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 )
                                          = 0,       if       d~l × ω
                                                                    ~ = 0 at    t = 0,           (1)
                                 dt

where d/dt is the Lagrangian time derivative defined by the vector field ~uf and
expressed in Eulerian variables by d/dt ≡ ∂/∂t + ~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.

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

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

                                ~
                                u            m e d ~
                                                   J   1 ~ ~     1 ~
                             E    ~ = η J~ +
                             ~ + ×B                  +   J × B −    ∇Pe             (2)
                                c              2
                                             ne dt nec           ne

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            
                                              ~ × ~u × B
                                             =∇        ~                            (3)
                                          ∂t

which has an algebraic structure analogous to that of the vorticity equation in a
fluid.

Dipartimento di Fisica                 Università di Pisa   pegoraro@df.unipi.it
Collisionless reconnection                                                             [9]

      From Eq.(3) we obtain

                         d(d~l × B)
                                 ~
                                    = (d~l × B) ~ · ~u) − [(d~l × B)
                                             ~ (∇                 ~ × ∇]
                                                                      ~ × ~u          (4)
                              dt

i.e., using the continuity equation for the plasma particle density n,

                             d[(d~l × B)/n]
                                       ~
                                            = −{[(d~l × B)/n]
                                                        ~       ~ × ~u
                                                              × ∇}                    (5)
                                    dt


    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
                                  Z
                                      ~ · dS
                                      B    ~=0                          (6)
                                dt S
if the surface S moves together with the plasma.


                             d
                               Z
                                              ∂
                                              Z ~
                                                B
                                                          Z               
                                  ~ · dS
                                  B     ~=           ~+
                                                  · dS         ~ · ~u × d~l =
                                                               B                        (7)
                             dt S           S ∂t            ∂S
                                                                 !
                                      Z
                                          ∂B~                 
                                   =          +∇ ~ × B  ~ × ~u        ~
                                                                   · dS,
                                       S   ∂t
etc.......



Dipartimento di Fisica                  Università di Pisa      pegoraro@df.unipi.it
Collisionless reconnection                                                             [11]

    Similarly one can prove the conservation of the magnetic helicity A ~·B ~ with
A~ the vector potential (integrated inside a closed magnetic flux tube moving with
the fluid):

                                          
                                   ∂t A~·B      ~ ·B
                                         ~ = (∂tA) ~ +A
                                                      ~ · (∂tB)
                                                             ~                         (8)
     ~ = −∇φ
From E          ~ we obtain:
          ~ − ∂tA/c

                             1 ∂ ~ ~     ~ ·B
                                              ~ − ∇φ
                                                  ~ ·B
                                                     ~ −A
                                                          
                                                        ~· ∇~ ×E
                                                                 
                                                               ~ .
                                  A · B = −E                                           (9)
                             c ∂t
                                                       
Using ∇~ · A
           ~ ×E~ =E  ~· ∇~ ×A
                            ~ −A ~· ∇~ ×E~ =E
                                            ~ ·B
                                               ~ −A  ~· ∇~ ×E
                                                            ~ we
obtain
              1 ∂  ~ ~  ~ h ~  ~ ~ i       ~ · B.
                                                   ~
                   A · B + ∇ · φB − A × E = −2E               (10)
              c ∂t
Dipartimento di Fisica                    Università di Pisa   pegoraro@df.unipi.it
Collisionless reconnection                                                                             [12]

                                               ~ ·B
From Ohm’s law with vanishing r.h.s. we obtain E  ~ = 0.
Then we must compute the contribution of the flux term through the surface S
of a magnetic flux
                 R tube occupying the volume V moving with velocity ~u.
We write KV = V A    ~·B~ dV and obtain

  d       d
                         Z                   Z                          Z            
     KV =                        ~ ·B
                                 A  ~ dV =            ~·B
                                                   ∂t A ~          dV +            A~·B
                                                                                      ~ ~u · dS
                                                                                              ~ = ... (11)
  dt      dt                 V                V                               S


and use Eq. (10) where E   ~ is re-expressed in terms of ~u and B
                                                                ~ through Ohm’s
                    ~ · dS
law, recalling that B    ~ = 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.

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

                     ~ ~u ~  1 ~ ~     ~  ~ue ~     me d~ue
                     E+ ×B−     J ×B ≡ E+    ×B = −         ,                   (12)
                        c   nec            c        e dt
Dipartimento di Fisica          Università di Pisa      pegoraro@df.unipi.it
Collisionless reconnection                                                          [15]

can be accounted for by introducing the vector fields

                             B    ~ − (mec/e)∇ × ~ue = ∇ × A
                             ~e ≡ B                        ~e                      (13)

(subscripts e denote electron quantities and ~ue is the electron fluid velocity) and

                     E    ~ + me∇u2e /(2e) + me∂t~ue/e = −∇ϕe − ∂tA
                     ~e ≡ E                                       ~ e/c,           (14)

where the generalized vector potential A ~ e is proportional to the fluid electron
canonical momentum and ϕe to the total electron energy and reduce to vector
          ~ and to the electrostatic potential ϕ in the limit of massless electrons.
potential A
   The vector fields B               ~ e(~x, t) satisfy the homogeneous Mawxell’s
                      ~ e(~x, t) and E
equations and the ideal Ohm’s law in the form

                                        ~    ~ue ~
                                        Ee +    × Be = 0,                          (15)
                                              c
Dipartimento di Fisica             Università di Pisa      pegoraro@df.unipi.it
Collisionless reconnection                                                                [16]

which leads to the generalized liking condition

                   ∂t (δ~l × B
                             ~ e) + (~ue · ∇) (δ~l × B
                                                     ~ e) + (∇~ue) · (δ~l × B
                                                                            ~ e) = 0.    (16)


Similarly, all the ideal MHD theorems (magnetic flux conservation, magnetic
                                                                          ~ e for B
helicity conservation, linking number etc,) are recovered by substituting 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.

Dipartimento di Fisica                Università di Pisa         pegoraro@df.unipi.it
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
                       ~ = B0~ez + ∇ψ(x, y, t) × ~ez ,
                       B                                               (17)
                   ~ = −∇ϕ(x, y, t) + ~ez ∂tψ(x, y, t)/c,
                   E                                                   (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.

Dipartimento di Fisica       Università di Pisa     pegoraro@df.unipi.it
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 A      ~ e, 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) + d2e Jz ,
where de = c/rωpe is the collisionless electron skin depth and Jz = −∇2ψ 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
             ~ + ~u × B/c
resistivity, E        ~ = 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.

Dipartimento di Fisica       Università di Pisa       pegoraro@df.unipi.it
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 ] = %2s [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).


Dipartimento di Fisica                 Università di Pisa                pegoraro@df.unipi.it
Collisionless reconnection                                                      [24]

Here F = ψ + d2e J corresponds to the canonical fluid electron momentum and
J = −∇2ψ 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 = ∇2ϕ 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 = (mic2Te/e2B 2)1/2 is comparable to the collisionless electron skin depth
de = c/ωpe.

   The fluid vorticity ∇2ϕ 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
                                  Z
                                      2              2          2
                                                                    d2e J 2       %2s U 2
                                                                                            
                             H=       d x |∇ψ| + |∇ϕ| +                       +                 /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.

Dipartimento di Fisica                    Università di Pisa                 pegoraro@df.unipi.it
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
                1/3                                                 2/5
resitive [γ ∝ η ] regime, as opposed to the ”tearing” [γ ∝ 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
d2e J ± de%sU which has small coefficients but involves higher spatial derivatives.


Dipartimento di Fisica       Università di Pisa       pegoraro@df.unipi.it
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
                                                        R 2       2
In the reconnection model adopted, magnetic energy d x|∇ψ|     R 2 is transformed,
in principle reversibly, into two forms of kineticRenergy, one, d x|∇ϕ|2, related
to the plasma motion in the x-y plane and one, d2xd2e J 2, toR the  electron current
along z and, for %s 6= 0, into electron parallel compression d2x%2s U 2.


Dipartimento di Fisica       Università di Pisa       pegoraro@df.unipi.it
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                                                    2
   We adopt the following normalizations ϕ = eϕ/me vthe     , ψ = eψ/me cvthe , x, y = x/L, y/L,              t=
     2      2
tme vthe c/L 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
                                + [ϕ − ψp|| − ψ 2/2, f ] = 0 − [ϕkin, f ],                                   (22)
                             ∂t

with
                                      ϕkin = ϕ − ψp||/c − ψ 2/2.
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)]2f (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.

Dipartimento di Fisica                      Università di Pisa          pegoraro@df.unipi.it
Collisionless reconnection                                                                   [43]

Then Ampere’s equation reads

                                                     d2e ∇2ψ = nu,                          (24)

and, as in the fluid case, the ion equation of motion together with quasineutrality
give
                                 (n − n0) = ρ2s ∇2ϕ,                          (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

                                          d2 x 2
                                      Z
                             Hkin =           [de (∇ψ)2 + ρ2s (∇ϕ)2 + nu2 + Π||||]          (26)
                                           2

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 v||2 /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]
                                                    13
equilibrium distribution is given by

                               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 = −n0 v ∗ exp (−2v ∗ ψ),
where j is normalized on no evthe and v ∗ is the standard parameter related to the diamagnetic fluid motion.


Dipartimento di Fisica                 Università di Pisa                 pegoraro@df.unipi.it
Collisionless reconnection                                                          [46]

      Evolution of the p|| = const foils

We
R     write the distribution function with f (x, y, t, p||) as f (x, y, t, p||) =
  dp̄||δ(p̄|| − p||)f (x, y, t, p̄||). 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 (−2ψ p̄|| − ψ 2) = exp [p̄2|| − v̂||(x)2],
where
                v̂||(x) ≡ v||(ψ, p̄||) = p̄|| + (1/v ∗) ln (cosh x).


Dipartimento di Fisica        Università di Pisa        pegoraro@df.unipi.it
Collisionless reconnection                                                          [47]

For negative values of p̄|| the maximum of the argument of the exponent is
located at x = ±arccosh[exp (−v ∗p̄||)] 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|| − ψ 2/2) = ~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 + p̄|| = 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 v̂|| > 0 or where v̂|| < 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.




Dipartimento di Fisica       Università di Pisa       pegoraro@df.unipi.it
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 p̄|| = −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]

  Y                                   Y                                   Y                                    Y
    4                                 4                                   4                                    4




    0                                 0                                   0                                    0




  -4                                  -4                                  -4                                   -4

                              X                                   X                                    X                                    X
        -3        0       3                 -3        0       3                 -3        0        3                 -3        0        3
             -3       0       3                  -3       0       3 0.41                                   0.06
     -4                                -4
     0                                 0
y




                                  y




   4                                 4                                0.30                                 0.04
0.05                              0.41

0.10                                                                  0.19                                 0.02
                                  0.45
0.15                                                                  0.08                                 0.00
                                                                         4                                    4
                                  0.49                                     0                                    0
0.20




                                                                                                           y
                                                                      y




                                                                           -4                                   -4
0.25                              0.53                                               -3       0x       3                  -3       0x       3




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.

Dipartimento di Fisica                                                           Università di Pisa                                            pegoraro@df.unipi.it
Collisionless reconnection                                                                                [52]
Y
    4                    4                 4                      4



    0                    0                 0                      0



 -4                      -4                -4                    -4


        -3   0     3          -3   0   3        -3    0      3        -3   0   3
             X                     X                   X                   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.
                                                          14
       Additional points to be analyzed                         :

   • 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

                                                                    ∂ϕ
                             ∂tF + [ϕ, F ] −         ρ2s [U, ψ]   =    − ρ2s ∂z U ,               (31)
                                                                    ∂z

                                   ∂tU + [ϕ, U ] − [J, ψ] = −∂z J,                                (32)

where F ≡ ψ + d2e J, 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
Collisionless reconnection                                                                                    [58]

      Electron inertia provides the mechanism which breaks the frozen-in condition.

      Equations (31,32) can be rewritten as

                             ∂G±              ∂[ϕ± ∓ (%s/de)G±]
                                 + [ϕ±, G±] =                   ,                                        (33)
                              ∂t                     ∂z

where again G± = ψ − d2e ∇2⊥ψ ± de%s∇2⊥ϕ and ϕ± = ϕ ± (%s/de)ψ.

      The energy, defined as the integral over z of the 2D energy density Ez (x, y, z),

                               1
         Z                         Z
                                                             2   2
                                                                        d2e |∇2⊥ψ|2             %2s |∇2⊥ϕ|2
                                                                                                              
 E=           dzEz (x, y, z) =          dxdydz |∇ψ| + |∇ϕ| +                                +                     ,
                               2
                                                                                                         (34)
is conserved in time.


Dipartimento di Fisica                 Università di Pisa           pegoraro@df.unipi.it
Collisionless reconnection                                                                              [59]

    The 3D effects originate from the fact that the generalized velocities, v± =
ez × ∇ϕ±, with which the functions G± are advected, are z-dependent.
At the same time, a z-dependent source term for G± appears at the right hand
side of Eqs. (33). G± are no longer Lagrangian invariants 16.

     As in the 2D case, the absence of any physical dissipation poses some
difficulties, from the computational point of view, in dealing with the very small
scales that develop due to nonlinear interactions.
A numerical code based on a finite volume scheme was developed where filters,
based on a Fast Fourier Transform (FFT) algorithm, have been introduced acting
only on typical length scales much smaller than any other physical length scale of
the system.
  16                                                                R
     The number of Casimirs in the 3D case reduces to two C± = dxdydzG±
The cross helicity defined by the functional C = dxdydz(G2+ − G2− ) is also a constant of motion.
                                                  R

In the 2D case the x-y integral of any arbitrary function of G ± is a Casimir (conserved functional).


Dipartimento di Fisica                 Università di Pisa                 pegoraro@df.unipi.it
Collisionless reconnection                                                           [60]

    These filters smooth out the small scales below a chosen cut-off, while leaving
unchanged the large scale dynamics even on long times, 17.
In the code, the average values of the G± fields are advanced in time using
an explicit, third-order Adams-Bashforth scheme. In the simulations presented
here, due to the periodic boundary conditions, a FFT method can be adopted to
reconstruct the fields at the grid points. In order to address the three dimensional
problem, the code has been parallelized adopting the MPI libraries.
The simulations have been carried out in a 3D, triple-periodic slab, starting from
a static equilibrium configuration with ψeq (x) = A cos(x).
The integration domain is defined by −Lx < x < Lx, −Ly < y < Ly and
−Lz < z < Lz , with Lx = π, Ly = 2π and Lz = 16π, and runs have been
performed using nx = ny = nz = 128 modes along x, y, z, respectively.
For all the runs presented a fixed value of the microscopic parameters was chosen
%s = de = 0.24.
  17
       S.K. Lele J.Comp. Phys., 103, 16 (1992)


Dipartimento di Fisica                  Università di Pisa   pegoraro@df.unipi.it
Collisionless reconnection                                                         [61]

    The presence of unstable modes with different helicity modifies the nonlinear
behavior of the system significantly.
In the linear phase each mode evolves independently from the others as a single
helicity mode.
Nonlinearly a strong interaction occurs between the different helicity modes.

   Numerical results for a symmetric initial perturbation consisting of two modes
with helicities that differ only in sign:

      δJ(x, y, z; t) = Jˆ1(x) exp (iky y + ikz1z) + Jˆ2(x) exp (iky y + ikz2z).   (35)

with wave numbers (1, ±1), i.e. with wave vectors ky = 1/2, kz1 = 1/16 and
kz2 = −1/16: ψeq = 0.48 cos(x). With this choice the resonant surfaces are
located at xs1 = 0.26, π − 0.26 and xs2 = −0.26, −π + 0.26.


Dipartimento di Fisica         Università di Pisa        pegoraro@df.unipi.it
Collisionless reconnection                                                       [62]

   The nonlinearities in Eqs. (31,32) are quadratic and in the initial phase the
growth of perturbations with helicities different from those initially excited is in
good agreement with a quasilinear estimate.
Subsequently, the current layers related to the two main modes start to move
towards each other and, when their interaction becomes highly nonlinear, the two
current peaks merge.
Just before merging, the width of each of the two peaks is of the order of de: the
width of the resulting peak goes on decreasing from about 2de to below de.

    The change in the structure of the current density and vorticity layers during
the nonlinear evolution is shown in Fig. 7 which gives a three-dimensional view
of the topological modification of the current density during the transition from
the linear to the nonlinear phase.
This figure shows a current density isosurface, corresponding to a value close to
the maximum of J(x, y, z; t) at that time.


Dipartimento di Fisica       Università di Pisa       pegoraro@df.unipi.it
Collisionless reconnection                                                      [63]




Figure 7: Three-dimensional view of the current density and vorticity layers during
the nonlinear evolution.
Dipartimento di Fisica       Università di Pisa       pegoraro@df.unipi.it
Collisionless reconnection                                                                                       [64]

    This graphical representation allows us to identify the current layers with
positive current 18. Fig. 7 also shows the contour plots of the current density on
the sides of the box frame.
The left frame shows that in the linear phase at t = 170τA there are two separate
positive current layers oriented along the wave vectors of the initial perturbations.
These layers were initially located at x = ±0.26. Due to the periodicity conditions
along z, each layer appears as divided in two parallel branches.
The right frame shows that in the nonlinear phase at t = 205τA the two current
layers start to merge in the regions around the four points, where they have the
same values of y and z.




  18
    The current isosurfaces corresponding to a value of J close to the minimum, that arise because of the periodicity
in x, are not shown in this figure.


Dipartimento di Fisica                  Università di Pisa                  pegoraro@df.unipi.it
Collisionless reconnection                                                                   [65]

                                  Magnetic Field Stochasticity


    The equations for the magnetic field lines, dx/Bx = dy/By = dz/Bz , can be
cast in Hamiltonian form in which the Hamiltonian function is proportional to the
flux function H = ψ(x, y, z)/B0, x and y are conjugate variables, z is a “time”
variable, and the true time t is a parameter.
Then, the trajectories corresponding to the magnetic field lines are the solutions
of Hamilton’s equations:

                             dx   1 ∂ψ Bx                     dy    1 ∂ψ By
                                =       =    ,                   =−       =    .            (36)
                             dz   B0 ∂y   B0                  dz    B0 ∂x   B0

In the presence of non ideal modes with more than one helicity, the Hamiltonian
for the magnetic field lines is in general non integrable, since it describes a
dynamical system with 1+1/2 degrees of freedom.

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