The attractive “hybrid” tokamak scenario combines comparatively high q95 operation with improved confinement compared with the conventional H98,y2 scaling law. Somewhat unusually, hybrid discharges often exhibit multiple neoclassical tearing modes (NTMs) possessing different mode numbers. The various NTMs are eventually observed to phase lock to one another, giving rise to a significant flattening, or even an inversion, of the core toroidal plasma rotation profile. This behavior is highly undesirable because the loss of core plasma rotation is known to have a deleterious effect on plasma stability. This paper presents a simple, single-fluid, cylindrical model of the phase locking of two NTMs with different poloidal and toroidal mode numbers in a tokamak plasma. Such locking takes place via a combination of nonlinear three-wave coupling and conventional toroidal coupling. In accordance with experimental observations, the model predicts that there is a bifurcation to a phase-locked state when the frequency mismatch between the modes is reduced to one half of its original value. In further accordance, the phase-locked state is characterized by the permanent alignment of one of the X-points of NTM island chains on the outboard mid-plane of the plasma, and a modified toroidal angular velocity profile, interior to the outermost coupled rational surface, which is such that the core rotation is flattened, or even inverted.

A magnetic confinement device is designed to trap a thermonuclear plasma on a set of toroidally nested magnetic flux-surfaces. Heat and particles flow around flux-surfaces relatively rapidly due to the free streaming of charged particles along magnetic field-lines. On the other hand, heat and particles are only able to diffuse across flux-surfaces relatively slowly, assuming that the magnetic field-strength is sufficiently high to render the particle gyroradii much smaller than the minor radius of the device. This article will concentrate on the tokamak, which is a type of toroidally axisymmetric magnetic confinement device in which the magnetic field is dominated by an approximately uniform toroidal component whose energy density is much larger than that of the plasma.1 

Tokamak plasmas are subject to a number of macroscopic instabilities that limit their effectiveness. Such instabilities can be divided into two broad classes. So-called ideal instabilities are non-reconnecting modes that disrupt the plasma in a matter of microseconds.2 However, such instabilities can usually be avoided by limiting the plasma pressure and/or by tailoring the toroidal current profile.3Tearing modes, on the other hand, are relatively slowly growing instabilities that are more difficult to prevent.3,4 These instabilities tend to saturate at relatively low levels,5–8 in the process reconnecting magnetic flux-surfaces to form helical structures known as magnetic islands. Magnetic islands are radially localized structures centered on so-called rational flux-surfaces that satisfy k·B=0, where k is the wave vector of the instability, and B is the equilibrium magnetic field. Magnetic islands degrade plasma confinement because they enable heat and particles to flow very rapidly along field-lines from their inner to their outer radii, implying an almost complete loss of confinement in the region lying between these radii.9 

Conventional tearing modes are driven unstable by inductive current gradients within the plasma and are relatively straight-forward to avoid in non-pathological tokamak discharges.3 On the other hand, so-called neoclassical tearing modes are driven unstable by the perturbed bootstrap current10,11 and are virtually impossible to avoid in high-β discharges.12 Neoclassical tearing modes are conveniently classified in terms of the poloidal and toroidal mode numbers, m and n, respectively, of the associated magnetic island chains. The mode numbers of neoclassical tearing modes typically observed in high-β tokamak discharges are m,n=5,4; 4, 3; 3, 2; and 2, 1.13 Note, however, that neoclassical tearing modes with different mode numbers do not usually occur simultaneously in such discharges.13–16 In other words, the presence of a 5, 4 neoclassical tearing mode usually implies the absence of a 4, 3 mode, and vice versa.

The so-called hybrid scenario combines comparatively high q95 operation with improved confinement compared with the conventional H98,y2 scaling law.17,18 If this scenario could be reproduced on ITER then it would enable high-Q operation at reduced plasma current, which would be highly advantageous.19,20 Somewhat unusually, hybrid discharges often exhibit multiple neoclassical tearing modes possessing different mode numbers. For example, 2, 1 and 3, 2 neoclassical modes have been observed simultaneously in both DIII-D and JET hybrid discharges.21,22 In addition, 4, 3 and 5, 4 modes have been observed simultaneously in JET hybrid discharges.23 In all cases, the various modes are eventually observed to phase lock to one another, giving rise to a significant flattening, or even an inversion, of the core toroidal plasma rotation profile.22,23 This behavior is highly undesirable because the loss of core plasma rotation is known to have a deleterious effect on plasma stability (because it facilitates locked mode formation).

The aim of this paper is to present a simple, single-fluid, cylindrical model of the phase locking of two neoclassical tearing modes with different poloidal and toroidal mode numbers in a tokamak plasma. Such locking takes place via a combination of nonlinear three-wave coupling24–26 and conventional toroidal coupling.27–29 It is of particular interest to establish the final phase relation of the phase-locked islands, as well as the effect of the phase locking on the plasma toroidal rotation profile.

This paper is organized as follows. In Sec. II, we introduce the fundamental building blocks of our theory. In Sec. III, we describe a simple model plasma equilibrium that is adopted in this paper to facilitate calculations. In Sec. IV, we analyze the nonlinear three-wave coupling of three neoclassical tearing modes with different poloidal and toroidal mode numbers. In Sec. V, we describe the conventional toroidal coupling of two neoclassical tearing modes with a common toroidal mode number. In Sec. VI, we combine the analysis of Secs. IV and V to explain how two neoclassical tearing modes with different poloidal and toroidal mode numbers can phase lock to one another. In Sec. VII, we investigate the phase-locking thresholds associated with the three different phase-locking mechanisms discussed in Secs. IVVI. Finally, we summarize the paper in Sec. VIII.

Consider a large-aspect-ratio tokamak plasma whose magnetic flux-surfaces map out almost concentric circles in the poloidal plane. Such a plasma is well approximated as a periodic cylinder. Suppose that the minor radius of the plasma is a. Let us adopt standard cylindrical coordinates: r, θ, z. The system is assumed to be periodic in the z-direction, with period 2πR0, where R0 is the simulated plasma major radius. It is helpful to define the simulated toroidal angle ϕ=z/R0, as well as the normalized radial coordinate r̂=r/a.

The equilibrium magnetic field is written

(1)

where

(2)

is the inverse aspect-ratio of the plasma. Here, Bϕ>0 is the (approximately uniform) toroidal magnetic field-strength. The equilibrium plasma current takes the form

(3)

where

(4)

The function σ(r̂) parameterizes the toroidal current profile. The safety-factor profile is defined

(5)

Moreover, the safety-factor and current profiles are related via

(6)

Note that a conventional tokamak plasma is characterized by a positive, monotonically increasing (in r̂), O(1), safety-factor profile. This implies that σ(r̂) is positive, monotonically decreasing, and O(1).

We can write the total magnetic field as

(7)

where

(8)

and |ψm,n||ψ0,0| for all m,n0,0. This representation is valid as long as mnϵ for all m,n0,0.26 

For the special case of the 0, 0 harmonic, we have

(9)

For the other harmonics, the linearized equations of marginally stable, ideal-MHD, which govern small-amplitude, helical magnetic perturbations throughout virtually all of the plasma (and, also, in the vacuum region surrounding the plasma), yield30 

(10)

where

(11)

and d/dr̂. Close to the magnetic axis, the well-behaved solutions of Eq. (10) take the form

(12)

where A(t) is arbitrary. If we assume that there is a perfectly conducting wall at r/a=b, where b1, then the appropriate boundary condition at r̂=b is30 

(13)

We can simulate a small Shafranov shift of the magnetic flux-surfaces by applying a static, 1, 0 perturbation to the equilibrium. Thus, if

(14)

then the Shafranov shift (normalized to a) is

(15)

where

(16)

It is convenient to normalize the ψ1,0(r̂) eigenfunction such that

(17)

where ψ̂1,0(r̂) and Δa1,0 are real, qa=q(1), and

(18)

It follows that Δ1,0(1)=Δa1,0. In other words, Δa1,0 is the Shafranov shift at the edge of the plasma. Assuming that Δa1,0>0, the simulated outboard mid-plane of the plasma corresponds to θ = 0 (because the Shafranov shift is in the direction of the outboard mid-plane in conventional tokamak plasmas29).

Equation (10) becomes singular at the rational surface, r̂=r̂m,n, where

(19)

indicating a local breakdown of marginally stable, ideal-MHD.4 Defining

(20)

the most general solution to Eq. (10) in the vicinity of the rational surface takes the form26,29

(21)

where

(22)

and

(23)

According to conventional asymptotic matching theory, the coefficient of the large solution, CLm,n, must be the same on both sides of the rational surface, whereas the coefficient of the small solution, CSm,n, can be different on either side.31 Let

(24)
(25)

Here, Ψm,n(t) is the (normalized) reconnected magnetic flux at the rational surface, whereas ΔΨm,n(t) parameterizes the strength of the m, n current sheet at the same surface.30,32

It is helpful to normalize resonant perturbations (i.e., perturbations with rational surfaces lying within the plasma) such that

(26)

where

(27)
(28)

and ψ̂m,n(r̂) is real. The conventional tearing stability index for the m, n mode (i.e., Δ normalized to the radius of the rational surface4) then becomes29 

(29)

Thus, if mode coupling effects are neglected then ΔΨm,n=Em,nΨm,n. Note that, for the sake of simplicity, we are neglecting any dependence of the stability index, Em,n, on the island width.6–8 

Let

(30)

where Ψ̂m,n(t) is real and positive, and φm,n(t) is real. Making use of the well-known constant-ψ approximation,4 which is valid provided that |ΔΨm,n|Wm,n|Ψm,n|,33 the reconnected magnetic flux at the m, n rational surface gives rise to a magnetic island chain whose full radial width (normalized to a) is

(31)

The island X-points are located at angular positions that satisfy

(32)

where j is an integer (i.e., at positions where Re{ψm,n(r̂m,n,t)exp[i(mθnϕ)]} attains its maximum positive value). Likewise, the island O-points are located at

(33)

(i.e., at positions where Re{ψm,n(r̂m,n,t)exp[i(mθnϕ)]} attains its maximum negative value).

The well-known Rutherford island width evolution equation takes the form5,11

(34)

where sm,n=s(r̂m,n), and τRm,n=(μ0a2/η)r̂m,n is the resistive evolution timescale at the m, n rational surface. Here, η(r̂) is the plasma (parallel) resistivity profile, whereas p(r̂) is the plasma pressure profile.

The final term on the right-hand side of Eq. (34) represents the destabilizing influence of the perturbed bootstrap current.10 Here, we are assuming that the magnetic island width exceeds the threshold value required to flatten the pressure profile within the island's magnetic separatrix.11 We are also assuming that the island width exceeds the ion banana width.34 Note that the stabilizing effect of magnetic curvature (which is generally smaller than the destabilizing effect of the perturbed bootstrap current) is neglected in this paper.35 

Finally, adopting the standard no-slip assumption (that a magnetic island chain is convected by the local plasma, which is effectively trapped within its magnetic separatrix),30 we can write

(35)

where Ωθm,n=Ωθ(r̂m,n,t) and Ωϕm,n=Ωϕ(r̂m,n,t). Here, Ωθ(r̂,t) and Ωϕ(r̂,t) are the plasma poloidal and toroidal angular velocity profiles, respectively.

It is easily demonstrated that zero net toroidal electromagnetic torque can be exerted in any region of the plasma governed by the equations of marginally stable, ideal-MHD.26,30 However, the equations of marginally stable, ideal-MHD break down in the immediate vicinities of the various rational surfaces inside the plasma. Hence, it is possible for localized torques to develop at these surfaces. The net toroidal electromagnetic torque that develops in the vicinity of the m, n rational surface is written30,32

(36)

where

(37)

It is assumed that the poloidal velocity profile is fixed, due to the presence of strong poloidal flow damping in the plasma.30,36 If ΔΩϕ(r̂,t) is the modification to the plasma toroidal angular velocity profile caused by the localized electromagnetic torques acting at the various rational surfaces then we can write30 

(38)

where

(39)

Here, ρ(r̂) and μ(r̂) are the plasma mass density and (perpendicular) viscosity profiles, respectively. The localized viscous toroidal torque that develops at the m, n rational surface, as a consequence of the modification to the toroidal angular velocity profile, is30 

(40)

Torque balance at the m, n rational surface requires that30,32

(41)

which reduces to

(42)

where τHm,n=(R0μ0ρ/Bϕ)r̂m,n is the hydromagnetic timescale at the m, n rational surface, whereas τMm,n=(a2ρ/μ)r̂m,n is the corresponding momentum diffusion timescale.

Consider the nonlinear (i.e., emanating from the δJ×δB term in the marginally stable, ideal-MHD, force-balance equation) coupling of the m1, n1, the m2, n2, and the m3, n3 modes. Of course, such coupling is only possible when the mode numbers satisfy the standard three-wave coupling constraint24,25

(43)
(44)

According to the analysis of Ref. 26 (adopting the conventional, large aspect-ratio, tokamak orderings qO(1),ϵ1, and mnϵ), the coupling is such that

(45)
(46)
(47)

where

(48)
(49)

In the following, to facilitate calculations, we adopt a model plasma equilibrium in which

(50)

where 0<c<1. It follows from Eq. (6) that

(51)

from Eq. (23) that

(52)

and from Eq. (11) that

(53)

Furthermore

(54)

and

(55)

Equations (10), (26), (53), and (55) reduce to

(56)

with

(57)

For the 1, 0 mode (which parameterizes the Shafranov shift of the equilibrium magnetic flux-surfaces), Eqs. (17), (18), (56), and (57) give

(58)

for 0r̂c, and

(59)

for c<r̂1. Thus, it follows from Eqs. (16), (51), and (54) that

(60)

In other words, the Shafranov shift is uniform across the plasma. It is also clear that

(61)

and

(62)

For a resonant mode, assuming that cr̂m,n1, Eqs. (11), (19), (51), and (54) yield

(63)

Furthermore, Eqs. (27), (28), (56), and (57) give

(64)

for 0r̂c, and

(65)

for c<r̂r̂m,n, and

(66)

for r̂m,n<r̂b. It follows from Eqs. (29), (65), and (66) that

(67)

where qb=qab2=q(b). Note that E1,n= for the special case of m = 1 internal-kink modes. For conventional m > 1 tearing modes, it is easily demonstrated that Em,n0 provided

(68)

Finally, it is also clear that

(69)

and

(70)

Consider the nonlinear coupling of three modes, resonant at three different rational surfaces within the plasma, whose mode numbers are m1, n1; m2, n2; and m3, n3, where m3=m1+m2 and n3=n1+n2. The fact that the modes are resonant implies that n1,n2,n3>0. It is assumed that m11 and m21. In other words, none of the coupled modes are m = 1 internal-kink modes. It follows that the modes in question are all tearing modes. Without loss of generality, we can write m1/n1<m2/n2, which implies that m1/n1<m3/n3<m2/n2. Hence, according to Eq. (63), r̂m1,n1<r̂m3,n3<r̂m2,n2. It is assumed that all three tearing modes are intrinsically stable—i.e., Em1,n1,Em2,n2,Em3,n3<0—but are maintained in the plasma by the destabilizing influence of the perturbed bootstrap current. Such modes are conventionally termed neoclassical tearing modes.

Equations (48), (49), (53)–(55), and (70) yield

(71)

where

(72)
(73)

Thus, it follows from Eq. (69) that

(74)

where

(75)

It is easily demonstrated that Am,n>0 provided that nq0<m, as must be the case for a resonant tearing mode. Moreover, m¯1n¯q0>0 provided that

(76)

as is assumed to be the case.

According to Eqs. (30), (31), (34), (45)–(47), (52), (63), and (74), the Rutherford island width evolution equations of the three coupled tearing modes take the form

(77)
(78)
(79)

where

(80)

and

(81)

Note that Ψ̂m,n=(wm,n)2/qa. Furthermore, τR=0.8227(23/2μ0a2/η) is the effective resistive evolution timescale of the plasma. Here, for the sake of simplicity, the plasma resistivity, η, is assumed to be uniform across the plasma. Finally

(82)
(83)

where, for the sake of simplicity, the pressure profile is assumed to take the form p(r̂)=p0(1r̂2).

Equations (30), (31), (37), (45)–(47), (52), (63), (74), and (80) imply that the normalized toroidal electromagnetic torques that develop at the three rational surfaces are

(84)
(85)
(86)

where

(87)

The analysis of Sec. II F reveals that

(88)

where

(89)

and ΔΩϕ(r̂,t) is the modification to the plasma toroidal angular velocity profile due to the aforementioned electromagnetic torques. Here, τM=a2ρ/μ is the plasma momentum diffusion timescale. Note that, for the sake of simplicity, the plasma mass density, ρ, and viscosity, μ, are assumed to be uniform across the plasma. Finally

(90)
(91)
(92)

where

(93)

and τH=R0μ0ρ/Bϕ is the plasma hydromagnetic timescale. Here, use has been made of Eqs. (84)–(87).

According to Eq. (81), we can write

(94)

where

(95)

is the so-called slip frequency.30 It follows from Eq. (35) that

(96)

Hence

(97)

where ω0 is the unperturbed slip frequency (i.e., the slip frequency in the absence of electromagnetic torques), ΔΩϕm1,n1(t)=ΔΩϕ(r̂m1,n1,t), etc.

Consider, first, the non-phase-locked state in which the slip frequency, ω, is non-zero.

Assuming that the slip frequency is sufficiently large [see Eq. (110)], we can neglect the terms in Eq. (77)–(79) that depend on cosφ (and, therefore, oscillate at the angular frequency ω), to lowest order, to give

(98)
(99)
(100)

where

(101)
(102)
(103)

To next order, we can write

(104)
(105)
(106)

where δwm1,n1/w¯m1,n11, etc. It follows from Eqs. (77)–(79) and (94) that

(107)
(108)
(109)

The constraints δwm1,n1/w¯m1,n11, etc., imply that

(110)

etc. In other words, Eqs. (104)(106) are valid provided that the slip frequency is sufficiently large.

To first order in small quantities, Eqs. (93) and (104)(106) yield

(111)

where

(112)

and

(113)

or, making use of Eqs. (107)(109)

(114)

Let us assume that the plasma toroidal rotation profile only responds appreciably to the time-averaged component of the electromagnetic torque

(115)

which turns out to be the case provided that τM is sufficiently large [see Eq. (160)]. It thus follows that ΔΩϕ(r̂,t)=ΔΩϕ(r̂). Hence, from Eqs. (88)–(92)

(116)

where

(117)

and

(118)
(119)
(120)

The solution of Eqs. (116)(120) is

(121)

for 0r̂r̂m1,n1, and

(122)

for r̂m1,n1<r̂r̂m3,n3, and

(123)

for r̂m3,n3<r̂r̂m2,n2, and

(124)

for r̂>r̂m2,n2, where

(125)
(126)
(127)

which implies, from Eq. (97), that

(128)

Note that the modification to the plasma toroidal angular velocity profile is localized to the region of the plasma lying within the outermost coupled rational surface.

Combining Eqs. (114) and (128), we obtain the torque balance equation

(129)

where ω̂=ω/ω0, and

(130)

with

(131)

For 0F<1, the torque balance equation, (129), possesses two solutions. One, which is characterized by 1/2<ω̂0, is dynamically stable, whereas the other is dynamically unstable.30 When F reaches the critical amplitude unity, at which point the slip frequency has been reduced to one half of its unperturbed value, the stable and unstable solutions annihilate, and there is a bifurcation to a phase-locked state.30,32

Incidentally, it is clear, from the previous discussion, that there is considerable overlap between the theory of the nonlinear mode coupling of three rotating tearing modes in a tokamak plasma and that of the locking of a single rotating tearing mode to a resonant error-field.30,32 In the former case, two tearing modes in the triplet beat together to form an effective error-field that acts to phase lock the third mode. The amplitude of the effective error-field is proportional to the product of the amplitudes of the two beating modes, whereas its rotation frequency is either the sum or the difference of the rotation frequencies of the beating modes.

In Sec. IV G, we simplified our calculation by assuming that the plasma toroidal rotation profile only responds appreciably to the time-averaged component of the electromagnetic torques that develop at the three coupled rational surfaces. Let us now consider the response of the rotation profile to the much larger amplitude time-varying components of these torques. According to Eq. (111), we have

(132)

where we have only retained the time-averaged component of the first-order term. Let us write

(133)

It follows from Eqs. (88)–(92) that ΔΩϕ0 satisfies Eqs. (116)(120). Furthermore, because ω=dφ/dt

(134)
(135)

where

(136)

and

(137)

Finally

(138)
(139)
(140)

In the limit ωτM1, the solution of Eqs. (134)(140) is such that

(141)
(142)
(143)

Thus, it follows from Eqs. (97) and (133) that

(144)

where use has been made of Eq. (128). It can be seen that the time-varying components of the electromagnetic torques that develop at the three coupled rational surfaces give rise to a modulation of the slip frequency. Given that ω=dφ/dt, we can integrate the previous equation to give

(145)

where

(146)

and

(147)

is the steady-state slip frequency.

The time-averaged component of the Rutherford island width evolution equation, (77), can be rearranged to give

(148)

where w¯m1,n1 is specified in Eq. (101). However, according to Eq. (145)

(149)

Hence

(150)

Equation (148) then yields

(151)

where

(152)

If we now take time variation into account in the Rutherford equation, as described in Sec. IV G, then we obtain

(153)

where δwm1,n1 is specified in Eq. (107) (with ω replaced by ω¯). We can perform similar analysis for the Rutherford equations (78) and (79).

Thus, taking the time-varying, as well as the steady-state, components of the electromagnetic torques acting at the three coupled rational surfaces into account in our analysis; we deduce that the normalized island widths vary as

(154)
(155)
(156)

where w¯m1,n1,w¯m1,n1, and w¯m3,n3 are specified in Eqs. (101)(103), whereas δwm1,n1,δwm2,n2, and δwm3,n3 are given by Eqs. (107)(109) (with ω replaced by ω¯). Here, φ¯ and ω¯ are defined in Eqs. (146) and (147), respectively. Finally

(157)
(158)
(159)

where use has been made of Eq. (112). It can be seen, from Eqs. (154)(156), that the time-varying components of the torques act to reduce the steady-state island widths of the three coupled tearing modes (i.e., w¯m1,n1w¯m1,n1δw¯m1,n1). The origin of this effect is the previously mentioned modulation in the slip frequency, which causes the three coupled modes to spend slightly more time in mutually stabilizing phase relations than in mutually destabilizing relations.30 We can safely neglect this average mutual stabilization effect provided that δw¯m1,n1/w¯m1,n11, etc., when the bifurcation to the phase-locked state, described in Sec. IV G, occurs. The bifurcation occurs when F = 1, where F is specified in Eq. (130). Thus, a comparison of Eq. (130) and the previous three equations reveals that the neglect of the average mutual stabilization effect in the non-phase-locked state is valid provided that

(160)

where w¯ is the typical value of w¯m1,n1,w¯m2,n2, and w¯m3,n3. In the following, for the sake of simplicity, we shall assume that the previous inequality is satisfied (i.e., we shall neglect δw¯m1,n1,δw¯m2,n2, and δw¯m3,n3).

The phase-locked state is characterized by zero slip frequency, i.e., ω = 0.30,32 Thus, it follows from Eq. (94) that

(161)

where φ0 is a constant, and from Eq. (97) that

(162)

In the phase-locked state, we expect ΔΩϕ(r̂,t)=ΔΩϕ(r̂). Thus, the modified plasma toroidal angular velocity profile is governed by Eqs. (116) and (117), where

(163)
(164)
(165)

and

(166)

Here, use has been made of Eqs. (90)–(93). The solution of Eqs. (116), (117), and (163)(166) is specified by Eqs. (121)(124), where

(167)
(168)
(169)

Thus, it follows from Eqs. (130), (162), and (166) that

(170)
(171)
(172)

and

(173)

where

(174)

Now, we know that F1 in the phase-locked state. Moreover, Eq. (110) implies that G1. Hence, Eq. (173) yields

(175)

There are two possible solutions to this equation. The first

(176)

is dynamically stable (because the electromagnetic torques that develop at the three coupled rational surfaces have the same signs as the corresponding torques in the non-phase-locked state, and are, therefore, acting to maintain the phase locking), the second, φ0=π, is dynamically unstable.

According to Eqs. (77)(79), (161), and (176), the Rutherford island width evolution equations of the three coupled modes in the phase-locked state take the form

(177)
(178)
(179)

Thus, writing

(180)
(181)
(182)

we obtain

(183)
(184)
(185)

where use has been made of Eqs. (101)(103).

Note that the modes phase lock in the most mutually destabilizing possible phase relation.30 Thus, whereas, in the non-phase-locked state, the nonlinear interaction of the three coupled modes gives rise to a relatively small amplitude (because ω0τR1) oscillation in the island widths, as described in Sec. IV G, the same interaction in the phase-locked state produces a time-independent increase in the three island widths that is O(ω0τRw¯)1 larger that the amplitude of the aforementioned oscillation.

Incidentally, in the theory presented in this paper, it is impossible for phase locking to occur without the slip frequency being reduced to zero. The ultimate reason for this is the no-slip constraint introduced in Sec. II E. According to this constraint, each magnetic island is convected by the plasma at its rational surface. The no-slip constraint holds whenever the island width significantly exceeds the linear layer width.37 In the opposite limit, the island can slip through the plasma at its rational surface, which would permit phase locking to occur without the slip frequency being reduced to zero. There is evidence from both experiment and numerical modeling that, under these circumstances, phase locking can give rise to a mutual, steady-state, stabilizing effect.38 In this paper, it is assumed that the island widths of all coupled modes remain sufficiently large that the no-slip constraint is always applicable.

Let θ,ϕ be simultaneous solutions of

(186)
(187)

It follows from Sec. II E that one of the X-points of the m1, n1 island chain coincides with one of the X-points of the m2, n2 island chain at the angular position characterized by the poloidal and toroidal angles θ and ϕ, respectively. However, making use of Eqs. (81), (161), and (176), we can add the previous two equations to give

(188)

This implies that one of the X-points of the m3, n3 island chain lies at the angular position θ,ϕ. In other words, the phase-locked state is such that one of the X-points of all three coupled island chains coincides permanently at the angular position specified by the simultaneous solution of Eqs. (186) and (187). It is easily demonstrated that this position rotates poloidally and toroidally at the angular velocities

(189)
(190)

respectively. Likewise, it can be shown that the one of the O-points of all three island chains coincides permanently at the angular position specified by the simultaneous solution of

(191)
(192)

and that this position rotates poloidally and toroidally at the angular velocities (189) and (190), respectively. Note that, at the special angular positions at which the island chain X-and O-points coincide, the radial plasma displacements associated with the three tearing modes add in phase at the edge of the plasma. This behavior is reminiscent of the so-called “slinky mode” generated by the nonlinear phase locking of tearing modes in reversed-field pinches.39–42 

Consider the coupling of two modes, resonant at two different rational surfaces within the plasma, via the 0, 1 perturbation associated with the Shafranov shift of the equilibrium magnetic flux-surfaces. Let the associated mode numbers be m, n and m+1,n, where n > 0. This type of coupling is generally known as toroidal coupling (because it couples tearing modes with a common toroidal mode number).27–29 It follows from Eq. (63) that r̂m,n<r̂m+1,n. As before, it is assumed that Em,n,Em+1,n<0, and m > 1, so that both resonant modes are conventional neoclassical tearing modes.

According to the analysis of Secs. II G and IV B, we can write

(193)

where A1,0 takes the special value

(194)

and

(195)
(196)

Here, use has been made of Eqs. (17), (54), and (61). Note that m¯1n¯q0>0 provided that

(197)

as is assumed to be the case.

Similarly to the analysis of Sec. IV C, the Rutherford island width evolution equations of the two coupled tearing modes take the form

(198)
(199)

where

(200)

Similarly to the analysis of Secs. II F and IV D, the normalized electromagnetic torques that develop at the two coupled rational surfaces are

(201)
(202)

where

(203)

Similarly to the analysis of Sec. IV E, the modified toroidal angular velocity profile satisfies Eqs. (88) and (89), where

(204)
(205)

and

(206)

Similarly to the analysis of Sec. IV F, we can write dφ/dt=ω, where

(207)

and

(208)

In the non-phase-locked state, similarly to the analysis of Sec. IV G, we can write

(209)
(210)

where δwm,n/w¯m,n1, etc. Here, w¯m,n and w¯m+1,n are specified by Eq. (101), and

(211)
(212)

The constraints δwm,n/w¯m,n1, etc., imply that

(213)

etc.

The time-averaged component of the electromagnetic torque can be written T=δT, where

(214)

As before, assuming that the plasma toroidal rotation profile only responds appreciably to the time-averaged torque, we can write ΔΩϕ(r̂,t)=ΔΩϕ(r̂), where ΔΩϕ(r̂) satisfies Eqs. (116) and (117), as well as

(215)
(216)

The solution to these equations is

(217)

for 0r̂r̂m,n, and

(218)

for r̂m,n<r̂r̂m+1,n, and

(219)

for r̂>r̂m+1,n, where

(220)
(221)

which implies that

(222)

Note that the modification to the plasma toroidal angular velocity profile is localized to the region of the plasma lying within the outermost coupled rational surface.

Combining Eqs. (214) and (222), we obtain the torque balance equation

(223)

where ω̂=ω/ω0, and

(224)

with

(225)

As described in Sec. IV G, when F reaches the critical value unity, at which point the slip frequency has been reduced to one half of its unperturbed value, there is a bifurcation to a phase-locked state.

Following the analysis of Sec. IV I, the phase-locked state is characterized by ω = 0, and

(226)

where φ0 is a constant. Thus, Eqs. (207) and (208) yield

(227)
(228)

respectively. If the plasma rotates poloidally as a solid body [i.e., Ωθ(r̂)=Ωθ], as is likely to be the case in the plasma core, then Eq. (227) implies that

(229)

In other words, the phase-locked state is characterized by a toroidal velocity profile in the plasma core that is either flattened (if Ωθ is negligible) or inverted (assuming that Ωθ and Ωϕ have the same sign). This entails a substantial loss of core momentum confinement.

In the phase-locked state, we expect ΔΩϕ(r̂,t)=ΔΩϕ(r̂). Thus, the modified plasma toroidal angular velocity profile is governed by Eqs. (116) and (117), where

(230)
(231)

and

(232)

The solution of Eqs. (116), (117), (230), and (231) is given by Eqs. (217)(219), where

(233)
(234)

Thus, it follows from Eqs. (224), (228), and (232) that

(235)
(236)

and

(237)

where

(238)

Now, we know that F1 in the phase-locked state. Moreover, Eq. (213) implies that G1. Hence, Eq. (237) yields

(239)

As before, the dynamically stable solution of this equation is

(240)

According to Eqs. (198), (199), (226), and (240), the Rutherford island width evolution equations of the two coupled tearing modes in the phase-locked state take the form

(241)
(242)

Thus, writing

(243)
(244)

we obtain

(245)
(246)

where use has been made of Eq. (101).

As previously, the two tearing modes phase lock in the most mutually destabilizing possible phase relation. Furthermore, whereas, in the non-phase-locked state, the nonlinear interaction of the two coupled modes gives rise to a relatively small amplitude oscillation in the island widths, the same interaction in the phase-locked state produces a time-independent increase in the two island widths that is much larger that the amplitude of the aforementioned oscillation.

Let θ,ϕ be simultaneous solutions of

(247)
(248)

It follows that one of the X-points of the m, n island chain coincides with one of the X-points of the m+1,n island chain at the angular position characterized by the poloidal and toroidal angles θ and ϕ, respectively. However, making use of Eqs. (200), (226), and (240), we can solve the previous two equations to give

(249)
(250)

In other words, in the phase-locked state, one of the X-points of the two coupled island chains coincides permanently on the outboard mid-plane (i.e., θ = 0). Moreover, the two X-points rotate toroidally at the angular velocity

(251)

Likewise, it can be shown that one of the O-points of the two coupled island chains also coincides permanently on the outboard mid-plane at the toroidal angle

(252)

and that the two O-points rotate toroidally at the angular velocity (251). The prediction that toroidally coupled tearing modes phase lock in a configuration in which their X- and O-points coincide permanently on the outboard mid-plane is in accordance with experimental observations.43 

It is possible for two tearing modes possessing different toroidal mode numbers to phase lock via a combination of non-linear and toroidal coupling. Let 2m1,2n and m, n be the associated mode numbers, where m > 1 and n > 0. The coupling proceeds in two stages. First, the m, n mode couples nonlinearly to itself to produce a 2m,2n mode. Second, the 2m,2n mode couples toroidally to the 2m1,2n mode. It follows from Eq. (63) that r̂2m1,2n<r̂m,n. As before, it is assumed that E2m1,2n,Em,n<0, and m > 1, so that both coupled modes are neoclassical tearing modes. It is also assumed that E2m,2n<0 (because this is inevitably the case if Em,n<0).

According to the analysis of Sec. II G, we can write

(253)
(254)
(255)

where Ψ1,0=Δa1,0/qa. Setting ΔΨ2m,2n=0, which ensures that there is no 2m,2n current sheet at the m, n rational surface, we obtain

(256)

as our expression for the equilibrium 2m,2n magnetic flux driven at the rational surface due to nonlinear coupling with the 2m1,2n and m, n modes. Note that |Ψ2m,2n|/|Ψm,n|1, assuming that |Ψm,n|,|Ψ2m1,2n|,|Ψ1,0|1 (i.e., the 2m1,2n and m, n island widths, as well as the edge Shafranov shift, are all relatively small compared to the plasma minor radius), which ensures that the magnetic island at the m, n rational surface predominately has the helicity m, n, rather than 2m,2n. Equations (253), (254), and (256) yield

(257)
(258)

where

(259)
(260)

However

(261)
(262)

where

(263)
(264)

Note that m¯1n¯q0>0 provided that

(265)

as is assumed to be the case.

Similarly to the analysis of Sec. IV C, the Rutherford island width evolutions equations of the two coupled tearing modes take the form

(266)
(267)

where

(268)

and

(269)

Similarly to the analysis of Secs. II F and IV D, the normalized electromagnetic torques that develop at the two rational surfaces are

(270)
(271)

where

(272)

Similarly to the analysis of Sec. IV E, the modified toroidal angular velocity profile satisfies Eqs. (88) and (89), where

(273)
(274)

and

(275)

Similarly to the analysis of Sec. IV F, we can write dφ/dt=ω, where

(276)

and

(277)

In the non-phase-locked state, similarly to the analysis of Sec. IV G, we can write

(278)
(279)

where δw2m1,2n/w¯2m1,2n1, etc.. Here, w¯2m1,2n and w¯m,n are specified by Eq. (101) (except that E2m1,2n is replaced by Ẽ2m1,2n, etc.), and

(280)
(281)

The constraints δw2m1,2n/w¯2m1,2n1, etc., imply that

(282)

The time-averaged component of the electromagnetic torque can be written T=δT, where

(283)

As before, assuming that the plasma toroidal angular velocity profile only responds appreciably to the time-averaged torque, we can write ΔΩϕ(r̂,t)=ΔΩϕ(r̂), where ΔΩϕ(r̂) satisfies Eqs. (116) and (117), as well as

(284)
(285)

The solution to these equations is

(286)

for 0r̂r̂2m1,2n, and

(287)

for r̂2m1,2n<r̂r̂m,n, and

(288)

for r̂>r̂m,n, where

(289)
(290)

which implies that

(291)

Note that the modification to the plasma toroidal angular velocity profile is localized to the region of the plasma lying within the outermost coupled rational surface.

Combining Eqs. (283) and (291), we obtain the torque balance equation

(292)

where ω̂=ω/ω0, and

(293)

with

(294)

As described in Sec. IV G, when F reaches the critical value unity, at which point the slip frequency has been reduced to one half of its unperturbed value, there is a bifurcation to a phase-locked state.

Following the analysis of Sec. IV I, the phase-locked state is characterized by ω = 0, and

(295)

where φ0 is a constant. Thus, Eqs. (276) and (277) yield

(296)
(297)

respectively. If the plasma rotates poloidally as a solid body [i.e., Ωθ(r̂)=Ωθ], as is likely to be the case in the plasma core, then Eq. (296) implies that

(298)

In other words, the phase-locked state is characterized by a toroidal velocity profile in the plasma core that is either flattened or inverted. This entails a substantial loss of core momentum confinement.

In the phase-locked state, we expect ΔΩϕ(r̂,t)=ΔΩϕ(r̂). Thus, the modified plasma toroidal angular velocity profile is governed by Eqs. (116) and (117), where

(299)
(300)

and

(301)

The solution of Eqs. (116), (117), (299), and (300) is given by Eqs. (286)(288), where

(302)
(303)

Thus, it follows from Eqs. (293), (297), and (301) that

(304)
(305)

and

(306)

where

(307)

Now, we know that F1 in the phase-locked state. Moreover, Eq. (282) implies that G1. Hence, Eq. (306) yields

(308)

As before, the dynamically stable solution of this equation is

(309)

According to Eqs. (266), (267), (295), and (309), the Rutherford island width evolution equations of the two coupled tearing modes in the phase-locked state take the form

(310)
(311)

Thus, writing

(312)
(313)

we obtain

(314)
(315)

where use has been made of Eq. (101) [with Em,n replaced by Ẽm,n, etc.].

As previously, the two tearing modes phase lock in the most mutually destabilizing possible phase relation. Furthermore, whereas, in the non-phase-locked state, the nonlinear interaction of the two coupled modes gives rise to a relatively small amplitude oscillation in the island widths, the same interaction in the phase-locked state produces a time-independent increase in the two island widths that is much larger that the amplitude of the aforementioned oscillation.

Let θ,ϕ be simultaneous solutions of

(316)
(317)

It follows that one of the X-points of the 2m1,2n island chain coincides with one of the X-points of the m, n island chain at the angular position characterized by the poloidal and toroidal angles θ and ϕ, respectively. However, making use of Eqs. (269), (295), and (309), we can solve the previous two equations to give

(318)
(319)

In other words, in the phase-locked state, one of the X-points of the two coupled island chains coincides permanently on the outboard mid-plane (i.e., θ = 0). Moreover, the two X-points rotate toroidally at the angular velocity

(320)

Likewise, it can be shown that one of the O-points of the two coupled island chains also coincides permanently on the inboard mid-plane at the toroidal angle

(321)

and that the two O-points rotate toroidally at the angular velocity (320).

The type of nonlinear toroidal coupling described in this section has been observed in both the JET and the DIII-D tokamaks.21,22 To be more exact, in both cases, the 3, 2 and 2, 1 neoclassical tearing modes are observed to phase lock to one another. This phase locking is inferred to take place in two stages. First, the 2, 1 mode couples to itself to produce a 4, 2 mode. Second, the 4, 2 mode couples toroidally to the 3, 2 mode. In accordance with the previous analysis, the observed phase-locked state is such that φ3,2=2φ2,1, which necessitates a toroidal angular velocity profile in the plasma core that is either flattened or inverted (as is observed to be the case). In further accordance, the phase-locking bifurcation is observed to take place when the slip frequency has been reduced to about one half of its original value, the modification to the toroidal velocity profile is localized to the region interior to the 2, 1 rational surface, and the phase-locked state is such that one of the X-points of the 3, 2 and 2, 1 island chains coincides permanently on the outboard mid-plane.

It is helpful to define the conventional plasma beta parameter

(322)

where denotes a volume average, as well as the normal beta parameter

(323)

where Iϕ=2πBϕa2/(μ0R0qa) is the toroidal plasma current.1 Finally, the magnetic Prandtl number of the plasma is defined

(324)

Consider the non-linear phase locking of three neoclassical tearing modes whose mode numbers are m1, n1; m2, n2; and m3, n3, where m3=m1+m2 and n3=n1+n2. As discussed in Sec. IV G, the phase-locking criterion is F1, where

(325)

Here, ω̂0=ω0τH, and

(326)

Furthermore

(327)

and

(328)

Finally, r̂m,n is specified in Eq. (63), Em,n in Eq. (67), m¯ in Eq. (72), n¯ in Eq. (73), and Am,n in Eq. (75).

According to Eq. (325), the nonlinear phase locking of three neoclassical tearing modes with different poloidal and toroidal mode numbers is greatly facilitated by increasing βN, but is impeded by increasing differential plasma rotation (parameterized by ω̂0) and plasma viscosity (parameterized by P). Figure 1 shows F̂1(m1,n1;m2,n2;q0,qa,qb) evaluated as a function of q0 and qa for the nonlinear phase locking of the 2, 1, the 3, 2, and the 5, 3 neoclassical tearing modes in a fixed boundary plasma (i.e., qb = qa). It can be seen that F̂1 increases very strongly with increasing q0. Moreover, at high q0, F̂1 also increases strongly with increasing qa. This indicates that the nonlinear phase locking of three neoclassical tearing modes in a tokamak plasma is greatly facilitated by increasing q0, and, facilitated to a lesser extent by increasing qa.

FIG. 1.

F̂1(m1,n1;m2,n2;q0,qa,qb) calculated as a function of qa for m1=2,n1=1,m2=3,n2=2, and qb = qa. The various curves correspond to q0=0.8, 0.9, 0.95, and 1.0, respectively, in order from the bottom to the top.

FIG. 1.

F̂1(m1,n1;m2,n2;q0,qa,qb) calculated as a function of qa for m1=2,n1=1,m2=3,n2=2, and qb = qa. The various curves correspond to q0=0.8, 0.9, 0.95, and 1.0, respectively, in order from the bottom to the top.

Close modal

Consider the toroidal phase locking of two neoclassical tearing modes whose mode numbers are m, n and m+1,n. As discussed in Sec. V G, the phase-locking criterion is F1, where

(329)

Here

(330)

where

(331)

Finally, r̂m,n is specified in Eq. (63), Em,n in Eq. (67), Am,n in Eq. (75), m¯ in Eq. (195), n¯ in Eq. (196), and ŵm,n in Eq. (328).

According to Eq. (329), the toroidal phase locking of two neoclassical tearing modes with the same toroidal mode number is facilitated by increasing βN, and increasing Shafranov shift (parameterized by Δa1,0), but is impeded by increasing differential plasma rotation (parameterized by ω̂0) and plasma viscosity (parameterized by P). Figure 2 shows F̂2(m,n;q0,qa,qb) evaluated as a function of q0 and qa for the toroidal phase locking of the 2, 1 and the 3, 1 neoclassical tearing modes in a fixed boundary plasma (i.e., qb = qa). As before, it can be seen that F̂2 increases very strongly with increasing q0, and, at high q0, also increases strongly with increasing qa, which indicates that the toroidal phase locking of two neoclassical tearing modes is greatly facilitated by increasing q0, and, facilitated to a lesser extent by increasing qa. Note that F̂2 is significantly larger than F̂1, which simply indicates that conventional toroidal coupling is a much stronger effect in tokamak plasmas than nonlinear three-wave coupling.

FIG. 2.

F̂2(m,n;q0,qa,qb) calculated as a function of qa for m = 2, n = 1, and qb = qa. The various curves correspond to q0=0.8, 0.9, 0.95, and 1.0, respectively, in order from the bottom to the top.

FIG. 2.

F̂2(m,n;q0,qa,qb) calculated as a function of qa for m = 2, n = 1, and qb = qa. The various curves correspond to q0=0.8, 0.9, 0.95, and 1.0, respectively, in order from the bottom to the top.

Close modal

Consider the nonlinear toroidal phase locking of twoneoclassical tearing modes whose mode numbers are 2m1,2n and m, n. As discussed in Sec. VI G, the phase-locking criterion is F1, where

(332)

Here

(333)

where

(334)

Finally, r̂m,n is specified in Eq. (63), Em,n in Eq. (67), Am,n in Eq. (75), m¯ in Eq. (263), n¯ in Eq. (264), and ŵm,n in Eq. (328). Note that we are neglecting the distinction between Ẽ2m1,2n and E2m1,2n, as well as that between Ẽm,n and Em,n [see Eqs. (259) and (260)]. This is a reasonable approximation provided that the island widths, as well as the Shafranov shift, are small compared to the plasma minor radius.

According to Eq. (332), the nonlinear toroidal phase locking of two neoclassical tearing modes with different poloidal and toroidal mode numbers is greatly facilitated by increasing βN, and facilitated to a lesser extent by increasing Shafranov shift (parameterized by Δa1,0), but is impeded by increasing differential plasma rotation (parameterized by ω̂0) and plasma viscosity (parameterized by P). Figure 3 shows F̂3(m,n;q0,qa,qb) evaluated as a function of q0 and qa for the nonlinear toroidal phase locking of the 2, 1 and the 3, 2 neoclassical tearing modes in a fixed boundary plasma (i.e., qb = qa). As before, it can be seen that F̂3 increases very strongly with increasing q0, and, at high q0, also increases strongly with increasing qa, which indicates that the nonlinear toroidal phase locking of two neoclassical tearing modes is greatly facilitated by increasing q0, and, facilitated to a lesser extent by increasing qa. Note that F̂3 is significantly smaller than F̂2, which simply indicates that conventional toroidal coupling is a much stronger effect in tokamak plasmas than nonlinear toroidal coupling.

FIG. 3.

F̂3(m,n;q0,qa,qb) calculated as a function of qa for m = 2, n = 1, and qb = qa. The various curves correspond to q0=0.8, 0.9, 0.95, and 1.0, respectively, in order from the bottom to the top.

FIG. 3.

F̂3(m,n;q0,qa,qb) calculated as a function of qa for m = 2, n = 1, and qb = qa. The various curves correspond to q0=0.8, 0.9, 0.95, and 1.0, respectively, in order from the bottom to the top.

Close modal

In this paper, we have investigated three distinct coupling mechanisms that can lead to the eventual phase locking of multiple neoclassical tearing modes in a tokamak plasma.

The first mechanism is the nonlinear three-wave coupling of three neoclassical tearing modes whose mode numbers are m1, n1; m2, n2; and m3=m1+m2,n3=n1+n2 (see Sec. IV). We find that there is a bifurcation to a phase-locked state when the mode amplitudes exceed a threshold value such that the frequency mismatch between the three modes is reduced to one half of its unperturbed value. The modes phase lock in the most mutually destabilizing possible phase relation. Moreover, the phase-locked state is such that one of the X-points of all three of the magnetic island chains associated with the different modes coincides permanently at a particular angular location that rotates both poloidally and toroidally. Note that this type of phase locking is fairly unlikely to occur in tokamaks, because it requires three neoclassical tearing modes (that just happen to satisfy the three-wave coupling constraint m3=m1+m2,n3=n1+n2) to be present simultaneously in the plasma, which would be a very unusual occurrence.

The second mechanism is the conventional toroidal coupling of two neoclassical tearing modes whose mode numbers are m, n and m+1,n (see Sec. V). This type of coupling can be treated as a form of nonlinear three-wave coupling in which one of the three coupled modes is the 1, 0 component of the plasma equilibrium associated with the Shafranov shift. We find that there is a bifurcation to a phase-locked state when the mode amplitudes exceed a threshold value such that the frequency mismatch between the m, n and the m+1,n modes is reduced to one half of its unperturbed value. The modes phase lock in the most mutually destabilizing possible phase relation. Moreover, the phase-locked state is such that one of the X-points of the magnetic island chains associated with the m, n and the m+1,n modes coincides permanently at an angular position on the outboard mid-plane that rotates toroidally. Finally, the phase-locked state is characterized by a modified toroidal angular velocity profile, internal to the m+1,n rational surface, which is either flattened or inverted (depending on the amount of poloidal flow present in the plasma). Toroidal coupling of tearing modes with the same toroidal mode number, and poloidal mode numbers differing by unity, is frequently observed in tokamak plasmas.43 Moreover, in accordance with our analysis, the observed phase-locked state is such that one of the X-points of the coupled island chains coincides permanently on the outboard mid-plane at an angular location that rotates toroidally.43 

The third mechanism is the nonlinear toroidal coupling of two neoclassical tearing modes whose mode numbers are m, n and 2m1,2n (see Sec. VI). This coupling takes place in two stages. First, the m, n mode coupled nonlinearly to itself to produce the 2m,2n mode. Second, the 2m,2n mode couples toroidally to the 2m1,2n mode. We find that there is a bifurcation to a phase-locked state when the mode amplitudes exceed a threshold value such that the frequency mismatch between the 2m1,2n and the 2m,2n modes is reduced to one half of its unperturbed value. The modes phase lock in the most mutually destabilizing possible phase relation. Moreover, the phase-locked state is such that one of the X-points of the magnetic island chains associated with the m, n and the 2m1,2n modes coincides permanently at an angular position on the outboard mid-plane that rotates toroidally. Finally, the phase-locked state is characterized by a modified toroidal angular velocity profile, internal to the m,n rational surface, which is either flattened or inverted (depending on the amount of poloidal flow present in the plasma). The nonlinear toroidal coupling of the 2, 1 and the 3, 2 neoclassical tearing modes has been observed on both JET and DIII-D.21,22 In accordance with our analysis, the phase-locking bifurcation is observed to take place when the frequency mismatch between the 3, 2 and the 4, 2 modes has been reduced to about one half of its original value. Moreover, the phase-locked state is characterized by a toroidal angular velocity profile, internal to the 2, 1 rational surface, which is either flattened or inverted. In further accordance, the phase-locked state is such that one of the X-points of the 3, 2 and 2, 1 island chains coincides permanently on the outboard mid-plane at an angular location that rotates toroidally.

This research was funded by the U.S. Department of Energy under Contract No. DE-FG02-04ER-54742.

1.
J. A.
Wesson
,
Tokamaks
, 4th ed. (
Oxford University Press
,
Oxford
,
2012
).
2.
J. P.
Freidberg
,
Ideal Magnetohydrodynamics
(
Springer
,
Berlin
,
1987
).
3.
4.
H. P.
Furth
,
J.
Killeen
, and
M. N.
Rosenbluth
,
Phys. Fluids
6
,
459
(
1963
).
5.
P. H.
Rutherford
,
Phys. Fluids
16
,
1903
(
1973
).
6.
A.
Thyagaraja
,
Phys. Fluids
24
,
1716
(
1981
).
7.
D. F.
Escande
and
M.
Ottaviani
,
Phys. Lett. A
323
,
278
(
2004
).
8.
R. J.
Hastie
,
F.
Militello
, and
F.
Porcelli
,
Phys. Rev. Lett.
95
,
065001
(
2005
).
9.
Z.
Chang
and
J. D.
Callen
,
Nucl. Fusion
30
,
219
(
1990
).
10.
R.
Carrera
,
R. D.
Hazeltine
, and
M.
Kotschenreuther
,
Phys. Fluids
29
,
899
(
1986
).
11.
R.
Fitzpatrick
,
Phys. Plasmas
2
,
825
(
1995
).
12.
R. J.
Buttery
,
S.
Günter
,
G.
Giruzzi
,
T. C.
Hender
,
D.
Howell
,
G.
Huysmans
,
R. J.
La Haye
,
M.
Maraschek
,
H.
Reimerdes
,
O.
Sauter
,
C. D.
Warrick
,
H. R.
Wilson
, and
H.
Zohm
,
Plasma Phys. Controlled Fusion
42
,
B61
(
2000
).
13.
Q.
Yu
,
S.
Günter
,
K.
Lackner
,
A.
Gude
, and
M.
Maraschek
,
Nucl. Fusion
40
,
2031
(
2000
).
14.
Z.
Chang
,
J. D.
Callen
,
E. D.
Fredrickson
,
R. V.
Budny
,
C. C.
Hegna
,
K. M.
McGuire
,
M. C.
Zarnstorff
, and
the TFTR Group
,
Phys. Rev. Lett.
74
,
4663
(
1995
).
15.
R. J.
La Haye
,
L. L.
Lao
,
E. J.
Strait
, and
T. S.
Taylor
,
Nucl. Fusion
37
,
397
(
1997
).
16.
The JET Team
,
Nucl. Fusion
39
,
1965
(
1999
).
17.
O.
Gruber
,
R. C.
Wolf
,
R.
Dux
,
C.
Fuchs
,
S.
Günter
,
A.
Kallenbach
,
K.
Lackner
,
M.
Maraschek
,
P. J.
McCarthy
,
H.
Meister
,
G.
Pereverzev
,
F.
Ryter
,
J.
Schweinzer
,
U.
Seidel
,
S.
Sesnic
,
A.
Stäbler
,
J.
Stober
, and
the ASDEX Upgrade Team
,
Phys. Rev. Lett.
83
,
1787
(
1999
).
18.
T. C.
Luce
,
M. R.
Wade
,
P. A.
Politzer
,
S. L.
Allen
,
M. E.
Austin
,
D. R.
Baker
,
B.
Bray
,
D. P.
Brennan
,
K. H.
Burrell
,
T. A.
Casper
,
M. S.
Chu
,
J. C.
DeBoo
,
E. J.
Doyle
,
J. R.
Ferron
,
A. M.
Garofalo
,
P.
Gohil
,
I. A.
Gorelov
,
C. M.
Greenfield
,
R. J.
Groebner
,
W. W.
Heidbrink
,
C.-L.
Hsieh
,
A. W.
Hyatt
,
R.
Jayakumar
,
J. E.
Kinsey
,
R. J.
La Haye
,
L. L.
Lao
,
C. J.
Lasnier
,
E. A.
Lazarus
,
A. W.
Leonard
,
Y. R.
Lin-Liu
,
J.
Lohr
,
M. A.
Mahdavi
,
M. A.
Makowski
,
M.
Murakami
,
C. C.
Petty
,
R. I.
Pinsker
,
R.
Prater
,
C. L.
Rettig
,
T. L.
Rhodes
,
B. W.
Rice
,
E. J.
Strait
,
T. S.
Taylor
,
D. M.
Thomas
,
A. D.
Turnbull
,
J. G.
Watkins
,
W. P.
West
, and
K.-L.
Wong
,
Nucl. Fusion
41
,
1585
(
2001
).
19.
M. R.
Wade
,
T. C.
Luce
,
R. J.
Jayakumar
,
P. A.
Politzer
,
A. W.
Hyatt
,
J. R.
Ferron
,
C. M.
Greenfield
,
M.
Murakami
,
C. C.
Petty
,
R.
Prater
,
J. C.
DeBoo
,
R. J.
La Haye
,
P.
Gohil
, and
T. L.
Rhodes
,
Nucl. Fusion
45
,
407
(
2005
).
20.
A. C. C.
Sips
,
G.
Tardini
,
C. B.
Forest
,
O.
Gruber
,
P. J.
Mc Carthy
,
A.
Gude
,
L. D.
Horton
,
V.
Igochine
,
O.
Kardaun
,
C. F.
Maggi
,
M.
Maraschek
,
V.
Mertens
,
R.
Neu
,
A. G.
Peeters
,
G. V.
Pereverzev
,
A.
Stäbler
,
J.
Stober
,
W.
Suttrop
, and
the ASDEX Upgrade Team
,
Nucl. Fusion
47
,
1485
(
2007
).
21.
T. C.
Hender
,
D. F.
Howell
,
R. J.
Buttery
,
O.
Sauter
,
F.
Sartori
,
R. J.
LaHaye
,
A. W.
Hyatt
,
C. C.
Petty
,
the JET EFDA Contributors, and the DIII-D Team
,
Nucl. Fusion
44
,
788
(
2004
).
22.
B.
Tobias
,
B. A.
Grierson
,
C. M.
Muscatello
,
X.
Ren
,
C. W.
Dornier
,
N. C.
Luhmann
, Jr.
,
S. E.
Zemedkun
,
T. L.
Munsat
, and
I. G. J.
Classen
,
Rev. Sci. Instrum.
85
,
11D847
(
2014
).
23.
E.
Alessi
,
P.
Buratti
,
E.
Giovannozzi
,
G.
Pucella
, and
the JET EDFA Contributors
, in
Proceedings of 14th European Physical Society Conference on Plasma Physics
(
2014
), No. P1.028.
24.
C. C.
Hegna
,
Phys. Plasmas
3
,
4646
(
1996
).
25.
R. M.
Coelho
,
E.
Lazzaro
,
M. F.
Nave
, and
F.
Serra
,
Phys. Plasmas
6
,
1194
(
1999
).
26.
R.
Fitzpatrick
,
Phys. Plasmas
6
,
1168
(
1999
).
27.
J. W.
Connor
,
S. C.
Cowley
,
R. J.
Hastie
,
T. C.
Hender
,
A.
Hood
, and
T. J.
Martin
,
Phys. Fluids
31
,
577
(
1988
).
28.
J. W.
Connor
,
R. J.
Hastie
, and
J. B.
Taylor
,
Phys. Fluids B
3
,
1532
(
1991
).
29.
R.
Fitzpatrick
,
R. J.
Hastie
,
T. J.
Martin
, and
C. M.
Roach
,
Nucl. Fusion
33
,
1533
(
1993
).
30.
R.
Fitzpatrick
,
Nucl. Fusion
33
,
1049
(
1993
).
31.
R.
Fitzpatrick
,
Phys. Plasmas
1
,
3308
(
1994
).
32.
R.
Fitzpatrick
,
Phys. Plasmas
5
,
3325
(
1998
).
33.
R.
Fitzpatrick
,
Phys. Plasmas
10
,
2304
(
2003
).
34.
E.
Poli
,
A. G.
Peeters
,
A.
Bergmann
,
S.
Günter
, and
S. D.
Pinches
,
Phys. Rev. Lett.
88
,
075001
(
2002
).
35.
M.
Kotschenreuther
,
R. D.
Hazeltine
, and
P. J.
Morrison
,
Phys. Fluids
28
,
294
(
1985
).
36.
T. H.
Stix
,
Phys. Fluids
16
,
1260
(
1973
).
37.
R.
Fitzpatrick
,
Phys. Plasmas
21
,
092513
(
2014
).
38.
Q.
Hu
,
Q.
Yu
,
B.
Rao
,
Y.
Ding
,
X.
Hu
,
G.
Zhuang
, and
the J-TEXT Team
,
Nucl. Fusion
52
,
083011
(
2012
).
39.
R. J.
La Haye
,
T. N.
Carlstrom
,
R. R.
Goforth
,
G. L.
Jackson
,
M. J.
Schaffer
,
T.
Tamano
, and
P. L.
Taylor
,
Phys. Fluids
27
,
2576
(
1984
).
40.
T.
Tamano
,
W. D.
Boyd
,
C.
Chu
,
Y.
Kondoh
,
R. J.
La Haye
,
P. S.
Lee
,
M.
Saito
,
M. J.
Schaffer
, and
P. L.
Taylor
,
Phys. Rev. Lett.
59
,
1444
(
1987
).
41.
K.
Hattori
,
Y.
Hirano
,
T.
Shimada
,
Y.
Yagi
,
Y.
Maejima
,
I.
Hirota
, and
K.
Ogawa
,
Phys. Fluids B
3
,
3111
(
1991
).
42.
R.
Fitzpatrick
and
P.
Zanca
,
Phys. Plasmas
9
,
2707
(
2002
).
43.
W.
Suttrop
,
K.
Büchl
,
J. C.
Fuchs
,
M.
Kaufmann
,
K.
Lackner
,
M.
Maraschek
,
V.
Mertens
,
R.
Neu
,
M.
Schittenhelm
,
M.
Sokoll
,
H.
Zohm
, and
the Asdex Upgrade Team
,
Nucl. Fusion
37
,
119
(
1997
).