2.2 Differential rotation and rotational splitting

The Sun’s rotation lifts the degeneracy between modes of the same l and different m, resulting in “rotational splitting” of the frequencies as waves propagating with and against the direction of rotation (prograde and retrograde) have higher and lower frequencies. To first order, the splitting δν ≡ ν − ν m,l − m,l +m,l is proportional to the rotation rate multiplied by m.

Figure 7View Image shows a typical m − ν acoustic spectrum of GONG data at l = 100. The effect of the rotation causes the ridges at each n to slant away from the ν = 0 axis; closer examination reveals that the ridges have an S-curve shape that arises from the differential rotation, and also shows the ridge structure due to leakage, which will be discussed below in Section 2.3.

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Figure 7: Spectrum for l = 100 in the ν,m plane (top) and detail (bottom) of a single ridge (radial order) showing the curvature due to differential rotation and the multiple-ridge structure arising from spherical harmonic leakage.

Because modes of different m values sample different latitude ranges, with the sectoral (|m | = l) modes confined to a belt around the equator and the zonal or m = 0 modes reaching to the poles, as illustrated in Figure 8View Image, we can measure the rotation as a function of latitude.

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Figure 8: Spherical harmonic patterns for l = 50: left, m = 0; center, m = 45; right, m = 50.

A given (n,l) multiplet consists of 2l + 1 frequency measurements if each (l,m) spectrum is analyzed separately, though some fraction of these frequencies may be missing in any given data set. This amount of data was somewhat unwieldy in the early days of helioseismology. It is therefore common to express νnl(m ) as a polynomial expansion, for example,

jm∑ax (l) νnlm = νnl + aj(n, l) 𝒫j (m ), (2 ) j=1
where the basis functions are polynomials related to the Clebsch–Gordan coefficients Clm j0lm by
∘ -------------------- (l) l--(2l −-j-)!(2l +-j-+-1)! lm 𝒫j (m ) = (2l)!√ 2l + 1 C j0lm (3 )
(Ritzwoller and Lavely, 1991). Indeed, in many analysis schemes coefficients of the expansion are derived by fitting directly to the acoustic spectrum and the individual frequencies are not measured. This approach can improve the stability of the fits, perhaps at the cost of imposing systematic errors. Early work used Legendre polynomials; however, most modern work uses either Clebsch–Gordan coefficients or the Ritzwoller–Lavely formulation, which come closer to being truly orthogonal for the solar rotation problem. Only the odd-order coefficients encode the rotational asymmetry, while the even-order coefficients contain information about the structural asphericity. Roughly speaking, the a1 coefficient describes the rotation rate averaged over all latitudes, and the a3 and higher coefficients describe the differential rotation.
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