The solenoidality of the magnetic field \(\underline{B}\) guarantees the existence of a vector potential \(\underline{A}\), so that
$$\begin{aligned} \underline{B}=\partial _{\underline{x}}\times \underline{A} \end{aligned}$$
(1)
holds, where \(\partial _{\underline{x}}\) is the spatial derivative. Using spherical coordinates with radius \(r\in \left[ 0,\infty \right)\), longitude \(\lambda \in \left[ 0,2\pi \right]\) and colatitude angle \(\theta \in \left[ 0,\pi \right]\) the vector potential can be seperated into its component \(\Psi _T\underline{r}\) parallel to \(\underline{r}=r\,\underline{e}_r\), where \(\underline{e}_r\) is the unit vector in radial direction, and its components \(\underline{F}\times \underline{r}\) perpendicular to \(\underline{r}\), yielding
$$\begin{aligned} \underline{A}=\Psi _T\underline{r}+\underline{F}\times \underline{r}+\partial _{\underline{x}}\varphi \end{aligned}$$
(2)
with the scalar functions \(\Psi _T\) and \(\varphi\) as well as a vector field \(\underline{F}\) (Jacobs 1987; Krause and Rädler 1980). Because of \(\partial _{\underline{x}}\times \partial _{\underline{x}}\varphi =0\), the function \(\varphi\) can be chosen properly so that \(\partial _{\underline{x}}\times \underline{F}=0\) holds, and therefore
$$\begin{aligned} \underline{F}=\partial _{\underline{x}}\Psi _P \end{aligned}$$
(3)
with a scalar function \(\Psi _P\) without changing the magnetic field (Jacobs 1987; Krause and Rädler 1980). The vector potential results in
$$\begin{aligned} \underline{A}&=\Psi _T\underline{r}+\partial _{\underline{x}}\Psi _P\times \underline{r} \end{aligned}$$
(4)
$$\begin{aligned}&=\Psi _T\underline{r}+\partial _{\underline{x}}\times \left( \Psi _P\underline{r} \right) . \end{aligned}$$
(5)
Substituting \(\underline{A}\) into Eq. (1) delivers
$$\begin{aligned} \underline{B}=\partial _{\underline{x}}\times \left( \Psi _T\underline{r} \right) +\partial _{\underline{x}}\times \left[ \partial _{\underline{x}}\times \left( \Psi _P\underline{r} \right) \right] , \end{aligned}$$
(6)
which is called the Mie representation of the magnetic field (Backus 1986; Backus et al. 1996).
The first term on the right hand side in Eq. (6)
$$\begin{aligned} \underline{B}_T=\partial _{\underline{x}}\times \left( \Psi _T\underline{r} \right) \end{aligned}$$
(7)
is the toroidal part of the field and the second term
$$\begin{aligned} \underline{B}_P=\partial _{\underline{x}}\times \left[ \partial _{\underline{x}}\times \left( \Psi _P\underline{r} \right) \right] \end{aligned}$$
(8)
is the poloidal part of \(\underline{B}\).
From the definition of \(\underline{B}_T\) and \(\underline{B}_P\) it is clear that \(\underline{B}_T\) is perpendicular to \(\underline{B}_P\) and also perpendicular to \(\underline{r}\). Therefore, the toroidal part of the field does not have a radial component. Furthermore, poloidal magnetic fields are generated by toroidal currents and vice versa:
$$\begin{aligned} \partial _{\underline{x}}\times \underline{B}_T=\partial _{\underline{x}}\times \left[ \partial _{\underline{x}}\times \left( \underline{r}\Psi _T \right) \right] \end{aligned}$$
(9)
is a poloidal vector and
$$\begin{aligned} \partial _{\underline{x}}\times \underline{B}_P=\partial _{\underline{x}}\times \left[ \partial _{\underline{x}}\times \left[ \partial _{\underline{x}}\times \left( \underline{r}\Psi _P \right) \right] \right] =\partial _{\underline{x}}\times \left[ \left( \partial _{\underline{x}}^2\Psi _P\right) \underline{r} \right] \end{aligned}$$
(10)
is a toroidal vector.
In curlfree regions where especially the poloidal current density \(\underline{j}_P\) vanishes, Ampère’s law reads as follows:
$$\begin{aligned} \mu _0 \underline{j}_P=\partial _{\underline{x}}\times \left[ \partial _{\underline{x}}\times \left( \Psi _T \underline{r} \right) \right] =\underline{r}\partial _{\underline{x}}^2\Psi _T+\partial _{\underline{x}}\partial _r\left( r\Psi _T \right) =0 \end{aligned}$$
(11)
or equivalently
$$\begin{aligned} \partial _{\underline{x}}\partial _r\left( r\Psi _T \right) =\underline{r}\partial _{\underline{x}}^2\Psi _T. \end{aligned}$$
(12)
Therefore, the gradient of \(\partial _r\left( r\Psi _T \right)\) has only a radial component, leaving us with
$$\begin{aligned} \partial _\lambda \Psi _T=0=\partial _\theta \Psi _T , \end{aligned}$$
(13)
so that the function \(\Psi _T\) solely depends on the radial distance from the center \(\left( \Psi _T=\Psi _T(r)\right)\) and thus, the toroidal magnetic field vanishes
$$\begin{aligned} \underline{B}_T=\partial _{\underline{x}}\times \left( \Psi _T \underline{r} \right) =\left( \partial _{\underline{x}}\Psi _T \right) \times \underline{r}=\partial _r\Psi _T\underline{e}_r\times \underline{r}=0. \end{aligned}$$
(14)
On the other hand, when the toroidal current density vanishes
$$\begin{aligned} \mu _0\underline{j}_T&=\partial _{\underline{x}}\times \partial _{\underline{x}}\times \left[ \partial _{\underline{x}}\times \left( \Psi _P\underline{r} \right) \right] =\partial _{\underline{x}}\times \left[ r\left( \partial _{\underline{x}}^2\Psi _P\right) \underline{e}_r \right] \end{aligned}$$
(15)
$$\begin{aligned}&=\frac{1}{r\sin (\theta )}\partial _\varphi \left[ r\partial _{\underline{x}}^2\Psi _P \right] \underline{e}_\theta \frac{1}{r}\partial _\theta \left[ r\partial _{\underline{x}}^2\Psi _P \right] \underline{e}_\varphi \end{aligned}$$
(16)
$$\begin{aligned}&=0 \end{aligned}$$
(17)
or equivalently
$$\begin{aligned} \partial _\varphi \partial _{\underline{x}}^2\Psi _P=0 \end{aligned}$$
(18)
and simulanously
$$\begin{aligned} \partial _\theta \partial _{\underline{x}}^2\Psi _P=0 \end{aligned}$$
(19)
the function \(\partial _{\underline{x}}^2\Psi _P\) solely depends on the radius. Therefore, the poloidal field
$$\begin{aligned} \underline{B}_P=\partial _{\underline{x}}\times \left[ \partial _{\underline{x}}\times \left( \Psi _P\underline{r} \right) \right] =\underline{r}\partial _{\underline{x}}^2\Psi _P+\partial _{\underline{x}}\partial _r\left( r\Psi _P \right) \ne 0 \end{aligned}$$
(20)
in general remains finite in currentfree regions.
Relation to the Gauss representation
When magnetic field data in curlfree regions (where \(\partial _{\underline{x}}\times \underline{B}=0\) holds) are analyzed, there exists a scalar potential \(\Phi\) so that the magnetic field can be written as
$$\begin{aligned} \underline{B}=\partial _{\underline{x}}\Phi =\partial _r\Phi \,\underline{e}_r\frac{1}{r}\partial _\theta \Phi \,\underline{e}_\theta \frac{1}{r\sin (\theta )}\partial _\lambda \Phi \,\underline{e}_\lambda , \end{aligned}$$
(21)
which is known as the Gauss representation of the magnetic field (Gauß 1839; Glassmeier and Tsurutani 2014). Simultanously, the Mie representation in due consideration that the toroidal magnetic field vanishes in curlfree regions is given by
$$\begin{aligned} \underline{B}&=\partial _{\underline{x}}\times \left[ \partial _{\underline{x}}\times \left( \Psi _P\underline{r} \right) \right] \end{aligned}$$
(22)
$$\begin{aligned}&=\frac{1}{r\sin (\theta )}\left[ \partial _\theta \left( \sin (\theta )\partial _\theta \Psi _P \right) \frac{1}{\sin (\theta )}\partial _\lambda ^2\Psi _P \right] \underline{e}_r \end{aligned}$$
(23)
$$\begin{aligned}&\quad +\frac{1}{r} \partial _\theta \partial _r\left( r\Psi _P \right) \underline{e}_\theta +\frac{1}{r\sin (\theta )} \partial _\lambda \partial _r\left( r\Psi _P \right) \underline{e}_\lambda \end{aligned}$$
(24)
Comparison of coefficients with Eq. (21) for \(\underline{e}_\theta\) shows that
$$\begin{aligned} \frac{1}{r}\partial _r\partial _\theta \left( r\Psi _P \right) =\frac{1}{r}\partial _\theta \Phi \end{aligned}$$
(25)
and analogously for \(\underline{e}_\lambda\)
$$\begin{aligned} \frac{1}{r\sin (\theta )} \partial _r\partial _\lambda \left( r\Psi _P \right) =\frac{1}{r\sin (\theta )}\partial _\lambda \Phi . \end{aligned}$$
(26)
Consequently, when \(\Psi _P\) is known, the scalar potential is given by
$$\begin{aligned} \Phi =\partial _r\left( r\Psi _P \right) . \end{aligned}$$
(27)
Comparison of the \(\underline{e}_r\)coefficients delivers
$$\begin{aligned} \frac{1}{r\sin (\theta )}\left[ \partial _\theta \left( \sin (\theta )\partial _\theta \Psi _P \right) +\frac{1}{\sin (\theta )}\partial _\lambda ^2\Psi _P \right] =\partial _r\Phi \end{aligned}$$
(28)
or equivalently
$$\begin{aligned} \partial _S^2\Psi _P=\frac{1}{r}\partial _r\Phi , \end{aligned}$$
(29)
where
$$\begin{aligned} \partial _S^2=\frac{1}{r^2\sin (\theta )}\partial _\theta \left( \sin (\theta )\partial _\theta \ \right) +\frac{1}{r^2\sin ^2(\theta )}\partial _\lambda ^2 \end{aligned}$$
(30)
is the angular part of the Laplacian. For a given \(\Phi\), the function \(\Psi _P\) can be found by solving Eqs. (27, 29) simultanously.
As a consequence, the scalar function \(\Psi _P\) and the scalar potential \(\Phi\) are equivalent in curlfree regions and the Mie representation transists into the Gauss representation. Thus, the Gauss representation can be understood as a special case of the Mie representation.
Parameterization of the magnetic field
Assuming that the conductivity of Mercury’s mantle is negligibly small (like lunar regolith (Zharkova et al. 2020)), the planetary contribution to the field outside Mercury is purely poloidal. The currents flowing in the magnetosphere generate poloidal and toroidal magnetic fields that superpose with the curlfree planetary magnetic field. To be able to separate the planetary magnetic field out of the measured field and to parameterize it via the Gauss coefficients, a combined parametrization composed of the Mie and the Gauss representation (Gauss–Mie representation), which is based on the works of Backus (1986) and Olsen (1997) is necessary.
Suppose that the magnetic field in the vicinity of Mercury is measured within a spherical shell \(S(a,c)\) with inner radius \(a>R_\text {M}\), where \(R_\text {M}\) indicates the radius of Mercury, outer radius c and mean radius \(b=\frac{1}{2}\left( a+c \right)\) as displayed in Fig. 1. The shell can be constructed independently of the orbit’s geometry by conceptually covering the orbit of the spacecraft. Furthermore, the shell may include current carrying regions. Although the Mie representation enables us to analyze those currents, we focus on the analysis of Mercury’s internal magnetic field.
Due to the underlying geometry, the space around Mercury can be decomposed into three disjoint radial zones:

points in the region \(r<a\) below the shell

points in the region \(r>c\) above the shell

points in the region \(a \le r \le c\) inside the shell layer
Making use of the superposition principle the total magnetic field \(\underline{B}\) measured inside the shell layer (\(a \le r \le c\)) is a composition of the field \(\underline{B}_{\underline{j}\in \left[ a,c \right] }\) generated by currents flowing inside the shell and the field \(\underline{B}_{\underline{j}\notin \left[ a,c \right] }\) generated by currents flowing outside the shell. Again considering the underlying geometry, the second part can be divided into an internal part \(\underline{B}^i\) resulting from currents flowing in the region \(r<a\) and an external part \(\underline{B}^e\) resulting from currents flowing in the region \(r>c\), so that
$$\begin{aligned} \underline{B}_{\underline{j}\notin \left[ a,c \right] }=\underline{B}^i+\underline{B}^e. \end{aligned}$$
(31)
As \(\underline{B}^i\) and \(\underline{B}^e\) have their sources beyond the shell they are purely poloidal and especially nonrotational within the shell. Thus, there exist scalar potentials \(\Phi ^i\) and \(\Phi ^e\) so that the field can be parameterized in the shell via the Gauss representation resulting in
$$\begin{aligned} \underline{B}^i= \; \partial _{\underline{x}}\Phi ^i \end{aligned}$$
(32)
and
$$\begin{aligned} \underline{B}^e=  \; \partial _{\underline{x}}\Phi ^e, \end{aligned}$$
(33)
where the scalar potentials are given by (Gauß 1839; Glassmeier and Tsurutani 2014)
$$\begin{aligned} & \Phi ^i=R_\text {M}\sum _{l=1}^\infty \sum _{m=0}^l\left( \frac{R_\text {M}}{r} \right) ^{l+1} \left[ g_l^m\,\cos (m \lambda )+ h_l^m\,\sin (m \lambda )\right] P_l^m\left( \cos (\theta ) \right) \end{aligned}$$
(34)
and
$$\begin{aligned} &\Phi ^e=R_\text {M}\sum _{l=1}^\infty \sum _{m=0}^l\left( \frac{r}{R_\text {M}} \right) ^{l}\left[ q_l^m\,\cos (m \lambda )+ s_l^m\,\sin (m \lambda )\right] P_l^m\left( \cos (\theta ) \right) . \end{aligned}$$
(35)
The expansion coefficients \(g_l^m\) and \(h_l^m\) are the internal Gauss coefficients, the coefficients \(q_l^m\) and \(s_l^m\) are the external Gauss coefficients and \(P_l^m\) are the Schmidt normalized Legendre polynomials of degree l and order m. Since Mercury’s internal magnetic field is dominated by the internal dipole, quadrupole and octupole fields, the series expansions in Eqs. (34, 35) will be truncated at the degree \(l=3\) for the practical application.
It should be noted that the internal field \(\underline{B}^i\) is canonically described in a MercuryBodyFixed corotating coordinate system (MBF), whereas the external field \(\underline{B}^e\) is canonically described in a MercurySolarOrbital system (MSO) with the xaxis orientated towards the sun, the zaxis orientated parallel to the rotation axis, i.e. antiparallel to the internal dipole moment, and the yaxis completes the righthanded system (Heyner et al. 2020). Let \(\underline{x}_{\text {MSO}}=\left( x_{\text {MSO}},y_{\text {MSO}},z_{\text {MSO}}\right) ^T\) define the MSO coordinate system and let \(\underline{x}_{\text {MBF}}=\left( x_{\text {MBF}},y_{\text {MBF}},z_{\text {MBF}}\right) ^T\) be the coordinates of the corotating MBF system. Then, the internal parts of the field are given by
$$\begin{aligned} \underline{B}^i(\underline{x}_{\text {MBF}})=\partial _{\underline{x}_{\text {MBF}}}\Phi ^i(\underline{x}_{\text {MBF}})=\underline{\underline{H}}^i(\underline{x}_{\text {MBF}})\,\underline{g}^i, \end{aligned}$$
(36)
whereas the external parts are described in the MSO system, i.e.
$$\begin{aligned} \underline{B}^e(\underline{x}_{\text {MSO}})=\partial _{\underline{x}_{\text {MSO}}}\Phi ^e(\underline{x}_{\text {MSO}})=\underline{\underline{H}}^e(\underline{x}_{\text {MSO}})\,\underline{g}^e, \end{aligned}$$
(37)
where the terms of the series expansion are arranged into the matrices \(\underline{\underline{H}}^i\) and \(\underline{\underline{H}}^e\) and the corresponding Gauss coefficients are summarized into the vectors \(\underline{g}^i\) and \(\underline{g}^e\).
The corotating MBF system can be transformed into the MSO system via
$$\begin{aligned} \underline{x}_{\text {MSO}}=\underline{\underline{A}}\, \underline{x}_{\text {MBF}}, \end{aligned}$$
(38)
where \(\underline{\underline{A}}\) describes a rotation matrix around the zaxis depending on the angular velocity of Mercury’s selfrotation measured within the MSO system.
For the practical application it is convenient to describe both parts of the field in one coordinate system, for example the MSO system. The transformed data are given by
$$\begin{aligned} \underline{B}\left( \underline{x}_{\text {MSO}}\right)&=\underline{\underline{A}}\,\underline{B}^i\left( \underline{x}_{\text {MBF}}\right) +\underline{B}^e\left( \underline{x}_{\text {MSO}}\right) \end{aligned}$$
(39)
$$\begin{aligned}&=\underline{\underline{A}}\,\underline{\underline{H}}^i(\underline{x}_{\text {MBF}})\,\underline{g}^i+\underline{\underline{H}}^e(\underline{x}_{\text {MSO}})\,\underline{g}^e \end{aligned}$$
(40)
in the MSO system.
Since for the first validation the model will be applied to simulated magentic field data, it is useful to match the coordinate system of the parametrization with the coordinate system of the simulation. Therefore, in the following all parts of the magnetic field are described in a MercuryBodyFixed antisolar orientated coordinate system (MASO) with coordinates \(\underline{x}=(x,y,z)^T\), where the xaxis is orientated towards the nightside of Mercury (away from the sun), the zaxis is orientated antiparallel to the internal dipole moment and the yaxis completes the righthanded system, so that
$$\underline{x} = \left[ {\begin{array}{*{20}c} {  1} & 0 & 0 \\ 0 & {  1} & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]\underline{x} _{{{\text{MSO}}}} .$$
(41)
As already mentioned in the introduction ("Introduction" section) there is no currentfree shelllike region around Mercury (Olsen et al. 2010). The currents flowing in the shell generate toroidal \(\underline{B}_T^{sh}\) and poloidal \(\underline{B}_P^{sh}\) magnetic fields which superpose with \(\underline{B}^i\) and \(\underline{B}^e\). Thus, the total measured field within the shell is composed of four parts given by
$$\begin{aligned} \underline{B}&=\underline{B}_{\underline{j}\notin \left[ a,c \right] }+\underline{B}_{\underline{j}\in \left[ a,c \right] } =\underline{B}^i+ \underline{B}^e+\underline{B}^{sh}_T+\underline{B}^{sh}_P \end{aligned}$$
(42)
$$\begin{aligned}&=\partial _{\underline{x}}\Phi ^i\partial _{\underline{x}}\Phi ^e+\partial _{\underline{x}}\times \left[ \partial _{\underline{x}}\times \left( \underline{r}\Psi _P^{sh} \right) \right] +\partial _{\underline{x}}\times \left( \underline{r}\Psi _T^{sh} \right) , \end{aligned}$$
(43)
where each part of the field is described either by a scalar potential \(\Phi ^i\), \(\Phi ^e\) or a scalar function \(\Psi _P^{sh}\), \(\Psi _T^{sh}\).
The scalar potentials \(\Phi ^i\) and \(\Phi ^e\) are already parameterized by the Gauss coefficients. In the following, a proper parameterization for the scalar functions \(\Psi _T^{sh}\) and \(\Psi _P^{sh}\) is required. Because of the underlying spherical geometry it is straightforward to expand the functions into spherical harmonics
$$\begin{aligned} \Psi _T^{sh}=\sum _{l=1}^\infty \sum _{m=0}^l\left[ a_l^m(r)\,\cos (m \lambda )+ b_l^m(r)\,\sin (m \lambda )\right] P_l^m\left( \cos (\theta ) \right) \end{aligned}$$
(44)
and
$$\begin{aligned} \Psi _P^{sh}=R_\text {M}\sum _{l=1}^\infty \sum _{m=0}^l\left[ c_l^m(r)\,\cos (m \lambda )+ d_l^m(r)\,\sin (m \lambda )\right] P_l^m\left( \cos (\theta ) \right) , \end{aligned}$$
(45)
where \(a_l^m(r)\), \(b_l^m(r)\), \(c_l^m(r)\) and \(d_l^m(r)\) are the expansion coefficients which in general depend on the radius r and again \(P_l^m\) are the Schmidt normalized Legendre polynomials. Since the toroidal and poloidal fields can be locally generated by currents flowing in the shell the radial dependences of the fields and the expansion coefficients, respectively, are unknown.
Series expansion of the coefficients
Since the exact radial dependence of the expansion coefficients \(a_l^m(r)\) and \(b_l^m(r)\) is unknown, it is useful to expand these functions into a Taylor series in the vicinity of the mean radius b of the shell. Within this series expansion it is advisable not to incorporate the effect of all components of the poloidal current density to the toroidal magnetic field at once. Here, we first concentrate on the radial component of the current density and consider the horizontal components in higher orders of the Taylor series.
The toroidal magnetic field \(\underline{B}_T^{sh}\) is generated by poloidal currents \(\underline{j}_P\) (cf. Eq. (11)). Ampère’s law yields
$$\begin{aligned} \mu _0\underline{j}_P&=\partial _{\underline{x}}\times \underline{B}_T^{sh}=\partial _{\underline{x}}\times \left[ \partial _{\underline{x}}\times \left( \underline{r}\Psi _T^{sh} \right) \right] \end{aligned}$$
(46)
$$\begin{aligned}&=\frac{1}{r\sin (\theta )}\left[  \partial _\theta \left( \sin (\theta )\partial _\theta \Psi _T^{sh}\right) \frac{1}{\sin (\theta )}\partial _\lambda ^2\Psi _T^{sh} \right] \underline{e}_r \end{aligned}$$
(47)
$$\begin{aligned}&\quad +\frac{1}{r}\partial _\theta \partial _r\left( r\Psi _T^{sh} \right) \underline{e}_\theta +\frac{1}{r\sin (\theta )}\partial _\lambda \partial _r\left( r\Psi _T^{sh} \right) \underline{e}_\lambda . \end{aligned}$$
(48)
The components of the horizontal \(\underline{e}_\theta\) and \(\underline{e}_\lambda\)direction are proportional to \(\partial _r\left( r\Psi _T^{sh} \right)\). Therefore, the ansatz
$$\begin{aligned} a_l^m(r)=\frac{R_\text {M}}{r}\,\left[ a_l^m+a_{\,l}^{\prime \,m}\rho +{\mathcal {O}}(\rho ^2) \right] \end{aligned}$$
(49)
and
$$\begin{aligned} b_l^m(r)=\frac{R_\text {M}}{r}\,\left[ b_l^m+b_{\,l}^{\prime \,m}\rho +{\mathcal {O}}(\rho ^2) \right] , \end{aligned}$$
(50)
where \(\rho =\frac{ rb }{R_\text {M}}\) and \(a_l^m\), \(a_{\,l}^{\prime \,m}\), \(b_l^m\), \(b_{\,l}^{\prime \,m}\) are constants for each pair of l and m, is utilized. In the first order of the Taylor series expansion in the vicinity of the mean radius b, where \(\Psi _T^{sh}\sim \frac{1}{r}\), the horizontal components of \(\underline{j}_P\) vanish and only the contributions of the radial currents driving the toroidal magnetic field are considered (Olsen 1997). Using higher orders of the Taylor series, also the contributions of the horizontal components of \(\underline{j}_P\) to the toroidal magnetic field in the vicinity of the mean radius b can be incorporated.
The scalar function of the toroidal magnetic field results in
$$\begin{aligned} \Psi _T^{sh}=\frac{R_\text {M}}{r}\sum _{l=1}^\infty \sum _{m=0}^l\left[ \alpha _l^m\ +\alpha _{\,l}^{\prime \,m}\,\rho +{\mathcal {O}}(\rho ^2)\right] P_l^m\left( \cos (\theta ) \right) , \end{aligned}$$
(51)
where \(\alpha _l^m=a_l^m\,\cos (m \lambda )+ b_l^m\,\sin (m \lambda )\) and \(\alpha _l^{\prime \,m}=a_{\,l}^{\prime \,m}\,\cos (m \lambda )+ b_{\,l}^{\prime \,m}\,\sin (m \lambda )\) (Olsen 1997). Thereby, each order of the Taylor series is linked with an additional set of expansion coefficients \(a_l^m\), \(b_l^m\), \(a_{\,l}^{\prime \,m}\), \(b_{\,l}^{\prime \,m}\) and so on which can be reconstructed from the data in analogy to the Gauss coefficients.
From a mathematical point of view the scalar function \(\Psi _P^{sh}\) of the poloidal magnetic field \(\underline{B}_P^{sh}\) can be parametrized analogously to the toroidal counterpart. But within the reconstruction procedure the poloidal fields that are generated by toroidal currents flowing inside the shell cannot be distinguished from the internally and externally driven poloidal fields, since these fields follow the same topological structure. But when the half thickness of the shell, defined by \(h=(ca)/2\) is smaller than the length scale on which the toroidal currents change in radial direction, the shell is called a thin shell and the scalar function \(\Psi _P^{sh}\) of the poloidal field \(\underline{B}_P^{sh}\) vanishes within this thin shell approximation (Backus 1986; Backus et al. 1996) as illustrated in the following section.
The thin shell approximation
The thin shell approximation (Backus 1986; Backus et al. 1996) finally allows the separation of the poloidal field into its internal and external contributions.
Conferring to Eq. (6), the Mie representation for the magnetic field in the whole space \({\mathbb {R}}^3\) is given by
$$\begin{aligned} \underline{B}=\partial _{\underline{x}}\times \left( \Psi _T\underline{r} \right) +\partial _{\underline{x}}\times \left[ \partial _{\underline{x}}\times \left( \Psi _P\underline{r} \right) \right] . \end{aligned}$$
(52)
Following Ampère’s law the current density \(\underline{j}\) is also solenoidal and can as well be parameterized via the Mie representation resulting in
$$\begin{aligned} \mu _0\underline{j}=\partial _{\underline{x}}\times \left( \Gamma _T\underline{r} \right) +\partial _{\underline{x}}\times \left[ \partial _{\underline{x}}\times \left( \Gamma _P\underline{r} \right) \right] \end{aligned}$$
(53)
with related scalar functions \(\Gamma _T\) and \(\Gamma _P\).
Since the poloidal part of the current density corresponds with the curl of the toroidal magnetic field, the comparision of Eq. (53) with Eq. (9) shows that the scalar function \(\Psi _T\) and \(\Gamma _P\) are the same
$$\begin{aligned} \Psi _T=\Gamma _P. \end{aligned}$$
(54)
Analogously, the toroidal part of the current density corresponds with the curl of the poloidal magnetic field and the comparision of Eq. (53) with Eq. (10) shows that the functions \(\Psi _P\) and \(\Gamma _T\) are related via
$$\begin{aligned} \partial _{\underline{x}}^2\Psi _P=\Gamma _T, \end{aligned}$$
(55)
so that the function \(\Psi _P\) is given by the Green’s function method
$$\begin{aligned} \Psi _P(\underline{r})=\frac{1}{4\pi }\int _{{\mathbb {R}}^3}\frac{\Gamma _T(\underline{r}^{\prime })}{\big  \underline{r}\underline{r^{\prime }} \big }\,\text {d}^3r^{\prime }. \end{aligned}$$
(56)
Due to the underlying geometry the toroidal current density \(\underline{j}_T\) flowing in the whole space can be written as the sum of the toroidal currents \(\underline{j}_T^i\) flowing in the region \(r<a\), the toroidal currents \(\underline{j}_T^e\) flowing in the region \(r>c\) and the toroidal currents \(\underline{j}_T^{sh}\) flowing inside the spherical shell in the region \(a\le r \le c\), so that
$$\begin{aligned} \underline{j}_T=\underline{j}_T^i+\underline{j}_T^e+\underline{j}_T^{sh}. \end{aligned}$$
(57)
Thereby, the Mie representation of each part is given by
$$\begin{aligned}&\underline{j}_T^i=\partial _{\underline{x}}\times \left( \Gamma _T^i\underline{r}\right) \end{aligned}$$
(58)
$$\begin{aligned}&\underline{j}_T^e=\partial _{\underline{x}}\times \left( \Gamma _T^e\underline{r}\right) \end{aligned}$$
(59)
$$\begin{aligned}&\underline{j}_T^{sh}=\partial _{\underline{x}}\times \left( \Gamma _T^{sh}\underline{r}\right) \end{aligned}$$
(60)
with \(\Gamma _T^i=\Gamma _T\,\chi _{\left[ r<a \right] }\), \(\Gamma _T^e=\Gamma _T\,\chi _{\left[ r>c \right] }\) and \(\Gamma _T^{sh}=\Gamma _T\,\chi _{\left[ a<r<c \right] }\), where
$$\begin{aligned} \chi _I(x)=\left\{ \begin{array}{ll} 1 &{} ,\, x\in I\\ 0 &{} ,\, x\notin I \end{array}\right. \end{aligned}$$
(61)
is the indicator function of the interval I. Using this segmentation the scalar function of the poloidal field can be rewritten as
$$\begin{aligned} \Psi _P(\underline{r})&= \frac{1}{4\pi }\int _{V(r{\prime }<a)}\frac{\Gamma _T^i(\underline{r}^{\prime })}{\big  \underline{r}\underline{r^{\prime }} \big }\,\text {d}^3r^{\prime }\nonumber \\&\quad +\frac{1}{4\pi }\int _{V(r{\prime }>a)}\frac{\Gamma _T^e(\underline{r}^{\prime })}{\big  \underline{r}\underline{r^{\prime }} \big }\,\text {d}^3r^{\prime }\nonumber \\&\quad +\frac{1}{4\pi }\int _{S(a,c)}\frac{\Gamma _T^{sh}(\underline{r}^{\prime })}{\big  \underline{r}\underline{r^{\prime }} \big }\,\text {d}^3r^{\prime }. \end{aligned}$$
(62)
Thus, the part of the scalar function that corresponds to the poloidal magnetic field which is generated inside the shell is given by
$$\begin{aligned} \Psi ^{sh}_P(\underline{r})=\frac{1}{4\pi }\int _{S(a,c)}\frac{\Gamma _T^{sh}(\underline{r}^{\prime })}{\big  \underline{r}\underline{r^{\prime }} \big }\,\text {d}^3r^{\prime }. \end{aligned}$$
(63)
Since \(2b=a+c\) and \(2h=ca\) the bounds of integration can be rewritten as \(a=bh\) and \(c=b+h\), so that
$$\begin{aligned} \Psi _P^{sh}(\underline{r})=\frac{1}{4\pi }\int _{S(bh,b+h)}\frac{\Gamma _T^{sh}(\underline{r}^{\prime })}{\big  \underline{r}\underline{r^{\prime }} \big }\,\text {d}^3r^{\prime }. \end{aligned}$$
(64)
Analogously to the scalar function \(\Psi _T\), the function \(\Gamma _T\) can be expanded into a Taylor series in the vicinity of the mean radius b, resulting in
$$\begin{aligned} \Gamma _T^{sh}(\underline{r}^\prime )=\sum _{n=0}^\infty \frac{1}{n!}\partial ^n_{r^\prime } \Gamma _T^{sh}\big _{r^\prime =b}(r^\prime b)^n. \end{aligned}$$
(65)
Substituting the Taylor series into the function \(\Psi _P^{sh}\) delivers
$$\begin{aligned} \Psi _P^{sh}(\underline{r})=\frac{1}{4\pi }\sum _{n=0}^\infty \frac{1}{n!}\int _{S(bh,b+h)}\partial ^n_{r^\prime } \Gamma _T^{sh}\big _{r^\prime =b}\frac{(r^\prime b)^n}{\big  \underline{r}\underline{r^{\prime }} \big }\,\text {d}^3r^{\prime }. \end{aligned}$$
(66)
For the further evaluation of the integral it is assumed that the derivatives of the toroidal currents with respect to r are bounded, i.e., there exists a constant \(L>0\), so that
$$\begin{aligned} L^n \big  \partial ^n_r \underline{j}_T \big \le \big  \underline{j}_T \big  \end{aligned}$$
(67)
for \(n\in {\mathbb {N}}\). Thus, L represents the length scale on which the toroidal currents change in radial direction. From
$$\begin{aligned} \underline{j}_T&=\partial _{\underline{x}}\times \left( \Gamma _T\underline{r} \right) \\ &=\frac{1}{\sin (\theta )}\partial _\lambda \Gamma _T\,\underline{e}_\theta \partial _\theta \Gamma _T \,\underline{e}_\lambda\\ & =\left( \frac{1}{\sin (\theta )}\underline{e}_\theta \partial _\lambda \underline{e}_\lambda \partial _\theta \right) \Gamma _T \end{aligned}$$
(68)
and therefore
$$\begin{aligned} \partial _r^n\underline{j}_T=\left( \frac{1}{\sin (\theta )}\underline{e}_\theta \partial _\lambda \underline{e}_\lambda \partial _\theta \right) \partial _r^n \Gamma _T \end{aligned}$$
(69)
it follows that
$$\begin{aligned} L^n \big  \partial ^n_r \Gamma _T^{sh} \big \le \big  \Gamma _T^{sh} \big  \end{aligned}$$
(70)
or equivalently
$$\begin{aligned} \big  \partial ^n_r \Gamma _T^{sh} \big \le \frac{\big  \Gamma _T^{sh} \big }{L^n}. \end{aligned}$$
(71)
Since \(r^\prime \in \left[ bh,b+h \right]\), each summand within the Taylor series can be estimated upwards via
$$\begin{aligned} \big \partial ^n_{r^\prime } \Gamma _T^{sh}(b)\big \cdot \big r^\prime b\big ^n\le \frac{\big  \Gamma _T^{sh}(b)\big }{L^n}\cdot h^n \end{aligned}$$
(72)
delivering for Eq. (66)
$$\begin{aligned} &\big  \Psi _P^{sh}(\underline{r}) \big \le \frac{1}{4\pi }\sum _{n=0}^\infty \frac{1}{n!}\left( \frac{h}{L} \right) ^n\,\max _{\theta ^\prime ,\lambda ^\prime }\big \{\big  \Gamma _T^{sh}(b)\big  \big \}\int _{S(bh,b+h)}\frac{1}{\big  \underline{r}\underline{r^{\prime }} \big }\,\text {d}^3r^{\prime }. \end{aligned}$$
(73)
The integral in Eq. (73) may be evaluated for any \(\underline{r}\in {\mathbb {R}}^3\) but we restrict to \(\underline{r}\) inside the shell as only there the magnetic field is measured. Then, the remaining integral results in
$$\begin{aligned}\begin{aligned} \int _{S(bh,b+h)}\frac{1}{\big  \underline{r}\underline{r^{\prime }} \big }\,\text {d}^3r^{\prime }=&\frac{4\pi }{3}\,\frac{1}{r}\left[ r^3(bh)^3 \right] +2\pi \left[ (b+h)^2r^2 \right] , \end{aligned}\end{aligned}$$
(74)
utilizing that the coordinate system can be chosen properly so that \(\theta ^\prime\) defines the angle between \(\underline{r}\) and \(\underline{r}^\prime\). Therefore
$$\begin{aligned} \Psi _{P}^{{sh}} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{r} ) \le \sum\limits_{{n = 0}}^{\infty } {\frac{1}{{n!}}} \left( {\frac{h}{L}} \right)^{n} {\mkern 1mu} \max _ {\theta ^{\prime } ,\lambda ^{\prime }} \left\{ {\Gamma _{T}^{{sh}} (b)} \right\}\left\{ {\frac{1}{3}\left[ {r^{2}  \frac{{(b  h)^{3} }}{r}} \right] + \frac{1}{2}\left[ {(b + h)^{2}  r^{2} } \right]} \right\} . \\ \end{aligned}$$
(75)
For all \(r\in \left[ bh,b+h \right]\), the function
$$\begin{aligned} f(r)=\frac{1}{3}\left[ r^2\frac{(bh)^3}{r} \right] +\frac{1}{2}\left[ (b+h)^2r^2\right] \end{aligned}$$
(76)
is nonnegative and reaches its maximum value at \(r=bh\) with the related function value \(f(bh)=2bh\). Therefore, an upper bound for \(\Psi _P^{sh}\) can finally be estimated as
$$\begin{aligned} \big  \Psi _P^{sh}(\underline{r}) \big &\le 2bh \sum _{n=0}^\infty \frac{1}{n!}\left( \frac{h}{L} \right) ^n\,\max _{\theta ^\prime ,\lambda ^\prime }\big \{\big  \Gamma _T^{sh}(b)\big  \big \} \\ &=2bL\sum _{n=0}^\infty \frac{1}{n!}\left( \frac{h}{L} \right) ^{n+1}\,\max _{\theta ^\prime ,\lambda ^\prime }\big \{\big  \Gamma _T^{sh}(b)\big  \big \} . \end{aligned}$$
(77)
When \(h\ll L\), the spherical shell is called a thin shell (Backus 1986; Backus et al. 1996) and the scalar function \(\Psi _P^{sh}\) of the poloidal magnetic field \(\underline{B}_P^{sh}\) vanishes. The scalar function \(\Psi _T^{sh}\), however, remains finite for all h as shown in Appendix A. Thus, if the shell may be regarded as thin, then the contribution of the toroidal currents in this shell to the poloidal magnetic field may be neglected. The poloidal magnetic field is mainly driven by currents beyond the shell. The contribution of the poloidal currents in this shell to the toroidal magnetic field may not be neglected.
From a first point of view the thin shell approximation is not an intuitive approximation. Considering the above presented nature of poloidal and toroidal magnetic fields it can be understood as follows:
The toroidal magnetic field only exists within current carrying regions (cf. Eq. 14) and thus, it is measurable only within these regions. Therefore, the spatial extent of the regions where the poloidal currents flow does not influence the strength of the toroidal magnetic field. It solely depends on the strength of the poloidal current density. In contrast to the toroidal magnetic field, the poloidal magnetic field is also measurable in currentfree region. Thus, the poloidal field is a superposition of fields generated by currents flowing inside and outside the shell as well as currents flowing within the shell. This superposition is verified in Eq. (62). Therefore, the amount of the poloidal field generated by currents flowing within the shell has to be compared with the amount of the internal/external contributions. Furthermore, the poloidal field does not solely depend on the strength of the toroidal current density, since for the evaluation of the integrals also the volume where the current density flows is vital. Thus, a small toroidal current density that flows within a large volume outside/inside the shell can have a larger contribution to the field measured within the shell than a stronger current flowing within the thin shell.