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.
Introduction
The magnetopause model proposed by
(hereafter the Shue model) is,
to the authors' knowledge, one of the most successful structure
models in space science. The Shue model can be given
in a simple analytic way by combining a parabolic shape
with a power law, and has successfully been tested against
the magnetopause of the Earth and the other planets,
such as Mercury .
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
either as a parabola or as a hyperbola; see,
and the analytic expression for the conformal map is known
.
The magnetopause shape (such as in the Shue model)
is, on the other hand, not a conic section,
and the existence of conformal mapping remained
a question for a long time. We tackle the question
by incorporating various conformal mappings.
Construction of conformal map
We start with the magnetopause model in polar coordinates
after ,
R=Rmp21+cosθα,
where R is the radial distance to the planet,
θ is the zenith angle (measured from the planet center),
and α is the power index
to designate the magnetopause shape in the tail region, e.g.,
a parabolic shape (corresponds to α=1),
an elongated shape (given by α=1/2),
or damped, converging shape (α<1/2).
Rmp denotes the magnetopause stand-off distance
at the subsolar point.
In our work, we choose α=0.5, which is
statistically representative .
By introducing the transformation,
2x=RRmpcosθ3y=RRmpsinθ,
the magnetopause location is given in the Cartesian form as
x2=44-y2-y2.
The derivation of Eq. () is shown
in Appendix.
Note that x and y are normalized to the magnetopause stand-off
distance Rmp for simplicity.
The magnetopause model (Eq. ) has
the following boundary conditions and asymptotic behavior:
The stand-off distance is restored at the subsolar point, i.e.,
x=1 at y=0.
The distance to the planet is y=±2 at the terminator (x=0).
The distance to the Sun–Earth axis (or the x axis in geocentric solar ecliptic coordinates, GSE)
is y=±2, when x→∞.
Now we express Eq. () in the complex plane
using the variable z=u+iv,
so that the magnetopause location is given as
f(z)=x+iy.
In other words,
6x=Ref(z)7y=Imf(z),
in the Cartesian representation.
The complex-valued function f(z) is an extension of the
magnetopause location. The magnetopause is restored
when choosing v=vmp=1
(or vmp to be evaluated as Rmp
when not normalized). The task is thus to find the suitable
function f(z).
To our task, we first transform the y coordinates onto
the imaginary axis as iy
(where i is the imaginary unit),
so that the denominator
in Eq. ()
is formulated from 4/(4-y2) into 4/(4+(iy)2).
Now we perform the analytic continuation of the right-hand side
of Eq. (), and replace iy by z.
We find out that the combination of four sequential conformal mappings
is a reasonable analytic continuation of the magnetopause model:
(t1) square transformation,
(t2) Joukowsky transformation (with shift),
(t3) square root transformation, and (t4)
scaling and shifting (for the matching with the boundaries).
Each transformation is discussed below.
Square transformation
In the first conformal mapping,
the square transformation is used with
a unit coefficient and no shift. The transformation
is expressed as
t1:z→z2.
The transformation yields the parabolic coordinates as
9Re(z2)=u2-v210Im(z2)=2uv,
which can be arranged into a parabolic equation
when eliminating u as
Re(z2)=Imz224v2-v2.
In fact, the parabolic model of magnetopause is introduced
by , which is equivalent to
the following transformation:
tK:z→-12z2+12.
Here, v=vmp corresponds to the magnetopause location.
Figure in the top left panel displays
the mapping of u=const lines (in gray) and v=const
lines (in black) for the transformation t1.
The “nose” of magnetopause is located on the negative x side.
Constant u lines (in gray) and
constant v lines (in black) for the conformal
mappings t1, t2, t3, and t4.
Shifted Joukowsky transformation
In the second conformal mapping, the parabolic shape of the mapped curves
are stretched using the poles at z=±i2.
The transformation is a variant of the Joukowsky transformation,
which deforms circles into ellipses .
We perform the Joukowsky transformation
by retaining the pole terms z2+4 as
t2:z2→z2+4+4z2+4.
Figure in the top right panel displays
the mapping for the transformation t2.
The overall structure of v=const lines
still retains the parabolic shape, but the focal point
shifts to a larger value of x, and the distance from
the x axis (the y=0 line) is larger.
Shifted square root transformation
In the third conformal mapping,
the Joukoswky-transformed function t3 is compared to
the magnetopause model (Eq. ).
The comparison yields a subtraction by 4
and a square root operation as
t3:z2+4+4z2+4→(z2+4)+4z2+4-41/2.
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 bottom left panel displays
the mapping for the transformation t3.
The tail shape is elongated by this transformation
The focal point is moved close to the origin.
Scaling and shifting
In the final conformal mapping, the mapping is
scaled by a factor a and also shifted by an offset of f0.
The transformation reads as
t4:z2+4z2+41/2→az2+4z2+41/2+f0,
where the scale factor a=-2 is determined by
the asymptotic behavior in the tail
(distance of 2Rmp to the axis),
and the shift f0=1+2/3 is determined by the
stand-off distance at the subsolar point.
Combining the four transformations, the scalable
magnetopause shape is expressed as a conformal mapping with
f(z)=-24z2+4+z21/2+1+23.
Figure in the bottom right panel displays
the mapping for the transformation t4.
The magnetopause nose is flipped to the positive x side.
and is scaled to match the magnetopause asymptotic behavior
in the tail, and the lines are shifted by f0
along the x axis to meet the stand-off distance.
Magnetopause location
The magnetopause location is restored when choosing v=1
in z=u+iv.
It is also worth noting that the function obtained by
the transformation
t3 for v=1 can analytically be evaluated as
4z2+4+z2=4u2+3u2+32+(2u)2+u2-1+i2u1-4u2+32+(2u)2,
which is used to determine the scale factor a and the shift f0
in the transformation t4 by comparing with the square of f(z) as
f-f02=a24z2+4+z2.
Applications and limitsAccuracy check
The function f(z)=x+iy using Eq. ()
at v=1 overall reproduces the shape of the Shue model.
Figure shows the comparison between
the magnetopause model using Eq. ()
and the Shue model.
The subsolar point (x=1 at y=0)
and the asymptotic behavior (y→±2 at x→-∞)
are reproduced as well.
However, it should be noted that the difference occurs
from the Shue model at the terminator (x=0).
Our function shows the magnetopause
distance at the terminator at y=±1.3504, which is
slightly underestimating that of the Shue model,
y=±2=±1.4142. The difference between
the two models is about 4.7 %.
This mismatch indicates that the analytic continuation
is not exact but is of approximate nature. Thus, care
should be exercised when working on the magnetopause around
the terminator with our conformal mapping.
Magnetopause location generated by
Eq. (), with v=1
in z=u+iv (in black) and
the magnetopause model by Shue et al. (1997)
(dotted gray).
x and y coordinates are normalized to
the magnetopause stand-off distance Rmp.
Curvilinear grid generation
The analytic nature of our function (Eq. )
can be used for the curvilinear grid generation around the magnetopause
for various numerical studies. Figure displays
the curvilinear grid generated by Eq. )
for values of u={0.5,0.7,…,1.4}
(the C-shaped curves) and
v={0.0,±0.2,…,±1.4}
(radial to the planet or perpendicular to the x axis).
The curves of constant u values are orthogonal
to that of constant v values. This property comes
from the fact that Eq. () is
an analytic function, which is one of the solutions
of the Laplace equation. In other words, Eq. ()
solves the Laplace equation for the given
magnetopause position (imposed by v=1).
Curvilinear grids generated
by the conformal map (Eq. )
around the magnetopause (v=1).
The C-shaped curves represent lines of
constant v values. The innermost curve corresponds
to a line of v=0.5. The v value for the curves are
shifted as 0.5, 0.7, …, 1.4 (10 curves are shown).
The radial curves represent constant u values,
and the curves are orthogonal to the curves of v values.
The subsolar direction Y=0 is given by u=0.
The curves are plotted for u values of
0, 0.2, 0.4, …, 4.4 (45 curves are shown).
Variation of tail shape
Qualitatively speaking, different tail shapes can also be
obtained by generalizing the square root operation
in t3 into a power with the index α as
t3′:z2+4+4z2+4→z2+4+4z2+4-4α.
The magnetopause coordinates are plotted
as grids for α of 0.2, 0.4, 0.6, and 0.8
in Fig. by using the scale factor and
the shift in t4. A converged tail shape is obtained
for α<0.5 and a divergent tail shape
for α>0.5, which is in agreement with the
Shue model (Eq. ).
Magnetopause grids generated for different values
of the power index α in t3′ transformation.
Values of u are 0.6 (innermost C-shaped curve),
0.8, …, 1.4 (outermost curve).
Summary and outlook
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
and the set of velocity potential and stream function
by
onto a more realistic magnetosheath domain
cf..
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.
Magnetopause location in Cartesian
In the case of α=0.5, the magnetopause position
in the Shue model is given by
R=Rmpℓ1+cosθ,
where ℓ=2.
Equation () is transformed
using the conversion rule in
Eqs. () and () into
the following normalized form:
r=ℓ1+xr,
where r=R/Rmp.
After squaring and exchanging r with 1+x/r,
Eq. () is expressed as
1+xr=ℓx2+y2.
We compute square of x/r in Eq. () and obtain
x2x2+y2=ℓx2+y2-12,
which can be arranged into a fourth-order
algebraic equation with respect to y as
y4+x2y2-2ℓy2-2ℓx2+ℓ2=0.
The factorized form of Eq. () reads
x2+y2y2-2ℓ+ℓ2=0.
Equation () delivers the Cartesian
representation of the Shue model in a convenient form
(Eq. ).
Code and data availability
No code or data were used in this paper.
Author contributions
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.
Competing interests
The contact author has declared that none of the authors has any competing interests.
Disclaimer
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Review statement
This paper was edited by Elias Roussos and reviewed by one anonymous referee.
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