have RLSF.

The OLA and the RLSF, including their variations, are currently the two

most often used inversion methods. For more details and discussion, see e.g.

Sekii (1997).

One of the often overlooked issues is correlation of data errors (Gough

1996, Gough & Sekii 1998) and correlation of the estimates (Howe & Thomp-

son 1996).

Correlation of data errors is mainly caused by mode leakage (see also

Schou 2003). Spherical harmonics are orthogonal to each other on a complete

sphere, but since we do not observe the entire surface of the sun, or even

the half of it in a uniform manner, spherical-harmonic decomposition of the

surface wave¬eld leads to leakage of power in one set of (lm) to other sets.

Sekii

268

This is in principle tractable, but even so the sheer size of the covariance

matrix, comprising M 2 /2 independent elements (although not all of them

of signi¬cant magnitude) renders it challenging in real cases to estimate and

then utilize the covariance matrix. However, as Gough (1996) pointed out,

a certain type of data correlation can lead to spurious features in inversions.

It is necessary to proceed with caution.

ˆ

From equation (18.3) we have the covariance of „¦:

ˆ ˆT

δ„¦δ„¦ = R eeT RT ,

where · · · denotes a statistical average. Even if there is no correlation in

the data errors, i.e., eeT is diagonal, it does not mean that the covariance

ˆ

matrix of „¦ is diagonal, irrespective of how RK or KR resemble identity

ˆ

matrices. In general, there is correlation between elements of „¦, that is to

say, between the estimates of „¦ at a certain part of the sun and another.

As Howe & Thompson (1996) pointed out, this correlation can also lead to

a spurious feature in inversion that looks like a systematic trend but is in

fact an artifact.

18.4 Solar internal rotation observed by helioseismology

18.4.1 Observational data

The observational data that are currently used widely for seismology of

solar rotation are obtained mainly through Doppler measurements, such

as GONG (http://www.gong.noao.edu/), a ground-based network obser-

vation, and SOI/MDI (http://soi.stanford.edu/), on-board the SOHO

satellite. It has turned out that measuring individual frequencies {ωnlm }

is rather hard. The GONG project determined ∼200,000 mode frequencies

around 1995 with a mixed result, although they have recently restarted their

attempt (Hill, private communication). On the other hand, so far there has

not been any such attempt from the SOI/MDI project. Instead of measuring

individual frequencies, a polynomial expansion is used:

Np

∆ωnlm = ak (n, l)Pk (m; l) ,

k=1

where Pk (m; l) is, e.g., a degree-k polynomial of m/l. The coe¬cients are

called a coe¬cients. The ¬tting for the ak (n, l) is done in Fourier space

(or power) directly, rather than painstakingly producing a poorly measured

table of {ωnlm } ¬rst. Since the rotationally induced component of ∆ωnlm is

an odd function of m (see previous section), the e¬ect of rotation appears

Seismology of solar rotation 269

only in the odd coe¬cients a1 , a3 , a5 , . . ., while the even coe¬cients a2 , a4 ,

a6 , . . . are measures of asphericity in solar structure.

18.4.2 How to tackle 2-dimensional (2D) inversions

The current datasets from GONG or SOI/MDI comprise a few thousands of

(n, l) multiplets of degree l up to ∼ 250. The a coe¬cients are most reliably

measured in the frequency range of ∼ 2 mHz to ∼ 4 mHz. The number of

terms in expansion is ∼ 20 or more.

In the previous section various inversion methods were brie¬‚y mentioned.

The classi¬cation there was based on what each method does mathemati-

cally. When discussing 2D inversions, we can also classify inversion methods

in terms of strategy they take in tackling the 2D problem.

The ¬rst is the most obvious ” direct 2D approach whose only fault is

that it is computationally most expensive (Sekii 1990, 1991, Schou 1991,

Christensen-Dalsgaard et al. 1995).

In the so-called 1.5D approach, „¦(r, θ) is expanded to reduce the 2D

problem to a set of one-dimensional (1D) problems (Brown et al. 1989):

„¦(r, θ) = „¦k (r)Qk (cos θ) ,

k

which, if Qk and Pk (m; l) have been chosen properly, leads to a set of 1D

problems

˜ nl

ak (n, l) = Kk (r)„¦k (r)dr (k = 1, 3, 5, . . .) .

As Sekii (1995) pointed out, with this approach one does not have control

over the angular resolution.

Then there is the 1D—1D approach. By exploiting the fact that, for p

modes,

Knl (r)[Plm (cos θ)]2 ,

Knlm (r, θ)

the 2D inverse problem is separated into radial and latitudinal 1D problems

(Sekii 1993, Pijpers & Thompson 1996). It is then possible to control both

the angular and radial resolution. Note that the 1D—1D approach is not

necessarily limited by the accuracy of the approximate formula above.

There are many combinations of choices of which basic method one uses

and which strategy one takes for a 2D case. A good news is that when

applied sensibly, they all agree reasonably well (cf. Schou et al. 1998). A

bad news is that none of these combinations can perform magic; irrespective

Sekii

270

Fig. 18.1. Di¬erential rotation in the sun inferred from inversion of MDI data.

Adapted from Schou et al. (1998).

of the way it is solved, the central region and the high-latitude region are

di¬cult to access because few splitting kernels have any amplitude there.

18.4.3 What we have learned

Figure 18.1 shows an example of rotation inversion of MDI data (adapted

from Schou et al. 1998). Let us summarize what we have learned from such

inversions of the current helioseismic data.

(i) The surface di¬erential rotation pattern holds through most of the the

solar convection zone. In particular, there is no Taylor-Proudman-

type cylindrical pro¬le in the convection zone, which compels numer-

ical modelling of a dynamical dynamo to be reconsidered.

(ii) Di¬erential rotation is weak, if present, in the radiative interior.

(iii) In low- to mid-latitude, the rotation rate is maximum around r/R

0.95, with a sub-photospheric shear layer immediately above. It has

been suggested that this layer supports a small-scale dynamo.

Seismology of solar rotation 271

(iv) There is another shear layer at the base of the convection zone, named

the tachocline (Spiegel & Zahn 1992, Gough & McIntyre 1998).

The sound-speed anomaly found also at the base of the solar convection

zone has been interpreted as evidence for extra mixing associated with the

tachocline (Gough et al. 1996). The classical ±-ω dynamo has di¬culty

in explaining equatorward migration of a dynamo wave, both due to the

magnitude and the sign of the radial gradient of angular velocity in the bulk

of the solar convection zone. An interface dynamo mechanism at the base

of the convection zone has been suggested (Choudhuri 1990, Parker 1993).

There are some other features such as high-latitude jet (Schou et al. 1998),

rotation slowing down near the pole (Schou et al. 1998) and zonal ¬‚ow

(Kosovichev & Schou 1997), which experts agree to be as real as the data,

in spite of their potentially controversial nature. The physical implications

of these features are yet to be understood. The variation over the solar cycle

has also been investigated (Howe et al. 2000). It seems that banded zonal

¬‚ows are moving towards the equator as a cycle progresses, in the region

down to ∼ 0.1R depth.

18.5 Rotation in the the solar convection zone