Dx H VxDr

Dx is positive for Dr > 0. The first member of (6.29) is not equal to zero. The cell is in pressure equilibrium and (6.30) still applies. The change DT/T with time of the cell in equilibrium is thus dt

8 Hp

8 Hp

By replacing in (6.29), one obtains

This is a generalization of (6.31) and it gives the sinking velocity of a fluid element moving down as a result of cooling in a medium with a x gradient [349]. The motions occur with the thermal timescale. This expression is more useful than (6.31) since salt fingers may occur in chemically inhomogeneous media. Not accounting for the ambient inhomogeneity gives wrong results. This velocity can be used to derive an appropriate diffusion coefficient.

6.2.3 Diffusion Coefficient for Semiconvection

Thermohaline and semiconvection are similar processes both involving Np and N?, but with a different sign of the p gradient (Fig. 6.5). In order to analyze semiconvection at small scales, one considers a small displaced fluid element with both p and T excesses with respect to the surroundings. Thus, locally this small fluid element is exactly in the situation of a thermohaline layer. This means that the above velocity vp is appropriate for both processes in their own context. The coefficient of semiconvection (6.26) was derived in the adiabatic limit, although semiconvection is a non-adiabatic effect. Here we derive this coefficient without making the adiabatic assumption.

6.2.3.1 From the "Sinking" Velocity

In a semiconvective zone (like in thermohaline convection), an upper fluid element is hotter and denser. The thermohaline velocity (or velocity excess in semiconvection) is determined by the heat losses (6.34). For a blob in equilibrium, one has (6.30), thus

The temperature difference between an eddy moving over a distance Dr and the medium is

T Hp for Dr > 0 one has DT > 0 as usual in a semiconvective region and vp is positive. The diffusion coefficient is DSC = (1/3) |vpDr| according to the definition (B.50). With the expression of the thermal time for a spherical blob itherm = d2/ (6K) from (B.66), one gets

which is positive. Eliminating Vint with (5.67) one gets

In the adiabatic limit, i.e., for r ^ one finds again the coefficient by Langer et al. (6.26). For small values of r, the diffusion coefficient is much reduced, which brings a better agreement with observations. Typically, r is of the order of 10-2-10-3 in stellar interiors (Fig. 5.6).

6.2.3.2 From Non-adiabatic Gravity Waves

The diffusion coefficient can be derived by taking into account the non-adiabatic effects in oscillatory motions at the Brunt-Vaisala frequency. The energy lost DUlost > 0 by an eddy of diameter e during a time Dt (5.62) leads to a decrease of T and thus an increase of q

For a sphere, the ratio of the surface to the volume is Z/V = 6/i and the excess of temperature is given by (6.36), thus we get dq = 6 K 8 (V - Vint) Dt q Hp I ' (

where we have put Dr = e. This relative excess of density leads to a corresponding increase of the velocity of the gravity oscillation Dv = g (dQ/Q) Dt

where the quantities are average over the interval of time Dt. The motions are oscillatory around an equilibrium position. The maximum of Dv is reached after 1/4 of an oscillatory period (2n/Nad). Let us consider that the average Dv is reached between 0 and the quarter of the period, say at 1/8 of the period. Thus Dt2 = n2/(16Na2d) and Dv becomes with (5.59)

Thus, the coefficient for semiconvective diffusion becomes with (5.67)

Apart from a small difference in the numerical factor (1.23 instead of 2), this is the same coefficient as that found above. The calculation of r as given by (5.66) is thus critical to determine the appropriate diffusion coefficient.

6.3 Time-Dependent Convection

During very fast evolution phases, mixing in the convective zone cannot be considered as instantaneous. This occurs when the nuclear timescales (9.9) are of the same order of magnitude as the convection turnover time (5.49). If so, the nuclear species have the time to be partially transformed into other nuclear species during their convective transport over a mixing length. Thus, an appropriate treatment of this situation must be performed. This applies, for example, to the deuterium burning in the pre-main sequence phase and also to the fast nuclear burning in the convective cores of pre-supernovae.

For such cases, a diffusion treatment may be used with a diffusion equation of the form (10.29)

where Dconv is a diffusion coefficient appropriate to convection. The general form of a diffusion coefficient is given by (10.19), it is Dconv = 1 v t (see also Appendix B.4.3). The velocity v is the average convective velocity given by (5.45). For the appropriate mixing-length t, we may take the usual expression t = a HP with a of the order of unity. The diffusion coefficients in convective zones are generally very large, of the order of 1015 or 1016 cm2 consistently with the value of the convective velocities and the short turnover time.

The boundary conditions for (6.44) need some special care (see 10.31). In particular, if there is some mixing in the zone adjacent to the convective region, a good solution is to apply a diffusion scheme to both regions with their appropriate diffusion coefficients, this avoids the use of many boundary conditions. However, the interpolation of diffusion coefficients when they vary rapidly needs special care [414], see also Sect. 10.2.2.

6.4 Effects of Rotation on Convection

Stellar rotation influences the size and properties of the convective zone. In turn, convection produces some internal coupling of rotation. Many aspects of these interactions are still uncertain.

6.4.1 Oscillation Frequency in a Rotating Medium

The Brunt-Vaisala frequency, expressed by (5.23) or (5.59) and leading to the Schwarzschild and Ledoux criteria (see Sect. 5.1.2), was derived in the absence of rotation. However, a displaced element (Fig. 6.6) is also submitted to the change of centrifugal force which modifies the oscillation frequency N2 and consequently the criterion for convective instability.

Fig. 6.6 The Rayleigh-Taylor and Solberg-Hoiland criteria. The displacement of a fluid element in a rotating medium

Ho

Let us write the equation of motion of a fluid element displaced from an equilibrium position at distance r0 from the center to a nearby position r in a medium in rotation with an angular velocity Q at r0. Cylindrical symmetry is assumed. The equation is similar to (5.1), with a centrifugal acceleration in addition,

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