Interaction of a Magnetic Island Chain in a Tokamak Plasma with a Resonant Magnetic Perturbation of Rapidly Oscillating Phase Richard Fitzpatrick Institute for Fusion Studies Department of Physics University of Texas at Austin Austin, TX 78712 An investigation is made into the interaction of a magnetic island chain, embedded in a tokamak plasma, with an externally generated magnetic perturbation of the same helicity, whose helical phase is rapidly oscillating. The analysis is similar in form to the classic analysis used by Kapitza [Soviet Phys. JETP 21, 588 (1951)] to examine the angular motion of a rigid pendulum whose pivot point undergoes rapid vertical oscillations. The rapid phase oscillations of the magnetic perturbation are found to give rise to new terms in the equations that govern the secular evolution of the island chain’s width and helical phase. It is also found that a magnetic island chain preferentially locks to an external magnetic perturbation of the same helicity, whose phase oscillates rapidly at a sufficiently high amplitude, in phase quadrature: i.e., such that the chain is neither stabilized nor destabilized by the perturbation. The implications of this result are discussed.
Transcript
Interaction of a Magnetic Island Chain in a Tokamak Plasma with
a Resonant Magnetic Perturbation of Rapidly Oscillating Phase
RichardFitzpatrick
Institute for Fusion Studies
Department of Physics
University of Texas at Austin
Austin, TX 78712
An investigation is made into the interaction of a magnetic island chain, embedded
in a tokamak plasma, with an externally generated magnetic perturbation of the same
helicity, whose helical phase is rapidly oscillating. The analysis is similar in form to
the classic analysis used by Kapitza [Soviet Phys. JETP 21, 588 (1951)] to examine
the angular motion of a rigid pendulum whose pivot point undergoes rapid vertical
oscillations. The rapid phase oscillations of the magnetic perturbation are found
to give rise to new terms in the equations that govern the secular evolution of the
island chain’s width and helical phase. It is also found that a magnetic island chain
preferentially locks to an external magnetic perturbation of the same helicity, whose
phase oscillates rapidly at a sufficiently high amplitude, in phase quadrature: i.e.,
such that the chain is neither stabilized nor destabilized by the perturbation. The
implications of this result are discussed.
2
I. INTRODUCTION
A tokamak is a device that is designed to trap a thermonuclear plasma on a set of
axisymmetric, toroidally-nested, magnetic flux-surfaces.1 Heat and particles are able to flow
around the flux-surfaces relatively rapidly due to the free streaming of charged particles
along magnetic field-lines. On the other hand, heat and particles can only diffuse across the
flux-surfaces relatively slowly, assuming that the magnetic field-strength is large enough to
render the particle gyroradii much smaller than the device’s minor radius.2
Tokamak plasmas are subject to a number of macroscopic instabilities that limit their
effectiveness.3 So-called tearing modes are comparatively slowly growing instabilities 4 that
saturate at relatively low amplitudes,5–9 in the process reconnecting magnetic flux-surfaces to
form helical structures known as magnetic island chains. Magnetic island chains are radially
localized structures centered on so-called rational magnetic flux-surfaces which satisfy the
resonance criterion k · B = 0, where k is the wave-number of the instability, and B 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.10
In many previous tokamak experiments, an externally generated, rotating, magnetic per-
turbation, resonant at some rational surface that lies within the plasma, was employed in
an attempt to reduce the amplitude of a saturated magnetic island chain that was localized
at the same rational surface (see, for instance, Refs. 11, 12, and 13). A major obstacle to
this method of tearing mode suppression is existence of the so-called phase instability, which
causes the helical phase of a magnetic island chain that has a stabilizing phase relation to an
external magnetic perturbation to spontaneously flip, such that the chain has a destabilizing
phase relation, on a comparatively short timescale.14 Rotating magnetic perturbations can
also be used to modify the flow profile in a tokamak, because they exert an electromagnetic
torque on the plasma in the immediate vicinity of a saturated magnetic island chain of the
same helicity (see, for instance, Ref. 15). Unfortunately, the existence of the phase instabil-
ity ensures that the island chain always phase-locks to the rotating perturbation in such a
manner that the chain is destabilized, which has a deleterious effect on plasma energy and
particle confinement.
3
Tokamak plasmas are highly sensitive to externally generated, non-rotating, magnetic
perturbations, resonant at some rational surface that lies within the plasma. Such per-
turbations, which are conventionally termed error-fields, are present in all tokamak exper-
iments because of imperfections in magnetic field-coils. An error-field can drive magnetic
reconnection in an otherwise tearing-stable plasma, giving rise to the formation of a non-
rotating magnetic island chain, localized at the associated rational surface, which is locked
in a destabilizing phase relation to the error-field.16–18 Such chains, which are generally
known as locked modes, severely degrade global energy confinement,10 and often trigger
major disruptions.16–18
The electromagnetic torque exerted by a resonant external magnetic perturbation on the
plasma in the vicinity of a magnetic island chain of the same helicity varies as the sine of
the helical phase difference between the chain and the perturbation.19 This torque modifies
the plasma flow in the vicinity of the island chain, which, in turn, changes the chain’s
helical phase, because the chain is constrained to co-rotate with the plasma at the rational
surface.19 Now, the gravitational torque exerted on a rigid pendulum depends on the sine
of the angle subtended between the pendulum and the downward vertical. It follows that
there is a analogy between the phase evolution of a magnetic island chain interacting with
a resonant external magnetic perturbation and the angular motion of a rigid pendulum.
In the case of a rigid pendulum, there are two equilibrium states. The first, in which
the pendulum’s center of mass is directly below the pivot point, is dynamically stable. The
second, in which the center of mass is directly above the pivot point, is dynamically unstable.
Likewise, in the case of a magnetic island chain interacting with a resonant external magnetic
perturbation of the same helicity (in the absence of plasma flow), there are two equilibrium
states.19 The first, in which the helical phase difference between the island chain and the
external perturbation is zero, is dynamically stable. The second, in which the helical phase
difference between the island chain and the external perturbation is π radians, is dynamically
unstable. Unfortunately, the dynamically stable equilibrium state is such that the island
chain is maximally destabilized by the external perturbation, and vice versa.19
It is well known that if the pivot point of a rigid pendulum is made to execute a small-
amplitude, vertical oscillation of sufficiently high frequency then the equilibrium state in
which the pendulum’s center of mass lies directly above the pivot point can be rendered dy-
namically stable, whilst the equilibrium state in which the center of mass lies directly below
4
the pivot point is rendered dynamically unstable.20,21 The analogy that exists between the
phase evolution of a magnetic island chain interacting with a resonant external magnetic
perturbation of the same helicity and the angular motion of a rigid pendulum leads us to
speculate that if the helical phase of the external perturbation were subject to a small am-
plitude, high frequency oscillation then it might be possible to induce the island chain to
lock to the perturbation in the maximally stabilizing phase relation. Such a phenomenon,
if it existed, would greatly facilitate the magnetic control of tearing modes in tokamak plas-
mas, and would also prevent any confinement degradation associated with the modification
of the plasma flow profile by rotating external magnetic perturbations. Furthermore, the
phenomenon in question might constitute a mechanism for expelling error-field driven locked
modes from tokamak plasmas, and could possibly even prevent their formation in the first
place. The aim of this paper is to explore the aforementioned possibilities.
II. PRELIMINARY ANALYSIS
A. Plasma Equilibrium
Consider a large aspect-ratio, low-β, 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. Standard cylindrical
coordinates (r, θ, z) are adopted. The system is assumed to be periodic in the z-direction,
with periodicity length 2π R0, where R0 a is the simulated plasma major radius. It is
convenient to define the simulated toroidal angle φ = z/R0.
The equilibrium magnetic field is written B = [0, Bθ(r), Bφ], where Bθ ≥ 0 and Bφ > 0.
The associated equilibrium plasma current density takes the form j = [0, 0, jφ(r)], where
µ0 jφ(r) =1
r
d(r Bθ)
dr, (1)
and jφ ≥ 0. Finally, the safety factor,
q(r) =r Bθ
R0 Bφ
, (2)
parameterizes the helical pitch of equilibrium magnetic field-lines. In a conventional tokamak
plasma, q(r) is positive, of order unity, and a monotonically increasing function of r.
5
B. Plasma Response to External Helical Magnetic Perturbation
Consider the response of the plasma to an externally generated, helical magnetic pertur-
bation. Suppose that the magnetic perturbation has m > 0 periods in the poloidal direction,
and n > 0 periods in the toroidal direction. It is convenient to express the perturbed mag-
netic field and the perturbed plasma current density in terms of a magnetic flux-function,
ψ(r, θ, φ, t). Thus,
δB = ∇ψ × ez, (3)
µ0 δj = −∇ 2ψ ez, (4)
where
ψ(r, θ, φ, t) = ψ(r, t) exp[ i (mθ − nφ)]. (5)
This representation is valid provided that 19
m
n a
R0. (6)
As is well known, the response of the plasma to the external magnetic perturbation is
governed by the equations of perturbed, marginally-stable (i.e., ∂/∂t ≡ 0), ideal magnetohy-
drodynamics (MHD) everywhere in the plasma, apart from a relatively narrow (in r) region
in the vicinity of the rational surface, minor radius rs, where q(rs) = m/n.4,19
It is convenient to parameterize the external magnetic perturbation in terms of the so-
called vacuum flux, Ψv(t) = |Ψv| e−iϕv , which is defined to be the value of ψ(r, t) at radius rs
in the presence of the external perturbation, but in the absence of the plasma. Here, ϕv is
the helical phase of the external perturbation. Likewise, the response of the plasma in the
vicinity of the rational surface to the external perturbation is parameterized in terms of the
so-called reconnected flux, Ψs(t) = |Ψs| e−iϕs , which is the actual value of ψ(r, t) at radius rs.
Here, ϕs is the helical phase of the reconnected flux.
The linear stability of the m, n tearing mode is governed by the tearing stability index,4
∆′ =
[d ln ψ
d ln r
]rs+rs−
, (7)
where ψ(r) is a solution of the marginally-stable, ideal-MHD equations for the case of an m,
n helical perturbation that satisfies physical boundary conditions at r = 0 and r = a (in the
6
absence of the externally generated perturbation). According to standard resistive-MHD
theory,4,5 if ∆′ > 0 then the m, n tearing mode spontaneously reconnects magnetic flux at
the rational surface to form a helical magnetic island chain.
C. Time Evolution of Island Width
The time evolution of the radial width of the m, n magnetic island chain is governed by
the Rutherford equation,5
I1 τRd
dt
(W
rs
)= ∆′(W ) + 2m
(Wv
W
)2
cos(ϕs − ϕv), (8)
where I1 = 0.8227. Here,
W = 4
(R0 qs |Ψs|ssBφ
)1/2
(9)
is the full (radial) width of the chain that forms at the rational surface, qs = q(rs), and
ss = (d ln q/d ln r)r=rs is the local magnetic shear. The nonlinear dependence of ∆′ on the
island width is specified in Refs. 7–9. The quantity
Wv = 4
(R0 qs |Ψv|ssBφ
)1/2
(10)
is termed the vacuum island width. It is assumed that W/a 1 and Wv/a 1. Finally,
τR = µ0 r2s σ(rs), (11)
is the resistive diffusion time at the rational surface, where σ(r) is the equilibrium plasma
electrical conductivity profile.
D. Electromagnetic Torque
It is easily demonstrated that zero net electromagnetic torque can be exerted on magnetic
flux surfaces located in a region of the plasma that is governed by the equations of marginally-
stable, ideal MHD.19 Thus, any electromagnetic torque exerted on the plasma by the external
perturbation develops in the immediate vicinity of the rational surface, where ideal MHD
breaks down. In fact, the net toroidal electromagnetic torque exerted in the vicinity of the
rational surface by the external perturbation takes the form 19
TφEM =4π2 nmR0
µ0|Ψs| |Ψv| sin(ϕs − ϕv). (12)
7
E. Plasma Toroidal Equation of Angular Motion
The plasma’s toroidal equation of angular motion is written 19
4π2R 30
[ρ r
∂∆Ωφ
∂t− ∂
∂r
(µ r
∂∆Ωφ
∂r
)]= TφEM δ(r − rs). (13)
Here, ∆Ωφ(r, t), ρ(r), and µ(r) are the plasma toroidal angular velocity-shift (due to the
electromagnetic torque), mass density, and (perpendicular) viscosity profiles, respectively.
The physical boundary conditions are 19
∂∆Ωφ(0, t)
∂r= ∆Ωφ(a, t) = 0. (14)
F. Time Evolution of Island Phase
The time evolution of the helical phase of the m, n magnetic island chain is governed by
the so-called no-slip constraint,19
dϕs
dt= −n [Ωφ(rs) +∆Ωφ(rs, t)], (15)
according to which the chain is forced to co-rotate with the plasma at the rational surface.
Here, Ωφ(r) is the unperturbed (by the external perturbation) plasma toroidal angular
velocity profile. The no-slip constraint holds provided that the island width exceeds the
linear layer width.22 Note that we are neglecting poloidal plasma rotation in this study,
which is reasonable because in tokamak plasmas such rotation is strongly constrained by
poloidal flow damping.19,23 We are also neglecting two-fluid effects; however, such effects can
easily be incorporated into the analysis because, to lowest order, they merely introduce a
constant diamagnetic offset between the island chain’s velocity (in the diamagnetic direction)
and that of the guiding center fluid at the rational surface.24–26
G. Normalization
Suppose, for the sake of simplicity, that both the mass density and (perpendicular) vis-
cosity are spatially uniform across the plasma, and also constant in time. It is helpful to
define the momentum confinement time,
τM =ρ a 2
µ, (16)
8
the hydromagnetic time,
τH =
(R0
n ss
)(µ0 ρ
B 2φ
)1/2
, (17)
the island width evolution time,
τW =I12m
Wv
rsτR, (18)
the island phase evolution time,
τϕ =2 4
√m
qsεa
(a
Wv
)2
τH , (19)
where εa = a/R0, and the natural frequency,
ωs = −nΩφ(rs). (20)
The latter quantity is the angular frequency of the m, n tearing mode in the absence of the
externally generated perturbation.
Let r = r/a, rs = rs/a, t = t/τM , τW = τW/τM , τϕ = τϕ/τM , W = W/Wv, ωs = ωs τϕ,
and Ω(r, t) = −n∆Ωφ τϕ. It follows that
τWdW
dt=
∆′
2m+ W−2 cos(ϕs − ϕv) (21)
τϕ
[∂Ω
∂t− 1
r
∂
∂r
(r∂Ω
∂r
)]= W 2 sin(ϕs − ϕv)
δ(rs − r)
r, (22)
∂Ω(0, t)
∂r= Ω(1, t) = 0, (23)
τϕdϕs
dt= ωs +Ω(rs, t). (24)
H. Derivation of Phase Evolution Equations
Let
uk(r) =
√2 J0(j0,k r)
J1(j0,k), (25)
where k is a positive integer, and j0,k denotes the kth zero of the J0 Bessel function. It is
easily demonstrated that 27 ∫ 1
0
r uk(r) uk′(r) dr = δkk′, (26)
and 28
δ(r − rs)
r=∑
k=1,∞uk(rs) uk(r). (27)
9
Let us write 29
Ω(r, t) =∑
k=1,∞hk(t)
uk(r)
uk(rs), (28)
which automatically satisfies the spatial boundary conditions (23). Substitution into Eq. (22)
yields the m, n island phase evolution equations: 29
τϕ
(dhk
dt+ j 2
0,k hk
)= −[uk(rs)]
2 W 2 sin(ϕs − ϕv), (29)
τϕdϕs
dt= ωs +
∑k=1,∞
hk. (30)
III. RESPONSE TO MAGNETIC PERTURBATION WITH RAPIDLY
OSCILLATING PHASE
A. Introduction
Suppose that
ϕv(t) = ωv t+ ε cos(ωf t), (31)
where 0 < ε 1, ωv = ωv τM , and ωf = ωf τM . This implies that, on average, the external
perturbation is rotating steadily at the angular velocity ωv, but that its helical phase is
also oscillating sinusoidally about its mean value with (relatively small) amplitude ε and
(relatively large) frequency ωf . To first order in ε, the Rutherford equation, (21), and the
phase evolution equations, (29) and (30), yield
τWdW
dt=
∆′
2m+ W −2 cosϕ+ ε W −2 sinϕ cos(ωf t), (32)
τϕ
(dhk
dt+ j 2
0,k hk
)= −[uk(rs)]
2 W 2 sinϕ+ ε [uk(rs)]2 W 2 cosϕ cos(ωf t), (33)
τϕdϕ
dt= ωo +
∑k=1,∞
hk, (34)
where
ϕ = ϕs − ωv t, (35)
ωo = (ωs − ωv) τϕ. (36)
In the following, we shall refer to ωo = ωs − ωv, which is the difference between the natural
frequency of the m, n tearing mode and the mean frequency of the externally applied m, n
magnetic perturbation, as the offset frequency.
10
B. Two Time-Scale Analysis
Following Kapitza,21,30 we can write
W = W + W , (37)
ϕ = ϕ+ ϕ, (38)
hk = hk + hk. (39)
Here, W , ϕ and hk are periodic functions of t with period 2π/ωf 1, whereas W , ϕ and
hk vary on a much longer timescale. Let 〈· · · 〉 = (ωf/2π)∫ 2π/ωf
0(· · · ) dt denote an average
over the rapid phase-oscillation time. It follows that 〈W 〉 = 〈φ〉 = 〈hk〉 = 0. It is assumed
that |W | |W |, |ϕ| |ϕ|, and |hk| |hk|.Equations (32)–(34) yield
τW
(dW
dt+
dW
dt
) ∆′
2m+W
−2
(1− 2
W
W
)(cos ϕ− sin ϕ ϕ) (40)
+ εW−2
(1− 2
W
W
)(sin ϕ+ cos ϕ ϕ) cos(ωf t),
τϕ
(dhk
dt+
dhk
dt+ j 2
0,k hk + j 20,k hk
) −[uk(rs)]
2W2
(1 + 2
W
W
)(sin ϕ+ cos ϕ ϕ)
+ ε [uk(rs)]2W
2
(1 + 2
W
W
)(cos ϕ− sin ϕ ϕ)
× cos(ωf t), (41)
τϕ
(dϕ
dt+
dϕ
dt
)= ωo +
∑k=1,∞
(hk + hk
). (42)
The rapidly-oscillating components of the previous three equations (which can be equated
separately 30) give
τWdW
dt −2W
−3cos ϕ W −W
−2sin ϕ ϕ+ εW
−2sin ϕ cos(ωf t), (43)
τϕ
(dhk
dt+ j 2
0,k hk
) −2 [uk(rs)]
2W sin ϕ W − [uk(rs)]2W
2cos ϕ ϕ
+ ε [uk(rs)]2W
2cos ϕ cos(ωf t), (44)
τϕdϕ
dt=∑
k=1,∞hk. (45)
11
Finally, averaging Eqs. (40)–(42) over the rapid oscillation,30 we obtain
τWdW
dt=
∆′
2m+W
−2cos ϕ + 2W
−3sin ϕ
⟨W ϕ
⟩− 2 εW
−3sin ϕ
⟨W cos(ωf t)
⟩+ εW
−2cos ϕ
⟨ϕ cos(ωf t)
⟩, (46)
τϕ
(dhk
dt+ j 2
0,k hk
)= −[uk(rs)]
2W2sin ϕ− 2 [uk(rs)]
2W cos ϕ⟨W ϕ
⟩+ 2 ε [uk(rs)]
2W cos ϕ⟨W cos(ωf t)
⟩− ε [uk(rs)]
2W2sin ϕ
⟨ϕ cos(ωf t)
⟩, (47)
τϕdϕ
dt= ωo +
∑k=1,∞
hk. (48)
If we assume that ωf τW 1 and ωf τ2ϕ 1 then we can find the following approximate
solutions of Eqs. (43)–(45):
W (
ε
ωf τW
)W
−2sin ϕ sin(ωf t), (49)
ϕ (
ε
ωf τ 2ϕ
)[−Σ1(rs, ωf) cos(ωf t) + Σ2(rs, ωf) sin(ωf t)]W
2cos ϕ, (50)
where
Σ1(rs, ωf) =∑
k=1,∞
[uk(rs)]2 ωf
ω 2f + j 4
0,k
, (51)
Σ2(rs, ωf) =∑
k=1,∞
[uk(rs)]2 j 2
0,k
ω 2f + j 4
0,k
. (52)
The variation of the functions Σ1(rs, ωf) and Σ2(rs, ωf) is illustrated in Figs. 1 and 2,
respectively. Note that both functions scale as 1/ω1/2f when ωf 1. It follows that
⟨W ϕ
⟩ 1
2
[ε 2 Σ2(rs, ωf)
(ωf τW ) (ωf τ 2ϕ )
]cos ϕ sin ϕ, (53)⟨
W cos(ωf t)⟩ 0, (54)⟨
ϕ cos(ωf t)⟩ −1
2
[εΣ1(rs, ωf)
ωf τ 2ϕ
]W
2cos ϕ. (55)
12
Thus, to lowest order [in (ωf τW )−1 and (ωf τ2ϕ )
−1], Eqs. (46)–(48) reduce to
τWdW
dt ∆′
2m+W
−2[1− ε 2Σ1(rs, ωf)
2 (ωf τ 2ϕ )
W2cos ϕ
]cos ϕ, (56)
τϕ
(dhk
dt+ j 2
0,k hk
) −[uk(rs)]
2W2[1− ε 2Σ1(rs, ωf)
2 (ωf τ 2ϕ )
W2cos ϕ
]sin ϕ, (57)
τϕdϕ
dt= ωo +
∑k=1,∞
hk. (58)
C. Phase-Locked Solutions
Let us search for a solution of Eqs. (56)–(58) which is such that ϕ is constant in time.
This implies that, on timescales much longer than the rapid phase-oscillation time, 1/ωf ,
the island chain co-rotates with the external magnetic perturbation: i.e., its phase is locked
to that of the external perturbation. We find that
hk [uk(rs)]2
τϕ j 20,k
W2[− sin ϕ+
ε 2 Σ1(rs, ωf)
2 (ωf τ 2ϕ )
W2cos ϕ sin ϕ
]. (59)
Hence, we obtain
τϕdϕ
dt= ωo +
ln(1/rs)
τϕW
2[− sin ϕ+
ε 2 Σ1(rs, ωf)
2 (ωf τ 2ϕ )
W2cos ϕ sin ϕ
]= 0, (60)
where use has been made of the identity 31∑k=1,∞
[uk(rs)]2
j 20,k
= ln
(1
rs
). (61)
Equation (60) can be written
f(ϕ) = 0, (62)
where
f(ϕ) =ωo τϕ
ln(1/rs)+W
2[− sin ϕ+
ε 2Σ1(rs, ωf)
2 (ωf τ 2ϕ )
W2cos ϕ sin ϕ
]. (63)
We shall refer to Eqs. (62) and (63) as the time-averaged torque balance equations, because
they state that, in order for the island chain to be phase-locked to the external perturbation
(on timescales much longer than 1/ωf), the mean electromagnetic locking torque exerted on
the plasma in the vicinity of the rational surface [i.e., the components of f(ϕ) that involve
sin ϕ] must balance the mean viscous restoring torque (i.e., the remaining component).19 An
examination of Eq. (60) reveals that if ϕ0 is a solution of Eqs. (62) and (63) then this solution
is dynamically stable provided (df/dϕ)ϕ=ϕ0 < 0, and dynamically unstable otherwise.
13
IV. RESPONSE OF SATURATED MAGNETIC ISLAND CHAIN TO
ROTATING RESONANT MAGNETIC PERTURBATION WITH RAPIDLY
OSCILLATING PHASE
A. Introduction
Consider the response of a tearing-unstable tokamak plasma, containing a saturated m,
n magnetic island chain, to a rotating magnetic perturbation of the same helicity with a
rapidly oscillating helical phase. In this study, we shall treat the time-averaged (normalized)
island width, W , as a quantity that is essentially independent of the time-averaged island
phase, ϕ.
The torque balance equations, (62) and (63), yield
f(ϕ) = 0, (64)
where
f(ϕ) = ζ − sin ϕ+ ε 2 cos ϕ sin ϕ, (65)
and
ε =ε√2
[Σ1(rs, ωf)
ωf τ 2ϕ
] 1/2
W, (66)
ζ =ωo τϕ
ln(1/rs)W2 . (67)
Moreover, the time-averaged Rutherford equation, Eq. (56), can be written
τWdW
dt ∆′
2m+W
−2H(ζ, ε), (68)
where
H(ζ, ε) = cos ϕ0 − ε 2 cos2 ϕ0. (69)
Here, ϕ0(ζ, ε) denotes a dynamically stable root of Eqs. (64) and (65).
B. Zero Offset Frequency Solutions
Suppose that ζ = 0, which corresponds to the simple case in which the offset frequency
is zero: i.e., the mean frequency of the external magnetic perturbation matches the island
chain’s natural frequency.
14
The time-averaged torque balance equations, (64) and (65), possess the obvious solutions
ϕ = 0 and ϕ = π, and possess the additional solution ϕ = cos−1(1/ε 2) when ε > 1. It is
easily demonstrated that the solution ϕ = 0 is dynamically stable when ε < 1, but becomes
dynamically unstable when ε > 1. Moreover, the solution ϕ = π is always dynamically
unstable, whereas the solution ϕ = cos−1(1/ε 2) is always dynamically stable (provided that
it exists).
We deduce that if the offset frequency is zero then the dynamically stable root of Eqs. (64)
and (65) is such that
ϕ0(0, ε) =
0 0 ≤ ε < 1
cos−1(1/ε 2) ε > 1. (70)
In other words, provided that the normalized amplitude of the external magnetic pertur-
bation’s rapid phase oscillation, ε, is less than the critical value unity, the mean phase of
a locked magnetic island chain exactly matches the mean phase of the external magnetic
perturbation to which it is locked (i.e., ϕ0 = 0). However, if ε exceeds the critical value
unity then a spontaneous difference develops between the mean island phase and the mean
phase of the magnetic perturbation (i.e., ϕ0 = 0). As the ε → ∞, this phase difference
asymptotes to π/2. Hence, we deduce that, when the offset frequency is zero, a sufficiently
large-amplitude rapid oscillation in the phase of the external perturbation causes the island
chain to lock to the perturbation in phase quadrature.
Substituting Eq. (70) into Eq. (69), we find that
H(0, ε) =
1− ε 2 0 ≤ ε < 1
0 ε > 1. (71)
In other words, as the normalized amplitude of the external magnetic perturbation’s rapid
phase oscillation, ε, increases from zero, the destabilizing effect of the perturbation on a
locked island chain of the same helicity is gradually reduced [see Eq. (68)]. Eventually, when
ε exceeds the critical value unity, the destabilizing effect disappears all together, despite the
fact that the island chain is still locked to the external perturbation (i.e., its mean frequency
matches that of the perturbation).
15
C. Finite Offset Frequency Solutions
Suppose that ζ > 0, which corresponds to the case in which the offset frequency is non-
zero: i.e., the mean frequency of the external magnetic perturbation differs from the island
chain’s natural frequency.
In the simple case in which the amplitude of the external magnetic perturbation’s rapid
phase oscillation is zero (i.e., ε = 0), the time-averaged torque balance equations, (64) and
(65), possess the dynamically stable solution 19
ϕ0(ζ, 0) = sin−1(ζ) (72)
when 0 ≤ ζ < 1. On the other hand, the torque balance equations do not possess any
solution when ζ > 1. Physically, as the offset frequency, which is parameterized by ζ ,
increases, an increasing viscous torque develops on the plasma in the vicinity of the rational
surface. This torque acts to increase the helical phase difference between the island chain
and the external perturbation. Eventually, when ζ attains the critical value unity, and
the phase difference becomes π/2 radians, the electromagnetic locking torque acting on the
plasma in the vicinity of the rational surface can no longer balance the viscous torque, and
a phase-locked solution becomes impossible (i.e., the island chain starts to rotate relative to
the external perturbation).19
Substituting Eq. (72) into Eq. (69), we find that 19
H(ζ, 0) = (1− ζ 2) 1/2 (73)
when 0 ≤ ζ < 1. In other words, as the offset frequency increases, the destabilizing effect
of the perturbation on a locked island chain of the same helicity is gradually reduced [see
Eq. (68)]. In fact, the destabilizing effect becomes zero at the point at which the island
chain unlocks from the perturbation (i.e., ζ = 1).
When ζ > 0 and ε > 0, the time-averaged torque-balance equations, (64) and (65),
can only be solved numerically. Figure 3 shows the mean helical phase difference between
the island chain and the external magnetic perturbation, ϕ0(ζ, ε), calculated from these
equations. It can be seen that, even in the presence of a rapid phase oscillation, the island
chain still unlocks from the external perturbation when ζ = 1 and ϕ0 = π/2. In other
words, phase-locked solutions are possible when 0 ≤ ζ < 1, but for ζ > 1 the island chain
16
is forced to rotate with respect to the external perturbation. The mean phase difference
asymptotes to π/2 as the normalized amplitude of the rapid phase oscillation, ε, approaches
infinity. Hence, we deduce that, even when the offset frequency is non-zero, a sufficiently
large-amplitude rapid oscillation in the phase of the external perturbation cause the island
chain to lock to the perturbation in phase quadrature.
Figure 4 shows the function H(ζ, ε). It can be seen that, as the normalized amplitude of
the external magnetic perturbation’s rapid phase oscillation, ε, increases from zero, the
destabilizing effect of the perturbation on a locked island chain of the same helicity is
gradually reduced [see Eq. (68)]. In fact, the destabilizing effect is significantly reduced
at ε = 1, and tends to zero when ε → ∞.
V. RESPONSE OF TEARING-STABLE PLASMA TO STATIC RESONANT
ERROR-FIELD WITH RAPIDLY OSCILLATING PHASE
A. Introduction
Consider the response of a tearing-stable tokamak plasma to a non-rotating, resonant,
error-field with a rapidly oscillating helical phase. In this study, we shall treat the time-
averaged (normalized) island width, W , of any magnetic island chain localized at the associ-
ated rational surface, as a function of the time-averaged island phase, ϕ, because the island
chain is ultimately maintained in the plasma by the error-field. Note that, for the case of a
non-rotating error-field, the offset frequency, ωo, simply reduces to the island chain’s natural
frequency, ωs.
B. Phase-Locked Solutions
If we search for a steady-state solution of the time-averaged Rutherford equation, (56),
and the time-averaged torque-balance equations, (62) and (63), then we obtain
0 = −X 2 +(1− e 2X 2 cos ϕ
)cos ϕ, (74)
f(ϕ) = 0, (75)
f(ϕ) = ξ − 2X 2(1− e 2X 2 cos ϕ
)sin ϕ, (76)
17
where
X =W
W full
, (77)
W full =
(2m
−∆′
) 1/2
, (78)
e =ε√2
[Σ1(rs, ωf)
ωf τ 2ϕ
] 1/2
W full, (79)
ξ =2 ωs τϕ
ln(1/rs)W2
full
. (80)
Here, W full is the value of W when the natural frequency and the amplitude of the error-
field’s rapid phase oscillation are both zero. As before, if ϕ0 is a solution of Eq. (75) then
the solution is dynamically stable provided that (df/dϕ)ϕ=ϕ0 < 0.
Equation (74) can be solved to give
X 2 =cos ϕ
1 + e 2 cos 2 ϕ. (81)
Hence, Eq. (76) yields
f(ϕ) = ξ − g(ϕ), (82)
where
g(ϕ) =2 cos ϕ sin ϕ
(1 + e 2 cos 2 ϕ) 2. (83)
It is easily demonstrated that the time-averaged torque balance equations, (75), (82),
and (83), possesses a stable solution for 0 ≤ ξ < ξc, which is such that 0 ≤ ϕ < π/2, and do
not possess any solution for ξ > ξc. Here, ξc = g(ϕc), and (dg/dϕ)ϕ=ϕc = 0. It follows from
Eq. (83) that
cos ϕc =
(2 + 3 e 2 − [(2 + 3 e) 2 − 8 e 2] 1/2
4 e 2
)1/2
. (84)
Thus, when ξ < ξc, the mean electromagnetic locking torque acting at the rational surface
is large enough to ensure that the island chain remains phase-locked to the error-field in
a destabilizing phase relation, which permits the error-field to maintain the chain in the
plasma. However, when ξ > ξc, the mean viscous torque acting at the rational surface
overwhelms the mean electromagnetic locking torque, and forces the island chain to rotate
with respect to the error-field, which implies that the chain cannot be maintained in the
plasma by the error-field.17,19
18
Figure 5 shows the critical mean phase difference, ϕc, between the island chain and the
error-field, above which the chain unlocks from the error-field. It can be seen that, in the
absence of any rapid oscillation in the error-field’s phase (i.e., e = 0), the parameter ϕc
takes the value π/4.19 However, as the normalized amplitude of the error-field’s rapid phase
oscillation, e, increases from zero, ϕc also increases, and approaches π/2 asymptotically as
e → ∞. Thus, as is consistent with our previous findings, we deduce that a sufficiently
large-amplitude rapid oscillation in the phase of the error-field cause the island chain to lock
to the error-field in phase quadrature.
The quantity ξ is the normalized natural frequency of the island chain. Figure 6 shows the
critical value of this parameter, ξc, above which the island chain unlocks from the error-field.
In can be seen that, as the normalized amplitude of the error-field’s rapid phase oscillation,
e, increases from zero, the parameter ξc decreases, and approaches zero asymptotically as
e → ∞. Hence, we conclude that rapid phase oscillations of a resonant error-field can greatly
facilitate the expulsion of an associated locked magnetic island chain from the plasma,
(because if ξc falls below ξ then the island chain will unlock, spin up, and then decay
away 17). Obviously, if the phase oscillations are of sufficiently large amplitude to expel a
locked island chain from the plasma then they are also large enough to prevent the island
chain from forming in the first place.
VI. SUMMARY AND CONCLUSIONS
In Sect. I, we remarked on the analogy that exits between the phase evolution of a mag-
netic island chain, embedded in a tokamak plasma, which is interacting with an externally
generated, rotating magnetic perturbation of the same helicity, and the angular motion of
a rigid pendulum. For the case of a pendulum, there are two equilibrium states. The first,
which is dynamically stable, is such that the pendulum’s center of mass lies directly below
the pivot point. The second, which is dynamically unstable, is such that the pendulum’s
center of mass lies directly above the pivot point. Likewise, in the case of a magnetic is-
land chain interacting with an external magnetic perturbation, there are two equilibrium
states.19 The first, which is dynamically stable, is such that the helical phase of the island
chain matches that of the external perturbation—this is also the phase relation in which
the island chain is maximally destabilized by the perturbation. The second, which is dy-
19
namically unstable, is such that the helical phase of the island chain differs from that of
the external perturbation by π radians—this is also the phase relation in which the island
chain is maximally stabilized by the perturbation. The fact that the stabilizing equilibrium
state is dynamically unstable is related to the existence of the so-called phase instability,
by which the phase of a magnetic island chain that possesses a stabilizing phase relation
to an external magnetic perturbation of the same helicity rapidly evolves until the converse
is true.14 The phase instability greatly complicates any attempt to reduce the width of a
magnetic island chain, or modify the flow profile, in a tokamak plasma via the application
of an externally generated, rotating magnetic perturbation.11–13,15 The phase instability also
ensures that a magnetic island chain embedded in a tearing-stable plasma always locks to
a static error-field of the same helicity in a destabilizing phase relation, and is therefore
maintained in the plasma by the error-field.16–18
It is well known that if the pivot point of a rigid pendulum is made to execute a small-
amplitude, vertical oscillation of sufficiently high frequency then the equilibrium state in
which the pendulum’s center of mass lies directly above the pivot point can be rendered
dynamically stable, whilst the state in which the center of mass lies directly below the pivot
point is rendered dynamically unstable. This phenomenon was first described by Stephen-
son in 1908.20 Moreover, in 1951, Kapitza introduced a convenient method of analyzing the
dynamics of a rigid pendulum with a vertically oscillating pivot point in which the over-
all motion is described as the sum of separate rapidly-varying periodic and slowly-varying
secular motions.21
The analogy that exists between the phase evolution of a magnetic island chain interacting
with an external magnetic perturbation of the same helicity and the angular motion of a
rigid pendulum led us to speculate (in Sect. I) that if the helical phase of the external
perturbation were subject to a small amplitude, high frequency oscillation then it might be
possible to induce the island chain to lock to the perturbation in the maximally stabilizing
phase relation (i.e., in anti-phase). In order to explore this possibility, the dynamics of a
magnetic island chain interacting with a magnetic perturbation of rapidly oscillating phase
was analyzed using Kapitza’s method in see Sect. III. The rapid phase oscillations were
found to give rise to new terms in the equations governing the secular evolution of the island
width and the island phase—see Eqs. (56)–(58). An examination of the properties of the
new secular evolution equations (see Sects. IV and V) revealed that a magnetic island chain
20
preferrentially locks to an external magnetic perturbation of the same helicity, whose phase
oscillates rapidly at a sufficiently high amplitude, in phase quadrature: i.e., such that the
helical phase difference between the island chain and the perturbation is π/2. This implies
that, on average, the magnetic island chain is neither stabilized nor destabilized by the
external perturbation.
The fact that it does not seems to be possible to cause a magnetic island chain, embedded
in a tokamak plasma, to lock in anti-phase to an external magnetic perturbation of the same
helicity, whose phase is rapidly oscillating, is a slightly disappointing result. Nevertheless,
our finding that the island chain actually locks to such a perturbation in phase quadrature
is significant. For instance, it implies that rotating resonant magnetic perturbations with
rapidly oscillating phase could be used to manipulate the flow profile in a tokamak plasma
(e.g., by introducing flow shear) without penalty (i.e., without making the magnetic islands
in the plasma any wider). More significantly, as described in Sect. V, a static resonant
magnetic perturbation with rapidly oscillating phase could be used to expel an error-field
driven locked mode of the same helicity from the plasma.
ACKNOWLEDGEMENTS
This research was funded by the U.S. Department of Energy under contract DE-FG02-
04ER-54742.
1 J.A. Wesson, Tokamaks, 3rd Edition. (Oxford University Press, 2004).
