The steady-state magnetosheath model has various applications for studying the plasma and magnetic field profile around the planetary magnetospheres. In particular, the magnetosheath model is analytically obtained by solving the Laplace equation for parabolic boundaries (bow shock and magnetopause). We address the question, how can we utilize the magnetosheath model by transforming into a more general, empirical, non-parabolic magnetosheath geometry? To achieve the goal, we develop the scalar-potential mapping method which provides a semi-analytic estimate of steady-state flow velocity and magnetic field in the empirical magnetosheath domain. The method makes use of a coordinate transformation from the empirical magnetosheath domain into the parabolic magnetosheath domain and evaluates a set of variables (shell variable and connector variable) to utilize the solutions of the Laplace equation obtained for the parabolic magnetosheath domain. Our model uses two invariants of transformation: the zenith angle within the magnetosheath with respect to the direction to the Sun and the ratio of the distance to the magnetopause and the thickness of magnetosheath along the magnetopause-normal direction. The use of magnetopause-normal direction makes a marked difference from the earlier model construction using the radial direction as reference. The plasma flow and magnetic field can be determined as a function of the upstream condition (flow velocity or magnetic field) in a wide range of zenith angles. The scalar-potential mapping method is computationally inexpensive, using analytic expressions as much as possible, and is applicable to various planetary magnetosheath domains.

Steady-state plasma flow and magnetic field can be regarded as a realization of potential field in the planetary magnetosheath region when the vorticity and the electric current are ignored. In such a case, the potential is obtained by solving the Laplace equation, which was elegantly and analytically solved by

The KF potential is obtained using the assumption that the planetary bow shock and magnetopause have a parabolic shape sharing the same focal point. Empirical models of the bow shock and magnetopause (fitted to the spacecraft data), on the other hand, are not necessarily parabolically or co-focally shaped. For example, the empirical Earth bow shock model by

Naively speaking, one wishes to find a conformal mapping (angle-preserving mapping) from the KF parabolic magnetosheath onto a non-parabolic empirical magnetosheath shape such as the analytic extension of magnetopause shape

Here we address the question, how can we utilize the KF magnetosheath model by transforming into a more general, empirical, non-parabolic magnetosheath geometry? To achieve the goal, we develop a mapping method which provides a semi-analytic estimate of steady-state flow velocity and magnetic field in the empirical magnetosheath domain. Our scalar-potential mapping method is computationally inexpensive by using the analytic expression as much as possible, and it is applicable to various planetary magnetosheath domains.

This work is organized in the following fashion. After reviewing the magnetosheath model constructed by

In the KF parabolic coordinates, the shell variable

In the frame of potential field theory, the flow velocity

The symbol

Hereafter, one may set

The magnetic field in the magnetosheath is derived from the scalar potential in the same fashion as the flow velocity; that is,

The magnetic potential is a function of the shell variable

Our task is to find the shell variable

A practically useful mapping procedure to utilize the KF potential is proposed by

Although the method introduced by

Grid pattern generated by the radial mapping for the Kobel–Flückiger parabolic magnetosheath

Velocity potential in the Kobel–Flückiger model

Our mapping method differs from the radial mapping method in that the magnetopause-normal direction is used as a reference to the magnetopause. Our method guarantees the grid orthogonality around the magnetopause both on the dayside and in the flank region. The magnetopause-normal grids are shown in Fig.

Mesh pattern used in the magnetopause-normal mapping in this work for the parabolic boundaries

The magnetopause-normal mapping is performed with two transformations. In the first transformation, the position vector is mapped from the empirical magnetosheath

In the second transformation, the mapped position vector is used to compute the shell variable

Figure

Variables used in the magnetopause-normal mapping with the zenith angle

We begin with a position vector in the empirical magnetosheath domain, and we express the position vector as

The empirical bow shock position expressed in GSE (Geocentric Solar Ecliptic) coordinates proposed and discussed by

The empirical magnetopause position by

We use a specific exponent for the Shue model (with an alpha exponent of 0.5) in an effort to show that the analytic model is “simple”. The solar wind conditions for which this exponent is applicable is not often encountered (e.g., interplanetary magnetic field has the

In our setup, the radial distance from the planet to the bow shock is expressed as

By introducing the zenith angle

Equation (

The radial distance to the magnetopause is given conveniently by Eq. (

In the first step, the distance from the given position in the magnetosheath to the nearest magnetopause is computed (see Fig.

The nearest magnetopause position is obtained by searching for the zenith angle

Measuring the distance to the empirical magnetopause (step 1).

Using the minimum distance to the magnetopause

The magnetopause-normal direction is obtained by normalizing the gradient vector

In the second step, the magnetosheath thickness is computed using the position vector and the magnetopause normal direction (Fig.

In the subsolar direction (

Equation (

Computing the magnetosheath thickness in the empirical model (step 2).

In the third step, the magnetosheath thickness is computed in the KF model (Fig.

The radial distance from the planet to the KF bow shock is

The KF magnetopause is defined in

From Eq. (

To obtain the magnetopause-normal direction in the KF system, we compute the gradient of the magnetopause shape function:

The gradient is analytically given as

The magnetopause-normal direction

Computing the magnetosheath thickness in the KF model (step 3). The same zenith angle as that in step 2 is used.

The thickness in the KF system

Equation (

In the fourth step, the mapping of the position vector is performed from the empirical magnetosheath onto the KF system (Fig.

Mapping the position vector onto the KF magnetosheath model (step 4).

In the fifth step, the shell variable

Evaluating the shell variable

Figure

Iso-contour lines with

The scalar potentials (velocity potential and magnetic potential) and the stream function are obtained from the shell

Velocity potential

The magnetic potential and the derived magnetic field are displayed in Fig.

Magnetic potential for the upstream magnetic field with an angle of 135° to the

It is straightforward to extend our method to different shapes of the Shue magnetopause model. The following form of gradient can be used for a general value of the Shue exponent

Mesh pattern applied to different values of the Shue exponent for an open-type magnetopause

Mesh pattern applied to different values of the Shue exponent for an open-type magnetopause

Also, our method can be extended to a three-dimensional, non-axisymmetric geometry of magnetosheath

It is possible to obtain the steady-state magnetosheath potential in different ways.

First, one may numerically solve the Laplace equation for a given set of boundary shapes (bow shock and magnetopause). Various numerical solvers are known for solving the Laplace equation such as the Jacobi method, the Gauss–Seidel method, and the successive over-relaxation (SOR) method. These Laplace solvers are numerically more expensive than the mapping method, but the computation in 3-D is feasible with the contemporary computational resources. On the other hand, the magnetosheath is not bounded but extends in the tail direction. The challenge here is thus to construct a properly bounded area for the Laplace equation.

Second, one may expand the magnetosheath magnetic field in different orthogonal functions. The KF solution makes use of the Bessel functions

Third, one may introduce a suitable conformal mapping by limiting the magnetosheath modeling to a complex plane (two-dimensional domain). The harmonic functions such as the KF solution are transformed from the parabolic magnetosheath shape into the non-axisymmetric magnetosheath shape. The problem here is that finding the conformal mapping is not an easy task, because the magnetosheath is not a spatially-closed domain and one has to set the boundary in the magnetosheath to complete the domain bounded by the bow shock, the subsolar axis, and the magnetopause.

Fourth, one may solve the Rankine–Hugoniot relations and track the streamline stepwise by referring to the KF solution, as is done in

Our potential mapping method may be regarded as an updated version of the radial mapping method

The advantages of our methods are as follows:

The method makes extensive use of the exact solution of the Laplace equation (the Kobel–Flückiger potential and the Guicking stream function). The plasma flow and magnetic field can be determined semi-analytically in a wide range of zenith angles in the magnetosheath when the solar wind conditions and the boundary shapes are given.

The method is applicable to a non-parabolic shape of magnetosheath domain, opening the door to develop a tool to assist numerical simulations and spacecraft observations of not only the Earth but also the planetary magnetosheath domain.

The method is computationally inexpensive. In particular, if the shapes of bow shock and magnetopause are analytically given, most of the computational steps in the potential mapping method have an analytic expression.

As stated in Sect.

Our method of computing the plasma flow and magnetic field should be compared against the observations and simulations. For example, THEMIS and ARTEMIS spacecraft

The IDL (Interactive Data Language) codes are available upon request to the authors.

YN, DS, and ST developed the idea of the potential mapping method; checked mathematics; and wrote the manuscript. YN prepared the figures. All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

The contact author has declared that none of the authors has any competing interests.

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This open-access publication was funded by Technische Universität Braunschweig.

This paper was edited by Minna Palmroth and reviewed by Steven Petrinec, Octav Marghitu, and two anonymous referees.