Ecole doctorale de Physique de la région Parisienne (ED107)

114 Inertial modes in slowly rotating stars : An evolutionary description
Numbers Definition Analytical expression
Ekman Ratio between P and Tv or between the viscous term
and the Coriolis force
E ∼ ν/ (nΩR2 )
Rossby Ratio between the typical velocities of the mode and
of the background fluid or between the nonlinear term
and the Coriolis force
Ro ∼ W/ (RΩ)
Reynolds Ratio between the Rossby and Ekman numbers or
between the nonlinear and viscous terms
Re ∼ nW R/ν
Chandrasekhar Ratio between Tv and Tg or between the RR force
and the viscous term
Ch ∼ n 2 GR 9 Ω 6 / (c 7 ν)
Tableau 4.1 – Characteristic numbers implied in the dynamics of inertial modes of
rotating NS. Note that the definition of Chandrasekhar number is adapted to be of the
order of 1 for linear l = m = 2 r-modes.
Note finally that with this factor ahead of the physical parameters, we ensure the
bifurcation value of Ch to be of the order of the unity, at least for the linear l = m = 2
r-mode. Indeed, this point is easily illustrated by looking at NSE for the linear l = m = 2
r-mode with time-dependent amplitude and a shear viscosity of the form s = 1 − ξ 2 . This
viscosity that vanishes at the surface implies no need to add more boundary conditions
(BC) and gives the exact (at the lowest order for the RR force) differential equation for
the amplitude :
∂tA[t] = 1
Tg
A[t] 1 − 2
Ch
where one easily sees that with this shape, the viscosity wins the battle against RR force
for Ch < 2. All the previous numbers are gathered in Table 4.1.
To end with this short discussion, we should insist on the most important conclusion
that is summarized in Table 4.2 : the above figures show how stiff is the numerical problem
of finding a dynamical solution of the NSE in this framework. This is a physical situation
in which several very different time scales appear. In order to get an efficient and accurate
code, some approximations have to be made.