## 3. The conditions for shear instability and the diffusion coefficient in semiconvective zonesIf there exists at least a purely real, positive solution for the
growth rate , 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
with positive real part, then one speaks of
overstable convection. Of the two natural timescales
and characterizing these
latter secularly unstable regions, the second one gives the phase
velocity of the disturbances in the vertical direction,
. 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
If one is searching for at least a purely real, positive solution of a generic cubic , with the real coefficients, then it is easily verified that the following sufficient conditions hold 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 Note that if the complex (shear) term is set to zero, the algebra is simplified since we find in place of (11) which follows from the expression on the imaginary component . Applying conditions (10) on (12), we eventually obtain as criterion for the onset of overstable convection ## 3.1. The case with no shearIf we apply criterion (10) on the dispersion equation (8) with , then we recover for the onset of convection the well known inequalities (the solution is rejected since unphysical). Semiconvective motions are excited wherever condition (13) is fulfilled, and leads to (Kato 1966) In this case, the overstable mode is
solution of (12) with frequency . We point out
that all these latter criteria are independent from the accessible
spectra in wavenumber In radiative regions stable according to the Ledoux criterion, we can recast Eq. (12) into a cubic for the quantity in terms of the timescales ratio , where , . In stars , and more often . However, if we were to solve for on the hypothesis (the adiabatic case), then one would lack of any growing solution (), 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 With the help of the expressions for in which the wavenumber where , ## 3.2. A new criterion for convective instability in shearing mediaIf we demand for a purely real growing solution, then the cubic (8) splits into the two equalities The root is of no interest here. We next find in place of the condition for a Taylor instability Since , with a geometrical factor of order unity, one gets where Likewise, the previous Ledoux inequality modifies as In the deep interior, and in terms of the Brunt-Väisälä frequency , Eq. (20) can be written in the most usual form In the limit of very large An interesting aspect of the inequalities (19)-(20) consists in the
absence of corrective terms of the form . 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 the corresponding criteria
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 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 . 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 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 ( is
the break-up angular velocity at the surface) and metallicity
. 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 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 and
. 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 into convective
equilibrium. An additional correction consists in a general mild
softening of the
## 3.3. Semiconvective shear zonesThe 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 is convectively unstable, even if the Ledoux criterion is not verified. For an intermediate situation with , 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 , criterion (13) for overstable convection would bear Schwarzschild inequality, as in many others situations (for example, if ), and one would thereby recover in a natural way the working hypothesis of Maeder in the regime . Actually, the shear term 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 (see Eq. 20). To exactly solve (11) for , with A useful limiting case can be studied analytically. Note in fact that in the linear coefficients (9) there appear terms . As we approach the surface, decreases and it becomes worthwhile to study the asymptotic solution . One then recovers the Schwarzschild criterion for overstable convection, and a modified diffusion coefficient in presence of shear Since the corresponding Ledoux stability criterion now reads
for semiconvection, the coefficient (22) is
consistently positive defined even when the © European Southern Observatory (ESO) 1997 Online publication: May 26, 1998 |