Ecole doctorale de Physique de la région Parisienne (ED107)

110 Inertial modes in slowly rotating stars : An evolutionary description
- n0 = M⋆/( 4
3 πR3 ). For the full slow rotation limit approximation (only terms linear
in Ω and spherical shape), R would be the radius of the star and then n0 the mean
density.
- s(hear) and b(ulk) dimensionless functions, and ν and λ pure numbers, all chosen in
such a way that if there is any viscosity, s and b are equal to 1 in a specific position.
Typically, for a single fluid model, this would be the center of the star. Nevertheless,
it is worth pointing out that to build a more realistic model of NS, several different
layers could be made to coexist. In this case, there would be different ν and λ, hugely
depending on the shell [see, for instance, Haensel et al. (2000), (2001) et (2002)].
With our definitions, those numbers ν and λ are twice the usual Ekman numbers,
and they then quantify the ratios between the viscosities and the Coriolis force.
- ˜ H and Σ, respectively, the dimensionless enthalpy and the dimensionless external
accelerations, both scaled by the inverse of α.
Concerning the latter quantities, we cut them in two parts and use for variables the
difference between their present values and their values in the steady state. For the background
parts, we then have
Σ0 = − ∇Ũ + (ξ sin ϑ)2
∇
2 α
with ∇ ˜ H0 = Σ0. (4.7)
This equality enables the background to be a solution of the NSE and can be solved
separately. From now until the end, we consider that this condition is realized, and we
forget about Σ0 and ˜ H0 (except in the mass conservation equation where the latter still
appears as an external parameter).
Finally, some words about the Newtonian gravitational field. Inertial modes are current
oscillations associated with small density variations. In this context, it is natural to use
the Cowling approximation [Cowling (1941)] which consists of forgetting fluctuations of
the gravitational field [see the relevance of this approximation for Newtonian r-modes in
Saio (1982)]. We will apply it for the nonlinear study. Nevertheless, for the Newtonian
linear study, we shall see later that depending on the way mass conservation is treated,
it may be not necessary (see Section 4.3 for more details). In this case, we have
Σ = Σ0 + σ with σ = − ∇δŨ (4.8)
where δŨ is the Eulerian perturbation of the dimensionless gravitational potential.