Improved fluid models for runaway generation and decay Ola Embr - - PowerPoint PPT Presentation

Improved fluid models for runaway generation and decay

Ola Embr´ eus

Adam Stahl, Linnea Hesslow, T¨ unde F¨ ul¨

• p

Chalmers University of Technology, Gothenburg, Sweden

Ola Embr´ eus | embreus@chalmers.se 1/19

CHALMERS The runaway fluid

jRE = −enREv

v =

1 nRE

• dp vfRE.

The runaway current evolution is given by djRE dt

= −ecnRE

• Γ(E, t)u(E, t) + d

dt u(E, t)

• ,

Γ(E, t) ≡

1 nRE dnRE dt , u(E, t) ≡ v/c.

Ola Embr´ eus | embreus@chalmers.se 2/19

CHALMERS The runaway fluid

Kinetic simulations unnecessary whenever

Γ(E, t) = Γ(E(t))

u(E, t) = u(E(t)), i.e. when the system is in momentaneous steady-state. 1 jRE djRE dt

= Γ(E) +

1 u(E)

∂u ∂E

dE dt

Ola Embr´ eus | embreus@chalmers.se 3/19

CHALMERS The runaway fluid

Requires slowly varying parameters

→ May be the case during the current quench

Collision time of relativistic electron :

τc = 4πε2

0m2 ec3

ln Λnee4

= mec

eEc Decay time of current :

τdecay ∼

jRE

∂jRE/∂t ≈ jREˆ

L Ec

τdecay τc ∼ eµ0IRE

2πmec ≈ IRE 8.5 kA ≫ 1

Ola Embr´ eus | embreus@chalmers.se 4/19

CHALMERS

Example: Let’s compare time-dependent vs steady state growth rates E(t) =

• 2 − 0.8 t

t∆E

• Ec

Ola Embr´ eus | embreus@chalmers.se 5/19

CHALMERS

E(t) =

• 2 − 0.8 t

t∆E

• Ec

1.2 1.4 1.6 1.8 2

E/Ec

• 15
• 10
• 5

5

τc Γ

×10−3 1.2 1.4 1.6 1.8 2

E/Ec

0.75 0.8 0.85

v/c

t∆E = ∞ t∆E = ∞

Ola Embr´ eus | embreus@chalmers.se 5/19

CHALMERS

Self-similar evolution occurs both for growth and decay:

20 40 60 80

Kinetic energy Ek [MeV]

10−25 10−20 10−15 10−10 10−5

dnRE/dEk E = 1.2Ec

t = 0 t = 300τc t = 600τc t = 900τc 20 40 60 80

Kinetic energy Ek [MeV]

10−9 10−7 10−5

dnRE/dEk E = 2Ec

t = 0 t = 100τc t = 200τc t = 300τc

Ola Embr´ eus | embreus@chalmers.se 6/19

CHALMERS

E(t) =

• 2 − 0.8 t

t∆E

• Ec

1.2 1.4 1.6 1.8 2

E/Ec

• 15
• 10
• 5

5

τc Γ

×10−3 t∆E = 10τc = 100τc = 1000τc = ∞ 1.2 1.4 1.6 1.8 2

E/Ec

0.75 0.8 0.85

v/c

t∆E = 10τc = 100τc = 1000τc = ∞

Ola Embr´ eus | embreus@chalmers.se 7/19

When quasi-steady state is valid, the mission of kinetic theory is only to determine Γ(E, ...) (and to a lesser extent v) How do we determine Γ as accurately as possible?

CHALMERS Avalanche generation

To describe knock-on collisions we add a (simplified) Boltzmann

• perator:

dfe dt = CFP{fe} + Cboltz{fe}, Cboltz{fa, fb}(p) =

• dp1
• dp2

∂σab ∂p vrelfa(p1)fb(p2) − fa(p)

• dp′ vrelσab(p, p′)fb(p′)

Generally we can linearize (nRE ≪ ne) Cboltz{fe, fe} ≈ Cboltz{fe, fe0}

• test-particle

+ Cboltz{fe0, fe}

• field-particle

.

Ola Embr´ eus | embreus@chalmers.se 9/19

CHALMERS Avalanche generation

The two most established knock-on models today: Cknock-on = Cboltz{neδ(p), fe} (only field-particle term) Rosenbluth-Putvinski: fe(p) = nRE lim

p0→∞

1 p2 δ(p − p0)δ(cos θ − 1) Chiu-Harvey: fe(p) = F(p)δ(cos θ − 1)

• F(p) =

1

−1 fe(p) d(cos θ)

• [Rosenbluth, Putvinski NF 1997; Chiu, Rosenbluth, Harvey NF 1998]

Ola Embr´ eus | embreus@chalmers.se 10/19

CHALMERS Avalanche generation

So how do these operators behave?

Ola Embr´ eus | embreus@chalmers.se 11/19

CHALMERS Avalanche generation

Both models have limitations:

• Double counting collisions
• Non-conservation of momentum and energy

– Rosenbluth-Putvinski even creates infinite energy and momentum!

• Chiu-Harvey model ignores pitch-angle distribution
• Arbitrary cut-off affecting solutions

Ola Embr´ eus | embreus@chalmers.se 12/19

CHALMERS Avalanche generation

We solved this, by

• Accounting for full fe(p)
• Including the test-particle term

[restores conservation laws]

• Modify ln Λ in Fokker-Planck operator [avoids double counting]

Ola Embr´ eus | embreus@chalmers.se 13/19

CHALMERS Avalanche generation

Ola Embr´ eus | embreus@chalmers.se 14/19

CHALMERS Avalanche generation

We can now revisit a classical calculation [R-P

, NF 1998]:

The steady state avalanche growth rate

Γ =

1 nRE dnRE dt

10 20 30

E Ec − 1

0.035 0.04 0.045

Γ/( E

Ec − 1) Full Boltzmann Field-particle Rosenbluth-Putvinski R-P theory

Ola Embr´ eus | embreus@chalmers.se 15/19

CHALMERS Avalanche generation in a near-threshold electric field

An interesting situation occurs when E ∼ Ec, as radiation losses become important.

[P . Aleynikov and B. N. Breizman, PRL 114, 155001 (2015)]

Ola Embr´ eus | embreus@chalmers.se 16/19

CHALMERS Near-threshold electric field

Approximate Γ calculated from the avalanche cross-section

Γ(E) ≈ v γmax

γmin

∂σ ∂γ dγ.

Negative growth for small E: Reverse knock-ons predicted!

[P . Aleynikov and B. N. Breizman, PRL 114, 155001 (2015)]

Ola Embr´ eus | embreus@chalmers.se 17/19

CHALMERS Near-threshold electric field

• Significant reverse knock-on

however not observed in kinetic simulations

• Runaway decay is described

mainly by Fokker-Planck dynamics when Γ 0.

1.5 2 2.5

E/Ec

• 0.01

0.01 0.02 0.03

Γ [arb. units]

No avalanche Full Boltzmann Aleynikov-Breizman theory

Ola Embr´ eus | embreus@chalmers.se 18/19

CHALMERS Summary

• Runaway fluids

— Strictly valid when background variations slow (for example current quench) — Accuracy then only limited by the kinetics used to find Γ(E, ...) — Runaway dissipation can be described in the fluid picture

• Avalanche runaway modelling

— Conservative knock-on operator from Boltzmann — Formally eliminates double counting collisions, and describes reverse knock-on

Runaway kinetic theory is here to stay.

Ola Embr´ eus | embreus@chalmers.se 19/19