aa r X i v : . [ a s t r o - ph . S R ] F e b Solar Interior Rotation and its Variation
Abstract
This article surveys the development of observational understandingof the interior rotation of the Sun and its temporal variation over ap-proximately forty years, starting with the 1960s attempts to determinethe solar core rotation from oblateness and proceeding through the devel-opment of helioseismology to the detailed modern picture of the internalrotation deduced from continuous helioseismic observations during solarcycle 23. After introducing some basic helioseismic concepts, it covers,in turn, the rotation of the core and radiative interior, the “tachocline”shear layer at the base of the convection zone, the differential rotationin the convection zone, the near-surface shear, the pattern of migratingzonal flows known as the torsional oscillation, and the possible temporalvariations at the bottom of the convection zone. For each area, the articlealso briefly explores the relationship between observations and models. Introduction
The internal rotation of the Sun is intimately related to the processes that drivethe activity cycle. Brown et al. (1989) stated that, “Knowledge of the internalrotation of the Sun with latitude, radius, and time is essential for a completeunderstanding of the evolution and the present properties of the Sun,” and thisremains true today.The Sun rotates on its axis approximately once every twenty-seven days;however, the rotation is not uniform, being substantially slower near the polesthan at the equator. This superficial aspect of the solar differential rotationwas well known from sunspot observations as early as the seventeenth century.However it is only within the last thirty years that it has become possible toobserve the rotation profile in the solar interior, and mostly within the mostrecent solar cycle that its subtle temporal variations have become evident. He-lioseismology – the study of the waves that propagate within the Sun and theinference from their properties of the solar interior structure and dynamics – isthe most important tool we have to measure this internal rotation.In this review, we start by introducing some of the basic concepts of helioseis-mology ( §
2) and the inversion problem ( §
3) as it applies to the internal solarrotation. Next, after a brief historical overview ( §
4) of the observations, weconsider what we have learned from helioseismology about the rotation profileand its variation with depth.We consider first the time-invariant part of the solar rotation profile. Themain features of interest are (Figure 1):1. the radiative interior and core, which appear to rotate approximately asa solid body, though the innermost core may behave differently; ( § § § . R ⊙ and the surface. ( § §
9) and the possible variations at the base of the convection zone( § The raw data of helioseismology consist of measurements of the photosphericDoppler velocity – or in some cases intensity in a particular wavelength band– taken at a cadence of about one minute and generally collected with as littleinterruption as possible over periods of months or years; the measurements canbe either imaged or integrated (“Sun as a Star”). An overview of the observationtechniques can be found in Hill et al. (1991a). Figure 2 shows a typical singleDoppler velocity image of the Sun, and Figure 3 a portion of an l = 0 time series,derived by averaging the velocity over the visible disk for each successive imagein a set of observations. The five-minute period and the rich beat structureare clearly visible in the time series. For an example of an integrated-sunlightspectrum from a long series of observations, see Figure 15.As was first discovered by Deubner (1975), the velocity or intensity variationsat the solar surface have a spectrum in k − ω or l − ν space that reveals theirorigin in acoustic modes propagating in a cavity bounded above by the solarsurface and below by the wavelength-dependent depth at which the waves arerefracted back towards the surface. These “ p modes” can be classified by theirradial order n , spherical harmonic degree l , and azimuthal order m ; as discussed,for example, in § l = 0 time series of Doppler velocity observations,showing the dominant five-minute period and the rich beat structure.5lement at time t , latitude θ and longitude φ can be written in the form δr ( r, θ, φ, t ) = l X m = − l a nlm ξ n l ( r ) Y ml ( θ, φ ) e iω nlm t , (1)where ξ nlm is the radial eigenfunction of the mode with frequency ω nlm and Y ml θ, φ is a spherical harmonic. As seen in Figure 4, the power in the spectrumfalls along distinct “ridges” in the l − ν plane, each ridge corresponding toone radial order. The modes making up the n = 0 ridge are the so-called f modes, which are surface gravity waves. The p modes, so called because theirrestoring force is pressure, are excited at the surface and have their largestamplitudes there. Another class of modes, the g modes with gravity as therestoring force, excited in the core and with amplitudes vanishing at the surface,are hypothesized to exist but have so far not been definitely observed ( § l − ν spectrum from one day of GONG observations. (Imagecourtesy NSO/GONG.)The longer the horizontal wavelength – and the lower the degree – the moredeeply the modes penetrate, with the radial l = 0 mode going all the wayto the core of the Sun (but providing no rotational information), while modeswith l ≥
200 or so penetrate only a few megameters below the surface andare generally too short-lived to form global standing waves; these are the modesused for local helioseismology. The lower turning point radius, r t , is a monotonicfunction of the phase speed ν/L , where L = p l ( l + 1) ≈ l + 1 /
2, as shown inFigure 5. The varying penetration depth with degree, as illustrated in Figure 6,makes it possible to deduce the rotation and other properties of the solar interiorprofile as a function of depth.The lowest-degree modes are observed in integrated sunlight, but for thepurposes of measuring the interior rotation profile we are mostly concernedwith what are termed medium-degree ( l ≤ ≈
10 arcsec) resolution. The6igure 5: Lower turning point of acoustic modes as a function of phase speed ν/L , based on Model S of Christensen-Dalsgaard et al. (1996).power in the modes peaks at about 3 mHz, or a period of 5 minutes; usefulmeasurements can be made for modes between about 1.5 and 5 mHz, withthe frequency determination becoming more challenging at the extremes due tosignal-to-noise issues and, at the high-frequency end, to the increasing breadthof the peaks.
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 propagatingwith and against the direction of rotation (prograde and retrograde) have higherand lower frequencies. To first order, the splitting δν m,l ≡ ν − m,l − ν + m,l isproportional to the rotation rate multiplied by m .Figure 7 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 shapethat arises from the differential rotation, and also shows the ridge structure dueto leakage, which will be discussed below in § m values sample different latitude ranges, withthe sectoral ( | m | = l ) modes confined to a belt around the equator and thezonal or m = 0 modes reaching to the poles, as illustrated in Figure 8, we canmeasure the rotation as a function of latitude.A given ( n, l ) multiplet consists of 2 l + 1 frequency measurements if each( l, m ) spectrum is analyzed separately, though some fraction of these frequencies7igure 6: Locations of modes in the l, ν plane for a typical MDI mode set. Thecolored shading represents the radial regions in which the modes have their lowerturning points; the core, r ≤ . R ⊙ , the radiative interior, 0 . ≤ r/R ⊙ ≤ . . ≤ r/R ⊙ ≤ .
75, the bulk of the convection zone, 0 . ≤ r/R ⊙ ≤ .
95, and the near-surface shear layer, r/R ⊙ ≥ .
95; the dashed line onthe lower right corresponds to r/R ⊙ = 0 . l = 100 in the ν, m plane (top) and detail (bottom) ofa single ridge (radial order) showing the curvature due to differential rotationand the multiple-ridge structure arising from spherical harmonic leakage.Figure 8: Spherical harmonic patterns for l = 50: (left, m = 0; center, m = 45,right; m = 50). 9ay be missing in any given data set. This amount of data was somewhatunwieldy in the early days of helioseismology. It is therefore common to express ν nl ( m ) as a polynomial expansion, for example, ν nlm = ν nl + j max X j =1 a j ( n, l ) P ( l ) j ( m ) , (2)where the basis functions are polynomials related to the Clebsch-Gordan coef-ficients C lmj lm by P ( l ) j ( m ) = l p (2 l − j )!(2 l + j + 1)!(2 l )! √ l + 1 C lmj lm (3)(Ritzwoller and Lavely, 1991). Indeed, in many analysis schemes coefficients ofthe expansion are derived by fitting directly to the acoustic spectrum and the in-dividual frequencies are not measured. This approach can improve the stabilityof the fits, perhaps at the cost of imposing systematic errors. Early work usedLegendre polynomials; however, most modern work uses either Clebsch–Gordancoefficients or the Ritzwoller–Lavely formulation, which come closer to beingtruly orthogonal for the solar rotation problem. Only the odd-order coefficientsencode the rotational asymmetry, while the even-order coefficients contain in-formation about the structural asphericity. Roughly speaking, the a coefficientdescribes the rotation rate averaged over all latitudes, and the a and highercoefficients describe the differential rotation. Spherical harmonic masks are used to separate the contributions from modes ofdifferent degree and azimuthal order into complex time series, which can thenbe transformed to acoustic Fourier spectra.The radial component of the velocity at the solar surface from a mode witha given degree l , azimuthal order m and radial order n is given by V n,l,m ( φ, θ, t ) = Re[ a n,l,m ( t ) P | m | l (cos θ ) e imφ )] , (4)where Re[] denotes the real part, φ is longitude and θ is latitude. (See, for exam-ple, Schou and Brown 1994a.) The masks used separate the different sphericalharmonics take the form M l,m ∝ Y l,m ( θ, φ ) A ( ρ ) , (5)where A is an apodization function and ρ ≡ p cos θ + sin θ sin φ representsthe fractional distance from disk center in the solar image. The line-of-sightprojection factor is V = p − ρ .Because only part of the solar surface is visible at any time, the masks arenot completely orthogonal and the modes “leak” into neighboring spectra. This10eakage complicates the analysis and can cause systematic errors in the measuredfrequencies if it is not correctly taken into account. For a detailed discussion ofthe calculation of the so-called “leakage matrix,” see Schou and Brown (1994a);Hill and Howe (1998). Briefly, the leakage matrix element s ( l, m, l ′ , m ′ ) for leak-age from the l ′ , m ′ mode to the l, m spectrum can be computed as s ( l, m, l ′ , m ′ ) = 1 π Z − Z π/ − π/ P ml ( x ) P m ′ l ′ ( x ) cos( mφ ) cos( m ′ φ ) V ( ρ ) A ( ρ ) dxdφ. (6)Symmetry properties in this expression lead to some simple exclusion rules; forexample, leaks with odd | δl + δm | (where δm ≡ m − m ′ and δl ≡ l − l ′ ) are notallowed.One example of the importance of the leakage is in the contribution of theso-called m -leaks ( δl = 0 , δm = ±
2) to the estimation of low-degree splittings.As pointed out, for example, by Howe and Thompson (1998), these leaks arestrongest for small | m | ; they are also asymmetrical, especially for | m | = l , wherethe m = l peak has an m = l − m = l + 2 leak on the other. Especially for l = 1, this can introduce a serioussystematic error into the estimate of the splitting if not properly accounted for.Leakage also means that integrated-sunlight instruments (which effectivelyuse the l = 0 mask) can detect modes of 0 ≤ l ≤
5, though the sensitivity fallsoff rapidly for l >
1. All these modes appear in a single acoustic spectrum; theinstruments are sensitive to odd- m modes for odd l and to even- m modes foreven l , with the sectoral, or | m | = l , modes most strongly detected.In general, the leakage has effects throughout the acoustic spectrum, but themost deleterious effects arise when the leaks cannot be resolved from the targetpeaks. This occurs for m -leaks at frequencies above about 2 mHz; for higher-degree modes the leakage between modes of adjacent l becomes a problem, asthe ridges become both broader, and more closely spaced in frequency, at around l = 150. Beyond this point the peaks cannot be fitted independently, and somemodeling of the leakage is essential in order to estimate the mode parameters. It is possible to make some inferences about the rotation profile without carryingout a full-scale inversion. Simple examination of the odd-order coefficients,sorted by the lower turning-point radius of the modes, reveals the existenceof the near-surface shear, the differential rotation within the convection zone,and a discontinuity in the differential rotation at the base of the convectionzone, as shown in Figure 9. More sophisticated analysis is also possible. Forexample, Wilson and Burtonclay (1995) gave approximate expressions for therotation profile at different latitudes as sensed by a particular n, l multiplet,¯Ω nl , as follows: ¯Ω nl ≈ a nl + a nl + a nl , (7)11igure 9: a (top) and a (bottom) coefficients for (left), 1986 BBSO observa-tions, (middle) 108 days of GONG observations in 1996, (right) the mean of 35consecutive 108-day periods of GONG observations from 1995 – 2005, plotted asa function of phase speed with the turning point radius marked on the upperaxis. 12Ω nl ≈ a nl − a nl − a nl , (8)¯Ω nl ≈ a nl − a nl − a nl , (9)¯Ω nl ≈ a nl − a nl a nl . (10)These estimates, where the subscripts on the LHS refer to the latitude in de-grees, are noisy for individual multiplets, but Wilson and Burtonclay (1995)were able to build up a picture of the internal rotation from BBSO data byforming cumulative averages with the input data sorted in ascending order of ν/L . 13 Inversion Basics
Various inversion techniques are used to infer the internal rotation profile fromthe observed frequency splittings. The aim of the inversion procedure is toform linear combinations of the data that give well-localized inferences of therotation at a particular location within the Sun. We will discuss only linearinversion methods, as non-linear approaches are not needed for the relativelylow velocities involved in the global rotation
The basic 2-dimensional rotation inversion problem can be stated as follows: wehave a number M of observations d i , from which we wish to infer the rotationprofile Ω( r, θ ) where r is distance from the center of the Sun, and θ is (conven-tionally) colatitude. Each datum is a spatially weighted average of the rotationrate: d i = Z R ⊙ Z π K i ( r, θ )Ω( r, θ ) drdθ + ǫ i , (11)where R ⊙ is the solar radius, the error term ǫ corresponds to the noise andmeasurement error in the data, and K is a model-dependent spatial weightingfunction known as the kernel (Hansen et al. , 1977; Cuypers, 1980). For the two-dimensional rotation inversion, the radial part is related to the eigenfunctionof the mode and the latitudinal part to the associated Legendre polynomial;Schou et al. (1994) give the expression for the kernel as K nlm ( r, θ ) = mI nl (cid:26) ξ nl ( r ) (cid:20) ξ nl ( r ) − L η nl ( r ) (cid:21) P ml ( x ) (12)+ η nl ( r ) L "(cid:18) dP ml dx (cid:19) (1 − x ) − P ml dP ml dx x + m − x P ml ( x ) (cid:21)) ρ ( r ) r sin θ, where I nl = Z R ⊙ [ ξ nl ( r ) + η nl ( r ) ] ρ ( r ) r dr, (13) x = cos θ , L = l ( l + 1), ξ nl is the radial displacement for the eigenfunction ofthe mode, L − η nl is the horizontal displacement, and ρ ( r ) is the density. (SeeFigure 10 for illustrations of sample kernels.)The aim of the inversion is to find¯Ω( r , θ ) = M X i =1 c i ( r , θ ) d i , (14)where ( r , θ ) is the location at which the inferred rotation rate ¯Ω is to be foundand the c i are the coefficients to be used to weight the data; the inversion processcan be thought of as the search for the best values for these coefficients.14igure 10: Sections through rotation kernels for selected azimuthal orders for l = 3 , n = 9 (top) and l = 20 , n = 5 (bottom).15 .2 Averaging kernels By substituting Equation (11) into the RHS of Equation (14) we obtain¯Ω( r , θ ) = Z R ⊙ Z π K ( r , θ ; r, θ )Ω( r, θ ) drdθ + ǫ i , (15)where K ( r , θ ; r, θ ) ≡ M X i =1 c i ( r , θ ) K i ( r, θ ) (16)is the averaging kernel for the location ( r , θ ). The averaging kernels are in-dependent of the values of the data. However, because the uncertainties in thedata are used to weight the inversion calculation that generates the coefficients c i , as described below in §§ et al. ,1992, 1994). If the errors on the input data are uncorrelated and properly described by anormal distribution whose width corresponds to the quoted uncertainty σ i , theformal uncertainty on the inferred profile is given by σ [Ω( r , θ )] = X i [ c i ( r , θ ) σ i ] . (17)In the (usually unrealistic) case where the errors on the input data are all equal,we can write σ [Ω( r , θ )] = Λ( r , θ ) σ, (18)where the “error magnification” is given byΛ( r , θ ) = X i [ c i ( r , θ ) ] / . (19)As discussed, for example, by Christensen-Dalsgaard et al. (1990), a quantita-tive choice of regularization parameters can then be made by finding the “knee”of a tradeoff curve where the error magnification is plotted against the width ofthe averaging kernel. However, in the two-dimensional case this does not alwaysgive a clear result, and this formulation of the error magnification is not veryuseful for modern data sets where the the uncertainties on the parameters areanything but uniform. Instead, one can consider the uncertainty on the inferredquantity at a particular location.Even when the errors on the input data are uncorrelated, the errors on theinferred profile will not be, as discussed by Howe and Thompson (1996). (As16 simple way to understand this, consider the case where one measurement issignificantly “off”; this will affect the inferred profile at every location wherethe inversion coefficient c i for that datum is non-zero.) In the one-dimensionalcase, the correlation between the errors for two points r and r is given by C ( r , r ) = P c i ( r ) c i ( r ) σ i [ P c i ( r ) σ i ] / [ P c i ( r ) σ i ] / ; (20)this can easily be generalized to the two-dimensional case. Howe and Thompson(1996) found that the spatial scale over which the inversion errors are signifi-cantly correlated is usually similar to that for the averaging kernels, though forsome cases where the inversion parameters have been badly chosen the resultscan be correlated over long distances even when the averaging kernels appearwell formed.Error correlations by definition should not distort the inferred profile beyondthe distribution predicted by the formal uncertainty on the inferences, providedalways that the input uncertainties are correct. However, the finite width ofaveraging kernels also gives rise to a systematic error that can be much larger.Consider, for example, the case where a thin shear layer is not resolved; then allthe estimated rotation rates on one side of the shear could be underestimated,and those on the other side overestimated, by several times the formal uncer-tainty. Such systematic errors and their relationship to the averaging kernelshave been discussed, for example, by Christensen-Dalsgaard et al. (1990).Gough et al. (1996) pointed out that it is not sufficient for the rotation ratesat two locations to have non-overlapping errors as calculated in Equation (17),and described a method for increasing the error estimates on inversions to al-low truly significant differences between the inferred rotation rate at differentlocations to be determined. This method, however, has not been widely used.Because the input data are noisy and of finite resolution, the inversion prob-lem does not have a unique solution; there will always be a tradeoff betweennoise and good localization. Two widely-used approaches to balancing these cri-teria are “regularized least squares” (RLS) and “optimally localized averaging”(OLA). The RLS approach to the inversion problem is to find (essentially through aleast-squares fit) the model profile that best fits the data, subject to a smooth-ness penalty term, or regularization. More regularization – a larger weight-ing for the penalty term – results in poorer spatial resolution (and potentiallymore systematic error) but smaller uncertainties. In one such implementation(Schou et al. , 1994), we minimize X i [ d i − R R R π ¯Ω( r, θ ) K i ( r, θ ) drdθ ] ( σ i / ¯ σ ) + µ r Z R Z π ( d ¯Ω dr ) drdθ + µ θ Z R Z π ( d ¯Ω dθ ) drdθ (21)17ith µ r and µ θ being the radial and latitudinal tradeoff parameters. The RLSinversion has the advantages of being computationally inexpensive and always(thanks to the second-derivative regularization, which amounts to an a prioriassumption of smoothness) providing some kind of estimate of the quantity ofinterest even in locations that are not, strictly speaking, resolved by the data.In this method, the averaging kernels K can (but need not be) calculated fromthe coefficients in a separate step. They are not guaranteed to be well localized,though they are forced to have a center of mass at the specified location r , θ .Figure 11 illustrates typical averaging kernels for a 2dRLS inversion of an MDIdata set. In the Subtractive OLA (SOLA) approach (Backus and Gilbert, 1968, 1970), theminimization is applied to the difference between the actual averaging kernels K and a target kernel T , for example a 2-dimensional Gaussian or Lorentzian func-tion. In this case (Pijpers and Thompson, 1992, 1994) the function minimizedis Z R Z π [ T ( r , θ ; r, θ ) − K ( r , θ ; r, θ )] rdrdθ + λ M X i =1 [ σ i c i ( r , θ )] . (22)Both the tradeoff parameter λ and the radial and latitudinal resolution of theinversions must be chosen before running the inversion. If the choice of targetkernel is poor – too narrow or too wide for the quantity and quality of the data– the reliability of the inversion will suffer. In OLA inversions, setting targetlocations outside the regions that can be resolved using the data will result inaveraging kernels displaced from their targets, and this should be taken intoaccount when interpreting the results. Figure 12 illustrates typical averagingkernels for a 2dSOLA inversion of an MDI data set.Another approach, older, and more computationally expensive, is the Multi-plicative OLA (MOLA) described by Pijpers and Thompson (1992, 1994). Here,no target form is imposed on the averaging kernel, but it is multiplied by a termwhich penalizes large values away from the target location. Alternatives to full 2-dimensional inversions are the so-called “1.5-dimensional”approach, in which 1-dimensional radial inversions are carried out separatelyfor each of the coefficients describing the latitudinal rotation variation, and“1 d ⊗ d ” inversions in which the radial and latitudinal variations of the rotationrate are integrated separately. For details of many of these methods, please seeSchou et al. (1998) and references therein.18igure 11: Averaging kernels for a typical RLS inversion of MDI data, fortarget latitudes 0 (a), 15 (b), 30 (c), 45 (d), 60 (e) and 75 (f) degrees as markedby the dashed radial lines, and target radii 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95,0.99 R ⊙ indicated by colors from blue to red as denoted by the dashed concentriccircles. Contour intervals are 5% of the local maximum value closest to thetarget location, with dashed contours indicating negative values.19igure 12: As Figure 11, for a SOLA inversion.20 .7 Limitations It is important to bear in mind the limitations of the inversion process whenconsidering the results. The deepest and shallowest depths that can be resolved,for example, are limited by the deepest and shallowest turning-point radii ofthe available modes. The rotational splitting at a given m is to first orderproportional to the rotation rate multiplied by | m | ; since the only mode whoselatitudinal kernel reaches the pole is the m = 0 mode, which has no longitudinalstructure and so can convey no rotational information, and the modes of small | m | /l have only a few nodes around the equator and hence have low sensitivityto the rotation, the 2d inversion becomes progressively less reliable at highlatitudes. Furthermore, since only modes of relatively low degree ( l ≤ Observations: A Brief Historical Overview
Systematic helioseismic observations stretch back nearly thirty years, as illus-trated in the schematic chart in Figure 13.Figure 13: Schematic time line of helioseismic observations in the last threesolar cycles (top panel), with the filled part of each bar representing approximateduty cycle, plotted on the same temporal scale as the butterfly diagram (bottompanel) of the gross magnetic field strength from Kitt Peak observations.Prior to the identification of global low-degree modes by Claverie et al. (1979), observing runs were usually short and carried out at a single site.However, the advantages of more extended observations (to obtain better fre-quency resolution), and of observations not modulated by the day-night cy-cle, were soon recognized. Grec et al. (1980) and Duvall Jr and Harvey (1984);Duvall Jr et al. (1984, 1986) carried out important observations at the SouthPole during the Austral summer, but for long time series it is more practical toobserve either from a network of sites spaced around the world, or from space.Some of the first long-term sets of low-degree observations came from theActive Cavity Irradiance Monitor (ACRIM) experiment (Woodard and Noyes,1985, 1988) aboard the Solar Maximum Mission spacecraft, which took helioseis-22ic measurements in 1980 and 1984 – 1985, the Mark I instrument in Tenerife(Pall´e et al. , 1989), and the precursors of the Birmingham Integrated Solar Net-work (BiSON) (Elsworth et al. , 1990a). Meanwhile, resolved-Sun observationswere carried out at the South Pole by Duvall and collaborators, and by variousother observers in the USA; these observations will be discussed in more detaillater.Libbrecht and Woodard (1990) observed the medium-degree modes fromBig Bear Solar Observatory (BBSO) in the 1986 – 1990 rising phase of so-lar cycle 22. The first observations from widely separated sites were carriedout by the Birmingham/Tenerife group in 1981 (Claverie et al. , 1984), and by1992 the six-station BiSON network was complete; it has been operating eversince. Another network of integrated-sunlight instruments, the French-basedIRIS (Fossat, 1995), operated from 1989 – 2003.The Global Oscillation Network Group (GONG) (Harvey et al. , 1996) hasbeen collecting continuous, high-duty-cycle observations of the medium-degree p modes since 1995, using a six-station worldwide network, and the MichelsonDoppler Imager (MDI) instrument (Scherrer et al. , 1995) aboard the SOHOspacecraft has been in operation since 1996, so that these two projects haveessentially complete coverage of solar cycle 23. SOHO also carries instrumentsdedicated to the study of low-degree oscillations; LOI (Luminosity OscillationsImager) (Frohlich et al. , 1995), and GOLF (Global Oscillations at Low Frequen-cies) (Gabriel et al. , 1995). This wealth of high-quality data has given us theopportunity to study the solar interior rotation and its solar-cycle changes inmore detail than ever before.Also worth noting are the LOWL-ECHO project (Tomczyk et al. , 1993)which made medium-degree observations from one or two sites from 1994 to2004. and the high-degree Taiwanese Oscillations Network (Chou et al. , 1995)deployed over the 1993–1996 period.All these observations will be considered in more detail as we proceed toexamine the results pertaining to the interior rotation.23 The Core and Radiative Interior
Interest in the rotation of the deep solar interior predates systematic helioseismicobservation. One other possible diagnostic of the internal rotation is providedby the solar oblateness; because the Sun is not a solid body, both gravitationaland rotational effects cause a very slight flattening. The lowest-order term inthis effect is related to the quadrupole moment J ; confusingly, the next-highestterm, J , is sometimes called octopole and sometimes hexadecapole. Accordingto Rozelot and Roesch (1997), who give a useful review of attempts to measurethe solar oblateness, for a non-rotating Sun the oblateness ∆ r = r eq − r pol isgiven by ∆ rr = 32 J , (23)where r eq and r pol are the equatorial and polar radii, respectively, and r is theradius of the best sphere passing through r eq and r pol . If there is an additional δr contribution from the surface rotation this expression becomes∆ r − δrr = 32 J . (24)The units of δr and ∆ r are conventionally arc ms.Dicke (1964) noted that, if the Sun were oblate because of fast interior ro-tation, the effect on its gravitational potential might destroy the agreementbetween the predictions of General Relativity and the observations of the per-ihelia of the inner planets, (specifically Mercury, though Venus could in princi-ple experience a smaller effect) potentially leaving room for alternate theoriesof gravitation. Dicke set out to determine the solar oblateness from ground-based measurements – a challenging endeavor that produced controversial re-sults. Models (e.g., Brandt 1966) suggested that the interior of the Sun couldstill be spinning at the rapid rate at which it originally formed, while the exte-rior had been slowed down by the torque of the solar wind. (As will be furtherdiscussed in Section 6, in the absence of direct observations of the solar interiorthe picture of solar interior dynamics was not at all clear, although the exis-tence of something like what we now call the tachocline could be inferred fromtheoretical arguments.) Dicke and Goldenberg (1967b) reported finding a solaroblateness value of 5 × − , which would be sufficient to create an 8% discrep-ancy between observations and the Einsteinian prediction for the precession ofthe perihelion of Mercury, and would imply a fast-rotating core.The results, and the inferences Dicke and collaborators drew from them,raised a storm of controversy that may well have helped to stimulate interest inthe Sun’s interior rotation profile. The criticisms and Dicke’s responses to themwould fill a lengthy review article by themselves; we give only a few exampleshere. Roxburgh (1967) suggested that the result might be explained by thesolar differential rotation, an idea rejected by Dicke and Goldenberg (1967a).Howard et al. (1967) concluded, on the basis of a variety of simple models of the24olar “spin-down,” that the Sun should have reached a state of uniform rotationquite quickly after its initial formation. Sturrock and Gilvarry (1967) pointedout that the presence of magnetic field in the solar interior might well com-plicate the issue, and in an accompanying article Gilvarry and Sturrock (1967)suggested using a space probe in a highly eccentric orbit as a more direct testof general relativity – or, alternatively, that “more complete theoretical andobservational knowledge of the visible layers and the interior of the Sun” wasneeded.At least partly inspired by the controversy, Kraft (1967) studied the rota-tional velocities of young solar-type stars in the Pleiades and concluded thatangular momentum was lost on a timescale of about half a billion years, butnoted in his conclusion that “it is wrong to conclude that the present work inany way supports the Dicke result.” Goldreich and Schubert (1968) consideredthe stability of differentially rotating stars and concluded that it was possi-ble but not likely that a radial rotation gradient such as that required by theDicke and Goldenberg (1967b) result might exist.H. Hill, a former colleague of Dicke who had helped build the instrumentwith which the 1964 observations were made (Dicke, 1964), and collaborators,also attempted to measure the solar oblateness, using an instrument, SCLERA[Santa Catalina Observatory for Experimental Relativity by Astrometry], whichwas later to play a role in the early days of helioseismology. This measurement,carried out in 1973, (Hill and Stebbins, 1975), found a 9 . × − value for theoblateness, much smaller than that of Dicke and Goldenberg (1967b); Hill et al. (1974) also pointed out a time-varying difference between the brightness of thesolar limb and poles that might account for the anomalously high oblatenessmeasurement.Ulrich and Hawkins (1981a,b) made an early attempt to deduce what the J and J terms should be based on a simple differential rotation profile deducedfrom surface measurements, obtaining predicted values of between 1 and 1 . × − for J and between 2 and 5 × − for J depending on the size of theconvective envelope.Dicke et al. (1986, 1987) repeated the 1966 measurements with an improvedinstrument, and obtained significantly smaller values for the oblateness, withsome weak evidence for a solar-cycle variation. Lydon and Sofia (1996) mademeasurements using a balloon-based instrument and obtained values of 1 . × − for J and 9 . × − for J . By this point, however, the focus in thesolar oblateness studies had moved away from trying to infer the core rotation.Mecheri et al. (2004) used more realistic models of the internal rotation profile tosuggest that the J term should be particularly sensitive to the subsurface shear.Recent work on determining the oblateness from the shape of the solar limbhas taken into account considerations of near-surface temperature or magneticvariations. Kuhn et al. (1998); Emilio et al. (2007) used observations from MDIduring rolls of the SOHO spacecraft and Fivian et al. (2008) used the RHESSIX-ray telescope. The work with SOHO revealed a temporal variation in theshape of the solar limb, with greater apparent oblateness at solar maximum,suggesting that hotter, brighter activity belts have greater apparent diameter.25his poses an apparent contradiction to the results obtained from helioseismicinferences of the asphericity. Indeed, Fivian et al. (2008) suggest that all thetemporally-varying, excess oblateness found in the observations can be correctedaway by removing an ad-hoc term related to magnetic elements in the enhancednetwork.Meanwhile, a much more flexible tool – helioseismology – had become avail-able for probing the interior solar rotation. Around the early 1970s there were numerous attempts to search for global p -mode oscillations, with interest at first focusing on longer-period oscillations, thelow-order, low-degree modes. Various theoretical predictions (Scuflaire et al. ,1975; Iben Jr, 1976; Christensen-Dalsgaard and Gough, 1976) of the periodswere available, offering the hope that global oscillations could be used to probethe rotation and structure deep inside the Sun. At first most of the results(Livingston et al. , 1977; Musman and Nye, 1977; Grec and Fossat, 1977), werenegative, except for the 160-minute period of Severnyi et al. (1976); Brookes et al. (1976), which was later (Elsworth et al. , 1989) determined to be spurious andwill not be further discussed here. The SCLERA group (Brown et al. , 1978;Hill and Caudell, 1979; Caudell and Hill, 1980) found a variety of longer-periodfluctuations in their solar-diameter data, but these results were not universallyaccepted; for example Fossat et al. (1981a, see also references therein) claimedthat the SCLERA results were consistent with pure noise.Low-degree helioseismology became a reality when the Birmingham group(Claverie et al. , 1979) identified oscillations in the five-minute frequency band inintegrated sunlight as low-degree global modes, using observations from Tenerifeand Pic du Midi during the summers of 1976 – 1978; these initial data wereadequate only to identify the spacing between modes of the same l and different n , without resolving separate l = 0 and l = 1 peaks.A French-American team (Grec et al. , 1980; Fossat et al. , 1981b) obtainedfive days of continuous observations at the South Pole in the austral summerof 1979 – 1980, and were able to identify peaks of degree 0, 1, 2, and 3 andeven a weak l = 4 peak by superposing sections of the acoustic spectrum withdifferent radial order. These modes were identified as being of radial orderaround 12 – 30, as opposed to the very low-order modes that had been soughtin the low-frequency spectrum; both the noise characteristics of the spectrumand the low amplitude of the lower-order modes mean that the fundamental( l = 0 , n = 0) mode remains unobserved to this day, although some low-degreemodes with single-digit n have been identified (Chaplin et al. , 1996b).Soon, the Birmingham team (Claverie et al. , 1981), using 28 days of integrated-sunlight data from the Tenerife site and an analysis that involved “collapsing”segments of the acoustic spectrum so as to average together modes of the samedegree and different radial order, reported finding three rotationally split com-ponents in the l = 1 modes and five in l = 2, with an average separation of0.75 µ Hz. If correct, this would have implied a solar core rotation substantially26aster than the surface. Isaak (1982) suggested that the excess component peaks(when two and three would be the expected number for l = 1 and l = 2 respec-tively) could be explained if the solar core were rotating on an oblique axis andhad a very strong magnetic field; this idea, which was also mooted by Dicke(1983) to explain an oscillation of about half the solar rotation period seen inthe oblateness data (Dicke, 1976), was rebutted in some detail by Gough (1982).Fossat et al. (1981b) reported that initial results from 5 days of low-degreeobservations at the South Pole suggested quite short lifetimes, about 2 days; the l = 0 peaks appeared narrower than those of l = 1 and l = 2. Grec et al. (1983)later identified about 80 normal modes in the South Pole data, but did notconfirm the Claverie et al. (1981) rotational splitting result, instead reportingthat the l = 1 peak seemed too narrow to accommodate the reported splitting.Claverie et al. (1982) reported a periodicity of approximately 13 days in theradial solar velocity, as measured using the resonant-scattering technique andthe potassium D-line, and interpreted this as an effect of the solar core rotation;however, this effect was quickly explained away (Durrant and Schroeter, 1983;Andersen and Maltby, 1983; Edmunds and Gough, 1983; Duvall Jr et al. , 1983)as an artifact caused by the rotation of surface features – sunspots and plage –across the disk.Meanwhile, the low-degree five-minute acoustic spectrum had also been ob-served using the Active Cavity Irradiance Monitor (ACRIM) aboard the SolarMaximum Mission (SMM) spacecraft (Woodard and Hudson, 1983a). Woodard and Hudson(1983b) agreed with Fossat et al. (1981b) in finding that the modes had life-times of about two days, too short for the rotational splitting reported byClaverie et al. (1981) to be real.Later work (Libbrecht, 1988a; Elsworth et al. , 1990b; Chaplin et al. , 1997)revealed that the width of the peaks – inversely proportional to the mode life-times – was strongly dependent on frequency across the five-minute spectrum,with lifetimes of a few days in the middle of the five-minute band and weeks ormonths at low frequencies where, unfortunately, the amplitudes of the modesare also small. Reliable direct measurement of the low-degree splittings wouldhave to wait for some years, while sufficiently long, high-quality time series ofdata accumulated. In the meantime, resolved-Sun observations provided some information aboutthe rotation in the radiative interior. Duvall Jr and Harvey (1984) reportedobservations at Kitt Peak, from 10–26 May 1983, for degrees 0 ≤ l ≤ l = 6, with an unexplainedbump at l = 11, followed by an increase at lower degrees up to a value of 660 nHzfor l = 1. These data, inverted by Duvall Jr et al. (1984), yielded a rotationprofile with much of the radiative interior rotating at or below the surface rate,but with a modest increase in the interior. A similar pattern was found byBrown (1985), using 6 days of observations from the newly-developed Fourier27achometer, a true 2-dimensional imaging instrument that gave access to allthe azimuthal orders for degrees between 8 and 50; however, the coincidence ofthe l = 11 bump seems to have been merely a coincidence of noise, as it wasnot reproduced in the early observations from the Big Bear Solar Observatory(Libbrecht, 1986). Hill et al. (1982) derived splittings from the SCLERA low-frequency peaks,and from those inferred a core rotating at 6 times the surface rate; however,Woodard (1984) used ACRIM data to place an upper limit of 2.2 times the sur-face rate on the interior rotation rate, inconsistent with these splittings. Later,Hill (1985) identified low-degree rotational splittings in the five-minute band ofthe SCLERA acoustic spectrum, but Libbrecht (1986) and Brown et al. (1989)found that these results were inconsistent with the other evidence and wereprobably the result of misidentification of the modes. Given the complexity ofthe spectrum in question, whose derivation from measurements sampled at afew points on the solar limb made it difficult to separate out spectra of differentdegree, this seems a likely explanation.
The next several years were active ones for low-degree helioseismology, withthe development of the BiSON (Birmingham-based) and IRIS (based in Nice)networks. Together with the IPHIR instrument that rode the PHOBOS space-craft on its cruise phase to Mars, and the ground tests of the LOI (LuminosityOscillations Imager) instrument that would later be mounted on the SOHOspacecraft, these brought a succession of estimates of the low-degree splitting,as summarized in Table 1 and Figure 14. In addition to the MDI instrumentfor medium and high-degree observations, the SOHO spacecraft carried bothLOI and GOLF (Global Oscillations at Low Frequencies) specifically for ob-serving low-degree modes. Even though GOLF malfunctioned and could notbe operated in its intended differential mode, instead being confined to makingDoppler observations on one side of an absorption line, it provided some of thebest available long-term, low-degree observations.The reported results show considerable variation, but apart from the earlyTenerife result, which was based on much shorter and lower-duty-cycle observa-tions than most of the others, they all cluster around the surface rotation rate,some (particularly the IRIS results) pointing to a core rotation faster than thesurface rate and some (in particular the BiSON results) to one substantiallybelow it, perhaps as low as zero. As we approach the present time and theobservation and analysis improve, the values tend to converge on a splittingquite close to that which would correspond to the surface rate. Early in thisperiod, there was room to speculate (e.g., Chaplin et al. , 1996a) that the dif-ferences reflected a temporal variation, but this could not explain away all thediscrepancies. 28igure 14: l = 1 splitting estimates as a function of publication date.Table 1: Summary of l = 1 splitting measurements, 1988 – 2002 Reference Project δν ( µ Hz)
Comment
Pall´e et al. (1988) Tenerife 0.75 summers of 1981-1986Toutain and Fr¨ohlich (1992) IPHIR 0 . ± .
017 Intensity measurementson PHOBOS spacecraftLoudagh et al. (1993) IRIS 0.494 Based on 3 low-frequencymodes.Jim´enez et al. (1994) Tenerife 0 . ± . et al. (1994) Tenerife 0 . ± . . ± . et al. (1995) BiSON 0 . ± . et al. (1995) LOI 0 . ± . l = 2Chaplin et al. (1996a) BiSON 0 . ± . et al. (1996) IRIS 0 . ± . et al. (1997) GOLF 0 . ± . et al. (1997) IRIS 0 . ± . et al. (2000) GOLF 0 . ± . et al. (2000) MDI 0 . ± .
011 Asymmetric profileChaplin et al. (2001) BiSON 0 . ± . et al. (2002) GOLF 0 . ± . l = 1 peak (red). 30 .6 Pitfalls of low-degree splitting measurements Unfortunately, all the measurements described in Section 5.5 suffer from similarproblems, as summarized below.1. The two components of the l = 1 mode are so close together (probably lessthan one microhertz apart) that they are resolved only for modes belowabout 2.2 mHz. This has implications for the measurements:(a) Estimates of the splittings of unresolved components are highly proneto systematic error (Appourchaux et al. , 2000a).(b) The components that can be resolved have small amplitudes (Fig-ure 15) and therefore require both observations over extended periodsand high signal-to-noise ratios.(c) On the other hand, these low-frequency modes have the advantagethat they show very little frequency shift with the solar cycle, whichsimplifies the analysis of long time series.2. Even though the low-degree modes penetrate deep into the solar interior,they spend most of their time in the outer layers of the Sun and are notvery sensitive to the core; conversely, estimates of the core rotation arevery sensitive to small errors in the splitting measurements.3. In order to properly estimate the rotation profile in the deep interior it isnecessary to combine the low-degree splittings with medium-degree onesin an inversion. However, because the low-degree modes are so few –a few dozen at most, compared to a couple of thousand medium-degreemultiplets with tens of thousands of individual frequencies or coefficients –the need for extremely precise measurements is even more pressing. Also,combining data from different instruments with different systematic errorsmay cause problems, particularly if the observations were made at differentepochs of the solar cycle.Point 1 above was noted by Loudagh et al. (1993) and Elsworth et al. (1995),and point 2 by Loudagh et al. (1993) and Lazrek et al. (1996), who point outthat “An accuracy of about 30 nHz, or (1 year) − on the measurement of the l = 1 rotation splitting does not really permit, then, to discriminate between asolar core rotating twice as fast as the rest or not rotating at all!” An approachto addressing point 3 was made by Tomczyk et al. (1995) with the newly-builtLOWL instrument, an imaging instrument optimized for lower degrees. Theyobtained splittings for 1 ≤ l ≤ . R ⊙ , finding a rotation rate that barely varied with radius between 0 . R ⊙ and 0 . R ⊙ , apart from a low-significance bump around 0 . R ⊙ .Eff-Darwich and Korzennik (1998) further addressed point 3 when they com-bined results from several different instruments, including GONG, BiSON, MDI,and GOLF. They give a nice illustration of the tendency of higher-frequencylow-degree mode splittings to be biased upward by the mode width, a point31hat was further illustrated by Chaplin et al. (2001), and conclude that withthe then-available data it is not possible to rule out fast rotation in the corebelow 0 . R ⊙ .Charbonneau et al. (1998) used a genetic forward-modeling approach to an-alyze the LOWL data, with results favoring a rigidly-rotating core. Starting around the turn of the century, there was a move towards more collab-orations and comparisons between different projects in an effort to understandthe systematic errors and better constrain the solar core dynamics. By this time,multi-year observations were available from GONG and the SOHO instruments,as well as good-quality observations from BiSON stretching back to 1991.Chaplin et al. (1999) combined the LOWL higher-degree splittings with thevery precise low-frequency BiSON splittings for the lowest-degree modes, andconcluded that the data were consistent either with rigid rotation or with aslight downturn in the rotation rate in the core (the latter being at best a 1- σ result); on the other hand, Corbard et al. (1998b) had used a very similaranalysis of GOLF and MDI data to deduce a slight increase in the rotationrate below 0 . R ⊙ , but Garc´ıa et al. (2003), also using MDI and GOLF data,obtained rather low splitting values from a 2243-day time series and tentativelyconcluded that they could rule out a high rotation rate in the core.Eff-Darwich et al. (2002), following on from the work of Eff-Darwich and Korzennik(1998), again combined BiSON, GOLF, GONG and MDI data and found a verysmall downturn in rotation in the core, while Couvidat et al. (2003) found a flatrotation profile down to 0 . R ⊙ using combined GOLF, MDI and LOWL data.Fletcher et al. (2003) investigated the problem of fitting the poorly-resolvedhigher-frequency low-degree mode splittings to integrated-sunlight observationssuch as those from BiSON. Using genetic fitting algorithms, they were ableto reduce, though not eliminate, the bias towards higher splittings for thesefits. They also found, in common with previous work, a strong anticorrelationbetween the estimated splitting value and its formal error, which would tend tocause overestimated splittings to be more heavily weighted in inversions.Garc´ıa et al. (2004) considered two years of “sun-as-a-star” observationsfrom early in the solar cycle, obtained from GOLF, GONG, MDI, VIRGO andBiSON, and were able to extract not only sectoral splittings but also a and a coefficients from the data, suggesting that it may be possible to infer differentialrotation even in stars from which we will never have resolved data.Chaplin et al. (2004) used artificial data to address the question the de-tectability of a rotation-rate gradient in the core. They concluded that, basedon the best available data from ten years of observations, the difference betweenthe rotation rate at 0 . R ⊙ and 0 . R ⊙ would be detectable only if it exceeded110 nHz.Chaplin et al. (2006) carried out an exhaustive “hare-and-hounds” exercise,in which one participant (the “hare” supplies the same set of artificial data to32he others, the “hounds,” who then apply their various fitting methods with-out knowing the “true” answer, and compare the results. They obtained goodagreement between the different techniques for l = 1, but systematic differencesfor the l > | m | < l ) components. To summarize, the best evidence we have so far seems to imply that the ro-tation rate between about 0 . R ⊙ and the base of the convection zone is mostlikely approximately constant with radius and spherically symmetric. It is notpossible to rule out a different rotation rate for the inner core, but there is noevidence from p -mode observations to support such a difference. Between about0 . R ⊙ and the base of the tachocline, no significant departure from rigid-bodyrotation has been found. As discussed by Eff-Darwich et al. (2002), for examplethe available constraints already seem to rule out the simplest models of hydro-dynamic spin-down, which would show a detectable increase in the rotation ratebelow 0 . R ⊙ . Understanding both of the relationship between p mode splittingsand the interior rotation, and of the care needed to measure them, has greatlyadvanced since the early days of helioseismology, but the rotation rate of theinnermost nuclear-burning core remains uncertain. One possible way to improve the constraints on the core rotation would be touse g modes, or gravity waves, instead of p modes. Because these modes havetheir greatest amplitude in the solar interior, they should be much more sensi-tive to the core properties. Unfortunately, they also have very small amplitudesat the surface. The history of helioseismology is littered with unconfirmed re-ports of g -mode identification; see, for example, Delache and Scherrer (1983);Van der Raay (1988); Thomson et al. (1995), and the review by Hill et al. (1991b).The most promising recent work has been carried out using long time series fromthe GOLF instrument aboard SOHO. Appourchaux et al. (2000b) placed an up-per limit of 10mm/s on g -mode amplitudes based on two years of observations,and Gabriel et al. (2002) reduced this limit further, to 6mm/s, using 5 yearsof data. Most recently, Garc´ıa et al. (2007) report finding a pattern of peakswith constant spacing in period corresponding to the model-predicted spacingfor l = 2 g modes with δl = 0 , δn = 1, and with a splitting that they interpretas corresponding to a core rotation rate of 3 – 5 times the surface rate; however,this is still a preliminary result in need of confirmation.In a related paper, Mathur et al. (2007) point out that the current predic-tions for low-order g -mode frequencies are much more consistent than was thecase a decade earlier, resulting in a period for the fundamental g -mode between34 – 35 minutes. This finding does make one wonder about the usefulness ofthe g -mode observations for discriminating among models; on the other hand,it lends somewhat more credence to the current identification.33 The Tachocline
While the bulk of radiative interior appears to rotate almost as a solid body,the base of the convection zone at 0 . R ⊙ coincides with a region of strongradial shear, above which the convection zone exhibits a differential rotationpattern that depends strongly on latitude and only weakly on depth. Thisshear layer is known as the tachocline , a term introduced to the literature bySpiegel and Zahn (1992), who attribute to D.O. Gough the correction of theearlier term “tachycline” (Spiegel, 1972). As is evident from the date of thelatter reference, the notion of a shear layer at the bottom of the convection zonehad been present in models for some time prior to its observational discovery,though its exact location was somewhat uncertain.The existence of a layer of radial shear around the base of the convectionzone, with approximately solid-body rotation below it, was first demonstratedby Brown et al. (1989), using the data of Brown and Morrow (1987); however,the significance of their results was quite low and they were at pains to point outthat other interpretations of the data were possible. Dziembowski et al. (1989)used BBSO data to improve the picture of rotation at the base of the convec-tion zone, again finding that the low-latitude rotation rate increased, and thehigh-latitude rate decreased, towards a common value at the base of the convec-tion zone. The position of the base of the convection zone was determined byChristensen-Dalsgaard et al. (1991) using sound-speed inversions of helioseismicfrequencies from the work of Duvall Jr et al. (1988) and Libbrecht and Kaufman(1988); their value of 0 . R ⊙ , confirmed by Basu and Antia (1997), has beenaccepted ever since.The discovery of this shear layer (as pointed out by Brown et al. ) offered asolution to the puzzle of the apparent absence of a radial gradient of rotationin the convection zone that could drive a solar dynamo, leading to speculationthat the dynamo must operate in the tachocline region instead of in the bulk ofthe convection zone.The tachocline lies in the region where modes of l ≈
20 have their lowerturning points, and the resolution of the inversions is quite low – about 5 – 10%of the solar radius in the radial direction. The thickness of the shear layeris therefore likely not to be resolved in inversions, and some ingenuity (andforward modeling) is required to estimate it and account for the effect of thefinite-width averaging kernels in smoothing out the inversion inferences. Theresults of various efforts to parameterize the tachocline shape at the equator aresummarized in Table 2. They mostly concur in placing the centroid of the shearlayer slightly below the seismically-determined base of the convection zone, andits thickness at around 0 . R ⊙ . The largest value for the thickness, that ofWilson et al. (1996b), was obtained using forward calculation and direct com-bination of splitting coefficients rather than a true inversion, while the very lowvalue of Corbard et al. (1999) was obtained using an inversion technique specifi-cally designed to allow a discontinuous step in the rotation rate at the tachocline.The analysis of Elliott and Gough (1999) was somewhat different from the oth-ers, in that it involved calibrating a particular model of the tachocline against34able 2: Tachocline radius r and width Γ. Reference r/R ⊙ σ r /R ⊙ Γ /R ⊙ σ Γ /R ⊙ Project
Kosovichev (1996) 0.692 0.005 0.09 0.04 BBSOWilson et al. (1996a) 0.68 0.01 0.12 – BBSOBasu (1997) 0.705 0.0027 0.0480 0.0127 GONGAntia et al. (1998) 0.6947 0.0035 0.033 0.0069 GONGCorbard et al. (1998a) 0.695 0.005 0.05 0.03 LOWLCorbard et al. (1999) 0.691 0.004 0.01 0.03 LOWLCharbonneau et al. (1999) 0.693 0.002 0.039 0.002 LOWLElliott and Gough (1999) 0.697 0.002 0.019 0.001 MDIBasu and Antia (2003) 0.6916 0.0019 0.0162 0.0032 MDI, GONGthe inferred sound-speed rather than against a rotation profile.Antia et al. (1998) and Corbard et al. (1999) found no significant evidencefor a variation in the position or thickness of the tachocline with latitude, butCharbonneau et al. (1999) found a significant prolateness, with the tachocline(0 . ± . R ⊙ shallower at latitude 60 ◦ than at the equator. Basu and Antia(2003) also found a slightly thicker and shallower tachocline at high latitudes,and speculated that the tachocline location might be discontinuous at the lati-tude (around 30 ◦ ) where the shear vanishes and changes sign. Even the most generous estimates for the observed tachocline thickness are smallenough to pose an interesting theoretical question: what prevents the shear fromspreading further into the radiative interior, destroying the observed uniformrotation? The literature on tachocline modeling is extensive, far beyond thescope of this review. In brief, three main candidate mechanisms have been pro-posed: turbulent flows (Spiegel and Zahn, 1992); “fossil” magnetic fields (e.g.,Gough and Mcintyre 1998); and gravity waves, known to observational helio-seismologists as g modes (e.g., Zahn et al. et al. (2007). 35 Rotation in the Bulk of the Convection Zone
The surface differential rotation, with the equator rotating faster than the poles,was known from, for example, sunspot tracking, long before helioseismologyopened up the solar interior. Most models in the pre-helioseismology era pre-dicted or assumed a rotation rate constant on cylinders parallel to the axis ofrotation. This is a consequence of the so-called Taylor-Proudman constraint, awell-known result in fluid dynamics.Duvall Jr and Harvey (1984); Duvall Jr et al. (1984) observed from the SouthPole, using only sectoral modes; their instrument used intensity images in a Cal-cium absorption line, scanning rather than imaging the whole Sun at once. Theirmain conclusion was that, “Most of the Sun’s volume rotates at a rate close tothat of the surface.”Brown (1985) had a different instrument, the Fourier Tachometer, whichproduced 100 ×
100 pixel velocity images. Brown’s initial crude analysis of fivedays of data used cross-correlation, and expanded the multiplet frequencies usinglow-order polynomial fits; the results showed little sign of any depth variationin the differential rotation.Duvall Jr et al. (1986), again using data from South Pole observations butnow covering the full range of azimuthal orders, found values of the a coeffi-cient (the first-order measure of differential rotation) consistent with the surfacerotation and rather larger than was consistent with the results of Brown (1985).Brown and Morrow (1987), with 15 days (not all consecutive) of FourierTachometer data, could not distinguish between rotation on cylinders and latitude-dependence, but found that there was definitely less differential rotation in theradiative interior below the convection zone; their a values were now closerto those of Duvall Jr et al. (1986), and they declared the previous ones erro-neous. Brown et al. (1989) carried out a much more detailed analysis of theBrown and Morrow (1987) data, strengthening the evidence for mostly depth-independent rotation in the convection zone, as shown in Figure 16.Both the South Pole observations and those of Brown and collaboratorswere relatively noisy and of poor resolution; although they strongly hinted ata picture with little radial differential rotation in the convection zone and littledifferential rotation at all below it, other interpretations were possible.Libbrecht (1989) published splittings from 100 days of BBSO observationsin summer 1986, broadly confirming the results of Brown et al. (1989) withsubstantially smaller uncertainties. Dziembowski et al. (1989) inverted thesedata, and inferred a sharp radial gradient at the base of the convection zoneand roughly constant rotation at each latitude above that. They also found abump in the rotation rate in the middle of the convection zone, to which wewill return below. Other inversions of the same data set were presented byChristensen-Dalsgaard and Schou (1988) and Libbrecht (1988b), with similarresults, though not all the early inversions (c.f. Korzennik et al. et al. (1989), reproducedby permission of the AAS.by Schou et al. (1992), who illustrated their averaging kernels; these were ratherbroad, but adequate to rule out a rotation-on-cylinders model. This paper wasalso the first to make the important point that the so-called “polar” rotationrate inferred from inversions is actually localized somewhat away from the pole.Gough et al. (1993) continued to challenge the observers to completely ex-clude rotation on cylinders, pointing out that it was possible to construct a cylin-drical model that satisfied the constraint of the BBSO data, but Schou and Brown(1994b) showed that such a model could not be made consistent with both theFourier Tachometer data and the gravitational stability of the rotating Sun.Bachmann et al. (1993) analyzed Fourier Tachometer observations from 1989and pointed out a “wiggle” in the splitting coefficients at ν/L ≃ µ Hz, (corre-sponding to a turning-point radius of about 0 . R ⊙ ); attributed to daily modu-lation of the observations, this now well-known effect accounts for the “feature”seen in the middle of the convection zone in many inversions of single-site data.Better data, with long time series free from daily modulation, were obvi-ously needed before much more progress could be made, and with the adventof the GONG network in 1995 and the MDI instrument aboard SOHO in 1996such data became available. Preliminary rotation profiles were presented byThompson et al. (1996) for GONG and by Kosovichev et al. (1997) for MDI,both showing the now familiar pattern of almost-constant rotation in the con-vection zone, with shear layers both at the base of the convection zone andbelow the surface.Schou et al. (1998) carried out a comprehensive analysis of the rotation pro-file based on the first 144 days of observations from MDI, using and comparing37igure 17: Rotation profile based on analysis of BBSO splittings, (Schou et al. ,1992), reproduced by permission of the AAS.38everal different rotation inversion techniques with an input data set consistingof coefficients up to a for p modes up to l = 194 and f modes up to l = 250.They were able to obtain consistent and robust results from the surface to about0 . R ⊙ at low latitudes; at higher latitudes the domain of reliability was shal-lower. Roughly speaking, the inversions could not be well localized within about0 . R ⊙ of the rotation axis. The results (Fig. 18) showed that the rotation inthe bulk of the convection zone, below 0 . R ⊙ , had a slow increase with radiusat most latitudes, but was definitely incompatible with rotation on cylinders.Figure 18: Four inferred rotation profiles from the first 144 days of MDI ob-servations (Schou et al. , 1998); (a) 2DRLS, (b) 2DOLA, (c) 1D ×
1D SOLA, (d)1.5d RLS, from Schou et al. (1998), reproduced by permission of the AAS.
In addition to the other general features described here, Schou et al. found someevidence for a “jet” of faster rotation at about 75 ◦ latitude and 0 . R ⊙ ; al-though this was more obvious in some inversions than in others, it did seem39o have a signature in the coefficients themselves (see also Howe et al. et al. , 2000b), or even in inversions of MDI data analyzed with the GONGpipeline (Schou et al. , 2002), and it is now believed to be an artifact related tothe MDI data analysis. Once both GONG and MDI had been running for a few years, it became ev-ident that the two projects were producing inferences of the interior rotationprofile that were different in some significant details, particularly at high lat-itudes within the convection zone. Schou et al. (2002) carried out a carefulcomparison, taking data from three epochs at different phases of the solar cy-cle from each project and deriving rotational splittings or splitting coefficientsfrom each, both with the usual algorithms and with those regularly used for theother project’s data, before using both RLS and OLA inversions. The resultsclearly showed that most of the discrepancies arose from the analysis pipelinesrather from the data themselves. The “CA” peak-fitting algorithm used for theMDI data was able to extract modes from the GONG data to somewhat higherdegrees and lower frequencies than the “AZ” algorithm could manage with ei-ther GONG or MDI input data. However, for both MDI and GONG data, the“CA” algorithm introduced an anomaly in the splitting coefficients centered ataround 3.3 mHz, which in turn caused the inversion inferences to show a higherrotation rate deep in the convection zone at higher latitudes. Excluding thesedata brought the GONG and MDI data (analyzed with the “AZ” and “CA”pipelines respectively) into much better agreement, at the cost of somewhatdegraded resolution. Restricting both data sets to the common mode set below3 mHz reduced the discrepancies even farther, but did not remove the “jet” inthe MDI data. Since the “jet” feature was only seen in the MDI data analyzedwith the CA pipeline, however, the authors concluded that this feature wasprobably spurious.
Although much of the debate in the early 1990s centered on discriminating be-tween rotation constant on cylinders and rotation constant along radial lines,neither picture gave a complete description of the data. Gilman and Howe(2003); Howe et al. (2005) pointed out that the differential rotation in the bulkof the convection zone, at least at low- to mid-latitudes, could be quite welldescribed by saying that the contours of constant rotation lay at about a 25 ◦ angle to the rotation axis, as illustrated in Figure 19.Figure 20 compares idealized rotation profiles for the cylindrical, radial, andslanted-contour configurations. 40igure 19: Mean rotation profile from GONG data; contours of constant ro-tation (left), showing lines at 25 ◦ to the rotation axis as dashed lines, afterHowe et al. (2005), and cuts at constant latitude as a function of radius (right),after Howe et al. (2000b). Another interesting feature revealed by the early GONG and MDI observations(Schou et al. , 1998; Birch and Kosovichev, 1998) was that, while the surfacerotation rate was mostly well described by the usual three-term expansion inthe cosine of the colatitude θ , Ω( θ ) = A + B cos θ + C cos θ , (e.g., Snodgrass1984) the rotation rate close to the poles was significantly slower than that.The authors speculated that this might be a result of drag from the solar wind,and that the effect might therefore disappear or become less marked at epochsof higher activity. In fact, though the inferred high-latitude rate did speed upduring solar maximum – as seen, for example, in Howe et al. (2005) and inFigure 26 – it remained at all times lower than the extrapolation of the three-term fit. The interior rotation is only one part of the complex system that drives the solarcycle, but it is perhaps still the easiest part to measure in the solar interior; themeridional circulation can be directly measured only in the shallower subsurfacelayers, and buried magnetic fields can at best only be inferred indirectly. Thedifferential rotation in the convection zone must arise from the interaction ofconvection cells and Coriolis forces, with the meridional motions playing animportant part.Early depictions of the solar dynamo (see, for example, K¨ohler 1974; Durney1975) required a rotation rate increasing inward, and a meridional flow rising atthe poles and sinking at the equator, in order to drive the solar cycle migrationof the activity belts in the observed sense. This picture, taken together with41igure 20: Idealized rotation profiles for rotation constant on cylinders (left),radial lines (middle) and lines at 25 ◦ to the rotation axis (right). The toprow shows contours of constant rotation, while the lower row shows rotationrate as a function of radius at constant latitude for latitudes at 15 ◦ intervalsfrom the equator (top) to 75 ◦ (bottom). The rotation rate is matched to theGONG inferences at 0 . R ⊙ and smoothed to simulate the broadening effect ofinversion resolution on the tachocline; the near-surface shear was not included.42otation on cylinders, would have meant that the observed surface differentialrotation was a superficial phenomenon, with the dynamo operating in the un-observable deeper layers. At this stage, there does not seem to have been aclear distinction made between the direction of the meridional circulation atthe surface and the direction of migration of the magnetic activity belts dur-ing the solar cycle, which are of course now understood to operate in oppositedirections; the poleward meridional flow at the surface was first measured byDuvall Jr (1979).The models of Glatzmaier (1985) and Gilman and Miller (1986), which wereamong the first numerical simulations of solar rotation and the dynamo, havebeen cited, for example by Wilson (1992) as dating from “Prior to the advent ofhelioseismology,” but this is not quite correct. In fact, both these papers referto the Duvall and Harvey data, and Gilman and Miller (1986) also mentions theobservations of Brown (1985), suggesting that the model results could be consis-tent with the helioseismic observations if there were a layer of inward-increasingvelocity below the surface and above the domain of the simulation. The simula-tions in both cases, like their precursors over the previous several years such asthat described by Gilman and Miller (1981), produced rotation approximatelyconstant on cylinders and increasing outward, which would result in a dynamowave propagating poleward if the dynamo were operating in the bulk of theconvection zone. The main message that modelers in the late 1980s seem tohave taken from the observations was that the rotation rate was increasing out-ward, in agreement with the simulations of Gilman and Miller (1986) but indisagreement with the α -effect dynamo picture, which required a rotation rateincreasing inward; see Parker (1987) for a review representing a theorist’s per-spective on the state of play at this stage. This led Gilman and Miller (1986)to suggest (not for the first time; see also, for example, Galloway and Weiss1981) that the dynamo might be operating in a thin layer at the bottom of theconvection zone; this speculation was further reinforced by the later helioseismicinferences that clearly showed this shear layer, or tachocline (see Section 6) andthe approximately radial configuration of the rotation in the convection zone.Even quite recent global simulations of convection (Brun et al. , 2004, for ex-ample, ) still show some tendency towards rotation on cylinders, but the higher-resolution calculation of Miesch et al. (2008) mostly eliminates the cylindricaleffect and produces a rotation pattern, based on giant convection cells, that aftersuitable temporal averaging looks quite solar-like, as illustrated in Figure 21.43igure 21: Three temporally averaged rotation profiles from the spherical-shell simulations of (a) Brun et al. (2004), (b) Browning et al. (2006), and (c)Miesch et al. (2008), reproduced by permission of the AAS.44 The Near-Surface Shear
One persistent puzzle in the measurements of rotation at the photosphere hadbeen that direct Doppler measurements consistently gave somewhat slower ro-tation rates than the measurements made by tracing surface features. For ex-ample, Brown et al. (1989) summarized the results of Snodgrass (1983, 1984)as Ω m π = 462 − µ − µ nHz (25)for magnetic features andΩ p π = 452 − µ − µ nHz (26)for the surface plasma, respectively, where µ is the sine of the latitude. Foran overview of such measurements, see Beck (2000). The usual explanationfor the discrepancy is that while the Doppler techniques measure the velocityat the surface, the tracers such as sunspots are anchored in a faster-rotatinglayer deeper down. For example, Gilman and Foukal (1979) noted that theobservations implied a subsurface shear layer and suggested that this mightarise from angular momentum conservation in the supergranular layer.An extremely early attempt to measure the subsurface rotation was madeby Rhodes Jr et al. (1979), when the identification of the 5-minute oscillationswith p modes was still a relatively recent discovery. These authors used high-degree modes, probing about the upper 20 Mm (0 . R ⊙ ) of the convectionzone, and detected an inwards-increasing gradient. If these measurements arereliable, they represent the first detection of the subsurface shear. However,most of the early helioseismic measurements of the internal rotation profilewere restricted to a degree range that did not allow the near-surface shear tobe resolved in inversions. Rhodes Jr et al. (1990), attempting to measure therotation in the bulk of the convection zone, also saw hints of a gradient, oppositeto that seen at the base of the convection zone, below the surface, and Wilson(1992) used forward calculation techniques on the data of Brown and Morrow(1987) and Libbrecht (1989) to deduce that the rotation rate must increaseinward immediately below the surface. We should remember, however, that atthis time the picture of the internal rotation profile was not as clear as it istoday, and it is not always obvious whether interpretation of the observationsas gradients of rotation refers to the near-surface shear, the shear at the base ofthe convection zone, or some unresolved amalgamation of the two. Wilson, forexample, was not arguing for a near-surface shear layer but against the modelwith rotation constant along radii.With the advent of GONG and MDI, measuring modes to higher degreesthan had previously been possible, the near-surface shear could be seen in globalinversions; it is visible in the early results presented by Thompson et al. (1996)for GONG and by Kosovichev et al. (1997) for MDI, in both cases apparentlychanging sign at higher latitudes. 45chou et al. (1998) found clear evidence of the near-surface shear in inver-sions of MDI data. All the inversion methods agreed well on the shear at lowlatitudes, but at high latitudes the picture was complicated by the proximity ofthe submerged “jet” feature and the methods agreed less well. The disagreementmay have been partly due to systematic errors in the splitting coefficients. In thecomparisons of MDI and GONG data and analysis carried out by Schou et al. (2002), the high-latitude reversal of the shear is seen only in data analyzed withthe “CA” pipeline; this may be partly because the “AZ” pipeline mostly failsto recover the splittings of the (narrow, low-amplitude) f -mode peaks, but thereversal persists in the MDI data even for the restricted common mode set.The near-surface shear (down to about 15Mm) was studied in detail byCorbard and Thompson (2002), using f modes from MDI data. They measuredthe slope of the rotation rate, close to the surface at low latitudes, as about −
400 nHz/ R ⊙ , decreasing to a very small value by about 30 ◦ latitude andpossibly reversing in sign at higher latitudes (though this result, seen in onlythe outer 5 Mm, was dependent on only the highest-degree modes, those with l ≥ ◦ . Basu et al. (1999) and Howe et al. (2006a) compared results from local ring-diagram analysis and global inversionsand found, at latitudes ≤ ◦ , quite good agreement between the d Ω /dr valuesobtained from local and global inversion results. However, although the slopefrom local measurements does show some variation with latitude (Figure 22), itby no means vanishes at 52.5 ◦ , the highest latitude at which the measurement ismade. The ring-diagram results allow us to consider the northern and southernhemispheres separately, but Basu et al. (1999) found very little difference in theshear between the two hemispheres.Some attempts have been made to use the near-surface shear to drive or atleast contribute to a solar dynamo, for example by Brandenburg (2005), butDikpati et al. (2002) showed that any dynamo contribution from the shear ofthe outer layers could only provide a fraction of the effect needed to power thesolar cycle. 46igure 22: Radial variation of the mean rotation rate after subtraction of thetracking rate, for global inversions (blue) and north – south averaged local in-versions of MDI (green) and GONG (red) data at latitudes 0 ◦ (a), 15 ◦ (b), 30 ◦ (c) and 45 ◦ (d); similar to Howe et al. (2006a).47 The Torsional Oscillation
The so-called Torsional Oscillation is a pattern of migrating bands of faster- andslower-than-average zonal (i.e., parallel to the equator) flow associated with theequatorward drift of the activity belts during the solar cycle. It was first de-scribed by Howard and Labonte (1980), who used twelve years (1966 – 1978)of full-disk velocity observations from the 150-foot tower at the Mount Wil-son observatory and found evidence of a pattern of flow bands migrating to-wards the equator; the greatest concentration of active regions is associatedwith the poleward edge of the main equatorward-moving band. They initiallyinterpreted the high-latitude variations as consisting of bands of faster rotationstarting at the poles and taking a full 22-year Hale cycle to drift to the equator.Scherrer and Wilcox (1980a); Scherrer et al. (1980), observing at the StanfordSolar Observatory, found no evidence of changes in the equatorial rotation ratefor data from 1976 – 1979, but as this period was close to a solar minimum, andthe resolution of the Stanford instrument was not high, this is neither surprisingnor inconsistent with the results of Howard and Labonte. Labonte and Howard(1982) note that Scherrer and Wilcox (1980b) (at a AAS meeting), had “con-firmed the existence of the global velocity field,” though this is not apparentfrom the latter’s published abstract.A somewhat different pattern of velocity variations is seen when magneticfeatures rather than Doppler measurements are used to determine the surfacerotation rate, as described for example by Komm et al. (1993a), who found thatthe pattern derived from magnetograms lay equatorward of that from Dopplermeasurements, with the slower-than-average bands coinciding with the zones ofgreater magnetic flux.Mount Wilson Doppler observations since 1986, clearly showing the patternof migrating zonal-flow bands, were presented by Ulrich (1998, 2001); see alsoHowe et al. (2006a) for updated results. The bands extend over about 10 ◦ inlatitude, and have zonal velocities a few meters per second faster or slower thanthe surrounding material, corresponding to excess angular velocity of less than0 .
5% of the overall rotation, or a few nanohertz.
The first hints of the signature of the migrating flow bands in helioseismic datacan be seen in the BBSO data (Woodard and Libbrecht, 1993), as was pointedout by Howe et al. (2000c), but these measurements do not give much informa-tion on the radial extent of the flows. Kosovichev and Schou (1997) found evi-dence of the flows, a few meters per second faster than the general rotation pro-file, in f -mode measurements from early MDI data; Giles et al. (1998) found asimilar pattern using the time-distance technique of local helioseismology, whileSchou and The SOI Internal Rotation Team (1998) and Schou (1999) clearlyshowed that these flows were migrating in a manner consistent with the MountWilson Doppler observations. The first radially-resolved evidence of zonal flowmigration was reported by Howe et al. (2000c) for GONG and by Toomre et al. et al. (2000a) combined MDI and GONG data andconcluded that the equatorward-migrating part of the flow pattern (at latitudesbelow about 40 ◦ ) penetrated to at least 0 . R ⊙ (56 Mm below the surface).Antia and Basu (2000) also reported similar findings. Antia and Basu (2001)studied the evolution of the variations poleward of 50 ◦ , which had much higheramplitudes than the equatorward-moving flows and which showed signs of prop-agating poleward over time. The larger amplitude of the high-latitude signalmay be related to the smaller angular momentum closer to the rotation axis. As more data accumulated, the signature of the torsional oscillation pattern inthe helioseismic observations became clearer. Vorontsov et al. (2002) studiedthe evolution of the flows in MDI data from 1996 through 2001. They con-cluded that at least the high-latitude region of changing rotation involves thewhole depth of the convection zone. The results on the radial extent of the flowsat lower latitudes were less clear, with evidence that the bands of slower rota-tion might penetrate close to the base of the convection zone, while the bandsof faster rotation appeared to reach about 0 . R ⊙ but no deeper. Another inter-esting feature of that paper was the introduction of the use of 11-year sinusoidsto characterize the variation of the rotation rate at any given location. Thisinnovation had the useful effect of clarifying the pattern, making obvious thepoleward propagation of the high-latitude flows even with data from little morethan half a cycle. The existence of a weak third-harmonic component to theeleven-year cycle, however, was not confirmed in later work.Basu and Antia (2003) found similar results in MDI and GONG data up to2002, as seen in Figure 23. These results also hint at another subtlety; at lowlatitudes, the phase of the flow pattern is not constant along radial lines. In fact,the variation in the lower part of the convection zone appears to lead that closeto the surface by a year or two, with the low-latitude band of faster rotation fol-lowing roughly the same 25 ◦ slant as the rotation contours. This tendency wasfurther studied by Howe et al. (2005, 2006b), who compared inversions of MDIand GONG data with forward-modeled profiles based on different flow configu-rations, including some derived from dynamo models. Although some detail waslost and distorted due to the resolution and uncertainties in the inversions, theauthors were able to conclude that the low-latitude branch probably penetratesthrough much of the convection zone, but is sufficiently displaced in phase atgreater depths that the correlation between the surface pattern and that deeperdown almost vanishes. In this work, the 11-year sinusoid analysis showed evi-dence of a second-harmonic component rather than the third harmonic reportedby Vorontsov et al. .Figures 25, 26, and 27 show the variations in rotation rate, based on theresults and figures in Howe et al. (2005, 2006b), but brought up to date withthe most recent GONG and MDI observations available at the time of writing.The plots were prepared using the same 2-D RLS inversion codes for both MDIand GONG medium-degree data, and 2-D SOLA for MDI, that were used for49igure 23: Contour diagrams of constant rotation velocity residuals at 0 . R ⊙ ,obtained using two dimensional RLS inversion of the GONG data, fromBasu and Antia (2003), reproduced by permission of the AAS.50igure 24: Zonal flow pattern derived from MDI f -mode measurements, withsmooth profile subtracted. Based on a figure from Schou (1999), updated andused by kind permission of J. Schou (2008, private communication.)the work of Howe et al. (2000a) and the other related papers. Figure 28 showsthe phase and amplitude profiles for 11-year sine functions fitted to the rotationvariations. The torsional oscillation pattern, at least at lower latitudes and closer to thesurface, is also suitable for measurements using the techniques of local helio-seismology, in which short-wavelength, short-lived waves are used to infer thestructure and dynamics of localized areas of the Sun. Because these waves donot penetrate very far below the surface, such techniques are restricted to theouter few megameters of the solar envelope, but this region can be studied inmuch greater detail and with shorter averaging times than is possible with globalhelioseismology.Basu and Antia (2000) detected the zonal flow migration using MDI dataand the ring-diagram technique (Hill, 1988), in which the displacement of three-dimensional acoustic power spectra derived from small areas of the solar disk isused to infer horizontal flows in both the zonal and meridional directions. Later,Haber et al. (2002) measured both the zonal flows and a corresponding mod-ulation of the meridional flow pattern, as seen in Figure 29 (left). Beck et al. (2002), using the time-distance technique, which considers the correlations be-tween oscillations at spatially separated locations, also found bands of merid-51igure 25: Rotation rate after subtraction of a temporal mean at each loca-tion, as a function of latitude and time at selected depths, for OLA (top) andRLS (middle) inversions of MDI data, and for RLS inversions of GONG data(bottom). 52igure 26: Rotation rate after subtraction of a temporal mean at each loca-tion, as a function of depth and time at selected latitudes. Latitudes are0 , , , , ◦ from left to right; inversions are MDI OLA (top), MDI RLS(middle) and GONG RLS (bottom). 53igure 27: Rotation rates at selected latitudes and depths as a function of time,after subtraction of a temporal mean. The results are from GONG RLS (black),MDI RLS (red), and MDI OLA (blue) inversions.54igure 28: Phase (left) and amplitude (right) of 11-year sine functions fittedto temporal variation of the rotation rate for OLA (top) and RLS (middle)inversions of around 11 years of MDI observations and for RLS inversions ofGONG data (bottom). 55igure 29: Local helioseismic inferences of zonal flows close to the surface, fromHaber et al. (2002) (left) and Zhao and Kosovichev (2004) (right), reproducedby permission of the AAS. 56igure 30: Zonal flows since 1986, from Mount Wilson Doppler measurements(top), global helioseismic measurements from BBSO and MDI (middle) andMDI ring-diagram analysis (bottom). The color scale is in nHz.57onal flow away from the activity belts associated with the zonal flow bands.Chou and Dai (2001); Chou and Ladenkov (2005), using data from the TaiwanOscillations Network [TON]), also found diverging meridional flows associatedwith the activity belts. Zhao and Kosovichev (2004) measured the zonal (Fig-ure 29 right) and meridional flows with the time-distance technique, and re-ported meridional flow converging on the activity belts above a depth of 12Mm,with diverging flows below 18Mm, forming circulation cells around the activ-ity belts. The presence of inflows into the activity belts was also observed atthe surface by Komm et al. (1993b); Komm (1994). Komm et al. (2005) stud-ied the flows in about a year of high-resolution GONG (“GONG+”) data, andconcluded that the overall flow pattern existed whether or not active regionswere included in the analysis; in other words, the zonal flow bands and theirassociated converging/diverging meridional flows appear to exist independentlyof the flows in the immediate vicinity of strong active regions.Howe et al. (2006a) compared the results from ring-diagram analysis of theMDI data, global analysis of MDI and GONG data, and the Mount WilsonDoppler observations. They found very similar results for the north–south sym-metrized flow pattern close to the surface in all three observations. Both theglobal and local helioseismic data indicated that the strength of the flow patterndid not fall off steeply below the surface.It should be noted that the local helioseismic observations are somewhatprone to systematic errors, some of which follow the changing B angle, or tiltof the solar rotation axis relative to the observer, as shown for example byZaatri et al. (2006). This can result, for example, in a pronounced and almostcertainly non-solar north–south variation of the zonal flow measurements, whichis generally corrected for by subtracting suitable averages.Some further features of the torsional oscillation pattern as we know it froma full cycle of observations from GONG and MDI (and nearly two cycles ofsurface Doppler observations) are worth noting.1. The exact appearance of the pattern is quite sensitive to the backgroundterm that is subtracted. For example, compare the f -mode results shownin Figure 24, which were plotted as the difference from a smooth 3-termexpansion of the rotation rate, with the plots in Figure 25, which wereplotted by subtracting the temporal mean at each location.2. Although the pattern repeats – of course not precisely – with each (ap-proximately) eleven-year activity cycle, each equatorward-migrating flowband exists for about eighteen years, emerging at mid-latitudes soon afterthe maximum of one cycle and finally disappearing at the equator a cou-ple of years after the minimum of the following cycle; thus, the band offaster rotation associated with the activity of cycle 22 was still visible atthe beginning of GONG and MDI observations in early cycle 23, and theband that is expected to accompany cycle 24 became visible around 2002(if we look at the mean-subtracted residuals), or 2005 – 2006 (if we use thesmooth-function subtraction). On the other hand, each poleward-moving58ranch seems to last only about nine years, appearing a year or so aftersolar minimum and moving to the pole before the next minimum.3. Although the equatorward-migrating bands of faster rotation are clearlyassociated with the migrating activity belts of the magnetic butterflydiagram, the relationship is not completely straightforward. The newequatorward-propagating branch is clearly visible some years before no-ticeable new cycle active regions begin to erupt, and the phase/latitudeprofiles of the magnetic index and the velocity are very different. Also,as was noted by Labonte and Howard (1982) and by Howe et al. (2006a),the strength of the torsional oscillation signal has not shown much changeover the last few solar cycles, while the level of magnetic activity variesmuch more from one cycle to another.4. Although the equatorward branch of the zonal flow migration patternshows some relationship to the pattern of enhanced activity in the Fe xiv corona going back to 1973 (Altrock, 1997), the “extended solar cycle” seenin these observations starts at a much higher latitude, apparently about70 ◦ , before migrating to the equator over about eighteen years; thus eventhe equatorward edge of these coronal activity bands seems to be at higherlatitude than the observed new branch in the zonal flows that starts atabout the same time.5. Finally, we note that because the angular velocity changes associated withthe torsional oscillation signal are relatively small compared to the differ-ence in angular velocity between the surface and the bottom of the near-surface shear layer, while the amplitude of the signal does not decreaserapidly with depth, the magnitude of the shear at a given location variesby only a fraction of its value during the solar cycle. However, the frac-tional change in the shear is much greater than the fractional change inthe rotation rate. While observers, for example Howard and Labonte (1980) and Ulrich (2001)have speculated that the torsional oscillation pattern might itself be part of thedriving mechanism for the solar cycle, perhaps generating activity by shearingmagnetic loops, modelers have generally seen it rather as a side-effect of themagnetic fields.Schuessler (1981) and Yoshimura (1981) modeled the torsional oscillation asa result of the Lorentz force due to dynamo waves; according to the latter paper,the phenomenon would be important only close to the surface, and would haveonly equatorward, not poleward, moving bands. Labonte and Howard (1982)objected to the Yoshimura model on the grounds that it would predict a strongcorrelation between the strength of the surface magnetic field and that of thevelocity signal, which did not seem to be the case in the observations.59ueker et al. (1996) used a different mechanism to generate the torsionaloscillation signal in their model, considering it as the response of the Reynoldsstress on the time-dependent dynamo magnetic field rather than a direct effectof the large-scale Lorentz force. This model gave a very weak poleward branchfor the torsional oscillation signal.Once the flows had been shown observationally to penetrate well below thesurface, Durney (2000) suggested that, “the pattern of torsional oscillations ap-pear to have the potential of critically discriminating between different dynamomodels as, e.g., the Babcock-Leighton and interface models.”Covas et al. (2000) used a model in which the observed rotation profile wasimposed and the rotation variations arises from the action of the Lorentz forceof the dynamo-generated magnetic field on the angular velocity. They were ableto simulate approximately solar-like patterns of zonal flow bands and magneticactivity. In subsequent papers they focused on the the possibility of so-called“spatio-temporal fragmentation” allowing cycles of different periods in differentregions, and in calculations with no density stratification in the convection zonethey found this to be feasible (Covas et al. , 2001a). The effect was not too sen-sitive to uncertainties in the rotation law (Covas et al. , 2001b, 2002), and some-what sensitive to the boundary conditions at the outer surface (Tavakol et al. ,2002). Adding density stratification (Covas et al. , 2004) did not substantiallychange the results, though the amplitude of the oscillations in the deeper layersof the convection zone did decrease as the density gradient increased. However,they did find that introducing quite a small amount of α -quenching (magneticfeedback on turbulent convection) would suppress the torsional oscillation effect.Spruit (2003) modeled the torsional oscillation pattern as a “geostrophicflow” driven by temperature variations near the surface associated with mag-netic activity, and therefore having its greatest amplitude at the surface andfalling to 1/3 of its surface value at 0 . R ⊙ . This model also accounts for theobserved inflows into the activity belts. There are some problems in reconcilingthis model with the observations; it is difficult to see how the observed depth-dependent phase pattern could arise from a surface-originated cause, and theexistence of the flows even at epochs where there are no active regions is alsohard to explain, though Spruit suggested that the flows might be produced byunobserved small-scale and short-lived magnetic regions.Rempel (2007) used a mean-field flux-transport dynamo model, with a model-derived differential rotation profile and meridional flow, to investigate the ef-fects of various driving mechanisms for the torsional oscillation. The authorconcluded that the poleward-propagating branch of the pattern could be ex-plained by a periodic forcing at mid-latitudes without any underlying migra-tion of buried polar field. On the other hand, in this type of model the ob-served equatorward-propagating branch could not be reproduced without addinga thermal forcing after the manner of the Spruit (2003) model. Howe et al. (2006b) compared such a model with the observations, and found it not to becompletely consistent with the observed interior behavior of the flows at lowerlatitudes. 60 Figure 31: Rotation-rate residuals at the equator at 0 . R ⊙ (top) and 0 . R ⊙ (bottom), for RLS (filled) and OLA (open) inversions of MDI (red triangles)and GONG (black circles) data.Howe et al. (2000b) reported finding variations of the equatorial rotation rateclose to the tachocline with a 1.3 year period during the early years (1995 – 1999)of GONG and MDI observations. The strongest signal was seen at 0 . R ⊙ , witha weaker anticorrelated signal below the tachocline at 0 . R ⊙ . At higher lat-itudes, there was also an apparent 1-year periodicity. The signal was moreclearly seen in the GONG data, and due to the different temporal sample of the61DI data it was difficult to make a quantitative comparison, but the visual ap-pearance of similar variations in both data sets was quite persuasive. Figure 31extends the data up to the present for the equatorial locations just above andbelow the tachocline.Because of the role of the tachocline region in the dynamo, as well as the coin-cidence of the period with that seen in some heliospheric and geomagnetic obser-vations (Silverman and Shapiro, 1983; Richardson et al. , 1994; Paularena et al. ,1995), this claim attracted considerable interest, inspiring modelers such asCovas et al. (2001a) to try to build models in which different periods could existat the top and bottom of the convection zone. However, Antia and Basu (2000)and Basu and Antia (2001), with a slightly different analysis of the same MDIand GONG data, reported finding no significant variations. (Basu and Antia(2001) did see a signal somewhat similar to that reported by Howe et al. (2000b)but did not consider it significant.)Moreover, the periodic signal disappears in the post-2001 data even in theoriginal authors’ analysis (Toomre et al. , 2003; Howe et al. , 2007), as shownin Figure 32, and it seems likely that the high-latitude 1-year period was anartifact. Intermittency in short-period variations is a known phenomenon inthe geomagnetic-index data, (Silverman and Shapiro, 1983), and does not initself imply that the phenomenon was not real. It will be interesting to seewhether the oscillation will reappear in the new solar cycle. Christensen-Dalsgaard et al. (2004) searched for evidence of jets close to thetachocline, which are predicted, for example, by the model of Dikpati et al. (2004). Using GONG data they reported finding possible evidence of a jet atthe tachocline, migrating equator-wards by about thirty degrees in two yearsbut not at the same latitude as the surface activity belts. The significance andmeaning of this finding remain unestablished.
Given estimates of both density and rotation as functions of depth and latitude,one can calculate the solar angular momentum locally or globally. Of course,such calculations will reflect, and in some cases enhance, any errors in the inputdata, and should therefore be approached with caution.Komm et al. (2003) investigated the angular momentum variation based onthe inversions of GONG and MDI data used by Howe et al. (2000b,a) and foundvariations reflecting the torsional oscillation well into the convection zone and1.3 year variations close to the tachocline. Because the density increases steeplywith decreasing radius, variations at greater depths will be more strongly seen inthe angular momentum than in the rotation rate, but it should be rememberedthat no new information has been added to the data.Lanza (2007) approached the problem from the other direction, consider-ing the role of angular momentum transport in the modeling of the torsional62igure 32: Sine-wave power in the rotation rate residuals from RLS inversions ofGONG data, at 0 . R ⊙ , ◦ , plotted as a function of frequency for a) 1995 – 2000,b) 1995 – 2003, c) 1995 – 2005, d) 2000 – 200563scillation.Antia et al. (2008) investigated temporal variations of the solar kinetic en-ergy, angular momentum and higher-order gravitational multipole moments asderived from helioseismic inferences of the internal rotation rate; they foundvariations on the time scale of the solar cycle (but not the 1.3 year cycle), withsome discrepancies between MDI and GONG results. They also speculate thatthe kinetic-energy changes might contribute to the observed irradiance varia-tions during the solar cycle; however, it is not clear that such a contributionis needed, as the usual view is that the solar-cycle variation in irradiance canbe modeled simply from the effects of sunspots and plage on the surface, asdiscussed, for example, by Jones et al. (2008).64 Since the 1970s, helioseismology has provided several insights into the interiorsolar rotation: the approximately-rigid rotation of the radiative interior; thedifferential rotation throughout the convection zone; the thin tachocline; theextension of the surface torsional oscillation throughout the convection zone.More than once, these discoveries have overturned theoretical expectations, in-spiring modelers to improve their calculations in an effort to reproduce theobserved behavior. Because of the surprising nature of many of the findings, ithas been important to have more than one source of observations, so that it ispossible to distinguish between real solar features – especially the unexpectedones – and systematic error.It may be that in the future solar cycle 23, with MDI and GONG operatingin parallel, will be seen as a golden age of helioseismology. At the time ofwriting, we eagerly anticipate the launch of the Solar Dynamics Observatory[SDO] with its Helioseismic and Magnetic Imager [HMI], a successor to MDIthat will provide near-continuous helioseismic observations at higher resolutionsthan ever before and may help in unraveling the relationships between activeregion flows, magnetic fields, and geoeffective solar activity as well as providing acontinued watch on the longer-term variations in the solar velocity fields. Sadly,however, current plans call for both GONG and MDI to cease to collect datasoon after the successful launch of SDO, which would leave HMI without anyindependent cross-checks, while on the low-degree front the BiSON network hasrecently lost its funding and there are no new dedicated low-degree space-basedinstruments currently scheduled.There are still areas – such as the strength of the near-surface shear at highlatitudes, the rotation of the inner core, and any inhomogeneities and changesin the tachocline – that remain unclear. Furthermore, a complete numericalmodel of the solar dynamo – vital for any long-term predictive capability – isstill lacking, and helioseismic observations still have an important part to playin constraining such models as they develop.65
This review has made use of NASA’s Astrophysics Data System.This work utilizes data obtained by the Global Oscillation Network Group(GONG) program, managed by the National Solar Observatory, which is op-erated by AURA, Inc. under a cooperative agreement with the National Sci-ence Foundation. The data were acquired by instruments operated by the BigBear Solar Observatory, High Altitude Observatory, Learmonth Solar Obser-vatory, Udaipur Solar Observatory, Instituto de Astrof´ısica de Canarias, andCerro Tololo Interamerican Observatory. The Solar Oscillations Investigation(SOI) involving MDI is supported by NASA grant NAG5-13261 to StanfordUniversity.
SOHO is a mission of international cooperation between ESA andNASA. NSO/Kitt Peak magnetic data used here are produced cooperatively byNSF/NOAO, NASA/GSFC and NOAA/SEL.The Mt. Wilson observations have been supported over several decades bya series of grants from NASA, NSF and ONR and are currently supportedby NSF/ATM. The Mt. Wilson observatory is managed by the Mt. WilsonInstitute under agreement with the Observatories of the Carnegie Institution ofWashington.BiSON has been funded by the U.K. Particle Physics and Astronomy Re-search Council. 66 eferences
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