Validation and Benchmarking of a Practical Free Magnetic Energy and Relative Magnetic Helicity Budget Calculation in Solar Magnetic Structures

Abstract

In earlier works we introduced and tested a nonlinear force-free (NLFF) method designed to self-consistently calculate the coronal free magnetic energy and the relative magnetic helicity budgets of observed solar magnetic structures. In principle, the method requires only a single, photospheric or low-chromospheric, vector magnetogram of a quiet-Sun patch or an active region and performs calculations without three-dimensional magnetic and velocity-field information. In this work we strictly validate this method using three-dimensional coronal magnetic fields. Benchmarking employs both synthetic, three-dimensional magnetohydrodynamic simulations and nonlinear force-free field extrapolations of the active-region solar corona. Our time-efficient NLFF method provides budgets that differ from those of more demanding semi-analytical methods by a factor of approximately three, at most. This difference is expected to come from the physical concept and the construction of the method. Temporal correlations show more discrepancies that are, however, soundly improved for more complex, massive active regions, reaching correlation coefficients on the order of, or exceeding, 0.9. In conclusion, we argue that our NLFF method can be reliably used for a routine and fast calculation of the free magnetic energy and relative magnetic helicity budgets in targeted parts of the solar magnetized corona. As explained in this article and in previous works, this is an asset that can lead to valuable insight into the physics and triggering of solar eruptions.

Appendices

Appendix A: Error Estimation for the Semi-analytical Method

From the vector potentials A, A _p we define the reconstructed fields B ^∗=∇×A, $\mathbf{B}_{\mathrm{p}}^{*}=\nabla\times\mathbf{A}_{\mathrm{p}}$. We then assign an uncertainty to each field simply as δ B=B−B ^∗ and $\delta \mathbf{B}_{\mathrm{p}}=\mathbf{B}_{\mathrm{p}}-\mathbf{B}_{\mathrm{p}}^{*}$. Of course, uncertainties and errors in our calculations have various origins, such as the solution of Laplace’s equation or the numerical integrations for obtaining the vector potentials. Since we cannot quantify these errors trivially, we simply treat them as zero and define a single error from the difference between the given and reconstructed fields, which can serve as a lower limit to the actual error.

The error in the total relative helicity can be written as

$$ \delta H=\sqrt{(\delta H_{\rm mut})^2+(\delta H_{\rm self})^2}, $$

(22)

assuming that the two errors are independent. Since all quantities of interest are volume integrals of their respective density, that is, of the form $X=\int{\rm dV}\,\mathcal{X}$, the error in them will be

$$ \delta X=\lambda^3 \sqrt{\sum_{i}( \delta\mathcal{X}_i)^2}, $$

(23)

where λ ³ represents the volume element and summation is made over all points in the volume. The standard error propagation for the mutual-helicity density error leads to

$$ \delta\mathcal{H}_{\mathrm{mut},i}= \bigl(4A_{px}^2\bigl( \delta B_x^2+\delta B_{px}^2 \bigr)+4A_{py}^2\bigl(\delta B_y^2+ \delta B_{py}^2\bigr) \bigr)^{1/2}, $$

(24)

where all quantities in the right-hand side are to be taken at point i. In a similar fashion, we can find the error in self-helicity density as

$$ \delta\mathcal{H}_{\mathrm{self},i}= \bigl((A_x-A_{px})^2 \bigl(\delta B_x^2+\delta B_{px}^2 \bigr)+(A_y-A_{py})^2\bigl(\delta B_y^2+\delta B_{py}^2\bigr) \bigr)^{1/2}. $$

(25)

Likewise, the error in free energy can be obtained from

$$ \delta E_\mathrm{c}=\sqrt{(\delta E_{\rm t})^2+( \delta E_{\rm p})^2}, $$

(26)

where the error in total-energy density is

$$ \delta\mathcal{E}_{\mathrm{t},i}=\frac{1}{4\pi} \bigl((B_x\delta B_x)^2+(B_y\delta B_y)^2+(B_z \delta B_z)^2 \bigr)^{1/2} $$

(27)

and a similar formula holds for the error in potential-energy density, $\delta\mathcal{E}_{\mathrm{p},i}$, if we replace B, δ B by B _p, δ B _p.

Another stringent error-estimation stems from the difference between the two theoretically equivalent free-energy formulas of Equations (3), (4), i.e. the imperfect solenoidal property of B. In this respect, an alternative error in free magnetic energy can be defined by $\delta E'_{\mathrm{c}}=\frac{1}{2}\vert E_{\mathrm{c}}-E'_{\mathrm{c}}\vert$ and then the final error will be $\max(\delta E_{\mathrm{c}},\delta E_{\mathrm{c}}')$. Where $\delta E_{\mathrm{c}}'>\delta E_{\mathrm{c}}$, and to be consistent, the error in total energy is recalculated from Equation (26) with E _c, E _t interchanged.

Taking into account at this point the LFF energy-helicity study of Georgoulis and LaBonte (2007) and assuming their simplified relation between the free magnetic energy and the relative magnetic helicity, the error $\delta E'_{\mathrm{c}}$ in free energy gives rise to an error δH′ in the relative helicity of the form

$$ \delta H'=H\frac{\delta E'_\mathrm{c}}{E_\mathrm{c}}. $$

(28)

The final selection of the error for relative magnetic helicity will thus be taken as max(δH,δH′), while the mutual- and self-helicity errors are calculated by Equations (23), (24) and (25).

Appendix B: Performance of the Semi-analytical Method in Simulated Data with Known Energy/Helicity Budgets

Here we test our potential-field and vector-potential methods against the well-known analytical nonlinear force-free field model of Low and Lou (1990). More specifically, we compare with the second Low and Lou’s case, namely the one with n=m=1 (in their notation) and source location parameters l=0.3, Φ=π/4. For this we used a rectangular box of size 2×2×1.6 (in arbitrary units) that was converted into a grid of 160×160×128 pixels. After calculating the vector potential A from Equations (10) – (13), we reconstructed the field B as B ^∗=∇×A. A visual comparison between the original and reconstructed fields at the lower boundary of the volume is shown in Figure 14. The method reproduces the given field quite well.

Figure 14

Comparison of the Low and Lou (1990) magnetic field B (first row) with the reconstructed one B ^∗=∇×A (second row) at the bottom boundary (x,y,z=0). The third row shows the difference B−B ^∗. The left, middle, and right columns show the x, y, and z components of the magnetic fields.

It is necessary to quantify the agreement between fields B and B ^∗. To this purpose we used the linear correlation coefficients for each component of the two fields, as well as the metrics described by Schrijver et al. (2006). Starting with the latter, the vector correlation of B, B ^∗ is $C_{\rm vec}=0.9967$, the Cauchy-Schwarz metric is $C_{\rm CS}=0.9984$, the complement of the normalized vector error is $E_{\rm n}'=0.9747$, the complement of the mean vector error is $E_{\mathrm{m}}'=0.9496$, and the total magnetic energy normalized to the input case is ϵ=1.013. The correlation coefficients for the x and y components are practically 1, while the one for B _z is slightly lower, ≈ 0.9935. This is a general rule, with the z-component of the field reproduced less well than the x and y-components, owing to the setup of the method. Indeed, with the used gauge the calculation of B _z involves the most numerical operations, therefore it is susceptible to more accumulated errors. The corresponding metrics for the potential fields B _p, $\mathbf{B}_{\mathrm{p}}^{*}=\nabla\times\mathbf{A}_{\mathrm{p}}$ have similar values, indicating a good agreement for these fields as well.

Moreover, if we take as reference the top plane (at z=z ₂) in the calculation of the vector potentials, the agreement between the two fields is slightly better. As an example, the vector correlation of B, B ^∗ for the top level is $C_{\rm vec}=0.9997$ and the complement of the mean vector error is $E_{\rm m}'=0.9951$. Improved precision if the top plane is used as reference seems a general characteristic of the method that also is seen in other examples.

If we calculate the total relative helicity from Equation (6) using the volume-calculation method, we obtain H=−0.495 H _LL, where $H_{\mathrm{LL}}=|\int_{\mathcal{V}} \mathrm{d}V\,\mathbf{A}_{\mathrm{LL}}\cdot\mathbf{B}|$ is a non-invariant helicity. A _LL is the vector potential that produces the Low and Lou (1990) field, given by

$$ \mathbf{A}_\mathrm{LL}=\frac{a}{r^2}\int_{-1}^\mu \mathrm{d}\mu'\,\frac{P^2(\mu ')}{1-\mu'^2}\hat{r}+\frac{P(\mu)}{r^2\sin\theta}\hat{\phi}, $$

(29)

where a ²=0.425 and P=P _1,1 in Low and Lou’s notation. One can easily verify that B=∇×A _LL. If we replace A in Equation (6) with $\mathbf{A}_{\mathrm{LL,}}$ we obtain H=−0.482 H _LL, which means that our implementation of the Valori, Démoulin, and Pariat (2012) method produces the actual helicity of the Low and Lou model with an accuracy of ≈ 2.7 %.

The energy of the field, as calculated by Equation (3), is E _t=45.0 in the arbitrary units used so far in this section. Similarly, the potential energy is E _p=34.8, and so the free magnetic energy is E _c=0.226 E _t, a fraction that agrees with that found by Low and Lou (1990), although for a slightly different case (Φ=π/2).