Stochastic modeling of scrape-off layer fluctuations

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Stochastic modeling of scrape-off layer fluctuations

R. Kube¹ O. E. Garcia¹ A. Theodorsen¹ D. Brunner² A. Kuang² B. LaBombard² J. Terry²

1Department of Physics and Technology, UiT - The Arctic University of Norway and²MIT Plasma Science and Fusion Center

August 24, 2017

R. Kube et al. (UiT) Stochastic modeling of scrape-off layer fluctuations August 24, 2017 1 / 33

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Bursts in single point measurements correspond to traversing blobs

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1 Stochastic model of data time series

2 Comparison to experimental measurements

3 Conclusions

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Superpose uncorrelated pulses to model data time series

Superposition of K pulses in a time interval [0 :T]

Φ_K(t) =

K(T)

X

k=1

A_kφ

t−t_k τ_d

where k labels a pulse and

Ak denotes the pulse amplitude t_k denotes pulse arrival time φdenotes a pulse shape τ_d denotes pulse duration time Intermittency parameter: γ =τ_d/τ_w

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Pulses arrive uncorrelated and form a Poisson process

Choose distribution for all random variables P_K(K|T) gives the number of bursts in time interval [0;T] P_A(A_k)→ distribution of pulse Amplitudes.

Pt(t_k)→ distribution of pulse arrival times.

Consider a Poisson process:

1 Pulses arrive uncorrelated: Pt(tk) = 1/T

2 Avg. rate of pulse arrival is 1/τ_w

P_K(K|T) = exp −T

τ_w T τ_w

K

1 K!

Exponentially distributed pulse amplitudes: hAiP_A(A_k) = exp (A_k/hAi) We often normalize the process as

Φ =e Φ− hΦi Φ_rms

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Intermittency parameter governs pulse overlap

100 105 110 115 120 125 130 135 140

t/ d

2 0 2 4

(t)

2 = 1 0 2 4

(t)

2 = 2 0 2 4

(t)

= 5

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Model experimental data with double-exponential pulses

Experimental data is approximated by a double-exponential pulse shape

φ(θ) = Θ (−θ) exp θ

λ

+ Θ (θ) exp

− θ 1−λ

In physical units: θ= (t−tk)/τ_d,τ_d≈10µs.

λdefines pulse asymmetry:

τ_r=λτ_d τ_f = (1−λ)τ_d

Notation: In= R∞

−∞

dθ[φ(θ)]ⁿ

Normalization: I₁ = 1 ⁴ ² ⁰ ² ⁴

0.0 0.2 0.4 0.6 0.8 1.0

()

= 0.00

= 0.25

= 0.50

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Correlation and power spectral density depend on pulse asymmetry

Correlation function of the pulse shape is given by

ρ_φ(θ) = 1 I2

Z∞

−∞

dχφ(χ)φ(χ+θ)

= 1

1−2λ

(1−λ) exp

− |θ|

1−λ

−λexp

−|θ|

λ

Wiener-Khinchin theorem states that the power spectral density is the Fourier-transform of the autocorrelation function

σ_φ(ω) = Z∞

−∞

dθρ_φ(θ) exp (−iωθ)

= 2

[1 + (1−λ)²ω²][1 +λ²ω²]

O.E. Garcia and A. Theodorsen, Phys. Plasmas24032309 (2017).

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The mean of the process can be computed analytically

Averaging the process over all random variables and neglect finite box effects by extending time integration to the entire real axis:

hΦ_Ki= Z∞

−∞

dA1PA(A1) Z∞

−∞

dt1

T . . . Z∞

−∞

dAKPA(AK) Z∞

−∞

dtK

T

K

X

k=1

Akφ

t−tk

τ_d

= K Tτ_dhAi

Average over number of pulsesK:

hΦi= τ_d τ_whAi

Mean value of the process increases with pulse overlap and average pulse amplitude.

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The variance can be computed analytically

hΦ²_Ki= Z∞

−∞

dA1PA(A1) Z∞

−∞

dt1

T . . . Z∞

−∞

dAKPA(AK) Z∞

−∞

dtK

T

K

X

k=1

A_kφ

t−t_k τ_d

K

X

l=1

A_lφ

t−t_l τ_d

This results in K(K −1) terms withk 6=l,K terms with k =l.

hΦ²_Ki=τ_dI₂hA²iK

T +τ_d²I₁²hAi²K(K −1) T²

⇒ hΦ²i= τ_d

τ_wI2hA²i+hΦi² where hK(K −1)i=hKi² has been used.

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Auto-correlation is determined by the pulse shape

Auto-correlation function is computed from hΦ(t)Φ(t+k)i

RΦ(r) =hΦi²+ Φ²_rmsρ_φ r

τ_d

=hΦi²+ Φ²_rms 1−2λ

(1−λ) exp

− |r|

(1−λ)τ_d

−λexp

−|r|

τ_d

0 1 2 3 4 5

r/d

0.0 0.2 0.4 0.6 0.8 1.0

(r)

= 0= 0.1

= 0.5

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Power spectral density

Ω_Φ(ω) = 2πhΦi²δ(ω) + Φ²_rmsτ_dσ_φ(τ_dω)

= 2πhΦi²δ(ω) + 2Φ²_rms τ_d h

1 + (1−λ)²τ_d²ω²i

1 +λ²τ_d²ω²

10¹ 10⁰ 10¹ 10² 10³

d

10⁶ 10⁵ 10⁴ 10³ 10² 10¹ 10⁰

()/2d

= 0= 0.1

= 0.5

λ= 0: Power law tail, ∼ω⁻² λ= 1/2: Power law tail,∼ω⁻⁴ Else: broken power law, curved spectrum.

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Probability distribution function

For exponentially distributed amplitudes and exponential wave forms is the process Gamma distributed:

hΦiP_Φ(Φ) = γ Γ(γ)

γΦ hΦi

γ−1

exp

−γΦ hΦi

0 1 2 3

/ 10⁶

10⁵ 10⁴ 10³ 10² 10¹ 10⁰

P

= 1.0

= 5.0 = 10.0

= 50.0

O.E. Garcia, Phys. Rev. Lett.108265001 (2012).

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SOL fluctuations measured in a density scan

Ohmic L-mode plasma

Lower single-null magnetic geometry Density varied fromn_e/n_G= 0.12..0.62 Probe head dwelled at the limiter radius 4 electrodes with Mirror Langmuir probes Approximately 1s long data time series in steady state

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Mirror Langmuir Probe allows fast I

s

, T

e

, and V

f

sampling

0.0 2.5 5.0 7.5 10.0

t/ s 10

V/Vf 0

20 30 Te/eV 20

30 40 Isat/mA

50 0

Iprobe/mA 50

0

Vprobe/V asp_mlp_show_mlp_uifit.py

MLP biases electrode to 3 voltages per microsecond.

Voltage range is dynamically adjusted

Probe current measured in each voltage state Fit input voltage and current is subject to 12pt smoothing (running average)

Fit U-I characteristic on (U,I) samples Largest error onT_e.

Resolves fluctuations on µs time scale

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Low density discharge, n

e

/n

G

= 0.12

0.02 0.04 Is/A

0.5 1.0 1.5

ne/1019m−3

10 15 Te/eV

0.0 0.2 0.4 0.6 0.8 1.0

time/ms

−20

−10 0

Vf/V

Intermittent, large amplitude bursts in Is.

Bursts in n_e andT_e appear correlated

Timescale approximately 25µs Irregular potential waveform

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High density discharge, n

e

/n

G

= 0.62

0.25 0.50 0.75

Is/A

5 10 15

ne/1019m−3

10 20 30

Te/eV

0.0 0.2 0.4 0.6 0.8 1.0

time/ms

−20 0 20

Vf/V

Bursts appear more isolated Average density larger by factor of 10

Average electron temperature approx. 8eV

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Ion saturation current histograms are well described by a Gamma distribution

4 2 0 2 4 6 8 10

Is

10⁴ 10³ 10² 10¹ 10⁰

PDF(Is)

ne/nG= 0.12 : = 2.68, = 0.032 ne/nG= 0.28 : = 1.60, = 0.048 ne/nG= 0.59 : = 0.68, = 0.059

A.

Theodorsen, O.E. Garcia, and M. Rypdal, Phys. Scr.92054002 (2017)

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Electron temperature histograms are well described by a Gamma distribution

4 2 0 2 4 6 8 10

Te

10⁴ 10³ 10² 10¹ 10⁰

PDF(Te)

ne/nG= 0.12 : = 11.82, = 0.048 ne/nG= 0.28 : = 6.07, = 0.000 ne/nG= 0.59 : = 0.75, = 0.077

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PSD of I

s

shows broken power law

10³ 10² 10¹ 10⁰

f/MHz 10⁵

10⁴ 10³ 10² 10¹ 10⁰ 10¹ 10²

PSD(Is)

f ²

ne/nG= 0.12 : d= 15.91 s = 0.0 ne/nG= 0.28 : d= 12.14 s = 0.0 ne/nG= 0.59 : d= 15.64 s = 0.0

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PSD of T

e

shows broken power law

10³ 10² 10¹ 10⁰

f/MHz 10⁵

10⁴ 10³ 10² 10¹ 10⁰ 10¹ 10²

PSD(Te)

f ²

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I

s

shows exponential autocorrelation function

0 5 10 15 20 25

/ s 0.0

0.2 0.4 0.6 0.8 1.0

Is()

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T

e

shows exponential autocorrelation function

0 5 10 15 20 25

/ s 0.0

0.2 0.4 0.6 0.8 1.0

Te()

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Bursts in I

s

are approximated by double-exponential waveform

20 10 0 10 20

/ s 0.0

0.2 0.4 0.6 0.8 1.0

Is()|Is(0)>2.5

ne/nG= 0.12 : d= 13.21 s, = 0.4 ne/nG= 0.28 : d= 10.28 s, = 0.4 ne/nG= 0.59 : d= 8.24 s, = 0.4

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Bursts in T

e

are approximated by double-exponential waveform

20 10 0 10 20

/ s 0.0

0.2 0.4 0.6 0.8 1.0

Te()|Is(0)>2.5

ne/nG= 0.12 : d= 17.31 s, = 0.4 ne/nG= 0.28 : d= 14.16 s, = 0.4 ne/nG= 0.59 : d= 12.24 s, = 0.4

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Time between bursts in I

s

signal is exponentially distributed

Exponential distribution describes the time between events in a Poisson process.

250 500 750 1000 1250 1500

w/ s 10⁵

10⁴ 10³ 10²

PDF(w)

ne/nG= 0.12 : w= 233.7 s ne/nG= 0.28 : w= 169.4 s ne/nG= 0.59 : w= 171.3 s

asp_mlp_mp800_tauwait_burstamp_scan.py

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Time between bursts in T

e

signal is exponentially distributed

250 500 750 1000 1250 1500

w/ s 10⁵

10⁴ 10³ 10²

PDF(w)

ne/nG= 0.12 : w= 279.6 s ne/nG= 0.28 : w= 240.7 s ne/nG= 0.59 : w= 198.1 s

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Burst amplitude distribution - Isat

2.5 5.0 7.5 10.0 12.5 15.0

A 10³

10² 10¹ 10⁰

PA(A)

ne/nG= 0.12 : A = 1.0 ne/nG= 0.28 : A = 1.1 ne/nG= 0.59 : A = 2.1

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Burst amplitude distribution - Te

2.5 5.0 7.5 10.0 12.5 15.0

A 10³

10² 10¹ 10⁰

PA(A)

ne/nG= 0.12 : A = 0.8 ne/nG= 0.28 : A = 0.8 ne/nG= 0.59 : A = 1.8

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Conclusions

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Overview of estimated parameters

n_e

nG γ (PDF) γ

Φ_rms hΦi

τ_d (PSD) τ_d,R τ_d(CA) τ_w hAi

Is 0.12 2.68 8.0 15.0µs 15.0µs 13.2µs 234µs 1.0

Is 0.28 1.60 5.7 12.1µs 11.3µs 10.3µs 169µs 1.1

Is 0.59 0.68 4.4 15.6µs 12.8µs 8.24µs 171µs 2.1

Te 0.12 11.82 25 15.4µs 14.9µs 17.3µs 280µs 0.8

Te 0.28 6.07 13 13.2µs 12.6µs 14.2µs 241µs 0.8

Te 0.59 0.75 4.6 23.4µs 16.7µs 12.2µs 198µs 1.8

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Conclusions

Theory Experimental data

Process is Gamma distributed Is andTe time series are Gamma distributed Pulses arrive uncorrelated Waiting time between bursts in

IsandTe is exponential distributed Exponential distributed pulse amplitude Burst amplitudes inIs

andTe are expon. distributed Double-exponential pulse shape PSD, autocorrelation function and

cond. avg. ofIsandTe time series agree

Less burst overlap at high densities

Burst duration time changes little with n_e/n_G. Burst amplitude increases with n_e/n_G

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Thank you for your attention.