Least-Squares Fitting Methods for Estimating the Winding Rate in Twisted Magnetic-Flux Tubes

Abstract

We investigate least-squares fitting methods for estimating the winding rate of field lines about the axis of twisted magnetic-flux tubes. These methods estimate the winding rate by finding the values for a set of parameters that correspond to the minimum of the discrepancy between vector magnetic-field measurements and predictions from a twisted flux-tube model. For the flux-tube model used in the fitting, we assume that the magnetic field is static, axisymmetric, and does not vary in the vertical direction. Using error-free, synthetic vector magnetic-field data constructed with models for twisted magnetic-flux tubes, we test the efficacy of fitting methods at recovering the true winding rate. Furthermore, we demonstrate how assumptions built into the flux-tube models used for the fitting influence the accuracy of the winding-rate estimates. We identify the radial variation of the winding rate within the flux tube as one assumption that can have a significant impact on the winding-rate estimates. We show that the errors caused by making a fixed, incorrect assumption about the radial variation of the winding rate can be largely avoided by fitting directly for the radial variation of the winding rate. Other assumptions that we investigate include the lack of variation of the field in the azimuthal and vertical directions in the magnetic-flux tube model used for the fitting, and the inclination, curvature, and location of the flux-tube axis. When the observed magnetic field deviates substantially from the flux-tube model used for the fitting, we find that the winding-rate estimates can be unreliable. We conclude that the magnetic-flux tube models used in this investigation are probably too simple to yield reliable estimates for the winding rate of the field lines in solar magnetic structures in general, unless additional information is available to justify the choice of flux-tube model used for the fitting.

Appendices

Appendix A: Testing the Fitting Methods with Twisted Magnetic Fields that Vary in the Vertical Direction

To show how the fitting methods perform when the observed flux tube has a magnetic field that varies in the z-direction, we consider an axisymmetric, linear force-free, magnetic field (e.g. Schatzman, 1965; Browning and Priest, 1983; Petrie, 2008), governed by the magnetic-flux function

$$ \Psi(r, z) = \Psi_0 r J_1 ( k r ) \exp( - m z ) / \bigl( R J_1 ( k R )\bigr) , $$

(13)

where J is the Bessel function of the first kind, m is a parameter, k ²=m ²+α ², and α is the (constant) force-free parameter. We assume that m is real and positive so the magnetic-flux function described by Equation (13) decays exponentially with increasing height z. The magnetic-field lines lie in the surfaces of constant Ψ. We assume that the external boundary of the flux tube is made up of the field lines on the flux surface Ψ=Ψ₀, and R is the radius of the flux tube at z=0 [Ψ(R,0)=Ψ₀]. The magnetic-field components are

(14)

(15)

(16)

This magnetic field is useful for constructing synthetic data to test the fitting methods because it has some features that are consistent with the fitting model (i.e. the magnetic field is axisymmetric, and the flux-tube axis is straight and vertical). This allows us to demonstrate the influence of the various features of this magnetic field that are not included in the fitting model, which are: the magnetic field varies in both the vertical and radial directions, the radial component of the magnetic field is non-zero, and the radius of the flux tube varies in the vertical direction (i.e. the radial location of the flux surface with Ψ=Ψ₀ varies in the z-direction).

For α≠0 this magnetic field is twisted (i.e. B _θ ≠0, see Equation (15)). At a given position, the winding rate of a field line about the flux-tube axis, per unit length along the axis, is

$$ q_{\mathrm{LFF}} = \frac{B_{\theta}}{ r B_z } = \frac{\alpha J_1 ( k r )}{ k r J_0 ( k r ) } , $$

(17)

which varies in the radial direction but not in the vertical direction. At the flux-tube axis (r=0), the winding rate is q _0,obs=lim _r→0 q _LFF=α/2.

As the winding rate varies only in the radial direction, we can test the various fitting methods with error-free synthetic data generated with this magnetic field using the same approach used in Sections 4 and 5. To this end, in Equation (1) we set w _obs,i =r _i q _LFF(r _i ). For synthetic data generated with this magnetic field it can be shown that the best-fit on-axis and average winding rates are not directly proportional to the respective true values. Therefore, we test the performance of the fitting methods over a range of winding rates, choosing a typical range of values as follows. For a flux loop of length l, the critical twist typically quoted for the onset of the kink instability is |ql|≈2π radians (e.g. Raadu, 1972; Hood and Priest, 1979; Einaudi and van Hoven, 1983; Mikic, Schnack, and van Hoven, 1990; Velli, Hood, and Einaudi, 1990; Lionello et al., 1998; van der Linden and Hood, 1998, 1999; Baty et al., 1998; Baty, 2001; Fan and Gibson, 2003, 2004; Török, Kliem, and Titov, 2004). For the magnetic-flux tube described by Equation (13), we consider a vertical section of length δl=4 Mm, taken to represent part of a larger magnetic-flux system. We construct 20 synthetic data sets with 20 equally spaced values for q _0,obs that yield partial twist values at the axis in the range −π/2≤q _0,obs δl≤π/2; the case where the winding rate at the axis is zero is not considered. We set m=0.08 Mm⁻¹ so that the magnetic-flux function decreases by approximately 27 % from z=0 to z=δl=4 Mm (at a fixed r). For this range of parameter values it can be confirmed that B _z is single signed in the volume bounded by the flux surface Ψ=Ψ₀ and 0≤z≤δl; we have restricted our attention to this volume to avoid locations where the winding rate for this field is infinite (corresponding to the zeroes of the Bessel function J ₀, i.e. B _z =0). Synthetic measurements are generated at 20 equally spaced observation points in the interval 0≤r _i ≤R, all of which are used for the fitting, the average winding rate is calculated for the interval [0,R], and we set R=2 Mm. For this choice of R the spacing between observation points (in the radial direction) is approximately 100 km, consistent with the spatial resolution provided by data from instruments such as the Solar Optical Telescope onboard Hinode (e.g. Kosugi et al., 2007; Tsuneta et al., 2008).

The results of this exercise (see Figure 5) are qualitatively similar to those discussed in Section 5. For example, we find that both the on-axis and average winding rates retrieved by the best-fit polynomial basis-function approximations are generally more accurate (and the χ ² values are smaller) than those retrieved with fitting models with a fixed radial variation for the winding rate; although for the case shown in Figure 5 the magnitude of the discrepancy between the inferred and true winding rates is generally smaller than in Tables 1 and 2. We also find that the various fitting models which make fixed assumptions about the radial variation of the winding rate can produce different estimates for the winding rates given the same synthetic data, as in Table 2. Several additional points are worth noting: i) For the best-fit polynomial basis-function approximations the magnitude of the discrepancy tends to increase with increasing magnitude of the winding rate at a fixed value for m; the same is also true for the fitting model that assumes the winding rate is constant. This can be understood by referring to Figure 6(a) which shows how the winding-rate profiles (for a fixed value for m) vary with q _0,obs. For small values of q _0,obs the winding-rate profile is almost constant over the flux-tube interior, but as q _0,obs increases the change in the winding rate from the axis to the external boundary increases. This is because higher order terms in Taylor-series expansion for q _LFF(r) about r=0 become more important as q _0,obs (or α) increases. Consequently, low-order polynomial basis-function approximations (and the fitting model with a uniform winding rate) are less accurate at large values of q _0,obs. ii) In other experiments with different values for m but the same range of values for q _0,obs we find that the magnitude of the discrepancy between the inferred and true winding rates tends to increase slightly as m increases.

Figure 5

(a) The ratio of the inferred and true winding rates at the flux-tube axis [$q_{0}^{\ast}/ q_{0,\mathrm{obs}}$] as a function of the true winding rate at the axis [q _0,obs] for different fitting models. In these tests we use the linear force-free magnetic field (see Equation (17)) to generate 20 synthetic data sets with a range of winding rates q _0,obs with fixed m (see text). Each point represents the result from a single fitting experiment; the curves joining the points are only included as a guide. The purple curve is the best-fit case for a fitting model with q _m (r)=q _c , the blue curve is for q _m (r)=q _c (1−(r/R)²)², the green curve is for q _m (r)=q _c (1+2(r/R)²−3(r/R)⁴), the gray curve is for q _m (r)=Q ₁+Q ₂(r/R)², and the orange curve is for q _m (r)=Q ₁+Q ₂(r/R)²+Q ₃(r/R)⁴. (b) Same as (a) except for the corresponding ratio of the inferred and true average winding rates [$q_{\mathrm{av}}^{\ast}/ q_{\mathrm{av},\mathrm{obs}}$] as a function of the true average winding rate [q _av,obs], with the average taken over the range used for the fitting. (c) The corresponding value of $\chi^{2} / (R^{2} q_{0,\mathrm{obs}}^{2})$ (see Equation (1)) for the best-fit model as a function of q _0,obs.

Figure 6

(a) Winding rate [q _LFF] as a function of r (see Equation (17)) for a flux tube with m=0.08 Mm⁻¹. The different curves correspond to four different values for the winding rate at the flux-tube axis: q _0,obs=α/2={0.05,0.075,0.1,0.125}π radians Mm⁻¹. (b) Winding rate [q _LFF] as a function of Ψ/Ψ₀ for a flux tube with m=0.08 Mm⁻¹ and α=0.2π radians Mm⁻¹. The different curves correspond to five different heights: z=0,1,2,3,4 Mm, with greater heights corresponding to curves with larger values for q _LFF.

If the observed magnetic field varies in the z-direction, the results in Figure 5 indicate that reasonable estimates for the winding rates can be obtained by fitting methods that infer the radial variation of the winding rate, for the plane where the magnetic field is measured, provided that the observed field is axisymmetric and has a flux-tube axis is straight and vertical. However, because the magnetic field varies in the z-direction the following considerations must be emphasised regarding the validity of the winding-rate estimates away from the plane where the magnetic field is measured.

For an axisymmetric magnetic-flux tube, with an axis that is straight and vertical, and with a magnetic field that may vary in the z-direction, the winding rate at the flux-tube axis does not vary in the z-direction (see, e.g., Ferriz-Mas and Schüssler, 1989, their Equation (4.15)). Thus, if an accurate estimate for the winding rate at the flux-tube axis can be retrieved at one height (by any method), it can be used to estimate the winding rate at the axis at other heights, provided that the observed flux tube is indeed axisymmetric with an axis that is straight and vertical.

Away from the flux-tube axis, an axisymmetric magnetic-flux tube can generally have a winding rate that varies in the z-direction (see, e.g., Browning and Priest, 1983, their Equation (5.7)). Therefore, estimates for the winding rate away from the flux-tube axis, and thus the average winding rate, retrieved with fitting methods are generally valid only for the plane where the magnetic field is measured. The magnetic field described above is a special case where the winding rate has no z-dependence (see Equation (17)) and, therefore, the radial average of the winding rate computed over a fixed averaging interval is independent of z, but this is not expected in general for fields that vary in the vertical direction.

From a different perspective, if the magnetic field varies in the vertical direction, the winding rate for a particular field line over a range of heights may be more important than the winding rate at a given radial location. For an axisymmetric magnetic-flux tube with a field that varies in the z-direction, the radial locations of the surfaces of constant Ψ generally vary in the z-direction (away from the axis), see, e.g., Equation (13). Therefore, the winding rate for a particular field line at one height does not generally correspond to the winding rate of the same field line at some other height (away from the axis), see Figure 6(b). Again, from this perspective if the observed magnetic field varies in the vertical direction, the winding-rate estimates retrieved by fitting methods away from the flux-tube axis are generally expected to be valid only for the plane where the magnetic field is measured.

As mentioned above, the radial component of the magnetic field (see Equation (14)) is non-zero (for m≠0). This suggests that the fitting model is not strictly appropriate for synthetic data constructed with this magnetic field. However, the radial component is single-valued for a given value of r at fixed height, which would indicate to the observer that the magnetic field is axisymmetric (as assumed by the fitting model). The non-zero radial component of the magnetic field does not directly affect the winding-rate estimates. This is because the winding rate at a given position is defined as the angle that a field line rotates about the flux-tube axis per unit length along the axis and, thus, we are fitting for the ratio w=B _θ /B _z , which does not involve B _r . If a non-zero radial component of the magnetic field is detected in the measurements this indicates that the magnetic field may be varying with height and, therefore, the caveats discussed above apply (provided that the observed flux tube is axisymmetric with an axis that is straight and vertical).

Appendix B: Testing the Fitting Methods with Twisted Magnetic Fields that Vary in the Azimuthal Direction

To show how the fitting methods perform when the observed flux tube has a magnetic field, and corresponding winding rate, that varies in the azimuthal direction, we consider a magnetic field with

(18)

(19)

where q _c,obs is a winding rate, B ₀ and B _a are magnetic-field strengths, R is the radius of the flux tube, and n _a is an integer. At r=0 the magnetic field and corresponding current density are both parallel to the z-direction, and the Lorentz force is zero. Away from r=0, the azimuthal component of this magnetic field [B _θ ] varies only in the radial direction, whereas the vertical component [B _z ] can vary in both the radial and azimuthal directions (for B _a ≠0 and n _a ≠0). This magnetic field is useful for testing the fitting methods because it has some features that are consistent with the fitting model (e.g. the flux-tube axis is straight and vertical, and the magnetic field does not vary in the vertical direction), which allows us to determine the influence of the variation of the magnetic field in both the azimuthal and radial directions on the winding-rate estimates retrieved with fitting methods.

For B _a =0 or n _a =0, the magnetic field described by Equations (18) and (19) is axisymmetric and the winding rate is q(r)=q _c,obsexp(−(r/R)²), which has the same radial dependence as the synthetic data used in Table 2. On the other hand, for B _a ≠0 and n _a ≠0, the winding rate of a magnetic-field line at the position (r,θ) about the flux-tube axis, per unit length along the axis,

(20)

varies in both the r- and θ-directions. In either case, the winding rate at the flux-tube axis (r=0) is q _0,obs=q _c,obs.

To generate synthetic data, we sample the magnetic field described by Equations (18) and (19) at a set of discrete locations on an x–y-plane, with a grid spacing of 100 km in both the x- and y-directions. We set B ₀=3.5 kG, R=20 Mm, and n _a =25; these parameter values are chosen to be roughly consistent with a typical sunspot where azimuthal variations in the magnetic field are expected. We use all points with r _i ≤R for the fitting, we calculate the radial average of the winding rate for the interval [0,R], and we choose the various parameter values for the synthetic data so that B _z >0 within the flux tube. This magnetic field may not closely resemble the types of field expected in sunspots, but we emphasise that the point of this exercise is to quantify how azimuthal variations in this field affect the estimates for the on-axis and average winding rates retrieved by fitting methods that use axisymmetric fitting models.

Because the magnetic field and the winding rate vary in the azimuthal direction, the ratio w _obs,i derived from the synthetic measurements is not single-valued for a fixed value of r (see Figures 7 and 8). The parameter B _a is related to the amplitude of the variation of the winding rate in the azimuthal direction (see Equation (20)), and we find that the scatter in the values of w _obs,i for a fixed value of r tends to increase as B _a increases. These features indicate that the fitting model is not strictly appropriate for synthetic data constructed with this magnetic field. It may be possible to overcome this issue by extending the basis-function approximation approach of Section 5 to fit for both the azimuthal and radial dependence of the winding rate, but this is beyond the scope of the present investigation. Instead, we apply the fitting methods discussed so far without modification (see, e.g., Figure 7). Because fitting methods that use axisymmetric fitting models cannot retrieve the azimuthal dependence of the winding rate, to test the efficacy of the fitting methods we use the azimuthally averaged, radially averaged winding rate of the observed magnetic field, along with the winding rate at the flux-tube axis.

Figure 7

The ratio, w=B _θ /B _z , as a function of r. The black points correspond to the ratio w _obs,i derived from the synthetic measurements of a non-axisymmetric magnetic field (see Equations (18) and (19)) with q _c,obs=0.1 radian Mm⁻¹, B _a /B ₀=1, R=20 Mm, and n _a =25. The purple curve is the best-fit case for a fitting model with q _m (r)=q _c , the blue curve is for q _m (r)=q _c ((1−(r/R)²)²), the green curve is for q _m (r)=q _c (1+2(r/R)²−3(r/R)⁴), the red curve is for q _m (r)=q _c exp(−(r/R)²), the gray curve is for q _m (r)=Q ₁+Q ₂(r/R)², and the orange curve is for q _m (r)=Q ₁+Q ₂(r/R)²+Q ₃(r/R)⁴, where R is the radius of the fitting model (here R=20 Mm).

Figure 8

(a) Winding rate [q(r,θ)] as a function of θ at fixed r for the non-axisymmetric magnetic field described in the text (see Equation (20)), with B ₀=3.5 kG, R=20 Mm, B _a /B ₀=1, n _a =25, and q _c,obs=0.1 radian Mm⁻¹. The red, green, blue curves correspond to r=5, 10, 15 Mm, respectively. (b) Winding rate [q(r,θ)] as a function of r at fixed θ for the same flux tube as (a). The red, blue, green curves correspond to θ=0, π/50, 3π/50 radians, respectively.

For synthetic data generated with this magnetic field, it can be shown that the best-fit on-axis and radially averaged winding rates are directly proportional to the true on-axis and true azimuthally averaged, radially averaged winding rates, respectively. In Table 3 we show the ratio of the inferred and true winding rates at the flux-tube axis [$q_{0}^{\ast}/ q_{0,\mathrm {obs}}$] along with the ratio of the inferred radially averaged winding rate to the true azimuthally averaged, radially averaged winding rate [$q_{\mathrm{av}}^{\ast}/ q_{\mathrm{av},\mathrm{obs}}$] for three different values of B _a /B ₀. We include the axisymmetric case (with B _a =0) for reference and because the magnetic field for these tests is sampled differently to the case shown in Table 2. We find that the results of these tests are not strongly sensitive to the value of n _a (although this depends on how the magnetic field is sampled). On the other hand, it is clear from Table 3 that the results are sensitive to the value of B _a /B ₀.

Table 3 Results for tests of several fitting models applied to synthetic data constructed with the non-axisymmetric magnetic field described by Equations (18) and (19), with B ₀=3.5 kG, R=20 Mm, and n _a =25, for three different values of B _a /B ₀.

For the case with B _a /B ₀=1, despite the presence of moderate azimuthal variations in the magnetic field and winding rate, we find the same general trend as in the cases that use axisymmetric synthetic data [B _a =0]. For example, we find that both the on-axis and average winding rates retrieved by the best-fit polynomial basis-function approximations are generally more accurate (and the χ ² values are smaller) than those retrieved with fitting models with a fixed radial variation for the winding rate. The exception is for the fitting model with q _m (r)=q _c exp(−(r/R)²), which retrieves accurate winding-rate estimates since it matches the winding-rate profile for the axisymmetric version of these data. We also find that the various fitting methods that make fixed assumptions about the radial variation of the winding rate can produce different estimates for the winding rates given the same synthetic data.

For a given fitting model, the discrepancy between the inferred and true winding rates does not necessarily increase as the value of B _a /B ₀ increases (as one may expect), although the magnitude of the χ ² values does generally increase (see Table 3). For the best-fit polynomial basis-function approximations applied to synthetic data with B _a /B ₀=2, we find that the discrepancy between the inferred and true winding rates increases as n _b increases, contrary to the trend found at lower values of B _a /B ₀. Nevertheless, we generally find for the best-fit polynomial basis-function approximations that the discrepancy between the inferred and true winding rates is less than 10 % for B _a /B ₀=2 (see Table 3). Those discrepancies are quite small considering that the winding rate varies approximately by a factor of three from its minimum to maximum value for some values of r for B _a /B ₀=2; for the case with B _a /B ₀=1 the winding rate can vary by a factor of roughly two (see Figure 8(a)). We find similar results in other experiments with different functional forms for both the azimuthal and radial dependence of the magnetic field as those used in Equations (18) and (19) (results not shown).

Appendix C: Testing the Role of the Location of the Fitting-Model Axis

Some methods that are commonly used to estimate the location of the axis of a magnetic-flux tube are: the peak of |B|, the peak of |B _z |, the flux-weighted centroid, and the peak of the force-free parameter α _z (e.g. Leamon et al., 2003; Leka, Fan, and Barnes, 2005; Nandy et al., 2008). As discussed by Leka, Fan, and Barnes (2005) some of these methods do not consistently recover the correct location of the flux-tube axis; we have confirmed this finding using synthetic data constructed with a model for a toroidal magnetic-flux loop (e.g. Fan and Gibson, 2003, 2004, and see Section 6.1), but do not show the results for the sake of brevity. To demonstrate how the winding-rate estimates are affected by an error in the location of the fitting-model axis (in the absence of other sources of error), we generate synthetic data using a magnetic field with

$$ B_{\theta} ( r ) = q_{{c,\mathrm {obs}}}r B_0 \exp \bigl[ - 2 (r / R)^2 \bigr] \quad\mbox{and} \quad B_z ( r ) = B_0 \exp \bigl[ - (r / R)^2 \bigr] , $$

(21)

where q _c,obs is a winding rate, B ₀ is a magnetic-field strength, and R is the radius of the flux tube. To generate synthetic data we set q _c,obs=0.1 radians Mm⁻¹, B ₀=3.5 kG, and R=20 Mm, and sample the magnetic field at a set of discrete locations on an x–y-plane, with a grid spacing of 100 km in both the x- and y-directions. We introduce an error into the location of the flux-tube axis by computing synthetic data with the axis of the flux tube located at x=4 Mm and y=0. We apply the various fitting methods without modification assuming that the fitting-model axis is located at x=0 and y=0. To estimate the winding rates we use only observation points that lie within 20 Mm of the fitting-model axis and have B _z >0. We calculate the average winding rate over the interval used for the fitting (i.e. 0≤r≤max(r _i )).

When there is an error in the location of the fitting-model axis we find that w is not single-valued for a fixed value of r (see Figure 9), which indicates that the fitting model is not strictly appropriate for these synthetic data. Nevertheless, it can be shown that the best-fit on-axis and average winding rates are directly proportional to the respective true values for this test. In Table 4 we provide the ratios $q_{0}^{\ast}/ q_{0,\mathrm{obs}}$ and $q_{\mathrm{av}}^{\ast}/ q_{\mathrm {av},\mathrm{obs}}$, along with the corresponding value for $\chi^{2} / (R^{2} q_{{c,\mathrm {obs}}}^{2})$ for the various best-fit models shown in Figure 9. The magnetic field described by Equations (21) is the same as for the axisymmetric case (with B _a =0) used in Appendix B; hence, for reference, the results in Table 3 for the case with B _a =0 are those that would be retrieved if the fitting-model axis were correctly located for this synthetic data. Evidently, the accuracy of the winding-rate estimates retrieved by the fitting methods is affected by an error in the location of the fitting-model axis. However, for the better-performing cases (such as the polynomial basis-function approximations with n _b =3 and the case with the correct winding-rate profile) the magnitude of the relative discrepancy caused by the error in the location of the fitting-model axis is less than 10 %. We find that the magnitude of the discrepancy in the inferred winding rates generally gets progressively larger as the error introduced into the axis location is increased. In experiments, using synthetic data generated with winding-rate profiles and magnetic fields different from Equation (21), we find that an error in the location of the fitting-model axis generally produces an error in the winding-rate estimates retrieved by the fitting methods of comparable magnitude to the case discussed here.

Figure 9

The ratio w=B _θ /B _z as a function of r (the radial distance from the fitting-model axis) for a test where an error is introduced into the location of the fitting-model axis, see Appendix C for details. The black points correspond to the ratio w _obs,i derived from the synthetic measurements. The purple curve is the best-fit case for a fitting model with q _m (r)=q _c , the blue curve is for q _m (r)=q _c ((1−(r/R)²)²), the green curve is for q _m (r)=q _c (1+2(r/R)²−3(r/R)⁴), the red curve is for q _m (r)=q _c exp(−(r/R)²), the gray curve is for q _m (r)=Q ₁+Q ₂(r/R)², and the orange curve is for q _m (r)=Q ₁+Q ₂(r/R)²+Q ₃(r/R)⁴, where R is the radius of the fitting model (here R=20 Mm).

Table 4 Summary of results for a test where an error is introduced into the location of the fitting-model axis, see Appendix C for details.