Local helioseismology seeks to probe the near surface regions of the Sun, and in particular of active regions. These are distinguished by their strong magnetic fields, yet current local techniques do not take proper account of this. Here, we first derive appropriate gravito-magneto-acoustic dispersion relations, and then use these to examine how acoustic rays entering regions of strong field split into fast and slow components, and the subsequent fates of each. Specifically, two types of transmission point, where wave energy can transfer from the fast to slow branch (or vice versa) are identified; one close to the equipartition level where the sound and Alfvén speeds coincide, and one higher up near the acoustic cutoff turning point. This second type only exists for rays of low frequency or low l though. In accord with recent studies of fast-to-slow mode conversion from the perspective of p-modes, magnetic field inclination is found to have significant consequences for wave splitting.
Modern local helioseismology is addressed in large part to exploring the near-surface layers beneath active regions, sunspots especially. Time–distance helioseismology (Kosovichev 1999, 2002; Kosovichev et al. 2000; Zhao & Kosovichev 2003) and acoustic holography (Braun & Lindsey 2000a,b) in particular have yielded impressively detailed views of active region sub-photospheres.
However, the role of magnetic field in both determining ray paths and influencing interpretation has not yet been adequately evaluated. Recently, Schunker et al. (2005) found strong observational evidence that active region fields do in fact significantly modify helioseismic signals, suggesting that we ignore these effects at our peril.
In this paper, we set out to develop the necessary theory to perform forward modelling of rays in regions of strong magnetic field, including mode conversion from fast to slow magnetoacoustic rays and vice versa. Ray theory forms the basis of ‘classical’ time–distance helioseismology (D'Silva 1996; Kosovichev 1999), but is now being partly supplanted by Born approximation (Birch & Kosovichev 2000) and Rytov approximation (Jensen & Pijpers 2003) techniques. These give a slightly deeper region of sensitivity, and allow the modelling of scattering effects, but otherwise yield very similar results (Birch & Felder 2004; Couvidat et al. 2004). Acoustic holography does not formally utilize a ray description (see Lindsey & Braun 2004) for a comparison of results from holographic and eikonal imaging, though it does use the Fermat principle acoustic ray path along which to propagate its wave-mechanical ingression and egression. In the interests of simplicity, the classical ray description is adopted here.
2. Dispersion relations
The ‘classical’ acoustic dispersion relation is , where c is the sound speed, ω the (circular) frequency, and kh and kz the horizontal and vertical components of the wavevector. In a non-uniform medium, it results from a zeroth order WKB (Wentzel–Kramers–Brillouin) analysis (eikonal) in which the coefficients are frozen. It is strictly correct in gravitationally stratified atmospheres only in the high frequency limit. If we then consider a simple atmosphere in which c increases monotonically with depth, acoustic rays have a lower turning point, where kz=0, i.e. at . However, in such an atmosphere there is no upper turning point. Reflection may take place at a boundary, e.g. c=0, in such a model, but there is no internal refraction which turns the ray over.
However, it is well known that finite frequency rays in these simple models (and in the real Sun) do turn over as they approach the surface, due to an acoustic cutoff effect. The standard and neatest way to explicitly incorporate the acoustic cutoff frequency ωc into the wave equations in a non-magnetic atmosphere in a manner suitable for zeroth order WKB analysis is to introduce the dependent variable(2.1)where ξ is the displacement vector (Deubner & Gough 1984). This is briefly reviewed in §2a.
The analysis is then extended to the magnetic case in §2b. However, our prime focus is on large magnetic field concentrations, such as active regions, where the length scale over which the field varies is very large, typically several megametres at least. The gravitational density scale height near the surface may be only one or two tenths of a megametre though. It is, therefore, essential to include density scale height effects (which produce the cutoff) in the acoustic terms, but the variation of magnetic field may be safely neglected. For that reason, the field B is assumed uniform in §2b. Slow variations in magnetic field are still accounted for within the WKB method. As is common in local helioseismology, and in the interests of simplicity, solar curvature effects are also neglected so that Cartesian coordinates may be used throughout.
(a) Non-magnetic case
With Ψ defined as in equation (2.1), the linear adiabatic wave equations in a plane stratified atmosphere may be expressed in the convenient form(2.2)where N is the Brunt–Väisälä (buoyancy) frequency and ωc the acoustic cutoff frequency, defined respectively by and(2.3)H(z) the density scale height, and represents the horizontal part of the Laplacian operator.
Assuming an time and horizontal space dependence, equation (2.2) reduces to an ordinary differential equation , where(2.4)and . In this form, it is amenable to WKB analysis, which is a singular perturbation technique of (almost) global applicability (see Bender & Orszag 1978 for a particularly cogent exposition). The ‘small parameter’ of the perturbation method in this instance is ω−1, but with ωc and N being considered of comparable size to ω. A solution is sought in the form(2.5)and is valid away from the turning points Q=0. A hierarchy of solutions for S0, S1, …, may then be built-up by equating powers of ω−1. The lowest order (eikonal or geometrical optics) approximation is obtained by truncating the expansion at n=0, leaving only . Within the cavity, where the solution is vertically propagating or trapped, Q is negative, and it is conventional to introduce the vertical wavenumber . The dispersion relation is then just a restatement of ; viz.(2.6)where . Progressing to the next order, n=1, is trivial in this case, giving , though we shall not require that result here.
An alternative derivation of dispersion relations in weakly inhomogeneous media based on the Weyl symbol (effectively a ‘local Fourier transform’; Friedland & Kaufman 1987; Tracy & Kaufman 1993) provides a convenient Hamiltonian formulation, which is particularly suited to ray theory and mode conversion. This approach is not used explicitly here, though its central philosophy of working in space–time/frequency–wavevector space underpins linear mode conversion theory, and the central result (3.3) derives from it.
The Brunt–Väisälä frequency in the solar convection zone is entirely negligible everywhere except in the top few 100 km, and even there it is only significant for waves of rather low frequency (Barnes & Cally 2001). However, the acoustic cutoff frequency is more important. It rises monotonically from zero in the top few millimetres (apart from some wild behaviour of dubious significance in the top 200 km; see Barnes & Cally 2001, fig. 3), before settling onto a photospheric/chromospheric plateau of about 5.5 mHz. Neglecting N, we see that kz=0, where . Even purely vertical waves (kh=0) are reflected if their frequency does not exceed the maximum ωc.
(b) Two-dimensional magnetic case
Although, magnetohydrodynamic (MHD) dispersion relations in gravitationally stratified atmospheres have been derived before (e.g. McLellan & Winterberg 1968; Thomas 1983), none (to our knowledge) has explicitly incorporated the acoustic cutoff and Brunt–Väisälä frequencies in the self-adjoint form required in ray splitting calculations (see §3a). The self-adjoint (Hermitian) formulation directly expresses the dissipationless character of the problem. We derive the necessary formulation here by extending the formalism above to the case of two-dimensional propagation in a plane stratified plasma with inclined but uniform magnetic field .
Assume an dependence on x and t, and define the Lorentz force per unit mass as(2.7)where(2.8) is the displacement vector, and is the Alfvén speed. Primes denote differentiation with respect to z. A great deal of algebra (carried out using the computer algebra package Mathematica) then leads to(2.9)where ρ(z) is the equilibrium density and is the ratio of the f-mode frequency to the wave frequency. In most helioseismic applications, ϵ may be small to moderate, i.e. frequencies being used exceed the f-mode frequency for the given kx. The term on the right-hand side represents coupling to magnetic oscillations. Similarly,(2.10)describes the acoustic influence on magnetic oscillations. The left-hand sides of equations (2.9) and (2.10) represent, respectively, uncoupled acoustic and magnetic propagation.
At this stage, we apply the WKB method by formally setting throughout, as in the non-magnetic case, and similarly for Y in equations (2.7) and (2.10). Guided by the necessity of a self-adjoint formulation, and in keeping with the definition of Ψ, it is then found best to set . The coupling terms on the right-hand sides of equations (2.9) and (2.10) thereby both introduce density derivatives, in the form of the scale height H.
Equations (2.9) and (2.10) may now be expressed in a convenient Hermitian matrix form, , where(2.11)and . Here, and are the components of the wavevector parallel and perpendicular to the magnetic field, and .
The resulting dispersion function is(2.12)where ωci=c/2H is the ‘isothermal’ acoustic cutoff frequency (cf. equation (2.3)). In the limit ϵ→0, the dispersion relation is(2.13)D2 reduces to (1−ϵ4)ω2D1 in the non-magnetic limit, and to the usual magnetoacoustic dispersion function if ϵ, ωc and N vanish. In the strong field limit a≫c, neglecting both ϵ and N, the slow wave dispersion relation reduces to(2.14)making clear that inclined magnetic field reduces the effective acoustic cutoff frequency in low beta plasmas (De Pontieu et al. 2004).
3. Ray conversion and transmission
Fast-to-slow wave conversion within a modal description has been analysed in great detail (Cally & Bogdan 1993; Cally et al. 1994; Bogdan & Cally 1997), and might explain both the absorption and phase shift of p-modes observed to occur in sunspots (Braun 1995; Cally et al. 2003). However, transmission and conversion of acoustic rays has been relatively neglected in solar applications (though see Frisch (1964), who addresses the coronal case where the spatial variation in the magnetic field is the dominant coupling mechanism). The ray description is more in keeping with most modern local helioseismic techniques, at least in philosophy if not in detail. Recently though, a ray-like (but non-eikonal) stationary phase perturbation method was applied to the case of vertical magnetic field in an adiabatic polytrope of index m (Cally 2005), finding that conversion occurs at the point at depth L, where the sound and Alfvén speeds coincide, a=c=ceq, and that the acoustic-to-acoustic transmission coefficient (fast-to-slow) through this point of a near-vertical upcoming incident fast wave is(3.1)where α is the ‘attack angle’—assumed small—of the wavevector on the magnetic field lines at the conversion point, i.e. the angle to the vertical in this case. However, the calculation is not readily extensible to more general magnetic fields or atmospheres. In this section, a flexible ray-based method discussed recently by Tracy et al. (2003) and based on a considerable body of development of the phase space formulation of mode conversion (e.g. Friedland & Kaufman 1987; Friedland et al. 1987; Tracy & Kaufman 1993 and references therein) is applied to gravitationally stratified MHD to determine transmission and conversion coefficients.
(a) General method
For the moment, only the two-dimensional system will be considered. Extensions to the three-dimensional case will be addressed elsewhere. The fundamental result of the theory, as it applies to two-dimensional problems, relies on the identification of a Hermitian ‘dispersion matrix’,(3.2)such that , where y is a two-vector specifying the perturbation field. The dispersion function is then . Here, Da and Db are the ‘uncoupled’ dispersion functions applying to the distinct modes (acoustic and magnetic) far from the transmission region, and η is the coupling term, which is ‘small’ in some sense. The WKB ansatz breaks down, and transmission occurs, where (in (z, kz) space) Da and Db become similarly small. Then, the transmission coefficient, representing the fraction of energy tunnelling from one WKB branch to the other, is given by(3.3)where(3.4)is the Poisson bracket of the uncoupled dispersion functions. The star indicates that T must be evaluated at the ‘star point’ (z★, kz★), where Da and Db simultaneously vanish. This summarizes the general theory as set out in Tracy et al. (2003). In the MHD case, with our specific dispersion matrix (2.11), equation (3.1) suggests that the attack angle, , may be identified as the small parameter.
Identifying for the moment, and assuming the frequency is sufficiently high that ωc, N, H−1 and ϵ may be neglected, yields a=c, i.e. z*=−L and . Then, equation (3.3) becomes(3.5)Transmission is clearly enhanced by a ‘sharp’ interaction region, i.e. large.
(b) Example: adiabatic polytrope
To illustrate the processes at work, a simple adiabatic polytropic model is adopted for the solar convection zone (Cally et al. 1994). With uniform magnetic field, the sound and Alfvén speeds, the acoustic cutoff frequency, and the density scale height are specified by(3.6)where z≤0, g is the gravitational acceleration, m=1/(γ−1) is the polytropic index, and L is the equipartition depth at which a=c. For simplicity, , i.e. , will be adopted in all numerical calculations. It is also convenient to define a dimensionless frequency . Typically, L may range from several 100 km up towards a megametre in a solar active region, depending on field strength and temperature structure. Since, the polytropic equation of state is a power law, which has no intrinsic length scale, wavenumbers k enter physically only in the form kL, and frequency ω only through , i.e. ν. The Brunt–Väisälä frequency N vanishes identically. Specifically, note that c increases with depth, while a and ωc both decrease from asymptotes at z=0. The unbounded increase in ωc as the surface is approached does not match the true solar behaviour, but mimics its effect at least qualitatively at frequencies below the chromospheric cutoff.
(i) Simple case: vertical field with ωc=N=ϵ=0
For comparison with the result (3.1), we first examine the case of vertical magnetic field with ωc and ϵ neglected. Then, and , where α is the angle of the wavevector to the vertical. It is easily shown that Da=Db=0 implies that a=c, so z*=−L and . Substituting these into equation (3.3) or (3.5) yields(3.7)to leading order in sin α, assumed small. Noting thatin the polytrope, and that K=ω/c for an acoustic wave, (3.1) is recovered.
The significance of this result is clear.
Acoustic transmission increases with decreasing frequency, as the WKB approximation becomes worse; and
acoustic transmission increases with decreasing attack angle, as the longitudinal acoustic wave more nearly slips along the field lines without disturbing them.
Both results make good physical sense.
(ii) More complex examples
We now include the effect of the acoustic cutoff frequency ωc. However, for simplicity, ωa and the other scale height terms in the off-diagonal elements of are neglected. These only serve to shift the position of the cutoff very slightly, and in view of the simplicity of our polytropic model, their inclusion only complicates the analysis with little benefit.
Various cases are illustrated in figures 1 and 2. These propagation diagrams plot kz against depth, showing both the fast branch (inner lobe) and the slow branch (outer). The dotted curves depict Da=0 and Db=0. An upcoming fast ray corresponds to a point moving leftward along the top of the inner lobe. However, here we have chosen to set , where(3.8)rather than itself. The transformation maintains the Hermitian structure, and is equivalent to a change of dependent variable y. It has the virtue that the resulting Da and Db recognize the field-aligned nature of the slow waves in the regimes a≫c and c≫a, whereas and are clearly isotropic in wavenumber space. Specifically, we now have(3.9)
Specific points to note from the figures include.
The diagrams are symmetric about kz=0 for vertical magnetic field (e.g. figure 1a), but not for inclined field (figure 1b). In the case of figure 1b, the field inclination of 30° tends to strengthen transmission on the upper side, but weaken it on the lower. This is reversed if the field is inclined the other way.
The slow branch turns over at small dimensionless depth s=−z/L, passing through kz=0 at some value sc. This is the upper turning point induced by the acoustic cutoff. A ray initially moving leftward along the top of the slow branch reaches this point and then reverses, passing downwards along the lower half of the branch. This is distinct from the case ωc=0, where the upper and lower slow branches asymptote to ±∞, respectively, as s→0+ (z→0−).
Higher frequencies push sc closer to the surface, allowing more of an ‘upswing’ in the slow branch before the acoustic cutoff takes effect (figure 2b).
The star points are displaced slightly from s=1 (a=c) by the acoustic cutoff. These star points, close to the equipartition level, shall be termed Type 1 transmission points. We use the term transmission to indicate wave energy tunnelling from the fast to slow branch or vice versa.
Fast rays emerging from greater depth (figure 2a) are more vertical, and hence penetrate closer to the surface. In so doing, they may impinge upon the slow branch at sc, generating two avoided crossings, and hence two more star points. This also happens if the frequency is lowered, since then the acoustic cutoff turning point drops lower to meet the top of the fast mode cavity (figure 2c). This second class of star point, associated with the acoustic cutoff, shall be called Type 2.
Figures 3–5 depict the ray paths corresponding to the dispersion diagrams, derived from the dispersion relation using the usual Hamilton equations (e.g. Barnes & Cally 2001, eqn (17–20)). The following points are evident:
Fast mode ray paths differ substantially from the purely acoustic paths that would exist in the absence of magnetic field, primarily in the near-surface layer where the Alfvén speed is significant. Both the shape of the path and, more significantly, the speed of propagation (figure 3, lower panel) are very different.
Far below z=−L, we have c≫a, and the fast wave is essentially acoustic (the usual non-magnetic ray), while the slow wave is a short wavelength nearly incompressive transverse oscillation propagating along the field lines (thereby resembling an Alfvén wave). On the other hand, well above the equipartition depth, where a≫c, the fast wave propagates freely in any direction (actually, somewhat faster across field lines than along them; against a), while the slow wave is predominantly acoustic in nature and progressively more field-aligned as a/c increases.
Slow rays generated by tunnelling from an upward propagating fast ray near the equipartition level initially propagate upwards but soon turn back downwards in response to the acoustic cutoff effect. As they pass through a=c their nature changes from acoustic to magnetic. They become progressively more field-aligned, and their speed decreases with depth in response to the decreasing Alfvén speed (figures 3–5).
Slow rays generated by tunnelling from a downward propagating fast ray near the equipartition level propagate downwards from the start. They are also field-aligned and slowing.
Whereas transmission points and coefficients are symmetric about a vertical axis in vertical magnetic field, this is not the case in inclined field (figure 4). In particular, transmission is greatly enhanced for fine ray attack angle and suppressed for large angle, as suggested by equation (3.5).
While significant energy may transfer from the fast branch to the slow branch at say the first Type 1 point, and subsequently at other Types 1 and 2 points along the fast ray, it is also true that energy can just as easily transfer the other way, from slow to fast, at those points. The results of such transfers have been explicitly shown in the ray diagrams figures 3–5. It is not a simple matter of energy going back into the original fast ray. It will, in fact, go into a shifted fast ray displaced some small distance in x, since the slow ray has taken a different path after excitation. This means that the fast ray emerging from an active region after undergoing such multiple transitions will be somewhat defocussed, with potential helioseismic implications.
Figure 5 depicts a case where there are four strong transmission points, which will not happen for inclined field. An even number of such points must result in substantial shifted fast-wave emergence.
Figure 6 shows how much energy is retained in the original fast ray after one pass through the array of transmission points near the surface. This fraction is plotted as a function of the lower turning point zbot of the ray, scaled by L, and the frequency f in millihertz, scaled by L−1/2 for different values of the magnetic field inclination θ. The figures are valid for arbitrary positive L.
In each case, energy retention is complete for mHz Mm1/2, since the fast ray turns over at too low a level to produce any transmission points (figure 2d). As frequency increases beyond this range, the Types 1 and 2 transmission points suddenly switch-on (figure 2c), and are generally extremely effective at extracting energy from the ray.
For vertical magnetic field (figure 6a), shallow skimming rays (d small) cross the field lines at a large angle, and therefore do not lose much energy. This effect is amplified as frequency increases (see equation (3.1)). As d increases, the rays are progressively more vertical near the surface, and fast-to-slow losses increase.
For 30° inclined field, fast-to-slow transmission is very strong for because the upcoming ray's attack angle on the field lines at is very fine. For lesser or greater d, it is larger, and, therefore, there is reduced energy loss.
The ‘black blobs’ near the base of both diagrams are due to both Types 1 and 2 transmissions being in effect. Typically, Type 2 losses are very strong. As frequency increases, only Type 1 remains (compare figure 2b,c).
Finally, figure 7 shows the total emergent flux in all fast rays combined.1 It is notable that much of the black blob has disappeared, indicating that Type 2 transmissions often ultimately result in transferal to another fast ray. The, generally, increased flux from the vertical field case is a result of two identical strong Type 1 transitions bringing most of the energy back into the fast mode. Of the four cases shown, the moderate (30°) inclined field clearly ‘absorbs’ most energy, especially for d=3–7, in the sense of transforming it to slow waves which are lost downwards. This is in accord with Crouch & Cally (2003). Highly inclined field (case d) is clearly almost immune to mode coupling because of the large attack angle of any incident helioseismic ray.
4. Discussion and conclusions
The purpose of this paper is to point out the significance of strong magnetic fields to the propagation of waves used in helioseismology. At this stage, the discussion has been restricted to the two-dimensional case, and specific examples to a polytropic model. However, many important lessons have been learned.
In particular, much can be discerned from plotting a simple dispersion diagram. Avoided crossings are sites of energy transfer between fast and slow magnetoacoustic modes. We have found that these are of two types: Type 1, situated near the a=c equipartition level and not dependent on the acoustic cutoff effect (though their position moves slightly in response to it), and Type 2, which is inherently related to ωc. Once these avoided crossings, or more particularly their associated ‘star points’, have been identified, it is a trivial matter to calculate the fast-to-slow (or vice versa) conversion there using equation (3.3). We have done this here for the case of a polytropic model, but the transmission coefficient is easily evaluated for any choice of c(z), a(z), ωc(z) and N(z), including tabulated ‘realistic’ solar models.
Examples such as that depicted in figure 5, in particular, suggest that solar active regions my split and defocus acoustic images emerging from them. This has implications for both time–distance helioseismology and acoustic holography. The effect is somewhat akin to scattering from small-scale inhomogeneities, though it occurs in a smooth model. It is tempting to suggest a link with the ‘acoustic showerglass’ effect discussed recently by Lindsey & Braun (2005).
The analysis we have presented here does not take account of interference effects, which can occur when there are multiple transmission points (Brizard et al. 1998). Work is currently underway to correct that omission, but in any case, a full analysis requires knowledge of the individual transmission coefficients as presented here.
The validity of WKB-based ray theory per se in this context may be called into question. We draw some comfort though from fig. 9 of Cally (2005), which compares the WKB formula (3.1) with a non-WKB calculation of (Type 1) transmission in vertical magnetic field. The two are indistinguishable for above 7 mHz Mm1/2, and close enough for practical purposes down to at least 1 mHz Mm1/2. Even if strict quantitative accuracy is not attained, the concepts and qualitative behaviours encountered here are surely of helioseismic interest.
Although it appears that Type 1 points are quite robust, and will occur for any feasible solar model, Type 2 points are more fragile, as they depend sensitively on the precise depth dependence of the acoustic cutoff frequency. This will be investigated elsewhere using more realistic active region models.
Other possible helioseismic implications include doubts about the identification of observed surface oscillations in sunspots: are they the ‘original’ fast wave, or are they instead slow waves which have split from these? The results of (Schunker et al. 2005) suggest that at least a substantial component of the observed oscillations is field aligned, indicating that they are largely the slow wave, but this may depend on frequency and details of the field. When observing the acoustic wave subsequently in a surrounding pupil in the quiet Sun then, it may be that the supposedly connected internal and external oscillations are due to different rays, though with the same progenitor.
The author is grateful to Alina Donea, Hannah Schunker, Gene Tracy, Tom Bogdan and Ashley Crouch for useful discussions and suggestions during the development of this work.
One contribution of 20 to a Discussion Meeting Issue ‘MHD waves and oscillations in the solar plasma’.
↵This is found most conveniently by calculating where T1, T2, …, Tn are the transmission coefficients associated with each of the individual star points (taken in any order). Then, M1,1=M2,2 is the total fast mode flux, and M1,1=M2,2 the total slow mode flux.
- © 2005 The Royal Society