Sparse Reconstruction of Electric Fields from Radial Magnetic Data

Because the inductive potential Φ solves the Poisson Equation (4), the solution may be expressed in terms of the Green's function as

$$\begin{eqnarray}{\rm{\Phi }}(x,y,t) & = & \displaystyle \frac{1}{2\pi }{\displaystyle \int }_{-\infty }^{\infty }{\displaystyle \int }_{-\infty }^{\infty }{\partial }_{t}{B}_{z}(x^{\prime} ,y^{\prime} ,t)\\ & & \times \mathrm{log}{| {(x-x^{\prime} )}^{2}+{(y-y^{\prime} )}^{2}| }^{1/2}\,{dx}^{\prime} \,{dy}^{\prime} .\end{eqnarray} \tag{ 5 }$$

Here we have taken Cartesian coordinates and an infinite plane for simplicity, but similar solutions hold on a spherical surface. For a localized source ${\partial }_{t}{B}_{z}$, we therefore have that ${\rm{\Phi }}\sim \mathrm{log}(r)$ for large distance r from the source. The corresponding inductive electric field ${{\boldsymbol{E}}}_{\perp 0}$ therefore decays only as ${r}^{-1}$. In particular, it may be nonzero well outside the regions of nonzero B_z.

As an explicit example, consider a bipolar distribution

$$\begin{eqnarray}&&{\partial }_{t}{B}_{z}=\exp \left(-\displaystyle \frac{{(x+\rho )}^{2}+{y}^{2}}{{\delta }^{2}}\right)-\exp \left(-\displaystyle \frac{{(x-\rho )}^{2}+{y}^{2}}{{\delta }^{2}}\right).\end{eqnarray} \tag{ 6 }$$

In this case, one may solve Equation (4) exactly to obtain the closed-form solution

$$\begin{eqnarray}&&{\rm{\Phi }}(x,y)=\displaystyle \frac{{\delta }^{2}}{2}\mathrm{log}\left(\displaystyle \frac{{r}_{+}}{{r}_{-}}\right)+\displaystyle \frac{{\delta }^{2}}{4}{E}_{1}\left(\displaystyle \frac{{r}_{+}^{2}}{{\delta }^{2}}\right)-\displaystyle \frac{{\delta }^{2}}{4}{E}_{1}\left(\displaystyle \frac{{r}_{-}^{2}}{{\delta }^{2}}\right),\end{eqnarray} \tag{ 7 }$$

where ${r}_{\pm }=\sqrt{{(x\pm \rho )}^{2}+{y}^{2}}$ and ${E}_{1}(x):= {\int }_{x}^{\infty }{{\rm{e}}}^{-t}/t\,{dt}$ is the exponential integral. The logarithmic behavior is clear from this expression, and the corresponding electric field components are plotted in Figure 1.

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Another way to characterize the inductive solution ${{\boldsymbol{E}}}_{\perp 0}$ is by the fact that it minimizes the L₂-norm $\parallel {{\boldsymbol{E}}}_{\perp }{\parallel }_{2}:= {\left({\int }_{S}| {{\boldsymbol{E}}}_{\perp }{| }^{2}{dS}\right)}^{1/2}$ among all possible solutions to (2). This is straightforward to see by inserting the expression (3) into the norm, which leads to

$$\begin{eqnarray}&&\parallel {{\boldsymbol{E}}}_{\perp }{\parallel }_{2}^{2}=\parallel {{\boldsymbol{E}}}_{\perp 0}{\parallel }_{2}^{2}+\parallel {{\rm{\nabla }}}_{\perp }{\rm{\Psi }}{\parallel }_{2}^{2}+2{\oint }_{\partial D}{\rm{\Psi }}{{\boldsymbol{E}}}_{\perp 0}\cdot {\boldsymbol{n}}\,{dS}.\end{eqnarray} \tag{ 8 }$$

On a periodic or infinite domain, the boundary term vanishes, and it follows that $\parallel {{\boldsymbol{E}}}_{\perp }{\parallel }_{2}$ is minimized by choosing ${\rm{\Psi }}\equiv 0$. In this sense, the inductive electric field ${{\boldsymbol{E}}}_{\perp 0}$ is the smoothest possible ${{\boldsymbol{E}}}_{\perp }$ satisfying Faraday's law for a given ${\partial }_{t}{B}_{z}$.

In practice, we work with ${{\boldsymbol{E}}}_{\perp }$ defined on a discrete numerical grid, so it is useful to formulate the analogous discrete problem. In this paper, we discretize ${{\boldsymbol{E}}}_{\perp }$ on a staggered grid (Yee 1966), where E_x is defined on the horizontal cell edges, E_y on the vertical cell edges, and ${\partial }_{t}{B}_{z}$ at the cell centers (Figure 2). Such grids are commonly used in MHD simulations. We discretize Faraday's law (2) using Stokes' theorem, so that the equation

$$\begin{eqnarray}{\rm{\Delta }}x{\rm{\Delta }}y\displaystyle \frac{\partial {B}_{z}^{i,j}}{\partial t} & = & {\rm{\Delta }}{{xE}}_{x}^{i,j+1/2}-{\rm{\Delta }}{{xE}}_{x}^{i,j-1/2}\\ & & +{\rm{\Delta }}{{yE}}_{y}^{i-1/2,j}-{\rm{\Delta }}{{yE}}_{y}^{i+1/2,j}\end{eqnarray} \tag{ 9 }$$

must hold for each grid cell $i,j=1,\ldots ,n$. (In more general curvilinear coordinates, the edge lengths and cell areas would be different for each cell.) This constitutes a system of linear equations for the unknowns E_x and E_y on each edge. However, the system is highly under-determined, because there are $2n(n+1)$ unknowns but only n² equations. This nonuniqueness of solution reflects our freedom in choosing the noninductive component.

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In the discrete case, the inductive solution may be computed by writing

$$\begin{eqnarray}&&{E}_{x0}^{i,j+1/2}=({{\rm{\Phi }}}^{i,j}-{{\rm{\Phi }}}^{i,j+1})/{\rm{\Delta }}y,\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{E}_{y0}^{i+1/2,j}=({{\rm{\Phi }}}^{i+1,j}-{{\rm{\Phi }}}^{i,j})/{\rm{\Delta }}x,\end{eqnarray} \tag{ 11 }$$

where ${{\rm{\Phi }}}^{i,j}$ is defined at the cell centers. Then (9) becomes

$$\begin{eqnarray}\displaystyle \frac{\partial {B}_{z}^{i,j}}{\partial t} & = & \displaystyle \frac{1}{{\rm{\Delta }}{y}^{2}}(2{{\rm{\Phi }}}^{i,j}-{{\rm{\Phi }}}^{i,j+1}-{{\rm{\Phi }}}^{i,j-1})\\ & & +\displaystyle \frac{1}{{\rm{\Delta }}{x}^{2}}(2{{\rm{\Phi }}}^{i,j}-{{\rm{\Phi }}}^{i-1,j}-{{\rm{\Phi }}}^{i+1,j}),\end{eqnarray} \tag{ 12 }$$

which is simply the standard five-point stencil for the Poisson equation. In the examples below, we solve this using a standard fast-Poisson solver (Press et al. 1992).

However, it is instructive to think of the system of Equation (9) in the more abstract form $A{\boldsymbol{x}}={\boldsymbol{b}}$, where ${\boldsymbol{x}}$ is the vector of unknowns (E_x, E_y), ${\boldsymbol{b}}$ is the vector of ${\partial }_{t}{B}_{z}$ values, and A is the ${n}^{2}\times 2n(n+1)$ matrix corresponding to Equation (9). The inductive solution is then the solution to $A{\boldsymbol{x}}={\boldsymbol{b}}$ that minimizes the discrete ℓ₂-norm $\parallel {\boldsymbol{x}}{\parallel }_{2}^{2}:= {\sum }_{i}{x}_{i}^{2}$. In other words, it is the (unique) least-squares solution to this under-determined system of equations. As such, the solution may be written in terms of the Moore–Penrose pseudo-inverse as ${\boldsymbol{x}}={A}^{\top }{({{AA}}^{\top })}^{-1}{\boldsymbol{b}}$, although in practice it is much more efficient to use the fast-Poisson solver. Nevertheless, viewing the inductive solution as the ℓ₂-minimum makes the connection with the sparse solution that we will describe in Section 3. The sparse solution is found by minimizing a different discrete norm of ${\boldsymbol{x}}$.

Our first test case is based on the simplest configuration that may arise from Ohm's law. In fact, due to the linearity of both Ohm's law and Equation (2), more general test configurations may readily be obtained by superimposing multiple copies of this basic configuration. For simplicity, the computation is done in Cartesian coordinates in a square domain $| x| \lt 3$, $| y| \lt 3$, using the discretization described in Section 2.3. Periodic boundary conditions are applied in both directions.

Our basic solution is the bipolar distribution

$$\begin{eqnarray}&&{\partial }_{t}{B}_{z}=x\exp \left(-\displaystyle \frac{{x}^{2}+{y}^{2}}{{\delta }^{2}}\right).\end{eqnarray} \tag{ 14 }$$

Physically, this could arise from the uniform advection of a single magnetic polarity ${B}_{z}=\exp [-({x}^{2}+{y}^{2})/{\delta }^{2}]$ under an ideal Ohm's law ${{\boldsymbol{E}}}_{\perp }=-{{\boldsymbol{v}}}_{\perp }\times ({B}_{z}{{\boldsymbol{e}}}_{z})$ with velocity ${{\boldsymbol{v}}}_{\perp }={\delta }^{2}/2{{\boldsymbol{e}}}_{x}$, in which case

$$\begin{eqnarray}&&{{\boldsymbol{E}}}_{\perp }=\displaystyle \frac{{\delta }^{2}}{2}\exp \left(-\displaystyle \frac{{x}^{2}+{y}^{2}}{{\delta }^{2}}\right).\end{eqnarray} \tag{ 15 }$$

(In this case we are taking a snapshot at t = 0, because the polarity is centered on the origin.) Alternatively, the same ${\partial }_{t}{B}_{z}$ (with different B_z) might correspond to the emergence of a bipolar magnetic field from the solar interior. The target ${{\boldsymbol{E}}}_{\perp }$ given by (15), along with the ${\partial }_{t}{B}_{z}$ in (14), are illustrated in the top row of Figure 3.

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The middle row of Figure 3 shows the inductive solution ${{\boldsymbol{E}}}_{\perp 0}$ for this ${\partial }_{t}{B}_{z}$, which is qualitatively similar to the illustrative solution in Section 2. Not only is ${{\boldsymbol{E}}}_{\perp 0}$ not localized within the region of nonzero ${\partial }_{t}{B}_{z}$, but the topology is wrong: there is a substantial E_x0 component, despite the fact that E_x = 0 for the target solution. The sparse solution using basis pursuit, shown in the bottom row of Figure 3, is substantially more accurate. To demonstrate convergence, Figure 4 shows the absolute errors $\max \,| {E}_{x}-{E}_{x}^{\mathrm{target}}|$ and $\max \,| {E}_{y}-{E}_{y}^{\mathrm{target}}|$ for the sparse solution, as a function of both grid resolution and the tolerance parameter in the basis pursuit algorithm. First, the error in ${\partial }_{t}{B}_{z}$ is independent of tolerance or grid resolution, indicating that the constraint $A{\boldsymbol{x}}={\boldsymbol{b}}$ is preserved to high accuracy in all cases. Second, there is convergence in both E_x and E_y as the tolerance is reduced. The error in E_x is independent of resolution, whereas that in E_y saturates at a (higher) level depending on grid resolution. This simply reflects the truncation error in the numerical approximation (9) for the curl, which is quadratic in ${\rm{\Delta }}x$. The saturation is not seen in E_x because of our particular target solution E_x = 0.

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To demonstrate how the sparse reconstruction performs on a more realistic example, we have taken a snapshot from a flux-transport simulation of ${B}_{r}(\theta ,\phi ,t)$ in spherical coordinates, covering the full solar surface. The reason for using a simulated map rather than an observed synoptic magnetogram is to ensure perfect flux balance; the consequences of not having flux balance will become evident in Section 5.

We used the flux-transport model described by Yeates et al. (2015), for which the numerical code is freely available (https://github.com/antyeates1983/sft_data). As input data, we used synoptic maps of B_r from the Global Oscillation Network Group (GONG, gong.nso.edu/data/magmap/), starting in Carrington rotation 2073 and generating the output snapshot in rotation 2109. As described by Yeates et al. (2015), the input maps are used (a) to initialize B_r at the beginning of the simulation, and (b) to extract the strong flux regions used to update B_r during the evolution. The computation used a 360 × 180 grid, equally spaced in longitude (ϕ) and sine latitude ($\cos \theta$).

For context, Figure 5(a) shows the B_r map from the flux-transport simulation, while Figure 5(b) shows the ${\partial }_{t}{B}_{r}$ map from the same time. Because the simulation uses a supergranular diffusion rather than imposing convective velocities, the map appears smoother than would a real observed map. To make the ${\partial }_{t}{B}_{r}$ map look more realistic, we have added random noise to produce the map in Figure 5(c). The noise is carefully constructed to preserve local flux balance, and is generated by adding a bipolar distribution

$$\begin{eqnarray}&&{\beta }^{{i}_{0},{j}_{0}}(i-{i}_{0})\exp \left(-\displaystyle \frac{{(i-{i}_{0})}^{2}+{(j-{j}_{0})}^{2}}{{\delta }^{{i}_{0},{j}_{0}}}\right)\end{eqnarray} \tag{ 16 }$$

centered at each pixel $({i}_{0},{j}_{0})$ on the $(s,\phi )$ grid. The magnitude ${\beta }^{{i}_{0},{j}_{0}}$ at each pixel is chosen from a normal distribution with mean zero and $\sigma =0.005{\max }_{{i}_{0},{j}_{0}}| {\partial }_{t}{B}_{r}|$, and the size ${\delta }^{{i}_{0},{j}_{0}}$ is either 1 or 2 pixels, with equal probability. In addition, we rotate the pattern by 90° at each particular pixel with equal probability. This generates a more realistic map as shown in Figure 5(c).

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Figure 6 shows the inductive and sparse reconstructions of ${{\boldsymbol{E}}}_{\perp }$ from the map in Figure 5(c). To account for the spherical coordinates, Equations (10)–(12) have been modified to include the necessary coordinate factors. The right-hand column of Figure 6 shows that both methods preserve ${\partial }_{t}{B}_{r}$ to high accuracy, as in the Cartesian test. Unlike in the Cartesian test, however, we no longer compare to a target ${{\boldsymbol{E}}}_{\perp }$, since the ${{\boldsymbol{E}}}_{\perp }$ corresponding to our added noise is unknown. Nevertheless, we can still see that the sparse reconstruction is superior to the inductive reconstruction. This is because, although the ${\partial }_{t}{B}_{r}$ distribution is more complex than in the previous test, the sparse ${{\boldsymbol{E}}}_{\perp }$ is still much better localized within regions of strong ${\partial }_{t}{B}_{r}$ than is the inductive ${{\boldsymbol{E}}}_{\perp }$. Moreover, it has a weaker E_ϕ than E_θ, which is consistent with the fact that the dominant Ohm's law contribution in the flux-transport simulation was from differential rotation (v_ϕ). This is not the case in the inductive solution. Thus we are confident that the sparse solution is superior to the inductive solution not just for simple test cases, but also for more complex ${\partial }_{t}{B}_{r}$ distributions that are qualitatively similar to those on the Sun.

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The sparse solution makes the physical assumption that ${{\boldsymbol{E}}}_{\perp }$ should be localized because it satisfies Ohm's law for a localized ${\boldsymbol{B}}$. If the magnetic field distribution is too diffuse, then we cannot expect the assumption of localization to recover the correct (diffuse) ${{\boldsymbol{E}}}_{\perp }$.

To illustrate what happens if we do try to apply the sparse reconstruction in such a situation, consider the horizontal diffusive spreading of an initial Gaussian magnetic polarity

$$\begin{eqnarray}&&{B}_{z}(x,y,t)=\displaystyle \frac{{a}^{2}}{{a}^{2}+4\eta t}\exp \left(-\displaystyle \frac{{x}^{2}+{y}^{2}}{{a}^{2}+4\eta t}\right).\end{eqnarray} \tag{ 17 }$$

From the resistive Ohm's law ${{\boldsymbol{E}}}_{\perp }=\eta {\rm{\nabla }}\times ({B}_{z}{{\boldsymbol{e}}}_{z})$, we get

$$\begin{eqnarray}&&{E}_{x}=-\displaystyle \frac{2{a}^{2}y}{{({a}^{2}+4\eta t)}^{2}}\exp \left(-\displaystyle \frac{{x}^{2}+{y}^{2}}{{a}^{2}+4\eta t}\right),\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{E}_{y}=\displaystyle \frac{2{a}^{2}x}{{({a}^{2}+4\eta t)}^{2}}\exp \left(-\displaystyle \frac{{x}^{2}+{y}^{2}}{{a}^{2}+4\eta t}\right).\end{eqnarray} \tag{ 19 }$$

Faraday's law then gives

$$\begin{eqnarray}{\partial }_{t}{B}_{z} & = & 4{a}^{2}\eta \left[\displaystyle \frac{{x}^{2}+{y}^{2}}{{({a}^{2}+4\eta t)}^{3}}-\displaystyle \frac{1}{{({a}^{2}+4\eta t)}^{2}}\right]\\ & & \times \exp \left(-\displaystyle \frac{{x}^{2}+{y}^{2}}{{a}^{2}+4\eta t}\right).\end{eqnarray} \tag{ 20 }$$

For this exercise, we fix a = 0.5 and $\eta t=0.1$. The solution is shown in the top row of Figure 7.

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It is important to note that, in this case, Ohm's law is compatible with a purely inductive solution, because ${{\boldsymbol{E}}}_{\perp }$ from Ohm's law has the inductive form with ${\rm{\Phi }}(x,y,t)\,=\eta {B}_{z}(x,y,t)$. Indeed, the middle row of Figure 7 shows the inductive solution to reproduce the target ${{\boldsymbol{E}}}_{\perp }$ in this case. However, the bottom row shows that the sparse solution using basis pursuit does not converge to the target. Rather, it favors a concentration of E_x along vertical lines, and E_y along horizontal lines. This behavior is typical when the target ${{\boldsymbol{E}}}_{\perp }$ is too diffuse. It is not possible for the total spatial extent of ${{\boldsymbol{E}}}_{\perp }$ to be more localized than ${\partial }_{t}{B}_{r}$, owing to the constraint of satisfying (2). But it is possible for more of the electric field to be concentrated in thinner regions, and this is what minimizing the ℓ₁-norm does.

This limitation is an obvious one, and limits the sparse solution technique to input maps where ${\boldsymbol{B}}$ is sufficiently localized. Fortunately, this situation is typical on the Sun, given high enough resolution. The flux-transport test case in Section 4.2 showed the method to work for realistic solar magnetic maps at the resolutions of present-day simulations. However, there is another more subtle problem with real data, which we describe in the following section.

For a localized ${{\boldsymbol{E}}}_{\perp }$ to exist, it is necessary not only that ${\partial }_{t}{B}_{r}$ is localized, but that the net flux ${\int }_{S}{\partial }_{t}{B}_{r}\,{dA}$ vanishes over the region S of localization. To see this, apply Stokes' theorem on some closed curve that encircles S. If the net flux in S is nonzero, then there must be nonzero ${{\boldsymbol{E}}}_{\perp }$ somewhere on the bounding curve, and indeed on any curve enclosing nonzero net flux.

This condition of vanishing net flux is satisfied by both of our simple examples in (14) and (20), and by every local flux concentration in the flux-transport test case (Section 4.2). But it need not be satisfied in an observationally derived map of ${\partial }_{t}{B}_{r}$, even if a flux correction has been applied. We will demonstrate this problem first in a controlled test and second in a "real" data-assimilative model.

For the controlled test, we modify the input map from Section 4.2 to simulate the typical situation where new magnetogram observations are assimilated only within a finite region on the visible face of the Sun. The most severe problems of flux imbalance occur when a new active region has emerged on the far-side of the Sun, and is gradually assimilated into the magnetic map as it rotates into view.

Figure 8 shows our test. Here the original ${\partial }_{t}{B}_{r}$ map, from Figure 5(c), has been modified to simulate the gradual assimilation of a new bipolar active region. In the right-hand column of Figure 8, the assimilation region is indicated on the modified ${\partial }_{t}{B}_{r}$ maps. Each row corresponds to a different time as the new active region rotates into view, with the times differing by one day of solar rotation (i.e., $27\buildrel{\circ}\over{.} 2753$ in these Carrington maps).

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When the new region is fully contained within the assimilation window (bottom row of Figure 8), it makes only a localized contribution to the sparse ${{\boldsymbol{E}}}_{\perp }$, correctly reproducing ${{\boldsymbol{E}}}_{\perp }$ as in Section 4.1. But when the region is only partially observed, there is an unbalanced flux of ${\partial }_{t}{B}_{r}$, both in the assimilation window and in the entire map. Correspondingly, it is impossible to find a localized electric field and the sparse solution fails. In the area of the new region, we see similar linear structures to the diffusive example in Figure 7, where the flux was also not locally balanced. But notice that the electric field is modified not only near the new region, but also at farther distances.

Of course, this is a rather unrealistic test. If ${{\boldsymbol{E}}}_{\perp }$ was being used to drive a coronal MHD model, for example, one would correct the ${\partial }_{t}{B}_{r}$ map for flux balance, before trying to compute ${{\boldsymbol{E}}}_{\perp }$. In Figure 9 we show the effect on E_θ of first making a flux-balance correction to the ${\partial }_{t}{B}_{r}$ map (from the top row of Figure 8). The middle and right plots show the results with two different methods of flux correction. In the "additive" method, an equal amount is subtracted from each pixel in the 360 × 180 grid to remove the imbalance. In the "multiplicative" method, all the pixels where ${\partial }_{t}{B}_{r}\gt 0$ are multiplied by a constant factor ${f}_{+}$, and all pixels where ${\partial }_{t}{B}_{r}\lt 0$ are multiplied by another constant factor ${f}_{-}$. The factors ${f}_{+}$, ${f}_{-}$ are chosen so that the net flux vanishes after scaling, and is equal to the mean of the positive and negative fluxes before scaling. Comparing the results in Figure 8, it is evident that neither method of flux correction gives much improvement in the reconstructed E_θ (or E_ϕ). Only by limiting the flux correction to the active region itself could the solution be improved.

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Similar behavior is found when we apply the sparse reconstruction technique to a time sequence of B_r maps generated by the Air Force Data-Assimilative Photospheric Flux Transport (ADAPT) model (Arge et al. 2010; Henney et al. 2012). This model assimilates observed magnetograms from the visible face of the Sun into a surface flux-transport simulation, so as to approximate the global distribution of B_r on the solar surface, as a function of time. The data assimilation means that the electric field ${{\boldsymbol{E}}}_{\perp }$ is not known everywhere in the map, so must be reconstructed if these maps are used to drive coronal MHD simulations.

As an illustration, we choose an ADAPT run driven by GONG magnetograms, and extract ${\partial }_{t}{B}_{r}$ for 2014 November 16, 00:00 UT. This particular date was chosen because Weinzierl et al. (2016) found a significant nonlocalized ${{\boldsymbol{E}}}_{\perp 0}$ caused by reassimilation of a large active region (see their Figure 11). Here, we process the ADAPT data by (i) applying a multiplicative correction for flux balance; (ii) applying a spatial smoothing to the maps; and (iii) remapping to a 360 × 180 grid in ϕ and $\cos \theta$. The time derivative ${\partial }_{t}{B}_{r}$ is then estimated using a Savitzky–Golay smoothing filter (of total width 18 hr), generating the map shown in Figure 10. The position of the data-assimilation window is evident in the map of ${\partial }_{t}{B}_{r}$, with the largest contribution coming from the main active region AR12209. (This is not newly emerging, but has significantly changed its structure since it was last observed on the previous Carrington rotation.)

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In Figure 11, we show the inductive and sparse solutions for ${{\boldsymbol{E}}}_{\perp }$ in this case. Because there is such a dominant source of ${\partial }_{t}{B}_{r}$, there are significant nonlocal electric fields in the inductive solution, which is in contravention of Ohm's law. The sparse E_ϕ is rather better, but the sparse E_θ is still not correctly localized. The cause of this is the flux imbalance in the original ADAPT map for ${\partial }_{t}{B}_{r}$, as in the previous test (Figure 8). Again the global flux correction has not really helped the problem. We conclude from these tests that lack of local flux balance is the main obstacle to driving coronal MHD models with data-assimilative magnetic maps.

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