Volume 123, Issue 8 p. 6307-6325
Research Article
Free Access

Global Three-Dimensional Simulation of the Earth's Magnetospheric and Ionospheric Responses to Small-Scale Magnetic Flux Ropes in the Solar Wind

Kyung Sun Park

Corresponding Author

Kyung Sun Park

Department Astronomy and Space Science, Chungbuk National University, Cheongju, South Korea

Correspondence to: K. S. Park,

kspark@chungbuk.ac.kr

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Dae-Young Lee

Dae-Young Lee

Department Astronomy and Space Science, Chungbuk National University, Cheongju, South Korea

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Myeong Joon Kim

Myeong Joon Kim

Department Astronomy and Space Science, Chungbuk National University, Cheongju, South Korea

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Cheong Rim Choi

Cheong Rim Choi

Department Astronomy and Space Science, Chungbuk National University, Cheongju, South Korea

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Rok Soon Kim

Rok Soon Kim

Korea Astronomy and Space Science Institute, Daejeon, South Korea

University of Science and Technology, Daejeon, South Korea

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Kyungsuk Cho

Kyungsuk Cho

Korea Astronomy and Space Science Institute, Daejeon, South Korea

University of Science and Technology, Daejeon, South Korea

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Kyu-Cheol Choi

Kyu-Cheol Choi

Selab, Inc., Seoul, South Korea

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Jaehun Kim

Jaehun Kim

Korean Space Weather Center of the National Radio Research Agency, Jeju, South Korea

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First published: 14 July 2018
Citations: 1

Abstract

The orientation of magnetic flux ropes in the solar wind is an important component that affects interactions with the Earth's magnetosphere and ionosphere. In this study, we performed global magnetohydrodynamic (MHD) simulations on the responses of the magnetosphere and ionosphere to the impact of small-scale magnetic flux ropes (SMFRs). We considered four types of SMFR structures according to the alignment direction of the flux rope axis in the plane perpendicular to the Sun-Earth line. The flux rope axis of the two types is oriented in the north-south direction, while the flux rope axis of the other two types is oriented along the dusk-dawn direction. Accordingly, the By and Bz profiles of the SMFR types vary as the SMFR passes through the Earth. The main features of the response are as follows: (i) The magnetic reconnection on both the dayside and nightside is well organized by the specific profiles of By and Bz. (ii) One type of SMFRs where Bz turns from south to north and By remains duskward leads to plasmoid formation in the tail, distinguishing it from the other types. (iii) The temporal responses of the tail plasma flow, cross-tail electric field, tail plasma pressure, and cross-polar cap potential depend on the specific profiles of By and Bz, causing different response times. (iv) The evolution of ionospheric convection pattern sensitively depends on the magnetic field variation within SMFRs. (v) The peak value of cross-polar cap potential ranges from 25 kV to >50 kV, providing energy storage suitable for substorm expansion.

Key Points

  • MHD simulations are undertaken on the magnetospheric and ionospheric responses to four types of small-scale magnetic flux rope (SMFR)
  • Magnetospheric responses in terms of reconnection, plasma flow, electric field, and pressure are well distinguished among four SMFRs
  • The ionospheric convection pattern and cross-polar cap potential evolve in different ways depending on the impact of specific SMFRs

1 Introduction

Solar activity, such as the coronal mass ejections from the solar atmosphere, has a significant influence on the Earth's magnetosphere and ionosphere. The coronal mass ejections arrive at the Earth's magnetosphere approximately 1 to 4 days after the initial solar eruptions, resulting in a strong geomagnetic disturbance. Some of the interplanetary coronal mass ejections observed in the solar wind exhibit a large-scale magnetic flux rope structure with a smooth rotation of the magnetic field direction and an enhancement of the magnetic field strength compared to the background interplanetary magnetic field (IMF). Such interplanetary coronal mass ejections are referred to as the magnetic cloud (MC), which is also characterized by proton temperature and plasma beta (the ratio of the plasma pressure to the magnetic pressure) values within the magnetic flux rope that are lower than those in the background solar wind. The typical diameters of MCs are approximately 0.1 to 0.5 AU near the Earth's orbit (Bothmer & Schwenn, 1998; Crooker et al., 2004).

Conversely, small-scale magnetic flux ropes (SMFRs) in the solar wind were first reported by Moldwin et al. (1995, 2000). While the SMFR sizes are approximately 0.005 to 0.1 AU, much smaller than those of the MCs, their properties are similar to those of the usual large-scale magnetic flux ropes (such as those associated with MCs); namely, they exhibit a smooth rotation of the magnetic field direction, an increase in the magnetic field strength, and a decrease in the proton temperature and plasma beta (Cartwright & Moldwind, 2008; Feng et al., 2007, 2008; Moldwin et al., 1995, 2000). A recent observational study based on a large number of SMFRs by our group (Kim et al., 2017) confirmed this and further found that the average magnetic field intensity of SMFRs is weaker (about 7.4 nT) than that of MCs (about 10.6 nT), and the average duration time and expansion speed of SMFRs are ~2.5 hr and 2.6 km/s, respectively, which are both smaller by a factor of ~10 than those of MCs.

Additionally, Feng et al. (2010) noted the geoeffectiveness of SMFRs by reporting that 18 (69%) of their 26 studied SMFRs triggered magnetospheric substorms. Based on a much larger study of SMFR events (309), Kim et al. (2017) showed that for 28.5% of the total SMFRs, substorms occur after the SMFR impacts. Therefore, the significance of SMFRs as causes of substorms is nonnegligible and warrants further study.

A key component of both MCs and SMFRs is a southward magnetic field component, as both are characterized by the rotation of the magnetic field. It is well known that when the IMF turns southward, the energy transfer from the solar wind to the magnetosphere is most efficiently done by magnetic reconnections on both the dayside and the nightside in the magnetosphere where plasma convection and electrical currents increase. Certainly, both the dayside and nightside reconnections affect substorm occurrence. In addition, the convection pattern at high latitudes appears as a two-cell formation with a tailward flow over the polar cap and a return sunward flow at the lower latitudes in the ionosphere. In contrast, during an extended northward IMF, the ionospheric convection is much weaker than during a southward IMF (Cowley & Lockwood, 1992; Dungey, 1961; Fairfied & Cahill, 1966; Park et al., 2010; Reiff & Bursh, 1985; Zhang et al., 2013; Zhang et al., 2015). The two-cell convection pattern exhibits various dawn-dusk asymmetries whose sense depends on the IMF By component. In the northern polar region, the dusk convection cell becomes crescent-shaped, and the dawn cell expands to higher latitudes for By < 0 and Bz < 0, while the dusk cell shrinks for By > 0 and Bz > 0 (Burch et al., 1985; Heelis, 1984; Ogino et al., 1986; Park et al., 2006; Zhang et al., 2016) (a schematic demonstration is given in Zhang et al., 2016).

Therefore, the impact of an interplanetary flux rope structure that contains rotation of the magnetic field, such as in the cases of MCs and SMFRs, affects all these responses in the magnetosphere and ionosphere in a complicated way. Monitoring detailed processes during this kind of interaction is difficult observationally because of the limit to simultaneous in situ observation from Sun to magnetosphere and ionosphere. An appropriate simulation is needed to examine the specific responses of the magnetosphere and ionosphere to MCs and SMFRs. We note that no previous study on the simulation of interactions between SMFRs and the Earth's magnetosphere and ionosphere exists, which is the focus of the present paper. In this study, we performed a three-dimensional global magnetohydrodynamic (MHD) simulation to understand the interactions between four different types of SMFRs and the Earth's magnetosphere. Mainly, we aimed to address differences in the responses to the impacts of four types of SMFRs.

In section 2, we briefly describe the simulation model and characteristic SMFR types. We present the simulation results in section 3. A summary and discussion of the results are presented in section 8.

2 Simulation Model

The simulation in this work is based on solving the normalized resistive MHD and Maxwell's equations as an initial value problem by using a modified version of the Leap-Frog scheme. We give only a brief description of the simulation model below since its details are described elsewhere (Ogino et al., 1992; Park & Ogino, 2006).

We use a simulation box with dimensions of −60RE ≤ X ≤ 30RE, 0 ≤ Y ≤ 30RE, and −30RE ≤ Z ≤ 30RE in Cartesian solar magnetospheric coordinates. The number of grid points is (nx, nx, nz) = (300, 100, 200) with a uniform grid spacing of 0.3RE. The internal ionospheric boundary conditions are set by forcing a static equilibrium at r = 2.5RE. A mirror dipole field is applied in the solar wind at time t = 0 to take up the shape of the magnetosphere. The smoothing function damps out all perturbations near the ionosphere, including parallel currents.

We first set a uniform solar wind with initial conditions defined by a number density, nsw = 5 cm−3, a velocity, Vsw = 400 km/s, and a temperature, Tsw = 2×105 K, with a pure northward IMF of Bz = 2 nT, to obtain a quasi steady state of the magnetospheric configuration during the first 2 hr. Then, we apply a set of conditions representing the SMFR structure, as described in detail below. The rotation of IMF is given by BIMF = (0, BIMFcosθ, BIMFsinθ) nT, where is θ is a counterclockwise angle from the Y axis.

In this work, we consider four types of SMFRs according to the way their axis is aligned in the plane perpendicular to the Sun-Earth line, along either Y or Z axis. Figure 1 shows four types of magnetic field structures within the SMFRs, which we assume satisfies a force-free condition. They are distinguished by the direction of core magnetic field, which is indicated by the sign in front of the letters Y and Z. They are also according to the direction of the toroidal current relative to the core magnetic field direction. If the relative directions are parallel (antiparallel) to each other, SMFRs are regarded as RH (LH) types, representing right handedness (left handedness) of the field twist. For the simulations performed here, we chose to study SMFRs with the RH sense.

Details are in the caption following the image
Schematic diagram showing the four different types of small-scale magnetic flux ropes (SMFRs) in a view from the Earth. They are classified according to magnetic field Bz direction of SMFR. The RH and LH are defined by the conditions that the toroidal current is parallel and antiparallel to the direction of the core magnetic field, respectively.

The magnetic flux rope structure within SMFRs is identified by a smooth rotation of the field direction as shown in Figure 2, if a satellite passes through the center of the SMFR. Here black circle line is the magnitude of magnetic field (B). Yellow, blue, and red circle lines are the X, Y, and Z components of the magnetic field within the flux rope, respectively. The magnetic field strength increases from 4 to 7 nT, and then it decreases to 4 nT for all types of SMFR. The basic features of the magnetic field for the four SMFRs are summarized in Table 1.

Details are in the caption following the image
The magnetic field profiles corresponding to the small-scale magnetic flux ropes (SMFRs) with RH sign in Figure 1 when assuming that the Earth encounters center of SMFR. The black circle line is magnetic field strength, B, the yellow circle line is Bx, the blue circle line is By, and the red circle line is Bz components within SMFR.
Table 1. Summary of the Magnetic Field Parameters for the Four Types of Small-Scale Magnetic Flux Ropes (SMFRs) Simulated in the Present Work
Physical parameter (1) SMFR type 1 (2) SMFR type 2 (3) SMFR type 3 (4) SMFR type 4
IMF angle θ (deg) 360° to 180° 180° to 0° 270° to 90° 90° to 270°
Bz (nT) Bz < 0 Bz > 0 Bz < 0 to Bz > 0 Bz > 0 to Bz < 0
By (nT) By > 0 to By < 0 By < 0 to By > 0 By < 0 By > 0
|B| (nT) Increases slowly from 4 to 7 and then decreases from 7 to 4
  1. Type 1: The axis of the SMFR (i.e., the core magnetic field) lies along the −Z direction. The IMF angle (θ) changes from 0° (or 360°) to 180°. The flux rope structure of this type provides a southward Bz during the rotation of the magnetic field from duskward to dawnward, as seen by the Earth. For the simulations, we use a Bz profile that decreases from 0 to −7 nT and again increases to 0 nT, while the By rotates from 4 to −4 nT.
  2. Type 2: The axis of the SMFR is along the +Z direction. The IMF angle (θ) changes from 180° to 0°, meaning that the bipolar magnetic field of By rotates from dawnward to duskward. This type of flux rope structure provides a northward Bz, as seen by the Earth. For the simulations, we use a By profile that changes from −4 to 4 nT and a Bz profile that changes from 0 nT through 7 to 0 nT.
  3. Type 3: The axis of the SMFR lies along the −Y direction such that the core magnetic field is dawnward. This type of flux rope structure first leads to a southward Bz followed by a northward Bz, and the IMF angle (θ) decreases from 270° to 90°, as seen by the Earth. For the simulations, we use a Bz profile that changes from −4 to 4 nT and a By profile that changes from 0 nT through −7 to 0 nT.
  4. Type 4: The axis of the SMFR is aligned with the +Y direction such that the core magnetic field is duskward. The IMF angle (θ) decreases slowly from 90° to 270°. This type of flux rope structure leads to the rotation of the magnetic field from north to south, as seen by the Earth. For the simulations, we use a By profile that increases from 0 to 7 nT and then decreases to 0 nT and a Bz profile that changes from 4 to −4 nT.

In order to model typical SMFRs that are appropriate for the purpose of the simulations, we take average plasma and magnetic field values of the observed SMFRs that we examined recently in a separate paper (Kim et al., 2017), as mentioned in section 1, where the average solar wind condition with a density of nsw = 5 cm−3 and velocity of Vsw = 400 km/s is maintained. The spatial size (i.e., diameter) of the SMFRs used in this simulation is approximately 678RE (0.028 AU).

3 Simulation Results

3.1 Magnetospheric Responses

In this section, we examine the reconnection, plasma flow, electric field, and plasma pressure within the magnetosphere and compare them among the four types of SMFRs.

3.1.1 Comparison of Reconnection Among the Four Types of SMFRs

In this subsection, we focus on the magnetospheric field configurations on which we based an attempt to identify reconnection features for the four types of SMFR. Figure 3 shows the simulation results for SMFR type 1, where the IMF has southward Bz and duskward to dawnward By components. Figure 3a shows projections of the region of minimum magnetic field magnitude from the simulation in a view from the Sun. To determine the regions of the minimum B values defined by the condition |B| < 0.2BIMF, we included values for X > −9RE at a fixed (Y, Z). The green area in Figure 3a corresponds to the location of the minimum B. The regions of minimum magnetic field magnitude are initially located at the high-latitude flanks in both hemispheres at t = 15 min, around the equatorial region when Bz is most southward and By is zero at t = 95 min, and again at the high-latitude flanks in both hemispheres at t = 180 min, although oriented in the opposite direction due to the change in the By direction. Figure 3b shows the three-dimensional configuration of the magnetic field lines at the three times in a view from the Sun (upper panels) and in a view from the duskside (bottom panel). The green, blue, and red lines refer to the closed field lines that connect to the Earth in both directions, the open field lines that connect to the ionosphere at one end and to the distant IMF at the other, and the field lines that are not connected to the Earth at all, respectively.

Details are in the caption following the image
The simulation results for the small-scale magnetic flux rope (SMFR) type 1 conditions when the interplanetary magnetic field (IMF) has a southward Bz component and a dusk to dawn By component. (a) Projections of the regions with a minimum magnetic field magnitude values for X > −9RE, as viewed from the Sun. (b) Three-dimensional configurations of the magnetic field lines.

We identify the reconnection sites that lie between the regions with the highly kinked open field lines. The dayside reconnection initially occurs at the high latitudes in the northern dusk and southern dawn sites (left panels in Figure 3b, e.g., t = 15 min). The splitting of the reconnection sites is purely an effect of the By component. When the SMFR has a strong southward Bz and a weak dawnward By component, the reconnection site moves from the high-latitude region to the magnetic equator (or subsolar region) where the SMFR magnetic field encounters the weakest field along the geomagnetic field line and the magnetosheath magnetic field line is antiparallel to the geomagnetic field (center panels in Figures 3a and 3b at t = 95 min). Then, the dayside reconnection site splits again into two regions in the northern dawn sector and the southern dusk sector due to the now dawnward By at t = 180 min. From the bottom panel, we identify the tail reconnection, which begins after the IMF Bz decreases to −6 nT and continues until the Bz increases to zero.

In the same way that Figure 3 shows the results for SMFR type 1, Figure 4 shows the results for SMFR type 2. Figure 4a shows how the minimum B region changes as the SMFR magnetic field topology changes. Recall that the SMFR of this type is characterized by a continuously northward Bz and a dawnward to duskward changing By. Initially, the dayside reconnection at the magnetopause site splits into two high-latitude regions in both hemispheres. The minimum magnetic field regions are more broadly distributed when Bz is weakly northward (at t = 15 min and t = 180 min in Figure 4a). When the Bz value is strongly northward with a weaker By component, the reconnection sites move closer to the Z axis in both hemispheres, as shown in the center panels of Figure 4a at t = 75 min and t = 110 min. From the bottom right panel, we can identify the red field lines, which indicates that the tail reconnection occurs despite the northward Bz (e.g., t = 75 min).

Details are in the caption following the image
Simulation results for the small-scale magnetic flux rope (SMFR) type 2 conditions with northward Bz and dawnward to duskward By. This figure uses the same formatting and criteria as Figure 3. IMF = interplanetary magnetic field.

The results for type 3 are shown in Figure 5 with the format described for Figures 3 and 4. Again, the minimum region is controlled by the evolution of the magnetic field topology of the SMFR. It is initially near the equator (at t = 5 min in Figure 5a) when Bz is the most southward and By is negligible, and it moves to higher latitudes (at t = 90 and 180 min in Figure 5a) as By increases. The reconnection first occurs near the magnetic equator (subsolar region) during a strong southward Bz (at t = 5 min in Figure 5a). When the Bz turns from southward to northward with a dawnward By, the reconnection site splits into two regions in the northern dawn and southern dusk sectors (at t = 90 min in Figures 5a and 5b). Then, it moves to high-latitude flanks in both hemispheres (e.g., at t = 180 min in Figure 5a) as the magnetic field becomes entirely dawnward. Figure 5c shows the formation of a plasmoid in the near midnight (−40RE < X < −20RE) when the SMFR field is purely southward. The twisted field lines (pink) refer to the plasmoid structure. The magnetic field lines of the plasmoid are connected to either the Northern Hemisphere or the Southern Hemisphere. The signature of the plasmoid is a north to south Bz variation. The plasmoid moves tailward with a velocity of 50 up to 200 km/s when the tail reconnection continues to the lobe field lines. Figure 5b demonstrates the tail reconnection when the SMFR magnetic field is dominated by a dawnward By component with no Bz component. At this time, both ends of the magnetic field lines of the plasmoid become completely disconnected from the Earth and attached to the IMF.

Details are in the caption following the image
The simulation results for the small-scale magnetic flux rope (SMFR) type 3 conditions when the Bz turns from southward to northward with a dawnward By. This figure uses the same formatting and criteria as Figure 3. IMF = interplanetary magnetic field.

The significance of a plasmoid in the tail region has long been recognized by many previous authors. There have been many observational studies on plasmoids in association with the near-Earth neutral line, indicating a north to south rotation of Bz (Baumjohann & Haerendel, 1990; Fairfield et al., 1989; Hones et al., 1984; Lepping et al., 1995; Lin et al., 1990; Miyashita et al., 2009; Moldwin & Hughes, 1993, 1994). These studies have reported that the formation of a plasmoid due to tail reconnection is associated with a substorm onset. Additionally, there are simulation studies that indicate that plasmoids develop on closed plasma sheet field lines (Birn & Hesse, 1990; Hesse & Birn, 1991; Hesse & Kivelson, 1998; Ogino et al., 1990; Ogino & Walker, 1998). Ogino et al. (1990) investigated the effect of the IMF By on substorms and found that plasmoids had configurations like twisted flux ropes. Ogino and Walker (1998) examined the case when the IMF turns from northward to southward and found a strong earthward flow, a tailward flow, and a plasmoid ejection because of the tail reconnection. The plasmoid structure in their study is similar to that of our simulation results for type 3, specifically when the SMFR magnetic field is mainly southward with a weak By component, although the formation and structure of plasmoids should depend on the initial state of the simulation.

Lastly, Figure 6 shows the simulation results for type 4, where the IMF Bz turns from northward to southward, with a duskward By. Figure 6a shows the minimum magnitude B regions, which indicate that the dayside reconnection initially starts at a high latitude when the SMFR magnetic field is purely northward (e.g., at t = 15 min). The reconnection region splits into two regions, a northern dusk sector and southern dawn sector (at t = 65 and 115 min in Figure 6a), and then moves from the high latitudes to the subsolar region as the SMFR magnetic field becomes more southward until t = 180 min. The tail reconnection also begins after the SMFR magnetic field turns from northward to southward.

Details are in the caption following the image
The simulation results for the small-scale magnetic flux rope (SMFR) type 4 conditions when the interplanetary magnetic field (IMF) turns from northward to southward Bz with a duskward By. This figure uses the same formatting and criteria as Figure 3.

3.1.2 Comparison of Flow Velocity, Electric Field, and Plasma Pressure in the Magnetospheric Tail Among Four Types of SMFR

In this subsection, we examine the plasma flows, electric field, and plasma pressure in the tail. The main results are summarized in Figures 7 and 8. First, Figure 7 shows the spatial profiles at a specific time of (a) the x components of the plasma flow velocity, Vx, (b) the y component of the electric field (Ey = Vx × Bz), and (c) the plasma pressure along the Sun-Earth line for the four SMFR types, distinguished by the different line colors (red, blue, orange, and green for types 1, 2, 3, and 4, respectively).

Details are in the caption following the image
(a) The x component of the velocity, Vx, (b) the y-component of the tail electric field, Ey, and (c) the plasma pressure, (P), along the Sun-Earth line on the nightside. The red, blue, orange, and green lines indicate data for small-scale magnetic flux rope (SMFR) types 1, 2, 3, and 4, respectively.
Details are in the caption following the image
Time evolution of the maximum value within a specific x range of (a) earthward plasma flow (Vx) and (b) plasma pressure (P) for the four small-scale magnetic flux rope (SMFR) types.

Figure 7a indicates that a strong earthward plasma flow appears for types 1, 3, and 4 at near-Earth region with a peak flow of approximately 300 km/s at t = 90 min, 261 km/s at t = 90 min, and 257 km/s at t = 115 min, respectively. This occurs when the tail reconnection region is at X ~ −11 to −13RE. For type 2, the overall earthward flow speed is notably weaker, which is about 68 km/s at t = 75 min. The locations of the peak flows differ and are closest to the Earth for type 1 and farthest from the Earth for type 2.

Figure 7b shows that the cross-tail electric field, Ey, increases up to approximately 2–2.6 mV/m for types 1 (at t = 90 min), 3 (at t = 95 min), and 4 (at t = 110 min). The profile for type 2 is much different than those of the other three types, indicating a much weaker electric field. The penetration of the solar wind electric field is most significant for type 1, as ~90% of the solar wind in the inner plasma sheet (−10RE < X < −3RE, that is, earthward side of the near-Earth neutral line) and ~26% of the solar wind in the outer plasma sheet (X < −10RE, that is, tailward side of the near-Earth neutral line). For type 3, these percentages are ~71% in the inner plasma sheet and ~15% in the outer plasma sheet, and they are ~78% in the inner plasma sheet and ~30% in the outer plasma sheet for type 4. Thus, a convective electric field over 70% of that of the solar wind is expected in the inner plasma sheet when types 1, 3, and 4 SMFRs impact the Earth's magnetosphere. However, caution is needed in taking this result since our global MHD model does not include inner magnetospheric physics such as the shielding process. This means that the penetration electric field in the inner plasma sheet is likely overestimated in our model (more discussion is given in section 8).

Figure 7c shows the plasma pressure profiles in the near-Earth tail region. The pressure at X = −6 to −8RE is the greatest, and this is most significant for type 1 and least significant for type 2. We find these peaks occur after a strong earthward flow. For type 2, a small earthward plasma flow (~68 km/s) associated with a weak duskward electric field of 0.62 mV/m appears during the tail reconnection at X ~ −14RE. For the other three types, a sharp pressure gradient at the earthward edge of the plasma sheet (at −8RE < X < −5RE for type 1, at −9RE < X < −5RE for type 3, and at −9RE < X < −6RE for type 4) is notable and can be critical in determining the dynamics there. Ogino and Walker (1998) suggested that a localized earthward flow in the tail region creates a distortion of the magnetic field lines to generate region 1 type field-aligned currents and a significant pressure gradient in the high inner plasma sheet to generate region 2 type field-aligned currents.

Figure 8 shows the time evolutions with a specific location of (a) the maximum earthward plasma flows (Vx) and (b) plasma pressure (P) with distinctions among the four SMFR types. Figure 8a indicates that, for SMFR type 1, a strong earthward flow starts to appear after the tail reconnection begins at t = 60 min. It reaches a maximum velocity just before the SMFR Bz is the most southward, and it remains elevated during the passage of the SMFR until Bz increases to 0 nT. In Figure 8b, we see that the plasma pressure peak for this type is slightly delayed relative to the peak time of Vx. We note that the steepest increase of pressure occurs at t = 60–70 min, which roughly corresponds to the time when Vx increases most rapidly.

Figure 8 shows that for type 2, while overall much weaker, the flow reaches a maximum at t = 75 min, which is prior to when SMFR Bz is the most northward and the SMFR By becomes zero. However, the plasma pressure increase is small as associated with the weak earthward flow due to the weak tail reconnection.

For SMFR type 3, Figure 8a indicates that the earthward flow is enhanced after t = 75 min, but the tail reconnection on the lobe field lines starts earlier, at t = 55 min. Additionally, Figure 8b indicates an interesting feature, that the early plasma pressure (t = 25 to 60 min) is approximately 20% to 40% higher for type 3 than for the other types. This is due to the plasmoid generation in the tail region. After the SMFR magnetic field turns from southward to northward, both the earthward flow and plasma pressure decrease rather rapidly.

Lastly, for type 4, where the SMFR magnetic field rotates from the north to south, a strong earthward flow and an increase in the plasma pressure occur only after the tail reconnection when the SMFR magnetic field is fully southward. Specifically, the major increase is delayed by ~20 min relative to the southward turning of the SMFR magnetic field.

3.2 Ionospheric Responses

In this section, we present the evolution of the ionospheric convection pattern and cross-polar cap potential and demonstrate the differences among the four types of SMFR. The results are shown in Figures 9-12. First, Figure 9 shows the time evolution of the convection cell pattern in response to the IMF rotation corresponding to each type of SMFR. In each panel, the numbers refer to the number of convection cell, and a schematic demonstration of time evolution of the IMF within SMFR is given on the right side of each panel (the IMF rotation sense is clockwise for all the four types as marked by the black arrows). We remind that all the simulations start from the quasi steady state, which is achieved by the 2-hr northward Bz as an initial condition (as mentioned in section 2). Accordingly, the initial quasi steady state is characterized by the well-known four-cell convection pattern as indicated in each panel in Figure 9.

Details are in the caption following the image
Time evolution of the number of the convection cell in the polar region in response to interplanetary magnetic field (IMF) changes corresponding to the four types of small-scale magnetic flux rope. The schematic demonstration on the right side of each panel refers to the IMF rotation (all in clockwise sense as marked by the black arrows).
Details are in the caption following the image
The simulation results of two-dimensional patterns of the plasma convection and electric potential at specific times with an open-closed field boundary in the polar region for small-scale magnetic flux rope (SMFR) types 1 and 2. In the panels of the potential, the blue and red contours indicate negative and positive potential, respectively. The blue and red contours indicate the negative and positive potentials, respectively. A double green line delimits the boundary between open and closed field lines.
Details are in the caption following the image
The same as Figure 10 except for small-scale magnetic flux rope (SMFR) types 3 and 4.
Details are in the caption following the image
Time evolution of the cross-polar cap potential in the polar region, shown in the same format as Figure 8. SMFR = small-scale magnetic flux rope.

Figure 9 indicates that the convection pattern in the polar ionosphere actively responds to the impact of the SMFRs in different ways. First, the main characteristics for type 1 is that the convection pattern during the first half of the entire period changes over a timescale of a few minutes to a few tens of minutes, whereas for the remaining half of the period, a two-cell convection pattern dominates. Specifically, for the first half of the period, the initial four-cell convection pattern changes first into a two-cell from t = 10 min (θ = 346°), followed by a three-cell after t = 35 min (θ = 316°), a four-cell for a short period, 5 min, and a three-cell again. Note that this fast evolution occurs even under southward Bz.

For type 2 in Figure 9, the main characteristics is that the convection pattern changes over a timescale of a few tens of minutes during the entire period of the northward Bz. Specifically, the initial four-cell convection pattern evolves first into a two-cell from t = 10 min (θ = 166°), followed by a three-cell from t = 40 min (θ = 132°), a four-cell from t = 80 min (θ = 98°), a three-cell pattern again from t = 125 min (θ = 66°), and finally a two-cell pattern from t = 165 min (θ = 21°).

For type 3 shown in Figure 9, the convection pattern is characterized mainly by a two cell-pattern for most fraction of the period. Specifically, the initial four-cell convection pattern rapidly changes to a two-cell one from t = 5 min (θ = 256°) and remains so until t = 155 min (θ = 129°) when a three-cell pattern appears, which is soon followed by a short-lasting (~10 min) four-cell pattern.

For type 4 shown in Figure 9, the convection pattern during the first ~80 min changes rather rapidly (on a timescale of a few tens of minutes) and is dominated by a two-cell for the remaining ~100-min period. Specifically, the four-cell convection pattern evolves first into a three-cell that survives only for a short time (~10 min) from t = 20 min (θ = 63°), which is followed by a short-lasting two-cell from t = 35 min (θ = 46°) and by another short-lasting three-cell pattern from t = 55 min (θ = 28°) during the northward Bz. Then, from t = 75 min (θ = 12°), it becomes a two-cell pattern, which persists for the remaining period (mostly southward Bz).

Figures 10 and 11 show two-dimensional patterns of the plasma convection and electric potential in the polar region for four types of SMFR condition at selected times. The physical quantities are mapped onto the ionosphere along the magnetic field line by using urn:x-wiley:21699380:media:jgra54421:jgra54421-math-0001, where ƒstands for each of the physical quantities, dl is the line segment along the field line, and B is the magnitude of the magnetic field. This expression generally gives an averaged value along the magnetic flux tube. The ionospheric potential is calculated by using the relationship ∇2ϕ = ∇·(−V × B). The velocity, V, or electric field, E, is mapped directly from the inner boundary of the magnetosphere to the ionosphere along the magnetic field line. The potential does not exactly match that due to convection since incompressibility condition was not assumed. If we impose ∇·V = 0, the potential pattern exactly matches the convection pattern. In the potential plots, the blue and red contours indicate negative and positive potentials, respectively. A double green line delimits the boundary between open and closed field lines.

The left two columns in Figure 10 show the convection flow vectors and potential contour lines at t = 15, 65, 95, and 180 min for type 1 of SMFR. A few interesting features are identified from these plots. At t = 15 min when the IMF is primarily duskward, we identify the strong convection flow in the cusp throat region (near noon), which is an antisunward flow with a duskward component. By the time of t = 65 and 95 min when the Bz is mainly southward, the convection flow intensifies and is mostly antisunward in the polar region. At t = 180 min when the IMF is purely dawnward, the convection flow direction in the polar region as well as at the cusp throat region is skewed toward the dawnward direction while still antisunward. The open-closed field boundary extends to a lower latitude at 67° at 24:00 magnetic local time (MLT), 75° at 12:00 MLT, and 71° at 06:00 and 18:00 MLTs (e.g., at t = 95 min). At this time, the cross-polar cap potential differences is ~47 kV (the peak value is ~52 kV and occurs at t = 100 min; see more details of its evolution in Figure 12).

The right two columns in Figure 10 show the results at t = 15, 75, 105, and 180 min for type 2 of SMFR. At t = 15 min when the IMF is largely dawnward, we identify the main convection flow in the dayside polar region and in the cusp throat region, which is primarily antisunward with a dawnward component. Later at t = 75 and 105 min when the IMF becomes primarily northward with a little By component, the flow in the polar region becomes mainly sunward. At t = 180 min when the IMF is purely duskward, the flow is mainly duskward in the polar region and roughly antisunward in the cusp throat region. At this time, the cross-polar cap potential difference is ~25 kV (see more details of its evolution in Figure 12). The open-closed field boundary remains at higher latitudes throughout the entire time interval compared to the case of type 1. However, the polar cap size is still nonnegligible and never shrinks to zero, and in fact, the open-closed field boundary extends close to about 70° on the nightside (24:00 MLT) from 55 to 75 min even under this northward IMF conditions.

The left two columns in Figure 11 show the results at t = 5, 40, 90, and 160 min for type 3 of SMFR. At t = 5 min when the IMF is mainly southward, we identify the strong antisunward flow in the cusp throat region. As the IMF turns toward the dawnward direction, the convection flow in the polar region becomes skewed dawnward (see plots at t = 40 and 90 min). At t = 160min when the IMF becomes northward but still with a significant dawnward By component, the polar convection is largely dawnward. At this time, a reverse cell pattern is seen near noon. Zhang et al. (2016) reported that the reverse convection cell inside the normal convection cell was generated by high-latitude lobe reconnection during the strong duskward By and weak northward Bz based on the DMSP F 18, SuperDARN, and PFISR observation. The open-closed field boundary extends close to ~67.7° at 24:00 MLT from 60 to 80 min. At t = 90 min, the open-closed boundary moves to 68°. At this time the cross-polar cap potential is largest among the presented four times, but more details of its time evolution are given in Figure 12.

Lastly, the right two columns in Figure 11 show the results at t = 15, 65, 115, and 180 min for type 4 of SMR. At t = 15 min when the IMF is still primarily northward, the convection flow did not develop much except near the cusp region. At t = 65 min when the IMF is now given a large duskward By component, the polar convection intensifies mainly in the duskside ionosphere and is mainly in the duskward direction. At t = 115 min when the IMF turns southward but still with large By components, the polar convection is even more significant at all over the polar region. It is both duskward and antisunward. In particular, we identify the convection in the ionosphere cusp throat region that is antisunward with a duskward component. At t = 115 min, while the convection pattern is mainly a two-cell, the negative potential value (on duskside cell) is higher than the positive potential value (on dawnside cell). The cross-polar cap potential difference is larger at t = 180 min than those at the earlier three times, but more details on the time evolution of the cross-polar cap potential are given in Figure 12. The open-closed field boundary extends to ~68° at 24:00 MLT at t = 125 min (plot not shown). At t = 180 min when the IMF is now purely southward, the polar convection flow is primarily antisunward.

Figure 12 shows the evolution over time of the electric potential mapped onto the polar region. The results are shown for all four types, as distinguished by the line colors and symbols. For type 1, the cross-polar cap potential increases gradually to 52 kV and then decreases to 30 kV. The potential is mainly controlled by the magnitude of the southward IMF Bz component within the type 1 SMFR. The peak value of the cross-polar cap potential at 100 min is closely associated with a minimum southward Bz of SMFRs at 90 min.

For the type 2 SMFR, where the northward IMF dominates, the cross-polar cap potential first increases to 25.4 kV and then decreases to a minimum value of ~15 kV, which is followed by a second peak of 25.3 kV. The cross-polar cap potential has a minimum value at 100 min after the largest northward Bz of SMFR conditions. The two peaks occur at the times of the maximum By magnitudes. Overall, the By profile controls the evolution of the potential of the type 2 SMFR.

For type 3, where the Bz turns from south to north, the cross-polar cap potential increases to 43 kV at about the time of the northward turning, and then it decreases to 15 kV. The cross-polar cap potential sharply increases to ~30 kV during the first 30 min due to the formation of a plasmoid in the near-Earth neutral line. Overall, the potential in this type is controlled by the Bz component.

Lastly, for type 4, the cross-polar cap potential increases gradually to 40 kV during the dayside reconnection, when the SMFR Bz is northward, and then the cross-polar cap potential increases further because of the dayside and tail reconnection due to southward turning of Bz, after which it eventually saturates. The potential in this type is controlled mainly by the magnitude of the magnetic field and the southward Bz component. Note that the peak occurs much later than the peak in type 3 due to the way that Bz changes from north to south instead of south to north.

In summary, while the evolution of the potential is well organized by the profiles of the SMFR magnetic field, the details differ among the four types due to the different internal structure of each SMFR type. The peak value of the cross-polar cap potential ranges between 25 (for type 2) and 52 kV (for type 1).

4 Summary and Discussions

In this work, we studied the effects of four different types of SMFR magnetic field conditions on the Earth's magnetosphere and ionosphere by using the three-dimensional global MHD simulations. The main features of the magnetospheric and ionospheric responses to the four SMFR types are summarized in Table 2.

Table 2. A Summary of the Major Features of the Magnetospheric and Ionospheric Responses to the Four Types of Small-Scale Magnetic Flux Ropes (SMFR) Magnetic Field Conditions
SMFR magnetic field conditions
(1) Type 1 (2) Type 2 (3) Type 3 (4) Type 4
IMF angle θ 360° to 180° 180° to 0° 270° to 90° 90° to 270°
IMF Bz < 0 Bz > 0 Bz < 0 to Bz > 0 Bz > 0 to Bz < 0
By > 0 to By < 0 By < 0 to By > 0 By < 0 By > 0
Response of the magnetosphere
Dayside Switches from region (1, 1′) to region (2) on to region (3, 3′) Switches from region (1, 1′) to region (2)
RR image
Tail region of −10RE < X < −3RE
Vx (km/s) 298 68 261 257
Ey (mV/m) 2.6 (90%) 0.62 (20%) 2.0 (71%) 2.2 (78%)
P (10−9 N/m) 1.09 0.43 0.82 0.61
Response of the ionosphere
CPa 4, 2, 3, 4, 3, 2 4, 2, 3, 4, 3, 2 4, 2, 3, 4 −4, 3, 2, 3, 2
OCFB 67° 70° 67.6° 68°
CPCP (kV) 52 ~25 43 40
  • Note. The peak value of earthward plasma flow (Vx), duskward electric field (Ey), and plasma pressure (P) in the tail region as shown in Figure 7. Here RR refers to the reconnection region, CP is the convection pattern evolution in the polar region, OCFB is the open-closed field boundary at 24:00 MLT when the polar cap is largest, and CPC means the peak values of the cross-polar cap potential.
  • a The numbers refer to the number of the convection cell in the order of time sequence as the convection pattern evolves during the SMFR impact period (details in Figure 9).
  1. Type 1 SMFR, where the SMFR Bz remains southward and the By turns from duskward to dawnward.

    • (i-1)

      During the passage of the SMFR, the dayside reconnection sites move from the high-latitude regions in the northern dusk and southern dawn hemispheres (marked by 1 and 1 in Table 2) to the subsolar region (marked by 2 in Table 2), and then it again moves to the high latitudes in the northern dawn and southern dusk hemispheres (marked by 3 and 3′ in Table 2).

    • (i-2)

      The maximum convective electric field in the inner plasma sheet is over 90% of that of the solar wind and ~26% of that in the outer plasma sheet. It is associated with a strong earthward plasma flow (with a peak of 298 km/s) and a steep pressure gradient due to the tail reconnection. The maximum cross-polar cap potential in the ionosphere is 52 kV with a two-cell convection pattern. For the first half period of the SMFR impact, the convection pattern evolves rather rapidly over timescales of a few minutes to a few tens of minutes; it evolves into a two-cell from the initial four cell pattern, followed by a three-cell pattern, four-cell pattern, and three-cell pattern again during the duskward By period. For the remaining half period, it finally becomes a two-cell pattern after dawnward turning of By. The open-closed field boundary extends to the low latitude of 67° on the midnight. The enhancements of all these quantities are stronger in the type 1 of SMFR than in the other SMFR types.

  2. Type 2 SMFR, where the SMFR magnetic field Bz is continuously northward and By rotates from dawnward to duskward.

    • (ii-1)

      During the passage of the SMFR, the dayside reconnection site first splits into two high-latitude regions in the northern dawn and southern dusk hemispheres (1 and 1′), moves closer to the Z axis (2 and 2′), and finally switches to the regions in the northern dusk and southern dawn hemispheres (3 and 3′). Despite the northward Bz, the tail reconnection occurs.

    • (ii-2)

      The flow speed, convective electric field, and pressure are weaker in the type 2 of SMFR than in the other SMFR types. The electric field in the inner plasma sheet is only 20% of that in the solar wind, the pressure gradient in the tail region is insignificant, and the maximum cross-polar cap potential in the ionosphere is approximately 25 kV, which is slightly higher than the typical value of a quiet time. Even though the Bz is continuously northward, the evolution of the convection pattern is complex and dramatic such that a series of different convection patterns occur on a timescale of a few tens of minutes from a two-cell pattern, a three-cell pattern, a four-cell pattern, a three-cell pattern again, and finally to a two-cell pattern. The open-closed field boundary extends to 70° latitude on midnight.

  3. Type 3 SMFR, where the SMFR Bz turns from south to north while the By remains duskward.

    • (iii-1)

      During the passage of the SMFR, the dayside reconnection occurs first at the subsolar region and moves to the high latitudes in the northern dawn and southern dusk hemispheres (2 and 2′). Notably, the formation of a plasmoid is seen, and its main features are similar to those of a plasmoid found by Ogino et al. (1986).

    • (iii-2)

      The enhancements of Vx, Ey and P are significant, though somewhat lower than those of type 1. They exhibit rather rapid increases associated with the plasmoid formation during the tail reconnection. The polar convection pattern is primarily a two-cell during the longest period of the SMFR impact. The open-closed field boundary extends at 67.6° latitude on midnight.

  4. Type 4 SMFR, where the SMFR Bz turns from north to south while the By remains dawnward.

    • (iv-1)

      During the passage of the SMFR, the dayside reconnection sites move in the opposite sense of those of type 3.

    • (iv-2)

      The enhancements of Vx, Ey, P, and the cross-polar cap potential are all comparable to those of type 3. The main difference is caused by the opposite way that the SMFR Bz changes its sign: Bz is first northward and then southward, while it changes from south to north in type 3. This leads to delayed responses in all the quantities for type 4 compared to type 3. Similar to the result of type 3, the open-closed field boundary extends to 68° in latitude, but for the early period of the SMFR impact when the IMF Bz is northward, the convection pattern evolves into a three-cell, a two-cell, and a three-cell pattern again over a short timescale (a few tens of minutes). Then for the remaining period of the SMFR impact, the convection flow becomes a two-cell pattern that continually exists in the polar region when Bz turns southward.

The recent study by Rong et al. (2015) analyzed the satellite data showing that the response time of the magnetotail to IMF and solar wind dynamic pressure is between 60 and 90 min. In their case, the IMF has a strong By (increase up to ~|20| nT) with a southward IMF Bz, and it maintained the polarity and amplitude for 1 hr at least while the solar wind dynamic pressure dramatically jumped in two cases. In contrast, in our simulations with SMFRs, we have taken constant weak solar wind values with a slowly rotating of magnetic field within SMFRs throughout the simulation periods. The delay times can vary depending on the specific types of the impacting SMFR. The plasma flow (and convection electric field) in the magnetotail responds with a time scale of ~10–15 min after the SMFR entered sufficiently the Earth. In addition, the ionospheric responses to IMF can have different delay times from dayside to nightside. For example, Zhang et al. (2016) examined the formation and evolution of the polar cap patches and found that the cross-cap transit time ranges from ~1.6 to 2.1 hr for the polar cap patches that exit at the midnight. In our study, it is no surprise that the ionospheric response timescales differ among the four SMFR types as they are distinguished by different IMF By and Bz profiles (Figures 9-12).

The main results of this work imply that the detailed responses of the magnetosphere and ionosphere can differ depending on the SMFR structure types. The separate and combined effects of the IMF Bz and By on the magnetosphere and ionosphere are predictable and already well known. Nevertheless, it is still worthwhile to study the impact of SMFRs, considering the various situations caused by different flux rope structures and impact angles relative to the Sun-Earth line. Note that our simulations treated only the idealized four types of flux ropes with an assumption that the flux rope axis is perpendicular to the Sun-Earth line. While our current work is the first to conduct MHD simulations on this subject where we incorporated the observed SMFR structure in an average sense, more realistic simulations can be done by considering the specific flux rope structure and realistic impact angle of an observed SMFR event, with the inclusion of the dipole tile angle. We leave this to future work.

While specific responses differ among the four SMFR types, we confirmed that the cross-polar cap potential is governed by a combination of components and the magnitude of the SMFR magnetic field. For all four types, the cross-polar cap potential increases above 20 kV. In particular, the peak potential is substantial enough to set a growth phase condition for the later release of energy via substorm occurrence. The relation between the polar cap potential and substorms has long been studied (e.g., Andalsvik et al., 2012; Liu et al., 2011; Lockwood et al., 2009; Milan, 2004). Lockwood et al. (2009) determined the transpolar voltage as a function of the IMF Bz for quiet times and the growth phase of a substorm, reporting values of 10 to 60 kV for −10 nT < Bz < 10 nT. Liu et al. (2011) reported an increase ranging from 16 to 29 kV of the cross-polar cap potential associated with substorm occurrences. Andalsvik et al. (2012) reported a substorm occurrence during an interplanetary coronal mass ejection passage where Bz was within −5 to −7 nT while By increased from −10 to −3 nT. They found that the cross-polar cap potential obtained by a SuperDARN spatial convection plot was 72 kV, and they found that substorm activities caused a 50% increase in the total cross-polar cap potential. The cross-polar cap potential obtained in our simulations increased by about 1.7 to 3.5 times the pre-impact value. The cross-polar cap potential value obtained by this study is within a similar range as the results of the previously described studies, supporting that the impact of SMFRs can set a condition appropriate for substorm energy storage and eventual release. Indeed, the observational studies by Feng et al. (2010) and Kim et al. (2017) indicate that a nonnegligible percentage of SMFRs might have led to substorms, as mentioned in the introduction.

One drawback in the present study is that our global MHD model does not include inner magnetospheric physics. There are other global MHD simulations that include the inner magnetosphere physics such as drift physics (Claudepierre et al., 2016; De Zeeuw et al., 2004; Zaharia et al., 2010). De Zeeuw et al. (2004) reported that the overshielding (dusk to dawn) electric field in the inner magnetosphere appears about 10 min after northward turning of IMF and the sudden decrease and shrinkage of the polar cap potential are due to the overshielding. Zaharia et al. (2010) showed that the cross-polar cap potential decreases by about 8% with inclusion of the inner magnetosphere model (e.g., Rice Convection Model). Claudepierre et al. (2016) compared the MHD simulations (using Lyon-Fedder-Mobarry (LFM) LFM model) with and without inner magnetospheric physics (using Rice Convection Model model). Clearly, inclusion of the inner magnetosphere physics is important for a more precise determination of the magnetospheric and ionospheric responses to SMFR. We expect that the penetration electric field shown in Figure 7 is likely overestimated compared to what would be obtained by a model that includes inner magnetosphere physics. The magnitude of the plasma properties in the magnetotail region obtained by our global MHD simulation is likely lower than the one that would be obtained by a model that includes inner magnetosphere physics. Also, the cross-polar cap potential in our model is possibly overestimated. To resolve all these issues, a future work will have to include inner magnetospheric physics in a simulation tool.

Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP) (NRF-2015R1C1A2A01051745), by the Korea Astronomy and Space Science Institute under the R&D program (project 2018-1-850-04) supervised by the Ministry of Science, ICT, by a grant from the Korean Space Weather Center of National Radio Research Agency to Chungbuk National University, and by the National Research Foundation of Korea (2016 M1A3A3A02017017). All the data used in this work were obtained from numerical simulations using a global MHD code, details of which are described in section 2 of the text.