For a first validation of the Gauss–Mie representation suggested here to reconstruct the current density \(\underline{j}^{sh}\) in the vicinity of the planned MPO orbits, simulated stationary magnetic field data and electric currents system information based on using the AIKEF hybrid code (electron fluid, kinetic ions) (Müller et al. 2011) are analyzed. This code has successfully been applied to several problems in Mercury’s plasma interaction (e.g., Exner et al. 2018, 2020). From the simulated magnetic field data, the current system is derived using the above described new approach and the results are compared with the electric currents directly derived from the simulation.

Mercury’s internal magnetic field is modeled as a dipole field with an internal dipole moment of \(-190\,{\rm nT}\), which is shifted northward by \(0.2\,R_{\rm M}\). This field can equivalently be described as a multipole field with the internal Gauss coefficients \(g_1^0=-190\,{\rm nT}\) for the dipole field, \(g_2^0=-78\,{\rm nT}\) for the quadrupole field and \(g_3^0=-20\,{\rm nT}\) for the octupole field (Anderson et al. 2012; Wardinski et al. 2019). The internal Gauss coefficients determining the internal stationary magnetic field are implemented in the simulation code and the plasma interaction of Mercury with the solar wind is simulated. The interplanetary magnetic field with a magnitude of \(B_{{\rm IMF}}=20\,{\rm nT}\) (Winslow et al. 2013) is orientated along the vector \(\left( x,y,z \right) ^T=\left( 0, 0, -1\right) ^T\) in the Mercury-Anti-Solar-Orbital coordinate system (MASO), i.e., the *x*-axis is orientated towards the nightside of Mercury (away from the sun), the *z*-axis is orientated parallel to the rotation axis (i.e., antiparallel to the internal dipole moment) and the *y*-axis completes the right hand system. The solar wind velocity of \(u_{{\rm sw}}=400\,{\rm km/s}\) points along the *x*-axis and the solar wind proton density was chosen to \(n_{{\rm sw}}=30\,{\rm cm}^{-3}\) (cf. Winslow et al. (2013)). Mercury’s outer mantle (\(0.7\,R_{{\rm M}}\le r\le 1\,R_{{\rm M}}\)) is modeled with a radially symmetric planetary resistivity profile with a resistivity of \(\eta _S\approx 840\,{\rm k}\Upomega {m}\) (or conductivity of \(10^{-6}\,{\rm S/m}\)) at the surface and the core possesses a vanishing resistivity (Jia et al. 2015; Exner et al. 2018, 2020). The influence of Mercury’s exosphere is neglected, since the plasma interaction is only affected by the exosphere in the case of an extremely high ion density and the absolute value of the density is unknown (Exner et al. 2020). The resulting simulated magnetic field data are evaluated along the ellipsoid

$$\begin{aligned} {\mathcal {E}}=\left\{ \left( x,y,z \right) ^T\,\Big |\,\frac{(x-\,0.2\,R_{{\rm M}})^2}{(1.4\,R_{{\rm M}})^2}+\frac{y^2}{(1.2\,R_{{\rm M}})^2}+\frac{z^2}{(1.16\,R_{{\rm M}})^2}=1\right\} , \end{aligned}$$

(20)

which describes the envelope of the elliptical MPO orbit rotated in longitudinal direction from \(-50^\circ\) (afternoon/post-midnight sector) over \(0^\circ\) (noon/midnight, *x*–*z*-plane) to \(50^\circ\) (morning/pre-midnight sector) around the rotation axis (*z*-axis).

For the reconstruction of the current density \(\underline{j}^{sh}\) from the magnetic field data \(\underline{B}\), the scalar potentials \(\Phi ^i\), \(\Phi ^e\) of the internal and external fields are expanded into spherical harmonics up to the third degree and order representing the internal/external dipole, quadrupole and octupole field. The scalar functions \(\Psi _P^{sh}\), \(\Psi _T^{sh}\) of the poloidal and toroidal magnetic fields are expanded into spherical harmonics up to the fourth degree and order. Furthermore, the scalar function \(\Psi _P^{sh}\) is expanded into a Taylor series with respect to the radius *r* in the vicinity of the mean radius *b* of the spherical shell up to the first order. The function \(\Psi _T^{sh}\) is cut off at the zeroth order for the radius describing the influence of the radial currents (Toepfer et al. 2021a) since it is expectable that the radial currents are the dominating poloidal currents in the vicinity of the surface. Thus, the magnetic field is modeled with 102 expansion coefficients in the data analysis, i.e., 15 internal Gauss coefficients, 15 external Gauss coefficients, 24 toroidal coefficients, 48 poloidal coefficients. As a proof of concept, the maximum orders of the series expansions are chosen to achieve a reasonable qualitative agreement between the simulated and the reconstructed currents. For a detailed quantitative analysis, it is worthwhile to incorporate higher orders of the series expansions. The wanted expansion coefficients are estimated with Capon’s method (Capon 1969; Motschmann et al. 1996; Toepfer et al. 2020a, b, 2021b) from the simulated magnetic field data. Afterwards, the estimated expansion coefficients \(\underline{g}_P^{sh}\) for the poloidal magnetic field and the coefficients \(\underline{g}_T^{sh}\) for the toroidal magnetic field are used to reconstruct the current density \(\underline{j}^{sh}\) (cf. Equation 15) within the shell.

The simulated and the reconstructed current density \(\underline{j}^{sh}\) along the ellipsoid \({\mathcal {E}}\) on the nightside and dayside of Mercury are displayed in Figs. 1 and 2, respectively. The simulated current system is dominated by equatorial currents flowing from dawn (\(y>0\)) to dusk (\(y<0\)) at the nightside (red arrows in Fig. 1a) and vice versa at the dayside (red arrows in Fig. 2a). The reconstructed equatorial currents (yellow arrows in Fig. 1b and green arrows in Fig. 2b) follow this geometry. At the nightside, the equatorial currents split into a current flowing towards Mercury at the dawnside and depart from the planet at the duskside. The polar regions are characterized by horizontal currents. At the northern pole, the currents penetrate the surface in the region \(y>0\) (green arrows in Fig. 2a, yellow arrows in Fig. 2b) and leave the planet at \(y<0\) (red arrows in Fig. 2a, red arrows in Fig. 2b). It should be noted that the reconstructed currents are calculated analytically and therefore, the reconstruction is smoother than the numerical simulation. However, the reconstruction method reproduces the structure of the simulated currents (cf. Figs. 1b and 2b) and thus, the Mie representation is capable of describing the geometrical nature of the currents flowing in the vicinity of the MPO orbit.

Furthermore, the simulated and the reconstructed current densities are of the same order. The magnitude of the reconstructed current density with a maximum amplitude of about \(20\,{\rm nA}/{\rm m}^2\) up to \(25\,{\rm nA}/{\rm m}^2\) in the polar regions differs only by the factor two from the simulated magnitude with a maximum value of about \(40\,{\rm nA}/{\rm m}^2\) in the polar regions. Since the scalar functions \(\Psi _P^{sh}\) and \(\Psi _T^{sh}\) are expanded into spherical harmonics up to the fourth degree and order, it is expectable that the extension of the series expansion by degrees \(l>4\) will close the void between the reconstructed and the simulated magnitude of the current density. Figure 3 displays the poloidal \(\underline{j}_P^{sh}\) and toroidal parts \(\underline{j}_T^{sh}\) of the reconstructed current density. The toroidal currents are characterized by closed loops with the divergence-free nature of \(\underline{j}_T^{sh}\) being immediately visible. Since the radial (poloidal) currents cross the ellipsoid \({\mathcal {E}}\), from the first point of view it is not obvious how these currents are closed. Moreover, the magnitude of the toroidal current density with a maximum value of \(\big |\underline{j}_T^{sh} \big |\approx 15\,{\rm nA}/{\rm m}^2\) at the northern pole is smaller than the magnitude of the poloidal current density (\(\big |\underline{j}_P^{sh} \big |\approx 20\,{\rm nA}/{\rm m}^2\) up to \(25\,{\rm nA}/{\rm m}^2\) at the northern pole), but both the magnitudes are of the same order.

### Poloidal and toroidal current systems under different IMF-orientations

The simulated current density has reasonably been reconstructed from the simulated magnetic field data. Now, the question arises how the currents are related to the current system around Mercury presented in the section “Current system at Mercury” and how the poloidal currents are closed.

Besides the geometry of the internal magnetic field the current system depends on the direction of the interplanetary magnetic field (IMF) (Ganushkina et al. 2015; Milan et al. 2017; Ganushkina et al. 2018). To investigate the qualitative structure of the current system around Mercury, simulated stationary magnetic field data and current densities resulting from the plasma interaction of Mercury with the solar wind under different IMF-orientations, i.e., \(\underline{B}_{{\rm IMF}}=\pm 20\,{\rm nT}\,\underline{e}_x\), \(\underline{B}_{{\rm IMF}}=\pm 20\,{\rm nT}\,\underline{e}_y\), \(\underline{B}_{{\rm IMF}}=\pm 20\,{\rm nT}\,\underline{e}_z\), where \(\underline{e}_x\), \(\underline{e}_y\) and \(\underline{e}_z\) are the unit vectors of the corresponding main axes in the MASO system, are analyzed. Although Mercury’s magnetosphere is a highly dynamic system, the interplanetary magnetic field in the vicinity of Mercury’s orbit can be regarded as stationary for a time period of 20–40 min in times of a calm upstream solar wind (He et al. 2017; James et al. 2017). The resulting poloidal and toroidal current systems on the nightside and on the dayside of Mercury for each IMF-direction are sketched in Figs. 4 and 5.

The dayside current system is dominated by the Chapman–Ferraro currents \(\underline{j}_{{\rm cf}}\). These currents decompose into poloidal (orange) and toroidal currents (green). The toroidal part of the Chapman–Ferraro currents is characterized by closed loops so that this part of the current system is closed within the magnetospheric plasma. The nightside current system is dominated by the neutral sheet current \(\underline{j}_{{\rm ns}}\), simplified by just a single dawn–dusk-directed arrow. The magnetopause current \(\underline{j}_{{\rm mp}}\) as well as the neutral sheet current are connected via poloidal currents at the dawn (\(y>0\)) and duskside (\(y<0\)) so that both the currents are characterized by a poloidal topology. Since the structure of the magnetopause current and the neutral sheet current is determined by the internal magnetic field, these currents remain qualitatively unchanged for all IMF-directions. In the polar regions the poloidal part of the Chapman–Ferraro currents transits into poloidal Region 1 currents \(\underline{j}_{{\rm R1}}\). Due to the high conductivity within Mercury’s core, the Region 1 currents flowing in radial direction towards the planet at the dawnside and depart from the planet at the duskside are able to penetrate the surface and partially close via the core–mantle boundary, as proposed by Anderson et al. (2014). The currents flowing depart from the planet are closed within the nightside magnetosphere. Within the simulation presented here, no exosphere has been adopted. Considering the influence of Mercury’s exosphere on the current system, a significant portion of the Region 1 currents can be closed within a sodium exosphere of sufficient density, in similarity to Earth’s ionosphere, as shown by Exner et al. (2020).

Analogously to the plasma interaction of the Earth’s magnetic field with the solar wind, there occur field-aligned currents \(\underline{j}_{{\rm fac}}\) at the northern and at the southern pole in the case of a non-vanishing \(B_y\)-component of the IMF (cf. Figs. 4b and 5b) (Leontyev and Lyatsky 1974; Trondsen et al. 1999; Liou and Mitchell 2019). Due to the frozen-in theorem the motional electric field is given by

$$\begin{aligned} \underline{E}_{{\rm IMF}}=-\underline{u}_{{\rm sw}}\times \underline{B}_{{\rm IMF}}. \end{aligned}$$

(21)

In the case of \(B_y\ne 0\) (\(B_x=B_z=0\)), the electric field is orientated (anti-)parallel to the *z*-axis, since the solar wind velocity is orientated along the *x*-axis. We find that the field-aligned currents flow towards the planet at the northern pole and depart from the planet at the southern pole in the case of \(B_y>0\) and vice versa in the case of \(B_y<0\).

Furthermore, the IMF-direction determines the symmetry of the magnetosphere. A non-vanishing \(B_y\)-component results in dawn–dusk asymmetries within the tail, whereas the \(B_x\)-component influences the north–south symmetry of the magnetosphere.

In the case of \(B_{{\rm IMF}}=20\,{\rm nT}\,\underline{e}_x\) as well as \(B_{{\rm IMF}}=\pm 20\,{\rm nT}\,\underline{e}_y\) (cf. Figs. 4a and b, 5b), within the reconstruction procedure there occur toroidal currents with an amplitude of about \(10\,{\rm nA}/{\rm m}^2\) that are oppositely directed to the (poloidal) magnetopause current at the dayside of Mercury and oppositely directed to the (poloidal) neutral sheet current at the nightside. This behavior is founded on the mathematical decomposition of the current density as a vector field. First of all, the quantities \(\underline{j}_P^{sh}\) and \(\underline{j}_T^{sh}\) are mathematical vector fields, which do not mandatorily exist as physical quantities. Thus, a vanishing current at a point \(\underline{x}_0\) can equivalently be described as a superposition of oppositely directed non-vanishing poloidal and toroidal currents with the same magnitude so that

$$\begin{aligned} \underline{j}_P^{sh}(\underline{x}_0)+\underline{j}_T^{sh}(\underline{x}_0) =0 \end{aligned}$$

(22)

is valid locally as schematically sketched in Fig. 6a and b. Due to the finite spatial extent of the currents, at least partially they can flow independently of each other in regions this side of \(\underline{x}_0\) or beyond \(\underline{x}_0\) (cf. Fig. 6c). Although the total (physical) current \(\underline{j}\) is determined by the superposition of the poloidal and the toroidal current, the mathematical decomposition of the current density enables us to predict the potential origin of the current and to analyze the potential trajectories of the particles carrying the current. Thus, the toroidal current flowing antiparallel to the magnetopause current as well as antiparallel to the neutral sheet current can be interpreted as a ring current which superposes with the poloidal neutral sheet current and the magnetopause current. This ring current should be partially trackable along the planned MPO orbits.

In terms of the poloidal–toroidal decomposition, the current system around Mercury as sketched in Figs. 4 and 5 can be summarized as follows. The poloidal current (orange) flows towards Mercury at the dawnside (\(y>0\)) and splits into the neutral sheet current \(\underline{j}_{{\rm ns}}\), the Region 1 current \(\underline{j}_{{\rm R1}}\) and the dayside magnetopause current \(\underline{j}_{{\rm mp}}\). The Region 1 currents are able to penetrate the surface. These currents close via the core–mantle boundary and leave the planet at the duskside (\(y<0\)), where they reconnect with the magnetopause current and the neutral sheet current. This poloidal current system remains qualitatively similar for all IMF-directions. The toroidal Chapman–Ferraro currents \(\underline{j}_{{\rm cf}}\) are characterized by closed loops (green). The position of these loops varies for different IMF-orientations. In the case of \(B_{{\rm IMF}}=20\,{\rm nT}\,\underline{e}_x\) as well as \(B_{{\rm IMF}}=\pm 20\,{\rm nT}\,\underline{e}_y\) the equatorial parts of the loops at the dayside are orientated antiparallel to the magnetopause current as well as antiparallel to the neutral sheet current at the nightside. For a non-vanishing \(B_y\)-component, the (poloidal) field-aligned currents \(\underline{j}_{{\rm fac}}\) at the northern and at the southern pole also penetrate the surface and close via the core–mantle boundary.

Although the plasma interaction of Mercury with the solar wind does not linearily depend on the IMF-orientation, it is expectable that the resulting current system for any other IMF-orientation can be constructed by superposing the cases presented above.