The double-gradient magnetic instability: Stabilizing ...

PHYSICS OF PLASMAS 22, 012904 (2015)

The double-gradient magnetic instability: Stabilizing effect of the guide field D. B. Korovinskiy,1,a) A. V. Divin,1,2 N. V. Erkaev,3,4 V. S. Semenov,1 A. V. Artemyev,5 V. V. Ivanova,1 I. B. Ivanov,1,6 G. Lapenta,7 S. Markidis,8 and H. K. Biernat9,10 1

Saint Petersburg State University, 198504, Ulyanovskaya 1, Petrodvoretz, Russia Swedish Institute of Space Physics, SE-751 21 Uppsala, Sweden 3 Institute of Computational Modelling, Russian Academy of Sciences, Siberian Branch, 660036 Krasnoyarsk, Russia 4 Siberian Federal University, 660041 Krasnoyarsk, Russia 5 Space Research Institute RAS, Profsoyuznaya 84/32, Moscow 117997, Russia 6 Theoretical Physics Division, Petersburg Nuclear Physics Institute, 188300 Gatchina, Russia 7 Centrum voor Plasma-Astrofysica, Departement Wiskunde, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium 8 PDC Center for High Performance Computing, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden 9 Space Research Institute, Austrian Academy of Sciences, 8042 Graz, Austria 10 Institute of Physics, University of Graz, 8010 Graz, Austria 2

(Received 1 November 2014; accepted 28 December 2014; published online 13 January 2015) The role of the dawn-dusk magnetic field component in stabilizing of the magnetotail flapping oscillations is investigated in the double-gradient model framework (Erkaev et al., Phys. Rev. Lett. 99, 235003 (2007)), extended for the magnetotail-like configurations with non-zero guide field By. Contribution of the guide field is examined both analytically and by means of linearized 2-dimensional (2D) and non-linear 3-dimensional (3D) MHD modeling. All three approaches demonstrate the same properties of the instability: stabilization of current sheet oscillations for short wavelength modes, appearing of the typical (fastest growing) wavelength kpeak of the order of the current sheet width, decrease of the peak growth rate with increasing By value, and total decay of the mode for By 0:5 in the lobe magnetic field units. Analytical solution and 2D numerical simulations claim also the shift of kpeak toward the longer wavelengths with increasing guide field. This result is barely visible in 3D simulations. It may be accounted for the specific background magnetic configuration, the pattern of tail-like equilibrium provided by approximated solution of the conventional Grad-Shafranov equation. The configuration demonstrates drastically changing radius of curvature of magnetic field lines, Rc. This, in turn, favors the “double-gradient” mode (k > Rc) in one part of the sheet and classical “ballooning” instability (k < Rc) in another part, which may C 2015 AIP Publishing LLC. result in generation of a “combined” unstable mode. V [http://dx.doi.org/10.1063/1.4905706]

I. INTRODUCTION

Flapping oscillations are large-scale low-frequency quasiperiodic oscillations of magnetotail current sheets, registered in numerous in-situ observations both in the Earth magnetotail2–11 and in the Saturn and Jupiter magnetospheres.12,13 Typically (for the Earth magnetotail conditions), flapping waves are observed in the central part of the tail at the distances of 10–30 Earth radii (Re) downtail, propagating toward the flanks with the speed of tens km/s (10 times less than Alfven velocity), amplitude and wavelength of several Re and quasiperiod of several minutes. Such motions are often observed during the substorm growth phase several minutes before the onset,10,14,15 which may imply some interrelation of these phenomena. Thus, these motions as well as ballooning/interchange (BICI) ones and magnetic reconnection that they often accompany16,17 are associated with the debated problem of the substorm triggering, where candidates are either the interplanetary magnetic a)

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field variations or some internal magnetospheric instability.18 The latter mechanism is argued to be more prominent substorm trigger in several recent works.19–21 In any case, flapping oscillations of current sheet contribute significantly to low-frequency part of the spectrum of magnetic field fluctuations in the magnetotail.22,23 Due to widespread of these oscillations, their theoretical investigations represent an important problem of plasma geophysics. Numerous analytical treatments of the flapping oscillations may be split roughly in two groups. First group considers kinetic drift instabilities, which may produce current sheet motions of such type. Among them are the drift-kink instability (DKI),24,25 which may develop due to the relative drift of ions and electrons, and the analogous ion-ion kink mode developing due to the shear into the ion velocity profile providing for the relative streaming between the cold lobe ions and the current-carrying hot plasma sheet ions.26–29 A similar approach was considered in Ref. 30, where background ions velocity shear introduces Kelvin-Helmholtz instability resulting in current sheet flapping motions. The kinetic approach is an evident advantage of the mentioned

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models, and some recent observations31–34 confirmed the presence of low-frequency, azimuthally periodic kinetic perturbations in the magnetotail. In particular, such approach is needed for accurate description of the thin (embedded) current sheets,35–37 where finite Larmor radius of ions or electrons and their decoupling should be considered. Though, in other cases, when the typical size of the system exceeds significantly the ion Larmor radius (and skin depth), MHD approximation can be used to investigate the large-scale plasma processes. Furthermore, drift modes support currentaligned propagation of the oscillations only, while observations confirm the bidirectional propagation to be more abundant,6,7 which means that another physical mechanisms of the flapping waves generation could be also involved. Another group of models considers the MHD ballooning mode,38 which may be characterized as “configurational” instability.39 This mode is related to the buoyancy effect, which may arise in the magnetotail due to the strong magnetic tension of the curved magnetic field lines. It is worth mentioning that plasma instabilities supported by magnetic tension against some force field are generally categorized as Rayleigh-Taylor instability (in many instances, it is also called the flute instability or interchange instability).40 It was shown that the instability of such type may develop when amount of the magnetic flux accumulated in the tail provides for the configuration with earthward growing entropy.41,42 In this regard, magnetic configurations with a local minimum of the normal magnetic field component (Bz) in radial direction (x) are of special attention.17,43 Such configurations with Bz below 1–2 nT are commonly found in THEMIS observations44 prior to substorm onset near midnight at distances 11 Re. In later cluster observations,45 tailward growing Bz component was not observed to 14 Re, while in the midtail gradient @Bz =@x was found to be fluctuating around zero with time scales of 5–15 min, making tailward direction of @Bz =@x ubiquitous. It was shown that configurations with a minimum of Bz(x) favor a number of plasma instabilities, such as lower hybrid drift (LHD) instability,46 kinetic BICI mode,43,47,48 tearing49–51 instability, and MHD ballooning mode. The latter was studied intensively in application to the near-Earth magnetic configurations in a number of works (see, e.g., Refs. 52–55, and references therein) and corresponding interpretation of the magnetotail flapping motions was developed in Ref. 56. The analytical treatment of ballooning mode utilizes a system of coupled equations for poloidal Alfven and slow magnetosonic waves propagating in curvilinear magnetic configurations.55 It was shown that in stretched current sheets, the instability is favored by the sharp earthward pressure gradient, and it may develop also at the boundary between the hot plasma sheet and cold lobe plasma.57 Thus, the wavelength band of the ballooning mode is rather wide, spreading from a few tenths of Re up to Rc, where Rc is the magnetic field line curvature radius (e.g., in Ref. 57, Rc was estimated of 20 Re). It corresponds well to the satellite data, where both long- and short-wave kink-like perturbations of the magnetotail current sheet are detected.58 The double-gradient model1 is less general, because it considers specific wavelength band Rc < k < L, where L is

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the typical system size (length). Here, Rc is assumed to be of the order of current sheet halfwidth, D, or less. These assumptions and, consequently, the model itself break down in the near-Earth region (see Fig. 1); however, the model is more relevant for the flapping oscillations in the midtail. Furthermore, careful analysis59 of the midtail conditions showed that its magnetic configuration may be often characterized by two small parameters, ¼ D=L 1 and ¼ Bz max =Bx max 1, where . Stability of such current sheets was studied in MHD approximation in Ref. 1. It was found out that the midtail conditions are favorable for perturbations of two kinds depending on the sign of the magnetic gradient @Bz =@x. The first is a transversely propagating wave (for earthward growing Bz), whose parameters (frequency, group velocity) are comparable to flapping oscillations.1,60 The second corresponds to instability (for tailward growing Bz), which was called “flapping” or “double-gradient” instability. The latter term is accounted for the product of two magnetic gradients appeared in derived expression for the typical frequency (growth rate) xf s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 @Bx @Bz ; (1) xf ¼ 4pq @z @x z¼0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi kD ; (2) x ¼ xf kD þ 1 where B and q denote the magnetic field and the mass density, respectively, Eq. (2) is the dispersion relation, and k ¼ ð0; k; 0Þ is the wave vector. Hereinafter, we adopt the reference system, where origin is placed in the center of the Earth, x axis points tailward, y axis is dawn-directed, and z axis completes the righthanded reference system (see Fig. 1). It is seen that the mode, described by Eqs. (1) and (2), being intrinsically the branch of the classical ballooning mode (see Sec. II in Ref. 61 and Sec. V in current paper), occupies, however, its separate niche (wavelength range, specific magnetic configuration), where the term “double-gradient” seems to be more descriptive. The dispersion relation (2) covers both wave propagation and wave growing regimes. It suits the Earth

FIG. 1. Sketch of magnetotail magnetic configuration, where two different types are marked by numbers. (1) Dipole-like configuration revealing strong normal (vertical) magnetic field component, Bz, and large radius of curvature of magnetic field lines, Rc L, where L is the system typical size. (2) Midtail-like configuration with weak Bz and small Rc D L, where D and L are the system typical sizes in vertical and horizontal directions, respectively. The value of Rc is estimated in equatorial plane. The system of coordinates is GSM-like (here, GSM stands for “Geocentric Solar Magnetospheric”) rotated 180 around the z axis.


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magnetotail conditions, where the quantity @Bx =@z is of fixed sign, while the sign of @Bz =@x may change. In particular, the double-gradient instability developing in the current sheet with tailward growing Bz may provide an initial perturbation for the sheet, switching to the propagating flapping wave when @Bz =@x changes its sign.62 Although such mechanism was not confirmed yet directly, in several recent observational works,63,64 the double-gradient model was found to be the most relevant to observational data. This model allows for two branches of solution, namely, faster-growing kink-type mode and slower sausage mode motions of the current sheet. Perturbations of both types were detected in magnetotail,32,65 though kink-type perturbations are more abundant. This kink branch is in the focus of our study. We emphasize here that the term “kink” brings geometrical sense only, having nothing common with the socalled “(driven) kink mode” driven by the parallel current.66 While generation of the double-gradient mode was studied extensively in some recent works,59–61,67–69 the mode stabilization mechanism was not addressed so closely. Meanwhile, whatever model of the flapping oscillations we choose, we face the same problem: regions with tailward growing Bz and/or ion/electron flows are present nearly always, which should make the current sheet unstable. However, quiet current sheets are not uncommon in observations. It means that understanding the stabilization mechanisms is of importance and they should be studied in a great detail. In particular, in addition to the earthward growing Bz component, the double-gradient mode can be stabilized by reducing of the magnetic field line curvature: the larger is Rc, the lower is the instability growth rate.61 In thin current sheets, the Hall effect also may hinder and even damp totally the mode in the direction opposite to the current velocity.60 A mechanism of stabilization by non-zero guide (out-ofplane) magnetic field was predicted in Artemyev and Zimovets,70 where stability of stretched current sheets was studied in MHD approximation in application to the Solar corona conditions. While two limiting cases of very weak and very strong (as compared to the normal magnetic field component) guide field were investigated, the intermediate case, which is more relevant to the Earth magnetotail conditions, was not. The case of spatial non-uniform distribution of the mass density was not considered either. Thus, on one hand, this stabilizing mechanism seems to be rather natural for the magnetotail conditions. On the other hand, analytical predictions of Ref. 70 are hardly applicable to the Earth magnetotail, and on the third part, analytical results obtained by using quasi-one-dimensional (1D) model can be applied to inherently 3-dimensional (3D) process with caution. That is, reliability of investigated analytical model is questionable and should be verified. In Secs. II–V, we present the careful analytical and numerical study of the guide field contribution in the development of the kink double-gradient magnetic instability. In Sec. II, we derive the equation for Vz velocity perturbation by using linearized MHD equations for plain transversally propagating waves in a current sheet with non-zero guide field and spatial non-uniform number density. This equation allows for obtaining the instability dispersion relation in

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Sec. III, where we compare the analytical solution and numerical linearized 2-dimensional (2D) MHD simulations. In Sec. IV, we present the results of fully 3D MHD simulations with the same background configuration. In Sec. V, we summarize and discuss the results. II. ANALYTICAL TREATMENT

Following Erkaev et al.68 and adopting their system of notations, we consider elongated magnetotail-like magnetic configuration with a uniform out-of-plane magnetic component and current sheet scaled by the halfwidth D in normal (z) direction and by length Lx, assuming ¼ D=Lx 1. We assume also ¼ Bz max =Bx max to be the second small parameter (relevance of such parametrization is proved in Ref. 59). We start with incompressible ideal MHD equations71 @V 1 þ ðV rÞV þ rP ¼ ðB rÞB; (3) q @t 4p @B þ ðV rÞB ¼ ðB rÞV; @t

(4)

ðr BÞ ¼ 0;

(5)

ðr VÞ ¼ 0:

(6)

Here, V and B are the plasma bulk velocity and magnetic field, respectively, and P ¼ p þ B2 =ð8pÞ is the total pressure, while p is the plasma pressure. In the following, we utilize dimensionless variables, which are introduced as follows: ~ B ¼ B B;

~; q¼q q

~ V ¼ VA V;

P ¼ ðq

r ¼ D~r ;

~ VA2 ÞP;

t ¼ ðD=VA Þ~t ;

(7)

where B* and q* are some fixed values, say, B ¼ maxðBx Þ pffiffiffiffiffiffiffiffiffiffi ðx; zÞ and q ¼ maxðqðx; 0ÞÞ, and VA ¼ B = 4pq . The reasonable values for the substorm growth phase could be adopted as Bx ¼ 20 nT; n ¼ 0:2 cm3 ; Ti ¼ 4Te ¼ 4keV, and D ¼ 1Re. Under these values, we have B ¼ 20 nT, q ¼ 0:33 1024 g cm3 ; VA ¼ 103 km=s, q VA2 ¼ 0:33 nPa, and D=VA ¼ 6:4 s. By taking additionally Bz ¼2 nT and Lx ¼5 Re, and estimate @Bx =@z B =D and @Bz =@x Bz =Lx , we obtain the typical frequency estimate by Eq. (1), xf 0:02s1 . Looking for a linear solution for small perturbations propagating in y direction in equilibrium plasma, we represent all quantities as a sum of the initial value and a small perturbation: B ¼ ðBx þ bx ; By þ by ; Bz þ bz Þ, V ¼ ðvx ; vy ; vz Þ; q ¼ q0 þ q1 ; P ¼ P0 þ p (hereinafter, tildes are omitted). The background magnetic field is specified as B0 ¼ ðBx ðzÞ; By ; Bz ðxÞÞ. Substituting the Fourier harmonics exp ðiky ixtÞ for perturbations and supposing perturbations to be independent of the x coordinate, and neglecting all terms of the order of 2 and 2, the linearized system of Eqs. (3)–(6) takes the form (compare to Eqs. (11)–(15) in Ref. 68) ixq0 vx ¼ Bz

@bx @Bx þ bz þ ikBy bx ; @z @z

ixq0 vy ¼ ikp þ ikBy by ;

(8) (9)


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ixq0 vz ¼

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@p @Bz þ ikBy bz ; þ bx @x @z

(10)

ixbx ¼ ikBy vx vz

@Bx ; @z

(11)

ixby ¼ ikBy vy þ Bz

@vy ; @z

(12)

@vz @Bz vx ; @z @x

(13)

ixbz ¼ ikBy vz þ Bz ikby þ

@bz ¼ 0; @z

(14)

ikvy þ

@vz ¼ 0: @z

(15)

Expressing vy from Eq. (15), by from Eq. (12), bx from Eq. (11), and bz from Eq. (13), we derive the equation for vz 1 d dvz U d3 vz 2 2 dvz ~ þ k vz 2 1 ¼ iv ; (16) q k ~ dz dz dz3 dz q x ~ , function U, where we introduce the effective mass density q and function v as follows: ~ ðx; z; By ; k; xÞ ¼ q0 ðx; zÞ ðkBy =xÞ2 ; q U ðx; z; By ; k; xÞ ¼

(17)

1 @Bx @Bz ; ~ @z @x q

(18)

kBy Bz : ~ x2 q

(19)

vðx; z; By ; Bz ; k; xÞ ¼

For q0 ¼ 1, Eq. (16) turns to Eq. (7) of Artemyev and ~ ¼ q0 , the righthand side Zimovets.70 For By ¼ 0, we have q of Eq. (16) turns to zero and equation itself turns to Eq. (16) in Ref. 68. III. THE DISPERSION CURVE

While the general solution of Eq. (16) may be achieved only numerically, qualitatively the effect of the guide field may be demonstrated as follows. Let us consider the simplest magnetic configuration studied by Erkaev et al.,1 where q0 ¼ 1, and derivatives @Bz =@x and @Bx =@z are constants. As far as the magnetic component Bz has the form Bz ¼ a bx, where a and b are small positive values, this component drops to zero and stays arbitrarily small in the vicinity of a point x* ¼ a=b, so that the righthand part of Eq. (16) tends to zero and may be neglected in this region. Then, our equation turns immediately into Eq. (11) of Ref. 1 and solution (19, Ref. 1) takes the form vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u x2f 2 ; xk;s ¼ kt þ VAy (20) k2 þ k2k;s 2 ¼ B2y =q0 is the where xf is defined by Eq. (1), while VAy squared Alfven velocity in magnetic field By (the factor 4p disappears in dimensionless units); subscripts k and s mark the solutions for kink and sausage modes, respectively, and kk;s ¼ kk;s ðkÞ are the solutions of the corresponding dispersion

equations [see Ref. 1]. Solution (22) in Ref. 68 is the subject 2 of the same modification, x2 ¼ x2f k=ðk þ 1Þ þ k2 VAy . Since the frequency is complex for unstable configurations, x2f < 0, the absolute value of the subradical expression in Eq. (20) reduces with growing k, and the dispersion curve =½x ðkÞ takes the bell-like shape. Such behavior of the dispersion curve also follows from the fact that the typical growth rate is specified by a function U (18), in which the denominator grows with k according to Eq. (17). The contribution of the righthand side of Eq. (16) may be estimated analytically by using the approach of Ref. 1. Considering U to be a piecewise continuous function independent of the z coordinate Uðx; By ; k; xÞ; jzj 1 (21) U¼ 0; jzj > 1; Eq. (16) for vz in the regions jzj > 1 takes the form d2 d d2 2 2 k vz ¼ iv k vz : dz2 dz dz2

(22)

Here, q0 and v specified by Eq. (19) are assumed to be independent of the z coordinate, while the condition U ¼ 0 means that magnetic components Bx or Bz or both are assumed to be constant. This equation can be solved by making a substitution qðx; z; By ; k; xÞ ¼ d 2 vz =dz2 k2 vz , for which Eq. (22) takes the form q ¼ ivdq=dz. Then, the solution for vz for jzj > 1 reads v2 i ð Þ z71 ; exp vz ¼ C1 exp½kðjzj 1Þ þ C2 1 þ k2 v2 v (23) where the rightmost term sign is negative in the upper semiplane and positive in the lower one. Comparing this expression with the solution (13) in Ref. 1, we see that the real part of the solution acquires an additional oscillating term; also the imaginary term is introduced. However, the assumption of U ¼ 0 for jzj > 1 means that the configuration is antiparallel (Bx ¼ const, Bz ¼ 0), the condition which turns the second term of solution (23) to zero as v Bz . Thus, the solution of Erkaev et al.1 does not change. It allows us to accept the assumption of Artemyev and Zimovets,70 who supposed that in general case the righthand part of Eq. (16) contributes mostly to the real part of the unstable mode frequency. Meanwhile, in the current paper, we focus on the influence of the guide field By on the instability growth rate, i.e., imaginary part of frequency. For this reason, we solve Eq. 6 0, this (16) numerically omitting the righthand part (for By ¼ approach should be the more accurate the less is ratio Bz=By). We consider a kink mode with the boundary condition dvz =dzð0Þ ¼ 0 and exponential decay vz expðkjzjÞ for z ! 1. Equation (16) is solved as the initial value problem (vz ðz Þ ¼ expðkðz 1ÞÞ; dvz =dzðz Þ ¼ k expðkðz 1ÞÞ for z 1) for a given wave number k and various x. Thus, the solution having dvz=dz(0) ¼ 0 corresponds to the correct value of x (see details of this approach in Ref. 72). Having obtained the dispersion curves for several fixed values of By


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from the boundary condition dvz=dz(0) ¼ 0, we compare the solution with dispersion curves computed by means of 2D linearized MHD simulations. 2D simulations are performed with the technique and background configuration described in Ref. 61. Namely, we use Lax-Friedrichs explicit finite difference method73 and the Richardson-like extrapolation74 to obtain the secondorder accuracy solution of the system of linearized compressible ideal MHD equations for perturbations dU @ ðdUÞ @Fx @Fz þ ¼ S; þ @x @z @t

(24)

where U ¼ fq; qV; B; Eg is the vector of plasma parameters, and E is the total energy density (the polytropic index value is 5/3). Expressions for the flux densities Fx;z and for the source function S are given in Ref. 69. The initial configuration is specified by the approximate solution43 of the Grad-Shafranov equation for the magnetic potential A ¼ ð0; A0y ; 0Þ of the in-plane magnetic field A0y ¼ ln

cosh½Fð xÞz

; F ð xÞ

(25)

1 p ¼ q ¼ ð1 þ expð2A0y ÞÞ; 2

(26)

where function F has a special form, providing for the region with a tailward growing Bz component 8 2 > < exp½e0 ðx x1 Þ þ e1 ðx x1 Þ =ð2x1 Þ ; FðxÞ ¼ exp½e0 ðx x1 Þ ; > : exp½e0 ðx x1 Þ e2 ðx x2 Þ2 =ð2ðx3 x2 ÞÞ ; (27) where the three expressions in the righthand part of Eq. (27) correspond to intervals x0 x x1 ; x1 x x2 , and x2 x x3 , respectively. Here, we used values x0 ¼ 0, x1 ¼ 1, x2 ¼ 3, x3 ¼ 15; e0 ¼ 0.1, e1 ¼ 0.11, and e2 ¼ 0.4. Thus, the only difference with configuration studied in Ref. 61 is the reduced parameter x3 (15 vs. 23), which simply increases the magnetic field gradient @Bz =@x. Configuration of the in-plane magnetic field lines is shown in Fig. 2, and profiles q0 ðxÞ; Bz ðxÞ and the typical growth rate cf ðxÞ ¼ =½xf ðxÞ are plotted in Fig. 3. For numerical simulations, we choose the segment of the sheet 4 x 8, where @Bz =@x is just tailward and the curvature

FIG. 3. Background magnetic configuration: profiles along the current sheet center z ¼ 0. 0.5q0 (dashed-dotted), Bz (dashed), and typical growth rate cf ¼ =½xf (solid) calculated by Eq. (1) for normalized quantities (without the factor 4p).

of the magnetic field lines stays moderate (see Fig. 13). Solving the system (24) with a seed perturbation vz jt¼0 ¼ exp ðz2 Þ for different values of wave number k, we obtain the dispersion curves cðkÞ ¼ =½xðkÞ for several fixed values of the guide field By. In Fig. 4, these curves are compared to the numerical solutions of Eq. (16) without the righthand part, where the mass density q0(x) is approximated by the expression 0:5½1 þ g2 =cosh2 ðgzÞ and g is the F(x) average value, hFix ¼ 0:5. It is seen that the curve peaks and the peak values of c obtained analytically and numerically are in good agreement; however, at the rightmost flanks of the curves deviations become considerable. In Fig. 5, the dependence of the normalized peak growth rate on By is shown accompanied with the fitting function f ¼ p=ðBy þ qÞ. The constants p ¼ 0.3121 and q ¼ 0.3152 demonstrate the surprising proximity, so that we can suggest the approximation cmax =c0 1=ð1 þ pBy Þ, which is valid for studied range of the guide field values, By 0:3. Here, c0 ¼ supðcf ÞjBy ¼0 0:17. Fig. 6 shows the highest marginally unstable wave numbers kmax(By) and peak growth rate wave numbers, kpeak(By). This figure indicates that the larger is the guide field value, the larger is the shift of kpeak and kmax toward the origin (longer wavelengths). The numerical scheme limitations did not allow us to calculate the dispersion curves for By > 0.3, but

4 2

z

0 −2 −4 0

5

x

10

15

FIG. 2. Background magnetic configuration: in-plane magnetic field lines. Thin vertical lines mark the linearized MHD simulation box.

FIG. 4. Dispersion curves obtained by linearized MHD numerical simulations (solid) and by numerical solution of simplified Eq. (16) (dashed) are shown by different colors for different values of By: By ¼ 0 (red), By ¼ 0.05 (green), By ¼ 0.1 (black), By ¼ 0.15 (blue), and By ¼ 0.2 (orange).


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FIG. 5. Linearized MHD simulations: the normalized peak growth rate values cmax =c0 for different values of guide field are shown by diamond markers; solid curve plots the fitting function f ðBy Þ ¼ p=ðBy þ qÞ. Here, p ¼ 0.3121, q ¼ 0.3152, while c0 ¼ 0:17. The approximation f~ ¼ 1=ð1 þ pBy Þ is shown by dashed curve.

extrapolation of the curves kmax(By) and kpeak(By) suggests that these two curves should intersect at By 0.5, which means that the double-gradient mode should decay totally for the guide field values of such order. IV. 3D SIMULATIONS

Results of the fully 3D MHD simulations of the flapping instability are presented in this section. We use MHD version of the code iPIC3D,75 which solves compressible MHD equations by means of the Total Variance Diminishing LaxFriedrichs (TVDLF) scheme of second order.76,77 The normalization is the same as in Secs. II–III of the paper. The simulations share much in common with our previous study,61 hence we provide a brief overview of the numerical techniques next. Simulations are initialized with the magnetic configuration (Eqs. (25)–(27)) used in Sec. III. In order to ensure numerical force balance at t ¼ 0 and reduce the numerical effects, the relaxation method78 is applied to the initial configuration. The method solves friction MHD equations prior to the main 3D simulation, minimizing the

FIG. 6. Linearized MHD simulations: the highest marginally unstable wave numbers kmax ðBy Þ are shown by solid curve, and peak growth rate wave numbers kpeak ðBy Þ are plotted by dashed curve.

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residual numerical force density j rp þ j Bj inside the computational domain. The computational domain size is Lx Ly Lz ¼ 15 22:5 11:25, and the grid resolution is Dx ¼ Dy ¼ Dz ¼ 0.0391, amounting for Nx Ny Nz ¼ 384 576 288 grid points. Grid coordinates vary in interval from 0 to Lj (j ¼ fx; y; zg), so that the current sheet center is localized in the plane z ¼ Lz=2. Four runs R1–R4 with the different guide field values By jt¼0 ¼ f0; 0:05; 0:1; 0:2g were performed. Notably, addition of a guide field to the configuration ((25)–(27)) does not modify the force balance, because the current flows initially along the y direction only. The initial configuration is invariant along the y dimension. A small Vz velocity perturbation is added to start the instability development. Keeping in mind that during the linear stage the different oscillative modes do not interact, we take the following expression for the Vz(x,y,z) velocity at t ¼ 0: 60 h i ð2z Lz Þ2 X sin ky ðmÞy þ m1:5 ; Vz0 ¼ f ðxÞexp 2 m¼1 " # 4 5 10 Lx 3Lx f ð xÞ ¼ tanh x tanh x ; 2 4 4 (28) where ky ðmÞ ¼ 2pm=Ly . The initial condition (28) represents a superposition of 60 modes 0.28 < ky < 16.8, each having an amplitude of 5 104 and being localized in x (Fig. 7(a)) and z (Fig. 7(b)) directions. The term m1.5 introduces a pseudo-random phase shift. Numerical simulations were performed up to 200 dimensionless time units. 2D slices of Vz velocity in the equatorial plane are presented in Fig. 7 (left panels) for run 1. The perturbation amplitude is uniform within 5 < x < 10 at t ¼ 0 (Fig. 7(a)) and peaks at maxðVz Þ 0:005. By the time t ¼ 40.14, the peak velocity reaches maxðVz Þ 0:01 (Fig. 7(c)) and at t ¼ 79.12 the Vz amplitude exceeds 0.05 (Fig. 7(e)). The time t ¼ 79.12 corresponds to late linear stage, and finally at t ¼ 117.1 (Fig. 7(g)) the separate waves start to interact, producing an irregular pattern of Vz (y > 17). Right panels of Fig. 7 depict 2D slices of Vz in the plane x ¼ Lx=2. The initial perturbation is localized near the equatorial plane z ¼ Lz=2 (Fig. 7(b)). During the linear stage (Figs. 7(d) and 7(f)), the instability spatial profile expands to 3 < z < 7.5. As the time progresses, the mode starts to shift whole flux tubes along the z direction, which is indicated by an extended spatial profile in Fig. 7(h). During the linear stage (t ¼ 40.14, t ¼ 79.12), the instability spatial profile demonstrates a gap in the interval 10 < y < 17, which we attribute to the specific choice of the phase term m1.5 in Eq. (28) and the fact that longer wavelengths possess smaller growth rates. Indeed, auxiliary simulations with the phase term m3 show more uniform distribution of Vz(y) plotted in Fig. 8. The particular choice of the phase function does not affect the instability growth rate, as it is seen in Fig. 12. Fig. 9 shows Vz distribution in the z ¼ Lz=2 plane for runs R1–R4. For each run, we select a time at which the


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FIG. 7. Slices of the Vz velocity in run 1 (guide field By ¼ 0) as viewed from 3D MHD simulations: z ¼ Lz=2 (left panels) and x ¼ Lx=2 (right panels). Time t ¼ 0.0 (a) and (b). Time t ¼ 40.14 (c) and (d). Time t ¼ 79.12 (e) and (f). Time t ¼ 117.1 (g) and (h).

peak Vz velocity in the equatorial plane is close to 0.01: t ¼ 40.14 (R1), t ¼ 53.23 (R2), t ¼ 104.4 (R3), and t ¼ 200.3 (R4). Obviously, the guide field impedes the growth of the instability. At the first glance, Vz structures are rather similar in runs R1, R2, and R3 during the linear stage. However, we notice that stabilizing effect of the guide field is stronger for the short wavelengths, which fade gradually with increasing By and disappear completely for By ¼ 0.2. And then, increase

of guide field forces some drift of the instability spatial localization in x direction. For By ¼ 0, unstable region is located around x 6, while for By ¼ 0.2 the instability develops a narrow region at x 11. For less values of By, this drift is less pronounced though can be observed also. Thus, for larger guide field values, the instability shifts to the large-Bz part of the current sheet (see Fig. 2), supporting thuswise a weak guide field configuration (i.e., a combination of small

FIG. 8. Slices of the Vz velocity (By ¼ 0) in 3D simulations with the phase term m3 in Eq. (28). z ¼ Lz=2 (left panels) and x ¼ Lx=2 (right panels). Time t ¼ 0.0 (a) and (b). Time t ¼ 38.84 (c) and (d). Time t ¼ 77.61 (e) and (f). Time t ¼ 114.3 (g) and (h).


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FIG. 9. Slices of the Vz velocity. (a) Run 1, guide field By ¼ 0, time t ¼ 40.14. (b) Run 2, By ¼ 0.05, time t ¼ 53.23. (c) Run 3, By ¼ 0.1, time t ¼ 104.4. (d) Run 4, By ¼ 0.2, time t ¼ 200.3.

By and large Bz). To keep the consistency of our synopsis, we place the more detail discussion of this effect in Sec. V. ~ we assume In order to estimate the growth rates cðkÞ, that the wave vectors are oriented along the y direction. Fast Fourier Transform (FFT) analysis is applied to Vz along the y direction. The intensities I(ky,t) are computed as I ðk y ; t Þ ¼

xmax X Lz 1 X jV~ð x; ky ; z; tÞjDxDz: Lx Lz xmin z¼0

(29)

Here, Lx ¼ xmax xmin , and V~z are the amplitudes of Fourier components of Vz. For the runs R1, R2, and R3 we take xmin ¼ 5, xmax ¼ 12, and for run 4 the limits are [8, 12]. The time evolution of the intensities I(ky,t) is shown in Fig. 10 for run R1 and in Fig. 11 for R4. For R1, it is seen that all modes experience the growth with time in the interval 40 < t < 90. The modes m 4 grow almost exponentially, whereas the long wavelength modes 1 m 3

display a deep fall in intensity and subsequent unsteady behavior. The time evolution of I(ky,t) is rather different in R4, as it is seen in Fig. 11. Long wavelengths (m 5; ky 1:4) and short wavelengths (m 19; ky 5:3) do not evolve clearly, in a qualitative agreement with 2D results shown in Fig. 4. A clear exponential growth is found for modes 5 < m < 19. It continues by the end of the simulation at t 200, so that saturation is not reached. Next, we estimate the growth rates by applying the linear fit to lnIðky ; tÞ during the linear growth interval (i.e., 40 < t < 90 for R1 and 100 < t < 200 for R4). Growth rates computed for the runs R1–R4 are summarized in Fig. 12. The central parts of the dispersion curves (1 < ky < 6) demonstrate the bell-shaped form of c(ky) peaking at kpeak 3, the value of kpeak decreases slightly with growing By. Such behavior agrees qualitatively with the linear theory (Fig. 4). The peak c 0:06 is found for the no guide field case, and much smaller values c 0:02 are typical for the run R4. The guide field By ¼ 0.2 damps the short wavelengths (ky > 5.3) completely. The evaluated growth rate value at the flanks (small ky’s, large ky’s) is affected by the numerical scheme limitations. The highest and the lowest resolved wave numbers are determined by the grid resolution Dx and the box size Ly, respectively. Short wavelength modes are damped numerically, whereas the long wavelength modes can be severely influenced by, e.g., boundary conditions. That is, reliable growth rate estimates are possible for the wavelengths, which satisfy: 0:0391 ky < 22:5, or 0:25 < ky 20. Hence, the estimates of cðky Þ for large ky are limited by the grid resolution. Therefore, one should not be surprised with the growth rate decreasing with growing ky, observed in the rightmost part of the dispersion curve for the case By ¼ 0, while according to the linear theory, c ! const as ky ! 1 (see Fig. 4). We interpret that as an effect of the numerical dissipation, which depends on the grid resolution and will be considered in a separate study. V. DISCUSSION AND CONCLUSIONS

In this paper, we have investigated the natural mechanism for the stabilization of magnetotail current sheet

FIG. 10. Time evolution of mode intensities I(ky) (ky ¼ 2pm=Ly ) for the modes (a) m ¼ {1, 2, …, 10} and (b) m ¼ {11, 13, …, 29} in run 1 (guide field By ¼ 0) as viewed from 3D MHD simulations.


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FIG. 11. Time evolution of mode intensities I(ky) (ky ¼ 2pm=Ly ) for the modes (a) m ¼ {1, 2, …, 10} and (b) m ¼ f11; 13; :::; 29g in run 4 (guide field By ¼ 0.2) as viewed from 3D MHD simulations.

relative to the kink oscillations by the non-zero dawn-dusk magnetic field component. Though qualitatively this mechanism was predicted in Ref. 70, it was not studied in application to the conditions of the Earth magnetotail. In particular, the non-uniform spatial distribution of the mass density, which is essential for the magnetotail modeling, was not considered in Ref. 70. In the current paper, we generalized analytical treatment for this case and obtained the analytical dispersion curve of the double-gradient mode by solving the one-dimensional equation for perturbation of Vz velocity. Comparison with results of 2D numerical linearized MHD simulations showed that averaging of inhomogeneities along the current sheet (in x-direction) yields analytical estimations matching numerical results (see Fig. 4). In 2D simulations, the double-gradient instability was studied in favorable magnetotail-like configuration, i.e., configuration with tailward growing Bz component, small magnetic field line curvature radius (0:3 < Rc < 3), and high b parameter value (15 < b < 90) shown in Fig. 13. It is found out that non-zero guide field modifies the dispersion curve of instability introducing the fastest growing wavelength kpeak,

FIG. 12. 3D MHD simulations: dispersion curves for different guide field runs: By ¼ 0 (black), By ¼ 0.05 (red), By ¼ 0.1 (green), and By ¼ 0:2 (blue). Error bars correspond to the error of the linear regression between lnIðky ; tÞ and the time t. Results of simulations with the phase term m1:5 in Eq. (28) are shown by thin markers. Dispersion curves obtained in simulations for By ¼ 0 with the phase terms m2:25 and m3 are shown by thick black and grey markers, respectively.

matching with the current sheet thickness and increasing in stronger By fields, while the extremal increment value cpeak decreases drastically. The fastest growing perturbation wave number, kpeak, reduces with increasing By and amounts the value of 0.5 for By 0:2. Taking into consideration that real flapping oscillations in the magnetotail are not pure periodical, we can just estimate the order of magnitude of corresponding wavelength kpeak 1=kpeak 2. For higher values of By 0:3, the value of kpeak decreases by 30%, while for smaller values of By 0.1 the value of kpeak doubles. Thus, for moderate values of By 0:260:1 (in lobe magnetic field units), the typical wavelength is found to be of the order of the current sheet width, kpeak 2D. Extrapolating the curves kpeak ðBy Þ and kmax ðBy Þ plotted in Fig. 6, we conclude that for By 0:5 the mode should be stabilized completely (become hardly observable at least). This result is important for interpretation of various spacecraft observations of magnetotail flapping motion accompanied by strong currents jz.5–11,79 Previous spacecraft observations11,80 suggest that magnetotail oscillations and formation of strong jz correspond to small amplitudes of By magnetic field component: By =Bz < 0:3. Therefore, the proposed model of kink oscillations with By 6¼ 0 can explain this effect: for large enough By, the magnetotail current sheet becomes stable, while oscillations of unstable current sheet can be observed only in the case of small enough By.

FIG. 13. Background magnetic configuration: profiles of the magnetic field curvature radius (dashed) and b parameter (solid) at the current sheet center.


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The last step of our study was conducted by means of fully 3D MHD numerical simulations. The computational box was expanded in x direction to make the initial configuration even closer to the real conditions. Namely, the box included subregions of earthward and tailward growing Bz with strongly varying Rc and b parameter, shown in Fig. 13. Four runs with different values of By were started with a special initial condition containing 60 small-amplitude sinusoidal modes ky ¼ 2pm=Ly ; m ¼ 1; 2; :::; 60. Dispersion curves cðk; By Þ were plotted using the growth rates, calculated for each mode at the linear stage of instability. The shapes of dispersion curves cðk; By Þ obtained in 3D simulations agree qualitatively with the linear theory and 2D simulations. However, the values of c for small or large ky are notably different from the analytical considerations, indicating that the effects of, e.g., finite resistivity or numerical dissipation should be studied in a more detail in the future. In accordance with our previous study,61 the peak growth rate value cmax 0:06 observed in Fig. 12 for run 1 (By ¼ 0) is close to the analytical estimate (Eq. (1)) averaged along the sheet hcf ix ¼ 0:057. This result marks the difference between the fully non-linear 3D and linearized 2D MHD studies, where the peak growth rate matches the peak analytical estimate maxðcf ðxÞÞ. Meanwhile, such averaging reduces the actual value of the growth rate more than twice, and this may be a reason why flapping motions seen in MHD simulations61 were not observed in kinetic simulations of Pritchett and Coroniti,43,48 where BICI mode was registered in the same initial configuration. Say, in Ref. 48, the observed value of BICI was 0.13, while the peak analytical estimate of cf calculated for that configuration amounts 0.15. However, the averaged estimate of cf ( 0:06), being much lower than the observed BICI growth rate value, makes double-gradient mode invisible. The second reason for such discrepancy of MHD and kinetic simulation results may consist in the closed x-boundary conditions used in kinetic simulations.43,48 This conclusion is supported by recent 3D kinetic simulations17 of explosive processes in magnetotaillike configuration utilized open x-boundaries and revealed both flapping and BICI-like buoyancy motions in a single run. In 3D runs, the stabilizing effect of By is found to be twice stronger than in 2D simulations and analytics (see Fig. 12); here, the peak value of growth rate cpeak for By ¼ 0.2 drops 3 times as compared to cpeak ðBy ¼ 0Þ value (vs. 1.5 times in 2D simulations). At the same time, the expected decrease of kpeak 3 with increasing By is scarcely noticeable. This yields the typical wavelength kpeak ¼ 2p=kpeak 2D in dimensional units (in 3D simulations, the pure periodic waves are studied, so that k is expressed in terms of k precisely), the value of which seems to be independent of By. We can suggest two explanations for this result. First, it may be attributed to purely numerical effects in the 3D MHD simulation, such as numerical dissipation. In addition, we need to stress the fact that the growth rate estimate (Eq. (29)) gets rather uncertain for the localized modes (changing xmin, xmax affects the overall estimate of Iðky ; tÞ, see Fig. 7). This notion, however, cannot explain high growth rates found for By ¼ 0:2 and By ¼ 0:1 for large ky.

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The second possible reason is more physical, accounted for the rather different configurations used in 2D and 3D runs. As it is seen in Fig. 9, in 3D runs, instability shifts toward the larger x values (and larger Bz values consequently) with increasing By, so that for By ¼ 0:2 it is localized around x 11, while the region x < 10 is stable. In itself, such behavior is not surprising, while localization of perturbations in the region By < Bz agrees with results of in-situ data analysis, discussed above. At the same time, for By ¼ 0, we observe unstable region at x < 11, while the segment x > 11 is calm. These two results contradict somewhat to our knowledge that non-zero By stabilizes double-gradient mode. This contradiction may be settled by taking into consideration that, strictly speaking, the mode developing in the rightmost part of the sheet, x > 10, cannot be described by the doublegradient model. Indeed, at the point x ¼ 10, the curvature radius Rc increases up to 10, reaching the value of 103 at x ¼ 15. It means that the double-gradient wavelength range Rc < k < Lx disappears. On the other hand, large values of Rc and Bz are relevant to the dipole-like magnetic configurations, where MHD ballooning mode may develop39,54 (relatively low values of b parameter, shown in Fig. 13, are also supportive). In Fig. 14, the growth rate of the ballooning mode, cb, calculated by Eqs. (5) and (6) of Golovchanskaya et al.57 is compared to the double-gradient mode estimate. It is seen that current sheet stays stable respectively to the ballooning mode for x < 9.75, while the rightmost part of the sheet becomes unstable with the peak value of cb 0:033 at x 10:5. So, the instability seen in our 3D modeling may be interpreted as a combination of the “double-gradient” and “ballooning” mode. On the one hand, it confirms the general idea that “double-gradient” and classical “ballooning” modes are intrinsically the different descriptions developed for the same plasma instability (generally, ballooning instability) arising in rather different magnetic configurations. On the other hand, this means that dispersion curves obtained in 3D simulations are not pure “double-gradient” curves, which impedes their comparison with 2D and analytical curves. Obviously, the suggested “mixed-mode” interpretation requires more rigorous study. Unfortunately, our simulation data do not allow us to separate somehow “double-gradient” and “ballooning” modes neither by the growth rate value,

FIG. 14. Growth rates of the double-gradient mode (dashed) by Eq. (1) and ballooning mode (solid) by Eqs. (5) and (6) of Golovchanskaya et al.57 for kjj ¼ 0 in normalized units.


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which are almost identical (see Fig. 14), nor by the type of perturbation, which are pure transversal for both modes (while generally ballooning mode may have non-zero parallel component, the maximal value of kjj for which the mode stays unstable in our configuration amounts 0.067 only according to Eqs. (5) and (6) in Ref. 57). Thus, the problem of the “double-gradient” and “ballooning” modes matching proposes the challenging subject for the future investigation. However, results of quasi-1D analytical study, linearized 2D, and fully 3D MHD simulations share in common the most important features of the double-gradient instability guide field stabilization: formation of the bell-shaped dispersion curve and appearance of the typical wavelength of the order of the current sheet width, decrease of the peak growth rate with increasing By value, and total suppression of the mode for By 0:5. Finally, we would like to stress that the problem of the double-gradient mode stabilizing/excitation goes beyond the guide field mechanism considered in this paper. Particularly, intense sheared plasma flows may be expected to contribute the double-gradient mode development substantially. As it was shown in Refs. 60 and 70, the uniform (@Vy =@x ¼ 0) drift in the out-of-plane direction just introduces the Doppler shift of the double-gradient frequency (growth rate) and does not contribute to the mode stabilization/excitation. At the same time, shear flows (Vy ¼ Vy ðxÞ) are known to be effective stabilizing factor for the short-wavelength band of the ballooning-type instabilities (see, e.g., Refs. 81 and 82). On the contrary, drift ballooning modes may be destabilized by the decelerated earthward plasma flows,83 which may be associated with the so-called bursty bulk flows (BBFs). Such flows are often observed in the close vicinity of the magnetotail central region (see, e.g., Ref. 84); they are found to be strongly localized in space both in dawn-dusk and northsouth directions,85 i.e., exactly in the region of excitation of flapping oscillations. In the framework of the doublegradient model, it was shown59 that BBF may produce kink type oscillations of the current sheet without considerable loss of its energy. We can note also that wave vector of the flapping wave produced by BBF (or the wave embedded in the earthward plasma flow) acquires a non-zero x component. This could be one of possible explanations of observed flapping wave fronts with kx 6¼ 0 reported in Ref. 10. ACKNOWLEDGMENTS

Study of the current sheet stability was supported by Russian Scientific Fund (RSF) Grant No. 14-17-00072. The work of H.K.B. was supported by the Austrian Science Fund (FWF): I193-N16. 3D simulations were conducted using resources provided by the Swedish National Infrastructure for Computing (SNIC) at KTH, Stockholm, Sweden, Grant Nos. 2014-1-176 and 2014-8-38. N.V.E. and D.B.K. thank also ISSI for hospitality and financial support. Authors thank the reviewer for his substantial help in improving the manuscript. 1

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