INDEPENDENT SIGNALS FROM THE INFLUENCE OF INTERNAL MAGNETIC LAYERS ON THE FREQUENCIES OF SOLAR p-MODES

The 11 year cyclic variation of magnetic activity is the most obvious manifestation of solar variability. Since most forms of solar activity are magnetic in origin, they also follow this 11 year cycle, which refers to a quasi-periodic oscillation in the surface distribution of the solar magnetic field. A large body of helioseismic data convincingly shows that a cyclical variation in p-mode frequencies has been observed on the timescale of the solar activity cycle (see Howe 2008, for a recent review). During different time epochs, several groups have shown that low-degree modes (0 ⩽ l ⩽ 4) at increasingly higher frequencies become more sensitive to solar cycle effects (Woodard & Noyes 1985; Fossat et al. 1987; Pallé et al. 1989; Elsworth et al. 1990, 1994; Anguera Gubau et al. 1992; Chaplin et al. 1998; Jiménez-Reyes et al. 1998). The solar cycle dependence of the intermediate-degree (5 ⩽ l ⩽ 150–300) frequencies in the p-mode spectrum was established later (Libbrecht & Woodard 1990; Rhodes et al. 1993; Ronan et al. 1994; Bhatnagar et al. 1999; Howe et al. 1999). These observations report, over a solar cycle, peak shifts at a frequency of 3900 μHz, which are of order 750 nHz for low-degree modes (Chaplin et al. 1998) and 900 nHz for intermediate-degree modes (Libbrecht & Woodard 1991; Howe et al. 1999).

Bachmann & White (1994) have found hysteresis shapes among seven indices of solar activity in their relative variations during a 20 year period. Such relations were also found in p-mode frequencies of low degree by Jiménez-Reyes et al. (1998). In some cases, these observed hysteresis patterns start to repeat over more than one solar cycle, giving evidence that this is a normal feature of solar variability. Bachmann & White (1994) argue that hysteresis represents a real delay in the onset and decline of solar activity. In this view, Jiménez-Reyes et al. (1998) suggest that the hysteresis shape could be due to time-delayed responses to one single phenomenon, which can be located deep in the Sun, while Moreno-Insertis & Solanki (2000) find that the hysteresis in low-degree shifts is consistent with near-surface magnetic flux distributions. Latest analysis of intermediate-degree p-modes data shows the presence of hysteresis effects in near-surface toroidal magnetic fields (Baldner et al. 2009).

Only if one could remove contributions due to the near-surface perturbations on the p-mode spectrum, one can hope to evaluate the independent signals from the solar interior (Dziembowski & Goode 1997). The recent unusually long solar minimum provides such test ground. As of 2009 September, the start of the new cycle 24 is underway, but in the declining phase of cycle 23 and the following solar minimum, Broomhall et al. (2009b) have found unusually large differences between the frequencies observed in data from the Birmingham Solar Oscillations Network (BiSON) and the surface activity levels traditionally considered as proxy for the "surface frequency shifts". The discrepancy, of the order of 100 nHz, is reported for low-degree modes. Similar discrepancies are reported by Jain et al. (2009) in data from the Global Oscillation Network Group (GONG) and Michelson Doppler Imager (MDI) aboard the Solar and Heliospheric Observatory (SOHO) spacecraft and by Salabert et al. (2009) with data from the Global Oscillations at Low Frequencies (GOLFs) instrument aboard SOHO. The anomalous frequency shifts (not correlated with the surface activity) are therefore suggested to result from effects in the solar interior. They appear also to be a continuation of a possible "quasi-biennial" variation in the frequency shifts, superimposed on the solar cycle variation (Broomhall et al. 2009b). The 2 year period variation may be explained in terms of two types of dynamos operating at the base and in the subsurface of the convection zone (Benevolenskaya 1998). Alternatively, Salabert et al. (2009) attributes shift differences between low-degree modes to latitudinal effects.

There might be a source of confusion between the effect of magnetism and the effect of radial stratification at the base of the convection zone, which is in some sense an abrupt transition for the modes. Through theoretical modeling, Foullon & Roberts (2005) concluded that the effect of a buried magnetic layer on the frequencies of solar p-modes is too small to account for the changes in frequency over the solar cycle. For a realistic magnetic field at the base of the convection zone or in the anchoring zones of sunspots, the observed shifts (over the solar cycle) cannot be reproduced, while an explanation must be found through surface effects, such as the influence of the magnetic atmosphere (e.g., Campbell & Roberts 1989; Evans & Roberts 1992; Jain & Roberts 1996). The sunspot anchoring zone might still be relevant but a more thorough modeling of this region would be needed.

We consider a pressure perturbation on the p-mode spectrum, due to a bundle of magnetic flux localized in the stratified convection zone, which can be modeled as a horizontal layer of magnetic field. Thus, following the theory and a tractable analytical approximation developed by Foullon & Roberts (2005), we compile frequency shifts by surveying parameter conditions in the solar interior. In addition, we treat frequency shifts in the fashion of Broomhall et al. (2009b) in order to demonstrate the expected p-mode spectral signatures of thermal or magnetic perturbations that could be responsible for the anomalous shifts at solar minimum. This work may ultimately provide a diagnostic tool, which may help to locate the depth(s) of the perturbation(s) and possible departures from standard solar models.

Foullon & Roberts (2005) provide an analytical estimate of the frequency shifts Δν on the p-mode frequencies ν of radial order n and degree l, which result from the effect of a thin non-magnetized isothermal layer of thickness h and lying between depths z = z_p and z = z_q within a polytrope (see Figure 1(a)). The layer may reside for instance in the convective overshoot region, at the base of the convection zone, or the anchorage zone of sunspots some 50 Mm below the surface. The polytropic index of the adiabatically stratified field-free media external to the layer is m = 3/2. We use the horizontal wavenumber k_h, related to the degree l through $k_h=\sqrt{l(l+1)}/\,{R}_\odot$, for solar radius  R_☉. The fractional frequency shift, f = Δν/ν, can be approximated for h = z_q − z_p ≪ z_p as (Foullon & Roberts 2005, Equation (63))

$$\begin{equation} f =-\frac{h}{k_h}\cdot \frac{(2k_hz_p)^{m+2}}{e^{2k_h z_p} }\cdot \frac{Re(\Pi) (k_h L_1)^2 (n-1)!}{2(m+2n)\Gamma (n+m+1)}, \end{equation} \tag{ 1 }$$

where

$$\begin{equation} \Pi =\left[\left(1+2\frac{L_2}{L_1}\right)-\frac{1}{2k_hH_o}\right]^2-\left[\frac{\kappa }{k_h}\right]^2, \end{equation} \tag{ 2 }$$

and L₁, L₂ are generalized Laguerre polynomials,

$$\begin{equation} L_1=L_{n-1}^{m+1}(2k_hz_p)\,,\qquad L_2=L_{n-2}^{m+2}(2k_hz_p). \end{equation} \tag{ 3 }$$

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The density scale height, H_o, and κ are elements of the magnetoatmospheric wave equation (Goedbloed 1971; Nye & Thomas 1976; Foullon 2002; Foullon & Roberts 2005). In the field-free case, they can be fully developed as a function of m and k_hz_p, namely,

$$\begin{equation} H_o=\frac{z_p}{m+1}\, \end{equation} \tag{ 4 }$$

and

$$\begin{equation} \left(\frac{\kappa }{k_h}\right)^2=\frac{m+2n+\frac{m}{m+2n}}{k_h z_p} -\frac{(m+1)^2}{4 (k_h z_p)^2}-1. \end{equation} \tag{ 5 }$$

f(h, z_p) in Equation (1) is an approximation to the shifts between the model including the isothermal layer and the simple polytrope. Such an analytical approximation avoids essentially the numerical task to find roots to a dispersion relation that includes confluent hypergeometric functions. Since the model extends to infinite depth, the shifts include the effect of equilibrium pressure changes at depths z > z_q. To estimate the shifts due solely to the thermal effect of a thin isothermal layer, one can first consider the analytical shifts f(h/2, z_p) for a field-free layer of thickness h/2 located at depth z_p and subtract from them the analytical shifts f(h/2, z_p + h/2) for a layer of same thickness located immediately below the first layer (at depth z_p + h/2). The result

$$\begin{equation} \frac{\Delta \nu }{\nu } = f \left(\frac{h}{2}, z_p\right) - f \left(\frac{h}{2}, z_p+\frac{h}{2}\right), \end{equation} \tag{ 6 }$$

represented in Figure 1(b), is an attractive analytical solution to the effect of a thin isothermal layer of thickness h/2 rising from depths z_p + h/2 to z_p and can be used to give an overall perturbation of thickness h to a polytropic profile of the solar interior, without the effect of equilibrium pressure changes at infinite depths. At depth z_p + h/2, the associated change in the square of the sound speed, c²_s, is

$$\begin{equation} \frac{\Delta c_s^2}{c_s^2} = -\frac{1}{1+\frac{2z_p}{h}}, \end{equation} \tag{ 7 }$$

and the associated change in density, ρ, is

$$\begin{equation} \frac{\Delta \rho }{\rho } = \frac{e^{(m+1)\frac{h}{2z_p}}}{\left(1+\frac{h}{2 z_p}\right)^m}-1. \end{equation} \tag{ 8 }$$

Note from Equation (1) that this model yields zero shifts for degree l = 0. Hence, in the two-dimensional parameter space of varying depths, z_p, and layer thickness, h, and for any degree l > 0 and radial order n, we obtain separate fractional frequency shifts Δν/ν as a result of the presence of a field-free layer. The effect of magnetic fields structured in such a way as to produce a constant Alfvén  speed may easily be introduced. However, since realistic magnetic fields can be neglected in this model (Foullon & Roberts 2005), we will not treat them here.

To allow comparison of the results with the helioseismic observations, we first obtain the frequency shift δν_nl for a given l and n by multiplication of Δν/ν to the corresponding average values ν_nl pertinent to a quiet Sun as listed by Broomhall et al. (2009a, their Table 3). We then follow the mode inertia weighting procedure of Chaplin et al. (2004; see also Chaplin et al. 2007a),

$$\begin{equation} \delta \nu =\left[\sum _\nu \frac{\delta \nu _{nl}F(\nu _{nl})}{Q_{nl}}\right]\left[\sum _\nu \frac{F^2(\nu _{nl})}{Q_{nl}^2} \right]^{-1}, \end{equation} \tag{ 9 }$$

which involves the mode inertia ratio, Q_nl (Christensen-Dalsgaard & Berthomieu 1991), and the scaling function, F, to the frequencies ν_nl (Chaplin et al. 2004)

$$\begin{equation} F(\nu _{nl})=a_i \nu _{nl}^i,\quad i \in [0, 1,2, 3,4, 5], \end{equation} \tag{ 10 }$$

where a_i∈ [−2.37116, 5.91549 × 10⁻³, −5.54457 × 10⁻⁶, 2.37824 × 10⁻⁹, −0.45240 × 10⁻¹², 0.03275 × 10⁻¹⁵], with ν_nl in μHz.

3.1. Mean Frequency Shifts

We first compile a mean frequency shift 〈δν〉 for ν_nl ranging between 2100 and 3500 μHz, to allow comparison with the BiSON observations (Broomhall et al. 2009b). The discrepancy between the observed frequency shifts and the surface activity proxies is of the order of 100 nHz (Broomhall et al. 2009b, their Figure 2). Figure 2(a) presents plots of 〈δν〉 in the h–z_p parameter space. One feature of particular interest is a sign change in 〈δν〉 near depth z_p ∼ 0.03 R_☉ for every thickness h, from negative shifts in the deeper convection zone to positive shifts nearer the surface. Effects on frequencies expected from model changes in global mode helioseismology (e.g., Gough & Thompson 1991) indicate that the density profile changes may yield shifts of opposite sign to those expected from sound speed changes alone. As z_p → 0, the density change associated with the layer model increases with an exponential factor compared to the sound speed change (see Equations (7) and (8)). This effect is due to the isothermal profile used in the layer. Thus, it is not surprising to find, for this particular model, a sign reversal zone near the surface.

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The thermal effect of an overall layer as thin as 0.003 R_☉ ∼ 2100 km at subsurface depths is able to yield a positive 100 nHz shift, while a layer as thick as 0.03 R_☉ ∼ 21 Mm near the base of the convection zone could yield a negative 100 nHz shift. Note that the sign of 〈δν〉 can be reversed by considering a downward motion or perturbation in the layer model (i.e., reversing the sign in Equation (6)). Considering a fixed thickness h, the shift increases in absolute value as z_p decreases (except near the zone of sign reversal), which could be used to represent an upward motion of the layer. Thus, it is possible to obtain the observed anomalous shifts via the upward motion of a perturbative layer of thickness 0.01 R_☉ ∼ 7000 km through the sign reversal zone (z_p ∼ 0.03 R_☉) and over a 2 year period. An expansion of the layer during such rising motion, which corresponds to an increase in thickness h while z_p decreases, can also yield a non-negligible shift.

3.2. Frequency Shifts for Individual Degrees

3.3. Correlation with Solar Activity from the Interior

The discovery that p-mode frequencies of low degree do not follow changes of solar surface activity during the recent solar minimum is potentially rich of information regarding the structure of the Sun, which has been neglected in the standard model of the solar interior. To tackle this problem, we have transformed a useful approximation to the effect of a thermal layer in a plane-stratified model of the Sun, initially developed to consider the effect of buried magnetic fields. The analytical approach provides a useful alternative to the perturbation theory approach. For any orientation of the (horizontal) magnetic field, the effect of magnetism itself on the central p-mode frequencies can be neglected in comparison with the thermal effect of the perturbative layer.

Our parametric study demonstrates the range of plausible diagnostics to the frequency shifts observed. Reported mean frequency shifts can be reproduced with a layer as thin as 2100 km at subsurface depths, while individual low-degree shifts can be explained with a layer of size ∼4200 km increasing by a small amount at depths near 0.08 R_☉. It is also possible to obtain the mean shifts via the upward motion through the convection zone of a rising perturbative layer of thickness ∼7000 km. In this later scenario, the layer, presumably signaling new cycle magnetic flux, is expected to rise over a 2 year period through depths near 0.03 R_☉. The scenario of a rising layer is consistent with the basic ideas for a dynamo based on magnetic flux tubes and previous observations of variations of the sound speed distribution with time at the base of the convective zone. Given that the inferred anomalous shifts are rough estimates and the model is a simplified one, we do not require to have mean frequency shifts and individual low-degree shifts explained by one single scenario, although this would be desirable for validation in future developments.

The discussed scenarios place the source of the anomalous shifts in the upper regions of the convection zone. They differ from the latitudinal effects proposed by Salabert et al. (2009), which are not treated here and may play an important role. Still, helioseismic inversions of data in the recent solar minimum would be desirable to locate more precisely the source of temporal variations of the interior stratification. The discussed diagnostics are not affected by absolute determinations of the base of the convection zone. While recent solar photospheric abundance analyses put new constraints on solar models, which in turn predict the base of the convection zone to be at a shallower depth than previously thought (e.g., Guzik et al. 2005), the results are still under intense debate (e.g., Basu et al. 2007; Chaplin et al. 2007b). Further analysis of the p-mode multi-modal parameters in the minimum epochs of solar activity is required to improve the diagnostics and locate the independent signals of the buried layers. Of particular interest are the details of theoretical oscillatory patterns in the frequency shifts of individual modes (Foullon & Roberts 2005), which could be used, if found in the helioseismic data, for further diagnostics. Modeling of the perturbative field-free layer could be improved analytically by an investigation of a polytropic layer with variable indices. Ultimately, such signatures of the responsible pressure perturbation in the wave guiding medium may help to understand departures from standard solar models and to address the cause of the recent unusually long solar minimum.

C.F. thanks A.-M. Broomhall from the BiSON group for helpful discussions and B. Roberts, with whom earlier work inspired this study.