A MECHANISM FOR THE DEPENDENCE OF SUNSPOT GROUP TILT ANGLES ON CYCLE STRENGTH

One of the unsolved problems of the solar activity cycle is the physical nature of the mechanism(s) underlying the observed variations in cycle amplitude (Charbonneau 2010). Among several possibilities, reduction of poloidal flux generation by reducing the average tilt angle of bipolar magnetic regions has recently been considered as a plausible candidate. Analysis of tilt angle data from Mt. Wilson and Kodaikanal observatories between solar cycles 15–21 by Dasi-Espuig et al. (2010) has led to the discovery that the cycle-averaged sunspot group tilt angle was inversely correlated with the cycle strength. In terms of the Babcock–Leighton dynamo process, this means that the surface source for the poloidal field becomes weaker for stronger cycles, potentially limiting the strength of the next cycle.

A possible explanation for the observed anti-correlation is based on the effective reduction of the tilt angle by inflows toward activity belts, which are observed by local helioseismic techniques (González Hernández et al. 2008). Incorporation of such inflows into surface flux transport models has shown the efficiency of this mechanism in limiting the solar axial dipole moment (Cameron et al. 2010; Jiang et al. 2010; Cameron & Schüssler 2012).

As already discussed by Dasi-Espuig et al. (2010), systematic changes in the tilt angle can also be led by changes in the internal structure of the lower convection zone, a potential location for the origin of magnetic flux loops which produce sunspot groups. An observational hint came from global helioseismology of low-degree oscillation modes by Baldner & Basu (2008), who found a statistically significant reduction in the acoustic wave speed near the base of the convection zone between the minimum and maximum of cycle 23. A temperature perturbation mainly in the same direction (cooling) was predicted by Rempel (2003), who considered a magnetic layer near the base of the convection zone and obtained time-dependent solutions for radial heat transport by including radiative heating from below, in the presence of an imposed horizontal magnetic field reaching 10⁵ G.

In addition to radiative effects on stratification, stronger cycles possibly involve more frequent flux tube explosions in the midst of the convection zone (Moreno-Insertis et al. 1995; Rempel & Schüssler 2001; Hotta et al. 2012). This can also lead to a decrease in the radial entropy gradient, hence a decrease in the (negative) superadiabaticity in the lower convection zone.

In both the convection quenching and the entropy mixing scenarios, the convection zone would be increasingly stabilized for stronger magnetic fields. Consequently, the critical field strength for the onset of flux tube instability would be raised with the cycle strength. Flux tubes would then become unstable at higher field strengths, emerge at the surface with smaller tilt angles owing to stronger tension force.

Motivated by the helioseismic observations and the theoretical arguments summarized above, I determine the variation in the thermal perturbation required to account for the observed changes in the cycle-averaged tilt angle. The results indicate that a thermodynamic cycle in phase with the activity cycle at the base of the convection zone can be responsible for the nonlinear saturation of the solar dynamo.

I use a one-dimensional stratification model of the solar convection zone (Skaley & Stix 1991), which uses the non-local mixing length formalism of Shaviv & Salpeter (1973). This model allows for a weakly subadiabatic lower convection zone below 0.775 R_⊙, extending down to where the convective heat flux changes sign at about 0.736 R_⊙. The convective overshoot region extends from this location down to 0.721 R_⊙, with a thickness of about 10⁴ km.

2.1. Perturbations to the Stratification

To approximate the effect of radiative heating of a magnetic layer in the overshoot region as estimated by Rempel (2003), I model the change in the stratification simply as a decrease in the temperature with an asymmetric piecewise Gaussian perturbation of the form

$$\begin{eqnarray}&&{T}_{1}={T}_{{\rm{m}}}\mathrm{exp}\left[\displaystyle \frac{-{(r-{r}_{p})}^{2}}{{\sigma }_{\pm }^{2}}\right],\end{eqnarray} \tag{ 1 }$$

where T_m is the amplitude of the perturbation, centered at r_p = 5 × 10¹⁰ cm (0.718 R_⊙), and σ_± is the characteristic width of the distribution, with σ₋ = 400 km for r < r_p, and σ₊ = 4000 km for r ≥ r_p. Denoting the background thermodynamic variables by index 0 and the perturbations by index 1, I assume that the perturbations satisfy hydrostatic equilibrium,

$$\begin{eqnarray}&&\displaystyle \frac{{{dp}}_{1}}{{dr}}={-\rho }_{1}g.\end{eqnarray} \tag{ 2 }$$

For linear perturbations the ideal gas relation takes the form

$$\begin{eqnarray}&&{\rho }_{1}={\rho }_{0}\left(\displaystyle \frac{{p}_{1}}{{p}_{0}}-\displaystyle \frac{{T}_{1}}{{T}_{0}}\right).\end{eqnarray} \tag{ 3 }$$

Using Equation (3) in (2), I obtain

$$\begin{eqnarray}&&\displaystyle \frac{{{dp}}_{1}}{{dr}}=-\displaystyle \frac{{p}_{1}}{{H}_{p0}}+{\rho }_{0}g\displaystyle \frac{{T}_{1}}{{T}_{0}},\end{eqnarray} \tag{ 4 }$$

where ${H}_{p0}(r):= {p}_{0}/({\rho }_{0}g)$ is the pressure scale height in the unperturbed stratification. For simplicity, I assume that flux tubes leading to sunspot groups have a sufficiently low filling factor within a diffuse background field, so that the contribution of magnetic pressure to the hydrostatic equilibrium is neglected against the other terms in Equation (4).

The perturbation in specific entropy, s₁, can be determined by writing energy conservation in the thermodynamic notation

$$\begin{eqnarray}&&{s}_{1}\equiv {ds}=\displaystyle \frac{1}{T}\left[{du}+{pd}\left({\rho }^{-1}\right)\right],\end{eqnarray} \tag{ 5 }$$

where u is the internal energy. Writing du and dρ in terms of dT and dp and expressing the differential quantities as perturbations leads to

$$\begin{eqnarray}&&{s}_{1}={c}_{p}\left(\displaystyle \frac{{T}_{1}}{{T}_{0}}-{{\rm{\nabla }}}_{{\rm{ad}}}\displaystyle \frac{{p}_{1}}{{p}_{0}}\right),\end{eqnarray} \tag{ 6 }$$

where ${{\rm{\nabla }}}_{{\rm{ad}}}:= {(\partial \mathrm{ln}{T}_{0}/\partial \mathrm{ln}{p}_{0})}_{s}=1-{\gamma }^{-1}$ is the adiabatic temperature gradient.

The most critical quantity which determines the mechanical stability of magnetic flux tubes in the overshoot region is the superadiabaticity $\delta := {\rm{\nabla }}-{{\rm{\nabla }}}_{{\rm{ad}}},$ whose perturbation reads

$$\begin{eqnarray}&&{\delta }_{1}=-\displaystyle \frac{{H}_{p0}}{{c}_{p}}\displaystyle \frac{{{ds}}_{1}}{{dr}}.\end{eqnarray} \tag{ 7 }$$

Using Equations (3), (6), and (7), the perturbation in the superadiabaticity is found to be

$$\begin{eqnarray}&&{\delta }_{1}={H}_{p0}\left(\displaystyle \frac{1}{{\rho }_{0}}\displaystyle \frac{d{\rho }_{1}}{{dr}}-\displaystyle \frac{1}{\gamma {p}_{0}}\displaystyle \frac{{{dp}}_{1}}{{dr}}-\displaystyle \frac{{\rho }_{1}}{{\rho }_{0}^{2}}\displaystyle \frac{d{\rho }_{0}}{{dr}}+\displaystyle \frac{{p}_{1}}{\gamma {p}_{0}^{2}}\displaystyle \frac{{{dp}}_{0}}{{dr}}\right).\end{eqnarray} \tag{ 8 }$$

2.2. Linear Stability of Magnetic Flux Tubes

2.3. Nonlinear Dynamics of Magnetic Flux Tubes

3.1. Effects on Stratification

I now solve Equation (4) numerically, using a fourth-order Runge–Kutta scheme, by taking the equilibrium quantities as a function of radius from the stratification model, and the corresponding perturbations from Equations (1)–(8). The radial profile of the pressure perturbation is then obtained by setting p₁ = 0 at r = 0.56 R_⊙ as the initial value. The radial profiles of the perturbations T₁, p₁, ρ₁, and δ₁ are shown in Figure 1, for ${T}_{{\rm{m}}}=-50$ K. Despite the simplifications made in Section 2.1, the resulting profile of δ₁ has a similar shape and amplitude to the result of Rempel (2003).

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The profile δ₁ of Figure 1(d) is shown in more detail in Figure 2, along with the unperturbed $| \delta |$ profile. The effect of the thermal perturbation is such that the stratification is destabilized within a narrow layer in the radiative zone, though its relative effect on the highly subadiabatic environment is insignificant (the yellow region). However, in the blue-shaded region between about 0.72 R_⊙ and 0.74 R_⊙, the stratification is considerably stabilized, mainly in the overshoot region (arrowed line).

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3.2. Instability of Flux Tubes

How would the modified stratification affect the mechanical stability of magnetic flux tubes in the convective overshoot region? I first calculate the influence of the thermal perturbation (Section 2.1) on the linear instability map of thin toroidal flux tubes subject to different strengths of thermal perturbation within the layer.

I have set up stratification models corresponding to five values for the amplitude of the temperature perturbation in Equation (1): T_m = 0, −5, −10, −20, and −50 K (labeled T0, T5, T10, T20, and T50). The stability diagrams resulting from the linear stability analysis (Section 2.2) are presented in Figure 3. As $| {T}_{{\rm{m}}}|$ is increased, magnetic buoyancy instability sets in at gradually higher field strengths, compared to the unperturbed stratification. For T50 (not shown here), flux tubes would have to be 3–5 times stronger to become unstable, compared to the unperturbed case, T0.

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3.3. Simulating Joy's Law

To obtain the average values and latitude dependence of the tilt angle, I have carried out a grid of simulations for all the cases T0-T50, where the initial latitudes and field strengths of the tubes are chosen with 5° intervals in latitude and for linear growth times between 40 and 60 days, with 5-day intervals. The initial location of the flux rings is taken at 0.728 R_⊙, corresponding to the middle of the overshoot region (same as for Figure 3). The cross-sectional radius of the tube is set to 2000 km, which leads to a magnetic flux of 1.26 × 10²² Mx for a field strength of 10⁵ G. The tilt angle as a function of the emergence latitude (Joy's law) is plotted in Figure 4 for all the cases. To fit the simulation data, I choose the following functions, which are commonly used in observational studies:

$$\begin{eqnarray}&&\alpha (\lambda )=a\lambda ,\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\alpha (\lambda )={\gamma }_{0}\mathrm{sin}\lambda ,\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\alpha (\lambda )=T{\lambda }^{1/2},\end{eqnarray} \tag{ 11 }$$

where λ is the emergence latitude and α is the tilt angle in degrees, and a, γ₀, and T are the fit coefficients corresponding to each function. The functions have been fitted using the nonlinear Levenberg–Marquardt algorithm. The form (9) was used by Dasi-Espuig et al. (2010). The sinusoidal function (Equation (10)) was used by Stenflo & Kosovichev (2012). Their data set was based on bipolar magnetic regions from magnetograms, which include plage regions alongside spots, which is the possible reason for their systematically higher tilt angles. The form (11) was used by Cameron et al. (2010) when fitting cycle-dependent tilt angles of Dasi-Espuig et al. (2010), to use in surface flux transport simulations.

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Joy's law coefficients resulting from the simulations are given in Table 1, which includes standard and latitude-normalized averages of tilt angles, the fitted parameters for different forms of Joy's law, and also the superadiabaticity at the initial location of the flux tube. The mean tilt angle and Joy's law coefficients are inversely proportional to the amplitude of the thermal perturbation. As a result of the stabilized environment, the tilt angles are systematically lower, owing to increasing magnetic tension between the legs of emerging flux loops. Changing the amplitude of cooling in the middle of the overshoot region from 5 to 20 K roughly accounts for the observed amplitude of cycle-averaged tilt angles for solar cycles 15–21. The assumption behind this conclusion is that the average depth from which sunspot region producing flux tubes originate does not change significantly as a function of cycle strength.

Table 1.  Mean Tilt Angles and Joy's Law Parameters

Tm (K)	δ (×10−5)	$\langle \alpha \rangle$	$\langle \alpha \rangle /\langle \lambda \rangle$	a	γ0	T
0	−0.098	6.69	0.23	0.25	15.2	1.39
−5	−0.636	5.34	0.21	0.23	13.7	1.22
−10	−1.16	4.29	0.17	0.19	11.2	1.03
−20	−2.24	3.63	0.14	0.15	9.0	0.86
−50	−54.9	2.91	0.11	0.13	7.7	0.72

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It should be noted that taking into account the radiative heating of flux tubes has recently been shown to have a mild effect on Joy's law (higher slopes), in the presence of turbulent convective flow fields (Weber & Fan 2015). In future studies, it would be of interest to include radiative diffusion in flux tube simulations, in conjunction with a cycle-dependent thermal perturbation.

The magnitude of the change of sound speed at the base of the convection zone found in the helioseismic analysis of Baldner & Basu (2008) is about δc²/c² = (7.23 ± 2.08) × 10⁻⁵, expressed as the difference in the squared sound speed between the solar minimum and maximum, normalized to the minimum value. Assuming that the reduction in wave speed is solely due to a temperature drop, the corresponding cooling amplitude amounts to about −150 ± 45 K. Following the approach taken in Baldner & Basu (2008) and assuming that the change in the sound speed between cycle minimum and maximum is purely due to the change in the local Alfvén speed, an estimate for the magnetic field strength reads from

$$\begin{eqnarray}&&B\simeq {\left(4\pi \gamma p\displaystyle \frac{\delta {c}^{2}}{{c}^{2}}\right)}^{1/2},\end{eqnarray} \tag{ 12 }$$

which is about 3.6 × 10⁵ G, using the sound speed perturbation from the helioseismic result, and the local gas pressure from the structure model used here. From another perspective, Rempel (2003) found that when a magnetic field of 10⁵ G quenches the convective heat conductivity by a factor of 100, a local cooling of about 40 K would be reached within 6 months, and 50 K within a few years, yielding a profile similar to Figure 1(a). For a quenching factor of 10⁴, he found nearly 200 K within 6 months.

The base location of the convection zone is different in the model used here (0.736 R_⊙) and the helioseismic inversions (0.713 R_⊙; Basu & Antia 1997). However, the instability and eruption properties of flux tubes in the overshoot region are most sensitively determined by the local superadiabaticity. Therefore, when a helioseismically calibrated stratification (e.g., Christensen-Dalsgaard et al. 2011; Zhang et al. 2012) is used, the superadiabaticity values given in Table 1 would lead to similar tilt angles, but the initial depth of flux tubes would be shifted inwards, leading to a similar inverse correlation with the perturbation strength.

In the simulations presented in the previous section, r_p in Equation (1) has been taken at the radiative zone boundary (0.718 R_⊙), where δ changes very steeply (Figure 2). I have made this choice to represent the conditions set up in the model of Rempel (2003). However, the central location of the sound speed reduction observed by Baldner & Basu (2008) is at the base of the convection zone, which is at 0.736 R_⊙ in the stratification I have used here. Unfortunately, the detailed radial profile or even the extent of the sound speed perturbation cannot be drawn from the helioseismic signal, owing to limited resolution provided by the low frequency waves used to probe the region. Taking r_p at 0.736 R_⊙, one finds that T_m on the order of −100 K results in a thin superadiabatic layer, whose thickness is determined by σ₋ in Equation (1), located in the midst of the stably stratified regions. This means either that the stratification where the temperature perturbation occurs would be completely restructured, or that the profile of the sound speed perturbation during solar maximum must have fine structure that could not be detected by Baldner & Basu (2008), possibly involving a region of enhanced sound speed immediately below the reduction region, owing to excess heating from the radiative zone. There are indications of such a layer, though within the upper radiative zone. Owing to such ambiguities, I have not attempted to seek a best fit to the helioseismic and tilt angle measurements in this preliminary study.

By numerical simulations of thin flux tubes within a thermally perturbed stratification, I have shown that temperature variations of only 5–20 K in the overshoot region among different cycles can reproduce the reported anti-correlation between the tilt angles of sunspot groups and the cycle strength (Dasi-Espuig et al. 2010), provided that the emerging flux loops originate from about the same depth in the overshoot region in different cycles. Thermal fluctuations with such amplitudes among different cycle maxima are possible, because their magnitudes are well below the level of cooling indicated by the sound speed reduction from cycle minimum to maximum (Baldner & Basu 2008). Whether this is a relevant mechanism for the saturation of the solar dynamo should be tested, through (i) helioseismic observations of sound speed variations at the base of the convection zone for several cycles, in conjunction with (ii) measurements of the cycle-averaged tilt angle of sunspot groups and (iii) theoretical models of the growth of the toroidal magnetic field and its interrelation with the evolving stratification (e.g., Cossette et al. 2013), which would be essential to explain the physics of the possible anti-correlation between the observed tilt angles and the sound speed perturbation.

I am thankful to R. H. Cameron and M. Schüssler for useful discussions which motivated this study. I acknowledge V. R. Holzwarth and Th. Granzer for help with issues related to the stratification model, M. Rempel for valuable comments, C. Sağıroğlu for support with simulations, and the anonymous referee for suggestions that helped to improve the manuscript. This work has been supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK), under project grant 113F070.