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Keywords:

  • Current systems;
  • Lunar wake;
  • Hybrid simulations

Abstract

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model
  5. 3 Model results
  6. 4 Discussion
  7. Acknowledgements
  8. References

We present the lunar wake current systems when the Moon is assumed to be a non-conductive body, absorbing the solar wind plasma. We show that in the transition regions between the plasma void, the expanding rarefaction region, and the interplanetary plasma, there are three main currents flowing around these regions in the lunar wake. The generated currents induce magnetic fields within these regions and perturb the field lines there. We use a three-dimensional, self-consistent hybrid model of plasma (particle ions and fluid electrons) to show the flow of these three currents. First, we identify the different plasma regions, separated by the currents, and then we show how the currents depend on the interplanetary magnetic field direction. Finally, we discuss the current closures in the lunar wake.

1 Introduction

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model
  5. 3 Model results
  6. 4 Discussion
  7. Acknowledgements
  8. References

Direct impact of the solar wind plasma with the Moon, with no atmosphere and no global magnetic field, neutralizes the plasma in the lunar dayside, leaves a plasma void, and forms an expanding rarefaction region [Lyon et al., 1967], confined into a plasma Mach cone downstream [Whang and Ness, 1970].

While the interplanetary magnetic field (IMF) penetrates through the non-conductive body of the Moon relatively undisturbed [Ness et al., 1968; Sonett, 1982], the plasma pressure gradient across the void generates a diamagnetic current sheet around the void boundary [Colburn et al., 1967; Michel, 1968a]. This results to compress the magnetic field lines within the void and to depress them at the wake flanks [Colburn et al., 1967; Ness et al., 1968]. Owen et al. [1996] illustrated the diamagnetic current flow around the void for an IMF perpendicular to the solar wind. They assumed the Moon to be a part of the void and showed that the current around the void moves perpendicular to the IMF, points in opposite directions on either side of the void, and closes through diffuse currents across the plasma shadow downstream and through the lunar surface on the dayside.

The magnitude of the magnetic field perturbations in the lunar wake depends on the angle between the IMF and the solar wind [Whang, 1968b; Colburn et al., 1971; Sibeck et al., 2011]. As this angle decreases, the plasma pressure decreases inside the void, where the magnetic field is increased to keep the pressure balance (magnetic pressure + plasma pressure = constant) [Michel, 1968a; Ness et al., 1968]. However, any magnetic field enhancement inside the void must be accompanied by a field reduction outside (∇ ⋅ B = 0) that depresses the magnetic fields at the void surroundings [Colburn et al., 1967]. When there is any magnetic field component perpendicular to the solar wind flow, plasma fills in the void and decreases both the plasma pressure gradient and the magnetic field compression there [Michel, 1968a; Ness et al., 1968].

The lunar plasma wake has been an interesting scientific subject for numerical modeling. The general features of the Moon-solar wind plasma interaction have been studied several times using the magnetohydrodynamics [Michel, 1968a; Wolf, 1968; Spreiter et al., 1970; Xie et al., 2012] and kinetic/particle [Whang, 1968a; Lipatov, 1976; Farrell et al., 1998; Birch and Chapman, 2001; Kallio, 2005; Wang et al., 2011] approaches. Recently, Holmström et al. [2012] using a self-consistent three-dimensional (3D) hybrid model of plasma confirmed the earlier mentioned features of the Moon-solar wind interaction in the lunar wake for the different IMF angles.

Here we present a new perspective of the current systems in the lunar wake using a 3D hybrid model of plasma [Holmström, 2010]. The main scientific questions are how these currents flow and close in the lunar wake. In the following sections, we use Holmström et al. [2012] simulation results to describe the current systems in the lunar wake. We show that the current is confined not only around the void region but also at the rarefaction boundaries. We discuss the characteristics of these three currents and their closure in the wake.

2 Model

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model
  5. 3 Model results
  6. 4 Discussion
  7. Acknowledgements
  8. References

Hybrid model of plasma is a kinetic modeling approach that involves solving Maxwell's equations for positively charged particles, while the electrons are treated as a fluid. Hybrid model electric and magnetic fields and proton number density for the solar wind interaction with the Moon have been shown in the earlier publication by Holmström et al. [2012]. In this model, the Moon is assumed to be a solar wind plasma absorber without any intrinsic magnetic field and no conductivity, as described in Holmström [2010, 2011]. The electrons are an ideal gas, and then electron pressure pe = nekbTe, where kb is the Boltzmann's constant, ne is the electron number density and is equal to the proton number density (np) due to the charge neutrality, and Te is the electron temperature. pe is assumed to be adiabatic with an adiabatic index γ = 5/3 [Holmström, 2010; Holmström et al., 2012]. Then pe ∝ |np|γ; therefore, pressure gradient is comparable to the density gradient in our model.

Here we use Holmström et al. [2012] magnetic field simulation results for the two extreme IMF angles, anti-parallel and perpendicular to the solar wind, to compute the current density inside the lunar Mach cone using the general Ampere's law,

  • display math(1)

where J and B are the current and the magnetic flux density, respectively, and μ0 is the vacuum permeability. The displacement current (∂ E/∂ t) is neglected in the hybrid model due to the Darwin approximation [Holmström, 2010] and is not considered here in equation ((1)).

We use a right-handed coordinate system centered at the Moon, with the + x axis directed toward the Sun. Here the solar wind flows along the − x axis, and the IMF is in the xy plane. The simulation cell size is 160 km ≃ 0.09RL, where RL ≃ 1730 km is the lunar radius.

3 Model results

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model
  5. 3 Model results
  6. 4 Discussion
  7. Acknowledgements
  8. References

Solar wind upstream parameters are the same as those used by Holmström et al. [2012], which are summarized in Table 1. We consider two simulation runs using two different IMF angles: anti-parallel (along the + x axis) and perpendicular (along the + y axis) to the solar wind as shown by arrows in Figures 1 and 2. Upstream solar wind parameters and IMF angle remain constant during the simulation runs.

Table 1. Solar wind parameters used in the simulations
ParameterSymbolValueUnit
Solar wind velocityvsw−450 inline imagekm/s
IMF magnitude|Bsw|7.0nT
Proton number densitynsw7.1cm−3
Proton temperatureTp10.35eV
Electron temperatureTe12.05eV
Sonic Mach angleΘs∼ 7.5deg
Magnetosonic Mach angleΘms∼ 10.5deg
image

Figure 1. IMF is anti-parallel to the solar wind, and the arrows in Figures 1a and 1b indicate the solar wind and the IMF directions. Black circles represent the Moon in each panel. (a and b) Relative proton number density in logarithmic scale, (c and d) current density along the y axis, (e and f) current density along the z axis, and (g and h) magnetic field perturbations (BX/BSW − 1) along the x axis. Figures 1a, 1c, and 1g show cuts in the y = 0 plane seen from the − y, and Figure 1e shows cut in the z = 0 plane seen from the + z axis. Panels on the right show cuts in the x = − 7RL plane seen from the + x axis marked with the vertical dashed lines in the panels on the left. The white regions in Figures 1a and 1b show zero density in the void.

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image

Figure 2. IMF is perpendicular to the solar wind, and the arrows in Figures 1a and 1b indicate the solar wind and the IMF directions. Black circles represent the Moon in each panel. (a and b) Relative proton number density in logarithmic scale and (c and d) current density along the x axis, (e) magnetic field perturbations along the y axis (BY/BSW − 1), and (f) magnetic field perturbations along the z axis (BZ/BSW) in the x = − 7RL plane seen from the + x axis. Stream lines in Figure 2e show the direction of the induced magnetic field. The geometry of the cuts is the same as in Figure 1.

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Since currents are computed from derivatives of the magnetic field (Equation (1)), they will be noisy due to the statistical nature of particle simulations. To enhance the visual appearance of the currents, we apply a 2D symmetric Gaussian (normal) low-pass filter [Crowley and Stern, 1984] to the magnetic field before computing the currents from equation ((1)). The width of the filter (the standard deviation) was chosen as one simulation cell size, since this shows the currents better, without too much smearing.

In Figure 1, the IMF is anti-parallel to the solar wind. Figures 1a and 1b show the relative plasma number density. They show that the plasma void, shown with white color, cannot be entirely filled in by the solar wind and the plasma flows cylindrically symmetric around the Moon-Sun axis (+ x axis). This forms the lunar plasma void, indicated with zero density, the rarefaction region with lower plasma density than the interplanetary plasma, and the re-compression region enveloped between the void and the inner boundary of the rarefaction [Johnson and Midgley, 1968; Wolf, 1968], all confined within the lunar Mach cone and form the lunar plasma wake.

Magnetic field perturbations between the earlier mentioned regions are the result of three current systems around the boundaries of these regions. Figures 1c–1f show these currents inside and around the lunar Mach cone: around the void (J1), around the rarefaction region (J2), and around the re-compression region (J3), all labeled in Figure 1c. These panels only show the most significant components of each currents (JY and JZ) which are flowing on a perpendicular plane to the IMF direction. JX is nearly zero and therefore not shown here. Figures 1d and 1f indicate the counter-clockwise flow of J1 and J3 and clockwise flow of J2 currents as viewed from the Moon-Sun axis.

Figures 1g and 1h show the x component of the induced magnetic field (BX/Bsw − 1), parallel to the IMF, as a result of the generated currents. Since both J1 and J3 are flowing in the same direction, the magnetic field enhancement in the void region increases more after J3 is added to J1 at X ≃ − 4RL (Figure 1g). Magnetic field reduction outside the void is inevitable due to the divergence free property of the magnetic fields. However, magnetic field perturbations in the lunar nightside are confined within the lunar Mach cone, and this is governed by J2.

In Figure 2, the IMF is perpendicular to the solar wind; hence, the upstream plasma refills the wake along the IMF and through the thermal expansion (Figure 2b). Figure 2a shows the regions of rarefaction and re-compression that surround the plasma void in the wake. Similar to the case of an anti-parallel IMF, three current systems form around the boundaries of these regions but with different flow directions.

Figures 2c and 2d show the main component (JX) of the currents. They show that above (below) the IMF plane, J1 and J3 flow toward (away from) the Moon and J2 flows away from (toward) the Moon. All flow in a plane perpendicular to the IMF and are all labeled in Figure 2c. The other two components of the currents (JY and JZ) are almost an order of magnitude smaller than JX (not shown here). The three currents induce magnetic fields parallel and anti-parallel to the IMF inside and outside the void region, respectively, (Figure 2e) and form magnetic field rotations perpendicular to the IMF (Figure 2f). Consequently, the magnetic fields are compressed in the void and depressed outside the void region.

3.1 Current closure in the wake

When IMF is parallel to the solar wind, none of the currents are flowing parallel to the IMF direction. This, together with the circular flow of the currents shown in Figures 1c–1f, indicate that each of the three currents contain current loops, each of which close with themselves. Figure 4a illustrates these current loops in the lunar Mach cone.

When the IMF is perpendicular to the solar wind, J1 flows perpendicular to the IMF along the x axis in opposite directions on either side of the void. Figure 2c shows connections between J1 and J2 in the vicinity of the lunar poles. Moreover, we see that J3 connects to J1 at the wake-refill distance (X ≃ − 4RL) and considerably changes the J1 current density at that connection. However, current closure far downstream cannot be determined from Figure 2. Since the lunar wake signature extends far away from the Moon (≫ 20RL) [Michel, 1968b; Clack et al., 2004] and the field perturbations in the wake reduces gradually, 3D modeling of a very large domain is computationally expensive, and the field perturbations are not considerably enough at large distances to see the currents in our simulations from equation ((1)). Instead, we qualitatively examine the lunar wake evolution in Figure 3 for an early time before the wake crosses the outflow boundary of our simulation domain. Figure 3 shows that J1 and J3 have upward and J2 has downward wake crossing flow downstream in the wake. This is consistent with Owen et al. [1996] illustration downstream in the wake for J1. We illustrate the main component of the current systems in the lunar wake for the IMF perpendicular to the solar wind in Figure 4b. J2 and J3 current closures far downstream in the wake are shown with dashed lines.

image

Figure 3. Current density along the x axis for an early time (t = 50 s, while t = 170 s for the results shown in Figures 1 and 2). Otherwise, the cut is the same as in Figure 2c.

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image

Figure 4. An illustration of the main components of the current systems in the lunar wake when IMF is (a) anti-parallel and (b) perpendicular to the solar wind.

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4 Discussion

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model
  5. 3 Model results
  6. 4 Discussion
  7. Acknowledgements
  8. References

Previous works on the current systems around the lunar wake had only considered the plasma density discontinuity between the void region (diamagnetic cavity) and the ambient solar wind plasma [Colburn et al., 1967; Michel, 1968a; Ness et al., 1968; Owen et al., 1996]. Since the plasma density gradient across the void is much larger than the magnetic field gradient in the void, the diamagnetic property of the plasma supports formation of a diamagnetic current around the void (J1). However, Figures 1 and 2 show that two more current systems (J2 and J3) are present around the rarefaction and re-compression regions in addition to the main current around the plasma void (J1). These two currents are non-diamagnetic currents and form to confine the magnetic field perturbations within the rarefaction region and eventually inside the Mach cone. Existence of such currents had not been noticed in the theoretical works by Ness et al. [1968] and Whang [1968a, 1968b] because they have only considered the magnetic field perturbations on a plane parallel to the IMF direction, while these currents are a results of the confinement of magnetosonic wave propagation perpendicular to the IMF direction by the solar wind flow.

All the currents are confined within the lunar Mach cone, flowing on circular closed loops perpendicular to the IMF axis and rotate with the IMF direction as the IMF angle changes. These currents perturb the magnetic fields in the lunar nightside (Figures 1g, 1h, 2e, and 2f), enhancing the magnetic fields in the void region and reducing them outside [Colburn et al., 1967; Ness et al., 1968; Whang, 1968a]. Comparing Figure 1h with Figure 2e confirms that the magnitude of the magnetic field compression in the void is a function of the IMF angle with the solar wind flow direction and is maximum when the angle is minimum.

Here we discussed the current systems and their closures in the lunar wake for two extreme IMF directions. However, the IMF direction changes by time and is nominally oriented at ~45° to the solar wind flow at the Moon-Sun distance [Smith and Wolfe, 1979]. Since the current system planes are perpendicular to the IMF axis, any IMF directions which neither are parallel nor perpendicular to the solar wind result in a superposition of the parallel and perpendicular cases. However, the geometry of the current systems will be far more complex than discussed here, if the IMF is not parallel or perpendicular to the solar wind.

When the IMF is perpendicular to the solar wind, the direction of J1 in our simulation is similar to the Owen et al. [1996] illustration in the nightside. Owen et al. [1996] considered the Moon to be a part of the plasma void; therefore, the pressure gradient across a void-Moon generates a current on the lunar dayside. However, the existence of the dayside current could depend on the lunar conductivity. Here, in our model, we assumed that the Moon to be a non-conductive body, while Blank [1969] theoretically calculated the magnitude of this current as well as the intensity of the induced magnetic field in the conductive layers of the Moon. The time-varying IMF results to induce currents on the conductive surface of the Moon and perturb the magnetic fields there [Sonett, 1982; Dyal et al., 1974, and references therein]. Therefore, the lunar wake current system may also close on the lunar dayside, if we consider the conductivity of the Moon. However, dayside current is not seen in our model for a non-conductive Moon.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model
  5. 3 Model results
  6. 4 Discussion
  7. Acknowledgements
  8. References

The work of S.F. was supported by the National Graduate School of Space Technology (NGSST), Luleå University of Technology, the Swedish National Space Board (SNSB), and the National Graduate School of Scientific Computing (NGSSC), Uppsala University, Sweden. This research was conducted using resources provided by the Swedish National Infrastructure for Computing (SNIC) at the High Performance Computing Center North (HPC2N), Umeå University, Sweden. The software used in this work was in part developed by the DOE-supported Flash Center for Computational Science at the University of Chicago. The authors thank the reviewers for their valuable comments to improve this manuscript.

References

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model
  5. 3 Model results
  6. 4 Discussion
  7. Acknowledgements
  8. References