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Magnetic double gradient instability and flapping waves in a current sheet

A new kind of magnetohydrodynamic instability and waves are analyzed for a current sheet in the presence of a small normal magnetic field component varying along the sheet. These waves and instability are related to existence of two gradients of the tangential and normal magnetic field components along the normal and tangential directions with respect to the current sheet. The current sheet can be stable or unstable if the multiplication of two magnetic gradients is positive or negative. In the stable region, the kink-like wave mode is interpreted as so called flapping waves observed in the Earth's magnetotail current sheet. The kink wave group velocity estimated for the Earth's current sheet is of the order of a few tens kilometers per second. This is in good agreement with the observations of the flapping motions of the magnetotail current sheet.


INTRODUCTION
Thin current layers are typical structures in the Heliosphere, including the solar corona, solar wind and planetary magnetospheres. We address some of the magnetohydrodynamic aspects concerning the stability of current layers which are still poorly understood. In particular, CLUSTER observations in the Earth's magnetotail current sheet indicated the appearance of strong wave perturbations propagating across the current sheet. Many event studies indicated very large current sheet variations and a predominant wave propagation in the transverse direction with respect to the magnetic field plane. The existence of such kind of waves associated with flapping motions was confirmed in many statistical studies [1,2,3,4,5,6,7] which allowed one to identify them as the "kink"-like perturbations. The plasma sheet flapping observations are interpreted as crossings of a quasiperiodic dynamical structure produced by almost vertical slippage motion of the neighboring magnetic flux tubes. The frequency of the flapping motions, estimated from observations is w f ∼ 0.035 s −1 [1]. For a majority of the observed events [4], a group speed of the flapping waves was found to be in the range of a few tens (30-70) kilometers per second. The wavelengths and spatial amplitudes are estimated to be of the order of 2 -5 R E (R E is the Earth's radius) [7].
A preferential appearance of one ("kink"-like) mode of the flapping motion was reported by [3]. CLUSTER observations give rise to the assumption that the flapping motions are notably more frequent in the central part of the tail than near the flanks. In the near-flank tail regions the motions of flapping waves are predominantly from the center to the flanks [2]. These experimental results confirm an internal origin of the flapping motions, due to some processes (like magnetic reconnection) localized deep inside the magnetotail. On the basis of CLUSTER observations of reconnection events, a relationship between the flapping motion and the reconnection process was investigated by [8]. During the reconnection events the current sheet exhibits strong flapping motions that propagate towards the flank of the tail.
With regard to a theoretical aspect of the problem, the Ballooning-type mode in the curved current sheet magnetic field was claimed to be able to propagate azimuthally in flankward directions from the source [9]. This ballooning theory was applied in the WKB approximation implying the condition that the wave length scale is much less than the curvature radius. This condition can hardly be fulfilled in the plasma sheet with a small normal component of the magnetic field. Another point is that according to the theory of [9], both "kink"-like and "sausage"-like deformations of the current sheet are equally possible, and the question arises about a reason, why the observed flapping perturbations of the current sheet are mainly associated with the "kink"-like wave modes.
In this paper, we propose a new approach to explain the existence of the "kink"-like flapping wave oscillations propagating across the current sheet. In a framework of a rather simple magnetohydrodynamic consideration, we elucidate a physical reason of the flapping wave oscillations of the current sheet, which is related with gradients of the tangential and normal magnetic field components with respect to the normal and tangential directions, respectively.

STATEMENT OF PROBLEM
A geometrical situation of the problem and coordinate system are illustrated in Fig. 1. We apply a system of incompressible ideal magnetohydrodynamics for nonstationary variations of plasma sheet parameters Here V, B, ρ, P are the velocity, magnetic field, density and total pressure, respectively. The total pressure is defined as the sum of the magnetic and plasma pressures. We consider specific wave perturbations propagating across the magnetic field lines, which are much slower than the magnetosonic modes. In this case the incompressible approximation seems to be appropriate. We focus our study on the very slow wave modes existing only in the presence of a gradient of the B z component in the magnetotail current sheet along the x direction. The background conditions are considered to be rather simple with a slow dependence of the B z component on the x coordinate Here ∆ is a thickness of the current sheet, and L x is a length scale of the B z variation along the current sheet.
We introduce normalized small perturbations marked by sign "tilde" which are considered to be functions of time and two spatial coordinates (y, z) Here P 0 is the background total pressure, the parameter ε means the ratio of the background normal and maximal tangential components of the magnetic field, and the parameter ν characterizes the gradient of the normal magnetic field component. For the background conditions considered in our model (B z (x), B x (z)), equation ∇·B = 0 is fulfilled for arbitrary independent parameters ε and ν.
Substituting Fourier harmonics (∝ exp(iωt − ikȳ)), we obtain finally a system of equations for Fourier amplitudes In this system of equations the derivative db z /dx is assumed to be constant, and all other quantities are considered to be not dependent on the x coordinate. Therefore Eqs. (6 -9) are treated as a system of ordinary differential equations with respect to thez coordinate. Excludingb x andb z in Eq. (6), we derivẽ Generally, for a nonconstant U (z), Eq. (10) yieldsṽ x = 0. From Eqs. (7-9), we finally obtain a second order ordinary differential equation for theṽ z velocity perturbation Further for simplicity we consider a piecewise constant function U (z) which means that the current density is assumed to be constant within the current sheet.

RESULTS
A choice of the piecewise constant function U (z) allows us to find analytical solutions which are of two kinds, "kink"-like and "sausage"-like modes. The "kink"-like mode is characterized by displacement of the current sheet center, and even functionṽ z (z) An odd functionṽ z (z) is relevant to the "sausage"-like mode characterized by variations of the thickness of the current layer without a displacement of its center Applying continuity conditions forṽ z and the first derivative dṽ z /dz at the current layer boundaries, we obtain algebraic system corresponding to the "kink" mode C = D cos(λ),kC = λD sin(λ), and also we find a system for the "sausage" mode C = D sin(λ), −kC = λD cos(λ).
Setting the determinants to vanish, we derive two equations corresponding to the "kink" and "sausage" modes, respectively tan(λ) =k λ ("kink"); tan(λ) = − λ k ("sausage"). (18) These equations have discrete sequences of roots λ 1 , λ 2 , ...λ n , ... . The main root is the minimal λ which corresponds to the maximal frequency. By numerical solving these equations, we obtain two main roots λ k,s which determine the dimensional frequencies ω k,s as functions of wave number for the "kink" and "sausage" modes Here ω f means a characteristic flapping frequency proportional to the square root of the multiplication of two gradients of the background magnetic field components, ∂B x /∂z and ∂B z /∂x. The dimensionless functions ω k,s /ω f are presented at the top panel in Fig. 2. Frequencies are monotonic functions of wave number, and they increase to the maximal asymptotic value ω f for k∆ → ∞. The group wave velocity is shown in Fig. 2 as functions of wave number (the second panel). It decreases monotonically to zero for increasing wave numbers. The flapping wave perturbations become unstable when the multiplication of two magnetic gradients becomes negative. In particular, for the Earth's plasma sheet this condition corresponds to the case of decreasing B z component towards Earth. The growth times of the instability for the "kink" and "sausage" modes are given by formulas The instability growth times (τ k,s /τ f ) are shown in Fig. 2 (bottom panel) as functions of wave number for the two wave modes. One can see from the figure that the unstable "kink" mode develops much faster than the sausage mode. In particular, for k∆ = 0.7 the ratio of growth times is τ s /τ k = 2. Fig. 3 illustrates a perturbation of the current sheet and the directions of plasma motion corresponding to the "kink" mode flapping. A qualitative explanation of the flapping instability and waves corresponding to the obtained solution is the following. Let us consider a plasma element of a unit volume at the center of the current layer as shown in Fig. 4.
Along the z direction the resulting force F z acting on this plasma element is a difference of two forces caused by the magnetic stress and the total pressure gradient. In equilibrium state, the resulting force F z vanishes, and the total pressure gradient compensates the magnetic stress In the new position of the magnetic tube element, the resulting force will be This force accelerates plasma in the z direction This equation yields the characteristic flapping frequency ω f which is proportional to the square root of the gradients of the magnetic field components. This qualitative explanation of the instability is illustrated in Fig. 4 where panels (a) and (b) correspond to the stable and unstable situations, respectively. For example, we estimate this frequency for the parameters which seem to be reasonable for the conditions of the current sheet in the Earth's magnetotail, For these parameters we find the characteristic flapping frequency ω f ∼ 0.03 s −1 , and also the group velocity V g = 60 km/s.

SUMMARY
The flapping instability and waves are analyzed for a current sheet in a presence of two gradients of the B x and B z magnetic field components along the z and x directions, respectively. These both gradients play a crucial role for the stability of the current sheet. The instability occurs in the regions of the current layer where the multiplication of two gradients is negative. In particular, the instability can arise in a vicinity of a localized thinning of the current sheet (Fig. 4b). In stable regions, the flapping waves are associated with the so called "Bursty Bulk Flows" or BBF's [3], which are the magnetic tubes rapidly moving through the center of the current sheet towards the Earth. These BBF's are considered to be the sources of the flapping wave oscillations propagating from the center of the current sheet towards the flanks in the ±y directions.
The analytical solution is obtained for the simplified model of the current layer with a constant current density. The frequency and the growth rate for the "kink" mode are found to be much larger than those for the "sausage" mode. For both modes, the frequencies are monotonic increasing functions of the wave number. The corresponding wave group velocities are decreasing functions of the wave number, and they vanishes asymptotically for high wave numbers.
For the typical parameters of the Earth's current sheet, the group velocity of the "kink"-like mode is estimated as a few tens of kilometers per second that is in good agreement with the CLUSTER observations. A strong decrease of the group velocity for high wave numbers means that the small scale oscillations propagate much slower than the large scale oscillations. Because of that, the propagating flapping pulse is expected to have a smooth gradual front side part, and a small scale oscillating backside part.
The neglected second order terms O(ε 2 ) are responsible for the small effects related to the Alfvén waves propagating in the z direction. These second order effects are subjects for future study. For the double gradient flapping waves studied in our model, magnetic tension is not pronounced, because the flapping waves propagate in the direction perpendicular to the plane of the background magnetic field lines. The magnetic field planes are just shifting with respect to each other.