plasmoid instability and fast magnetic reconnection

Plasmoid Instability and Fast MagneticReconnection

Luca Comisso

Department of Astrophysical Sciences, Princeton Universityand

Princeton Plasma Physics Laboratory

Astroplasmas Seminar - February 26, 2016

Acknowledgements: D. Grasso, F.L. Waelbroeck, Y.-M. Huang,

A. Bhattacharjee

Luca Comisso Astroplasmas Seminar

Outline

Brief Introduction

Sweet-Parker Model

Resistive Plasmoid Instability (Linear Regime)

Implications for Fast Resistive Reconnection

Visco-Resistive Plasmoid Instability (Linear Regime)

Nonlinear Regime of the Plasmoid Instability

Implications for Fast Visco-Resistive Reconnection

Condition to trigger collisionless reconnection

Summary of the main new results

If there is enough time: Plasmoid-Mediated Reconnectionin Taylor’s Model


Brief Introduction

Magnetic connections: if the ideal Ohm’s law is satisfied,two plasma elements connected by a magnetic field line at agiven time will remain connected by a magnetic field line at anysubsequent time [Newcomb, Ann. Phys (1958)].Recent relativistic generalizations:Pegoraro, EPL (2012); Asenjo and Comisso, PRL (2015).


Brief Introduction

Magnetic reconnection: the process whereby the connectionof the magnetic field lines is modified due to the presence of alocalized diffusion region.

It results in conversion of magnetic energy into bulk kineticenergy, thermal energy and super-thermal particle energy.

It causes a topological change of the macroscopic magneticfield configuration.


Brief Introduction

Magnetic reconnection is thought to play a key role in manyphenomena in laboratory, space and astrophysical plasmas.

Sawtooth crashes

Yamada et al., PoP (1994)

Magnetospheric substorms

Open magnetosphere morphology


Brief Introduction


Solar flares and CMEs

Multiwavelength image of the Sun

taken by Solar Dynamics Observatory

Gamma-ray flares

X-ray image of the Crab Nebula

taken by Chandra X-ray Observatory


Brief Introduction


Magnetar giant flares

Masada et al., PASJ (2010)


Brief Introduction

In many situations of interest, magnetic reconnectionoccurs within thin and strong current sheets.

We will first recall a simple model (Sweet-Parker) ofmagnetic reconnection within current sheets that isconsidered to be too slow to explain the fast energy releaserates observed in nature.

Then we will see how a plasma instability (plasmoidinstability) changes the standard Sweet-Parker picture,leading to fast magnetic reconnection rates.


Sweet-Parker Model

The Sweet-Parker model gives the reconnection rate andother relevant quantities for a quasi-stationary, 2-D,incompressible, resistive current sheet:

outflow velocity vd ≈ vAdownstream magnetic field Bd ≈ S−1/2B0

current sheet width δcs ≈ S−1/2Lcsreconnection rate Ez ≈ S−1/2vAB0 ← slow for high S

where S = LcsvA/η.


Sweet-Parker Model (with viscosity)

Park et al. [PoF (1984)] generalized the Sweet-Parkermodel, obtaining:

outflow velocity vd ≈ (1 + Pm)−1/2vAdownstream magnetic field Bd ≈ S−1/2(1 + Pm)1/4B0

current sheet width δcs ≈ S−1/2(1 + Pm)1/4Lcsreconnection rate Ez ≈ S−1/2(1 + Pm)1/4vAB0

where Pm = ν/η.


Brief Introduction

The Sweet-Parker model gives good predictions formoderately large Lundquist numbers S = LcsvA/η

Ji et al., PoP (1999)Ji et al., PoP (1999)


Sweet-Parker Model

...but its assumptions does not hold for larger Lundquistnumbers S = LcsvA/η > Sc.

Numerical Results

Biskamp, PoF (1986)


Resistive Plasmoid Instability (Linear Regime)

T. Tajima and K. Shibata (“Plasma Astrophysics”,Addison-Wesley, 1997) showed that very elongatedSweet-Parker sheets are subject to a fast plasmoid(tearing-like) instability with:

- growth rate γmaxLcsvA∼ S1/4

- wavenumber kmaxLcs ∼ S3/8

Important caution!

Note that in Tajima & Shibata scalings γmax →∞ for η → 0This cannot be true! A more general theory is required(Comisso et al., in preparation)In this presentation we will set this (important) issue aside



Fast reconnection in collisional plasmas is possible because ofthe plasmoid instability!

104

105

106

10−3

10−2

10−1

SL

(dψ

/dt)

/BV

A

SL

−1/2

Bhattacharjee et al., PoP (2009)



The reconnection layer may be considered as a chain ofplasmoids connected by marginally stable current sheets[Huang and Bhattacharjee, PoP 2010].

np ∼LcsLc∼ Sc−1S R ∼ ηB0

δc∼ S−1/2c B0vA


Other important effects of the plasmoid formation

The formation of plasmoids has other crucial implications:

particle accelerationself-generated turbulent reconnection............

Drake et al., Nature 2006

Sironi & Spitkovsky 2014

Guo et al. 2014/15/16

Werner et al. 2016

Daughton et al., Nature 2011

Oishi et al. 2015

Huang & Bhattacharjee 2016

.....



We generalize the linear analysis by Loureiro et al. [PoP(2007)] by considering arbitrary Pm.

Main steps:Consider a linear flow profile inside the current sheet

v0 =

(− vdLcs

x,vdLcs

y

)(1)

Evaluate the equilibrium magnetic field B0 self-consistentlyLinearize the visco-resistive MHD equations and search forperturbations with growth rate

γ � vdLcs

(2)

Perform a tearing stability analysis by considering

λ =γ

kvA� 1 , ε =

δcsLcs� 1 , κ = kLcs � 1 (3)

but also κε� 1.



Main results [Comisso and Grasso, to appear, PoP]:

Growth rate

γmaxLcsvA∼ α3/2S1/4

(1 + Pm)5/8

, (4)

Wavenumber

kmaxLcs ∼α5/4S3/8

(1 + Pm)3/16

, (5)

Inner layer width

δin,max

δcs∼ (1 + Pm)

1/16

α3/4S1/8. (6)

Therefore, plasma viscosity has the effect of:

decreasing the growth rate γmax

decreasing the wavenumber kmax

increasing (slightly) the inner layer width δin,max


Critical threshold

The stabilizing effect of the flow becomes ineffective forcurrent sheets exceeding the critical aspect ratio

Lcδc

=1

εc(7)

It follows that [Comisso and Grasso, to appear, PoP]

Sc = ε−2c (1 + Pm)1/2 (8)

Due to the convective nature of the plasmoid instability, itis not possible to obtain a clear-cut value of εc [see, e.g.,Huang and Bhattacharjee, PoP (2013)].

Note that plasma viscosity allows us to extend the validityof the (viscous) Sweet-Parker equilibrium assumption(related to the previously mentioned issue).



The previous linear analysis breaks down when

w & δin,max (9)

To ascertain the proper nonlinear regime, we note that forthe fastest growing mode [Comisso and Grasso, to appear,PoP]

∆′maxδcs ≈2α2

κmaxε∼ 2α3/4S1/8

(1 + Pm)1/16(10)

The plasmoid half width is w ∼ δin,max at the beginning ofthe nonlinear evolution, therefore

∆′maxw ∼ 2 (11)

when the plasmoids enter into the nonlinear regime.This implies that the plasmoids evolve according to:× Rutherford regime [PoF (1973)] (∆′maxw � 1)X Waelbroeck regime [PoF B (1989)] (∆′maxw & 1)



Waelbroeck regime implies that the inter-plasmoidsX-points collapse to form thin inter-plasmoids currentsheets soon after entering the nonlinear regime.

The nonlinear growth of the plasmoids can be determinedby matching a model of reconnection within current sheetswith Waelbroeck’s solution [PoF B (1989)] for the magneticconfiguration of rapidly reconnecting islands.



The temporal rate of change of magnetic flux at theseparatrix of a plasmoid can be evaluated as

dψsdt≈ η

B∗yδ∗cs

, (12)

Estimating δ∗cs as in [Park et al. PoF (1984)] it follows

dψsdt≈ η1/2

(1 + Pm)1/4

(B∗3yL∗cs

)1/2

. (13)

For w � 1/∆′max (but w < δcs), Waelbroeck’s solution forrapidly reconnecting islands tells us that the plasmoidsbecome self-similar with a magnetic configuration given by

w ∼(ψsα

)1/2

, B∗y ∼ αw , L∗cs ∼1

kmax. (14)



Substituting these relations into Eq. (13), we obtain theplasmoid-width nonlinear evolution equation

dw

dt∼ 1

2

[ηα

(1 + Pm)1/2kmaxw

]1/2

(15)

This equation yields the algebraic growth law [Comisso andGrasso, to appear, PoP]

w

δcs∼ α9/4

16

S3/8

(1 + Pm)19/16

(t

τA,L

)2

(16)

The growth of the plasmoids slows down from theexponential growth of the linear stage to an algebraic(quadratic) growth in time.



The obtained growth law in the nonlinear regime sets thetime scale for plasmoids to growth from w ∼ δin,max tow ∼ δcs:

τNL ∼ S−3/16(1 + Pm)19/32τA,L (17)

Note that for extremely large S-values also the secondarycurrent sheets connecting the plasmoids may themselves besubject to the plasmoid instability if

δ∗csL∗cs∼

[δcsw

(1 + Pm)5/16

S5/8

]1/2

< εc . (18)

This cascade process speeds up the current sheet disruption


Implications for fast magnetic reconnection

The global reconnection rate may be evaluated as the rateof change of the flux reconnected at the main X-point.

In the plasmoid-dominated regime the reconnection processis strongly time dependent, with plasmoids constantlybeing generated, ejected and merging each others.

We may assume a statistical steady-state with a marginallystable current sheet located at the main X-point [Huangand Bhattacharjee PoP (2010), Uzdensky et al. PRL(2010)].



In statistical steady state⟨dψ

dt

∣∣∣∣X

⟩≈ ηBy

δc(19)

For a visco-resistive sheet

δc = εcLc =

(ηLcvd

)1/2

, vd = vA(1 + Pm)−1/2 (20)

We can use the global reconnecting magnetic field Bu as anestimation of By.

Recalling also that Sc depends on Pm as

Sc = ε−2c (1 + Pm)1/2 , (21)

the global (time-averaged) reconnection rate become⟨dψ

dt

∣∣∣∣X

⟩≈ εc

vABu

(1 + Pm)1/2. (22)



In the limit Pm � 1 we recover Huang and BhattacharjeePoP (2010) and Uzdensky et al. PRL (2010) result:

⟨dψ

dt

∣∣∣∣X

⟩≈ εcvABu

εc ∼ 0.01

104

105

106

10−3

10−2

10−1

SL

(dψ

/dt)

/BV

A

SL

−1/2

Bhattacharjee et al., PoP 2009



In the more general case of non-negligible plasma viscosity,the global reconnection rate depends on the microscopicplasma parameters through Pm = ν/η:

⟨dψ

dt

∣∣∣∣X

⟩≈ εc

vABu

(1 + Pm)1/2

εc ∼ 0.01

Comisso, Grasso, Waelbroeck, PoP 2015


Condition to trigger collisionless reconnection

The fractal-like cascade process caused by the plasmoidistability produces local reconnection layers that may be inthe collisionless regime even if

δcs � lk , where lk =

{di = c/ωpiρτ = cs/ωci

(23)

This transition occurs if [Daughton et al., PRL (2009)]

δc . lk . (24)

By rewriting δc [Eq. (20)] as

δc ≈ Lcs(1 + Pm)1/4(Sc/S2)1/2 , (25)

the condition to trigger the collisionless regime becomes[Comisso and Grasso, to appear, PoP]

S &Lcsεclk

(1 + Pm)1/2 . (26)



In a visco-resistive MHD plasma, the linear plasmoidinstability is characterized by

γmaxLcsvA∼ α3/2S1/4

(1 + Pm)5/8(27)

kmaxLcs ∼α5/4S3/8

(1 + Pm)3/16(28)

δin,max

δcs∼ (1 + Pm)1/16

α3/4S1/8(29)

The threshold for the plasmoid instability is

Sc = ε−2c (1 + Pm)1/2 (30)



The nonlinear growth of the plasmoids is quadratic in time

w

δcs∼ α9/4

16

S3/8

(1 + Pm)19/16

(t

τA,L

)2

(31)

In the plasmoid dominated regime, the (time-averaged)reconnection rate is⟨

dψ

dt

∣∣∣∣X

⟩≈ εc

vABu

(1 + Pm)1/2(32)

The condition to trigger the collisionless regime is

S &Lcsεclk

(1 + Pm)1/2 (33)


Plasmoid-Mediated Reconnection in Taylor’s Model

⇒ L. Comisso, D. Grasso, F.L. Waelbroeck, Phys. Plasmas 22, 042109 (2015)

Magnetic reconnection in a given system is conventionallycategorized as spontaneous or forced/driven.

Spontaneous magnetic reconnection refers to the cases inwhich the reconnection arises by some internal instabilityof the system or loss of equilibrium.

Most typical paradigm: Tearing mode

Forced/Driven magnetic reconnection refers to the cases inwhich the reconnection is driven by some externallyimposed flow or magnetic perturbation.

Most typical paradigm: Taylor problem



Assume a tearing-stable slab plasma with an equilibriummagnetic field of the form B = Bzez +B0 (x/L) ey

Suppose that the conducting walls are subject to a suddendisplacement xw → ±L∓ Ξ0 cos(ky)

Determine the evolution of the forced reconnection process



It is possible to show that a nonlinear current sheet occursif [Fitzpatrick, PoP (2003), Comisso et al., PoP (2015) andJPP (2015)]

Ψ0 & ΨW = B01

∆′sτ−1/3η

(1 +

τντη

)−1/6( τAkL

)1/3

︸︷︷︸ΞW

(34)

where

τA =L

vA, τη =

L2

η, τν =

L2

ν(35)

∆′s =2k

sinh(kL)

(dψ1

dx

∣∣∣∣0+0−

= ∆′0ψ1(0) + ∆′sΨ0

)(36)



It is also possible to show that the reconnecting currentsheet is sufficiently narrow to undergo the plasmoidinstability if the amplitude of the perturbation is such that[Comisso et al., PoP (2015) and JPP (2015)]

Ψ0 > Ψc = B0CkL

∆′s

τAτη

(1 +

τητν

)1/2

︸︷︷︸Ξc

(37)

where

C ∼ 2ε−2c , εc =

δcLc∼ 10−2 (38)



The dynamical evolution of the forced reconnection processleads to a plasmoid-dominated regime if

Ψ0

{> Ψc, if Ψc & ΨW

& ΨW , if Ψc < ΨW(39)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5k0.000

0.002

0.004

0.006

0.008

0.010

YW , Yc

Comisso, Grasso, Waelbroeck, PoP (2015)



Depending on the external source perturbation and themicroscopic plasma parameters, three possible evolutionscan occur in an MHD plasma:

Ψ0 � ΨW

HK scenario (1985)

ΨW . Ψ0 < Ψc

WB scenario (1992)

Ψ0 & ΨW ∧ Ψ0 > Ψc

CGW scenario (2015)

The (time-averaged) reconnection rate in statisticalsteady-state is [Comisso, Grasso, Waelbroeck, PoP (2015)]⟨

dψ

dt

∣∣∣∣X

⟩≈ εcB0L(∆′sΞ0)

2τ−1A

(1 +

τητν

)−1/2(40)


plasmoid instability and fast magnetic reconnection

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