An axi-symmetric two-dimensional magnetopause model is constructed by making use of the conformal mapping in the complex plane. The model is an analytic continuation of the power-law damped (or asymptotically elongated) parabolic shape. The complex-plane expression of the magnetopause opens the door to properly map the magnetopause and magnetosheath coordinates from one model to another.

The magnetopause model proposed by

Here we report our finding that the magnetopause model can be formulated as a conformal mapping in the complex plane. This mapping preserves local angles. Any analytic function satisfies the conformal (angle-preserving) character in the complex plane as long as there is a non-zero derivative. Expression of the magnetopause as a conformal map is ideal when dealing with different magnetopause models.

Our study is motivated to fill the gap between
the property of the bow shock models
and that of the magnetopause models.
The bow shock is often modeled as a conic section

We start with the magnetopause model in polar coordinates
after

By introducing the transformation,

The stand-off distance is restored at the subsolar point, i.e.,

The distance to the planet is

The distance to the Sun–Earth axis (or the

Now we express Eq. (

To our task, we first transform the

We find out that the combination of four sequential conformal mappings
is a reasonable analytic continuation of the magnetopause model:
(

In the first conformal mapping,
the square transformation is used with
a unit coefficient and no shift. The transformation
is expressed as

Constant

In the second conformal mapping, the parabolic shape of the mapped curves
are stretched using the poles at

Figure

In the third conformal mapping,
the Joukoswky-transformed function

Again, the poles are retained in this transformation.
The mapped function has a shape of magnetopause, but
the focal point is located in the far tail region,
and the distance to the magnetopause is smaller
than the stand-off distance.
Figure

In the final conformal mapping, the mapping is
scaled by a factor

Figure

The magnetopause location is restored when choosing

The function

Magnetopause location generated by
Eq. (

The analytic nature of our function (Eq.

Curvilinear grids generated
by the conformal map (Eq.

Qualitatively speaking, different tail shapes can also be
obtained by generalizing the square root operation
in

The magnetopause coordinates are plotted
as grids for

Magnetopause grids generated for different values
of the power index

Conformal mapping is a useful method in the model
construction when the axi-symmetry holds
and the boundary is modeled in the two-dimensional
spatial domain.
Our magnetopause model completes
the scenario that both dayside boundaries
(bow shock and magnetopause) can be modeled
by conformal mapping, which opens the door
to analytically or semi-analytically map
the magnetosheath scalar potential by

The easiest approach of magnetosheath coordinate mapping would be to introduce the transfinite interpolation in the complex plane. Or, one could numerically solve the Laplace equation for the given boundaries in order to generate strictly orthogonal curvilinear coordinates.

In the case of

We compute square of

The factorized form of Eq. (

Equation (

No code or data were used in this paper.

YN, ST, and DS developed the idea of conformal mapping applications, 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 paper was edited by Elias Roussos and reviewed by one anonymous referee.