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TitleThree-dimensional peeling-ballooning theory in magnetic fusion devices
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Page 1

3D Ideal Linear Peeling Ballooning Theory in
Magnetic Fusion Devices

Ph.D. Thesis presented by:

Toon Weyens

Thesis supervisor & promoter:

Luis Raúl Sánchez Fernández (Universidad Carlos III de Madrid)

Thesis co-promoters:

Guido Huijsmans (Eindhoven University of Technology)
Luis García Gonzalo (Universidad Carlos III de Madrid)

Doctoral Program in Plasmas and Nuclear Fusion
(Universidad Carlos III de Madrid; Physics Department)

Joint European Doctorate and Network in Fusion Science and Engineering
(Eindhoven University of Technology; Department of Applied Physics)

Leganes, 16th of December 2016

Page 2

3D Ideal Linear Peeling Ballooning Theory in
Magnetic Fusion Devices

Author: Toon Weyens

Thesis directors: L. R. Sánchez Fernández, G. Huijsmans & L.
García Gonzalo

Examination committee:
signature:

President: N. J. Lopes Cardozo

Vocal: C. Hidalgo Vera

Secretary: E. Ahedo Galilea

mark:

Leganes, 16th of December 2016

Page 68

50 Chapter 3 : 3-D Edge Theory

Therefore, the operatorQ0m becomes:

Q0m =
nq−m

n

1



(




∂θ
− Bθ



∂α

)
Θθ +

∂Θθ

∂θ
+




(
∂Θα

∂θ
+ q ′

)
+

+
2



∂Bψ

∂θ
+

1



(
Θα



∂α



∂ψ
+Θθ



∂θ

)
Bθ+

+
(
−Θζ +Θθ

m

n

)
Q1m

=
nq−m

n

J


B � rψ � rΘ θ + q ′




+

1



(
∂Bψ

∂θ

∂Bθ
∂ψ

)
+

+
(
−Θζ +Θθ

m

n

)
Q1m

=
Bαq

′ + Jµ0p



+
(
−Θζ +Θθ

m

n

)
Q1m+

+
nq−m

n

JB � rψ � rΘ θ


,

(3.73)

with Θζ = Θα + q ′θ+ qΘθ. using the same technique, the operatorQ1m
simplifies to:

Q1m = −i (nq−m)



. (3.74)

The axisymmetric limit of these equations corresponds to the work done
by [CHT79] and is discussed in subsection 2.

Minimization of plasma potential energy

The series of equation 3.18 is introduced into the plasma potential energy,
given by equation 3.4, making use of the expressions for the adjoint opera-
tors UTk andDU

T
k of 4. This is done here term by term.

Line bending
term

The stabilizing magnetic terms were described in subsection 1 by the
term 1

µ0
|Q⊥|

2. The parallel component, also called the magnetic compres-
sion term, was minimized to zero by the condition of equation 3.25 and the
two perpendicular components, also called the line bending terms, are to
be calculated independently from

1

µ0

(
1

|rψ |2

∣∣∣∣1J ∂X∂θ
∣∣∣∣2 + |rψ |2B2

∣∣∣∣1J ∂U∂θ − SX
∣∣∣∣2
)

, (3.75)

Page 69

Section 3.5 : Appendices 51

Inserting the series of equation 3.18 then results in a contribution

1

µ0

1

J2 |∇ψ|2

∣∣∣∣∣∑
m

i (nq−m)Xme
i[nα+(nq−m)θ]

∣∣∣∣∣
2

, (3.76)

from the normal component, which directly leads to

1

µ0


k,m

X∗ke
i(k−m)θ

[{
(nq− k) (nq−m)

1

J2 |∇ψ|2

}]
Xm , (3.77)

and

1

µ0

|∇ψ|2

J2B2

∣∣∣∣∣∑
m

[
DU0m − JS+DU

1
m

i

n



∂ψ

]
(Xm) e

i[nα+(nq−m)θ]

∣∣∣∣∣
2

,

(3.78)
from the geodesic component. Extracting the different orders in the ψ
derivatives:

1

µ0


k,m

X∗ke
i(k−m)θ

[(
DU

T ,0
k

− JS+DU
T ,1
k

i

n



∂ψ

)
(
|∇ψ|2

J2B2

(
DU0m − JS+DU

1
m

i

n



∂ψ

))
(Xm)

]

=
1

µ0


k,m

X∗ke
i(k−m)θ

[{
DU

T ,1
k

|∇ψ|2

J2B2
DU1m

}(
i

n

)2
∂2

∂ψ2
+

+

{(
DU

T ,0
k

− JS
)
|∇ψ|2

J2B2
DU1m +DU

T ,1
k

|∇ψ|2

J2B2

(
DU0m − JS

)
+

+DU
T ,1
k

i

n



∂ψ

(
|∇ψ|2

J2B2
DU1m

)}(
i

n

)


∂ψ
+

+

{(
DU

T ,0
k

− JS
)
|∇ψ|2

J2B2

(
DU0m − JS

)
+

+DU
T ,1
k

i

n



∂ψ

(
|∇ψ|2

J2B2

(
DU0m − JS

))}]
(Xm) ,

(3.79)
with the surface term, discussed in equation 3.32 for the adjoint operator

Page 135

117

Index

ballooning mode, 9
ballooning transform, 10
Boltzmann equation, 18
Braginskii equations, 20

CAS3D, 14
CASTOR, 14
closure, 19
current-driven mode, 27
curvature, 32
cyclotron motion, 3

edge, 9
Edge Localized Mode, see ELM
eikonal, 8
ELITE, 11
ELM, 11
energy principle, 22
equilibrium, 6
essential boundary conditions, 30
extended energy principle, see energy

principle

field-line average, 44
fluid approach, 4
fluted perturbation, 7, 32
flux coordinates, 31
flux surface, 5
force formalism, 6
Fourier representation, 35

H-mode, 11
high confinement, see H-mode
high-n, 7
hot plasma, 18

JOREK, 15

kinetic theory, 19

L-mode, 11
linearization, 5
Liouville’s theorem, 18
low confinement, see L-mode

magnetic field, 31
magnetohydrodynamics, 4
Mercier criterion, 9
MHD, 21
MISHKA, 14

normal direction, 5
normal mode analysis, 6

parallel current, 30
peeling mode, 9
perturbation, 6
phase space, 18
plasma perturbation, 31
plasma potential energy, 30
pressure-driven mode, 27

Rayleigh quotient, 23, 30

Page 136

118 Index

resonant magnetic perturbation, see
RMP

RMP, 11

Saha ionization equation, 18
scale analysis, 5
shear, 31
short wavelength, see high-n
stellarator, 3

TBM, 13
TERPSICHORE, 15
test blanket module, see TBM
tokamak, 3
toroidal field ripple, 13

vacuum, 42
vacuum potential energy, 42

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