To E or Not to E: Numerical Nuances of Global Coronal Models

The ideal MHD solver used in this paper is COCONUT, a solver based on the Computational Object-Oriented Libraries for Fluid Dynamics (COOLFluiD) platform. The latter has been developed over many years for scientific multiphysics simulations (Kimpe et al. 2005; Lani et al. 2006, 2013), including high-speed flows, radiation, thermochemical nonequilibrium, and plasma flows (see, e.g., Lani et al. 2011, 2014; Santos & Lani 2016; Alonso Asensio et al. 2019, and references therein).

The details of the ideal MHD model for global coronal modeling, its verification, and initial validation are presented in Perri et al. (2022), where more details about the numerical setup can be found. Unlike the majority of global coronal models, this solver relies upon an implicit scheme, i.e., the backward Euler time discretization scheme, for steady-state simulations. This allows it to converge much faster compared to explicit schemes, as the Courant–Friedrichs–Lewy (CFL) number can largely exceed the value of 1 (up to 1000 or more, depending on the test case) during the computation. This feature makes it a promising tool for real-time data-driven solar simulations. The solver also works with unstructured grids, which allows us to experiment with grid resolutions and topologies to optimize the solver performance (see Brchnelova et al. 2022). In the future, we also plan to integrate adaptive mesh refinement (Ben Ameur & Lani 2021) and high-order flux reconstruction algorithms (Vandenhoeck & Lani 2019) into this model, which would dramatically increase the accuracy (potentially up to 10th order) on coarser grids and possibly accelerate the convergence of the solver even further.

The code uses a second-order accurate finite volume discretization along with the hyperbolic divergence cleaning to ensure that the divergence of the B -field stays zero. The MHD equations are solved on arbitrarily unstructured grids in a nondimensional form. The full form can be found in Perri et al. (2022), along with the formulation of the baseline BCs. In this study, however, the rotation is not considered. The reference values are set to

$$\begin{eqnarray}&&{\rho }_{\mathrm{ref}}=1.67\cdot {10}^{-13}\,\mathrm{kg}\,{{\rm{m}}}^{-3},\quad {B}_{\mathrm{ref}}=2.2\cdot {10}^{-4}\,{\rm{T}},\end{eqnarray} \tag{ 1 }$$

with V_ref computed to be the Alfvén velocity (4.5 · 10⁵ m s⁻¹), while the length scales and coordinates are nondimensional according to the solar radius.

All of the simulations in this paper, unless otherwise specified, were performed without the use of a limiter and with the fifth subdivision level icosahedron-based spherical mesh with prisms (see Brchnelova et al. 2022 for details) with roughly 300,000 elements. The surface of the spherical shell domain, starting from 1.1 to 21.5 R_⊙, is shown in Figure 1 along with a cut-through showing the radial discretization. When other grids are used in the paper (e.g., the grid designed to minimize outer BC effects introduced later), it will be mentioned explicitly.

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Finally, before proceeding to the physics, the convergence of the solver should be touched upon. When discussing residual levels in the text, the following definition of a residual is used:

$$\begin{eqnarray}&&\mathrm{res}(a)=\mathrm{log}\sqrt{\sum _{i}{\left({a}_{i}^{t}-{a}_{i}^{t+1}\right)}^{2}},\end{eqnarray} \tag{ 2 }$$

where a is the relevant physical quantity, i is the spatial index, and t is the temporal index during the pseudo–time stepping. In general, the velocity and the magnetic field components are monitored during convergence. The simulation is typically considered converged when the velocity residual drops below −3 to −4. For comparison analyses in this paper, however, the residual was brought down to −10 in V_x to ensure that the differences observed in the flow field are indeed due to the investigated aspects (e.g., the BCs) and not to insufficient convergence. It could be argued that looking at the squares of absolute value differences might bias our focus to regions where these values are by default higher, e.g., the outer boundaries, when it comes to the velocity field. For this reason, however, we also monitor the other residuals, like the pressure, density, and magnetic field, where the higher values are generally near the inner boundary instead. Via ensuring that all of the residuals are sufficiently small, the evaluation of convergence is more or less balanced over the domain.

Now that the basic numerics have been discussed, the physics can be elaborated on in further detail.

To outline the role of the electric field and why it should be addressed in the inner BC design, it is instructive to show the full MHD equations considering both ion (subscript "i") and electron (subscript "e") fluids separately, as they both contribute to the generation and removal of the electric field owing to their charges. The resulting two-fluid model, in a dimensional form, can be formulated as follows (see, e.g., Meier & Shumlak 2012). The (pre-)Maxwell equations read

$$\begin{eqnarray}&&{\boldsymbol{\nabla }}\times {\boldsymbol{E}}=-\displaystyle \frac{\partial {\boldsymbol{B}}}{\partial t},\quad {\boldsymbol{\nabla }}\times {\boldsymbol{B}}={\mu }_{0}{\boldsymbol{J}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{\boldsymbol{\nabla }}\cdot {\boldsymbol{B}}=0,\quad {\boldsymbol{\nabla }}\cdot {\boldsymbol{E}}=\sigma /{\epsilon }_{0},\end{eqnarray} \tag{ 4 }$$

where σ is the charge density (σ = n_e q_e + n_i q_i ), and J is the electric current density ( J = n_i q_i V _i + n_e q_e V _e ). For our two species, here ions and electrons, the continuity equations can be written as

$$\begin{eqnarray}&&\displaystyle \frac{\partial {n}_{i}}{\partial t}+{\boldsymbol{\nabla }}\cdot \left({n}_{i}{{\boldsymbol{V}}}_{i}\right)=0\quad \mathrm{and}\quad \displaystyle \frac{\partial {n}_{e}}{\partial t}+{\boldsymbol{\nabla }}\cdot \left({n}_{e}{{\boldsymbol{V}}}_{e}\right)=0,\end{eqnarray} \tag{ 5 }$$

respectively. The momentum conservation is, for electrons,

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial }{\partial t}\left({m}_{e}{n}_{e}{{\boldsymbol{V}}}_{e}\right)+{\boldsymbol{\nabla }}\cdot \left({m}_{e}{n}_{e}{{\boldsymbol{V}}}_{e}{{\boldsymbol{V}}}_{e}+{P}_{e}\right)\\ \quad =\,-{{en}}_{e}\left({\boldsymbol{E}}+{{\boldsymbol{V}}}_{e}\times {\boldsymbol{B}}\right)+{m}_{e}{n}_{e}{\boldsymbol{g}}+{{\boldsymbol{R}}}_{i}^{{ie}},\end{array}\end{eqnarray} \tag{ 6 }$$

and for ions,

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial }{\partial t}\left({m}_{i}{n}_{i}{{\boldsymbol{V}}}_{i}\right)+{\boldsymbol{\nabla }}\cdot \left({m}_{i}{n}_{i}{{\boldsymbol{V}}}_{i}{{\boldsymbol{V}}}_{i}+{P}_{i}\right)\\ \quad =\,{{en}}_{i}\left({\boldsymbol{E}}+{{\boldsymbol{V}}}_{i}\times {\boldsymbol{B}}\right)+{m}_{i}{n}_{i}{\boldsymbol{g}}+{{\boldsymbol{R}}}_{e}^{{ei}},\end{array}\end{eqnarray} \tag{ 7 }$$

where P is the pressure, g is the gravitational acceleration, and ${{\boldsymbol{R}}}_{i}^{{ie}}$ and ${{\boldsymbol{R}}}_{e}^{{ei}}$ are the collisional momentum terms. If the collisional frequency between ions and electrons can be approximated through

$$\begin{eqnarray}&&{\nu }_{{ie}}\sim \displaystyle \frac{{\nu }_{{ei}}}{{N}_{c}}\sim \displaystyle \frac{{m}_{e}}{{m}_{i}}{\nu }_{{ei}}\sim \displaystyle \frac{{Z}^{2}{e}^{4}{n}_{i}{m}_{e}^{1/2}\mathrm{ln}{{\rm{\Lambda }}}_{{ei}}}{{\left(4\pi {\epsilon }_{0}\right)}^{2}{m}_{i}{T}_{e}^{3/2}},\end{eqnarray} \tag{ 8 }$$

with $\mathrm{ln}{\rm{\Lambda }}$ being the Coulomb logarithm and a Z of 1, then the momentum collisional exchange terms can be computed through

$$\begin{eqnarray}&&{{\boldsymbol{R}}}_{i}^{{ie}}=\displaystyle \frac{{m}_{i}{m}_{e}}{{m}_{i}+{m}_{e}}{n}_{i}{\nu }_{{ie}}({{\boldsymbol{V}}}_{e}-{{\boldsymbol{V}}}_{i})\quad \mathrm{and}\quad {{\boldsymbol{R}}}_{i}^{{ie}}=-{{\boldsymbol{R}}}_{e}^{{ei}}.\end{eqnarray} \tag{ 9 }$$

Finally, the energy conservation reads

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial {{\rm{\Psi }}}_{e}}{\partial t}+{\boldsymbol{\nabla }}\cdot \left({{\rm{\Psi }}}_{e}{{\boldsymbol{V}}}_{e}+{{\boldsymbol{V}}}_{e}\cdot {P}_{e}+{{\boldsymbol{h}}}_{e}\right)\\ \quad =\,{{\boldsymbol{V}}}_{e}\cdot \left(-{{en}}_{e}{\boldsymbol{E}}+{m}_{e}{n}_{e}{\boldsymbol{g}}+{{\boldsymbol{R}}}_{e}^{{ei}}\right)+{Q}_{e}^{{ei}},\end{array}\end{eqnarray} \tag{ 10 }$$

and

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial {{\rm{\Psi }}}_{i}}{\partial t}+{\boldsymbol{\nabla }}\cdot \left({{\rm{\Psi }}}_{i}{{\boldsymbol{V}}}_{i}+{{\boldsymbol{V}}}_{i}\cdot {P}_{i}+{{\boldsymbol{h}}}_{i}\right)\\ \quad =\,{{\boldsymbol{V}}}_{i}\cdot \left({{en}}_{i}{\boldsymbol{E}}+{m}_{i}{n}_{i}{\boldsymbol{g}}+{{\boldsymbol{R}}}_{i}^{{ie}}\right)+{Q}_{i}^{{ie}},\end{array}\end{eqnarray} \tag{ 11 }$$

in which Ψ_e and Ψ_i are the total internal energies of the electrons and ions, and ${Q}_{e}^{{ei}}$ and ${Q}_{i}^{{ie}}$ are the collisional work terms between the two species,

$$\begin{eqnarray}&&{Q}_{i}^{{ie}}=\displaystyle \frac{1}{2}\left({{\boldsymbol{R}}}_{i}^{{ie}}\cdot ({{\boldsymbol{V}}}_{e}-{{\boldsymbol{V}}}_{i})\right)+3{n}_{i}{\nu }_{{ie}}{k}_{B}({T}_{e}-{T}_{i}),\end{eqnarray} \tag{ 12 }$$

and

$$\begin{eqnarray}&&{Q}_{e}^{{ei}}=\displaystyle \frac{1}{2}\left({{\boldsymbol{R}}}_{e}^{{ei}}\cdot ({{\boldsymbol{V}}}_{i}-{{\boldsymbol{V}}}_{e})\right)+3{n}_{i}{\nu }_{{ei}}{k}_{B}({T}_{i}-{T}_{e}).\end{eqnarray} \tag{ 13 }$$

In the model presented above, the electric field E is typically one of the primitive variables that is solved for. In Equations (10) and (11), we can see, for example, the tendency of the two-fluid MHD to respond to existing electric fields via their dynamics, as the two fluids react to the present E -field in opposite ways. Charge separation, which can result from this response, is, by itself, a mechanism to create an opposing electric field; see Equations (3) and (4). Looking at Equations (10) and (11), the E -field can also be a major contributor to the fluids' energy budgets if it has a nonzero value and is not perpendicular to the fluids' velocity field.

However, the form of the MHD equations that we generally use in global coronal models is simplified through a variety of assumptions to the single-fluid ideal MHD equations. In particular, assuming nonrelativistic flows, the displacement current and electrostatic force are ignored in MHD. As a result, the electric field and current density adopt a secondary role, as they can be eliminated from the equations; i.e., they can simply be computed from the other variables without solving a differential equation ( E = − V × B and J = ∇ × B /μ₀). The MHD equations then read (here in a nondimensional form, as also used in our code)

$$\begin{eqnarray}&&\displaystyle \frac{d\rho }{{dt}}+{\boldsymbol{\nabla }}\cdot (\rho {\boldsymbol{V}})=0,\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d(\rho {\boldsymbol{V}})}{{dt}}+{\boldsymbol{\nabla }}\cdot \left(\rho {\boldsymbol{V}}\otimes {\boldsymbol{V}}+{\boldsymbol{I}}\left(P+\displaystyle \frac{{{\boldsymbol{B}}}^{2}}{8\pi }\right)-\displaystyle \frac{{\boldsymbol{B}}\otimes {\boldsymbol{B}}}{4\pi }\right)=\rho {\boldsymbol{g}},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d{\boldsymbol{B}}}{{dt}}+{\boldsymbol{\nabla }}\times \left(-{\boldsymbol{V}}\times {\boldsymbol{B}}\right)={\bf{0}},\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{d}{{dt}}\left(\rho \displaystyle \frac{{{\boldsymbol{V}}}^{2}}{2}+\rho { \mathcal E }+\displaystyle \frac{{{\boldsymbol{B}}}^{2}}{8\pi }\right)\\ \quad +\,{\boldsymbol{\nabla }}\cdot \left[\left(\rho \displaystyle \frac{{{\boldsymbol{V}}}^{2}}{2}+\rho { \mathcal E }+P\right){\boldsymbol{V}}-\displaystyle \frac{1}{4\pi }({\boldsymbol{V}}\times {\boldsymbol{B}})\times {\boldsymbol{B}}\right]\\ \quad =\,\rho {\boldsymbol{g}}\cdot {\boldsymbol{V}},\end{array}\end{eqnarray} \tag{ 17 }$$

where ${ \mathcal E }$ is the nondimensional internal energy, and I is the identity dyadic.

Assuming plasma with negligible resistivity, using the definition of E as being the cross product of the magnetic and velocity fields, Equations (16) and (17) of ideal MHD can be turned into

$$\begin{eqnarray}&&\displaystyle \frac{d{\boldsymbol{B}}}{{dt}}+{\boldsymbol{\nabla }}\times \left({\boldsymbol{E}}\right)={\bf{0}}\end{eqnarray} \tag{ 18 }$$

and

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{d}{{dt}}\left(\rho \displaystyle \frac{{{\boldsymbol{V}}}^{2}}{2}+\rho { \mathcal E }+\displaystyle \frac{{{\boldsymbol{B}}}^{2}}{8\pi }\right)\\ +\,{\boldsymbol{\nabla }}\cdot \left[\left(\rho \displaystyle \frac{{{\boldsymbol{V}}}^{2}}{2}+\rho { \mathcal E }+P\right){\boldsymbol{V}}-\displaystyle \frac{1}{4\pi }(-({\boldsymbol{E}}))\times {\boldsymbol{B}}\right]\\ \quad =\,\rho {\boldsymbol{g}}\cdot {\boldsymbol{V}}.\end{array}\end{eqnarray} \tag{ 19 }$$

Equations (18) and (19) show the implicit inclusion of the E -field in ideal MHD. While in this sense, the E -field is included in the model, these MHD equations do not account for its absolute value but only for its curl and the divergence of its cross product with the B -field. Unlike the full multifluid model, in the ideal MHD equations without resistivity, there are no mechanisms included that would set constraints on the amplitude of the electric field in case it is accidentally artificially generated in the simulation during convergence, as already pointed out by Sokolov et al. (2021).

Starting with a zero E -field at the beginning of the simulation, the E -field during the run will relax such that Equations (18) and (19) are met. The contribution to the E -field will thus come from its curl and the divergence of its cross product with the B -field. In principle, in steady cases, the former should not contribute to the rise of the E -field, since ∂ B /∂t is zero. Equivalently, if the E -field was nonexistent and remained nonexistent during convergence, the (−( E )) × B term would also be zero, regardless of the B -field orientation. This would remove the contribution of this term to the energy budget (and thus possibly minimize the system's energy).

However, a steady magnetic field is not something that time-stepping codes compute with, even for steady cases. The magnetic field starts at some distribution given by the initial state and keeps adjusting and changing during the time stepping while converging to the steady state, giving rise to ∂ B /∂t in between the iteration steps. This also means that the electric field will attain a certain distribution to satisfy the changing magnetic field. With the ideal MHD formulation, as shown above, the curl of this resulting E -field will be close to zero to satisfy Equation (18), and its orientation with respect to the B -field will be such that Equation (19) is met, but there are no constraints on its actual value.

It should be emphasized that since the E -field in this model represents the misalignment between the magnetic field and the velocity field, its value affects how well the plasma flow will follow the resolved magnetic field lines. Since the V × B product enters the energy equation, ultimately, this does not only directly affect the B - and V -fields but, as will be shown later, also other hydrodynamic variables, such as temperature.

Let us first look at the major sources of the E -field in the simulations. Since the domain investigated here is spherical, the analysis from here on will be interpreted using the spherical coordinate system, r, θ, and φ. During our analysis, the main possible sources were identified as follows:

• 1.  
  the initial conditions,
• 2.  
  the inner and outer BCs,
• 3.  
  internal physical processes such as reconnection, and
• 4.  
  the mesh topology and domain discretization.

The first two points will be primarily discussed in this paper. The third point represents an actual physical process and is thus not something numerical adjustments can (or should) influence. As will be shown later in this paper, in some cases, premature magnetic reconnection can actually be triggered by the other aspects, and in such a case, it can also be considered as the result of the artificial electric field rather than its source. The last point refers to the fact that the currently applied mesh, created as a radially outward extruded icosahedral surface, has local patches of larger distortion, skewness, and nonorthogonality. This can lead to the generation of artificial fluxes, giving rise to mesh-related remnants in the solution. To first separate these mesh-related artifacts, a technique was employed in which after the first simulation, the mesh was rotated (here by 30° in the positive φ direction) and the simulation repeated (see the procedure in Brchnelova et al. 2022). This means that in the second simulation, the mesh artifacts occur at different locations in the domain and thus can be easily identified and corrected for, if needed, using interpolation from the two solutions at the problematic locations.

In our case, through this technique, it was found that the radial and azimuthal components of the electric field in the solution of the steady dipole were mostly related to the mesh. An example of the E_θ remnants is shown in Figure 2, where the electric field changes polarity once the mesh is rotated by 30° in the positive φ direction. The same was observed for E_r . In general, both of these components had very small values compared to the total E -field magnitude. This is expected when looking at the magnetic field of the nonrotating dipole (which has a negligible φ-component) and Equation (17). These two components will thus not be analyzed extensively from here on.

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However, the E_φ (poloidal) component was a hundredfold larger and did not change polarity with the rotation of the mesh (see Figure 3), demonstrating that this was created due to things other than the mesh. As it is the dominant component, it is the focus of our analysis in the rest of the paper, as far as the dipolar field is concerned.

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The evolution of E_φ during the transient in one of the simulations is shown in Figure 4. The features that were introduced at the zeroth iteration (left; green arrows), created at the initial state, largely remain in the simulation even after 100 iterations (as shown in the right panel). In this case, the reason the initial state had a nonzero E -field is because the initial velocity profile came from a piecewise linear fit of Parker's wind solution, while the initial B -field came from a potential field source surface (PFSS) solution, and these two were not parallel. Furthermore, an additional E_φ contribution coming from the BC develops around the inner boundary during the iterative process toward steady state, as highlighted in the right panel of Figure 4 with blue arrows. What is shown in Figure 4 already indicates two possible strategies for reducing the artificial E -field, namely by (1) ensuring that the initial state is completely E -field-free (set V ₀ such that V ₀ × B _PFSS = 0) and (2) adjusting the inner BC to produce a minimum contribution to the E -field.

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While the fact that the E -field exists and evolves in the simulations has been shown, what it actually means for the steady-state solution has not yet been demonstrated. Whether or not some E -field exists at the initial state depends heavily on the solver, and it is an issue that can be easily fixed in most cases. Therefore, this aspect is not discussed further. Instead, focus will be placed on the formulation of the inner BC.

In this subsection, we will consider two separate cases. The simulation that was shown in the previous paragraphs, with the BC producing the E -field, is the first case, indicated here as the "large E " simulation. The BC used here was taken from the code Wind-Predict (Réville et al. 2015; Perri et al. 2018) and is one of the standard formulations for global coronal models. In this setup, on the inner boundary, it is assumed that the radial and azimuthal components of the B -field can be derived from the PFSS solution, originally expressed in Cartesian coordinates:

$$\begin{eqnarray}&&{B}_{r,b}=\displaystyle \frac{{x}_{b}}{{r}_{b}}{B}_{x,\mathrm{PFSS}}+\displaystyle \frac{{y}_{b}}{{r}_{b}}{B}_{y,\mathrm{PFSS}}+\displaystyle \frac{{z}_{b}}{{r}_{b}}{B}_{z,\mathrm{PFSS}},\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{B}_{\theta ,b}=\displaystyle \frac{{x}_{b}{z}_{b}}{{\rho }_{b}{r}_{b}}{B}_{x,\mathrm{PFSS}}+\displaystyle \frac{{y}_{b}{z}_{b}}{{\rho }_{b}{r}_{b}}{B}_{y,\mathrm{PFSS}}-\displaystyle \frac{{\rho }_{b}}{{r}_{b}}{B}_{z,\mathrm{PFSS}},\end{eqnarray} \tag{ 21 }$$

where the b subscript refers to the boundary location, and, in the instances of coordinate transforms, ρ_b represents $\sqrt{{x}_{b}^{2}+{y}_{b}^{2}}$. Then, the velocity condition is formulated such that the poloidal flux is removed. Thus, with

$$\begin{eqnarray}&&{V}_{r,b}^{* }=\displaystyle \frac{1935.07}{({B}_{\mathrm{ref}}/\sqrt{{\mu }_{0}{\rho }_{\mathrm{ref}}})},\quad {B}_{\mathrm{pol}}=\sqrt{{B}_{r,b}^{2}+{B}_{\theta ,b}^{2}}\,,\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&\mathrm{and}\quad {V}_{| | }={V}_{r,b}^{* }\displaystyle \frac{{B}_{r,b}}{{B}_{\mathrm{pol}}^{2}},\end{eqnarray} \tag{ 23 }$$

the radial, azimuthal, and poloidal velocities on the boundary are

$$\begin{eqnarray}&&{V}_{r,b}={V}_{| | }{B}_{r,b},\quad {V}_{\theta ,b}={V}_{| | }{B}_{\theta ,b},\quad \mathrm{and}\quad {V}_{\phi ,b}=0.\end{eqnarray} \tag{ 24 }$$

Herein, the value of 1935.07 m s⁻¹ was the prescribed radial outflow on the inner boundary, as coming from Parker's wind profile. This value was chosen because it is the default value used in the Wind-Predict code, through which our model was originally validated (Perri et al. 2022).

The other simulation (i.e., "low E ") was run with the same setup and initial conditions, apart from the fact that on the inner boundary, the E -field was enforced to be zero in all of its components via

$$\begin{eqnarray}&&{{\boldsymbol{V}}}_{b}\times {{\boldsymbol{B}}}_{b}={\bf{0}},\end{eqnarray} \tag{ 25 }$$

from which the imposed direction of the velocity on the boundary V _b was derived. In addition, while in the previous case, B_θ was set according to the PFSS solution (see Equation (21)), in this setup, this constraint was omitted, and B_θ was determined through a Neumann condition just like B_φ .

These seemingly small adjustments resulted in considerable differences between the results. The radial velocity contours of the two cases are shown in Figure 5. It is apparent that the "large E " case reaches higher velocities on the outer boundary and also creates more diffused profiles of the equatorial streamers. Furthermore, the "low E " simulation has additional features of locally enhanced radial velocity patches near the outer boundary in the equatorial regions. This is a subtle but important difference, since it fundamentally affects the results on the 21 R_⊙ surface, which is where the heliospheric EUHFORIA code is supposed to be coupled and take its input data from. These two simulations would have fundamentally different velocity distributions passed to EUHFORIA, one of them with larger velocities near the poles and the other with enhanced velocity in the region of the heliospheric current sheet.

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The observed velocity patches in the equatorial regions are due to magnetic reconnection. The expected profile of the magnetic field lines around a solar dipole with an outflow can be found, for example, in Pneuman & Kopp (1971). While most of the magnetic field lines remain open, depending on the magnetic field strength, there is a magnetic reconnection forming a current sheet at distances of 1–3 R_⊙ in the equatorial region. This collimated sheet then propagates throughout the heliosphere. In our simulations, however, we see an additional reconnection in this current sheet.

Due to the finite discretization and the presence of some unavoidable artificial diffusion, when the current sheet reaches a certain minimum thickness, it reconnects, creating an "X-point" in the B -field lines, opening these lines up behind it. Where exactly (and if) this happens depends on a variety of factors, such as the BCs, the B -field strength, and the grid resolution. The presence of this X-point and its behavior will be briefly touched upon after the end of this subsection.

How the flow field responds to this X-point is the cause of the qualitative differences in the velocity profiles in Figure 5. In the "large E " case, the velocity field does not follow the magnetic field lines well. Thus, even though the B -field changes topology, the velocity profile remains more or less undisturbed as it propagates outward. As a result, the magnetic reconnection does not show in the final velocity field. For the "low E " run, this is not the case, as here the velocity field follows the magnetic field, so it also responds to the change in its topology. This can be more clearly seen in Figure 6, where the edge of the domain is plotted with V_r in the background and with the B -field lines plotted on the top. Behind the X-point, the B -field lines start to open up.

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This showcased "ballooning" of the streamers due to numerical diffusion is not physical, though it might be stabilizing for most simulations. However, it can present problems later on, when, for example, determining magnetic connectivity; if we trace a magnetic field line from the Earth (or another point of interest in the heliosphere) back to the Sun, it could get trapped in this diverging region instead of finding its way to the solar surface and the corresponding photospheric features. In addition, if we project this B_r on the outer surface, in the regions where the current sheet started to diverge, the sign of the B_r field might be incorrect. Additional postprocessing of CFD results might be required to avoid these problems when coupling to heliospheric codes as a result.

The more diffused profile of the "large E " case, especially around the central region, is shown better as a close-up in Figure 7. Considering that the two simulations use exactly the same grid, and none of them uses a flux limiter (which would include excessive numerical dissipation), minimizing the E -field leads to a substantial improvement in the resolution of the features. This difference can mean that perhaps a coarser grid could be used in production runs while achieving the same accuracy, which would save computational resources.

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In order to visualize how much the E -field has changed between the two cases via the adjustment of the inner BC, in Figure 8, the E_φ is shown. The color bar of each subplot is adjusted to show the main features.

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Some regions of a locally larger E -field are still observed in the "low E " case, mainly following the mesh on the surface. Despite that, the E -field is everywhere in the domain at least 50 times smaller than the E -field of the simulation without the adjusted BC (right). For the improved case, in the domain, the E -field follows the structure of the streamers, which is where it is most difficult for the velocity and magnetic fields to align. In the other case, the E -field structure in the domain is mostly spherical, since it is mostly the inner spherical BC generating it.

The sharpness and magnitude of the velocity field are, however, not the only aspects that were found to change significantly through the reduction of the E -field. It was also found that the density and pressure profiles (and thus also the derived temperature profile) were different both qualitatively and quantitatively. Before the adjustment, the temperature (here computed as the ratio of the nondimensional pressure and density) was enhanced in the regions of strong magnetic field, i.e., around the poles for the case of a dipole; see the right panel of Figure 9. After the adjustment, the temperature was instead following the streamers around the equatorial regions (see the left panel of the same figure).

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The density profile was also found to be more pronounced for the "low E " simulations; see Figure 10. This might significantly aid the accuracy of procedures such as white-light imaging, where the density is the parameter to be integrated along the line of sight.

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Lastly, convergence impacts should be discussed, as computational speed is, next to physical accuracy, a crucial feature of global coronal modeling and any space weather simulation in general. In general, with the new formulation, the steady dipole case required two to three times more iterations in order to reach the same final residual. As we are mostly interested in realistic magnetic map–driven applications, however, a worse convergence of the dipole case is of relatively little importance. When running a variety of real magnetic map–based cases as discussed in Kuzma et al. (2022), it was observed that the adjusted BC led to easier convergence when using maps, especially those of the higher ${l}_{\max }$. For a constant CFL of 2 for the 2008 eclipse used later in this paper, the convergence is compared for the two inner BC formulations in terms of a V_x residual in Figure 11. After the initial jump in the residual, the two simulations converge to the same residual of −4 after roughly the same number of iterations. The initial larger residuals in the run with the adjusted inner BC are due to the changing B_θ , as it is released from the PFSS solution value.

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Before moving into further investigation of the BCs, the issue of the presence of the X-point in our CFD simulations, as mentioned in the discussion of Figure 5, should be addressed. It is also important to discuss this phenomenon because it provides a context for the analysis of the outer BC formulations, mentioned later on in the paper.

The (additional) reconnection of the B -field lines that we have observed is also normal in other numerical codes, not just COCONUT (e.g., see the comparison between our code and Wind-Predict in Perri et al. 2022, where the same behavior is observed in the solution of Wind-Predict). As found during our numerical experiments, beyond the BCs and the limiter, it is mostly dependent on the grid resolution; with increasing resolution, the reconnection point occurs further away from the Sun. Table 1 shows the location of this reconnection point (in terms of solar radii) for four different grids.

• 1.  
  The coarse/coarse 300,000 grid: 5120 surface elements, 65 radial steps.
• 2.  
  The coarse/fine 980,000 grid: 5120 surface elements, 192 radial steps.
• 3.  
  The fine/coarse 1.3 million grid: 20,480 surface elements, 65 radial steps.
• 4.  
  The fine/fine 3.9 million grid: 20,480 surface elements, 192 radial steps.

In Table 1, levels are used in order to express the longitudinal/latitudinal discretization. Level 5 refers to the fifth-level subdivision of the elementary icosahedron from which the mesh is derived with 5120 surface elements. Level 6 refers to the sixth-level subdivision with 20,480 surface elements.

By simply refining the domain (without any changes to the physics of the simulation), from Table 1, it can be observed that the X-point moves almost twice the distance away from the solar surface. Thus, it can be inferred that the reconnection occurs mainly due to the finite discretization, with its location depending on the amount of numerical diffusion in the domain.

In addition, it was found that the behavior and shape of the current sheet (and thus also the location of the reconnection point) also depend on the amount of the B -field divergence in the flow field. By default, in our simulations, for the B -field divergence cleaning constant, we use a value of 1, leading to the amount of divergence of B to be on the order of −4 to −5 (nondimensional). With lower values of the cleaning constant, it was observed that the amount of divergence of the B -field decreased; see Figure 12, where the logarithm of the absolute value of ϕ (the divergence cleaning parameter) is shown. The BC treatment for ϕ (homogeneous Neumann or Dirichlet) on the inner boundary does not significantly affect this distribution, and in the default configuration, ϕ is set to zero (i.e., homogeneous Dirichlet).

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However, decreasing the value of the cleaning constant below a certain threshold destabilizes the simulation. The particular value that causes this most likely depends on the specific scheme, the mesh, and the BCs; in our case, it is around a value of 0.3. The simulation reaches a certain minimum residual, following which the residual increases again and starts to oscillate indefinitely. What this means for the simulation is that the current sheet becomes unstable. The reconnection region starts to move forward and backward between iterations (on the current mesh, generally oscillating between 5 and 10 R_⊙), creating lumps of mass that are ejected outward. To illustrate this, the projection of the radial velocity on the outer surface (where the passing lumps are the most visible) is shown in Figure 13. We should note that this kind of unstable structure is actually visible in the observations of the solar wind, although they appear for more physical reasons (reconnection along the Harris current sheet that ejects flux ropes and density structures; see Réville 2020).

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Thus, it is apparent at this point that both the numerical dissipation due to the finite discretization and the amount of ∇ · B in the domain affect the shape and behavior of the current sheet. As we aim at developing software that will eventually have standardized and automated handling of magnetic map–based simulations, the abovementioned oscillating behavior in the residual is not desirable, even though at certain phases, the reconnection point is located much further away from the Sun than without it. When the residual starts oscillating, it becomes difficult to infer the state of convergence of the solution without any visual inspection. Since the higher values of the cleaning constant give quantitative results comparable to the time-averaged values of the oscillating solution, it was decided to keep the value of the cleaning constant to 1 for all of our simulations.

Even though we can now explain the behavior and origin of the X-point, its presence still complicates matters for the reasons outlined earlier, such as coupling to EUHFORIA or magnetic connectivity studies. It is thus important to lower the amount of numerical diffusion as much as computationally feasible and move this reconnection point as close to the outer boundary as we can, which can, to some extent, be realized through the outer BCs.

As pointed out in the Introduction, the formulation of the outer BC in the available literature is often reduced to calling it a simple extrapolation. It is indeed clear that, since the outer boundary is supersonic, the hydrodynamic variables cannot be prescribed but must be extrapolated from the inner states. However, this extrapolation can be formulated in various ways. The prescription can simply be done through a zero-gradient extrapolation (i.e., homogeneous Neumann condition) or the extrapolation of the given variable via a function of the radius or density. To our knowledge, the effects that these different formulations have on the behavior of the code or shape of the solution have not yet been extensively discussed in the available literature. For this reason, a parametric study was performed to analyze these effects in a systematic way.

All of the primitive variables of the code, ρ, B_x , B_y , B_z , V_x , V_y , V_z , p, and ϕ (the divergence cleaning parameter), were extended via either zero gradient (the ghost state being equal to the inner state, where the distances of the inner cell center and the ghost cell center to the boundary are equal), a function of radius or density, or both. The formulations of the specific functions for the velocity and magnetic field were taken (apart from the last expression in Equation (26)) from Talpeanu et al. (2022). The following expressions for density

$$\begin{eqnarray}&&\rho =\mathrm{cst}.\quad \mathrm{or}\quad \rho {r}^{2}=\mathrm{cst}.\quad \mathrm{or}\quad \rho {r}^{3}=\mathrm{cst}.,\end{eqnarray} \tag{ 26 }$$

velocity

$$\begin{eqnarray}&&{V}_{r}=\mathrm{cst}.\quad {V}_{\theta }=\mathrm{cst}.\quad {V}_{\varphi }=\mathrm{cst}.\quad \end{eqnarray} \tag{ 27 }$$

or

$$\begin{eqnarray}&&{V}_{r}{r}^{2}\rho =\mathrm{cst}.\quad {V}_{\theta }=\mathrm{cst}.\quad {V}_{\varphi }r=\mathrm{cst}.,\end{eqnarray} \tag{ 28 }$$

magnetic field

$$\begin{eqnarray}&&{B}_{r}=\mathrm{cst}.\quad {B}_{\theta }=\mathrm{cst}.\quad {B}_{\varphi }=\mathrm{cst}.\quad \end{eqnarray} \tag{ 29 }$$

or

$$\begin{eqnarray}&&{B}_{r}{r}^{2}=\mathrm{cst}\quad {B}_{\theta }=\mathrm{cst}.\quad {B}_{\varphi }r=\mathrm{cst}.\end{eqnarray} \tag{ 30 }$$

and ϕ

$$\begin{eqnarray}&&\phi =\mathrm{cst}.\quad \mathrm{or}\quad {\phi }_{b}=0,\end{eqnarray} \tag{ 31 }$$

were combined into nine different cases, as summarized in Table 2. In the Table, "G" refers to the ghost state, "I" to the inner state, and "B" to the location on the boundary. To achieve continuity in temperature, the pressure was extrapolated in the same way as the density. The magnetic configuration was again dipolar for easier evaluation. All of these cases were simulated with the same CFL law (starting from 2 with the CFL doubling every few hundred iterations up to 128) for controlling the convergence to steady state, making comparisons between them possible.

Table 2. The Setup of Nine Different Dipole Simulation Cases with Varying Outer BC Settings

No.	ρ	Br	Bθ	Bφ	Vr	Vθ	Vφ	p	ϕ
1	G = I	G = I	G = I	G = I	G = I	G = I	G = I	B = I	B = 0
2	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	G = I	G = I	G = I	G = I	G = I	G = I	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	B = 0
3	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	G = I	G = I	G = I	G = I	G = I	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	B = 0
4	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	G = I	G = I $\tfrac{{r}_{I}}{{r}_{G}}$	G = I	G = I	G = I	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	B = 0
5	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	G = I	G = I $\tfrac{{r}_{I}}{{r}_{G}}$	G = I	G = I	G = I	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	G = I
6	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	G = I	G = I $\tfrac{{r}_{I}}{{r}_{G}}$	G = I $\tfrac{{r}_{I}^{2}{\rho }_{I}}{{r}_{G}^{2}{\rho }_{G}}$	G = I	G = I $\tfrac{{r}_{I}}{{r}_{G}}$	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	B = 0
7	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	G = I	G = I	G = I $\tfrac{{r}_{I}^{2}{\rho }_{I}}{{r}_{G}^{2}{\rho }_{G}}$	G = I	G = I $\tfrac{{r}_{I}}{{r}_{G}}$	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	B = 0
8	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	G = I	G = I	G = I	G = I $\tfrac{{r}_{I}^{2}{\rho }_{I}}{{r}_{G}^{2}{\rho }_{G}}$	G = I	G = I $\tfrac{{r}_{I}}{{r}_{G}}$	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	B = 0
9	G = I $\tfrac{{r}_{I}^{3}}{{r}_{G}^{3}}$	G = I $\tfrac{{r}_{I}^{2}}{{r}_{G}^{2}}$	G = I	G = I $\tfrac{{r}_{I}}{{r}_{G}}$	G = I $\tfrac{{r}_{I}^{2}{\rho }_{I}}{{r}_{G}^{2}{\rho }_{G}}$	G = I	G = I $\tfrac{{r}_{I}}{{r}_{G}}$	G = I $\tfrac{{r}_{I}^{3}}{{r}_{G}^{3}}$	B = 0

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For each simulation, the V_r profiles were compared, as well as the maximum V_r value and the location of the X-point in the B -field lines, since its location generally directly corresponded to the divergence of the field lines near the outer boundary. The evaluation of the exact location of the X-point has an uncertainty of approximately ±0.05 R_⊙, since it was determined by eye inspection of the solution in visualization software. In addition, the total time of each simulation required to reach the V_x residual of −10 was also measured to compare the convergence. The results for the nine cases are given in Table 3.

Table 3. Results for the X-point Location, Maximum V_r , and Time to Converge Down to the Residual of −10 in V_x for the Various Outer BC Simulation Cases

No.	X-point (R⊙)	${V}_{r,\max }$ (m s−1)	${T}_{{V}_{x}\mathrm{res}\lt -10}$ (s)
1	5.8	3.09E+05	7135.12
2	6.7	2.87E+05	7174.89
3	7.2	2.86E+05	7150.49
4	7.2	2.84E+05	7162.78
5	6.5	2.84E+05	N/A
6	7.2	2.84E+05	7171.25
7	7.2	2.84E+05	7157.08
8	7.1	2.86E+05	7161.79
9	7.2	2.79E+05	7157.28

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From observing the convergence alone, it can be quickly concluded that the divergence cleaning parameter ϕ must be set to zero via the Dirichlet BC instead of being extrapolated via zero gradient, given that the fifth simulation never converged. The results presented in Table 3 were taken at the V_x residual of roughly −2.5. It was further tested to see which Dirichlet formulation is better—setting ϕ_g = 0 or ϕ_b = 0—but both led to roughly the same convergence and solution, with the former giving a slightly larger divergence of the B -field lines near the outer boundary. The convergence of the other cases, in terms of the computational resources needed, was almost the same.

According to the results in Table 3, if we consider the other two evaluated parameters alone (the maximum V_r and the location of the X-point), the following points can be further raised.

• 1.  
  From comparing cases 3 and 4 and 6 and 7. Whether we extrapolate B_φ via zero gradient or radius is of no consequence (at least for the nonrotating dipole, where B_φ is negligible).
• 2.  
  From comparing cases 7 and 8 and 8 and 9. The B_r should be extended via a function of radius (here r⁻²) instead of a zero gradient (this will be revisited in the subsection below).
• 3.  
  From comparing cases 2 and 8 and 4 and 6. Whether we extrapolate V via zero gradient or the given laws is of barely any consequence.
• 4.  
  From comparing cases 1 and 2. The density (and pressure) should be extrapolated as a function of radius (here r⁻²).
• 5.  
  From comparing cases 6 and 9. The density (and pressure) extended via r⁻³ give lower maximum speeds.

Obviously, simply observing the location of the X-point alone is insufficient to justify the variable prescription to make the case physical. Nevertheless, these results are a good indication of which conditions the behavior of the solution is very sensitive to and which it is not. Especially the extrapolation of the radial magnetic field component seem to have a profound effect on the magnetic field line topology, judging from the shifting of the reconnection location. In addition, the formulation of the extrapolation of density also affects the maximum speeds in the domain. Thus, these two aspects will be elaborated on in more detail in the following subsections.

While Table 3 indicates that the density extrapolation affects the maximum speed in the domain, it is still not clear whether a more physically accurate result is given by the inverse square, inverse cube, or perhaps some other law. Most of the available coronal models use the inverse square extension (see, for example, Talpeanu et al. 2022), as this represents a flow that is expanding spherically with a uniform radial speed. However, from the V_r contours shown in, for example, Figure 5, it is apparent that the flow is still accelerating near the outer boundary, making the uniform-flow assumption incorrect. In addition, it will be shown later that in some regions of the domain of data-driven cases, the flow is expanding superradially. For these reasons, a steeper descent of ρ could be expected.

In order to determine which formulation is more accurate, an extended grid analysis was performed, where the computational domain radius was set to 30 R_⊙, and the cells around 21.5 R_⊙ were more refined radially to capture the local gradients better. This way, the outer BC was relatively far away from this region; thus, the undisturbed density profile at 21.5 R_⊙ that would exist at this location without the presence of the boundary could be investigated. The setup was otherwise equivalent to case 9 from the parametric study described above.

Initially, the 1/r² and 1/r³ profiles were compared to the nondimensional density. For easier comparison of the local slopes, the logarithms of the profiles were used for the analysis. These logarithms were plotted around the region of interest at 21.5 R_⊙, where they were made to intersect with the logarithmic nondimensional density profile as extracted from the CFD solution. The results are shown in Figure 14, where the green profile is the inverse square distance function, and the red profile is the inverse cubic distance function.

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It was observed that the actual CFD density slope lay roughly in the middle between the inverse cube and inverse squared profiles, and thus also a new profile, the function of 1/r^−2.5 was added, shown as the blue line in Figure 14. It can be easily concluded that this profile matches the density slope at the given radius almost perfectly. With this power law used as the outer BC for density, the maximum radial velocity reached for the steady dipole (now again modeled in the 21.5 R_⊙ domain) is 2.82E+05 m s⁻¹, considering the same resolution as used for the cases in Table 2.

The same analysis was performed on the magnetic field BC. For a flow that is not expanding superradially, the magnitude of B_r should be roughly decreasing with r². Capturing the correct gradient of B_r is important, as it also strongly affects how B_θ and B_ϕ are resolved. To demonstrate the difference that the extrapolation of B_r alone causes, we compare B_r and B_θ between cases 2 and 3 from Table 2 by evaluating the differences B_r,diff = B_r,2 − B_r,3 and B_θ,diff = B_θ,2 − B_θ,3 in Figure 15. From this figure, two observations can be made. First, as a result of the outer BC formulation, B_r changes not only near the boundary but also inside of the domain. Second, it is apparent that B_θ also changes as a result of the B_r extrapolation, despite the fact that the condition for B_θ remained unchanged.

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The consequences this has on the flow field and topology can be easily visualized when comparing the resulting magnetic field lines in Figure 16. One of the field lines from case 2 is outlined in red and then projected onto the domain of case 3, where it is compared to a case 3 field line in blue that starts at the same point on the boundary. This illustrates that the field lines of case 3 are less divergent, especially close to the outer edge.

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The fact that the outer BC affects the magnetic topology so extensively even inside of the domain is concerning, especially since we cannot always ensure that there are no regions where the flow would expand superradially (as will be shown later in a magnetogram-driven case).

As shown in the previous two sections, correct formulations of the BCs are not always straightforward and can affect the flow in the entire domain. Especially for cases where the BC formulations cannot be formulated universally, it would be useful to have additional tools to reduce the extrapolation error.

The larger the region of extrapolation (the ghost cell), the larger the extrapolation errors are, which can then also more considerably spoil the solution in the rest of the domain. In the original grid that has been used so far, the last cell has a radial size of roughly 1 R_⊙, meaning that the center of the ghost cell is 0.5 R_⊙ away from the boundary. If the last grid cell is made even larger, the effects become even more pronounced. Inverting this logic, with a very small edge cell, the ghost cell center is also very close to the boundary; thus, the extrapolation errors should be limited.

Moreover, the results on the outer boundary are very important from the perspective of coupling and transferring the CFD results to other software. Thus, not knowing how much of an effect the BC formulation can have on these results (at least locally), we no longer deem it safe to apply these BCs exactly at the location from which the solution is to be read. It would thus be better practice to extend the domain beyond the coupling location of 21.5 R_⊙ to make sure that the solution at the coupling location is not significantly affected in case the BC formulation is inadequate.

For this reason, an experimental grid was set up, which extends up to 25 R_⊙ and has a very small cell on the outer edge to demonstrate that the extrapolation error depends on the extrapolation distance. With this new grid, cases 1 and 9 (i.e., the cases with arguably the largest differences between them in outer BC formulations) were rerun and compared. The new mesh is shown in Figure 17, along with the 21.5 R_⊙ location and the boundary detail.

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For both grids (the original and the new one), the relative differences (in absolute scale) in V_r between the two cases were evaluated. These relative differences are plotted in Figure 18, on the left for the original grid and the right for the new grid. Once again, we can see that on the former mesh with the large ghost cell (on the left), the outer BC formulation affects the solution even very close to the Sun. The differences with the new mesh are not even properly visible with the color scale adjusted for the old grid, as they are 10–100 times smaller in magnitude everywhere in the domain. Clearly, with this new mesh design with a small ghost cell, the relative differences between the cases due to different extrapolation formulations are much lower. In Figure 19, we can also see that with the new mesh design, the artificial divergence of the magnetic field lines near the outer boundary decreased.

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It should be noted that this grid was designed in this way to merely demonstrate the effects of the size of the ghost cell. The fact that the last cell is so much smaller than the neighboring cell, with such a large aspect ratio difference, could, in practice, cause problems with convergence. If that happens, a more gradual decrease in cell size near the outer boundary might be required at the cost of having to include more cells and thus increasing the computational demands. In our case, however, with this grid, we have so far not observed any convergence problems even when resolving solar maxima, nor we have noticed any significant reduction in the computation speed.

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The abovementioned analysis of the inner and outer BCs considered a dipolar case only. To see how well the results of this analysis hold for a real case, the same analysis was repeated on a magnetogram-driven simulation.