## Abstract

The helioseismic investigation of the solar interior to elucidate the behaviour of the Sun's magnetic field and the interior's role in the solar interior is considered. Such study may be called magnetohelioseismology (from the Greek roots *magnetis, helios* and *seismos*).

## 1. Introduction

The interior of the Sun houses the solar dynamo; all the Sun's magnetic flux emerges from there; and in a fundamental sense, the Sun's interior is the driver of the solar cycle. It is therefore of great interest and importance to use helioseismology to probe and constrain the physics of the solar dynamo, the dynamics of the solar interior, magnetic flux emergence and, if possible, the internal magnetic field itself.

The Sun oscillates simultaneously in more than one million global resonant modes, set up by predominantly acoustic waves, which are generated by acoustic emission from the turbulent convection in the near-surface layers of the Sun, and which propagate through the solar interior. The oscillations are detectable *inter alia* through Doppler velocity measurements of the motion of the Sun's surface. Helioseismology, the seismology of the solar interior, proceeds by the analysis of observations of these oscillations. In particular, the frequencies of the resonant modes depend on conditions in the solar interior that affect the wave propagation, and hence can be used to make inferences about those properties. Since the Sun is nearly spherically symmetric, the horizontal structure of the eigenfunctions of the resonant modes are well described by spherical harmonics , where *θ* and *ϕ* are heliocentric colatitude and longitude, respectively. Integer *l* is the degree of the mode and integer *m* is its azimuthal order (*m*=−*l*, …, *l*). The classification of the modes is completed by a third integer *n*, the radial order of the mode, whose absolute value is approximately equal to the number of nodes in the radial direction in, say, the pressure eigenfunction inside the Sun. The mode frequency at fixed *l* and *m* is an increasing function of *n*. The observed modes are mostly *p*-modes, for which pressure provides the dominant restoring force and for which *n*>0. Also observed is the *f*-mode, which at high degree has the character of a surface gravity wave and for which *n*=0. Also supportable by the solar interior, though their actual observation is still debated, is the low-frequency internal *g*-modes (*n*<0) for which the restoring force is gravity.

Since the *p*-modes are the most important observationally, it is worth further remarking that they propagate inside an acoustic cavity between two turning points. The lower turning point is located, where the horizontal phase speed of the mode, , is equal to the local adiabatic sound speed *c*. Here, *ω* is the angular frequency of the mode—I shall also use the cyclic frequency —and is the horizontal wavenumber, *r* being the radial coordinate (i.e. distance from the Sun's centre) and . The upper turning point is located near the surface essentially (except for very high-degree modes), where the mode frequency equals the acoustic cutoff frequency *ω*_{c}: the cutoff frequency increases towards the surface and so lower-frequency *p*-mode waves turn further beneath the surface than do their higher-frequency counterparts. Outside the acoustic cavity the mode is evanescant, and so its frequency and other properties are relatively insensitive to conditions outside its acoustic cavity, particularly far from the turning points. Specifically, modes with low frequency (which have upper turning point deeper beneath the surface) or with low degree *l* (which penetrate deep into the Sun) are harder to perturb by a near-surface perturbation. This is quantified by the *mode inertia*(1.1)where *ξ*_{r} and *ξ*_{h} are the radial and horizontal components of the displacement eigenfunction * ξ*,

*ρ*is the local density,

*r*is the radial coordinate and

*R*and

*M*are the photospheric radius and total mass of the Sun. The term

*mode mass*is also used: we define this to be equal to

*I*

_{nl}times

*M*. The mode inertia is shown for various modes in figure 1.

In a spherically symmetric star, the mode frequencies for given *n* and *l* would be independent of *m*. Asphericities such as rotation and magnetic fields raise the *m*-degeneracy of the mode frequencies. The observed frequencies of modes within a given *n*, *l* multiplet are often described using a polynomial expansion(1.2)where the basis functions are polynomials related to the Clebsch–Gordan coefficients by(1.3)(Ritzwoller & Lavely 1991). The are odd or even polynomials in *m* according to whether *j* is odd or even. The coefficients from this expansion are referred to as *a*-coefficients. The odd-order coefficients arise from rotation. The even-order coefficients arise from magnetic fields and any other asphericities (including centrifugal distortion) which do not distinguish between eastward- and westward-propagating waves.

Another technique of helioseismology uses not frequencies, but rather the travel times between points on the surface. Travel times depend on conditions in the region travelled. Travel times are inferred by cross-correlating the detected oscillations at different points on the solar surface. This method of inference is called time–distance helioseismology (Duvall *et al*. 1993).

## 2. Global oscillations and magnetic fields

The linearized adiabatic wave equation for global oscillations of a non-rotating, non-magnetic star can be written as(2.1)where operator is defined to be(2.2)(*p* being the pressure). Here, the Cowling approximation has been made, which ignores the Eulerian perturbation to the gravitational potential. This is reasonable except for modes where the degree and radial order are both small.

In a rotating, magnetized star the wave equation is modified. Here I simply sketch the treatment presented by Gough & Thompson (1990) for oscillations of a star with a buried magnetic field (see also Dziembowski & Goode 1989). It is presumed that there is a frame, possibly rotating, in which the structure of the star and the magnetic field are stationary. In general, one needs to make a transformation to work in that frame (see Gough & Thompson 1990 for details), but I omit explicit mention of the extra terms such a transformation generates in the equations below.

The wave equation in the rotating, magnetized star can be written as(2.3)The first-order effect of rotation through advection and the Coriolis force is contained in the term(2.4)where * v* is the rotation velocity field. The term(2.5)contains the explicit second-order effects of rotation. The explicit effects of the magnetic field are contained in the term(2.6)where

*is the equilibrium magnetic field and(2.7)is the Eulerian perturbation to the magnetic field, assuming the solar plasma to be a perfect conductor.*

**B**Note that the operator is also implicitly changed by the presence of rotation and magnetic field, because the equilibrium pressure and density fields are modified. These changes affect the oscillations at the same order as and and must be taken into account in any calculation. See Gough & Thompson (1990) for details.

The wave equation is solved with suitable boundary conditions to yield the eigenfrequencies and eigenfunctions. For the Sun, the effects of rotation or a buried magnetic field have been solved with a perturbation method, positing(2.8)where **ξ**_{0}, *ω*_{0} are the solution for a non-rotating, non-magnetized solar model and **ξ**_{1}, *ω*_{1} are small perturbations.

Note that one must use degenerate perturbation theory, as the zero-order eigenfrequencies have the *m*-degeneracy within each multiplet (and near-degeneracies between different multiplets may enter as well, depending on the magnitude of the perturbations). In the case of axisymmetric rotation and magnetic field, one can ‘guess’ that the relevant zero-order eigenfunctions to use as **ξ**_{0} are those corresponding to a single value of *m* with respect to polar axes aligned with the axis of symmetry. For example, retaining just the perturbation gives the usual first-order rotational perturbation to the frequencies used in helioseismic inference.

If the system is axisymmetric, *m* is still a well-defined quantum number. Then an observer will see (at most) 2*l*+1 frequencies for given *n* and *l* (i.e. one for each value of *m*). (Depending on the observation method and vantage point, not all the modes may be observed.) If magnetic field breaks the axisymmetry, e.g. if the field is inclined to the rotation axis, then each eigenfunction is a mixture of azimuthal orders. This gives what is known as hyperfine splitting. In principle, the observer can then see up to (2*l*+1)^{2} apparent frequencies when viewing the system from an inertial frame.

Hyperfine splitting may be one explanation for the multiplicity of peaks reported for one candidate *g*-mode multiplet (Turck-Chieze *et al*. 2004). If the individual peaks in such a multiplet cannot be fully resolved in the frequency domain, hyperfine splitting may appear as excessive mode line widths (Goode & Thompson 1992).

To summarize, effects of a magnetic field in the solar interior can enter in various ways. There are the direct dynamical effects of the magnetic field, encapsulated in the term in equation (2.3). There are also indirect effects, due to changes in the solar structure. Specifically, the hydrostatic equilibrium is modified, leading to changes in pressure, density and possibly the adiabatic exponent which enter through the modification to the term. Further effects may enter similarly: these can arise from the magnetic field's modification of the energy transport, for example through inhibiting turbulent convective motions *in situ* or via the thermal shadow in the convection zone of a field at its base. The presence of a magnetic field may also have an effect on the rotation, for example giving rise to toroidal jets (Dikpati & Gilman 1999), and, thus, indirectly affects the mode frequencies through the rotational terms.

## 3. Solar results

Below we discuss results of analysing the observed even-order *a*-coefficients and the temporal variations of these and the mean-multiplet frequencies over the solar cycle. Before doing that we discuss briefly some other results which also pertain to the seismic probing of the magnetized solar interior.

Helioseismology has been very successful in mapping the internal rotation of the Sun (e.g. Thompson *et al*. 1996, 2003), and in recent years there have been numerous investigations of the temporal variation of the internal rotation over the solar cycle. The so-called torsional oscillations—bands of weak but apparently coherent faster and slower rotation that migrate along with the active latitudes over the solar cycle (Howard & LaBonte 1980)—have been established to penetrate at least one-third of the way into the convection zone (Howe *et al*. 2000*a*). The temporal variations of the rotation may involve the whole convection zone, particularly at higher latitudes (Vorontsov *et al*. 2002). These variations take place on a solar-cycle time-scale. There have also been reported quasi-periodic variations on a shorter time-scale of about 1.3 years at low latitudes near the tachocline region at the base of the convection zone (Howe *et al*. 2000*b*). The variation appears to extend into the radiative interior, but there it is in anti-phase with the variation at the bottom of the convection zone, suggesting perhaps an exchange of angular momentum between the convection zone and the outer radiative interior. Although such a short time-scale compared with the solar cycle may at first sight be surprising, variations on nearly the same time-scale have been reported from time to time in the photospheric magnetic flux, the solar wind and in auroral activity (e.g. Richardson *et al*. 1995). One might wonder whether there is a causal link connecting these apparently disparate observations.

Another area in which helioseismology has made rapid progress recently is in time–distance seismic probing under active regions and sunspots. The measured perturbations to wave travel times through sunspots are generally interpreted as arising from wave-speed perturbations under the spots, though other explanations are possible (see Brüggen & Spruit 2000; also Cally, this volume). The results commonly indicate a region of slower wave propagation in a shallow region of about 3 Mm depth beneath the sunspot with a region beneath that of faster wave propagation, relative to the horizontal mean at that depth over the rest of the Sun. Other investigations, with temporal resolution of about 8 h, show the development of subsurface wave-speed anomalies as active regions emerge (see Kosovichev 2003 and references therein), including a very striking sequence as active region AR10488 developed alongside AR10486 in late 2003 (A. Kosovichev, personal communication, 2004). Flows under sunspots have also been mapped using time–distance techniques (Zhao & Kosovichev 2003). The technique and observations are extremely promising for increasing our understanding of the emergence and evolution of active regions and the connection between the magnetic interior and the magnetic field in the solar atmosphere.

### (a) Looking for acoustic asphericity in the solar interior

The even-order *a*-coefficients, which arise from acoustic asphericity of the Sun, are significantly non-zero and vary systematically over the solar cycle (Antia *et al*. 2001). The even-order coefficients can be inverted to infer something about the aspherical properties of the solar interior. These properties have generally been presented to date as an isotropic perturbation to the wave speed at which the predominantly acoustic waves propagate (Gough *et al*. 1996; Antia *et al*. 2003), which can be interpreted as, for example, due to an aspherical temperature perturbation or magnetic field. The coefficients can be averaged in time and inverted to obtain a mean asphericity, or inverted epoch by epoch to infer how the aspherical perturbation varies over time.

After subtracting out the contribution due to centrifugal distortion and other second-order effects of rotation, the even-order *a*-coefficients can be represented as having contributions from the solar interior and from the near-surface and atmosphere(3.1)With the factor of the inverse mode mass having been explicitly taken out, the surface contribution may be a function of frequency, but it is presumed independent of degree. The interior contribution, on the other hand, depends on mode frequency and degree. This enables the two to be separated out, and the interior contribution can then be inverted using standard techniques (e.g. Antia *et al*. 2001).

Results of such an inversion of the time-averaged *a*-coefficients are shown in figure 2, for data from two projects—the ground-based Global Oscillation Network Group (GONG) and the Michelson Doppler Imager (MDI) aboard the Solar and Heliospheric Observatory satellite. The inferred maps of wave-speed asphericity appear to show a positive anomaly at 60° latitude, spread over a range of depths.

The amplitude of the perturbation is clearer if one takes cuts through the inversion result at different latitudes, as shown in figure 3. As a fraction of the spherical component of the (squared) sound speed, the perturbation is largest at 60° and a radius of about 0.9*R*, i.e. 70 Mm beneath the surface of the Sun, about one-third of the way down into the convection zone. However, the sound speed increases with depth, and so does the density. Suppose that the aspherical perturbation to wave speed at 60° is interpreted as arising from an additive perturbation of to the squared sound speed, where *v*_{A} is the Alfvén speed. Then the magnetic field strength at 60° thus inferred increases monotonically with depth from the surface right down to the base of the convection zone at 0.7*R*, where it reaches a value of about 300 kG (Antia *et al*. 2003).

Antia *et al*. (2003) find no significant variation of the inferred 60° anomaly with time. It would appear to be surprising if there were such a subsurface anomaly at 60°, far from the active latitudes, although the lack of variation over the solar cycle might in any case suggest no direct link with low-latitude cyclic activity. The result could be due to some as yet unidentified systematic error either in the *a*-coefficient data or in their interpretation. One might though imagine that the anomaly arises from some confluence of large-scale meridional convective motions at 60° latitude, which could create a thermal anomaly there or could act either to drag up new flux from the tachocline or to subduct old flux from the surface down into the interior.

### (b) Near-surface solar-cycle effects on the modes

The largest part of the temporal variation of the even-order coefficients is actually not the contribution from the interior but the surface term in equation (3.1). The temporal variation of the surface contribution, and hence of the *a*-coefficients themselves, is strongly correlated with the photospheric magnetic field and reflects, in particular, the migration of the active latitudes over the solar cycle. Antia *et al*. (2001; see also Howe *et al*. 1999) demonstrate that there is a very tight correlation between each even-order coefficient *a*_{2k} and the corresponding projection , where *B* is the photospheric magnetic field strength, *P*_{2k} is a Legendre polynomial and *x*=cos *θ*. Such a relation follows if we assume that the temporal variation of an individual mode frequency is proportional to . Note that the observed dependence of the *a*-coefficients is on the absolute value of the measured magnetic field strength, not on its value squared.

To elucidate further the physical cause of the solar-cycle changes of mean multiplet frequencies, Dziembowski & Goode (2005) fit the temporal variations as(3.2)which defines the new quantities *γ*_{nl} as that part of the frequency variation that is left after the inverse mode mass has been factored out. The mean multiplet frequency depends only on a spherically symmetric mean of any perturbation. Owing to the different physics of the *f*- and *p*-modes, Dziembowski & Goode analyse their frequency variations separately. The factors *γ*_{nl} of both *f*- and *p*-modes—which we refer to as *γ*_{f} and *γ*_{p}—are correlated with a measure of solar activity such as sunspot number (figure 4).

Having factored out the inverse mode mass, the functions *γ*_{nl} should be principally functions of frequency *ν* and depend only weakly on *l* at fixed frequency, because this is true of the eigenfunctions far above the lower turning point of the modes. Dziembowski & Goode's analysis of the *γ*_{nl} as a function of frequency is shown in figure 5 (see also similar results for low-degree *p*-modes in the paper by Chaplin *et al*. 2001). It should be noted that normalization of mode inertia adopted by Dziembowski & Goode may differ from that in equation (1.1). Dziembowski & Goode conclude from fitting these *f*- and *p*-mode variations as functions of frequency that: (i) the rise in *γ*_{f} with *ν* (and hence with degree) is caused by a subphotospheric field reaching 500–700 G at 5 Mm depth; (ii) this field can only account for the behaviour of the *p*-modes at low frequency, not at higher *ν*; (iii) the most plausible explanation for *γ*_{p} is a 2% decrease in the radial component of turbulent velocities in the outer layers, from solar mininimum to solar maximum.

There have been a number of theoretical investigations of the effects of near-surface and atmospheric magnetic fields and associated phenomena on the frequencies of global oscillations. Roberts and collaborators investigated the effects of a chromospheric magnetic field (e.g. Campbell & Roberts 1989; Evans & Roberts 1991, 1992; Jain & Roberts 1994). Zweibel & Bogdan (1986) calculated effects of fibril magnetic fields on *p*-mode frequencies. Effects of near-surface changes in magnetic field and entropy were investigated by Goldreich *et al*. (1991) (see also Gough & Thompson 1988; Jain & Roberts 1993; Balmforth *et al*. 1996). Effects of photospheric flows were considered by Murawski & Roberts (1993). More recently, effects on mode frequencies and frequency splittings due to flows and magnetic fields, both organized and random, have been investigated by Erdélyi and collaborators (e.g. Erdélyi & Taroyan 2001; Pintér *et al*. 2001; Erdélyi *et al*. 2005). Investigations such as that by Dziembowski & Goode (2005) give fresh observational impetus to such work, and it is certainly worth revisiting these models in the light of the most recent observational results.

## 4. Conclusions

The helioseismic investigation of the solar interior to elucidate the behaviour of the magnetic field and the interior's role in the solar cycle, which we may call magnetohelioseismology, is proceeding on a number of fronts. Global solar frequencies are sensitive (directly and indirectly) to the internal magnetic field. Mean frequencies and *a*-coefficients vary over the solar cycle. The dependence of these variations on mode degree and frequency indicates that the dominant cause of the cycle variations is located close to the surface. The variations are linearly related to the measured strength of the unsigned photospheric flux. Analysis of *f*- and *p*-mode mean-frequency shifts indicate a 500 G field at 5 Mm depth, and suggest a 2% modulation of turbulent convective velocities from solar minimum to maximum. There is an apparent interior wave-speed anomaly at 60° latitude through the convection zone. Angular momentum variations, both the torsional oscillations and quasi-periodic oscillations near the tachocline, may indicate a role for magnetic field. Time–distance studies show wave-speed anomalies under active regions are extremely promising for future understanding of the linkages between the magnetic interior and the magnetic field in the solar atmosphere.

## Discussion

S. M. Jefferies (*Department Maui Scientific Research, University of New Mexico, Kihei, HI, Postcode 86761, USA*). Could the penetration of the photosphere–convection zone boundary by the *β*∼1 surface (the location where the gas and magnetic pressures are similar) around active regions with strong fields cause a shortening of the acoustic cavity by providing a reflecting layer for the acoustic waves (thus providing an alternative mechanism for the observed changes in *p*-mode frequencies with solar cycle)?

M. J. Thompson. I think there can be some effect. I believe Paul Cally will discuss such an effect in his talk.

R. Erdelyi (*Department of Applied Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK*). Kosovichev's reconstruction of sub-surface structures in sunspot only cites sounds waves. Could you comment on the errors possibly introduced by this neglect of magnetic effect?

M. J. Thompson. All such work in helioseismology to date, not just Kosovichev's, makes the assumption that the magnetic field simply modifies the effective isotropic sound speed. It takes no account of anisotropies introduced by the field direction, mode conversion, etc. Some groups are beginning to look at such effects, including our own and Paul Cally's in Monash. After making relevant forward modelling, we shall then be able to assess the effects on the inversions.

M. Goossens (*Department of Mathematics, KU Leuven, Leuven B-3001, Belgium*). (i) Is the internal magnetic field toroidal and/or poloidal? (ii) If the magnetic field is of poloidal and/or dipolar structure, can you still talk about the value of *l*?

M. J. Thompson. (i) Effects of both toroidal and poloidal field have been modelled (e.g. Gough & Thompson 1990) but in most analysis of observed frequencies we have only been interested in a local effect of the field on the effective sound speed and so have not sought to distinguish poloidal and toroidal geometries. In my work with Antia and Chitre, for example I believe we used results from calculations with a toroidal field. (ii) You are right, in that even if the field is axisymmetric so *m* is still a well-defined quantum number, *l* is not. However, for a weak field (in a global sense), there is still a dominant *l* value.

B. Roberts (*Mathematical and Computational Sciences, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK*). (i) Is the correlation of *a*_{k} coefficients to *B* being proportional to the strength of the magnetic field, rather than *B*^{2}, a reflection of the magnetic filling factor? (ii) The inferred field strength of 200–300 kG at the base of the convection zone, at 60° latitude, is comparable to that deduced by theorists modelling the emergence of magnetic flux tubes from the base of the convection zone. Can you comment on this?

M. J. Thompson. (i) That is a credible explanation of the findings, yes. Then the measured field strength (not resolving individual fibrils) is *f*×*B*_{f}, where *f* is the filling factor and *B*_{f} the individual fibril field strength, and the frequency shifts are proportional to *f*×*B*_{f2}. If *B*_{f} doesn't change over the cycle, the observed linear relation follows, as you suggest. (ii) I am not aware that anyone has commented on this before; but the latitude is rather high compared to active latitudes.

P. S. Cally (*Department of Mathematics, Monash University, VIC Postcode 3800, Australia*). Do you interpret the 60° latitude wave anomaly as a surface or deep effect? How could it feed into the sunspot band at much lower latitudes?

M. J. Thompson. It appears from the inversions to be a deep effect. It could be a systematic of some sort from surface modulation of the modes, and for that reason should be regarded at present with caution; but we have not identified as yet how such a systematic error would arise. I don't have a ready answer to the second part!

M. J. Aschwanden (*Lockheed Martin Department, Solar and Astrophysics Laboratory, Palo Alto, CA 9430, USA*). Why does the tachocline feature not show up in your diagram of the difference of the sound speed as function of depth between theoretical models and helioseismic inversion? Is there an inversion problem?

M. J. Thompson. The tachocline is almost certainly too narrow for us to resolve it, but I don't think this is the issue here. The results I showed (not included in the write-up) were of sound-speed differences between Sun and model, not the sound speed itself. The sound speed shows a distinct break in gradient near the tachocline, consistent with a discontinuity in first or second-derivative at the base of the convection zone.

## Acknowledgments

I thank P. R. Goode for providing a preprint of Dziembowski & Goode (2005), and P. R. Goode and H. M. Antia for providing figures for this paper.

## Footnotes

One contribution of 20 to a Discussion Meeting Issue ‘MHD waves and oscillations in the solar plasma’.

- © 2005 The Royal Society