ON ESTIMATING FORCE-FREENESS BASED ON OBSERVED MAGNETOGRAMS

Generally, it is assumed that the magnetic field is force-free in the corona (Wiegelmann et al. 2014), because model results suggest that the plasma β (the ratio of the gas pressure to the magnetic pressure) is much less than unity ($\beta \ll 1$) in the corona (Gary 2001). In this case, the magnetic field satisfies the following equations (see the reviews by Wiegelmann & Sakurai 2012):

$$\begin{eqnarray}&&\bigtriangledown \times {\boldsymbol{B}}=\alpha \,{\boldsymbol{B}},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\bigtriangledown \cdot {\boldsymbol{B}}=0,\end{eqnarray} \tag{ 2 }$$

where ${\boldsymbol{B}}$ is the vector magnetic field and α is the force-free coefficient.

Among all possible force-free fields, if α = 0, then there is no electric current in the space and the field is a potential field; if α is a constant, then the field is called a linear force-free field. In the real Sun, α usually varies spatially and the field is a general nonlinear force-free field (see Wiegelmann & Sakurai 2012 for details).

According to Low (1985), in an isolated magnetic structure, a necessary condition for a force-free field to exist above a measured layer is

$$\begin{eqnarray}&&{F}_{x}\ll {F}_{p},\,{F}_{y}\ll {F}_{p},\,{F}_{z}\ll {F}_{p},\end{eqnarray} \tag{ 3 }$$

where F_x, F_y, and F_z are the components of the net Lorentz force and F_p is the characteristic magnitude of the total Lorentz force that can be brought to bear on the atmosphere if the magnetic field is not force-free. Assuming that the magnetic field above the plane z = 0 (photosphere) vanishes as z goes to infinity, the volume's boundaries other than the lower one do not contribute in the half-space $z\gt 0$, so the Maxwell stress can be written as the following surface integrals:

$$\begin{eqnarray}{F}_{x} & = & -\,\displaystyle \frac{1}{4\pi }\displaystyle \int {B}_{x}\,{B}_{z}\,{dx}\,{dy},\\ {F}_{y} & = & -\,\displaystyle \frac{1}{4\pi }\displaystyle \int {B}_{y}\,{B}_{z}\,{dx}\,{dy},\\ {F}_{z} & = & -\,\displaystyle \frac{1}{8\pi }\displaystyle \int ({B}_{z}^{2}\,-{B}_{x}^{2}\,-{B}_{y}^{2}\,)\,{dx}\,{dy},\\ {F}_{p} & = & \displaystyle \frac{1}{8\pi }\displaystyle \int ({B}_{z}^{2}\,+{B}_{x}^{2}\,+{B}_{y}^{2}\,)\,{dx}\,{dy},\end{eqnarray} \tag{ 4 }$$

where B_x, B_y, and B_z are the three components of the vector magnetic field, B_z is the vertical magnetic field, and B_x and B_y are the two components of the horizontal magnetic field B_t.

The above approach to estimating the force-freeness using measured vector magnetograms as the boundary condition at the plane z = 0 has become a common practice in the community. Whereas F_x, F_y, and F_z must all be zero for an entirely force-free magnetic field, a magnetic field may be considered to be nearly force-free if the magnitudes of F_x/F_p, F_y/F_p, and F_z/F_p are sufficiently small (Low 1985). Metcalf et al. (1995) suggested values of $| {F}_{x}/{F}_{p}|$, $| {F}_{y}/{F}_{p}|$, and $| {F}_{z}/{F}_{p}|$ less than 0.1 as a criterion for a measured magnetic field to be considered to be force-free. Following Metcalf et al. (1995) and others (e.g., Moon et al. 2002; Tiwari 2012; Liu et al. 2013), we will adopt this criterion in the following.

We will use three magnetic fields as representative fields to discuss the possible influences of measurement limitations on the estimate of force-freeness.

The first one is taken from analytical solutions provided in Low & Lou (1990). Under a few assumptions, such as the fields being axisymmetric and the solutions separable in r and θ directions, Low & Lou (1990) reformulated Equations (1) and (2) into a second-order partial differential equation in spherical coordinates with constant n and m, where the integer n describes the power-law decreasing index in the r-direction and m defines the number of node points in the θ-direction. Finding eigenvalue solutions of this second-order partial differential equation generates a series of analytical nonlinear force-free fields. These fields can be transformed into Cartesian coordinates by arbitrarily positioning a plane, characterized by the parameters l and Φ, to produce 3D force-free magnetic fields that represent the magnetic fields over active regions with a striking geometric realism. Here l is the distance between the surface plane and the location of a point source, and Φ is the angle between the axis of symmetry of the magnetic field and the z-axis in the Cartesian coordinate system. These analytical force-free solutions are quite useful in testing the properties of force-free fields. For example, they have been extensively used to test the reliability and accuracy of extrapolation algorithms for force-free fields (e.g., Schrijver et al. 2006).

We use one of these analytical force-free solutions as a representative field. The field is generated by $n=1,m=1,l=0.3$, and ${\rm{\Phi }}=\pi /2$ and is shown in Figure 1(a). Here the contours show the vertical field strength with solid contours indicating a field ${B}_{z}\gt 0$ and dashed contours showing ${B}_{z}\lt 0$. The contour interval is 180 G. Green squares outline the ranges of different FOVs to simulate different sizes of FOVs in real observations. The smaller the red number is, the larger the FOV is.

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To make the field more comparable to real magnetograms, we have rescaled the field to make the maximum of $| {B}_{z}|$ be 2000 G. The value of 2000 G is not specifically chosen from among the possible values from 1000 to 5000 G of active regions. The rescaling is done merely to make our studied fields comparable. The particular value chosen would not influence our results qualitatively because the estimate of force-freeness is based on the ratio, not the absolute value, of F_x, F_y, or F_z to F_p. Some properties of the field are listed in Table 1.

Table 1.  Information on the Three Representative Magnetic Fields

			FOV Number = 0
	Date and Time (UT)	Position	Fx/Fp	Fy/Fp	Fz/Fp	$\max ({B}_{z})$ (G)	$\min ({B}_{z})$ (G)
Low and Lou	...	...	$1.57\times {10}^{-8}$	−0.00036	−0.00468	2000.00	−699.31
AR11072	2010 May 23: 0500	S14W00	0.00149	0.00169	−0.02141	2000.00	−1130.34
AR10960	2007 June 07: 0304	S07W07	0.07035	−0.03191	−0.51746	1506.42	−2000.00

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The second representative field is a modeled solar-like nonlinear force-free field that might exist on the Sun. In other words, we treat this modeled field as a more or less theoretical field but with a geometry more similar to a realistic one on the Sun, if such exists, than the analytical one given by Low & Lou (1990). We use a vector magnetogram of NOAA 11072 obtained by the Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO) (Schou et al. 2012) as the z = 0 boundary and carry out a 3D force-free field extrapolation. We select a time when the active region is near the center of the disk, at a position of S14W00. We use the magnetogram from the hmi.sharp_cea_720s series, which has been inverted by the HMI team using the Milne–Eddington (ME) inversion algorithm of Borrero et al. (2011), solved for the $180^\circ$ ambiguity using the minimum energy method (Metcalf 1994), and processed using a cylindrical equal-area projection. We then further preprocess the magnetogram using methods described in Wiegelmann et al. (2006, 2008) to make it suitable for force-free extrapolation. The extrapolation is done with the help of an optimization code described in Wiegelmann (2004).

We use the layer about 1 Mm above the z = 0 photosphere in the extrapolated field as the second representative field: the modeled solar-like nonlinear force-free field. Note that 1 Mm above the z = 0 photosphere already puts this field into the layer of the chromosphere. However, our purpose here is not to discuss what the true chromospheric magnetic field might be, but to get a modeled solar-like force-free field. So the particular height, 1 Mm or even 2 Mm above, makes no difference to the results of our study here. In addition, to make all our representative fields comparable, the maximum value of $| {B}_{z}|$ of this field has also been normalized to be 2000 G. Some properties of this field are also shown in Table 1.

To quantify how good our force-free extrapolation is, we have calculated a few numbers like those in DeRosa et al. (2015). They are $\langle \mathrm{CWsin}\theta \rangle =0.37$ and $\langle | {f}_{i}| \rangle =7.9\times {10}^{-4}$. These numbers are of the same magnitudes as those in DeRosa et al. (2015). In particular, it shows the success of the force-free extrapolation when we get very small numbers of ${F}_{x}/{F}_{p}=0.00149$, ${F}_{y}/{F}_{p}=0.00169$, and ${F}_{z}/{F}_{p}=-0.02141$ for the modeled solar-like field. The original observed magnetogram has these numbers as ${F}_{x}/{F}_{p}=-0.00207$, ${F}_{y}/{F}_{p}=0.08495$, and ${F}_{z}/{F}_{p}=-0.03457$. We see that F_y/F_p has reduced to be 2% of its original value.

Figures 1(b) shows the B_z map of this modeled solar-like force-free field. As before, colored rectangles in Figure 1(b), labeled with sequential numbers, outline ten different FOVs, to mimic limited FOVs in real observations. Again, the smaller the labeled number is, the larger the FOV is.

The third representative field is a non-force-free field. According to Tiwari (2012), the fields of active region NOAA 10960 are non-force-free in that ${F}_{x}/{F}_{p}=0.137$, ${F}_{y}/{F}_{p}=0.093$, and ${F}_{z}/{F}_{p}=-0.482$. We used a single vector magnetogram of this active region, obtained by the Solar Optical Telescope/Spectro-Polarimeter (SP) on Hinode (Kosugi et al. 2007). The SP magnetograms are inverted by the SP team from Stokes profiles using the MERLIN ME inversion algorithm (Skumanich & Lites 1987; Lites et al. 2007) and the inherent $180^\circ$ azimuth ambiguity is resolved in the same way as for HMI data. Again, we choose a time, 03:04 UT on 2007 June 7, when the active region is near the center of the disk. Also, the maximum of $| {B}_{z}|$ is rescaled to be 2000 G, to make the three representative fields comparable to each other. Information on this field can be found in Table 1 and a B_z map of it is shown in Figure 1(c), with the colored rectangles again representing different FOVs.

As already stated in Canfield et al. (1991), a limited FOV may not be appropriate for Maxwell stress integration, for the use of which the flux balance is a prerequisite. To minimize this effect, Moon et al. (2002) considered only magnetograms whose magnetic imbalance (MI) is within 10%. Metcalf et al. (1995) as well as Tiwari (2012) used this approach too. However, the effect of limited FOV on the force-free measure has not been studied systematically. Aiming to discuss how the FOV could influence the judgment of force-freeness, we shrink the FOV from the original flux-balanced one to get a series of magnetograms with different FOVs, as indicated by the colored rectangles in Figure 1.

A discussion of the effect of a limited FOV is actually related to the effect of flux imbalance (MI). We follow Moon et al. (2002) to calculate the MI associated with different FOVs as the index to represent them. MI is defined as

$$\begin{eqnarray}&&\mathrm{MI}=\displaystyle \frac{| {F}^{+}-{F}^{-}| }{{F}^{+}+{F}^{-}}\times 100,\end{eqnarray} \tag{ 5 }$$

where ${F}^{+}$ and ${F}^{-}$ are upward (${B}_{z}\gt 0$) and downward (${B}_{z}\lt 0$) magnetic fluxes respectively.

Observed magnetograms also suffer from issues of instrument sensitivity and measurement noise. The random errors (noises) in the measurement of horizontal fields are particularly large, typically ten times those for vertical fields. For example, random errors in HMI are about 5 G in the line-of-sight component whereas the uncertainty in the transverse field is between 70 and 200 G (Wiegelmann et al. 2012; Hoeksema et al. 2014).

To deal with this situation, in previous studies usually only data points whose field strengths are larger than a certain value have been used. For example, Metcalf et al. (1995) only use data points with magnetic field strength greater than 150 G (1σ noise level in a transverse magnetic field). Moon et al. (2002) used a field strength larger than 100 G as a criterion. Liu et al. (2013) used data points with $| {B}_{z}| \gt 20\,{\rm{G}},| {B}_{x}| \gt 150\,{\rm{G}}$, and $| {B}_{y}| \gt 150\,{\rm{G}}$. However, this "cutting" method is more equivalent to setting a low level of sensitivity; the large measurement error (noise) is still buried in the remaining data points.

We mimic the sensitivity and noise separately to divide the effects of these two issues. First, to simulate the different levels of sensitivity, set ${B}_{t}^{0}$ and ${B}_{z}^{0}$ as the horizontal and vertical field sensitivities respectively. If $\sqrt{{B}_{x}^{2}+{B}_{y}^{2}}\leqslant {B}_{t}^{0}$ or $| {B}_{z}| \leqslant {B}_{z}^{0}$, then omit these pixels. Taking knowledge from previous studies, we have assumed ${{B}_{x}^{0}=B}_{y}^{0}=10{B}_{z}^{0}$ and hence ${B}_{t}^{0}=10\sqrt{2}{B}_{z}^{0}$. We have studied the model for ${B}_{z}^{0}$ increasing from 0 G, with a constant step size of 1 G. Only the magnetogram with the largest FOV is studied. To quantify this undertaking, a number NP, defined as the percentage of the data points that have been omitted, is calculated for each ${B}_{z}^{0}$ level.

Similarly, given σ_x, σ_y, and σ_z as the white noise in B_x, B_y, and B_z respectively, to mimic the different levels of noise in the observed magnetograms, we have replaced the value of each pixel in the magnetogram using the following method: B_x is replaced by ${B}_{x}+{\sigma }_{x}$, B_y by ${B}_{y}+{\sigma }_{y}$, and B_z by ${B}_{z}+{\sigma }_{z}$. σ_z is created by multiplying ${\sigma }_{z}^{0}$ by a normally distributed random number, and similarly for σ_y from ${\sigma }_{y}^{0}$ and σ_x from ${\sigma }_{x}^{0}$ . Again, we have assumed ${\sigma }_{x}^{0}$ = ${\sigma }_{y}^{0}$ = 10${\sigma }_{z}^{0}$, with ${\sigma }_{z}^{0}$ increasing from 0 G with a constant step of 1 G.

Figures 2 shows how the different sizes of FOVs could influence the estimation of the force-freeness. Plotted in the left panels are the values of F_x/F_p (blue lines), F_y/F_p (red lines), and F_z/F_p (black lines), obtained from Equation (4), as a function of the FOV number. Meanwhile, the variation of corresponding MI with each FOV, calculated from Equation (5), is shown in the right panels.

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Panel (a) shows the results for the analytical force-free field. It shows that with increasing FOV number (that is, with decreasing size of the FOV) the values of F_x/F_p, F_y/F_p, and F_z/F_p can increase from the theoretical zero of force-free fields (FOV Number 0) to a magnitude larger than 0.1 (FOV Number 9). This suggests that a true force-free field (where ${F}_{x}/{F}_{p}=0$, ${F}_{y}/{F}_{p}=0$, and ${F}_{z}/{F}_{p}=0$) may be mistakenly estimated as being non-force-free (where $| {F}_{x}/{F}_{p}| \gt 0.1$, $| {F}_{y}/{F}_{p}| \gt 0.1$, or $| {F}_{z}/{F}_{p}| \gt 0.1$) at FOV Number 9, where MI is larger than 90% (see panel (b)). However, before FOV Number 7, the changes in F_x/F_p, F_y/F_p, and F_z/F_p are all small (with magnitude less than 0.1). This suggests that the wrong judgment may not happen even for an MI value as large as 43% (FOV Number 7) and a criterion setting as MI less than 10% is fairly safe.

As in panels (a) and (b), panels (c) and (d) show the results for the modeled solar-like force-free field based on the magnetogram of NOAA 11072. Before FOV Number 8, the changes in F_x/F_p, F_y/F_p, and F_z/F_p are small and within a magnitude of 0.1, again suggesting that setting the MI within 10% is safe. Also, if the FOV is too small where the MI is too large, a wrong judgment can be made, as is the case for FOV Number 8. It is interesting that under some circumstances, such as for FOV Number 9, the severe flux imbalance (MI ∼ 40%) does not seem to influence the measure, which implies that the MI is not the only factor that controls the estimate of force-freeness.

Panels (e) and (f) are for the observed non-force-free field (NOAA 10960). This field shows a large variation in MI with increasing FOV number. However, a fair judgment can still be made if we confine the MI to within 10%, as before FOV Number 3 for this field. So, again we see 10% of MI as a good criterion.

In summary, for the three representative fields, we see that a limited FOV with a large MI value indeed can have certain effects on the results of an estimation of force-freeness as previous studies have suggested (Canfield et al. 1991; Moon et al. 2002), and that setting a criterion such as using magnetograms whose MI is within 10% as suggested by many previous studies is a safe approach.

Figure 3 presents the results for the influence of instrument sensitivity on measurement of force-freeness. Variations of F_x/F_p, F_y/F_p, and F_z/F_p are shown in the left panels. The right panels show the variations of MI (lines with cross symbols) and of NP (lines with triangle symbols). The lower x-axis is ${B}_{z}^{0}$ in units of gauss and the upper x-axis is ${B}_{x}^{0}$ or ${B}_{y}^{0}$ also in units of gauss.

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Again panels (a) and (b) are for the analytical force-free field, and panels (c) and (d) are for the modeled solar-like force-free field. Here we see that the variations in F_x/F_p, F_y/F_p, and F_z/F_p are all within a magnitude of 0.1 for these two force-free fields. This suggests that the problem of instrument sensitivity does not influence the estimation of force-freeness seriously, at least for these two cases. For the analytical force-free field, even when MI has reached 52% and even when 90% of the data points have been omitted (case with ${B}_{z}^{0}=25\,{\rm{G}}$ in Figure 3(b)), the changes in F_x/F_p, F_y/F_p, and F_z/F_p are all within 0.05 in magnitude. The same is true for the modeled solar-like force-free field: even when MI has increased to 30% and more than 90% of data points have been omitted (${B}_{z}^{0}=25\,{\rm{G}}$ in Figure 3(d)), the changes in F_x/F_p, F_y/F_p, and F_z/F_p are all within 0.06 in magnitude.

For the observed non-force-free field as shown in panels (e) and (f), however, the problem of instrument sensitivity would not obviously influence the force-free measures only if ${B}_{z}^{0}$ is less than 12 G or MI is less than 10%. Therefore, we see that the instrument sensitivity would not influence the estimates of force-freeness too seriously as long as MI is controlled within the safe criterion of 10%.

Figure 4 shows the effect of measurement noise for the three representative fields. Again panels (a) and (b) are for the analytical force-free field, panels (c) and (d) for the modeled solar-like force-free field, and panels (e) and (f) for the observed non-force-free field. Here the left panels are for F_x/F_p and F_y/F_p and the right panels are for F_z/F_p. The lower x-axis is the added noise level of ${\sigma }_{z}^{0}$ and the upper x-axis is the added noise level of ${\sigma }_{x}^{0}$ or ${\sigma }_{y}^{0}$, all in units of gauss. Since the noise we added is white noise, it would not change the total flux and hence MI. A calculation of MI shows that the changes here are all less than 0.1%, so we do not plot it here.

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In Figure 4, it can be seen that the measurement noises have little influence on F_x/F_p and F_y/F_p for the three representative fields. The variations in F_x/F_p and F_y/F_p, with different noise levels, are all within 0.1 in magnitude. However, F_z/F_p increases monotonically with increasing noise level. Panel (b) shows that when ${\sigma }_{z}^{0}$ is larger than 10 G (100 G for ${\sigma }_{x}^{0}$ and ${\sigma }_{y}^{0}$), F_z/F_p has increased to a value larger than 0.1 for the analytical force-free field. For the modeled solar-like force-free field (panel (d)), even when ${\sigma }_{z}^{0}=5\,{\rm{G}}$ (${\sigma }_{x}^{0}$ = ${\sigma }_{y}^{0}$ = 50 G), F_z/F_p has already increased to above 0.1. These suggest that a truly force-free field may be estimated as non-force-free by the calculation of F_z/F_p.

For the non-force-free field (panel (f)), the value of F_z/F_p also increases monotonically with increasing noise level. This results in a situation where, when ${\sigma }_{z}^{0}$ is between 10 and 15 G, the originally non-force-free field may appear to be force-free ($| {F}_{x}/{F}_{p}| \leqslant 0.1$, $| {F}_{y}/{F}_{p}| \leqslant 0.1$,  and $| {F}_{z}/{F}_{p}| \leqslant 0.1$). Note that a noise level between 10 and 15 G in ${\sigma }_{z}^{0}$ (100–150 G in ${\sigma }_{x}^{0}$ or ${\sigma }_{y}^{0}$) is right in the range of current measurement errors (70–200 G for transverse fields of HMI for example). So these plots deliver a serious warning against using F_z/F_p to judge the force-freeness in the presence of measurement noise.

Figure 5 shows a further study of the influence of the noise level. Here we have added both the white noise and the sensitivity cutting. We first added the white noise at a level of σ_z (again with σ_x = σ_y = 10σ_z) and then omitted the data points whose magnitudes are below 1σ or 2σ. This situation is closer to a real observation: being influenced by the instrument sensitivity and measurement noise at the same time.

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Plotted in the left panels of Figure 5 are the variations of F_z/F_p with different noise levels. Again the lower x-axis is the added noise level of ${\sigma }_{z}^{0}$ and the upper x-axis is the added noise level of ${\sigma }_{x}^{0}$ or ${\sigma }_{y}^{0}$, all in units of gauss. The black lines show the same result as those in Figure 4, that is, without considering the influence of sensitivity. The red lines show the results of cutting at the 1σ level, and the blue lines for cutting at the 2σ level. Similarly to Figure 3, the values of MI and NP are plotted in the right panels, with the same color coding for the sensitivity cutting levels.

From the left panels of Figure 5, we see that the addition of the influence of instrument sensitivity to the influence of white noise does not change the results too much from what we have already seen in Figure 4. Still, a true force-free field may be estimated as being non-force-free, and a non-force-free one as force-free, if the noise level is high enough. Note that the changover point, where a wrong judgment may be made, is high only compared to those results without any measurement noise. These changover points of noise level are actually right within the range of current measurement noise levels.

If we take the non-force-free fields as an example, then when ${\sigma }_{x}^{0}$ or ${\sigma }_{y}^{0}$ is larger than 130 G, in the case of 2σ cutting, even though 95.6% data points have been omitted, the noise in the remaining data points can still lead to a wrong estimation.

These results call for serious caution in interpreting the F_z/F_p measures when using observed magnetograms. Even if cutting at a 2σ level, the noises in the remaining 10% of data points can still affect the estimation of force-freeness. In particular, note that most previous studies (Metcalf et al. 1995; Moon et al. 2002; Tiwari 2012; Liu et al. 2013) show that most active regions have their magnitudes of F_z/F_p larger than those of F_x/F_p and F_y/F_p, and the judgment of force-free nature mainly depends on the former.