OVERSHOOT AT THE BASE OF THE SOLAR CONVECTION ZONE: A SEMIANALYTICAL APPROACH

Transcription

1 The Astrophysical Journal, 607: , 2004 June 1 # The American Astronomical Society. All rights reserved. Printed in U.S.A. OVERSHOOT AT THE BASE OF THE SOLAR CONVECTION ZONE: A SEMIANALYTICAL APPROACH M. Rempel High Altitude Observatory, Advanced Study Program, National Center for Atmospheric Research, 1 P.O. Box 3000, Boulder, CO 80307; rempel@hao.ucar.edu Received 2003 September 8; accepted 2004 February 16 ABSTRACT Despite the importance of overshoot at the base of the solar convection zone for the storage of strong toroidal magnetic field produced there by the solar dynamo, uncertainties concerning the depth and mean subadiabatic stratification remain large. Overshoot models based on the nonlocal mixing-length theory generally produce a shallow, weakly subadiabatic region with a sharp transition to the radiative interior, whereas several numerical simulations lead to significantly subadiabatic overshoot with penetration depth of more than a pressure scale height. We present a semianalytical convection zone/overshoot region model based on the assumption that the convective energy flux is governed by coherent downflow structures starting at the top of the domain and continuing all the way down into the overshoot region, which allows for modeling both the parameter regime addressed by nonlocal mixing-length approach and the regime addressed by numerical simulations. It turns out that the main differences between the nonlocal mixing-length approach and numerical simulations (nearly adiabatic vs. strongly subadiabatic overshoot) are caused by the much larger energy flux used in numerical simulations as a consequence of larger thermal diffusivities required by numerical constraints. The depth of the overshoot region is determined predominantly by the mixing between downflows and upflows in the convection zone. Furthermore, our model shows that the sharp transition between the nearly adiabatic overshoot and radiative interior, a typical result of the nonlocal mixing-length approach, can be avoided by assuming an ensemble of downflows with different strength. Subject headings: convection Sun: helioseismology Sun: interior 1. INTRODUCTION A proper description of convective overshoot at the base of the solar convection zone is crucial for understanding the structure of the solar tachocline and solar magnetic activity (Spruit & van Ballegooijen 1982; Ferriz-Mas & Schüssler 1993; Rempel et al. 2000; Rempel & Dikpati 2003; Dikpati et al. 2003). Even though there have been many attempts in the past to model the solar overshoot region, there is to date no general agreement among these models. Early models of van Ballegooijen (1982), Schmitt et al. (1984), and Pidatella & Stix (1986) showed very similar features, such as a nearly adiabatic stratification, a steep transition toward the radiative zone, and an overshoot depth of 20% 40% of a pressure scale height. At least for these models the results looked very robust, since the three models assumed completely different flow structures. Whereas the model of Pidatella & Stix (1986) was based on the nonlocal mixinglength theory (MLT) proposed earlier by Shaviv & Salpeter (1973), van Ballegooijen (1982) assumed overturning convective rolls, and Schmitt et al. (1984) based their model on explicit modeling of plumes in the overshoot region. During the last two decades the overshoot problem has been addressed also by direct numerical simulations of fully compressible convection in a local domain with Cartesian geometry (Roxburgh & Simmons 1993; Hurlburt et al. 1994; Singh et al. 1995, 1998; Saikia et al. 2000; Brummell et al. 2002). The main emphasis of these studies was to verify scaling relations of the overshoot depth with the stiffness of the 1 The National Center for Atmospheric Research is sponsored by the National Science Foundation subadiabatic stratification and the downflow velocity at the base of the convection zone that were found previously in the model of Schmitt et al. (1984) and an analytical approach of Zahn (1991). Whereas the approach of Roxburgh & Simmons (1993) and Hurlburt et al. (1994) was two-dimensional, Singh et al. (1995, 1998) and Saikia et al. (2000) used threedimensional models; however, only the most recent approach of Brummell et al. (2002) can be considered as a marginally turbulent three-dimensional simulation, as a result of the larger numerical resolution possible. Except for the latter, all other studies found, depending on their parameters (mainly the stiffness of the subadiabatic layer), both nearly adiabatic overshoot and extended subadiabatic overshoot. (In this paper we do not follow the nomenclature using penetration for the nearly adiabatic overshoot, since at least for our model the physics behind the overshoot process are independent of the value of the subadiabaticity finally reached on average.) In contrast to this, the solutions found by Brummell et al. (2002) show only strongly subadiabatic overshoot with a depth of 40% 200% of a pressure scale height and a smooth transition toward the radiation zone. The physical reason for the discrepancy between the different approaches, including the nonlocal mixing-length approach mentioned earlier, is still unclear. On the one hand, simplified models include some crude assumptions, as pointed out for the MLT by Renzini (1987); on the other hand, numerical simulations are not solar-like, since numerical constraints require unrealistic parameters. As a result of the large thermal conductivity required typically for numerical stability, the total energy flux exceeds the solar energy flux by orders of magnitude, leading to a very vigorous convection. Thus, discrepancies could be a consequence of crude approximations in

2 OVERSHOOT AT BASE OF SOLAR CONVECTION ZONE 1047 simplified models or introduced by the unrealistic parameter range in numerical simulations. In this paper we present a semianalytical model, which encompasses elements found in simplified models like the MLT approach, as well as properties found in numerical simulations. As a consequence, we are able to study the full parameter range from solar-like to the values typical for numerical simulations. We want to emphasize two papers dealing with the solar overshoot problem, since their approach is closely related to our work. Schmitt et al. (1984) used a semianalytical model for solar overshoot in which they use a set of equations that describe the properties of downflow plumes for a given hydrostatic reference state. They allow for mass exchange between the plumes and the surrounding upflow region using an entrainment model and use Gaussian profiles for describing the internal structure of the plumes. They solve their equations starting at the base of the convection zone by iteratively adjusting the reference state until the total energy flux is conserved. For the initial values of their integration at the base of the convection zone they choose values close to the predictions of mixing-length approaches, meaning velocities of the order of 100 m s 1 and filling factors in the range Their results show typically nearly adiabatic overshoot with a depth around 0.3 pressure scale heights. Besides differences in the treatment of entrainment and the assumptions concerning the internal structure of plumes (for details see the following section), we apply our model to the whole convection zone and overshoot region. Furthermore, we do not restrict the parameter range of our model to conditions close to the mixing-length results, which allows modeling situations that are closer to the results of numerical simulations. We incorporate the momentum equation for the upflow region so that our model is not restricted to applications with small downflow filling factors. Our model also allows for a distribution of downflows with different strength, which is closer to what is expected for time-dependent convection. Zahn (1991) used analytically derived scaling relations in order to estimate the depth and the departures from adiabaticity for overshoot beneath stellar convection zones and above convective cores. Close relations to their approach can be found in x 3, where we also use scaling relations and estimates for discussion of our model results. The main results of our model are in full agreement with the analysis of Zahn (1991). The paper is structured as follows: In x 2 we discuss the model assumptions, derive the relevant equations, and discuss the free model parameters. Section 3 is subdivided into four parts. We first present general solution properties in x 3.1 and focus then on single downflow models in x 3.2. After a discussion of the properties of ensemble models (models with a distribution in the downflow strength) with special emphasis on the steepness of the transition region at the base of the overshoot region in x 3.3, we compare our model to the numerical overshoot simulations of Brummell et al. (2002) in x 3.4. In x 4 we give a summary of the main results of this model and discuss them in x 5 in the context of solar overshoot and numerical overshoot modeling. 2. MODEL Numerical simulations of convection in a strongly stratified medium show, as a very robust feature, strong downflows mainly driven by a thin surface boundary layer. Plume-dominated convection was also proposed for the solar convection zone by Spruit (1997) as an alternative to the classical MLT approach. The simple structure of plume-dominated convection allows the construction of a semianalytical convection/overshoot model. Even though we are mainly interested in the overshoot beneath the convection zone, we derive a model that also treats the whole convection zone. This is necessary because of the high degree of nonlocality introduced by coherent downflow plumes. We give in the following a summary of the basic assumptions of our model and discuss the justification for parameterizations and simplifications used. Our goal is here to derive a simplified model that gives enough insight into the basic physics of the problem rather than a highly sophisticated convection/zone overshoot model addressing detailed problems. The basic assumptions of this model can be summarized as follows: 1. The convection is dominated by strong downflows starting in a surface boundary layer and continuing down into the overshoot region. 2. The downflows are surrounded by a broad upflow region. Downflows exchange momentum and energy with this upflow region as a result of entrainment and mixing. 3. The velocity of the downflow is determined by buoyancy driving and braking resulting from a mixing of momentum between upflow and downflow regions. 4. The enthalpy of the downflow is given by a balance between advection, radiative heating, heating caused by dissipation of kinetic energy resulting from mixing of momentum between upflow and downflow regions, and heat exchange resulting from a mixing of enthalpy between upflow and downflow regions. 5. There is no interaction between individual downflows. 6. Downflows and the surrounding upflow region are in horizontal pressure balance. The horizontal pressure balance is an assumption we have to make in order to be able to construct a simple one-dimensional downflow model. 7. The downflows and the surrounding upflow region have no internal structure. Material entering from the upflow region mixes completely with the downflow. This assumption is one of the shortcomings of the model, and we discuss it later in more detail. 8. The downflow filling factor is prescribed rather than found by solving an entrainment model. Whereas assumptions 1 4 are rather general and supported by numerical simulations of compressible convection, assumptions 5 8 are chosen in order to keep complexity of the model in a range that can be addressed by a semianalytical approach. Assumption 5 affects the convection mostly near the surface, where the filling factor of downflows is not small compared to 1 so that interactions between downflows cannot be neglected. It is nevertheless no major restriction for the deeper convection zone where the filling factor of the downflows becomes significantly smaller so that the separation between downflows is much larger than their diameter. The same applies to assumption 6, since near the surface horizontal and vertical motions are of the same magnitude. In the bulk of the convection zone the vertical velocity dominates so that the horizontal momentum balance can be neglected. Assumption 7 is a major simplification, since numerical simulations (see, e.g., Brummell et al. 2002) show that downflows consist of a strong core and a mixing region surrounding the core with a larger filling factor. Since we do not allow

3 1048 REMPEL Vol. 607 for an internal structure, we have to assume that the material entering the downflow from the upflow region mixes completely. Nevertheless, our model is capable of describing an ensemble of downflows with different strength, which mimics to some extent also an internal downflow structure. The main difference is that the single downflows of the ensemble solution only interact with the upflow region, whereas in a structured downflow each layer interacts with the next adjacent layer. Numerical simulations suggest also an internal structure for the upflow, especially in the overshoot region, where downflows are surrounded by fast upflows containing high-entropy material detrained from the downflow. Our model can be generalized by including both the internal downflow and upflow structure, but we restrict the discussion to the simpler approach as outlined above, since each of these improvements also increases the number of free parameters. Since many of these additional parameters cannot be determined from first principles, it would be required to either make a thorough parameter study or perform numerical simulations to constrain these additional degrees of freedom. In our model we prescribe the filling factor as a function of depth rather than solving an entrainment model, since there is no generally accepted model for highly stratified compressible convection. It is unclear if models used successfully for upward plumes in the Earth atmosphere with a very moderate density stratification can be applied to downward plumes in the solar convection zone with a density contrast of 10 6.We therefore decided to prescribe the filling factor for which there is some guidance from numerical simulations Basic Equations The depth z ¼ 0 is the upper boundary, and z increases downward into the convection zone. Gravity is assumed to be constant, and we normalize the pressure, temperature, and density with the values at the base of the convection zone (defined by 9 rad ¼ 9 ad,where9¼dln T=d ln p; 9 ad is the logarithmic temperature gradient for an adiabatic stratification, and 9 rad is the logarithmic temperature gradient for a stratification in radiative flux balance). In these units the heat capacity is normalized by R= ¼ p bc =(% bc T bc ), leading to a dimensionless value of c p ¼ =( 1) ¼ 1=9 ad ¼ 2:5 (with ¼ 5=3), where the index bc denotes the values at the base of the convection zone. The unit of the velocity is the isothermal sound speed at the base of the convection zone ( p bc /% bc ) 1/2,and the length unit is the pressure scale height at the base of the convection zone p bc /(% bc g bc ), leading to a dimensionless value for the gravity of g ¼ 1. The equation of state reduces to p ¼ %T. For the energy flux we use the dimensionless measure ¼ p bc F tot bc ; ð1þ 1=2 ðp bc =% bc Þ which leads to for solar values (F tot bc 1:2 108 W m 2, p bc Pa, and % bc 200 kg m 3 ). The properties of the downflows (indexed with a subscript d in the following) and upflow region (indexed with a subscript u in the following) follow from the momentum and energy equation including the effects of mixing as outlined in Figure 1. In our model we compute the downflows as stationary flows, considering in the momentum equation a balance between advection, buoyancy driving (and braking), Fig. 1. Model assumptions about the mixing process between down- and upflows. A mass flux m e enters the downflow with upflow properties and mixes with the downflow mass flux m d.onlyanamountofm d remains in the downflow, whereas the mass flux m e m d returns with downflow properties to the upflow and mixes there. We have the constraints m e 0and m e m d. pressure gradient force, and mixing of momentum with the upflow. In order to apply the equations of hydrodynamics to our plume geometry and include the mass exchange between the different regions, it is advantageous to start with the conservative formulations for the time-independent momentum and energy equations: div ð%vvþ ¼ %g grad p; ð2þ div ½vðe þ pþš ¼ %v = g þ H rad ; ð3þ where e is the total energy density including internal and kinetic energy density e ¼ p 1 þ % v2 ð4þ 2 and H rad ¼ div F rad is the radiative heating term. The energy fluxcanbeexpressedas p vðe þ pþ ¼ v% 1 % þ v2 2 ¼ m c p T þ v2 2 where we used the mass flux m ¼ %v: With the definition ; ð5þ " ¼ c p T þ v 2 ð7þ 2 we can write the vertical momentum and energy balance in the form ð6þ div ðmv z Þ ¼ %g dp dz ; ð8þ div ðm" Þ ¼ %v z g þ H rad : ð9þ

4 No. 2, 2004 OVERSHOOT AT BASE OF SOLAR CONVECTION ZONE 1049 For a fully consistent model we have to solve these two equations for the downflows and the upflow region. Alternatively, we can solve the momentum equation for a reference state including corrections due to the dynamical pressure of the downflows and upflow region and solve momentum and energy equation for the downflows. As we outline later, it is not necessary to solve an energy equation for the upflow region since the set of equations can be closed using the total energy flux balance considering convective and radiative fluxes. In the following we derive a convection zone/overshoot region model based on our assumptions 1 8, which allows us to describe an ensemble of downflows with different strength. For this purpose we have to index the downflow quantities by an additional number that counts the downflows in the ensemble, e.g., v d (z; i) for the downflow velocity. For reasons of clarity we drop the dependence on z in our notation in most cases and just use v d (i) orusev d (z) ifthez dependence needs to be emphasized more (similar with all other variables) Reference State The downflows and upflow region are in mass flux balance given by m tot d ¼ XN m d (i) ¼ XN f d (i)% d (i)v d (i) " # ¼ m u ¼ 1 XN f d (i) % u v u ; ð10þ with the total downflow mass flux m tot d and the number of different downflows N. In equation (10) f d (i), % d (i), and v d (i) denote the filling factor, density, and velocity for individual downflows, respectively. The mass flux of an individual downflow is given by m d (i) ¼ f d (i)% d (i)v d (i). Equation (10) leads to a relation between upflow and downflow velocity of the form with (i) ¼ v u ¼ XN (i)v d (i); ð11þ f d (i) 1 P % d (i) N k¼1 f : ð12þ d(k) % u We want to emphasize that N is the number of downflows with different properties. Thus, N ¼ 1 means that the energy is transported by downflows with identical properties, but it does not necessarily imply that there is only one downflow in the whole convection zone. For computing the total convective energy flux, only the total filling factor of the downflows with identical properties matters, and not the distribution among individual downflows with the same strength. We derive an equation for the reference state (indexed in the following with a subscript m ) by taking the horizontal average of equation (8). Assuming periodicity in the horizontal direction, the horizontal components of the divergence operator cancel, leading to dp m dz ¼ % mg d dz p dyn; ð13þ wherethedynamicpressurep dyn is defined by p dyn ¼ XN ¼ XN m d (i)v d (i) m tot d v u m d (i) þ (i)m tot d vd (i) ð14þ and the mean density by " # % m ¼ XN f d (i)% d (i) þ 1 XN f d (i) % u : ð15þ In the following we use the dimensionless dynamic pressure correction ¼ 1 dp dyn ð16þ % m g dz and the reference state temperature defined by T m ¼ p m =% m, leading to, together with equation (15) and assumption 6 of horizontal pressure balance [ p m ¼ p d (i) ¼ p u ], the relation 1 T m ¼ XN " # f d (i) XN 1 þ 1 f d (i) : T d (i) T u ð17þ This relation can be used to eliminate the dependence on T u in equation (12), leading to (i) ¼ f d (i) ½T m =T d (i) Š 1 P N k¼1 f d(k) ½T m =T d (k) Š : ð18þ The upflow temperature can be computed from the downflow and reference state temperatures using T u ¼ T m XN (i) ½T d (i) T m Š: ð19þ For solving equation (13) we need to specify an additional equation for T m. With the definition 9 ¼ 9 ad þ ¼ p m T m dt m dp m ð20þ (with 9 ad ¼ 0:4 and the superadiabaticity ) the reference state follows from an integration of the system dp m dz ¼ p m g(1 ); T m ð21þ dt m dz ¼ g ð 9 ad þ Þ(1 ): ð22þ As we show later, the whole system of equations can be written as an implicit equation of the form F() ¼ 0, which will be solved iteratively and thus will provide the superadiabaticity self-consistently Momentum Flux Balance To derive an equation for the vertical velocity from equation (8), we need to incorporate the mixing model described in Figure 1 in order to express the horizontal components of

5 1050 REMPEL Vol. 607 the divergence operator in terms of known quantities. We allow a mass exchange between up- and downflow regions in both directions. Entrainment of upflow material into a downflow changes the properties of the downflow by mixing of enthalpy and momentum, whereas the opposite process, leaking of downflow material into an upflow, changes the upflow properties butkeeps thedownflow properties unchangedexcept for thetotal mass flux of the downflow. To describe both processes, we define two quantities: the quantity m e measures the total amount of mass flux entering the downflow within the depth interval z, whereas m d measures the change in the mass flux of the downflow. Thus, m e m d is the mass flux returning back to the upflow region. In order to apply the mixing model described in Figure 1, we use the integral form of equation (8) (theorem of Gauss) and express the divergence by the surface integral of the momentum flux over the mixing region between z and z þ z (for reasons of clarity we drop the index i for the different downflows): ½m d (z) þ m e (z) Šv d (z þ z) m d (z)v d (z) ¼ % d (z)g dp m f d (z)z þ m e (z)v u (z); dz ð23þ where the downflow filling factor f d is related to the mass flux by m d ¼ f d % d v d : ð24þ In equation (23) we used the horizontal pressure balance to express p d by p m. Since in this step the horizontal mass exchange is expressed by the variations of the vertical mass flux component (m d ) and the total entrainment (m e ) with depth, the three-dimensional problem is reduced to a one-dimensional problem in z. The first term on the left-hand side of equation (23) represents the momentum flux leaving the mixing region at z þ z, and the second term is the momentum flux entering at z. The first term on the right-hand side represents local momentum change due to the buoyancy force, whereas the second term on the right-hand side is the momentum entering from the upflow region and thus represents the horizontal exchange between downflows and upflows. The amount of momentum that returns back to the upflow region after the mixing does not change the downflow velocity and need not be included in equation (23). This effect will be considered by adjusting the downflow mass flux accordingly (m d m e ). As already mentioned in assumption 7, this treatment assumes a complete mixing with the fluid entering from the upflow, before a fraction of the downflow material returns back into the upflow region. In the limit z! 0 equation (23) is equivalent to the differential equation (using again the index i for addressing different downflows) m d (i) dv d(i) dz ¼ % d (i)g dp m f d (i) dm e(i) dz dz which can be written as ½v d (i) v u Š; ð25þ d vd 2(i) dz 2 ¼ % d(i)g dp m 1 dz % d (i) v d(i) ½v d (i) v u Š ; ð26þ r u:d (i) where the relaxation length r u:d (i) causedbyfluidentering from region u to d is defined by 1 r u:d (i) ¼ 1 m d (i) dm e (i) : ð27þ dz The first term on the right-hand side of equation (26) describes the buoyancy driving and braking of the downflow, whereas the second term represents the reduction of the kinetic energy due to mixing of momentum with the upflow material. Since the mixing process is based on conservation of momentum, this term also addresses implicitly the dissipation of kinetic energy required by this mixing process. From the mixing model described in Figure 1, r u:d (i) is not completely independent of m d (i), since we have the constraint m e (i) m d (i) leading to 1 r u:d (i) 1 m d (i) dm d (i) ; r u:d (i) > 0: ð28þ dz This constraint is fulfilled with the following approach: where we defined 1 r u:d (i) ¼ 1 þ 1 d (i)h p m d (i) ½Š x >0 ¼ x x > 0 0 x 0 and used the pressure scale height H p ¼ p m % m g ¼ T m g : dm d (i) dz ; ð29þ >0 ð30þ ð31þ In the case of no entrainment the relaxation length is given by r u:d (i) ¼ d (i)h p, which corresponds to local downflow/upflow mixing. Following the mixing-length approach, we set the length scale associated with this process proportional to the pressure scale height introducing a mixing parameter d (i). Typical values used in the mixing-length approach are in the range 1 2 (Pidatella & Stix 1986; Skaley & Stix 1991). The second term on the right-hand side of equation (29) is the mixing from entrainment caused by an increase of the downflow mass flux, which is the reason for considering this term only if dm d (i)=dz > 0. As already mentioned, detrainment does not influence the relaxation length, since detrainment does not cause an unphysical inverse mixing process. Downflow detrainment only changes the downflow mass flux and returns material to the upflow region, which mixes there in a complimentary mixing process. It can be shown that the resulting equation for the upflow velocity is similar to equation (26) after the exchange u $ d, changing the sign of the relaxation term (upflow velocity in negative z direction) and adding a sum over all exchange terms with the downflows (whereas each downflow interacts only with one upflow region, the upflow region gets the contributions of all different downflows). The complimentary relaxation lengths for the downflow material entering the upflow region are given by 1 r d:u (i) ¼ m d(i) m tot d 1 r u:d (i) 1 m d (i) dm d (i) dz : ð32þ

6 No. 2, 2004 OVERSHOOT AT BASE OF SOLAR CONVECTION ZONE 1051 We do not have to solve the momentum equation for the upflow, since it is already considered implicitly in equation (21) Energy Flux Balance Because of the similar mathematical form of the energy and momentum balance, we can apply the mixing model in a similar way to the energy equation by substituting in equation (26) v d (i) with" d (i), v u with " u, and substituting the source terms d" d (i) dz ¼ g þ H rad % d (i)v d (i) " d(i) " u ; ð33þ r u:d (i) where the radiative heating is given by H rad ¼ d dt m : ð34þ dz dz Using the reference state temperature for computing the radiative heating is an approximation that does not introduce much error, since the relative difference between T m and T d does not exceed a few percent and is significantly less for most of the models we discuss. So far we have computed only the properties of the downflows and the reference state. Since the dynamic pressure correction for the reference state implicitly already includes the momentum equation for the upflow, the energy equation for the upflow is the only equation not addressed yet. Instead of solving this equation directly, we use to total energy flux balance F tot ¼ dt m dz þ XN m d (i) ½" d (i) " u Š ¼ : ð35þ Differentiation of equation (35) leads, together with equation (33), to d" u dz ¼ g þ H rad % u v u þ XN " u " d (i) ; ð36þ r d:u (i) where the relaxation length r d:u (i) caused by material returning from the downflow i to the upflow region is already defined in equation (32). Note the opposite sign of this term compared to equation (33), which arises from the upflow velocity in the negative z direction Summary of Relevant Equations For the iterative numerical solution it is advantageous to subtract the reference state quantities from " d (i) and" u since the convective flux is determined by small deviations. To this end we define and express all equations in terms of " m ¼ c p T m ð37þ " d (i) ¼ " d (i) " m ; ð38þ " u ¼ " u " m : ð39þ Inserting the pressure gradient of the reference state equation (21) into equation (26) and using the horizontal pressure balance to express the dependencies on density by p m, T d (i), and T m, the full set of model equations can be written as a set of 2N þ 2 coupled ordinary differential equations (where N is the number of downflows considered), dp m dz ¼ p m g(1 ) ð40þ T m dt m dz ¼ g ð 9 ad þ Þ(1 ) ð41þ d vd 2(i) dz 2 ¼ 1 T d(i) g þ T d(i) g v d(i) ½v d (i) v u Š ð42þ T m T m r u:d (i) d" d (i) ¼ T d(i)h rad " d(i) " u þ g gc p (1 ); dz p m v d (i) r u:d (i) ð43þ a set of additional relations T d (i) ¼ T m 1 þ " d(i) c p T m v2 d (i) 2c p T m ; ð44þ H rad ¼ d ½ dz g ð 9 ad þ Þ(1 ) Š; ð45þ m d (i) ¼ f d (i)v d (i) p m T d (i) ; 1 r u:d (i) ¼ g þ 1 d (i)t m m d (i) (i) ¼ v u ¼ XN f d (i) ½T m =T d (i) Š dm d (i) dz ð46þ ; ð47þ >0 1 P N k¼1 f d(k) ½T m =T d (k) Š ; ð48þ T u ¼ T m XN (i)v d (i); ð49þ (i) ½T d (i) T m Š; ð50þ " u ¼ c p ðt u T m 2 ; ð51þ " # ¼ T m d X N X N m d (i)v d (i) v u m d (i) ; ð52þ p m g dz Þþ v2 u and the constraint for the total energy flux F tot () ¼ dt m dz þ XN m d (i) ½" d (i) " u Š ¼ ; ð53þ which is an implicit equation for the unknown superadiabaticity of the reference state Boundary Condition For solving the differential equations (40) (43) we have to specify boundary values at z ¼ 0: T m (0) ¼ T 0 ; ð54þ p m (0) ¼ p 0 ; ð55þ " d (0) ¼ 0; ð56þ v d (0) ¼ 0: ð57þ The boundary values for the reference state T 0 and p 0 are determined iteratively in such a way that the values of T m and p m at the base of the convection zone stay the same. This minimizes the changes in the reference state at the base of the

7 1052 REMPEL Vol. 607 convection zone and makes it easier to compare different solutions with each other; it is also more consistent with the fixed radiative conductivity profile we use (see x 2.4). In contrast to this, many numerical models fix the temperature at the top and the surface value of the pressure follows indirectly from mass conservation. Fixing the temperature at the top forces the whole domain to adjust if the convection requires a large superadiabaticity near the top. By fixing the temperature at the base of the convection zone, we allow for this adjustment by lowering the surface temperature while leaving the largest fraction of the domain unaffected. The difference between fixing the density at the base of the convection zone and conserving the total mass turns out to be insignificant. Physically there is no preference for either, since the mass conservation only applies to the whole star and not just to the outermost layers; in addition, the radial position of the base of the convection zone is not fixed, since the whole star adjusts to any change introduced in the outer layers. This ambiguity can only be resolved by solving a fully self-consistent stellar model, which is beyond the scope of this paper Iterative Solution Given the parameters, (i), and f d (i) and as function of z, the system of equations (40) (53) defines an implicit solution for the superadiabaticity given by F tot () ¼. After a discretization of ¼f 1 ::: n g and F tot ¼fF tot1 :::F totn g, the resulting n-dimensional system of nonlinear equations can be solved by a Newton-Raphson iteration. Since the iteration converges only if the initial guess is close to the solution, another iterative scheme is used in the beginning: nþ1 ¼ n þ " 1 F totð n Þ ; ð58þ where " is a relaxation parameter that needs to be chosen as a compromise between convergence and relaxation speed. The physical idea behind this iteration is to change the superadiabaticity based on the flux mismatch. Even though the convective flux is highly nonlocal for plume-dominated convection, the local relationship between a change of and the flux mismatch expressed by equation (58) leads to fast convergence. Using values for, r u:d (i),, v u,and" u from the previous iteration step, we solve equations (40) (43), where we already use the relations given by equations (44) and (45) for the temperature ratios and radiative heating. From the new reference state and downflow quantities we compute updates for r u:d (i),, v u,and" u using the relations given by equations (46) (52). Using equation (53), together with equation (58), we compute the new superadiabaticity and start the next iteration step. With this procedure there are two terms that introduce singularities at locations of vanishing velocity v d (i), at the surface z ¼ 0 and in the overshoot region, where v d (i) approaches zero again. These terms are the expression for the relaxation length given by equation (47) and the radiative heating term in equation (43). To avoid these singularities, we compute these terms in the region in between with v d (i) > 0 and use a linear extrapolation for values at the boundaries. We want to emphasize that these terms are not real physical singularities. The relaxation term in the momentum equation (42) vanishes at the boundary, since the velocity itself vanishes there, even though r u:d diverges. The heating term is more complicated, since both H rad and v d vanish at the boundary, but the ratio of the two terms remains regular. Nevertheless, these terms introduce numerical problems, which require the treatment described above Free Parameters To solve the system of equations given by equations (40) (53), together with the initial conditions given by equations (54) (57), we must specify f d (i) and as functions of depth. Additional parameters are the total imposed flux and the mixing parameters d (i). The specification of also defines the initial state. Since the radiative conductivity determines the imposed energy flux, we can write as (z) ¼ g bc 9 ad 0 (z); ð59þ where 0 (z) is a dimensionless profile function with 0 (z bc ) ¼ 1. Since g bc 9 ad is the adiabatic temperature gradient (without dynamic pressure correction), the imposed energy flux can be transported by radiation with an adiabatic temperature gradient at locations with 0 (z) ¼ 1. Values of 0 (z) > 1lead to a subadiabatic stratification, whereas 0 (z) < 1 requires a superadiabatic stratification and thus lead to convective instability Radiative Conductivity For the radiative conductivity profile, 0 (z), we use (the index ph refers to values at the top of the domain photosphere ) 2 0 (z) ¼ 1 þ exp½ðz z bc Þ=d bc Š ; " # z 2 0 (z) ¼ ph 0 (z) exp þ 0 (z); ð60þ where we enlarge 0 at the upper boundary in such a way that the energy flux can be transported by a nearly adiabatic temperature gradient. The value of ph is chosen in such a way that for the relaxed solution ph ¼ 0:5max(). This ensures a superadiabatic gradient at the top required for driving the downflows. The parameters d ph and d bc determine the width of the transition regions at the upper boundary photosphere and at the bottom of the convection zone, respectively. We have chosen 0 in the deep interior in such a way that 9 rad approaches 0.2, similar to solar values. We prefer to use a fixed 0 as a function of z compared to the solar radiative conductivity depending on temperature and density, since this dependence can only be described consistently in a full stellar model, which is beyond the scope of this study. The main goal of this model is to understand the differences between different approaches (including numerical simulations using step function like profiles of ) rather than constructing a detailed solar model. For the results shown below we use the parameters d ph ¼ 0:2 andd bc ¼ 0:5. The value of d ph has no significant influence on the solution as long as it is small compared to the depth of the convection zone. For d bc we used a value leading toaprofileof9 rad comparable to the Sun. InFigure2weshowtheinitialstate(temperaturegradient and radiative energy flux) obtained from equation (60), together with equations (40) and (41), using ¼ 0. For the resulting stratification we have 9 rad < 9 ad for z > 2 and d ph

8 No. 2, 2004 OVERSHOOT AT BASE OF SOLAR CONVECTION ZONE 1053 values could result from downflows, which are driven by radiative cooling in the solar photosphere and travel down to the base of the convection with only little additional entrainment. As a result of the density contrast of 10 6 in the solar convection zone, values as low as 10 7 could be possible for the filling factor at the base of the convection zone. Since we treat f d (z bc ) as a free parameter of our model, we cannot answer the question of which of these extreme scenarios is closer to the Sun, but we can study the consequences for the structure of the overshoot region for each of these scenarios. Fig. 2. (a) Initial temperature gradient 9 (solid line), adiabatic temperature gradient (dotted line), and radiative temperature gradient (dashed line). (b) Radiative energy flux for the case d ph ¼ 0:2 andd bc ¼ 0:5, which will be used in most of the following discussion. The radiative energy flux is normalized with the total energy flux. 9 rad > 9 ad for z < 2, so z ¼ 2 is the base of the convection zone. Within the convection zone (9 rad > 9 ad ) we choose initially 9 ¼ 9 ad þ 0 with 0 > 0. The value of 0 is chosen to be of the order of in the relaxed solution Downflow Filling Factor For the filling factor f d as a function of depth we assume the form f d (z; i) ¼ f d ðz bc ; i Þ % mðz bc Þ ; ð61þ % m (z) with the value f d (z bc ) at the base of the convection zone and a depth dependence based on the reference state density profile. For reasons of simplicity we use the same depth dependence for the filling factors of different downflows and allow only for different values for f d (z bc ; i). We show later that the overshoot properties depend mainly on f d (z bc ; i) andare less sensitive to. For, most reasonable values are in the range 0 1. This covers reasonable assumptions about the solar convection zone, as well as numerical results. The filling factor depth dependence obtained in numerical simulations (Brummell et al. 2002) depends on the downflow strength. Whereas the total downflow filling factor shows only a very weak depth dependence, the strong downflow filling factor is close to ¼ 1. In the end of x 3.4 and in x 4we briefly discuss the implications of such a downflow strength relation. In our model we treat f d (z bc ) as a free parameter and allow for values significantly smaller than the values 0.1, which are typically used in the nonlocal mixing-length approach and also found in numerical simulations. Significantly smaller Single Downflow Models We call models using only downflows of one specific strength in the following single downflow models, even though it does not mean that the whole convection zone has only one downflow. For computing the total energy flux, it does not matter if there is one big downflow or several small downflows with the same properties and same total filling factor. For a given profile of the radiative conductivity the free parameters are: total energy Cux f d (z bc ) Blling factor at the base of the convection zone d mixing upcow, downcow proble of f d. We show in x 2.5 that in the limit f d T1 the solutions follow a self-similar scaling, dependent only on the parameter combination ¼ =f d (z bc ). This reduces the free parameters of the model for this case to three, of which only two, and d, have a significant influence on the solution. It should be noted that the free parameters are basically the same as in the mixing-length approach: the mixing length ( d ) and the filling factor. Nevertheless, our model is quite different, as a result of the large degree of nonlocality arising from the assumption of downflows extending through the whole convection zone and overshoot region Ensemble Models For models with an ensemble of downflows we still have to specify a relation between f d (z bc ; i) and d (i). To this end we define a downflow strength distribution function!(i) ¼!( d (i)) d (i) with 1 ¼ P N!(i) R!() d (theoretically the integral should be normalized, but since we have to use discrete values for d, we normalize the discrete distribution function), which specifies the relation between the filling factor of individual downflows and the total downflow filling factor: f d ðz bc ; i Þ ¼ f tot ð d z bc Þ! ð d (i) Þ d (i): ð62þ The functional form of! cannot be determined within the framework of this model. In order to demonstrate the effect, we use an analytical approximation for! of the form "! ð d Þexp ln # 2 d ln (100) 0 (ln) 2 ; ð63þ which leads to a Gaussian in ln d with!( 0 =) ¼!( 0 ) ¼ 0:01!( 0 ). The two parameters 0 and are the chosen location of the maximum and the chosen width of the distribution, respectively. We use values for d (i) intherange [ 0 /, :::, 0, :::, 0 ].

9 1054 REMPEL Vol. 607 For a given profile of the radiative conductivity, the free model parameters are in this case: total energy Cux fd tot(z bc) total Blling factor at the base of the convection zone 0 maximum of distribution width of distribution proble of f d. Similar to the previous case, these models show in the limit f d T1 a self-similar scaling dependent only on the combination ¼ =fd tot(z bc). For the ensemble solutions shown later in this paper we use 51 individual downflows to represent the ensemble. The ensemble solutions require typically more than 15 downflows in order to resolve the structure of the overshoot region sufficiently. Increasing the number of downflows increases the smoothness of the solution but does not change the physical properties of the solution Self-Similar Scaling Relation The downflow filling factor enters the equations in two ways: directly by scaling the convective energy flux and indirectly by scaling the magnitude of the upflow quantities v u and " u, which enter the equations via the dynamic pressure correction and the terms describing the momentum and energy exchange between up- and downflows. It turns out that the latter indirect dependence is only a correction, which has only significant influence if f d > 0:1 (we prove this later numerically). Formally we can express this dependence in the form F c ¼ f d ðz bc ÞˆF c ½ f d ðz bc ÞŠ; ð64þ where we separated the direct linear scaling from the latter one, which introduces higher order corrections. Using equation (59), we can express the radiative flux in the form F r ¼ ˆF r ; ð65þ which leads to an energy flux balance of the form 1 ˆF c ½ f d ðz bc ÞŠþ ˆF r ¼ 1; ð66þ with ¼ f d ðz bc Þ : ð67þ The indirect dependence of ˆF c on f d (z bc ) becomes negligible if we have (see eqs. [40] [51]) T1; ð68þ jv u jtjv d j; ð69þ j" u jtj" d j; ð70þ j T T d T m j ; T m ð71þ T: ð72þ Whereas the first three conditions are fulfilled for f d T1, the latter two are not that obvious. Assuming that the implicit dependence on f d is only a correction term (we prove this numerically later) and the solution depends mainly on the ratio, this is also the case for and jt d T m j=t m. Accordingly the latter two conditions are fulfilled in the limit f d (z bc )! 0if is kept constant. Under these conditions also the radiative heating term in equation (43) vanishes since H rad f d (z bc ). Once the indirect dependence on f d (z bc ) is negligible, the solutions become quasi self-similar if the ratio is kept constant. This is also true for ensemble solutions, since equation (62) also ensures the linear scaling of the convective energy flux with fd tot(z bc). Because of the highly nonlinear character of our model, it is difficult to predict from order-of-magnitude estimates when this self-similar scaling will break down. In x 3.1 we therefore show numerical results computed for a fixed value of but different values of f d (z bc ). The higher order corrections introduced by f d turn out to be negligible for f d < 0:01. The corrections become significant only for f d > 0:1. In all cases they modify the results but do not change the general properties of the solution. 3. RESULTS 3.1. General Properties Figure 3 shows the temperature gradient, energy fluxes, downflow velocity, and Mach number Ma ¼ v=(p=%) 1=2,as well as downflow and slow upflow entropy for a model with the parameters ¼ 10 3, d ¼ 1:5, and ¼ 1 for a single downflow solution. In the solution only the top part of the convection zone (z P 1:2) shows a superadiabatic gradient, which is a typical feature caused by the nonlocality introduced by the assumption of very coherent downflows. Since the entropy in the downflow is significantly lower than the entropy in the surrounding material, driving of the flow by a local instability of the stratification is not required in the lower convection zone. The entropy within the downflow has a superadiabatic gradient in the whole convection zone (monotonic increase of entropy with depth) and follows roughly the entropy of the slow upflow region, as a result of the coupling caused by the mass exchange. This coupling is strongest near the surface, where the second term on the right-hand side of equation (29), which considers the mixing with material entraining into the downflow from the upflow region as a result of the increase of the downflow mass flux, is the dominant contribution. Once the mass flux reaches the maximum in the middle of the convection zone, the coupling is reduced to the first term on the right-hand side of equation (29), which considers local downflow/upflow mixing. Even though the lower third of the convection zone is already subadiabatic, the entropy difference between up- and downflow and thus also the enthalpy flux and buoyancy force change sign around z ¼ 2:1 in the overshoot region. Nevertheless, the total convective flux changes sign close to z ¼ 2 as a result of the significant contribution of the kinetic energy flux in the coherent downflows. It should be mentioned that the fact that the entropy difference between up- and downflows changes sign significantly below the point where the stratification turns subadiabatic is a typical feature of nonlocal convection, in contrast to the local approach, where the entropy difference is proportional to the superadiabaticity. Since there are many definitions of the overshoot depth, we want to emphasize that we always use the definition based on the radiative energy flux (jf rad j > jf tot j). This means that the overshoot starts at z 2 and ends where the velocity of the fastest downflow approaches zero. We do not include the lower subadiabatic part of the convection zone in our definition of overshoot.

10 No. 2, 2004 OVERSHOOT AT BASE OF SOLAR CONVECTION ZONE 1055 Fig. 3. (a) Temperature gradient (9: solid line; 9 ad : dotted line; 9 rad : dashed line). (b) Energy fluxes (normalized with total flux; F rad : solid line; F ent : dotted line; F kin : dashed line). (c) Downflow velocity and Mach number (v d : solid line; Ma:dashed line). (d) Entropy in reference state and downflow (s m : solid line; s d : dashed line). The model parameters used are ¼ 10 3, d ¼ 1:5, and ¼ 1. The initial state is given in Fig. 2. Since we used f d 1=% d, the profile of the mass flux is identical with the profile of v d. In Figure 4 we show the sensitivity of the model on the choice of in the range between 0 and 1, as well as the corrections arising from large downflow filling factors. Figures 4a and 4c show j9 9 ad j for both cases, while Figures 4b and 4d show the corresponding profiles of the Mach number. In order to make small changes of j9 9 ad j more visible, we use a logarithmic scale in Figures 4a and 4c. Note that the logarithmic plots show a singularity in the convection zone, where the stratification changes from superadiabatic (upper part of convection zone) to subadiabatic (lower part of convection zone, overshoot region, and radiative zone). We have also shown in Figures 4a and 4c j9 rad 9 ad j of the initial state. For illustration of the effects we use values of ¼ 10 3 and d ¼ 1:5 in Figure 4, but the results are not very sensitive to this particular choice. According to the way we defined the downflow filling factor in equation (61), a change of influences the filling factor mainly near to the surface, while leaving it unchanged at the base of the convection zone. Thus, lowering (lowering the surface filling factor) leads to a more vigorous convection near the surface (larger superadiabaticity and Mach number). Since near to the surface the relaxation length given by equation (29) is very small, the influence on the deep convection zone and overshoot region is rather limited. For evaluating the corrections caused by large downflow filling factors, we used a model with ¼ 0:5 and varied the filling factor at the base of the convection zone between 10 7 and 0.1; the corresponding value at the top of the domain is about 3 times larger in each case. Solutions with a larger downflow filling factor show larger deviations from adiabaticity in the bulk of the convection zone and the overshoot region and also a reduced overshoot depth. Only in the upper boundary layer is the superadiabaticity decreasing slightly. Whereas the latter is mainly caused by the larger influence of the radiative heating term in the downflows, the behavior in the bulk of the convection zone and the overshoot region is caused mainly by the influence of v u and " u in the relaxation terms in equations (42) and (43). Since v u and " u have an opposite sign of v d and " d, this corresponds to a decrease of the relaxation length (see also discussion of influence of d in x 3.2).Itturnsoutthatforf d < 0:01 the corrections can be neglected and the solutions fulfill the self-similar scaling discussed in x Single Downflow Solutions It turns out that ¼ =f d (z bc )and d are the most important parameters determining the properties of the overshoot region. To demonstrate their influence, we use models with only one downflow strength in order to keep the discussion simple. Ensemble solutions are discussed in the next subsection. We further use models with f d T1 so that the self-similar scaling applies. Figures 5a and 5b show the influence of on superadiabaticity and Mach number for a fixed value of d ¼ 1:5; Figures 5c and 5d show the influence of d for a fixed value of ¼ The change of leads mainly to a change of the superadiabaticity of the solution, while the depth of the overshoot is only marginally affected. A significant influence on the depth is apparent only when the overshoot layer is