[PPT] - Dynamics of transport barrier relaxations in tokamak edge plasmas P PowerPoint Presentation

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Dynamics of transport barrier relaxations in tokamak edge plasmas

P . Beyer, S. Benkadda, G. Fuhr-Chaudier Laboratoire PIIM, Equipe Dynamique des Syst` emes Complexes CNRS – Universit´ e de Provence, Marseille, France

X. Garbet, Ph. Ghendrih, Y. Sarazin

Association Euratom – CEA, CEA/DSM/DRFC CEA Cadarache, France

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Introduction

The operational regime of future fusion

reactors is characterized by – an edge transport barrier, – relaxation oscillations of the barrier (Edge Localized Modes, ELMs).

Explanations for relaxations are usually

based on MHD instability,

q=2 q=2.5 q=3 50 100 150 minor radius mean pressure

with barrier w/o barrier

– analysis of linear stability properties, no dynamics.

Most existing dynamical models are phenomenological,

– not based on 1st principles, i.e. turbulence simulations.

Frequency, crash time and energy release are central issues.

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Outline

Overview of existing reduced dynamical models

for transport barrier relaxations.

3D fluid turbulence simulations.
Subsequent reduced 1D model.
Systematic reduction
0D model.

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Reduced models for barrier dynamics: not straightforward

simple model (1D):

transport eqn. coupled to instability amplitude eqn. at plasma edge

∂t ¯ p

=

∂x

✁

χ0 ¯ π

✂

χ1

✄

ξ

✄

2 ¯

π

Γ

☎

∂tξ

=

γ0

✆

¯ π

αc

✝

ξ

✂

ν0∂2

xξ

¯ π

✞

∂x ¯

p ¯ p: pressure profile, ξ: perturbation ampl., Γ: incoming energy flux, x: minor radius

x p Γ

no oscillations, stable fixed point, robust property

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Possible modifications to obtain oscillations or relaxations

Introduction of S–curve

– for dependency of flux vs gradient (due to ExB shear flow), – in dynamical eqn. for perturbation amplitude (explosive instability), – in dynamical eqn. for ExB shear flow (multiple states: L–H).

Introduction of characteristics of ideal MHD eigenmodes

– vanishing growth rate below threshold, – radial shape of global modes.

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S–curve for flux vs gradient produces relaxations

Introduce ambient turbulent flux Γturb

due to drift waves, etc.

˜

Φ

✆

¯ π

✝ ✞

Γturb

✆

¯ π

✝ ✂

χ0 ¯ π : S-curve due

to turb. stabilization by ExB shear flow.

∂t ¯ p

✞

∂x
˜

Φ

✆

¯ π

✝ ✂

χ1

✄

ξ

✄

2 ¯

π

Γ

✁

∂tξ

✞

γ0

✆

¯ π

αc

✝

ξ

✂

ν0∂2

xξ

¯ π

✞

∂x ¯

p

Relaxations, frequency

✂ ✂ ✂

with power.

More sophisticated models available.

p

✄

¯ π

Lebedev, Diamond, PoP 95

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Explosive instability

Account for non-linear terms in amplitude equation:

first is destabilizing, second is stabilizing.

∂t ¯ p

=

∂x

✁

χ0 ¯ π

✂

χ1

✄

ξ

✄

2 ¯

π

Γ

☎

∂tξ

=

γ0

✆

¯ π

αc

✝

ξ

✂

µξ2

νξ3

¯ π

✞

∂x ¯

p

Dynamics close to Van der Pol oscillations.

Cowley, Wilson, PPCF 03

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Multiple states for shear flow

L–H transition: multiple states for ExB shear flow ¯

u

✆

¯ π

✝

, and: effective flux χeff

✆

¯ u

✝

¯ π depends on shear flow. ∂t ¯ p

=

∂x
χeff

✆

¯ u

✝

¯ π

Γ

✁

∂t ¯ u

=

✆

¯ π

αc

✝

µ1 ¯

u3

✂

µ2 ¯ u

✂

ν∂2

x ¯

u ¯ π

✞

∂x ¯

p

Ginzburg–Landau type, limit cycle oscillations.
No perturbation amplitude, more appropriate for “dithering”.
Generalization to ELMs available.

Itoh, Itoh, PRL 91, PRL 95

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Linear ideal MHD instability model

Linear ideal MHD eigenmodes:

– growth rate

0 below threshold,

– global mode structure.

Modeled by

– Heaviside funct. H on growth rate, – Gaussian shape G in eff. diffusivity

∂tξ

✞

γ0

✆

¯ π

αc

✝

H

✆

¯ π

αc

✝

ξ

ν

✆

ξ

ξ0

✝

+ transp. code with χeff ∝

✄

ξ

✄

2G

✆

x

✝

Relaxations, frequency

✂ ✂ ✂

with power.

More sophisticated models (peeling).

L¨

nnroth, Parail, PPCF 04

B´ ecoulet, Huysmans, EPS 03

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State of the art

Most existing models are phenomenological.
Difficult to reproduce relaxations with frequency
with power.
Turbulence simulations of relaxations exist,

based on turbulent ExB flow generation, no barrier.

Need for 1st principles based model, i.e. 3D turbulence simulations,

reproducing i) transport barrier ii) complete relaxation cycle.

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3D edge turbulence simulations with transport barrier

turbulence model: resistive ball. modes,

reduced MHD equations

∂t∇2

φ

✂ ✁

φ

✂

∇2

φ

✄ ✞

∇2

☎

φ

Gp

✂

ν∇4

φ

∂t p

✂ ✆

φ

✂

p

✝ ✞

δcGφ

✂

χ

☎

∇2

☎

p

✂

χ

∇2
p

✂

S

3D toroidal geometry at plasma edge
driven by incoming flux Γin

✞ ✞

r rmin Sdr

✟

,

press. profile evolves self-consistently
barrier generated by imposed flow U,

locally sheared, ωEext

✞ ✆

∂rU

✝

max

0rmin rq=2 rq=2.5 rq=3 rmax S(r)

radial profile of source S

rmin rq=2 rq=2.5 rq=3 rmax U(r)

radial profile of imposed flow U

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Strong local ExB shear

formation of barrier

q=2 q=2.5 q=3 10 20 30 rayon flux turbulent

ωEext=7.3 ωEext=5.5 ωEext=3.7 ωEext=0

q=2 q=2.5 q=3 200 400 600 rayon pression moyenne

ωEext=7.3 ωEext=5.5 ωEext=3.7 ωEext=0

turbulent flux profile time averaged pressure profile

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Barrier relaxation oscillations appear

0.5 1 normalized pressure gradient 5 10 15 normalized turbulent flux 3000 5000 7000 9000 −0.6 −0.4 −0.2 normalized velocity shear fluctuations time

1.

✄

∂x ¯ p

✄

✆

Γin

χ
✝
2. Γturb
Γin

3.

✆

ωE

ωEext

✝

ωEext
all evaluated at barrier center
bserved in a range of Γin, ωEext
robust property

scenario:

turb. state

✁

relaxations

✁

quiescent st.

✂

ωE

no barrier

✁

barrier

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Fixed input power: frequency decreases with shear

5 10 15 normalized turbulent flux

ωE = 14

5 10 15

ωE = 12

5 10 15

ωE = 10

5 10 15

ωE = 8

3000 6000 9000 12000 5 10 15 time

ωE = 6

input power: Γ

✞

36 shear layer width: 28.8%

2 4 6 8 10 12 400 800 1200 1600 2000 # time lag time lag ωE = 12 ωE = 10 ωE = 8 ωE = 6

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Fixed input power: frequency decreases with shear

10 20 normalized turbulent flux

ωE = 14

10 20

ωE = 12

10 20

ωE = 10

10 20

ωE = 7.3

10 20

ωE = 5.5

3 8 13 18 10 20 time / 103

ωE = 3.7

input power Γ

✞

36 shear layer width: 34.4%

4 8 12 16 400 800 1200 1600 2000 # time lag time lag ωE = 12 ωE = 10 ωE = 7.3 ωE = 5.5 ωE = 3.7

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Fixed flow shear: frequency increases with power

2 4 normalized turbulent flux

Γ = 12

2 4

Γ = 11

2 4

Γ = 10

3 8 13 18 2 4 time / 103

Γ = 9

flow shear: ωE

✞

2 shear layer width: 34.4%

4 8 12 16 400 800 1200 1600 # time lag time lag Γ = 12 Γ = 11 Γ = 10 Γ = 9

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Frequency dependence: two opposite trends

freq.

✂ ✂ ✂

with Γin for ωE fixed (ωE

2)

9 10 11 12 2 4 6 Γ (time lag)−1 × 103 9 10 11 12 250 500 Γ standard deviation of time lag

freq.

with ωE for Γin fixed (Γin
36)

4 8 12 2 4 6 shear layer width ωE (time lag)−1 × 103 0.288 0.313 0.344 4 8 12 250 500 ωE standard deviation of time lag

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Frequency dependence: two opposite trends

if ωE increases fast enough with Γin

frequency decreases with Γin.

0.5 1 normalized pressure gradient 3 8 13 18 2 4 normalized turbulent flux time/103

Γin

10
ωE
2

0.4 0.8 1.2 normalized pressure gradient 3 8 13 18 5 10 15 20 normalized turbulent flux time / 103

Γin

36
ωE
12

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Possible mechanisms excluded

Relaxations persist even if ExB

shear flow is frozen

mechanism
✞

turbulent shear flow generation.

No significant variation of modes

localized outside barrier

mechanism
✞

toroidal mode coupling.

All fluctuations die out when sup-

pressing curvature

Kelvin–Helmholtz stable.

Γin

36
ωE
8

0.5 1 normalized pressure gradient 5 10 15 normalized turbulent flux 3000 5000 7000 9000 −0.6 −0.4 −0.2 normalized velocity shear fluctuations time

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Relaxation governed by mode at barrier center

snapshots of potential fluctuations between relaxations during relaxation

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1D model for central mode amplitude ˜

p

x

✁

t

✂

˜ p

x

✁

t

✂

˜ p

x

✁

t

✂

& profile ¯

p

x

✁

t

✂

¯ p

x

✁

t

✂

¯ p

x

✁

t

✂

∂t ¯ p

✞

2γ0∂x

✄

˜ p

✄

2

✂

χ

∂2

x ¯

p

✂

S ∂t ˜ p

✞

γ0

✆

∂x ¯

p

α0

✝

˜ p

iω

✟

Ex ˜

p

χ

✟ ☎

x2 ˜ p

✂

χ

∂2

x ˜

p x

✞

r

r0: radial dist. from barrier center

m: poloidal wavenumber of central mode

reproduces relaxation oscillations
ExB shear ω

✟

E

✞

ωEm

r0
nonlinear short-term dynamics.
Not described by linear modes

(long-term dynamics).

0.5 1 normalized pressure gradient 3 6 9 12 15 10 20 normalized turbulent flux time / 103

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Description by linear modes is not appropriate

For ∂x ¯

p

✞

α, evolution equation for ˜

p is linear: ∂t ˜ p

✞

γ0

✆

α

α0

✝

˜ p

iω

✟

Ex ˜

p

χ

✟ ☎

x2 ˜ p

✂

χ

∂2

x ˜

p

most unstable eigenmode:

˜ peimθ

exp
x2

2σ2

✂

imθ

iω

✁

Ex

2 χ

✁ ✂

χ

✄

with σ2

☎ ✆

χ

✝ ✞

χ

✟ ✠

growth rate:

γ

✞

γ0

✆

α

α0

✝

ω

✁

2 E

4γ

✁ ✂

χ
χ

✟ ☎

when iω

✟

Ex term replaced by shift of

instability threshold

no oscillations

−20 20 −20 20 (r − r0) / ξ r0 θ′ / ξ

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Time delay in stabilization by ExB shear flow

Short term dynamics of initial pulse

˜ p

✆

x

✂

t

✞ ✝

∝ δ

✆

x

✝

with

∂x ¯

p

✞

α and χ

✟ ☎

term neglected.

Solution :

˜ p ∝ exp

γ

✟

0t

t3
✆

3τ3

D

✝ ✁

γ

✟ ✞

γ0

✆

α

α0

✝

, τD

✞ ✁

1 4χ

ω

✟

2 E

☎

1

✁

3

.

Transient growth before stabilization.
τD large for small χ
(barrier) and low m.
Clearly observed in simulations (— curve).

10 20 10 20 30 40 time mode amplitude

- - curve

γ

✟ ✞ ✂

42 τD

✞

10

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0D model reproduces oscillations Radial mode structure

✁

linear mode

From 1D model
relevant radial structures: ˜

pR

✆

x

✝

, ˜

pI

✆

x

✝

, ¯

p0

✆

x

✝ ✂

linear mode in case ∂x ¯

p

✞

const: ˜

plin

✞

˜ pR

✂

i ˜ pI

✄

Projection: ˜

p

✆

x

✂

t

✝ ✞

aRpR

✂

iaIpI

✂

¯ p

✆

x

✂

t

✝ ✞

αx

✂

a0 ¯ p0

Amplitude equations:

˙ aR

✞ ✆

Γ

δ1a0

✝

aR

✂

Ω1

✆

aR

aI

✝

˙ aI

✞ ✆

Γ

δ2a0

✝

aI

Ω2

✆

aI

aR

✝

˙ a0

✞

γ0a0

✂

2δ1a2

R

✂

2δ2a2

I Γ

γ0

☎

α

α0

✆

γs
γE

γ2

s

χ

✁ ✂

χ

✄

Ω1

✝

Ω2

✝

2γEe

✞

γE

✞

γs

γE

ω

✁

E 2

✁ ☎

4χ

✁ ✂ ✆

400 800 1200 −2 2 Time aR

γE/γs = 0.20 γE/γs = 0.70 γE/γs = 1.24

same frequency dependence on ExB shear as in 3D simulations

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Conclusions

3D nonlinear turbulence simulations based on 1st principles show

– onset of transport barrier, – barrier relaxation oscillations.

Mechanism based on effective time delay for stabilization by ExB shear,

no obvious S–curve, no global mode.

Mean features are reminiscent of type III ELMs:

– frequency dependence, – resistive ballooning mode model (low temperature plasma), – sensitivity to shear flow.

turb. state

✁

relaxations

✁

quiescent st.

✂

ωE

no barrier

✁