Global Energetics of Solar Flares. IX. Refined Magnetic Modeling

The conceptually simplest representation of a magnetic potential field that fulfills Maxwell's divergence-free (solenoidal) condition (${\rm{\nabla }}\cdot {\boldsymbol{B}}=0$) is a unipolar magnetic charge j that is buried below the solar surface, which predicts a spherically symmetric magnetic field ${{\boldsymbol{B}}}_{j}({\boldsymbol{x}})$ that points away from the buried unipolar charge, and its field strength falls off with the square of the distance r_j,

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{{\boldsymbol{j}}}({\boldsymbol{x}})={B}_{j}{\left(\displaystyle \frac{{d}_{j}}{{r}_{j}}\right)}^{2}\displaystyle \frac{{{\boldsymbol{r}}}_{j}}{{r}_{j}},\end{eqnarray} \tag{ 1 }$$

where B_j is the magnetic field strength at the solar surface above a buried magnetic charge, (x_j, y_j, z_j) is the subphotospheric position of the buried charge, d_j is the depth of the magnetic charge,

$$\begin{eqnarray}&&{d}_{j}=1-\sqrt{{x}_{j}^{2}+{y}_{j}^{2}+{z}_{j}^{2}},\end{eqnarray} \tag{ 2 }$$

and the vector ${{\boldsymbol{r}}}_{j}$ between an arbitrary location ${\boldsymbol{x}}=(x,y,z)$ in the solar corona (were we desire to calculate the magnetic field) and the location (x_j, y_j, z_j) of the buried charge vector has a distance ${{\boldsymbol{r}}}_{j}$ of

$$\begin{eqnarray}&&{{\boldsymbol{r}}}_{j}=[x-{x}_{j},y-{y}_{j},z-{z}_{j}].\end{eqnarray} \tag{ 3 }$$

Of course, the concept of a magnetic charge is only valid as a far-field approximation, which avoids the complexity of the near-field details for the sake of mathematical convenience. We choose a Cartesian coordinate system $(x,y,z)$ with the origin in the Sun center, and we use units of solar radii, with the direction of z chosen along the line of sight from Earth to Sun center. For a location near disk center (x ≪ 1, y ≪ 1), the magnetic charge depth is ${d}_{j}\approx (1-{z}_{j})$. Thus, the distance r_j from the magnetic charge is

$$\begin{eqnarray}&&{r}_{j}=\sqrt{{(x-{x}_{j})}^{2}+{(y-{y}_{j})}^{2}+{(z-{z}_{j})}^{2}}.\end{eqnarray} \tag{ 4 }$$

The absolute value of the magnetic field B_j(r_j) is simply a function of the radial distance r_j (with B_j and d_j being constants for a given magnetic charge),

$$\begin{eqnarray}&&B({r}_{j})={B}_{j}{\left(\displaystyle \frac{{d}_{j}}{{r}_{j}}\right)}^{2}.\end{eqnarray} \tag{ 5 }$$

In order to obtain the Cartesian coordinates $({B}_{x},{B}_{y},{B}_{z})$ of the magnetic field vector ${{\boldsymbol{B}}}_{j}({\boldsymbol{x}})$, we can rewrite Equation (1) with Equations (2)–(5) as

$$\begin{eqnarray}\begin{array}{rcl}{B}_{x}(x,y,z) & = & {B}_{j}{\left({d}_{j}/{r}_{j}\right)}^{2}(x-{x}_{j})/{r}_{j}\\ {B}_{y}(x,y,z) & = & {B}_{j}{\left({d}_{j}/{r}_{j}\right)}^{2}(y-{y}_{j})/{r}_{j}\\ {B}_{z}(x,y,z) & = & {B}_{j}{\left({d}_{j}/{r}_{j}\right)}^{2}(z-{z}_{j})/{r}_{j}.\end{array}\end{eqnarray} \tag{ 6 }$$

We proceed now from a single magnetic charge to an arbitrary number N_m of magnetic charges and represent the general magnetic field with a superposition of N_m buried magnetic charges, so that the potential field can be represented by the superposition of N_m fields ${{\boldsymbol{B}}}_{j}$ from each magnetic charge $j=1,\ldots ,\,{N}_{{\rm{m}}}$,

$$\begin{eqnarray}&&{\boldsymbol{B}}({\boldsymbol{x}})=\displaystyle \sum _{j=1}^{{N}_{{\rm{m}}}}{{\boldsymbol{B}}}_{j}({\boldsymbol{x}})=\displaystyle \sum _{j=1}^{{N}_{{\rm{m}}}}{B}_{j}{\left(\displaystyle \frac{{d}_{j}}{{r}_{j}}\right)}^{2}\displaystyle \frac{{{\boldsymbol{r}}}_{{\boldsymbol{j}}}}{{r}_{j}}.\end{eqnarray} \tag{ 7 }$$

The radial vector ${{\boldsymbol{r}}}_{j}$, the magnetic charge location (x_j, y_j, z_j), the intersection of the vertical symmetry axis with the photosphere at (x₀, y₀, z₀), and an arbitrary coronal location (x, y, z) are depicted in Figure 1. The potential field of a point charge is solely defined by the radial vector component B_r (Figure 1), by setting B_φ = 0 and B_θ = 0.

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In order to improve the accuracy of the old VCA-NLFFF code, we searched for more accurate analytical solutions of the equation system for the divergence-free,

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\boldsymbol{B}}=0,\end{eqnarray} \tag{ 8 }$$

and force-free magnetic field,

$$\begin{eqnarray}&&{\rm{\nabla }}\times {\boldsymbol{B}}=\alpha ({\boldsymbol{x}}){\boldsymbol{B}},\end{eqnarray} \tag{ 9 }$$

where the nonlinear force-free parameter $\alpha ({\boldsymbol{x}})$ is spatially varying but remains constant along a given magnetic field line.

We can derive an NLFFF solution by writing the divergence-free condition (Equation (8)) and the force-free condition (Equation (9)) of a magnetic field vector ${\boldsymbol{B}}=({B}_{r},{B}_{\theta },{B}_{\varphi })$ in spherical coordinates (r, θ, φ). The location of the magnetic charge and the spherical symmetry axis are aligned with the vertical direction to the local solar surface. Equations (8)–(9) expressed in spherical coordinates are then

$$\begin{eqnarray}\begin{array}{rcl}({\rm{\nabla }}\cdot {\boldsymbol{B}}) & = & \displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{\partial }{\partial r}({r}^{2}{B}_{r})+\displaystyle \frac{1}{r\sin \theta }\displaystyle \frac{\partial }{\partial \theta }({B}_{\theta }\sin \theta )\\ & & +\,\displaystyle \frac{1}{r\sin \theta }\displaystyle \frac{\partial {B}_{\varphi }}{\partial \varphi }=0,\end{array}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{\left[{\rm{\nabla }}\times {\boldsymbol{B}}\right]}_{r}=\displaystyle \frac{1}{r\sin \theta }\left[\displaystyle \frac{\partial }{\partial \theta }({B}_{\varphi }\sin \theta )-\displaystyle \frac{\partial {B}_{\varphi }}{\partial \varphi }\right]=\alpha {B}_{r},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{\left[{\rm{\nabla }}\times {\boldsymbol{B}}\right]}_{\theta }=\displaystyle \frac{1}{r}\left[\displaystyle \frac{1}{\sin \theta }\displaystyle \frac{\partial {B}_{r}}{\partial \varphi }-\displaystyle \frac{\partial }{\partial r}({{rB}}_{\varphi })\right]=\alpha {B}_{\theta },\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{\left[{\rm{\nabla }}\times {\boldsymbol{B}}\right]}_{\varphi }=\displaystyle \frac{1}{r}\left[\displaystyle \frac{\partial }{\partial r}({{rB}}_{\theta })-\displaystyle \frac{\partial {B}_{r}}{\partial \theta }\right]=\alpha {B}_{\varphi }.\end{eqnarray} \tag{ 13 }$$

For a simple approximative nonlinear force-free solution we require axisymmetry with no azimuthal dependence (∂/∂φ = 0). This requirement simplifies Equations (10)–(13) to

$$\begin{eqnarray}&&\displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{\partial }{\partial r}({r}^{2}{B}_{r})+\displaystyle \frac{1}{r\sin \theta }\displaystyle \frac{\partial }{\partial \theta }({B}_{\theta }\sin \theta )=0,\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\displaystyle \frac{1}{r\sin \theta }\displaystyle \frac{\partial }{\partial \theta }({B}_{\varphi }\sin \theta )-\alpha {B}_{r}=0,\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&-\displaystyle \frac{1}{r}\displaystyle \frac{\partial }{\partial r}({{rB}}_{\varphi })-\alpha {B}_{\theta }=0,\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&\displaystyle \frac{1}{r}\displaystyle \frac{\partial }{\partial r}({{rB}}_{\theta })-\displaystyle \frac{1}{r}\displaystyle \frac{\partial {B}_{r}}{\partial \theta }-\alpha {B}_{\varphi }=0.\end{eqnarray} \tag{ 17 }$$

An approximate solution of this system of four differential equations has been analytically derived for the radial field component B_r(r, θ) (Equation (18)), the azimuthal field component ${B}_{\varphi }(r,\theta )$ (Equation (19)), and the nonlinear parameter α(r, θ) (Equation (21)), for the special case of a vanishing poloidal field component B_θ(r, θ) = 0 (Aschwanden 2013b). Here we find a new (approximative) analytical solution for the poloidal field component B_θ(r, θ) $\ne$ 0, as expressed in (Equation (20)),

$$\begin{eqnarray}&&{B}_{r}(r,\theta )={B}_{0}\left(\displaystyle \frac{{d}^{2}}{{r}^{2}}\right)\displaystyle \frac{1}{(1+{b}^{2}{r}^{2}{\sin }^{2}\theta )},\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{B}_{\varphi }(r,\theta )={B}_{0}\left(\displaystyle \frac{{d}^{2}}{{r}^{2}}\right)\displaystyle \frac{{br}\sin \theta }{(1+{b}^{2}{r}^{2}{\sin }^{2}\theta )},\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&{B}_{\theta }(r,\theta )={B}_{0}\left(\displaystyle \frac{{d}^{2}}{{r}^{2}}\right)\displaystyle \frac{{b}^{2}{r}^{2}{\sin }^{3}(\theta )}{(1+{b}^{2}{r}^{2}{\sin }^{2}\theta )}\displaystyle \frac{1}{\cos \theta },\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&\alpha (r,\theta )\approx \displaystyle \frac{2b\cos \theta }{(1+{b}^{2}{r}^{2}{\sin }^{2}\theta )},\end{eqnarray} \tag{ 21 }$$

where the constant b is defined in terms of the number of full twisting turns N_twist over the loop length L, i.e., $b=2\pi {N}_{\mathrm{twist}}/L$. In the old VCA-NLFFF code we neglected the poloidal magnetic field component (i.e., B_θ = 0), because the magnitude of this component is of second order $\propto {({br}\sin \theta )}^{2}$ like the residuals of the VCA approximation. In the new VCA-NLFFF code presented here, however, we increase the accuracy of the nonpotential field solutions by taking an approximate solution of B_θ $\ne$ 0 (Equation (20)) into account. Inserting this new solution (Equations (18)–(21)) into the divergence-free term (Equation (14)) and the force-free terms (Equations (15)–(17)), we find that the solution is accurate to second order in ${\rm{\nabla }}\cdot {\boldsymbol{B}}$ (Equation (10)), to third order in ${[{\rm{\nabla }}\times {\boldsymbol{B}}]}_{\varphi }$ (Equation (13)), to fifth order in ${[{\rm{\nabla }}\times {\boldsymbol{B}}]}_{\theta }$ (Equation (12)), and being exactly zero for the radial component ${[{\rm{\nabla }}\times {\boldsymbol{B}}]}_{r}=0$ (Equation (11)). A proof of this approximative analytical solution is provided in the Appendix.

In the previous derivation we derived the solution in terms of spherical coordinates (r, θ, φ) in a coordinate system where the rotational symmetry axis is aligned with the vertical to the solar surface intersecting a magnetic charge j (Figure 1), expressed with the 3D vectors ${B}_{r}(r,\theta )$, ${B}_{\varphi }(r,\theta )$, and ${B}_{\theta }(r,\theta )$. For a model with multiple magnetic charges at arbitrary positions on the solar disk, we have to transform the individual coordinate systems $({r}_{j},{\theta }_{j},{\varphi }_{j})$ associated with magnetic charge j into a Cartesian coordinate system (x, y, z) that is given by the observer's line of sight (in z-direction) and the observer's image coordinate system (x,y) in the plane of the sky. The geometric relationships of the spherical coordinate vectors are shown in Figure 1.

Defining the radial vectors ${{\boldsymbol{r}}}_{j}$ (between (x_j, y_j, z_j) and (x, y, z)) and ${\boldsymbol{R}}$ between the solar center (0, 0, 0) and a magnetic charge (x_j, y_j, z_j), as well as the vector products between the orthogonal spherical components $({B}_{r},{B}_{\varphi },{B}_{\theta })$, we obtain the following directional cosines of three spherical vector components (see Figure 1):

$$\begin{eqnarray}&&{{\boldsymbol{r}}}_{j}=[x-{x}_{j},y-{y}_{j},z-{z}_{j}],\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{\boldsymbol{R}}=\left[{x}_{j},{y}_{j},{z}_{j}\right],\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{{\boldsymbol{B}}}_{r}}{{B}_{r}}=\left[\displaystyle \frac{x-{x}_{j}}{{r}_{j}},\displaystyle \frac{y-{y}_{j}}{{r}_{j}},\displaystyle \frac{z-{z}_{j}}{{r}_{j}}\right]=\left[{\cos }_{r,x},{\cos }_{r,y},{\cos }_{r,z}\right],\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{{\boldsymbol{B}}}_{\varphi }}{{B}_{\varphi }}=\displaystyle \frac{{\boldsymbol{R}}\times {{\boldsymbol{B}}}_{r}}{| {\boldsymbol{R}}\times {{\boldsymbol{B}}}_{r}| }=\left[{\cos }_{\varphi ,x},{\cos }_{\varphi ,y},{\cos }_{\varphi ,z}\right],\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{{\boldsymbol{B}}}_{\theta }}{{B}_{\theta }}=\displaystyle \frac{{{\boldsymbol{B}}}_{r}\times {{\boldsymbol{B}}}_{\varphi }}{| {{\boldsymbol{B}}}_{r}\times {{\boldsymbol{B}}}_{\varphi }| }=\left[{\cos }_{\theta ,x},{\cos }_{\theta ,y},{\cos }_{\theta ,z}\right].\end{eqnarray} \tag{ 26 }$$

The vector product allows us also to extract the inclination angle θ_j between the radial magnetic field component ${{\boldsymbol{B}}}_{r}$ and the local vertical direction ${\boldsymbol{R}},$

$$\begin{eqnarray}&&{\theta }_{j}={\sin }^{-1}\left(\displaystyle \frac{| {\boldsymbol{R}}\times {{\boldsymbol{B}}}_{r}| }{| {\boldsymbol{R}}| \ | {{\boldsymbol{B}}}_{r}| }\right).\end{eqnarray} \tag{ 27 }$$

The transformation of the nonpotential field from spherical coordinates (B_r, B_φ, B_θ) into Cartesian coordinates (B_x, B_y, B_z), which takes the sphericity of the solar surface fully into account, amounts to

$$\begin{eqnarray}\begin{array}{rcl}{B}_{x} & = & {B}_{r}({r}_{j},{\theta }_{j}){\cos }_{r,x}+{B}_{\varphi }({r}_{j},{\theta }_{j}){\cos }_{\varphi ,x}+{B}_{\theta }({r}_{j},{\theta }_{j}){\cos }_{\theta ,x}\\ {B}_{y} & = & {B}_{r}({r}_{j},{\theta }_{j}){\cos }_{r,y}+{B}_{\varphi }({r}_{j},{\theta }_{j}){\cos }_{\varphi ,y}+{B}_{\theta }({r}_{j},{\theta }_{j}){\cos }_{\theta ,y}\\ {B}_{z} & = & {B}_{r}({r}_{j},{\theta }_{j}){\cos }_{r,z}+{B}_{\varphi }({r}_{j},{\theta }_{j}){\cos }_{\varphi ,z}+{B}_{\theta }({r}_{j},{\theta }_{j}){\cos }_{\theta ,z}.\end{array}\end{eqnarray} \tag{ 28 }$$

This is a convenient parameterization that allows us directly to calculate the magnetic field vector of the nonpotential field ${{\boldsymbol{B}}}_{j}$ = (B_x, B_y, B_z) associated with a magnetic charge j that is characterized by five parameters: (B_j, x_j, y_j, z_j, α_j), where the force-free α-parameter is related to the twist parameter in terms of the number of twists N_twist and the loop length l by ${b}_{j}=2\pi {N}_{\mathrm{twist}}/l$.

The total nonpotential magnetic field from all $j=1,\ldots ,{N}_{{\rm{m}}}$ magnetic charges can be approximately obtained from the vector sum of all components j (in an analogous way to what we applied in Equation (7) for the potential field),

$$\begin{eqnarray}&&{\boldsymbol{B}}({\boldsymbol{x}})=\displaystyle \sum _{j=1}^{{N}_{{\rm{m}}}}{{\boldsymbol{B}}}_{j}({\boldsymbol{x}}),\end{eqnarray} \tag{ 29 }$$

where the vector components ${{\boldsymbol{B}}}_{j}=({B}_{x,j},{B}_{y,j},{B}_{z,j})$ of the nonpotential field of a magnetic charge j are defined in Equation (28), which can be parameterized with 5N_m free parameters (B_j, x_j, y_j, z_j, α_j) for a nonpotential field, or with 4N_m free parameters for a potential field (with α_j = 0).

Let us consider the condition of divergence-freeness. Since the divergence operator is linear, the superposition of a number of divergence-free fields is divergence-free also,

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\boldsymbol{B}}={\rm{\nabla }}\cdot (\displaystyle \sum _{j}{{\boldsymbol{B}}}_{j})=\displaystyle \sum _{j}({\rm{\nabla }}\cdot {{\boldsymbol{B}}}_{j})=0.\end{eqnarray} \tag{ 30 }$$

Now, let us consider the condition of force-freeness. A force-free field has to satisfy Maxwell's equation (Equation (9)). Since we parameterized both the potential field and the nonpotential field with a linear sum of N_m magnetic charges, the requirement would be

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\nabla }}\times {\boldsymbol{B}} & = & {\rm{\nabla }}\times \displaystyle \sum _{j=1}^{{N}_{{\rm{m}}}}{{\boldsymbol{B}}}_{j}=\displaystyle \sum _{j=1}^{{N}_{{\rm{m}}}}({{\rm{\nabla }}}_{j}\times {{\boldsymbol{B}}}_{j})\\ & = & \displaystyle \sum _{j=1}^{{N}_{{\rm{m}}}}{\alpha }_{j}({\boldsymbol{r}}){{\boldsymbol{B}}}_{j}=\alpha ({\boldsymbol{r}}){\boldsymbol{B}}.\end{array}\end{eqnarray} \tag{ 31 }$$

Generally, these three equations of the vector ${\rm{\nabla }}\times {\boldsymbol{B}}$ cannot be fulfilled with a scalar function $\alpha ({\boldsymbol{r}})$ for a sum of force-free field components, unless the magnetic field volume can be compartmentalized by spatially separated subvolumes. Since the magnetic field of each point source decreases with the square of the distance, the force-free field solution of one particular magnetic charge is not much affected by the contributions from a spatially well-separated secondary magnetic charge, and thus Equation (31) is approximately valid for spatially well-separated sources. Our derivation of the analytical solution of the magnetic field (shown in the Appendix and summarized in Table 1) proves that the force-free condition (Equation (31)) is fulfilled with an accuracy of the third-order term of $({br}\sin \theta )$. In practice, this means that the force-free field at a location with $({br}\sin \theta )\leqslant 0.1$ has an accuracy of $({br}\sin \theta )\leqslant 0.001$.

Table 1.  Order of Accuracy in the Divergence-freeness and Force-freeness Conditions in the Analytical Approximation of the (Old) VCA-NLFFF and (New) VCA3-NLFFF Codes

Equation	Old Code	New Code
	VCA-NLFFF	VCA3-NLFFF
${\rm{\nabla }}\cdot {\boldsymbol{B}}$	${({br}\sin \theta )}^{2}$	${({br}\sin \theta )}^{2}$
${[{\rm{\nabla }}\times {\boldsymbol{B}}]}_{r}$	0	0
${[{\rm{\nabla }}\times {\boldsymbol{B}}]}_{\theta }$	${({br}\sin \theta )}^{3}$	${({br}\sin \theta )}^{5}$
${[{\rm{\nabla }}\times {\boldsymbol{B}}]}_{\varphi }$	0	${({br}\sin \theta )}^{3}$

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We demonstrate the improvement of the new VCA3-NLFFF code in various ways. One test consists of calculating the figures of merit for the divergence-freeness and the force-freeness, integrated over the simulated computation box (Equations (48)–(49) in Aschwanden 2013b). Such tests yield very similar values for the old VCA-NLFFF and the new VCA3-NLFFF code. A second test is the display of poloidal field lines, which should show a curvature in poloidal planes, as is expected for a correction from B_θ = 0 to B_θ $\ne$ 0, which is indeed the case and confirms the expected change in the geometry of magnetic field lines (Figure 2, panel (b) vs. panel (d)).

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A particularly instructive test is the reproduction of a Gold–Hoyle flux rope (Gold & Hoyle 1960), which is thought to have a large number of multiple helical windings, as inferred from eruptive flux ropes propagating along with coronal mass ejections (CMEs) through the heliosphere (Amari et al. 1996; Gibson & Low 2000; Wang et al. 2017). An attempt to reproduce such flux ropes with one to five full turns was carried out in Aschwanden (2013b, Appendix A), which revealed large distortions from a uniformly thick flux tube geometry (Figure 3 left panels). With the new VCA3-NLFFF code we find that the number of helical windings does not exceed more than about one full turn (Figure 3, right panels), no matter how large the nonpotential force-free α parameter (or b) is chosen to be, which implies that helical geometries with more than one full turn cannot be reproduced with the new VCA3-NLFFF code. The most likely explanation for this limit is that helical magnetic field lines with a twist of more than one turn are not force-free, as is theoretically expected from the kink instability criterion (Hood & Priest 1979; Török et al. 2003; Kliem & Török 2006; Kliem et al. 2014; Priest 2014), as well as from observational evidence (N_twist ≲ 0.5) of the helical twisting number (or the Gauss braiding linkage number) of coronal loops in solar active regions (Aschwanden 2019b). Helical flux ropes as envisioned by Gold & Hoyle (1960) may still exist in interplanetary space but are transient and require a time-dependent solution of the non-force-free magnetic field.

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Over the duration of the SDO mission (Pesnell et al. 2011) from 2010 to 2019, some changes in the scaling of the EUV brightness of AIA images (from a factor of unity to a factor of 0.1) occurred, which are now automatically corrected by the AIA and Helioseismic and Magnetic Imager (HMI) data-reading software, based on the FITS descriptor value BSCALE.

For the absolute calibration of the magnetic energy we compare the line-of-sight magnetic field component ${B}_{z}^{\mathrm{map}}(x,y)$ of the observed HMI magnetogram with the modeled component ${B}_{z}^{\mathrm{model}}(x,y)$ obtained from fitting each local peak in the magnetogram with the expected field from a buried unipolar magnetic charge. We obtain equivalence of the corresponding magnetic energy component for an empirical correction factor of q_B ≈ 0.8, i.e.,

$$\begin{eqnarray}&&{q}_{e}^{\mathrm{model}}=\displaystyle \frac{{\displaystyle \sum }_{x,y}{({q}_{B}\times {B}_{z}^{\mathrm{model}})}^{2}}{{\displaystyle \sum }_{x,y}{({B}_{z}^{\mathrm{map}})}^{2}},\end{eqnarray} \tag{ 32 }$$

where the relationship E ∝ B² is used for the relationship between the magnetic energy E and the magnetic field strength B. This correction factor of q_B ≈ 0.8 accounts for missing magnetic flux in the model that results from overlapping magnetic field domains where opposite magnetic polarities (positive and negative magnetic fields) cancel out parts of the field. Note that a correction factor of q_B = 0.8 of the average field strength corresponds to a factor of q_e ≈ 0.8² = 0.64 in magnetic energy. This correction propagates in the various forms of uncorrected magnetic energies (${E}_{p}^{* },{E}_{\mathrm{np}}^{* }$) as

$$\begin{eqnarray}&&{E}_{p}(t)={E}_{p}^{* }(t)/{q}_{e}^{\mathrm{model}},\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}&&{E}_{\mathrm{np}}(t)={E}_{\mathrm{np}}^{* }(t)/{q}_{e}^{\mathrm{model}},\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&{E}_{\mathrm{free}}(t)={E}_{\mathrm{np}}(t)-{E}_{p}(t)=[{E}_{\mathrm{np}}^{* }(t)-{E}_{p}^{* }]/{q}_{e}^{\mathrm{model}}.\end{eqnarray} \tag{ 35 }$$

This empirical correction supersedes a geometric twist correction factor ${q}_{\mathrm{iso}}\approx {(\pi /2)}^{2}\approx 2.5$ used in previous work (Equation (2) in Paper I), which represents another approach to recover underestimated free energy.

A specialty of the VCA3-NLFFF code is the usage of EUV images (from AIA/SDO), in addition to the line-of-sight magnetograms (with B_z(x, y) from HMI/SDO), in order to constrain the NLFFF solution, rather than using the transverse (B_x, B_y) magnetic field components. We use AIA images in all six coronal wavelengths. An automated pattern recognition code, which is specialized to measure the projected 2D coordinates [x(s), y(s)] of curvi-linear features, is then employed to extract coronal loop segments (mostly in the lowest-density scale height of the corona) at altitudes of 2–50 Mm (see yellow curves in Figures 4 and 5). Several hundred loop features are extracted in a set of six wavelength images, which are further subdivided into seven positions per segment, and for each location the line-of-sight coordinates z(s) are optimized by varying the nonlinear force-free α-parameter for each unipolar magnetic charge (typically 30 values α_i, i = 1, ..., 30).

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The VCA3-NLFFF code converges to a best-fit solution of the average misalignment angle between the observed magnetic field directions and the theoretical magnetic field model for every time step (indicated with μ₂ in Figures 4 and 5, found in a range of μ₂ ≈ 59–206 here). The numerical convergence can be improved if the solution from a previous time step t_i is used as an initial condition in the iterative convergence of subsequent time steps ${t}_{i+1}$. The convergence behavior of our VCA3-NLFFF code is found to be stable but slightly oscillates between two subsequent time steps. In order to take advantage of this numerical behavior, we smooth the time evolutions of the energies by averaging two subsequent solutions, which produces a smoother time evolution E(t) in each of the numerically obtained energies (Equations (33)–(35)),

$$\begin{eqnarray}&&E(t)=\displaystyle \frac{E({t}_{i})+E({t}_{i+1})}{2}.\end{eqnarray} \tag{ 36 }$$

The most important correction of the improved VCA3-NLFFF code is the introduction of an analytical expression for the poloidal magnetic field vector B_θ (r, θ) (Equation (20)), which was neglected in the original code, i.e., B_θ (r, θ) = 0), as well as the related coordinate transformations of the spherical magnetic field components of ${{\boldsymbol{B}}}_{\theta }(r,\theta )$ (Equation (26)) into Cartesian coordinates ${\boldsymbol{B}}=({B}_{x},{B}_{y},{B}_{z})$, which depend on spherical coordinates (r, θ, φ) as derived in Equation (28).

The 3D magnetic field solutions obtained with the VCA3-NLFFF code are shown in the form of magnetic field lines in Figures 4 and 5, while we present in Figures 6 and 7 the time evolution, the free energy E_free(t) (red diamonds), and the time derivative (black hatched areas) of the GOES flux F_GOES(t) (dashed curve), which is a proxy for the hard X-ray time profile. We compare the energies obtained with the VCA3-NLFFF code (red curves with diamonds) with those from the Wiegelmann (W-NLFFF) code (blue curves with crosses). The time evolution of the free energy E_free(t) is computed from the cumulative negative decreases (dashed red curves) in the time interval between the flare start time and end time (black dotted lines in Figures 6 and 7). The time intervals of negative decreases are also marked with red hatched areas in Figures 6 and 7. Note that they often coincide with the peak time of the GOES time derivative (solid vertical black line in Figures 6 and 7). The uncertainties of the dissipated energies E_diss were estimated from the median values of energy decreases in subsequent time steps.

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We see different degrees of agreements between the two codes. For example, an excellent agreement between the VCA3-NLFFF and the W-NLFFF codes is obtained for flare #67 (Figure 6(d)), which shows a dissipated energy of E_diss[W-NLFFF] = $86\times {10}^{30}$ erg and E_diss[VCA3-NLFFF] = (76 ± 7) × 10³⁰ erg. In another example, the expected step function of the free energy prominently shows up in flare #384 (Figure 7(e)), with a value of E_diss[VCA3-NLFFF] = (639 ±31) 10³⁰ erg, but the W-NLFFF code yields a 10 times smaller step with E_diss[W-NLFFF] = 62 × 10³⁰ erg. There is one case (flare #148, Figure 6(f)) with no energy decrease detected with E_diss[W-NLFFF], while E_diss[VCA3-NLFFF] = (88 ± 12) shows clearly a significant magnetic energy decrease. Hence, there are obvious discrepancies between the W-NLFFF and the VCA3-NLFFF codes, but at this point we cannot determine which code is more accurate.

We show the forward-fitted magnetic field solutions for the 11 X-class flares (out of the 173 M- and X-class events) in Figures 4 and 5. We see that major sunspots with opposite magnetic polarities (white and black parts of the magnetograms in Figures 4 and 5) occur in each of the flaring active regions, but the interface between opposite magnetic polarities can be quite complex. Examples of isolated leading sunspots are observed in flares #66, 349, 351, and 384 (Figures 4, 5), while the other cases manifest mixed-polarity sunspots, often with sharp gradients in the magnetic field, such as in flares #12, 147, and 148 (Figures 4, 5). These details are important because the degree of complexity in mixed magnetic polarities determines how well the data are suited to fit the theoretical model of buried unipolar magnetic charges. The more isolated, isotropic, and axisymmetric a sunspot is, the better it can be fitted with a buried unipolar charge, and a higher accuracy of the fitted magnetic field strengths is expected. In other words, the sharper the magnetic field gradients become between positive and negative magnetic polarities, the less accurate is the modeling with unipolar point charges. We will come back to this issue in the next section on the accuracy of absolute magnetic field strengths.

A first quantitative test of our study is how well the VCA3-NLFFF code can retrieve the absolute magnetic field value in an active region. A widely used alternative W-NLFFF code has been created by Wiegelmann et al. (2006, 2012), which we used for comparison in Paper I also. For the subset of all (11) X-class flares, a mean ratio of ${E}_{p}^{\mathrm{phot}}/{E}_{p}^{\mathrm{cor}}=1.05\pm 0.33$ was found. Here we rename the photospheric NLFFF code of Wiegelmann by ${E}_{W}={E}_{p}^{\mathrm{phot}}$ and the coronal (VCA) code by ${E}_{\mathrm{VCA}3}={E}_{p}^{\mathrm{cor}}$, which yields an inverse ratio of ${E}_{\mathrm{VCA}}/{E}_{W}\,=0.95\pm 0.31$ for the potential field energy. Using the improved new VCA3-NLFFF code, we find a mean ratio of

$$\begin{eqnarray}&&\displaystyle \frac{{E}_{\mathrm{VCA}3}}{{E}_{W}}=0.99\pm 0.21\end{eqnarray} \tag{ 37 }$$

for the 11 cases shown in Figures 4–5, measured at the peak time of the free energy (which generally occurs between the flare start and flare peak in soft X-rays, as defined by GOES). Our new result is consistent with the old results of ${E}_{\mathrm{VCA}}/{E}_{W}=0.95\pm 0.31$ and yields an overall mean value that is closer to unity (as expected for a perfect absolute calibration of the code), and moreover it has a smaller scatter (σ = ±0.21) than the old code (σ = ±0.31). Therefore, the new VCA3-NLFFF code appears to retrieve the absolute magnetic field more accurately than the previous VCA-NLFFF code, thanks to the inclusion of the poloidal field component (B_θ) and the empirical calibration factor q_B = 0.8 (Equation (32)).

However, it is not entirely clear what affects the accuracy in the measurement of the magnetic field strength. Since we compare the potential field strength only, which excludes the nonlinear effects of any NLFFF code, we suspect that the accuracy is affected by the decomposition of the magnetograms into unipolar magnetic charges. For axisymmetric magnetic field structures, such as they occur above a sunspot with spherical symmetry, the model of buried unipolar charges is expected to be most accurate. If two sunspots with opposite magnetic polarity appear to be spatially separated on the solar surface, the magnetic flux should be perfectly conserved. However, if two sunspots with opposite magnetic polarity overlap each other, part of the oppositely directed magnetic polarity cancels out, so that the signed flux is not conserved. This interpretation predicts that the unsigned magnetic flux should be conserved for separated sunspots, while the unsigned flux is likely to be underestimated for spatially overlapping sunspot pairs. We investigated whether systematic errors in the magnetic field reconstruction depend on the separation of the main sunspots, or on the heliographic longitude, but did not find a systematic pattern. Nevertheless, the empirical correction factor ${q}_{e}^{\mathrm{model}}={0.8}^{2}=0.64$ (Equation (32)) for magnetic energies, which corresponds to a correction factor of ${q}_{B}^{\mathrm{model}}=0.8$ for magnetic fields (Equation (32)), yields a statistically accurate correction that agrees with other NLFFF codes within a few percent (Equation (37)).

A key result of this study is the more accurate measurement of the four types of magnetic energies, i.e., the potential energy E_p(t), the nonpotential energy E_np(t), the free energy E_free(t), and the total dissipated magnetic energy E_diss during a flare. As representative values we use the time averages of the potential energy,

$$\begin{eqnarray}&&{E}_{p}=\sum {E}_{p}(t)/{n}_{t},\end{eqnarray} \tag{ 38 }$$

and the nonpotential energy,

$$\begin{eqnarray}&&{E}_{\mathrm{np}}=\sum {E}_{\mathrm{np}}(t)/{n}_{t}.\end{eqnarray} \tag{ 39 }$$

The free energy, which is generally defined as

$$\begin{eqnarray}&&{E}_{\mathrm{free}}(t)={E}_{\mathrm{np}}(t)-{E}_{p}(t),\end{eqnarray} \tag{ 40 }$$

contains two time-dependent components: one is the time intervals with positive increases of the free energy, and one is the time intervals with negative decreases of the free energy. We interpret the positive increases as injection of free energy from the photosphere or chromosphere (or new emerging current-carrying flux), while the negative decreases represent the dissipation of magnetic energy in the solar corona during a flare magnetic reconnection process. Thus, we can define the time evolution of magnetic energy dissipation by a cumulative function that contains only time intervals with negative decreases (similar to the method in Paper I),

$$\begin{eqnarray}&&{E}_{\mathrm{cum}}({t}_{i})={E}_{\mathrm{cum}}({t}_{i-1})-\left([{E}_{\mathrm{free}}({t}_{i})-{E}_{\mathrm{free}}({t}_{i-1})]\gt 0\right).\end{eqnarray} \tag{ 41 }$$

This characterization in terms of a cumulative time evolution function, which is monotonically decreasing with time, allows us to separate the internal energy dissipation from the external energy injection of additional energy. The time intervals of energy dissipation are rendered with hashed areas in the panels of Figures 6 and 7. As representative values of the free energy we use the initial maximum of the cumulative time evolution function,

$$\begin{eqnarray}&&{E}_{\mathrm{free}}=\max [{E}_{\mathrm{cum}}(t)]),\end{eqnarray} \tag{ 42 }$$

and as a representative value of the dissipated energy we use the difference between the initial maximum and the final minimum of the cumulative time evolution function,

$$\begin{eqnarray}&&{E}_{\mathrm{diss}}=\max [{E}_{\mathrm{cum}}(t)]-\min [{E}_{\mathrm{cum}}(t)].\end{eqnarray} \tag{ 43 }$$

This definition limits the dissipated energy to E_diss ≤ E_free at the upper end and to E_diss > 0 at the lower end, therefore avoiding unphysical solutions with E_diss > E_free.

In Figure 8 we show the various energy correlations as a function of the potential energy (Figures 8(a)–(c)) or the free energy (Figure 8(d)), where we obtain the following linear regression fits in log–log space:

$$\begin{eqnarray}&&\left(\displaystyle \frac{{E}_{\mathrm{np}}}{{10}^{30}\ \mathrm{erg}}\right)=0.96{\left(\displaystyle \frac{{E}_{p}}{{10}^{30}\mathrm{erg}}\right)}^{1.019},\end{eqnarray} \tag{ 44 }$$

$$\begin{eqnarray}&&\left(\displaystyle \frac{{E}_{\mathrm{free}}}{{10}^{30}\ \mathrm{erg}}\right)=0.018{\left(\displaystyle \frac{{E}_{p}}{{10}^{30}\mathrm{erg}}\right)}^{1.261},\end{eqnarray} \tag{ 45 }$$

$$\begin{eqnarray}&&\left(\displaystyle \frac{{E}_{\mathrm{diss}}}{{10}^{30}\ \mathrm{erg}}\right)=0.0065{\left(\displaystyle \frac{{E}_{p}}{{10}^{30}\mathrm{erg}}\right)}^{1.320},\end{eqnarray} \tag{ 46 }$$

$$\begin{eqnarray}&&\left(\displaystyle \frac{{E}_{\mathrm{diss}}}{{10}^{30}\ \mathrm{erg}}\right)=0.43{\left(\displaystyle \frac{{E}_{\mathrm{free}}}{{10}^{30}\mathrm{erg}}\right)}^{1.044}.\end{eqnarray} \tag{ 47 }$$

These nonlinear relationships can be approximated by averaged energy ratios, for which we find the following means and standard deviations (Figure 8, dotted lines):

$$\begin{eqnarray}&&\left(\displaystyle \frac{{E}_{\mathrm{np}}}{{E}_{p}}\right)\approx 1.10\pm 0.03,\end{eqnarray} \tag{ 48 }$$

$$\begin{eqnarray}&&\left(\displaystyle \frac{{E}_{\mathrm{free}}}{{E}_{p}}\right)\approx 0.12\pm 0.03,\end{eqnarray} \tag{ 49 }$$

$$\begin{eqnarray}&&\left(\displaystyle \frac{{E}_{\mathrm{diss}}}{{E}_{p}}\right)\approx 0.07\pm 0.03,\end{eqnarray} \tag{ 50 }$$

$$\begin{eqnarray}&&\left(\displaystyle \frac{{E}_{\mathrm{diss}}}{{E}_{\mathrm{free}}}\right)\approx 0.60\pm 0.26.\end{eqnarray} \tag{ 51 }$$

Note that the uncertainties of these energy ratios are significantly smaller than in the previous study (Paper I), which indicates less scatter due to systematic errors, especially for those with unphysical solutions E_diss > E_free in the previous study. As a rule of thumb, we obtain the main results that the nonpotential energy is roughly 110% of the potential energy, the free energies and dissipated energies amount typically to ≈10% of the potential energy, and the dissipated energy is about half the free energy. These results are consistent with those in the previous study (Figure 13 in Paper I) but in addition display tighter correlations, smaller spreads in the energy ratios, and elimination of unphysical solutions.

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One of the most important tasks in our study is the energy partition and energy closure in flares. Quantitative information on different forms of energies and their partition in flares and CMEs became more available lately (Emslie et al. 2012; Aschwanden et al. 2014b, 2015a, 2016b, 2017, 2019; Aschwanden 2016a; Warmuth & Mann 2016a, 2016b; Aschwanden & Gopalswamy 2019). Virtually no statistical study on flare energies existed 5 yrs ago (Aschwanden 2004, 2019a). In Figure 9 we show scatterplots of the various forms of energies as a function of the dissipated magnetic energy, which supposedly is the upper limit of any energy that can be dissipated in a flare. Figure 9(a) shows a comparison between the old (VCA-NLFFF) and the new (VCA3-NLFFF) code, being a factor of 3 lower with the new code. Nevertheless, the averaged ratios of the other forms of energies are all below unity (see Equations (48)–(51)), which means that solar flares provide sufficient free (magnetic) energy to supply the nonthermal energy in accelerated electrons and ions, as well as direct heating and expulsion of CMEs (Equations (48)–(51)), yielding the mean ratios

$$\begin{eqnarray}&&\left(\displaystyle \frac{{E}_{\mathrm{nt},e}}{{E}_{\mathrm{diss}}}\right)\approx 0.34\pm 0.09,\end{eqnarray} \tag{ 52 }$$

$$\begin{eqnarray}&&\left(\displaystyle \frac{{E}_{\mathrm{nt},i}}{{E}_{\mathrm{diss}}}\right)\approx 0.11\pm 0.03,\end{eqnarray} \tag{ 53 }$$

$$\begin{eqnarray}&&\left(\displaystyle \frac{{E}_{\mathrm{dir}}}{{E}_{\mathrm{diss}}}\right)\approx 0.36\pm 0.09,\end{eqnarray} \tag{ 54 }$$

$$\begin{eqnarray}&&\left(\displaystyle \frac{{E}_{\mathrm{cme}}}{{E}_{\mathrm{diss}}}\right)\approx 0.31\pm 0.08.\end{eqnarray} \tag{ 55 }$$

We can now test the energy closure by summing the primary energy release processes in E_tot, where the thermal energy E_th is defined as the sum of nonthermal heating (by electron and ion precipitation) and direct heating,

$$\begin{eqnarray}&&{E}_{\mathrm{tot}}={E}_{\mathrm{nt},e}+{E}_{\mathrm{nt},i}+{E}_{\mathrm{dir}}+{E}_{\mathrm{cme}}={E}_{\mathrm{th}}+{E}_{\mathrm{cme}},\end{eqnarray} \tag{ 56 }$$

for which we obtain a ratio of (Figure 9(f))

$$\begin{eqnarray}&&\left(\displaystyle \frac{{E}_{\mathrm{tot}}}{{E}_{\mathrm{diss}}}\right)=0.79\pm 0.12.\end{eqnarray} \tag{ 57 }$$

A pie chart of this energy partition is shown in Figure 10(c), which significantly differs from our previous study (Figure 10(b)) or the study of Emslie et al. (2012) (Figure 10(a)). In Figure 11 we depict the relative energy partitions of 12 individual flares, which have an energy closure within a factor of $1/4\lesssim {E}_{\mathrm{tot}}/{E}_{\mathrm{diss}}\,\lesssim 4.0$. These examples document large differences between individual flares that need to be explored in more detail in future studies.

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In Figure 12 we show the size distributions of the same six forms of energies as in the scatterplots of Figure 9. Ignoring the flattening turnover at the lower side of the logarithmic size distributions, which results from undersampling of weak flares with class M < 1.0, all six size distributions exhibit a power-law distribution $N(E)\propto {E}^{-\alpha }$, with a slope of a ≈ 1.6–1.8, which agrees well with predictions of self-organized criticality (SOC) models. A statistical fractal-diffusive avalanche model of a slowly driven SOC system with Euclidean dimension S = 3 and fractal dimension ${D}_{S}=(1+S)/2=2$ predicts a power-law slope of (Aschwanden 2012)

$$\begin{eqnarray}&&{\alpha }_{P}=2-\displaystyle \frac{1}{S}=\displaystyle \frac{5}{3}\approx 1.67\end{eqnarray} \tag{ 58 }$$

for the peak energy dissipation rate P and

$$\begin{eqnarray}&&{\alpha }_{E}=1+\displaystyle \frac{(S-1)}{({D}_{S}+2)}=\displaystyle \frac{3}{2}=1.5\end{eqnarray} \tag{ 59 }$$

for the dissipated energy distribution. The accuracy of the power-law slope is on the order of σ_α ≈ 0.1–0.2, limited by the relatively small number of analyzed events (N ≈ 10²), which yields inertial ranges of one to two decades in the size of energies only. The values, however, are close to power-law slopes in larger samples, e.g., α_E = 1.53 ± 0.02 for nonthermal (electron) energies in flares (Crosby et al. 1993).

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Since we have more accurate measurements of the time evolution of free energy, before, during, and after flares, we briefly revisit the old issue of energy storage in the solar corona. Rosner & Vaiana (1978) envisioned a unified model of cosmic flare transients, where the time evolution of energy storage and release can be derived from the event occurrence frequency distribution. Simply put, energy is stored in the solar corona with an exponential growth rate, which produces a power-law distribution of flare energies, if the time intervals between two subsequent flares are governed by a stochastic (random) waiting time distribution. However, the postulated correlation between the waiting time and the intermittently released flare energies has never been confirmed by observational statistics in solar flares (Crosby et al. 1998; Wheatland 2000; Lippiello et al. 2010), and timescale arguments were brought forward that indicate that such a storage model is inconsistent with solar flare energy storage in coronal magnetic fields (Lu 1995).

If the coronal energy storage model were true, we would expect a high level of stored free energy shortly before the flare, which then reduces to a lower level after the flare, manifesting an approximate step function in the free energy E_free(t). Although we do find an unambiguous step function in the free energy in some cases, with equal magnitude measured by both the W-NLFFF and VCA3-NLFFF codes (e.g., flare #67 in Figure 6(d)), the step function often does not have the same magnitude in both codes (e.g., flare #384 in Figure 7(e) or flare #220 in Figure 7(a)), or the step function is accompanied by other impulsive magnetic energy increases after the flare peak time (e.g., flare #66 in Figure 6(c), or flare #344 in Figure 7(b)). These erratic fluctuations in the free energy E_free(t) indicate that a simple energy storage model with a monotonic increase before the flare peak, and with an irreversible decrease after the flare peak time (defined by the maximum of the GOES time derivative), is not always consistent with the observations, although we detect some level of energy decrease in all cases (Figures 6 and 7). Consequently, we should generalize the step function scenario to a more comprehensive model that allows coronal energy storage and dissipation on timescales that are even shorter than the flare duration.

We illustrate the impulsiveness of free energy storage and dissipation in Figure 13 for the case of the longest observed flare in our sample, with a total duration of τ_flare = 4.1 hr. In this case we see numerous decreases of the free energy (13 red hatched areas in Figure 13, middle panel). Clearly the storage and dissipation of free energy vary on much shorter timescales than the flare duration, in contrast to the coronal storage model of Rosner & Vaiana (1978), where energy storage times are required to be much longer than the flare duration, continuously accumulating during two subsequent large flares (which may not repeat until timescales of days to weeks according to observations).

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The previous discussion raises the issue of timescale separation (see also Section 2.13 in Aschwanden 2016b). SOC models assume a timescale separation between the energy buildup or storage timescale τ_storage and the duration τ_diss, i.e., τ_storage ≫ τ_diss, in order to enable a slowly driven system that produces a power-law distribution of dissipated energies, as originally proposed for sandpile avalanches by Bak et al. (1987). The waiting times between subsequent avalanches are assumed to be a stochastic (random) process and to produce a Poisson distribution (Wheatland et al. 1998). The same criterion of timescale separation has been applied to the energy storage model of Rosner & Vaiana (1978), i.e., τ_storage ≫ τ_diss, but the size distribution of the avalanches is assumed to follow some scaling with the waiting times, which would constitute a deterministic system, where the flare magnitude could be predicted by the waiting time, which, however, is not the case (see the following discussion in Section 5.5). Moreover, the timescales of energy buildup or storage, as well as the timescales of energy dissipation, are observed to be much shorter (on the order of minutes) than the flare duration (up to 4 hr for the case shown in Figure 13), and thus the observed timescale ranges are opposite to the model of Rosner & Vaiana (1978), as well as opposite to slowly driven SOC models, which leaves us with "fast-driven" or "strongly driven" SOC systems (Aschwanden et al. 2016a).

Earlier estimates of flare energies assumed that the free (magnetic) energy has a high level before the flare and then drops like a step function to a lower level after the flare, which implies energy conservation of the free energy in the solar corona, i.e.,

$$\begin{eqnarray}&&{E}_{\mathrm{free}}({t}_{\mathrm{before}})-{E}_{\mathrm{diss}}={E}_{\mathrm{free}}(t={t}_{\mathrm{after}}).\end{eqnarray} \tag{ 60 }$$

Since we developed tools to calculate the nonpotential energy and the free energy E_free(t) (e.g., with the W-NLFFF or VCA3-NLFFF code), we can apply the principle of energy conservation in order to determine the combined energy dissipation of the primary processes operating in flares (Equation (56)), which we estimated from the cumulative energy decrease function E_cum(t) (by extracting it from the free energy E_free(t) as shown in Figures 6–7). However, since the energy dissipation often does not follow a single-step function, but rather multiple step functions (up to ≈14 episodes in the long-duration flare shown in Figure 13), we should introduce a more generalized energy conservation formalism that includes multiple step functions,

$$\begin{eqnarray}&&\sum {E}_{\mathrm{free},\mathrm{pos}}(t){\rm{\Delta }}t-{E}_{\mathrm{diss}}=\sum {E}_{\mathrm{free},\mathrm{neg}}(t){\rm{\Delta }}t,\end{eqnarray} \tag{ 61 }$$

where ${E}_{\mathrm{free},\mathrm{pos}}(t)$ and ${E}_{\mathrm{free},\mathrm{neg}}(t)$ encompass time intervals with positive and negative changes of the free energy. In this study we measured the cumulative negative changes of the free energy, ${E}_{\mathrm{cum}}(t)={E}_{\mathrm{free},\mathrm{neg}}(t)$, but did not measure the sum of the positive free energy changes, since there is some ambiguity about the flare duration and the definition of the flare-associated pre-flare and post-flare phases.

Nevertheless, our analysis indicates that the flare-associated changes of the free (magnetic) energy are often more complex than a single-step function. Moreover, since there is no simple monotonic increase of stored energy in the corona, but rather intermittent pulses of energy injection with rapid dissipation during flare phases, probably of chromospheric origin, energy conservation cannot simply be deduced from the coronal (nonpotential) magnetic field alone, but has to include also energy input from the chromosphere. In this sense magnetic energy is not conserved for coronal contributions alone but requires the knowledge of the energy contributions from the chromosphere also.

A perhaps surprising result is the small ratio of the free magnetic energy to the potential energy, i.e., ${E}_{\mathrm{free}}/{E}_{p}=0.12\,\pm 0.03$ (Figure 8(b); Equation (49)). Why is the available free energy an order of magnitude smaller than the total potential field energy? According to the helically twisted geometry of a flux tube, this energy ratio can be translated into a geometric angle between the potential and nonpotential field components, which is also known as helical (or azimuthal) twist angle,

$$\begin{eqnarray}&&\displaystyle \frac{{E}_{\mathrm{free}}}{{E}_{p}}=\displaystyle \frac{{B}_{\varphi }^{2}}{{B}_{p}^{2}}={(\tan \mu )}^{2}.\end{eqnarray} \tag{ 62 }$$

According to our measurements with ${E}_{\mathrm{free}}/{E}_{p}\approx 0.12\pm 0.03$, the energy ratio corresponds to a helical twist angle of

$$\begin{eqnarray}&&\mu =\arctan \left[{\left(\displaystyle \frac{{E}_{\mathrm{free}}}{{E}_{p}}\right)}^{1/2}\right],\end{eqnarray} \tag{ 63 }$$

yielding a value of μ = 19° ± 2°. Thus, we suspect that helical twisting is occurring until a critical angle of μ_crit ≈ 19° is reached, while further accumulation of free energy by helical twisting is inhibited by some stabilizing force, or by the onset of an instability.

The number of twisting turns in a flux tube can be expressed by

$$\begin{eqnarray}&&\tan \mu =\displaystyle \frac{{N}_{\mathrm{twist}}\ 2\pi r}{L}\approx 6.3\ {N}_{\mathrm{twist}}\left(\displaystyle \frac{r}{L}\right),\end{eqnarray} \tag{ 64 }$$

where r is the flux tube radius and L is the loop length. For instance, a critical twist angle of μ = 19° and an aspect ratio of $(r/L)=(1/20)$ yield a number of one twist turn, i.e., N_twist ≈ 1.0, which is the approximate threshold for the kink instability (Hood & Priest 1979; Török et al. 2003; Kliem & Török 2006; Kliem et al. 2014; Priest 2014). Hence, a constant value of the critical helical twist angle can explain the proportionality between the free energy E_free and the potential energy E_p found here (Equation (49)), as well as the prediction that the free energy scales with the helical twist angle μ (Equation (63)).

The performance of solar flare forecasting methods has been diligently tested, but there is no consensus about the best method (Falconer et al. 2003, 2011, 2012; Leka & Barnes 2003; Cui et al. 2006; Barnes et al. 2007, 2016; Bélanger et al. 2007; Georgoulis & Rust 2007; Li et al. 2007; Schrijver 2007; Barnes & Leka 2008; Colak & Qahwaji 2009; Mason & Hoeksema 2010; Bloomfield et al. 2012; Ahmed et al. 2013; Strugarek & Charbonneau 2014; Bobra & Ilonidis 2016; Nishizuka et al. 2017; Jonas et al. 2018). A question of general interest is whether the evolution of the free energy, E_free(t), can be used for flare forecasting. The free energy has been found to be one of the best-correlated flare indicators (Bobra & Couvidat 2015). However, the free energy calculated in Bobra & Couvidat (2015) is derived from photospheric magnetograms, which measure the longitudinal (line-of-sight) component B_z and the transverse components B_x and B_y in the non-force-free photosphere (Metcalf et al. 1995). Another well-correlated parameter that can be calculated with our VCA3-NLFFF code includes the total unsigned vertical current.

Our statistics with enhanced accuracy provides some useful information for forecasting of the magnitude of solar flares. The tight correlations between the four magnetic energy parameters found here (i.e., the potential, nonpotential, free energy, and flare-dissipated energies; Figure 8) enable us to predict the flare magnitude from the knowledge of the potential field of an active region. The potential field is easy to calculate and is slowly varying in an active region, often being stable over several days (Sun et al. 2012). From another statistical study it was concluded that currents associated with coronal nonpotentiality have a characteristic growth and decay timescale of ≈10–30 hr (Schrijver et al. 2005; Schrijver 2016). The free energy is confined to a relatively small fraction of 12% ± 3% of the potential energy (Equation (49)). Also, the flare-dissipated energy is found in a relatively small range of 7% ± 3% (Equation (50)), from which we can predict an upper limit of the dissipated energy a few days ahead of the flare. We cannot predict the specific onset time of the flare, but we can provide an upper limit of the largest flare occurring over the next few days. The uncertainty of our prediction method, based on the statistical ratio found here for M- and X-class flare events, i.e., ${E}_{\mathrm{diss}}/{E}_{p}=7 \% \pm 3 \%$ (Equation (50)), amounts to a factor of 10%/7% = 1.4. The largest nonpotential energy we measured in this study is E_np ≈ 6 × 10³³, from which we predict a maximum dissipated flare energy of 7% ± 3%, or E_diss = (4.2 ± 1.8) × 10³² erg, similar to our actual measurement of ${E}_{\mathrm{diss}}^{\mathrm{obs}}=6.4\times {10}^{32}$ erg (Table 2).

Table 2.  Energy Parameters of the First 10 of the 174 M- and X-class Flare Events in the Longitude Range of [−45°, +45°], Computed with the New VCA3-NLFFF Code

Nr	Date	Time	GOES	Helio	Ep	Enp	Efree	Ediss	${E}_{\mathrm{nt},e}$	${E}_{\mathrm{nt},i}$	Edir	Eth	Ecme	Esum
3	2010 Aug 7	17:55	M1.0	N13E34	402.2	459.5	59.5	22.4	1.1	0.4	6.8	8.2	0.0	8.2
4	2010 Oct 16	19:07	M2.9	S18W26	164.6	173.6	12.7	11.7	1.4	0.5	17.4	19.2	0.1	19.3
10	2011 Feb 13	17:28	M6.6	S21E04	802.3	866.0	66.5	23.8	6.8	2.3	11.6	20.8	0.7	21.4
11	2011 Feb 14	17:20	M2.2	S20W07	998.6	1058.6	71.4	35.3	0.0	0.0	13.5	13.5	0.5	13.9
12	2011 Feb 15	01:44	X2.2	S21W12	966.8	1039.2	94.1	38.8	25.1	8.5	48.6	82.2	9.7	91.9
13	2011 Feb 16	01:32	M1.0	S22W27	959.2	1037.1	101.8	79.2	2.5	0.9	3.9	7.3	2.0	9.3
14	2011 Feb 16	07:35	M1.1	S23W30	1100.3	1209.6	125.0	59.8	0.0	0.0	4.4	4.4	0.0	4.4
15	2011 Feb 16	14:19	M1.6	S23W33	1054.5	1163.8	146.2	89.5	4.6	1.6	0.2	6.4	0.0	6.4
16	2011 Feb 18	09:55	M6.6	N15E05	727.3	779.1	87.3	83.1	11.8	4.0	0.1	4.2	0.0	15.9
17	2011 Feb 18	10:23	M1.0	N17E07	743.3	808.5	70.5	8.9	0.0	0.0	3.4	3.4	0.0	3.4

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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