Ecole doctorale de Physique de la région Parisienne (ED107)

132 Inertial modes in slowly rotating stars : An evolutionary description
S xx
Sp(S xx )
5
0
-5
2
1
0 50 100 150
t Ω
0
0 2 4 3 ω / Ω 6 8 10
Figure 4.16 – Time evolution of one of the two independent components of the Sij[t]
tensor that appears in the RR force. This calculation was done during the same run as
the results in Figures 4.14 and 4.15. We can see the almost monochromatic associated
spectrum with the same frequency as in the previous spectra of the unstable mode.
ratio between the polar energy and the total energy and the time evolution of the radial
velocity at the point of coordinates ξ = 1 π , ϑ = 2 2 , ϕ = 0 . This results comes from a calculation
done with a spatial lattice of the shape (64, 48, 4), the anelastic approximation
(with a free surface) and a rotation law given by the Equation (4.21) with βn = 0.4. For
reasons that will be explained later, we included degenerate viscosity (cf. Section 4.3.4)
with an Ekman number Es = 5 × 10−5 in order to regularize the solution. The evolution
was done to the last 30 periods of the linear r-mode with 100 steps per oscillation. In the
Figure 4.17, we see that, after a while, the ratio between the energy in the polar part of
the mode and the total energy reaches a kind of stationary state with a coupling between
different modes. The existence of this “hybrid final state” was verified during other runs
with other physical conditions and is once again to compare with results achieved in GR
by Lockitch et al. (2001).
Concerning the existence of modes, looking simultaneously at Figures 4.18, 4.19 and
4.20 shows that apparently one single mode mainly appears in the reaction force (or in
the component of the corresponding tensor) even if both the axial and polar parts of the
velocity contain several modes. Yet, the spectrum of the Sij tensor is quite noisy due to
the fact that the RR force is not switched on and that the run is short. We verified this
feature during other calculations. The conclusion is that for small values of the β parameter
(these values are depending of the chosen rotation law), the main effect of differential
rotation on a “free linear r-mode” is to give it a polar counterpart and to widen its spec-