Relativistic runaway electrons in a near-threshold ... Theory...Relativistic runaway electrons in a near-threshold electric field ... • Historical background & RE Fokker-Planck Equation

TSD Workshop 2015, PPPL July 14 2015

Page 1

Relativistic runaway electrons in a near-threshold electric field

Pavel Aleynikov1,2 and Boris Breizman3

1 Max-Planck-Institut für Plasmaphysik, Teilinstitut Greifswald, Germany 2 ITER Organization, Route de Vinon sur Verdon, 13115 St Paul Lez Durance, France

3 Institute for Fusion Studies, UT Austin, USA


Page 2

Outline

•  Summary of numerical modelling–  Non-monotonic RE distribution function

•  Historical background & RE Fokker-Planck Equation •  New Kinetic Theory for RE in a near threshold electric field

–  Analytical Solution of the Fokker-Planck Equation–  Physics details of the theory

§  Non-monotonic distribution function§  Sustainment electric field§  Avalanche onset electric field§  New avalanche growth rate§  Hysteresis

•  Mitigation/current decay regime•  Summary


Page 3

Bounce averaged Fokker-Planck + current channel modeling (0D & 1D)*•  Kinetic model:

•  Current channel modeling:

–  0D chain equation and radiation balance –  GTS. 1D current diffusion, heat diffusion, densities diffusion equations

Non-simplified (conservative) knock-on

source (significantly affects avalanche rate)

Synchrotron and bremsstrahlung

(enhanced scattering and radiation limits the

RE energy)

Scattering and friction accounts for an

interaction with not fully ionized impurities

(significantly higher scattering on High-Z

impurities)ZAr(1.6eV, 5 MeV)~12

Numerical calculations

dtdILRE −=π2

* [P. Aleynikov, K. Aleynikova, B. Breizman, G. Huijsmans, S. Konovalov, S. Putvinski, and V. Zhogolev, 25th IAEA Fusion Energy Conference, St. Petersburg, Russian Federation, 2014, pp. TH/P3–38.]

Prad T( )= PΩ T( )


Page 4

Mitigation in ITER

0

2

4

6

8

10

12

14

0 50 100 150 200 250 300 350 400

E av,

MeV

t, ms

nAr [1020 m-3]0.1 1.0 3.0

0

1

2

3

4

5

6

7

0 50 100 150 200 250 300 350 400

I re, M

A

t, ms

nAr [1020 m-3]0.1 1.0 3.0

Mitigation of the RE current and energy with different Ar MGI.

•  Start from a given value of RE seed current (≈ 70kA) after the TQ•  Ar density required for TQ mitigation is ≈ 1019m−3

•  Red curves – not mitigated RE decay•  Green and Blue– Ar density is introduced at 30ms


Page 5

RE distribution function

0

2

4

6

8

10

12

14

0 50 100 150 200 250 300 350 400

E av,

MeV

t, ms

nAr [1020 m-3]0.1 1.0 3.0

RE 2D distribution function at t=30ms and t=100ms. nAr = 3·1020m−3

•  Exponential distribution function during/after avalanche (agrees with [RP])

•  Fast energy drop after MGI•  Plateau on the average energy evolution

•  Peculiar distribution during the plateau

•  High E ~ 2Ec during the decay

Ar


Page 6

Solutions of RE kinetic equation •  The effect was identified in 1925 by Charles Wilson (inventor of the cloud chamber) [C.T.R. Wilson, Proc. Cambridge Philos. Soc. 42, (1925) 534] •  Early experimental observation in tokamaks in 50th and 60th and later studied in [Bobrovski 1970, Vlasenkov 1973, TFR group 1973, Alikaev 1975]

•  The first analysis of runaway phenomena has been carried our by Harry Dreicer [Proceedings of 2nd Geneva conf 1958, 31, 57; Phys Rev., 1959, 115, 238]

•  Frequently cited theory has been derived by Alexander Gurevich [JETF 1960, 39, p1296] •  Relativistic case by Jack Connor and Jim Hastie [J. W. Connor, R. J. Hastie, Nucl. Fusion 15, 415 (1975)]

•  Discovery of the avalanche phenomena by Yuri Sokolov [Yu. A. Sokolov, JETP Lett., 29, No. 4 (1979)]

•  Marshall Rosenbluth, Sergei Putvinski “Theory for avalanche of runaway electrons in tokamaks” [M.N. Rosenbluth, S.V. Putvinski, Nucl. Fusion 37, 1355, 1997]

•  Studies with “Rosenbluth-Putvinski” avalanche source and synchrotron [Martin-Solis (1998); Andersson, Helander and Eriksson, (2001-…); Stahl et al., PRL (2015)]

Gen

esis

Diff

usiv

e co

llisio

ns

Aval

anch

e io

ns


Page 7

Kinetic equation ∂F∂t+ eE 1

p2∂∂p

p2 cosθF − 1psinθ

∂∂θ

sin2θF$

%&

'

()= CF + RF + SF

Small-angle collisions:

CF = mcτ

1p2

∂∂p

p2 +m2c2( )F + (Z +1)2sinθ

mc p2 +m2c2

p3∂∂θ

sinθ ∂∂θ

F$

%&&

'

())

Synchrotron radiation reaction:

RF = mcτ rad

1m2c2p2

∂∂p

p3 m2c2 + p2 sin2θF + 1psinθ

∂∂θ

pcosθ sin2θ

m2c2 + p2F

+

,--

.

/00

Large-angle collisions (Möller source): SF

SF = ncoldcγ0

2 −1γ0

δ cosθ − cosθ p+, ./

2πre2

γ02 −1

γ0 −1( )2γ0

2

ε −1( )2γ0 −ε( )2 −

2γ02 +2γ0 −1

ε −1( ) γ0 −ε( )+1

567

87

9:7

;7∫ b

2F p2dpdλ

1−bλ

δ cosθ − cosθ p+, ./ =

1π

1

b2λλ0 − 1−bλ 1−bλ0 −ε −1ε +1

γ0 +1γ0 −1

$

%&

'

()

2

Time scales:

τ rad ≡3m3c5

2e4B2

τ ≡ m2c3

4πnee4 lnΛ


Page 8

Kinetic equation in Rosenbluth-Putvinski ∂F∂t+ eE 1

p2∂∂p


∂∂θ

sin2θF$

%&

'

()= CF + RF + SF


CF = mcτ

1p2

∂∂p

p2 +m2c2( )F + (Z +1)2sinθ

mc p2 +m2c2

p3∂∂θ

sinθ ∂∂θ

F$

%&&

'

())


RF = 0

Time scales:

τ rad ≡3m3c5

2e4B2

The equation in this form is solved analytically in [M. N. Rosenbluth, S. V. Putvinski, Nucl. Fusion 37, 1355 (1997)]

However this source appropriate well above the avalanche threshold, but needs to be generalized in the near-threshold regime.

τ ≡ m2c3

4πnee4 lnΛ

Large-angle collisions (Möller source):SF


Page 9

Kinetic equation Time scales:

τ rad ≡3m3c5

2e4B2

1) Möller (avalanche) source ( ) is weaker than electron drag by Coulomb logarithm.2) The small parameter is , i.e. electric file is close to the threshold.

SF

ε = E − EaEa

τ ≡ m2c3

4πnee4 lnΛ

∂F∂t+ eE 1

p2∂∂p


∂∂θ

sin2θF$

%&

'

()= CF + RF + SF


CF = mcτ

1p2

∂∂p

p2 +m2c2( )F + (Z +1)2sinθ

mc p2 +m2c2

p3∂∂θ

sinθ ∂∂θ

F$

%&&

'

())


RF = mcτ rad

1m2c2p2

∂∂p

p3 m2c2 + p2 sin2θF + 1psinθ

∂∂θ

pcosθ sin2θ

m2c2 + p2F

+

,--

.

/00

Large-angle collisions (Möller source): SF

SF = ncoldcγ0

2 −1γ0

δ cosθ − cosθ p+, ./

2πre2

γ02 −1

γ0 −1( )2γ0

2

ε −1( )2γ0 −ε( )2 −

2γ02 +2γ0 −1

ε −1( ) γ0 −ε( )+1

567

87

9:7

;7∫ b

2F p2dpdλ

1−bλ

δ cosθ − cosθ p+, ./ =

1π

1

b2λλ0 − 1−bλ 1−bλ0 −ε −1ε +1

γ0 +1γ0 −1

$

%&

'

()

2


Page 10

The separation of timescales between small-angle collisions and knock-on collisions suggests a two-step approach to the problems of runaway production:

1)  Ignore the large-angle collisions and study the behavior of pre-existing runaways

2)  Use the distribution function of the accumulated runaways to predict their production and loss

Dimensionless kinetic equation:

Solution of the Fokker-Planck equation

∂F∂s

+ ∂∂p

Εcosθ −1− 1p2

− 1τ rad

p 1+ p2 sin2θ⎡

⎣⎢

⎤

⎦⎥F =

1sinθ

∂∂θsinθ Ε sinθ

pF + (Z +1)

2p2 +1p3

∂F∂θ

+ 1τ rad

cosθ sinθ1+ p2

F⎡

⎣⎢⎢

⎤

⎦⎥⎥

[P. Aleynikov, B. Breizman, Theory of Two Threshold Fields for Relativistic Runaway Electrons, Phys. Rev. Lett., 114, 155001 (2015)]


Page 11

In the near-threshold case the time-scale for pitch-angle equilibration is much shorter than the momentum evolution time-scale.

Thus, the lowest order kinetic equation is:

solution:

Integration of the exact kinetic equation over all pitch-angles eliminates the lowest order terms and gives a one-dimensional continuity equation for the momentum flow:

Fast pitch-angle equilibration

ΕpF + (Z +1)

2p2 +1p3

1sinθ

∂F∂θ

= 0 F = G t; p( ) A2sinhA

exp Acosθ[ ]

A p( ) ≡ 2Ε(Z +1)

p2

p2 +1

∂G∂t

+ ∂∂p

U(p)G = 0

U(p)≡ − 1A p( )

−1

tanh A p( )( )#

$%%

&

'((Ε −1− 1

p2+Z +1Eτ rad

p2 +1p

1A p( )

−1

tanh A p( )( )#

$%%

&

'((


Page 12

The flow velocity

-0.4

-0.2

0

0.2

0.4

0 1 2 3 4 5 6 7 8

Flow

vel

ocity

, U(p

)

Electron momentum, p

pmin pmax

The roots pmin and pmax merge at a certain electric field E=E0. This is the minimal electric field required for RE sustainment.

Peaking of the distribution function around pmax

U(p) ≡ Ε cosθav −1−1p2−Z +1Eτ rad

p2 +1pcosθav; cosθav =

1tanh A p( )( )

−1A p( )


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The sustainment threshold field is always higher than Connor’s Ec field.

Contours of the sustainment threshold field (solid)

The sustainment threshold field

E0 = Ec 1+Z+1( )τrad / τ

18

+Z+1( )2

τrad / τ6

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟


Page 14

1)  All primaries are at pmax with2)  Both electrons have p>pmin after the collision3)  Energy conservation requires4)  The cross-section for such collisions gives the growth rate

5)  The integral is evaluated analytically for the Möller cross-section:

Avalanche growth rate

Γ ≡ 1nre

∂nre∂t

= necτ2

γ 02 −1γ 0

dσdγγ min

γ 0+1−γ min

∫ dγ

Γ = 14Λγ 0 γ 0

2 −1

γ 0 +1− 2γ min( ) 1+ 2γ 02

γ min −1( ) γ 0 −γ min( )⎛

⎝⎜⎞

⎠⎟

− 2γ 0 −1γ 0 −1

2 ln 1+ γ 0 +1− 2γ minγ min −1

⎛⎝⎜

⎞⎠⎟

⎧

⎨

⎪⎪

⎩

⎪⎪

⎫

⎬

⎪⎪

⎭

⎪⎪

γ min < γ < γ 0 +1−γ min

γ 0 = pmax2 +1

-0.3-0.2-0.1

0 0.1 0.2 0.3

0 1 2 3 4 5 6 7 8

Flow

vel

ocity

, U(p

)


pmin pmax


Page 15

Avalanche onset field

γ 0 +1− 2γ min = 0 -0.3-0.2-0.1

0 0.1 0.2 0.3

0 1 2 3 4 5 6 7 8

Flow

vel

ocity

, U(p

)


pmin pmaxThe avalanche onset condition:

The avalanche onset field Ea is greater than the runaways sustainment field E0 Contours of the avalanche onset field (dashed)


Page 16

-0.01 0

0.01 0.02 0.03 0.04 0.05 0.06

1 1.5 2 2.5 3 3.5 4

Aval

anch

e gr

owth

rate

, Γ

Electric field, E

Near-threshold avalanche growth rate

Rosenbluth-Putvinski theory

1jre∂ jre∂t≈

1τ lnΛ

π3 Z+5( )

E−1( )

Rosenbluth-Putvinski

Z=5τrad= 70


Page 17

Hysteresis

Ea

E0

Growing inductive field Decaying inductive field

No runaways

Attractor holds seed electrons

Runaway avalanche

Rapid decay of fast electron current

Attractor holds residualfast electrons

Runaway avalanche stops at E = Ea

Attractor disappears at E = E0

Phase space attractor forms at E = E0


Page 18

Mitigation regime

0

2

4

6

8

10

12

14

0 50 100 150 200 250 300 350 400

E av,

MeV

t, ms

nAr [1020 m-3]0.1 1.0 3.0

EaE0

•  The avalanche is switched-off•  The field stays between Ea and E0 •  The runaways remain peaked at pmax•  The runaway density and current

decrease in step with the dissipation of the magnetic energy

0

1

2

3

4

5

6

7

0 50 100 150 200 250 300 350 400

I re, M

A

t, ms

nAr [1020 m-3]0.1 1.0 3.0


Page 19

Current decay time-scale

d(Ire + IΩ )dt

≈−2πR

(L−Lwall )E0

1)  - function only of plasma parameters

2)  The total energy loss is

3)   The current decay is linear and the decay rate is given by

4)  This estimate is insensitive to the distribution function

∂ j∂t

= 1µ01r∂∂rr ∂E0∂r

E0 ≈ Ea

~ ( jΩ + jre )E0 ≈ ( jΩ + jre )Ea


Page 20

Reduced RE kinetic model for fluid-like codes

E, n, Z, B

Distribution function (pmin, pmax)

Avalanche “Resistivity”

∂ jre∂t E=const

∂ jre∂E

Current equation RE energy


Page 21

Reduced RE kinetic model for fluid-like codes

0 1 2 3 4 5 6 7

I re, M

A

NumericalAnalitical

0 5

10 15 20 25 30 35 40

E av,

MeV

0 5

10 15 20 25 30

0 50 100 150 200 250 300 350

E, V

/m

t, ms

0 5

10 15 20 25 30

0 50 100 150 200 250 300 350

E, V

/m

t, ms

nAr = 3.0 ⋅1020m−3

[IAEA_FEC_2014] results with Reduced RE kinetic module.

E, n, Z, B

Distribution function (pmin, pmax)

Avalanche “Resistivity”

∂ jre∂t E=const

∂ jre∂E

Current equation RE energy


Page 22

Summary 1)  Rigorous kinetic theory for relativistic RE in

the near-threshold regime 2)  The electric field for runaway avalanche

onset is higher and the avalanche growth rate is lower than previous predictions

3)  The new theory predicts peaking of the runaway distribution function near pmax

4)  Existence of two different threshold fields produces a hysteresis in the runaway evolution

5)  Particular features of this regime allows for evaluation of the current decay rate

E0 ≈ Ec 1+Z+1( )2

τrad / τ3

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

-0.01 0

0.01 0.02 0.03 0.04 0.05 0.06

1 1.5 2 2.5 3 3.5 4

Aval

anch

e gr

owth

rate

, Γ

Electric field, E

p cos θ


Page 23

Benchmark against numerical solution •  Numerical solution exhibits peaked distribution function •  Launching two groups of electrons from p>pmax and pmin<p<pmax

p pcos θ cos θ


Page 24

2

4

6

8

10

12

14

20 40 60 80 100 120 140 160 180 200

Stab

le p

oint

pm

ax=p

min

τrad/τ

Z=1Z=5

Z=10


Page 25

Questions

Calculations Rosenbluth & Putvinski

•  Not monotonic distribution

•  Linear RE current decay

•  Plateau on the average energy evolution

1jre∂ jre∂t

=1τ lnΛ

π3 Z+5( )

E−1( )

Documents

Relativistic runaway electrons in a near-threshold ... Theory...Relativistic runaway electrons in a near-threshold electric field ... • Historical background & RE Fokker-Planck Equation