The Plasma-prescribed Active Region Static Extrapolation (PARSE) Dataset: A Machine-Learning-Ready Collection of Magnetohydrostatic Coronal Active Regions

The SHARP dataset labels each active region as it rotates onto the face of the solar disk wtih a SHARP number. These active regions are then imaged with a six-minute cadence at high resolution. However, many statistical or machine-learning-based use cases of the PARSE dataset will require the samples to be independent of each other, and the SHARP images of a particular active region are closely correlated in time. To guarantee robustness against this temporal correlation in the dataset, we restrict ourselves to only one time frame from each SHARP number. This may result in more than one of a given active region, since they are re-numbered if they survive rotation across the far side of the sun, but we anticipate such a long cadence in potential repetitions to remediate any potential temporal correlation in the active region.

For each active region, we first discount any time frames whose flux-weighted centers are more than $60\degree$ (Stonyhurst) latitude or longitude away from disk center. Of the remaining frames (if any), the upper quartile are considered by total unsigned flux. Finally, the timeframe for modeling is chosen to be the one of these which is closest to disk center. In this way, we hope to obtain the best possible representative from each active region element to image and extrapolate.

The numerical model uses a cartesian coordinate system wherein $\hat{z}$ is the vertical direction ($z=0$ is defined as the photosphere), and $\hat{x}$ and $\hat{y}$ are the transverse components. We use the Disambiguated Lambert Cylindrical Equal-Area Projection vector field, which gives $B_{r}$, $B_{\theta}$ and $B_{\phi}$, which is provided by the SHARP repository directly. This disambiguation can include errors and assumptions, but for the purposes of this dataset they will be taken as given to allow easier comparison for the user with the original dataset.

The coordinate system in the SHARP is mirrored from the one in the model. To accommodate this, first the SHARP image is flipped vertically. Then $B_{r}$ is mapped directly to $B_{z}$, $B_{\theta}$ is mapped to $-B_{x}$ and $B_{\phi}$ to $-B_{y}$.

A great power of the forward model we leverage for the extrapolation is its complete agnosticism with respect to computational node layout. It is not restrained to any kind of grid layout. We aim to take advantage of this property to cluster nodes near the spatially dynamic areas in the active region volume. To that end, we scatter nodes within the domain nearer high-flux regions of the boundary, while simultaneously scaling node density exponentially with height to cluster them near the lower boundary.

A novel methodology to scatter nodes variably in space while retaining quasiuniform clustering has been the focus of recent research, and we apply state of the art advancing-front technique per van der Sande & Fornberg (2021). A zoom-in on a portion of the domain of one simulation is given in Figure 1.

The clustering density of the method depends on an exclusion radius; the higher the function value, the sparser the node placement. We scale this quantity according to

where $\tilde{B}_{z}$ is a smoothed version of $|B_{z}-\text{median}(B_{z})|$ constructed via a maximal binning window and normalized to have a maximum value of $1$, and $L$ is the ratio of the length of the longer transverse side of the domain to the shorter one (the domain is computationally normalized for the shortest side to be length $1$). Note that this algorithm generates nodesets with variable total numbers of nodes; $100,000$ is an upper bound for this quantity which is occasionally obtained, but most members of the PARSE dataset fall slightly below that threshold.

The active region is extrapolated as a solution to the magnetohydrostatic equations, namely

We consider this a heterogeneous forced equation in terms of the conservative plasma forcing field $\mathbf{F}:=\nabla P+\rho g\hat{z}$.

The numerical model used for the extrapolations in this repository is described in detail in Mathews et al. (2022), and the curious reader is directed to that work. Here we discuss in detail only the model setup and determination of tunable parameters.

• •

  The photospheric boundary is informed by the SHARP, as described in Section 2.2. Furthermore, a radiative condition is enforced at the upper boundary, $\partial_{z}B_{z}+B_{z}=0$. And the side boundaries have wave-permissible bounary conditions, $\partial_{nn}B_{n}+B_{n}=0$, $\hat{n}$ the normal vector to the given boundary.

• •

  The numerical model uses a fixed hyperviscocity parameter to remediate numerical noise in the solution; a value of $\gamma=10^{-2}$ has been selected for this purpose.

• •

  The model resolves and computes a set of nonphysical “ghost nodes” below the solar surface. These nodes are necessary for the extrapolation technique, but their fields are not considered physical, and they have been omitted from the published data.

• •

  Finally, we find that the auxiliary equation solution included in the original method to remove any small divergence in the field converged poorly on the scattered domain, and has been omitted from calculations. This means that the domain is not necessarily totally free of magnetic divergence, but we note in Section 3 that the effect of this omission is negligible.

The numerical model allows for different selections of $\mathbf{F}$ to be chosen and so poll the different possible topologies of the magnetic field configurations which can be extrapolated from the photospheric observations. However, the choice of plasma is not well-determined from the observations available. Indeed, determination of the correct magnetic field configuration is likely to be partially the task of the machine learning algorithm based on available auxiliary information.

Instead, we select a suite of possible plasma distributions, determined by 100 scattered volumetric collocation points. Such a determination is considered possible if it satisfies the nullspace equation

at the $z=0$ surface. This is accomplished by taking the QR factorization of the transpose of the linear system in (3) and taking a random subset of the columns of $Q$ which correspond to diagonal elements of $R$ below a threshold ($\varepsilon=10^{-3}$ was determined suitable for this purpose). This has the benefit of selecting plasmas which are orthonormal in terms of their volumetric collocation points, allowing more representative sampling of the model output space.

For purposes in which the plasma can be discounted, or the variations in magnetic topology not sufficiently interesting, a non-linear force-free extrapolation is included for each SHARP, obtained with an otherwise identical algorithm (the pressure is just set uniformly to $0$).