulsating stars are presented with results for oscillation properties, such as the effects of convection dynamics on the oscillation frequencies, or the stability of pulsation modes, in classical pulsators and in stars supporting solartype oscillations.
Keywords
Stellar convection Timedependent convection Mode physics1 Introduction
Transport of heat (energy) and momentum by turbulent convection is a phenomenon that we experience on a daily basis, such as the boiling of water in a kettle, the circulation of air inside a nonuniformly heated room, or the formation of cloud patterns. Convection may be defined as fluid (gas) motions brought about by temperature differences with gradients in any direction (Koschmieder, 1993). It is not only important to engineering applications but also to a wide range of astrophysical flows, such as in galaxycluster plasmas, interstellar medium, accretion disks, supernovae, and during several evolutionary stages of all stars in the Universe. The transport of turbulent fluxes by convection is mutually affected by other physical processes, including radiation, rotation, and any kind of mixing processes. In stars turbulent convection affects not only their structure and evolution but also any dynamical processes with characteristic time scales that are similar to the characteristic time scale of convection in the overturning stellar layers. One such important process is stellar pulsation, the study of which has become the field of asteroseismology. Asteroseismology and, when applied to the Sun, helioseismology, is now one of the most important diagnostic tools for testing and improving the theory of stellar structure and evolution by analysing the observed pulsation properties. It is, therefore, the aim of this review to provide an uptodate account on the most widely used stellar convection models with emphasis on the formalisms that describe the interaction of the turbulent velocity field with the stellar pulsation.
The temperature in a star is determined by the balance of energy and its gradient depends on the details how energy is transported throughout the stellar interior. Red giants and solarlike stars exhibit substantial convection zones in the outer stellar layers, which affect the properties of the oscillation modes such as the oscillation frequencies and mode stability. Among the first problems of this nature was the modelling of the red edge of the classical instability strip in the HertzsprungRussell diagram which, for intermediatemass stars with about 1.5–2.0 M_{⊙}, is located approximately at surface temperatures between 7200–6600 K. The first pulsation calculations of classical pulsators without any pulsationconvection modelling predicted red edges which were much too cool and which were at best only neutrally stable. What followed, 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 (e.g., Deupree, 1977b; Baker and Gough, 1979; Gonczi, 1982b; Stellingwerf, 1984). Later, several authors, e.g., Bono et al. (1995, 1999), Houdek (1997, 2000), Xiong and Deng (2001, 2007), Dupret et al. (2005a,b) were successful to model the red edge of the classical instability strip, and mode lifetimes in stars supporting solarlike oscillations (e.g., Gough, 1980; Balmforth, 1992a; Houdek et al., 1999a; Xiong et al., 2000; Houdek and Gough, 2002; Dupret et al., 2004a; Chaplin et al., 2005; Dupret et al., 2006a; Houdek, 2006; Dupret et al., 2009; Belkacem et al., 2012).
Thermal heat transport in convective regions is governed by turbulent motion of the underlying fluid or gas. To determine the average of vertical velocity, temperature and momentum fluctuations, the full structure of the turbulent flow is needed. This is until today not a tractable theoretical problem without the introduction of some hypothetical assumptions in order to close the system of equations describing the turbulent flow. Such closure models can be classified basically into four categories: (i) ‘algebraic models’, including the mixinglength approach (e.g., Prandtl, 1925; Vitense, 1953; BöhmVitense, 1958), (ii) ‘oneequation models’, which use a modified turbulent kinetic energy equation with highorder moments closed approximately by means of a locally defined mixing length (e.g., Rodi, 1976; Stellingwerf, 1982), (iii) ‘twoequation models’, such as the K−ϵ_{t} model, where K denotes the turbulent kinetic energy and ϵ_{t} the associated viscous dissipation of turbulent energy (e.g., Jones and Launder, 1972, 1973), and (iv) ‘Reynolds stress models’, which use transport equations for all secondorder moments (typically five) including the turbulent fluxes of heat and momentum, and appropriate approximation for the thirdorder moments to close the equations (e.g., Keller and Friedmann, 1924; Rotta, 1951; Castor, 1968; Xiong, 1977; Canuto, 1992; Grossman, 1996; Canuto and Dubovikov, 1998; Kupka, 1999; Montgomery and Kupka, 2004; Xiong and Deng, 2007).
Theories based on the mixinglength formalism (Prandtl, 1925) still represent the main method for computing the stratification of convection zones in stellar models. An alternative convection formulation, based on the EddyDamped QuasiNormal Markovian approximation to turbulence (e.g., Orszag, 1977), was introduced by Canuto and Mazzitelli (1991) which, however, still requires a (local) mixing length for estimating the convective heat (enthalpy) flux. The EddyDamped QuasiNormal Markovian approximation is characterized as a twoequation model and is sometimes referred to as twopoint closure, because it describes correlations of two different points in space, or two different wave numbers k and k′ in Fourier space. Although twoequation models have a reasonable degree of flexibility, they are restricted by the assumption of a scalar turbulent viscosity and that the stresses are proportional to the rate of mean strain. The Reynolds stress models are, in principle, free of these restrictions and were discussed, for example, by Xiong (1989) and Canuto (1992, 1993) for the application in stellar convection. Xiong’s model was applied successfully to various types of pulsating star, and Canuto’s model was applied to nonpulsating stars with relatively shallow surface convection zones (Kupka and Montgomery, 2002; Montgomery and Kupka, 2004).
Timedependent convection models are required to describe the interaction between the turbulent velocity field and the oscillating stellar background. Semianalytical models for pulsating stars were proposed, for example, by Schatzman (1956), Gough (1977a), Unno (1967), Xiong (1977), Stellingwerf (1982), Gonczi (1982a), Kuhfuß (1986), and Grossman (1996).
The present unprecedented computer revolution enables us to perform fully hydrodynamical simulations of largescale turbulent flows (large eddy simulation) of stellar surface convection (e.g., Stein and Nordlund, 1989, 2000; Nordlund et al., 1996; Trampedach et al., 1998; Kim et al., 1996; Chan and Sofia, 1996; Freytag et al., 2002; Robinson et al., 2004; Wedemeyer et al., 2004; Muthsam et al., 2010; Magic et al., 2013; Trampedach et al., 2013, 2014a; Magic et al., 2015). A review of threedimensional (3D) hydrodynamical simulations of the Sun, together with their shortcomings, was presented by Miesch (2005). Such numerical simulations represent a fruitful tool for investigating the accuracy and hence the field of application of phenomenological prescriptions of convection such as the mixinglength approach.
In this review, we summarize the two timedependent convection models by Gough (1965, 1977a) and Unno (1967, 1977) for estimating stellar stability properties in classical pulsators and solartype stars. In Section 2, we start from the equations of fluid motion to derive first the mean and fluctuating equations within the commonly adopted Reynolds separation approach. Section 3 discusses first the timedependent convection equations by Gough (1965, 1977a) and Unno (1967, 1977) for radially pulsating stellar envelopes, followed by a summary of Gough’s (1977b) nonlocal equations, before embarking on a discussion on a generalization of Unno’s (1967) model to nonradial stellar oscillations by Gabriel et al. (1975) and Grigahcène et al. (2005). A summary of Reynolds stress models adopted to stellar convection is provided in Section 4. Applications of the two timedependent convection models by Gough (1977a,b) and Grigahcène et al. (2005) are provided in Sections 5, 6, and 7, starting with the role of convection dynamics on the oscillation frequencies, the socalled surface effects, followed by a summary of our current understanding of mode physics in classical pulsators and stars supporting solarlike oscillations. Final remarks and prospects are given in Section 8.
2 Hydrodynamical Equations
2.1 Mean equations
2.2 Boussinesq mean equations for radially pulsating atmospheres
One of the first questions to ask is how one would go about the separation of the velocity field into a component that is associated with the stellar pulsation and into another component that is related to the convection. The answer is not necessarily straightforward (for a recent discussion see, e.g., Appourchaux et al., 2010, §3.1). This separation of the velocity field is probably best known for radial pulsations, for which the horizontal motion is uniform (the convective motion is not). By adopting Eq. (6) for averaging the horizontal motion of the convective velocity field the radial pulsations can be separated in an (mathematically) obvious way (e.g., Gough, 1969), in which the smallscale convective Eulerian fluctuations (u) are advected by the largescale radial Lagrangian motion (U) of the pulsation.
The second term on the lefthand side of Eq. (22) results from taking the horizontal average of the radial component of the nonlinear advection term ρv · ∇v: with the definition of the Reynolds stress tensor (10) and velocity anisotropy (26) the last term of Eq. (9), \({(\nabla \cdot {\sigma _{\rm{t}}})_{\rm{r}}} = (3  \overline \Phi){p_{\rm{t}}}/r\), assuming axisymmetric turbulence about the radial direction. From a physical point of view this term arises because horizontal motion, in spherical coordinates, transfers momentum in the radial direction, resulting from a difference in the net radial force between the horizontal component of \(\overline {\rho {u_i}{u_j}}\) for \(\overline \Phi \neq 3\), and the radial component of magnitude p_{t} for \(\overline \Phi = 3\) (Gough, 1977a).
In the mean equations, the turbulent pressure (Reynolds stress) p_{t} and convective heat flux F_{c} are the quantities that must be determined from the equations for the convective fluctuations. To solve these equations, a model for the convective turbulence is required, which is discussed in the next section.
3 Timedependent MixingLength Models
3.1 Introduction
The simplest closure model of turbulence is the early one by Boussinesq (1877), who suggested that turbulent flow could be considered as having an enhanced viscosity, a turbulent (or eddy) viscosity ν_{t}. Boussinesq assumed t to be constant, in which case the equations of mean motion become identical in structure with those for a laminar flow. This assumption, however, does become invalid near the convective boundary layers, where the turbulent fluctuations vanish, and so does ν_{t}, at least in a local convection model.
The simplest turbulence model able to account for the variability of the turbulent mixing with the use of only one empirical constant is the mixinglength idea, introduced independently by Taylor (1915) and Prandtl (1925). Based on Boussinesq’s approach and considering the turbulent fluid decomposed into socalled eddies, parcels or elements, Prandtl obtained, for the case of shear flow, from dimensional reasoning, an expression for the turbulent viscosity or exchange coefficient of momentum (“Austauschkoeffizient”). This expression is in the form of a product of the velocity fluctuation perpendicular (transverse) to the mean motion of the turbulent flow and the mixing length ℓ. The mixing length is characterized by the distance in the transverse direction which must be covered by a fluid parcel travelling with its original mean velocity in order to make the difference between its velocity and the velocity in the new layer equal to the mean transverse fluctuation in the turbulent flow. Inherent in this physical picture is the major assumption that the momentum of the turbulent parcel is assumed to be constant along the travel distance ℓ, which is analogous to neglecting the streamwise pressure forces and viscous stresses. Prandtl’s concept of a mixing length may be compared, up to a certain point, with the mean free path in the kinetic theory of gases. A somewhat different result was obtained by Taylor (1932) who assumed that the rotation (vorticity) during the transverse motion of the parcel remains constant, yielding a mixing length which is larger by a factor \(\sqrt 2\) compared with Prandtl’s momentumtransfer picture.
The first physical picture interprets the turbulent flow by direct analogy with kinetic gas theory. The motion is not steady and one imagines the overturning 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 but are taken to be responsible for the creation and destruction of the convective eddies (Spiegel, 1963; Gough, 1977a,b). By retaining only the acceleration terms the equations become linear and the evolution of the fluid properties carried by the turbulent parcels can be approximated by linear growth rates. The mixing length ℓ enters in the calculation of the eddy’s survival probability for determining the convective heat and momentum fluxes (see Appendix A).
In the second physical picture the fluid element maintains exact balance between buoyancy force and turbulent drag by continuous exchange of momentum with other elements and its surrounding (Prandtl, 1932). Thus the acceleration terms are unimportant in a static atmosphere and the evolution of the convective fluctuations are independent of the initial conditions. The nonlinear advection terms (i.e., momentum exchange) provide dissipation (of kinetic energy) that balances the driving terms, and are approximated appropriately (e.g., Kraichnan, 1962; Unno, 1967), leading to two nonlinear equations which need to be solved numerically together with the mean equations of the stellar structure.
The two physical pictures are complementary in envelopes that do not pulsate (Gough, 1977a). However, in a timedependent formulation additional information is required how the initial state of a convective element depends on conditions at the time of its creation. Hence, the different versions of mixinglength models yield different formulae for the turbulent heat and momentum fluxes when applied to pulsating stars (Unno, 1967; Gough, 1977a, 2012a).
In the above discussed models, the overturning fluid parcels were still considered to move adiabatically. Öpik (1950) suggested to treat radiative heat exchange between the element and the background fluid in a similar way as for the momentum exchange. Based on this assumptions Vitense (1953) and BöhmVitense (1958) established a mixinglength description which is still widely used for calculating the convective heat flux in stellar models.
The perhaps simplest description to model the temporal modulation of the convection by the oscillations, put forward in the 1960s, is to presume that the convective fluxes simply relax exponentially on a timescale τ_{c} towards the timeindependent formula \({\rm{d}}{F_{\rm{c}}}/{\rm{d}}t = ({F_{{\rm{c}}0}}  {F_{\rm{c}}})/{\tau _{\rm{c}}}\), where F_{c} is a component of any turbulent flux and F_{c0} is the formula for F_{c} in a statistically steady environment. The constant t_{c} is a multiple of w/ℓ with ℓ being the mixing length and w a characteristic convective velocity.
In the past, various timedependent convection models were proposed, for example, by Schatzman (1956), Gough (1965, 1977a), Unno (1967, 1977), Xiong (1977, 1989), Stellingwerf (1982), Gonczi (1982a), Kuhfuß (1986), Unno et al. (1989), Canuto (1992), Gabriel (1996), Grossman (1996), and Grigahcène et al. (2005). Here, we shall review and compare the basic concepts of two, currently in use, convection models. The first model is that by Gough (1977a,b), which has been used, for example, by Baker and Gough (1979), Balmforth (1992a), Houdek et al. (1995), Rosenthal et al. (1995), Houdek (1997, 2000), Houdek et al. (1999a), and Chaplin et al. (2005). The second model is that by Unno (1967, 1977), upon which the generalized models by Gabriel (1996) and Grigahcène et al. (2005) are based, with applications by Dupret et al. (2005c,a,b, 2006a,b,c, 2009), Belkacem et al. (2008, 2009, 2012), and Grosjean et al. (2014).
3.2 Two timedependent convection models for radially pulsating stars
Unno (1967) and Gough (1965, 1977a) generalized the mixinglength formulation for modelling the interaction of the turbulent velocity field with radial pulsation. Both authors adopted the Boussinesq approximation. The mean equation of motions were already discussed in Section 2.2 for a radially pulsating atmosphere. Therefore, we start here with the Boussinesq approximation for the convective fluctuations. This approximation is based on a careful scaling argument and an expansion in small parameters, i.e., the ratio of the maximum density variation across the layer over the (constant) spatial density average, and the ratio of the fluid layer height to the locally defined smallest scale height (Spiegel and Veronis, 1960; Mihaljan, 1962; Gough, 1969). In this subsection we follow the discussion by Gough (1977a).
3.2.1 Boussinesq fluctuation equations
In these fluctuation equations geometrical terms, which distinguish Cartesian from the spherical coordinates q_{ i }, are neglected, i.e., it is assumed that the convective velocity field is located in stellar layers where ℓ ≪ r. It is also assumed, in accordance with the Boussinesq approximation, that ℓ ≪ H, where H represents any locallydefined scale height.
The third term on the lefthand side of Eq. (33) comes from substituting the mean continuity equation into the mean radial component of the nonlinear advection term of the mean momentum equation. With the help of Eq. (21), to relate timederivatives in Eulerian convective fluctuations to the Lagrangian coordinates q_{ i }, one obtains \({\partial _3}U =  \partial (\ln {r^2}\overline \rho)/\partial t\). The third term of the lefthand side of Eq. (34) is a result of having taken into account the pulsationally induced time dependence of the mean temperature \(\overline T\) and gas pressure \(\overline p\) in a pulsating atmosphere.
3.2.2 Local mixinglength models for static atmospheres
Linear pulsation calculations perturb the stellar structure equations around a timeindependent (on a dynamical time scale) equilibrium model, which must be constructed first from, e.g., stellar evolutionary calculations. We start the discussion of two versions of the mixinglength formulation first for a static stellar envelope before embarking on the model description for radially pulsating envelopes.
In Section 3.1 we introduced two physical pictures of mixinglength models, both of which are based on the picture of an overturning convective cell (see Figure 1). In both pictures, the convective cell is created as a result of instability with the same average properties than its immediate surroundings. The overturning motion of a convective cell is then accelerated by the imbalance between buoyancy forces, nonlinear advection processes, pressure gradients, and heat losses by radiation. Various guises of convection models can be obtained by approximating these processes in different ways and even neglecting some of it. Also, different assumptions about the geometry of the turbulent flow does lead to different results in the turbulent fluxes. Two of the convection models will be described below which, to some extent, make different assumptions about the dynamics of the turbulence.
3.2.2.1 Convection model 1: Kinetic theory of accelerating eddies
3.2.2.2 Convection model 2: Balance between buoyancy and turbulent drag
The nonlinear Eqs. (62) and (63) are solved numerically for w and T′ from which the turbulent fluxes \({F_{\rm{c}}} \simeq \overline {\rho cp} \overline {w{T^{\prime}}}\) and \({p_{\rm{t}}} \simeq \overline \rho \overline {w{w^{\prime}}}\) are constructed. Unno (1967) neglects, however, the turbulent pressure p_{t} in the mean momentum equation (22).
3.2.3 Local mixinglength models for radially pulsating atmospheres
In the previous section, we discussed two mixinglength models in a static atmosphere. In a static atmosphere the (mean) coefficients ρ, c_{ p }, and \(\rho, \;{c_p}\) are independent of time, which had led to Eqs. (53) and (54) for the convection model of accelerating eddies. What follows is a discussion of the timedependent treatment of the two convection models in a radially pulsating atmosphere.
3.2.3.1 Convection model 1: Kinetic theory of accelerating eddies
3.2.3.2 Convection model 2: Balance between buoyancy and turbulent drag
3.3 A nonlocal mixinglength model
One of the major assumptions in the above described local mixinglength theory is that the characteristic length scale ℓ must be shorter than any scale length associated with the structure of the star. This condition is violated, however, for solarlike stars and red giants where evolution calculations reveal a typical value for the mixinglength parameter α = ℓ/H_{p} of order unity, where H_{p} is the pressure scale height. This implies that fluid properties vary over the extent of a convective element and the superadiabatic gradient can vary on a scale much shorter than ℓ.
The nonlocal theory takes some account of the finite size of a convective element and averages the representative value of a variable throughout the eddy. Spiegel (1963) proposed a nonlocal description based on the concept of an eddy phase space and derived an equation for the convective flux which is familiar in radiative transfer theory. The solution of this transfer equation yields an integral expression which would convert the usual ordinary differential equations of stellar model calculations into integrodifferential equations. An approximate solution can be found by taking the moments of the transfer equation and using the Eddington approximation to close the system of moment equations at second order (Gough, 1977b). The next paragraphs provide a brief overview of the derivation of the nonlocal convective fluxes, following Gough (1977b) and Balmforth (1992a).
3.3.1 Formulation for stationary atmospheres
The nonlocal equations discussed above were derived in the physical picture in which the convective elements are accelerated from rest and whose evolutions along their trajectories are described by linear growth rates, as already discussed in the local theory. Obviously the nonlocal equations may also be discussed in the view of the second picture, where the eddies are regarded as cells with the size of one mixinglength and centred at some fixed height, again, similar as in the local treatment of mixing length theory. This is the picture in which Gough (1977b) discussed the derivation of the nonlocal equations, which corresponds to treating the finite extent of the eddy and the nonlocal transfer of heat and momentum across it by using the averaging idea which had led to the equation for \(\mathcal{B}\) described above. The integral expression (86) may then be interpreted such that an eddy centred at z_{0} samples \(\mathcal{B}\) over the range determined by the extend of the eddy, i.e. (z_{0} − ℓ/2. z_{0} + ℓ/2). Moreover, the averaged convective fluxes \({\mathcal{F}_{\rm{c}}}\) and \({\mathcal{P}_t}\) are constructed not only by eddies located at z_{0} = z, but by all the eddies centred between z_{0} − ℓ/2 and z_{0} + ℓ/2. Hence the two additionally parameters a and b (three, if the kernels for the convective heat flux and turbulent pressure are treated differently) control the spatial coherence of the ensemble of eddies contributing to the total heat and momentum flux (a), and the degree to which the turbulent fluxes are coupled to the local stratification (6). Theory suggests values for these parameters, but the quoted values are approximate and to some extent these parameters are free. These parameters control the degree of “nonlocality” of convection, where low values imply highly nonlocal solutions and in the limit a, b → ∞, the system of equations reduces to the local theory. Balmforth (1992a) explored the effect of a and b on the turbulent fluxes in the solar case very thoroughly and Tooth and Gough (1988) calibrated a and b against laboratory convection.
Dupret et al. (2006a) proposed to calibrate the nonlocal convection parameters a and b against 3D largeeddy simulations (LES) in the convective overshoot regions.
3.3.2 Formulation for radially pulsating atmospheres
Gough’s nonlocal generalization was adopted, in a simplified form, by Dupret et al. (2006c) for Grigahcène et al.’s (2005) convection model. It was implemented only in the pulsation calculations and, instead of perturbing the nonlocal equations as shown by Eq. (92), Dupret et al. (2006c) replaced the turbulent fluxes, \(({F_{\rm{c}}},{\mathcal{F}_{\rm{c}}})\) and \(({p_{\rm{t}}},{\mathcal{P}_{\rm{t}}})\) in Eqs. (87) and (91) by their Lagrangian perturbations.
3.4 Unno’s convection model generalized for nonradial oscillations
In this section, we summarize the model by Grigahcène et al. (2005), who adopted and generalized Unno’s (1967) description for approximating the nonlinear terms in the fluctuating convection equations and Gabriel et al.’s (1975, see also Gabriel, 1996) approach for describing timedependent convection in nonradially pulsating stellar models.
For completeness we summarize the nonradial pulsation equations of the stellar mean structure in Appendix C.
3.4.1 Equations for the convective fluctuations
As in the previous section of radially pulsating stars, we also adopt the Boussinesq approximation to the convective fluctuation equations for nonradially pulsating stars. The detailed discussion of the derivation is presented in Appendix D. Here, we introduce and discuss the final equations that are used in the stability computations.
Note that the radial component \({(\bar \rho \bar T\nabla \bar s \cdot u)_r} =  \bar \rho {\bar c_p}\beta w\), where the superadiabatic lapsrate β is defined by Eq. (38) and ω is the vertical component of u.
3.4.2 Perturbation of the convection
In order to determine the pulsational perturbations of the terms linked to convection we proceed as follows. We perturb Eqs. (93), (94) and (101). Then we search for solutions of the form \(\delta ({X^{\prime}}) = \delta {({X^{\prime}})_k}{{\rm{e}}^{ik \cdot r}}{{\rm{e}}^{ {\rm{i}}\omega t}}\), assuming constant coefficients, where δ denotes a linear pulsational perturbation in a Lagrangian frame of reference, and ω is the (complex) eigenfrequency of the pulsations. These particular solutions are integrated over all wavenumber values of k_{ θ } and k_{ ϕ }such that \(k_\theta ^2 + k_\varphi ^2 = k_r^2/(\Phi  1)\), assuming Φ to be constant, and that every direction of the horizontal component of k has the same probability. We have to introduce this distribution of k values to obtain an expression for the perturbation of the Reynolds stress tensor which allows the proper separation of the variables in the equation of motion (Gabriel, 1987).
Horizontal averages are computed on a scale larger than the size of the eddies but smaller than the horizontal wavelength of the nonradial oscillations (r/l).
We note, however, that a nonlocal treatment of the convective fluxes, such as the model discussed in Section 3.3, does not necessarily need the adhoc introduction of the additional parameter \(\hat \beta\).
The expressions for the remaining perturbed quantities are listed in Appendix E.
3.4.3 Perturbation of the convective heat flux
3.4.4 Perturbation of the turbulent pressure
3.4.5 Perturbation of the rate of dissipation of turbulent kinetic energy into heat
As shown by Ledoux and Walraven (1958) and Grigahcène et al. (2005), it is important to emphasize that the turbulent pressure variation and the turbulent kinetic energy dissipation variations have an opposite effect on the driving and damping of the modes. This can be seen clearly by considering the contributions of these terms to the work integral.
3.4.6 Perturbation of the mixing length
In Section 3.2.3, we discussed two descriptions for the pulsationally distorted convective eddy shape and therefore also for the pulsationally modulated mixing length. One of the earliest suggestions was provided by Cowling (1934), who proposed δℓ/ℓ = ξ_{r}/r. Cowling’s suggestion was adopted by Boury et al. (1975), and by Unno (1967) in the limit ωτ_{c} ≫ 1 [see also Eq. (80)], where τ_{c} is the convective turnover time scale. In this limit l would vary with the local Lagrangian scale of the mean flow, a result similar to rapid distortion theory (see Section 3.2.3.1).
Expressions for the perturbation of the nondiagonal components of the Reynolds stresses were reported by Gabriel (1987, see also Houdek and Gough 2001, and Smolec et al. 2011).
3.5 Differences between Gough’s and Unno’s local convection models
Section 3.2 discussed the detailed equations of Gough’s (1965, 1977a) and Unno’s (1967) local, timedependent mixinglength models.
Gough’s model puts much attention on the dynamics of the linearly growing convective elements by means of an eddy creation and annihilation model. In particular, the phase between the pulsating background state and the convective fluctuations are considered by adopting a quadratic distribution function for the convective temperature fluctuations at the time of the eddy creation (zero velocity of the eddies) in order to describe more realistically the initial conditions of the convective elements. This turns out to be crucial for the damping and driving of the stellar pulsations and consequently for their stability properties. Although the nonlinear effects are taken into account by the instantaneous eddy disruption (annihilation) after the eddy’s mean lifetime \(\tau \propto \sigma _{\rm{c}}^{ 1}\), where σ_{c} is the linear convective growth rate [see Eq. (56) and Appendix A], the continuous damping effects of the smallscale turbulence are omitted, which are expected to limit both the velocity and the temperature fluctuations of an eddy and consequently the convective velocity.
Unno’s convection model includes the nonlinear advection terms, though in a simplified manner, by means of a scalar turbulent viscosity [Eqs. (60) and (61), see also discussion in Section 4], but the evolution of the turbulent fluctuations is independent from any initial conditions. Additional simplifications in Unno’s model are the omission of the timederivatives of the mean quantities in the fluctuating convection equations, i.e., the third terms on the lefthand side of Eqs. (33) and (34), and of the (mean) turbulent pressure i in the equation of hydrostatic support (22).
Another substantial difference between the two convection models by Gough and Unno is the treatment of the anisotropy of the turbulent velocity field (or eddy shape) in both the static and pulsating stellar model. The way how Unno eliminates the fluctuating pressure gradient ∇p′ in Eq. (33) leads to \(k_{\rm{v}}^2 = k_{\rm{h}}^2\), i.e., to an (fixed) anisotropy parameter Φ = 2 (this is also the value adopted by BöhmVitense 1958). Gough parametrizes Φ, i.e., how the pressure fluctuations couple the horizontal to the vertical motion. The most important difference is, however, the modelling of the pulsationally modulated eddy shape Φ and consequently also mixing length ℓ. While Gough adopts rapid distortion theory for describing the variation of both Φ and ℓ [Eq. (67), see also discussion in Sections 3.2.3 and 3.4.6], assumes Unno the eddy shape to be invariant despite of an pulsating background, and adopts Eq. (80) for describing the pulsational variation of ℓ (see also Section 3.4.6). These differences affect the stability of the pulsations (Gough, 1977b; Balmforth, 1992a).
Radiative losses of the convective elements play also a role in determining convective efficacy and dynamics. Unno adopts the diffusion approximation to radiative transfer [Eqs. (63) and (64)]. Gough describes the radiative losses by means of the Eddington approximation by Unno and Spiegel (1966) [Eqs. (35) and (36); see also Eqs. (51 and (52)].
It should also be noted that Gough’s model has only been applied to linear radial pulsations. Efforts to generalize this model to nonradial oscillations have been reported by Houdek and Gough (2001), Gough and Houdek (2001), and Smolec et al. (2011, 2013).
3.6 Differences between Unno’s and Grigahcène et al.’s local convection models
Gabriel (1996), and Grigahcène et al.’s (2005) (G9605) models are a generalization of Unno’s (1967) approach to nonradial oscillations. They treat the momentum equation in its fully vectorial form for the oscillations and for the convection.
There are also more subtle improvements in the treatment of the closure terms in the momentum and energy equations for convection. In the momentum equation for the convective fluctuations, the linear part of the advection term is treated rigorously in G9605 [term − ρu · ∇U in Eq. (94)], while it is neglected in Unno (1967). G9605 models use an approximation for the nonlinear terms similar to Unno (1967) [Eq. (96)], but G9605 use Λ = 8/3 instead of 2 in order to be consistent with their equilibrium structure models. Concerning the energy equation for the convective fluctuations (101), the first term \({(\rho T)^{\prime}}/\overline {\rho T} {\rm{d}}\bar s/{\rm{d}}t\) is included in G9605 while it is neglected in Unno (1967). G9605 models use the same approximation for the nonlinear terms as Unno (1967) [Eqs. (97) and (99)] but again with different dimensionless geometrical factors in order to be consistent with their equilibrium structure models. Finally, Grigahcène et al. (2005) introduced an additional parametrization for the perturbation of the closure term of the energy equation (107). The complex parameter \(\hat \beta\) introduced in this last approach allows to avoid unphysical short wavelength oscillations of the mean entropy perturbation. It also requires calibration in order to fit the solar damping rates as discussed in more detail in Section 6.3.
We now consider the pulsation equations. In addition to the fact that the G9605 theory can deal with nonradial oscillations, it includes several improvements. The perturbation of the turbulent pressure [Eq. (116)] is included in the momentum equation for the pulsations [Eqs. (179) and (180)]. The perturbation of the nondiagonal components of the Reynolds stress can also be obtained (Gabriel, 1987). The perturbation of the dissipation rate of turbulent kinetic energy into heat [Eq. (120)] is included in the energy equation for the pulsations [Eq. (181)]. All these terms are neglected in Unno (1967).
4 Reynolds Stress Models
Instead of adopting approximations for the secondorder moments, dedicated transport equations can be constructed, for example for \(\overline {u{T^{\prime}}}\), from multiplying Eq. (40) by T′, Eq. (41) by u, summing the results followed by averaging, in a similar way as we did for the averaged, turbulent kinetic energy equation (13). The soconstructed transport equations for the secondorder moments constitute the Reynolds stress approach, as proposed first by Keller and Friedmann (1924), and first completely derived by Chou (1945). The transport equation for the secondorder moments, however, include terms of thirdorder moments which need, as discussed above for the secondorder moments, to be represented by appropriate approximations or by additional transport equations, which will contain terms of fourthorder moments. This can, in principle, be continued to ever higherorder moments, but there will always be more variables (higherorder moments) than equations, representing the socalled closure problem of turbulence.
Xiong’s model was applied to stability computations of solar oscillations (Xiong et al., 2000) and of classical pulsators (Xiong et al., 1998a; Xiong and Deng, 2007). These calculations could successfully reproduce the location of the cool edge of the classical instability strip (see discussion in Section 6.2), but report for a solar model overstable (unstable) radial modes with radial order n = 11–23, which is in disagreement with the observed finite mode lifetimes discussed in Section 6.3.
Canuto (1992, 1993) went beyond Xiong’s treatment by proposing, additionally to the secondorder transport equations, including also nonlocal expressions for ϵ_{t} and ϵ_{ t′ }, separate transport equations for the thirdorder moments, which imply fourthorder moments. Canuto adopts the EddyDamped QuasiNormal approximation (Orszag, 1977; Hanjalic and Launder, 1976), which is based on the quasinormal approximation by Millionshtchikov (1941), to close the fourthorder moments. This approximation assumes the fourthorder moments to be Gaussian random variables, leading to an expression of products and sums of secondorder moments. The fourthorder pressure correlation terms are approximated by thirdorder damping terms. In this approximation, all six thirdorder terms are expressed by six, partial differential equations which now include only secondand thirdorder moments with five closure coefficients (see Canuto and Dubovikov, 1998, Eq. 37g). For the stationary case (∂/∂t = 0) the thirdorder terms form a set of six linear algebraic equations from which the thirdorder moments can be solved analytically as functions of loworder moments. If the dissipation rate ϵ_{ t′ } of thermal potential energy is approximated by a local expression, the whole turbulent convection problem is described by five coupled partial differential equations for the secondorder moments \(\overline {w{T^{\prime}}},\overline {{T^{\prime}}{T^{\prime}}},\overline {ww},\overline {u \cdot u}\) and ϵ_{t}, where the velocity field u = (u, v, w) in a planeparallel geometry. The five transport equations for the secondorder moments use five empirical closure coefficients (see Canuto and ChristensenDalsgaard, 1998, Eqs. (13)–(16)), additionally to the five closure coefficients for the thirdorder moments.
Canuto and Dubovikov (1998) extended Canuto’s (1993) Reynolds stress model by deriving improved expressions for the dissipation terms ϵ_{t} and ϵ_{ t′ }, and for the empirical constants that were used in Canuto’s (1993) model, using renormalization group techniques. Canuto and Dubovikov’s model, together with a simplified version of the thirdorder moments in the stationary limit, was implemented by Kupka (1999) and applied to nonpulsating (stationary) envelope models of Astars and white dwarfs by Kupka and Montgomery (2002) and Montgomery and Kupka (2004).
5 Convection Effects on Pulsation Frequencies
Convection modifies pulsation properties of stars principally through two effects: (i) dynamical effects through the additional turbulent pressure term p_{t} (12) in the mean momentum equation (22), and its perturbation δp_{t} (73), (92), where δ denotes here a linear perturbation in a Lagrangianmean frame of reference, in the pulsationally perturbed mean momentum equation; (ii) nonadiabatic effects, additional to the perturbed radiative heat flux δF_{r}, through the perturbed convective heat (enthalpy) flux δF_{c} (72), (92) in the pulsationally perturbed mean thermal heat (energy) equation.
5.1 The effect of the Reynolds stress in the equilibrium stellar model
Rosenthal et al. (1995), for example, investigated the effect of the contribution that p_{t} makes to the mean hydrostatic stratification on the adiabatic solar eigenfrequencies. They examined a hydrodynamical simulation by Stein and Nordlund (1991) of the outer 2% by radius of the Sun, matched continuously in sound speed to a model envelope calculated, as in a ‘standard’ solar model, with a local mixinglength formulation without p_{t}. The resulting frequency shifts of adiabatic oscillations between the simulations and the ‘standard’ solar reference model, Model S, are illustrated in Figure 2b. The frequency residuals behave similarly to the solar data depicted in the left panel of that figure, though with larger frequency shifts at higher oscillation frequencies.
5.2 The effects of nonadiabaticity and momentum flux perturbation

L.a A local mixinglength formulation without turbulent pressure p_{t} was used to construct the mean envelope model. Frequencies were computed in the adiabatic approximation assuming δp_{t} = 0.

NL.a Gough’s (1977a,b) nonlocal, mixinglength model, including turbulent pressure, was used to construct the mean envelope model. Frequencies were computed in the adiabatic approximation assuming δp_{t} = 0.

NL.na The mean envelope model was constructed as in NL.a. Nonadiabatic frequencies were computed including consistently the Lagrangian perturbations of the convective heat flux δF_{c}, additionally to δF_{r}, and turbulent momentum flux δp_{t}.
Additional care was necessary when frequencies between models with different convection treatments were compared, such as in the models L.A and NL.a. In order to isolate the effect of the nearsurface structures on the oscillation frequencies the models had to posses the same stratification in their deep interiors. This was obtained by requiring the models to lie on the same adiabat near the base of the (surface) convection zone and to have the same convectionzone depth. Varying the mixinglength parameter α = ℓ/Hp and hydrogen abundance by iteration in model L.a, the same values for temperature and pressure were found at the base of the convection zone as those in models NL.a and NL.na. The radiative interior of the nonlocal models NL.a and NL.na were then replaced by the solution of the local model L.a, and the convectionzone depth was calibrated to 0.287 R_{⊙} (ChristensenDalsgaard et al., 1991). Further details of the adopted physics in the model calculations can be found in Houdek et al. (1999a).
If, however, the positive frequency shifts between models NL.na and NL.a (dashed curve) are interpreted as the nonadiabatic and momentum flux corrections to the oscillation frequencies then their effects are to bring the frequency residuals of the hydrodynamical simulations (Figure 2b) in better agreement with the data plotted in Figure 2a.
The effects of the nearsurface regions in the Sun were also considered by Rosenthal et al. (1999) and Li et al. (2002) based on hydrodynamical simulations.
A similar conclusion as in the solar case was found for the solarlike star η Boo by ChristensenDalsgaard et al. (1995) and Houdek (1996), demonstrated in Figure 4b, and more recently by Straka et al. (2006). Grigahcène et al. (2012) studied the surface effects in the Sun and in three solartype stars with the conclusion that the use of the local timedependent convection treatment of Section (3.4) reduces the frequency residuals between observations and stellar models. In these calculations, however, the hydrostatic equilibrium model was corrected a posteriori by the effect of the mean turbulent pressure with some consequent inconsistencies in the thermal equilibrium structure.
The nearsurface frequency corrections also affect the determination of the modelled mean large frequency separation \(\Delta \nu: = \langle {{\nu _{n + 1l}}  {\nu _{nl}}} \rangle\) (angular brackets indicate an average over n and l). In both models for the Sun (ChristensenDalsgaard and Gough, 1980) and for η Boo the resulting corrections to ∇ν are about −1 μHz. Although this correction is less than 1% it does affect the determination of the stellar radii and ages from the observed values of ∇ν and small frequency separation δν_{02} in distant stars.
A simple procedure for estimating the nearsurface frequency corrections was suggested by Kjeldsen et al. (2008). It is based on the ansatz that the frequency shifts can be scaled as a(ν/ν_{0})^{ b } (ChristensenDalsgaard and Gough, 1980), where ν_{0} is a suitable reference frequency, b is obtained from solar data, and the surfacecorrection amplitude a is determined from fitting this expression to the observed frequencies.
Kjeldsen et al.’s empirical powerlaw has been applied to the modelling of a large number of solartype Kepler stars (e.g., Metcalfe et al., 2012; Mathur et al., 2012; Gruberbauer et al., 2013; Metcalfe et al., 2014). Mathur et al., for example, determined statistical properties of the surfacecorrection amplitude from 22 Kepler stars. The model frequencies were, however, obtained in the adiabatic approximation neglecting, as did Metcalfe et al. (2012) and Metcalfe et al. (2014), any convection dynamics in both the equilibrium and pulsation calculations. Mathur et al. (2012) concluded that the surfacecorrection amplitude is nearly a constant fraction of the mean largefrequency separation ∇ν. Information like this could provide additional insight into the physical processes responsible for the high radialorder frequency shifts between observations and stellar models. Gruberbauer et al. (2013, see also Gruberbauer and Guenther, 2013) analysed the surface effects in 23 Kepler stars with a Bayesian approach neglecting, however, convection dynamics in both the equilibrium and in the nonadiabatic eigenfrequency calculations.
ChristensenDalsgaard (2012) suggested an improved functional form for the highorder frequency shifts between observations and stellar models. This improved functional form can be determined for the Sun from the surface term in Duvall’s differential form for the asymptotic expression for frequencies using a large range of mode degrees l. This leads to a better representation of the solar frequency residuals brought about by the very surface layers. By adopting the acoustic cutoff frequency as the relevant frequency scale the scaled solarsurface functional form can also be applied to other stars that are not too dissimilar to the Sun. ChristensenDalsgaard (2012) applied it to Kepler data for the solartype star 16 Cyg A and reported a better representation of the frequency surface correction compared to the empirical power law by Kjeldsen et al. (2008). Another empirical approach was recently reported by Ball and Gizon (2014), which is based on the scaling relation for mode inertia by Gough (1990).
Although these empirical approaches offer some description of the surface effects, they do not provide the much needed insight for describing the relevant physical processes. The most promising approach today for a better understanding of these surface effects is the use of the latest implementations of threedimensional (3D) hydrodynamical simulations, and their results, for developing improved onedimensional (1D) convection models. Several international groups are now pursuing this approach.
6 Driving and Damping Mechanisms
The question of whether the amplitude (or energy) of a particular oscillation mode in a star is growing or declining with time is related to the problem of vibrational stability. Because vibrational stability (or instability) is characterized by the existence of a periodicity in the temporal behaviour of the perturbations, a reasonable useful criteria is the sign of the total energy change (thermal and mechanical) over one pulsation period assuming that the system returns precisely to its original state at the end of the period. This is the definition of the work integral W.
6.1 The work integral
6.1.1 Expressions for radial pulsations
6.1.2 Expressions for nonradial pulsations
It should be noted that in Gough’s (1977a,b) convection model the viscous dissipation by turbulent kinetic energy into heat is neglected in the thermal energy equation, as suggested by Spiegel and Veronis (1960) for a (static, i.e., U = 0) Boussinesq fluid.
In the following sections, we shall review and compare results of stability calculations between the two timedependent convection formulations by Gough (1977a,b) and Grigahcène et al. (2005) for various classical pulsators and for stars with stochastically excited oscillations.
6.2 Intrinsically unstable pulsators
One of the most prominent stability computations in stars has been the modelling of the location of the classical instability strip in the HertzsprungRussell diagram. Since the seminal work by Baker and Kippenhahn (1962, 1965) for modelling linear stability coefficients in Cepheids, various attempts have been made to reproduce theoretically the observed location of the instability strip. The properties of the hotter, blue edge of the instability strip could be explained first (e.g., Castor, 1970; Petersen and Jørgensen, 1972; Dziembowski and Kozlowski, 1974; Stellingwerf, 1979, and references therein), mainly because for these hotter stars the rather thin surface convection zone does not affect pulsation dynamics too severely. The modelling of the return to pulsational stability at the cooler, red edge, however, has been less successful, despite the first promising attempts by, e.g., Deupree (1977a), who solved the nonlinear hydrodynamic equations, using a timevarying eddy viscosity, for studying the stability properties of RR Lyrae stars. The need for a timedependent convection treatment for modelling the lowtemperature, red edge of the instability strip was recognized by Baker and Kippenhahn (1965, see also Cox, 1974).
6.2.1 Cepheids and RR Lyrae stars
The first theoretical studies describing successfully the location of the cool edge of the classical Cepheid instability strip were reported by Baker and Gough (1979) for RR Lyrae stars, using linear stability analyses of radial modes and the local timedependent convection formulation of Section 3.2.3.1. Shortly thereafter, Xiong (1980) was successful with Cepheid models, using his own local timedependent convection model (Xiong, 1977, see Section 4). In the same year Gonczi and Osaki (1980) used Unno’s (1967, see Section 3.2.3) timedependent convection model for analysing stability properties of Cepheid models, but only with the inclusion of an additional scalar turbulent viscosity, brought about by the smallscale turbulence [see Eq. (138)], could Gonczi (1981) successfully model the return to stability near the cool edge of the instability strip.
Later, Stellingwerf (1986), using Stellingwerf’s (1982) turbulence formulation with a simplified extension for onezone pulsation models (Baker, 1966; Baker et al., 1966), conducted linear and nonlinear Cepheid stability analyses. He reported that the coupling between pulsation and convection can describe a return to stability for cooler Cepheid models. In this study, however, the effect of turbulent pressure was omitted in the calculations, but later included by Munteanu et al. (2005), who concluded that the turbulent pressure appears to be a driving mechanism. Nonlinear pulsation modelling of Cepheids, using the nonlocal, timedependent, oneequation, convection formulation by Kuhfuß (1986), were reported by Smolec and Moskalik (2008); Buchler (2009), and Smolec and Moskalik (2010).
Linear stability analyses of radial Cepheid pulsations were also conducted by Balmforth and Gough (1988), using Gough’s (1977a) local convection model of Section 3.2.3. Houdek et al. (1999b) discussed linear stability analyses and nonadiabatic pulsationperiod ratios in doublemode Cepheids, using Gough’s (1977b) nonlocal convection formulation of Section 3.3.2. Both studies reproduced the cool edge of the classical instability strip, with the pulsationally perturbed turbulent pressure being the main contributor for stabilizing the pulsation modes. Yecko et al. (1998), on the other hand, found the damping effect of the smallscale turbulent eddy viscosity (see Eq. 138) to be the main agent for making the pulsation modes stable at the cool side of the instability strip. The authors adopted the convection model by Gehmeyr (1992), which is based on Stellingwerf’s (1982) turbulence model, for their linear stability computations.
6.2.2 Mira variables
Mira variables are longperiod variables (LPV) with radial pulsation periods P ≳ 80 days located to the red of the classical instability strip with typical surface temperatures between 2500 and 3500 K and luminosities between ∼ 10^{3} and ∼ 7 × 10^{3} L_{⊙}. The detailed driving mechanism of these loworder radial oscillations depends crucially on the treatment of the coupling of the pulsations to the convection. Several attempts have been made in the past to model this coupling in both linear and nonlinear calculations with rather oversimplified descriptions (e.g., Kamijo, 1962; Keeley, 1970; Langer, 1971; Cox and Ostlie, 1993). The first attempt to describe the coupling in a more realistic way was conducted by Gough (1966, 1967), who included the pulsational perturbations of both the convective heat (enthalpy) flux δF_{c} and momentum flux δp_{t} in the linear stability analyses. Gough concluded that in particular the momentum flux perturbation δp_{t} has a stabilizing effect on the pulsations if the pulsation period is much shorter than the characteristic time scale of the convection, whereas for longperiod pulsations, such as in Mira variables or supergiants, the turbulent pressure fluctuations δp_{t} destabilizes (drives) the stellar pulsations. It is perhaps interesting to note that a similar effect was noticed in linear Delta Scuti stability computations by Houdek (1996) and more recently by Antoci et al. (2014) (see Section 6.2.3), in which δp_{t} was found to drive highorder radial pulsations, in agreement with observations. Using the local timedependent formulation by Gough (1977a), Balmforth et al. (1990) concluded that including the turbulent pressure in the mean model of Mira variables modifies the equilibrium structure such as to make the observed radial pulsations overstable in the pulsation computations which is, however, partially offset by the stabilizing influence of δp_{t}. Xiong et al. (1998b), on the other hand, found δp_{t}, together with the turbulent eddy viscosity [see Eq. (138)], to be the main stabilizing contribution to linear Mira pulsations. Munteanu et al. (2005) and Olivier and Wood (2005) conducted nonlinear pulsation models using the oneequation turbulence models by Stellingwerf (1986) and Kuhfuß (1986) respectively, and reported about the destabilizing effect of δp_{t}, i.e., in accordance with the earlier findings by Gough (1966, 1967). It appears that further progress on modelling the interaction between convection and pulsations in Mira variables is required.
6.2.3 Delta Scuti stars
Already before the successful space missions CoRoT (Baglin et al., 2009) and Kepler (ChristensenDalsgaard et al., 2009a) several observing campaigns, e.g., the Delta Scuti Network (DNS) or the Whole Earth Telescope (WET), have been providing excellent oscillation data of Delta Scuti stars. For example, Breger et al. (1999) identified 24 pulsation frequencies in the Delta Scuti star FG Vir. Such highquality seismic data also provided welldefined observed locations of the lower part of the classical instability strip (e.g., Rodríguez et al., 2000), which modellers could use to test their timedependent convection models.
Dupret et al. (2005c) used the local timedependent convection treatment of Grigahcène et al. (2005, see Section 3.4), to study the stability properties of radial and nonradial pulsations in δ Sct stars. In these calculations the perturbations of both the convective heat flux δF_{c} and turbulent pressure δp_{t} were included in the linear pulsation computations, but the (mean) turbulent pressure p_{t} was omitted in the construction of the equilibrium structure. Dupret et al. (2005c) found well defined red edges of the instability strip for both radial and nonradial modes using Grigahcéne et al.’s timedependent convection model. The authors found that the δ Sct loworder p modes become stable again with decreasing T_{eff} when the two thin convective zones, associated with the partial ionization of hydrogen and helium, merge to form one large surface convection zone. For a solarcalibrated mixinglength parameter α = ℓ/H (ℓ is the mixing length and H is the pressure scale height) the return to stability occurs at the observed location in the HertzsprungRussell diagram. For smaller values of the calculated cool edge of the instability strip is shifted towards cooler surface temperatures T_{eff} (Dupret et al., 2005c), in accordance with the findings by Houdek (2000), i.e., the observed location of the red edge could be used to calibrate the mixinglength parameter.
It is clear from the discussion above that all three timedependent convection models (Gough, 1977a; Xiong, 1989; Grigahcène et al., 2005) are able to reproduce theoretically the red edge of the instability strip, and about at the same location as observations suggest. The very detailed physical processes, however, that lead to the definition of the red edge are different in all three convection models: Gough’s model predicts that it is the perturbed Reynolds stress, Xiong (1989) the viscous dissipation by the smallscale turbulence, and the model by Grigahcène et al. (2005) predicts that it is the perturbed convective heat flux, which is responsible for the return to stability.
Form these results it is obvious that further research is necessary to identify the correct processes that define the location of the cool edge of the classical instability strip.
6.2.4 Gamma Doradus stars
γ Dor stars are Ftype gmode pulsators located near the red edge of the δ Scuti instability strip. A driving mechanism for these modes was proposed by Guzik et al. (2000), who used a standard timeindependent, or frozen, convection model. The time scale associated to convective motions is, however, shorter than the pulsation periods in most of the convective envelope γ Doradus stars, and the validity of frozen convection models for estimating stability properties of oscillations is therefore doubtful in these stars. This motivated Dupret et al. (2004b) and Dupret et al. (2005a,b) to use the timedependent convection treatment of Grigahcène et al. (2005) for studying the driving mechanisms in γ Dor stars. The important result was that unstable g modes are also obtained with this timedependent convection treatment, with a range of frequencies (from ∼ 0.3 to 3 days) in agreement with typical observations. The theoretical instability strip could be computed and good agreement with observations was obtained for certain values of the mixinglength parameter α.
In the study of nonadiabatic processes we generally define the transition region in a star as the region where the thermal relaxation time is of the same order as the pulsation period. This region generally plays the major role in the driving or damping (see e.g., Cox, 1974). Efficient driving of γ Dor g modes occurs when much of the region lies just above the base of the convective envelope, for there the mode of heat transport changes dramatically. Because convection typically carries most of the heat, yet the flux is presumed to be frozen, it dams up heat when the radiative flux from below is relatively high and transmits more when the incident flux is low. The radiative component of the flux in the convection zone is essentially unchanged, aside from that resulting directly from the modification by convection of the mean thermal stratification. The process can drive the pulsations, and is often called “convective blocking”, a terminology that could be misleading. A more accurate term would be “convective shunting”, because convection essentially redistributes the radiative flux, thereby reducing the relative modulation by radiation of the total flux.
For the mixinglength parameter α = 2, and adopting Grigahcène et al.’s convection model, the transition region and bottom of the convective envelope coincide for stellar models that are located in the HertzsprungRussell diagram where γ Dor stars are observed. For smaller values of α, stellar models with lower effective temperatures are required to have a sufficiently deep convective envelopes, i.e., the location of the theoretically determined instability strip is shifted to lower temperatures in the HertzsprungRussell diagram. An important issue that has not yet been fully studied is the role of the nondiagonal components of the Reynolds stress in the driving. Preliminary studies using the formulation of Gabriel (1987) indicate that it is important, but numerical instabilities make this problem very delicate (see also Gough and Houdek, 2001).
6.2.5 Rapidly oscillating Ap stars
Rapidly oscillating Ap stars (hereafter roAp stars) are mainsequence stars with typical masses between 1.5 and 2.0 M_{⊙} and with effective temperatures T_{eff} between 6800 and 8400 K. They are the coolest stars amongst the chemically peculiar Atype (Ap) stars with high overabundances of Sr, Cr and Eu. They show strong, predominantly dipolar, largescale magnetic fields with magnitudes varying typically from 1 to about 25 kG, leading to antipodal spots. The roAp stars have in general rotation periods larger than about two days. The periods of the light variability range from roughly 5 to 21 minutes and are interpreted as highorder, lowdegree acoustic modes. The first roAp star was discovered photometrically by Kurtz (1978) and their number has increased today to about 43 (Kurtz et al., 2011). Recent reviews on roAp stars were given by Gough (2005); Cunha (2007), Shibahashi (2008) and Kochukhov (2009).
The observed pulsation properties in roAp stars suggest that the pulsation axis is not aligned with the rotation axis. This had led to the socalled oblique pulsator model (Kurtz, 1982), in which the observed cyclically varying oscillation amplitudes are explained by dipole oscillations being aligned with the magnetic axis, which itself is oblique to the rotation axis of the star. The pulsation eigenfunction differs, however, from a simple spherical harmonic (e.g., Takata and Shibahashi, 1994, 1995; Montgomery and Gough, 2003; Saio and Gautschy, 2004). By taking into account the effects of rotation and magnetic field, Bigot and Dziembowski (2002) generalized the oblique pulsator model, suggesting that the pulsation axis can be located anywhere between the magnetic and rotation axis.
Several models were suggested for the mechanism that drives the lowdegree highorder acoustic modes to the relatively low (up to 6 mmag) observed amplitudes (for a review see, e.g., Houdek, 2003). In the first theoretical paper on roAp stars by Dolez and Gough (1982), the authors assumed a strong dipolar magnetic field which inhibits convection totally in the polar spotlike regions, whereas in the equatorial region the convection is unaffected. The highorder acoustic oscillations are excited by the mechanism in the hydrogen layers of the radiative polar spotlike regions. This model was adopted by Balmforth et al. (2001) using updated opacity tables and the nonlocal, timedependent convection model by Gough (1977a,b). Depending on the assumed size of the polar spotlike regions Balmforth et al. (2001) did find overstable, highorder, axisymmetric dipole modes and other overstable modes with increasing spot size.
This encouraging result has led Cunha (2002) to model the instability strip for roAp stars, but the author concluded that the models cannot explain the presence of observed oscillations in the coolest roAp stars. Even if the metallicity is varied (Théado et al., 2009) the agreement between theory and observation could not be improved.
Dolez and Gough (1982) also addressed the question why the axisymmetric oscillations should always be nearly aligned with the spots, even if those spots are located near the (rotational) equator. They proposed that the (essentially standing) eigenmodes of oscillation suffer retrograde Coriolis precession in a frame of reference rotating with the star, and are therefore excited to observable amplitudes by the mechanism only if they are nearly aligned with the spots. A more detailed discussion on this model was recently presented by Gough (2012b).
The theory of roAp stars is further complicated by the still not fully understood mechanism that limits the pulsation amplitudes to the rather small values of ≲ 6 mmag, compared to the amplitudes of classical pulsators such as Cepheids or Delta Scuti stars. A possible explanation could be energy dissipation in the higher atmospheric layers brought about by shocking characteristics leading to steepening of the eigenfunctions which can then be thought of as a temporally harmonic series (Gough, 2013), with the high harmonics propagating farther out into the atmosphere where they dissipate the energy. Obviously, there is still much more to investigate in this type of stars.
6.3 Mode lifetimes in stars supporting solarlike oscillations
It is now generally accepted that stochastically excited oscillations are intrinsically damped.^{2} The excellent data from the Kepler spacecraft of solarlike oscillations in many distant stars have further strengthened this picture (e.g., Appourchaux et al., 2014). Nonadiabatic effects contribute, however, to the destabilization of stochastically excited modes, known as the “general kappamechanism” (Balmforth, 1992a), which are believed to be responsible, at least in part, for the local depression in the linear mode damping rates at an oscillation frequency near the maximum of the spectral mode heights in the Fourier power spectrum (see also discussion about Figure 10). Some early studies about solar mode stability discussed the possibility that stochastically excited modes could be overstable (Ulrich, 1970; Antia et al., 1988). This idea was reconsidered recently by Xiong and Deng (2013), but no convincing explanation was given by these authors about a mechanism that could limit the amplitudes to the observed values. If solarlike acoustic modes were indeed overstable some nonlinear mechanism must limit their amplitudes. The only convincing mechanism, reported until today, that could limit the growth of overstable modes is nonlinear mode coupling proposed by Kumar and Goldreich (1989). For the rather small amplitudes of stochastically excited oscillations only threemode coupling is important. Kumar and Goldreich (1989) studied the threemode coupling analytically and concluded that this nonlinear mechanism cannot limit the growth of unstable modes within the observed amplitude values. The remaining discussion on the properties of stochastically excited modes will therefore interpret the full width at half maximum (FWHM), or linewidth, of the spectral peaks in the Fourier power spectrum as (approximately) twice the linear damping rate, 2η, and τ:= η^{−1}, where η is in units of angular frequency, as the lifetime of the mode amplitude.
6.3.1 Solartype stars
Important processes that influence the thermal energy balance are nonadiabatic processes attributed to the modulation of the convective heat flux by the pulsation. This contribution is related to the way that convection modulates largescale temperature perturbations induced by the pulsations which, together with the conventional κmechanism, influences pulsational stability.
Current models suggest that an important contribution that influences the momentum balance is the exchange of energy between the pulsation and the turbulent velocity field through dynamical effects of the perturbed Reynolds stress. In fact, it is the modulation of the turbulent fluxes by the pulsations that seems to be the predominant mechanism responsible for the driving and damping of solartype acoustic modes. It was first reported by Gough (1980), using his local timedependent convection model of Section 3.2.3, that the dynamical effects, arising from the turbulent momentum flux perturbation δp_{t}, contribute significantly to the damping Γ = 2η. Detailed analyses by Balmforth (1992a), Houdek et al. (1999a), and Chaplin et al. (2005) revealed how damping is controlled largely by the phase difference between the momentum and density perturbations. Those authors used the nonlocal generalization (Section 3.3) of Gough’s convection model including consistently the Reynolds stresses (turbulent pressure) in both the equilibrium and pulsation calculations. Damping arising from incoherent scattering, η_{scatt}, (Goldreich and Murray, 1994, see Figure 8) was not modelled in these calculations.
The analysis by Dupret et al. (2004a) also included the pulsational perturbations of both the turbulent pressure and the convective heat in the pulsation computations using the local timedependent convection formulation by Grigahcène et al. (2005). The mean turbulent pressure in the hydrostatic equilibrium model was, however, omitted. Interestingly, Dupret et al. (2004a) found the perturbed convective heat flux δF_{c} as the main mechanism for making solar oscillations stable, similarly to the results found in Delta Scuti stars by Dupret et al. (2005c) (Section 6.2.3). The turbulent momentum flux perturbation δp_{t}, however, acts as a driving agent in these calculations. Obviously, turbulent pressure perturbations must not be neglected in stability analyses of solartype p modes.
Although both calculations provide similar results, the very physical mechanism for defining the frequencydependent function of the estimated linear damping rates is very different between these two calculations: whereas Dupret et al. (2004a, right panel of Figure 10) reports that it is predominantly the perturbed convective heat flux that stabilizes the solar p modes, the results from Chaplin et al. (2005, left panel of Figure 10; see also Figure 9) suggest that it is the perturbed turbulent pressure (Reynolds stress) that makes all modes stable.
Estimates of linear damping rates in other solartype stars were reported by Houdek (1996), Houdek et al. (1999a); Chaplin et al. (2009) and more recently by Belkacem et al. (2012). Houdek (1997) and Houdek et al. (1999a) discussed the frequencydependence of linear damping rates in mainsequence models with masses (0.9–2.0) M_{⊙}. Chaplin et al. (2009) discussed mean linear damping rates and linewidths around the maximum pulsation mode height in several solartype stars. Belkacem et al. (2012) compared linear damping rates at the maximum pulsation mode height with linewidth measurements from the CoRoT (Convection and RoTation) and Kepler space crafts.
Beside from testing and improving timedependent convection models, the comparison of damping rate estimates with measured linewidths may also provide general scaling relations for mode linewidths (or lifetimes) of solarlike oscillations. A first attempt was made by Chaplin et al. (2009), using a limited number of estimated damping rates and measured linewidths from predominantly groundbased instruments in solartype stars. The authors reported that the largest dependence of the linewidths is given by the star’s surface temperature and proposed the scaling relation \(\eta \propto {\Delta _{nl}} \propto T_{{\rm{eff}}}^4\). This scaling relation was challenged later by measurements from the highquality Kepler data. Appourchaux et al. (2012, see also Baudin et al., 2011) measured linewidths at both the maximum mode amplitude and mode height in 42 Kepler stars, supporting solarlike oscillations, and reported a steeper surfacetemperature dependence of \({\Delta _{nl}} \propto T_{{\rm{eff}}}^{13}\). Belkacem et al. (2012) compared these Kepler measurements with theoretical estimates, Γ = 2η, and reported reasonable agreement between observations and model computations (see left panel of Figure 11). Using a nonlocal generalization (Dupret et al., 2006c, see also Section 3.3.2) of Grigahcéne et al.’s timedependent convection model in the pulsation computations only, Belkacem et al. (2012) found a surfacetemperature dependence of \(\Gamma = 2\eta \propto T_{{\rm{eff}}}^{10.8}\), which is in reasonable agreement with the measurements by Appourchaux et al. (2012). The hydrostatic equilibrium models were constructed with the local mixinglength formulation of Section (3.4.1) neglecting the mean turbulent pressure p_{t} in the equation of hydrostatic support. Moreover, Belkacem et al.’s (2012) computations suggest that in the stability computations the main contribution to mode damping is now the turbulent pressure perturbation δp_{t}. The use of a nonlocal treatment of the turbulent fluxes, though still only in the pulsation computations and in a simplified manner (see Section 3.3.2), has changed the effect of δp_{t} from a driving agent in Grigahcène et al.’s local convection model to a damping agent in Belkacem et al.’s nonlocal stability analyses. The damping effect of δp_{t} is in accordance with the previously reported findings by Gough (1980), Balmforth (1992a), Houdek et al. (1999a), and Chaplin et al. (2005). In Belkacem et al.’s calculations a new strategy was adopted for selecting a value for the parameter \({\hat \beta}\) [see Eq. (107)]: it was calibrated such as to make the frequency of the local reduction (depression) in the linear damping rate η (see, e.g., Figure 9 for solar model) coincide with the frequency ν_{ max } at which the power in the oscillation Fourier spectrum is largest, using the linear scaling relation by Kjeldsen and Bedding (1995, see also Brown et al., 1991) between ν_{ max } and (isothermal) cutoff frequency.
Preliminary results by Houdek et al. (in preparation), using Gough’s (1977a,b) nonlocal convection model and with the Reynolds stresses consistently included in both the equilibrium and pulsation calculations, suggest a less steep surfacetemperature dependence of \(\Gamma = 2\eta \propto T_{{\rm{eff}}}^{7.5}\) (see right panel of Figure 11). The Kepler data suggest a steeper surfacetemperature dependence of about 13 (Appourchaux et al., 2012) in the considered temperature range 5300 < T_{eff} < 6400 K. It is, however, interesting to note that the observed mode linewidths may be affected by a shortperiodic (magnetic) activity cycle, which modulates (periodically shifts) the mode frequencies and thereby effectively augments the mode linewidths when measured over a period longer than the activity cycle. Possible evidence of such an effect was recently reported by R. García and T. Metcalfe (personal communication, see also García et al., 2010) for the Kepler star KIC 3733735 with a preliminary estimated effective linewidth broadening of up to a factor of two. If this is indeed the case, a substantial amount of active stars would then have smaller intrinsic linewidths than those plotted in Figure 11 by the bluefilled triangles, thereby bringing the observations closer to the model estimates (open rectangles) in the right panel of this figure as illustrated, for example, by the redfilled triangle for the Kepler star KIC 3733735. The remaining discrepancy between theory and observation indicate that most likely a physical mechanism is still missing in our current theory. One such crucial mechanism is incoherent scattering at the inhomogeneous upper boundary layer (Goldreich and Murray, 1994, see also Figure 8), which becomes increasingly more important for stars with higher effective temperatures (Houdek, 2012).
6.3.2 Redgiant stars
From the scaling relations for stochastically excited modes (e.g., Kjeldsen and Bedding, 1995; Houdek et al., 1999a; Houdek, 2006; Samadi et al., 2007, see also ChristensenDalsgaard and Frandsen, 1983) it is expected to observe such modes with even larger pulsation amplitudes in redgiant stars. First evidence of stochastically excited oscillations in redgiant stars were reported by Smith et al. (1987); Innis et al. (1988) and Edmonds and Gilliland (1996). A comprehensive review about asteroseismology of red giants was recently provided by ChristensenDalsgaard (2014).
Detailed structure modelling of ξ Hydrae was carried out by, e.g., Teixeira et al. (2003), who concluded that ξ Hydrae could either be in the ascending phase on the red giant branch or in the later phase of stable heliumcore burning, i.e., located in the socalled ‘red clump’ in the HertzsprungRussell diagram. Because the stable heliumcore burning phase lasts by far much longer than the ascending phase, it is more likely that ξ Hydrae is a ‘red clump’ star. Regardless of its detailed evolutionary phase, the model’s mean large frequency separation ∆ν:= 〈ν_{ n }_{+1l} − ν_{nl}〉 was identified to be similar to the frequency separation between two consecutive peaks in the observed Fourier power spectrum, i.e., identifying all the observed modes to be of only one single spherical degree, presumably of radial order. Supported by previous arguments by Dziembowski (1977b) and Dziembowski et al. (2001), ChristensenDalsgaard (2004) discussed qualitatively the possibility that all nonradial modes in redgiant stars are strongly damped and therefore have small amplitudes and peaks in the Fourier power spectrum. Adopting this idea, Stello et al. (2004, 2006) developed a new method for measuring mode lifetimes from various properties of the observed oscillation power spectrum and reported mode lifetimes of only about 2–3 days for the star ξ Hydrae. This is in stark contrast to the predicted values of 15–17 days by Houdek and Gough (2002, see Figure 12).
This discrepancy was resolved later by more detailed observations of redgiant stars. Spectroscopic observation of oscillation modes in redgiant stars by Hekker et al. (2006) reported first evidence of the presence of nonradial pulsation modes and Kallinger et al. (2008) reported possible nonradial oscillations in a redgiant star using data from the Canadian spacecraft MOST (Microvariability and Oscillations of STars). It was, however, the highquality data from the CoRoT satellite that showed clear evidence of nonradial oscillations in several hundreds of redgiant stars (De Ridder et al., 2009, see also Mosser et al., 2011) and later also from the NASA spacecraft Kepler (Huber et al., 2010). Lifetime measurements from these highquality space data provided values of about 15 days (Carrier et al., 2010; Huber et al., 2010; Baudin et al., 2011) which are in good agreement with the earlier predictions for radial modes by Houdek and Gough (2002).
7 Multicolour Photometry and Mode Identification
The quantities f_{ T } and ψ_{ T } can only be rigorously obtained from nonadiabatic computations. Mode identification methods based on multicolour photometric observations are thus model dependent. This is particularly important for stars with convective envelopes (e.g., δ Sct and γ Dor stars). For these stars the nonadiabatic predictions are very sensitive to the treatment of convection and its interaction with oscillations.
7.1 Delta Scuti stars
Mode identification based on multicolour photometry has been widely applied to δ Sct stars. First studies considered f_{ T } and ψ_{ T } as free parameters (e.g., Garrido et al., 1990). Later, nonadiabatic computations were performed but with a timeindependent convection treatment (frozen convection approximation), predominantly with local mixinglength models (Balona and Evers, 1999; DaszyńskaDaszkiewicz et al., 2003; Moya et al., 2004), and with Full Spectrum of Turbulence models (Montalbán and Dupret, 2007). The frozen convection approximation, however, is often not justified in δ Sct stars. Dupret et al. (2005a) used the local timedependent treatment of Grigahcène et al. (2005) to determine the nonadiabatic photometric observables in δ Sct stars and compared their theoretical results with the observations for several stars (see also Houdek, 1996, who used Gough’s (1977a) nonlocal timedependent convection model to predict the complex f quantity and phases in the δ Scuti star FG Vir). Dupret et al. (2005a) found that from the middle to the red border of the instability strip models with the timedependent convection treatment provide significantly different predictions for the photometric amplitudes and phases compared to models in which the perturbation of the turbulent fluxes were neglected (frozen convection). The largest differences are found for models with values for the mixinglength parameter α of the order of the solarcalibrated value. With the frozen convection and a large value for α a significant phase lag is obtained in the hydrogen ionization zone. This phase lag is related to the huge time variations of the temperature gradient in this region. Using models with timedependent convection, large variations of the entropy gradient (and thus the temperature gradient) are not allowed because of the control by the convective flux, and smaller phaselags in the hydrogen zone are predicted. Therefore, a timedependent treatment of the turbulent fluxes is required in the stellar model calculations for photometric mode identification in cooler δ Sct stars.
7.2 Gamma Doradus stars
Mode identification based on multicolour photometry can also be considered for γ Dor stars. Dupret et al. (2005b) showed that frozenconvection models give phaselags in complete disagreement with observations. Timedependent convection models give a better agreement with observations and are thus required for photometric mode identification. In frozenconvection models the κmechanism plays some role in the He and H ionization zones, implying the wrong phaselags. In timedependent convection models, the control by the convective flux does not allow significant phaselags inside the convective zone, which leads to a better agreement with observations. However, it must be mentioned that rotation through the action of the Coriolis force could affect significantly the geometry of the modes in α Dor stars, because their long pulsation periods are not much smaller than the rotation periods. Moreover, the Reynolds stress tensor perturbations, which were not included in Dupret et al. (2005b), could also significantly affect the nonadiabatic predictions. Hence, we must not be surprised that some disagreements between theoretical and observed amplitude and phases can be found when the effects of rotation and Reynolds stress perturbations are neglected.
8 Brief Discussion and Prospects
This review provides only a small cross section of the complex physics of how stellar pulsations are coupled to the convection and how simplified convection models can describe most of the relevant processes of this interaction. The discussions concentrated preliminarily on onedimensional (1D) modelling, yet we know that convection is an inherently threedimensional process, such as vortexstretching which is believed to be the major nonlinear mechanism for transferring turbulent kinetic energy from larger to smaller scales, at least within the socalled inertial range of the turbulent kinetic energy spectrum. Threedimensional (3D) hydrodynamical simulations do become more accessible now, thanks to the ever increasing calculation speed of modern computers, with many astrophysical applications such as modelling star formation, accretion disks, or supernovae explosions. In the context of stellar structure and dynamics, several 3D numerical codes are now available to simulate either the outer atmospheric stellar layers in a rectangular box (e.g., Stein and Nordlund, 2000; Wedemeyer et al., 2004; Trampedach et al., 2014a; Magic et al., 2015), conelike geometries (e.g., Muthsam et al., 2010; Mundprecht et al., 2015), or even the whole star (e.g., Elliott et al., 2000; Brun et al., 2011, 2014).
A promising approach today, and also for the near future, is the use of 3D simulation results in 1D stellar calculations. For example, an interesting approach is to replace the outer layers of 1D (equilibrium) model calculations by 3D simulations after applying appropriate averages (in space and time) to the 3D results. Such a procedure was adopted, for example, by Rosenthal et al. (1995, 1999) for estimating the socalled surface effects (see Section 5) on the solar acoustic oscillation frequencies, and is now being applied also to other stars by various research groups with interesting results to be expected soon. Another promising approach is to calculate a grid of 1D stellar atmospheric layers as a function of surface gravity and (effective) temperature obtained from properly averaged 3D simulations (Trampedach et al., 2014a,b). Here the atmospheric structure is provided as a T − τ relation (T being temperature and τ the frequencyaveraged optical depth) together with a calibrated value of the mixing length for a particular version of the mixinglength formulation. This allows a relatively simple integration of 3D simulation results into 1D stellar evolutionary calculations together with the selection of the correct adiabat in the deeper convectively unstable surface layers through the adoption of the 3Dcalibrated mixing length. A first application of this approach was recently reported by Salaris and Cassisi (2015). Mundprecht et al. (2015) used 2D hydrodynamical simulation results to constrain the convection parameters of a 1D nonlinear stability analyses of (shortperiod) Cepheid pulsations by adopting the 1D timedependent convection model by Kuhfuß (1986). The interesting outcome of this simulation study is that constant assumed convection parameters in 1D models can vary up to a factor of 7.5 over the pulsation cycle. The 2D simulation results can then be included in the 1D nonlinear stability computations by varying the (otherwise constant assumed) 1D convection parameters over the pulsation cycle according to the simulations. Hydrodynamical simulations of pulsations in classical variable stars were also conducted by Geroux and Deupree (2013). These are a few of the examples that point towards the direction in which this complex field of convectionpulsation interaction is heading.
Although the move to 3D hydrodynamical simulations for describing stellar convection is the most promising path to go, we must remain aware of its current shortcomings. Firstly, the spatial dimensions of stellar simulations in a box are typically of the order of 10 Mm (e.g., Trampedach et al., 2014a), which therefore makes it difficult to describe the coupling of turbulent convection to oscillation modes of the lowest radial order. Secondly, the typical Reynolds numbers R_{e} ≃ 10^{12} in stars suggest that the ratio \({l_{\rm{L}}}/{l_\eta}\sim R_{\rm{e}}^{3/4}\) of the largest (l_{L}) to the smallest (l_{ η }) scales would require a total meshpoint number N ≃ 10^{27} in the 3D numerical simulations. With today’s super computers, however, the maximum achievable number of meshpoints N_{max} ≃ 10^{12}, and is therefore some 15 magnitudes too small for what is required to resolve all the turbulent scales of stellar convection. Consequently, only a very limited range of scales are resolved by today’s 3D hydrodynamical simulations, which are therefore called largeeddy simulations or just LES. LES require socalled subgrid models for describing the dynamics of the numerically unresolved smaller scales of the turbulent cascade. Various models are available. The most commonly used models are hyperviscosity and Smagorinsky models. All subgrid modes assume that the turbulent transport is a diffusive process. Hyperviscosity models, for example, use higher derivatives, a purely mathematical device, for the diffusion operator in the momentum equation, thereby extending the inertial range, which also leads to a better representation of the dynamics of the larger scales. More detailed discussions on largeeddy simulations and subgrid models in the context of stellar convection can be found in, e.g., Elliott (2003) and Miesch (2005). It is also important to note that the Prandtl number in 3D simulations is currently about 0.01–0.25 (e.g., Elliott et al., 2000; Miesch et al., 2008). It is therefore substantially larger than the Prandtl number in solartype stars.
3D hydrodynamical simulations are the best tools we currently have at hand for extending our knowledge of stellar convection and for calibrating semianalytical 1D convection models. 1D models will still be necessary, for many years to come, for stellar evolutionary calculations and for both linear and nonlinear stability analyses of stellar pulsations.
Footnotes
Notes
Acknowledgements
We thank Jørgen ChistensenDalsgaard and Douglas Gough for many helpful discussions. GH is grateful to Neil Balmforth for providing a running version of the computer programme for Gough’s nonlocal mixinglength model. We also thank the two anonymous referees for their very useful comments which improved the manuscript substantially. Funding for the Stellar Astrophysics Centre is provided by The Danish National Research Foundation (Grant agreement no.: DNRF106). The research is supported by the ASTERISK project (ASTERoseismic Investigations with SONG and Kepler) funded by the European Research Council (Grant agreement no.: 267864).
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