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Keywords:

  • MHD;
  • dipolarization;
  • fronts

Abstract

  1. Top of page
  2. Abstract
  3. Acknowledgments
  4. References

[1] A simplified MHD model is proposed that explains characteristic features of dipolarization fronts observed by the five-probe THEMIS mission, and in particular the recurrent or multiple fronts, as structures arising from the nonlinear evolution of the interchange instability of the initial reconnection ejecta in the terrestrial magnetotail. Modeling the effects of the magnetic field curvature and plasma braking by an effective gravity and imposing an initial seed perturbation consistent with the observed dawn-dusk scale of fronts is shown to reproduce the observed variations of the north-south magnetic field, bulk flow plasma velocity, number density and pressure.

[2] Dipolarization fronts are sharp jumps in the Bz component of the magnetic field in the terrestrial magnetotail observed in earthward flowing plasmas, following presumably the onset of magnetic reconnection. Their observations were reported earlier [Slavin et al., 1997; Fairfield et al., 1998; Tu et al., 2000; Ohtani et al., 2004] but have gained more attention due to recent observations on multiple satellites of the CLUSTER [R. Nakamura et al., 2002, 2005, 2009; Eastwood et al., 2005; Forsyth et al., 2008; Walsh et al., 2009] and THEMIS mission [Angelopoulos et al., 2008; Runov et al., 2009; Zhou et al., 2009]. The latter observations have demonstrated unambiguously that such fronts can propagate earthwards over ten earth radii with both a finite extent in the plane of the current sheet as well as transverse to the sheet. Furthermore recent observations show that multiple fronts can be recurrent [Nakamura et al., 2009; Zhou et al., 2009].

[3] Recent full particle simulations [Pritchett, 2006; Sitnov et al., 2009] have clearly shown the formation of dipolarization fronts following the Near-Earth-Neutral Line-reconnection. They confirmed and improved on earlier MHD [Ohtani et al., 2004] and hybrid simulations [Hesse et al., 1998; M. S. Nakamura et al., 2002; Krauss-Varban and Karimabadi, 2003] by reproducing the observed scales of these fronts along the Sun-Earth direction, their north-south magnetic field asymmetry and other features. Moreover, structuring of fronts along the dawn-dusk direction was reproduced in some 3D hybrid [M. S. Nakamura et al., 2002] and full-particle [Pritchett, 2006] simulations, consistent with the interchange instability of the magnetotail equilibria with the tailward gradient of the equatorial magnetic field [Pritchett and Coroniti, 2010]. Similar structuring was observed in MHD simulation of plasma bubbles [Birn et al., 2004].

[4] However, some important issues regarding the front structure remain poorly understood. Fronts are often recurrent and it is unclear if this is caused by the recurrence of the reconnection source (for example, multiple ejected plasmoids) or by structuring fronts in the (X,Y) GSM plane because of the interchange instability. The latter, when simulated by hybrid or full-particle codes [e.g., Pritchett and Coroniti, 2010] often results in too narrow fingers-like structures (500–700 km) inconsistent with the observed scale of fronts in the dawn-dusk direction (1–3 Re) as well as their recurrent feature.

[5] In this paper we propose a simple model of the front structure with the emphasis on their interchange evolution, which appear to resolve the above mentioned issues. To simplify the consideration we model the effects of the magnetic field line curvature and plasma braking by an effective gravity. The latter simplification allows one to limit the MHD simulation region to a relatively small box capable of resolving important small-scale structures developing in the nonlinear phase of the instability. On the other hand, in contrast to earlier hybrid and particle simulations, which considered the instability of the front which was homogeneous in the dawn-dusk direction, and similar to the MHD simulations of plasma bubbles [Birn et al., 2004] we start with a seed perturbation of the observed dawn-dusk scale, assuming that this scale is determined by the global geometry of the magnetotail current sheet and the preceding reconnection process which triggers the dipolarization.

[6] The basic geometry for the model is shown in the schematic diagram in Figure 1. Here the x coordinate points away from the earth and the y direction is in the equatorial plane orthogonal to x. Its direction is chosen such that a right handed coordinate system gives a vertically upwards z direction. The post near-earth-neutral-line (NENL) reconnection dynamics of the field lines earthward of the reconnection point, typically at about 20 RE, undergo rapid dipolarization. The braking of the earthward flow together with the curvature of the vertical field leads to an effective gravity g away from the earth. Thus a new quasi-stationary equilibrium is established in which the plasma pressure balances the magnetic field and the effective gravity.

image

Figure 1. Schematic of the model.

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[7] The basic 2D ideal MHD equations used in this investigation are the continuity, momentum and induction equation given below.

  • equation image

The normalizations used are

  • equation image

together with an isothermal equation of state equation image.

[8] The quasi-equilibrium state is determined by

  • equation image

For this study, a model equilibrium for the density is used. By specifying the density profile in x, equation (2) can be solved for the magnetic field profile for a given value of the gravitational acceleration.

  • equation image

Here ρL is the density closer to the earth (to the left) and ρR, the density to the right away from the earth. Shown in Figures 2 (top) and 2 (bottom) are the magnetic field and density equilibrium profiles respectively. The characteristic scale l = 0.2 RE is about 1000 Km and length of the simulation domain in x and y is 3 RE. The equilibrium is qualitatively similar to the one used in the work of Pritchett and Coroniti [2010]. Although our density profile is steeper the basic instability is the same. The interchange instability in their study develops from initial noise, while in our study we seed the simulation with an initial perturbation, which then grows and develops the subsequent nonlinear structure. It is the combination of this profile and the initial seed perturbation which determines the time scale for the preliminary evolution of the instability. However in the late phase the time scale is determined by the nonlinearly developing structure. The parameters for the equilibrium are ρL = 1.25, ρR = 1.0, β = 2 and ĝ = 0.13. The choice of the gravity is justified as follows. The effective gravity associated with a curved field line is cs2/R, where cs is the sound speed and R is the radius of curvature of the field line. By using the normalizations of space (L) and time(L/vA) this makes the effective gravity ∼βL/2R. equation image = L/R∼0.1 is a reasonable approximation, since in our simulations we have assumed that L1 RE and the radius of curvature of the field line can be about 10 RE. The use of 0.13 is a slight enhancement which could be attributed to the braking process which contributes an effective gravity in the same direction as the curvature in this case. For a magnetic field of 20 nT and a density of 0.65/cm3, the Alfven velocity vA∼1000 km/s. Thus time is normalized to approximately 6 s.

image

Figure 2. (top) Bz(x) vs x. (bottom) ρ(x) vs. x, ρL = 1.25, ρR = 1.00 L = 0.2, ĝ = 0.13, β = 2.

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[9] The equilibrium density profile is unstable to the interchange mode. Thus to investigate the interchange instability and its nonlinear evolution, the system of MHD equations (equation (1)) with gravity are solved. As mentioned earlier, the simulation is carried out on a 2D grid in x and y of size 3 RE. The choice of this scale length is quite arbitrary in the present study, since MHD equations do not display any intrinsic scale. The number of grid point in both these directions are nx = ny = 101 In the x direction all perturbed quantities are assumed to be zero on the boundary, while on the boundaries in the y direction, they are taken to be periodic.

[10] Shown in Figure 3 are two panels for the density (Figures 3a3d) and the Bz component of the magnetic field (Figures 3e3h) is the nonlinear evolution of the interchange instability at four different instants of time. In a time scale of about two minutes, a twenty percent amplitude perturbation in the density with two wavelengths in the y direction evolves into a full blown “mushroom” structure characteristic of the interchange instability and the subsequent Kelvin Helmholtz instability which creates the “rolls” [Cattaneo and Hughes, 1988]. Of course this does depend on the choice of the parameters especially the value of the gravity. What is clearly seen is that two “bubble” structures propagate earthward (to the left) while the high density plumes propagate away from the earth. The magnetic field (Figures 3e3h) on the other hand shows that the stronger field is concentrated in the density bubble region while the weaker fields are localized in the region of high density. Since the flows are subsonic, the total pressure balance is approximately maintained across the front. The change in the density across the front is about 50% while the magnetic field changes by a factor of three as seen in the color scales.

image

Figure 3. Contours of (a–d) ρ(x,y) and (e–h) Bz(x,y) at tVA/L = 4,12,16,20, respectively. The parameters used are the same as in Figure 2.

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[11] Recently there have been very clear observation of “multiple” or recurrent fronts. If each front is associated with a NENL reconnection one possible explanation of multiple fronts can be bursty period reconnection events. Here we offer an alternate explanation based on the one reconnection event and the subsequent multiple fronts being caused by the interchange instability.

[12] Shown in Figure 4 (left) is a multiple front event recorded on satellite P4 of the THEMIS cluster on Feb 15 2008. Figure 4 (left) displays the three different components of the magnetic field (Bx(blue), By(green) and Bz(red)), the three components of the ion velocity (vx(blue), vy(green) and vz(red)), the ion density Ni(black) and the plasma pressure(red), the magnetic pressure (blue) and total pressure (black) respectively in the four plots.

image

Figure 4. (left) (1) (Bx(blue), By(green) and Bz(red)), (2) (vx(blue), vy(green) and vz(red)), (3) Ni(black) and (4) the plasma pressure(red), the magnetic pressure (blue) and total pressure (black) respectively as a function of x observed on P4. (right) Similar quantities (with same color) obtained from the simulation with parameters same as in Figure 2.

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[13] Displayed in Figure 4 (right) are similar quantities (matching the colors in each panel on the left) obtained from the simulation by taking a cut through the mushroom structure at t = 20 (Figure 3h). In Figure 3h it is assumed that the satellite intersects the mushroom structure at y = 0.6. The multiple jumps are evident in all the quantities except in the velocities. The Vx velocity displays a mono-polar structure, while the Vy velocity displays a dipolar structure both in the observations and the simulations. Recalling that vA = 1000 Km/s, the maximum flow velocities from the simulation would be about 150 Km/s for the present choice of parameters. This is indeed subsonic. The interesting feature in the Vx velocity is that its maximum value is in the region between the two jumps as expected from the interchange structure. The magnitude of the jumps in the density and Bz component of the magnetic field obtained from the simulations are also consistent with the observations. Hence many gross features of the MHD simulation are in structural and quantitative agreement with the observed characteristics in density, magnetic field and velocity.

[14] In this paper, a mechanism for the formation of multiple dipolarization fronts is advanced within the framework of ideal MHD interchange instability. It complements the earlier reconnection based models by considering the front evolution in the (X,Y) plane and it identifies the fronts to those of plasma bubbles [Chen and Wolf, 1993]. It has often been conjectured that plasma bubbles propagate earthwards following a NENL reconnection. We build a model to mimic this process. Post NENL reconnection dynamics can trigger the nonlinear development of an interchange instability which can unify the occurrence of plasma bubbles, bursty bulk flows and dipolarization fronts with multiple structures. The key features predicted by this model can provide a qualitative and quantitative explanation of many features observed in satellite data of dipolarization front events.

[15] However there are many interesting questions related to the variety of different hybrid and particle-in-cell simulations [M. S. Nakamura et al., 2002; Pritchett, 2006; Pritchett and Coroniti, 1999, 2010] which have investigated the interchange mode with non-MHD effects (like diamagnetic drift and finite Larmor radius (FLR)) which do observe development of finger-like structures on much smaller scales, that need to be addressed in the context of the present work. Typically within the MHD framework, interchange modes have the growth rates which are independent of the wave-number (in the dawn-dusk direction). Thus the wavelength at which the initial perturbation is seeded is the dominant wavelength in the nonlinear evolution. However in the hybrid and particle-in-cell simulations there is indeed a maximally growing mode determined by FLR or diamagnetic effects because of the stabilizing effects at large wave-numbers. Thus with these additional non-MHD physics effects, starting with a noise source which allows for a very broad range of scale-lengths the maximally growing short wavelength modes (compared to the box size) grow and dominate the nonlinear state. Furthermore since these modes are already at short scales the secondary Kelvin Helmholtz instability which leads to the “mushroom” structure can be stabilized by the non-ideal effects (diamagnetic and FLR). This may explain why the structures in these simulations retain there finger-like structures even in the late nonlinear phase. However if an initial perturbation is externally imposed on the system on large MHD scales, the nonlinear evolution will be dictated by this scale. Before the shorter modes with comparable growth rates can amplify to any appreciable level, the background plasma has changed so significantly by the growth of the seeded mode that the stability in this new evolving “equilibrium” can weaken their growth. Thus our MHD simulations seem to suggest that an externally imposed structure (on the order of a few RE) seeds the interchange unstable plasma and this evolves into the observed mushroom-like bubble which agrees with the observed feature. The finite size of the plasma in the dawn dusk direction in the closed field line region after reconnection is in the range of tens of RE. Also the effective gravity has a spatial variation in this direction since the curvature of the field line (which determines the gravity) varies along this direction. Thus the combined effect of the finite size and the variation in the gravity can be responsible for seeding the interchange instability on global scales. This is indeed a conjecture yet a plausible explanation for the observations which indicate that the dipolarization fronts in the dawn-dusk direction are of the order of a couple of earth radii.

[16] There are obviously other limitations to our present model. The model assumes the presence of an effective gravity. A high resolution fully three-dimensional MHD simulation of the post reconnection dynamics of the closed field line region should be able to address the problem self-consistently. Also within the framework of the MHD model issues related to particle energization cannot be addressed. Nevertheless, the current MHD model captures the spatial features and quantitative variations of the observed macroscopic plasma parameters quite well.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Acknowledgments
  4. References

[17] Work supported by grants from NASA and NSF.

References

  1. Top of page
  2. Abstract
  3. Acknowledgments
  4. References