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1.3. A SEMI-EPIRICAL CRITERION ON COIL CURRENT REQUIREMENT TO CONTROL EDGE LOCALIZED MODES IN TOKAMAK PLASMAS1
Yueqiang Liu, General Atomics, USA; Southwestern Institute of Physics, China. E-mail: liuy@fusion.gat.com
Li Li, College of Science, Donghua University, China
Abstract: Recent encouraging results, showing the correlation between the toroidally computed plasma displacement near the X-point on one hand, and the experimentally achieved mitigation or suppression of the type-I edge localized modes (ELMs) by the applied resonant magnetic perturbation (RMP) fields on the other hand, motivates the development of an X-point displacement based criterion, to guide the choice of the magnitude of the ELM control coil current. Full toroidal simulations of the ELM control discharges in four existing devices (MAST,ASDEX Upgrade, has to trigger a minimal level of the has DIII-D and JET), based on the single fluid, resistive magnetiohydrodynamic model with plasma flow, reveal that the applied RMP field has to trigger a minimal level of the X-point displacement, of 1-3 mm, in order for the ELM to be mitigated or suppressed. This plasma response based criterion can be useful for predicting the coil current requirements in future such as ITER.
Index terms: fusion, tokamak, ELM control, resonant magnetic perturbation, plasma response.
ПОЛУЭМПИРИЧЕСКИЙ КРИТЕРИЙ ОПРЕДЕЛЕНИЯ ТОКА КАТУШКИ КОНТРОЛЛИРУЮЩЕЙ НЕУСТОЙЧИВОСТЬ ПЛАЗМЫ ЛОКАЛИЗОВАННОЙ НА ГРАНИЦЕ ПЛАЗМЫ ТОКАМАКА
Юсцян Лю, General Atomics, США; Southwestern Institute of Physics, Китай. E-mail: liuy@fusion.gat.com
Ли Ли, College of Science, Donghua University, Китай
Аннотация: В проблеме создания управляемого термоядерного синтеза в плазме установок токамак важное место занимает разработка высоко производительных режимов (так называемых H-мод). Однако, в этом случае на границе плазмы возникает скачок макроскопических параметров, что приводит к неустойчивости погранично-локализованных мод (ELM). Стабилизация этой неустойчивости проводится с помощью системы внешних винтовых проводников. Для моделирования таких режимов требуется создание цепочки моделей нелинейных макро и микро-процессов, что до сих пор не выполнено. В настоящей работе приводится создание полу-эмпирической математической модели для описания высокопроизводительных H-мод процессов, правильно описывающей работу серии современных токамаков. Основой модели является система линейных магнитогидродинамических уравнений, в которой решение согласуется экспериментально измеряемым смещением плазмы в диверторной точке границы плазмы. Показано хорошее согласие с проведенными экспериментами на различных установках. Приводятся данные, полезные для будущих экспериментов на установке ITER.
Ключевые слова: термоядерный синтез, токамак, контролирование ГЛМ, резонансное магнитное возмущение, реакция плазмы.
I. Introduction
With sufficient power input and good boundary conditions, the fusion plasmas in tokamaks can enter into a highperformance regime, the so-called H-mode regime, which manifests itself with substantially improved energy and particle confinement near the plasma edge, forming an edge pedestal. The H-mode regime offers much higher potential for the fusion energy production, and thus has been a favorite operational regime for the next generation fusion experiments such as ITER.
Unfortunately, the large gradients associated with sharp variation of the plasma equilibrium profiles, in particular that of the pressure and the toroidal current density, in the pedestal region of the H-mode plasmas, often drive macroscopic instabilities called the edge localized modes (ELMs).
1 The work is supported by National Natural Science Foundation of China with Grant No. 11605046, 11505050, and National Magnetic Confinement
Fusion Science Program with the Grant No. 2015GB105004 and
2015GB104004. Work is also supported by US DoE Office of Science under
Contract DE-FG02-95ER54309 and DE-FC02-04ER54698.
These instabilities, being localized near the plasma edge, can be severely dangerous to the operation of the tokamak devices, in particular in future devices such as ITER [1]. This is because large bursts of such instabilities, in particular the so-called type-I ELMs, can cause a significant amount of heat and particle flux loads on the plasma facing components, thus potentially damaging the machine. It is by this reason, the edge localized mode, and the major disruption, are currently regarded as two of the most dangerous transient events, that have to be controlled in future large scale tokamak experiments.
One promising method to control the type-I ELMs is to use long wavelength, three-dimensional (3D) resonant magnetic perturbations (RMP), externally generated by currents flowing in magnetic coils surrounding the plasma. These 3D magnetic field perturbations, though normally of a very small fraction (of order of 10"4-10"3) compared to the toroidal equilibrium field, have been shown to be capable of profoundly affecting the confinement and stability of the tokamak plasmas, which are designed as essentially a 2D
configuration with axi-symmetry. One important effect is that these RMP fields can either mitigate or suppress type-I ELMs, as shown in present day experiments [2-7]. Here the mitigation refers to the increase of the ELM bursting frequency, normally accompanied by the reduction of the amplitude of each individual burs. This amplitude reduction is crucial, since it reduces the peak load of the heat flux on the plasma facing components such as the divertor surface. The ELM suppression refers to a process where the type-I ELMs are completely eliminated during the H-mode operation.
The physics understanding of the ELM mitigation or suppression - the ELM control in general - is currently an active research area. First principle modeling of the ELM control process, which is a substantially non-linear phenomenon involving both macro- and microscopic physics, is still at initial stages. On the other hand, linear or quasi-linear models have so far been reasonably successful in explaining many aspects associated with ELM control [8-10]. In particular, it has been found that the linear plasma response model, based on macroscopic magneto-hydrodynamic (MHD) description of the plasma, can produce results that are in quantitative agreement with experimental measurements [11]. The ELM controllability has been found to be in close correlation with a special type of the plasma response, i.e. the edge-peeling response [12-13]. Furthermore, the edge-peeling response, in turn, often produces large plasma surface displacement near the X-point of a divertor plasma. In fact, the computed X-point displacement, by a toroidal code, can be used as a good figure of merit, in order to judge the success of ELM control in experiments [12, 14-15].
Based on the aforementioned success, this work exploits the possibility of using the plasma displacement near the X-point as a criterion, to predict the RMP coil current requirement for obtaining the type-I ELM mitigation or even suppression.
II. Formulation and computational model
We compute the linear resistive plasma response to the applied RMP field using the MARS-F code [16], which solves the full toroidal, single fluid MHD equations in the presence
of toroidal plasma flow
í(Hrmp + = v + • (1)
ÍPeqC^RMP + Wn)v = -Vp + j X Beq + Jeq X b -
-peq[2fiZ X V +(v-Vn)fi$] (2)
i(fiRMP + nfi)b = V X (v X Beq)
+(b • vn)fi0 - V X (^j) (3)
i(nfiMP + nfl)p = -v • VPeq - r?eqV • V (4)
j = V X b (5)
where R is the major radius of the torus, Z the unit vector along the vertical direction. The plasma displacement, perturbed velocity, magnetic field, current and pressure are denoted by v, b, j, p respectively, whilst peq, Beq,Jeq and Peq represent the equilibrium plasma density, magnetic
field, current and pressure. n is the plasma resistivity, for which the Spitzer model is used in this work. n is the toroidal mode number of the applied RMP field. In ELM control experiments, normally low-n field perturbations are applied. Generally speaking, a rotating field, with the toroidal frequency of QRMP, can be applied, though static field is often used in the ELM control experiments, i.e. with QRMP=0. In our model, the plasma is assumed to have an equilibrium toroidal flow, with flow speed of V0 = and the angular rotation frequency of Q.
In MARS-F, the applied vacuum RMP field is generated by a source current (with current density jRMP) flowing in magnetic coils located in the vacuum region outside the plasma. Thus, in the region occupied by the RMP coils, the following equation
jRMP = V X b (6)
is also solved by MARS-F. The source current is modeled as a surface current at the radial location of the RMP coils. The above Eq. (6), together with the vacuum field equations, is consistently solved with the MHD equations (1-5). Because of the presence of a source term jRMP, the overall plasma response problem formulated here is a driven problem. The eventual amplitude of the plasma response scales linearly with the applied coil current amplitude, due to the linear nature of the problem. The resulting perturbed field including the plasma response, when superposed with the equilibrium field, forms the so called perturbed 3D equilibrium, with the steady state plasma response.
The MHD equations are solved in a curve-linear coordinate system based on the equilibrium magnetic flux surfaces. The solution variables, i.e. essentially all the perturbed quantities, are decomposed in Fourier harmonics along both the toroidal and poloidal angles. For divertor plasmas, the flux based coordinates cannot resolve the exact X-point. Therefore, in MARS-F computations, we usually slightly smooth the plasma boundary shape (the separatrix) near the X-point, before pursuing the equilibrium reconstruction. This procedure normally does not sensitively modify the computed plasma response [17]. Along the plasma minor radius, a finite element method is used in MARS-F, which allows strong mesh packing near resonant surfaces associated with the perturbation. Such strong mesh packing is critical for accurate computation of the resistive plasma response.
The MARS-F resistive response model has been bench-marked with analytic theory [18-19], other codes developed for computing the plasma response to 3D RMP fields [20], as well as validated against experimental measurements [11]. In particular, an extensive effort has been devoted to examining the validity of the single, linear fluid model for computing the plasma RMP response in realistic tokamak experiments, as reported in Ref. [11].
III. Computational results
The initial evidence, that the plasma X-point displacement can be used as a criterion to indicate the ELM mitigation threshold, has been reported for MAST [21] and ASDEX Up-
grade [22]. More importantly, despite the fact that different threshold amplitude of the coil current was required to achieve ELM control in these two devices, as shown in Table 1, the critical magnitude of the plasma surface displacement near the X-point was computed to be similar, i.e. between about 1-2 mm level. This also holds irrespective of the toroidal harmonic numbers (n=4 or 6 for MAST, and n=2 for ASDEX Upgrade) used for the RMP fields.
Table 1
List of reference discharges from various tokamak devices: MAST, ASDEX Upgrade (AUG), DIII-D and JET, where type-I ELMs
were found to be marginally mitigated or suppressed, at the threshold coil current amplitude I^p. Listed is also the MARS-F computed plasma displacement ^n(~X) near the X-point.
On the other hand, the MAST and ASDEX Upgrade plasmas have similar minor radius, though rather different machine size (major radius, see Fig. 1). In order to identify any potential machine size dependence of the critical X-point displacement, as well as to establish a more robust criterion, in this study, we expand our investigation by including into consideration other devices, i.e. DIII-D and JET.
1.5 1
0.5
I 0
N
-0.5
-1 -1.5
0 12 3 4
R[m]
Fig. 1. Plasma boundary shapes for discharges listed in Table 1, from MAST, ASDEX Upgrade (AUG), DIII-D and JET. Indicated are also the locations of the ELM control coils for MAST, ASDEX Upgrade and DIII-D. The error field correction coils, located at
R = 5.5 m, Z = ±3.0 m, are used to control ELMs in JET. Figure 1 compares the plasma boundary shapes from four typical ELM control discharges (listed in Table 1) in these four devices. We note a substantial variation of the plasma major and minor radii in this choice of dataset. Furthermore, the locations of the RMP coils are also different depending on the device, as shown in Fig. 1. Only a single row of coils, located below the mid-plane, was used in these MAST experiments, whilst two rows of coils, located above and below the mid-plane, respectively, were used in ASDEX Upgrade and DIII-D. JET uses the error field control coils (EFCC), located far from the plasma and near the outboard mid-plane, to control type-I ELMs. The toroidal spectrum of the applied coil current also differ as reported in Table 1.
The equilibrium profiles for these four plasmas are also different. As examples, figure 2 compares the radial profiles for two of the key equilibrium quantities, i.e. the safety factor q and the plasma pressure. We observe notable variations in the q-profiles, and even larger variation in the normalized pressure profiles. In particular, the large normalized pressure in MAST is essentially due to the low toroidal field, which is typical for spherical tokamaks.
0 0.2 0.4 0.6 0.8 1
a)
0.12
0 0.2 0.4 0.6 0.8 1
b)
Fig. 2. Radial profiles for (a) the safety factor q and (b) the equilibrium pressure normalized by B0 /\i0, where B0 is the toroidal vacuum field on the magnetic axis, for four discharges listed in Table 1.
For the given plasma equilibria and coil current configurations, we then compute the plasma response, by solving the resistive MHD equations as a driven problem as discussed in Dection II. Our primary interest is in the plasma displacement caused by the RMP fields, as a result of the plasma response to these fields. Figure 3 plots the MARS-F computed radial displacement amplitude, for the MAST plasma with n=4 and n=6 RMP configurations, respectively. The experimentally observed threshold current amplitude is assumed in the computations. A somewhat similar pattern of the plasma displacment is found between the n=4 and 6 configurations, but the overall magnitude differ. Note also that the largest displacement occurs above the mid-plane of the plasma, although the RMP coils are located below the mid-plane. This shows that the plasma response is not local. On the other hand, most of the displacement is located near the edge of the plasma, indicating a peeling-like nature of the underlying response mode.
device discharge R0 [m] a [m] coils n wcrit 'RMP [kAt] ¡U~X) [mm]
MAST 25075.450 0.89 0.53 Lower-I 4 ~1.5 0.76
MAST 25075.450 0.89 0.53 Lower-I 6 ~3.2 1.91
AUG 31128.3500 1.72 0.52 B-coils 2 ~6.0 1.84
DIII-D 157308.4200 1.70 0.60 I-coils 3 ~3.0 0.90
JET 82469.5350 3.01 0.87 EFCC 2 ~38.4 2.79
Fig. 3. The MARS-F computed magnitude of the normal plasma displacement, as a result of the plasma response to the applied (a) n=4, and (b) n=6, RMP fields, with the coil current amplitued at the corresponding threshold values (1.5 kAt for n=4 and 3.2 kAt For n=6) for the type-I ELM mitigation in MAST.
The pattern of the computed plasma displacement varies with devices, as shown by further examples from ASDEX Upgrade, DIII-D and JET, in Fig. 4(a-c), respectively. We point out that the pattern of the displacement depends on many factors. Even for the same device, the pattern can significantly vary with the plasma conditions (e.g. toroidal flow speed [12]) as well as the coil configurations (e.g. relative coil current phasing between different rows of coils [13-15,23]). However, roughly two types of patterns can always be identified [12,9], one where the displacement is the most pronounced near the ourboard mid-plane, as shown in Fig. 3 and Fig. 4(a,c) here; the other where the largest displacement occurs near the top and bottom of the machine (Fig. 4(b)). Note that in the JET example shown here (Fig. 4(c)), the displacement near the X-point is also relatively large.
It has been found, by comparing the MHD predicted plasma displacement and the experimentally observed RMP effects of plasma (i.e. ELM mitigation or suppression, and sometimes also the so-called density pumpout effect), that the plasma displacement near the X-point plays an important role [12-13, 24, 4, 15, 25].
Fig. 4. The MARS-F computed magnitude of the normal plasma displacement, as a result of the plasma response to the applied RMP fields, for (a) ASDEX Upgrade with the n=2 coil configuration and the threshold current amplitude of 6 kAt, (b) DIII-D with n=3 and the threshold current of 3 kAt, and (c) JET with n=2 and the threshold current of 38.4 kAt. Type-I ELM mitigation was achieved in ASDEX Upgrade and JET, whilst type-I ELM suppression was
achieved in DIII-D. Besides the difference in the computed pattern of the plasma displacement, we note the variation of the overall amplitude of the displacement in different devices. Note also that, in our linear response model, the displacement amplitude is proportional to the applied current. In the results resported in Figs. 3 and 4, the marginal plasma current amplitude, as found in experiments for achieving ELM mitigation or suppression, has been assumed. On the other hand, despite the substantial difference in the
threshold current amplitude (by a factor of more than 25) found on different devices as listed in Table 1, the computed plasma displacement amplitude does not vary much - by a factor of about 4 among all 4 devices considered here (Figs. 3 and 4). Figure 5 shows a more direct comparison by plotting the displacements for MAST, DIII-D and JET in the same figure.
Fig. 5. Comparison of the computed magnitude of the normal displacement of the plasma in three devices (MAST, DIII-D, JET) of different size, in the presence of the ELM control coil currents at the corresponding threshold values for ELM mitigation or suppression.
In order to better illustrate the qualitative difference of the plasma displacement between different devices, we plot in Fig. 6 the normal surface displacement magnitude versus the geometric poloidal angle of the plasma surface. The peak values of the displacement near the X-point, defined as the shaded region in Fig. 6, are also listed in Table 1. We conclude that, based on these numerical results for four devices with rather different machine size, the near X-point displacement, computed assuming the threshold coil current for ELM control, does not significantly vary (between ~ 1-3 mm).
-100 0 100 200 geometric 0
Fig. 6. The computed magnitude of the plasma surface displacement, plotted along the geometric poloidal angle, for the five cases listed in Table 1. The maximal displacement in the shielded region (near the X-point of the separatrix) is taken as an indicator for ELM controllability, with the values being also listed in Table 1.
V. Summary
It has recently been demonstrated, via both extensive toroidal modeling and ELM control experiments on multiple tokamak devices, that the edge-peeling response of the plasma plays a significant role during the ELM mitigation or suppression, by the externally applied, long wavelength 3D magnetic field perturbations. The edge-peeling response causes large plasma surface displacement near the X-point of the separatrix. Therefore, computation of the plasma displacement near the X-point has provided a useful guidance for optimizing the RMP coil configurations (e.g. the so-called coil phasing), in order to achieve the best ELM control, for a given coil current.
A fundamental question, however, remains on the coil current amplitude requirement for the ELM control. At present, it appears that the first principle prediction of the coil current threshold, required for achieving the ELM mitigation or suppression for a given plasma condition, is still far from mature. Even qualitative understanding of the ELM control physics, by 3D fields, is still under active investigation.
On the other hand, the desire of having certain predictive capability of the coil current requirement for future devices, such as ITER and DEMO, strongly motivates the development of physics based criteria. The island overlap criterion [26], based on the vacuum field approximation, may be of limited usefulness. The plasma response based criteria, in particular that involves the plasma displacement near the X-point, offers alternative, and hopefully also more robust way of predicting the RMP coil current threshold for ELM control. We point out two key merits of the plasma displacement based criterion: (I) it is less sensitive to the coil geometry, such as the coil size or the distance of the coils to the plasma surface; (II) such a criterion naturally encompasses physics factors, such as the edge safety factor, the pedestal collisionality, the plasma flow, that have been experimentally shown to be of critical importance in achieving ELM control.
This work provides the first attempt to develop a robust criterion for predicting the RMP coil current threshold required for ELM control, based on a semi-empirical approach. More specifically, we compute and compare the plasma X-point displacement, for a range of ELM control experiments in multiple devices (MAST, ASDEX Upgrade, DIII-D, JET), where the threshold condition is met. This gives us the critical magnitude of the plasma X-point displacement, above which we know that the ELM control is achieved in experiments. We find that this critical amplitude is between 1-3 mm, valid for all devices that we modeled. This is despite the vast variation of the coil current threshold found between different devices, ranging from 1.5 kAt in MAST to 38.4 kAt in JET.
Such an X-point displacement based criterion, or its improved version (e.g. by taking into certain machine size scaling, still to be found), can be useful in predicting the ELM control coil current requirement in future devices. This predictive possibility is enabled by the linear relation between
the computed displacement and the applied coil current amplitude, for a given plasma configuration.
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