Ecole doctorale de Physique de la région Parisienne (ED107)

4.5.1 Modifications
4.5 Differential rotation 129
Assuming differential rotation of the background star involves slight modifications of
what we said in the Section 4.2. First, terms coming from the spatial derivatives of the
non constant Ω[ξ, ϑ, ϕ] must be added in the NSE. Second, we shall now specify what
exactly Ω means in the definition of dimensionless variables [cf. Equation (4.6)].
Concerning Ω, we chose to take for a time unit the inverse of its value at the equator.
This is uniquely determinate due to the fact we have always assumed that the rotation
law is of the form Ω[r, ϑ, ϕ] = Ω[r sin[ϑ]] or of the form Ω[r, ϑ, ϕ] = Ω[r]. The first case
corresponds to what must be this law for the background to be stable with respect to the
Newtonian EE, and the second case is in a way inspired by GR even if it is not a solution
of the full Newtonian EE. For more details see Section 4.6.
In the dimensionless NSE, the modifications induced by differential rotation are quite
simple. First, ∂ϕ is replaced with ˜ Ω[r, ϑ]∂ϕ where ˜ Ω[r, ϑ] is the dimensionless profile of
rotation. Then, we have to add new terms coming from eϕ r sin[ϑ] V ·
∇ ˜Ω[r, ϑ] that are
in the spherical orthonormal basis
0
0
Vrr sin[ϑ]∂r ˜ Ω + Vϑ sin[ϑ]∂ϑ ˜ (4.20)
Ω.
4.5.2 Noise with huge RR force
Once again, our goal was to stay as close as possible to the basic and well understood
situation to minimize the number of unknowns. Thus, we took noisy initial data, put it in
a differentially rotating background, switched on the RR force and looked for what was
to happen. In this way, the question was not to look for the existence of r-modes, but
simply to look for the existence of modes driven to instability by the RR force in a non
rigidly rotating background.
The only difference with the basic study done in the case of rigid rotation was that
we had to choose a law for Ω. The first choice was very simple and corresponded to a
background stable with respect to the Newtonian EE. It was of the form
˜Ω[r, ϑ] = W 1 + βn r 2 sin[ϑ] 2
(4.21)
where βn was a constant depending on the run and W calculated to have ˜ Ω 1, π
= 1. 2
As explained above, we also tried a law inspired by GR :
˜Ω[r, ϑ] = W 1 + βgr r 2
(4.22)
or laws coming from the already quoted article by Karino et al. (2001) :
˜Ω[r, ϑ] =
W βL 2
r 2 sin[ϑ] 2 + βL 2
(4.23)