A code that simulates fast-ion Dα and neutral particle measurements

TL;DR: In this article, a code that predicts the efflux to a neutral particle analyzer (NPA) diagnostic and the photon radiance of Balmer-alpha light to a fast ion Dα (FIDA) diagnostic is described.

Abstract: A code that models signals produced by charge-exchange reactions between fast ions and injected neutral beams in tokamak plasmas is described. With the fast- ion distribution function as input, the code predicts the efflux to a neutral particle analyzer (NPA) diagnostic and the photon radiance of Balmer-alpha light to a fast- ion Dα (FIDA) diagnostic. Reactions with both the primary injected neutrals and with the cloud of secondary "halo" neutrals that surround the beam are treated. Accurate calculation of the fraction of neutrals that occupy excited atomic states (the collisional- radiative transition equations) is an important element of the code. Comparison with TRANSP output and other tests verify the solutions. Judicious selection of grid size and other parameters facilitate efficient solutions. The output of the code has been validated by FIDA measurements on DIII-D but further tests are warranted.

Summary

1 Introduction

• Supra-thermal populations of energetic ions play an important role in magnetic fusion research.
• Both NPA and FIDA diagnostics provide valuable information about the fast-ion distribution function but also depend sensitively on other plasma parameters and on atomic cross sections.
• A least-squares minimization scheme that utilizes a weight function can determine which model distribution function agrees best with the data.
• Section 3 describes tests that verify that the code correctly solves the desired equations.

2 Model

• The code has four main sections (Fig. 1).
• The first section prepares the data and the second calculates the neutral populations.
• The third and fourth sections both rely on the first two sections but are independent of each other.
• One section computes the NPA flux and the other computes the FIDA radiance.

2.1 Input data and coordinate mapping

• The code begins by collecting the input data.
• The code uses the conventions of the NUBEAM module [8] of the TRANSP code [9] to describe the geometry of the viewed neutral beam source (or sources).
• For installations that do not use EFIT, a post-processor that is part of the TRANSP distribution can convert TRANSP output files into the desired format.
• Plasma parameters are one-dimensional functions of flux coordinates.
• Depending on the source of the theoretical fast-ion distribution function, the mapping into the (x,y,z) coordinates can be fairly complicated.

2.2 Injected and halo neutral densities

• The second major section of FIDASIM is devoted to calculation of the injected and halo neutral distributions in real space, velocity space, and energy levels.
• The third major simplification in the treatment of the collisional-radiative transitions is to assume that the speeds of the various species follow the ordering ve ≫v f ∼vn ∼vi ≫vI , where the subscripts represent electrons, fast ions, hydrogenic neutrals (both fast and thermal), thermal hydrogenic ions, and impurity ions, respectively.
• From the known source, the code computes the cloud of halo neutrals that surround each injected beam.
• The neutral is followed until it ionizes.
• (For typical parameters, this is true for 99% of the halo neutrals, so this is an excellent approximation.).

2.3 NPA flux

• The NPA flux is found in the third stage of the code.
• The detector geometry determines the diagnostic viewing cone but the detected particles have guiding centers that are a gyroradius from the viewing cone.
• The distribution function used in Eq. (2.6) is evaluated at the guiding center position.
• The attenuation of neutrals is computed in the last part of the NPA calculation.

2.4 FIDA radiance

• The fourth stage of the code uses a weighted Monte Carlo routine to calculate the FIDA radiance.
• The product n f ∑nn provides a convenient estimate of the probability of a charge exchange reaction (that neglects the computationally intensive dependence of the reaction rate on the relative velocity), so this product is used to determine how many fast neutrals to launch from each cell.
• Next, the trajectory of the fast neutral through the cells is computed by TRACK.
• The code also performs a simple integration over these emissivities ǫi along the specified sightlines.
• Note that the computed spectra neglect instrumental broadening.

3.1 Atomic physics

• Atomic physics cross sections are important in two places in the code: in the calculation of neutralization probability and in the solution of Eq. (2.3) in COLRAD.
• The required cross sections and reactivities are available in the literature and in the Atomic Data and Analysis Structure (ADAS) compilation [20, 21].
• An alternative compilation of many of the rates appears in [26] but the effect of these differences is small compared to other uncertainties so the current version of the code uses the older compilation by Janev [24].
• In COLRAD, rates for energy levels up to n=7 are normally employed.
• For the initial neutralization probability, cross sections for the charge-exchange reactions between fast ions and neutrals in states n = 1−4 are given in ADAS [20].

3.2 Beam deposition

• The calculation of the injected neutral density was compared with TRANSP for an NSTX case.
• The neutral profiles along the y axis (perpendicular to the neutral beam source) and along the z axis (vertical to the source) axis agree well with the TRANSP simulation results (Fig. 4) and with neutral beam calibration data [27].
• Fig. 5(b) shows the attenuation factors for 60keV neutrals along an NPA sightline for the TRANSP and FIDASIM simulations.
• Fig. 6 compares spectra after integration over the sightline.
• At low density, the agreement is satisfactory when halo neutrals are neglected in FIDASIM.

3.3 Halo neutrals

• The halo neutral simulation portion of the code was verified by comparing with a onedimensional diffusion model.
• The leftmost term in Eq. (3.2) represents the halo neutral diffusion.
• It is not easily solved analytically for a spatially varying beam neutral profile but a solution exists for a constant source.
• Fig. 7 shows the halo neu- tral densities calculated from the diffusion model and the Monte-Carlo halo simulation subroutine for identical plasma profiles.

3.4 FIDA spectra

• The SPECTRUM subroutine used to compute the Dα spectra and the weighted Monte Carlo scheme were verified as follows.
• As part of the initial investigation of the feasibility of FIDA, a simplified model of the expected spectra was developed that ignores atomic physics and assumes that the magnetic field is purely toroidal; Fig. 2 of [3] shows a result calculated by this code.
• To test the main FIDA simulation loop, the authors replaced the magnetic field with a toroidal field and modified the cross sections to be independent of velocity.
• The resulting spectra were consistent with the output of the simple model.

4 Numerics

• Numerical input parameters affect both the accuracy and computational expense of a calculation.
• Uncertainties in plasma parameters (particularly electron density) introduce uncertainties in the predicted radiance or efflux of 20% or more , so extremely fine grids are wasteful and unnecessary.
• The neutral-beam injection energy is 80keV in this plasma.
• The Fortran version of the code is an order of magnitude faster than the IDL version.
• For a typical FIDA simulation with 107 reneutrals this translates to about 28 hours.

4.1 Number of Monte Carlo reneutrals launched

• There is a linear dependence between the number of launched neutralized fast ions (or ”reneutrals”) and the computational time in the fourth section of the code.
• Fig. 8(a) compares the spectra at the location of peak Dα emissivity (R = 187.5cm) for several simulations with varying number of launched reneutrals.
• To eliminate the influence of the random number seed choice, the same seed value was used in all five simulations.
• Considering the resulting spectra as the most accurate, it is instructive to compare the ratios of the spectra from simulations with less particles to the spectra from this particular simulation (Fig. 8(b)).
• If the integration starts at half the beam injection energy, ∼3 times more particles are needed.

4.2 Simulation volume

• The computational grid must enclose most of the interacting beam ions and neutral particles.
• Further reduction of the grid in the x-direction to 90cm truncates a sizable fraction of the halo neutrals but the effect on the calculated spectra is still small because Further down the x-axis, the plasma density increases and, as more injected neutrals charge exchange with thermal deuterium ions, the beam halo profiles spread in both transverse directions (Fig. 11(c)).
• The authors baseline simulation uses values of the grid half-width of 30 and 40cm in the y- and z-directions, respectively.
• For FIDA chords that view horizontally, expanding the grid along the y-axis is more important than expanding it along the z-axis.

4.3 Cell size

• Once the simulation volume in the (x,y,z) space is defined, the size of each cell needs to be determined.
• Physically, the emissivity profiles depend on numerous quantities including the beam injection geometry and the plasma parameters; gradients in any of these quantities impact transverse emissivity gradients.
• Generally speaking, owing to the line integration, the cell size in the approximate direction of the sightlines can be twice as large as in transverse directions.
• To study the effect of coarser or finer grids, the authors use the standard 120×60×80cm computational domain and 107 particles.
• The corresponding computational times are 63% and 22% higher.

5 Validation

• The calculation of the injected neutrals was compared with experimental measurements of the beam-emission light in a DIII-D experiment [32].
• After passing through a bandpass filter, two-dimensional images of the light were measured with a CCD camera.
• The code predictions are in good agreement with measurements of the vertical extent of the beam and of the beam penetration as a function of density.
• In MHD-quiescent DIII-D plasmas, code predictions based on the fast-ion distribution function predicted by NUBEAM have the same spectral shape as experiment and the intensity of the FIDA signal agrees to within 25%.
• Comparison of the relative intensity of these four features is a useful check that is independent of any experimental errors in the intensity calibration.

6 Outlook

• With plasma profiles and a fast-ion distribution function as input, FIDASIM predicts the flux measured by NPAs and the radiance measured by a Dα spectrometer.
• One possible area of improvement is a post-processor that replaces the approximation of infinitesimal sightlines with an accurate treatment of the collection optics.
• A more challenging upgrade is needed to treat plasmas where the fastion density is comparable to the thermal-ion density.
• A complementary reduced model that is sufficiently fast to make predictions between discharges (for example) is needed.

Acknowledgments

• This work has benefited from the contributions of a large number of scientists.
• Keith Burrell, Bill Davis, Rainer Fischer, Manuel Garcı́a-Muñoz, Doug McCune, and Mike Van Zeeland gave valuable advice.
• The authors are also indebted to their experimental collaborators on DIII-D and NSTX.
• The originating developer of ADAS is the JET Joint Undertaking.

Figures

Figure 4: Comparison of injected neutral density profile along the y and z axes from the FIDASIM code (symbols) and from TRANSP (lines) at the beam tangency point (R=70cm) for NSTX Source A.

Figure 5: Comparison between the FIDASIM and TRANSP simulations in NSTX discharge #122631 at t=0.1s. (a) Neutral density, (b) attenuation factor for 60 keV neutrals and (c) differential charge-exchange flux along the NPA sightline.

Figure 11: Vertical (y,z) planes of (a) injected neutral density near the plasma edge, (b) halo neutral density near the plasma edge, and (c) halo neutral density in the plasma interior.

Figure 10: (a) Midplane (z = 0) cross section of a quadrant of the DIII-D tokamak, showing the intersection of the vertical FIDA sightlines (triangles) and the (x,y) coordinate system associated with the active neutral beam. (b) Midplane halo neutral density. The distribution is asymmetrical due to the plasma flow. (c) Beam-ion density. (d) Injected neutral density.

Figure 14: (a) FIDA emissivity in an (x,z) plane, with the standard (33×21) grid boundaries overlaid. The red lines illustrate the range of sightline angles graphed in the lower figure. (b) Calculated radiance vs. sightline angle for different choices of grid size in the (x,z) plane. The flat portions of the curves are caused by pixelation.

Figure 12: Dependence of the calculated FIDA radiance on the volume of the simulation. The spectra are integrated over wavelengths above Eλ =10keV.

Figure 13: (a) Ratio of the blue-shifted FIDA radiance for simulations without halo neutrals and with halo neutrals vs. major radius. The spectra are integrated over wavelengths above Eλ = 10keV. (b) Ratio of the FIDA spectral radiance for simulations without halo neutrals and with halo neutrals for the FIDA channel at R=187.5cm. Two values of Eλ are indicated.

Figure 3: Solution of Eq. (2.3) immediately following neutralization of the fast ion for a typical case (density of 4×1013cm−3, temperature of 4keV). The total time represented by the abscissa is 11.1ns; the fast neutral travels 3cm during this time. The evolution of the n=3 state determines the intensity of FIDA light.

Figure 1: Flow diagram for the FIDASIM code.

Figure 8: (a) Comparison of FIDA spectra from simulations with 106 to 9×107 Monte Carlo particles. (The reduction in FIDA light for small Doppler shifts is an artifact caused by truncating the fast-ion distribution function at Emin = 10keV.) (b) FIDA spectra normalized to the spectrum computed with 90 million particles. The dashed vertical lines relate Doppler shifts to equivalent energies along the line-of-sight Eλ.

Figure 9: Comparison of FIDA profiles for simulations with 106 to 9×107 Monte Carlo particles. The spectra are integrated over Doppler shifts larger than (a) Eλ≥10keV and (b) Eλ≥30keV.

Figure 7: Comparison of halo neutral densities calculated from the 1-D diffusion model (Eq. (3.3)) and the FIDASIM halo diffusion subroutine for plasma profiles with ni=ne=1.1×1014cm−3, Te=2.5keV and Ti=1.25keV. The three dashed steps represent the densities of the full, half, and third energy components of injected neutrals used in both codes.

Figure 2: Plan view of NSTX. Geometrical neutral beam and detector input to the code is in (u,v,z) coordinates. Neutral beam parameters (upper case labels) follow the TRANSP conventions. The code transforms quantities into (x,y,z) coordinates along the selected beam.

Figure 6: NPA energy spectra for a sightline with Rtan = 70cm from the TRANSP and FIDASIM simulations with and without halos at three different times in NSTX discharge #122631. The density increased from approximately 2×1013cm−3 to 5×1013cm−3 between 0.1 and 0.5s.

Full text

UC Irvine
UC Irvine Previously Published Works
Title
A code that simulates fast-ion Dα and neutral particle measurements
Permalink
https://escholarship.org/uc/item/12f712f4
Journal
Communications in Computational Physics, 10(3)
ISSN
1815-2406
Authors
Heidbrink, WW
Liu, D
Luo, Y
et al.
Publication Date
2011
DOI
10.4208/cicp.190810.080211a
Copyright Information
This work is made available under the terms of a Creative Commons Attribution License,
availalbe at https://creativecommons.org/licenses/by/4.0/
Peer reviewed
eScholarship.org Powered by the California Digital Library
University of California

Commun. Comput. Phys.
doi: 10.4208/cicp.190810.080211a
Vol. 10, No. 3, pp. 716-741
September 2011
A Code that Simulates Fast-Ion D
α
and Neutral
Particle Measurements
W. W. Heidbrink
1,∗
, D. Liu
1,3
, Y. Luo
1,4
, E. Ruskov
1
and
B. Geiger
2
1
Department of Physics and Astronomy, University of California, Irvine,
California, CA 92697, USA.
2
Max-Planck Institute f¨ur Plasmaphysik, Garching, Germany.
3
Department of Physics, University of Wisconsin-Madison, Madison,
WI 53706, USA.
4
Tri Alpha Energy Corporation, 27211 Burbank, Foothill Ranch, CA 92610, USA.
Received 19 August 2010; Accepted (in revised version) 8 February 2011
Available online 1 June 2011
Abstract. A code that models signals produced by charge-exchange reactions between
fast ions and injected neutral beams in tokamak plasmas is described. With the fast-
ion distribution function as input, the code predicts the efflux to a neutral particle
analyzer (NPA) diagnostic and the photon radiance of Balmer-alpha light to a fast-
ion D
α
(FIDA) diagnostic. Reactions with both the primary injected neutrals and with
the cloud of secondary ”halo” neutrals that surround the beam are treated. Accurate
calculation of the fraction of neutrals that occupy excited atomic states (the collisional-
radiative transition equations) is an important element of the code. Comparison with
TRANSP output and other tests verify the solutions. Judicious selection of grid size
and other parameters facilitate efficient solutions. The output of the code has be en
validated by FIDA measurements on DIII-D but further tests are warranted.
PACS: 52.55.Pi, 52.65.Pp, 52.70 .Kz
Key words: Fast ions.
1 Introduction
Supra-thermal populations of energetic ions play an important role in magnetic fusion
research. These ”fast ions” are created by neutral-beam injection, by RF heating, and in
fusion reactions. The distribution function that describes t h ese populations ge n erally is a
∗
Corresponding author. Email addresses: Bill.Heidbrink@uci.edu (W. W. Heidbrink), dliu29@wisc.edu ( D.
Liu), yluo@trialphaenergy.com (Y. Luo), eruskov@uci.edu (E. Ruskov), bgeiger@ipp.mpg.de (B. Geiger)
http://www.global-sci.com/ 716
c
2011 Global-Science Press

W. W. Heidbrink et al. / Commun. Co mput. Phys., 10 (2011), pp. 716-741 717
complicated function of velocity and configuration-space variables. Measuring the fast-
ion distribution function in the harsh magnet ic fusion environment is a major diagnostic
challenge.
One approach is to exploit charge exchange reactions between energetic deuterium
ions and an injected neut r al beam. Collection of escaping n eutrals is t h e basis of neutral
particle analysis (NPA) [1], a te chnique that has been applied to tokamak plasmas for
nearly five decades [2]. A more recent technique is to analyze the visible photons emitted
by hyd rogenic fast ions that neutralize in the injected beam [3]. A review of these fast-ion
D
α
(FIDA) me asurements was recently published [4].
Both NPA and FIDA diagnostics provide valuable information about the fast-ion dis-
tribution function but also depend se n sitively on other plasma parameters and on atomic
cross sections. One way to relate the measured signals to theory is to construct a phase-
space weight function for each me asurement [5]; the signal is the convolution of the fast-
ion distribution function with the weight function. As illustrated by the examples in [4],
this approach is quite useful for rapid qualitative interpretation of the measurements. It
can also be the basis for an inversion algorithm. Although the processes are too com-
plicated for a unique inversion [6], a least-squares minimization scheme that utilizes a
weight function can dete r mine which model distribution function agrees best with the
data. An example of inference of the d istribution function from collective Thomson scat-
tering data was recently published [7].
Alternatively, one can use forward modeling. In this approach, the distribution func-
tion is a given quantity supplied by theory. The code described in this paper, dubbed
FIDASIM, takes this approach. FIDASIM accepts a theoretical distribution function as
input and predicts FIDA and NPA spectra for comparison with the data. The code is
designed to comput e ”active” s ign als produced by an injected neutral beam. (In real-
ity, collisions with edge neutrals also produce FIDA and NPA signals but the code does
not treat these ”passive” reactions.) To date, the code has been used to model measure-
ments on the DIII-D and ASDEX-Upgrade conventional tokamaks and on the NSTX and
MAST spherical t okamaks. An early version of the code was described in the Appendix
of [3]. This paper describes version 3.0 and is organized as follows. Section 2 presents
the assumptions and organization of the code. Section 3 describes tests that verify that
the code correctly solves the desired equations. Section 4 explains the optimal selection
of n umerical parameters in terms of phy sical processes. Section 5 summarizes validation
by experiment. Section 6 provides an outloo k for further tests and improvements.
2 Model
The code has fou r main sections (Fig. 1). The first section prepares the data and the
second calculates t h e neut r al populations. The third and fourth sections bo th rely on the
first two sections but are indepen dent of each other. One s ection comp utes the NPA flux
and the other computes the FIDA radiance.

718 W. W. Heidbrink et al. / Commun. Comput. Phys., 10 (2011), pp. 716-741
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Figure 1: Flow diagram for the FIDASIM code.
2.1 Input data and coordinate mapping
The code begins by collecting the input data. The geome try o f the source of injected neu-
trals is specified first. In some devices (such as NSTX) the detector sightlines intersect
several beams, so the code can accommodate multiple beam lines. The code uses the
conventions of the NUBEAM module [8] of the TRANSP code [9] to describe the geome-
try of the viewed neutral beam source (or sources). Each tok amak has its own subroutine
called, e.g., BEAM GEOMETRY D3D. As in NUBE AM, the neutral beam is described by
rectangular source and aperture dimensions and by fo cal lengths and d ivergences in both
the horizontal and ve r tical directions. The beam energy, power, and species mix between
full-energy, h alf-energy, and third-energy components are also input parameters.

W. W. Heidbrink et al. / Commun. Co mput. Phys., 10 (2011), pp. 716-741 719
Next, th e code collects information about the detector locations and sightlines. For
FIDA, the ”detector” location is actually t h e position of the primary lens (or mirror) of the
collection optics, s ince it is this position that det ermines the Doppler shift of th e emitted
radiation. For an NPA, bot h the sightlines and the solid angles are specified.
Information on the equilibrium is input using the so-called ”eqdsk” format produced
by the EFIT equilibrium code [10]. For installations that do not use EFIT, a post-processor
that is part of the TRANSP distribution can convert TRANSP output files into the desired
format.
The code requires profiles of electron density and temp erature, ion temperature and
toroidal rotation, and impurity de n sity as a function of flux sur face. (Th ese quantities are
all assumed to be flux functions.) A subroutine exists that conver ts TRANSP o utput into
the d esired format.
The final major piece of input data is the theo retical fast-ion distribution function,
which can have a complicated dependence on energy E, pitch p =v
k
/v, and space r. (As
in TRANSP, pos itive p is defined by the direction of the plasma current rather than by
the d irection of the toroidal field.)
Three distinct coordinate systems are utilized in t h e initial stages of t h e code (Fig. 2).
The beam and dete ctor geometries are specified in right-handed Cartesian (u,v,z) coor-
dinates w ith origin the center of the tokamak and z the vertical direction. Plasma pa-
rameters are one-dimensional functions of flux coordinates. Because neutrals travel in
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Neutral beam parameters (upper case labels) follow the TRANSP conventions. The code transforms quantities
into (x,y,z) co ordinates along the selected beam.

Frequently asked questions

Q1. What are the contributions in this paper?

With the fastion distribution function as input, the code predicts the efflux to a neutral particle analyzer ( NPA ) diagnostic and the photon radiance of Balmer-alpha light to a fastion Dα ( FIDA ) diagnostic. The output of the code has been validated by FIDA measurements on DIII-D but further tests are warranted.