cylindrical rotation pro¬le in the solar convection zone. Some discussed that

it might be an artifact of the seismic analysis due to lack of spatial resolution

(Gough et al. 1993, Sekii, Gough & Kosovichev 1995). With improved data

and more spatial resolution, however, it now seems an unlikely explanation.

So why does the rotation rate in the solar convection zone not follow the

Taylor-Proudman behaviour? Let us start from the momentum equation for

the ¬‚uid velocity v in a rotating frame:

∇P

‚v

+ (v · ∇)v = ’ ’ ∇¦ ’ 2„¦0 — v ,

‚t ρ

where P and ¦ are pressure and gravitational potential, respectively, and „¦0

denotes the angular velocity of the rotating frame, which is to be identi¬ed

with the local angular velocity of the point under consideration. We drop

the LHS for a stationary ¬‚ow and with a small inertia term, and take the

curl of the RHS which has to vanish. From its φ component we have

(∇P — ∇ρ)φ (∇T — ∇P )φ

‚vφ

2„¦0 = = (18.4)

ρ2

‚z ρT

(vφ is the φ component of v), where a simple equation of state P ∝ ρT is

Sekii

272

Fig. 18.2. Rotation rate obtained by a non-linear model ¬tting to MDI data. From

top to bottom, angular velocities at the latitudes of 0—¦ , 30—¦ , 45—¦ , 60—¦ and 90—¦ .

assumed. If the ¬‚uid is barotropic, ∇T and ∇P are parallel and ‚vφ /‚z

must vanish. In fact, if we take a rather naive view of a perfectly spherical

sun, ∇T and ∇P are parallel irrespective of the barotropic nature of the

¬‚uid. However, the statement that ‚vφ /‚z vanishes, because of the adiabatic

thermal structure brought about by convection, is more robust and has to

be valid irrespective of how fast the sun might be rotating.

In deriving equation (18.4) we dropped a few terms. Also, there are terms

missing in the ¬rst place, such as those due to magnetic ¬eld or Reynolds

stress, which are certainly important in some parts of the sun, and may

be important in many other places in the current context of considering a

¬ne balance between quantities. Even so, let us suppose that we take the

equation seriously. Throughout the convection zone, ‚vφ /‚z is negative,

and so therefore is (∇T — ∇P )φ (in the northern hemisphere). This means

that the temperature gradient is inclined to the pressure gradient in such

a way that if we move up towards the pole along an isopressure surface,

temperature increases; the pole is warmer than the equator, which might

have been caused by anisotropy of the convective ¬‚ux. Coming back to the

missing terms in the equation, the baroclinicity may of course be due to a

magnetic ¬eld. Or, it may be the e¬ect of Reynolds stress and this may be

a way to start observational study of turbulent convection at high Re (see,

Seismology of solar rotation 273

Fig. 18.3. Nominal latitudinal temperature gradient along isopressure surface at

various radii, calculated from rotation rate presented in Figure 18.2.

however, Elliott 2003). As for the terms we dropped, note that vφ , the main

component of the ¬‚ow on a global scale, cannot produce, on its own, any

additional term in equation (18.4) through the time-dependent term or the

inertia term. The r- and θ-components of a small-scale velocity ¬eld can

contribute via the inertia term.

Let us go further to see how big is this baroclinicity. We maintain ax-

isymmetry but there is an angle between the temperature gradient and the

pressure gradient; we call it χ. We assume that the asphericity is so small

that for everything other than the angle χ and what are derived from this an-

gle, we can use a value of a spherical model of the sun. Then, the latitudinal

gradient of temperature, along local isopressure surface, is

1 ‚ log T ‚ log T sin χ

≈’ sin χ = ,

r ‚φ ‚r HT

where HT denotes temperature scale height. On the other hand, from equa-

tion (18.4) we have (the subscript ˜0™ has been dropped from „¦)

(∇T — ∇P )φ

‚vφ 1 ‚T ‚P P sin χ

≈

2„¦ = sin χ =

‚z ρT ρT ‚r ‚r ρHP HT

Sekii

274

(HP is density scale height). From these two equations it follows that

ρrHP ‚vφ

‚ log T

≈2 „¦ .

‚φ P ‚z

From

r2P

HP = ’ ,

ρGMr

where G is the gravitational constant and Mr the mass contained in a sphere

of radius r, and using vφ = r„¦ cos φ, we ¬nally have

’1

2R3

‚ log T Mr ‚„¦

≈ x4 „¦ cos φ (18.5)

‚φ GM M ‚ζ

(x = r/R and ζ = z/R ). In the middle of the convection zone around

the mid-latitude,

‚„¦

x ∼ 0.8 , Mr /M ∼ 1 , „¦ ∼ 400 nHz , ∼ 100 nHz ,

‚ζ

and by dropping (but not completely forgetting) cos φ, we have a rough

estimate of the temperature gradient:

‚ log T

∼ 10’7 .

‚φ

If we measure the latitudinal gradient along an isobar, the only change is

an additional multiplying factor that is the ratio of the density scale height

to the pressure scale height.

Figure 18.2 presents a result of a non-linear model-¬tting to MDI rota-

tional splitting data. A model with a rigidly rotating core, di¬erentially

rotating (of the form „¦0 (r) + „¦1 (r) cos2 θ + „¦2 (r) cos4 θ) outer zone and

a smooth transition between the two, has been ¬tted to the MDI data.

Figure 18.3 shows latitudinal variation of ‚ log T /‚φ at three radii in the

convection zone calculated using equation (18.5) from the rotation rate pre-

sented in Figure 18.2.

Taken at their face value, the results above imply a relative di¬erence

in temperature, between the pole and the equator, of the order of 10’7 .

This does not change much in the outer layers, as can be guessed from

equation (18.5) and from inversions. This is very small compared to the

relative di¬erence of the order of 10’4 , which was obtained from a surface

temperature observation (Kuhn 1988). The number is so small because,

essentially, the sun rotates so slowly.

Seismology of solar rotation 275

18.6 Line-blending problem

Among the global p modes, low-degree and high-frequency modes are the

ones that penetrate most deeply into the interior of the sun, and therefore

the most useful probes into the deep interior. However, for average splitting

measurements at low degree there was a disagreement between full-disk ex-

periments: some experiments came up with an average splitting of around

450 nHz (Toutain & Kosovichev 1994, Elsworth et al. 1995, Appourchaux et

al. 1995), which is basically the same as the rotation rate in the outer part

of the sun, but some came up with numbers close to or exceeding 500 nHz

(Toutain & Fr¨hlich 1992, Loudagh et al. 1993, Jim´nez et al. 1994). The

o e

former implies a more-or-less ¬‚at rotation pro¬le in the core, while the latter

requires a rapidly spinning core.

So which numbers are correct? Indeed, are any of them correct? There

was some concern because the discrepancies arose in the high-frequency

range where the mode linewidth became comparable with the rotational

splitting, which they were trying to measure. The analysis technique might

have been inadequate, or an accurate measurement might be simply impos-

sible.