The devil is often hidden in details or in what is looking at first as a detail and then reveals itself as a point of prime importance. Convection theory is full of such "details" able to critically change the results. We may mention the problem of overshooting, i.e., where is the exact edge of a convective zone, the role of the ^ gradients, the heat losses by convective fluid elements, semiconvection and thermohaline mixing, the effects of rotation on convection, convection at sonic velocities, etc.

The Schwarschild and Ledoux criteria for fixing the boundaries of convective regions give in fact the dynamical limit rAT in the star, where the average temperature excess AT and thus the acceleration of the convective cells are zero. In reality, convective motions extend up to the kinematical limit rv, where the velocity of the fluid elements is zero. The difference between the kinematical and the dynamical edges, rv and rAT, is the distance of overshooting dover = Vv - rAT | • (6.1)

The value of dover is the average distance up to which convective mixing extends beyond the formal limit, defined by the Schwarzschild or Ledoux criteria. Overshooting may occur above convective cores or below convective envelopes. The overshooting from convective core determines the amount of nuclear fuel available for the star. The overshooting below the solar envelope is an "observable" parameter in helioseismology. We examine three different approaches to this problem.

Overshooting is a non-local process, namely the extent of overshooting critically depends on the properties of the adjacent convective layers. Thus, it cannot be

A. Maeder, Physics, Formation and Evolution of Rotating Stars, Astronomy and Astrophysics Library, DOI 10.1007/978-3-540-76949-1-6, © Springer-Verlag Berlin Heidelberg 2009

treated in a local theory. Even in the mixing-length theory (MLT) a non-local approach is possible, where one follows the motion of an average fluid element from deep in the convective zone up to the place where its velocity is zero. The ratio of the total flux Fot = Fconv + Frad to the conductive (here radiative) flux is equal to the Nusselt number nu. The total flux is

Fot = "¡ ""ó = NuFad =----Nu— , (6.2)

4 KT2 3kq dr where the acoustic transport has been neglected. From (5.31), one has dp _ gm T _L v (63)

dMr _ 4 nr4 P nu Vrad ' ( ) By comparison with (5.57), one has in an adiabatic convective zone

Figure 6.1 shows the variations of 1 /Nu in a 2 M0 star.

Fig. 6.1 The velocity v, excess AT and ratio f = 1 /Nu = Frad/(Frad + Fconv) in the convective core of a 2 M0 star on the ZAMS. From the author [335]

Fig. 6.1 The velocity v, excess AT and ratio f = 1 /Nu = Frad/(Frad + Fconv) in the convective core of a 2 M0 star on the ZAMS. From the author [335]

6.1.1.1 Critical Levels at the Edge of a Convective Core

Several critical levels at the edge of a convective core can be defined [354]. In addition to levels rAT and rv, one also has a level rNu where Nu = 1 and a level rV where Vad = Vrad. In the present approach, the levels rNu and rAT coincide since AT = 0 ^ Fconv = 0 and Nu = 1. The levels rV and rNu do not coincide since (6.4) is not strictly realized, i.e., the real gradient V differs from Vad by about 10—8. Due to this small difference, the distance between these two levels is negligible. Thus, at the external edge of a convective core, one has

In the non-local MLT [335, 527], the ballistic trajectory of an average fluid element is followed to study up to which level it goes. The T excess of a fluid element starting from a level rt and moving to a level r is

In a stationary situation, the acceleration of the cell with a density excess Aq/q = -SAT/T is v (dv/dr) = —g(Aq/q). In a non-local case, one has

The factor 1/2 in front of the integral means that the half of the work of buoyancy forces is dissipated by friction. The distance between the upper level r and the lower bound of integration rt in (6.6) and (6.7) is limited by the mixing length. For consistency with local developments (Sect. 5.2), one takes r — rt < t/2, where t = aHP. The relative density excess is expressed as a function of the T and 1 excesses and the convective flux is given by (5.46).

Some results are illustrated in Fig. 6.1. The velocity reaches a maximum inside the core, then it decreases because the range of integration (from where the cell "remembers" acceleration) is limited. The value v = 0 is reached beyond the formal limit Vrad = Vad. The excess AT decreases outward, it is zero at the formal limit rAT and then negative, which decelerates the cell up to the dynamical limit rv. A fraction of the energy is transported by radiation. At the formal limit, the convective flux is zero, the whole energy being carried by radiation. In the overshooting region, since AT < 0 the convective flux is negative, the radiative flux must compensate for it making the Nusselt number smaller than 1. The relative departure from the adiabatic gradient is very small in the overshooting region as in the core (< 10—7).

The overshooting distance is about 15% of the mixing-length t, with little dependence on the ratio a = t/HP. The distance of overshooting is independent of the value of the fraction (e.g., 1/2 or 1) of buoyancy forces which is converted to kinetic energy. However, a severe simplification of this approach is that no account is given to the spectrum of velocities and sizes of the convective elements.

6.1.2 The Roxburgh Criterion for Convective Overshoot

Another approach, in the form of an integral condition, has been proposed to estimate the distance of overshooting from convective core [500]. Let us express it in a simplified form. One starts from the energy equation (3.43) for a stationary situation and writes the radiative flux Frad = -%VT, e £ 1 ^ 0

where the term 0/T represents the rate of energy dissipation (in erg cm-3 s-1), it is positive and contributes locally to the entropy increase. One has

The second term on the right is zero according to the equation of continuity (B.3) in a stationary case. Thus, e £ 1 0

This equation is integrated over a sphere V which contains the convective core and the overshooting zone. Outside this volume the average velocity u = 0 on a surface Z and thus

The first two terms give with integration by parts

/yT V • Frad dV = T j Frad • d a -J Lrad df1j , (6.13)

^rad

^nuc

Over the considered volume, the two luminosities Lrad and Lnuc must be equal (see below) and (6.12) becomes,

If the dissipation rate 0 = 0, we get the Roxburgh criterion,

The outer edge of the core with overshooting lies where this expression is satis-fled (Fig. 6.2). Close to center, the nuclear (total) luminosity increases rapidly, then it tends toward a constant. In the core, the radiative luminosity is lower, because some flux is carried by convection (Fig. 6.1). At the edge of the core given by Schwarzschild's criterion the two luminosities are equal. In the overshooting region, the convective flux is negative since the T excess of the cells is negative, this implies that the radiative flux is larger than the total flux. Outside the overshooting zone, both fluxes are identical, since there is no other transport of energy. The criterion says that the integrals of the two luminosities as a function of 1/T are equal at the outer edge of the overshooting region (Fig. 6.2). In turn, this means that the integral of the positive convective flux (as a function of 1 /T) below the Schwarzschild limit must have the same size as the integral of the negative convective flux in the overshooting region.

The neglect of dissipation is not satisfactory [633]. Convection is highly turbulent and the turbulent viscosity, which is ~ 1011 times larger than the molecular viscosity, has to be considered (Appendix B.4.1). Criterion (6.16) largely overestimates the overshooting distance, giving values at least twice as required by observational constraints (Sect. 6.1.4). Thus, although there is formally no free parameter, the uncertainty about dissipation is a problem for this criterion.

center

Fig. 6.2 Schematic representation of the Roxburgh criterion for convective overshooting. The various luminosities are represented as a function of 1 /T. The Schwarzschild limit is indicated. The overshooting distance is such that the two gray areas are equal. Adapted from [633]

center

Fig. 6.2 Schematic representation of the Roxburgh criterion for convective overshooting. The various luminosities are represented as a function of 1 /T. The Schwarzschild limit is indicated. The overshooting distance is such that the two gray areas are equal. Adapted from [633]

The insufficiencies of the mixing-length theory are well known: it is local, it ignores the size and velocity distributions of the eddies and considers turbulence as incompressible. However, these properties are essential for a correct estimate of the amount of overshooting as shown by Canuto and Dubovikov [90]. Let us briefly give some insight into how the velocity and temperature fluctuations may influence the modeling of convective flux and overshooting.

Let w be the vertical velocity of fluid elements and 9 (= AT) the temperature fluctuations of the turbulent medium. The convective flux (5.46) is of the form Fconv = CPq w 9. In the MLT, the flux is proportional to the T gradient, d T

Dt is an appropriate turbulent diffusivity (see 6.24 below). However, in reality w 9 is not just depending on the T gradient, but also on the fluctuations of turbulence. The momentum and heat transport equations for the fluctuating vertical velocity w and for the temperature fluctuations 9 are [91], consistently with Sects. 1.1 and 3.2, dw =ga9-, (6.18)

where a is the volume expansion coefficient a = — ^^jn^) , which is 1/T for a perfect gas and K is the thermal diffusivity (3.46). The first term in the second member of (6.19) represents the advection of T fluctuations, and the second term is the radiative heat conductivity (cf. 3.45). Multiplying (6.18) by 9 and (6.19) by w and taking the averages, then by summing the two expressions one has d w9 = —w2 ^ + g a92 — + (6.20)

dt dr q dr

Further equations for w2 and 92 can be established [91] from (6.18) and (6.19). The dots, here and below, represent terms not essential for our discussion. The term (9/q) dP/dr can be written as

where c\ is a constant. tp9 is the timescale of the temperature-pressure correlation term (9/q) dP/dr, tp9 being some function of the local thermal timescale (3.47) and of a dissipation timescale. Although (6.21) has some similarity with (6.19), its exact derivation is different (see [87] Eq. (43a); [88] Eq. (53)), it expresses that the temperature-pressure correlations are related to both the velocity-T correlations and the T fluctuations. Introducing (6.21) into (6.20), one has d s — dT

dt dr wQ = -w2 — - TPlw 9 + (1 - C1) g a92 +... . (6.22)

In a stationary state of turbulence, this becomes

The comparison of this relation with (6.17) is interesting, one notes:

- The turbulent diffusivity Dt is given by

which is to be compared to the MLT expression Dt « (1/3) wI (Appendix B.4.1).

- The interesting point is the presence of other terms in addition to the above one. The convective flux contains in particular a contribution from the T fluctuations [91] (in some cases, this may give a positive convective flux, even if dT/dr > 0). The evaluation of the T fluctuations requires the solution of the equations of motion and energy conservation, with an account of the dissipation rate.

- The turbulent velocity w2 is determined by the equations of motions and its moments. The knowledge of tp9 depends on the coupling of thermal and dynamical effects.

Equations such as (6.22) expressing the relations between the various fluctuations are important for the treatment of overshooting. Another expression for the flux Fw of kinetic energy (1/2) w3 may also be written [91]. When the convective flux becomes negative, kinetic energy from Fw may still be available to sustain convection (cf. Figs. 6.1 and 6.2). Thus an accurate description of the convective flux requires the knowledge of the velocity and T fluctuations and of their relations with the perturbations of pressure.

Various convection theories have been developed. In particular, Canuto and Mazzitelli [93] account for the spectrum of turbulence in velocities and sizes. A mixing-length, taken as equal to the distance to the top of the convective zone, is nevertheless adopted. Various improvements have been further made, in particular in a form which allows calculations of stellar structure [92]. Other non-local developments can be found in the literature [624].

The future in this field is coming from 2D and 3D numerical simulations, as illustrated in Fig. 6.3 (e.g., also [420]). After an initial transient situation, a stationary situation is reached with an input of kinetic energy which produces an extension of the convective core with respect to Schwarzschild's criterion [631]. Such models may provide indications on the appropriate treatment required for overshooting. Different approximations are often made in numerical simulations. For example, in the approximation of Boussinesq, the medium is considered as incompressible: the density variations are not considered, except when multiplied by gravity to give

Fig. 6.3 2D numerical simulations of convection in the advanced phase of central O burning. Here, the mixed region (cloudy) overshoot by about 30% of the size of the convective region defined by Schwarzschild's criterion (white curve). Courtesy from Patrick Young [631]

Fig. 6.3 2D numerical simulations of convection in the advanced phase of central O burning. Here, the mixed region (cloudy) overshoot by about 30% of the size of the convective region defined by Schwarzschild's criterion (white curve). Courtesy from Patrick Young [631]

a buoyancy force. (With the Boussinesq approximation, a system of five coupled differential equations for five variables can be established [91]: these are turbulent kinetic energy, average 92 of turbulent T fluctuations, turbulent pressure, convec-tive flux and energy dissipation. The inclusion of the compressibility would require a system of 18 coupled differential equations.)

There are various constraints on the amount of overshooting from convective cores. The distance of the top of the MS (where central H content is zero) to the ZAMS in the HR diagram of open clusters is a sensitive test [360] about the real size of

Fig. 6.4 The mass fraction of the convective cores without and with an overshooting dover/Hp = 0.25. If the size rc of the core is smaller than Hp, dover = 0.25 rc. At 1.15 M®, dover/Hp = 0.125. From the author [363]

Fig. 6.4 The mass fraction of the convective cores without and with an overshooting dover/Hp = 0.25. If the size rc of the core is smaller than Hp, dover = 0.25 rc. At 1.15 M®, dover/Hp = 0.125. From the author [363]

the convective cores, thus about overshooting. The core extension supported by observations corresponds to an overshooting distance dover = (0 - 0.10) HP below 1.4 M®, dover - 0.10 Hp at about 1.5-2 M0 and 0.3 Hp around 15 M0 [159, 361]. This is in agreement with other determinations in this range of masses [158, 553], which suggest a value of dover of about 0.2 HP. This is comparable to the results of Sect. 6.1.1.

Figure 6.4 shows the size of the convective core as a function of stellar mass without overshooting and with an overshooting of 0.25 HP. For small masses, the relative increase of the core size is large. As suggested above, the overshooting distances predicted by Roxburgh's criterion are too large. The value of 0.2 HP found from cluster observations of various ages corresponds to about 50% of the distance predicted by Roxburgh's criterion [592]; for low masses of about 1.3 M0 in the cluster M67, the observations support an overshooting distance equivalent to a fraction of 7% of the value predicted by Roxburgh's criterion (6.16). Helioseismic observations also support the existence of some overshooting below the external convective zone of the Sun, of about 7-10% of HP. Asteroseismic observations will allow us to better estimate the overshooting from convective cores [163].

In this context, one may note that 1- other effects may extend the convective core, in particular rotational mixing (Chaps. 11 and 12); 2- the overshooting distance is usually referred to the core defined by Schwarzschild's criterion and 3- the use of the Ledoux criterion, the account for semiconvection (see Sect. 6.2) or of the Solberg-Hoiland criterion (Sect. 6.4) when rotation is present, could also influence what is defined as overshooting.

Let us consider a zone in the stellar interior where the gradient Vrad is intermediate between the stability predicted by the Ledoux criterion (5.55) and the instability predicted by the Schwarzschild criterion (5.54)

Vint < V < Vint + ^Vv, with Vint « Vad and V « Vrad • (6.25)

The instability occurring in this zone is called semiconvection. Formally, the result of a stability analysis (Sect. 5.1) indicates that there is no convection. This is true if convection is perfectly adiabatic. This is closely, but not exactly, the case even in the deep stellar interior.

Conditions (6.25) imply that an upward displaced convective eddy is denser (Ledoux criterion) than the surrounding medium, thus it is brought back by gravity. The eddy is also hotter (Schwarzschild's criterion) than the surrounding medium. Thus, it will radiate in the surrounding medium, this increases the internal density of the eddy. It thus goes down faster and its oscillations around the equilibrium position progressively become larger. The mixing produced by these growing oscillations is semiconvection (Fig. 6.5). Semiconvection occurs when Vv is positive, i.e., when v increases toward the interior, as is currently the case. Such a situation is sometimes called an overstability. The growth of the amplitudes of the oscillations is determined by the timescale of the thermal adjustment of a fluid element.

In some numerical models, the application of Schwarzshild's criterion may lead to a kind of "sandwich" of radiative and convective layers. The region above the core

semiconvection v Of >. |
radiative stability NT.ad2 |

\ V |

Fig. 6.5 Schematic illustration of the regions of convection, semiconvection, thermohaline convection and radiative stability in the plane showing the thermal and composition terms Nj ad and N2 of the Brunt-Vaisala frequency as defined in (5.60). The hatched region schematically indicates where stabilization by rotation may occur due to the Solberg-Hoiland (SH) criterion (see Sect. 6.4.3)

Fig. 6.5 Schematic illustration of the regions of convection, semiconvection, thermohaline convection and radiative stability in the plane showing the thermal and composition terms Nj ad and N2 of the Brunt-Vaisala frequency as defined in (5.60). The hatched region schematically indicates where stabilization by rotation may occur due to the Solberg-Hoiland (SH) criterion (see Sect. 6.4.3)

breaks up into a series of shells where convective zones alternate with radiative ones. If there is a gradient of x, this makes successive discontinuities of composition. The treatment of such zones with an appropriate diffusion coefficient describing the mixing, as suggested below, generally reduces or suppresses the stepwise chemical distribution.

It may also occur, more rarely, that the medium has a negative Vx, i.e., with x growing outward. This may result from accretion in a binary system. Such a situation remains stable as long as a bubble of fluid with a x higher than the surroundings is hot enough to be lighter. However, radiative losses reduce T and at some point the bubble becomes denser than the surroundings and sinks. This is thermohaline convection. It occurs in the region indicated in Fig. 6.5. The occurrence of thermohaline motions depends on both the thermal diffusivity K and the particle diffusion v, this is why such instabilities are called double diffusive. As the Prandtl number Pr = v/K is very small in stars (cf. Appendix B.5.2), the thermal diffusion generally dominates and the bubbles of higher x eventually descend in the medium. This is especially the case if the geometry of the unstable region favors the heat leakage by forming "salt fingers".

6.2.1 Various Approaches

Several methods have been applied to treat the semiconvective regions, let us briefly discuss some of them.

In the zone defined by (6.25) it is assumed [524] that matter is redistributed changing X until Vrad = Vad is reached. This implies a large mixing. Later, it was proposed [505] that the matter redistribution rather leads to Vrad = Vad + (/8)Vx, which implies less mixing. The consequences of these two assumptions were examined by Chiosi and Summa [128]. They showed that in the second case (Ledoux) the He-burning stage of massive stars (20 M0) starts in the red, while in the first case (Schwarschild) it starts in the blue. This is quite consistent with the results of Sect. 27.3.6, where it is shown that more mixing leads to a longer blue evolution in the He-burning phase.

6.2.1.2 The Method of Langer, Sugimoto and Fricke

The growth rate of the overstable oscillatory motions has been calculated by Kato [278]. From this estimate, Langer et al. [313] have expressed a corresponding diffusion coefficient DSC allowing them to treat semiconvection as a diffusion process

(Sect. 10.2). The oscillations are of the form Sf ~ expi(kx + st). The diffusion coefficient is expressed as DSC = (1/3)£ v, with v = £ s and £ = 2n/k, aK V - Vad

To be consistent with Kato's work, the numerical coefficient in (6.26) should rather be (2naK)/6. Here, K is the thermal diffusivity and a a numerical factor of the order of unity. Comparisons with observations [308] suggest that the above coefficient (6.26) should be multiplied by a factor of 0.01-0.04. If we account for the difference in the numerical factor mentioned above, the necessary reduction should reach a factor of 150-600! This is a problem for the above theory.

The origin of the problem is likely that the above solution is not the general solution of the dispersion relation, but the solution for a Peclet number (Appendix B.5.3) P e — ttherm/tdyn ^ i.e., it assumes as perfectly adiabatic a process which is by essence non-adiabatic. This hypothesis leads to a large overestimate of the coefficient of semiconvective diffusion (see Sect. 6.2.3).

6.2.1.3 Stevenson's Method

The unstable oscillations drive resonance instabilities which feed the growth of smaller scale instabilities, which in turn break down and produces mixing on small scales. The diffusion coefficient [552] has an upper bound

in the case of a perfect gas. The p gradient being generally rather large in semicon-vective region, this means that this coefficient is generally much smaller than the one by Langer et al.

The assumption [544] is that layered convection takes place as is observed in the laboratory. The mixing is due to the overturn of cells in the medium separated by stable thin layers, across which transport proceeds by microscopic diffusion. The diffusion coefficient in a medium of perfect gas behaves like

Vp where *dif is the coefficient of microscopic diffusion. As this coefficient is much smaller than K (e.g., by 10-8), the predicted transport is very small. However, some numerical models of semiconvective zone do not support the layered structure and show that it is rapidly destroyed [399] in a medium where the Prandtl number is small (ratio v/K of the viscosity to the thermal diffusivity), which is precisely the stellar case.

Numerical 2D or 3D simulations of the kind shown in Fig. 6.3 are likely the best way to derive what happens exactly in a stellar semiconvective zone. Some 2D simulations of semiconvection were performed by Merryfield [399], they exhibit some qualitative similarities with the above picture by Langer et al. However, the numerical simulations were not performed on timescales and spatial extent large enough to allow tests of the various analytical coefficients. The non-linear effects limiting the amplitude of the overstable oscillations have also been studied [224]. In regions where V is sufficiently different from Vad, the solutions are also qualitatively similar to that of Langer et al. [313].

6.2.2 Kato Equation, Thermohaline Convection

Let us consider a blob of helium in equilibrium at some level in a radiative H-rich stellar layer. To be in equilibrium in a medium of lower mean molecular weight, the blob must be hotter than the surroundings. Progressively the blob cools, its density increases and it slowly moves down. The velocity, say vof the descending motion is determined by the heat leakage from the blob. A similar situation in laboratory experiments or in the sea is that of the "salt fingers" or thermohaline convection as seen above. The instability is slow and called a secular instability, it is governed by the thermal adjustment timescale.

Let us write the equation governing the change of the excess temperature DT (or A T) with time of a fluid element during its motion through an homogeneous medium. The change results from the thermal loss rate -DT/¿therm and from the difference between the internal and the external T during the vertical displacement over a distance Dr during a time Dt, DT = (V - Vint) (T/HP)Dr (5.41). However, since the heat losses are already explicitly accounted for, the difference to be considered during the motion is only the difference between the external and the adiabatic gradients. Thus, we write Vad instead of Vint in this last expression,

This equation was first written by Kato (Eq. (7) in [278]), although with different notations and by Kippenhahn and Weigert (Eq. (6.27) in [285]). The thermal adjustment time ttherm is given in Appendix (B.5.3). Let us consider a stationary situation for a blob with Dx > 0. The blob is in dynamical equilibrium with the medium, i.e., both DP and Dq are zero. From the equation of state (3.60), one gets f=n- (-30)

where Dx/x is constant due to the homogeneity of the ambient medium. The first member of (6.29) is zero and one gets for the sinking velocity [285]

In a radiative medium Vad > V, thus for Dx > 0 one has vx < 0 and the blob is sinking.

Let us now consider thermohaline instability in an ambient medium with a x gradient Vx, which is more appropriate in stars. In such a medium, the difference Dx varies with the location of the fluid element considered. The x excess after a small trip over a distance Dr is

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