High Lundquist Number Resistive
MHD Simulations of Magnetic
Reconnection: Searching for
Secondary Island Formation
C. S. Ng and S. Ragunathan
Geophysical Institute, University of Alaska Fairbanks
5th Annual International Conference on Numerical Modeling of Space Plasma Flows (ASTRONUM-2010)
San Diego, California, U.S.A., June 14, 2010
Abstract
Recently, secondary island formation due to the tearing instability of the Sweet-Parker
current sheet was identified as a possible mechanism that can lead to fast reconnection
(less sensitive dependence on Lundquist number S) both in numerical simulations using
Particle-in-Cell (PIC) method [Daughton et al. 2009], as well as using resistive
magnetohydrodynamics (MHD) [Lapenta 2008; Bhattacharjee et al. 2009]. This
instability is thought to appear when S is greater than a certain threshold. These recent
results prompt us to perform more resistive MHD simulations of a basic reconnection
configuration based on the island coalescence instability, using much higher resolutions
and larger S, since secondary island formation has not been observed before for S
smaller than the order of 104. Our simulations are based on a fairly standard pseudo
spectral code, which has been well tested for accuracy, convergence, and compared
well with codes using other methods [Ng et al. 2008]. Based on the experience in our
simulations, we argue that a highly accurate simulation is important since secondary
islands have been observed to form due to numerical inaccuracy. Latest results based
on simulations with S up to 2 × 105 will be presented, as well as simulations
superimposed with turbulence.
This work is supported by a NASA grant NNX08BA71G and a NSF grant
AGS-0962477.
“Slow” Sweet-Parker reconnection
From [Gurnett and Bhattacharjee, 2005]
 Slow reconnection rate: U in = 2VA /S1/ 2
(S = µ0VA L /η)
 Thin and long current sheet: Δ = 2L /S1/ 2
 Reconnection rate might be higher in Hall MHD or kinetic theory,
€
but is it possible to have fast reconnection€within resistive MHD?
€
Recent results on fast reconnection in resistive MHD
From [Lapenta, 2008]
Recent results on fast reconnection in resistive MHD
From [Loureiro et al., 2009]
Recent results on fast reconnection in resistive MHD
From [Bhattacharjee et al., 2009]
Secondary island (plasmoid) formation in resistive MHD
From [Samtaney et al., 2009]
Secondary island (plasmoid) formation in resistive MHD
From [Bhattacharjee et al., 2009]
Reconnection in 2D island coalescence instability
 Resistive MHD:
∂Ω
+ [φ, Ω ] = [A,J] + ν∇ 2⊥ Ω ,
∂t
∂A
+ [φ, A] = η∇ 2⊥ A
∂t
 Similar to the system simulated in [Bhattacharjee et al. 2009].
€
€
Resistive 2D incompressible MHD equations
A few cases with η >> ν has been used that show very
similar dynamics.
Boundary and initial conditions
Simulation method
High accuracy in the pseudo spectral method
The accuracy of this code
is checked with respect to
how well it preserves
conservation laws. In this
run, the fractional
difference between the
two sides of the following
energy conservation
equation is below 10-5:
d
( EK + E M ) =
dt
−ν Ω2 − η J 2
t
A run with η = ν = 5 × 10-5 using a
resolution of 20482.
€
High accuracy in the pseudo spectral method
The fractional difference
between the two sides of
the follow conservation
equation for the magnetic
helicity is below 10-6:
d A2
dt
t
A run with η = ν = 5 × 10-5 using a
resolution of 20482.
€
= −ηE M
Resolution and resistivity used in resolved runs
Note that
S = Bmax L / η
= 2 πA0 / η ≈ 2.51/ η
Highest S so far is
∝η
€
−1/2
Smax ≈ 2 ×10 5
using a resolution of
8192 2.
€
€
∝ η −1
€
Reconnection rate vs. resistivity η
The reconnection rate
(peak value) scales
roughly with η1/2,
consistent with the
Sweet-Parker theory.
∝ η1/2
€
No sign of leveling
off or increasing of
the reconnection rate
at the small resistivity
limit.
A typical time series of the reconnected flux The reconnection rate
(peak value) is the
maximum of the slope of
the time series.
t
A run with η = ν = 5 × 10-5 using a
resolution of 20482.
Even finer scales appears at small resistivity
At larger resistivity, the
current sheet width is the
finest scale of the system
and it scales with η1/2 as
predicted by the SweetParker theory.
lJ
∝ η1/2
€
At smaller resistivity
below ~ 10-4, fine
structures thinner than the
current sheet width
appear and scale roughly
with η (or even thinner).
Even finer scales appears at small resistivity
The same is true for the
small scales in the
vorticity field.
This means that even
higher linear resolution
(proportional to η-1) is
needed to resolve all
fine scales.
lΩ
∝ η1/2
€
Finer structures in the reconnection out flow
J-contours
The finest structures are
found at the location
when the reconnection
out flow collides with
another island.
Note also that these
structures can have
orientations different
from the reconnection
current sheet.
y
x
A run with η = ν = 1.25 × 10-5 (S ~ 2 × 105)
using a resolution of 81922.
No secondary island has
been found along the
reconnection current
sheet in resolved runs.
Absence of secondary island
[Courtesy of Yi-Min Huang]
Our run with η = ν = 1.25 × 10-5 (S ~ 2 × 105) using a resolution of 81922.
No secondary island is found in our run.
Secondary islands in under-resolved cases
We have seen many cases of “secondary island” formation (along with increased
reconnection rate) when the runs are not well resolved. No secondary island is
found when resolution is sufficiently increased using the same η and ν.
Secondary islands in turbulent cases
dA0/dt
(c)
(a)
t
(b)
If we add random noise throughout the simulation,
secondary islands do form as shown in (a) for a case
with η = ν = 3 × 10-4 (S ~ 8.4 × 103) using 2562, while
there is no secondary island for the no-noise case in
(b). The reconnection rate can spike up at times as
shown in the black curve in (c) as compare with the
case with no-noise (red). However, the total
reconnection time for the two cases are about the
same, as shown in (d).
A0
(d)
t
Secondary islands in turbulent cases
dA0/dt
(c)
(a)
t
(b)
Similarly, adding random noise for higher S,
secondary islands also form as shown in (a) for a case
with η = ν = 2 × 10-4 (S ~ 1.3 × 104) using 5122, while
there is no secondary island for the no-noise case in
(b). The reconnection rate can spike up at times as
shown in the black curve in (c) as compare with the
case with no-noise (red). Now, the total reconnection
time for the case with noise is shorter.
A0
(d)
t
Secondary islands in turbulent cases
dA0/dt
(c)
(a)
t
(b)
Again, adding random noise for even higher S,
secondary islands form as shown in (a) for a case with
η = ν = 10-4 (S ~ 2.5 × 104) using 10242, while there is
no secondary island for the no-noise case in (b). The
reconnection rate spikes up even more as shown in the
black curve in (c) as compare with the case with nonoise (red). Now, the total reconnection time for the
case with noise is getting even shorter comparatively.
A0
(d)
t
Conclusions
 Numerical simulations of magnetic reconnection in the configuration of island
coalescence instability have been performed systematically using a well-tested
pseudo spectral 2D MHD code for a range of resistivity and resolution, with the
highest Lundquist number at about 2 × 105, using a resolution of 81922.
 The reconnection rate at the small resistivity limit is found to be consistent
with the Sweet-Parker theory.
 No secondary island is found in all resolved runs, although the highest
Lundquist number is at the similar level as in other studies that showed
secondary island formation and increased reconnection rate.
 Secondary islands are found in cases with insufficient resolution. Such
formation is absent when the resolution is sufficiently increased.
 Fine scales that scale roughly with η is found in the small η limit. This means
that even higher linear resolution (proportional to η-1) is needed to have resolved
runs.
 Secondary islands can form in simulations when random noise is imposed. For
higher S cases, reconnection rate becomes higher and reconnection time shorter.