Formation of the lunar wake in quasi-neutral hybrid model

Transcription

1 GEOPHYSICAL RESEARCH LETTERS, VOL. 32, L06107, doi: /2004gl021989, 2005 Formation of the lunar wake in quasi-neutral hybrid model E. Kallio Space Research Unit, Finnish Meteorological Institute, Helsinki, Finland Received 12 November 2004; revised 20 February 2005; accepted 10 March 2005; published 31 March [1] We study the formation of the lunar wake by a quasineutral hybrid (QNH) model. In the model ions are particles while electrons form a massless change neutralizing fluid. The model is three dimensional enabling us to study nonaxisymmetric filling of the tail with solar wind plasma resulting from non-axisymmetric electromagnetic forces. We find that already a not fully kinetic QNH model can reproduce some of the basic observed features, namely (1) a long (over 10 lunar radii) tail of depleted plasma density, (2) enhanced magnetic field within the optical shadow and depressed magnetic field near the edge of the optical shadow and (3) non-maxwellian plasma beams with high temperature anisotropy. The analysis also supports previous studies which have emphasized the role of the direction of the magnetic field and kinetic effects in the lunar wake. Citation: Kallio, E. (2005), Formation of the lunar wake in quasi-neutral hybrid model, Geophys. Res. Lett., 32, L06107, doi: /2004gl Introduction [2] The Moon does not have an atmosphere nor a significant intrinsic magnetic field and therefore the solar wind can freely impact its surface. Despite the vicinity of the Moon, not much is known about the properties of the plasma and the magnetic field in the lunar wake. In situ data from the lunar wake are based mostly on the measurements made by Explorer 35 in 60 s [Colburn et al., 1967; Lyon et al., 1967; Ness et al., 1968; Siscoe et al., 1969] and several flybys made by Wind starting at mid 90 s [Bosqued et al., 1996; Ogilvie et al., 1996; Owen et al., 1996; Clack et al., 2004]. [3] The plasma and magnetic field observations have been interpreted by using a drift approximation [Ness et al., 1968], gas dynamic (GD) and magnetohydrodynamic (MHD) models [e.g., Colburn et al., 1967; Siscoe et al., 1969] and recently 1D and 2D fully kinetic models [Farrell et al., 1998; Birch and Chapman, 2001, 2002]. The purpose of this paper is to take the first step to derive a selfconsistent three dimensional (3D) picture of the properties of the plasma and the magnetic field in the lunar wake with a quasi-neutral hybrid (QNH) model. The QNH model differs from a fully kinetic model so that in the QNH model the plasma is assumed to be quasi-neutral. Fully kinetic models have demonstrated the role of charge separation [Farrell et al., 1998; Birch and Chapman, 2001] and it is therefore not self-evident in what respect a QNH approach can increase our understanding of the formation of the lunar wake. [4] This paper is organized as follows. First, the basic features of the applied QNH are described. Then the properties of the solar wind protons and the magnetic field in the lunar wake are presented. 2. Description of the Lunar QNH Model [5] The runs were performed using an improved version of the 3D QNH model that has previously been used to study how the plasma interacts with Mars [Kallio and Janhunen, 2001], Mercury, Titan and Venus. [6] In the simulation the undisturbed solar wind flows along X, the component of the interplanetary magnetic field (IMF) perpendicular to the X-axis is along +Y, and the Z-axis completes the right handed coordinate system. The size of the simulation box was X = [ 9.8, 1.8] R L,Y= [ 4.2, 4.2] R L and Z = [ 4.2, 4.2] R L. The average number of particles per cell was 10. The time step for the propagation of the particles and the electric field, dt ion,e, was 0.1 s. In the presented run the solar wind velocity, U sw, the solar wind density, n sw, and the interplanetary magnetic field (IMF), B sw, was chosen to approximate upstream parameters during the crossing on the lunar wake by Wind in Dec. 27, 1994 [Bosquedpetffiffiffial., 1996]: U sw =[ 500, 0, 0] km s 1, B sw = [7, 7, 0]/ 2 nt and nsw = 5 cm 3. The ion temperature, T p, during the flyby was K[Bosqued et al., 1996]. Thisptemperature ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi corresponds to ion thermal velocity, U ith (= 2kT p =m proton ), of 36 km s 1 and, consequently, the sonic Mach number, M s (= U sw /U ith ), of 14. In this study we adopted M s = The electron temperature, T e, was taken to be 3 ev. Runs were performed using constant cubic grid cells of size 0.1R L = 173 km = 73l e in ptheffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi solar wind (l e = electron inertia length = 5.31 km/ n e ½cm 3 Š,R L = radius of the Moon = 1730 km). [7] In the QNH model ions are treated as particles and they are accelerated by the Lorenz force. Electrons form a massless charge neutralizing fluid. A quasi-neutral hybrid model cannot model self-consistently the ambipolar electric field associated with the charge separation when electrons start to fill the Lunar wake. Therefore in this paper the electric field is a sum of two electric fields: E moment and E ambipolar. The former is derived from the electron momentum equation E moment ¼ u e B: Here u e and B are the electron bulk velocity and the magnetic field, respectively. The ambipolar electric field, E ambipolar, was modelled by the equation E ambipolar ¼ kt ð o = jj e ÞrðÞ=n n ð1þ ð2þ Copyright 2005 by the American Geophysical Union /05/2004GL were T o is a constant temperature, n is the ion density and e is the elementary charge. The adopted functional form L of5

2 Figure 1. The lunar wake in a quasi-neutral hybrid model: (a) n(h + )incm 3,(b)U x (H + )inkms 1,(c)jBjin nt and (d) U z in km s 1 on X = 2 R L,X= 6.8 R L, Y = 0 and Z = 0 planes. The vectors on Figure 1b and Figure 1c show the normalized bulk velocity and the magnetic field vectors, respectively. The vectors are plotted in the four planes at every fourth pffiffi grid point. In the run the upstream parameters were n sw =5cm 3, U sw =[ 500, 0, 0] km s 1 and B sw = [7, 7, 0]/ 2 nt. The white lines in Figure 1a represent density contours n(h + ) = 1, 2, 3, and 4 cm 3. One R L corresponds to about 730 l e in the solar wind. captures basic features of the ambipolar electric field seen in a fully kinetic 2D model [see Birch and Chapman, 2002], namely (1) it points radially inwards, (2) it is strongest near the optical shadow and (3) its magnitude decreases with increasing distance from the Moon. Furthermore, in this paper T o = T e (= 3 ev) was adopted which result to the minimum electric potential associated with E ambipolar at the center of wake of 44V. This value is close to the minimum potential ( 50 V) obtained from a fully kinetic model [see Birch and Chapman, 2001, Figure 3]. It is worth noting that runs without the ambipolar electric field (i.e. when T o was put to zero) resulted qualitatively similar Lunar tail region than presented in this paper although certain differences exist, especially in the ion velocity distribution function. Finally, the magnetic field is propagated by Faraday s law and the resistivity is assumed to be zero everywhere, both outside and within the Moon. [8] The initial and the boundary conditions were the following. Initially, the magnetic field in the simulation box was constant: B(r, t=0)=b sw. Maxwellian protons are injected into the simulation box at the front wall (X = 1.8R L ). An ion was taken away from the simulation if it hits the back wall (X = 9.4R L ), the front wall or the surface of the Moon. Periodic boundary conditions were applied at the side faces of the simulation box (Y,Z = + 4.2R L ). The basic difference between the present QNH model and our previous QNH models is that in this study the obstacle, the Moon, was assumed to be a perfect insulator. In the simulation such a situation was implemented by applying a constant electric field inside the body, that is, E(r, t) = E sw = U sw B sw inside the Moon. [9] From a global MHD or QNH modelling point of view the lunar wake near the surface is a problematic region due to the low ion density and, consequently, due to the high Alfvén speed. A stable self-consistent solution could be obtained by either using a large grid cell size or by a small time step (Courant-Friedrich-Levy condition). However, in this preliminary study the simulation was to run with reasonable spatial resolution and in a feasible time. Therefore, the following two constraints were introduced. First, the Alfvén speed and, consequently, the fast magnetosonic speed, was limited by implementing a minimum density threshold value, n min, which is used in a grid cell if the plasma density within it, n cell, was smaller than the thresh- 2of5

3 Figure 2. The magnetic field on the X = 6.8 R L plane: (a) jbj,(b)b x,(c)b y and (d) B z, all in nt. White circles show the position of the optical shadow. The six solid lines in Figure 2d are added to help an eye to catch basic draping pattern and the direction of the magnetic field lines when viewed from the tail toward the Moon (draping not to scale). old value, that is, n cell = max(n cell,n min ). In this respect the most problematic and probably the least accurately modelled region is a (vacuum) region at the midnight connected to the surface of the Moon which practically never contains particles in the QNH model. Second, in the analyzed run the magnetic field was kept constant during the first 80 seconds. Then, after this pure electrostatic running period, the magnetic field was propagated only dt B = 0.01 s at every step when ions and electric field are propagated by dt ion,e = 0.1s. As a consequence, at this running period the magnetic field is propagated ten times slower than ions and electric field. Finally, at t = 140 s the magnetic field was left to propagate self-consistently (dt B = dt ion,e = 0.1 s) and the solution presented in this paper was recorded at t = 160s when the system has not yet reached a fully stationary solution. The applied two constraints enables us to study some basic features when the system starts to evolve from a stationary electrostatic solution toward a fully self-consistent solution. 3. Results [10] Figure 1 displays some basic 3D plasma and magnetic field features based on the QNH model. Planes X = 2 R L and x = 6.8 R L are displayed because they show parameters at the distances where Explorer 35 in 60 and Wind in 1994 crossed the lunar tail, respectively. The low ion density region was found to extend beyond the simulation box (Figure 1a). The magnitude of the magnetic field is enhanced within the tail and depressed near the edge of the optical shadow (Figure 1c). Note that the value of ju x j is close, but not exactly equal to ju sw j: ju x j < ju sw j in the z > 0 hemisphere and ju x j > ju sw j in the z < 0 hemisphere implying that the properties of the plasma in the wake are not exactly axially symmetric with respect to the X-axis (Figure 1b). U z is negative (positive) on the Z > 0(Z < 0) hemisphere on the XZ plane, illustrating the convergence of the flow within the tail (Figure 1d). Note also that U z is not zero on the XY-plane providing another example of a pslightly ffiffiffi non-axially symmetric flow when IMF = [7, 7, 0]/ 2 nt. [11] Properties of the magnetic field on the X = 6.8 R L plane are depicted in detail in Figure 2. The total magnetic field is enhanced within the optical shadow and depressed near the edge of the optical shadow especially near the XZ plane (Figure 2a). Magnetic field x-component decreases quite axisymmetric near the edge of the optical shadow (Figure 2b) which is a consequence of a relatively axially symmetric convergence of the flow near the edge of the optical shadow towards the X-axis. The magnitude of B y is on its minimum near the edge of the optical shadow on the XZ plane (Figure 2c). The magnitude of B z can be seen to form several regions on the XY plane (Figure 2d). Variations of B y and B z components can be interpreted to present the magnetic field configuration sketched in Figure 2d: The magnetic field lines are moved near the optical shadow toward the X-axis by the converting plasma and the field 3of5

4 Figure 3. Ion velocities (v x,v y ) near the point pffiffiffi r =[ 6.8, 0, 0] R L : (a) a run made forpimf ffiffi = [7, 7, 0]/ 2 nt and (b) a run for IMF = [0, 7, 0]/ 2 nt. The solid lines show the direction of the IMF and the dashed lines the direction perpendicular to the IMF. The solid and dotted lines were drawn to cross at the ion bulk velocity U =(U x,u y,u z ). lines are packed together within the optical shadow and slightly twisted around the X-axis. [12] The ion velocity distribution is non-maxwellian within the lunar tail. Figure 3 depicts ion velocities near the point [ 6.8, 0, 0] R L. The ion velocities are derived by recording the velocity and the position of an ion when it moves inside a sphere with a radius of 0.1 R L centered at r detector =[ 6.8, 0, 0] R L during t [100s, 130s]. Figure 3a shows a run that contains a non-zero IMF x-component while Figure 3b corresponds to a run for identical upstream plasma parameters but a zero IMF x-component. At the center of the tail the plasma is a mixture of two ion populations moving against each other. The plasma temperature is highly anisotropic. For example, at [ 6.8, 0.5, 0] R L T perp /T par 5(T perp = ion temperature perpendicular to IMF; T par = temperature parallel to the IMF) and the velocity distribution becomes even more anisotropic at the center of the tail. 4. Discussion and Concluding Remarks [13] This paper represents the first step to study the formation of the lunar wake with a 3D QNH model. Despite the limitations of the QNH model the main lesson to be learned from this study is that already a not fully kinetic model can reproduce some basic features observed in the lunar wake. First, a long (>10 R L ) tail of depleted plasma density is formed (Figures 1a and 3a), much as observed [Ogilvie et al., 1996; Bosqued et al., 1996]. Second, an enhanced magnetic field within the optical shadow and a decreased field near the optical shadow (Figures 1c and 2) are qualitatively in agreement with observations [Ness et al., 1968; Owen et al., 1996]. Third, non-maxwellian plasma beams with high temperature anisotropy are formed within the central tail, much as anticipated to be the case according to Wind observations [Ogilvie et al., 1996; Clack et al., 2004]. [14] Furthermore, the filling of the lunar wake has been argued to depend on the orientation of the IMF [e.g., Owen et al., 1996]. Wind observations also suggest that converging plasma results in a density enhancement at the central wake [Clack et al., 2004]. According to Wind data, ion beams coming from the opposite direction have also generally unequal densities [Clack et al., 2004]. Similar plasma features can also be seen pffiffi in the two analysed QNH runs: In the IMF = [0, 7, 0]/ 2 nt run the density increase was observed already at X 7R pffiffiffi L (figure not shown) while in the IMF = [7, 7, 0]/ 2 nt run no noticeably density increase can be detected at the same distance (Figure 1a). The difference between these two QNH model runs can also be identified if pone ffiffiffi compares Figures 3a and 3b. In the IMF = [0, 7, 0]/ 2 nt case the central wake is filled with two ion populations about the same pffiffi number of hits to the detector. In the IMF = [7, 7, 0]/ 2 nt case, in contrast, the ion population which has U y > 0 (i.e. ions coming from the Y < 0 hemisphere) seems to result in less hits to the detector than the ion population which has U y < 0 (ions coming from the Y > 0 hemisphere). [15] In addition, Wind observations suggest that the lunar wake has 3D plasma [e.g., Bosqued et al., 1996; Ogilvie et al., 1996; Clack et al., 2004] and magnetic field features [e.g., Owen et al., 1996]. The two runs analysed in this paper support the idea of a 3D lunar tail. First, Figure 1b suggests that U x is asymmetric in the XZ-plane with respect to the Z-axis ju x j being slightly (about the thermal speed km s 1 ) smaller(larger) than ju sw j on the Z > 0(Z < 0) hemisphere. That is most likely manifestation of diamagnetic effects. It is also worth noting that while the converging flow results in the decrease of the B x component quite symmetrically with respect to the X-axis (Figure 2a), the converging flow results in a non-axisymmetric B y and B z variations (Figures 2c and 2d) and, consequently, a nonaxisymmetric jbj (Figure 2a). [16] To summarise, this preliminary study shows that a QNH model can reproduce several basic observed features of the lunar wake providing new insight to the Moon-solar wind interaction. [17] Acknowledgments. The author thank P. Janhunen and I. Sillanpää for a careful inspection of the manuscript and the two referees for numerous valuable suggestions to improve the paper. References Birch, P. C., and S. C. Chapman (2001), Particle-in-cell simulations of the lunar wake with high phase space resolution, Geophys. Res. Lett., 28(2), , (Correction, Geophys. Res. Lett., 28(13), 2669, 2001.) Birch, P. C., and S. C. Chapman (2002), Two dimensional particle-in-cell simulations of the lunar wake, Phys. Plasmas, 9(5), Bosqued, J. M., et al. (1996), Moon solar wind interaction: First results from the Wind/3DP experiment, Geophys. Res. Lett., 23(10), Clack, D., J. C. Kasper, A. J. Lazarus, J. T. Steinberg, and W. M. 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5 Ogilvie, K. W., J. T. Steinberg, R. J. Fitzenreiter, C. J. Owen, A. J. Lazarus, W. M. Farrell, and R. B. Torbert (1996), Observations of the lunar plasma wake from the Wind spacecraft on December 27, 1994, Geophys. Res. Lett., 23(10), Owen, C. J., R. P. Lepping, K. W. Ogilvie, J. A. Slavin, W. M. Farrell, and J. B. Byrnes (1996), The lunar wake at 6.8R L : Wind magnetic field observations, Geophys. Res. Lett., 23(10), Siscoe, G. L., E. F. Lyon, J. H. Binsack, and H. S. Bridge (1969), Experimental evidence for a detached lunar compression wave, J. Geophys Res., 74, E. Kallio, Space Research Unit, Finnish Meteorological Institute, Vuorikatu 15A, FIN Helsinki, Finland. (esa.kallio@fmi.fi) 5of5