Model of bifurcated current sheets in the Earth's magnetotail: Equilibrium and stability

Authors


Abstract

[1] Recent observations of CLUSTER have reconfirmed the recurring presence of bifurcated current sheets. The present paper revisits the model of Schindler and Birn (2002) to describe analytically bifurcated current sheets. Our contributions are in order. First, we describe a number of analytical velocity distribution functions that lead to bifurcated current sheets. Second, we derive necessary and sufficient conditions that determine when current sheets can be produced within the mathematical model of Schindler and Birn (2002). Third, we present a class of bifurcated current sheets, and we describe their properties. Fourth, we study the stability of bifurcated current sheets to the tearing instability finding that bifurcated current sheets tend to be more stable. Finally, we investigate the stability of bifurcated current sheets to the lower hybrid drift instability (LHDI) and kinking instability proving their presence. The work reported here is intended to extend and investigate the properties of instabilities from the typical but academic case of the Harris current sheet to current sheet equilibria that are more realisticly representative of the real magnetotail.

1. Introduction

[2] Recent observations by CLUSTER showed that bifurcated current sheet can sometimes be observed in the magnetotail [Nakamura et al., 2002; Runov et al., 2003; Sergeev et al., 2003; Runov et al., 2004]. In a bifurcated current, the maximum of the magnetic field variation (corresponding to the maximum current, since Jy ∝ ∂Bx/∂z, in the standard geomagnetic coordinate system) does not coincide with the region of lowest magnetic field. In a Harris current sheet, the maximum current occurs at the center of the sheet where the magnetic field vanishes. In a number of observations, [Nakamura et al., 2002; Runov et al., 2003], instead, the maximum current occurs off center, in two symmetrically located regions respectively north and south of the neutral line.

[3] The presence of current sheet bifurcation has sometimes been observed to be linked with periods of activity of the magnetotail, corresponding to the onset of reconnection. For example, Runov et al. [2004] observed that the current sheets observed on August 29, 2001 by the CLUSTER mission, correspond to periods of intense flapping of the magnetotail and correspond to a substorm event measured by magnetometers on Earth.

[4] While CLUSTER observations have brought the topic to the immediate attention of the community, previous observations had already presented evidence for the existence of bifurcated current sheets. [Sergeev et al., 1993] using data from ISEE-1/2 observed evidence for current sheet bifurcation. Statistical evidence for current sheet bifurcation was given by [Hoshino et al. 1996], Asano [2001], and Asano et al. [2003, 2004a, 2004b].

[5] A number of theories have been suggested to explain the origin of bifurcated current sheets.

[6] The early theory by Cowley [1978, 1979] for magnetotail equilibria included bifurcated solutions. More recently, generalizations of the standard Harris current sheet equilibrium have been proposed to reproduce the bifurcation observed by satellites [Birn et al., 2004; Sitnov et al., 2003].

[7] Several mechanisms have been proposed to explain the origin of the current sheet bifurcation [Hoshino et al. 1996].

[8] Holland and Chen [1993] linked the presence of bifurcated equilibria with the existence of non-diagonal terms of the pressure tensor and with temperature anisotropy. Similarly, Daughton et al. [2005] observed that the non-linear evolution of the lower hybrid drift instability (LHDI) leads to non-isotropic temperature and to a bifurcated current sheet. Simulations confirm this suggestion as a possible mechanism [Karimabadi et al., 2003a, 2003b; Ricci et al., 2004c].

[9] Bifurcation of the current sheet in the plasma outflow region is a well-known signature of magnetic reconnection [Hesse et al., 1998; Lottermoser et al., 1998; Shay et al., 1998].

[10] Finally, turbulence concentrated in a quasineutral sheet can lead to bifurcation [Greco et al., 2002].

[11] While the study of the origins of the current sheet bifurcation remains a topic of great interest, in the present paper we assume a bifurcated equilibrium is already present and we ask the question of what differences exist in the evolution of a bifurcated current sheet compared with a classic Harris equilibrium.

[12] The spirit of the paper is not to obtain a faithful representation of actual magnetotail equilibria or to interpret the satellite observation. Instead, the goal is to observe, all else being equal, what is the effect of bifurcation alone. Our approach is to consider the classic 1D Harris equilibrium and modify it to become bifurcated. Our goal is to distill the effect of bifurcation from all other effects present in the real system, and most notably we want to eliminate the stabilizing effect of vertical magnetic fields present in 2D magnetotail equilibria.

[13] We approach the problem by starting from an 1D initial equilibrium that is already bifurcated and we observe what consequence the bifurcation has on the known dynamics of a current sheet.

[14] Recently, Schindler and Birn [2002]; Birn et al. [2004] have developed an extension of the theory of magnetotail equilibria, that includes bifurcated current sheet. We explore under what conditions the theory can yield bifurcated current sheets. We derive new conditions, necessary and sufficient, for the presence of bifurcated current sheets. We then settle on a number (6 in all) of different classes of bifurcated current sheets and we describe their properties and we characterize their configuration.

[15] Next, using our workhorse PIC code, CELESTE3D, proven in many previous studies and extensively benchmarked [Ricci et al., 2002a, 2004a; Lapenta et al., 2003; Ricci et al., 2002b] we have studied the dynamical evolution of bifurcated current sheets.

[16] We focus the analysis on two classes of dynamical modes.

[17] First, in the reconnection plane (x, z using the standard geomagnetic coordinate system), the presence of an initially bifurcated current sheet alters profoundly the evolution of the tearing instability. A bifurcated current tends to be more stable towards tearing. An initially unperturbed system, subject just to the random noise of the PIC simulations saturates at low amplitude. But the same happens for a Harris equilirium [Ricci et al., 2004a] as is predicted by a theory [Biskamp et al., 1970]. More remarkably, however, the bifurcated initial state does not progress towards the creation of large plasmoids, even when subjected to a large initial perturbation. When subjected to the same perturbation used in the now classic GEM challenge [Birn et al., 2001], or even to twice that much, the system does not respond starting a fast reconnection process. In lack of a complete theory of tearing in bifurcated current sheets, we present a heuristic mechanism, based on the linear theory of the tearing instability, that indeed explain such enhanced stability of bifurcated current sheets.

[18] Second, in the current aligned plane (y, z), the presence of a bifurcated current sheet alters the timing and the details of the sequence of mechanisms already present in a Harris sheet but does not modify qualitatively the known dynamics observed in previous simulations of Harris equilibria [Lapenta and Brackbill, 2002; Daughton, 2002; Lapenta et al., 2003]. Of particular relevance is the fact that the electrostatic branch of the lower hybrid drift instability (LHDI) is not present even in the reduced current region between the two peaks, and remains limited to the outer flanks. This latter finding is of considerable relevance to studies of anomalous resistivity, as it confirms that the electrostatic branch of the LHDI cannot produce significant resistivity in the center of the sheet.

[19] The paper is organized as follows. Section 2 summarizes the theory of current sheet equilibria due to Schindler and Birn [2002]. Section 3 describes the types of distribution functions considered in the present study. Section 4 describes a number of general properties of the equilibria covered by the theory of Schindler and Birn [2002] and discusses necessary and sufficient conditions for the formation of bifurcated current sheets. Section 5 presents a number of examples of bifurcated current sheets and describes their properties. Section 6 describes the simulation method used to study the stability of current sheets. Section 7 applies the method to the study of a specific class of bifurcated current sheets. Conclusions are drawn in section 8.

2. Analytical Model of Bifurcated Current Sheets

[20] We summarize briefly the results of the analytical theory by Schindler and Birn [2002] for the steady state equilibrium of a collisionless plasma, limiting the attention to a one dimensional model, in which all quantities are functions of only the z space coordinate. The steady state particle distribution function f(r, v) is a solution of the Vlasov equation:

equation image

in which the electric and magnetic fields are calculated self-consistently from the steady-state Maxwell equations in potential form:

equation image
equation image

with

equation image
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The charge density σ and current density j are obtained at each point in space from the integral of the distribution function:

equation image
equation image

In the present 1D approach where z is the only coordinate, we can assume that the only non-zero component of A is along y and will be simply called A = Ay. It follows that

equation image

A proven property of the Vlasov equation is that solutions of equation (1) can be formulated as any nonnegative function of the particle constants of motion [Krall and Trivelpiece, 1986].

[21] In the case of the system described above, two constants of motion can be identified with the canonical momentum with respect to the y coordinate:

equation image

and with the single-particle Hamiltonian

equation image

Thus the distribution function for each species is

equation image

Other invariants could be considered [Sitnov et al., 2000], but at the cost of complicating the approach.

[22] We follow the model suggested by Schindler and Birn [2002] and assume that Fs can be factored as

equation image

where kB is Boltzmann's constant, Ts the temperature of species s and Cs is a normalization constant such that number density is

equation image

While in general any arbitrary non negative function Fs(Hs, Py) could be considered, Schindler and Birn [2002] justify the choice in equation (12) based on an analysis of particle orbits.

[23] Following [Schindler and Birn, 2002], a number of dimensionless parameters are introduced to characterize the equilibria:

equation image

The pressure balance relation n0kBT = B02/2μ0 must be satisfied.

[24] Imposing the quasi-neutrality condition

equation image

and introducing the functionals:

equation image
equation image

a relationship between ϕ and A is found:

equation image

One can now use this relationship to express the current density jys(A, ϕ) and the number density ns as function of only A:

equation image
equation image
equation image

where all quantities are dimensionless and C = image

[25] Inserting now equations (19, 20) into Maxwell equation (3) the Grad-Shafranov equation follows:

equation image

hat provides the explicit dependence of A on z.

3. Choice of g(Py) for a Bifurcated Current Sheet

[26] In the framework outlined above, different choices of gs(Py) lead to different shapes of js in space, each of them exact solution of the Vlasov-Maxwell system.

[27] The choice of the distribution function can be motivated by observational constraints and by theoretical constraints based on the theories on the origin of the bifurcation.

[28] Below we consider a number of options considered in the previous literature and we propose a few new choices that are helpful in extending the ability of the theory to describe observed equilibria.

[29] We consider first some of the classic choices found in the literature. Harris [1962] choose a gauge with null electric field and assumes

equation image

It leads to a non bifurcated current sheet, and the magnetic field found is B ∝ tanh(z). Channell [1976] studies solutions with several g functions but still with no electric field. One of them of interest in present paper is

equation image

He uses a pseudo-particle approach, developing a formalism which allows the generation of a large class of equilibria, including wave-like solutions.

[30] Mottez [2003] shows that a continuous linear combination of equation (23) and (24) leads to an analytical equilibrium solution; in this case the plasma does not have to be isothermal, but, like in Channel, a null electric field is imposed.

[31] Schindler and Birn [2002] choose

equation image

with the c and d parameters chosen freely. They allow for different values of c for electrons and ions.

[32] Other authors take in consideration a third constant of motion. In this case the distribution function will contain a dependence from it.

[33] Roth et al. [1996] considers the canonical momentum with respect to the x coordinate a constant of motion alongside Py. He also factorizes the distribution function, being g a linear combination of complementary error functions of Py and Px:

equation image
equation image

Sitnov et al. [2000, 2003] includes the action variable across the sheet among the constants of motion and uses a non-factorized distribution function.

[34] We focus on studying the distribution function proposed by Schindler and Birn [2002], using the same parameters c and d:

equation image
equation image

[35] Moreover we studied slight deviations from Harris and Channel distributions, selecting the following cases:

equation image
equation image
equation image
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[36] Inserting g(Py) in equations (16, 17) and (19, 20) we can compute the current densities jys and then solve the Grad-Shafranov equation (22) numerically using a finite difference method. Note that while in cases (a), (b) a numerical solution is needed in computing current densities, in all other cases (c)–(f) an analytical expression can be reached for the current js. We point out that the Grad-Shafranov equation is a non-linear second order ordinary differential equation. Therefore, two boundary conditions are needed. We choose to fix A′(0) = 0 requiring a z symmetrical solution. The non linearity of the equation implies that different choices of the value of A(0) lead to different shapes of the current density.

[37] We found that with the appropriate choice of A(0) all of the distribution functions (a)–(e) show a bifurcated structure. The last choice, (f), yields a wave like solution, of the kind studied by Channell [1976].

4. Properties of the Current Sheet

[38] Before analyzing in detail each of the choices (a)–(f), some a priori considerations can be formulated regarding the conditions for the formation of bifurcated current sheets.

[39] The system is symmetrical in the z-coordinate and in order to have a bifurcated current structure, the current density jy must have a critical point in z ≠ 0. This is, of course, a necessary but not sufficient condition. The condition can be formulated mathematically as

equation image

For solutions not too far from Harris-type, A(z) does not present maxima or minima in z ≠ 0. Indeed the vector potential for the Harris equilibrium is

equation image

It follows that to investigate sufficient conditions for bifurcation, extrema in jy(A) are needed. If any such points exist, we indicate it as A0. We can always find a solution of equation (22) such that A0 belongs to A(z) choosing an appropriate boundary condition, e.g.,

equation image

If such condition can be proven the current density jy(z) will present a critical point in z = z0.

[40] To investigate the point further we consider first the case when jy(z0) > 0. If we want z0 to be the point where current density peaks, it follows jy(z) has a maximum in z0, or

equation image

By computing the second derivative of jy(z)

equation image

and observing that the second term in the first parenthesis is zero (since A0 is an extremum), the requested condition for the current to peak is

equation image

that is jy(A) has a maximum in A = A0. Similarly we can consider the case of jy(z0) < 0.

[41] In conclusion a sufficient condition for having a bifurcated current sheet is that a maximum exists in jy(A) if jy(z) > 0, and a minimum if jy(z) < 0.

[42] If we restrict the choice of gs(Py) to

equation image

and assume that g(Py) has a continuous second derivative, a necessary and sufficient criterion involving the choice of g(Py) can be obtained for the current density to be bifurcated.

[43] In fact in a situation of zero electric field, Channel observes that [Channell, 1976]

equation image

Later Mottez extended this relation for a more general situation [Mottez, 2003].

[44] In all cases, there is a proportionality relationship between the current density and the first derivative of number density. This can be easily demonstrated. In fact, taking into account only the ion contribution to number and current density, and computing the derivative of n with respect to A, from equation (21) we have

equation image

where

equation image

Using the last two equations we can recognize the definition (19) for ion current density. The same computation can be conducted for the electron contribution, and equation (35) follows immediately.

[45] As a consequence of equation (35), finding a solution for equation (28) is equivalent to finding a solution A0 for

equation image

Under the hypothesis (34), the following relation is derived straightforwardly from definitions (16, 17)

equation image

Recalling the definition of n(A)

equation image

it follows

equation image

Using the definition for Ii(A) and with straightforward algebraic manipulations it follows that

equation image

Now, since the first factor of the integrand is positive definite, the second term cannot be nonnegative, otherwise it would be identically equal to zero, for the equation to be satisfied. In other words d2g(mvy + A)/dA2 changes its sign somewhere. This is equivalent to affirm that g(Py) has an inflection point, since fixed a vy for which

equation image

being g(A) the same function g(mvy + A) shifted by mvy.

[46] The following criterion follows: A necessary condition for having a bifurcated current sheet is that the function g(Py) has an inflection point.

[47] The presence of an inflection point in g(Py) is, under certain limiting assumptions, not only necessary, but also sufficient for having a bifurcated structure. To prove this statement, let us consider the case where ρs is small compared to 1 for both ions and electrons. Furthermore we assume that g(Py) is such that exp(−mvy2/2ρi2) strongly localizes the integration in equation (42) to the neighborhood of vy = 0. Using the saddle point method [Morse and Feshbach, 1953], in that neighborhood the function g(Py) can be represented by a Taylor expansion in vy.

[48] The second derivative of g(mvy + A) in vy is

equation image

Noting that

equation image

we can insert equation (43) in equation (42) and obtain the following equation for A

equation image

The second term is zero, since the argument of the integral is odd. For the same reason if we had considered all the terms in the Taylor expansion, all terms involving odd derivatives of g would vanish. It follows that, with the previous assumptions

equation image

[49] In conclusion the following criterion can be stated: The presence of an inflection point in g(Py) is a necessary and sufficient condition for having a bifurcated current density, within an error of second order in vy. This approximation becomes exact if all even derivatives of g(Py) vanish.

[50] The necessary and sufficient conditions derived above have an intrinsic theoretical value in determining the fundamental requirements for the existence of bifurcated equilibria within the theory by Schindler and Birn [2002]. But their practical relevance emerges when an attempt is made to model with the theory actual current sheets observed in simulations or in satellite data. In that perspective, different forms of the distribution function g(Py) must be tried. And the conditions derived above can be used to rule out distributions which would not lead to bifurcated equilibria.

5. Choice of Characteristic Bifurcated Equilibria

[51] In this section we study the equilibria already mentioned and labeled as cases (a)–(f) and discuss whether these distribution functions can lead to a bifurcated structure in current density.

[52] A bifurcated structure in magnetotail current density is not only justified on the basis of the satellite data already mentioned, but is suggested by theoretical considerations based on the necessary and sufficient conditions derived above. Based on the criteria above, an extrema in jy(A) is necessary for a bifurcated current structure (equation (28)).

[53] In the case of Harris equilibria the function jy(A) is monotonic function, and it can be computed easily from equations (1619):

equation image

Hence the Harris distribution function cannot in any case lead to a bifurcated current density.

[54] However, we reach a different result if we consider perturbations of the Harris sheet due to the teraing instability. Based on the classical study of linear tearing instability [Nishida, 1982], and on more recent kinetic theories [Lapenta and Brackbill, 1997; Daughton, 1999], the perturbed A(z) in the tearing mode has extrema in z ≠ 0. Inserting such a function in equation (46) will lead to a bifurcated structure.

[55] This suggests to consider bifurcated current sheets as the result of the non linear saturation of a perturbation. We note that this interpretation of a bifurcated current sheet as the natural state following the saturation of the tearing instability suggests that a bifurcated equilibrium should be more stable to the tearing mode, as indeed is observed in section 6 below.

[56] For solving equation (1621) we need to set up the dimensionless parameters. We follow Schindler and Birn [2002] choosing:

equation image

We want to emphasize the fact that the choice of A(0), needed to solve equation (22), has no particular physical meaning, since in B = ∇ × A, A(0) can be always chosen changing the gauge. However, due to the non-linearity of the Grad-Shafranov equation, changing this value doesn't mean just a shift in the solution for A(z), but a change of the shape of the functions g(Py) and jy(A) and so it leads to a brand new solution for A(z) and consequently for B(z).

[57] Although Schindler and Birn [2002] developed a simplified approximated analytical model for computing jys(A) once given g(Py) in the assumption that ρs is small compared to 1 for both electrons and ions, this is not the case here, being this condition not satisfied for our choice of ρi. Then, as already outlined above, except for cases (a), (b), the computation of both Ii, Ie (16, 17) and jyi,jye (19, 20) has been completed exaclty without introducing any approximation.

[58] This first two cases are shown in Figures 1 and 2. The two cases differ only in the ion distribution function, taken as an Harris-distribution in (b). This difference leads to a small variation in the electron current density, and a more consistent in the ion current density. It is however sufficient to change the total current and bifurcating the current. One can also see how the choice of A(0) is determining for the shape of current density.

Figure 1.

Current densities for case (a): (top) A(0) = 0 (bottom) A(0) = 0.4.

Figure 2.

Current densities for case (b): (top) A(0) = 0 (bottom) A(0) = 0.4.

[59] Distribution functions both in the center of the current and far from it, are plotted for electrons as combined gray-scale and contour-plot in Figures 3 and 4 in the (vy, vz) plane. The fundamental feature of this kind of distribution is the clear presence of two parts to the distribution function, separated by a sharp cutoff, the left side having a much smaller concentration.

Figure 3.

Cases (a) and (b). Distribution functions for electrons in the center of the sheet (top) and far from the center (bottom).

Figure 4.

Distribution functions for electrons (top) and ions (bottom) in the peak of the sheet in case (b). Note from equation (28) that the shape of distribution function in the peak does not depend on z0, coordinate of the peak.

[60] In case (a) the ion and electron distribution functions are symmetric, under the transformation vy = −vy, while in case (b) ion distribution has always the same shape for all values of A(0) and z. It is in fact a Maxwellian distribution with a shift in vy and it is centered in vy = −1. This can be better understood looking at the expression of Fi(Hi, Py):

equation image
equation image

The factor exp(−vy) causes the shift while the factors containing z do not modify the shape in the (vy, vz) plane.

[61] Case (c) is a modification of the solution discussed in Channell [1976]. This distribution function leads to a clearly marked bifurcation in the current density (Figure 5), that decreases as A(0) increases, however the factor exp(−Py2) leads to an anisotropy in the plane (vx, vy). The anisotropy is not limited to the center of the current sheet and extends in the outer regions.

Figure 5.

Current densities for case (c): (top) A(0) = 0.5 (bottom) A(0) = 1.

[62] Case (d) is also anisotropic, but shows a double-bifurcated structure in current density (Figure 6). Distribution functions of cases (c) and (d) are compared in Figure 7.

Figure 6.

Current densities for case (d): (top) A(0) = 0.2 (bottom) A(0) = 1.5.

Figure 7.

Distribution functions for electrons in the center of the sheet for A(0) = 1. Case (c) (top) and case (d) (bottom).

[63] Case (e) leads to a bifurcated current sheet approximately for 1 < A(0) < 2, while a central dominating current density is present when A(0) < 1 or A(0) > 2. Far from the center, both the factor exp(−Py) and (1 + Py2) are weaker then the factor involving H. The distribution function presents two peaks in the current sheet but reduces to a shifted Maxwellian in the flanks.

[64] We show the two limits of A(0) and an intermediate value in Figures 8 and 11. Furthermore we show the ion and electron velocity in Figure 9, the electric potential in Figure 10 and distribution functions in Figure 11.

Figure 8.

Current densities for case (e): (top) A(0) = 1 (bottom) A(0) = 2.

Figure 9.

Ion and electron velocity for case (e): (top) A(0) = 1 (middle) A(0) = 1.5 (bottom) A(0) = 2.

Figure 10.

Electric potential ϕ and number density for case (e): (top) A(0) = 0.5 (middle) A(0) = 1.5 (bottom) A(0) = 2.

Figure 11.

Case (e).Distribution functions for electrons in the center of the sheet for A(0) = 1(top) and for A(0) = 2(bottom).

[65] Case (f) is not of interest for the study of Earth's magnetotail. We report it as an example of wave-like solutions, predicted by [Channell, 1976]. In fact for A(0) ≳ −0.2 the solution of the Grad-Shafranov equation is an electromagnetic wave. We show the relative current density in Figure 12 and the distribution function in Figure 13.

Figure 12.

Current densities for case (f): A(0) = −0.2.

Figure 13.

Case (f). Distribution functions for electrons in the center of the sheet for A(0) = −0.2.

[66] Comparing the distribution functions for all cases (a)–(f), two distinct features emerge. In the cases (a)–(b), the distribution has a distinct cut-off between two regions of distinctly different populations. In the cases (c)–(e), the distribution shows a temperature anisotropy sometimes accompanied by the presence of two distinct peaks, one around zero velocity and one around a drift speed. Both features have been observed in satellite data.

[67] The presence of a sharp transition between two distinct populations in the ion distribution is observed in the recovery phase of substorms [Nishida, 2000; Nagai et al., 2001, 2002], often accompanied by signature of reconnection. Anisotropy have been observed in conjunction with the presence of turbulence due to the lower-hybrid drift instability [Shinohara et al., 1998]. Simulations of the LHDI also show this feature and explain the presence of the peak at zero velocity in terms of scattering of ions between crossing and non-crossing orbits leading to an accumulation of ions at rest [Lapenta et al., 2003].

[68] The two features of the distributions can explain different observational evidence relative to different magnetospheric conditions. The cut-off distributions (cases (a)–(b)) can be more relevant to the case of bifurcated current sheets produced by reconnection, as observed in some simulations of the formation of bifurcated current sheets [Hesse et al., 1998; Lottermoser et al., 1998; Shay et al., 1998]. But non-isotropic distributions (cases (c)–(e)), can be instead more representative of current sheets caused by the non-linear development of the LHDI as observed in other simulations [Karimabadi et al., 2003a, 2003b; Ricci et al., 2004c].

[69] Future work should link the equilibria described above with actual satellite data identifying for each class of magnetotail events the most relevant class of equilibria.

[70] In the present work, however, we focus primarily on the cases where the bifurcation is caused by the non-linear evolution of the LHDI. In that class, case (e) is preferred over cases (c)–(d) because in case (e) the distribution function is anisotropic in the current sheet but becomes a drifting Maxweillian away from it.

6. Simulation Method

[71] To study the evolution of a perturbation to the equilibria described above, we use the implicit PIC code CELESTE3D based on the Vlasov-Maxwell model. A detailed description of the implicit moment method used in CELESTE 3D can be found in the review by Brackbill and Cohen [1985] and the details of the implementation can be found in Ricci et al. [2002b].

[72] The implicit moment method removes the stability constraint on the time step due to the Courant condition on the speed of light and the plasma frequency of Langmuir waves. The typical time step used in the implicit simulations is ωpiΔt = 0.1 which must be compared with the typical time step used in explicit PIC codes, ωpeΔt = 0.1, where ωpi and ωpe are the ion and electron plasma frequencies. The use of the implicit PIC method allows us to follow the ion time scale while retaining the relevant aspects of the electron dynamics. The larger time step allowed by the implicit method brings realistic mass ratios mi/me within the reach of modest computing facilities.

[73] Furthermore, the implicit moment method relaxes the finite grid instability constraint ΔxDe < π and allows one to use coarser grids that are chosen to resolve only the length scales of interest (current sheet thickness, skin depth) without having to resolve needlessly the electron Debye length. This feature reduces the cost of the simulations.

[74] The implicit moment method has been validated and verified in a great number of benchmarks and realistic applications. For the types of simulations described below the code has been compared extensively with the massively parallel explicit PIC code NPIC [Daughton, 2002] with state of the art resolution. Of particular relevance, mass ratios up to the physical mass ratios have been successfully tested in previous publications [Ricci et al., 2002a]. The performance in terms of energy conservation and solution fidelity has been remarkable [Ricci et al., 2002b, 2004a, 2004b; Lapenta et al., 2003] even for coarser grid spacings and larger time steps than used below. So the results presented below have a high degree of reliability.

[75] In the simulations below, a grid spacing at least 0.1cpi is used. This results in a great accuracy in resolving the ion skin depth and the current profile. For a mass ratio of mi/me = 25 the electron skin depth is also well resolved. For the mass ratio mi/me = 180, the resolution would be marginal and we have decreased Δz to 0.05 cpi to resolve it better. Nevertheless, previous convergence studies and previous comparisons with massively parallel explicit PIC codes proved that sufficient accuracy is retained even for the latter case [Ricci et al., 2004b]. The time step used is ωpiΔt = 0.03 corresponding to a energy conservation within a tolerance of only 2.2 · 10−4 [Ricci et al., 2004b].

[76] The simulations described below all share periodic boundary conditions in x and y. In the vertical z direction, the particles are reflected specularly and Dirichlet boundary conditions (E = 0, B = 0) are used. Even though these boundary conditions in the vertical direction are often used in simulations by various authors [Birn et al., 2001], they are appropriate only for a conductive wall, and certainly are not realistic for the real magnetotail. However, the important evolution here is limited to a narrow region around the current sheet, the plasma density decays exponentially away from the current sheet, and near vacuum conditions are found at the boundary. The effect of the vertical boundary conditions has been observed to be negligible.

[77] A peculiarity of the simulations described below is due to the complication of the particle distribution function. Typically PIC simulations rely on the very well-know normal distribution random number generator [Birdsall and Langdon, 1985], appropriate for Maxwellian distributions. Unfortunately, in the present case the distribution is very non Maxwellian and its non Maxwellian nature is crucial to the consistency of the initial equilibrium. Furthermore, the initial magnetic field and density profile is not given by the simple hyperbolic functions typical of the Harris equilibrium.

[78] To solve this problem, we have focused on the specific initial profile labelled as case (e) above and we have designed a numerical technique to handle both the fields profile generation and the particle loading. Within the choice of gi(Py) = ge(−Py) = exp(−Py)(1 + Py2), equations (19, 20) have an analytical solution, that inserted in equation (22) leads to a second order ordinary differential equation that is solved numerically using the fourth order Runge-Kutta method. [Press et al., 1986] We solved A(z) for several choices of A(0). The initial number density and electric field is obtained from the exact analytical solution of equations (18, 21).

[79] The generation of the particle velocity has been obtained using the rejection-method [Dupree and Fraley, 2002]. The distribution function at the point zp is

equation image

where A(zp) is calculated by linear interpolation of the solution A(z) from the grid used to solve the Grad-Shafranov equation.

7. Stability of Bifurcated Current Sheets

[80] The aim here is to study the effect of bifurcation on an idealized model of bifurcated current sheets. We are not attempting to model any specific observation. Rather, we consider a general class of idealized bifurcated sheets that share the general structure observed in some magnetotail current sheets [Shinohara et al., 1998] and in some simulations of the magnetotail [Karimabadi et al., 2003a, 2003b; Ricci et al., 2004c; Holland and Chen, 1993]: the particle distribution function has anisotropic temperature in the current sheet but the temperature becomes isotropic away from it. Case (e) satisfies these requirements.

[81] Of course, in other experimental or natural occurrences the other bifurcated equilibria might be better models but in the present work we focus on this class of bifurcated equilibria.

[82] To simplify the comparison with the recent literature we have used the same parameters of the GEM challenge. This choice sacrifices direct comparison with available magnetotail observations where bifurcated current sheets tend to be somewhat thicker [Sergeev et al., 2003] (although this is primarily due to the fact that published observations are based on a period when the four CLUSTER satellites where relatively far apart, more recent data still under investigation and relative to a subsequent period when the satellites were closer might be more relevant to the results shown below), but allows a direct comparison of the evolution and stability of the GEM equilibrium [Birn et al., 2001] with the results of the present work.

[83] Below, we report the results obtained comparing the stability of this class of equilibria with the classic Harris current sheet.

7.1. Simulations in 1D

[84] The simulations in 1D are run on a grid 1 × 1 × 128, with a Δz = 0.1 cpi and a time step ωpiΔt = 0.05. The code used is still fully 3D but only one cell us used in x and y and periodic boundary conditions are applied to effectively reduce the simulation to 1D. The total number of cycles is 9000, the mass ratio is 25. We show two simulations, one for A(0) = 1, the other for A(0) = 1.5 (Figure 14). The choice of mass ratio differs from the correct physical mass ratio for numerical convenience. The results show the time evolution of a bifurcated current sheet. In 1D simulations, none of the instabilities present in the field plane (x, z) or on the current plane (y, z) can be present and in absence of other effects the current should remain unaltered. This conclusion confirm numerically the soundness of the theory developed above. The theory developed above assumed 1D equilibria and must be tested appropriately in 1D where other instabilities are absent. The simulations shown in Figure 14 confirm at the same time the theory and its implementation in the simulation code CELESTE3D. Were any inconsistency present, a pressure imbalance would have been present in the simulations leading to a modification of the initial state. Instead, no macroscopic change develops confirming that the initial state is indeed an equilibrium.

Figure 14.

Simulation 1D: jy vs time. The time is normalized with cyclotron frequency ωc. A(0) = 1 (left), A(0) = 1.5 (right).

7.2. Simulations in 2D: (x, z) Plane

[85] The tearing instability develops in the (x, z) plane. We have conducted a series of simulations using mass ratios mi/me = 25 (as in the GEM challenge [Birn et al., 2001]) and mi/me = 180. We compare the results relative to the classical Harris sheet and to the bifurcated equilibria of case (e), with the specific choice A(0) = 2, referred to below as modified Harris for convenience. The size of the system is chosen to be Lx/L = 4π in x and Lz/L = 8π in z. The grid is 64 × 1 × 64 in the mi/me = 25 case and 64 × 1 × 128 in the mi/me = 180 (to better resolve the smaller electron skin depth and reconnection layer present in the higher mass ratio case). Except for the mass ratio, system size and the absence of a background density, we use the usual parameters of the GEM challenge: the temperature ratio is Te/Ti = 0.2, the current sheet thickness is L = 0.5cpi.

[86] In some cases, an initial perturbation is applied to B corresponding to the following vector potential (also chosen following the specification of the GEM challenge):

equation image

where Ay1 = pB0cpi is the amplitude of the initial perturbation.

[87] Below we compare two types of simulations: with and without the initial perturbation and we compare for each case the behavior of the bifurcated sheet with the the classic Harris case.

[88] For a classic Harris equilibrium, Figure 15 shows the evolution of the system when no initial perturbation is present. The natural noise of the simulation promotes the linear growth of the tearing mode, as predicted by the linear theory [Coppi et al., 1966; Laval et al., 1965; Galeev and Zelenyi, 1976]. As the island size reaches a relatively small amplitude (with an island width smaller than the current sheet thickness), the mode saturates and no further growth of the island is observed. The cause of the saturation was theoretically predicted to be the creation of a temperature anisotropy [Biskamp et al., 1970]. Recent simulations investigated the mechanism and confirmed the theoretical predictions, finding indeed the creation of temperature anisotropy [Ricci et al., 2004a]. We observe the same mechanism here. We remark that our simulations were continued up to a total time of ωcit = 83 and the conclusions reached above are valid within that limit. It cannot be ruled out that if the simulations could be run for much longer times, further growth of the islands could be observed, as argued in Drake et al. [2003].

Figure 15.

Simulation results for jy and Ay for mi/me = 25 in Harris case at several times: (top) ωcit = 0; (bottom) ωcit = 83.

[89] The saturated level of the tearing instability in a classic Harris equilibrium corresponds itself to a bifurcated equilibrium (see Figure 15) confirming one of the possible origins of bifurcation.

[90] When the initial state is already bifurcated, the tearing instability still grows and saturates at the same level observed in the Harris case. Figure 16 shows an initial bifurcated current sheet and the subsequent growth and saturation of the tearing instability. The initial current sheet is composed of two current sheets of roughly the same width of the initial sheet for the classic Harris case. This circumstance is by design, as we have used distribution (e) but all other parameters as in the Harris case. The linear growth phase for the bifurcated case can only be analyzed qualitatively due to the lack of a complete linear theory for bifurcated current sheets. The final island size reflects this larger total current sheet thickness, but the level of saturation is still consistent with the theoretical predictions by Biskamp et al. [1970] and the island width is smaller than the initial current sheet. Figure 17 shows the reconnected flux as a function of time. A short linear phase is followed by a long period of saturation where the island no longer grows (note that only the first part of the evolution is shown, but the simulations were continued until ωcit = 83, and the same level is kept to the end). In the linear phase, the reconnection rate (time derivative of the reconnected flux) is comparable in the two cases, but significantly faster in the Harris case.

Figure 16.

Simulation results for jy and Ay for mi/me = 25 in Harris modified case without initial perturbation at several times: (top) ωcit = 0; (bottom) ωcit = 83.

Figure 17.

The reconnected flux is plotted for Harris (solid line) and modified Harris (dots) initial equilibria. Case with mi/me = 25 and with no initial perturbation. The reconnected flux is normalized to B0cpi.

[91] When an initial perturbation is added, as prescribed by the GEM challenge (p = 0.1), a striking difference is observed in the Harris and bifurcated case. In the Harris case, the initial island sizes is already large enough to overcome the stabilization effects predicted by Biskamp et al. [1970] and the island growth can progress until filling the entire computational box. In the bifurcated case, instead, the island size is still smaller than the threshold and it does not grow. Figure 18 compares the reconnected flux for the case with a p = 0.1 initial perturbation for the classic Harris case (solid) and for the initial bifurcated equilibrium (dashed). Clearly, the behavior is opposite: in the Harris case reconnection progresses beyond the initial perturbation, but in the bifurcated case, the island never grows beyond its initial size.

Figure 18.

The reconnected flux is plotted for Harris (solid line) and Harris modified (dashed line) initial equilibria. Case with mi/me = 25 and with initial perturbation p = 0.1. The reconnected flux is normalized to B0cpi.

[92] The increased stability of bifurcated current sheets can be justified by two observations. First, bifurcated current sheets are known to follow after the saturation of the tearing instability. As an example Hesse et al. [1998]; Lottermoser et al. [1998]; Shay et al. [1998] show that the non-linear consequence of the tearing instability is a bifurcated current. It stands to reason that if a bifurcated current sheet is already the non-linear consequence of the tearing instability, taking it as an initial state would result in a condition more resistent to further growth of the tearing mode. Second, the initial bifurcated current sheet is larger than the reference Harris sheet, accounting for additional stability.

[93] The conclusions outlined above are not sensitive to the mass ratio. We have repeated the same simulations for mi/me = 180, albeit for shorter times, confirming the results above.

7.3. Simulations in 2D: (y, z) Plane

[94] The same cases studied above in the (x, z) plane, are considered next in the orthogonal plane (y, z). We compare the results relative to the classical Harris case and to the bifurcated equilibria of case (e), with the specific choice A(0) = 2, referred to again as modified Harris. The size of the system is chosen to be π in y and the grid is 1 × 128 × 128. The grid spacing in y is 4 times smaller than the grid spacing used for x in the previous subsection, because the instabilities expected to propagate along y have shorter wavelength. Periodic boundary conditions are used in y, and the same boundary conditions in z are used as above. No initial perturbation is added and the system evolves out of the natural noise of any PIC simulation.

[95] The evolution in the (y, z) plane is comparatively more active than in the (x, z) plane considered above. Below we compare the behavior of the bifurcated sheet with the the classic Harris case. We have conducted the study at different mass ratios (mi/me = 25 and mi/me = 180) obtaining similar conclusions. Results are presented only for the case of mass ratio mi/me = 180.

[96] The evolution of the Harris sheet is well know but is nevertheless repeated to provide a comparison for the bifurcated case. Previous literature has clarified the type and nature of the instabilities present. Initially the system is unstable to two types of instabilities: the lower hybrid drfit instability (LHDI) and the drift-kink instability (DKI). The LHDI has long been known [Davidson and Gladd, 1975], and it includes both shorter wavelength electrostatic modes (kyρe ∼ 1) and relatively longer wavelengths electromagnetic modes (kyequation image ∼ 1) [Winske, 1981; Daughton, 2003]. We observe both in the simulations.

[97] Figures 1920 show in order the evolution of the Ey component of the electric field and of the Jy component of the current. The shorter wavelength electrostaic component of the LHDI grows only at the edge of the current channel. Its growth is stabilized by high beta [Davidson et al., 1977] and cannot penetrate into the center. The electrostatic component has a larger growth rate and saturates first, before the slower electromagnetic components have had a chance of growing significantly. The presence of the electrostic components is primarily shown by Figure 19 where the electric field is shown. The electrostatic LHDI causes a microturbulence on the two sides of the current channel and is visible already at very early times (ωcit = 0.5) and remains present throughout the simulation.

Figure 19.

Plot of Ey for Harris case, mi/me = 180 in following times: (a) ωcit = 0.5; (b) ωcit = 2.

Figure 20.

Plot of jy for Harris case, mi/me = 180 in following times: (a) ωcit = 0; (b) ωcit = 2; (c) ωcit = 10; (d) ωcit = 46.

[98] The electromagnetic component of the LHDI, instead, affects also the magnetic field and can be observed at later times (ωcit = 2) in the ripples present in the center of the current sheet (see Figure 20b).

[99] The later stages of the evolution are dominated by the kinking of the current. Two instabilities can be responsible for it. First, the DKI is a linearly unstable mode, whose growth rate decays rapidly with the mass ratio. In the results relative to a mass ratio of 25 (not shown) the mode can be strongly active, but in the results shown in Figures 1920 (for mi/me = 180) the growth rate of the DKI has reduced to small values [Daughton, 1999]. The second possible cause of the kinking is the Kelvin-Helmholtz instability (KHI) driven by velocity gradients [Lapenta and Brackbill, 2002]. From a kinetic perspective the KHI can be interpreted as a ion-ion kink mode [Daughton, 2002; Lapenta et al., 2003]. The KHI is driven by velocity gradients; the initial state of a Harris equilibrium has a perfectly flat velocity profile and no KHI is present. However, the LHDI developing at the beginning of the simulation has been recently shown to produce two significant changes of the original equilibrium: current peaking and ion velocity shearing. The peaking is shown in Figure 21a and the shearing of the ion velocity is shown in Figure 22a. Both effects have been documented in previous works [Lapenta and Brackbill, 2002; Daughton, 2002; Lapenta et al., 2003] and recently a physical interpretation has been given in terms of electron acceleration due to the creation of a electrostatic potential induced by the LHDI [Daughton et al., 2005].

Figure 21.

Plot of average jy(z), mi/me = 180: (a) Harris case, ωcit = 0.5 (solid line), ωcit = 10 (dashdot), ωcit = 46 (dashed); (b) Harris modified case, ωcit = 0.5 (solid line), ωcit = 4.5 (dashdot), ωcit = 35 (dashed).

Figure 22.

Plot of average ion velocity vy(z), mi/me = 180: (a) Harris case, ωcit = 0 (dotted), ωcit = 5 (solid), ωcit = 15 (dashed); (b)Harris modified case, ωcit = 0 (dotted), ωcit = 5 (solid), ωcit = 15 (dashed).

[100] The results above constitute now the generally accepted behavior of a Harris current sheet and the results shown above are reported primarily for reference but similar results can be found in the literature [Lapenta and Brackbill, 1997, 2002; Daughton, 2002; Lapenta et al., 2003]. The issue at hand here, instead, is wether any or all of the processes above are affected by changing the initial state from a Harris equilibrium to a bifurcated equilibrium. And the answer is not, they are not modified. We note that a similar conclusion has also been reached in a recent paper by Sitnov et al. [2004] for a different class of bifurcated equilibria.

[101] This conclusion is highly relevant to the real magnetotail where simple Harris equilibria can at best be a rough approximation. Therefore proving that the scenario outlined above is robust enough to survive as radical a change as the shift from a Harris equilibrium to a bifurcated equilibrium consists in a significant step in understanding how the real magnetotail behaves.

[102] Figures 23 and 24 report the magnetic field, electric field and current corresponding to the same times shown above in Figures 1920 but now for an initially bifurcated equilibrium. The same stages are present, the faster electrostatic LHDI develops first, followed by the electromagnetic branch. The non linear evolution of the LHDI still produces a current sheet peaking and a shearing of the velocity (see Figure 21b and Figure 22b).

Figure 23.

Plot of Ey for Harris modified case, mi/me = 180 in following times: (a) ωcit = 0.5; (b) ωcit = 4.5.

Figure 24.

Plot of jy for Harris modified case, mi/me = 180 in following times: (a) ωcit = 0; (b) ωcit = 4.5; (c) ωcit = 20; (d) ωcit = 35.

[103] We observe that the rapid initial peaking of the current that rises by about a factor of 3 by time ωcit = 4.5 is actually slow when observed on the scale of evolution of the LHDI (ωcit = 4.5 corresponds to ωLHt = 60) and is not due to a lack of initial equilibrium. We have tested that the initial analytical solution is consistent with the initial setup of the code. And we have conducted 1D simulations where in absence of instabilities, the current sheet is in pressure equilibrium and remains unmodified indefinitely.

[104] The rapid evolution of the initial current sheet is due to the LHDI through the same mechanism observed above for the Harris case. The results reported in Figure 22 show that the peaking happens also in the case of an initial bifurcated current sheet.

[105] Of interest to the debate on the role of the LHDI in producing anomalous resistivity is the fact that in the bifurcated case the LHDI does not develop in between the two current peaks. Even in the bifurcated case the LHDI is limited to the outer edges.

8. Conclusions

[106] We have considered bifurcated equilibria based on the theory by Schindler and Birn [2002].

[107] We have considered first a number of different equilibria. The theory allows great freedom in selecting the particle distribution functions. We derived first necessary and sufficient conditions for the theory to generate bifurcated equilibria. We have then selected a number of classes of distribution functions satisfying the conditions and indeed leading to bifurcated equilibria.

[108] We studied the different equilibria considered. We obtain fundamentally two classes. First, (a)–(b) based on a distribution function composed by two populations of particles, separated by a sharp cutoff at a given vy (velocity component in the dawn-dusk direction). Second, the distributions (c)–(e) present a non-isotropic temperature and in some limits two beams, one centered around a drift speed and one at rest.

[109] Both classes lead to bifurcated equilibria, characterized also by an equilibrium velocity shear and by an equilibrium electrostatic potential. We discussed the relevance of the equilibria considered here to satellite observations.

[110] We find that cases (a)–(b) present the typical cutoff signature observed in the substorm recovery phase in presence of reconnection and Earth-ward flowing plasmas [Nishida, 2000; Nagai et al., 2001, 2002].

[111] Instead, cases (c)–(e) present the typical signature of anisotropic temperature (ellipsoidal particle distribution function) observed in presence of strong LHDI activity [Shinohara et al., 1998].

[112] Each of the two cases correspond to mechanisms suggested for the possible origin of bifurcated equilibria. Reconnection is observed to lead to bifurcated current sheets in a number of simulations [Hesse et al., 1998; Lottermoser et al., 1998; Shay et al., 1998]. The LHDI is also observed to lead to bifurcation even without any reconnection being allowed [Daughton et al., 2005]. Future work will be directed toward a detailed comparison of the particle distribution functions observed in the simulations of each of these two mechanisms for bifurcations, satellite data and the two different classes of equilibria tested here.

[113] We then moved to study the stability of the bifurcated equilibria of case (e). We selected case (e) primarily because it is in the best agreement with our recent simulations on current sheet bifurcation due to the non-linear evolution of the LHDI [Daughton et al., 2005].

[114] We studied first the stability of the tearing instability. We compared the evolution of classic Harris equilibria with comparable bifurcated equilibria. When the evolution is studied starting from an unperturbed equilibria (in presence of just the noise from the simulations), little difference is observed between Harris and bifurcated equilibria. In absence of linear theory results for bifurcated equilibria we focused on the saturation levels for the evolution from the unperturbed state, obtaining in both cases that the island caused by the tearing mode saturates at sizes smaller than the current sheet size.

[115] However, when a strong initial perturbation is added (equal to the perturbation considered in the GEM challenge [Birn et al., 2001], or twice that amplitude) the Harris case present a fast reconnection phase, followed by a saturation when the island thickness reaches the size of the computational box. But in the bifurcated case, the island does not grow in size from its initial perturbation size. This resilience of the bifurcated equilibria will need further theoretical investigation. A linear theory will be developed to study not only the initial bifurcated case but also the initially perturbed bifurcated case to investigate the region and the reason of stability.

[116] Next we compared the stability of Harris and bifurcated equilibria towards the LHDI. We find that the linear evolution and the non-linear evolution of both are rather similar. They follow the same steps. First, the electrostatic branch of the LHDI grows and quickly peaks the current layer. Next the electromagnetic branch of the LHDI ripples the edges of the current layer and enters it reaching the center of the layer and causing a short wavelength rippling of the whole current. Concurrently, the velocity profile is strongly modified by the LHDI leading to strong velocity shears and to the long wavelength kinking of the current sheet due to the Kelvin-Helmholtz instability. The same sequence has been observed for Harris equilibria [Lapenta and Brackbill, 2002; Daughton, 2002; Lapenta et al., 2003; Daughton et al., 2005].

[117] The most substantive result of the stability study of the LHDI for bifurcated current sheets is that the electrostatic branch of the LHDI remains limited in the flanks even in the bifurcated case. In a bifurcation equilibrium, the center of the current sheet has a relatively lower current and one could have wondered whether the LHDI would be allowed to grow there. If that were the case, the LHDI could create a strong turbulence and a strong anomalous dissipation right in the center of the sheet. However, this is not the case; the electrostatic LHDI remains on the outside of the current.

Acknowledgments

[118] The authors gratefully acknowledge useful discussions with J. Birn, W. Daughton, K. Schindler, P. Ricci, and M. Sitnov. This research is supported by the LDRD program at the Los Alamos National Laboratory, by the United States Department of Energy, under contract W-7405-ENG-36 and by NASA, under the Sun Earth Connection Theory Program and the Geospace Sciences Program.

[119] Lou-Chuang Lee thanks the two reviewers for their assistance in evaluating this paper.

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