j. ho¨o¨k div. fusion plasma physics, kth joh@kthszepessy/inversfor/josef.pdfintroduction...

IntroductionInferring current distribution

ExampleConclusion

Current tomography for axisymmetric plasmas

J. HookDiv. fusion plasma physics, KTH

[email protected]

December 3, 2009

J. Hook Div. fusion plasma physics, KTH [email protected] Current tomography for axisymmetric plasmas


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The Joint European Torus (JET) tokamak



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Background

Experimental data and theoretical models in nuclear fusionexperiments depends on the inference of the magnetictopology.

1A number of assumptions is used when solving the GS equation ;

Equilibrium assumption, zero plasma rotation or isotropic plasma pressure.



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Background


Nested flux surfaces provide a natural coordinate system inwhich to express physics.





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Background


Nested flux surfaces provide a natural coordinate system inwhich to express physics.

Flux surface geometry is estimated from magneticmeasurements by solving an equilibrium equation; theGrad-Shafranov equation1.





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Inverse problem

The paper

Current tomography for axisymmetric plasmas

J Svensson, A Werner and JET-EFDA Contributors

Max Planck Institute for Plasma Physics, Teilinstitut Greifswald, Germany

Received 21 August 2007, in nal form 7 April 2008

Published 29 May 2008

Online atstacks.iop.org/PPCF/50/085002

How much information of the internal current distribution isavailable in magnetic diagnostic signals (assuming onlymagnetostatics and axisymmetry)?



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Inverse problem

Magnetic measurement problem

Is it possible to determine the plasma current distributionfrom external magnetic measurements?

The community is dominated by naysayers.

Authors of the presented paper claim the question to beresolved.



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Inverse problem

Setup

Poloidal flux is given by,

Ψ(R, Z ) =

∮

A · dl = 2πRAφ

Aφ =µ0

4π

∫

Ω

jφ(r ′)

|r − r ′|d3V ′

Magnetic field can becalculated with B = ∇×A,

BR = −1

2πR

∂Ψ

∂Z

BZ =1

2πR

∂Ψ

∂R



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Inverse problem

Toroidal current model

Figure 1. Plasma toroidal current model. The volume enclosed by the rst wall is lled with a

grid of toroidal beams with rectangular cross sections, each carrying a uniform current.

A model for the toroidalplasma current is used.

Each beam carry a uniformcurrent.

With this model the vectorpotential and magneticfield is simplified,

Aφ(r j) =∑

N beams

aij Ii , (1)

Bp(r j) =∑

N beams

bij Ii (2)



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Inverse problem

Pickup and external -coils

-pickup coiltoroidal coils

poloidal coils



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Inverse problem

Measurements

Pickup coils measure local magnetic field,

Pi = BR(Ri , Zi ) cos(θi ) + BZ (Ri , Zi ) sin(θi )

where Ri , Zi is the position of the coil in poloidal plane.

Saddle coils measures the difference in flux between twopoloidal positions,

Si =di

8(Ψ(R2

i , Z 2i ) − Ψ(R1

i , Z 1i )).

Full flux loop2 measurement is given by,

Fi = Ψ(Ri , Zi ).

2Generally: A large, single-turn coil of wire, used to measure the magnetic

flux enclosed by it.



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Inverse problem

Forward problem

Measurements are collected in: DMag

= (Pi, Si, Fi)T .

The free beam-current parameters: I = (IPli , IExt

i )T .

Forward problem:

P = M I + C , (3)

P is the vector of predicted signals given I and MNI×NDis the

response matrix. Contributions from each pickup coil isspecified in C 3.

3Here C is a constant not a noise term!



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Inverse problem

Bayesian inference (inverse problem)

Bayes formula,

p(I |D) =p(D|I )p(I )

p(D)

Posterior = Likelihood × Prior

Regularization in Bayesian models, corresponds to specifyingprior knowledge p(I ) about the solution e.g. smooth solutionsare more likely than non-smooth solutions.

Ill-posed problems are characterized by having a non-peaked(uninformative) likelihood function.



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Inverse problem

Likelihood and prior distribution

Likelihood distribution, assuming normally distributed errorson all measurements:

p(DMag

|I ) =

exp

(

−1

2(M I + C − D

Mag)TΣ

−1

D (M I + C − DMag

)

)

(2π)ND/2|ΣD |1/2

where ΣD is the covariance matrix.

Assume a multivariate normal prior

p(I ) =

exp

(

−1

2(I − mI )

TΣ−1

I (I − mI )

)

(2π)NI /2|ΣI |1/2



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Inverse problem

Posterior estimates

Multiplying the likelihood with the prior we obtain:

p(I |DMag

) =

exp

(

−1

2(I − m)TΣ

−1

(I − m)

)

(2π)NI /2|Σ|1/2

where

m = mI + (MT

Σ−1

D M + Σ−1

I )−1MT

Σ−1

D (DMag

− C − MmI )

Σ = (MT

Σ−1

D M + Σ−1

I )−1



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Inverse problem

MAP estimator

Log likelihood of the posterior can be written as:

log(p(I |DMag

)) = −1

2(M I + C − D

Mag)TΣ

−1

D (M I + C − DMag

)

−1

2(I − m)TΣ

−1

I (I − m)

The maximum posterior (MAP) estimator is given by

I = (MT

Σ−1

D M + Σ−1

I )−1MT

Σ−1

D (DMag

− C )

When ΣD = σ2D I i , ΣI = σ2

I I j we obtain a solution of thesame form as obtained by Tikhonov regularization.

I = (MT

M + αI )−1MT

(DMag

− C ), α = (σD/σI )2



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Real test case: a JET pulse

A beam grid of about 1400 beams and a prior with a standarddeviation centered at zero such that total current is 10MA isused. Posterior distribution of current density:

Figure 3. (a) Cross section of toroidal current density at the magnetic axis position, from samples

drawn from the posterior distribution of a model with about 1400 free parameters.

b) The marginal distribution at R = 2.73m, Z = 0.33m shows little more than a

higher probability for a negative current density.

Prior not good enough. Regularization needed!



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Flux surfaces

Figure 4. (a) Boundary and internal ux surfaces for normalized radiusr = 0.2, 0.5 and 0.8

calculated from 200 samples from the posterior distribution used in !g3, using the nominal 2%

error on the measurements, a n d (b ) using in ated 6% error on all measurements.



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Conditional autoregressive prior

Neighbouring current beams are more likely to have similarvalues than beams far apart.

Spatial dependencies can be expressed with a so calledconditional autoregressive (CAR) model:

Ii |I−i ∼ N(∑

j

βij Ij , τi ), βii = 0, βij = 1/4, neighbours ⇒

p(I ) ∝ exp

(

−1

2IT

Q I

)

, Q =1

τ

(

1 −1

4W

)

where W is an adjacency matrix; Wij = 1 if i , j are neighboursand zero otherwise. Each beam is dependent only on its fournearest neighbours. Gives a multivariate normal posterior.



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Current distribution with CAR prior

Figure 5. (a) Posterior toroidal current distributions constructed using the CAR prior in

The solid line is MAP estimate and the dashed lines are the 1 σ –3σ level curves

of posterior distribution. ( b) Marginal distribution at R = 2.73 m, Z = 0.33 m.



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Internal flux surfaces with CAR prior

Figure 6. (a) Boundary and internal ux surfaces for normalized radiusr = 0.2, 0.5 and 0.8 using

the CAR prior and (b) only boundary, showing that the error in the reconstruction varies over the

boundary, being higher near the X-point and in the upper left corner, which is the region nearest to

the magnetized iron. Constructed from 200 samples from the posterior distribution.



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Time evolving flux surfaces

Figure 7. Flux surfaces calculated from MAP estimates of current distribution for the evolution

of JET pulse 66271. Colours correspond to values of the poloidal ux. The boundary line is from

EFIT for comparison.



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Total plasma current

Figure 8. Total plasma current as calculated with the current tomographymethod (blue line),

error bars are very small and correspond to approximately 0.5% error (blue dashed). The red curve

shows the total current as calculated using the MAGN program, normally used at JET for the total

toroidal current.



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Conclusion

With a proper choice of prior the current distribution can bereconstructed.

The method can deal fully with the problem of degeneratesolutions as well as uncertainty from measurement errors.

The CAR prior enable simple calculations; MCMC not needed.

Current tomography give a very low error on the total currentand give an almost perfect agreement with the standardcalculations.



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References

J Svensson, A Werner and JET-EFDA Contributors , “Currenttomography for axisymmetric plasmas”, Plasma Physics andControlled Fusion, 2008


j. ho¨o¨k div. fusion plasma physics, kth joh@kthszepessy/inversfor/josef.pdfintroduction...

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