Current Sheet and Vortex Singularities Drivers of Impulsive by coryelJudie

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									Current Sheet and Vortex Singularities:
  Drivers of Impulsive Reconnection

        A. Bhattacharjee, N. Bessho, K.
   Germaschewski, J. C. S. Ng, and P. Zhu
             Space Science Center
 Institute for the Study of Earth, Oceans, and
     Space University of New Hampshire

Isaac Newton Institute, Cambridge University, August
                       9, 2004
                 High-Performance Computing Tools
     • Magnetic Reconnection Code (MRC), based on extended two-
       fluid (or Hall MHD) equations, in a parallel AMR framework
       (Flash, developed at the University of Chicago).
     • EPIC, a fully electromagnetic 3D Particle-In-Cell code, with
       explicit time-stepping.
     MRC is our principal workhorse and two-fluid equations
       capture important collisionless/kinetic effects. Why do we
       need a PIC code?
     • Generalized Ohm’s law

                       1       2 dJ di            
            E  v  B  J  de      J  B    pe 
                     S         dt n
     • Thin current sheets, embedded in the reconnection layer, are
       subject to instabilities that require kinetic theory for a complete
       description.

Adaptive Mesh Refinement
                   Effectiveness of AMR
                                                   High effective resolution
Example: 2D MHD/Hall MHD


Efficiency of AMR

Level    # grids   # grid points
0              1          70225
1             83        146080
2           103         268666
3           153         545316
4           197        1042132
5           404        1926465
6           600        1967234



Grid points in adaptive simulation:      6976118
Grid points in non-adaptive simulation:268730449
Ratio                                     0.02
Impulsive Reconnection: The Trigger
Problem
Dynamics exhibits an impulsiveness, that is, a
sudden change in the time-derivative of the
reconnection rate.
 The magnetic configuration evolves slowly for a
 long period of time, only to undergo a sudden
 dynamical change over a much shorter period of
 time.
 Dynamics is characterized by the formation of
 near-singular current sheets in finite time.
Examples
Sawtooth oscillations in tokamaks and RFPs
Magnetospheric substorms
Impulsive solar and stellar flares
Sawtooth crash in tokamaks (Yamada et al.,
                   1994)
Sawtooth events in MST (Almagri et al., 2003)
Magnetospheric Substorms
Current Disruption in the Near-Earth
            Magnetotail




           (Ohtani et al., 1992)
Impulsive solar/stellar flares
               2D Hall MHD: m=1 sawtooth instability
Two-Field Reduced Model for large guide field and low
plasma beta




                                     (Schep/Pegoraro/Kuvshinov 1994)
                                     (Grasso/Pegoraro/Porcelli/Califano
                                                                 1999)

Equilibrium:
Resolving the current sheet




       zoom                   zoom
                                                   current density
                  Current sheet collapse, s = 0




1/current sheet
width




                               t
                   magnetic flux function




Island width




               t
                        Island equation
      d    cJ k 2 4
              Ý
         
                
                2   2 2
      dt  L x  4de  x

            Ý        2                         s 
                                                         2
             
      Ý    x   s  or  x   L exp 
      x         1                              c d 2 
                                                     1       
                  c d 2 
             L                              L 
                      J e                           J e 
                                                          2
                                                         de / 3 1/ 3
If x and L attain constant values and are of order of de (or s
), the island equation becomes (Ottaviani & Porcelli 1993 for s
= 0 ): 2
            ˆ
        d  1 ˆ
                c  ˆ4
        dtˆ 2 4
                                   1/ 3 2 / 3 ˆ
   with t   L t ,  L  kde or kde  s ,   /  L .
         ˆ
       Island equation c.f. simulation




Solid: simulation, dashed: island equation, cJ = 0.025,
s  0.2, de  0.1, k  0.5,  L  0.0024
    Scaling of the reconnection rate: Is it
                 Universal?
Consider scaling of the inflow       V in~ fVAd
velocity:
It has been argued that f ~ 0.1 [Shay et al., 2004], in a
universal asymptotic regime, independent of system
                              
size and dissipation mechanism.
Using the island equation in the asymptotic regime:
                          c J  L k3 / 2
                    f ~
                           10 de
                                     2
                                    de / 3 1/ 3
Note that L is of the order of de (or s        ).
f depends on parameters de, s      and k. It also depends
weakly on time through (t) ~ 1 in the nonlinear phase.
Numerically f is seen to be of the order of 0.1 for
                   
certain popular cloices of simulation parameters, but
this is not universal.
Magnetospheric Substorms
Observations (Ohtani et al. 1992)




 Hall MHD Simulation




  Cluster observations: Mouikis
  et al., 2004
               Bx  B0tanh(z /a)
               vA /c  0.2 a  L z /2 mi /me  25        J y y
               Ti  5Te di / a  0.1
               E yb  0.01B0 (vA /c)(1 cos  x / Lx )
Ey/B0 (c/vA)




               0.02
               0.01
                 0
                      0   10      20      30      40
                                 t/A
                  2
  /(B0a)




                  1

                  0
                      0    10      20      30      40
                                  t/A
Ux   Uz




          t=5.8   Highly Compressible
                  Ballooning Mode
                  in Magnetotail
                  (Voigt model)

                  x = -1 to -16 RE
                  z = -3 to 3 RE
                  ky= 25*2
                  e = 126
                  growth rate:0.2
          t=29
ky and  Dependence
Euler equation for incompressible flow:


   v
       v  v  p
   t
    v  0

The incompressibility condition produces a
Poisson’s equation for pressure

    2 p   (v  v)


Integro-differential equation for pressure

                1   (v  v)
    p(x,t)                   dx  + b.c.
               4   x  x 


Solutions to Euler are essentiallynonlocal.
                           t=.49
High-symmetry flow (Pelz
        1997)
   t=0            t=.33
                 Vorticity in the high-symmetry flow


Vorticity
2D and 1D cuts
           Growth of
           vorticity




Distortion of
vortices
vorticity   pressure
Sufficient condition for finite-time
singularity
Consider flow along x -axis ( y  z  0). By
Kida symmetry vy  vz  0,   0
Euler equation         v x   px
                       Ý
Define    x vx ,  =  y vy ,  =  z vz
(      0)
Differentiating Euler, obtain exact equation
  2   p xx
Ý
If p xx  0 following a Lagrangian element,
then  will be singular in finite time.
Sufficient condition (Eulerian)
Origin is a
    • velocity null point
    • point of intersection of vorticity null
lines


Kida symmetries ensure
p x (0)  pxx (0)  p xxx (0)  0
From the exact equation (along y  z  0),
by Taylor expansion
                           pxxxx (0) 2
  2   p xx (x)  
Ý                                   x  O(x 4 ),
                              2
where    x vx .
If there exists a range 0  x  X(t) in which
p xxxx is positive and increasing rapidly
enough, there is a finite-time singularity.
Resistive Tearing Modes in 2D Geometry
Equilibrium Magnetic field assumed to be either infinite
or periodic along z
                                          B  xBP tanhy / a  BT z
                                              ˆ                  ˆ
                   x
                                              Neutral line at y=0

                               y
Time Scales         A  a/VA  a41/2 /BP
                           
                    R  4a 2 /(c 2 )

          
Lundquist Number       S   R / A
          
Tearing modes      S 3 / 5 (slower) or S 1 / 3 (faster)
          
                            (Furth, Killeen and Rosenbluth,
                            1963)
     
      Collisionless Tearing Modes at the
   Magnetopause (Quest and Coroniti, 1981)

• Electron inertia, rather than resistivity, provided the
  mechanism for breaking field lines, but the magnetic
  geometry is still 2D.
• Growth rates calculated for “anti-parallel”T( 0
                                              B        ) and
  “component” ( T  0 ) tearing. It was shown that
                 B
  growth rates for “anti-parallel” merging were
  significantly higher.
                                     
• Model provided theoretical support for global models
        
  in which global merging lines were envisioned to be
  the locus of points where the reconnecting magnetic
  fields were locally “anti-parallel” (e.g. Crooker, 1979)
Magnetic nulls in 3D play the role of X-points in 2D
                                           Spine




                                     Fan




                               (Lau and Finn, 1990)
    Towards a fully 3D model of reconnection
• Greene (1988) and Lau and Finn (1990): in 3D, a topological
  configuration of great interest is one that has magnetic nulls with
  loops composed of field lines connecting the nulls.
• The null-null lines are called separators, and the “spines” and
  “fans” associated with them are the global 3D separatrices where
  reconnection occurs.
• For the magnetosphere, the geometrical content of these ideas
  were already represented in the vacuum superposition models of
  Dungey (1963), Cowley (1973), Stern (1973). However, the
  vacuum model carries no current, and hence has no spontaneous
  tearing instability. The real magnetopause carries current, and is
  amenable to the tearing instability.
Equilbrium B-field
Perturbed B-lines
Equilibrium




 Perturbed




              Field lines
              penetrating the
              spherical tearing
              surface
Equilbrium Current Density, J
 Features of the spherical tearing mode
• The mode growth rate is faster than classical 2D
  tearing modes, scales as S-1/4 (determined
  numerically from compressible resistive MHD
  equations).
• Perturbed configuration has three classes of field
  lines: closed, external, and open (penetrates into the
  surface from the outside).
• Tearing eigenfunction has global support along the
  separatrix surface, not necessarily localized at the
  nulls.
• The separatrix is global, and connects the cusp
  regions. Reconnection along the separatrix is
  spatially inhomogeneous. Provides a new framework
  for analysis of satellite data at the dayside
  magnetopause. Possibilities for SSX, which has
  observed reconnection involving nulls.
          Solar corona




astron.berkeley.edu/~jrg/ ay202/img1731.gif   www.geophys.washington.edu/
                                              Space/gifs/yokohflscl.gif
   Solar corona: heating
   problem
               photosphere          corona
                       3              6
Temperature     ~ 5 10 K         ~ 10 K
                                      12 3
Density         ~ 1023 m 3       ~ 10 m
                    4             ~ 20s
Time scale      ~ 10 s
Magnetic fields (~100G) --- role in heating?
     Alfvén wave
     current sheets
      Parker's Model (1972)
Straighten a
curved magnetic loop




 Photosphere
   Reduced MHD equations
               J
     [ ,  ]      [ A, J]     
                                    2
                                          low 
t               z
A                                      limit of
    [ , A]       2 A               MHD
t              z
B z  B  z   A  z --- magnetic field,
     ˆ          ˆ         ˆ
v     z --- fluid velocity,
              ˆ
    --- vorticity,
          2

         2
J   A --- current d ensity ,
 --- resistivity,  --- viscosity,
[ , A]   y Ax   x Ay
       Magnetostatic equilibrium
     J                             (current density
         [ A, J]  0 ,or B J  0 fixed on a field line)
     z
with  =   0. Field-lines are tied at z  0 , L .
                      
c. f. 2D Euler equation  [ ,  ]  0
                       t
 A , J , z t
 Existence theorem: If  is smooth initially, it is so for all
 Time. However, Parker problem is not an initial value
 problem, but a two-point boundary value problem.
      Footpoint Mapping
x  (z)  X[x  (0), z], x  (L)  X[x  (0), L],
     dX A                   dY    A
with       (X,Y, z) ,            ( X,Y,z)
     dz  y                  dz    x
Identity mapping:x  (L)  x  (0)
       e. g. uniform field B  z or A  const
                               ˆ
For a given smooth footpoint mapping, does
more than one smooth equilibrium exist?
     A theorem on Parker's
     model
For any given footpoint mapping
connected with the identity mapping,
there is at most one smooth equilibrium.
A proof for RMHD, periodic boundary condition in x
                      (Ng & Bhattacharjee,1998)
        Implication
An unstable but smooth equilibrium cannot relax to a
second smooth equilibrium, hence must have current sheets.
              In 3-D torii (such as stellarators), current
       singularities are present where field lines close on
                                  themselves
                                                        
                                                   
      • Solutions to the quasineutrality equation,   J  0,
                                   J||
          B      J
                            
                                        B
                   JB
                2  mn mne im in ,
                    B                  
              mn (m  n ) mne im in   p'  mnmn amne im in
                     
      •   General solution has two classes of singular currents at rational surface
          [ymn) = n/m] (Cary and Kotschenreuther, 1985; Hegna and
          Bhattacharjee, 1989).
                        mn amn    ˆ
           mn  p'             (y  y mn )
                       (m  n )
                                     mn



           – Resonant Pfirsch-Schluter current
           – Current sheet
 
 Reconnection without nulls or closed field
 lines
Formation of a true singularity (current sheet) thwarted
by the presence of dissipation and reconnection.
 Classical reconnection geometries involve:
 (i) closed field lines in toroidal devices
 (ii) magnetic nulls in 3D
Parker’s model is interesting because it fall under
neither class (i) or (ii). Questions:
• Where do singularities form? What are the geometric
properties of singularity sites and reconnection? Strategic
issue for CMSO research.
• Need more analysis and high-resolution simulations.
Incompressible spectral element MHD code (under
development by F. Cattaneo) may be very useful.
    Simulations of Parker's
    model
   Start with a uni form B field
   Apply constant footpoint t wisting
     (x  ,0)  0,  (x  , L)   0 (x  ) with
    [ 0 , 2  0 ]  0
            
   Current l ayers appear after large distortion.
   Quasi-equilibrium at first, becomes unst able
   J grows faster in the middl e  non
    equilibrium
         Simulations of Parker's
         model

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middle
                                   top




                        +
         Simulations of Parker's
         model

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middle
                                   top




                        +
        Simulations of Parker's
        model
t        J max   J max0 q 2        d   2
                                              qmax    dmax

118.5 9.91       9.86   0.233     0.00028     7.946   0.708
131.6 16.5 10.9         0.930     0.00238     16.04   2.272
145.6 28.3 14.6         4.361     0.1471      32.94   5.997
150.5 35.3 16.4         21.98     13.486      83.39   79.01

with q   J /  z  [ A, J ], d  q   2  .
                                          
More general topology
             Parker's optical
             analogy (1990)
Main current sheet



                        Qu i ckTi m e™ a nd a
                        GIF de co mp re s so r
               a re ne ed ed to se e th is pi c tu re.

								
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