Tearing and Surface Preserving Electron Magnetohydrodynamic Modes in A Current Layer
aa r X i v : . [ phy s i c s . p l a s m - ph ] N ov Tearing and Surface Preserving Electron MagnetohydrodynamicModes in A Current Layer
Gurudatt Gaur ∗ and Predhiman K. Kaw † Institute for Plasma Research, Bhat, Gandhinagar - 382 428, India (Dated: August 8, 2018)
Abstract
In this paper, we have carried out linear and nonlinear analysis of tearing and surface preservingmodes of two dimensional (2D) Electron Magnetohydrodynamics (EMHD). A linear analysis showsthat the perturbations parallel to equilibrium magnetic field B (characteristic tangent hyperbolicspatial profile), driven by the current-gradients, lead to two different modes. The first mode isthe tearing mode having a non-local behavior which requires the null-line in the magnetic fieldprofile. Whereas, the second mode is a surface preserving local mode which does not require thenull-line in the magnetic field. The quantity B − B ′′ should change sign for these modes to exist.In nonlinear simulations, for tearing case we observe formation of magnetic island at the null-linedue to the reconnection of magnetic field lines. However, for surface preserving mode, a channellike structure is observed instead of the island structure. ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION Stability of electron current layers is a long standing topic in theoretical plasma physics.The typical electron current layers are found to be formed in many physical situations like,fast z-pinches [1–3], collision less magnetic reconnection [4–12], fast ignition phenomena oflaser fusion [13, 14], plasma opening devices [15–17], inter planetary current-carrying plasmas[18] etc. These current layers having equilibrium length scale smaller than the ion skin depthare prone to various current-gradient driven instabilities under which they evolve, sometimesto the point of complete destruction. In typical electron current layers, current flows fasterthan the Alfven velocity and the Magnetohydrodynamic (MHD) model is not applicable.The stability of these current layers has been studied using Electron Magnetohydrodynamic(EMHD) model of plasmas [1]. EMHD is a single fluid description of plasma which describesthe dynamics of electron only by ignoring the ion response. EMHD model has proven to bevery convenient in describing numerous phenomena occurring at fast time and short lengthscale where MHD is not applicable. A rich literature on this model is available, which canbe found elsewhere. Here, we will discuss applicability of this model to describe the currentgradient driven instabilities.The current-gradient driven instabilities in the framework of EMHD have been previouslyconsidered by Califano et al. [19], where they have been broadly categorized as tearing andbending instabilities depending upon the orientation of perturbations relative to the equi-librium magnetic field. The classification can be understood from Fig. 1. The perturbationspropagating along the direction of flow (perpendicular to magnetic field) give rise to theexcitation of Kelvin Helmholtz (KH) like modes [20, 21], which bends the flow lines andleads to the formation of vortex structures. This mode is known to play the role in stabilityof vortices generated by the interaction of ultra intense laser pulse with a plasma [22], gen-eration of small scale turbulence [23], anomalous stopping of the energetic electron beam infast ignition [24] etc. The other choice of perturbations (i.e. propagating along the magneticfield) gives excitation to the collisionless tearing instability [4, 5] of thin current sheets whichleads to the magnetic reconnection in the presence of electron inertia.Apart from these tearing-bending instabilities, an inertial scale instability is known whichalso falls in the category of current-gradient driven instabilities [25–27]. This mode sharesthe geometry of tearing mode [Fig. 1], but unlike the tearing mode it is a local mode and2oes not require reversed equilibrium magnetic field configuration. This mode preserves thesurface of magnetic flux. Henceforth, this mode shall be referred to as non-tearing mode. Inthe study presented here, we suppress the KH mode by not considering the perturbationsalong the flow direction. Hence our study is two dimensional with perturbations confined inthe plane containing the magnetic field and gradient directions.In the literature, the tearing mode has been studied for a 1D magnetic field profile, B = tanh ( x/ǫ ), which is a Harris current sheet [28] with thickness ǫ . For this choice ofprofile, as we will see, one of the conditions for non-tearing mode, B B ′′ > B > B B ′′ > B < II. MODEL AND GOVERNING EQUATIONS
The EMHD model works for the phenomena involving the fast time scales ω ci,pi << ω < 0) and the condition of quasi-neutrality demands theincompressibility of electron fluid i.e. ( ∇ . ~v e = 0). Moreover, the electron ion collisions havealso been ignored.In two dimensions (with variation along x − z only) with the use of ∇ . ~B = 0 conditionthe total magnetic field can be expressed as, ~B = b ˆ y + ˆ y × ∇ ψ . The corresponding electronvelocity would be expressed as, ~v e = −∇ × ~B = ˆ y × ∇ b − ˆ y ∇ ψ . Thus, the above set ofEMHD equations [Eqs. (1)] can be cast in terms of the evolution of two scalars, ∂∂t ( ∇ b − b ) + ˆ y × ∇ b · ∇ ( ∇ b − b ) − ˆ y × ∇ ψ · ∇ ( ∇ ψ − ψ ) = 0 ∂∂t ( ∇ ψ − ψ ) + ˆ y × ∇ b · ∇ ( ∇ ψ − ψ ) = 0 (2)The equilibrium of Eq. (2) is defined as follows. We choose, b = constant . With thisand choice of one dimensional equilibrium, Eq. (2) becomes, d ψ dx − ψ = F ( ψ ) (3) ⇒ d ψ dx = G ( ψ ) (4)Here, G ( ψ ) ≡ F ( ψ ) + ψ . F and G are the arbitrary functions. Simplifying Eq. (4) furthergives, H ( ψ ) = x (5) ⇒ ψ = f ( x ) (6) ⇒ B = − dψ dx = f ′ ( x ) (7)4ere, H ( ψ ) = (1 / √ R I ( ψ ) dψ − K ; I ( ψ ) = 1 / [ R G ( ψ dψ + K )] / . K and K arethe constants and H and f are some arbitrary functions. Thus, we choose an equilibriumsheared magnetic field ~B ( x ) = B ( x )ˆ z . This corresponds to a sheared electron flow directedalong ˆ y axis as, ~v ( x ) = v ( x )ˆ y . This choice would exclude the KH modes in our system asthere are no variations along the equilibrium velocity.We linearize the EMHD equations [Eqs. (2)] about the above equilibrium. Since theequilibrium is the function of x only, we take Fourier transform in z and t to obtain thefollowing set of linearized equations, d b dx − (1 + k z ) b + k z B ω d ψ dx − k z ψ ! − k z B ′′ ω ψ = 0 d ψ dx − (1 + k z ) ψ − k z ( B − B ′′ ) ω b = 0 (8) III. LINEAR INSTABILITY In this section we analyze the set of coupled linearized equations (8) obtained in theprevious section to understand the growth rate and eigen functions of tearing and non-tearing modes.We solve the Eqs. (8) numerically as a matrix eigen value problem for B ( x ) = tanh ( x/ǫ ) + C (9)Where, ǫ is shear width and C is a uniform magnetic field added externally in thedirection of tangent hyperbolic field. The presence of this magnetic field does not disturbthe equilibrium, however, it shifts/removes the null-point in the profile depending upon itsmagnitude [Fig. 2]. We numerically obtain the growth rate as eigen values for differentchoices of C . We would like to point out here that the modes with value of C = 0 . C ≥ . < C < . k z . The three different curves correspondto different values of C . The growth rate for pure tearing case (i.e. C = 0) is nonlocal i.e.only modes with k z ǫ ∼ O (1) are unstable. All the local modes (with k z > /ǫ ) are stable.However, as the value of C is made finite, local modes are also become unstable. For the5ase C = 0 . 5, the growth rate curve shows a dip and then saturates at higher k z values.For the case C = 1 . 0, the growth rate curve shows no dip. This is the case when magneticfield has no null-line and the tearing mode is absent. The growth rate curve saturates inthe local region and becomes independent of k z . This behavior is consistent with studies ofLukin [26].In Fig. 4, we show the surface plot of growth rate as a function of k z ǫ and C . This plotshows that the growth rate of local modes first increases with C and then vanishes for some C = C stable .In Fig. 5 we plot the eigenfunctions of pure tearing and pure non-tearing modes in theleft and right panels respectively. The eigenfunction of tearing mode shows its standardspatial character, where ψ is even in x and slowly varying around x = 0, while b is odd in x and peaked around x = 0. Whereas, the eigen mode structure loses the symmetry for purenon-tearing mode C . The structure is asymmetric around x = 0 both in ψ and b . Investigation of local region Assuming that the perturbation scales ( k − z ) are sharper than the equilibrium scales, wecan take the Fourier transform of Eqs.(8) along x also and obtain the dispersion relation asfollows, ω (1 + k ) = k z ( B − B ′′ )( k B + B ′′ ) (10)Here k = ( k x + k z ) / . For instability, RHS should be negative. From the dispersion relationit is clear that for B ′′ (= − v ′ ) = 0, there is no instability, which implies that the modes arecurrent-gradient driven (in EMHD the current is directly proportional to electron velocitythrough the relation ~J = − ne~v ). Using the above dispersion relation [Eq. (10)], we obtain agrowth rate curve which matched the non-local growth rate curve at high k z values [Fig 6].The mismatch at small k z value this is because the local analysis is not valid there.In order to understand the role of C , we write B = B + C in the above dispersionrelation and obtain, ω (1 + k ) = k z ( B − B ′′ )( k B + B ′′ ) + k z C [2 k B + k C + B (1 − k )] (11)In the limit k z >> ω (1 + k ) = k z k ( B + C )( B − B ′′ + C ) (12)6elow we discuss some cases for different values of C : • Case (i): For C = 0, the dispersion relation reduces to, ω (1 + k ) = k z k B ( B − B ′′ ) = k z k ( B − B B ′′ ) (13)For profiles like tanh, where the quantity B B ′′ < B B ′′ > • Case (ii): For 0 < C < 1, the quantity B B ′′ is positive in some region and negativein the other. In this case local instability might be present. In Fig. 7 we show that inthe region where B B ′′ > 0, the local instability is present. This is the case of mixedtearing and non-tearing modes. • Case (iii): For C > 1, the null-point is removed, B B ′′ > • Case (iv): For C > max ( max | B | , max | B − B ′′ | ), the RHS becomes positive, andagain there is no instability. We show in Fig. 8 that for certain value of C there is noinstability, this value C = C stable as pointed out earlier in this section. At this valuethe quantity B − B ′′ has definite sign (positive). This infers that for instability thequantity B − B ′′ should change sign. Here, C stable = max ( | B − B ′′ | ) ∼ . IV. NONLINEAR SIMULATIONS The coupled set of 2D EMHD evolution Eqns. (Eqn.(2) in Section II) can be expressedin the form of generalized continuity equations with source terms which have been evolvedin slab geometry. A package of subroutines LCPFCT [31] has been used which uses theflux corrected transport algorithm [30]. The output of LCPFCT gives ∇ f − f where, f ≡ b, ψ at each time step. A 2D Helmholtz solver is employed for evaluating b and ψ at theupdated time. The components of velocity and magnetic fields can be calculated using therelations, ~B = b ˆ y + ˆ y × ∇ ψ and ~v e = ˆ y × ∇ b − ˆ y ∇ ψ . An equilibrium configuration givenas, ψ ( x ) = ǫ log { cosh( x/ǫ ) } − c x and b = const (= 0) has been chosen that would describean equilibrium magnetic field ~B = ˆ z (tanh( x/ǫ ) + C ). The equilibrium has been evolved7gainst very low amplitude random numerical noise. We carry out the three simulation runsfor C = 0 . , . . C = 0.0,0.5, and 1.0. During the initial phase of the simulation the total perturbed energy increasesexponentially. In the semilog plot of Fig. 9 this can be seen initially where the curve is astraight line. The slope of this line matches closely with twice the maximum growth rateobtained in Sec. III for each of the distinct values of C . The dashed line shown alongside thesimulation curve has twice the slope corresponding to the analytical value of the maximumgrowth rate. As the amplitude of the perturbed field increases, the nonlinear effects becomeimportant in the simulation resulting in the saturation of the perturbed energy seen at thelater stage.In Fig. 10 we show the contour plots of out of plane magnetic field b for the case C = 0.Also plotted are the contours of magnetic flux function ψ . It can be seen in the figure thatthe magnetic field lines which are straight initially get deformed at later times and show thereconnection of field lines. Consequently, an island type of structure is formed. The contourstructure of b is random initially which are initial low amplitude numerical noises. Laterthe field evolves into a quadrupole structure formed at X - point. This observed behavioris typical for tearing instability seen elsewhere also [29]. The non-tearing mode will not bepresent here, as pointed out in linear analysis carried out in the previous section.We now discuss the next simulation simulation run with C = 0 . 5. Here, the non-tearingmode will also be present. In Fig. 11, we show the evolution of ψ and b . The initiallystraight field lines evolve and show the reconnection of field lines again. Here we see theformation of two islands because the box size permits two wavelengths of the fastest growingmode. But unlike the case for C , the islands formed here are asymmetric. The locationof island is at x ∼ − . b also, the quadrupole formed is asymmetric. Thesefindings are in accordance with asymmetric reconnections [32]. Asymmetric reconnectionsare expected to occur at magnetopause, where density and magnetic field strength on twosides of dissipation region are different [33]. 8ventually, we study the case C = 1 . 0. For this case, there is no null-point in themagnetic field profile, and hence tearing instability will not be present. This is confirmedin the evolution of ψ in Fig. 12. Here, initially straight field lines show deformation at latertimes, but no reconnection is observed. In the final state, the magnetic filed line form achannel like structure. The evolution of b variable is also completely different from that oftearing instability. Here, instead of a quadrupole, we see the localized small scale patternswhich decay later. V. SUMMARY We have investigated tearing and surface preserving modes of Electron Magnetohydro-dynamics (EMHD) in a current layer. Both linear as well as nonlinear studies have beencarried out. The tearing instability breaks the magnetic field lines in the presence of electroninertia and leads to magnetic reconnection. Whereas, surface preserving mode unlike thetearing mode preserves the magnetic flux surface. We have called this mode non-tearingmode.Linear perturbation analysis for a tangent hyperbolic profile of equilibrium magneticfield shows the existence of tearing mode. To study the non-tearing mode we add a uniformmagnetic field C . Presence of C satisfies the condition of non-tearing mode B B ′′ > B − B ′′ , where B is the equilibriummagnetic field, is necessary for any of these instabilities to exist.For pure tearing case C = 0 . 0, we observe the formation of magnetic island at the null-line due to the magnetic reconnection in the nonlinear state. The out-of-plane magnetic fieldshows the formation of quadrupole. These observations are typical for tearing instability. Inthe simulation with C = 0 . 5, when both tearing and non-tearing mode are present we seethe island formed is asymmetric. The quadrupole pattern in out-of-plane magnetic field isalso asymmetric. These finding are in accordance with asymmetric reconnections in whichmagnetic field on two sides of dissipation region is asymmetric. In the case C = 1 . 0, when9here is no null-line present in the magnetic field, the tearing mode will not be present. Inthis case we do not observe the island structure, instead, we observe a channel-like patternin the nonlinear state. 10IGURE CAPTIONFig.1 The schematic describes tearing and bending modes depending upon the orientationof perturbations relative to one dimensional equilibrium magnetic field B ( x )ˆ z . Thismagnetic field is created by an equilibrium electron flow v ( x )ˆ y sheared along x di-rection. Perturbations lying in the vertical plane, containing magnetic field with anull-line, give rise to tearing instability. When the angle of perturbations is changedto lie in the horizontal plane of shear and flow, the instability changes from tearing typeto bending type. Both the instabilities are driven by velocity shear or equivalently,current shear in system where electron dynamical response is only of relevance.Fig.2 The figure shows the equilibrium magnetic field profile, B = tanh ( x/ǫ ) + C . Thenull-point is located at x = 0 for C = 0. The null-point shifts to the left for C = 0 . C = 1 . 0. 2 ǫ , being the shear width, remains same for all cases.Fig.3 Plot of growth rate vs k z ǫ for profiles given in Fig.1 with ǫ = 0 . 3. Different curves arefor different values of C .Fig.4 Surface plot of growth rate as a function of k z ǫ and C .Fig.5 Eigen function plots of pure tearing mode and of pure non-tearing mode in left andright panels, respectively. The other parameter values are ǫ = 0 . k z = 1 . ǫ = 0 . C = 1 . 0. The growth rate from two curves are seen to match at large k z values. The reason of mismatch at small k z values is that the local analysis is notvalid there.Fig.7 The figure shows that the local modes are unstable in the region where B B ′′ > ǫ = 0 . k z = 30, C = 0 . γ local is the growth rate obtainedfrom local analysis. Also plotted are B and B − B ′′ . M ax | B − B ′′ | = 9 . C . The other parameter valuesare ǫ = 0 . k z = 30.Fig.9 The evolution of perturbed energy for C = 0.0, 0.5, and 1.0 in subplots (a), (b), and(c), respectively. The dashed straight lines shown alongside each of the plots have11een drawn with a slope of 2 γ l , where γ l is the linear growth rate of the system. 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