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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
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
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
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
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
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
Velocity-space weight functions can be derived by averaging over the “unneeded” coordinates
11
[1] L. Stagner, W.W. Heidbrink Physics of Plasmas 24 (2017)
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
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
[1] L. Stagner, W.W. Heidbrink Physics of Plasmas 24 (2017)
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
[1] L. Stagner, W.W. Heidbrink Physics of Plasmas 24 (2017)
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
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
[1] L. Stagner, W.W. Heidbrink Physics of Plasmas 24 (2017)
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
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
[1] L. Stagner, W.W. Heidbrink Physics of Plasmas 24 (2017)
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
Orbits naturally correlate different viewing chords
Trapped and Co-Passing orbits are blue shiftedSmaller red shifted Ctr-Passing contribution
20
Blue Shifted Red Shifted
Orbits naturally correlate different viewing chords
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
The total FIDA spectra can be expressed as a linear combination of orbit weight functions
Consider the signal produced by a single orbit
This is the definition of an Orbit Weight Function
23
The total FIDA spectra can be expressed as a linear combination of orbit weight functions
Same form as Velocity-space Tomography
Consider the signal produced by a single orbit
This is the definition of an Orbit Weight Function
24
The total FIDA spectra can be expressed as a linear combination of orbit weight functions
Same form as Velocity-space Tomography
Consider the signal produced by a single orbit
This is the definition of an Orbit Weight Function
25
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
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
Prior information is added to improve reconstructions29
1. Magnetic Equilibrium
2. The distribution should be smooth
Information already encoded in the fast-ion orbits
Prior information is added to improve reconstructions30
1. Magnetic Equilibrium
2. The distribution should be smooth
3. A guess fast-ion distribution
Information already encoded in the fast-ion orbits
Prior information is added to improve reconstructions31
1. Magnetic Equilibrium
2. The distribution should be smooth
3. A guess fast-ion distribution
Assuming the data is normally distributed the posterior is analytic
Information already encoded in the fast-ion orbits
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
[1] L. Stagner, W.W. Heidbrink Physics of Plasmas 24 (2017)
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