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Understanding how coronal structure propagates and evolves from the Sun and into the heliosphere has been thoroughly explored using sophisticated MHD models. From these, we have a reasonably good working understanding of the dynamical processes that shape the formation and evolution of stream interaction regions and rarefactions, including their locations, orientations, and structure. However, given the technical expertize required to produce, maintain, and run global MHD models, their use has been relatively restricted. In this study, we refine a simple Heliospheric eXtrapolation Technique (HUX) to include not only forward mapping from the Sun to 1 AU (or elsewhere), but backward mapping toward the Sun. We demonstrate that this technique can provide substantially more accurate mappings than the standard, and often applied “ballistic” approximation. We also use machine learning (ML) methods to explore whether the HUX approximation to the momentum equation can be refined without loss of simplicity, finding that it likely provides the optimum balance. We suggest that HUX can be used, in conjunction with coronal models (PFSS or MHD) to more accurately connect measurements made at 1 AU, Stereo-A, Parker Solar Probe, and Solar Orbiter with their solar sources. In particular, the HUX technique: 1) provides a substantial improvement over the “ballistic” approximation for connecting to the source longitude of streams; 2) is almost as accurate, but considerably easier to implement than MHD models; and 3) can be applied as a general tool to magnetically connect different regions of the inner heliosphere together, as well as providing a simple 3-D reconstruction.

Plasma is heated in the corona and accelerates away from it to form the solar wind. It is convenient (although probably an oversimplification) to separate what we believe to be intrinsically spatial variations from temporal variations [

Being able to connect

[

Several studies have leveraged the HUX technique to improve space weather forecasts of solar wind streams. [

The HUX technique is also being incorporated into space weather programs. [

The HUX technique has also been generalized to study time-dependent phenomena [

In this study, we describe a simple refinement to the HUX allowing the user to map solar wind streams from 1 AU (or elsewhere) back to the Sun, which can have a substantial impact on the inferred source longitudes of the observed plasma. Additionally, using a data-driven sparse regression method, we explore whether there are any simple improvements that can be applied to the HUX approach. Finally, we explore the possible impacts of including differential rotation, as well as any constraints imposed by resolution.

Here, we introduce the three main modeling techniques that are applied in our study. Specifically: 1) ballistic mapping; 2) MHD modeling; and 3) the HUX technique.

The simplest (but least useful) approximation we could make for evolving the speed of the solar wind as it propagates away from the Sun is to assume that it does not change speed in response to dynamical interactions between adjacent parcels of plasma. This has a number of obvious problems, perhaps largest of which is that in mapping the speeds out, it becomes possible for faster parcels to outrun slower ones. By virtue of the fact that the solar wind can be reasonably approximated as a fluid, this is patently nonphysical. When the procedure is instead applied in the reverse direction, this can lead to the well known problems of “dwells”, where parcels of plasma observed at a later times in the solar wind map back to earlier launch times at the Sun, again by apparently “crossing paths” [

In spite of its simplicity, for completeness, we formally define the ballistic approximation as:

For our numerical experiments, we use the Magnetohydrodynamic Algorithm outside a Sphere (MAS) code, which solves the time-dependent resistive magnetohydrodynamic (MHD) equations (e.g., [

In an earlier study [

Briefly, the solar wind motion can be described as the fluid momentum equation in a corotating frame of reference:_{s} is the mass of the Sun, and

As discussed in more detail in paper 1, we can apply an acceleration term to the forward mapped speeds of the form:

Similarly, by inspection, we can define a deceleration term for the HUX-b mapping as follows:

As in paper 1, we use a global MHD solution as the “ground truth” for validating the more approximate HUX results. We focus again on CR 2068, but also add a number of other Carrington rotations to the analysis, to ensure that we do not over-generalize the results from one solution. In

Probability density (histogram) of the difference between the ballistically mapped longitudes and the MHD-mapped longitudes of the field lines from the inner boundary at 30R_{s} to 1 AU. The blue blocks show the actual frequency while the orange curve provides a Gaussian fit to the data. The Gaussian fit is used only as a practical way to estimate the spread in the values, and not to suggest that the errors are normally distributed.

In

To more quantitatively assess the difference between the MHD mapping and the ballistic mapping, in ^{−1}, the errors would be substantially larger. It should be noted, however, that when mapping out to different radial distances, the associated errors would be proportional to the distance mapped. Between, Solar Orbiter and Parker Solar Probe (PSP), for example, when separated by fractions of an AU, the errors may be considerably smaller.

Before doing this, however, it is useful to assess how much evolution has taken place from the inner radial boundary to 1 AU in the MHD model. In

We explore the errors more quantitatively in

Estimate of the error in the HUX-f mapping as compared to the MHD result at 1 AU. Positive errors are shown in blue, negative errors are red, and an exact match appears as white.

As a final consideration of the errors associated with the mapped speeds, in

Point by point scatter plot between MHD and HUX-f results.

We can also map out field lines using the HUX-f technique and compare them with the MHD results. In

Again, to more quantitatively assess the difference between the MHD mapping and the HUX-f mapping, in

Probability density (histogram) of difference between the ballistically mapped longitudes and the MHD-mapped longitudes of the field lines. The blue blocks show the actual frequency while the orange curve provides a Gaussian fit to these data.

Finally, we can repeat the analysis using the HUX-b technique. In this case, we start the mapping at 1 AU and draw the field lines back to the Sun. In

Considering the distribution in the errors, in

Probability density (histogram) of difference between the ballistically mapped longitudes and the MHD-mapped longitudes of the field lines. The blue blocks show the actual frequency while the orange curve provides a Gaussian fit to these data.

The HUX approach is simple. In fact, it is probably the simplest physically based improvement to a ballistic approximation that could be made. As noted earlier, several refinements have been made, principally to better address the mixing of temporal and spatial variations at the source. However, here we would like to ask specific questions. First, does differential rotation affect the tuning of the free parameters in the HUX model? Second, does the inclusion of more terms from the momentum equation would improve the approach? And third, is the HUX mapping sensitive to grid resolution?

In paper 1, we estimated the two free parameters of the HUX technique (

To estimate the error associated with these solutions, we define the residuals between the MHD model (the “data”, or “ground truth” in this case) and the HUX model to be:

As shown in paper 1, we could reduce the momentum equation to the inviscid Burgers’ equation under the assumption that the magnetic field, pressure gradient, and gravity can be neglected. But, to what extent is this a reasonable approximation, or, stated another way, what errors are likely introduced? As a secondary question, we can ask whether there is a more speculative formalism that might produce better mappings of the streams from the Sun to 1 AU?

To investigate this, we consider the following generalized expression for

To fully explore the parameter space defined by

The details of the analysis are provided in the supplemental information (SI, GitHub), but briefly, we used ridge sparse regression to find the optimal number of terms:

Finally, it is worth noting that PDE-FIND is a physics-agnostic approach, thus, our inclusion of various combinations of first and second-order partial derivatives of was a useful test for validating the approach since it correctly and independently identified the term

Although the HUX approach is extremely simple, it is still derived from a partial differential equation, and subject to potential convergence issues, such as the Courant–Friedrichs–Lewy (CFL) condition. For the HUX technique, this requires that:

Comparison of MHD model speeds with HUX-f solutions of different radial resolution. In this case

In this study, we have further developed the Heliospheric Upwinding eXtrapolation (HUX) technique. Specifically, we have: 1) Demonstrated how the approach can be used to map solar wind streams back to the Sun; and 2) Shown that the current formalism is probably as complicated as it needs to be to produce useful results. In addressing these questions, we were able to show how much the HUX technique would improve the accuracy of ballistic mapping studies, which are typically used to identify the source regions of solar wind streams or energetic particles, for example.

This study is not without limitations. In particular, we used MHD model results as the “ground truth”. Although this is reasonable in the sense that we have no other global dataset of solar wind speeds, it assumes that the MHD formalism is accurate enough to evolve solar wind streams through the heliosphere. This, of course breaks down at sufficiently high frequencies. Thus we can only claim that the HUX approach is reasonable on the largest spacial and temporal scales (i.e., macroscopic structure). Additionally, any artifacts introduced by the MHD model, such as numerical diffusion, which are also mimicked by the HUX technique would also contribute to a higher accuracy of the results, but which might not exist in practice. However, given the many studies that have validated the MHD approach for studying solar wind evolution (e.g., [

It is worth noting that our application of PDE-FIND included the viscous term (

In closing, we reiterate that in this study, we have focused on the procedure that could be applied to various

The model results used in this study can be found in the HUX repository, located at

PR developed the theoretical formalism for the ideas presented here, with contributions from OI. OI wrote the Python code to test the concepts with minor contributions from PR. PR wrote the paper with input from OI.

The authors gratefully acknowledge support from NASA (80NSSC18K0100, NNX16AG86G, 80NSSC18K1129, 80NSSC18K0101, 80NSSC20K1285, 80NSSC18K1201, and NNN06AA01C)

Authors PR and OI were employed by Predictive Science Inc.

Additionally, the authors would like to express their gratitude to both reviewers for constructive comments that improved the clarity of the manuscript.