Generation mechanism of Region 1 field-aligned current and energy transfer from solar wind to polar ionosphere

Abstract

Field-aligned current (FAC) is of importance in energy transfer from space to the Earth. The Region 1 current, which flows in the polar region, is the most significant one, but the generation region and mechanism are long-lasting questions. By using the global magnetohydrodynamics (MHD) simulation, we traced packets of the Alfvén waves traveling parallel to the magnetic field in the rest frame of moving plasma. The low-latitude magnetospheric boundary layer is found to be the major generation region, where the plasma pulls newly reconnected magnetic field lines. The generation region is far from the original magnetic field lines extending from the Region 1 FAC in the ionosphere because of low Alfvén velocity in the outer magnetosphere and beyond. The Region 1 FACs are surrounded by the integral curves of the Poynting vector (called S-curve). The S-curve shows a helix with its center moving toward the Earth. The Region 1 FACs, circular motion of plasma (probably related to the magnetospheric convection), and transfer of magnetic energy to the Earth are closely related with each other. The new method can be widely applied to search for the generation region of FACs and useful for better understanding of the dynamics of the magnetosphere.

1 Introduction

There are two types of electric currents flowing in the magnetosphere. One is the current flowing in the direction perpendicular to the magnetic field, and the other one flows in the field-aligned direction. The field-perpendicular currents are responsible to sustain the shape of the magnetosphere because of the action of the Lorentz force, which include, for example, the magnetopause current and the cross-tail current. The field-aligned currents (FACs) are important because they are accompanied with energy and charge transfer to the ionosphere, giving rise to electromagnetic disturbances. The currents flowing in the magnetosphere are well summarized and illustrated by Ganushkina et al. (2018).

The presence of the FACs was predicated by Birkeland (1908) on the basis of the requirement of the conservation of the current, and observational evidence for the FACs was first acquired in the 1960s. The FACs were observed by the ATS-1 satellite at geosynchronous orbit (Cummings et al. 1968). They interpreted that the FACs are connected to the ring current. The transverse magnetic disturbances observed at low altitudes in the polar region (Zmuda et al. 1966) were interpreted in terms FACs (Cummings and Dessler 1967). After substantial survey of the transverse magnetic disturbances, it became clear that two pairs of the FACs flow at low altitudes in the polar region (Iijima and Potemra 1976; Zmuda and Armstrong 1974). They are called Region 1 and Region 2 FACs, respectively. The Region 1 system consists of a downward current on the dawnside, and an upward current on the duskside. The Region 2 system, which is located equatorward of the Region 1 system, consists of the currents with their polarities opposite to those of the Region 1 system.

Observations have shown that the Region 1 FACs are intensified when the Kp index is high (Iijima and Potemra 1976), and the interplanetary magnetic field (IMF) is southward (Anderson et al. 2008; Juusola et al. 2014; Korth et al. 2010, 2014; Papitashvili et al. 2002; Weimer 2001, 2005). Figure 1 summarizes averaged FACs flowing at low altitudes in the polar region for different clock angles of IMF (Weimer 2005). The Region 1 and 2 currents are clearly shown, in particular, for the southward IMF conditions (negative z-direction). The magnitude of them increases with the southward component of the IMF.

Fig. 1

The averaged field-aligned currents (FACs) flowing in the polar ionosphere in the Northern Hemisphere for the solar wind speed of 450 km/s and the solar wind density of 4 cm⁻³. They are sorted by the clock angle of the interplanetary magnetic field (IMF). The top panels indicate the northward IMF cases, and the bottom ones indicate the southward IMF cases. The red color indicates the downward FACs, and the blue one indicates the upward FACs. The Sun is to the top. Total amount of the FACs is displayed at the right-top corner of each panel. The distribution is obtained by the empirical model developed by Weimer (2005)

The FACs are thought to be of importance in the energy transfer to the ionosphere (Strangeway et al. 2000a). According to observations, the parallel-component of the Poynting vector toward the Earth is accompanied with the FACs (Gary et al. 1994; Kelley et al. 1991; Strangeway et al. 2000b; Sugiura 1984). The energy is consumed as Joule heating in the ionosphere. The Joule heating rate in both hemispheres increases to ~ 0.5 TW and more during magnetic storms, or disturbed conditions (Lu et al. 1998; McHarg et al. 2005; Robinson and Zanetti 2021; Weimer 2005). To sustain the energy consumed in the ionosphere, a “generator” must exist somewhere in space.

Because of the continuity of the current, the FACs must be connected to the horizontal currents flowing in the ionosphere (Birkeland 1908). The Region 1 FAC supplies the positive space charge on dawnside, and negative charge on duskside while the plasma has to stay quasi-neutral. This results in the formation of two-cell pattern of the ionospheric electric potential that occupies the polar ionosphere (Fedder et al. 1998; Tanaka 1995). In the high-altitude ionosphere, plasma moves horizontally along the equipotential lines by the E × B drift in the counterclockwise direction on the dawnside and the clockwise direction on the duskside. In the low-altitude ionosphere, the ions tend to move in the direction of the electric field due to collision with neutral, whereas the electrons undergo the E × B drift. Consequently, the ionospheric Hall current flows in the direction opposite to that of the E × B drift. That is, it flows clockwisely on the dawnside and counterclockwisely on the duskside. The two-cell pattern of the Hall current is directly related to the large-scale equivalent current known as the DP2 system (Kikuchi et al. 1996; Nishida 1968; Obayashi and Nishida 1968). The DP2 system expands to the magnetic equator and the geomagnetic variations related to the DP2 take place coherently from pole to equator (Nishida 1968). This can be explained by prompt penetration of electric field from the polar region where the Region 1 FACs flows to the magnetic equator (Kikuchi et al. 1996). The expanded DP2 current to the magnetic equator causes strong electrojet flowing at magnetic equator (Kikuchi et al. 1996). The Pedersen current, which flows parallel to the ionospheric electric field, comes from the dawn cell to the dusk cell so as to connect the downward FAC and the upward FAC. The relation among the Region 1 FAC, the ionospheric Hall current, and the ionospheric Pedersen current is schematically drawn in Fig. 2.

Fig. 2

Relation among Region 1 FAC, the ionospheric Hall current (solid line), and the ionospheric Pedersen current (dashed line). The Hall current flows in the direction opposite to the E × B drift, whereas the Pedersen current flows in the direction of the electric field

The flow speed of the ionospheric plasma is extremely enhanced to 1 km/s in a latitudinally narrow region equatorward of the auroral oval. The fast flow is called a subauroral ion drift (SAID) (Anderson et al. 2001; Spiro et al. 1979), and a sub-auroral polarization stream (SAPS) (Foster and Vo 2002). The fast flow is related to the enhanced electric field in the region sandwiched by the Region 1 and Region 2 FACs (Anderson et al. 1993, 2001).

The origin of the Region 1 FAC is problematic. There are 3 distinct regions previously suggested as summarized in Fig. 3. One is the magnetospheric boundary layer at high latitudes and low latitudes (Bythrow et al. 1981; Coroniti and Kennel 1979; Cowley 1982; Eastman et al. 1976; Iijima 2000; Lee and Roederer 1982; Lotko et al. 1987; Sato and Iijima 1979; Siscoe et al. 1991; Siscoe and Sanchez 1987; Sonnerup 1980; Stern 1983; Strangeway et al. 2000a; Swift and Lee 1982; Troshichev 2003; Watanabe et al. 2019). In the boundary layer, the fast magnetosheath (solar wind) flow interacts with the slow magnetospheric flow, giving rise to the generation of the strong FACs. If this is the case, a question arises as to the dependence on the southward component of the IMF. Observations have shown that the overall magnitude of the Region 1 FACs increases with the magnitude of the southward IMF (Anderson et al. 2008; Juusola et al. 2014; Korth et al. 2010; Papitashvili et al. 2002; Weimer 2001). The second region is the bow shock (Fedder et al. 1997; Guo et al. 2008; Lopez et al. 2011; Wilder et al. 2015). The global magnetohydrodynamics (MHD) simulation enables us to draw the current line by line integral of the instantaneous current vector. Strangeway et al. (2000a) suggested that the antisunward flow near the reconnection site in the magnetopause is accelerated by the Lorentz force, driving the FACs. Some current lines extending from the Region 1 FACs are shown to reach the bow shock. The third is the plasma sheet in the magnetotail (Antonova and Ganushkina 1997). When the gradient of the plasma pressure is misaligned with the gradient of the magnitude of the magnetic field, the FACs appear to conserve the current continuity (Vasyliunas 1970). In other words, the remnant of the diamagnetic current is connected to the FACs. It is suggested that the Region 1 FAC appears when the specific condition is met in the plasma sheet.

Fig. 3

Previously suggested regions where the Region 1 FACs are generated; (1) high-latitude and low-latitude magnetospheric boundary layers, (2) the bow shock, and (3) plasma sheet in the tail region

Figure 4 summarizes methods that have been used to search for the generation region of the FACs. The most favorite method is to trace a magnetic field line from the ionosphere (Fig. 4a). It has been suggested that the amount of FACs can be evaluated by integrating the divergence of the field-perpendicular current J_⊥ along a magnetic field (Hasegawa and Sato 1979; Sato and Iijima 1979; Vasyliunas 1984). Since the Region 1 FACs are distributed widely in the polar ionosphere, the magnetic field lines extending from the Region 1 FACs pass through various regions, including the low-latitude and high-latitude boundary layers as well as the tail region. This leads to the conclusion that the origin of the FACs is located in these regions as summarized in Fig. 3.

Fig. 4

Methods to search for generation region of FAC; a tracing a magnetic field line, b tracing a current line, and c tracing a packet of Alfvén wave that is supposed to carry FACs. The cylinders represent the regions where the FACs are generated. The vertical lines indicate the magnetic field lines

Tracing the current lines (which are integral curves of the current density J) was recently made possible by the global MHD simulations (Fig. 4b). According to the MHD simulations, the current lines extending from the Region 1 FACs appear to pass through the inner part of the high-latitude boundary layer (Janhunen & Koskinen 1997; Siscoe et al. 2000; Tanaka 1995, 2000; Wilder et al. 2015). It has been thought that, on the analogy to an electric circuit, a generator (J·E < 0) and a load (J·E > 0) are attached to the current line, where E is the electric field. The ionosphere is a load, so that a generator must be present somewhere to supply the energy to the ionosphere. By tracing the current lines, the high-latitude boundary layer is suggested to be such a region where J·E < 0 (Janhunen and Koskinen 1997; Siscoe et al. 2000; Tanaka 1995, 2000; Wilder et al. 2015). In that region, the current is almost field-perpendicular, which implies that the field-perpendicular currents must be converted to the FACs somewhere else. Watanabe et al. (2019) pointed out that another generator process is required for the conversion from the field-perpendicular currents to the FACs, but the generator process is not clearly found along the current lines for the Region 1 FACs. The current line would be deflected by the field-perpendicular current, and there is a possibility that the current line does not always pass through the generation region of the FACs as illustrated in Fig. 4b.

The third method, in which a packet of the Alfvén wave is traced, is recently proposed by Ebihara and Tanaka (2022). When perturbations associated with the FACs are transported at the Alfvén velocity, the packet of the Alfvén wave travels parallel or antiparallel to the background magnetic field in the rest frame of the moving medium (Walker 2008). If the bulk velocity of plasma is zero, the packet will travel along the magnetic field line, and the result will be the same as the first mechanism, in which the magnetic field line is traced. If the bulk velocity of plasma is nonzero, the packet will be diverted from the original magnetic field line as illustrated in Fig. 4c. This implies that the generation region of the FAC is not necessarily attached to the magnetic field line extending from the FAC in the ionosphere. Using the global MHD simulation, Ebihara and Tanaka (2022) showed that the trajectory of the packet is totally different from the instantaneous magnetic field line extending from the Region 1 FAC in the ionosphere. That is because the Alfvén velocity is relatively low in the outer magnetosphere and beyond while the perpendicular plasma flow is fairly high. By tracing the packet, the generation region of the Region 1 FACs was found in the low-latitude boundary layer (flank magnetopause region) where the newly reconnected magnetic field lines are primarily pulled by the solar wind-originated plasma (Ebihara and Tanaka 2022). The involvement of the reconnection could explain reasonably the dependence of the magnitude of the Region 1 FACs on the polarity of IMF.

For the Region 1 FACs, it became clear that the first method (Fig. 4a) is not applicable because of nonnegligible bulk velocity of plasma. The second method (Fig. 4b) is probably useful for displaying the current system, but is not suitable for searching the generation region of the FACs. That is because the current line does not always pass through the generation region. The third method (Fig. 4c) seems to be the most suitable one so far. The aim of this paper is to overview the methodology and the results of the third method (Fig. 4c) for the problem of the origin of the Region 1 FACs. The associated energy transfer will be dealt with as well.

2 Simulation

2.1 General description of global MHD simulation

We used the global MHD simulation, REPPU (Tanaka 2015), which solves the MHD equation in 3-dimensional space by the finite volume method (FVM) (Tanaka 1994). The simulation domain extends to 200 R_E at midnight, 600 R_E at noon, and 400 R_E at dusk and dawn, where R_E is earth’s radii (≈ 6371 km). The inner boundary of the simulation domain is located at a sphere with a radius of 2.6 R_E. The sphere with the radius of 2.6 R_E was first divided into 12 pentagons. The dodecahedron is a basis of the grid system. Each pentagon was further divided into 5 triangles. We call this Level 1. There are 60 triangles in the inner boundary. We further divided each triangle into 4 triangles, which is called Level 2. There are 240 triangles. We repeated the division 4 times. The number of triangles is 61,440, and we call it Level 6. We stacked 320 triangular prisms outward from the inner boundary. The prisms are concentrated in the nightside plasma sheet so as to resolve the nightside plasma sheet with more grid points than on the dayside. We gave the solar wind parameters at the outer boundary of the simulation domain.

The inner boundary of the simulation domain is connected to the ionosphere by the dipole magnetic field lines. At the inner boundary, we calculated FACs by

$$ J_{||} = \frac{1}{{\mu_{0} }}\left[ {\nabla \times {\mathbf{B}}} \right]_{||} , $$

(1)

where B and μ₀ are the magnetic field and the magnetic constant, respectively. We mapped the FACs, the plasma pressure, and the temperature acquired at the inner boundary of the simulation domain to the ionosphere along the dipole magnetic field line. On the basis of these quantities as well as the solar zenith angle, we calculated the height-integrated ionospheric conductivities. For the ionospheric conductivity, we took into consideration of the following 3 sources. (1) The first one is the solar EUV that ionizes the dayside ionosphere. The conductivity is given as a function of the solar zenith angle. (2) The second one is the contribution from precipitating electrons that are associated with discrete aurorae, which are likely accelerated by field-aligned electric field. Since the MHD simulation cannot deal with kinetic processes and field-aligned electric field, we increased the ionospheric conductivity in accordance with the upward FACs. The contribution from the downward field-aligned current to the ionospheric conductivity is assumed to be 10 times lower than that from the upward field-aligned current. (3) The third one is the contribution from the precipitating particles that are associated with diffuse aurorae. These electrons are supposed to be scattered by waves. The MHD simulation cannot solve the scattering processes of trapped electrons and energy distribution of precipitating electrons. Instead, we calculated the conductivity in proportion to the square root of the plasma pressure and to the temperature of the magnetospheric plasma to the power of 1/4. With the diagonal and nondiagonal components of the conductivity tensor (Tsunomura 1999), we solved an elliptic partial differential equation for the continuity of the current and obtained the electric potential from pole to pole. The electric field was mapped from the ionosphere to the inner boundary of the simulation domain located at 2.6 R_E along the dipole magnetic field lines. Thus, the magnetosphere and the ionosphere are coupled with each other. The coupling and the procedure are described by Ebihara et al. (2014) in more detail.

There are some intrinsic limitations in the MHD simulation. First, microscopic processes are not taken into consideration. It has been suggested that in the reconnection site, the anomalous resistivity is increased by wave–particle interactions (Fujimoto et al. 2011; Treumann 2001), as well as the current-aligned instabilities, for instance, lower-hybrid drift instability and ion-acoustic waves (Ricci et al. 2004; Rowland and Palmadesso 1983). Secondly, in the MHD limit, the field-aligned electric field cannot be dealt with, and the displacement current is neglected. If the time scale of interest is much longer than the typical time scale of electron kinetics, the MHD approximation is probably not so far from the reality. Thirdly, we did not solve the kinetic equations for the electrons, and cannot calculate the ionization rate due to precipitating electrons correctly. Instead, we treated the contribution from the precipitating electrons to the ionospheric conductivity by functional forms as mentioned above. In spite of the limitations, global MHD simulations are capable of reproducing fundamental features that have been observed: (1) large- and meso-scale auroral patterns that are similar to sun-aligned arcs, a quiet arc, westward traveling surge, and beads (Ebihara and Tanaka 2016; Fedder et al. 1995; Palmroth et al. 2006; Raeder et al. 2012; Tanaka 2015), (2) intensification of auroral electrojets in the ionosphere (Ebihara et al. 2019; Lopez et al. 2001; Lyon et al. 1998; Raeder et al. 2001; Tanaka 2001; Wiltberger et al. 2000), (3) midlatitude positive bay of ground magnetic disturbance during the substorm (Tanaka 2015), (4) geomagnetic disturbances from pole to equator including overshielding condition during the substorm (Ebihara et al. 2014), (5) Region 1 and 2 current system (Tanaka 1995), (6) a pair of FACs associated with substorm expansion (Birn and Hesse 1991; Birn et al. 1999; Ebihara and Tanaka 2015, 2023; Tanaka 2015), and (7) formation of multiple near-Earth neutral lines (El-Alaoui et al. 2009).

2.2 Three criteria for generation of FACs

We have used the following three criteria to identify the generation region of the FACs (Ebihara and Tanaka 2022, 2023). The first criterion is based on the current continuity, that is,

$$ \nabla \cdot {\mathbf{J}}_{||} = - \nabla \cdot {\mathbf{J}}_{ \bot } . $$

(2)

The subscripts || and ⊥ denote the quantities parallel and perpendicular to the magnetic field, respectively. In the ideal MHD approximation, the field-perpendicular current consists of the diamagnetic current J_d and the inertial current J_i as

$$ \begin{aligned} {\mathbf{J}}_{ \bot } = & \frac{{{\mathbf{B}} \times \nabla P}}{{B^{2} }} + \frac{{\mathbf{B}}}{{B^{2} }} \times \left( {\rho \frac{{d{\mathbf{V}}}}{dt}} \right) \\ \equiv & {\mathbf{J}}_{d} + {\mathbf{J}}_{i} , \\ \end{aligned} $$

(3)

where P is the plasma pressure, V is the plasma bulk velocity, and ρ is the mass density of plasma. Equation (3) comes from the equation of momentum. With Eq. (2), we can specify the region where the field-perpendicular current is converted to the FACs and vice versa, but it does not tell us the reason why the FACs are generated.

The second criterion is a nonzero rate of change in the FACs. From Faraday’s and Ampère laws, the rate of change in the FACs is given by (Itonaga et al. 2000; Song and Lysak 2001a, b)

$$ \frac{{\partial J_{||} }}{\partial t} = - \frac{1}{{\mu_{0} }}\left[ {\nabla \times \nabla \times {\mathbf{E}}} \right]_{||} $$

(4)

which yields as (Itonaga et al. 2000)

$$ \frac{{\partial J_{||} }}{\partial t} = - \frac{{\nabla_{||} \left( {\nabla \cdot {\mathbf{E}}} \right)}}{{\mu_{0} }} + \frac{{\left( {\nabla^{2} {\mathbf{E}}} \right)_{||} }}{{\mu_{0} }}, $$

(5)

where E is the electric field. In the ideal MHD approximation, only E_⊥ is present. The first term on the right-hand side of Eq. (5) describes that when the field-aligned gradient of ∇⋅E_⊥ is present, the FACs are generated. The second term on the right-hand side arises when the magnetic field line is curved (Itonaga et al. 2000). The intuitive understanding of the second term is difficult. Suppose a cylindrical coordinate system. r, θ, and z refer to the distance from the z-axis of the cylinder, the azimuth angle in the plane orthogonal to z-axis, and the distance along the axis, respectively. Assume that the background magnetic field directs the θ direction. The θ-component of ∇²E in the cylindrical coordinate system is exactly given by

$$ \left( {\nabla^{2} {\mathbf{E}}} \right)_{\theta } = \nabla^{2} E_{\theta } + \frac{2}{{r^{2} }}\frac{{\partial E_{r} }}{\partial \theta } - \frac{{E_{\theta } }}{{r^{2} }}. $$

(6)

Because the magnetic field line directs the θ direction, the subscripts θ and r can refer to || and ⊥, respectively. Here, E_z is assumed to be zero. In the ideal MHD approximation, E_|| is absent, that is, E_θ = 0. In that situation, Eq. (6) yields

$$ \left( {\nabla^{2} {\mathbf{E}}} \right)_{\theta } = \frac{2}{{r^{2} }}\frac{{\partial E_{r} }}{\partial \theta }. $$

(7)

Equation (7) implies that (∇²E)_θ is nonzero when E_r varies in the θ direction. Considering the second term of the right-hand side of Eq. (5), we can say that the rate of change in the FACs is nonzero when the magnetic field is curved and E_⊥ varies along the magnetic field line. We rewrite Eq. (4) as

$$ \frac{{\partial J_{||} }}{\partial t} \approx - \frac{{\nabla_{||} \left( {\nabla \cdot {\mathbf{E}}_{ \bot } } \right)}}{{\mu_{0} }} + \frac{{2\nabla_{||} E_{ \bot } }}{{r_{c} \mu_{0} }}, $$

(8)

where r_c stands for the curvature radius of the magnetic field line. Here, we introduced an element of length of the magnetic field line ds (= rdθ), and ∇_||= ∂/∂s. The second term on the right-hand side of Eq. (8) is held when E_⊥ is dominated by the electric field in the normal direction of the curved magnetic field line, E_r. We used Eq. (4), not Eq. (8), to calculate the rate of change in the FACs. Note that the first and second criteria are not sufficient to specify the region where the FACs are generated because they can occur in the course of the propagation of the FACs.

The third criterion is related to the excitation of the Alfvén waves. We assume that the FACs are carried by the Alfvén waves (Kivelson 2004; Strangeway et al. 2000a), in which perpendicular perturbations of the magnetic field are transported parallel and antiparallel to the background magnetic field. To excite the Alfvén waves, magnetic energy perpendicular to the background magnetic field must be generated. Poynting's theorem states that the magnetic energy increases when J⋅E < 0. In the MHD approximation, it can be stated that

$$ \begin{aligned} {\mathbf{J}} \cdot {\mathbf{E}} = & {\mathbf{V}} \cdot \left( {{\mathbf{J}} \times {\mathbf{B}}} \right) \\ = & {\mathbf{V}} \cdot \left( {\frac{{\nabla \times {\mathbf{B}}}}{{\mu_{0} }} \times {\mathbf{B}}} \right) \\ = & {\mathbf{V}} \cdot \left[ {\frac{{B^{2} }}{{\mu_{0} }}\left( {{\mathbf{b}} \cdot \nabla } \right){\mathbf{b}} - \nabla_{ \bot } \left( {\frac{{B^{2} }}{{2\mu_{0} }}} \right)} \right] \\ \equiv & {\mathbf{V}} \cdot \left( {{\mathbf{F}}_{t} + {\mathbf{F}}_{m} } \right), \\ \end{aligned} $$

(9)

where b is the unit vector of the magnetic field (= B/B). Here, we used Ohm’s law, E =  − V × B. F_t and F_m are the magnetic tension force density and the magnetic pressure force density, respectively as

$$ \begin{gathered} {\mathbf{F}}_{t} \equiv \frac{{B^{2} }}{{\mu_{0} }}\left( {{\mathbf{b}} \cdot \nabla } \right){\mathbf{b}}, \hfill \\ {\mathbf{F}}_{m} \equiv - \nabla_{ \bot } \left( {\frac{{B^{2} }}{{2\mu_{0} }}} \right). \hfill \\ \end{gathered} $$

(10)

To excite the Alfvén waves, magnetic tension must be created. In other words, plasma must perform negative work against the magnetic tension force to excite the Alfvén waves, that is, V⋅F_t < 0. Note, however, that the excitation of the Alfvén waves does not always mean the generation of the FACs (Cravens 1997). Suppose a plasma slab perpendicular to the magnetic field. The slab has a finite thickness and moves in the perpendicular direction. The magnetic field lines are bent due to motion of the plasma slab, and the plasma performs negative work against the magnetic tension force, that is, V⋅F_t < 0. The bent magnetic field line implies the current flowing in the perpendicular direction. When the slab has finite width, FACs appear to conserve the current continuity. If the slab is extended to infinity, no FACs will flow because the divergent of the perpendicular current is zero. When V⋅F_m < 0, the magnetic field is compressed (Watanabe et al. 2019).

In summary, we assumed that the FACs are generated where the following criteria are satisfied. We call this an FAC dynamo.

• (i) Conversion from the field-perpendicular current to the FACs takes place. That is,

  $$ \nabla \cdot {\mathbf{J}}_{||} \ne 0. $$

  (11)

More specifically,

$$ \left\{ {\begin{array}{*{20}c} {\nabla \cdot {\mathbf{J}}_{||} > 0} & {{\text{for }}\left\{ {\begin{array}{*{20}c} {J_{||} > 0{\text{ and }}V_{A} > 0} \\ {J_{||} < 0{\text{ and }}V_{A} < 0} \\ \end{array} } \right.} \\ {\nabla \cdot {\mathbf{J}}_{||} < 0} & {{\text{for }}\left\{ {\begin{array}{*{20}c} {J_{||} < 0{\text{ and }}V_{A} > 0} \\ {J_{||} > 0{\text{ and }}V_{A} < 0} \\ \end{array} } \right.} \\ \end{array} } \right., $$

(12)

where J_|| =J_||·b, and V_A =V_A·b. V_A is the Alfvén velocity given by

$$ {\mathbf{V}}_{A} = \frac{{\mathbf{B}}}{{\sqrt {\mu_{0} \rho } }}, $$

(13)

where ρ is the charge density of plasma.

• (ii) Rate of change in the FACs determined by Faraday’s and Ampère’s laws is nonzero. That is,

  $$ \frac{{\partial J_{||} }}{\partial t} \ne 0. $$

  (14)

More specifically,

$$ \begin{array}{*{20}c} {\frac{{\partial J_{||} }}{\partial t} > 0} & {{\text{for }}J_{||} > 0,} \\ \\ {\frac{{\partial J_{||} }}{\partial t} < 0} & {{\text{for }}J_{||} < 0.} \\ \end{array} $$

(15)

∂J_||/∂t is given by Eq. (4).

• (iii) Plasma performs negative work against the magnetic tension force

  $$ {\mathbf{V}} \cdot {\mathbf{F}}_{t} < 0. $$

  (16)

The magnetic tension force density F_t is given by Eq. (10).

Earth’s intrinsic magnetic field lines come from the Southern Hemisphere, and go into the Northern Hemisphere. For example, the upward FAC (J_||< 0) in the Northern Hemisphere is supposed to be transported with the Alfvén waves parallel to the magnetic field (V_A > 0) from the FAC dynamo region that satisfies the following conditions: (i) ∇·J_||< 0, (ii) ∂J_||/∂t < 0, and (iii) V⋅F_t < 0.

2.3 Tracing packets of Alfvén waves

The group velocity of the Alfvén wave is parallel or antiparallel to the background magnetic field in the rest frame of the moving plasma (Walker 2008). We made an assumption that the perturbation associated with the FACs propagates at the characteristic velocity v (Neubauer 1980; Wright and Southwood 1987) as

$$ {\mathbf{v}}^{ \pm } = \pm {\mathbf{V}}_{A} + {\mathbf{V}}, $$

(17)

where V_A is the Alfvén velocity (= B/(μ₀ρ)^1/2). We introduced the packet that is supposed to carry the perturbation associated with the FACs, and traced the position of the packet r as

$$ {\mathbf{r}}^{ \pm } (t) = \int\limits_{0}^{t} {{\mathbf{v}}^{ \pm } ({\mathbf{r}},\tau )d\tau } + {\mathbf{r}}_{0} , $$

(18)

where r₀ is the initial position.

2.4 Solar wind condition

We imposed a northward IMF for the first 120 min to the boundary condition of the MHD simulation to obtain a quasi-steady magnetosphere. In this period, a solar wind density is 5.0 cm⁻³, a solar wind speed is 372 km s⁻¹, the y-component of the interplanetary magnetic field (IMF B) is − 2.5 nT, and the IMF B_z is 4.3 nT. IMF B_x was held at 0 throughout the calculation. After 120 min, IMF B_z was changed from 4.3 nT to − 3.0 nT.

3 Results

3.1 Region 1 FACs at ionosphere

Figure 5 shows the FACs at the ionosphere in the Northern Hemisphere at t = 244.8 min. The epoch time “t = 0” is referred to as the moment when the southward IMF reached x = 40 Re. IMF had been southward since t = 120 min, so that substorms frequently occurred. We chose this moment (t = 244.8 min) because the FACs were fairly stable during the interval form t ~ 220 to t ~ 270 min. In Fig. 5, two pairs of the FACs are seen, which resemble the Region 1 and 2 FACs shown in the bottom and middle panel of Fig. 1.

Fig. 5

Field-aligned currents at the ionospheric altitude in the Northern Hemisphere at t = 244.8 min. Negative values (blue color) indicate upward field-aligned currents. Sun is to the top, and the outermost circle represents the magnetic latitude of 60°. The contour lines indicate equipotential lines with an interval of 5 kV. The solid (dashed) contour lines represent positive (negative) potential, respectively. The circles indicate some selected points where the upward Region 1 FACs with amplitudes larger than 0.1 μA/m² at the ionosphere altitude. The white circles represent the points where the packets of the Alfvén waves come from the solar wind by way of the bow shock and the flank magnetopause. The gray circles represent the points where the packets originate from the distant tail (> 80 R_E), and that the generation regions are not clearly identified. The black circles represent that the packets come from the Southern Hemisphere, that is, closed field lines. P denotes the point at 18 h in magnetic local time (MLT) and 74.7 degrees in magnetic latitude (MLAT), and Q denotes the point at 14.25 h in MLT and 76.4 degrees in MLAT. The thick line indicates the "open-closed boundary" of the instantaneous magnetic field lines. After Ebihara and Tanaka (2022)

3.2 Magnetic field lines, current lines, and trajectories of packets of Alfvén wave associated with upward Region 1 FAC

Figure 6 shows the magnetic field lines extending from the upward Region 1 FACs in the Northern Hemisphere. They are obtained by taking the line integral of the instantaneous magnetic field vector B from the filled circles shown in Fig. 5. Hereinafter, we call them B-lines. Some of the B-lines reach the opposite hemisphere, and the others pass through the high-latitude magnetopause, the low-latitude magnetopause, and the distant tail of the magnetosphere. The variety of the B-lines might lead to the conclusion that the Region 1 FACs could originate in the various regions, such as the high-latitude and the low-latitude magnetospheric boundary layers, and the distant tail.

Fig. 6

Instantaneous magnetic field lines (B-lines) connecting to the upward Region 1 FACs indicated by the filled circles in Fig. 5. The color code on the B-lines indicates the value of V·F_t, where V is the bulk velocity of plasma, and F_t is the magnetic tension force density. The scale of the color codes is chosen to illuminate the small values of V·F_t. The green sphere indicates the Earth. The equatorial plane on the duskside is shaded by gray color

Figure 7 is the same as Fig. 6 except for the current lines extending from the upward Region 1 FACs. They were drawn by taking the line integral of the instantaneous current density vector J from the filled circles shown in Fig. 5. Hereinafter, we call them J-lines. The J-lines pass through the high-latitude boundary layer (mantle, or the tailward part of the cusp) where the magnetopause current flows. This is consistent with the previous studies (Janhunen and Koskinen 1997; Siscoe et al. 2000; Tanaka 1995, 2000; Watanabe et al. 2019; Wilder et al. 2015). The previous studies have shown that J·E is negative in the high-latitude boundary layer where the J-lines pass through. This has led to the conclusion that the Region 1 FACs are generated in the high-latitude magnetopause. It is shown in Fig. 7 that V.F_t is generally positive in the high-latitude boundary layer. This implies that the Alfvén waves are not efficiently excited. Instead, V.F_m is largely negative, where F_m is the magnetic pressure force density, suggesting plasma works negative work against the magnetic pressure force so as to compress the magnetic field. Watanabe et al. (2019) pointed out that V.F_m is negative in the high-latitude boundary layer, which is referred to as an expanding slow dynamo.

Fig. 7

Same as Fig. 6 except for instantaneous current lines (J-lines) connecting to the upward Region 1 FACs that are indicated by the circles in Fig. 5

Figure 8 shows the trajectories of the packets of the Alfvén waves that reach the upward Region 1 FACs. The trajectories were calculated by integrating v⁺ in Eq. (18) backward in time. Most of the trajectories are found to pass through the low-latitude boundary layer where V.F_t < 0. The trajectories are fully different from the B-lines and J-lines.

Fig. 8

Same as Fig. 6 except for the trajectories of the packets of the Alfvén waves that reach the upward Region 1 FACs indicated by the circles in Fig. 5

Figure 9 also shows the trajectories of the packets. The color code represents the V_⊥/V_A ratio, where V_⊥ is the plasma bulk velocity perpendicular to the magnetic field. It is clearly found that the ratio is nonzero in the outer part of the magnetosphere and beyond. This implies that the packet of the Alfvén waves is largely deflected from the original magnetic field line extending from the FACs observed in the ionosphere.

Fig. 9

Same as the right-bottom panel of Fig. 8 except the color code representing the V_⊥/V_A ratio, where V_⊥ is the perpendicular plasma flow speed, and V_A is the Alfvén speed

3.3 Trajectory of packet arriving at point P (18.00 MLT, 74.7 MLAT)

Figure 10 demonstrates the trajectory of the packet of the Alfvén waves that reach the point P. The point P is located at 18 h in magnetic local time (MLT) and 74.7 degrees in magnetic latitude (MLAT), and is indicated in Fig. 5. The magnetic field lines extending from each position of the packet are drawn by using the instantaneous magnetic field. It is found that the packet originates in the solar wind. In the solar wind, the bulk velocity of plasma (400 km/s) is much higher than the Alfvén velocity (~ 50 km/s). The packet moves mostly in the antisunward direction and encounters the bow shock, which is manifested by deceleration of plasma. As it traverses the low-latitude magnetopause, the magnetic field lines extending from the packet undergo the magnetic reconnection. As the packet proceeds into the magnetosphere toward the Earth, the Alfvén velocity increases because of the increase in the magnetic field. The elapsed time traveling from the bow shock to the Earth is ~ 20 min and that from the magnetopause to the Earth is ~ 15 min.

Fig. 10

Bird’s-eye view of the magnetosphere looking down the Earth from the dusk–noon sector (top) and from the dawn–noon sector (bottom). The small cylinders indicate the positions of the packets of the Alfvén waves that is supposed to supply the FACs at the point P (18 MLT and 74.7 MLAT) shown in Fig. 5. The instantaneous magnetic field line extending from each position of the packet is also shown. The color codes on the cylinders, the lines, and the vertical plane (y–z plane at x =  − 15 R_E) indicate the value of V⋅F_t. The color codes on the equatorial plane indicate the x-component of the plasma bulk velocity

Figure 11 summarizes some key quantities that are taken along the trajectory of the packet of the Alfvén waves that reach the point P. The trajectory of the packet with the velocity V_A + V (moving toward the Earth in the Northern Hemisphere) was traced backward in time from the point P where the upward FAC flows, namely, negative J_|| (= b·J). The packet travels from the solar wind (rightmost), toward the Earth (leftmost) by way of the bow shock and the flank magnetopause. Figure 11a indicates the FACs normalized by the area of the flux tube, which is given by (B/B_i)J_||, where B_i is the magnetic field at the ionosphere. The normalized FACs are fluctuated near the bow shock. As the packet passes through the low-latitude boundary layer (flank magnetopause), the FACs show a negative excursion. The large FACs flow in the low-latitude boundary layer, which will be shown later. After that, the FACs are slightly negative and are almost constant until it approaches the Earth. Figure 11b shows the divergent of the FACs (∇·J_||), the divergent of the inertial current (∇·J_i) and the divergent of the diamagnetic current (∇·J_d). The conversion of the perpendicular currents to the FACs is found to occur in the bow shock and the low-latitude boundary layer. Figure 11c shows the rate of change in the FACs normalized by the area of the flux tube. The black line indicates the normalized ∂J_||/∂t obtained by using Eq. (4), which suggests that the FACs are generated in the bow shock and the low-latitude boundary layer. The red and blue lines indicate the first and the second terms on the right-hand side of Eq. (5). Both the terms, − ∇_||(∇⋅E)/μ₀ and (∇²E)_||/μ₀, appear to be significant. The dashed black line indicates the sum of these two terms. The dashed black line is different from the black one because of numerical errors probably arising from the calculation of the 2nd order derivative of the electric field that are obtained at discrete grid points in the simulation domain. Figure 11d indicates J·E (black, dot product of the current density and the electric field), V·F_t (red, dot product of the velocity and the tension force density), and V·F_m (blue, dot product of the velocity and the magnetic pressure force density). These values describe the conversion of energy. V·F_t is largely negative near the bow shock and the low-latitude boundary layer, suggesting that the Alfvén waves are excited in these regions.

Fig. 11

Normalized FAC (B_i/B)J_|| (positive parallel and negative anti-parallel to the magnetic field), b ∇⋅J_||, ∇⋅J_d, and ∇⋅J_i, where J_||, J_d, and J_i are the FAC, the diamagnetic current, and the inertial current, respectively, c the rate of change in the FAC, ∂J_||/∂t, d (black), V⋅F_t (red), and V⋅F_m (blue), where F_t and F_m are the magnetic tension force density and the magnetic pressure force density, respectively, and e the position as a function of the distance D from the point of departure of the backward tracing. x and y point toward the Sun and dusk, respectively, and z is antiparallel to the Earth’s dipole moment. The point of departure is located at (0.00, 1.79, 3.12) R_E, which is connected with the point P (at the ionospheric altitude) along the dipole magnetic field line. G₁ and G₂ denote the possible regions where the Region 1 FAC is generated. After Ebihara and Tanaka (2022)

There are at least two noticeable regions where the Region 1 FACs are generated. One is located near the low-latitude boundary layer, and the other is located in the bow shock region. However, the contribution from the bow shock to the Region 1 system is probably minor because the FACs degrade significantly after the passage of the bow shock as shown in Fig. 11a. The low-latitude boundary layer is likely to be the FAC dynamo region for the Region 1 system. The FAC dynamo region in the low-latitude boundary layer may consist of two subregions, G₁ and G₂. G₁ is associated with the relatively large amplitude of FAC peaking at D ≅ 33–34 R_E. As the packet passes through G₁, the upward FACs are intensified, and soon after, degraded. When the packet enters G₂, the upward FACs are again intensified, and the magnitude of the normalized FACs is almost kept until the packet reaches close to the ionosphere. V·F_t is slightly negative in G₂.

Figure 11a and c show that the change in J_|| does not always correspond to ∂J_||/∂t as the packet moves inward. This discrepancy may come from the contributions from packets traveling in the different directions. One comes from the packet moving with the velocity V_A + V, and the other one comes from the packet moving with the velocity − V_A + V. Figure 12 shows the trajectories of such packets. The packet moving at the velocity of V_A + V comes from the solar wind. The other one moving at the velocity of − V_A + V comes from the inner part of the magnetosphere. The two packets intersect with each other at (− 3.3, 14.9, 7.6) R_E in G₁ at t = 233.8 min. Both packets had traveled in the region where V⋅F_t < 0, and should contribute to the FACs at the point of intersection. This may result in the poor correspondence between the local J_|| value and ∂J_||/∂t. This may also make it difficult to answer the question: Which is more important, G₁ and G₂? Of course, the Region 1 FACs in the ionosphere in the Northern Hemisphere are probably transported by the packet of the Alfvén waves with velocity of V_A + V. Previously, the generation of the FACs has been evaluated by the line integral of the local values of ∇·J_⊥ (= − ∇·J_||) along a magnetic field line (Sato and Iijima 1979). When the plasma bulk velocity V is nonzero, the local J_|| value will be determined by the packets of the Alfvén waves traveling with velocities of V_A + V and − V_A + V. In that case, the FACs cannot be simply determined by the line integral of the local value of ∇·J_||.

Fig. 12

Bird’s-eye view of the magnetosphere looking down the Earth from the noon–dusk sector. Trajectories of the two packets moving at the velocities V_A + V and − V_A + V are indicated, where V_A is the Alfvén velocity and V is the plasma bulk velocity. The former one comes from the solar wind, reaches the Earth, and contributes to the Region 1 FAC in the ionosphere. The other one comes from the Earth’s inner region and goes into the tail region. The two packets intersect with each other at (− 3.3, 14.9, 7.6) R_E at t = 233.8 min

3.4 Plasma flow involved in generation of FAC

The amplitude of the FACs is substantially large in G₁. However, it decreases as the packet moves toward the Earth and becomes almost zero quickly. It is uncertain if G₁ is responsible to the upward Region 1 FACs that appear at the ionosphere. There is a possibility that another region also participates in the generation. The region is located adjacent to G₁ and is labeled by G₂ in Fig. 13. In G₂, as shown in Fig. 11, V.F_t < 0 and the upward FACs are slightly developed, whereas the polarity of ∂J_||/∂t is unclear because of numerical difficulties. In Fig. 13, two streamlines are shown, which are obtained by line integral of the instantaneous bulk velocity of plasma V. One streamline passes through G₁, indicating that the plasma originates in the solar wind. It is clear from Fig. 13 that the solar wind-originated plasma pulls the newly reconnected magnetic field lines, which is represented by negative V.F_t and is indicated by the blue color. The thin lines represent the magnetic field lines extending from the packet. It is found that V.F_t > 0 just inside G₁, suggesting that the plasma is accelerated by the tension force as indicated by the red color. The accelerated plasma moves tailward and participates in the excitation of the Alfvén wave in G₂. As shown in Fig. 13, the plasma participating in the excitation of the Alfvén wave in G₂ originates in the magnetosphere. Thus, it can be said that the magnetosphere-originated plasma could also participate in the generation of the Region 1 FAC.

Fig. 13

The thick lines indicate streamlines of plasma (line integral of plasma velocity) passing through the generation regions G₁ and G₂. Color on the streamlines and the y–z plane at x =  − 12 R_E represents V⋅F_t, where V is the plasma velocity and F_t is the magnetic tension force density. The small cylinders indicate the positions of the packet that is supposed to carry perturbations associated with the FACs at the point P. The magnetic field lines extending from the packet are overlaid. Color on the equatorial plane indicates V_x. The white sphere indicates the Earth. After Ebihara and Tanaka (2022)

3.5 Trajectory of packet arriving at point Q (14.25 MLT, 76.4 MLAT)

Figure 14 shows the trajectory of the packet that reaches the point Q. The point Q is located at 14.25 h in MLT and 76.4 degrees in MLAT and is indicated in Fig. 5. The packet originating in the solar wind pulls the newly magnetic field lines as manifested by the negative V.F_t (blue color) in G₁. Just inside G₁, V.F_t > 0 (red color), indicating that plasma originating in the magnetosphere is accelerated tailward by the magnetic tension force. The accelerated plasma is supposed to participate in the excitation of the Alfvén wave as it propagates tailward. The mechanism is essentially the same as that for the point P.

Fig. 14

Bird’s-eye view of the magnetosphere from the noon–dusk sector. The short cylinders indicate the positions of the packet that reaches the point Q. The color code on the lines indicates the value of V⋅F_t and that on the equatorial plane indicates the x-component of the plasma bulk velocity. After Ebihara and Tanaka (2022)

3.6 Possible generation mechanism for Region 1 FAC

Figure 15 illustrates the possible generation mechanism for the Region 1 FACs. For the sake of simplicity, the mechanism for the upward Region 1 FACs is shown. The same mechanism also works for the downward Region 1 FAC on the dawnside.

• (1) The solar wind-originated plasma indicated by the red line pulls the newly reconnected magnetic field line B₁, that is, V.F_t < 0. This corresponds to G₁. The solar wind-originated plasma travels tailward (V₁) and pulls the magnetic field lines subsequently, that is, V.F_t < 0.

• (2) The pulled magnetic field line B₁ is bent just inside G₁. The magnetosphere-originated plasma is accelerated tailward by the magnetic tension force, that is, V.F_t > 0. After a while, the accelerated plasma (V₂) starts pulling the magnetic field line B₂, that is, V.F_t < 0. This corresponds to G₂.

• (3) The FACs are generated by two means. One is related to the first term on the right-hand side of Eq. (5), that is, − ∇_||(∇⋅E)/μ₀. This term is associated with shear of the plasma flow. The other term, (∇²E)_||/μ₀, also participates in the generation of the FACs. The contribution from the latter term is characterized by the field-aligned gradient E_⊥ in a curved magnetic field line.

Fig. 15

Possible generation mechanism of the Region 1 FAC. Only the dusk part of the Region 1 FAC (upward FAC) is drawn. There are two generation regions G₁ and G₂. The red line indicates the flow of solar wind-originated plasma (V₁), which pulls the newly reconnected magnetic field lines B₁ and B₂ (that is, V⋅F_t < 0). The solar wind-originated plasma (red line) is associated with G₁. The blue line indicates the flow of magnetosphere-originated plasma (V₂), which is accelerated by the magnetic tension of the field line B₁ (that is, V⋅F_t > 0). Namely, the solar wind-originated plasma (V₁) indirectly pulls the magnetosphere-originated plasma (V₂) by way of the magnetic tension force. The magnetosphere-originated plasma is accelerated, and is able to pull the magnetic field line B₂ (that is, V⋅F_t < 0). The magnetosphere-originated plasma (blue line) is associated with G₂. The yellowish arrows indicate the perpendicular electric field. After Ebihara and Tanaka (2022)

So far, we do not know which is more realistic, G₁, G₂, or both. Further studies are needed to reach the definitive conclusion on it.

3.7 Region 1 FAC, circular motion, and energy transfer in the magnetosphere

Finally, we demonstrate the relation between the Region 1 FAC and the energy transfer in terms of the Poynting vector. The bluish surface in Fig. 16 indicates the upward FACs at 1.5 μA/m² in the Northern Hemisphere on the duskside. There are two distinct regions where the strong FACs flows. One extends from the Earth to the dusk magnetosphere, which is directly associated with the Region 1 upward FAC. The other is located near the magnetopause, which corresponds to the large-amplitude upward FAC in G₁. The yellowish lines in Fig. 16 indicate the integral curves of the Poynting vector S related to the upward Region 1 FAC. The S-curve was calculated by the line integral as (Ebihara and Tanaka 2017)

$$ {\mathbf{r}}_{S} = \int\limits_{0}^{l} {\frac{{\mathbf{S}}}{{\left| {\mathbf{S}} \right|}}dl} + {\mathbf{r}}_{0} , $$

(19)

where S is the Poynting vector (= E × B/μ₀), r_s is the position vector, and r₀ is the initial position vector. We call the integral curves S-curves. r₀ was determined by the followings. We traced the upward Region 1 FACs indicated by the circles in Fig. 5 to the geocentric distance of 5 R_E along the dipole magnetic field line. We avoided tracing the S-curve near the inner boundary of the simulation domain (which is located at 2.6 R_E) because of fluctuations arising from a numerical problem. Figure 16 shows that the Region 1 FACs is clearly surrounded by the S-curves. The S-curve shows a helix with its center moving toward the Earth.

Fig. 16

Bird’s-eye views of the magnetosphere from the dusk–midnight sector. The yellow lines indicate the S-curves, which were obtained by line integral of the Poynting vector S

The relation between the FAC and the S-curve may be explained by the simple configuration illustrated in Fig. 17. Let us consider plasma in a cylinder elongated in the vertical direction, and the background magnetic field B₀ directing downward. We enforce plasma to move in the clockwise direction on the topside of the cylinder. In the ideal MHD condition, the corresponding electric field (= − V × B) directs the axis of the cylinder. The magnitude of the electric field decreases with the distance from the topside. The gradient of the electric field in the vertical direction gives rise to the generation of the magnetic field in the counterclockwise direction due to the Faraday’s law. The generated magnetic field ΔB is perpendicular to the background magnetic field. Thus, it does not contribute to the compression of the magnetic field. The generated magnetic field induces the upward FAC J_||, and the inertial current J_i that directs outward. The generated magnetic field ΔB propagates downward at the Alfvén speed. Consequently, the magnetic field lines (= B₀ + ΔB) are twisted as shown by the orange lines in the right panel. The S-curve, which is a line integral of the Poynting vector S, shows a helix with its center moving downward as shown by the blue line. In that condition, the FAC is surrounded by the S-curve. On the topside of the cylinder, the plasma performs negative work against the magnetic force to excite the Alfvén wave, that is, V·F_t < 0. Because the magnetic field line is straight, the FAC is generated in accordance with the first term on the right-hand side of Eq. (5). The magnetic energy converted from the kinetic energy of plasma on the topside of the cylinder propagates downward. This situation is similar to that shown in Fig. 16. From this consideration, we can draw the following inferences.

• 1. The Region 1 FAC, circular motion of plasma, and transfer of magnetic energy in the parallel and perpendicular directions are closely related to each other. They are probably equivalent when the energy transfer is dominated by the Alfvén waves.

• 2. According to the observations, the large-scale FACs are always present. This implies that the magnetosphere is more or less twisted all the time.

Fig. 17

Cartoon indicating the relation between the FAC (J_||), S-curve (line integral of the Poynting vector S), and motion of plasma V. E, B, and J_i represent the electric field, magnetic field, and the inertial current, respectively. The background magnetic field B₀ is downward

4 Discussion and summary

A new method was proposed to search for the generation region of the field-aligned currents (FACs). We assumed that the perturbations associated with the FACs propagate parallel or antiparallel to the background magnetic field at the Alfvén velocity in the rest frame of the moving medium. We traced the packet of the Alfvén wave by using the global MHD simulation. We also used the 3 criteria for the generation region of the FAC (FAC dynamo); (1) current continuity, (2) rate of change in the FACs in accordance with Faraday’s and Ampère’s laws, and (3) negative work of plasma against the magnetic tension force.

The following is a summary of this study on the generation region of the Region1 FACs.

• 1. The low-latitude magnetospheric boundary layer is found to be the major generation region of the Region 1 FACs. In this region, the solar wind-originated plasma pulls the newly reconnected magnetic field lines. The magnetosphere-originated plasma seems to participate in the generation. The low-latitude boundary layer has been pointed out by the previous studies (Bythrow et al. 1981; Eastman et al. 1976; Echim et al. 2007; Iijima 2000; Johnson and Wing 2015; Lotko et al. 1987; Lundin and Evans 1985; Sato & Iijima 1979; Sonnerup 1980; Troshichev 2003; Wing and Johnson 2015). The role of the newly reconnected magnetic field line has been pointed out by Strangeway et al. (2000a) and Strangeway et al. (2000b).

• 2. The generation region is totally different from that obtained using the instantaneous magnetic field lines and the instantaneous current lines. Motion of background medium cannot be neglected in finding the generation region of the Region 1 FACs.

• 3. Most of the previous studies pointed out that the solar wind plasma flowing in the boundary layer gives rise to the flow shear that generates the FACs. In the MHD simulation, both the terms on the right-hand side of Eq. (5), − ∇_||(∇⋅E)/μ₀ and (∇²E)_||/μ₀, are important for the generation of the FACs. The first term implies that the FACs are generated when the field-aligned gradient of ∇⋅E is present. That is, field-aligned gradient of space charge manifested by ∇⋅E gives rise to the generation of the FACs. The second term implies that the FACs are also generated when the magnetic field line is curved and E varies along the magnetic field line. Here, we neglected E_||.

• 4. Most of the previous studies used J·E to evaluate the energy conversion related to the excitation of the Alfvén wave. Instead, V·F_t is probably better to evaluate the excitation of the Alfvén wave. Because of the contribution from V·F_m, J·E is not always negative where the Alfvén waves are excited.

• 5. The Region 1 FACs are surrounded by the integral curves of the Poynting vector (S-curve). The S-curve is a helical shape, with its center moving toward the Earth. The Region 1 FACs, the circular motion of plasma, and the energy transfer are closely related to each other. It is probable that the magnetosphere is more or less twisted all the time because the large-scale FACs always present.

We considered that the condition that ∂J_||/∂t =  − (∇ × ∇ × E/μ₀)_||≠ 0 must be satisfied to generate the FACs. However, the Region 1 FACs are, in many cases, observed to be steady in the ionosphere. This can reasonably be explained in terms of propagation of the Alfvén wave and momentum balance in the ionosphere as described in Appendix A.

A question arises if the Alfvén wave is necessary to generate the FACs. When electrons are enforced to move in one direction along the magnetic field line by parallel acceleration or pitch angle scattering, the Alfvén wave will not be necessary to generate the FACs. If this is the case, the parallel acceleration or pitch angle scattering must take place somewhere in the magnetic field lines extending from the Region 1 FACs. As shown in Fig. 6, some magnetic field lines extending from the Region 1 FACs are open, and connected to the distant tail region. We do not know if parallel acceleration or pitch angle scattering takes place continuously along such magnetic field lines to sustain the Region 1 FACs. In addition, a new question arises as to if the processes involved in parallel acceleration or pitch angle scatter can sufficiently supply energy consuming in the ionosphere. From that sense, it is thought that the Alfvén wave is the most reasonable interpretation for, at least, the generation of the Region 1 FAC.

The circular motion associated with the Region 1 FACs is probably related to, in part, the magnetospheric convection. If so, the driving mechanism of the magnetospheric convection is shared with that of the Region 1 FACs. Future study is needed to confirm it.

As shown in the black circles in Fig. 5, some packets of the Alfvén waves come from the opposite hemisphere, not the low-latitude magnetospheric boundary layer. These packets are related to the Region 1 FACs near the open-closed boundary. The Alfvén waves seem to be generated at mid- and low altitudes, and be involved in the internal magnetospheric processes related to the convection. To date, we have no definitive conclusion for the generation mechanism for these particular FACs. The gray circles shown in Fig. 5 indicate that the packets originate from the distant tail (> 80 R_E). We cannot identify the generation region for them. In future, we will investigate the generation regions of the Region 1 FACs indicated by the black and gray circles in Fig. 5.

There are some issues arising from limitations of the global MHD simulation. The first issue is a high level of turbulent fluctuations in the magnetosheath (Rakhmanova et al. 2021), which might break the frozen-in condition. The fluctuations in magnetosheath may affect the reconnection on the dayside magnetopause, and the generation of the FACs. The global Vlasov simulation (Pfau-Kempf et al. 2020) will be useful to resolve this issue. The second issue is the kinetic effects (Song and Lysak 2001a). In the MHD limit, the displacement current and electron motion are neglected. The role of the kinetic effect is unclear. Considering the facts that the simulated distribution of the FACs is well consistent with the observation, we speculate that the contributions from the turbulent fluctuations and the kinetic effects to the large-scale Region 1 FACs could be minor. Further studies are necessary to confirm the region and mechanism for the generation of the Region 1 FACs.

The new method proposed can be generally applied to search for the generation region of FACs. It would help understand the current system as well as the structure and dynamics of the magnetosphere not only at the Earth but also other planets such as Jupiter and Saturn.

Appendix A

Following Cravens (1997), we try explaining the steady field-aligned current J_|| in the ionosphere by introducing two plasma slabs, a top slab (magnetosphere) and a bottom slab (ionosphere) as shown in Fig. 

Fig. 18

Plasma slabs, in which the top slab moves at the constant speed V, and ion-neutral collision dominates in the bottom slab

18. The magnetic field is initially uniform to be B₀ and downward (negative z-direction). The top slab moves in the x-direction with a constant velocity V. We assume that the electric field is given by E =  − V × B from Ohm’s law. At the lower edge of the top slab, the perpendicular magnetic field δB_⊥ is generated by Faraday’s law, that is, ∂B/∂t =  − ∇ × E. When the slab has a finite width, ∇ × (δB_⊥) ≠ 0 at both edges of the slab, that is, ∂J_||/∂t =  − (∇ × ∇ × E)/μ₀. δB_⊥ (together with J_||) propagates from the lower edge of the top slab to the bottom slab downwardly at the Alfvén speed.

See Fig. 18.

Next, we consider the momentum balance at the bottom slab where the ion-neutral collision dominates.

$$ \rho \frac{{D{\mathbf{V}}}}{Dt} = - \nabla p + {\mathbf{J}} \times {\mathbf{B}} - \rho \nu {\mathbf{V}}, $$

(20)

where ρ is the mass density, p is the plasma pressure, and v is the ion-neutral collision frequency. When the steady and uniform case, that is, D/Dt = 0 and p is absent, the Lorentz force is balanced with friction force, that is,

$$ {\mathbf{J}} \times {\mathbf{B}} = \rho \nu {\mathbf{V}}. $$

(21)

We assume that the Lorentz force consists of two terms as

$$ {\mathbf{J}} \times {\mathbf{B}} = - \nabla \left( {\frac{{B^{2} }}{{2\mu_{0} }}} \right)_{ \bot } + \frac{{B^{2} }}{{\mu_{0} }}\left( {{\mathbf{b}} \cdot \nabla } \right){\mathbf{b}}. $$

(22)

For the bottom slab, the second term on the right-hand side is

$$ \frac{{B^{2} }}{{\mu_{0} }}\left( {{\mathbf{b}} \cdot \nabla } \right){\mathbf{b}} \approx \frac{{B_{0}^{2} \delta b_{x} }}{{\mu_{0} H}}{\mathbf{e}}_{x} , $$

(23)

where H is the characteristic height and δb_x is the x-component of the unit vector of B, that is, δB_⊥. When the first term on the right-hand side of Eq. (22) is negligible, by equating Eqs. (21) and (23) we obtain

$$ \frac{{B_{0}^{2} \delta b_{x} }}{{\mu_{0} H}} = \rho \nu V_{x} $$

(24)

and

$$ \delta B_{x} \approx \frac{{\mu_{0} H\rho \nu V_{x} }}{{B_{0} }}. $$

(25)

Since the bottom slab has also a finite width, ∇ × (δB_⊥) = μ₀ J_||≠ 0 at both edges of the slab. The derivatives related to the bottom slab, including J_||, are zero, that is, they are steady. Thus, even though ∂J_||/∂t is nonzero in the magnetosphere, steady field-aligned current J_|| can exist in the ionosphere when the ionosphere is steady.