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Astron. Astrophys. 323, 831-838 (1997)

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3. The conditions for shear instability and the diffusion coefficient in semiconvective zones

If there exists at least a purely real, positive solution for the growth rate [FORMULA], the stellar layer is said to be convectively unstable. If on the contrary there are no real, positive solutions but, however, one (or more) complex solutions [FORMULA] with positive real part, then one speaks of overstable convection. Of the two natural timescales [FORMULA] and [FORMULA] characterizing these latter secularly unstable regions, the second one gives the phase velocity of the disturbances in the vertical direction, [FORMULA]. The resulting crossing time over the semiconvective thickness is several orders of magnitude shorter compared with the evolutionary timescale (Kato 1966), and should ensure that the amplitude of the disturbances can grow sufficiently for the medium to become turbulent 1. As to the diffusion coefficient, it is suitably written out of the growing amplitude timescale [FORMULA] as [FORMULA] (Langer et al. 1983), where [FORMULA], [FORMULA] in view of the definition adopted in (7) for the expansion in normal-mode.

If one is searching for at least a purely real, positive solution of a generic cubic [FORMULA], with the [FORMULA] real coefficients, then it is easily verified that the following sufficient conditions hold

[EQUATION]

If none of the previous inequalities is satisfied, the cubic may still possess complex solutions with positive real parts. Eq. (8) then splits into the couple

[EQUATION]

Note that if the complex (shear) term [FORMULA] is set to zero, the algebra is simplified since we find in place of (11)

[EQUATION]

which follows from the expression on the imaginary component [FORMULA]. Applying conditions (10) on (12), we eventually obtain as criterion for the onset of overstable convection

[EQUATION]

3.1. The case with no shear

If we apply criterion (10) on the dispersion equation (8) with [FORMULA], then we recover for the onset of convection the well known inequalities

[EQUATION]

(the solution [FORMULA] is rejected since unphysical). Semiconvective motions are excited wherever condition (13) is fulfilled, and leads to (Kato 1966)

[EQUATION]

In this case, the overstable mode [FORMULA] is solution of (12) with frequency [FORMULA]. We point out that all these latter criteria are independent from the accessible spectra in wavenumber k, and bear no explicit signature on the specific geometry of the problem.

In radiative regions stable according to the Ledoux criterion, we can recast Eq. (12) into a cubic for the quantity [FORMULA] in terms of the timescales ratio [FORMULA],

[EQUATION]

where [FORMULA], [FORMULA]. In stars [FORMULA], and more often [FORMULA]. However, if we were to solve for [FORMULA] on the hypothesis [FORMULA] (the adiabatic case), then one would lack of any growing solution ([FORMULA]), confirming the fact that semiconvection is really the outcome of the radiative leakages suffered by the perturbed fluid.

We now rely on the principle of maximum heat transport to select the anisotropy factor x which extremizes the growth rate, or equivalently y (Canuto & Hartke 1986). Let us then differentiate Eq. (14) with respect to x, and demand [FORMULA]. One obtains

[EQUATION]

With the help of the expressions for y, f, and the identity (4), the diffusion coefficient becomes

[EQUATION]

in which the wavenumber k plays no role. Two more expressions must however be supplemented in order to close the system for the unknowns D, [FORMULA] and [FORMULA]. These latter are (Maeder 1996)

[EQUATION]

[EQUATION]

where [FORMULA], A and V being the mean area and volume of an unstable fluid cell, and [FORMULA] (the peak wavenumber in the turbulent energy spectrum [FORMULA], or [FORMULA] in the framework of mixing length theories of convection). Within a numerical factor practically equal to unity ([FORMULA]), expression (16) for the diffusion coefficient is the same as that obtained by Maeder & Meynet (1996) from a totally different procedure. For a thorough discussion on the form taken by the solutions of the system (16)-(18), see Maeder (1996).

3.2. A new criterion for convective instability in shearing media

If we demand for a purely real growing solution, then the cubic (8) splits into the two equalities

[EQUATION]

The root [FORMULA] is of no interest here. We next find in place of the condition for a Taylor instability

[EQUATION]

Since [FORMULA], with [FORMULA] a geometrical factor of order unity, one gets

[EQUATION]

where

[EQUATION]

Likewise, the previous Ledoux inequality modifies as

[EQUATION]

In the deep interior, and in terms of the Brunt-Väisälä frequency [FORMULA], Eq. (20) can be written in the most usual form

[EQUATION]

In the limit of very large z, expression (21) reduces to [FORMULA], which is very near the same as the original Richardson criterion for plane-parallel, incompressible flows when the curvature is no longer important.

An interesting aspect of the inequalities (19)-(20) consists in the absence of corrective terms of the form [FORMULA]. In other words, the centrifugal contribution only enters the equations coupled to the shear field. As a consequence, one does not recover in the limit [FORMULA] the corresponding criteria 2 for the special case [FORMULA]. In fact, if we were considering the star as a solid body, rotation would affect the hydrostatic equilibrium configuration by lowering the effective gravity, and the thermic properties only through modification of the spherical radiative gradient (Endal & Sofia 1976). As for the physical meaning of the new term appearing in Eq. 21, it can be understood by recalling that [FORMULA] is proportional to the mean buoyant acceleration over a distance [FORMULA] of an unstable element, caused by thermal effects [FORMULA] and by chemical stratification [FORMULA]. In the same vein, the correction brought in by the inertial acceleration [FORMULA] in the linear regime is [FORMULA].

Condition (20) on the temperature gradients offers an algebraic criterion for establishing convective instability in a shearing medium. The minus sign in the two right hand terms reminds us that differential rotation has a destabilizing effect, and that convective boundaries may thus spread beyond what is commonly assumed in standard calculations. This is certainly not the case for a slow rotator like the Sun, whose minimum timescale [FORMULA] in the shear layer at the base of the convection zone is of the order of 250 days, compared to rotation and dynamical timescales of 26 days and 1 hour, respectively, giving shear coefficients [FORMULA]. In contrast, massive MS stars are known to rotate faster, and their reduced evolutionary timescale weakens the effectiveness of the angular momentum transport mechanisms resulting in steeper angular velocity profiles. Rotational periods of a few days are common among massive field stars, and the effects of shear on convection boundaries may then become significant, especially in their final nuclear stages where one expects rapid structural changes on timescales equal to a fraction of the hydrogen burning stage.

To test the possible role of shear in fixing the radial extent of the unstable convective layers, we applied criterion (20) on the internal structure solution of a 40 [FORMULA] star as obtained by Meynet & Maeder (1996). Their calculations account for both the hydrostatic corrections and first order effects introduced by rotational mixing. The starting model on the ZAMS is a uniformly rotating star with [FORMULA] ([FORMULA] is the break-up angular velocity at the surface) and metallicity [FORMULA]. During most of its MS lifetime, the angular velocity profile keeps sufficiently shallow so that no discernible differences are observed with respect to the standard analysis. However, nearing the end of the hydrogen burning phase the shearing terms [FORMULA] grow sufficiently in the semiconvective regions and the radiative core slightly affecting the predicted convective boundaries. On Fig. 1, we present the radial profiles of the relevant physical quantities inside the models with central hydrogen mass fraction [FORMULA] and [FORMULA]. In the more evolved model, the larger effect is observed at the upper boundary of the more external convective shell, where shear turns 0.3 [FORMULA] into convective equilibrium. An additional correction consists in a general mild softening of the µ-barriers in what are actually semiconvective regions (see next section).

[FIGURE] Fig. 1a and b. Profiles of various quantities inside a 40 [FORMULA], rotating model nearing (left-hand side, [FORMULA]) and at the end (right panel, [FORMULA]) of the H-burning phase: a Profiles of [FORMULA] (solid line) and of [FORMULA] (short dashed line). Arrows indicate semiconvective regions; b and c Profiles of the shear coefficients [FORMULA] and [FORMULA], respectively; d As for panel a, but where the solid line traces now the total effective gradient [FORMULA].

3.3. Semiconvective shear zones

The Richardson criterion imposes a necessary, but not sufficient condition on the horizontal velocity stratification for a radiative medium to become unstable to shear generated waves, which would then transfer energy vertically and thus modify the local entropy gradient. According to our linear stability analisys, a region that satisfies

[EQUATION]

is convectively unstable, even if the Ledoux criterion is not verified. For an intermediate situation with [FORMULA], the available shear energy is not sufficient to completely overturn the stable thermal gradient, but we can suspect that this excess energy may feed a turbulent field, and that some fraction of the turbulent kinetic energy production is used for mixing, the rest being dissipated by viscosity or transferred to global internal wave motions which do little mixing or none at all. Indeed, if it were not for the imaginary term [FORMULA], criterion (13) for overstable convection would bear Schwarzschild inequality, as in many others situations (for example, if [FORMULA]), and one would thereby recover in a natural way the working hypothesis of Maeder in the regime [FORMULA]. Actually, the shear term [FORMULA] slightly weakens Schwarzschild inequality. This result should not come as a surprise since the modified Schwarzschild and Ledoux criteria must converge in the limit for a vanishing [FORMULA] (see Eq. 20).

To exactly solve (11) for [FORMULA], with x such to maximize the growth rate, we must differentiate both equations with respect to x and assume [FORMULA]. The resulting four expressions with Eqs. (4) and (18) provide six equations for the unknowns [FORMULA], [FORMULA], [FORMULA], x, [FORMULA] and [FORMULA]. The case [FORMULA] provides the initial guess for a Newton-Raphson algorithm.

A useful limiting case can be studied analytically. Note in fact that in the linear coefficients (9) there appear terms [FORMULA]. As we approach the surface, [FORMULA] decreases and it becomes worthwhile to study the asymptotic solution [FORMULA]. One then recovers the Schwarzschild criterion for overstable convection, and a modified diffusion coefficient in presence of shear

[EQUATION]

Since the corresponding Ledoux stability criterion now reads [FORMULA] for semiconvection, the coefficient (22) is consistently positive defined even when the µ-barrier vanishes. Nearing the convective boundary one also obtains [FORMULA]. At the radiative boundary, however, since [FORMULA] is zero in this approximation, one should employ the classical Schwarzschild criterion to ensure [FORMULA]. As anticipated, the role of shear in lowering the µ-barriers is also reflected in an increased magnitude for the diffusion coefficient. As a first approximation however (the plane parallel limit), the expected correction is rather limited.

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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