Ebook Fusion physics Part 2

(BQ) Part 2 book Fusion physics has contents: Plasma heating and current drive by neutral beam and alpha particles, plasma–wall interactions, helical confinement concepts, inertial fusion energy, the broader spectrum of magnetic configurations for fusion.

PLASMA HEATING AND CURRENT DRIVE BY NEUTRAL

BEAM AND αLPHA PARTICLES

5.1.1 Basic processes of neutral beam injection

The purpose of plasma heating is to raise the plasma temperature enough

to produce a deuterium and tritium reaction (D + T → 4He + n) The required

plasma temperature T is in the range of 10–30 keV Since the high temperature

plasma is confined by a strong magnetic field, injection of energetic ions from outside to heat the plasma is difficult due to the Lorenz force The most efficient way to heat the plasma by energetic particles is to inject high energy “neutrals” which get ionized in the plasma Neutral beam injection (NBI) with a beam energy much above the average kinetic energy of the plasma electrons or ions is used (beam energy typically ~40 keV – 1 MeV) This heating scheme is similar

to warming up cold water by pouring in hot water

There are two types of neutral beam, called P-NBI and N-NBI (P- and N- means “positive” and “negative”, respectively) P-NBI uses the acceleration of positively charged ions and their neutralization, while N-NBI uses the acceleration

of negative ions (electrons attached to neutral atoms) and their neutralization Details are given in NBI technology Section 5.2 The first demonstration of plasma heating by P-NBI was made in ORMAK [5.1] and ATC [5.2] in 1974, while that by N-NBI was made in JT-60U [5.3] for the first time in 1996 ITER has also adopted the N-NBI system as the heating and current drive system with

a beam energy of 1 MeV Figure 5.1 shows a typical bird’s eye view of a tokamak with N-NBI and N-NBI (JT-60U)

Since the magnetic confinement system is a torus and the tokamak has a toroidal plasma current, there are three injection geometries, namely co-tangential, counter-tangential and perpendicular injection, as shown schematically in Fig 5.2 “Co-” means that beam is injected parallel to the toroidal plasma current, while “counter” means that beam is injected anti-parallel to the plasma current “Perpendicular” means that beam is injected (nearly) perpendicular to

the magnetic field A plasma can be heated in all injection geometries In addition

to plasma heating, co-injection can also drive the plasma current (see Sections 5.1.7–5.1.9) and also drive co-toroidal rotation through its momentum input, while counter-injection can drive a counter-plasma current and counter-toroidal rotation

FIG 5.2 Schematics of tokamak plasma geometry and NBI beam orientations.

After the injection, beam neutrals are ionized through various atomic processes such as charge exchange, ionization by ions and ionization by electrons, which will be described in detail in Sections 5.1.2 and 5.1.3 After the ionization, there is some possibility of re-neutralization and loss of fast ions due to charge exchange with residual neutrals in the plasma The main sources of neutrals in a high temperature plasma are edge warm neutrals and halo neutrals The source

of edge neutrals is neutrals from the wall and divertor, while the source of halo neutrals is charge exchange processes between the neutral beam and bulk plasma ions

FIG 5.1 Bird’s eye view of tokamak with P-NBI and N-NBI (JT-60U) [5.4] Reprinted from Ref [5.4] Copyright (2011), IOP Publishing Ltd.

Ionized ions are magnetically trapped in the plasma and their orbit follows the magnetic surface with only slight deviation There are two types of particle orbits, namely, passing particles and trapped particles (see Section 5.1.5) The magnetic field should be strong enough to confine these energetic ions until they transfer their energy to plasma electrons and ions The tokamak system

is axisymmetric but real machines have small non-uniformities in the toroidal direction because of the finite number of toroidal field coils; this is called toroidal field ripple This ripple causes loss of fast ions The detailed physics of particle orbits in a tokamak will be described in Sections 5.1.5 and 5.1.10

The energy transfer from ionized beam fast ions to thermal ions and electrons is basically through classical Coulomb collision processes The basic processes of Coulomb collision between fast ions and a thermal plasma are

slowing down and diffusion in the velocity space If the beam energy W b is

sufficiently high (W b > 15T e,), fast ions transfer their energy mainly to electrons,

while more energy is transferred to ions when W b< 14.8 for hydrogen and

W b < 19 for deuterium Details of classical beam–plasma Coulomb interactions will be given in Section 5.1.4

5.1.2 Physics of ionization of injected neutral beam

Here we discuss the basic atomic processes which are important during the ionization of a neutral beam in a high temperature plasma For simplicity

we consider the case of injection of a deuterium neutral beam into an electron–deuterium impurity plasma The processes are direct ionization of the ground state (1s) of the deuterium neutral beam through charge exchange (Cx) with bulk ions, ionization by ion impact, ionization by impurity and ionization by electron impact

Decay of the neutral beam intensity I b (t) is governed by the processes given

in Table 5.1

TABLE 5.1 BASIC ATOMIC PROCESSES DURING IONIZATION OF A

NEUTRAL BEAM IN A HIGH TEMPERATURE PLASMA

(The subscript ‘b’ stands for beam, and D and A for the deuterium and

impurity species, respectively)

Charge exchange Db0(1s) + D+ → Db+ + D0

Ionization by ions Db0(1s) + D+ → Db+ + D+ + e

Ionization by impurities Db0(1s) + Az+ → Db+ + Az+ + eImpurity Cx Db0(1s) + Az+ → Db+ + A(z–1)+

Ionization by electrons Db0(1s) + e → Db+ + 2e

Cx by impurities, the electron ionization rate coefficient and the beam speed, respectively Since the pioneering work of Riviere [5.5] on ionization of a neutral beam in a high temperature plasma, extensive efforts have been made to accumulate atomic data for fusion under the auspices of the IAEA

The analytical equations for the ionization cross-sections cx and i

of ground state hydrogen isotopic atoms by hydrogen isotopic ions under the condition of vth i, vb are given by Janev and Smith [5.6] (Table 5.2) Since the beam speed is dominant in the relative speed (vr i,  vb vth i, ~ vb), the cross-sections cx and i are simply functions of energy/mass number (u = W b /A b) and shown for the case of a deuterium beam in Fig 5.3

FIG 5.3 Beam energy dependence of charge exchange, ion, impurity and electron ionization cross-sections (calculated from Table 5.2).

Charge exchange is the dominant process in the low energy regime (W b /A b

< 45keV) and ionization by hydrogen ions is dominant in the high energy regime

(W b /A b > 45 keV) A fusion plasma is always accompanied by some impurities

such as carbon and helium The cross-section for ionization by impurities z

includes both charge exchange and ionization and scales as z ~Z W A Z f( b/ b/ )

(where Z f is the Z of the fast ions) as found by Olson [5.7] Janev [5.8] gave an

analytical formula for z (see Table 5.2), which is consistent with various charge state impurity measurements (He, C, O, Fe) In Fig 5.3, z for fully stripped

carbon (Z = 6) is shown Although the cross-section z is much larger than

  , its contribution to the stopping cross-section is comparable or smaller

if the impurity content is small The relative speed of ionization processes by electrons is dominated by the electron speed (vr e,  vbvth e, ~ vth e, ) So the reaction has to be averaged over the Maxwellian electrons Janev [5.9] gave an analytical formula for the electron ionization rate coefficient (see Table 5.2) This

rate coefficient becomes a maximum just above 0.1 keV and decreases with T e The contribution to the stopping cross-section from this process is < e ev / vb> (where ve stands for the electron velocity and for the average over velocity)

and is shown in Fig 5.3 for T e = 1 keV and 10 keV

TABLE 5.2 CROSS-SECTION AND RATE COEFFICIENT FORMULAS Charge exchange

[5.6]

s cx[m2]=[a1ln(a2/u+a6)]/[1+a3u+a4u3.5+a5u5.4];

[an]=[3.2345 × 10–20, 235.88, 0.038371, 3.8068 × 10–6, 1.1832 × 10–10, 2.3713]

and the e-folding length  2.5 m, which is comparable with the plasma minor

radius of ITER and the DEMO reactor But the above discussion is based on the ionization from the ground state and the inclusion of multi-step ionization through excited states changes the situation This effect will be discussed in the next section.

5.1.3 Multi-step ionization and Lorenz ionization

In the previous section, we considered ionization only from the ground state Ionization can also occur from excited states (n = 2–6, ) In this case,

we have to consider first excitation from the ground state (for example, D(1s) →  D*(2s,2p)), and then ionization from an excited state (for example, D*(2s,2p) →  D+) So, the ionization process becomes “multi-step” ionization (MSI) This MSI is important for high energy beams, especially those for ITER (1 MeV) and beyond [5.8, 5.10] Figure 5.4 shows a comparison of the measured neutral beam current profile and the calculated one (both with and without multi-step ionization process in the N-NBI experiment) [5.11] Good agreement

is obtained only for the calculation with multi-step ionization processes So, it is important to understand these processes

FIG 5.4 Experimental NBCD current profile compared with calculation with and without multi-step ionization [5.11] Here j NNB is the non-inductively driven current density by the injection of negative ion based neutral beam injection r (horizontal axis) is the plasma minor radius normalized to the plasma minor radius a.

Comprehensive atomic data including excitation from the ground state and ionization from excited states are compiled by the IAEA [5.6] The excitation cross-section ex from the ground state to the excited state with principal quantum number n is a decreasing function of n, while the ionization cross-section from an excited state increases with n as seen in Fig 5.5

FIG 5.5 Ionization and CX cross-sections from excited (n = 2~6) and ground (n = 1) states (calculated from ion and cx formulas given in Sections 2.2 and 2.3 of Ref [5.6] with obvious correction of W n = (n/3) 2 W on page 74 of Ref [5.6]).

Higher excited states are subject to ionization by the Lorenz field v

E  B The critical electric field for Lorenz ionization is given by

4 , (n) 0/ (16n )

n to n′, Ann′ is the radiative decay and vb is the beam speed, respectively The collisional transition rate is given as Kn'n  j j n n'nj v , where n j is the

density of particles j, n'nj is the cross-section of hydrogen for transition from

n′ to n through collision with particle j (= e, i, I) being electron, ion or impurity

respectively Equations (5.4) and (5.5) can be approximately rewritten by using the beam stopping cross-section s as

b

e s b

where I b =SIn, s  / vb e n , where  is the minimum eigenvalue of the

transition matrix {Qn’n} The contributions of multi-step processes and ionization

by the Lorentz field are given by the following enhancement factor  defined by

the order of 0.3–0.5 for an electron density n e = 1020 m–3 [5.8] The stopping cross-section s has a strong dependence on the beam energy W, the electron density n e , the electron temperature T e and the effective charge Z eff and has almost

no dependence on ion temperature T i and the magnetic field B An analytical fit of

the stopping cross-section based on recent data is given in Ref [5.12] and typical values of  are shown in Fig 5.6

An experimental measurement of the multi-step effect has been made in various tokamaks and was published in the ITER physics basis as shown in

Fig 5.6 The shine through rate (h = I b (L)/I b (0), where I b (x) is the beam intensity

at x) is compared with calculations with and without multi-step ionization in

JT-60U showing better agreement with the multi-step process as shown in Fig 5.6

FIG 5.6 Top: Comparison of measured and calculated enhancement [5.13] Below: Comparison of measured shine through rate h with calculation with and without multi-step ionization [5.14] Here L is the plasma length along the beam path.

5.1.4 Energy transfer to electrons and ions by neutral beam injection

A fundamental feature of heating by a neutral beam was clarified by Stix [5.15] and can be seen in introductory textbooks such as that by Wesson [5.16] Here, we discuss the issue using the correct impurity contribution Neutral beam “heating” occurs through energy transfer by the Coulomb collision of the energetic beams with bulk Maxwellian electrons and ions (deuterium,

tritium and impurities) with the electron temperature T e and ion temperature T i ,

respectively The beam speed vb is usually much larger than the ion thermal speed

where W b is the beam energy, and W c and se are called the critical energy and

beam electron slowing down time, respectively The formulas for W c and se are

Here, Z  n Z A j 2j b / (n A e j) The first term on the right hand side

of Eq (5.8) is the energy loss through beam–electron Coulomb collisions Since the electron mass is much smaller than beam ion (see Fig 5.7a),

m e /m D ~ 1/3672, beam ions lose energy through the friction with bulk electrons

proportional to the square of the speed This power 2W b/se is transferred from the beam to the bulk Maxwellian electrons

The second term is the energy loss through beamion/impurity Coulomb collisions (see Fig 5.7a) Since the field particles have equal or larger mass than the beam ions, the beam ions lose energy through friction with bulk ions,

in the parallel direction The energy decay time by the ion channel depends on the beam speed ( ~ v3b ) If the beam energy W b is higher than W c, the energy transfer to ions is smaller than that to electrons, while the energy transfer to ions

becomes dominant when W b becomes less than W c This is the reason why W c is called the “critical energy” The instantaneous ion heating fraction is given by

F i (W b /W c ) = 1/(1+(W b /W c)3/2).The fast ion distribution function f b (W b) is given by the flux conservation in energy space and is shown in Fig 5.7b as follows:

The integrated ion heating fraction F i (W b0 /W c) is shown in Fig 5.7c Consider

the typical ITER case (T e = 20 keV, W b0 = 1 MeV, n e = 1020 m–3, D0→D+), beam electron slowing down time se 0.5s0.5 s, critical energy W c18.6T e372 keV,

W b0 /W c = 2.7 and the thermalization time th0.28 s In this case, the initial 0.28sheating is dominated by the electron heating but the integrated heating is almost equal between ions and electrons

The Coulomb interaction of the beam with bulk electrons and ions also includes pitch angle scattering, which becomes significant when the beam energy is less than the critical energy The deflection time d (defined as

Here we note that d ~ 1/Z eff The ratio of the energy increase rate

in the perpendicular direction over the total energy loss is given by

When impurities exist, pitch angle scattering is enhanced since 1/d ~Z eff,

while Z has a weaker dependence on the impurity content

FIG 5.7 a) Schematics of beam electron, beam ion/impurity collision; b) normalized fast ion distribution function as a function of W b0 /W c ; c) integrated ion-heating fraction F i (Eq (5.11))

as a function of W b /W c .

5.1.5 Energetic particle orbits on the axisymmetric magnetic surfaces

Energetic particle orbits are well discussed in many textbooks such as Wesson [5.16] and Miyamoto [5.17] and also in Chapter 2, Section 2.2 of this book Here, we discuss the beam ion motion on the axisymmetric magnetic surfaces without collisions by using the conservation of canonical momentum and introduce longitudinal adiabatic invariants (Fig 5.8) In an axisymmetric

system without collisions, the energy W = m bv2/2 and the magnetic moment

If we define B x W / for a particle, this particle will undergo reflection

by the magnetic mirror at B B x(since v0 at B B x), and it is called a

trapped particle The maximum of B (Bmax) appears at the inboard side and the trapped particle with B x ~Bmax is called “a barely trapped particle” There is another conserved quantity in an axisymmetric system such as a tokamak, namely the “canonical angular momentum”

FIG 5.8 Schematic view of passing and trapped beam ion orbits for a circular tokamak The

2 2

v (di m(v v / 2) eRB) is the toroidal drift due to B and the curvature drifts.

The guiding centre of the trapped particle has a large scale periodic bounce motion in the magnetic mirror along a so-called banana orbit In this case, the

so-called second adiabatic invariant or longitudinal adiabatic invariant J is

conserved The bounce time of trapped particle bounce is given by

length along the magnetic field, respectively The subscript a stands for species a.

Banana orbits have a slow precession motion in the toroidal direction and the precession speed is given by

triton due to the D-D reaction produced at r/a = 0.2 in elliptically shaped JT-60U

plasma with normal shear (NS) and reversed shear (RS) The orbit excursion becomes larger for RS since B is smaller due to the hollow current profile

FIG 5.9 Orbits of 1 MeV triton born at r/a = 0.2 due to D-D reaction for NS (normal shear) and RS (reversed shear) plasmas with I p = 2.4 MA for various pitch angles of the triton in degrees [5.18].

5.1.6 Fast ion behaviour and high temperature production

with NB injection

The first experiments of neutral beam heating were reported from ATC and ORMAK devices at the 7th IAEA International Conference on Plasma Physics and Controlled Nuclear Fusion Research in 1974 Significant efforts in neutral beam system development were made and 2.4 MW injection heating into PLT

produced significant ion and electron heating (T i = 1 keV to 6.6 keV) in 1979 Based on this successful plasma heating, further implementation of the NBI system has been made worldwide, especially for three large tokamaks (TFTR, JT-60U and JET) NB injection of 40 MW was achieved in TFTR in 1994 and in JT-60U in 1996

A comprehensive review of fast ion behaviour in tokamaks is given in Heidbrink [5.19] The behaviour of fast ions agrees fairly well with the prediction

of classical Coulomb collision theory for a quiescent plasma where no significant MHD modes are observed The slowing down time of fast ions was measured

in various tokamaks, showing good agreement with classical slowing down formula Eq (5.8) Pitch angle scattering was also roughly consistent with the prediction of the Fokker–Planck equation The radial transport of fast ions under the turbulent fluctuations was very small compared with that for thermal ions and electrons Figure 5.10 shows TFTR experiments to measure the fast ion diffusion coefficient showing D fast~ 0 m ·s2 –1[5.20]

FIG 5.10 Fast ion diffusion coefficient measured in TFTR [5.20] The unit n/s means neutrons/ second and the three curves are plotted for different values of fast ion diffusion coefficient

D = 0 m 2 ·s –1 ; D = 1.0 m 2 ·s –1 and D = 0.1 m 2 ·s –1 Reprinted from Ref [5.20] Copyright (2011), American Institute of Physics.

The behaviour of fast ions following classical slowing down and virtually

no fast ion diffusion in an MHD-free tokamak led to good agreement of measured

D-D and D-T neutron emission rates with simulation using measured plasma parameters and a classical slowing down fast ion model with no fast ion diffusion TRANSP simulation of JET D-D and D-T experiments is shown in Fig 5.11 [5.21] Both D-D and D-T discharges are characterized by significant neutrons from beam–plasma reaction So, the agreement of total neutron emission rate between measurement and simulation provides fair confidence of the absence of anomalous slowing down and fast ion diffusion in axisymmetric tokamaks.

FIG 5.11 Measured and simulated D-D and D-T neutrons with significant beam–plasma reactions in JET [5.21].

Excellent simple physics led to the adoption of NBI heating as the main auxiliary heating system in tokamaks Figure 5.12 shows the results from JT-60U

under intensive neutral beam heating P abs = 27 MW (absorbed part of the injected NB) with the beam energy of 92 keV into the discharge with plasma current

I p = 2.4 MA and toroidal magnetic field B= 4.3 T The discharge went into the H-mode 0.6 s after the start of beam injection, and the plasma stored energy reached 8.6 MJ with a D-D neutron emission rate S n5.2 10 s 16  1, and a global energy confinement time E= 0.75 s An H-factor above L-mode of 3.3 was

achieved and a central ion temperature T i(0) of 45 keV (5.2 10 8 centigrade)

and a central electron temperature T e(0) of 10.6 keV were achieved as measured

by charge exchange recombination spectroscopy (CxRS) of fully stripped carbon impurities for the ion temperature and by an ECE Fourier spectrometer (ECE) for

the electron temperature [5.22, 5.23] In this case, the critical energy W c = 198 keV

and W b /W c = 0.46 So, about 90% of the total heating power goes to ions and

10% goes to electrons This preferential ion heating resulted in T i (0) ~ 4T e(0) and it is called a “hot ion regime” Although the ion temperature is measured for impurities, it is confirmed that classical energy exchange between deuterium and

carbon is enough to reach T I (0) ~ T i (0), where the subscript I stands for impurity

This central ion temperature was the world’s highest plasma temperature in 1996

as certified by the Guinness Book of Records

FIG 5.12 T i and T e profiles for the world’s highest central ion temperature in JT-60U and Guinness Book of Records on achievement of 45 keV in 1996 [5.22].

5.1.7 Physics of neutral beam current drive: fast ion distribution function

Neutral beam injection produces fast ion current circulating around the torus The current carried by a single beam is not large but multiple circulations around the torus give a large fast ion current Collision of these directional fast ions with bulk electrons produces an electron drift in the same direction as the fast ions, whose current is in the opposite direction to the fast ion current This electron current is called the “shielding current” and will be discussed in Section 5.1.8 It can cancel the fast ion current completely if there are no trapped electrons and impurities In reality, the existence of trapped electrons and impurities in the toroidal plasma provides great opportunity for current drive by neutral beam injection The fast ion trajectory is assumed to be on the magnetic surface () in the following theory of neutral beam current drive but trapping in the magnetic mirror is taken into account for beam ions

The velocity distribution function of fast ions f b is determined as a solution

of the Fokker–Planck equation valid for vth i, vbvth e, Since a tokamak

has variation of the magnetic field B along the fast ion trajectory, part of the

fast ion population is trapped in the magnetic mirror as discussed in the previous section The fast ion current is carried by the passing particles and the trapped particles do not contribute to the fast ion current since the return motion cancels the current under the assumption of a fast ion trajectory on the magnetic surface The degree of trapping can be measured by the constant of motion of a single

min

(1 B / )W

   , where m bv / 22 B is the magnetic moment,

W = m bv2/2 is the fast ion energy and Bmin is the minimum of the total magnetic field on the magnetic surface, usually at the outboard side of the torus The value of  varies from 0 (at W B/ min) to 1 (at  ) and the value at the 0trapped–passing boundary t is given by 1/2

where se is the beam electron slowing down time given by Eq (5.10),

vc is the critical velocity defined by vc = (2W c /m b)1/2 where W c is given

by Eq (5.9), vb is the beam velocity (= (2W b /m b)1/2), β is defined by

1 2

      , where ln and lne  are the i

Coulomb logarithms of the electron and ion species S(v,h) is the bounce averaged fast ion source rate per unit volume, Z eff is the effective charge defined

by Z eff  j j n Z2j /n e, v / v is given by v / v (2 / ) [( / ) ] K  t 2 and

v / v is given by v / v (2 / ) [( / ) ] E  t 2 for passing ions, where K and

E are complete elliptic integrals of the first and second kind, respectively The

solution of Eq (5.22) is given in Ref [5.25] as follows:

where S 0 , c n are neutral beam source intensity and eigenfunction of ,

respectively, and a n(v) is the analytical solution for a uniform magnetic field given

Here, S(v, ) S0(v v ) ( ) b k S0(v v ) b k c n n( ) , where δ is the Dirac delta function, k n the n’th source coefficient in the c n expansion of the source, and the fast ion distribution function above the beam energy (v > vb) comes from energy diffusion in velocity space The following equation defines the eigenvalue n and eigenfunction c n (h):

2 0

In reality, the existence of trapped electrons and impurities in the toroidal

plasma provides a great opportunity for current drive by neutral beam injection

The beam driven current j bd is a current in response to the external momentum

source S b, which is obtained by solving parallel momentum and heat momentum balance equations for electrons, ions, impurities and fast ions (only momentum balance for fast ions) [5.28], as follows:

Here, e a , n a, Mˆ and ˆL are the electrical charge, density, viscous and

friction matrixes, respectively, and S b is the momentum source from fast ions

Also u a is the flow velocity of species a (a = e,i,I,b) So, if we divide the beam

driven current j bd into the fast ion current j fast and the shielding current j shield

j B  j B  j B

, we obtain

Various physical processes have been included self-consistently in

Eq (5.39) the friction between the bulk electrons and fast ions produces shielding

of the fast ion current, while the electron viscosity gives a neo-classical drag force to the passing electrons from the trapped electrons, as a result of which the shielding current carried by the passing electrons is partially cancelled Friction between electrons and impurities also reduces the shielding current This stacking

factor F has Z eff and ε = r/R dependences but no v b /ve dependence if vb << ve The

parametric dependences of F on Z eff and ε for an arbitrary aspect ratio (0 ≤ ε ≤ 1)

can be calculated from Eq (5.39) and are shown in Fig 5.13

FIG 5.13 Shielding factor F as a function of r/R and Z eff [5.28] in comparison with Ref [5.30] Here, J/J b stands for j bd /j fast in the text.

At r/R = 0 (case with no trapped particles), the stacking factor F can be

derived from a simple momentum balance equation for the electrons, namely,

the shielding current with increasing r/R as seen in Fig 5.13

Wesson [5.16] gave a rough estimate of the beam current drive efficiency

by assuming j bd ~ j fast (see Eq (5.29)) divided by the driving power per unit

volume, P d = SW b0 , as follows,

1 3 1

Here u v / vb0 is the ratio of the normalized fast ion velocity

to fast ion injection speed The parametric dependence can be given by

dependence of the current drive efficiency on W b0 /W c (0, which stands for ‘initial’,

is omitted hereafter for W b) as given in Fig 5.14 Integration over the power deposition profile gives the following formula for the current drive efficiency:

respectively This equation tells us that higher electron temperatures and higher

beam energies to approach the optimum W b /W c ~ 5 are key to achieving the highest current drive efficiency

FIG 5.14 W b /W c dependence of normalized NBCD efficiency .

Viscosity and friction matrices valid for an arbitrary aspect ratio, all collisionality and multi-species (electron, ion, impurity and beam) at

where l and µ are friction and viscosity coefficients, respectively, and detailed

expressions are given in Ref [5.28]

5.1.9 Experimental observation of beam-driven current

First observations of a beam induced current in a tokamak were made in

DITE [5.31], where 1 MW of H beam with W b = 24 keV was injected tangentially

and T e(0) = 0.9 keV achieved, showing 33 kA driven non-inductively as shown in Fig 5.15 Since then, a wide range of experimental data have been accumulated including those from DIII-D, JET, TFTR and JT-60U In particular, the electron temperature was increased to 14 keV and the beam energy was increased to

360 keV in JT-60U experiments [5.32]

FIG 5.15 DITE tokamak showing existence of beam driven current (¾¾ experimental loop voltage V loop , - - - theory with beam driven current,¾ ¾ calculation with beam driven current) [5.31] Reprinted from Ref [5.31] Copyright (2011) by the American Physical Society.

Methods to calculate plasma equilibrium with inductive and non-inductive

(beam driven and bootstrap) current densities j NI have been established in various numerical codes such as ACCOME [5.27] and ASTRA [5.33] incorporating the physical processes described in previous sections A method for the experimental determination of non-inductively driven current was established by Forest [5.34] Ohm’s Law in general toroidal geometry is given by

j B  E B   j B

(5.46)where

such as EFIT [5.35] By calculating the ohmic current using the measured ¶y/¶t, density, temperature and Z eff profiles, the non-inductively driven current j NIB

can be “measured” as the difference NI

m

j B  j B  E B 

This current density profile can be compared with the numerical calculation of j NIB c

using the measured density, temperature and Z eff profiles A comparison is shown

in Fig 5.16 It is worth noting that “calculated” and “measured” non-inductively driven current profiles agree when multi-step ionization effects are taken into account in the calculation of j NIB

FIG 5.16 Left: Comparison of calculated and measured beam driven current Right: NBCD current drive efficiency as a function of central electron temperature T e (0)[5.14, 5.32] Here

I NNB stands for I CD with a negative ion based neutral beam.

Agreement between measured and calculated total driven current is good

in a range of 0.1–1 MA as shown in Fig 5.16 The theoretical NB current

drive efficiency increases with electron temperature T e (see Eq (5.43)) This dependence is also confirmed experimentally as shown in Fig 5.16 Maximum NBCD efficiency CD 1.55 10 A·W ·m 19 –1 –2 is achieved at T e(0) = 14 keV with

a beam energy of 360 keV The projected NBCD efficiency for W b = 1 MeV in ITER is 2 3 10 A·W ·m  19 –1 –2 for T e(0) = 10–20 keV For the DEMO, a higher

central temperature T e (0) ~ 30 keV and higher beam energy W b ~ 2 MeV might

be necessary to have a high NBCD efficiency of 5 10 A·W ·m 19 –1 –2 These experimental results are encouraging for steady state operation of ITER, DEMO and beyond

Redistribution of beam driven current and/or reduced driven current have been observed in various tokamaks [5.14] if the discharge is associated with MHD activities, such as toroidicity-induced Alfvén eigenmodes (TAEs), sawteeth, fishbones and tearing modes, while the characteristics of these modes are introduced in Chapter 3 of this book So, it is important to control MHD activity so that NBCD does not deteriorate Recent experiments on ASDEx-U show the appearance of a discrepancy with theory in high power off-axis NBCD experiments without any MHD activity [5.36] This subject is left for future investigation

5.1.10 Physics of ripple loss of fast ions: banana drift and ripple trapped losses

Ideally, a tokamak is an axisymmetric system But the finite number of toroidal field coils leads to the loss of perfect axisymmetry and produces ripple

loss [5.37] The toroidal variation of vacuum toroidal magnetic field B and ripple  are defined by

where R in and R out are the major radius of the outer and inner legs of the toroidal

field coils and the numerical constant β ~ 1.5 An analytical formula of the ripple

for a circular TF coil is given by Yushmanov [5.38]

FIG 5.17 Top: Variation of toroidal magnetic field with a ripple field and two types of trapped particle orbits Bottom: Schematics of banana drift orbit (blue) and ripple trapped orbit (red)

In Fig 5.17, the field strength along a magnetic field line is shown, which

contains two variations, namely the large scale cos ϕ and the small scale cos (Nϕ) variations The ripple well parameter α is defined by

where B B R( 0( )) and B B R( 0( ) ( , )cos( R Z N)) are axisymmetric and

non-axisymmetric fiel ds, respectively If α is less than 1, there will be a local

ripple well Using /  l (B R/ ) /B   for an axisymmetric field (here, R

B R is the R component of the magnetic field), and   / l R 1  for a / 

non-axisymmetric field, the ripple well parameter α can be simplified as follows:

Since α is proportional to I p /B ~ 1/q, the ripple well region becomes

wider for low current and high q operation.

There are two types of ripple loss mechanism in the presence of toroidal field ripple One is ripple trapped loss in which fast ions are trapped in the local magnetic mirror due to the toroidal field ripple, which is shown as a red orbit in

Fig 5.17 and occurs in the ripple well region α < 1 In this case trapped fast ions

drift vertically due to the toroidal drift Another mechanism is the banana drift loss in which axisymmetric banana orbit drifts horizontally due to the toroidal field ripple, which is shown as a blue orbit in Fig 5.17

Quantitative comparison of measured and calculated ripple loss was made

in JT-60U [5.39] The heat flux was estimated from the IR measurement of the temperature rise of the first wall and the magnitude and location of the hot spot agree with those from measurement and calculation for banana drift loss (Fig 5.18), while the effect of the radial electric field should be included for ripple trapped loss [5.40]

FIG 5.18 Left: Comparison of contour plots of measured and calculated heat flux due to ripple loss Calculation is done without including the radial electric field E r effect Right: Measured heat flux contour compared with calculations with and without E r [5.40] Reprinted from Ref [5.40] Copyright (2011), American Nuclear Society.

The effect of collisions can be simulated using the Monte Carlo technique [5.41] The change in the parallel and perpendicular velocities of a test particle from ( v , v ) to ( v , v  ) after Coulomb collision can be given by the following relations:

Eq (5.53) can be simulated by a uniform random number between 0 and 2π.

5.1.11 Physics of particle trajectories in non-axisymmetric fields

Fast ion trajectories in a non-axisymmetric magnetic field with toroidal ripple are much more complicated than those described in Section 5.1.5 In this case, the physics of a particle trajectory in a tokamak becomes very much similar

to that of a helical system Despite such complications, the magnetic moment

In the case   , this equation means that the turning point of a banana 0orbit ( v 0) is always on a B W /constant surface and drift of the

banana turning point occurs along the B = constant surface If we consider the

case   ( ) , the turning point of a banana particle drifts along a W = constant

surface (  B( , ) e a ( , ))  W, where  is the poloidal angle Usually, a

B = constant surface does not match the flux surface  = constant and is typically

a vertical line for the case of a tokamak, while it can be closed inside the closed

magnetic surface if  is strong enough so that the E B  drift ( vE E B/ )

is much larger than the toroidal drift vd ( vE vd) The effect of electrostatic potential is particularly important for helical magnetic configurations for improving particle confinement, where Boozer coordinates are used to analyse particle dynamics [5.43]

It is also important to note that the longitudinal adiabatic invariant v

a

J m  dl (Eq (5.19)) is conserved in most cases The longitudinal

adiabatic invariant J is an area of closed surface in ( ,v l ) phase plane as

shown in Fig 5.19(a) Conservation of J implies that the precession motion of

a banana is along the J = constant surface When the toroidal field ripple exists,

an equi-contour map of B has a local well as seen in Fig 5.19(b) For such a

case, the topological structure of a phase space ( ,vl ) orbit changed from simple conformal ellipses to separated ellipses with separatrix and peanut-shape phase

space orbit as seen in Fig 5.19(b) We define J s as the area of a local ripple well

It is obvious that J s = 0 means that there is no local ripple well close to the α = 1

boundary

FIG 5.19 Magnetic field strength B along particle orbit x  (l: orbit length along the magnetic

field ( dl ( / )B B dx  )) and the geometrical meaning of longitudinal adiabatic invariant J in

a tokamak without (a) and with (b) a rippled field Here, J corresponds to the area inside the separatrix B H and B L are the local maximum and minimum of the magnetic field, respectively Also, l 1 and l 2 correspond to orbit length l where B = B H .

Spatial variation of ripple well depth produces interesting phenomena called “collisionless ripple-trapping” of banana particles as shown in Fig 5.20a Consider a banana particle and its motion near its turning point ( v∼0) The remaining particle drift velocity is toroidal drift (and E B  drift) velocity, which

is in the Z direction in most cases If the effective ripple well gets larger with z,

a banana ion which barely passed the local ripple field is reflected at the turning

point and undergoes toroidal drift in the z direction and feels a larger magnetic

field ripple and may be reflected into the local magnetic mirror

Figure 5.20b shows the poloidal cross-section of the tokamak and banana

orbit The thick blue line shows the α = 1 condition When a banana ion is

bouncing near the separatrix in Fig 5.19b, it can be collisionlessly trapped into the local mirror if the effective well eff B H /B L is increasing in the 1direction of total drift (vdv )E  eff 0 where eff is given as a function of

α (Eqs (5.49), (5.51) and (5.52)) as follows [5.41, 5.44]:

So collisionless ripple-trapping can occur in the shaded region 2 of

Fig 5.20b bounded by α = 1 and (vd v )E  eff 0 As the particle moves along the banana orbit, it encounters a deeper magnetic well after turning from the banana tip and is trapped to the local magnetic well These locally trapped particles drift vertically and are lost, which is called ripple trapped loss Similar collisionless trapping occurs in the shaded area 1 but ripple trapped fast ions drift vertically to the top banana tip and can be detrapped to the banana orbit Yushmanov gave a condition for this collisionless ripple-trapping by using the

longitudinal adiabatic invariant J s as (vdv )E   J s 0 [5.38] The possible

collisionless ripple-trapped region is bounded by J s = 0 and (vd v )E   J s 0

FIG 5.20 a) Collisionless ripple trapping of banana particle in spatially varying ripple field [5.41] b) Region of possible collisionless ripple trapping region and poloidal cross-section of typical banana orbits.

Even without trapping, a non-axisymmetric field can modify the banana orbit of fast ions through stochastic processes The radial position of the banana tip is characterized by B W / and is always the same for the axisymmetric case But the radial position of the banana tip changes with the ripple field The radial displacement r is given by [5.38]

of B, respectively This displacement can be modelled as a radial kick at the

banana tip, leading to a stochastic diffusion of fast banana ions [5.45] This is the physical background of the banana drift loss described in the previous section

5.1.12 Alpha heating

When the plasma reaches a high temperature by strong auxiliary heating,

deuterium and tritium can fuse and energetic helium (Z = 2, A = 4) or an “alpha particle” with energy W α0 = 3.52 MeV is produced This “alpha” heating is called

“self-heating” since the plasma itself produces the “heating” The power density

of alpha heating P α is given by

Here T i is in keV More accurate reaction rates within 1% for 1 keV < T i <

100 keV can be given as follows [5.47]:

where n D , n T and T i are in 1020 m–3 and in keV respectively

FIG 5.21 a) Particle distribution function f(v) (T e = 20 keV); b) Comparison of α heating power and loss power densities with T (t E ~ 1/T, P α = P loss at 10 keV) Here, v c is the critical velocity defined for alpha particles and v a0 is the alpha birth velocity.

The steady state distribution functions of alpha particles



where S P W / 0 is the alpha particle production rate, se is the alpha–

electron slowing down time (Eq (5.10)), t th is the thermalization time and

2

v / 2

W m  , W c mv / 22c (W c is given in Eq (5.9)), respectively Figure 5.21a shows the steady state velocity distributionf(v) for a D-T plasma

at T = 20 keV If the empirical energy confinement time t E is proportional to

1/T, the loss power P loss is proportional to T2 The alpha heating and loss powers are plotted as a function of plasma temperature for a D-T density of

0.5 10 m

n n    in Fig 5.21b

A rough parametric dependence of the alpha particle parameters can

be given as particle production rate S ~n T2 2i S, particle thermalization

0

/ 2

B  Typical values for ITER are

P α ~ 0.3 MW·m–3, th, ~ 1s, n(0) / (0) ~ 0.3%n e , (0) ~ 0.7%, which were almost achieved in D-T experiments on TFTR and JET Since the alpha particle velocity is much larger than the critical velocity, alpha particles interact mainly with electrons and the slowing down is dominant over pitch angle scattering The particle distribution function was measured in TFTR as shown in Fig 5.22 and was consistent with collisional transport theory [5.48, 5.49]

FIG 5.22 Measurements of alpha velocity distribution by α-CHEARS (top) and pellet CX (bottom) compared with calculations [5.48, 5.49] Here three sources of alpha charge exchange are considered in alpha-CHERS One is neutral beam-alpha particle charge exchange; the second is charge exchange with halo neutrals produced by beam-ion charge exchange; and the third is electron and ion impact excitation of He + ions (called alpha plume) Partly reprinted from Ref [5.48] Copyright (2011) by the American Physical Society.

5.1.13 D-T experiments in large tokamaks (TFTR and JET)

Experimental demonstration of D-T fusion power production in a magnetically confined plasma was made in JET and TFTR There are number

of reviews on scientific achievements in D-T experiments[5.50–5.52] The first experiments using T in a magnetically confined plasma were carried out in JET at a JET Joint Undertaking in 1991 as the Preliminary Tritium Experiment

(PTE), in which fusion power over 1.5 MW and fusion energy of 2 MJ were produced with a 10% tritium concentration [5.53] TFTR at Princeton Plasma Physics Laboratory started D-T experiments in December 1993 Experimental observation of electron heating by fusion alpha particles was reported for the first time by TFTR [5.54] and systematically by JET [5.55] as shown in Fig 5.23.

FIG 5.23 Direct evidence of electron heating by alpha particles at TFTR (top) [5.54] (reprinted from Ref [5.54] Copyright (2011) by the American Physical Society) and JET (bottom) [5.55].

Significant fusion power production by the use of a D and T fuel mixture

was demonstrated in TFTR (P fusion = 10.7 MW) in 1994 [5.56] JET achieved the

largest fusion power production of P fusion = 16.1 MW in 1997 [5.50] as shown in Fig 5.24 This highest fusion power production was achieved with 25.4 MW

of additional heating, and central ion and electron temperatures of 28 keV and

14 keV were recorded The TFTR operation was completed in April 1997 There were a total of 300 D-T discharges A total of ~5 g of tritium was introduced into the TFTR vessel and a total energy of ~1.5 GJ of fusion power was produced

FIG 5.24 Significant D-T fusion power production in TFTR [5.56] and JET [5.50].

5.2 NEUTRAL BEAM HEATING

as 2x-IIB [5.67] and TMx [5.68], and in stellarators such as JIPP T-II [5.69] and Wendelstein VII-A [5.70] In the 1980s multi-megawatt positive ion sources were developed for large tokamak machines such as TFTR [5.71], JET [5.72] and JT-60 [5.73] to inject several tens of MW neutral beams at beam energies of 80–120 keV

In the 1990s a multi-ampere, multi-megawatt negative ion source was developed [5.74] and the construction of the first negative-ion-based neutral beam injector was carried out in JT-60 [5.75] followed by LHD [5.76, 5.77] At present a negative ion source producing 40 A, 1 MeV deuterium negative ion beam is being developed to realize neutral beam injection in ITER [5.78, 5.79].Figure 5.25 shows the schematics of the neutral beam injector (NBI) The heart of the NBI is the ion source, where hydrogen or deuterium ions are generated by a plasma generator and accelerated to a desired energy by an extractor/accelerator The ion source is followed by a neutralizer cell in which a fraction of the accelerated ions is converted into energetic neutrals typically by

a charge exchange process with collisions of the background gas It is necessary

to convert the ions into neutrals to inject the neutral beams into the magnetically confined plasma The energetic neutral beam goes straight towards the plasma through a drift duct which connects the injection port of the plasma device The un-neutralized ions are removed from the neutral beam and guided to ion beam dumps by a bending magnet High speed vacuum pumps such as a cryosorption

pump with pumping speeds approaching 1000 m3·s–1 are generally employed to exhaust the residual neutral gas A complex electrical system that repetitively switches high voltage electrical power is employed to accelerate the ions.

FIG 5.25 Schematic of a neutral beam injector.

Figure 5.26 shows the NBI system for LHD [5.76, 5.77] as an example

of the real NBI The LHD-NBI has a negative ion source producing energetic negative hydrogen ion beams which are converted to neutral beams during passage through a gas neutralizer The neutral beam power and its profile are measured by a movable calorimeter located downstream of the ion beam dumps During the beam injection, the calorimeter is removed from the beam axis

FIG 5.26 Cross-sectional view of negative ion based NBI for LHD [5.76].

The overall power efficiency of NBI is mainly determined by the ization efficiency in the neutralizer Figure 5.27 shows the neutralization efficiency as a function of the beam energy for the gas neutralizer (solid line), where the positive ions are neutralized by the charge exchange process in the gas neutralizer The efficiency decreases rapidly as the beam energy increases The efficiency is only 20% at a beam energy of 100 keV for hydrogen positive ion beams and decreases to almost zero at energies higher than 500 keV Higher beam energies are preferable for denser and bigger plasmas, but positive ion based NBI cannot be used because of the low power efficiency above energies

neutral-of 100 keV per nucleon To overcome this difficulty, negative ion beams have

to be used, whose neutralization efficiency is 60% even at energies higher than

100 keV per nucleon

FIG 5.27 Neutralization efficiency of positive and negative ions as a function of energy [5.80].

Table 5.3 summarizes typical NBI systems using positive ion sources (positive ion based NBI) and negative ion sources (negative ion based NBI) Positive ion based NBIs have been used for TFTR [5.71], JET [5.72] and JT-60 [5.73] at beam energies lower than 100 keV per nucleon Above this energy negative ion based NBIs have been developed for JT-60U [5.75], LHD [5.76, 5.77] and ITER [5.78, 5.79]

TABLE 5.3 MAIN CHARACTERISTICS OF NEUTRAL BEAM

INJECTORS

Positive-ion-based NBI Negative-ion-based NBI

Output 120kV/65A 125kV/60A 100kV/40A 500kV/22A 180kV/30A 1000kV/40A

Figure 5.28 shows the history of the development of high current positive and negative ion sources High current positive ion sources were developed in the 1960s to 1980s; then high current negative ion sources were developed beginning

in the late 1980s

FIG 5.28 History of the development of high current positive and negative ion sources.