determining the population of individual fast-ion orbits using ......determining the population of...

Determining the Population of Individual Fast-ion Orbits using Generalized Diagnostic Weight Functions

Luke Stagner1 & W.W. Heidbrink11University of California, Irvine

IAEA Technical Meeting on Energetic Particles in Magnetic Confinement Systems

September 6, 2017

1

F(E,Rm

) F(E,Rm

)

Knowing the fast-ion distribution function is key to developing strategies to mitigate fast-ion losses

4D Fast-ion Distribution Function● Energy [keV]● Pitch [unitless] v။/v● Gyro-angle (cyclic)● R [m]● Z [m]● Toroidal angle (cyclic)

F(R,Z)

F(E,p)

■ There is a significant non-thermal population

of ions i.e. fast ions

■ Fast ions can drive instabilities

■ Need to know fast-ion distribution function (F) to

understand instabilities

and prevent losses

NUBEAM Simulation of Fast-ion Distribution

2

Main points of this talk

1. We can infer a local fast-ion distribution function using an

approximate forward model of our diagnostics, but there are

problems

2. We can represent the full forward model as a linear

combination of orbits

3. Using Orbit Tomography we can better infer the full fast-ion

distribution function

3




problems





4

FIDA diagnostic is used to measure the fast-ion distribution

Fast-ion D-alpha (FIDA) detects Doppler-shifted light from the fast neutral

FIDA viewing chords are setup to measure a radial profile

Each chord collects FIDA signal over an extended region

Complicated physics and geometry require a forward model to interpret experimental data

5

FIDA spectra can be simulated by using a Full Forward Model (FIDASIM) or by using a local approximation

Full Fast-ion Distribution

Local Fast-ion Distribution

W(E,p) is a function that weights different parts of the distribution

Full Forward Model

Approximate Forward Model

6

Discretizing the approximate model creates a system of linear equations which can be solved

M. Salewski et. al. Nucl. Fusion 56 (2016)

W is a m × n matrix F is an n × 1 column vector

Current State of the ArtVelocity-space

Tomography

True DistributionReconstructed

Distribution

7

The lack of spatial information is a limiting factor in Velocity-space tomography

A.S. Jacobsen & L. Stagner P.P.C.F. 58 (2016)

Requires many spatially overlapping diagnostics which is not common

Approximate model performs poorly in regions where fast-ion gradients are large

8




problems





9

Full forward model can be put into the same form as the approximate model

S(x) is the expected diagnostic signal for a fast ion with position and velocity x (6-dimensional) [1]

MATH

[1] L. Stagner, W.W. Heidbrink Physics of Plasmas 24 (2017)

10



MATH

Velocity-space weight functions can be derived by averaging over the “unneeded” coordinates

11




MATH

Velocity-space weight functions can be derived by averaging over the “unneeded” coordinates

A weight function is just the average signal produced by a fast ion with given energy and pitch

12


Action-angle coordinates are used to reduce dimensionality without sacrificing model accuracy

Coordinates that do not appear in the Lagrangian are safe to average out

13




Action-angle coordinates naturally separate out the cyclic coordinates from the critical coordinates

Action Coordinates (J) ➢ Constants of motion

Angle Coordinates (Θ)➢ Cyclic Coordinates with period τ

14




Action-angle coordinates naturally separate out the cyclic coordinates from the critical coordinates

Action Coordinates (J) ➢ Constants of motion

Angle Coordinates (Θ)➢ Cyclic Coordinates with period τ

Full Forward Model Weight Function

15


In guiding center motion action coordinates define an orbit

EPR Coordinates

■ Three Action Coordinates (J)➢ Energy (E)

➢ Maximal R along orbit (Rm

)

➢ Pitch at Rm

(p)

■ Three Angle Coordinates (Θ)➢ Initial Toroidal angle: τɸ = 2π➢ Gyro-angle: τγ = 2π➢ Poloidal Transit Time: τ

p

■ Unique labeling of orbits■ Naturally bounded space■ Similar to coordinates used by Rome

and Others

JA Rome et. al. Nucl. Fusion 19 (1979)

16

Slice of EPR-space➢ Fixed Energy and R

m➢ Scan over Pitch

Easy to enumerate all possible orbits

In guiding center motion action coordinates define an orbit

EPR Coordinates

■ Three Action Coordinates (J)➢ Energy (E)

➢ Maximal R along orbit (Rm

)

➢ Pitch at Rm

(p)

■ Three Angle Coordinates (Θ)➢ Initial Toroidal angle: τɸ = 2π➢ Gyro-angle: τγ = 2π➢ Poloidal Transit Time: τ

p

■ Unique labeling of orbits■ Naturally bounded space■ Similar to coordinates used by Rome

and Others

JA Rome et. al. Nucl. Fusion 19 (1979)

17

Projections of the full fast-ion distribution in

EPR coordinates

F(E,Rm

)

F(Rm

,p)

F(E,p)

Orbits naturally correlate different viewing chords18

Orbits naturally correlate different viewing chords

Blue Shifted Red Shifted

Trapped and Ctr-Passing orbits are red shiftedSmaller blue shifted Co-Passing contribution

19


Trapped and Co-Passing orbits are blue shiftedSmaller red shifted Ctr-Passing contribution

20

Blue Shifted Red Shifted


All the lines of sight can be used to reconstruct the fast-ion distribution

21

Emission Integrated over Wavelength

The total FIDA spectra can be expressed as a linear combination of orbit weight functions

Consider the signal produced by a single orbit

22



This is the definition of an Orbit Weight Function

23


Same form as Velocity-space Tomography



24


Same form as Velocity-space Tomography



25




problems





26

Bayesian methods are used to regularize the ill-conditioned system of linear equations

27

Bayesian methods provide a systematic way of incorporating additional prior information to

constrain the possible set of solutions

Posterior: Encodes knowledge about the model parameters with respect to the data

Likelihood: Modifies prior with the data

Prior: Encodes knowledge about model parameters before data is collected

Prior information is added to improve reconstructions28

1. Magnetic Equilibrium Information already encoded in the fast-ion orbits


1. Magnetic Equilibrium

2. The distribution should be smooth

Information already encoded in the fast-ion orbits




3. A guess fast-ion distribution





3. A guess fast-ion distribution

Assuming the data is normally distributed the posterior is analytic


Reconstruction technique benchmarked using a synthetic FIDA data generated from a Non-Classical distribution

32

Non-Classical: D=1.0 m2/s ReconstructionReconstruction Parameters■ 1000 Orbits■ 9298 FIDA measurements from

53 viewing chords■ μ

N= 0

F(E,Rm

)

F(Rm

,p)

F(E,p)

F(E,Rm

)

F(Rm

,p)

F(E,p)

Nfast

= 1.09e19 Nfast

= 1.20e19

Reconstruction of a MHD-quiescent H-mode distribution is in reasonable agreement with Classical distribution

33

Classical Distribution ReconstructionReconstruction Parameters■ 1000 Orbits■ 9298 FIDA measurements from

53 viewing chords■ μ

N= N

class

F(E,Rm

)

F(Rm

,p)

F(E,p)

F(E,Rm

)

F(Rm

,p)

F(E,p)

Nfast

= 1.88e19 Nfast

= 1.87e19

Future Directions

■ Study a fast-ion distribution that is not well described classically■ Calculate orbit weight functions for more diagnostics

➢ FIDA, Neutral Particle Analyzer (NPA), and Neutron

diagnostics already implemented [1]

■ Make calculation of the orbit weight functions faster➢ Switch FIDASIM to MPI-based parallelism

➢ Replace bottlenecks with faster surrogates

34


Backup Slides35

The reconstructed number of fast ions in each orbit type are similar to the classical prediction

36

Potato Stagnation Trapped Ctr-Passing Co-Passing Total

Classical 1.27e17 3.28e17 3.51e18 2.74e18 1.21e19 1.88e19

Reconstruction 1.34e17 1.64e17 4.04e18 3.20e18 1.11e19 1.87e19

Velocity-space weight functions can be generalized to the full 6D phase space

Consider a fast ion with generalized phase-space coordinate x = [p,q]. Let S(x) be the expected signal produced by the fast ion. The total signal then the sum of the all the individual signals.

The sum can be expressed in terms of the frequency in which x occurs.

In the continuum limit the sum can be written in terms of an integral and the generalized PDF

The final result has the same form as the approximate model

A weight function is just the expected diagnostic signal for a given phase space coordinate

37

Hamilitonian Coordinates do not uniquely label orbits38

Simulation/Reconstruction in Energy-pitch space39

ReconstructionNUBEAM Simulation

Diagnostics are simulated using FIDASIM

Inputs:■ Plasma Parameters■ Electromagnetic Fields■ Fast-ion Distribution■ Diagnostic Geometry

Outputs:■ Bremsstrahlung■ Neutral Beam & Halo Density■ Beam & Halo D-alpha Emission■ Birth Profile■ FIDA Spectra■ NPA Energy Spectra■ Beam-Target Neutron Rate■ FIDA/NPA/Neutron Velocity-space Weight Functions

Source Code: https://github.com/D3DEnergetic/FIDASIM

Documentation: http://d3denergetic.github.io/FIDASIM

40

https://github.com/D3DEnergetic/FIDASIMhttp://d3denergetic.github.io/FIDASIM

Weight functions indicate which phase-space regions a diagnostic is most sensitive

■ Typically derived using probabilistic arguments

■ Calculated only in velocity-space

■ Spatial coordinates averaged over or ignored

Calculated using open source FIDASIM code

41

Orbit weights for FIDA diagnostic

Oblique FIDA: 660 nm and 1.9 m

42

Reconstruction of a MHD-quiescent H-mode distribution is in reasonable agreement with Classical distribution

43

F(E,Rm

) F(E,Rm

)

Reconstruction TRANSP/NUBEAM

determining the population of individual fast-ion orbits using ......determining the population of...

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