Ecole doctorale de Physique de la région Parisienne (ED107)

186 Spectral methods and vectorial equations
B.2.3 Boundary conditions
It was already explained how to impose BC for the divergence-free case. For the system
of Equations (B.20) or the Equation (B.24), it is worth distinguishing the two cases :
either the enthalpy vanishes or does not vanish at the boundary of the integration domain.
Here we will only consider the case of EE. Indeed, for NSE we chose to take a viscosity
that vanishes at the surface to avoid the need of further BC.
If H0 |r=1 > 0, the solution is quite easy : the Equation (B.24) or the system of Equations
(B.20) admit a homogeneous solution that can be used to satisfy one BC.
If H0(1) vanishes, the system of Equations (B.20) or the Equation (B.24) are degenerate
and therefore no BC can be imposed. But we shall show that, in this case, the correct
BC are automatically satisfied.
Indeed, consider first the case of the linearized exact system of Equations (B.20). On
the surface of the nonperturbed star, we have ∂th + W · ∇H0 which is the correct BC.
The surface H0 + h = 0 then defines the profile of the perturbed star. To show that the
solution in the anelastic approximation also satisfies the BC, it is more convenient to first
examine the nonlinear case. Here, h is not infinitesimal, and the Equation (B.23) must be
replaced with the equation
Γ(H0 + h) div W + W · ∇(H0 + h) = 0 (B.26)
that is satisfied if h is a solution of the nonlinearized Equation (B.24) :
Γ (H0 + h) ∆h − div F j+1/2 + ( ∇h − F j+1/2 ) · ∇(H0 + h) = 0. (B.27)
Once again, the surface of the star is defined by H0 + h = 0. But from the Equation
(B.26) we have the correct BC : W · ∇(H0 + h) |H0+h=0 = 0. The surface of the star is
then an unknown quantity that is determined by the relation H0 + h = 0 in solving the
Equation (B.27) by iteration. This technique was developed in a different context in the
already quoted paper Gourgoulhon et al. (2001).
In the linear case, we have Wr = 0 at r = 1 [cf. Equation (B.23)], and the enthalpy
does not vanish on the nonperturbed surface of the star. But once again the free surface of
the star can be obtained by looking for the position where the total enthalpy H0 + h = 0
vanishes. This surface coincides within first order quantities with the nonperturbed surface
r = 1. Note that if γ > 0 the pressure P |r=1 ∝ h γ+1 |r=1 vanishes within terms o(h).