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Grad–Shafranov reconstruction of magnetohydrostatic equilibria with nonisotropic plasma pressure: the theory
Earth, Planets and Space volume 70, Article number: 34 (2018)
The Correction to this article has been published in Earth, Planets and Space 2018 70:185
Abstract
The basic theory for reconstruction of twodimensional, coherent, magnetohydrostatic structures with nonisotropic plasma pressure is developed. Three fieldline invariants are found in the system. A new Poissonlike partial differential equation is obtained for this reconstruction, which can be solved as a spatial initialvalue problem in a manner similar to the socalled Grad–Shafranov reconstruction, without resort to auxiliary equations. Moreover, we find that with some simple substitutions this new equation can be applied for fieldaligned flow with isotropic plasma pressure. The numerical code for new reconstruction has been developed and is benchmarked with an exact analytical solution. Results show that the reconstruction works well with small errors in a rectangular region surrounding the spacecraft trajectory. Applications to in situ spacecraft measurements will be reported separately.
Introduction
The Grad–Shafranov (GS) reconstruction is a data analysis tool to produce a twodimensional (2D) field or flow map of a coherent structure observed by a single spacecraft. The original GS reconstruction is based on spatial initialvalue integration of the magnetohydrostatic GS equation in 2D geometry, i.e., \( \nabla^{2} A =  \mu_{0} {{{\text{d}}(p + {{B_{z}^{2} } \mathord{\left/ {\vphantom {{B_{z}^{2} } {2\mu_{0} }}} \right. \kern0pt} {2\mu_{0} }})} \mathord{\left/ {\vphantom {{{\text{d}}(p + {{B_{z}^{2} } \mathord{\left/ {\vphantom {{B_{z}^{2} } {2\mu_{0} }}} \right. \kern0pt} {2\mu_{0} }})} {{\text{d}}A}}} \right. \kern0pt} {{\text{d}}A}} =  \mu_{0} {{{\text{d}}p_{t} } \mathord{\left/ {\vphantom {{{\text{d}}p_{t} } {{\text{d}}A}}} \right. \kern0pt} {{\text{d}}A}} \) (Sonnerup and Guo 1996; Hau and Sonnerup 1999). Here \( A \), \( p \), and \( B_{z} \) are the vector potential, the plasma pressure, and the axial magnetic field, respectively, and \( \nabla^{2} = {{\partial^{2} } \mathord{\left/ {\vphantom {{\partial^{2} } {\partial x^{2} + }}} \right. \kern0pt} {\partial x^{2} + }}{{\partial^{2} } \mathord{\left/ {\vphantom {{\partial^{2} } {\partial y^{2} }}} \right. \kern0pt} {\partial y^{2} }} \). The value \( p_{t} \) is invariant along the field line for which the function \( {\text{d}}p_{t} /{\text{d}}A \) can be determined numerically. Once \( \partial^{2} A/\partial y^{2} =  \mu_{0} {{{\text{d}}p_{t} } \mathord{\left/ {\vphantom {{{\text{d}}p_{t} } {{\text{d}}A}}} \right. \kern0pt} {{\text{d}}A}}  {{\partial^{2} A} \mathord{\left/ {\vphantom {{\partial^{2} A} {\partial x^{2} }}} \right. \kern0pt} {\partial x^{2} }} \) is calculated, the vector potential can therefore be advanced in a small step along the y axis by use of the Taylor expansion to second order. This classical GS reconstruction has been successfully applied to studying magnetopause structures (e.g., Hau and Sonnerup 1999; Hu and Sonnerup 2000; Hasegawa et al. 2004), magnetic flux ropes and magnetic clouds (e.g., Hu and Sonnerup 2001, 2002; Hu 2017; Sonnerup et al. 2004). Sonnerup et al. (2006) have further generalized the reconstruction method to allow for fieldaligned flow with isotropic plasma pressure (Teh et al. 2007) and for plasma flow perpendicular to a magnetic field (Hasegawa et al. 2007). Moreover, a GSlike equation is derived to include fieldaligned flow with nonisotropic plasma pressure (Sonnerup et al. 2006), which can be reduced for magnetohydrostatic equilibria with nonisotropic plasma pressure.
The extensions of the GS reconstruction developed by Sonnerup et al. (2006) have a GSlike governing equation, in which some quantities that appear in the equation are not invariant along the field lines or streamlines. Those quantities need to be advanced by use of auxiliary equations, and thus the reconstruction requires more xderivative calculations in each integration step as compared to the classical GS reconstruction. These extra numerical differentiations will act as a catalyst for the error growing in each integration step, and that it turns out to suppress the reconstruction domain.
In this paper, we aim to develop a new GSlike equation for magnetohydrostatic equilibria with nonisotropic plasma pressure, which can be solved as a spatial initialvalue problem in a manner similar to the classical GS reconstruction, without resort to auxiliary equations. We find that with some simple substitutions this new GSlike equation can be applied for fieldaligned flow with isotropic plasma pressure. The paper is organized as follows. In “Theory” section, we describe the basic theory of the reconstruction of magnetohydrostatic equilibria with nonisotropic plasma pressure. In “Benchmark case” section, we develop an exact analytical solution to validate our new reconstruction code. Finally, discussion is given in “Discussion” section.
Theory
In this section, we derive the GSlike equation for magnetohydrostatic equilibria with nonisotropic plasma pressure. The derivation presented here is similar to that used in the paper of Sonnerup et al. (2006) for structures with steady fieldaligned flow. In the magnetohydrostatic conditions, twodimensional coherent structures are described by the balance between pressure forces and magnetic forces:
Here the pressure tensor \( {\mathbf{\rm P}} = p_{ \bot } {\mathbf{\rm I}} + {{(p_{\parallel }  p_{ \bot } ){\mathbf{BB}}} \mathord{\left/ {\vphantom {{(p_{\parallel }  p_{ \bot } ){\mathbf{BB}}} {B^{2} }}} \right. \kern0pt} {B^{2} }} \) where \( p_{ \bot } \) and \( p_{\parallel } \) are the plasma pressures perpendicular and parallel to the magnetic field, respectively. By use of the pressure anisotropy factor \( \alpha = {{\mu_{0} (p_{\parallel }  p_{ \bot } )} \mathord{\left/ {\vphantom {{\mu_{0} (p_{\parallel }  p_{ \bot } )} {B^{2} }}} \right. \kern0pt} {B^{2} }} \), Eq. (1) can be written as
Using \( \nabla \cdot {\mathbf{B}} = 0 \) and the notation \( {\mathbf{F}} = (1  \alpha ){\mathbf{B}} \), Eq. (2) becomes
Using the identity \( {\mathbf{F}} \cdot {\mathbf{\nabla F}} = ({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern0pt} 2}){\mathbf{\nabla }}F^{2}  {\mathbf{F}} \times ({\mathbf{\nabla }} \times {\mathbf{F}}) \) and \( B^{2} = {{F^{2} } \mathord{\left/ {\vphantom {{F^{2} } {(1  \alpha )^{2} }}} \right. \kern0pt} {(1  \alpha )^{2} }} \), Eq. (3) then becomes
where \( p_{T} = (1  \alpha )p_{ \bot } + \alpha (1  \alpha )^{  1} ({{F^{2} } \mathord{\left/ {\vphantom {{F^{2} } {2\mu_{0} }}} \right. \kern0pt} {2\mu_{0} }}) \).
Since \( {\mathbf{F}} = (1  \alpha ){\mathbf{B}} \), one can define \( {\mathbf{F}} = {\mathbf{\nabla }}A \times {\hat{\mathbf{z}}} + {\hat{\mathbf{z}}}(1  \alpha )B_{z} \) for which \( \nabla \cdot {\mathbf{F}} = 0 \). Here A is the modified vector potential, different from the one in the original GS equation mentioned in the introduction section. Note that \( {\mathbf{\nabla }} = {\hat{\mathbf{x}}}{\partial \mathord{\left/ {\vphantom {\partial {\partial x}}} \right. \kern0pt} {\partial x}} + {\hat{\mathbf{y}}}{\partial \mathord{\left/ {\vphantom {\partial {\partial y}}} \right. \kern0pt} {\partial y}} \) and \( {\partial \mathord{\left/ {\vphantom {\partial {\partial z}}} \right. \kern0pt} {\partial z}} = 0 \). Since \( \nabla \cdot {\mathbf{F}} = 0 \), one gets \( {\mathbf{F}} \cdot {\mathbf{\nabla }}\alpha = 0 \). Thus, \( \alpha \) is a fieldline invariant. Taking dot product of (4) with F yields \( {\mathbf{F}} \cdot {\mathbf{\nabla }}p_{T} = 0 \). Also, \( p_{T} \) is a fieldline invariant. Since the righthand side of (4) does not have a z component, one gets \( {\mathbf{F}}_{t} \times ({\mathbf{\nabla }} \times {\mathbf{F}})_{t} = 0 \), where \( ({\mathbf{\nabla }} \times {\mathbf{F}})_{t} = {\mathbf{\nabla }}F_{z} \times {\hat{\mathbf{z}}} \). Therefore, it turns out that the vector \( {\mathbf{\nabla }}F_{z} = {\mathbf{\nabla }}(1  \alpha )B_{z} \) is to be perpendicular to \( {\mathbf{F}}_{t} \), indicating that \( F_{z} \) is a fieldline invariant. Furthermore, one can find from the invariant \( F_{z} \) that the axial field \( B_{z} \) is also a fieldline invariant since \( \alpha \) is invariant along the field line.
In the transverse (x, y) part of (4), it can be written as
By substituting \( {\mathbf{F}}_{t} = {\mathbf{\nabla }}A \times {\hat{\mathbf{z}}} \), \( ({\mathbf{\nabla }} \times {\mathbf{F}})_{z} =  \nabla^{2} A \), and \( ({\mathbf{\nabla }} \times {\mathbf{F}})_{t} = {\mathbf{\nabla }}F_{z} \times {\hat{\mathbf{z}}} \), Eq. (5) becomes
Since \( F_{z} \), \( p_{T} \), and \( \alpha \) are functions of vector potential A alone, one can write \( {\mathbf{\nabla }}F_{z}^{2} = ({{{\text{d}}F_{z}^{2} } \mathord{\left/ {\vphantom {{{\text{d}}F_{z}^{2} } {{\text{d}}A}}} \right. \kern0pt} {{\text{d}}A}}){\mathbf{\nabla }}A \), \( {\mathbf{\nabla }}p_{T} = ({{{\text{d}}p_{T} } \mathord{\left/ {\vphantom {{{\text{d}}p_{T} } {{\text{d}}A}}} \right. \kern0pt} {{\text{d}}A}}){\mathbf{\nabla }}A \), and \( {\mathbf{\nabla }}\alpha = ({{{\text{d}}\alpha } \mathord{\left/ {\vphantom {{{\text{d}}\alpha } {{\text{d}}A}}} \right. \kern0pt} {{\text{d}}A}}){\mathbf{\nabla }}A \). By removing the factor \( {\mathbf{\nabla }}A \) from (6), one can then obtain the GSlike equation for magnetohydrostatic equilibria with nonisotropic pressure:
which can be further rewritten as
As a consistency check, when \( \alpha \) goes to zero, \( F_{z} \) becomes equal to \( B_{z} \) and \( p_{T} = p_{ \bot } = p \), and the last term on the right in (8) becomes zero. Therefore, as expected, Eq. (8) reduces to the classical GS equation, i.e., \( \nabla^{2} A =  \mu_{0} {{{\text{d}}\left( {p + {{B_{z}^{2} } \mathord{\left/ {\vphantom {{B_{z}^{2} } {2\mu_{0} }}} \right. \kern0pt} {2\mu_{0} }}} \right)} \mathord{\left/ {\vphantom {{{\text{d}}\left( {p + {{B_{z}^{2} } \mathord{\left/ {\vphantom {{B_{z}^{2} } {2\mu_{0} }}} \right. \kern0pt} {2\mu_{0} }}} \right)} {{\text{d}}A}}} \right. \kern0pt} {{\text{d}}A}} \). When \( \alpha \) is equal to one, Eq. (8) becomes singular. For reconstruction, one can use a spline function to calculate the solution at the grid point where \( \alpha = 1 \) occurs, from its value at the neighboring points. In addition, it is found that the MHD forcebalanced equation for steady fieldaligned flow with isotropic plasma pressure is similar to Eq. (2), in which \( \alpha \) and \( p_{ \bot } \) are replaced by \( M_{A}^{2} \) and p, respectively. Here \( M_{A}^{2} \) is the Alfvén Mach number. As a result, a new GSlike equation for fieldaligned flow with isotropic plasma pressure can be obtained from (8) by simply putting \( \alpha = M_{A}^{2} \) and \( p_{ \bot } = p \).
The numerical scheme used to integrate Eq. (8) is similar to that for the magnetohydrostatic GS reconstruction (e.g., Hau and Sonnerup 1999; Sonnerup et al. 2006). The values of the vector potential A at points along the x axis (the spacecraft trajectory) are computed as
where \( {\text{d}}x^{\prime} = V_{0} {\text{d}}t \) and \( V_{0} \) is the motion of the structure. While the functions of \( {{{\text{d}}F_{z}^{2} } \mathord{\left/ {\vphantom {{{\text{d}}F_{z}^{2} } {{\text{d}}A}}} \right. \kern0pt} {{\text{d}}A}} \), \( {{{\text{d}}p_{T} } \mathord{\left/ {\vphantom {{{\text{d}}p_{T} } {{\text{d}}A}}} \right. \kern0pt} {{\text{d}}A}} \), and \( {{{\text{d}}\ln (1  \alpha )} \mathord{\left/ {\vphantom {{{\text{d}}\ln (1  \alpha )} {{\text{d}}A}}} \right. \kern0pt} {{\text{d}}A}} \) are determined numerically, the value \( \partial^{2} A/\partial y^{2} \) on the left in (8) can then be calculated as \( {{\partial^{2} A} \mathord{\left/ {\vphantom {{\partial^{2} A} {\partial y^{2} = }}} \right. \kern0pt} {\partial y^{2} = }}{\text{RHS}}  {{\partial^{2} A} \mathord{\left/ {\vphantom {{\partial^{2} A} {\partial x^{2} }}} \right. \kern0pt} {\partial x^{2} }} \), where RHS is the righthand side of (8). Thus, the vector potential A can be advanced in small steps, \( \pm \;\Delta y \), as
where \( {{\partial A} \mathord{\left/ {\vphantom {{\partial A} {\partial y}}} \right. \kern0pt} {\partial y}} = F_{x} (x,y) = [1  \alpha (x,y)]B_{x} (x,y) \). Similarly, the new value of \( F_{x} \) is obtained as
The new value of \( F_{y} \) is then calculated as \( F_{y} (x,y \pm \Delta y) =  {{\partial A(x,y \pm \Delta y)} \mathord{\left/ {\vphantom {{\partial A(x,y \pm \Delta y)} {\partial x}}} \right. \kern0pt} {\partial x}} \). Finally, the new values of \( \alpha \), \( p_{T} \), and \( F_{z} \) can be obtained by use of the functions \( {{{\text{d}}\ln (1  \alpha )} \mathord{\left/ {\vphantom {{{\text{d}}\ln (1  \alpha )} {{\text{d}}A}}} \right. \kern0pt} {{\text{d}}A}} \), \( {{{\text{d}}p_{T} } \mathord{\left/ {\vphantom {{{\text{d}}p_{T} } {{\text{d}}A}}} \right. \kern0pt} {{\text{d}}A}} \) and \( {{{\text{d}}F_{z}^{2} } \mathord{\left/ {\vphantom {{{\text{d}}F_{z}^{2} } {{\text{d}}A}}} \right. \kern0pt} {{\text{d}}A}} \), respectively, with the new values of A.
Once the quantities \( F_{x} \), \( F_{y} \), \( F_{z} \), \( \alpha \), and \( p_{T} \) are known, the quantities \( B_{x} \), \( B_{y} \), \( B_{z} \), \( p_{ \bot } \), and \( p_{\parallel } \) can then be calculated, where \( {\mathbf{B}} = {{\mathbf{F}} \mathord{\left/ {\vphantom {{\mathbf{F}} {(1  \alpha )}}} \right. \kern0pt} {(1  \alpha )}} \), \( p_{ \bot } = (1  \alpha )^{  1} [p_{T}  \alpha (1  \alpha )^{  1} ({{F^{2} } \mathord{\left/ {\vphantom {{F^{2} } {2\mu_{0} }}} \right. \kern0pt} {2\mu_{0} }})] \), and \( p_{\parallel } = \alpha ({{B^{2} } \mathord{\left/ {\vphantom {{B^{2} } {\mu_{0} )}}} \right. \kern0pt} {\mu_{0} )}} + p_{ \bot } \). It is worth to point out that in the paper of Sonnerup et al. (2006) the GSlike equation of magnetohydrostatic equilibria with nonisotropic pressure can be obtained from (A7) (the equation (7) in their Appendix) by putting \( M_{A}^{2} = 0 \), \( G^{2} = 0 \), and \( v^{2} = 0 \). As compared to our Eq. (8), solving that resulting equation has to deal with a 6 × 6 sparse matrix, for which the integration scheme is different from the classical GS reconstruction.
Benchmark case
In this section, we derive an exact analytic solution of Eq. (8) to validate our numerical code for integration scheme described in “Theory” section. In the calculations, all the physical quantities are assumed to be functions of the cylindrical radius r alone. We assume that \( A = {\text{e}}^{  r} \), \( B_{z}^{2} = r^{2} {\text{e}}^{  r} \), and \( \alpha = 1  {1 \mathord{\left/ {\vphantom {1 r}} \right. \kern0pt} r} \). With these expressions for A, \( B_{z}^{2} \), and \( \alpha \), \( {{{\text{d}}(rp_{T} )} \mathord{\left/ {\vphantom {{{\text{d}}(rp_{T} )} {{\text{d}}r}}} \right. \kern0pt} {{\text{d}}r}} \) can thus be obtained from Eq. (8) as
where \( r_{0} \), \( B_{z0}^{2} \), and \( A_{0} \) are the normalized factors. By integrating (12) with respect to r, one gets
where \( \beta_{z0} = {{\mu_{0} p_{0} } \mathord{\left/ {\vphantom {{\mu_{0} p_{0} } {B_{z0}^{2} }}} \right. \kern0pt} {B_{z0}^{2} }} \) and \( \beta_{0} = {{\mu_{0} p_{0} r_{0}^{2} } \mathord{\left/ {\vphantom {{\mu_{0} p_{0} r_{0}^{2} } {A_{0}^{2} }}} \right. \kern0pt} {A_{0}^{2} }} \), and \( p_{0} \) is the reference value at \( r = 1 \). The benchmark solution is confined within \( r \ge 1 \).
Analytic results for \( \beta_{z0} = 1.0 \) and \( \beta_{0} = 1.0 \) are displayed in Figs. 1a and 2a. Figure 1a shows the inplane magnetic field lines of \( {\mathbf{F}}_{t} \) with \( \alpha \) in color. In Fig. 2a, black curves are the contour of \( F_{z} \) and color code is \( p_{T} \). The white horizontal line at \( y = 0.5 \) is the path of a virtual spacecraft through the structure. Data taken along this line are then used as initial values for integration. Figure 3 shows the plots of the three fieldline invariants versus the vector potential, where the data and fitting curves are shown as black and red curves, respectively. The reconstruction results are demonstrated in Figs. 1b and 2b. It is seen that the reconstruction maps agree well with the exact solution. Errors are shown in Figs. 1c and 2c, where solid and dashed curves indicate positive and negative errors, respectively. The contour of errors for vector potential in Fig. 1c and \( F_{z} \) in Fig. 2c are at 0.5% intervals, while the errors for \( \alpha \) and \( p_{T} \) are shown in color in Figs. 1c and 2c, respectively. It is seen that the errors are small and mainly in the corners of the reconstruction domain.
Discussion
In this paper, we have developed the theory of the reconstruction for magnetohydrostatic equilibria with nonisotropic plasma pressure, where three fieldline invariants are found in the system. A new GSlike equation is obtained for this reconstruction, which can be solved as a spatial initialvalue problem in a manner similar to the classical GS reconstruction, without resort to auxiliary equations. We have found that this new GSlike equation can be applied for fieldaligned flow with isotropic plasma pressure by simply putting \( \alpha = M_{A}^{2} \) and \( p_{ \bot } = p \). We have produced a new reconstruction code for integration and validated it against an exact analytical solution. The reconstruction results are in good agreement with the exact solution. Errors remain within a few percent in the corners of the reconstruction domain.
For simplicity, this study does not include plasma flow, gravity, and other effects (e.g., gyroviscosity and parallel heat flux) in the new formulation of the GSlike equation with pressure anisotropy. There are other general formulations of the GS equation with pressure anisotropy, for example, for ideal magnetohydrodynamic flows with the double adiabatic relations (e.g., Beskin and Kuznetsova 2000) and for highbeta tokamaks (e.g., Ito and Nakajima 2011). From the reconstruction point of view, such GS formulations remain challenged for reconstruction.
For applications to actual spacecraft measurements, important prerequisites for reconstruction are a moving frame of reference and the invariant axis of the 2D structure. Detailed discussions on this issue can be found in Sonnerup et al. (2006) and the references therein. It is known that the integration of the GS equation is numerically unstable. To suppress the spurious solutions, a smoothing operation is taken into account in each step of the integration. In the benchmark case, a Savitzky–Golay smoothing is implemented instead of the previous threepoint smoothing used by Hau and Sonnerup (1999). Pressure anisotropy is commonly seen in the magnetosheath and the reconnection region at the magnetopause as well as in the magnetotail. This type of reconstruction will be useful to provide new insight into the coherent structures being observed. Applications to in situ spacecraft measurements will be reported separately.
Change history
28 November 2018
After publication of this article (Teh 2018), the author has noticed some error with the statement in the abstract and discussion sections.
Abbreviations
 GS:

Grad–Shafranov
 2D:

two dimensional
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Acknowledgements
This work was supported by the Fundamental Research Grant Scheme (FRGS) from the Ministry of Higher Education Malaysia (FRGS/1/2016/STG02/UKM/03/1).
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Teh, W. Grad–Shafranov reconstruction of magnetohydrostatic equilibria with nonisotropic plasma pressure: the theory. Earth Planets Space 70, 34 (2018) doi:10.1186/s406230180802z
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Keywords
 Grad–Shafranov reconstruction
 Magnetohydrostatic equilibria
 Pressure anisotropy