Analytical models and numerical studies of magnetic reconnection

Analytical models and numerical studies of magnetic reconnection

Analytical models and numerical studies of magnetic reconnection

Analytical models and numerical studies of magnetic reconnection Francesco Pegoraro Magnetic field topology plays a central role in the development of magnetic field line reconnection in the laboratory. This is particularly the case for collisionless magnetic field line reconnection (Hamiltonian reconnection) where the features of magnetic field line redistribution and of energy dissipation are separated. Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Analytical models and numerical studies of magnetic reconnection

Collisionless reconnection [1] The conservation of magnetic field topology is directly related to the formation of current and vorticity layers and controls their shape.

In the laboratory two dimensional and quasi-three dimensional plasma configurations with a strong guide magnetic field are of interest and fluid like plasma descriptions “may” be applicable. Numerical simulations are used in order to elucidate the role of the magnetic field topology on the nonlinear evolution of Hamiltonian reconnection.

Numerical results will be presented for two dimensional regimes (single ”helicity” instabilities) and quasi-three dimensional regimes (multiple helicities). Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Analytical models and numerical studies of magnetic reconnection

Collisionless reconnection [2] Introduction Magnetic field line reconnection is one of the most general phenomena in magnetized plasmas and has been widely investigated in astrophysical and space plasmas, in laboratory magnetically confined plasmas and, more recently, in relativistic laser produced plasmas1 . Nevertheless it is not easy to define reconnection unequivocally, as its development combines different features ranging from the change of the topological structure (breaking of magnetic connections) and flux non-conservation of the magnetic field in the plasma, to the (irreversible) conversion of magnetic energy in other forms of plasma energy (e.g., in the form of particle acceleration) and the formation of spatially localized structures such as current layers.

1 G.A. Askar‘yan, et al., Comm. Plasma Phys. Contr. Fus., 17, 35 (1995); F. Califano, et al., Phys. Rev. Lett., 86, 5293 (2001). Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [3] The same difficulties apply to the experimental identification of a reconnection event. The very name “reconnection” is in the negative as it implies that it is absent over most of the plasma so that we can give a meaning to the concept of time evolution of field lines. Only locally, around the so called critical points, field lines break and ”reconnect” in a different pattern. This separation is valid only in plasma regimes where the processes leading to the breaking of the connections are per se weak, but are locally enhanced by the formation at the critical points of small spatial scales (singular perturbations).

Mathematically the time evolution of the field lines of a vector field can be defined by the condition that, if two points are connected by (an arc of) a field line at t = 0, at any subsequent time there exists (an arc of) a field line that connects these two same points at their new positions. Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [4] Such a ”connection” is physically meaningful only if we specify on dynamical grounds the lines of which vector field we are referring to and with which velocity field the points move in time.

In incompressible inviscid hydrodynamics the velocity field is the velocity uf of the fluid elements and the connection field lines are those of the fluid vorticity ~ ω ~ uf. In this case the connection condition can be expressed in differential form as d(d~ l × ~ ω) dt = 0, if d~ at t = 0, (1) where d/dt is the Lagrangian time derivative defined by the vector field ~ uf and expressed in Eulerian variables by d/dt uf · ∇ and d~ l(t) ≡ ~ x2(t) − ~ x1(t) is the difference in position between the two close points ~ x2(t) and ~ x1(t) which move with velocities ~ uf (~ x1(t)) and ~ uf (~ x2(t)) respectively.

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Collisionless reconnection [5] In ideal magnetohydrodynamics (MHD) the velocity field is the velocity up of the quasineutral plasma elements and the connection field lines are those of the plasma magnetic field B. However2 under not too restrictive conditions one can formally define a velocity field which is not directly related to the motion of the particles in the plasma but which preserves the topology of the magnetic field in time. On the other hand a redefinition of the velocity field may be physically significant: the introduction of the Hall term in Ohm’s equation in MHD does not violate magnetic connections, but accounts for the fact that the evolution of the magnetic field lines is more conveniently described following the electron instead of the plasma motion when the electron and ion velocities are different.

2 A.H., Boozer,Phys. Rev. Lett., 88, 215005 (2002). Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [6] Closely related to the topological nature of magnetic reconnection is its interpretation as a process that leads to magnetic energy conversion. This is best understood in the case of ideally stable MHD plasma equilibria with inhomogeneous magnetic fields and current gradients. These equilibria can become unstable when the (infinite number of) constraints arising from the conservation of the magnetic connections between plasma elements are removed and lower magnetic energy states become available to the plasma.

In fact the magnetic energy release and the associated particle acceleration are possibly the features of magnetic reconnection most relevant to astrophysical plasmas, whereas in the laboratory plasmas the most important feature is often related to the loss of plasma confinement due to the change of magnetic topology.

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Collisionless reconnection [7] This energy release can lead to the interpretation of the nonlinear development of magnetic reconnection as a transition, forbidden within the ideal MHD equations, between two MHD (equilibrium) states with different magnetic energies, the excess energy being eventually dissipated into heat (or transported away by accelerated particles). This relationship between dissipation and reconnection ceases to be valid in dilute high temperature plasmas. In such collisionless regimes we find that, even maintaining a fluid-like plasma description with a barothropic scalar pressure, magnetic connections are broken not by electron resistivity but by electron inertia3 .

Again, but with formulae. In ideal magnetohydrodynamics (MHD) the velocity field is the velocity ~ u of the quasineutral plasma elements and the connection field lines are those of the plasma magnetic field ~ B. 3 Pressure anisotropy and phase space effects such as Landau damping are important factors that lead to connection breaking but are not considered here Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [8] Start from Ohm’s law me ne2 d ~ J dt + 1 nec ne ~ ∇Pe (2) and assume that the r.h.s., which describes the effect of electron resistivity, of electron inertia and of the Hall and of the electron pressure terms, is negligible (or can be reduced to a gradient) Then Faraday’s law combined with Eq.(2) gives ∂ ~ B ∂t (3) which has an algebraic structure analogous to that of the vorticity equation in a fluid.

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Collisionless reconnection [9] From Eq.(3) we obtain d(d~ l × ~ B) dt = (d~ l × ~ B) (~ d~ l × ~ B) × ~ ~ u (4) i.e., using the continuity equation for the plasma particle density n, d[(d~ l × ~ B)/n] dt [ ( d~ l × ~ B)/n] × ~ ~ u (5) Equations (4,5) express the condition that within the ideal M.H.D. equations if two plasma points are initially connected by a magnetic field line, they remain connected by a magnetic field line at any subsequent time (regularity properties of the plasma flow have been obviously assumed) Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [10] Under the same conditions we can prove Alfvèn theorem (frozen magnetic flux) d dt d~ S = 0 (6) if the surface S moves together with the plasma. d dt d~ ∂t · d~ S + Z ∂S d~ l
= (7) ∂t d~ S, etc . . Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [11] Similarly one can prove the conservation of the magnetic helicity with ~ A the vector potential (integrated inside a closed magnetic flux tube moving with the fluid): ∂t (∂t ~ A) (∂t ~ B) (8) From ~ E = −~ ∇φ − ∂t ~ A/c we obtain: 1 c ∂ ∂t ∇φ (9) Using we obtain 1 c ∂ ∂t
i = −2 ~ E · ~ B.

(10) Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [12] From Ohm’s law with vanishing r.h.s. we obtain 0. Then we must compute the contribution of the flux term through the surface S of a magnetic flux tube occupying the volume V moving with velocity ~ u. We write KV dV and obtain d dt KV = d dt dV = Z V ∂t dV d~ S ( 11) and use Eq. (10) where ~ E is re-expressed in terms of ~ u and ~ B through Ohm’s law, recalling that ~ B · d~ S = 0 because S is the surface of a flux tube ... The connection theorem (4) is “violated” by the terms on the r.h.s of Eq.(2). The definition of magnetic field line reconnection requires that this breaking occurs only locally around critical points (or lines..

where field lines break and ”reconnect” in a different pattern.

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Collisionless reconnection [13] This separation between global regions where magnetic connections are preserved and localized regions where they are broken and reorganized is valid only in plasma regimes where the processes leading to the breaking of the connections are per se weak but are locally enhanced by the formation at the critical points of small spatial scales (singular perturbations). We are interested in the case where magnetic connections are broken not by electron resistivity but by electron inertia.

In these regimes the plasma dynamics is Hamiltonian and the transformation of magnetic energy into plasma energy is in principle reversible. Furthermore, contrary to the dissipative case, generalized magnetic connections are preserved by the Hamiltonian plasma dynamics. Topological properties are modified but not destroyed. Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [14] The new connection field lines are defined by the vector field Be ≡ B − (mec/e)∇ × ue (subscripts e denote electron quantities) which is proportional to the rotation of the fluid electron canonical momentum and reduces to B only in the limit of massless electrons.

It is evident that, contrary to the magnetic field B, the connection field Be is not directly relevant to the dynamics of the ions in the plasma. Again, but with formulae In a collisionless cold plasma model the effect of electron inertia and of the Hall term in Ohm.s law nec ue me e d~ ue dt , (12) Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [15] can be accounted for by introducing the vector fields ~ Be ≡ ~ B − (mec/e ~ ue ~ Ae (13) (subscripts e denote electron quantities and ~ ue is the electron fluid velocity) and ~ Ee ≡ ~ E + me∇u2 e/(2e) + me∂t~ ue/e = −∇ϕe − ∂t ~ Ae/c, (14) where the generalized vector potential ~ Ae is proportional to the fluid electron canonical momentum and ϕe to the total electron energy and reduce to vector potential ~ A and to the electrostatic potential ϕ in the limit of massless electrons. The vector fields ~ Be(~ x, t) and ~ Ee(~ x, t) satisfy the homogeneous Mawxell’s equations and the ideal Ohm’s law in the form ~ Ee + ~ ue c × ~ Be = 0, (15) Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [16] which leads to the generalized liking condition ∂t (δ~ l × ~ Be ( ~ ue ( δ~ l × ~ Be ∇ ~ ue) · (δ~ l × ~ Be) = 0. (16) Similarly, all the ideal MHD theorems (magnetic flux conservation, magnetic helicity conservation, linking number etc,) are recovered by substituting ~ Be for ~ B and ~ ue for ~ u. The concept of magnetic connections simplifies in the case of two-dimensional (2-D) configurations where all quantities depend on x, y and on time t only. The magnetic configurations of interest here are characterized by a strong, externally imposed, Bz field which is taken to be fixed and does not play the role of a dynamical variable and by an inhomogeneous shear field in the x-y plane associated with a current density J(x, y, t) along the z-axis.

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Collisionless reconnection [17] The field Bz plays a very important physical role in determining the model that is appropriate to represent the plasma dynamics in the x-y plane. Plasma configurations where Bz is absent display a different behaviour both in the fluid and in the kinetic description. In such a 2-D configuration, the magnetic and the electric field can be expressed as ~ B = B0~ ez + ∇ψ(x, y, t) × ~ ez, (17) ~ E = −∇ϕ(x, y, t) + ~ ez∂tψ(x, y, t)/c, (18) where the flux function ψ(x, y, t) is the z-component of the vector potential of the shear magnetic field and ϕ is the electrostatic potential Then, the conserved connections between plasma elements moving in the x-y plane take the form of Lagrangian invariants i.e., can be expressed in term of scalar quantities that are advected by the plasma motion and are constant along characteristics.

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Collisionless reconnection [18] In the ideal MHD limit this Lagrangian invariant corresponds to the z component Az of the magnetic vector potential i.e., to the flux function ψ. Plasma elements that lie initially on an ψ = const curve in the x-y plane and that move along the characteristics of the stream function ϕ remain at all times on the same ψ = const curve, i.e., ψ-connections are preserved. When electron inertia is taken into account in Omhs’ law, the 2-D counterpart of Eq.(16) is the conservation of the Lagrangian invariant that corresponds to the z component of the electron canonical momentum ~ Ae, i.e., aside for multiplication constants, and assuming for the sake of simplicity an almost uniform electron density, the conservation of ψe(x, y, t) = ψ(x, y, t) − (mec/e)uez ∼ ψ(x, y, t) + d2 eJz, where de = c/rωpe is the collisionless electron skin depth and Jz = −∇2 ψ is the z component of the plasma current 4 .

4 Ion motion along field lines is neglected and Jz is taken to be carried by electrons only. Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [19] Let us now go return to energy conservation in magnetic reconnection and transitions between magnetic equilibria As mentioned before the breaking of the magnetic connections allows the system to access configurations with lower magnetic energy. The possibility of a transition between two magnetic equilibria with different magnetic energies can be easily conceived in the case of dissipative reconnection, when the local decoupling between the magnetic field and the plasma motion is due to electric resistivity, B/c = η ~ J, since the excess magnetic energy that is released in the transition can be transformed into heat.

The possibility of such a transition between two equilibrium states is less obvious in the nondissipative case where energy can only be transferred into mechanical or (reversible) internal energy so that one could expect that the system cannot be ”stopped”in a new stationary equilibrium with a lower magnetic energy. Moreover generalized connections are preserved.

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Collisionless reconnection [20] Indeed this apparent difficulty is not very different from the one that occurs in the treatment of Landau damping in Vlasov’s equation for the distribution function f(~ x,~ v, t). In Vlasov’s equation no energy is dissipated and particle-points in phase space that lie initially on an f = const hypersurface and that move along the characteristics of the single-particle Hamiltonian H(~ x,~ v, t) lie at all times on an f = const hypersurface (with the same value of the constant).

This amounts to say that, in the absence of collisions, f-connections are preserved. This brings into play the role of current and vorticity layers Current layers are a generic feature of magnetic reconnection. The formation of spatially localized current structures is related to the fact that the magnetic connections in the plasma are broken only locally, around the critical points of the magnetic configuration where the current density accumulates. Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [21] In Hamiltonian plasma regimes the formation of current layers is an even more important feature since, as will be discussed below, their presence decouples the time evolution of the reconnecting magnetic field lines from that of the unbroken field lines of Be. As an illustration I will report some recent two dimensional results 5 obtained on the nonlinear evolution of collisionless magnetic field reconnection in a fluid-like two-dimensional (2-D) model. This model is mainly applicable to laboratory plasmas where the plasma is embedded in a strong highly ordered magnetic field and the configuration we analyze is constrained by boundary conditions that may be too restrictive for astrophysical plasmas.

5 E. Cafaro, et al., Phys. Rev. Lett., 80, 4430 (1998), D. Grasso, et al., Phys. Rev. Lett., 86, 5051 (2001), D., Del Sarto, Phys. Rev. Lett., 91, 235001 (2003), F. Pegoraro, et al., Nonlinear Processes in Geophysics, 11, 567-577 (2004), D. Del Sarto, et al., Phys. Plasmas, 12, 2317 (2005) Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [22] However this model gives a rather clear picture of the dynamical role played by the magnetic field topology in the nonlinear process of magnetic field line reconnection and in the magnetic energy redistribution.

I will also illustrate the relevance of this topological approach in a (drift) kinetic numerical investigation of collisionless reconnection6 , possibly more relevant to astrophysical conditions. Finally an extension to a quasi-three dimensional two fluid model of the nonlinear evolution of the instability will also be presented7 together with a discussion on the difficulties one encounters in defining a reconnection rate. 6 T.V. Liseikina, et al., Phys. Plasmas, 11, 3535 (2004) 7 D. Grasso, et al., Computer Physics Comm., 164, 23 (2004), D. Borgogno, et al., Phys. Plasmas, 12, 32309 (2005) Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [23] Two-Dimensional Fluid Hamiltonian Model We consider a 2-D magnetic field configuration uniform along z with B = B0ez + ∇ψ × ez, where B0 is taken to be constant and ψ(x, y, t) is the magnetic flux function. The plasma dynamics is described by the two-fluid dissipationless “drift-Alfvèn” model 8 which includes the effects of electron inertia in Ohm’s law: ∂F/∂t + [ϕ, F] = %2 s[U, ψ], ∂U/∂t + [ϕ, U] = [J, ψ]. (19) 8 This model is explicitly derived in Schep, T.J., et al., Phys. Plasmas, 1, 2843 (1994); Kuvshinov, B.N., et al., Phys. Letters A, 191, 296 (1994); Kuvshinov, B.N., et al., Journ.

Plasma Phys., 59, 4 (1998). and is closely related to the so called Reduced Magnetohydrodynamic equations B.B. Kadomtsev and O. P. Pogutse, Sov. Phys. JETP, 38, 283 (1974) H. R. Strauss, Phys. Fluids 19, 134 (1976).

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Collisionless reconnection [24] Here F = ψ + d2 eJ corresponds to the canonical fluid electron momentum and J = −∇2 ψ is the electron current density along z, ϕ(x, y, t) is the electron stream function, with ex×∇ϕ the incompressible electron velocity in the x-y plane, and U = ∇2 ϕ is the plasma fluid vorticity. The Poisson brackets [A, B] are defined by [A, B] = ez · ∇A × ∇B. The first of Eqs.(19) describes the electron motion along field lines and is equivalent to the parallel component of Ohm’s law, while the second of Eqs.(19) originates from the continuity equation and includes parallel electron compressibility.

This electron temperature effect becomes important if the sound Larmor radius %s = (mic2 Te/e2 B2 )1/2 is comparable to the collisionless electron skin depth de = c/ωpe. The fluid vorticity ∇2 ϕ is related to the ion density variation which is set equal to the electron density variation because of quasineutrality. Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [25] No equilibrium density and temperature gradients effects are included in the present analysis. The system of Eqs. (19) is Hamiltonian with energy H = Z d2 x |∇ψ|2 + |∇ϕ|2 + d2 eJ2 + %2 sU2
/2.

It can be cast in Lagrangian form ∂G±/∂t + [ϕ±, G±] = 0, (20) for the two Lagrangian invariants G± = F ± de%sU which are advected by the incompressible velocity fields ez × ∇ϕ± where ϕ± = ϕ ± (%s/de)ψ are generalized stream functions and the term ±(%s/de)ψ accounts for the thermal electron motion along magnetic field lines.

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Collisionless reconnection [26] In the cold electron limit (%s → 0) the Lagrangian invariants G± become degenerate and only F = (G++G−)/2 admits a Lagrangian conservative equation advected by the fluid velocity ez × ∇ϕ, while the rescaled difference term (G+ − G−)/(2de%s) ≡ U obeys the second of Eqs.(19). In 2-D configurations the advection of Lagrangian invariant quantities is equivalent to the conservation of field line connections. Plasma elements connected by magnetic field lines in the x-y plane lay on ψ = const curves.

If electron inertia is neglected, ψ is a Lagrangian invariant and magnetic field lines do not break.

For de 6= 0 magnetic field lines can break and reconnect but the structure of Eqs.(20) implies that in this Hamiltonian regime the development of magnetic reconnection is constrained by the conservation of the connections given by the field lines of G± (or of F in the degenerate %s = 0 case). Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [27] Nonlinear Reconnection Regimes In the above quoted articles Eqs.(19) were integrated numerically in order to investigate the long term nonlinear evolution of a fast growing (large de∆09 ) reconnection instability produced by electron inertia in a sheared magnetic equilibrium configuration with a null line.

Periodic conditions were taken along y and the configuration parameters were chosen such that only one mode can be linearly unstable, as of interest for laboratory plasmas.

9 This is an important technical point: in terms of resistive reconnection this regime corresponds to the socalled resitive [γ ∝
1/3 η ] regime, as opposed to the ”tearing” [γ ∝
2/5 η ] regime where the constant-ψ approximation holds. Inertial tearing modes have an excessively small growth rate that makes them physically uninteresting. The large ∆0 regime is encountered in the case of resistive internal kink modes in a cylinder or in a torus, or, in general, in the presence of steep equilibrium currents Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [28] The results shown here were obtained with a numerical code that advances the cell averaged values of F and U in time using a finite volume technique (i.e., calculating the cell fluxes).

A Fourier Transform method is then used to reconstruct the grid points values of F and U at the cell corners. Time is advanced using the explicit third order Adams-Bashforts scheme. Typical mesh sizes are Nx = 2048 and Ny = 512. Random perturbations were imposed on the equilibrium configuration ψ0(x) = −L/[2 cosh2 (x/L)] in a simulation box with Lx = 2Ly = 4πL, taking de = 3/10L and %s/de in the range 0-1.5. The accuracy of the integration has been verified by testing the effects of numerical dissipation on the conservation of the energy and of the Lagrangian invariants. Formation of small spatial scales in the nonlinear phase The Lagrangian invariants G± differ from the flux function ψ by the term d2 eJ ± de%sU which has small coefficients but involves higher spatial derivatives.

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Collisionless reconnection [29] As shown by the numerical results, magnetic reconnection proceeds unimpeded in the nonlinear phase because of the development near the X point of the magnetic island of increasingly small spatial scales that effectively decouple ψ from G±. In Hamiltonian regimes the formation of such scales does not stop at some finite resistive scalelength. This corresponds to the formation of increasingly narrow current and vorticity layers. Because of the conserved G± connections, the spatial localization and structure of these layers depends on the value of %s/de.

Mixing of the Lagrangian invariants and island growth saturation In the reconnection model adopted, magnetic energy R d2 x|∇ψ|2 is transformed, in principle reversibly, into two forms of kinetic energy, one, R d2 x|∇ϕ|2 , related to the plasma motion in the x-y plane and one, R d2 xd2 eJ2 , to the electron current along z and, for %s 6= 0, into electron parallel compression R d2 x%2 sU2 .

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Collisionless reconnection [30] The last two energies involve quantities with higher derivatives. Being the system Hamiltonian, it is not a priori clear whether a reconnection instability can induce a transition between two stationary plasma configurations with different magnetic energies, as is the case for resistive plasma regimes where the excess energy is dissipated into heat. Taking at first %s/de ∼ 1 we showed that, in spite of energy conservation, this transition is possible at a “macroscopic” level. A new coarse-grained stationary magnetic configuration can be reached because, as the instability develops, the released magnetic energy is removed at an increasingly fast rate from the large spatial scales towards the small scales that act a perfect sink.

This leads to the saturation of the island growth. Similarly, the conservation of the G± connections ceases to constrain the system at a macroscopic level. Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [31] The advection of the two Lagrangian invariants G± is determined by the stream functions ϕ±. The winding, caused by this differential rotation type of advection, makes G± increasingly filamented inside the magnetic island, leading to a mixing process similar to that exemplified by the “backer transformation” in statistical mechanics.

These filamentary structures of G± do not influence the spatial structure of ψ which remains regular.

There is an analogy with the Bernstein-Greene-Kruskal (BGK) solutions of the Vlasov equation for the nonlinear Landau damping of Langmuir waves. The evolution of G+ for %s/de = 1.5 is shown in Fig.1 together with the contours of ψ, J and U. The contours of G− are obtained from those of G+ by mirror reflection. Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [32] Figure 1: Laminar mixing: contours of G+ at t = 40, 60, 80 (from left to right) at t = 80 for %s/de = 1.5 Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [33] Figure 2: Laminar mixing: contours of ψ, J and U (from left to right) at t = 80 for %s/de = 1.5 Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [34] Onset of a secondary Kelvin Helmoltz instability: turbulent versus laminar mixing The advection, and consequently the mixing, of the Lagrangian invariants can be either laminar, as shown in Figs. 1,2, or turbulent depending on the value of the ratio %s/de. The transition between these two regimes was shown to be related to the onset of a secondary Kelvin Helmoltz (K-H) instability driven by the velocity shear of the plasma motions that form because of the development of the reconnection instability.

Whether or not the K-H instability becomes active before the island growth saturates, determines whether a (macroscopically) stationary reconnected configuration is reached and affects the redistribution of the magnetic energy. In the cold electron limit, %s/de = 0, the system (19) becomes degenerate and the generalized connections are determined by a single Lagrangian invariant F. Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [35] Initially, F is advected along a hyperbolic pattern given by the stream function ϕ which has a stagnation point at the O-point of the magnetic island.

This motion leads to the stretching of the contour lines of F towards the stagnation point and to the formation of a bar-shaped current layer along the equilibrium null line, which differs from the cross shaped structure found in the initial phase of the reconnection instability for %s/de 6= 0. Subsequently, F contours are advected outwards in the x-direction as shown Figs.3,4 at t = 90.

At this stage F starts to be affected by a K-H instability that causes a full redistribution of F, as shown at t = 103. In this phase the spatial structure of F is dominated by the twisted filaments of the current density which spread through the central part of the magnetic island. The contours of the vorticity U exhibit a well developed turbulent distribution of monopolar and dipolar vortices, while those of ψ remain regular although pulsating in time. Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [36] Figure 3: Turbulent mixing: contours of F at t = 90, 103, 112 for %s/de = 0 Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [37] Figure 4: Turbulent mixing: contours of ψ, J and U (bottom row, from left to right) at t = 112 for %s/de = 0 Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [38] The energy balance shows that part of the released magnetic energy remains in the form of plasma kinetic energy corresponding to the fluid vortices in the magnetic island and that an oscillatory exchange of energy persists between the plasma kinetic energy and the electron kinetic energy corresponding to the pulsations of the island shape.

This turbulent evolution of the nonlinear reconnection process also occurs in the non degenerate, finite electron temperature, case but as the ratio %s/de is increased, i.e. as the electron temperature effects become more important, the onset of the K-H instability occurs later during the island growth and its effect on the current layer distribution becomes weaker.

For %s/de ∼ 1, no sign of a secondary instability is detectable during the time the island takes to saturate its growth. Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [39] Need for a kinetic electron description The conservation of the generalized connections in the reconnection process leads to the formation of current and vorticity layers with spatial scales that, in the absence of dissipation, becomes increasingly small with time. In this nonlinear phase of the reconnection instability, the fluid approximation may become inconsistent inside the layers.

The generalized connections and the constraints that they exert on the plasma dynamics apply to the case of a fluid plasma, where fluid elements can be defined and the linking property between fluid plasma elements can be formulated. It thus becomes important to understand what is the role of the topological invariants in a kinetic electron description.

The role of a finite electron temperature on the topological properties of the plasma is already evident from the above results, since the contribution of the Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [40] parallel electron compressibility introduces two new Lagrangian invariants G± and two different streaming functions ϕ± instead of F and ϕ, and consequently changes the nonlinear evolution of reconnection in a significant way. Drift kinetic formulation Let F(x, y, v||, t) be the drift-kinetic electron distribution function, with v|| the electron velocity coordinate along field lines.

It is convenient to adopt the electron canonical momentum, divided by the electron mass, p||, defined by10 p|| ≡ v|| − ψ, (21) as the kinetic variable instead of v||. Since we consider two dimensional (z independent) fields and perturbations, p|| is a particle constant of the motion. 10 We adopt the following normalizations ϕ = eϕ/mev2 the, ψ = eψ/mecvthe, x, y = x/L, y/L, t = tmev2 thec/L2 eB0, p|| = p||/vthe, where L is a characteristic length and the other symbols are standard Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [41] In the x, y, p||, t variables the drift kinetic equation for the distribution function f(x, y, p||, t) ≡ F(x, y, v||, t) reads 11 ∂f ∂t + [ϕ − ψp|| − ψ2 /2, f] = 0 − [ϕkin, f], (22) with ϕkin = ϕ − ψp||/c − ψ2 /2. In Eq.(22) the spatial derivatives are taken at constant p|| and not at constant v||. For each fixed value of p||, the time evolution of f corresponds to that of a Lagrangian invariant “density” advected by the velocity field obtained from the generalized stream function ϕkin.

11 H.J. de Blank, Phys. Plasmas, 8, 3927 (2001); G., Valori, in Fluid, kinetic aspects of collisionless magnetic reconnection, ISBN 90-9015313-6, Print Partners Ipskamp, Enschede, the Netherlands (2001); H.J.

de Blank, G. Valori, Plasma Phys, Contr. Fus., 45, A309 (2003). Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [42] The advection velocity field is different on each p|| = const foil. Thus f consists of an infinite number of Lagrangian invariants, each of them advected with a different velocity, that take the place of the two fluid invariants G±. The fluid quantities are defined in terms of distribution function f as follows Z dp||f(x, y, p||, t) = n(x, y, t), (23) Z dp||p||f(x, y, p||, t) = [u(x, y, t) − ψ(x, y, t)] n(x, y, t), Z dp||[p|| − u(x, y, t) + ψ(x, y, t)]2 f(p||, x, y, t) = Π||||, where n(x, y, t) and u(x, y, t) are the normalized electron density and fluid velocity and Π||||(x, y, t) is the (z, z) component of the pressure tensor.

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Collisionless reconnection [43] Then Ampere’s equation reads d2 e∇2 ψ = nu, (24) and, as in the fluid case, the ion equation of motion together with quasineutrality give (n − n0) = ρ2 s∇2 ϕ, (25) where n0 = n0(x) is the initial normalized density and the density variations are supposed to remain small. The above system of equations admits a conserved energy functional Hkin Hkin = Z d2 x 2 [d2 e(∇ψ)2 + ρ2 s(∇ϕ)2 + nu2 + Π ( 26) Aside for the normalization, the main difference between these energy terms and Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [44] the corresponding ones derived in the fluid case is in the expression of the electron compression work: as natural in a kinetic theory, the pressure tensor Π|||| cannot be expressed in terms of the lower order moments of the distribution function.

Electron equilibrium distribution function The stationary solutions of Eq.(22) are of the form f = f(p||, ϕkin). Using the identity for the single particle energy v2 ||/2−ϕ = p2 ||/2−ϕkin, we can write a stationary distribution function that depends only on the particle energy as f = f(p2 ||/2 − ϕkin), while the well known 12 static (ϕ0 = 0) Harris pinch 12 E.G., Harris, Il Nuovo Cimento, 23, 115, (1962). Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [45] equilibrium distribution is given by 13 f = f0 exp [−(p2 || − 2ϕkin) − 2v∗ p||] (27) In order to have a less inhomogeneous plasma configuration we can add a pedestal (Maxwellian) distribution function of the form fped = f00 exp [−(p2 || − 2ϕkin)]. The corresponding self consistent vector potential ψ0(x) is given by ψ0(x) = (1/v∗ ) ln (cosh x) and the shear magnetic field has the standard hyperbolic tangent distribution. 13 In velocity variable v|| this distribution corresponds to F0 exp [−(v|| − v∗ )2 − 2v∗ ψ] and leads to a particle and current density of the form n = n0 exp (−2v∗ ψ) and j = −n0v∗ exp (−2v∗ ψ), where j is normalized on noevthe and v∗ is the standard parameter related to the diamagnetic fluid motion.

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Collisionless reconnection [46] Evolution of the p|| = const foils We write the distribution function with f(x, y, t, p||) as f(x, y, t, p||) = R dp̄||δ(p − p||)f(x, y, t, p . This is a foliation of the electron distribution function in terms of the infinite number of Lagrangian invariants obtained by taking the distribution function f at fixed electron canonical momentum. Within the drift-kinetic equation each p̄||-foil evolves independently, while all foils are coupled through Maxwell’s equations.

The total number of particles in each foil is constant in time. In the initial configuration, the spatial dependence of each p̄||-foil is given for the case of the Harris distribution by exp (2ϕ̄kin) = exp (−2ψp − ψ2 ) = exp [p̄2 || − v̂||(x)2 ], where v̂||(x) ≡ v||(ψ, p = p ( 1/v∗ ) ln (cosh x).

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Collisionless reconnection [47] For negative values of p̄|| the maximum of the argument of the exponent is located at x = ±arccosh[exp (−v∗ p ] i.e. the foil is localized in space within two symmetric bands, respectively to the right and to the left of the neutral line of the magnetic configuration. For positive values of p̄|| all the foils are centered around x = 0. Nonlinear twist dynamics of the foils In the adopted drift kinetic framework the p̄||-foils take the role of the Lagrange invariants G± of the fluid plasma description. In this perspective, the dynamics of the foils can be predicted by looking at the form of stream function ϕkin inside each foil.

The advection velocity can be written as ~ ez × ∇(ϕ − ψp|| − ψ2 /2) = ~ ez × ∇ϕ + (p|| + ψ) ∇ψ × ~ ez (28) which represents the particle E × B drift and their free motion along field lines. Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [48] At fixed p|| = p̄|| we see that depending on the sign of ψ + p = v̂||(x), the advection velocity field takes two counter oriented rotation patterns, reminiscent of those that advect G− (G+) in fluid theory. In the equilibrium configuration where all quantities are function of ψ = ψ(x) and ϕ = 0, this advection corresponds to the free particle motion along ψ = const surfaces inside each foil. However, when the instability starts to move the plasma along the x axis and ∂ϕ/∂y 6= 0, the portions of the foil where v > 0 or where v < 0 bend in opposite directions.

This leads to a distortion and twist of the foils and to their eventual spatial mixing, analogously to the mixing of G± in fluid theory.

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Collisionless reconnection [49] Numerical results: drift-kinetic regime For a Harris equilibrium the evolution of the reconnection instability is characterized by three dimensionless parameters that can be expressed as the dimensionless ion sound gyro-radius ρs and the electron skin depth de from Poisson’s and Ampere’s equations respectively, and n0. The size of the simulation box along y has been chosen equal to 4π such that the parameter ∆0 is positive only for the lowest order mode corresponding to ky = 1/2 so that only the ky = 1/2 mode can be linearly unstable. The simulation box is 40 long in the x direction, with periodic boundary conditions in y and first type boundary conditions in x.

We have taken fixed ρs = 1 and de = 1, v∗ = 4, corresponding to ψ0 = 1/4, n0 = 1/16, de = 1, v∗ = 2 (=> ψ0 = 1/2, n0 = 1/4), de = 0.5, v∗ = 2 (=> ψ0 = 1/2, n0 = 1/16). Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [50] Smaller values of v∗ correspond to larger instability growth rates i.e., to faster evolving instabilities where the saturation of the island growth is reached sooner. The growth rate increases with de faster than linearly. The instability saturation is shown in Figs.(5,6) for the case with de = 1 and v∗ = 4.

The evolution of the p̄||-foils f(x, y, p̄||, t), restricted to the interval −4 < x < 4 around the neutral line, is shown at t = 81 for p − 1.5, −0.5, 0.5, 1.5, together with the contour plots of the stream function ϕkin in x-y for the same values of p̄|| and the same interval in x.

Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [51] -3 0 3 -4 4 y 0.05 0.10 0.15 0.20 0.25 -3 0 3 -4 4 Y X -3 0 3 -4 4 y 0.41 0.45 0.49 0.53 -3 0 3 -4 4 Y X -3 0 3 x -4 4 y 0.08 0.19 0.30 0.41 -3 0 3 -4 4 Y X -3 0 3 x -4 4 y 0.00 0.02 0.04 0.06 -3 0 3 -4 4 Y X Figure 5: Contour plots (top) and 3D plots (bottom) of the p|| = constant foils of the electron distribution function at t = 81 for p − 1.5, −0.5, 0.5, 1.5 from left to right in the interval −4 < x < 4 around the neutral line. Note the different scales in the vertical axes of the 3D plots.

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Collisionless reconnection [52] -3 0 3 -4 4 Y X -3 0 3 -4 4 X -3 0 3 -4 4 X -3 0 3 -4 4 X Figure 6: Contour plots of the kinetic stream function ϕkin at t = 81 for p − 1.5, −0.5, 0.5, 1.5 from left to right in the interval −4 < x < 4 around the neutral line. Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [53] Foils corresponding to negative values of p̄|| were initially localized in two symmetric bands to the left and to the right of the neutral line and are thus modified by the onset of the reconnection instability only in their portion that extend into the reconnection region.

On the contrary foils corresponding to positive values of p̄|| were initially localized around x = 0 and are thus twisted by the development of the reconnection instability. The contour plots of the stream function ϕkin corresponds to a differential rotation in the x-y plane. The sign of the rotation is opposite for positive and for negative values of p̄|| leading to the mixing of the p||-foils. As in the fluid case, the mixing of the Langrangian invariants in x-y space is accompanied by the energy transfer towards increasingly small scales. Within the range of parameters explored in the simulations discussed in the present paper, we have not evidenced any onset of a secondary instability.

This result is fully consistent with the fluid simulations that show that the onset of the Kelvin-Helmoltz instability is impeded by increasing the electron temperature. Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [54] Quasi 3D fluid reconnection Reconnection processes in the laboratory are nearly two-dimensional but 3D effects are are important as they introduce regions of magnetic field line stochasticity centered around the separatrices of 2D magnetic islands, where current and vorticity sheets are localized. Additional points to be analyzed 14 :
3D effects can be expected to alter significantly the spatial structure of these sheets and hence the rate at which reconnection can proceed,
secondary instabilities driven by the very large gradients in these sheets require a fully 3D model in order to be properly treated, 14 D.

Borgogno, D. Grasso, F. Porcelli, F. Califano, F. Pegoraro, D. Farina, Phys. Plasmas, 12, 2309 (2005) Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [55]
in collisionless regimes, it has been shown that very narrow scale lengths are produced in the course of the nonlinear evolution of 2D magnetic islands. These scale lengths are a consequence of conservation properties of 2D collisionless fluid models. In 3D, these conservation properties are partly lost, and so it becomes of interest to assess how the formation of small scales is affected. We represent the magnetic field as: B(x, y, z, t) = B0ez + ∇ψ × ez, (29) where ψ(x, y, z, t) is the magnetic flux function and B0 is a strong constant guide field. The fluid velocity in the perpendicular plane is written in terms of the stream function ϕ(x, y, z, t) as: v⊥(x, y, z, t) = ez × ∇ϕ.

(30) Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [56] “Oblique modes” An oblique mode, although depending on all three spatial coordinates, has a well defined single helicity, i.e., is equivalent to a 2D mode after a proper coordinate rotation. The discussion of a single helicity mode is of interest since, in general, oblique modes have mixed parity around the corresponding resonant magnetic surfaces. Thus, the plasma flow associated with these perturbations is not stagnant at the island X and O-points. This gives rise to a nonlinear drift of the X-point of the magnetic island and indeed of the overall current sheet structure.

3D modes” In the irreducible 3D problem several helicities are present in the initial perturbation. In this case, the phase planes of the different modes have different orientations and cannot be simultaneously reduced to 2D modes by a coordinate transformation.

Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

Collisionless reconnection [57] The modified twofluid equations are ∂tF + [ϕ, F] − ρ2 s[U, ψ] = ∂ϕ ∂z − ρ2 s∂zU, (31) ∂tU + [ϕ, U] − [J, ψ − ∂ zJ, (32) where F ≡ ψ + d2 eJ, J ≡ −∇2 ⊥ψ and U ≡ ∇2 ⊥ϕ are the current density and the vorticity along the z-direction. The Poisson brackets are defined as [A, B] = ez · ∇⊥A × ∇⊥B, Eq. (31) originates from the generalized Ohm’s law and represents the electron momentum equation along the z-direction, Eq.

(32) is the continuity equation after substituting the ion density for the electron density by using the quasineutrality condition and expressing the ion density using the ion continuity and momentum balance equations15 .

15 see Schep, T.J., et al., Phys. Plasmas, 1, 2843 (1994); Kuvshinov, B.N., et al., Phys. Letters A, 191, 296 (1994). Dipartimento di Fisica Università di Pisa pegoraro@df.unipi.it

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