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Fast Secondary Reconnection in the Sawtooth Crash

Daniele Del Sarto 1, Maurizio Ottaviani 2

1 Université de Lorraine, Institut Jean Lamour UMR-CNRS 7198, Nancy (France)

2 CEA-IRFM, Saint-Paul-lez-Durance (France)

(8th IAEA Technical Meeting: Theory of Plasma Instabilities, Wien, 12th June 2017 )

Problem:

why is the sawtooth crash so “fast” at large Lundquist number S=τR/τA?

τR = a2/η Resistive timeτA = a/VAlfven Alfven time

a current sheet thickness(L current sheet length)

Outline of the presentation

Logic of the approach and heuristic assumptions.

Context of the problem : -) resistive internal kink mode and sawtooth crash. -) secondary (tearing-type) reconnecting instability in a current sheet generated by a primary tearing-type mode.

Application to a primary resistive m/n=1/1 mode (internal kink)

Conclusions

The m=1 mode and the sawtooth crash

The sawtooth crash time in JET is shorter than the collision time. [Edwards et al., (1986)]

This process is too fast to be explained by resistive MHD

The m=1 mode: an essential element of the sawtooth cycle [Von Goeler et al.(1974), Kadomtsev(1975)]

Process mediated by the restivity η. Becomes slow at large S, small η

Non-collisional physics (electron inertia and other effects) gives the right time scale (100 microseconds) for JET:

γ τA ~ (de /L) both in the linear [Porcelli, (1991)] and nonlinear phase [Ottaviani and Porcelli, (1993)]. Typically de /L~10-3

(de = skin depth, L ~a~R = machine scale length. τA = Alfven time).

Probably not sufficient to explain alone fast crashes in medium size tokamaks and in some simulations at moderate Lundquist number

Look for a new explanation...

What about the stability of m=1 islands?

Current sheets of large aspect ratio are developed during the nonlinear stage of primary internal-kink modes. How does their instability to secondary reconnecting modes relate to fast sawtooth crashes?

ψ Jφ

( Figures taken from [Q.Yu et al., Nucl. Fusion 54, 072005 (2014)] )

Δ' = ∞Ex :

What about the stability of m=1 islands?

Current sheets of large aspect ratio are developed during the nonlinear stage of primary internal-kink modes. How does their instability to secondary reconnecting modes relate to fast sawtooth crashes?

ψ Jφ

( Figures taken from [Q.Yu et al., Nucl. Fusion 54, 072005 (2014)] )

Δ' = ∞Ex :

CURRENT SHEET

What about the stability of m=1 islands?

Current sheets of large aspect ratio are developed during the nonlinear stage of primary internal-kink modes. How does their instability to secondary reconnecting modes relate to fast sawtooth crashes?

ψ Jφ

( Figures taken from [Q.Yu et al., Nucl. Fusion 54, 072005 (2014)] )

Δ' = ∞Ex :

Secondary islands

( Figures taken from [M. Ottaviani and F. Porcelli , Phys. Plasmas 2, 4104 (1995)] )

Large Δ' regime in slab geometry

Large Δ' regime in slab geometry

Current sheet

( Figures taken from [M. Ottaviani and F. Porcelli , Phys. Plasmas 2, 4104 (1995)] )

Secondary reconnecting instabilities to primary tearing-type modes

[W. Daughton et al., PoP 13, 072102

(2006)]

In relatively recent years (2001, 2005-2006) the power increase of computational resources allowed to identify secondary reconnecting instabilities on current sheets generated by primary tearing-type modes.

[J.F. Drake et al., GRL 33, L13105

(2006)]

[N.F. Loureiro et al., PRL 95, 235003

(2005)]

[S. Tanuma et al., Astr. J.

551, 312 (2001)]

Slab approximation for the current sheet

B0

Due to the current sheet's large aspect ratio L/δI >> 1 a quasi-continuum spectrum of unstable wave-numbers can be assumed.

The maximum growth rate occurs at short wavelength. This justifies the local assumption about the geometry

L

a ~ δI

Key assumptions

1) In the current sheet, the current density is comparable to the equilibrium current density which is what one gets at the end of the linear phase of the m=1 mode:

Jcs ~ J0

1) The above is reasonable if the instability of the current sheet (what we call the secondary instability) is fast enough that its growth rate exceeds the evolution rate of the “equilibrium” (i.e. the primary reconnecting mode):

γII > τcs -1 ~ γI

This is verified a posteriori. The m=1 island can be considered as static.

3) The width of the current sheet is assumed to be of the order of the m=1 layer width.

This is justified at the end of the linear phase/beginning of the nonlinear phase of the m=1 primary island

4) We perform an asymptotic analysis (i.e. Lundquist number S → ∞ , etc. )

a

B0

L ~ L0

width ~ L ~ L0

Example of Harris-pinch equilibrium, typical for a / L ≤ ka << 1

Size of the transverse magnetic field in the current sheet

In a tokamak L0 ~ R and L ~ R ~ L0.

And the end of the internal-kink linear phase a ~ δI = S-1/3 L

Note:in several papers BCS ~ B0 and JCS >> J0 NOT JUSTIFIED in this context!

For the fastest growing mode we can take the ka <<1 limit of the expression of Δ'(ka). This brings in explicit dependence on “a”:

Scaling of the reconnection rates (n/m=1/1 case of a tokamak in cylindrical approximation)

Primary internal-kink : Δ'δI = ∞

It has a broad range of unstable secondary modes with a broad range of wavenumbers k and a varying Δ' . The layer width is δ

II

The fastest growing mode always occours at Δ'δII ~ 1

Estimation :

Tokamak geometry :

Secondary tearing mode : Δ'δII = 1

Maximum tearing growth rate in slab geometry

In slab geometry also a fastest growing mode exists ([H. Furth, J.Killeen, M.N. Rosenbluth, PoF 1963]), which satisfies the condition Δ'δ = 1 [Battacharjee et al., PoP, 2009]. Imposing the condition γ

IK(k

M )

~ γ

T(k

M ) one can estimate its scaling also

for equilibria which differ from Harris' pinch case [Del Sarto et al., JGR, 2016].

Example (Harris-pinch) :

Δ'δ > 1

Δ'δ < 1

Δ'δ = 1

(Adaptation of a figure courtesy of Fulvia Pucci - related to [F.Pucci, M. Velli, ApJL 780, L19 (2014)])

Results : secondary collisionless reconnection, possibly at near Alfvenic rate, in tokamak devices with resistive

primary reconnection

Collisionless regime :γ

τ

0 > ν

c τ

0

Lundquist number

Rates, normalisedto the Alfventime

Results : secondary collisionless reconnection, possibly at near Alfvenic rate, in tokamak devices with resistive

primary reconnection

Collisionless regime :γ

τ

0 > ν

c τ

0

m=1 growth rate

Lundquist number

Normalisedrates

Results : secondary collisionless reconnection, possibly at near Alfvenic rate, in tokamak devices with resistive

primary reconnection

Collisionless regime :γ

τ

0 > ν

c τ

0

m=1 growth rate

Collision rate

Lundquist number

Normalisedrates

primary reconnection

Collisionless regime :γ

τ

0 > ν

c τ

0

m=1 growth rate

Collision rate

intersection

Lundquist number

Normalisedrates

primary reconnection

Collisionless regime :γ

τ

0 > ν

c τ

0

m=1 growth rate

Collision rate

Ottaviani & Porcelli, 93

Lundquist number

Normalisedrates

primary reconnection

Collisionless regime :γ

τ

0 > ν

c τ

0

Secondary instability, collisional

Collision rate

Ottaviani & Porcelli, 93

Lundquist number

Normalisedrates

primary reconnection

Collisionless regime :γ

τ

0 > ν

c τ

0

Collision rate

Ottaviani & Porcelli, 93

intersection

Secondary instability, collisional

Lundquist number

Normalisedrates

primary reconnection

Collisionless regime :γ

τ

0 > ν

c τ

0

Secondary instability, collisionless

Collision rate

Ottaviani & Porcelli, 93

Lundquist number

Normalisedrates

primary reconnection

N.B.: for JET S ~ 109 ~ ( L/de )3

( L ~ R)

Ottaviani & Porcelli, 93

New Theory

Lundquist number

Normalisedrates

primary reconnection

N.B.: for JET S ~ 109 ~ ( L/de )3

( L ~ R)

Ottaviani & Porcelli, 93

New Theory

Faster growth rate

Lundquist number

Normalisedrates

primary reconnection

N.B.: for JET S ~ 109 ~ ( L/de )3

( L ~ R)

Ottaviani & Porcelli, 93

New Theory

Faster growth rateBroader collisionless range

Lundquist number

Normalisedrates

primary reconnection

N.B.: for JET S ~ 109 ~ ( L/de )3

( L ~ R)

Ottaviani & Porcelli, 93

New Theory

Faster growth rateBroader collisionless range

Lundquist number

Normalisedrates

Minimum rate

Summary and conclusions

A current sheet generated by a primary instability with Δ'= ∞, such as the resistive internal kink mode, becomes unstable at an early stage in its nonlinear development because of subalfvenic modes.

In the case of sawtooth phenomenon in a purely resistive framework the current sheet becomes unstable to a secondary mode with growth rate γτ

A ~ S -1/6 , apparently in agreement

with the numerical results of [Yu et al., Nucl. Fusion 54, 072005 (2014)].

When finite electron inertia is included in the analysis, for (L/de)12/5 < S < (L/d

e)3, regime

relevant to most medium size tokamak devices, the secondary instability develops in the inertia-driven collisionless regime with a growth rate γ

II τ

A ~ (d

e /a)2 ~ (d

e /L)2 S 2/3 which can

become near-Alfvenic.

Paper: D. Del Sarto, M. Ottaviani “Secondary fast instability in the sawtooth crash”, Physics of Plasmas 24, 012102 (2017) http://dx.doi.org/10.1063/1.4973328

The overall reconnection rate is at least of order γ τ

A ~(d

e /L)2/5, always larger than the internal

kink growth rate.