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

01 Sep 2011-Communications in Computational Physics (Cambridge University Press)-Vol. 10, Iss: 3, pp 716-741

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 (4 min read)

Jump to: [1 Introduction] – [2 Model] – [2.1 Input data and coordinate mapping] – [2.2 Injected and halo neutral densities] – [2.3 NPA flux] – [2.4 FIDA radiance] – [3.1 Atomic physics] – [3.2 Beam deposition] – [3.3 Halo neutrals] – [3.4 FIDA spectra] – [4 Numerics] – [4.1 Number of Monte Carlo reneutrals launched] – [4.2 Simulation volume] – [4.3 Cell size] – [5 Validation] – [6 Outlook] and [Acknowledgments]

### 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.

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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

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) co ordinates along the selected beam.

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### Cites background or result from "A code that simulates fast-ion Dα a..."

...Our tomographic and theoretical results contradict the conventional wisdom that at least twoCTS or FIDAviewswould necessarily be required for tomography of fast-ion velocity distribution functions [12, 22–32]....

[...]

...It has since become conventional wisdom that a 2D velocity distribution function could not be found from one single 1D CTS or FIDA view and that at least two CTS or FIDA views with different projection directions would be necessary for that [12, 22–32]....

[...]

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TL;DR: In this paper, an efficient method is given to reconstruct the current profile parameters, the plasma shape, and a current profile consistent with the magnetohydrodynamic equilibrium constraint from external magnetic measurements, based on a Picard iteration approach.

Abstract: An efficient method is given to reconstruct the current profile parameters, the plasma shape, and a current profile consistent with the magnetohydrodynamic equilibrium constraint from external magnetic measurements, based on a Picard iteration approach which approximately conserves the measurements. Computational efforts are reduced by parametrizing the current profile linearly in terms of a number of physical parameters. Results of detailed comparative calculations and a sensitivity study are described. Illustrative calculations to reconstruct the current profiles and plasma shapes in ohmically and auxiliarily heated Doublet III plasmas are given which show many interesting features of the current profiles.

1,587 citations

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TL;DR: The NUBEAM module as mentioned in this paper is a comprehensive computational model for Neutral Beam Injection (NBI) in tokamaks, which is used to compute power deposition, driven current, momentum transfer, fueling, and other profiles.

Abstract: The NUBEAM module is a comprehensive computational model for Neutral Beam Injection (NBI) in tokamaks. It is used to compute power deposition, driven current, momentum transfer, fueling, and other profiles in tokamak plasmas due to NBI. NUBEAM computes the time-dependent deposition and slowing down of the fast ions produced by NBI, taking into consideration beam geometry and composition, ion-neutral interactions (atomic physics), anomalous diffusion of fast ions, the effects of large scale instabilities, the effect of magnetic ripple, and finite Larmor radius effects. The NUBEAM module can also treat fusion product ions that contribute to alpha heating and ash accumulation, whether or not NBI is present. These physical phenomena are important in simulations of present day tokamaks and projections to future devices such as ITER. The NUBEAM module was extracted from the TRANSP integrated modeling code, using standards of the National Transport Code Collaboration (NTCC), and was submitted to the NTCC module library (http://w3.pppl.gov/NTCC). This paper describes the physical processes computed in the NUBEAM module, together with a summary of the numerical techniques that are used. The structure of the NUBEAM module is described, including its dependence on other NTCC library modules. Finally, a description of the procedure for setting up input data for the NUBEAM module and making use of the output is outlined.

636 citations