[1] The sufficient stability criterion of the collisionless ion tearing mode in the magnetotail current sheet, which was first obtained by Lembege and Pellat in 1982, is considered. For many conventional 2D current sheet equilibria, this criterion is satisfied within the WKB approximation, which is commonly interpreted as stability of those equilibria with respect to tearing. However, this is not necessarily the case for equilibria with more than two characteristic spatial scales. An example for substantial tearing destabilization of an equilibrium with accumulation of the magnetic flux at the tailward end of a thin current sheet is presented. Similar equilibria are reported in Geotail and THEMIS observations prior to onsets of magnetospheric substorms and dipolarization fronts associated with bursty bulk flows.

[2] A distinctive feature of magnetic reconnection in the tail of Earth's magnetosphere is its bursty character, with a relatively long phase of the energy loading and its quick release in the form of substorm onsets and bursty bulk flows [Angelopoulos et al., 2008; Runov et al., 2009]. Many substorms [Hsu and McPherron, 2009] and underlying reconnection processes [Nagai et al., 2005] have no explicit solar wind triggers suggesting that they are caused by an internal instability of the magnetotail current sheet. At the same time, the mechanism of the corresponding plasma instability remains poorly understood. The original mechanism proposed by Coppi et al. [1966] was based on the electron tearing instability. In more realistic tail current sheet models Schindler [1974] took into account the finite magnetic field component B_{z} normal to the neutral plane [Fairfield and Ness, 1970], concentrating on ion dynamics. The onset problem became more complicated after the discovery that magnetized electrons could also change the sign of the tearing mode energy [Galeev and Zelenyi, 1976]. However, a real crisis appeared in the theory when Lembege and Pellat [1982] and then Pellat et al. [1991], Brittnacher et al. [1994], Quest et al. [1996], and Schindler [2007] showed that the ion tearing mode should be universally stable, based on a sufficient stability criterion obtained in the WKB approximation in the stability analysis. Thus the stability theory successfully explained why the magnetotail is able to accumulate the energy and magnetic flux, but it did not explain their spontaneous release.

[3] In this Letter we investigate the aforementioned sufficient stability criterion and show that the ion tearing mode is indeed stable for many conventional tail current sheet models. We also show that such conclusion cannot be drawn in general, and we describe a new class of equilibria that are potentially unstable in the sense that they show a strong destabilizing effect compared with the conventional models. In our approach we address the classical scenario where reconnection is the result of a tearing instability rather than its cause. Other scenarios are also possible and they are discussed in section 5.

2. Lembege-Pellat Stability Condition and Flux Tube Volume Formula: General Equations

[4] According to Lembege and Pellat [1982] and a more general analysis (Schindler [2007], see in particular, equation (10.240)), the sufficient stability condition for a tearing mode with the wavenumber k in a tail current sheet, which is locally described by a modified Harris equilibrium [Harris, 1962], can be written in the form

It is important, that this stability criterion involves both local values of the current sheet thickness L_{z}, normal magnetic field B_{z}, inhomogeneity scaling factor b = B_{z}/B_{0}, such that L_{z}/b determines the characteristic scale L_{x} along the x- (Sun-Earth) direction, and a global parameter, the flux tube volume V = dl/B.

[5] If the parameter C_{d} <∼ 1, the criterion (1) is satisfied within the WKB condition kL_{z}/(πb) ≫ 1 commonly extrapolated to kL_{z}/(πb) > 1. This corresponds to the stability analysis of the extended and inhomogeneous magnetotail locally in x. Thus the current sheet is tearing-stable. In the opposite case no definite statement on stability is available, but there is a destabilizing tendency.

[6] Let us specify the destabilization factor C_{d} for a class of 2D isotropic locally Harris equilibra weakly inhomogeneous along the x-axis (∂/∂x ≪ ∂/∂z). The corresponding approximate solution of the 2D Grad-Shafranov equation can be described by the following y-component of the vector potential [Schindler, 1972; Birn et al., 1975; Lembege and Pellat, 1982]

for any choice of β(x) slowly varying in x. The corresponding magnetic field components can be written as

so that b = B_{z}(x, 0)/B_{0} = Lβ_{x}′/β.

[7] Following Hurricane et al. [1996], we reduce the flux tube volume V to the integral over the x-coordinate

where the maximum excursion over the flux tube is given by the equation β (x_{max}) = exp (ψ/LB_{0}). Then using (2) and (3) we present the field B_{x} in the form

[8] Finally, substituting (6) into (5) and changing variables in that integral from x to u = 2ψ/LB_{0} − 2 ln β we obtain the following expression for the flux tube volume

which is a generalization of equation (16) of Hurricane et al. [1996] for an arbitrary function β = β (x(u)). Using the latter expression, the factor C_{d} in the local vicinity of x = x_{max} can be presented in the form

where we used the relations L_{z} (x_{max}) = Lβ (x_{max}) and B_{z} (x_{max}) = B_{z} (x_{max}, 0) = LB_{0}β_{x}′ (x_{max})/β (x_{max}) for the local values of the current sheet thickness and the normal component of the magnetic field, which follow from the 2D equilibrium model (2)–(4) and which enter the definition (1) of the destabilization factor C_{d}.

3. Stable Equilibria

[9] For the specific 2D equilibrium used by Lembege and Pellat [1982] and Hurricane et al. [1996] with the function β (x) = exp (ɛx/L) and b = ɛ ≪ 1 we obtain that β_{x}′/β = ɛ/L = const. Then taking into account, that du/ = π, we deduce from (8) that C_{d} = 1, indicating the universal stability of Lembege-Pellat equilibrium. Note, that this exact value of the destabilization factor is quite close to the original estimate C_{d} ∼ 2/π made by Lembege and Pellat [1982] using a simple approximation of the flux tube volume V ∼ 2L_{z}/B_{z} where the integration over the flux tube was limited to a sheet thickness ∣z∣ < L_{z} (the exact value of V was calculated for the first time by Hurricane et al. [1996], though not in the context of the tearing stability analysis). It is also interesting that a similar result can be obtained for another equilibrium with ψ = −A_{0} cos(πz/(2Δ)) exp(−αx) and the constant current sheet thickness L_{z} ∼ Δ [Voigt, 1986], which does not even belong to the general class of the locally Harris sheets (2). As follows from Wolf et al. [2006], for that model VB_{z}/(2Δ) = 1.

[10] Let us now consider another big family of equilibria with the parameter β_{x}′/β decreasing tailward. These include models of the class (2) with β (x) = (1 + ɛx/(νL))^{ν} and ν > 0 [Zwingmann et al., 1990] and β (x) = 1/(1 + 1/(2)) and 0 < ɛ 1 [Schindler and Birn, 2004]. Noting that for this class of models the function β_{x}′/β is monotonously decreasing with x, and therefore (β/β_{x}′) < β (x_{max})/β′_{x} (x_{max}) one can estimate the integral in (8) as

which means, that C_{d} < 1, and the current sheet is again tearing-stable. For further models that fall into the same category see Birn et al. [1975].

4. Multiscale Current Sheet Model

[11] According to (8), the magnitude of the destabilization factor C_{d} is determined by the balance between the local value of the parameter β_{x}′/β at x = x_{max} and an average of its reciprocal over the whole length of the flux tube (−, x_{max}). Therefore, C_{d} may be increased by increasing β_{x}′/β near the tailward end of the flux tube x = x_{max}. Such an increase introduces the third (meso)scale in the tail current sheet, in addition to the current sheet thickness L_{z} and the global scale L_{x}.

[12] Consider, for example, the following modification of the original Lembege-Pellat equilibrium

where

Then with ɛ_{1} ≪ ɛ_{2} ≪ 1 and α ≫ 0 the parameter β_{x}′/β ∼ B_{z}(x, 0) will have a local maximum near x = x_{0} = Lξ_{0} ≫ L in the transition region from a thin current sheet at x < x_{0} to a thicker sheet at x > x_{0}. Since the region of the B_{z} enhancement is localized, the corresponding integral in (8) can be estimated as

whereas

Therefore

which means that such an equilibrium may be potentially unstable. Note that the analytical estimate (14) is only valid in the limit ɛ_{1}/ɛ_{2} → 0. However, the explicit numerical evaluation of the destabilization factor C_{d}(α) using (8) and (10)–(11), whose results are shown in Figure 1, reveals the same tendency to the current sheet destabilization. Moreover, the analysis of (8) shows, that a significant destabilization takes place for a broad spectrum of magnetotail configurations with the ratio β_{x}′/β growing as a function of the parameter β/β(x_{max}) in the interval (0, 1).

[13] An example of the potentially unstable equilibrium of the type of (10)–(11) is provided in Figure 2. It shows that the destabilization occurs at the tailward edge of a relatively thin and long current sheet where the magnetic flux is locally accumulated. This is consistent with the conclusion based on the analysis of Geotail observations [Asano et al., 2004; Miyashita et al., 2009], that the magnetic reconnection in the tail starts near the tailward edge of an intense cross-tail current, which extends from X ∼ −5 to −20 R_{E} in the GSM coordinate system. Moreover, a distribution of the north-south magnetic field B_{z} similar to the specific profile shown in Figure 2a has recently been inferred from the statistical vizualization of the magnetotail Geotail data prior to the substorm onset (Machida et al. [2009], see in particular, their Figure 2c). The formation of a thin current sheet embedded into a thicker plasma sheet and extended from near geosynchronous orbit to the mid - magnetotail (∼30 R_{E}) during the growth phase of a substorm has been shown using multi-satellite measurements by Pulkkinen et al. [1999]. Recently the fleet of five THEMIS satellites distributed over the magnetotail between X ∼ −10 and −20 R_{E} [Runov et al., 2009] detected the propagation of bursty bulk flows and dipolarization fronts through that extended region of a strongly stretched magnetic field with B_{z} being only a few nanoteslas in magnitude.

[14] The equilibrium (10)–(11) also resembles thin current sheets [Birn and Schindler, 2002] that arise from finite boundary deformations with a profile of the plasma pressure parameter p(x) = 1/(2β(x)^{2}) similar to Figure 2b. A direct comparison with the present results is difficult because Birn and Schindler [2002] use MHD, while the present approach is based on kinetic theory.

[15] For collisionless plasmas of the terrestrial magnetosphere the current sheet thickness must be comparable to the thermal ion gyroradius to provide the ion Landau dissipation [Pritchett et al., 1991], which makes the equlibria of the class (10)–(11) quite an extreme case. This explains the long periods of the energy and flux accumulation in the tail current sheet with rather transient periods of their release. At the same time, it is the broad range of scales between the thermal ion gyroradius ρ_{0i} in the field outside the thin current sheet (L_{z} ∼ ρ_{0i} ∼ 0.1 R_{E}) and global scales (L_{z}/ɛ_{1} >∼ 10 R_{E}) that makes the multiscale model described above appropriate for the terrestrial magnetotail.

5. Conclusion

[16] In this paper we have addressed the Lembege-Pellat criterion (1) for stability of the magnetotail and applied it to a number of equilibrium models. For several equilibria of the standard two-scale type, stability is confirmed in agreement with earlier studies. In the relevant regime, due to the presence of the B_{z} component, the magnetized electrons have a stabilizing influence via compressibility [Pellat et al., 1991] and their Hall current [Schindler, 2007].

[17] The main purpose of this paper is to present a class of equilibria characterized by more than two spatial scales. We found that the B_{z}-stabilization can be considerably reduced compared with the two-scale models. In our example the destabilization may be possible at the tailward edge of a relatively thin and long current sheet where the B_{z} component is locally enhanced and the magnetic flux is accumulated. For reasonable tail parameters the effect reaches a level such that stability can no longer safely be inferred from the Lembege-Pellat criterion (1).

[18] For the realistic regime where the ion temperature dominates over the electron temperature, the B_{z}-stabilization can be understood as resulting from the electron Hall current (see pp. 265, 266 of [Schindler, 2007]). The electron Hall current density is directed oppositely to the driver current of the tearing mode and its leading WKB contribution is inversely proportional to the local value of the field B_{z} and the flux tube volume V (see, in particular, equations (10.249), (10.224), and (10.237) of Schindler [2007]). This explains the stabilizing effect of small B_{z}. But this also shows, that the stabilizing effect of the Hall current can be reduced if earthward of the considered point, subject to the local stability analysis, the values of B_{z} are less, although still large enough to magnetize electrons.

[19] As we use a version of the underlying variational principle that provides only sufficiency for stability, we cannot claim that this effect necessarily leads to the onset of an instability. It is well possible that additional destabilizing effects, such as the presence of nongyrotropic electrons arising from a local reduction of the normal field B_{z} [Hesse and Schindler, 2001; Pritchett, 2005] or modified boundary conditions [Sitnov et al., 2002, 2009; Daughton et al., 2006; Divin et al., 2007], are necessary to turn the tail unstable. However, even then destabilization due to the presence of a multiscale equilibrium structure of the type discussed in this paper would play an important role, so that it has an essential favoring influence on the onset of magnetospheric substorms. Our results are consistent with observations showing that substorm onset often occurs at the tailward end of a thin current sheet where magnetic flux accumulates.

[20] Apart from leaving out several stabilizing effects, which limits the Lembege-Pellat criterion to sufficiency, its main limitations are spatial two-dimensionality and pressure isotropy with respect to the (X, Z) plane. Matsui and Daughton [2008] have recently demonstrated that electron pressure anisotropy may have a significant influence on tearing of bifurcated current sheets, however with B_{z} = 0. The impact of ion anisotropy and agyrotropy may be even stronger, because ions often determine the multiscale structure of the current sheet, making it either embedded into a thicker plasma sheet [Sitnov et al., 2000, 2003; Schindler and Birn, 2002] or bifurcated [Sitnov et al., 2003; Birn et al., 2004]. Zelenyi et al. [2008] found tearing destabilization for a family of 1D embedded current sheet models, the so-called forced current sheets, where the magnetic field line tension is balanced by the ion inertia rather than by the pressure gradient. However, the relevance of forced current sheets to the actual magnetotail is rather limited, because those 1D models have no characteristic scale length in the X-direction and as a result the flux tube volume cannot be properly defined. Moreover, they have no limit of small plasma anisotropy [Sitnov et al., 2000]. At the same time, some of the recent models built as modifications of the Harris equilibrium (for instance, [Schindler and Birn, 2002; Sitnov et al., 2003]) assume small values of plasma anisotropy. They may be used to construct multiscale current sheet equilibria and help explain the reconnection onsets occuring relatively close to the Earth [Sergeev et al., 2008].

[21] Although it is tempting to visualize that the present destabilizing effect may cause substorm onset due to an ion tearing mode at a flux accumulation region, further studies, including full-particle simulations, are required to clarify whether within the present framework of a plasma with magnetized isotropic electrons an unstable regime exists, and if it does what its linear (growth rate and eigenfunctions) and nonlinear properties are.

Acknowledgments

[22] The authors acknowledge useful discussions with R. Wolf and participants of the 2009 Yellowstone Workshop on Thin Current Sheets. This work was supported by NASA grants NNX09AH98G and NNX09AJ82G as well as NSF grant AGS0903890.