TRANSP modelling of total and local neutron emission on MAST I. Klimek1 , M. Cecconello1 , M. Gorelenkova2 , D. Keeling3 , A. Meakins3 , O. Jones3,4 , R. Akers3 , I. Lupelli3 , M. Turnyanskiy5 , G. Ericsson1 and the MAST team3 1
Department of Physics and Astronomy, Uppsala University, SE 751 20 Uppsala, Sweden 2 Princeton Plasma Physics Laboratory, Princeton, NJ 08543-0451, USA 3 CCFE, Culham Science Centre,Abingdon, Oxon, OX14 3DB, UK 4 Department of Physics, University of Durham, Durham, DH1 3LE, UK 5 ITER Physics Department, EFDA CSU Garching, Boltzmannstrasse 2, D-85748 Garching, Germany E-mail:
[email protected] Abstract. The results of TRANSP simulations of neutron emissivity profiles measured by a collimated neutron flux monitor - neutron camera (NC) - for different plasma scenarios on MAST are reported. In addition, the effect of various plasma parameters on neutron emission is studied by means of TRANSP simulation. The fast ion redistribution and losses due to fishbone modes, which belong to a wider category of energetic particle modes, are observed by the NC and modelled in TRANSP.
2 1. Introduction The importance of measuring and modelling the neutron emission from nuclear fusion reactions rests on the fact that the unconfined neutrons carry information about the plasma properties such as the local deuterium-deuterium or deuterium-tritium reactivity, the fuel density and its temperature, all parameters that are crucial for assessing the progress towards the achievement of burning plasma conditions. In present day fusion devices, the commonly employed deuterium-deuterium fusion reactions give rise to a flux of 2.45 MeV neutrons which is typically dominated by the so-called beamthermal component, that is by those neutrons generated by fusion reactions between the thermal fuel ions and energetic (fast) deuterium ions. These energetic deuterium ions, with energies far in excess of that of the thermal ions, arise from the external additional heating systems such radio-frequency heating or Neutral Beam Injection (NBI). The thermal and beam-beam components contribute to the total neutron emission in different proportions depending on the particular fusion device and plasma scenarios. If these two components are negligible compared to the beam-thermal component, the unconfined neutrons can be used to diagnose the behaviour of the fast ions and how they are affected by Magneto-Hydro-Dynamic (MHD) instabilities. Of particular interest is the study of the excitation and damping of resonant MHD instabilites, due to the presence of a large population of fast ions, which might cause redistribution and losses of these fast ions with a reduction of the plasma performances and possible damage to the first wall [1]. These issues are of particular interest for ITER and DEMO and can be partially addressed in present day devices by an accurate design of the plasma scenarios. The Mega Ampere Spherical Tokamak (MAST) [2] is particularly suitable for addressing the role of super-Alfv´enic fast ions in driving MHD instabilities that resemble those likely to be driven unstable by α-particles in ITER in plasmas in presence of a large α particle pressure. With its relatively low magnetic field (BT ≈ 0.6 T on axis) and deuterons with energies up to 75 keV produced by NBI, MAST easily satisfy the condition vk > VA where VA is the Alfv´en velocity and v|| is the fast ion velocity parallel to the magnetic field. The MAST NBI system is capable of delivering a total additional power of 5 MW, thereby creating a fast ion population whose pressure is comparable to the thermal pressure. Under such circumstances, Toroidal Alfv´en Eigemodes (TAE) and fishbones are routinely excited in MAST [3, 4] and are responsible for a significant redistribution fast ions from the core to the outer regions of the plasma. In some cases, fast ion losses from the plasma are also observed. The net effect of these MHD instabilities is a reduction in the plasma energy content which is accompanied by a reduction in the total neutron rate. On MAST, the total neutron rate has been routinely measured using an absolutely calibrated 235 U Fission Chamber (FC) [5] with a time resolution of 10 µs. Recently, a prototype collimated neutron flux monitor, also known as Neutron Camera (NC) [6], has been extensively used to measure the local neutron emission with a time resolution of 1 ms. The neutron camera has an advantage over the FC in its ability to resolve redistribution or loss of fast ions from particular spatial
3 locations. The main disadvantage of this prototype is the limited number of lines of sight (LoS) which are available: two on the equatorial plane (EQ) which observe the plasma toroidally at different tangency radii; and two inclined diagonally (DIAG) to pass below the equatorial plane at the same tangency radii as the EQ LoS. Neutron count rate profiles are measured by moving the neutron camera between plasma discharges to allow the tangency radius p to be scanned from R/R0 ≈ 0.2 to 1.3 (R0 is the major radius, a typical position of the magnetic axis is RM ≃ 0.90 m and a high- and a low-fieldside positions of the last closed flux surface on the plasma midplane are RHFS ≃ 0.3 m and RLFS ≃ 1.5 m, respectively). Typically, from 4 to 6 plasma discharges are used to measure the neutron count rate profiles, taking advantage of the high repeatability of the plasma discharges in MAST. Additional fast ions diagnostic on MAST include a Fast Ion Dα (FIDA) system, which provides information on which part of the fast ion velocity distribution is affected by MHD instabilities [7], and a charged fusion product detector [8]. A detailed comparison of the observations from these three diagnostics is outside the scope of this paper: the interested reader can find an initial comparison in [9]. The total neutron rate and collimated neutron fluxes in tokamaks are typically modelled using the time dependent transport code TRANSP [10] combined with the NBI fast ion modelling code NUBEAM [11]. A classical fast ion behaviour is, for example, reported in [12, 13] and a good match between TRANSP simulations and experimental data is found. For different plasma scenario however, mismatches between the measured and TRANSP/NUBEAM predicted neutron rates have been observed. These have been accounted for by assuming a reduction of the beam power [13], an increase of plasma effective charge [14, 15] or an anomalous (i.e. larger than neoclassical) diffusion of the fast ions in the presence of strong MHD activity [13, 9]. The amplitude of this anomalous fast ion diffusion coefficient Da is indicative of the strength of the redistribution. Heuristic fast ion losses models are included in NUBEAM to model the effect of MHD instabilities on the fast ion population. These fast ion loss models, although rather simplistic, allow a first assessment of the effect of the MHD instabilities on the temporal and spatial evolution of the neutron emission and can therefore contribute to the understanding of the mechanisms responsible for the redistribution and loss of fast ions [9]. A different approach to modelling the effect of MHD activity on the fast ion distribution, and hence the neutron emission can be found in [16]. This work reports on the results of quantitative modelling of the neutron emission in MAST in plasma scenarios (described in section 2) which are typically used for the study of the interplay between MHD instabilities and fast ions. The approach used in this work is to systematically compare the results of TRANSP modelling with the total and local neutron rate measurements provided by the fission chamber and the neutron camera. The forward modelling framework that has been developed for modelling the neutron emission on MAST, and its validation, are discussed in section 3. In particular, in section 3 the effect of the anomalous fast ion diffusion coefficient on the total neutron rate and the local neutron emission is addressed. The dependence of the TRANSP
4 predicted total neutron rate on key global plasma parameters, and their uncertainty, is systematically studied. Finally, the fast ion loss models available in NUBEAM are assessed in the case of strong fishbone activity (section 4) by means of a systematic comparison of NC with FC chamber observations. 2. Reference plasma scenarios for TRANSP validation Three reference sets of Double Null Divertor plasma discharges have been chosen for this study. These sets are characterized by total neutron rates in the range of 0.3 − 1.2 × 1014 s−1 and different levels of MHD activity, and are representative of the vast majority of the scenarios used to study fast ion physics in MAST. Each reference set consists of several repeated plasma discharges to allow measurements of the neutron count rate profile. The first set consists of seven plasma discharges (29195, 29904 − 29906, and 29908−29910) referred to here as the high density (HD) set due to a high line-integrated electron density n ¯ e ≃ 2×1020 m−2 . These discharges are characterized by plasma current Ip ≃ 780 kA, NBI power PNBI ≃ 1.6 MW, core electron temperature Te ≃ 0.7 keV, core electron density ne ≃ 3.9 × 1019 m−3 , total neutron rate Yn ≃ 0.3 × 1014 s−1 and a long MHD-quiescent period between 0.210 ≤ t ≤ 0.265 s. The time traces of the aforementioned plasma parameters together with the neutron count rates measured by the NC for plasma discharge 29905 are shown in figure 1 panels a) - d). The second set contains five plasma discharges (29222, 29924, 29928, 29929 and 29931), which due to the line-integrated electron density n ¯ e ≃ 1 × 1020 m−2 is called the low density (LD) set. The LD set is characterized by Te ≃ 1.3 keV, ne ≃ 3.5 × 1019 m−3 , a moderate Yn ≃ 0.6 × 1014 s−1 and the same NBI power and plasma current as the HD set. Panels e) - h) of figure 1 show the same time traces as panels a) - d) but for a plasma discharge belonging to this set (29222). It is worth noting that injecting the same NBI power into a lower density plasma resulted in a moderate level of quasi-continuous MHD activity observed throughout the entire plasma discharge, and that a reduction in neutron rate is correlated with bursts of MHD activity. In this set, Yn is almost double that in the HD set as shown in figure 1 panels c) and g). This doubling is also observed by the NC in the LoS characterized by an impact parameter p = 0.95 m, which looks at the plasma core region where most neutrons are generated, while the neutron count rate measured in the LoS with an impact parameter p = 1.15 m is roughly the same on both sets of discharges as can be seen in figure 1 panels d) and h). The possibility of measuring such local variations in neutron emission emphasizes the advantage of the NC over the FC. An example of the neutron count rate profiles as a function of tangency radius p measured by the NC in the EQ LoS in the HD and LD sets is depicted in figure 2. Neutron count rates are typically averaged over 5 - 10 ms time intervals in order to average out small temporal fluctuations. Here, averaging was carried out over the time interval 0.225 ≤ t ≤ 0.260 s to obtain the neutron count rate profiles shown in figure 2. Large error bars in the LD neutron count rate profile are mainly due to the
5 3.0
P (MW)
ne (x1020 m−2)
a)
29905
e)
29222
2.0
1.0
0.0 b)
f)
Φ˙ (V)
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T
a
g)
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0.6
p1 = 0.95 m
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p2 = 1.15 m
p1 = 0.95 m p2 = 1.15 m
0.4 0.2 0
0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 time (s)
0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 time (s)
Figure 1: Comparison between plasma discharges 29905 and 29222 which belong to the HD and LD sets: a) and e) the line-integrated density and NBI power, b) and f) the time derivative of the poloidal magnetic flux at the location of a Mirnov coil on the outboard midplane, c) and g) neutron rate measured by FC (Yn ) and calculated by TRANSP (YT ), explained in detail in section 3, assuming two different values of anomalous diffusivity Da . The solid black line shows YT with Da = 0, while the dashed black line in g) shows YT with Da = 1.5 m2 s−1 for t < 0.27 s and of Da = 2.5 m2 s−1 for t ≥ 0.27. The grey shaded region indicates the limit of the uncertainty in YT derived according to the method discussed in section 3. d) and h) show neutron count rates recorded by NC in the position characterized by the tangency radius p1 and p2 substantial variation of the neutron count rates caused by MHD activity, rather than due to statistical uncertainty which is typically less than 15%. The third reference scenario comprises four plasma discharges (27525 − 27528) characterized by n ¯ e ≃ 1.8 × 1020 m−2 , Ip ≃ 740 kA and PNBI ≃ 3.5 MW, more than double the NBI power of the HD and LD sets. These plasma parameters resulted in a very high total neutron rate Yn ≃ 1.1 × 1014 s−1 and strong MHD activity. This scenario
Neutron rate (x1014 s−1)
c)
0.5
Y , D = 1.5 m2s−1 and D = 2.5 m2s−1
6 ×10 600
3
0.225 ≤ t ≤ 0.260 s
low density high density
Count rate (Hz)
500
400
300
200
100
0
0.5
0.6
0.7
0.8 0.9 p (m)
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Figure 2: Neutron count rate profiles for the EQ LoS as a function of tangency radius p averaged over time interval 0.225 ≤ t ≤ 0.260 s for the HD and LD sets is referred to as the high density high power (HDHP) set. As can be seen in figure 3 panel b), after the second NBI was switched on (t = 0.14 s) a noticeable increase in the MHD activity was observed indicating that the spatial fast ion pressure gradient, despite the high plasma density, was large enough to drive MHD instabilities. From the spectrogram of a Mirnov coil shown in figure 3 panel c), the MHD instabilities can be recognized as: toroidal Alfv´en eigenmodes (TAEs) at frequencies ∼ 100 kHz in the time interval 0.12 ≤ t < 0.15 s; bursting TAEs along with small fishbones with a duration of ∼ 1 − 2 ms and frequencies ∼ 50 kHz in the time interval 0.15 ≤ t < 0.195 s; TAEs and large fishbones with duration ∼ 4 ms and frequencies ∼ 35 kHz in the time interval 0.195 ≤ t ≤ 0.230 s; and a long-lived internal kink mode (LLM) whose frequency tracks the plasma rotation frequency and lasts up to t ≃ 0.325 s. Small fishbones and TAEs do not cause large drops in Yn , while substantial drops in Yn are clearly correlated with three large fishbones, characterized by a frequency chirp from approximately 35 to 20 kHz, occurring at t = 0.199 s, t = 0.212 s and t = 0.225 s. The modes visible at twice this frequency are n = 2 harmonic oscillations. Two of these large fishbones are accompanied by very small spikes in the Dα emission from the bottom divertor; the absence of strong Dα suggests that fast ions are being internally redistributed rather than lost from the plasma. 3. TRANSP modelling of the fast ion distribution and neutron emission Neutron rates and count rate profiles as those shown in figures 1 and 2 can be interpreted via forward modelling of the neutron emission using TRANSP and NUBEAM codes. Input plasma profiles required for TRANSP simulations are measured in MAST by
7
P (MW)
3.0
27527
a)
2.0 1.0 b)
Dα (a.u.)
4.0 2.0 0 2.0 −4.0
27526*
0.75 0.50 0.25 0
c)
Frequency (MHz)
˙ (V) Φ
ne (x1020 m−2)
4.0
0.20
d)
0.15 0.10
Neutron rate (x1014 s−1)
0.05 1.5
YT, Da = 1.5 m2s−1
e)
YT, Da = 0 m2s−1
1.0
Yn
0.5 0 0.05
0.1
0.15
0.2 time (s)
0.25
0.30
0.35
Figure 3: Plasma discharge 27527: a) the line-integrated electron density and NBI power, b) the time derivative of the poloidal magnetic flux recorded by an outboard Mirnov coil, c) bottom divertor Dα emission, d) spectrogram of the magnetic fluctuation signal depicted in b) showing the time evolution of MHD activity starting from TAEs at frequencies ∼ 100 kHz, evolving into chirping modes and finally developing into the long-lived mode (LLM) at around t ≃ 0.23 s, e) Yn and YT under the assumption of zero and spatially-uniform anomalous fast diffusion of Da = 1.5 m2 s−1 applied throughout the plasma discharge. *The line integrated density for plasma discharge 27526 is depicted as it was not available for plasma discharge 27527. a set of comprehensive plasma diagnostics: the electron density and temperature are measured by a Thomson scattering system (TS) [17, 18], the ion velocity and temperature are provided by a charge exchange recombination spectroscopy diagnostic [19], the deuterium neutral flux at the plasma edge is derived from a linear Dα camera [20] and finally Zeff is obtained from a two-dimensional visible bremsstrahlung camera
8 [21, 22] and TS measurements. The pre-processor code MC3 maps these experimental measurements on plasma equilibria provided by EFIT++ [23] (in this work modelled without including fast ion pressure) and prepares the input files for TRANSP and NUBEAM simulations. NUBEAM provides estimates of 2D flux surface averaged −2.4 3 −1.8
Z (m)
−0.6
2
0 1.5
(x107 cm−3s−1)
2.5
−1.2
0.6 1 1.2 0.5
1.8
2.4 0
0.5
1 1.5 R (m)
2
2.5
0
Figure 4: Non-flux surface averaged neutron emissivity (ξn ) in the poloidal cross section averaged over the time interval 0.245 ≤ t ≤ 0.250 s for plasma discharge 29195. The MAST vacuum vessel (black line), divertor and central column shielding (light grey), central column and poloidal field coils (dark grey) are also shown. thermonuclear hξTH (t, R, Z)i, beam-thermal hξBT (t, R, Z)i, beam-beam hξBB (t, R, Z)i and total hξn (t, R, Z)i neutron emissivities together with the corresponding neutron rates derived from 5D fast ion distribution fi (t, R, Z, E, v||/v). The neutron emissivity however is not a flux-surface quantity as fast ions can deviate significantly from any given flux surface. NUBEAM has been modified to estimate the local 2D beam-thermal ξBT (t, R, Z) and beam-beam ξBB (t, R, Z) neutron emissivities which are not averaged over the flux surfaces [24]. The total non flux averaged neutron emissivity ξn (t, R, Z) is then given by ξn (t, R, Z) = ξBT (t, R, Z) + ξBB (t, R, Z) + hξTH (t, R, Z)i
(1)
and shown in figure 4. The field of view geometry of the different NC LoS has to be taken into account in order to convert ξn into a simulated neutron count rate profile as a function of tangency radius p. Calculations of the solid angles for different LoS Ω(p; R, Z) have been carried
9 out using the specially developed full 3D Monte Carlo code LINE2 [25]. TRANSP predicted count rate CR(p) at a given tangency radius p is then calculated according to the following equation: X CR(p) = ̺εd ξn (Ri , Zj )Ω(p; Ri , Zj ), (2) i,j
where ̺ ≈ 0.91 takes into account the attenuation of the neutron flux by the thin flange through which the plasma is observed, εd is the detector efficiency and the sum is over the poloidal cross section. Under the assumption that all terms in equation 2 are correctly estimated, CRexp = κCR
(3)
with κ = 1 and CRexp being the measured count rate. Deviations of κ from unity suggest that one or more terms in equation 2 are incorrect. The relative uncertainty in κ can be estimated by taking into account the uncertainty in the predicted and measured neutron count rates. The uncertainty in CR arises from the uncertainties in ̺, εd and ξn . There is no uncertainty associated with Ω as it is a purely geometrical factor. In this work it was assumed that δ̺/̺ = 0.05, δεd /εd = 0.06, δξn /ξn = 0.15 (see subsection 3.3) and δCRexp /CRexp = 0.10 resulting in δκ/κ ≈ 0.20. This indicates that the deviations of κ of 20% from unity are compatible with the overall uncertainties with which the quantities in equation 2 are known. 3.1. Validation of TRANSP simulation in MHD-quiescent plasmas In the TRANSP simulation carried out for plasma discharge 29195 the predicted total neutron rate YT follows very closely Yn as shown in figure 1 panel c). A good agreement was also found between the experimental and calculated using ξn neutron count rate profiles as can be seen in figure 5. This agreement is further confirmed by the estimates of κ which was found to vary in the range 0.65 to 0.76 in the time interval 0.19 - 0.30 s, a range that is slightly larger than the experimental uncertainties in the quantities in equations 2 and 3. 3.2. Effect of an anomalous fast ion diffusion on fast ions and neutron emission In the LD scenario, in which moderate quasi-continuous MHD activity is observed throughout the plasma discharge, YT substantially overestimates Yn as shown in figure 1 panel g). In order to match YT to Yn , an ad hoc anomalous diffusion coefficient Da of the order of 1.5 m2 s−1 for t < 0.27 s and of 2.5 m2 s−1 for t ≥ 0.27 s was introduced in the TRANSP simulation. The effect of Da , here constant in time and uniform in space, R on fi (R, Z, E, v|| /v) and the fast ion density fi (R, Z) = fi (R, Z, E, v||/v)dEdv/v|| averaged in the time interval 0.295 ≤ t ≤ 0.3 s, is shown in figure 6. Enhanced radial diffusion of confined fast ions into plasma regions where the confined/loss boundaries in (E and v|| /v) space are quite different results in an overall reduction and broadening of fi (R, Z) as fast ions undergo collisionless transition onto unconfined orbits and are lost
10 ×10 400
χ2 = 1.38
350
Count rate (kHz)
× 10 400
a) 0.225 ≤ t ≤ 0.230 s
b) 0.235 ≤ t ≤ 0.240 s
350
red
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×10 400 c) 0.245 ≤ t ≤ 0.250 s 350 χ = 1.16 300 250 200 150 EQ 100 ξ 50 00.5 0.6 0.7 0.8 1.1 1.2 2 red
red
n
n
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Figure 5: HD set: experimental (points) and modelled using ξn (solid line) and hξn i (dashed line) neutron count rate profiles, averaged over 5 ms time intervals, as a function of tangency radius p for EQ LoS. The reduced-χ2 of the fits when ξn was used to calculate profiles are also shown. from the plasma. Setting Da = 2.5 m2 s−1 results in neutron emissivity a factor three lower than in the simulation without Da as shown in panel e) of figure 6. Satisfactory agreement between modelled and measured neutron count rate profiles for three different time intervals, depicted in figure 7, is found. In this case κ, estimated for 11 time intervals throughout the entire plasma discharge, varies in the range 0.57 to 0.77. TRANSP simulations have also been carried out for plasma discharge 27527 belonging to the HDHP set. As in the case of the LD set, YT significantly overestimates Yn and matching the two requires setting Da = 1.5 m2 s−1 throughout plasma discharge as can be seen in figure 3 panel e). It is worth noting that YT roughly follows Yn apart from the time interval 0.199 ≤ t ≤ 0.230 s during which three large MHD events occur. The modelling of these events is described in detail in section 4 of this work. Nevertheless, neutron count rate profiles can be calculated for the time intervals in which a match between Yn and YT is satisfactory. An example of simulated and experimental neutron count rate profiles for EQ and DIAG LoS is shown in figure 8. Very good agreement is found between measured and modelled profiles. In this set κ, calculated for only three time slices shown in figure 8, varies in the range 0.60 to 0.70. 3.3. Effect of plasma parameters on neutron emission In the previous section it was shown that in order to obtain agreement between YT and Yn for plasma discharges in which MHD activity is present it is necessary to introduce Da in TRANSP simulations: without this the discrepancy between YT and Yn in the LD and HDHP sets is as high as ∼ 85% and ∼ 32%, respectively. This section is dedicated to the verification that the discrepancy between YT and Yn cannot be explained as result of uncertainties in the plasma parameters used as input to TRANSP simulations. In a plasma, Yn is composed of a thermonuclear (YTH ), a beam-thermal (YBT ) and
11 1.00
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ε (R, Z=0) (x10 cm s )
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e)
Da = 2.5 m s
0.6 0.4 0.2 0 0.4
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1.4
R (m)
Figure 6: Plasma discharge 29222: a) and b) fast ion density fi (R, Z) and core fast ion distribution function fi (E, v|| /v) without any anomalous fast ion diffusion, c) and d) fast ion density fi (R, Z) and core fast ion distribution function fi (E, v|| /v) under the assumption of the spatially-uniform anomalous fast ion diffusion of Da = 2.5 m2 s−1 , e) radial neutron emissivity profiles for the two aforementioned levels of Da . All data averaged in the time interval 0.295 ≤ t ≤ 0.3 s.
12 ×10
× 10
a) 0.215 ≤ t ≤ 0.220 s χ2 = 3.38
Count rate (kHz)
600
× 10
b) 0.225 ≤ t ≤ 0.230 s
red
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χ 2red = 3.44
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χ 2red = 2.99
EQ ξn 0.6
0.7
0.8
0.9 p (m)
1
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Figure 7: LD set: experimental (points) and calculated using ξn (solid line) and hξn i (dashed line) neutron count rate profiles, averaged over 5 ms time intervals, as a function of tangency radius p for the EQ LoS. A constant in time and spatially-uniform anomalous diffusion coefficient of the order of 1.5 m2 s−1 was imposed on fast ions in modelling of ξn and hξn i. The reduced-χ2 of the fits when ξn was used to calculate profiles are also shown. ×10 900
× 10
a) 0.205 ≤ t ≤ 0.210 s χ2 = 1.06
900
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χ2
red, DIAG
= 1.04
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×10 c) 0.275 ≤ t ≤ 0.280 s 900 χ = 0.58 800 χ = 0.79 700 600 500 400 300 200 100 0 0.6 0.7 0.8 0.9 1.1 1.2 2 red, EQ
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800
b) 0.235 ≤ t ≤ 0.240 s χ2 = 1.03
0.8
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Figure 8: HDHP set: experimental (points) and simulated using ξn (solid line) and hξn i (dashed line) neutron count rate profiles, averaged over 5 ms time intervals, as a function of tangency radius p for the EQ (black) and the DIAG (grey) LoS. A constant in time and spatially-uniform anomalous diffusion coefficient of the order of 1.5 m2 s−1 was imposed on fast ions in modelling of ξn and hξn i. The reduced-χ2 of the fit when ξn was used to calculate the EQ and DIAG profiles are also shown. a beam-beam (YBB ) neutron rates, where 2 1 2 1 2 Zeff − ZC YTH = nD hσviTH ∝ ne hσviTH , 2 2 1 − ZC Zeff − ZC 3/2 hσviBT , YBT = nD nD,B hσviBT ∝ PNBI Te 1 − ZC
(4) (5)
13 YBB
1 1 = n2D,B hσviBB ∝ 2 2
3/2
PNBI Te ne
!2
hσviBB ,
(6)
with nD and nD,B being the thermal and fast ion densities, v being the relative velocity of interacting ions and σ being the DD reaction cross section [26]. Assuming that the only impurity in MAST is carbon [27], originating from the divertor and the central column, the thermal ion density can be expressed as Zeff − ZC ne , (7) nD = 1 − ZC
where Zeff is the effective plasma charge, ZC is the carbon atomic number and ne is the electron density. From eqs. (4) to (6) it follows that the difference in the total and local neutron rates observed in HD and LD sets can be mainly ascribed to the almost doubled Te , as core electron densities ne and Zeff were very similar. The higher Te results in the increase of YBT and YBB fusion reaction rates which are dominant in MAST. This increase can be explained by the fact that in a plasma characterized by higher Te the fast ion slowing down process on electrons takes longer due to the reduced electron drag force. The longer 3/2 slowing down time τse ∝ Te n−1 increases probability that a fast ion will undergo a e fusion reaction with a thermal ion, since the fusion reactivity is a strongly increasing function of the relative velocity between the reactants. There are two plasma parameters, namely Zeff and ne , which directly influence nD , hence YTH and YBT , following from eq. (7). Another plasma parameter which has an effect on YTH and YBT is the thermal ion temperature Ti which affects the relative velocity of interacting ions through hσviTH and hσviBT . A change in fast ion density nD,B , being 3/2 proportional to NBI power PNBI and fast ion slowing down time τse ∝ Te n−1 e , directly influences YBT and YBB . The effect of the aforementioned plasma parameters on the neutron emission has been investigated in a systematic way for plasma discharge 29195 belonging to the HD set. This plasma discharge is appropriate for such a study as there is a long period without MHD activity therefore any change in YT will be due solely a change in the selected plasma parameter. In this study, the plasma parameters have been varied in ranges which are comparable to or exceed the measurement errors: ne , Te and Ti have been varied by ±20% from their measured values, Zeff has been changed between 1 and 2.5 while nD,B has been modified by ±10%. From this study, it follows that Zeff , Te and nD,B have a strong effect on YT as shown in figure 9. Under the assumptions that i) the only impurity in this plasma discharge is carbon, ii) ne and nD,B are held fixed and iii) hσvi is also held fixed, then it can be shown that the variation of YT with Zeff is very well described by Zeff,ref − Zeff YT − YT,ref = . (8) YT,ref ZC − Zeff,ref The dependence of YT on Zeff is shown in panel a) of figure 9 where the dashed line represents the variation in YT calculated according to eq. (8). The simulations indicate that Zeff has a very strong impact on YT , since for a given ne , an increase in Zeff implies
14 0.4
0.1
0.3 0.0 T,ref T,ref
0.0 −0.1
)/Y
0.1
T
−0.1 −0.2
(Y − Y
(YT − YT,ref)/YT,ref
0.2
−0.3
T
i
T
−0.2 −0.4 −0.5 0.5
e
n
e
−0.3 −0.4 1
1.5
2 Zeff
2.5
3
nD,B −0.2
−0.1
0 0.1 (P − P )/P ref
0.2
0.3
ref
Figure 9: Relative change in YT as a function of a) Zeff and b) a relative change in the plasma parameter P : Ti , Te , ne and nD,B . a reduction in nD and thus in YBT and YTH . The large impact of Te on YT can be 3/2 explained by its strong effect on τse ∝ Te where a variation of Te by ±20% results in approximately 30% change in τse and YT . Variations in nD,B also cause significant changes in YT as they modify fast ion density and thus YBT and YBB as expected. A rather small effect on YT was observed for changes in ne and Ti . On MAST, the velocity of thermal ions is much lower than the fast ion velocity and so an increase in Ti only slightly modifies YBT ; YTH increases with Ti , but makes a negligible contribution to YT . YT was almost unchanged due to variations of ne : this can be explained by the fact that a rise of ne causes an increase in nD and a corresponding decrease in 3/2 nD,B ∝ PNBI τse ∝ PNBI Te n−1 e . As a result YBT , which is dominant in MAST, is directly sensitive to changes of PNBI and Te but not of ne . Of the three plasma parameters which have a significant effect on YT , namely Te , Zeff and nD,B , typically nD,B is well known parameter from the NBI system calibration, therefore only the combined effect of Te and Zeff on YT and neutron count rate profiles has been investigated. Spatial and temporal variations in uncertainties in measured Te and Zeff have been taken into account in these simulations. From the simulations, it follows that varying Te and Zeff within experimental uncertainties results in approximately a 15% change in YT (figure 1 panel c)) and neutron count rate profiles (figure 10 panel a)). The maximum expected uncertainty in YT estimated for the HD set has then been applied to YT in LD and HDHP sets as shown in figure 1 panel g) and figure 3 panel g). It is clear that the variation in YT due to experimental uncertainties in plasma parameters can not account for the discrepancies between Yn and YT observed during strong MHD activity in LD and HDHP sets. There is also disagreement between the measured and modelled neutron count rate profiles as presented in figure 10 panel b). Agreement between Yn and YT and between experimental and modelled neutron count rate profiles can be obtained only if an anomalous fast ion diffusion coefficient is introduced in the modelling as can be seen in figure 1 panel g) and figure 10 panel b).
15 × 10 1200 b) LD: 0.295 ≤ t ≤ 0.300 s
× 10 1200 a) HD: 0.235 ≤ t ≤ 0.240 s
800
EQ Da = 0 m2 s-1 1000 2 -1 Da = 2.5 m s max/min 800
600
600
400
400
200
200
Count rate (kHz)
1000
0.6
0.7
0.8
0.9 p (m)
1
1.1
0 1.2
0.6
0.7
0.8
0.9 p (m)
1
1.1
1.2
Figure 10: HD and LD sets: experimental (points) and simulated neutron count rate profiles under the assumption of zero anomalous fast ion diffusion (solid black line) and a spatially uniform anomalous fast ion diffusion of Da = 2.5 m2 s−1 (dashed black line) as a function of tangency radius p for EQ LoS. The dotted lines represent the maximum expected uncertainties in simulated neutron count rate profiles due to uncertainties in plasma parameters used in TRANSP simulations. 4. Modelling the effect of fast ion losses on neutron emission As shown in the previous section, the introduction of Da improves the overall agreement between Yn and YT . However, Da fixed in time and uniform in space cannot model the effect of large MHD bursts as shown in figure 3. A series of TRANSP/NUBEAM simulations for plasma discharge 27527 has been therefore performed using the fishbone loss model implemented in NUBEAM [28] to model the changes in Yn . In NUBEAM, the effect of fishbones on the fast ion population can only be modelled by selectively removing fast ions in user defined energy-pitch angle (Loss Model 1 - LM1) or energy-trapping depth regions (Loss Model 2 - LM2) regions. LM1 allows for a crude selection of passing and trapped fast ions by the choice of the pitch-angle while LM2 affects only trapped fast ions. In LM1, an ion is removed if its energy and either of its pitch angles evaluated at the last two midplane crossings fall into the specified energy and pitchangle ranges. A strong limitation of LM1 is that the ejection of the fast ions is applied uniformly across the plasma poloidal cross-section not taking into account the fact that the passing-trapped boundary is location dependent. In both models, the loss of fast ions is controlled by a series of common parameters: TLOSS , WFB and TFB . The effective ion life time in the plasma is controlled by TLOSS which is typically chosen to be TLOSS ≪ τse to model the effect of prompt ion losses. WFB defines the time interval during which fast ions are removed and this is typically made to coincide with the duration of the
16 80
1.4 Yn
70
−
4 60
−1
1.0
50 0.8 0.6
3
−3 −1
E (keV)
40 2
30
0.4
20
0.2
10
8
Neutron rate (x1014 s−1)
||
b) c) d)
1.2
5
b)−0.2 ≤ v /v ≤ 0.2
x10 cm s keV (v||/v) 1
27527
1
0.20
0.25
0.30
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time (s)
c) 0 ≤ E ≤ 37
5
70 4 60
E (keV)
50
3
40 30
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d) E ≥ 50
10
5
70 4 60 50
3
40 30
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0 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 v||/v
0
||
x108 cm−3s−1keV−1(v||/v)−1
80
0 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 v /v
x108 cm−3s−1keV−1(v||/v)−1
0.15
E (keV)
0 0.10
1 10
0
0 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 v||/v
0
Figure 11: Plasma discharge 27527: a) Yn and YT modelled using LM1 when the cuts presented in panels b), c) and d) were applied to fi (R, Z, E, v||/v). Panels b), c) and d) show f (R = 0.98 m, Z = −0.01 m, E, v|| /v) averaged over time interval 0.19 ≤ t ≤ 0.20 s when the following regions were selected for fast ion removal: b) v|| /v ǫ [−0.2, 0.2] regardless of their energy, c) E ǫ [0, 37] keV regardless of their pitch angle and d) E ǫ [50, 75] keV regardless of their pitch angle. fishbone burst. TFB determines the period with which the fishbone bursts occur during a time interval defined from the onset of the first fishbone burst. Ejection of fast ions is then carried out repeatedly every TFB for a duration equal to WFB until the loss model is turned off at a time also specified by the user. An attempt to model the overall trend of Yn in the time interval 0.124 ≤ t ≤ 0.24 s using LM1 has been performed in order to assess if its possible to obtain a similar agreement between Yn and YT as when the diffusive model has been used. TFB = WFB = 4 ms and TLOSS = 1 µs have been set in these simulations thereby implying the continuous removal of fast ions. LM1 is able to roughly reproduce Yn when all ions with pitch angle in the range −0.2 ≤ v|| /v ≤ 0.2 or with energy in the range 0 ≤ E ≤ 37 keV are expelled from the plasma as shown in figure 11. YT slightly overestimates Yn in the time intervals where TAEs together with small fishbones (0 ≤ t ≤ 0.195 s) and LLM (t ≥ 0.230 s) are present. However, better agreement was achieved in the same time intervals when the diffusive model with Da = 1.5 m2 s−1 was used as shown in panel f) of figure 3. YT drops dramatically when ions with the highest energies E ≥ 50 keV regardless of their pitch angle are removed from the plasma. The effects
17 of the cuts described above on the fast ion distribution function fi (t, R, Z, E, v||/v) are shown in figure 11 panels b), c) and d) respectively. The lack of fast ions in the specified regions in E and v|| /v is clearly visible. There are very few ions present in the region −0.2 ≤ v|| /v ≤ 0.2 in figure 11 b) due to pitch angle scattering of fast ions occurring after the LM1 was applied. Only ions born with one half or one third of the beam injection energy are present in the plasma plasma in panel d) of figure 11, since LM1 was applied for longer (t ≃ 120 ms) than the fast ion slowing down time (τse ≃ 63 ms calculated at time t = 0.125 s).
Neutron rate (x1014 s−1)
1.4 1.2
Y
n
a)
Y
n
d)
YT (LM1)
1.0
YT (LM2)
0.8 0.6 0.4 0.2 0.10
0.15
0.20 0.25 time (s)
0.30
0.35
0.15
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0.30
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Count rate (MHz)
1.0 0.8
b)
e)
0.6 0.4
EQ pre−FB EQ post−FB
EQ pre−FB EQ post−FB
LM1 pre−FB, χ2red = 2.38
0.2
χ2red
LM1 post−FB,
LM2 pre−FB, χ2red = 0.83 LM2 post−FB, χ2red = 5.42
= 7.57
Count rate (MHz)
0 0.4
c)
f)
0.3 0.2 0.1
DIAG pre−FB DIAG post−FB
DIAG pre−FB DIAG post−FB
LM1 pre−FB, χ2red = 3.86
LM2 pre−FB, χ2red = 2.63
2
2
LM1 post−FB, χred = 6.91
0 0.5
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0.9
p (m)
LM2 post−FB, χred = 4.27
1.1
1.3
0.7
0.9
1.1
1.3
p (m)
Figure 12: Comparison of a) Yn and YT , b) and c) pre/post-fishbone experimental (points) and TRANSP predicted (lines) neutron count rate profiles as a function of tangency radius p for EQ and DIAG LoS respectively, calculated using LM1. Panels d), e) and f) show the same data as panels a), b) and c) but for a TRANSP simulation in which LM2 was used. As noted in section 3.2, the purely diffusive anomalous transport model fails to match Yn in the time interval 0.195 ≤ t ≤ 0.230 s during which the fishbones are characterized by well separated bursts with an almost periodic behaviour. The large
18 drops in Yn associated with these fishbones suggest that in this case the fast ions might be lost rather than redistributed. To test this hypothesis, Da was set to zero and LM1, with TFB = 13 ms and WFB = 4 ms, was used in this interval. The diffusivity modelling outside time interval (0.195 ≤ t ≤ 0.230 s) and the purely LM1 are ”glued” together in a single TRANSP run to provide the result shown in figure 12 panel a). A reasonable agreement is found when all fast ions (both trapped and passing) with energy E ≥ 55 keV and pitch angle in the range −0.5 ≤ v|| /v ≤ 0.5 are expelled from the plasma. To quantify this agreement the parameter σLM defined as 2 k 1 X Yni − YTi , (9) σLM1,2 = k i=1 u(Yni )
is used, where u(Yni ) is the uncertainty of Yni and k is the number of data points in the given time interval. In this case σLM1 = 7.4 × 106 . Yn can also be modelled using LM2 with the same fishbone setting used for LM1 combined with a small diffusion Da = 0.5 m2 s−1 as shown in panel d) of figure 12. In this case σLM2 = 5.0 × 106 when all trapped fast ions with E ≥ 37 keV are removed. This better agreement suggests that these large fishbones cause both losses and redistribution of fast ions. The modelled neutron count rate profiles based on ξn provided by LM1 and LM2 can be used to verify which model better describes measured pre/post-fishbone neutron count rate profiles. The comparisons of the calculated and measured pre/post fishbone profiles for EQ and DIAG LoS are presented in figure 12. The overall reduction in the neutron count rate is seen for post-fishbone profiles. It is more pronounced in the core plasma region (0.80 < p < 1.10 m), indicating fast ion redistribution and losses from the plasma core. No increase of the neutron count rate is observed outside the plasma core. The changes in the pre/post fishbone neutron count rate profiles are much larger than the experimental uncertainties. The calculated pre-fishbone neutron count rate profiles agree rather well with the experimental profiles with κ varying between 0.63 and 0.68. The predicted post-fishbone profiles are clearly underestimated at the inboard plasma side. In this case the parameter κ is in the range 0.53 - 0.57. It is not possible to ascertain which fishbone loss model describes the experimental observation better as both models reproduce Yn and the neutron count rate profiles to a similar degree. 5. Summary and conclusions TRANSP/NUBEAM modelling of the total and local neutron emission has been validated in MHD-quiescent, moderate MHD and strong MHD plasma scenarios on MAST. In particular, for plasmas with no MHD activity very good agreement between the TRANSP predicted and measured total neutron rate is found. Satisfactory agreement is also obtained between the modelled and experimental neutron count rate profiles. These results are consistent with the modelling of the total neutron rates and count rate profiles carried out on ASDEX and JET [15, 13]. Accurate measurements of plasma profiles used in TRANSP simulations are very
19 important to obtain a good estimate of the neutron emission from the plasma; this point has been emphasised in [15]. The effects of uncertainties in the plasma profiles used in TRANSP simulations on the predicted total and local neutron emission have been investigated and found to be ≃ 15% on MAST. The disagreement observed between the experimental neutron rate and that calculated by TRANSP for plasma scenarios in which different levels of MHD activity are present is much larger than expected due to uncertainties in TRANSP simulations. This implies that this difference is due to additional transport caused by the MHD activity. The effect of MHD activity on fast ions and thus neutron emission can be modelled in TRANSP/NUBEAM by introducing additional fast ion diffusion or removing fast ions from the plasma. The diffusive model has been successfully applied to model the total and local neutron emission in the presence of small amplitude MHD instabilities with a short repetition time. However, applying time-dependent, spatially-varying and energy dependent anomalous diffusion coefficient has not yet been fully explored in predicting changes in the total and local neutron emission caused by large amplitude MHD events with a much longer repetition time. Instead, the semi-heuristic loss model available in NUBEAM has been applied to model the effect of the fishbone instability on the total and local neutron emission. Similar agreement between the measured and TRANSP predicted neutron rates and count rate profiles has been obtained by removing either passing and trapped ions with energy E ≥ 55 keV and pitch angle in the range −0.5 ≤ v|| /v ≤ 0.5 from all plasma regions or only trapped ions with energy E ≥ 37 keV. Although the total and local neutron emission can be modelled using either of these fishbone loss models, these results do not allow us to unambiguously identify which part of fast ions distribution function interacts with MHD instabilities. To resolve this issue, changes in neutron and FIDA emission profiles have been modelled using the loss models [9]. It was found that the model can reproduce either the total neutron rate and the neutron count rate profile by removing highly energetic trapped and barely passing ions, in which case the changes in FIDA profiles are not well reproduced, or the total neutron rate and the FIDA profile by removing ions from a very narrow pitch angle range 0.88 ≤ v|| /v ≤ 0.91 regardless of ion energy, in which case the model is not able to account for changes in the neutron count rate profiles. These results indicate a need for first-principles-based kinetic modelling of the wave-particle interaction between the fast particles and fishbone modes. Preliminary results from the modelling of the effects of fishbone instabilities on the total and local neutron emission, carried out using MISHKA [29], LOCUST [30] and HAGIS [31] codes, are consistent with the experimental observations [32]. Acknowledgments This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement number 633053, from the Swedish Research Council and from the RCUK Energy Programme [grant number EP/I501045].
20 The views and opinions expressed herein do not necessarily reflect those of the European Commission. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
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