Ecole doctorale de Physique de la région Parisienne (ED107)

130 Inertial modes in slowly rotating stars : An evolutionary description
or
˜Ω[r, ϑ] =
W βv
. (4.24)
r2 2 21/2
sin[ϑ] + βv
Since in all these laws, none of the free parameters β corresponds to a physical variable,
we will not give quantitative results. It will be done in the relativistic study (and
then in another article) where there is a parameter that physically quantifies the way the
equations are far from the Newtonian EE : the compactness of the star. Here we will only
discuss in a qualitative way the results obtained in all the previous cases and give a very
representative example : what happens in a γ = 2 polytrope with the anelastic approximation
(free surface) and the rotation law given by the Equation (4.21) with βn = 0.4. With
the exception of these choices, the calculation was done with exactly the same conditions
as in the preceding study with the noisy initial data in the previous section.
The main difference with the case of rigid rotation is that the mode that grows is
no longer purely axial. Indeed, we can see on Figure 4.14 that the radial velocity is
now also driven to instability. Moreover, comparing power spectra in this figure and in
Figure 4.15 and 4.16 shows that this is really a single mode with frequency very close to
the frequency of the r-mode. By a single mode, we mean that this is exactly the same
frequency for several physical quantities and several positions in the star. This is not a
sufficient condition to talk about “discrete spectrum”, but it is a necessary condition.
This unavoidable existence of a polar part of the velocity in a differentially rotating
Newtonian star with barotropic EOS should be compared with the results achieved by
Lockitch et al. (2001) in the relativistic framework. Indeed, in GR, the main reason for
the coupling between axial and polar parts of the velocity is the frame dragging that is
imitated in a Newtonian framework by differential rotation. Finally, note that the value
of the frequency in the Newtonian case is depending on the way we chose to normalize
Ω. Then, to summarize all our calculation, we can say that we observed that the polar
counterpart of the mode appears as soon as there is differential rotation, and whatever the
chosen law. Yet, the more the law for Ω is far from the rigid case, or in a more pragmatic
way the greater the free parameter is, the more the unstable mode has a polar counterpart.
4.5.3 Free evolution
The other question we asked was this : “What does happen to linear r-modes if they
are put into a differentially rotating background ?”. The idea in taking such kind of initial
data was to have something quite close to an eigenvector, assuming if there is one it should
be quite similar to the linear r-mode.
Once again, we did not make quantitative calculations and postponed it to the GR
case. As in the case of noisy initial data driven toward instability by an RR force, the
free evolution of the linear r-mode showed the growth of a polar part of the velocity. It
can be seen in Figures 4.17 and 4.18 that, respectively, show the time evolution of the