¬rst di¬culty of the problem. The second and main one comes from the

ill-posed nature of the mathematical problem: the operator governing the

spatial structure of the eigenmodes is of mixed type: in some regions of the

star it is hyperbolic and in others elliptic. As shown by Hadamard, hyper-

bolic operators require initial conditions while here boundary conditions are

imposed.

Rieutord

102

This mathematical property is not speci¬cally associated with the Coriolis

acceleration and is already present with the buoyancy force. However, in

a non-rotating star, due to the spherical symmetry of the system and the

ensuing separability of the variables, the “ill-posedness” does not generate

any singularity; but as soon as this symmetry is broken, both terms (Coriolis

and buoyancy) introduce hyperbolicity and thus “ill-posedness”.

To be more illustrative, let us consider the simpli¬ed set up of a rotating

spherical shell ¬lled with a stably and radially strati¬ed ¬‚uid. To avoid un-

necessary complications, we use the Boussinesq approximation. The reader

may ¬nd in Dintrans and Rieutord (2000) a ¬rst investigation of a more

realistic case applied to γ Doradus stars. The Boussinesq set up has been

used many times in the past (Chandrasekhar 1961, Friedlander and Sieg-

mann 1982, Dintrans et al. 1999, etc.). Neglecting di¬usion terms and

using non-dimensional variables with the time-scale (2„¦)’1 , perturbations

of in¬nitesimal amplitude obey the linear equations

iωu + ez — u = ’∇P + ω ’1 N (r)2 iur er ,

(7.1)

∇·u =0,

where er and ez are unit vectors in the radial direction and along the rota-

tion axis respectively, ω is the frequency of oscillation, u the velocity per-

turbation (of which ur is the radial component), and P is the pressure. We

recognize the Coriolis acceleration ez —u and the buoyancy term N 2 iur /ωer ,

N (r) being the Brunt-V¨is¨l¨ frequency; note that temperature has been

a aa

eliminated with the energy equation.

To see the mathematical nature of the operator, it is necessary to re-

duce this system to a single second order equation for the reduced pressure.

Keeping only highest order derivatives, one ¬nds

‚2P ‚2P

(ω ’ N (r) cos θ) 2 + 2N (r) sin θ cos θ

2 2 2 2

‚s ‚s‚z

‚2P

+(ω 2 ’ 1 ’ N 2 (r) sin2 θ) 2 + · · · = 0

‚z

as ¬rst derived by Friedlander and Siegmann (1982). In this equation s

and z are the radial and vertical cylindrical coordinates while θ is the polar

angle. This equation shows that the type of the operator changes on the

critical surface, whose equation is

ω 4 ’ (N 2 (r) + 1)ω 2 + N 2 (r) cos2 θ = 0

The oscillations of rapidly rotating stars 103

while the characteristics of the hyperbolic domain are lines (in a meridional

˜

plane) given by the equations (assuming N (r) = r N ):

zsN 2 ± ξ 1/2

˜

dz

ξ = ω 2 N 2 s2 + (1 ’ ω 2 )(ω 2 ’ N 2 z 2 ) .

˜ ˜

= , (7.2)

ω2 ’ N 2z2

˜

ds

Fig. 7.2. The trajectories of characteristics in the meridional plane of spherical

rotating incompressible ¬‚uid shell. Note how rapidly the characteristics converge

towards the attractor.

These properties have been known for a long time (Friedlander and Sieg-

mann 1982) but it is only recently that their consequences have been fully

appreciated (Dintrans et al. 1999, Rieutord et al. 2001). To have a taste

of them, it is necessary to understand the dynamics of characteristics which

propagate in the hyperbolic regions. These lines physically represent the

path of energy of a wave packet travelling inside the domain; they may be

compared to the trajectories of a point mass particle in a potential well

delimited by the boundaries of the hyperbolic domain. However, unlike a

particle which can move in any direction, characteristics at a given point of a

meridional plane, have only two possible directions. Thus, their dynamic is

much constrained: in fact, the phase space is one-dimensional and no chaos

is possible. We illustrate in Figure 7.2 the dynamics of characteristics in the

unstrati¬ed case.

Rieutord

104

Fig. 7.3. The kinetic energy of an axisymmetric inertial mode at frequency ω =

0.7822 associated with an attractor where the shear layers are clearly visible. The

¬‚uid is viscous and incompressible. In this numerical solution, · = 0.35 is the ratio

of the shell radii, L = 1000 is the number of spherical harmonics used, Nr = 400

is the number of radial grid points, and the Ekman number measuring viscosity is

E = 1.0 — 10’9 . Stress-free boundary conditions have been used.

In Rieutord et al. (2000, 2001), we have studied in some details the un-

strati¬ed case but we think that the properties thus uncovered are generic

and equally apply to the stably strati¬ed case. It turns out that character-

istics may follow three types of trajectories: strictly periodic trajectories,

quasi-periodic ones or trajectories converging towards an attractor. The

¬rst type occurs for some speci¬c frequencies and it can be shown (Rieutord

et al. 2001) that they are in ¬nite number; quasi-periodic ones have been

demonstrated to exist only in the full unstrati¬ed sphere. In fact, converg-

ing trajectories is the generic case. The attractor is usually a periodic orbit

but may be a point (in the meridional plane when a critical surface meets

a boundary, e.g. Dintrans et al. 1999). In all cases, attractors are sin-

gularities which make the solution neither integrable nor square-integrable

(neither the total momentum nor the total kinetic energy exist). Therefore

when attractors are present no eigenmode is possible (the point spectrum

of the operator is said to be empty). In fact, attractors control the whole

spectrum of gravito-inertial modes in a star: if a star had no di¬usion the

The oscillations of rapidly rotating stars 105

so-called r-modes would be the only ones to “survive”, the spectrum of oscil-

lation would be almost empty. But real stars have di¬usion and eigenmodes

are still possible: indeed, singularities associated with attractors are regu-

larized into shear layers as shown in Figure 7.3. However, shear layers do

appear only when the di¬usion is small enough given a length of the attrac-

tor. It turns out that long attractors require such a small di¬usion that even

in stars shear layers would not appear. In such cases, the singularity gen-

erated by the inviscid part of the operator is not able to overcome di¬usive

e¬ects and regular-like modes exist, but it is clear that such modes cannot

be obtained with an adiabatic approach.

The foregoing discussion shows that in rotating stars inertial gravity

modes should be studied carefully: ¬rst the strongest attractors should be

localized on the frequency axis as they will correspond to strongly damped

modes; surely, no observed period of star pulsation can be within such bands.

Secondly, eigenmodes of rotating stars do require di¬usion to be properly

computed, otherwise results may be resolution dependent.

To conclude this section on the e¬ects of the Coriolis acceleration, we

should emphasize that the ¬rst step in order to evaluate their importance

is to compare the Coriolis frequency 2„¦ to that of the lowest order gravity

modes ωg . If ωg 2„¦ then perturbative methods are valid for large-scale

modes; if ωg ∼ 2„¦ non-perturbative methods should be used. Presently,

γ Doradus stars are the only stars which fall in the second category but

others may join the club!

7.4 The part played by centrifugal acceleration

The centrifugal acceleration plays a much more subtle and less understood

rˆle in the dynamics of rotating stars. Its ¬rst e¬ect is obviously to change

o