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Numerical and semi-analytical treatments of neutral beam current drive in DEMO-FNS
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Numerical and semi-analytical treatments of neutral beam current drive in DEMO-FNS Alexey Yu. Dnestrovskiy a,∗ , Pavel R. Goncharov b a b
National Research Center ‘Kurchatov Institute’, Moscow, Russia Peter the Great Saint Petersburg Polytechnic University, Saint Petersburg, Russia
h i g h l i g h t s • Validation of the hot ion current source value in a tokamak based neutron source. • Benchmark of Monte Carlo code NUBEAM with the semi-analytic solution. • Important role of the hot ions gradient drift.
a r t i c l e
i n f o
Article history: Received 3 October 2016 Received in revised form 17 February 2017 Accepted 6 March 2017 Available online xxx Keywords: Neutral beam Neutron source Monte Carlo code Kinetic equation
a b s t r a c t Neutral Beam Current Drive (NBCD) is considered as a highly attractive mechanism for a steady state regime in such contemporary projects as a tokamak based Fusion Neutron Source (FNS) or a DEMO type thermonuclear reactor. In this report numerical calculations of NBCD with a Monte Carlo code NUBEAM are complemented by a semi-analytical treatment of fast ion velocity distribution function. Spectral distributions of fast ions injected with neutral beams to FNS plasma are obtained for DEMO-FNS and a Spherical Tokamak Neutron Source FNS-ST. A detailed comparison of NUBEAM and semi-analytical approaches is presented on the basis of ASTRA transport code provided equilibrium data for different plasma regimes. The applicability of the semi-analytical approach is discussed. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Neutral Beam (NB) injection impact on tokamak plasma results in the creation of the fast ion component, which heats the plasma and contributes to the plasma current and to the fusion reaction rate. These effects are highly important and will be employed in the future devices such as ITER or DEMO power plant. Calculations of these effects have to be validated and the benchmarking of different codes with different NB and fast ion description is a generally accepted procedure for result confirmation [1]. The purpose of the work is to strengthen the physics basis of the tokamak based neutron source DEMO-FNS [2] by cross-checking the two approaches. We present here fast ion energy spectra for DEMOFNS calculated with the NUBEAM code [3,4] implemented into the ASTRA transport code [5], in comparison with the results obtained with the semi-analytic approach [6–8]. The semi-analytic approach has advantages of the deeper physics understanding due to the
∗ Corresponding author. E-mail address: [email protected] (A.Yu. Dnestrovskiy).
presence of explicit formulae as well as the opportunity of saving the computation time; the NUBEAM code from the other side includes more physical effects than it is possible to consider semianalytically. A detailed description of the NUBEAM code approach and the semi-analytic approach is presented in Table 1. NUBEAM considers fast and thermal ions separately unlike the semi-analytic approach, which takes into account the fast ion thermalization and particle conservation. The semi-analytic approach does not treat the transport in the r-space. A more detailed description of the semi-analytic approach is provided in the next section. 2. Semi-analytic approach The results were obtained by solving the kinetic equation
∂ (n˛ f˛ ) n˛ f˛ , = C˛ + S˛ − ˛ ∂t
(1)
where the subscript ␣ denotes the injected particle species, S˛ is the source function, n˛ is the density and f˛ is the distribution function of velocities of particles ␣, ˛ is the lifetime of these particles in the system, which was assumed infinite in the calculations,
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Please cite this article in press as: A.Yu. Dnestrovskiy, P.R. Goncharov, Numerical and semi-analytical treatments of neutral beam current drive in DEMO-FNS, Fusion Eng. Des. (2017), http://dx.doi.org/10.1016/j.fusengdes.2017.03.023
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2 Table 1 Methods of two approaches.
NUBEAM [3,4]
Semi-analytic [6–8] Geometry
r-space v-space
, poloidal angle V|| , V⊥
r-space v-space
Monte Carlo Fast ion dynamics Monte Carlo including gradient drift Monte Carlo
V|| , V⊥ Beam deposition Monte Carlo – Kinetic equation (semi-analytic approach)
Velocity distribution function Unified for fast and thermal ions Separate for fast ions and thermal ions
C˛ =
v3c 1 s v2
∂ ∂v
v2c
a(v) ∂(n˛ f˛ ) ∂ c(v) 1 + b(v)(n˛ f˛ )) + vc sin ϑ ∂ϑ 2v ∂v
sin ϑ
∂(n˛ f˛ ) ∂ϑ
(2)
is Landau collision term for the case of Maxwellian target plasma, expressed in the convenient form suggested in [9]. The term containing a(v) describes the speed diffusion, and the term with c(v) describes the pitch angle scattering. Both of these terms originate from the diffusion tensor in velocity space, whereas the term with b(v), describing the slowing down process, is related to the dynamic friction force. Explicit formulae for the functions a(v), b(v), and c(v) in (2) are given in [6]. The parameters in (2) are
vc =
m 1/3 2T 1/2 e
e
m˛
me
and
s =
m˛ Z˛ eωpe
2
(3)
ωpe =
∞ S˛ d3 v = 2v3c
v3c me
1 u2 du
0
d S˛ (u, ) = S0 ‘.
(10)
−1
The dependence of S0 and Z( ) on the magnetic surface label is calculated using [7]. The analytic representation described above reduces to the following differential equation of the second order which is solved numerically [6] 2
,
(4)
4ne e2 me
(5)
is the electron plasma frequency, ne is the electron density, and e is the elementary charge. The distribution function = n˛ (r) f˛ (v) [cm−6 s3 ] of test particles of type ␣ is calculated as described in [6], bearing in mind Maxwellian background plasma in magnetic field and assuming to be azimuthally symmetric. Using spherical polar coordinates in velocity space of injected particles new dimensionless variables are introduced, namely the speed u = v/vc
(6)
and the pitch angle cosine = cosϑ
∂ n ∂n + q(u) + r(u)n (u) = f (u) ∂u2 ∂u
(11)
The electric current density j|| = Z˛ en˛ v|| of particles of type ␣ is determined by their parallel velocity averaged over the distribution function. The details regarding the semi-analytic approach to the calculation of the fast ion current density were described in [8]. 3. Results of calculations DEMO-FNS steady state equilibrium [2] (R/a = 3.2m/1m, elongation 2.1, triangularity 0.5, Btor = 5T, Ipl = 4.6MA, Te(0)–Ti(0)–12 keV) is calculated in the transport code ASTRA and used in both approaches. No impurities and no main gas neutrals are taken into account for the clearness. NB is taken to be monoenergetic with the energy E = 500 keV and power for D/T components PD /PT = 15MW/15MW. A simplified geometry is chosen with zero beam divergence angle and a uniform neutral beam injection current distribution over the rectangle shaped cross section of the beam. The horizontal direction for NB with the target parameter Rt = 3.5 m is taken. 3.1. Source function
(7)
The solution to Eq. (1) is obtained in the form of an expansion in Legendre polynomials (u, ) =
p(u)
where Te is the electron temperature, me is the electron mass, m␣ is the injected particle mass, Z␣ is the electric charge number, is Coulomb logarithm,
where ı(u − u0 ) designates Dirac delta function, u0 = v0 /vc is the dimensionless injection velocity, and Z( ) is the unity-normalized angular distribution of the source. The source function (9) is normalized to the source rate S0 [cm−3 s−1 ] at each of the injection energies, i.e. the number of particles of type ␣, occurring in the unit volume during unit time,
∞
n (u)Pn ( )
(8)
n=0
due to the nature of the angular part of the collision operator in (2). The local monoenergetic source function of particles of type ␣ at each of the injection energies E, E/2 and E/3 is expressed as S˛ (u, ) =
1 ı(u − u0 )Z( ), 2v3c u2 S0
(9)
The common model of the neutral beam penetration with the MC method for beam rays results in very similar ion source function profiles over the poloidal flux radius (see fast deuterium ions source in Fig. 1) in both approaches. Thus also the beam stopping cross section have been shown to be identical for the simplest case of a pure plasma. 3.2. Fast ion density and current density The gradient drift of fast ions to the central plasma direction and fast ion losses at the periphery, taken into account in the NUBEAM code, are the reason for the discrepancies in fast ion densities and
Please cite this article in press as: A.Yu. Dnestrovskiy, P.R. Goncharov, Numerical and semi-analytical treatments of neutral beam current drive in DEMO-FNS, Fusion Eng. Des. (2017), http://dx.doi.org/10.1016/j.fusengdes.2017.03.023
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Fig. 1. Fast ion source from NBI of D (black) and T (red) ions, calculated with the NUBEAM code (solid lines) and with the semi-analytic approach (dashed lines). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
3
Fig. 4. Deuterium ion energy spectra, obtained with the NUBEAM code (solid lines) and with the semi-analytic approach (dashed lines) for two magnetic surfaces
pol = 0.2 (black) and pol = 0.6 (red). Inset shows the same spectra in a linear scale. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. The density of ions with the energy above 150keV, calculated with NUBEAM code (solid lines) and with the semi-analytic approach (dashed lines).
Fig. 5. Deuterium current density distribution over the pitch angle for deuterium ions with the energy >150keV, obtained with the NUBEAM code (solid lines) and with the semi-analytic approach (dashed lines) for two magnetic surfaces pol = 0.5 (black) and pol = 0.75 (red). Inset shows the current distribution in a negative direction. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Fast ion current density, calculated with NUBEAM code (solid lines) and with the semi-analytic approach (dashed lines).
the fast ion current density profiles shown in Figs. 2 and 3. However the difference between the NUBEAM and the semi-analytic approach in the total fast ion current is less than 10%. The comparison of fast ion density profiles is shown for particle energies above 150 keV. Current density profile is calculated using the full distribution function in the semi-analytic approach, the contribution of the thermal fraction to the current density is automatically cancelled. 3.3. Fast ion spectra Dashed lines in Fig. 4 show the distribution function over the energy obtained in the semi-analytic approach describing thermal and fast ions together. NUBEAM code in its turn treats fast ions
separately from the thermal component. The plot of spectra of fast ions shown in the inset of Fig. 4 demonstrates a good agreement between the two approaches in the middle ( pol = 0.6) and a discrepancy in the plasma center ( pol = 0.2). However, the difference in the total neutron source rate does not exceed 20%. The current distribution over the pitch angle for ions with energies exceeding 150 keV is shown in Fig. 5. Hot ions with cos()–1 make the main contribution to the current in the central plasma and a slightly shifted to cos()–0.9 maximal contribution can be seen in the more peripheral plasma. This means that the beam mainly intersects magnetic surfaces along magnetic lines having an optimal geometry for current drive. The inset in Fig. 5 shows the negative current in NUBEAM calculations arising due to returns in banana orbits, having its maximum magnitude at cos() ∼ −0.5, corresponding to trapped particles in this tokamak. This negative current does not appear in the semi-analytic approach because it neglects excursions in the r-space, but the value of this current is negligible and so it has no considerable influence on the total current density. Again, the contribution of the central plasma to the current density in the NUBEAM results exceeds that in the semianalytic approach due to the gradient drift as it is shown in Fig. 5.
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inside the middle radius, that shows the role of the fast ion gradient drift taken into account in the NUBEAM approach. This critical point shows the difficulties in the comparison of the MC method of the NUBEAM code and the semi-analytic method used for the solution of the kinetic equation. The effect of the gradient drift can be clarified if the hot ion movement in the NUBEAM approach would be restricted by the magnetic surface. The introduction of first orbit losses to the semi-analytic model is also planned as the next step for the cross-checking of these approaches. A larger discrepancy between the two approaches has been obtained for a spherical tokamak neutron source FNS-ST with a lower aspect ratio. In this plasma configuration fast ion excursions across the magnetic surfaces are more important. Fig. 6. The ratio of NUBEAM calculated fast ion current density to its value calculated with the semi-analytic approach in DEMO-FNS device (black line) and in FNS-ST device (red line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
3.4. DEMO and ST comparison A spherical tokamak neutron source FNS-ST [10] (R/a = 0.5 m/0.3 m, elongation 3, triangularity 0.5, Btor = 1.5 T, Ipl = 1.5MA, Te(0)=Ti(0)=7 keV, NB energy E = 130keV, NB power for D/T beams PD /PT = 3MW/3MW), where plasma in a steady-state contains up to a half plasma energy in the hot ion component, has also been investigated. Radial profiles of the ratios of current densities are presented in Fig. 6. The deviation of the ratio from 1 shows the value of discrepancy between the two approaches. The highest discrepancy is obtained in the central and the edge plasma where the gradient drift in the center and orbit particle losses at the edge play the main role. These effects are taken into account in the NUBEAM code in contrast to the semi-analytic approach. In the spherical tokamak FNS-ST with aspect ratio A = 1.67 this deviation is found to be considerably stronger than in DEMO-FNS with an approximately twice higher aspect ratio A = 3.2. 4. Discussion and conclusions This work can be considered as the first step toward the code benchmarking with semi-analytic solutions, where these are applicable. Energy spectra of hot ions in two approaches are obtained to be rather similar in the energy range from 100 keV to 500keV. A rather good coincidence is achieved for density and current density profiles of hot ions outside the middle radius of the plasma. At the same time the discrepancy in these profiles appears for the plasma
Acknowledgments The work was partially supported by the State Atomic Energy Corporation ROSATOM [contract No. Н.4х.44.9Б.16.1022]; and by Ministry of Education and Science of Russian Federation [order No. 1014]. References [1] T. Oikawa et al. Proc. 22nd Int. Conf. on Fusion Energy Geneva, Switzerland, 2008 CD-ROM file IT/P6–5 http://www-naweb.iaea.org/napc/physics/FEC/ FEC2008/html/node313. htm#66422. [2] Yu.S. Shpanskiy, E.A. Azizov, B.V. Kuteev, Status of DEMO-FNS development, FNS/1-1, in: Paper Presented at 26th IAEA Fusion Energy Conference in Kyoto, Japan, October, 2016. [3] R.J. Goldston, et al., New techniques for calculating heat and particle source rates due to neutral beam injection in axisymmetric tokamaks, J. Comp. Phys. 43 (1981) 61. [4] A. Pankin, D. Mccune, R. Andre, G. Bateman, A. Kritz, The tokamak Monte Carlo fast ion module NUBEAM in the National Transport Code Collaboration library, Comp. Phys. Comm. 159 (2004) 157–184. [5] G.V. Pereverzev, P.N. Yushmanov, Preprint Max-Planck-Institut für Plasmaphysik ID 282186, 2002 http://edoc.mpg.de/282186. [6] P.R. Goncharov, B.V. Kuteev, T. Ozaki, S. Sudo, Analytical and semianalytical solutions to the kinetic equation with Coulomb collision term and a monoenergetic source function, Phys. Plasmas 17 (2010) 112313. [7] P.R. Goncharov, Certificate No. 2015663239 from 14.12.2015 on State Registration of the Computer Program Code for Calculation of the Source Function of Fast Particles in Plasma Due to Neutral Beam Injection, 2015 http://www1. fips.ru/Archive/EVM/2016/2016.01.20/DOC/RUNW/000/002/ 015/663/239/document.pdf. [8] P.R. Goncharov, B.V. Kuteev, V.Yu. Sergeev, T. Ozaki, S. Sudo, Semianalytical treatment of current density of particles injected by a monoenergetic source, Nucl. Fusion 51 (2011) 103042. [9] Yu. NP. Dnestrovskij, D.P. Kostomarov, A.P. Smirnov, Role of impurities and electric field in plasma heating by an injected neutral beam in tokamaks, Nucl. Fusion 17 (1977) 433–442. [10] B.V. Kuteev, et al., Steady-state operation in compact tokamaks with copper coils, Nucl. Fusion 51 (2011) 073013.
Please cite this article in press as: A.Yu. Dnestrovskiy, P.R. Goncharov, Numerical and semi-analytical treatments of neutral beam current drive in DEMO-FNS, Fusion Eng. Des. (2017), http://dx.doi.org/10.1016/j.fusengdes.2017.03.023
ARTICLE IN PRESS
FUSION-9203; No. of Pages 4
Fusion Engineering and Design xxx (2017) xxx–xxx
Contents lists available at ScienceDirect
Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Numerical and semi-analytical treatments of neutral beam current drive in DEMO-FNS Alexey Yu. Dnestrovskiy a,∗ , Pavel R. Goncharov b a b
National Research Center ‘Kurchatov Institute’, Moscow, Russia Peter the Great Saint Petersburg Polytechnic University, Saint Petersburg, Russia
h i g h l i g h t s • Validation of the hot ion current source value in a tokamak based neutron source. • Benchmark of Monte Carlo code NUBEAM with the semi-analytic solution. • Important role of the hot ions gradient drift.
a r t i c l e
i n f o
Article history: Received 3 October 2016 Received in revised form 17 February 2017 Accepted 6 March 2017 Available online xxx Keywords: Neutral beam Neutron source Monte Carlo code Kinetic equation
a b s t r a c t Neutral Beam Current Drive (NBCD) is considered as a highly attractive mechanism for a steady state regime in such contemporary projects as a tokamak based Fusion Neutron Source (FNS) or a DEMO type thermonuclear reactor. In this report numerical calculations of NBCD with a Monte Carlo code NUBEAM are complemented by a semi-analytical treatment of fast ion velocity distribution function. Spectral distributions of fast ions injected with neutral beams to FNS plasma are obtained for DEMO-FNS and a Spherical Tokamak Neutron Source FNS-ST. A detailed comparison of NUBEAM and semi-analytical approaches is presented on the basis of ASTRA transport code provided equilibrium data for different plasma regimes. The applicability of the semi-analytical approach is discussed. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Neutral Beam (NB) injection impact on tokamak plasma results in the creation of the fast ion component, which heats the plasma and contributes to the plasma current and to the fusion reaction rate. These effects are highly important and will be employed in the future devices such as ITER or DEMO power plant. Calculations of these effects have to be validated and the benchmarking of different codes with different NB and fast ion description is a generally accepted procedure for result confirmation [1]. The purpose of the work is to strengthen the physics basis of the tokamak based neutron source DEMO-FNS [2] by cross-checking the two approaches. We present here fast ion energy spectra for DEMOFNS calculated with the NUBEAM code [3,4] implemented into the ASTRA transport code [5], in comparison with the results obtained with the semi-analytic approach [6–8]. The semi-analytic approach has advantages of the deeper physics understanding due to the
∗ Corresponding author. E-mail address: [email protected] (A.Yu. Dnestrovskiy).
presence of explicit formulae as well as the opportunity of saving the computation time; the NUBEAM code from the other side includes more physical effects than it is possible to consider semianalytically. A detailed description of the NUBEAM code approach and the semi-analytic approach is presented in Table 1. NUBEAM considers fast and thermal ions separately unlike the semi-analytic approach, which takes into account the fast ion thermalization and particle conservation. The semi-analytic approach does not treat the transport in the r-space. A more detailed description of the semi-analytic approach is provided in the next section. 2. Semi-analytic approach The results were obtained by solving the kinetic equation
∂ (n˛ f˛ ) n˛ f˛ , = C˛ + S˛ − ˛ ∂t
(1)
where the subscript ␣ denotes the injected particle species, S˛ is the source function, n˛ is the density and f˛ is the distribution function of velocities of particles ␣, ˛ is the lifetime of these particles in the system, which was assumed infinite in the calculations,
http://dx.doi.org/10.1016/j.fusengdes.2017.03.023 0920-3796/© 2017 Elsevier B.V. All rights reserved.
Please cite this article in press as: A.Yu. Dnestrovskiy, P.R. Goncharov, Numerical and semi-analytical treatments of neutral beam current drive in DEMO-FNS, Fusion Eng. Des. (2017), http://dx.doi.org/10.1016/j.fusengdes.2017.03.023
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2 Table 1 Methods of two approaches.
NUBEAM [3,4]
Semi-analytic [6–8] Geometry
r-space v-space
, poloidal angle V|| , V⊥
r-space v-space
Monte Carlo Fast ion dynamics Monte Carlo including gradient drift Monte Carlo
V|| , V⊥ Beam deposition Monte Carlo – Kinetic equation (semi-analytic approach)
Velocity distribution function Unified for fast and thermal ions Separate for fast ions and thermal ions
C˛ =
v3c 1 s v2
∂ ∂v
v2c
a(v) ∂(n˛ f˛ ) ∂ c(v) 1 + b(v)(n˛ f˛ )) + vc sin ϑ ∂ϑ 2v ∂v
sin ϑ
∂(n˛ f˛ ) ∂ϑ
(2)
is Landau collision term for the case of Maxwellian target plasma, expressed in the convenient form suggested in [9]. The term containing a(v) describes the speed diffusion, and the term with c(v) describes the pitch angle scattering. Both of these terms originate from the diffusion tensor in velocity space, whereas the term with b(v), describing the slowing down process, is related to the dynamic friction force. Explicit formulae for the functions a(v), b(v), and c(v) in (2) are given in [6]. The parameters in (2) are
vc =
m 1/3 2T 1/2 e
e
m˛
me
and
s =
m˛ Z˛ eωpe
2
(3)
ωpe =
∞ S˛ d3 v = 2v3c
v3c me
1 u2 du
0
d S˛ (u, ) = S0 ‘.
(10)
−1
The dependence of S0 and Z( ) on the magnetic surface label is calculated using [7]. The analytic representation described above reduces to the following differential equation of the second order which is solved numerically [6] 2
,
(4)
4ne e2 me
(5)
is the electron plasma frequency, ne is the electron density, and e is the elementary charge. The distribution function = n˛ (r) f˛ (v) [cm−6 s3 ] of test particles of type ␣ is calculated as described in [6], bearing in mind Maxwellian background plasma in magnetic field and assuming to be azimuthally symmetric. Using spherical polar coordinates in velocity space of injected particles new dimensionless variables are introduced, namely the speed u = v/vc
(6)
and the pitch angle cosine = cosϑ
∂ n ∂n + q(u) + r(u)n (u) = f (u) ∂u2 ∂u
(11)
The electric current density j|| = Z˛ en˛ v|| of particles of type ␣ is determined by their parallel velocity averaged over the distribution function. The details regarding the semi-analytic approach to the calculation of the fast ion current density were described in [8]. 3. Results of calculations DEMO-FNS steady state equilibrium [2] (R/a = 3.2m/1m, elongation 2.1, triangularity 0.5, Btor = 5T, Ipl = 4.6MA, Te(0)–Ti(0)–12 keV) is calculated in the transport code ASTRA and used in both approaches. No impurities and no main gas neutrals are taken into account for the clearness. NB is taken to be monoenergetic with the energy E = 500 keV and power for D/T components PD /PT = 15MW/15MW. A simplified geometry is chosen with zero beam divergence angle and a uniform neutral beam injection current distribution over the rectangle shaped cross section of the beam. The horizontal direction for NB with the target parameter Rt = 3.5 m is taken. 3.1. Source function
(7)
The solution to Eq. (1) is obtained in the form of an expansion in Legendre polynomials (u, ) =
p(u)
where Te is the electron temperature, me is the electron mass, m␣ is the injected particle mass, Z␣ is the electric charge number, is Coulomb logarithm,
where ı(u − u0 ) designates Dirac delta function, u0 = v0 /vc is the dimensionless injection velocity, and Z( ) is the unity-normalized angular distribution of the source. The source function (9) is normalized to the source rate S0 [cm−3 s−1 ] at each of the injection energies, i.e. the number of particles of type ␣, occurring in the unit volume during unit time,
∞
n (u)Pn ( )
(8)
n=0
due to the nature of the angular part of the collision operator in (2). The local monoenergetic source function of particles of type ␣ at each of the injection energies E, E/2 and E/3 is expressed as S˛ (u, ) =
1 ı(u − u0 )Z( ), 2v3c u2 S0
(9)
The common model of the neutral beam penetration with the MC method for beam rays results in very similar ion source function profiles over the poloidal flux radius (see fast deuterium ions source in Fig. 1) in both approaches. Thus also the beam stopping cross section have been shown to be identical for the simplest case of a pure plasma. 3.2. Fast ion density and current density The gradient drift of fast ions to the central plasma direction and fast ion losses at the periphery, taken into account in the NUBEAM code, are the reason for the discrepancies in fast ion densities and
Please cite this article in press as: A.Yu. Dnestrovskiy, P.R. Goncharov, Numerical and semi-analytical treatments of neutral beam current drive in DEMO-FNS, Fusion Eng. Des. (2017), http://dx.doi.org/10.1016/j.fusengdes.2017.03.023
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Fig. 1. Fast ion source from NBI of D (black) and T (red) ions, calculated with the NUBEAM code (solid lines) and with the semi-analytic approach (dashed lines). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
3
Fig. 4. Deuterium ion energy spectra, obtained with the NUBEAM code (solid lines) and with the semi-analytic approach (dashed lines) for two magnetic surfaces
pol = 0.2 (black) and pol = 0.6 (red). Inset shows the same spectra in a linear scale. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. The density of ions with the energy above 150keV, calculated with NUBEAM code (solid lines) and with the semi-analytic approach (dashed lines).
Fig. 5. Deuterium current density distribution over the pitch angle for deuterium ions with the energy >150keV, obtained with the NUBEAM code (solid lines) and with the semi-analytic approach (dashed lines) for two magnetic surfaces pol = 0.5 (black) and pol = 0.75 (red). Inset shows the current distribution in a negative direction. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Fast ion current density, calculated with NUBEAM code (solid lines) and with the semi-analytic approach (dashed lines).
the fast ion current density profiles shown in Figs. 2 and 3. However the difference between the NUBEAM and the semi-analytic approach in the total fast ion current is less than 10%. The comparison of fast ion density profiles is shown for particle energies above 150 keV. Current density profile is calculated using the full distribution function in the semi-analytic approach, the contribution of the thermal fraction to the current density is automatically cancelled. 3.3. Fast ion spectra Dashed lines in Fig. 4 show the distribution function over the energy obtained in the semi-analytic approach describing thermal and fast ions together. NUBEAM code in its turn treats fast ions
separately from the thermal component. The plot of spectra of fast ions shown in the inset of Fig. 4 demonstrates a good agreement between the two approaches in the middle ( pol = 0.6) and a discrepancy in the plasma center ( pol = 0.2). However, the difference in the total neutron source rate does not exceed 20%. The current distribution over the pitch angle for ions with energies exceeding 150 keV is shown in Fig. 5. Hot ions with cos()–1 make the main contribution to the current in the central plasma and a slightly shifted to cos()–0.9 maximal contribution can be seen in the more peripheral plasma. This means that the beam mainly intersects magnetic surfaces along magnetic lines having an optimal geometry for current drive. The inset in Fig. 5 shows the negative current in NUBEAM calculations arising due to returns in banana orbits, having its maximum magnitude at cos() ∼ −0.5, corresponding to trapped particles in this tokamak. This negative current does not appear in the semi-analytic approach because it neglects excursions in the r-space, but the value of this current is negligible and so it has no considerable influence on the total current density. Again, the contribution of the central plasma to the current density in the NUBEAM results exceeds that in the semianalytic approach due to the gradient drift as it is shown in Fig. 5.
Please cite this article in press as: A.Yu. Dnestrovskiy, P.R. Goncharov, Numerical and semi-analytical treatments of neutral beam current drive in DEMO-FNS, Fusion Eng. Des. (2017), http://dx.doi.org/10.1016/j.fusengdes.2017.03.023
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inside the middle radius, that shows the role of the fast ion gradient drift taken into account in the NUBEAM approach. This critical point shows the difficulties in the comparison of the MC method of the NUBEAM code and the semi-analytic method used for the solution of the kinetic equation. The effect of the gradient drift can be clarified if the hot ion movement in the NUBEAM approach would be restricted by the magnetic surface. The introduction of first orbit losses to the semi-analytic model is also planned as the next step for the cross-checking of these approaches. A larger discrepancy between the two approaches has been obtained for a spherical tokamak neutron source FNS-ST with a lower aspect ratio. In this plasma configuration fast ion excursions across the magnetic surfaces are more important. Fig. 6. The ratio of NUBEAM calculated fast ion current density to its value calculated with the semi-analytic approach in DEMO-FNS device (black line) and in FNS-ST device (red line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
3.4. DEMO and ST comparison A spherical tokamak neutron source FNS-ST [10] (R/a = 0.5 m/0.3 m, elongation 3, triangularity 0.5, Btor = 1.5 T, Ipl = 1.5MA, Te(0)=Ti(0)=7 keV, NB energy E = 130keV, NB power for D/T beams PD /PT = 3MW/3MW), where plasma in a steady-state contains up to a half plasma energy in the hot ion component, has also been investigated. Radial profiles of the ratios of current densities are presented in Fig. 6. The deviation of the ratio from 1 shows the value of discrepancy between the two approaches. The highest discrepancy is obtained in the central and the edge plasma where the gradient drift in the center and orbit particle losses at the edge play the main role. These effects are taken into account in the NUBEAM code in contrast to the semi-analytic approach. In the spherical tokamak FNS-ST with aspect ratio A = 1.67 this deviation is found to be considerably stronger than in DEMO-FNS with an approximately twice higher aspect ratio A = 3.2. 4. Discussion and conclusions This work can be considered as the first step toward the code benchmarking with semi-analytic solutions, where these are applicable. Energy spectra of hot ions in two approaches are obtained to be rather similar in the energy range from 100 keV to 500keV. A rather good coincidence is achieved for density and current density profiles of hot ions outside the middle radius of the plasma. At the same time the discrepancy in these profiles appears for the plasma
Acknowledgments The work was partially supported by the State Atomic Energy Corporation ROSATOM [contract No. Н.4х.44.9Б.16.1022]; and by Ministry of Education and Science of Russian Federation [order No. 1014]. References [1] T. Oikawa et al. Proc. 22nd Int. Conf. on Fusion Energy Geneva, Switzerland, 2008 CD-ROM file IT/P6–5 http://www-naweb.iaea.org/napc/physics/FEC/ FEC2008/html/node313. htm#66422. [2] Yu.S. Shpanskiy, E.A. Azizov, B.V. Kuteev, Status of DEMO-FNS development, FNS/1-1, in: Paper Presented at 26th IAEA Fusion Energy Conference in Kyoto, Japan, October, 2016. [3] R.J. Goldston, et al., New techniques for calculating heat and particle source rates due to neutral beam injection in axisymmetric tokamaks, J. Comp. Phys. 43 (1981) 61. [4] A. Pankin, D. Mccune, R. Andre, G. Bateman, A. Kritz, The tokamak Monte Carlo fast ion module NUBEAM in the National Transport Code Collaboration library, Comp. Phys. Comm. 159 (2004) 157–184. [5] G.V. Pereverzev, P.N. Yushmanov, Preprint Max-Planck-Institut für Plasmaphysik ID 282186, 2002 http://edoc.mpg.de/282186. [6] P.R. Goncharov, B.V. Kuteev, T. Ozaki, S. Sudo, Analytical and semianalytical solutions to the kinetic equation with Coulomb collision term and a monoenergetic source function, Phys. Plasmas 17 (2010) 112313. [7] P.R. Goncharov, Certificate No. 2015663239 from 14.12.2015 on State Registration of the Computer Program Code for Calculation of the Source Function of Fast Particles in Plasma Due to Neutral Beam Injection, 2015 http://www1. fips.ru/Archive/EVM/2016/2016.01.20/DOC/RUNW/000/002/ 015/663/239/document.pdf. [8] P.R. Goncharov, B.V. Kuteev, V.Yu. Sergeev, T. Ozaki, S. Sudo, Semianalytical treatment of current density of particles injected by a monoenergetic source, Nucl. Fusion 51 (2011) 103042. [9] Yu. NP. Dnestrovskij, D.P. Kostomarov, A.P. Smirnov, Role of impurities and electric field in plasma heating by an injected neutral beam in tokamaks, Nucl. Fusion 17 (1977) 433–442. [10] B.V. Kuteev, et al., Steady-state operation in compact tokamaks with copper coils, Nucl. Fusion 51 (2011) 073013.
Please cite this article in press as: A.Yu. Dnestrovskiy, P.R. Goncharov, Numerical and semi-analytical treatments of neutral beam current drive in DEMO-FNS, Fusion Eng. Des. (2017), http://dx.doi.org/10.1016/j.fusengdes.2017.03.023