Q Jo Ju Ub

With (B.68), this becomes

Q( = ¡0*17=0Q2(r)Ps(cos ft) sin3 ftdft = ~ ( ( \$Ps(cos ft) sin3 ftdft \ () ¡nsin3ftdft s=0 s( V ¡onsin3ftdft J.

This defines Is; its denominator is equal to 4/3; thus

Let us express sin2 ft in terms of P2(cos ft); one has

P2(cosft) = 1 - 2 sin2 ft; thus sin2 ft = 2 [1 -P2(cosft)] . (B.74)

Is can be written as

1 rn

= - [P0(cos ft)Ps(cos ft) - P2 (cos ft)Ps(cos ft)] sin ft dft 20

where Ss,i is the Kronecker symbol. Thus, one has I0 = 1 and I2 = — 5 while all the other components Is = 0. We can write Q with (B.73),

We see that the average Q of the angular velocity on an isobar for the expression of the angular momentum is not equal to just the radial component of the first term of the development in spherical harmonics. One can also express the horizontal term Q as

Q(r, 6) = Q(r, 6) — Q(r) = £Qs(r) Ps(cos 6) — £Qs(r) Is s=0

Q0(r) + £Qs(r)Ps(cos 6) — Qi(r) — £ Qs(r)Is s>0 s>0

The development of Q(r, 6) is

Thus, if we limit the development to the term in P2(cos6), one has to take for the expression of the angular velocity in the angular momentum,

The noticeable point is that there is a fraction 1/5 to be added to P2(cos6) multiplying Q2(r). For the other terms in the developments, there is no additive constant. This fraction 1/5 only applies in the developments of Q(r, 6) for the angular momentum. Interestingly enough, this term 1/5 is most useful in writing the equation for the time evolution of Q2 in (12.6).

Since all the terms Is are equal to zero except for s = 0 and s = 2, this means from (B.73) that the only components of circulation able to carry angular momentum are the components in P0(cos 6) and in P2(cos 6). This is an argument to limit the development of the various functions to the second Legendre polynomial.

B.6.2 Rotational Splitting for Non-uniform Rotation

Some developments on the rotational splitting in helioseismology are given here in complement of Sect. 16.6.1 in order to account for the fact that Q may be a function of r and - and that the Coriolis force also influences the splitting [131, 227].

In the equation of motion (16.14), | is the perturbation displacement with respect to the fluid at rest and v the associated velocity. Now, we must consider the perturbation Sr with respect to the fluid moving at a velocity v0, supposed to be small. One has the following relation between the material derivative of the displacement and the Lagrangian velocity perturbation:

The material time derivative of the perturbation becomes with (1.17)

dt t

The acceleration term in the left member of (16.14, see joined remarks) written for the velocity perturbation Sv becomes with the above rule, d2Sr 2Sr Sr Sr 2

^ = -w + (v0 •V) -IT + (v0 •V) ■IT + (v0 •V)2Sr. (B.82)

For small velocities, we neglect the term in v20 and (16.14) becomes 2Sr Sr

Q0 -d^T + 2^0 (v0 • V)— = -VP' + q0g' + q'g0 . (B.83)

The oscillations around the equilibrium structure are of the form eimt; thus

- rn2Q0Sr + 2ifflQ0(v0 • V)Sr = -VP' + q0g' + q'g0 . (B.84)

Formally these are the amplitudes. The two terms on the left correspond to a perturbed frequency, say a', and we write their sum as

- (or2e0Sr = -(a + Sa>)2e0Sr « -(a2 + 2aSa)e0Sr . (B.85)

By comparison with (B.84), we identify -Q0SaSr <—> ¡Qo(v0 • V)Sr. Multiplying both sides of this expression by the complex conjugate Sr*, one has

so that by making the average over all layers, we get for the frequency shift due to the velocity field,

We evaluate this quantity for a motion of rotation with an angular velocity Q(r, 6), where as usual 6 is the colatitude. The velocity is v0 = Qrsin6ep. At a point of coordinates (r, 6, p), the vector Q can be expressed by its components along the unity vectors er and e6 (Fig. B.2), Q = Q(cos 6 er — sin 6e6), and one has v0 = Q x r. (B.88)

A quantity like (v0 • V) F is the gradient of F projected on the velocity vone has

(v0 • V)F = eF(v0 • V) \F\ + \F\ (v0 • V) eF . (B.89)

The gradient of the scalar \ F\ is

rsin 6 dp because for a perturbation the dependence on p is given by e(imp(. The last term in (B.89) represents the change of the unit vector eF along the velocity v0 produced by rotation. At some level in a rotating star the change of eF is just produced by the rotation motion, which makes the unit vector eF, after a small displacement, to point in a slightly new direction. After one full axial rotation, the unit vector eF is again the same. If eF would coincide with er, this projection would just be along v0 and we would have (v0 • V) er = ev0 = Q x er. More generally, one has

According to the property of the cross-product and because Q has no component along ep, one has

Q x F = Q{ — sin 6 Fper — cos6Fpe6 + (cos6F6 + sin 6 Fr) ev} . (B.93)

Now, we express the term in (B.87) using this last expression and (16.50) for Sr, in which we omit the tildas over the amplitudes B,r and

The numerator of (B.87) becomes then

Fig. B.2 The various vectors in the spherical coordinate system

QoSr * • (v0 • V) SrdV = im i qoQ \Sr\2 dV r Jv r ( dYm

We use (16.38) for the spherical harmonics Yem. The derivatives with respect to p give im for Yl", with a sign minus for the complex conjugate. Since in each term of the second integral, one has the product of Ym and its conjugate or the derivative of it, the exponential eimp disappears. The integration in p just gives a factor 2n. Thus, the previous expression can be written as with

Unim = j"sinftdft j*Qqor2 { \2Pm(cosft)2

dPftr) Pim(cos ft)2

dft 1 sin2 ft 1

The denominator of (B.87) becomes

* dYm with

Dn£m = fo "sin ftdft j* Q0 r2 {\2 P£m(cosft)

0 0