# 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.

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
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
   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

bottom

middle
top

+
Simulations of Parker's
model

bottom

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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