CC BY
13
1.9. ЗНАЧЕНИЕ РАДИАЛЬНОГО ЭЛЕКТРИЧЕСКОГО ПОЛЯ ДЛЯ ПЛАЗМЫ С МАГНИТНЫМ УДЕРЖАНИЕМ
Гидо Ван Оост, профессор кафедры прикладной физики Гентского университета, Бельгия; профессор-совместитель кафедры физики плазмы НИЯУ «МИФИ», Москва, Россия, профессор-совместитель НИУ «Московский энергетический институт», Москва, Россия. E-mail: guido.vanoost@Ugent.be
Аннотация. В работе обсуждается роль и значение радиальных электрических полей в устройствах с магнитным удержанием. На лимитерных и диверторных токамаках, стеллараторных устройствах и прямых ловушках с различными типами разряда и методами нагрева плазмы, а также в экспериментах с краевой поляризацией было показано, что улучшение удержания часто оказывается связанным с наличием сильной радиальной неоднородности радиального электрического поля Er, и что стабилизации турбулентности широм полоидальной скорости Eх B является надежным и универсальным механизмом, который играет важную роль в формировании и поддержании транспортных барьеров в устройствах с магнитным удержанием. Акцентируется внимание на связи между генерацией внутренних транспортных барьеров по электронному каналу и концепцией канонических профилей, разработанной Ю.Н. Днестровским, в которой профили давления и температуры плазмы имеют тенденцию организовываться в «универсальную» форму профиля, в соответствии с принципом минимума свободной энергии плазмы.
Ключевые слова: магнитное удержание, электрическое поле, шир скорости вращения, канонические профили.
1.9. IMPORTANCE OF RADIAL ELECTRIC FIELDS FOR MAGNETICALLY CONFINED PLASMAS
Guido Van Oost, Department of Applied Physics, Ghent University, Belgium; National Research Nuclear University «MEPHI», Moscow, Russia, National Research University «Moscow Power Engineering Institute», Moscow, Russia
Abstract. The importance of radial electric fields in magnetic confinement devices is outlined. It has been demonstrated in limiter - and divertor tokamaks, helical devices and mirror machines with a variety of discharge - and heating conditions as well as edge biasing schemes that improved confinement is often associated with strongly radially varying profiles of E, and that Eх B velocity shear turbulence stabilisation is a robust and universal mechanism which plays a major role in the formation and sustainment of transport barriers in magnetic confinement devices. Emphasis is put on the relation between the generation of electron internal transport barriers and the concept of profile consistency developed by Yu.N. Dnestrovskij, in which the plasma pressure and temperature profiles have a tendency to organize themselves into an 'universal' profile shape, in agreement with the plasma minimum free energy principle.
Keywords: Magnetic confinement, electric field, rotation velocity shear, canonical profiles.
I. Introduction
The importance of radial (i.e. perpendicular to the magnetic surface) electric fields was already recognised early in the research on controlled thermonuclear fusion. An initial description of electric field effects in toroidal confinement was given by Budker [1]. Such a configuration with combined magnetic and electric confinement («magnetoelectric confinement»), where the electric field provides a toroidal equilibrium configuration without rotational transform) was studied by Stix [2], who suggested that a reactor-grade plasma under magnetoelectric confinement (electric fields of order 1 MV/cm) may reach a quasi-steady-state with ambipolar loss of electrons and some suprathermal ions (e.g. 3,5 MeV a-particles). Experiments such as on the Electric Field Bumpy Torus EFBT [3, 4] provided quite favourable scaling for particle confinement. The possible importance of radial electric fields for transport was in the past repeatedly established [5, 6, 7, 8]. Since the early days the plasma potential has been measured in tokamaks such as ST [9], TM-4 [10] and ISX-B [11], but because no significant effects of the radial electric field Er on plasma transport were observed under the machine conditions at that time, no further research was conducted in tokamaks.
However, a renaissance came after the transition from a low confinement mode (L-mode) to a high confinement mode (H-mode) was discovered in ASDEX [12]. The interest was suddenly refreshed and a flurry of activity started with the experimental [13, 14] and theoretical recognition [15, 16, 17] of a possible link between Er and the H-mode phenomenon. Since then research on E has flourished and the H-mode has now been seen
r
in a wide variety of magnetic confinement devices. Many theories have pointed to the possible decisive role of Er in the creation of transport barriers (i.e. zones of finite radial extent where particle and/or heat diffusivity are depressed) and in the L-H bifurcation mechanism.
Typical features of an L-H-transition could also be obtained by externally inducing a controlled radial electric field in the plasma (independently of other plasma parameters) in the tokamaks CCT [13] and TEXTOR [18, 19] and later in many other machines (see e.g. reviews [20, 48, 74]). These electrode biasing experiments (induced H-modes) have contributed significantly to the understanding of the H-mode phenomenon and of the effects of Er on plasma transport [21].
Besides an important theoretical activity, many experiments have since been performed in the plasma edge
Guido Van Oost
and the SOL of limiter or divertor devices [20, 24, 25, 48]. Imposing electric fields independently of other machine parameters allows to manipulate the edge and SOL profiles and flows, to control impurities and to affect particle and power exhaust [20, 24, 25].
Radial electric fields have been studied in a variety of devices: tokamaks [66, 67], stellarators [68, 69] and other helical devices, reversed field pinches, mirrors, etc. In stellarators [70] where neoclassical transport dominates [71], the transport coefficients depend on Er. A radial electric field limits the excursions of the helically trapped particles due to E x B poloidal rotation, whereby neoclassical transport can be reduced to such an extent that stellarators become viable for a fusion reactor. The present paper concentrates on tokamaks in which E itself without shear cannot contribute
r
to confinement improvement because the ensuing rigid rotation which reduces orbit losses («orbit squeezing») and improves neoclassical transport, has no effect on microturbulence which is regarded as the dominating cause of anomalous transport in auxiliary heated tokamaks. Effects of Er on transport enter only through derivatives of Er.
As outlined by Burrell [21], one of the scientific success stories of fusion research is the development of the Ex B velocity shear turbulence stabilisation model to explain the formation of transport barriers in magnetic confinement devices. This model has the universality needed to explain turbulence reduction and confinement improvement under a variety of conditions in limiter- and divertor tokamaks, stellarators, torsatrons, reversed field pinches, mirror machines, etc.
Further details on radial electric fields and their role in plasma confinement and exhaust can be found in review articles [21, 22] and in the proceedings of Topical Workshops [23, 24, 25, 47].
II. Radial electric fields and rotation
Radial electric field and plasma rotation are connected through the radial momentum balance. Er can be determined from the single ion radial force balance equation (generalised Ohm's law):
1
E,
n, Z, e
~Vm B
(1)
dVEy.B (RBa )2 d Er
dr B dy RBe
where n. is the ion density, Z is the charge number of the ion, e is the electronic charge, P. is the ion pressure, vei and v^ are the poloidal and toroidal rotation velocities, respectively, of the ion species considered; and B0 and B^ are the poloidal and toroidal magnetic fields, respectively. This equation is valid at each point on any given flux surface, and the quantities involved are local quantities (Er itself is not a flux function).
It follows from Eq. (1) that Er is determined by three major driving forces: radial pressure gradient, poloidal and toroidal rotation. Because Er can be influenced by particle-, heat- and angular momentum input, and by changing the current profile (changing B0), various of these terms can be active in various machines with respect to Ex B shear flow reduction of turbulence and transport, which occurs regardless of the plasma rotation direction. This provides the possibility of active control of transport; E x B shear as a control mechanism for turbulence and transport has the major advantage of flexibility, in that the shear can be generated or enhanced in several ways. Particle-, heat-, and momentum transport are not independent of each other, but have a complex coupling. Therefore, research on Er can clarify complex plasma transport mechanisms.
III. Ex B Velocity shear reduction of turbulence
Ex B velocity shear reduction of turbulence in a plasma is a mechanism akin to the interaction between sheared velocity fields and turbulence in fluids. However, in a plasma Ex B velocity and fluid velocity due to Er can be quite different. The fundamental velocity is not the mass velocity, but rather the E x B velocity, the drift velocity at which all particles move - regardless of their charge or mass - and at which turbulent eddies are convected.
The fundamental physics involved in transport reduction is the effect of E x B shear on the growth, radial extent and phase decorrelation of the turbulent eddies. The identification of individual modes responsible for the observed turbulence may not be as important as the knowledge of turbulence drive suppression mechanisms, which provide a direct route to transport control.
An important point in plasmas are the synergistic effects between Ex B velocity shear and magnetic shear. In neutral fluid dynamics sheared velocity is a source of free energy which can drive turbulence through Kelvin-Helmholtz instabilities. In a plasma, shear in the magnetic field prevents coupling of the various modes across the velocity gradient so that they are unable to extract energy from the Ex B velocity shear and grow [21].
E x B flow velocity shear is a universal and robust mechanism for regulating and controlling entire classes of turbulent modes at all radii, and thus for controlling transport.
Turbulence is stabilised by the shear rate wf x in the Ex B flow velocity vE induced by E [26]
(2)
where R is the major radius, B0 is the poloidal magnetic field and ^ is the poloidal flux.
The Ex B shear rate enters quadratically into the various theories; accordingly, its sign is irrelevant. Indeed, H-mode edge barriers have been seen with both signs of Er and its derivative [27]. Equation (2) shows that both Er and Be contribute to the final result; Er/RBe is the toroidal angular speed due to the equilibrium flow driven by Er in standard neoclassical theory, suggesting that the basic shearing is in the toroidal direction. As illustrated in Fig. 1 [21], the differences between the shear in Er/B and Er/RBe have great practical significance; although the former vanishes locally, the latter is significant throughout this plasma which shows confinement improvement across the whole minor radius.
Equation (2) also shows that the shear rate is not constant on a given magnetic flux surface, being significantly larger on the low toroidal field side, where the flux surfaces are more dense (the electric potential being constant on a flux surface) Experimental data on H-modes have indeed demonstrated significant poloidal variation in the effect of E x B shear on turbulence.
Theoretically, there are two points of view [21]. The first (non-linear suppression) is that the turbulent eddies are distorted and the radial transport is reduced if the E x B shear rate exceeds the decorrelation rate of the ambient turbulence in the absence of Ex B shear; this is valid for entire classes of turbulent modes. The second is linear stabilisation, which is mode specific, and therefore the details depend on the turbulence driving mechanisms. The fluctuation spectra are Ex B-Doppler-shifted, and the stabilisation is mainly due to shear in this Doppler shift.
w
100 50
E
I of
ui : -50 :
I i I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_L
SHOT 84713 Time 2080 ms
IV.A. Ex B Shear Effects at the Plasma Edge
The largest data base for testing the Ex B velocity shear theory is due to the plenitude of edge transport barrier studies in spontaneously occurring H-modes (already discovered in ASDEX [12] in 1982) as well as in edge biasing-induced H-modes [48]. The edge transport barrier has been found in tokamaks, helical devices, mirrors and reversed field pinches. An overview of H-mode physics results can be found in Ref. 23.
Fig. 1. Plot of radial electric field E , toroidal angular speed E/RBg, and the Ex B shearing rate defined in Eq. (2) as a function of flux surface label p for a high performance, deuterium VH-mode plasma(see IV) in DIII-D. Here, p is proportional to the square root of the toroidal flux inside a given flux surface. Although the derivative of E vanishes near p = 0,5, the Ex B shearing rate is appreciable across the whole plasma. Plasma conditions are 1,2 MA plasma current, 1,6 T toroidal field, 9,8 MW injected deuterium neutral beam power, and 4,7 x 1019 m"3 line averaged density. Discharge is a double-null divertor [21]
A simple rule of thumb which is also used in comparisons between theory and experiment, is that for turbulence stabilisation wf x B is to be comparable to the linear growth rate of the most unstable mode in the plasma; however, theory and experiment show factors of 2-3 deviation from that rule.
IV. Transport barriers and confinement improvement
As outlined in the review paper of Burrell [21] (see also references therein) the E x B shear stabilisation model was originally developed to explain the transport barrier formation at the plasma edge at the L to H transition. Later, it has been applied to explain the wider edge transport barrier at the H-to VH- (very high) mode transition seen in some tokamaks such as DIII-D and JET, as well as to the core or internal transport barriers (ITB) formed in plasmas with modified (negative, optimised) central magnetic shear (DIII-D, TFTR, JT-60U, JET, ASDEX Upgrade, Tore Supra, etc), and to plasmas with transport reduction across the whole plasma radius (JT-60U and DIII-D). During electrode biasing experiments in TEXTOR the role of E x B flow shear in the confinement improvement and the temporal causality has been clearly established [28, 48, 51].
All these experimental results, some of which will be outlined below, show that the E x B shear stabilisation concept has the universality needed to explain transport barriers at different radii seen in limiter- and divertor tokamaks, helical devices, reversed field pinches, and mirror machines with a variety of discharge and heating conditions and edge biasing schemes.
0,12 I 0,08
E
0>
Q° 0,04
0,00
1450
1460
1470 1480 Time, ms
1490
1500
Fig. 2. The temporal evolution of two L to H transitions in DIII-D, where the L to H transition times are indicated by the vertical lines:
(a) Er from CER (Charge Exchange Recombination spectroscopy) of He II spectra 4 mm outside (crosses) and 8 mm inside (filled circles) the separatrix, showing the increase in Er shear 2 ms before each transition; (b) the main ion temperature gradient /e contribution to E r for the same radii as in (a); (c) two channels of the lithium BES (Beam Emission Spectroscopy) diagnostic showing the earliest rise (upper thick line) and earliest drop (lower thin line) in edge electron density; (d) Da emission from the channel showing the earliest rapid drop [29]
A key prediction of the Ex B velocity shear theory is that it causes the reduction in turbulence and transport. However, causality can be difficult to pin down in spontaneous H-modes. According to Eq. (1), Er can be sustained by plasma rotation and by the ion pressure gradient. In stationary H-modes, it is quite often found [30, 31] that Er can be upheld by the sole pressure gradient, such that the shear might actually be interpreted as the result of the improved confinement, not as its cause. Therefore, establishing the causal link has to rely on the time sequence in discharges where an electric field growth, not or only partly accountable by the ion pressure gradient, precedes the confinement improvement. The DIII-D team, using high time and space resolved measurements, could firmly reveal such a growth within a few milliseconds prior to an L to H transition. The temporal evolution (Fig. 2) of a pair of representative L
to H transitions in DIII-D [29] shows that the trigger to the L to H transition is the v x B term in Eq (1), and not the main ion pressure gradient. For these same discharges the fluctuation data [29] from far-infrared (FIR) scattering and reflectometry show that the radial electric field shear increases before the fluctuation suppression, consistent with increasing Er shear as the cause of the turbulence reduction.
Another way to approach the causality question is to look at H-modes produced by plasma biasing [48] (see also V) In TEXTOR radial electric fields were externally imposed by means of electrode biasing [19] to study the role of Ex B flow shear in improved confinement [28]. Whereas the ion pressure gradient is normally negative in a tokamak plasma, and according to Eq. (1) therefore creates a negative Er, the external imposition of a positive electric field allows us to avoid the afore mentioned, for causality studies inopportune link between Er and ion pressure gradient (a steepening of the pressure gradient would tend to counteract the imposed positive Er). The spiky behaviour on VE and other signals is due to ringing of the power supply. Under these experimental conditions changes in the local density gradient Vn can be interpreted as diffusivity changes and a zone of enhanced Vn is thus a measure for a transport barrier. A particle transport barrier is found to be built up as the electric field gradient increases (prior to the occurrence of bifurcation phenomena, t = 2,1 s). Figure 3 shows Er profiles measured with a rake probe for
Guido Van Oost
two times during the bias voltage ramp together with the corresponding density profiles which are seen to develop a pronounced steepening in the radial region where the electric field is located (the electrical layer). To characterise this steepening, we compare the local Vn with its pre-biasing value, a measure of which is ^ (r, t) = Vn (r, t)/Vn (r, t0). The temporal evolution of the plasma parameters plotted in Fig. 4 shows the two regions of enhanced where the density gradient steepens (43,9 and 45,0 cm; toroidal belt limiter ALT-II at 46,2 cm). Generally, the radially innermost maximum is 2 to 4 times higher than the outer one. The figure also demonstrates the temporal link between Vn and VE (and not Er or its curvature). Probe measurements of electrostatic turbulence have clearly demonstrated E x B shear flow turbulence stabilization in the double shear layer [33, 48, 51]. In conclusion, the experimental evidence points to Ex B shear as the decisive element for the confinement improvement (see Ne in Fig. 4, the main energy gain in these discharges being density related [19]): there is a strong spatial correlation between VE and transport barrier formation, the temporal causality is clearly established (VE leads Vn), and the magnitude of the shear at which the confinement improvement occurs, agrees rather well with theoretical models, both in the value of the critical shear and in its global shear dependence [32]. This critical VE (about 50 V/cm2) required to create a transport barrier seems to be the same in different machines (see also IV.B.III).
TEXTOR-discharge #67307
TEXTOR #67307
500 400 300 200 100 0
-100 3.0e +12
2.0e +12
1.0e +12
0.0e + 00
1 1 1 1 1 - - i i i 1 i i i t= 2.635 s 1 1 1 a -
- -
■ -
■ -
■ -
- -
1 1 b
■ \ \ \ - n ,.t = 2. ,.t = o. 535 s 795 s "
■ X xs -
■ Electrc , , , }de ,,, ALT-II , , , , , , ——
41 42 43 44 45 46 47 48 Radius, sm
Fig. 3. Radial profiles of (a) electric field and (b) electron density at two times during a voltage ramp (see Fig. 4). The electric field is pointing outward for positive biasing in contrast to an unbiased ohmic discharge [28]
600 400 200 0 0.6
0.5 1.6 1.4 1.2 1.0
2.0 1.0
0.0
-1-1-1-1— H-1-1-1-
- % yl
---- _ j* ¿Mf \ _
i 1 i 1 i i i i i
1 1 1 1 " -Vf, (r = 43.9 cm) i 1 i 1 i
----VE. (r = 45.0 cm) ; i i * i.
--Ç-1 (r = 43.9 cm) H-i-1-1—4-t==
---^ -1 (r = 45.0 cm)
____J '"N--J--N y
, 1 , 1 V
1.0
1.5 2.0 2.5 3.0 3.5 Time, s
Fig. 4. Time traces of the electrode voltage VE, electric field maximum E , total number of electrons N,., H -light, particle confinement
r max tot' a 0 ' r
time Tp, field gradient maxima VE and the relative change of the density gradient ^ - 1 at the corresponding radii [32]
IV.B. Ion Internal Transport Barriers (/-ITBs)
Exciting results have been obtained since 1995: reduced transport in the central region of tokamak plasmas and concomitant record fusion performance in DIII-D [34, 42], JT-60U [35] and JET [36]. The example of JT-60U in Fig. 5 demonstrates that impressively steep core gradients can be produced [32]. The formation of an ion ITB dramatically reduces ion heat and particle flux from the core
(sub-neoclassical ion thermal diffusivity obtained). In as much as neoclassical transport is usually considered to be as the minimum transport possible in a tokamak, these results represent a dramatic improvement in confinement and performance. Furthermore, the strong pressure gradient associated with ITBs drives a bootstrap current which can substantially contribute to overcome the limited pulse length in tokamaks.
e
Ol
O
r.* 4
>
fi
1 10
T-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1—
0.0
0.4
0.8
2.6 3.0
Time, ms
1 1 1 i d
At ERS transition time (2.6 s)
ERS / -
/"V /
rs nn. ^
\\ X
NN i i i i
0.0
0.2
0.4
r/a
0.6
0.8
1.0
Fig. 5 (left). Radial profiles for a high performance, negative central shear discharge in JT-60U:
a - electron density n measured by Thomson scattering. Solid curve is obtained by fitting interferometer data (one tangential and two vertical chords); b - T. from charge exchange recombination spectroscopy and T . In the T profile, closed points are measured by Thomson scattering and open points are by electron cyclotron emission; c - q-profile from motional Stark effect measurements [21, 37]
Although the target plasmas have modified central magnetic shear, it appears that the key physics for the transport reduction is the E x B velocity shear as illustrated in Fig. 6 where RS (reverse shear, unimproved confinement) and ERS (enhanced reverse shear, improved confinement) transition data in TFTR [38] are compared; the difference is the time evolution of the Ex B shear rate. Other evidence that negative magnetic shear is not the key factor in core (ion) transport barrier formation is that in most machines with negative magnetic shear no confinement improvement is reached until a power threshold is observed which is also consistent with the idea of Ex B shear stabilisation.
Fig. 6 (right). Plots from two discharges in TFTR showing that the q-profile is identical at the time of the RS-ERS transition:
a - Deuterium neutral beam input power; b - central electron density; c -density peaking factor all plotted as a function of time; d - q-profile at the time of the transition. The RS plasma trace is the dashed line [38]
A key causality test has been performed on TFTR [39]. The angular momentum input to the plasma was changed by varying the fraction of the beam power which came from neutral beams injecting parallel to (co) and antiparallel to (counter) the plasma current, showing clean temporal correlation between the decrease in E x B shear rate and the increase in the fluctuations and transport.
Poloidal rotation dynamics has been investigated in ITB regions of JET [52] using CER spectroscopy of carbon ions. The turbulence reduction and ITB formation (see also IV.B.IV) and sus-tainment can differ from machine to machine, and are most likely controlled by a combination of two or more of the following main mechanisms: (1) ex B flow shear; (2) magnetic shear and low order rational q-surface; (3) influence of the ratio Ti/ Te or strong electron density gradients (e.g. due to pellet injection) on instability growth rates; and (4) turbulence stabilization by self-generated poloidal ExB zonal flows [61].
IV.C. Electron Internal Transport Barriers (e-ITBs)
Research in the T-10 and TEXTOR devices [60] has concentrated on understanding the physical mechanisms that are responsible for the generation of electron internal transport barriers (e-ITBs) and also on finding out in which way they are related to the concept of profile consistency, in which the plasma pressure and temperature profiles have a tendency to organize themselves [53, 54] into an 'universal' profile shape, in agreement with the plasma minimum free energy principle. If Vp exceeds a certain critical value, instabilities connected with the pressure gradient will counteract the formation of an even steeper gradient. The radial distribution of transport coefficients is determined by the necessity to maintain the self-consistent pressure profile under different external impacts.
The property of the plasma to maintain the (optimal or 'canonical') profile of some plasma parameters is called «self-con-sistency» or simply self-organization of the plasma [54]. The basic iea is that transport in tokamak plasmas is dual in nature. The profiles of the plasma parameters and the this is fluxes of energy and particles are separated [54]. This is the real meaning of self-organization.
It has been shown 55 that e-ITBs are formed when dq/dr is low in the vicinity of rational magnetic surfaces with low m and n values. For the investigation of effects bound with ITB formation experiments with a rapid plasma current ramp up were performed in T-10. In this case, due to (Pp + l./2) ~ 1//p2 a rapid change of the magnetic surface densities in the central part of plasma takes place, while current penetration in this region occurs only after t > 50 ms. Thus, confinement changes observed in the plasma core are the result of a magnetic surface density change only [56].
Experiments have been conducted to investigate the interplay between the formation of electron Internal Transport Barrier (e-ITB) and the maintenance of self-consistent plasma profiles under the action of Electron Cyclotron Resonance Heating and Current Drive ECRH/ECCD. A joint analysis of T-10 and TEXTOR experimental results enabled to analyze effects bound with plasma self-organization. It was shown that the plasma pressure profiles obtained in different operational regimes and even in various tokamaks may be represented by a single typical curve, called the self-consistent pressure or canonical profile, also often referred to as profile resilience or profile stiffness [54].
The investigation of self-consistent pressure profile effects was carried out under different experimental conditions, such as regimes with plasma density near the Greenwald limit and regimes with deuterium pellet injection. It can be concluded that the effect takes place in a wide region of plasma density up to that, which leads to disruption. The conditions described by this self-consistent profile are realized in a very short time, less than the experimental time resolution At > 2-4 ms. During ECRH it is realized by a plasma density redistribution: ne decreases in the plasma heating zone. This implies that the famous «density pump out» is the result of plasma self-consistent organization. Experimentally this means that, when one tries to distort the self-consistent pressure profile, the heat (cold) pulse spreads much more quickly than can be expected from transport coefficients, calculated from a radial power balance. Since the self-consistency effect is exactly valid for the plasma pressure profile it is suggested that it is determined by the density of the turbulent cells, which have to be located at rational surfaces. The pressure gradient changes the distance
Guido Van Oost
between such cells and hence the turbulent flux. However, in ITB regions Vp can largely exceed that from the self-consistent pressure profile.
The T-10 results, as well as recent results from JET [57] and DIII-D [58] show that ITBs are formed in the vicinity of low order rational surfaces, where the gap between rational surfaces is large. The ITB formation is the result of the regulation of turbulent flux by changing the contact between the turbulent cells. It seems that so-called «hybrid» regimes can also be explained by such hypothesis: magnetic shear close to zero leads to more rarified rational surfaces, especially, when they are located near low order rational surface. Nevertheless, the plasma confinement has to depend on the width of the turbulent cells and its dependence on plasma parameters. Rational surfaces play a key role in the establishment of ITBs, as has been observed in stellarators, too [59]. However, this does not exclude a possible supporting role of Ex B shear in ITB formation near rational surfaces (interaction between neighbouring cells) [60]. Research on DIII-D and gyrokinetic simulations [58] hint at possible synergy between Ex B shear and effects of rational surfaces. Large profile corrugations in electron temperature gradients at lowest-order singular surfaces lead to the build-up of a huge zonal flow Ex B shear layer which provides a trigger for the low power ITB observed in DIII-D.
IV.D. Transport Reduction across the whole Plasma
The combination of H-mode edge confinement improvement and core confinement improvement has been achieved (see e.g. JT-60U [40] and DIII-D [34]). In these discharges, the transport is reduced throughout the plasma and the radial profiles show no sign of a local transport barrier. In JT-60U four types of transport barriers have been observed [41] as shown in Fig. 7. The high confinement modes are characterized by the combination of an ITB with an H-mode. Most of these high performance results were obtained transiently. Substantial progress has since been made to extend these discharges towards steady state [36, 41, 42].
V. Discussion and conclusions
The importance of radial electric fields is now widely recognised. It has been demonstrated in limiter- and divertor tokamaks, helical devices and mirror machines with a variety of discharge- and heating conditions as well as edge biasing schemes that improved confinement is often associated with strongly radially varying profiles of E, and that ExB velocity shear turbulence stabilisation is a robust and universal mechanism which plays a major role in the formation and sustainment of transport barriers in magnetic confinement devices. The experimental results show good qualitative agreement with theory which however still needs considerable improvement. An improved comparison between experiment and theory requires the development or improvement of plasma diagnostics such as for (1) Er measurements (see review [49]) using the motional Stark effect [43], the HIPB (heavy ion beam probe [44]), and novel types of probes [50]; (2) direct measurements of fluctuation-driven fluxes in the core plasma; (3) 2D plasma flow velocity field measurements using HIPB and high-speed frame cameras; (4) ion rotation measurements using CXRS with high time and space resolution; (5) microwave backscattering and correlation reflectometry to investigate plasma macroscopic and turbulent motions, such as the turbulence-generated zonal flows [61].
Internal Barriers
Central Pellet q = 1 surface
High-pp mode weak positive shear
Reverse Shear mode negative shear
T-1-1-1-1-1-1-1-1-1-rn-1-1-1-1-1-1-1--20 -1-1-1-1-1-1-1-1-1—1-1-1-1-r
Edge Barrier
H-mode
High-ßp ELMy H-mode
> OJ
10
Cr
1 I I 1 I I i i i i i i i i I i i 10
ITB ETB
"
1 foot
- T.
q -
' e 2
...... ..... ..... L-Sr o
T—J-1—I—I-
c
"W
r/a
RS ELMy H-mode
r/a
Fig. 7. Transport barriers observed in JT-60U. Internal transport barriers (ITB) in:
a - the central pellet injection mode with the ITB foot at the q = 1 surface; b - the high mode with the ITB foot (pfoot) in weak positive magnetic shear; c- the reversed shear (RS) mode with pfoot in negative or zero magnetic shear; d- edge transport barrier (ETB) in ELM-free H-mode; e, f - show profiles of T, T and q in the high 3 ELMy H-mode and the RS ELMy H-mode with ITB and ETB
There also exist synergistic effects between Ex B velocity shear and magnetic shear, although the key physics for the (ion) transport reduction is the E x B velocity shear mechanism [45, 48], which can be operational in various regions of the plasma because there are a number of ways to change the radial electric field. These synergistic effects and the extension of high performance discharges towards steady state are presently investigated on large tokamaks like JET. Ion thermal and particle diffusivities at or below the standard neoclassical values have been observed.
There is a clear relation between the generation of electron internal transport barriers and the concept of profile consistency developed by Yu.N. Dnestrovskij, in which the plasma pressure and temperature profiles have a tendency to organize themselves into an 'universal' profile shape, in agreement with the plasma minimum free energy principle. The only possibility of organizing a more peaked pressure profile than the self-consistent profile is the establishment of pronounced e-ITBs; the hypothesis is that it is determined by the density of turbulent cells in the vicinity of low-order rational surfaces.
Edge biasing experiments [48, 60, 62] have provided new and complementary evidence on the physics of the universal mechanism of Ex B velocity shear stabilization of turbulence, concomitant transport barrier formation and radial conductivity. In TEXTOR the causality between transport reduction and
induced electric fields in the edge has been for the first time clearly demonstrated. The high electric field gradients have been identified as the cause for the quenching of turbulent cells. The scaling of plasma turbulence suppression with velocity shear has been established. A reduction of the anomalous conducted and convected heat fluxes resulting in an energy transport barrier has been measured directly. In CASTOR the biasing electrode has been placed at the separatrix in a non-intrusive configuration which has demonstrated strongly sheared electric fields and consequent improvement of the global particle confinement, as predicted by theory. The impact of sheared Ex B flow on edge turbulent structures has been measured directly using a comprehensive set of electrostatic probe arrays as well as emissive probes. In CASTOR a periodic collapse of the transport barrier induced by edge biasing is observed: above a threshold bias voltage critical gradients are periodically achieved both on floating potential and plasma density, and followed by a relaxation phase. The transport barrier is periodically created and relaxes with a frequency of about 10 kHz. The observed relaxation events are found to be associated with a stream of density radially propagating towards the wall.
In T-10 edge biasing has clearly improved the global performance of ECR heated discharges. Reflectometry and heavy ion beam probe [72] measurements show the existence of a narrow plasma layer where strong changes of plasma electric potential
[73], turbulence levels and plasma rotation take place. On IST-TOK, the influence of alternating positive and negative electrode and (non-intrusive) limiter biasing has been compared. Electrode biasing is found to be more efficient in modifying Er and confinement, limiter biasing acting mainly on the SOL. In the RFX reversed field pinch it has been demonstrated that also in RFPs biasing can increase the local Ex B velocity shear in the edge region, and hence substantially reduce the local turbulence driven particle flux. Limiter biasing experiments in the TJ-II stellarator [63] have shown electric field induced improved confinement.
The influence of a magnetic perturbation field, generated by the Dynamic Ergodic Divertor (DED) [64], on the turbulence and transport properties has been studied and compared to plasmas without such a field perturbation. The external magnetic field breaks up the magnetic field lines structure and causes an ergodization of the plasma edge. The strength and radial range of the perturbation field can be widely varied. A main effect of the DED is the modification of the radial electric field. The ergodization of the magnetic field lines leads to an increased electron loss rate which charges the plasma edge more positively. The application of the DED increases the rotation in the SOL, where the original rotation is in the ion diamagnetic drift direction. Since the rotation at radii smaller than the limiter radius is in the electron diamagnetic drift direction, the DED slows down the rotation. The inversion point of the radial electric field (as well as the poloidal rotation velocity) is shifted further in-side.The combination of counter-current neutral beam injection and the DED can lead to the formation of a transport barrier at the plasma edge [65]. The turbulence rotation is decreased at the barrier, which again demonstrates the braking effect of the DED. The acceleration of rotation by counter neutral beam injection and braking by the DED yields an increase in the velocity shear at r/a = 0.9. At the barrier, the level of density fluctuations is constant, the turbulence decorrelation time is increased and the turbulence wavelength is decreased. The evaluation of turbulent diffusion using a random walk model yields the reduction of transport by about 50 % within the barrier.
Acknowledgement
The author acknowledges for the partial financial support from MEPhI in the framework of the Russian Academic Excellence Project.
References
1. Budker T. in Plasma Physics and the Problem of Controlled Thermonuclear Reactions, edited by M.A. Leontovich, Pergamon Press, New York, 1, p. 78 (1951).
2. Stix T. Phys. Fluids 14, 692 (1971).
3. Roth J.R. et al. Phys. Rev. Letters, 22, 1450 (1978).
4. Roth J.R. Proc. IAEA Technical Conference Meeting on Tokamak Plasma Biasing. Montreal. Vienna: IAEA, 1992. P. 132.
5. Gorman J.G. and Rietjens L.H. Phys. Fluids, 9, 2504 (1966).
6. Strait E.J. Nucl. Fusion, 21, 943 (1981).
7. Taylor R.J. et al. In Plasma Physics and Controlled Thermonuclear Research, 3 (IAEA Vienna), p. 251 (1982).
8. W7-A team et al. Proc. 3rd Joint Varenna-Grenoble Int. Symp. Heating in Toroidal Plasma, Grenoble, 2, p. 813 (1982).
9. Hosea J.C. et al. Phys. Rev. Letters, 30, 839 (1973).
10. Razumova K.A. Plasma Physics and Controlled Fusion, 26, 37 (1984); and Bugarya V.I., Gorshkov A.V., Grashin S.A. et al. Nucl. Fusion, 25, 1707 (1985).
11. Murakami M. et al. In Plasma Physics and Controlled Nuclear Fusion Research, 1 (IAEA Vienna), p. 87 (1984).
12. Wagner F. et al. Phys. Rev. Letters, 49, 1408 (1982).
13. Taylor R.J. et al. Phys. Rev. Letters, 63, 2365 (1989).
Guido Van Oost
14. Groebner R.J., Burrell K.H. and Seraydarian R.P. Phys. Letters, 64, 3015 (1990).
15. Itoh S.-I. and Itoh K. Phys. Rev. Letters, 60, p. 2276 (1988); K. Itoh, S.-I. Itoh, A. Fukuyama, «Transport and Structural Formation in Plasma», I.O.P. Publishing, Bristol (1999).
16. Shaing K.C., Crume E.C. JR. and Houlberg W.A. Phys. Rev. Letters, 63, 2369 (1989).
17. Tendler M. Plasma Physics and Controlled Fusion, 39, B371 (1997).
18. Weynants R.R. et al. Proc. 17th Eur. Conf. On Controlled Fusion and Plasma Physics, Amsterdam, 1, (Europhysics Conf. Abstr. 14B), p. 287 (1990).
19. Weynants R.R., Van Oost G., et al. Nucl. Fusion, 32, p. 837 (1992).
20. Weynants R.R. and Van Oost G. Plasma Physics and Controlled Fusion, 35, B177 (1993).
21. Burrell K.H. Phys. Plasmas, 4, 1499 (1997).
22. Ida K. Plasma Physics and Controlled Fusion, 40, 1429 (1998).
23. Proceedings of the Technical Committee Meeting on H-mode Physics, Kloster Seeon, Germany, September 22-24, 1997, in Plasma Physics and Controlled Fusion 40, Nr. 5.
24. Proceedings of the Workshop on «Role of Electric Fields in Plasma Confinement and Exhaust», Prague, July 6-8, 1998, in Czechoslovak J of Phys., 48, Suppl. S3, 1998; Proceedings of the Second Europhysics Workshop on «Role of Electric Fields in Plasma Confinement and Exhaust», Maastricht, June 19-20, 1999, Czechoslovak J. of Phys., 49, Suppl. S3, 1999; Proceedings of the Third Europhysics Workshop on «Role of Electric Fields in Plasma Confinement and Exhaust», Budapest, June 18-19, 2000, in Czechoslovak J. of Phys., 50, Nr. 12, 2000; Proceedings of the Fourth Europhysics Workshop on «Role of Electric Fields in Plasma Confinement and Exhaust», Funchal, Madeira, June 24-25, 2001, in Czechoslovak J. of Phys., 51, Nr. 10, 2001; Proceedings of the Fifth Europhysics Workshop on «Role of Electric Fields in Plasma Confinement and Exhaust», Montreux,, June 23-24,
2002, in Czechoslovak J. of Phys., 52, Nr. 10, 2002.
25. Proceedings of the Workshop on «Electric Fields, Structures and Relaxation in Edge Plasmas», St. Petersburg, June 13-14,
2003, in Czechoslovak J. of Phys., 53, Nr. 10, 2003; Proceedings of the Workshop on «Electric Fields, Structures and Relaxation in Edge Plasmas», Nice, October 26-27, 2004, in Czechoslovak J. of Phys., 55, Nr. 3, 2005; Proceedings of the Workshop on «Electric Fields, Structures and Relaxation in Edge Plasmas», Tarragona, July 3-4, 2005, in Czechoslovak J of Phys., 55, Nr. 12, 2005; Proceedings of the Workshop on «Electric Fields, Structures and Relaxation in Edge Plasmas», Rome, June 26-27, 2006, in Czechoslovak J. of Phys., 56, Nr. 12, 2006; Proceedings of the Workshop on «Electric Fields, Structures and Relaxation in Plasmas», Warsaw, July 8-9, 2007.
26. Hahm T.S. and Burrell K.H. Phys. Plasmas, 2, 1648 (1995).
27. Burrell K.H. Plasma Physics and Controlled Fusion, 36, A291 (1994).
28. Jachmich S. et al. Plasma Physics and Controlled Fusion, 40, 1105
(1998).
29. Moyer R.A. et al. Phys. Plasmas, 2, 2397 (1995).
30. Hugill J. Plasma Physics and Controlled Fusion, 36, B173 (1994).
31. BurrelL K.H. et al. Phys. Plasmas, 1, 1536 (1994).
32. Weynants R.R., Jachmich S. and Van Oost G. Plasma Physics and Controlled Fusion, 40, 635 (1998).
33. Boedo J. et al. Czechoslovak J of Phys., 48, Suppl. S3, 99 (1998).
34. Lazarus E.A. et al. Phys. Rev. Letters, 77, 2714 (1996).
35. Ushigusa K. and the JT-60 team, Plasma Physics and Controlled Nuclear Fusion Research, 1 (IAEA Vienna), 37 (1996).
36. Gormezano C. in Plasma Physics and Controlled Fusion, 41, B367
(1999).
37. Fujita T. and the JT-60 team, Phys. Rev. Letters, 78, 2377 (1997).
38. Mazzucato E. et al. Phys. Rev. Letters, 77, 3145 (1996).
39. Synakowski E. et al. Phys. Plasmas, 4, 1736 (1997).
40. Koide Y. et al. Plasma Physics and Controlled Fusion, 38, 1011 (1996).
41. Kamada Y. and the JT-60 team, Plasma Physics and Controlled Fusion, 41, B77 (1999).
42. Greenfield C.M., Murakami M., Zeng L. Fusion Engineering and Design 81, 2807 (2006).
43. Zarnstorff M.C. et al. Phys. Plasmas, 4, 1097 (1997).
44. Melnikov A.V. et al. Heavy ion beam probing - diagnostics to study potential and turbulence in toroidal plasmas Nucl. Fusion 57 (2017) 072004.
45. Razumova K.A. Czechoslovak J. of Phys., 49, Suppl. S3,7 (1999).
46. Van Oost G. et al. J. Plasma Fusion Res. Series, 4, 29 (2001).
47. Van Oost G. and Tendier M. Plasma Physics and Controlled Fusion, 44, 1761 (2002).
48. Van Oost G. et al. Plasma Physics and Controlled Fusion, 45, 621 (2003), and references therein.
49. Donné A.J.H., Melnikov A.V., Van Oost G. Czechoslovak J. of Physics, 52, 1077 (2002).
50. Van Oost G. "Advanced probe edge diagnostics for fusion devices", Transactions of Fusion Technology (February 2012) 61, 2T, 365-375.
51. Jachmich S., Van Schoor M., Weynants R.R. Proc. 29th Eur. Conf. On Controlled Fusion and Plasma Physics, Montreux, (Europhysics Conf. Abstr. 26B), O-1.01 (2002).
52. Crombe K. et al. Phys. Rev. Lett., 95, 155003 (2005).
53. Tendler M., Van Oost G. and Stöckel J. Comments on Modern Physics, 2, N6,C 203 (2002).
54. Dnestrovski YU.N. Self-Organization in hot Plasmas (The Canonical Profile Transport Model), Springer International Publishing (2014), and YU.N. Dnestrovski et al. Nucl. Fusion, 46 953 (2006).
55. Razumova K.A. et al. Nucl. Fusion, 44, 1067 (2004).
56. Razumova K.A. et al. Plasma Phys. Control. Fusion, 48, 1373 (2006).
57. Baranov YU.F. et al. Plasma Phys. Control. Fusion, 46, 1181 (2004).
58. Waltz R.E. et al. Phys. Plasmas, 13, 052301(2006).
59. Brakel R. et al. Nucl. Fusion, 42, 903 (2002).
60. Van Oost G. et al. Plasma Physics and Controlled Fusion, 49, A29 (2007).
61. Special issue on Experimental studies of zonal flow and turbulence, Plasma Phys. Control. Fusion, 48, 1181 (2006).
62. Van Oost G. et al. Nucl.Fusion, 47, 378 (2007), and references therein.
63. Hidalgo C., Pedrosa M.A., et al. Plasma Physics and Controlled Fusion, 46, p. 287, (2004).
64. Finken K.H. et al. Fusion Eng. Des., 37, 335(1997).
65. Kraemer-Flecken A. et al. Nucl.Fusion, 46, 730 (2006).
66. Бугаря В.И., Горшков А.В., Грашин С.А. и др. Электрический потенциал и скорость тороидального и полоидального вращения в токамаке // Письма в ЖЭТФ. 1983. Т. 38, № 7. С. 337-341.
67. Melnikov A.V., Andreev V.F., Grashin S.A., et al. Nucl. Fusion, 53, 093019 (2013).
68. Melnikov A.V., Hidalgo C., Eliseev L.G., et al. Nucl. Fusion, 51, 083043 (2011).
69. Мельников А.В. Электрический потенциал в плазме тороидальных установок. Монография. М.: НИЯУ МИФИ, 2015. 260 с.
70. Melnikov A.V. et al. Plasma Potential Evolution Study by HIBP Diagnostic During NBI Experiments in the TJ-II Stellarator, Fusion Science and Technology, 51, № 1, (2007), pp. 31-37.
71. Мельников А.В., Дябилин К.С., Елисеев Л.Г., Лысенко С.Е., Днестровский Ю.Н. Измерения и моделирование электрического потенциала в стеллараторе TJ-II // ВАНТ. Сер.: Термоядерный синтез. 2011. № 3. С. 54-73.
72. Dnestrovskij Yu.N., Melnikov A.V., Krupnik L.I., and Nedzelskij I.S. Development of Heavy Ion Beam Probe Diagnostics // IEEE Trans. Plasma Sci. 1994. V. 22, No. 4. P. 310-331.
73. Melnikov A.V. et al. Plasma Potential Measurements by the Heavy Ion Beam Probe Diagnostic in Fusion Plasmas: Biasing Experiments in the TJ-II Stellarator and T-10 Tokamak // Fusion Science and Technology. 2004. 46. Pp. 299-311.
74. Мельников А.В. Исследование электрического потенциала термоядерной плазмы с магнитным удержанием // Computational nanotechnology. 2017. Вып. 2. С. 13-23.
