Formation of a broad plasma turbulent layer by forward and inverse energy cascades of the Kelvin Helmholtz instability

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2009ja014637, 2010 Formation of a broad plasma turbulent layer by forward and inverse energy cascades of the Kelvin Helmholtz instability Yosuke Matsumoto 1 and Kanako Seki 1 Received 12 July 2009; revised 7 June 2010; accepted 23 June 2010; published 15 October [1] The 2 D MHD simulations of the Kelvin Helmholtz instability (KHI) with transverse magnetic field and highly asymmetric density configurations in a large simulation domain show that rapid formation of a broad plasma turbulent layer can be achieved by forward and inverse energy cascades of the KHI. The forward cascade is triggered by growth of the secondary Rayleigh Taylor instability excited during the nonlinear evolution of the KHI. The inverse cascade is accomplished by nonlinear mode couplings between the fastest growing mode of the KHI and other KH unstable modes. As a result of the energy transport by the inverse cascade, the growth rate of the largest vortex allowed in the system reaches 3.7 times as large as the linear growth rate. A PIC simulation under the similar initial configuration is also conducted and shows the quick growth of the largest vortex and the resultant spread of the turbulent layer in which plasmas are effectively mixed. The proposed mechanism enables rapid formation of a large scale mixing layer which resembles observational characteristics of the low latitude boundary layer of the magnetosphere. Citation: Matsumoto, Y., and K. Seki (2010), Formation of a broad plasma turbulent layer by forward and inverse energy cascades of the Kelvin Helmholtz instability, J. Geophys. Res., 115,, doi: /2009ja Introduction [2] It has been revealed by in situ observations that the plasma sheet in the magnetotail becomes cold and dense when the interplanetary magnetic field (IMF) becomes northward [Terasawa et al., 1997; Borovsky et al., 1998]. The cold, dense plasma population is distributed in the flanks of the magnetosphere, and their spatial extent is about 10 times the earth radius [Wing and Newell, 2002]. In those regions, observations often show a mixture of two populations of solar wind and magnetospheric origins on the closed magnetic field line [e.g., Fujimoto et al., 1998], and therefore the region is termed low latitude boundary layer (LLBL). The formation mechanism of such a broad boundary layer is a long standing issue in magnetospheric physics, and the observations of the LLBL have indicated possible entry processes of the solar wind plasma into the magnetosphere responsible for its formation. [3] One mechanism can be represented by the double lobe reconnection model [Song and Russell, 1992]. The model incorporates magnetic reconnection for a purely northward IMF condition. In that situation, the high latitude reconnection takes place in both hemispheres at the same time, and the newly closed magnetic field lines capture and transport the solar wind plasma into the nightside of the magnetosphere. The double lobe reconnection has been evidenced by 1 Solar Terrestrial Environment Laboratory, Nagoya University, Nagoya, Japan. Copyright 2010 by the American Geophysical Union /10/2009JA the Cluster observations [Lavraud et al., 2006], and was recently found to be responsible for the solar wind transport by the observations and the global MHD simulation [Øieroset et al., 2005; Li et al., 2008]. [4] Anomalous diffusion across the magnetopause is another possible mechanism, which is still under debate, however. The Kelvin Helmholtz instability (KHI) driven by the flow shear between the solar wind and the magnetosphere has been considered as one of the candidate diffusive mechanisms, since the observations have shown that the low latitude magnetopause is actually unstable to the KHI [Fairfield et al., 2000; Hasegawa et al., 2006]. To explain the formation of the broad boundary layer, recent numerical simulations have proposed promising solar wind transport mechanisms. [5] Nonlinear evolution of the KHI with the in plane component of the magnetic field revealed that a rolled up vortex triggered magnetic reconnection inside the vortex [Nykyri and Otto, 2001; Nykyri and Otto, 2004; Nakamura et al., 2008]. Then the reconnected field lines detach the plasma from the solar wind and transport it into the magnetosphere. Turbulent transport proposed by Matsumoto and Hoshino [2004] is another idea which allows particles move across the field lines and results in the formation of a broad mixing layer. They discussed the turbulent evolution of the KHI under a transverse magnetic field configuration and a large background density variation and found that the transition from a laminar flow to turbulence is initiated by a secondary instability excited at the density interface inside the vortex. It was identified that the secondary instability is a kind of the Rayleigh Taylor instability (RTI), where the 1of15

2 Figure 1. (a) Imaginary (growth rate) and (b) real parts of the angular frequency as a function of the wave number are obtained by the linear analysis of the KHI for the present initial configurations of a = 1.0 (solid line) and a = 0.1 (dashed line). The wave number and the angular frequency are normalized by the half shear width l and the transit time l/v 0, respectively. centrifugal force by the rotating fluid motion acts as a gravitational force. The following electromagnetic particlein cell simulations [Matsumoto and Hoshino, 2006] showed an efficient mixing of collisionless plasmas across the field lines and found that turbulent electrostatic fields excited by the secondary RTI play an important role in the ion and electron mixing. [6] While the recently proposed nonlinear theories of the KHI explain the solar wind transport across the magnetopause, it remains unclear whether they are responsible for the formation of the broad boundary layer much wider than the primal vortex size. In this context, successive pairings of vortices of the KHI have been considered to contribute to the broad boundary layer. The evolution of the KHI in a large simulation domain is characterized by vortex parings of the fastest growing mode due to the subsequent growth of the subharmonic modes [Miura, 1997, 1999]. The resultant width of the shear layer becomes much wider than the initial one. Its time scale is, however, much slower than that of the fastest growing mode as regulated by the linear theory of the KHI. A direct application of the vortex pairings to the LLBL formation therefore implies a difficulty in explaining the time scale. [7] In this paper, we show that rapid formation of a broad plasma turbulent layer is possible by a combination of the secondary RTI and a nonlinear growth of a large scale vortex in a large simulation domain which allows growth of multiple unstable modes. The results are based on highresolution, large scale 2 D MHD and electromagnetic particle in cell (PIC) simulations described in section 2. In section 3, we first focus on nonlinear mode couplings among KH unstable modes and a resultant inverse energy cascading process of the 2 D KHI. Then we show the secondary RTI coincident with the nonlinear mode couplings. The importance of both forward and inverse energy cascades is discussed in this section. The results are summarized, and applications of the proposed mechanism to the LLBL formation are discussed in the last section. 2. Numerical Model 2.1. MHD Simulation [8] The nonlinear evolution of the physical quantities, plasma number density N, pressure P, velocity V, and magnetic field B are followed on the basis of MHD equations. We numerically solve the equations by utilizing the CIP algorithm [Yabe et al., 2001; Matsumoto and Seki, 2007, 2008]. Although the scheme is not a conservative one, all simulation runs presented in this paper satisfy the total energy conservation within an error level of 0.1%. [9] We consider a 2 D simulation domain in which we impose a periodic boundary condition along the x direction and a reflecting boundary condition along the y direction. We initially set the background velocity as V x = 0.5V 0 {1.0 + tanh (y/l)}, where l is the half width of the initial shear layer. We set the homogeneous transverse magnetic field B 0^z so that the initial thermal pressure is constant in the system. The number density profile is provided by N = 0.5N 0 {(1 + a) + (1 a)tanh(y/l)}, where a is the asymptotic number density ratio. We set a = 1.0 to understand basic natures of the KHI in a large simulation domain. We also adopt a = 0.1 to model the magnetospheric boundary where the cold and dense solar wind plasma is interacting with the hot and tenuous magnetospheric plasma. We set M f = 1.0 and the plasma b = 1.0, where M f = V 0 /V f is the fast magnetosonic Mach number. For the purpose of solving a turbulent evolution, we specially added a hyperdiffusion term in the momentum equation in the form Figure 2. MHD simulation results for the homogeneous density case (a = 1.0). Snapshots of the z component of the vorticity profile are shown in a normalized unit of V 0 /l. The results for the simulation run with the perturbation of equation (1) are shown in the left column and for the reference run in the right column. The figures are taken at (a and e) T = 125.0l/V 0, (b and f) T = 175.0l/V 0, (c and g) T = 225.0l/V 0, and (d and h) T = 275.5l/V 0, respectively. 2of15

3 Figure 2 3 of 15

4 Figure 3. Time evolution of the Fourier amplitude of V y for the m = 1 mode (solid lines) and the FGM (dashed lines). The red lines show the results from the simulation run with the perturbation of equation (1). The black lines are the results of the reference run. Dashed dotted line indicates the linear growth rate of each mode. of n/nr 4 V, where n is provided through a dimensionless parameter of R = N 0 V 0 l 3 /n = [10] The system size in the x direction (L x ) is set so that the m = 4 mode becomes the fastest growing mode (FGM) of the KHI, which is given by the linear analysis for the present initial configurations. Figure 1a shows the growth rate as a function of the wave number. For a = 1.0 (solid line), we obtained the FGM as k x fgm l = 0.40 and its corresponding linear growth rate as gl/v 0 = For a = 0.1 (dashed line), the FGM and the maximum linear growth rate are larger than for a = 1.0. The FGM is k x fgm l = 0.56 and gl/v 0 = The simulation domain in the y direction ranges 40l < y < +40l with l resolved with 20 grid points. [11] For reference, we also note here that the linear growth rates of the largest vortex allowed in this system (m = 1) are gl/v 0 = for a = 1.0 and gl/v 0 = for a = 0.1. The average phase speeds of the KHI are obtained by taking the average slopes of the dispersion relations shown in Figure 1b, which give V ph = 0.50V 0^x for a = 1.0 and V ph = 0.65V 0^x for a = 0.1. [12] The simulation run is initialized by a perturbation in V y provided through V y ¼ X8 0:0025V 0 cos 2mx cosh 2 y : ð1þ L x m¼1 Equation (1) shows that the initial perturbation has a continuous spectrum as it would be observed in the solar wind and the magnetosphere, while Miura [1999] has examined it with the FGM and its harmonic modes (m = 1, 2, 4, and 8). The latter case is also examined in section 3 as a reference run. In addition to the perturbation in V y, we initially added a random noise in the number density whose amplitude is 1% of the background number density for simulation runs with a = EM PIC Simulation [13] We also examine with an electromagnetic particlein cell (EM PIC) code, which fully solves ion and electron dynamics along with electric and magnetic field developments [Birdsall and Langdon, 1991]. The code follows particle motions by the Buneman Borris method and the electric and magnetic field developments by the implicit FDTD method [Hoshino, 1986]. The charge conservation scheme [Umeda et al., 2003] is also implemented for the calculation of the current density. The code is parallelized via domain decomposition in the flow direction by using the Message Passing Interface (MPI) library. [14] With this EM PIC code, we examined the case with a = 0.1. The initial macroscopic (MHD) conditions are the same as those set in the MHD simulation, except that the plasma b = 0.2. The difference in b does not change the FGM and the normalized growth rate, however. In addition to the MHD parameters, the kinetic parameters of the mass ratio (M/m), the ratio of the electron plasma frequency to the gyrofrequency (w pe /W ge ), the temperature ratio (T i /T e ), and the Debye length (l D ) defined at y = +40l (in the magnetosheath) are respectively given as M/m = 16, w pe /W ge = 4.0, T i /T e = 1.0, and l D = 0.5Dh, where Dh is the spatial grid size. We also set the initial half shear width l =2r gi, where r gi is the ion thermal gyroradius in the magnetosheath. For this velocity shear layer width, the initial equilibrium and the linear growth of the KHI are only slightly influenced by the finite Larmor radius effect of the ion [Pritchett and Coroniti, 1984; Cai et al., 1990]. Particles are initially loaded so that the number density profile becomes N = 0.5N 0 {(1 + a) +(1 a)tanh(y/l)} with N 0 = 400particles/cell. Since r E 0 for the present magnetic field configuration, we also loaded additional particles of ion (electron) when E y / y > 0 (ale y / y < 0) so that the Gauss law r E =4pr e is fulfilled. Thus the system is not charge neutral. Total numbers of grid points and particles are and , respectively. The simulation run presented in this paper satisfies the total energy conservation within an error level of 0.2%. 3. Simulation Results 3.1. Homogeneous Density Case (a = 1.0) [15] In this subsection, we examine nonlinear evolutions of the KHI in a large simulation domain with the uniform background number density (a = 1.0) by 2 D MHD simulations. Simulation runs are initialized with the perturbations of equation (1) or with the fundamental mode (the fastest growing mode, m = 4) and its harmonic modes (m = 1, 2, and 8). The latter case has been examined by Miura [1997, 1999], and we call this run the reference run. From the previous works, it has been considered that the emergence of the larger scale vortices is due to the linear growth of the subharmonic modes. The time scale of their growth is 4of15

5 Figure 4. Time history of the growth rate obtained by taking slope of the growth curves in Figure 3 for the run initialized by equation (1). (a and b) Black lines are the results of the m = 1 and 2 modes, respectively. Green, blue, and magenta lines are the result of the m = 3, 4 (FGM), and 5 modes, respectively. The results of the m = 1 and m = 2 modes for the reference run are shown by black dotted lines. The obtained growth rate is normalized by the linear growth rate (Figure 1a). therefore regulated by the linear theory of the KHI, which predicts that they are slowly growing modes in time as shown in Figure 1a. [16] Figure 2 shows the time evolutions of the z component of the vorticity (r V z ). The snapshots are shown for the simulation run initialized by the continuous spectrum (equation (1)) on the left column (Figures 2a 2d) and for the reference run on the right column (Figures 2e 2h). At T = 125l/V 0, the m = 4 mode appears first in both cases (Figures 2a and 2e) as expected from the linear theory. The m = 2 mode subsequently appears in Figures 2b and 2f at T = 175l/V 0 and the m = 4 mode coalesces with each other. A clear difference is seen at T = 225l/V 0 when the m = 1 mode appears, and the largest vortex in the system is formed in Figure 2c, whereas the m = 2 mode still dominates in the reference run (Figure 2g). In the bottom images, the m = 1 mode seems to appear, but the m = 2 mode is still a dominant one in the reference run (Figure 2h), while the m = 1 mode dominates at T = 275l/V 0 in Figure 2d. [17] Figure 3 shows the time evolution of the Fourier amplitude of each mode. Here we focus on the growth of the FGM (m = 4) and the m = 1 mode, whose growth is clearly different depending on the initial seed perturbations. The FGM grows at a rate predicted by the linear theory in both simulation runs (dashed lines) as expected from Figure 2. On the other hand, the m = 1 mode behaves differently. In the reference run, the m = 1 mode obeys the linear theory (black solid line), whereas the m = 1 in the simulation run initialized by equation (1) rapidly grows after T =60l/V 0 at a rate much faster than we expect from the linear theory (red solid line). As a result, the m = 1 mode appears in a similar time scale of the FGM, resulting in fast formation of the largest vortex. [18] By taking the time derivative of the growth curves in Figure 3, we obtained a growth rate and compared it with the linear growth rate. Figure 4 shows the obtained growth rate for each mode (m = 1, 2, 3, 4, and 5) divided by the linear growth rate. Here we additionally focus on the m = 2, 3, and 5 modes. In Figure 4a, while the FGM (m = 4) and the m = 3 and m = 5 modes obey the linear growth rates until T =60l/V 0, the growth rate of the m = 1 mode reached 3.4 times as large as the linear growth rate at T = 75l/V 0 after the saturation of the m = 3, 4, and 5 modes. In contrast, the growth of the m = 1 mode in the reference run obeys the linear growth rate during this time interval (dotted line). This signature indicates that the rapid growth of the m = 1 mode is due to the nonlinear coupling of the m = ±4 modes with the m = 3 and the m = +5 modes, which are absent in the reference run, that is, an inverse energy cascade. (Note that the m = 1, 3, 4, and 5 modes also show the same growth, since they are complex conjugates of the m = 1, 3, 4, and 5 modes.) In Figure 4b, the growth rate of the m = 2 mode is also shown, which is maximized to 1.2 times as large as the linear growth rate at T =80l/V 0, while it obeys the linear growth rate until T = 140l/V 0 in the reference run (dotted line). [19] To understand the energy transport to the m = 1 mode (and also the m = 2 mode) by the inverse energy cascade, we introduce the second order perturbation theory by following Miura and Sato [1978]. The present simulations show that 5of15

6 the flow is always submagnetosonic. Thus incompressiblity can hold and is assumed in the following consideration. For the present transverse magnetic field configuration, the governing equation is greatly simplified to the vorticity equation for an inviscid fluid @ @ ; where is the stream function which relate to the velocity as follows: V ; V : We then divide the stream function to unperturbed 0 and perturbed d parts, and put them into equation (2). By expanding d in a Fourier series as follows: ðx; y; tþ ¼ X k ^ k ðy; tþ expðikxþ; we obtain the equation for d ^ k by retaining the secondorder terms as follows: kv k ^ k ^! 2 þ 2 ^ k " # ¼þ X ^ k 0 ðk k 0 Þ ^ k k ^ k 3 k 0 6¼k þ X ðk k 0 ^ " k 2 ^ # k 2 ðk k 0 Þ 2 ^ k k 0 ; ð5þ k 0 6¼0 where V ^ : The right hand side of equation (5) consists of the source terms due to the mode coupling between the modes k and k k, which is second order in the perturbation amplitude. [20] Now we focus on the mode m = ±1 and assume that only the m = ±1 ±8 modes can be mode coupled. Then equation (5) reads as V ^ 1 ^! 2 ^ 1 ¼þS ^ 1 ^ 2 þ S ^ 2 ^ 3 þ S ^ 3 ^ 4 þ S ^ 4 ^ 5 þ S ^ 5 ^ 6 þ S ^ 6 ^ 7 þ S ^ 7 ^ 8 ; ð7þ ð2þ ð3þ ð4þ ð6þ 8 where S k =2 S(d ^ ±1 k d ^ ±k ) represents the source terms on the right hand side of equation (5) consisting of the mode coupling term of d ^ ±1 k d ^ ±k. From equation (7) and Figure 1, most of the energy to the m = ±1 modes is fed by the source terms S(d ^ 3 d ^ ±4 ) and S(d ^ 4 d ^ ±5 ) because the m = ±3, m = ±4 (FGM), and m = ±5 modes are the first to enter the nonlinear stage. Other source terms are negligible, since the corresponding modes are in small amplitudes and products of their amplitudes are much smaller at T =75l/V 0 when the growth rate of the m = 1 mode is maximized. [21] If one focused on the m = 2 mode, equation (5) reads as 2V 4 ^ 2 ^! 2 2 ^ 2 ¼þS ^ 1 ^ 1 þ S ^ 1 ^ 3 þ S ^ 2 ^ 4 þ S ^ 3 ^ 5 þ S ^ 4 ^ 6 þ S ^ 5 ^ 7 þ S ^ 6 ^ 8 : ð8þ For the m = ±2 mode, only the source term S(d ^ 3 d ^ ±5 ) can contribute as the energy source, since the m = ±4 modes can be coupled with the m = ±2 and m = ±6 modes, which are still in the early linear stage. The slight increase in the growth rate of the m = 2 mode in Figure 4b can be attributed to this reason. [22] In the reference run, in which only the m = 1, 2, 4, and 8 modes are initially seeded, there is no mode to be coupled with the FGM (m = 4) when it entered the nonlinear stage to feed the energy to the m = 1 mode (and also to the m = 2 mode). Thus the modes follow the linear growth as described by the left hand side of equation (5) Inhomogeneous Density Case (a = 0.1) [23] In the previous subsection, we showed that the mode couplings among the KH unstable modes are important to generate a large scale vortex. In this subsection, we examine both inverse and forward energy cascades by setting the large background density variation across the velocity shear layer (a = 0.1) by 2 D MHD simulations. Under this configuration, we expect to see the growth of the secondary Rayleigh Taylor instability (RTI) excited during the nonlinear evolution of the FGM [Matsumoto and Hoshino, 2004, 2006]. [24] Figure 5 shows the time evolution of the number density color coded in a logarithmic scale for the simulation Figure 5. MHD simulation result for the inhomogeneous density case (a = 0.1). Snapshots of the number density profile are shown in a logarithmic scale. The figures are taken at (a) T =77.4l/V 0,(b)T = 89.1l/V 0,(c)T = 96.8l/V 0, (d) T = 135.5l/V 0, (e) T = 174.3l/V 0, and (f) T = 251.7l/V 0, respectively. The white dashed line indicates the original shear boundary position. 6of15

7 Figure 5 7 of 15

8 Figure 6. Time evolution of the Fourier amplitude of V y for the m = 1 mode and the FGM in the same format as Figure 3. run initialized by equation (1). Initially, we observed the growth of the FGM (m = 4) (Figure 5a) as expected from the linear theory. Then, in a vortex that appears around x =35l, the secondary RTI is excited due to the centrifugal force acting on the highly asymmetric density layer (Figure 5b). The characteristic is similar to what was reported by Matsumoto and Hoshino [2004]. During the growth of the secondary RTI, the vortices started to merge into larger scale vortices, which caused the broadening of the turbulent area in the y direction (Figures 5c and 5d). The secondary RTI is still active after the emergence of the largest vortex allowed in the system at the thin density boundary layer (Figure 5e), and the final state of the system is dominated by the turbulent region (Figure 5f). [25] Figure 6 shows the time evolution of the Fourier amplitude of each mode in the same format as Figure 3. From T =35l/V 0 to T =80l/V 0, the m = 1 mode grows with a much larger rate than its linear growth rate (red solid line) while the FGM grows with the linear growth rate (red dashed line). We again meet the inverse cascade under the large background density variation. Owing to the large growth rate, the m = 1 mode appears in a similar time scale to the FGM, resulting in the fast broadening of the turbulent layer shown in Figure 5. We also examined the reference run for a = 0.1. In this run, we reproduced that the m = 1 mode and the FGM obey the linear theory (black lines) as expected from Figure 3. This result contrasts with the recent work by Faganello et al. [2008], who reported that the secondary RTI interferes the growth of the subharmonic mode. Figure 7 shows the obtained growth rate from the growth curves in Figure 6 in the same format as Figure 4a. It is found that the maximum growth rate of the m = 1 mode reaches 3.7 time as large as the linear growth rate at T =50l/V 0. [26] Figure 8 shows the time history of the average wavelength (2p/hk x i) defined as follows: X ^V 2 jk hk x i¼ xj y jk xj X ; ð9þ ^V 2 jk xj y where ^V y is the Fourier amplitude of V y and S kx sums over the entire wave number space from k x = 0 to the Nyquist wave number. In both simulation runs, the average wavelength increases linearly until T =25l/V 0 reflecting the linear growth of the FGM and the m = 3 and 5 modes from the initial spectrum. In the reference run, the average wavelength becomes shorter after the saturation of the FGM (black line), indicating that the energy is transported toward smaller scales by the secondary RTI. After the time T = 200l/V 0, the average wavelength becomes longer as the m = 1 mode enters the nonlinear stage. The inverse energy cascade is also found in the reference run in the late nonlinear stage as has been reported previously [Miura, 1999]. Meanwhile, the simulation run starting with the continuous spectrum shows that the average wavelength becomes longer after T =50l/V 0 (red line), indicating the inverse energy cascade. After the saturation of the m = 1 mode at T = 160l/V 0, the energy is transported toward shorter scales, i.e., the forward energy cascade is still active by the secondary RTI. [27] As a result of the rapid growth of the largest vortex together with the forward energy cascade by the secondary RTI, the turbulent layer quickly broadens. Figure 9 shows the time evolution of the average number density profile in the y direction defined by hnðy; tþi ¼ 1 Z Nðx; y; tþdx: ð10þ L x Figure 7. Time history of the growth rate obtained by taking the slope of the growth curves in Figure 6 in the same format as Figure 4a. 8of15

9 It is clearly seen from Figure 9 that there are three stages of the broadening of the number density profile. The first stage (T = l/V 0 ) is mainly contributed by the saturated FGM (see Figure 6). This is a typical nature of the KHI. The second stage is represented by the profile between T = 100l/V 0 and T = 160l/V 0. This interval corresponds to the appearance of the m = 1 mode, whose amplitude exceeded the FGM as indicated in Figure 6. In the final stage after T = 160l/V 0, the turbulent layer further broadens after the saturation of the m = 1 mode. The width of the turbulent layer finally reached about 45l. Figure 8. Time history of the average wavelength defined by equation (9). The red line shows the result from the simulation run starting with the perturbation given by equation (1), and the black line is the result of the reference run Inhomogeneous Density Case by EM PIC Simulation (a = 0.1) [28] The MHD simulation shown in the previous subsection shows generation of small scale vortices that spread in the simulation domain. The energy transport toward small scale vortices causes an energy dissipation, resulting in mixing of the two populations. The dissipation mechanism itself is caused by the numerical viscosity and resistivity introduced in the MHD simulation, which are most effective in spatial scales comparable to the simulation mesh size. To deal with the mixing process self consistently, we conducted a 2 D EM PIC simulation under the similar MHD conditions described in section 2 and subsection 3.2. The simulation run is initialized by equation (1). With the fully kinetic simulation, the mixing processes is discussed in quantitative manners. Figure 9. Time history of the average profile of the number density in the y direction defined by equation (10). The profile is color coded in a logarithmic scale. The bottom and left axes indicate the time and the spatial scale in the y direction in normalized units, whereas the top and right axes indicate the actual spatial scales in the magnetosphere in the unit of the Earth radius (R E ). 9of15

10 Figure 10. PIC simulation result for the inhomogeneous density case (a = 0.1). Snapshots of the ion mixing rate profile are shown. The figures are taken at (a) T = 71.6l/V 0,(b)T =77.6l/V 0,(c)T = 89.5l/V 0, (d) T = 131.3l/V 0, (e) T = 173.0l/V 0, and (f) T = 256.6l/V 0, respectively. The white dashed line indicates the original shear boundary position. 10 of 15

11 Figure 11. Snapshots of the electron mixing rate profile are shown in the same format as Figure of 15

12 of the mixing layer is somewhat inhibited in the PIC simulation as compared to the MHD simulation result ( 35l < y < +15l). [31] The spread of the mixing layer is assisted by a quick growth of the largest vortex (the m = 1 mode). Figure 12 shows the time evolutions of the Fourier amplitude of the m = 1 mode and the FGM in the same format as Figure 3. While the FGM obeys the linear growth rate, it is also found from the PIC simulation that the m = 1 mode rapidly grows as the FGM entered the nonlinear stage as shown in the MHD simulation. Kinetic effects do not largely affect the inverse energy cascade to the m = 1 mode. [32] Time history of the average profile of the mixing rate in the y direction is obtained as we did for the number density in the MHD simulation (equation 10) by the following: hr mix ðy j ; tþi ¼ 1 Z R mix ðx i ; y j ; tþdx: ð12þ L x Figure 12. Time evolution of the Fourier amplitude of the y component of the ion velocity (V yi ) for the m = 1 mode and the FGM obtained by the PIC simulation in the same format as Figure 3. [29] Herewedefineamixingrate[Matsumoto and Hoshino, 2006] as follows: R mix ði; jþ ¼ 2 S ls ð1þ l ði; jþ 0:5 S l S l ði; jþ þ 1; ð11þ where S l (i, j) is the shape function of a superparticle l at a grid point (i, j), S l stands for summation for particles around the grid point (i, j), and S l (1) (i, j) is for the particles initially located in the region y > 0 (magnetosheath). The mixing rate is defined so that it is maximized when the particles (1) occupy the cell equally with the other population (y 0). [30] Figures 10 and 11 show the time evolution of the KHI by diagnosing a spatial distribution of the ion and electron mixing rates. Each snapshot is taken at a similar stage of the MHD simulation result (Figure 5). With the EM PIC code, we reproduced the nonlinear evolution of the KHI characterized by both forward and inverse energy cascades similar to that obtained by the MHD simulation. Mixed cells are found around the demarcation line which separates the two plasma populations (the magnetopause). The demarcation line, whose width is of the order of the gyroradius of each species, swirls during the linear growth of the KHI (Figures 10a and 11a). The secondary RTI excites a localized electrostatic field as reported by Matsumoto and Hoshino [2006], and the turbulent electrostatic field scatters ions (Figures 10b and 10c) and folds the electron demarcation line into fine scale structures as they fill the ion mixing cells (Figures 11b and 11c). This swirling and folding process continues as the m = 1 mode nonlinearly grows (Figures 10d and 10e and Figures 11d and 11e) and causes the mixed cells spread over the entire region in 30l < y < +15l at the end of the run (Figures 10f and 11f). The resultant width Shown in Figure 13 are the time profiles of the average ion and electron mixing rates. We obtained that the time and spatial scales of the mixing layer are similar to the MHD simulation result shown in Figure 9. In the PIC simulation, the mixing process is self consistently treated and the microscopic electric field is the key agent of the mixing. Nonetheless, the overall structure of the mixing layer is similar to the MHD simulation result. With the present PIC simulation, we found that the width of the mixing layer is mainly determined by the MHD processes, i.e., the forward and inverse energy cascades via the KHI evolution. 4. Summary and Discussion 4.1. Summary [33] We have examined high resolution 2 D MHD and electromagnetic PIC simulations of the KHI in a large simulation domain which allows growth of multiple KH unstable modes. It is shown that rapid formation of a broad plasma turbulent layer is possible by a combination of forward and inverse energy cascades of the KHI. [34] We first examined the inverse energy cascading process by the MHD simulation. In a large simulation domain, we showed that the inverse cascade is accomplished by nonlinear mode couplings among the KH unstable modes. While the fastest growing mode (FGM) obeys the linear growth rate, the growth rate of the largest vortex (m = ±1) allowed in the system reached 3.4 times as large as the linear growth rate owing to the mode couplings between the FGM (m = ±4), and the m = ±3 and m = ±5 modes. [35] Under the large background density variation, we examined both the inverse and forward energy cascades of the KHI by the MHD and PIC simulations. The forward cascade is triggered by the secondary Rayleigh Taylor instability (RTI) excited during the nonlinear evolution of the KHI. As studied for a single KH vortex [Matsumoto and Hoshino, 2004, 2006], the secondary RTI is operative in the larger scale vortices at the thin density boundary layer. The secondary RTI triggers a transition from the laminar flow to the turbulent one which enhances mixture of the plasmas. At the same time, the inverse cascade is also operative in this simulation. The growth rate of the largest vortex allowed in 12 of 15

13 Figure 13. Time history of the average profile of the mixing rate in the y direction defined by equation (12) for (a) the ion and (b) the electron in the same format as Figure 9. the system reached 3.7 times as large as the linear growth rate. Owing to the rapid growth of the largest vortex by the inverse energy cascade, the turbulent area broadens in a time scale much faster than we expect from the linear theory. Thus, the underlying mechanisms range over scales from the dissipation scale to that of the largest vortex and proceed at the same time. A cross scale coupling is essential in the present mechanism Application to the LLBL Formation [36] In this subsection, we apply the present mechanism to the actual magnetospheric boundary. We transform the nor- 13 of 15

14 Figure 14. Time history of (a) the mixing area defined by equation (14) and (b) the diffusion coefficient defined by equation (15). Black and red lines show the ion and electron profiles, respectively. malized time and the spatial scale to the actual scales by adopting l = 1000 km and V 0 = 400 km s 1. The transformed scales are indicated on the top and right axes in Figure 9. [37] On the top axis in Figure 9, the distance that the KHI travels along the magnetopause during the evolution is indicated by using the phase speed of the KHI (see section 2) as X ½R E Š¼Time ½=V 0 ŠjV ph j½v 0 Š 1000 ½kmŠ 400 ½km s 1 Š 400 ½km s 1 Š6380 ½kmŠ: ð13þ Here the unit of each term is denoted in brackets. From Figure 9, we obtained that a turbulent layer of 5 to 7 R E is formed during the KHI and travels a distance of 25 R E which corresponds to a time scale of 10 min. [38] The diffusion coefficient of the mixing layer is estimated from the PIC simulation result as follows. We integrate the mixing rate in the simulation domain to calculate the total mixing area as Z Z S mix ¼ R mix ðx i ; y j Þdxdy: ð14þ The mixing area is contributed by cells linearly weighted by the mixing rate. Then the mixing area is normalized by ll x. Thus the normalized mixing area represents the width (L mix ) of the mixing layer in a rectangular area of L x L mix in the unit of l. By taking the time derivative of the obtained time profile, we calculate a diffusion coefficient by D yy ðtþ ¼ dl mixðtþ 2 : ð15þ dt Note here with caution that the obtained diffusion coefficient is not defined locally across the deformed magnetopause but characterizes how the mixed cells spread in the y direction as the KHI develops. [39] Figure 14a shows the time evolution of the normalized mixing area. The ion and electron mixing areas monotonically increase after some fluctuations around the scale of each gyroradius. At the end of the simulation run, the width of the mixing layer reached 20l. Figure 14b shows the time history of the diffusion coefficient. It increases proportionally as D yy / T 2 until T = 160l/V 0. After the saturation of the m = 1 mode (T 170l/V 0 ), the diffusion coefficient again increases as D yy / T 2, exceeds D yy = 3.0lV 0, and saturates. Further growth of the diffusion coefficient is limited by the simulation box size. By transforming the simulation units to the actual scales, it is found that the present mechanism gives the diffusion coefficient of m 2 s 1 within 10 minutes. [40] The obtained spatial and time scales characterize the typical nature of the LLBL observations [e.g., Wing and Newell, 2002] and suggest that the KHI plays important roles in the LLBL formation under a purely northward IMF case. Some of important issues, however, remain to be discussed below Discussion [41] We presented by the MHD and PIC simulations that the overall structure of the turbulent layer is mainly controlled by the MHD processes of the forward and inverse energy cascades via the KHI evolution. We note that there is a difference between the MHD and PIC simulations in definition of the mixing layer. In the MHD simulation, we regarded the density transition (turbulent) layer as a mixing layer, while the mixing in the PIC simulation is defined from the particle populations in each cell. Thus a convective transport may have some contributions to the broadening of the transition layer in the MHD simulation, resulting in the broader mixing layer than that in the PIC simulation. 14 of 15

15 [42] We also note a difference in dissipation scale between the MHD and PIC simulations. In the PIC simulation, the dissipation scale is of the order of the ion scale [Matsumoto and Hoshino, 2006], while in the MHD simulation, the dissipation is due to the numerical one, which is most effective in scales comparable to the mesh size. In other words, the present PIC simulation is more dissipative than the MHD simulation. How the dissipation scale is determined in the kinetic treatment is not fully understood, because the present PIC simulation was examined with a low b (= 0.2) plasma and a reduced mass ratio of M/m = 16 and frequency ratio of w pe /W ge = 4.0. Although the weak dependence of the mixing efficiency on the mass ratio has been reported by Matsumoto and Hoshino [2006], the dependencies on these kinetic parameters are remaining issues. [43] Since the present mechanism is most effective under a purely northward IMF condition which is required to last only for 10 minutes, it may be somewhat idealized as compared to the realistic situation. In the actual magnetospheric boundary layer, the effect of the in plane component of the magnetic field in the magnetosheath and the magnetosphere, which results in excitation of magnetic reconnection inside the vortices [e.g., Nykyri and Otto, 2001], and 3 D effects such as the differential rotation [Brackbill and Knoll, 2001; Knoll and Brackbill, 2002], the tail like geometry [Takagi et al., 2006], the line tying effect of the ionosphere [Miura, 1996], and the growth of the secondary magnetorotational instability [Matsumoto and Seki, 2007] are to be considered. All these issues have stabilizing effects against the present mechanism. A highresolution global MHD simulation, which is able to capture the growth of the KHI at the magnetopause, is the way to incorporate all of these effects and to advance further understanding of roles of the KHI at the magnetopause. [44] Acknowledgments. This work was supported by grant in aid for JSPS Fellows 08J The computational resources for this study were supported by NEC SX 9 at the Institute of Space and Astronautical Science, JAXA, Japan, and Fujitsu FX1 and HX600 at Nagoya University, Japan. This work is supported by the Grant in Aid for Scientific Research (Category B., No ) from the Japan Society for the Promotion of Science. [45] Wolfgang Baumjohann thanks the reviewers for their assistance in evaluating this paper. References Birdsall, C. K., and A. B. Langdon (1991), Plasma Physics Via Computer Simulation, Adam Hilger. Borovsky, J. E., M. F. Thomnsen, and R. C. Elphic (1998), The driving of the plasma sheet by the solar wind, J. Geophys. Res., 103(A8), 17,617 17,639, doi: /97ja Brackbill, J. U., and D. A. Knoll (2001), Transient magnetic reconnection and unstable shear layers, Phys. Rev. Lett., 86(11), , doi: /physrevlett Cai, D., L. R. O. Storey, and T. Neubert (1990), Kinetic equilibria of plasma shear layers, Phys. Fluids, 2(1), Faganello, M., F. Califano, and F. Pegoraro (2008), Competing mechanisms of plasma transport in inhomogeneous configurations with velocity shear: The solar wind interaction with Earth s magnetosphere, Phys. Rev. Lett., 100, doi: /physrevlett Fairfield, D. H., A. Otto, T. Mukai, S. Kokubun, R. P. Lepping, J. T. Steinberg, A. J. Lazarus, and T. Yamamoto (2000), Geotail observations of the Kelvin Helmholtz instability at the equatorial magnetotail boundary for parallel northward fields, J. Geophys. Res., 105(A9), 21,159 21,173, doi: /1999ja Fujimoto, M., T. Terasawa, T. Mukai, Y. Saito, T. Yamamoto, and S. Kokubun (1998), Plasma entry from the flanks of the near Earth magnetotail: Geotail observations, J. Geophys. Res., 103(A3), , doi: /97ja Hasegawa, H., M. Fujimoto, K. Takagi, Y. Saito, T. Mukai, and H. Réme (2006), Single spacecraft detection of rolled up Kelvin Helmholtz vortices at the flank magnetopause, J. Geophys. Res., 111, A09203, doi: /2006ja Hoshino, M. (1986), Theoretical and computational studies of plasma kinetic phenomena: Tearing mode instability and foreshock cyclotron interaction, Ph.D. dissertation, Univ. of Tokyo, Tokyo, Japan. Knoll, D. A., and J. U. Brackbill (2002), The Kelvin Helmholtz instability, differential rotation, and three dimensional, localized, magnetic reconnection, Phys. Plasmas, 9(9), doi: / Lavraud, B., et al. (2006), Evidence for newly closed magnetosheath field lines at the dayside magnetopause under northward IMF, J. Geophys. Res., 111, A05211, doi: /2005ja Li, W., J. Raeder, M. F. Thomsen, and B. Lavraud (2008), Solar wind plasma entry into the magnetosphere under northward IMF conditions, J. Geophys. Res., 113, A04204, doi: /2007ja Matsumoto, Y., and M. Hoshino (2004), Onset of turbulence induced by a Kelvin Helmholtz vortex, Geophys. Res. Lett., 31, L02807, doi: / 2003GL Matsumoto, Y., and M. Hoshino (2006), Turbulent mixing and transport of collisionless plasmas across a stratified velocity shear layer, J. Geophys. Res., 111, A05213, doi: /2004ja Matsumoto, Y., and K. Seki (2007), The secondary instability initiated by the three dimensional nonlinear evolution of the Kelvin Helmholtz instability, J. Geophys. Res., 112, A06223, doi: /2006ja Matsumoto, Y., and K. Seki (2008), Implementation of the CIP algorithm to magnetohydrodynamic simulations, Comput. Phys. Commun., 179, , doi: /j.cpc Miura, A. (1996), Stabilization of the Kelvin Helmholtz instability by the transverse magnetic field in the magnetosphere ionosphere coupling system, Geophys. Res. Lett., 23(7), , doi: /96gl Miura, A. (1997), Compressible magnetohydrodynamic Kelvin Helmholtz instability with vortex pairing in the two dimensional transverse configuration, Phys. Plasmas, 4(8), Miura, A. (1999), Self organization in the two dimensional magnetohydrodynamic transverse Kelvin Helmholtz instability, J. Geophys. Res., 104(A1), , doi: /98ja Miura, A., and T. Sato (1978), Theory of vortex nutation and amplitude oscillation in an inviscid shear instability, J. Fluid Mech., 86, Nakamura,T.K.M.,M.Fujimoto,andA.Otto(2008),Structureofan MHD scale Kelvin Helmholtz vortex: Two dimensional two fluid simulations including finite electron inertial effects, J. Geophys. Res., 113, A09204, doi: /2007ja Nykyri, K., and A. Otto (2001), Plasma transport at the magnetospheric boundary due to reconnection in Kelvin Helmholtz vortices, Geophys. Res. Lett., 28(18), , doi: /2001gl Nykyri, K., and A. Otto (2004), Influence of the hall term on KH instability and reconnection inside KH vortices, Ann. Geophys., 22(3), Øieroset, M., J. Raeder, T. D. Phan, S. Wing, J. P. McFadden, W. Li, M. Fujimoto, H. Rème, and A. Balogh (2005), Global cooling and densification of the plasma sheet during an extended period of purely northward IMF on October 22 24, 2003, Geophys. Res. Lett., 32, L12S07, doi: /2004gl Pritchett, P. L., and F. V. Coroniti (1984), The collisionless macroscopic Kelvin Helmholtz instability 1. Transverse electrostatic mode, J. Geophys. Res., 89(A1), , doi: /ja089ia01p Song, P., and C. T. Russell (1992), Model of the formation of the lowlatitude boundary layer for strongly northward interplanetary magnetic field, J. Geophys. Res., 97(A2), , doi: /91ja Takagi, K., C. Hashimoto, H. Hasegawa, M. Fujimoto, and R. TanDokoro (2006), Kelvin Helmholtz instability in a magnetotail flank like geometry: Three dimensional MHD simulations, J. Geophys. Res., 111, A08202, doi: /2006ja Terasawa, T., et al. (1997), Solar wind control of density and temperature in the near Earth plasma sheet: WIND/GEOTAIL collaboration, Geophys. Res. Lett., 24(8), , doi: /96gl Umeda, T., Y. Omura, T. Tominaga, and H. Matsumoto (2003), A new charge conservation method in electromagnetic particle in cell simulations, Comput. Phys. Commun., 156, 73 85, doi: /s (03) Wing, S., and P. T. Newell (2002), 2D plasma sheet ion density and temperature profiles for northward and southward IMF, Geophys. Res. Lett., 29(9), 1307, doi: /2001gl Yabe, T., F. Xiao, and T. Utsumi (2001), The constrained interpolation profile method for multiphase analysis, J. Comput. Phys, 169, Y. Matsumoto and K. Seki, Solar Terrestrial Environment Laboratory, Nagoya University, Furo cho, Chikusa ku, Nagoya, Aichi , Japan. (ymatumot@stelab.nagoya u.ac.jp) 15 of 15