Comisso - Plasmoid Instability in Time-Evolving Current Sheets

Plasmoid Instability in Time-Evolving Current
Sheets
Luca Comisso
Collaborators: M. Lingam, Y.-M. Huang, A. Bhattacharjee
Department of Astrophysical Sciences, Princeton University
and
Princeton Plasma Physics Laboratory
Los Alamos Astrophysics Seminar - August 24, 2017
Luca Comisso Los Alamos Astrophysics Seminar

Outline
I Background
Motivations to work on this topic
Magnetic connections and reconnection
Some classical reconnection models
I Plasmoid Instability in Sweet-Parker Current Sheets
Scalings assuming Sweet-Parker
Problems with this approach
I A More Complete Theory of the Plasmoid Instability
Principle of least time for plasmoids
Scaling laws
I Concluding Remarks
I Supplement: e↵ects of plasma viscosity

Why are we interested in magnetic reconnection?
Magnetic reconnection is thought to play a key role in many
phenomena in laboratory, space and astrophysical plasmas.
Solar ﬂares and CMEs
Multiwavelength image of the Sun
taken by Solar Dynamics Observatory
Magnetospheric substorms
Aurora Australis recorded from the
International Space Station

Sawtooth crashes
Interior of a tokamak device
with a hot plasma
Gamma-ray ﬂares
X-ray image of the Crab Nebula
taken by Chandra X-ray Observatory
Many other systems where magnetic reconnection is important!

It could be important also in the vicinity of black holes...
[see Asenjo & Comisso, PRL (2017)]
Artist’s conception of a supermassive black hole and accretion disk

Magnetic connections
Magnetic connections: if the ideal Ohm’s law E + v ⇥ B = 0
is satisfied, two plasma elements connected by a magnetic field
line at a given time will remain connected by a magnetic field
line at any subsequent time (Newcomb, 1958).
B
A
t
t Δt
B
A
+
E+v×B = 0
Relativistic generalizations:
Pegoraro, EPL (2012); Asenjo & Comisso, PRL (2015)

Magnetic reconnection
=t1t =t2t
A
C
D
B
A
C
B
D
E+v×B ≠ 0 E+v×B ≠ 0
Magnetic reconnection: process whereby the connectivity of
the magnetic field lines is modified due to the presence of a
localized di↵usion region.
I It results in conversion of magnetic energy into bulk kinetic
energy, thermal energy and super-thermal particle energy.
I It causes a topological change of the macroscopic magnetic
field configuration.

Sweet-Parker model
2L
2a
vout
vin B
I The Sweet-Parker model assumes a resistive current sheet,
2-D, @/@t ⇡ 0 and r · v ⇡ 0.
I It gives some important quantities as a function of the
Lundquist number S := LvA/⌘ :
a ⇡ L/
p
S , vout ⇡ vA , vin ⇡ vA/
p
S . (1)
For solar corona parameters S ⇠ 1014
, ⌧A = L/vA ⇠ 1 s
) ⌧SP = L/vin ⇠ 107
s (solar ﬂares last 102
103
s)

Several other models
Other models have been proposed to explain “fast” reconnection
Petschek model
(also Priest and collaborators)
2L
2a
2L*
vout
vin B
Turbulent models (e.g. LV99)
Collisionless models
(many contributions...)

Sweet-Parker at large S is too slow to be true...
I Sweet-Parker scalings do not hold for S = LvA/⌘ > Scritical
because of the plasmoid instability !
10
4
10
5
10
6
10
7
10
0
10
1
10
2
S
trec
=1e−3
=1e−4
=1e−5
S1/2
Bhattacharjee et al., PoP 2009
Huang & Bhattacharjee, PoP 2010
(also Uzdensky et al., PRL 2010)

Breakdown of the Sweet-Parker model at large S
I Speed-up of the reconnection process due to plasmoid
formation has been shown by many other research groups
Daughton et al., PRL 2009
Loureiro et al., PoP 2012
Comisso et al., PoP 2015
Ebrahimi & Raman, PRL 2015

Other important e↵ects of the plasmoid formation
I The formation of plasmoids has other crucial implications:
particle acceleration
self-generated turbulent reconnection
............
Drake et al., Nature 2006
I Sironi & Spitkovsky 2014
I Guo et al. 2014/15/16
I Werner et al. 2016, .....
Daughton et al., Nature 2011
I Oishi et al. 2015
I Huang & Bhattacharjee 2016
I .....

An important issue to address
I We have seen that the formation of plasmoids plays a
crucial role in magnetic reconnection
BUT
I What is the dynamical picture behind the onset and
development of the plasmoid instability?
2L
2a
B
vout
vin
I We will see why the previous knowledge of the plasmoid
instability was unsatisfactory
AND
I We will see what are the properties of this instability.

Assuming a Sweet-Parker aspect ratio...
I Tajima & Shibata, Plasma Astrophysics (1997)
(Note that the same result has been re-obtained 10 years later
by Loureiro et al., PoP 2007)

Let’s do the exercise!
I Tearing mode dispersion relation (Coppi et al., 1976):
Δ�δ≪�
Δ�δ≫�
��
γτ�
5/4
⌧
1/2
H ⌧3/4
⌘
⇥
⇤3/2 1 /4
⇤
⇥
⇤3/2 + 5 /4
⇤ =
8
⇡
0
⇤ := ⌧
2/3
H ⌧
1/3
⌘ , ⌧H = 1
kvA
, ⌧⌘ = a2
⌘
I For 0a ⇠ 2/ka (FKR 1963), the fastest growing mode has
kmaxa ⇠ S 1/4
a , max
a
vA
⇠ S 1/2
a .
I Using a ⇠ LS 1/2 and Sa ⇠ (a/L)S, one easily obtain
kmaxL ⇠ S3/8
, max
L
vA
⇠ S1/4
.

But there is a problem with this result...
The instability growth rate is too fast for large S-values!
Sweet-Parker sheets cannot form in large S plasmas!

What if we take into account plasma viscosity?
I Introducing the (perpendicular) Prandtl number Pm = ⌫/⌘:
a ⇡
L(1 + Pm)
1/4
p
S
, vout ⇡
vA
p
1 + Pm
, vin ⇡
vA
p
S(1 + Pm)
1/4
.
I Plasma viscosity modiﬁes the previous scalings as (Comisso
& Grasso, PoP 2016)
growth rate ) ⌧A ⇠ S1/4
(1 + Pm) 5/8
wavenumber ) kL ⇠ S3/8
(1 + Pm) 3/16
I decreases for increasing Pm, but... for realistic values of
Pm (10 4 ÷ 104) and S >>> 1, is still super-Alfv´enic!
I The chosen equilibrium current sheet (Sweet-Parker sheet)
can not be attained as current layers disrupt earlier!

Since in reality current sheets form over time...
We need to consider a time-evolving current sheet
t1
t2
t3
t4
t5

A plot to keep well in mind
I The plasmoid instability remains quiescent for a certain
time, and the ﬂuctuation amplitude starts to grow only
when max⌧A > O(1).
I Fast reconnection occurs at t > td.
=
Huang et al., submitted to ApJ (2017), arXiv:1707.01863

General Theory [Comisso et al., PoP 2016]
Tearing modes become unstable at
di↵erent times and exhibit di↵erent
instantaneous growth rates (k, t)
�
γ
��
��
��
I Their amplitude changes in time according to
(k, t) = 0(k) exp
⇣ Z t
t0
(k, t0
)dt0
⌘
.
I Their evolution becomes nonlinear when
w(k, t) = 2
r
a
B
> in(k, t) =
⇥
⌘ a2
/(kvA)2
⇤1/4
.
2a
B
2w 2δin

�
��
��δ��(��)
�(��)
I Principle of Least Time for the Plasmoid Instability, i.e.,
the mode of the plasmoid instability that emerges from the
linear phase is the one that traverses it in the least time.
I To implement this formulation, we introduce the function
F(k, t) := in(k, t) w(k, t) .
I Then, the least time principle is formulated as
F(k, t)|k⇤,t⇤
= 0 , dt⇤/dk⇤ = 0 .

I From Eqs. (2)-(2) we obtain the least time plasmoid Eqs:
⇢✓
¯t
1
2
◆
@
@k
+
k
+ 2
w0
@w0
@k k⇤,t⇤
= 0
(
ln
in
w0
g1/2
f1/2
!
1
2
Z t
t0
(t0
)dt0
)
k⇤,t⇤
= 0
with:
w0 = 2
q
0a0
B0
, ¯t = @
@
R t
t0
(t0)dt0 , f = a(t)
a0
, g = B(t)
B0
.
I From these Eqs. it is possible to arrive at:
⇤ (final growth rate)
k⇤ (final wavenumber)
in⇤ (final inner layer)
L⇤/a⇤ (final aspect ratio)
t⇤ (elapsed time from t0)
⌧p (time from ˆ(ˆk⇤, ˆton) > 1/⌧)

Until now the equations are very general.
Let’s try to be more specific...
I We are interested in the case where:
L ⇡ const., B0 ⇡ const., a(t) = a0f(t).
I For the moment, we consider an exponentially thinning
current sheet (this will be generalized later) of the form
â(ˆt)2
= (â2
0 â2
1)e 2ˆt/⌧
+ â2
1 , with â1 = S 1/2
I Here and in the following:
lengths normalized by the current sheet half-length L
time normalized by the Alfvén time ⌧A = L/vA
magnetic field normalized by the upstream field B0

I With some algebra we can see that there is a Transitional S
ST =
1
⌧4
"
ln
⌧9(2 ↵)/2
26"3 ˜↵˜↵
!#4
above which the plasmoid instability change behavior!
I The slope of ˆk⇤ substantially decreases at large S !!!!!!!!!!!
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▲ ▲ �� =��-�
▲ �� =��-��
▲ �� =��-��
��
��
��
��
��
��
��
�
�*
∝ ��/�
For S < ST :
ˆk⇤ ⇠ S3/8
For S > ST :
ˆk⇤ ' ck
S1/6
⌧5/6

ln
✓
⌧
S2
ˆa3
0
ˆw6
0
◆ 5/6

I The behavior of ˆ⇤ is non-monotonic in S !!!!!!!!!!!
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▲ �� =��-�
▲ �� =��-��
▲ �� =��-��
��
�
�
��
��
��
�
γ*
γ*τ=�
For S < ST :
ˆ⇤ ⇠ S1/4
For S > ST :
ˆ⇤ '
c
⌧
ln
✓
⌧
S2
ˆa3
0
ˆw6
0
◆
I The earlier scaling (black dashed line) is not applicable for
large-S plasmas.
I ˆ⇤ can decrease with S because ˆin decreases with S
) less “space” to accelerate the perturbation growth

I What about the disruption of the current sheet?
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▲ �� =��-�
▲ �� =��-�
▲ �� =��-��
▲ �� =��-��
��
��-�
��-�
��-�
��-�
��
�
�*
�-�/�
�-�/�
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▲ �� =��-��
▲ �� =��-��
��
��
��
��
��
��
��
��
�
�*â⇤ '
⌧2/3
S1/3

ln
✓
⌧
S2
â3
0
ˆw6
0
◆ 2/3
, ˆt⇤ ' ⌧ ln
(
â0
S1/3
⌧2/3

ln
✓
⌧
S2
â3
0
ˆw6
0
◆ 2/3
)
I Current sheets disrupt before the Sweet-Parker state can
be achieved (as expected!)

�
��
��δ��(��)
�(��)
I The emergent mode e↵ectively starts growing when
ˆ(ˆk⇤, ˆton) > 1/⌧
(Occurs when ˆa < ˆk⇤S 1/2⌧3/2)
I This leads to the intrinsic time scale of the instability
⌧p ' ⌧ ln
(
ck

ln
✓
⌧
S2
ˆa3
0
ˆw6
0
◆ 3/2
)
.

I To generalize the previous scaling laws, we consider the
generalized current thinning function
â(ˆt)2
= (â2
0 â2
1)
✓
⌧
⌧ + 2ˆt/
◆
+â2
1 , with â1 = S 1/2
I The final aspect-ratio ( 1
â⇤
) depends on the thinning process!
= 2 , = 4 , = 10 , ! 1
104 107 1010 1013 1016 1019 1022
10-9
10-7
10-5
0.001
S
a
`
*
[⌧ = 1, ˆw0 = 10 12
, â0 = 1/2⇡]
â⇤ ⇠ â
2⇣
0
⌧⇣
S
⇣
2
2
4ln
0
@ ⌧
3⇣
2
S
6+3⇣
4
â
3⇣
0
26"3
1
A
3
5
⇣
where ⇣ := 2
4+3

I And finally... also the scaling laws of the plasmoid
instability at large S depend on the thinning process!
ˆ⇤ ⇠ S(3⇣ 2)/4 ln(✓R)
â
2/
0 ⌧
!3⇣/2
,
ˆk⇤ ⇠ S(5⇣ 2)/8 ln(✓R)
â
2/
0 ⌧
!5⇣/4
,
în⇤ ⇠ S (3⇣+2)/8 â
2/
0 ⌧
ln(✓R)
!3⇣/4
,
where
✓R :=
⌧3⇣/2
S3(2+⇣)/4
â
3⇣/
0
26"3
.

Connections with experiments and observations
I Now we have a theory that can potentially predict the
onset of fast magnetic reconnection.
Opportunities to check the theory in the real world...
I The “easiest” quantities to check should be ˆa⇤ and ˆt⇤.
Ji et al., PoP (1999)
Lin et al., SSRv (2015)

Concluding Remarks
I The scaling laws of the plasmoid instability are not simple
power laws, and depend on:
The Lundquist number (S)
The noise of the system ( 0)
The characteristic rate of current sheet evolution (1/⌧)
The thinning process ( )
Viscosity (Pm) and kinetic (to be included) e↵ects.
I In astrophysical systems, reconnecting current sheets break
up before they can reach the Sweet-Parker aspect ratio.
The scaling laws of the plasmoid instability obtained
assuming a Sweet-Parker current sheet are inapplicable to
the vast majority of the astrophysical systems.
I The plasmoid instability comprises of a quiescence period
(ˆt⇤ ⌧p) followed by rapid growth over a shorter timescale
(⌧p).

Supplement: plasma viscosity [arXiv:1707.01862]
Also plasma viscosity could be important in several systems.
I Plasma viscosity allows to extend the validity of the
Sweet-Parker based scalings to larger S-values (ST ").
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▲ �� =�
▲ �� =��
��
�
��
��
��
�
γ*
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▲ �� =�
▲ �� =��
��
��
��
��
��
��
�
�*
ˆ⇤ ' ⌧ ln
⇣
⌧P
1/2
m
S2
ˆa3
0
ˆw6
0
⌘
, ˆk⇤ ' k
S1/6P
1/3
m
⌧5/6
h
ln
⇣
⌧P
1/2
m
S2
ˆa3
0
ˆw6
0
⌘i5/6

Supplement: plasma viscosity [arXiv:1707.01862]
I At large S, plasma viscosity allows to reach larger aspect
ratios of the reconnecting current sheets.
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▲ �� =��
��
��-�
��-�
��-�
��-�
�
�*
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▲ �� =�
▲ �� =��
��
��
��
��
��
��
��
��
��*
â⇤ ' ⌧2/3
S1/3P
1/6
m
h
ln
⇣
⌧P
1/2
m
S2
â3
0
ˆw6
0
⌘i 2/3
, ˆt⇤ ' ⌧ ln
⇢
S1/3P
1/6
m
⌧2/3/â0
h
ln
⇣
⌧P
1/2
m
S2
â3
0
ˆw6
0
⌘i2/3

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