Disruption of magnetospheric current sheet by quasi-electrostatic field

Recent observational evidence has indicated that local current sheet disruptions are excited by an external perturbation likely associated with the kinetic ballooning (KB) instability initiating at the transition region separating the dipoleand tail-like geometries. Specifically a quasielectrostatic field pointing to the neutral sheet was identified in the interval between the arrival of KB perturbation and local current disruption. How can such a field drive the local current sheet unstable? This question is considered through a fluid treatment of thin current sheet (TCS) where the generalized Ohm’s law replaces the frozen-in-flux condition. A perturbation with the wavevector along the current is applied, and eigenmodes with frequency much below the ion gyrofrequency are sought. We show that the second-order derivative of ion drift velocity along the thickness of the current sheet is a critical stability parameter. In an E-field-free Harris sheet in which the drift velocity is constant, the current sheet is stable against this particular mode. As the electrostatic field grows, however, potential for instability arises. The threshold of instability is identified through an approximate analysis of the theory. For a nominal current sheet half-thickness of 1000 km, the estimated instability threshold is E∼4 mV/m. Numerical solutions indicate that the two-fluid theory gives growth rate and wave period consistent with observations.


Introduction
Thin current sheets (TCSs) are widely observed in natural plasma systems from the solar corona to Earth's magnetotail, especially as a precursor to eruptive episodes of global Correspondence to: W. W. Liu (william.liu@space.gc.ca) energy release.In the magnetosphere, TCSs mainly manifest themselves as embedded structures in the central plasma sheet (CPS) during the growth phase of substorm (Sergeev et al., 1993;Pulkkinen et al., 1999).As convection in the magnetotail intensifies, energy cannot flow around or through the largely dipolar inner magnetosphere in sufficient quantity to balance the input from the distant tail, and the accumulated energy in the CPS stretches the magnetic field tailward.Formation of thin current sheets starts out as a quasistatic process, taking tens of minutes to develop in the magnetosphere.Evolution and maintenance of TCSs has been a subject of extensive investigation through theory and simulation (Lee et al., 1995;Pritchett and Coroniti, 1995;Schindler and Birn, 2002;Sitnov et al., 2000;Birn et al., 2004).
TCSs are also known to be highly dynamic, collapsing rapidly in a matter of seconds and leading to the expansion of substorms.Why some TCSs are meta-stable while others not is a question lacking a satisfactory answer.Many researchers take the thickness of a TCS as the determining factor in the stability question.TCSs are often observed to have a thickness comparable to ion gyroradius (∼1000 km); at this scale, magnetic field can slip through the ion fluid owing to the breakdown of the frozen-in-flux condition.Yet, some TCSs can persist in this condition for many minutes, far longer than the time it takes for information to propagate through the current sheet via the fast mode, while other TCSs, for roughly the same integrated current strength and thickness, are unstable, losing equilibrium in tens of seconds.Therefore, there could be factors other than thickness at play.
Recent observations have shed new light on the problem.Saito et al. (2008a, b) showed that local current disruptions (CDs) are preceded by several minutes of wave activity in the 10-20 mHz range, which they attributed to the ballooning instability.Through multiple satellite observations, Donovan et al. (2008) and Liu et al. (2008) showed that the 10-20 mHz wave band propagates downtail at a substantially reduced speed compared to the fast mode.Through a wavelet analysis, Liu et al. (2008) and Liang et al. (2009) were able to identify that the mode directly responsible for local CD was not the KB mode but a new mode (referred to as the CD mode) occurring at a higher frequency in the 20-100 mHz range.Liu et al. (2008) demonstrated that the CD mode was quasi-electrostatic.The above studies give us sufficient reason to believe that waves in the 20-100 mHz range play a significant role in local CDs.Using independent techniques, Saito et al. (2008a) and Liang et al. (2008) estimated the azimuthal wavelength of the waves to be 600-3000 km, comparable to the ion thermal gyroradius in equatorial plane.
We first review some essential information and key arguments regarding the formation and nature of the quasielectric field (QEF) before onset.We then consider an idealized thin current sheet represented by the Harris distribution.We will show that this configuration is stable in the two-fluid regime no matter how thin it is.We then introduce an electrostatic field in the equation and illustrate its propensity to drive the system toward instability.The instability threshold is established through an approximate analysis of the eigenvalue equation.Numerical examples are presented to demonstrate that the predictions of the theory are in line with observed growth time and oscillating periods.Contrast is drawn between the new instability and previously known drift-kink instability.How the theory fits in the larger context and some of its limitations are described in the discussion section.

Quasi-electrostatic field in a thin current sheet
Numerous studies have shown that the thickness of the near-Earth current sheet can become comparable to the ambient ion gyroradius (Mitchell et al., 1990;Sergeev et al., 1993;Sanny et al., 1994).On this scale, ions are partially demagnetized and do not fully follow the magnetic field lines, while electrons, by and large, continue to be frozen in the B field.In this condition, charge separation may develop in the neutral sheet.
The existence of a QEF in a TCS has been studied theoretically (Schindler and Birn, 2002;Birn et al., 2004) and found in computer simulations (e.g., Pritchet and Coroniti, 1995).Its latest experimental identification, particularly its association with local CDs, was reported by Liu et al. (2008).Using different techniques, Liang et al. (2008) and Saito (2008a) showed that the wave mode associated with substorm onset has a dominant azimuthal wavelength on the order of 1000 km.In-situ and THEMIS ASI observations gave the growth timescale of the mode in the 10-30 s range (Liang et al., 2008).Coupled with in-situ magnetic measurements reported in Liu et al. (2008), the curl of the electric field can be estimated through Faraday's law.The resulting |∇×E| was found to be small compared to the measured electric field over the estimated length scale.The observed electric field had a neutral sheet-pointing bias, implying a negative charging of the neutral sheet.The electric field was found only in the interval between the arrival of the kinetic ballooning perturbation and onset of local CD and subsided quickly afterward, whereas the magnetic field perturbation persisted for many minutes further.The interpretation was that the local CD removed the condition for charge imbalance and quenched the electrostatic field.
From both theory and observations, the existence of a QEF before local CD appears to be well established.What is not clear is whether the QEF is an incidental property or an essential element in the destabilization of the local current sheet.In the following sections, we prove that the latter is possible.

Context
In order to see where our theory stands and that the result indeed represents a new mode, a brief survey of the extant literature is conducted.The survey is not meant to be comprehensive, but to draw out the necessary contrast with some known modes of TCS instability.Dalburgh et al. (1992) showed that the classical Harris sheet with B z =0 is stable against ideal MHD perturbations.However, in the presence of a finite B z , the problem of current sheet stability becomes more complicated.Various authors have studied the ballooning mode associated with field line curvature under the MHD and kinetic conditions (Hameiri et al., 1991;Roux et al., 1991;Liu, 1997;Cheng and Lui, 1998).As we will comment in the discussion, the presence of B z can further introduce resonant and non-local plasmas responses that may not be accounted for in a fluid treatment.
The two-fluid theory is a more accurate approximation of plasma physics in a TCS.A much studied two-fluid mode is the drift-kink instability (DKI).Under the fictitious condition m i =m e and T i =T e (where the subscripts i and e refer to the ion and electron species, respectively), Pritchett et al. (1996) derived the growth rate of DKI in the classical Harris sheet as where 2V 0 is the relative drift between the two species, k y the wavenumber parallel to the drift, and 0 the gyrofrequency in the asympotic field of the Harris sheet.Numerical calculations and simulations of these authors indicate that the growth rate decreases as more realistic mass ratio is used.Daughton (1999) studied the DKI mode under the condition of incompressibility for an unmagnetized inner region but without assuming identical ion and electron properties.The growth rate was derived to be Equation ( 2) suggests that the DKI is an electron inertial effect, as well as a relative drift effect.Lapenta and Brackill (1999) constructed a kinetic model of DKI, again under some approximations.Their results were largely consistent with those of Pritchett et al. (1996) and Daughton (1999).
In order to clearly distinguish our theory from DKI, we take the limit m e =0.By virtue of Eq. ( 2), this limit should eliminate the DKI from our consideration, and whatever mode that emerges should thus be different.
We note that numerous other TCS modes have been studied in the literature.They often involve kinetic processes in the velocity space.While these modes are potentially important, they do not have a close relationship to what interests us here, namely TCS modes arise in a fluid formulation of the plasma physics.

Hall MHD formulation
A key motivation for the present work is an observation of the implication of the limits made by Pritchett et al. (1996) and Daughton (1999).It is well-known that a linear combination of the momentum equations of the ion and electron species in the two fluid theory results in the generalized Ohm's law (GOL) (e.g., Krall and Trivelpiece, 1973).When the ion and electron masses are made equal artificially as in Pritchett et al. (1996), the Hall term in GOL vanishes, resulting in a loss of important physics.In the Daughton approximation, the ion and electron fluids are incompressible in the unmagnetized region.It can be verified that this condition is not satisfied generally in our treatment.Thus, the mode of interest here, which we call the Hall mode, was not captured by the Daughton limit.
In-situ observations have shown that the electron temperature T e is typically much lower than the ion temperature in TCSs in the magnetosphere.As a further simplification, we take the limit T e →0.The role of electrons in our theory is to provide the negative charge to achieve quasi-neutrality, but otherwise have no impact on the mode under study.The generalized Ohm's law under our approximations is written as where E the electric field, u is the plasma velocity, B the magnetic field, J the electric current density, n the plasma density, and e the unit electric charge.Equation (3) replaces the ideal MHD condition E=−u×B.Depending on one's preference, our theory can be regarded either as a reduced version of the two-fluid theory (through the m e →0 and T e →0 limits) or a generalized version of one-fluid MHD (through the Hall MHD condition 3).For mathematic convenience, Eq. ( 3) is combined with the momentum equation to give an alternate form of the GOL as The mass conservation, energy equation, and Faraday's law, written in the stated order below, round out the equations where ρ the plasma mass density (ρ=nm, with m being the ion mass), p the plasma pressure, and λ the polytropic index.In the ideal MHD, λ is usually assigned the value 5/3, corresponding to the isotropic adiabatic compression.In the two-fluid regime, ions are partially decoupled from the magnetic field and do not respond to its variation as rigidly as in the ideal MHD regime; a λ value closer to unity may be justified.In fact, the Harris sheet has the property p/n=const, consistent with a polytropic index of 1.Without affecting the generality of the argument and result, we set λ=1 in this study.
Using the vector identity u•∇u=∇ u 2 /2 −u×∇×u and defining a pseudo-magnetic field H =B+m∇×u/e, we obtain from Eqs. ( 5) and (8) the following Equations ( 4), ( 6), (7), and (9) form a closed set.In the Appendix, we describe the linearization of these equations and the derivation of the eigenvalue equation.Thevalidity of the fluid treatment will be justified in the discussion section.
The background magnetic configuration is the classical Harris sheet, with B z =0 and the following distributions: where L is the half-thickness of the current sheet, T 0 is the ion temperature, and p 0 =B 2 0 /2µ 0 .We assume that the wave mode is dominated by the azimuthal wavenumber k y .This approximation is motivated by the observations of Saito et al. (2008a) and Liang et al. (2008) that the azimuthal wavelength has the order of ∼1000 km, likely short in comparison with the radial extent of current disruption (≥1 R E ).For this reason, the linear mode under consideration has the harmonic form f (z) exp i k y y−ωt W. W. Liu and J. Liang: Electrostatic field and current disruption throughout this study.Note that DKI perturbations have the same harmonic form.
In the Harris sheet, the plasma has a zeroth-order velocity in the y direction given by where E z is the zeroth-order electrostatic field discussed in Sect. 2. In the classical Harris sheet, the ion and electron drifts are balanced in proportion to the temperature, i.e., U i /T i +U e /T e =0, and E z =0.However, we have noted in Sect. 2 that, when ion-scale effects become significant, this balance may be broken leading to a finite E z .
We begin the study by considering the Harris sheet without E z .This is a reasonable approximation during periods when the current sheet is relatively thick and consistent with the configuration underlying the DKI studies cited above.Since U =const (hence U =0) in this case, the linear mode is described by a relatively simple equation whose derivation and definition of terms therein is given in the Appendix.Multiplying the complex conjugate ρ u y * to Eq. ( 14) and integrating over z, we obtain the following equation, under the regularity condition u y (z=±∞) =du y (z=±∞) /dz=0, where The solution to Eq. ( 15) is Since A 1 is by definition positive, and A 3 by definition negative, the eigenfrequencies given by Eq. ( 19) are always real, and the two-fluid modes for the E-field-free Harris sheet are thus always stable.
It is pertinent to comment that both analytic and simulation results show that the DKI is unstable in a classical Harris sheet without E z .In this regard, one can conclude that the DKI is more easily excited than the Hall mode.
When U is not a constant of space, however, the Hall mode can become unstable and can achieve a faster growth rate than the DKI (as our later numerical calculation will attest).The eignevalue equation of the linear mode is more involved under this condition.The derivation of this equation is described in the Appendix, culminating in the form where Equation ( 20) lacks an energy-principle integral to give a general perspective on the stability problem as in Eq. ( 14).An approximate but instructive look at the problem, however, can be effected by examining the near-solution to the equations near the neutral sheet, namely, for z L. In this limit, odd functions B x , dp/dz, and dU/dz can be neglected in comparison to even functions such as dB x /dz, U , and where k 2 ⊥ =k 2 y +k 2 n .It is understood that all terms in Eq. ( 22) are evaluated at the neutral sheet z=0.Consider the symmetric mode u y (0) =1 and u z (0) =0.This boundary condition leads to unique solutions to Eq. ( 20).It can also be verified that asymptotically at z→∞, the solution varies as C 1 exp k y z +C 2 exp −k y z .Only for discrete values of k n can the physical constraint C 1 =0 be satisfied.As a rough guide, if we match the near solution u y =A cos (k n z) and far solution C 2 exp −k y z at z=L, we find k n +k y tan (k n L) =0, which gives a discrete set of k n values.
When the second derivative of the drift velocity is significant, the possibility of instability arises.The situation is visualized through Fig. 1.We note that G (ω d ) =0 has real solutions ω =0 and ω =±k ⊥ c f , the latter being the fast mode.For solutions to the equation G (ω d ) =C, however, it is clear that for C outside the stable range marked in Fig. 1, the equation has only one real solution, and the other two is a pair of conjugate complex numbers, one of which is unstable.It can be easily verified that the unstable situation occurs first when C exceeds the local maximum of G (ω d ), occurring at with value Ann. Geophys., 27, 1941Geophys., 27, -1950Geophys., 27, , 2009 www.ann-geophys.net/27/1941/2009/From the above argument, it is easily verified that is a necessary condition of instability.Equation ( 22) can be rearranged to the following form, where When the TCS distribution just passes the threshold of instability, the frequency can be written as ω =ω m +iγ , which, upon substitution into Eq.( 26), yields It can be confirmed that the unstable mode given in Eq. ( 28) exists only for intermediate values of k y ; namely, in both the k y →0 and k y →∞ limits, ω i =0.If we assume that k n =α/H is reasonably insensitive to k y , the stabilizing term on the right-hand side of Eq. ( 28) minimizes at At this k y value, Note that there is another branch of instability for the local minimum in Fig. 1.Using the same procedure as above, one can show that this branch has a growth rate given by Eq. ( 28) but with the sign of the d 2 U/dz 2 term reversed.From Eq. ( 13), we see that U can become a variable of space either due to the emergence of E z or due to the variation of the temperature along z.We consider the E z effect first.Assume the electric field has the form, Substituting Eq. ( 31) into (30) gives the critical value for k n ∼1/L.The maximum E z value according to Eq. ( 3) is E 0 / √ 2e.If we take B 0 =50 nT and neutral-sheet density n 0 =1 cm −3 , we obtain c F ∼700 km/s, giving E z ∼4 mV/m as the instability threshold.In Liu et al. (2008), the average E z value reached 4 mV/m in the few minutes prior to onset, with spikes at high as 10 mV/m.28).The figures shows the change of the solution to G (ω) =C.When C is sufficiently large (dashed line), there is a pair of complex solution, one of which is unstable.

Numerical solutions
Equations ( 20) are solved numerically as an eigenvalue problem to expand on the analysis given in the last section.The calculations were performed under realistic parameters, and the results are then compared to actual observations.We concentrate on the case of symmetric mode u z (z=0) =0 and d u z (z=0) /dz=0.The shooting method is used to search for the eigenvalue ω, under the constraint u z (z→∞) =0.As a test and for curiosity, we applied the algorithm to the field-free Harris sheet and indeed found no unstable mode.
The first case deals with the E z -driven mode, with the distribution given by Eq. ( 31).The background distributions are show in Fig. 2a. Figure 2b gives one example of eigenvalue solutions as a function of the azimuthal wavelength λ y =2π/k y , for the following parameters: n 0 =1 cm −3 , B 0 =50 nT, E z max =5 mV/m, and L=1000 km.The real part of the eigen-period has the range 20-40 s, and the imaginary part (expressed as growth time) is approximately constant, at 15-18 s.These values are consistent with the period of CD mode from wavelet analysis of Liu et al. (2008) and Liang et al. (2008), and the growth timescale of CD we reported earlier (Donovan et al., 2008;Liang et al., 2008).
We remark that the instability is sensitive to E z and L. Our search algorithm has an upper limit of 90 s for the growth time.For E z max <2 mV/m, and H >3000 km, there is no unstable mode returned by the algorithm within the search space.The numerical results are consistent with our analysis given in Sect.3.
In Fig. 3, we present a comparative case on how the instability is also possible when there is a temperature variation across the current sheet, in the absence of E z .The temperature model is given by We chose λ=1.3 in the example shown in Fig. 3.In Fig. 3a, the background distributions are plotted.In contrast to Fig. 2a, d 2 U/dz 2 <0 in the neutral sheet.We showed previously that both large positive and negative values of the second-order derivative of U can lead to instability.The variation of the eigenvalue as a function of λ y is shown in Fig. 3b, for the same set of background parameters as Fig. 2. The general behavior remains remarkably similar to Fig. 2b, with the exception that the wave period is about half of the first case.
It is pertinent to compare the predicted growth rates between the Hall mode and DKI.The range of k y corresponding to Figs. 2 and 3 is 1.5−3×10 −6 m −1 .The temperature T 0 corresponding to B 0 =50 nT and n=1 cm −3 is T 0 =10 −15 J.The drift speed is thus calculated to be 250 km/s, roughly 1/3 of the thermal speed.The growth rate predicted by Eq. ( 2) is then between 0.0087 and 0.0174 s −1 .The growth time range Fig. 3. Same as Fig. 2, but for the variable temperature model given in Eq. ( 39).The quasi-electrostatic field is absent in this calculation.
is 58 to 115 s, much longer than the 15-20 s range shown in Figs. 2 and 3, as well as the actually observed timescale of local current disruption.Note further that that the result from Eq. ( 2) is an upper limit to the actual DKI growth rate (see Fig. 2 of Daughton, 1999).Another comparison is with Fig. 3 of Lapenta and Brackbill (1997).The drift-to-thermal speed ratio underlying our Figs. 2 and 3 is at the low end of the curve.The m i /m e ratio is about 10 times the actual value.If we apply the scaling relationship in Eq. ( 2) to account for the real mass ratio and T e =0, an extrapolated γ ∼8×10 −3 i is obtained.For i ∼5 s −1 , the predicted e-folding time of DKI growth is ∼25 s.It is noted, however, that this corresponds to the k y value that maximizes the growth rate.The k y range in Figs. 2 and 3 is somewhat off this value (∼5×10 −5 s −1 according to Eq. ( 1) and the parameters used for the figures).It is therefore safe to say that the Hall mode, for conditions representative of the magnetospheric TCS, grows as fast as, if not faster than, the DKI.

Discussion
This work was motivated by the question why some TCSs can stay stable for many minutes while others disintegrate in seconds.Recent observation that an electrostatic field of several mV/m precedes local current disruptions further led us to investigate the role this field might play in the destabilization of TCS.Another motivation stemmed from our analysis of the drift-kink instability.It was found that analytic results widely used for the growth rate of DKI did not include the effect of the Hall term in the generalized Ohm's law.To make up for this neglect, we constructed our theory by eliminating the possibility of DKI but with a full account of the Hall effect, leading to the identification of the so-called the Hall mode, which is exclusively an ion-scale effect.The quickest way to verify that this mode is indeed a Hall effect is through Eq. ( 22); that is, the d 2 U/dz 2 term.If one inspects the original equations in the Appendix, he will find that this term stems from the dH x /dz term in (A1), which, upon further inspection, is a direct consequence of the Hall MHD as manifested in Eq. ( 9).
The fluid formalism is applicable under limited conditions.We confirm that the results shown in Figs. 2 and 3 reside within the bound of this applicability.First, the eigenfrequency ω must be small in comparison to the asymptotic ion gyrofrequency i =eB 0 /m.For the conditions underlying Figs. 2 and 3, ω∼0.1 s −1 , while i ∼5 s −1 .Second, the same eigenfrequency must be small compared to the ion bounce frequency in the magnetic field, given by b ∼V T i / √ LR i , where R i =mV T i /eB 0 is the asymptotic thermal ion gyroradius.For the parameters underlying Figs. 2 and 3, V T i ∼700 km/s, giving R i ∼140 km.The resulting b ∼2 s −1 indeed far exceeds ω.Third, the thermal ion gyroradius must be small compared to the thickness of the Harris sheet.The relevant comparison in Figs. 2 and 3 is between 140 km for R i and 1000 km for L. Lastly, the azimuthal wavelength must be long compared to the thermal gyroradius.For Figs. 2 and 3, the wavelength ranges between 2000 and 4000 km, again meeting the constraint.Therefore, at least for the examples we used to demonstrate the Hall mode, the fluid approach is valid.
It is noted that the presence of a finite B z can change the character of the problem and elicit new modes.In this situation, the bounce motion is no longer that associated with the meandering motion of particles about the neutral sheet, but that with mirror point reflection.The proper condition for the validity of a fluid treatment in this case would be ω>ω b .The ion bounce frequency for a magnetic bottle geometry is approximately B z V T i /2B 0 L. For the same ion thermal speed and half current sheet thickness as above and for B z /B 0 ∼0.1, the bounce frequency is ∼0.03 s −1 , less than ω∼0.1 s −1 .However, the electron bounce frequency is typically faster than the wave frequency.In this paper, we took the electron temperature to be zero, and this problem was suppressed.In reality, electrons have a finite temperature.
The effect of electron terms is beyond the scope of this study and remains to be evaluated.
It is also possible for a TCS to contain a finite guide field B y .The correction associated with B y is on the order B y /B 0 , hence more significant than B z .Inclusion of B y would generalize the eigenvalue problem (20) into a fourth-order differential equation, instead of the current order of two.Solution of the generalized equation is another future interest.
In this paper, we took the limit k x =0 along the tail.If k x is retained, the reduction of the problem would lead to another fourth-order differential equation.Some problems of interest to the substorm problem can only be studied with k x =0.For example, wave propagation speed in x cannot be calculated without k x ; the field-aligned currents vanish as well.A more general treatment with finite k x will be attempted elsewhere.
Some other potential future work includes the relaxation of the m e →0 limit to allow a unified treatment of the DKI and Hall mode and a kinetic study of the Hall mode based on the Vlasov equation.
In Fig. 1, the electric field was ascribed with a neutralsheet point sense.This was based on the expectation, often observed, that the drift balance condition should be broken with an electron bias (i.e., |U e /T e | >U i /T i ).However, there can be situations in which the neutral sheet is positively charged.For instance, ions can be trapped longer in the neutral sheet in some Speiser-type orbits, while electrons largely execute bounce motions.As we mentioned in respect to Eq. ( 30), the Hall mode can be destabilized with a negative sign of d 2 U/dz 2 , which corresponds to an electric field pointing away from the neutral sheet.In fact, Fig. 3, albeit of a different origin, has a second-order derivative of this sense.Hence, the Hall mode, in principle, can be excited in a positively charged neutral sheet as well.Finally we note that our result is invariant under the Galilean transformation, as adding a constant U 0 to U in our equations do not change dU/dz, and d 2 U/dz 2 .
The origin of the electric field that gives rise to d 2 U/dz 2 is ascribed to a separation of ion and electron current sheets.The parameters in Figs. 2 and 3 were chosen from THEMIS observations of current sheet disruption.At these values, the asymptotic ion gyroradius is about 14% of the half thickness of the current sheet.At this scale, the aforementioned drift balance can be broken in the electrons' favor, and a detailed calculation (which is out of the scope of this paper) shows that an electric field of a few mV/m can arise.
The result that a TCS can become unstable with a significant temperature gradient offers an interesting prediction for future verification: the stable TCS should have more or less uniform temperature profile.When, however, the TCS starts to interact with the colder embedding plasma sheet, strong temperature gradients may develop, leading to current disruption.
The role the Hall mode plays in the substorm dynamics holds significant interest.Observational evidence suggests that magnetospheric substorms originate in the transitional www.ann-geophys.net/27/1941/2009/Ann.Geophys., 27, 1941Geophys., 27, -1950Geophys., 27, , 2009Geophys., 27, 1948 W. W. Liu and J. Liang: Electrostatic field and current disruption region where the magnetic field geometry changes from dipole-like to tail-like (Lui and Burrows, 1978;Samson, 1992).In this region, magnetic field lines are not quite stretched into a thin current sheet configuration, and the condition is not quite ideal for the Hall mode.Instead, many researchers believe that the region is more susceptible to the destabilization of the ballooning mode.The destabilizing term in the ballooning mode is given by κ c dp/dx, where κ c is the curvature of radius of field line and dp/dx the pressure gradient in equatorial plane.This term is expected to maximize in the transition region.For example, the calculation of Cheng and Zaharia (2004) put the maximal growth rate of ballooning occurs at approximately 6 R E .It has been proposed in the literature that perturbations from ballooning propagate outward and destabilize the current sheet in a domino effect (e.g., Ohtani et al., 1992).In the substorm event of 13 March 2007 studied by Donovan et al. (2008), three THEMIS probes were spaced by a fraction of R E at ∼8.5 R E , allowing a direct estimate of the propagation speed at ∼100 km/s (or 1 R E /min), 1/10 of the fast mode speed.Tailward expansion of the ballooning mode at the slower propagation speed would take about 10 min to cover 10 R E , more consistent with the average duration of the expansive phase measured by geomagnetic indices and auroral activity.
An important future work is to study how the passage of the ballooning mode (i.e., rarefaction wave) interacts with the local current sheet, to give rise to the destabilizing electrostatic field.

Conclusions
A new mode of thin current sheet dynamics was investigated in this paper.The origin of the mode is closely connected to the Hall term in the generalized Ohm's law.For reasonable parameters representing the magnetospheric TCS, the mode has a growth timescale of 10-20 s, when the current sheet thickness and azimuthal wavelength approach 1000 km.It was verified, a posteriori, that the mode satisfies the condition of two-fluid treatment.Our analysis indicated that the Hall mode is distinct from and complementary to the driftkink instability.For the parameters chosen to reproduce the observed local current disruptions, the Hall mode appears to grow faster than the DKI, suggesting that it can be the dominant mechanism of local current disruption.Unlike the DKI, which grows in a classical Harris sheet, the Hall mode can be destabilized only when there is a significant departure from the Harris sheet, particularly a violation of the constant plasma drift across the thickness leading to a significant second-order derivative in z.This violation can be effected by either an electric field along z or a variation of the temperature across the current sheet.In the first scenario, the critical E z is estimated to be 4 mV/m.Observations have shown that this field indeed appears a few minutes prior to local current disruption.This electric field could be the effect of a rarefac-tion wave excited by instabilities such as ballooning occurring at an earthward location, which, in its wake, thins the current sheet to such an extent that charge separation starts to develop between the ion and electron current sheets.Details of this proposed linkage were not part of this study and requires future attention.

Fig. 1 .
Fig. 1.Diagram showing the development of instability.G (ω) is the function given in Eq. (28).The figures shows the change of the solution to G (ω) =C.When C is sufficiently large (dashed line), there is a pair of complex solution, one of which is unstable.

Fig. 2 .
Fig. 2. Numerical solution of the unstable eigenmode for the case given in Eq. (37) and parameters given the subsequent text.Plot (a) shows the variation of the background quantities of the model current sheet.Plot (b) gives the real (solid) and imaginary (dotted) parts of the eigenvalue as a function of azimuthal wavelength.