magnetic field in ergodic divertors
TRANSCRIPT
Appendix AMagnetic Field in Ergodic Divertors
In this chapter Iwould like to present a detailed analytical calculations of themagneticfield generated by a set of helical coils. Themethod is demonstrated for themodels oftheDEDof theTEXTORand theEDof theToreSupra. Themain idea of themethod isconcluded in the following.Typically a set of coils are located on the surface of a torus.The density of the current flowing in the coils can be described by the delta functionswith singularities on the coil locations. Using the Poisson summation rule whichgives the relation of the delta functions with the trigonometric functions one canpresent the current density on the surface of torus a sum of infinity number of helicalcurrents, jmn ∝ cos(mθ−nϕ). For the large aspect ratio tokamaks the magnetic fieldgenerated by these helical currents can be found by approximating the system by thecylinder (see, e.g., Morozov and Solov’ev (1966b)) and making corrections due toa toroidicity. This approach allows one to qualitative and quantitatively analyze thepoloidal and toroidal spectra of the magnetic field generated by a set of coils as wellas its radial dependence.
A.1 Magnetic Perturbation in the TEXTOR-DED
The magnetic field perturbations in the TEXTOR-DED by the set of helical coilsschematically shown in Fig. 9.4a, b. The geometrical locations of the helical coils inthe (ϕ, θ)-plane are plotted in Fig. A.1a for the ideal configuration and in Fig. A.1bfor the real configuration.
A.1.1 Density of Perturbation Currents
It is convenient to introduce the density of DED perturbation currents in order to findthe magnetic perturbations. The current density j(r, θ,ϕ) is introduced as
j = jr er + jθeθ + jϕeϕ, (A.1)
S. Abdullaev, Magnetic Stochasticity in Magnetically Confined Fusion Plasmas, 333Springer Series on Atomic, Optical, and Plasma Physics 78,DOI: 10.1007/978-3-319-01890-4, © Springer International Publishing Switzerland 2014
334 Appendix A: Magnetic Field in Ergodic Divertors
120
140
160
180
200
220
240
0 60 120 180 240 300 360
θ
ϕ
j=1 16 15 14 13 12 11 10 9 8 76 5 4 3 2=
2θc
120
140
160
180
200
220
240
0 60 120 180 240 300 360
θ
ϕ
j=4-1 j=16-13 j=12-9 j=8-5
Δθ
2θc
(a) (b)
Fig. A.1 Models of the DED coil configuration: a the ideal configuration, b the real configuration.The coils are numbered by j = 1, 2, . . . , 16
where er , eθ, eϕ are unit vectors along the coordinates r, θ,ϕ, respectively. The cor-responding components, jr , jθ, and jϕ, are defined as
( jr , jθ, jϕ) = (0, j (r, θ,ϕ) sinα0, j (r, θ,ϕ) cosα0),
where
j (r, θ,ϕ) = δ(r − rc)
16∑
j=1
I j r−1c δ(θ − θ j (ϕ)), (A.2)
and α0 = θcrc/πRc is an angle between current direction and toroidal axis ϕ, θ j (ϕ)
is a poloidal position of the j th coil at the toroidal section ϕ. The current I j on thej th coil can be expanded into a series
I j =∑
n
ιn I (n)j ,
I (n)j = Id sin
(n2π j
16+ χn
), (A.3)
where I (n)j is the basic current distributionwhich generates themagnetic perturbation
with toroidal modes n = n+4p, (p = 0, 1, 2, . . .). The coefficients ιn and the phasesχn can be determined by the currect distribution on coils I j . They will be listed inTable A.1. The details of calculations of ιn and χn can be found in Finken et al.(2005a).
For the given mode n Eq. (A.2) may be written as
j (n)(r, θ,ϕ) = δ(r − rc)g(θ,ϕ)Id
rc
∞∑
j=−∞sin
(n2π j
16+ χn
)δ
(θ − θ j (ϕ)
), (A.4)
where g(θ,ϕ) is a step function equal to 1 in the area covered by coils and zeroelsewhere (see Eqs. (A.14) and (A.19)), the poloidal position θ j (ϕ) of the j th coil
Appendix A: Magnetic Field in Ergodic Divertors 335
at the section ϕ is given by
θ j = θ(ϕ) − jδθ, θ(ϕ) = θ0 − θc
πϕ. (A.5)
The angle θ01 = θ0 + δθ is a poloidal position of the first coil at the toroidal sectionϕ = 0, δθ is the angular distance between neighboring coils.
A.1.2 Continuous Current Density
Using the coil positions (A.5), we present the current distribution given by Eq. (A.4)as
j (n)(r, θ,ϕ) = δ(r − rc)Id
rcg(θ,ϕ)
∞∑
j=−∞sin
(n2π j
16+ χn
)δ (θ − θ(ϕ) + jδθ) .
(A.6)Using the representation of delta function
δ(x) = 1
2π
∞∫
−∞eixpdp,
one obtains
j (n)(r, θ,ϕ) = Idδ(r − rc)
2πrcg(θ,ϕ)W,
W = Im
⎧⎨
⎩eiχn
∞∫
−∞dpeip(θ−θ(ϕ))
∞∑
j=−∞exp
[i j
(2πn
16+ pδθ
)]⎫⎬
⎭ . (A.7)
Then using the Poisson rule
∞∑
j=−∞ei2π j x = 2π
∞∑
s=−∞δ(s − x),
we have
336 Appendix A: Magnetic Field in Ergodic Divertors
W = Im
⎧⎨
⎩eiχn
∞∫
−∞dpei(θ−θ(ϕ))p
∞∑
s=−∞δ
(s − n
16− pδθ
2π
)⎫⎬
⎭
= 2π
δθ
∞∑
s=−∞sin
(2π(s − n/16)
δθ(θ − θϕ) + χn
). (A.8)
Using the dependence of θ(ϕ) on ϕ given in Eq. (A.5) we reduce (A.7) to
j (n)(r, θ,ϕ) = δ(r − rc)g(θ,ϕ)J0
×∞∑
s=−∞cos
(m0(16s − n)
4θ + n0(16s − n)
4ϕ + χ(n)
s
), (A.9)
where
χ(n)s = −m0(16s − n)
4θ0 + χn − π
2. (A.10)
In (A.9) the following notations are introduced:
J0 = Id
δθrc= 2m0 Id
πrc, m0 = π
2δθ,
n0 = m0θc
π= θc
2δθ, θ0 = θ01 + δθ. (A.11)
Because of periodicity of j (r, θ,ϕ) along ϕwith a period 2π follows that n0 mustbe an integer number equal to n0 = 4l, where l = 1, 2, . . .. Putting n0 = 4, the terms = 0 in Eq. (A.9) which gives the main contribution the perturbed magnetic field inthe plasma can be presented as
j (n)0 (r, θ,ϕ) = δ(r − rc)g(θ,ϕ)J0 cos
(nm0θ/4 + nϕ − χ
(n)0
). (A.12)
A.1.3 Fourier Expansion of the Current Density
For calculations of the magnetic field created by helical coils it is convenient topresent the current density j (n)
0 (r, θ,ϕ) in Fourier series in θ,ϕ:
j (n)(r, θ,ϕ) =∑
m,n′jmn′(r) cos
[mθ + n′ϕ + χmn′
],
where
Appendix A: Magnetic Field in Ergodic Divertors 337
jmn′(r)eiχmn′ = e−iχ(n)0
J0δ(r − rc)
(2π)2
×2π∫
0
2π∫
0
dθdϕg(θ,ϕ) exp[−i
(m − nm0
4
)θ − i(n′ − n)ϕ
].
(A.13)
The content of Fourier spectrum, jmn(r), depends on the function g(θ,ϕ) whichdetermined by the coil configurations. The ideal and real configurations of coils areshown in Fig. A.1a, b, respectively. Below we consider these cases separately.
A.1.3.1 Ideal Coil Configuration
For the ideal coil configuration (see Fig. A.1a) the function g(θ,ϕ) is given by:
g(θ,ϕ) ={1 for π − θc < θ < π + θc,
0 otherwise.(A.14)
The current density (A.9) can be presented as a Fourier series:
j (n)(r, θ,ϕ) = J0δ(r − rc)
∞∑
m=−∞
∞∑
s=−∞g(s)
m cos [mθ + (16s − n)ϕ + χns] ,
(A.15)where
g(s)m = (−1)m sin([m − m0(n/4 − 4s)]θc)
[m − m0(n/4 − 4s)]π ,
χns = m0(16s − n)
4(π − θ0) + χn − π
2. (A.16)
The main contribution to the magnetic field comes from the term s = 0 whichcan be rewritten as (by changing the summation over m to −m)
j (n)0 (r, θ,ϕ) = J0δ(r − rc)
∞∑
m=−∞gm cos (mθ + nϕ − χn0) , (A.17)
where
gm ≡ g(0)m = (−1)m sin[(m + m0n/4)θc]
(m + m0n/4)π,
χn0 = χn − m0n
4(π − θ0) − π
2. (A.18)
338 Appendix A: Magnetic Field in Ergodic Divertors
A.1.3.2 Non-ideal Coil Configuration
For the non-ideal configuration of coils (see Fig. A.1b) the step function g(θ,ϕ) isgiven by
g(θ,ϕ) ={1, for π − θc(ϕ) < θ < π + θc(ϕ),
0, elsewhere,(A.19)
where θc(ϕ) is the piece-wise function
θc(ϕ) = θc0 − 2Δθ
π(ϕ − ϕl) for ϕl < ϕ < ϕl+1,
ϕl = ϕc + (l − 1)π
2, 0 < ϕc <
π
2, l = 0, 1, 2, 3, 4. (A.20)
For the sake of simplicity we consider the term s = 0 in Eq. (A.9), i.e., Eq. (A.12).Furthermore, we need to estimate the integral
fm,n′ = 1
(2π)2
2π∫
0
2π∫
0
dθdϕg(θ,ϕ)e−imθ−in′ϕ.
It is not difficult to show that
fmn′ = −e−iπmδn′,4se−in′ϕc2 sin(mαπ/4)
π2m
2[in′ cos(mθc) + mα sin(mθc)]n′2 − (mα)2
,
where α = 2Δθ/π, θc = θc0−Δθ/2, δn,k is the Kronecker symbol, i.e., δn,k = 0 forn �= k and δn,n = 1. Substituting this expression into (A.13) we obtain the followingexpression for the Fourier components of the current density jmn′(r),
jmn′(r)eiχmn′ = δ(r − rc)eiχ(n)
0 J0 fm−nm0/4,n′−n . (A.21)
From Eq. (A.21) follows that due to non-ideal configuration the current distribution(A.9) creates the toroidal modes n = n + 4s, (s = 0,±1,±2, . . .).
The main contribution to the toroidal spectrum n′ comes from the terms n′ = n.In this case s = 0, and one obtains
fm,0 = e−iπm sin[πmα/4]πmα/4
sin(mθc)
mπ,
and from Eq. (A.21) we obtain
j (n)0 (r, θ,ϕ) = δ(r − rc)
∞∑
m=−∞
∞∑
s=−∞Jm,s cos(mθ + (4s + n)ϕ − χms). (A.22)
Appendix A: Magnetic Field in Ergodic Divertors 339
The Fourier coefficients, Jm,0, corresponding to the term s = 0 which gives the maincontribution to the perturbed field is given by
Jm,0 = J0gmCm, χn0 = χn − m0n
4(π − θ0) − π
2, (A.23)
where gm is given by Eq. (A.18), and
Cm = sin[(m + nm0/4)Δθ/2](m + nm0/4)Δθ/2
(A.24)
is a correction factor due to non-ideal configuration. For the ideal configurationΔθ = 0 and therefore Cm = 1.
One should note that the current distribution (A.3) with n = 4 creates also thetoroidal mode n′ = 0 (see Eq. (A.21)). This mode may disturb the plasma equilib-rium. For this reason in the m : n = 12 : 4 operational mode of the TEXTOR-DEDone applies the compensation coils which annuls the effect of the n′ = 0 mode.
A.1.4 Magnetic Field Perturbations
In this section we present the formulae for the magnetic field created by the surfacecurrent (A.1). Each term in the Fourier expansion (A.22) of this perturbation currentdescribes a helical current on the toroidal surface of radius r = rc. Consider a singlehelical current vector jmn corresponding to the (m, n) mode
jmn(r, θ,ϕ) = δ(r − rc) jmnemn cos(mθ + nϕ + φmn), (A.25)
emn =(0, eθ sinαmn, eϕ cosαmn
),
where eθ and eϕ are unit vectors along the poloidal and toroidal directions, respec-tively, and αmn = nrc/m Rc, (m �= 0), is a helicity, i.e., the angle between a helicalcurrent direction and toroidal axis.
The total current density (A.1) can be presented as a sumof helical currents (A.25),i.e.,
jh(r, θ,ϕ) =∑
mn
jmn(r, θ,ϕ). (A.26)
with the same toroidal components of the vector jmn but different poloidal compo-nents, i.e.,
jmn cosαmn = Jm,(n−k)/4 cosα0,
jmn sinαmn �= Jm,(n−k)/4 sinα0,
340 Appendix A: Magnetic Field in Ergodic Divertors
where k stands for a toroidal mode number n in the basic current distribution (A.3).For the coefficients jmn and the phases, φmn , of the helical current we have
jmn = Jm,(n−k)/4 cosα0
cosαmn, φmn = χm,(n−k)/4. (A.27)
The difference between jh(r, θ,ϕ) (A.26) and j(r, θ,ϕ) (A.1) can be neglected,since the sum of differences of poloidal modes is negligible small, i.e.,
∞∑
m=−∞
(jmn sinαmn − Jm, n−k
4sinα0
)=
∞∑
m=−∞Jm, n−k
4
sin(α0 − αmn)
cosαmn≈ 0.
A.1.5 Cylindrical Approximation
Here we consider the magnetic field created by a single component of the helicalcurrent jmn(r, θ,ϕ) (A.25) in a cylindrical geometry. The magnetic field B of thishelical current can be expressed by the scalar potentialΦ(r, θ,ϕ) (B = ∇Φ(r, θ,ϕ))(see e.g., Morozov and Solov’ev (1966b))
Φ =
⎧⎪⎪⎨
⎪⎪⎩
ai Im
(nrRc
)sin(mθ + nϕ + φmn), for r < rc,
ae Km
(nrRc
)sin(mθ + nϕ + φmn), for r > rc,
where Im(z) and Km(z) are modified Bessel functions. Coefficients ai , ae are foundby the boundary conditions at the r = rc:
Br
∣∣∣r=rc−0
− Br
∣∣∣r=rc+0
= 0,
Bθ
∣∣∣r=rc−0
− Bθ
∣∣∣r=rc+0
= μo jmn cos(mθ + nϕ + φmn) cosαmn .
Using the relations in Eq. (A.27) we have
ai = −μ◦ Jm,(n−k)/4rc cosα0nrc
m R0K ′
m
(nrc
Rc
).
Further we consider only the leading terms s = 0 (A.23) for helical currents.For them we have the following formula for the scalar potential Φ(r, θ,ϕ) of themagnetic field created by a set of helical currents (A.26) inside the toroidal surfacer < rc:
Appendix A: Magnetic Field in Ergodic Divertors 341
Φ(r, θ,ϕ) =∑
m
Φmn(r) sin(mθ + nϕ + χn0),
Φmn(r) = −BcCmgm fmn(r)rc
m, (A.28)
where a quantity
Bc = μ◦ Idm0 cosα0
πrc(A.29)
is the characteristic value of the DEDmagnetic field perturbation. For the TEXTOR-DED parameters, rc = 53.25 cm, Rc = 130 cm, m0 ≈ 20, Id = 15 kA and n = 4the value of Bc is 0.22535 T (or 2253.5 G).
The radial dependence of magnetic perturbations is described by the functionfmn(r):
fmn(r) = −2nrc
RcK ′
m
(nrc
Rc
)Im
(nr
Rc
).
Using the asymptotics of theBessel function Km(z), Im(z), one can show the functionfmn(r) and its radial derivative have the following asymptotics at r < rc,
fmn(r) ≈(
r
rc
)m
, f ′mn(r) ≈ m
rc
(r
rc
)m−1
.
The radial magnetic field Br (r, θ,ϕ) is given by
Br (r, θ,ϕ) = ∂Φ
∂r=∑
m
Bmn(r) sin(mθ + nϕ + χmn),
where
Bmn(r) = −BcCmgmrc
m
d fmn(r)
dr≈ −BcCmgm
(r
rc
)m−1
.
The ϕ-component of the vector potential Aϕ related the magnetic field as
Br (r, θ,ϕ) = 1
r
∂ Aϕ
∂θ, Bθ(r, θ,ϕ) = −∂ Aϕ
∂r,
is determined by
Aϕ(r, θ,ϕ) =∑
m
Amn(r) cos(mθ + nϕ + χmn),
where
Amn(r) = −m−1r Bmn(r) ≈ BcCmgmrc
m
(r
rc
)m
.
342 Appendix A: Magnetic Field in Ergodic Divertors
A.1.6 Toroidal Corrections
According toMorozov and Solov’ev (1966b) the effect of toroidicity on themagneticfield can be taken into account, multiplying the scalar potentialΦ(r, θ,ϕ) by a factor√
R0/R, if the small corrections (n0rc/2R0)m+3 are neglected for each poloidal
component m. In this approximations we have
Φ(r, θ,ϕ) =√
R0
R0 + r cos θ
∑
m
Φmn(r) sin(mθ + nϕ + χn0), (A.30)
where the amplitudes Φmn(r) are given by Eq. (A.28). Then, one can show that thevector potential Aϕ(r, θ,ϕ) is determined by
Aϕ(r, θ,ϕ) = εB0R0a(r, θ,ϕ),
a(r, θ,ϕ) =∑
m
amn(r, θ) cos(mθ + nϕ + χn0), (A.31)
where ε is a dimensionless perturbation parameter defined by
ε = Bc
B0, (A.32)
B0 is the toroidal magnetic field at the center of torus R0, and
amn(r, θ) = − 1
Bc R0
r
m
∂
∂r
(√R0
RΦmn(r)
)
≈ Cmgmrc
m R0
(r
rc
)m√
R0
R0 + r cos θ
(1 − r cos θ
2m(R0 + r cos θ)
).
(A.33)
For the radial component of the magnetic field, Br (r, θ,ϕ), we have
Br (r, θ,ϕ) = 1
r
∂ Aϕ(r, θ,ϕ)
∂θ=∑
m
Bmn(r, θ) sin(mθ + nϕ + χn0),
Bmn(r, θ) ≈ BcCmgm
(r
rc
)m−1√
R0
R0 + r cos θ
(1 − r cos θ
2m(R0 + r cos θ)
).
(A.34)
The radial dependence of the perturbation field is determined by the poloidal modespectra, gm . According (A.24) the latter is localized near the central mode mc =(2s − 1)m0 = nm0/n0, (s = 1, 2, . . .) and has a width Δm = π/Δθi . Therefore,
Appendix A: Magnetic Field in Ergodic Divertors 343
32
34
36
38
40
42
44
46
0 60 120 180 240 300 360
r
θ−1.5
−1.0
−0.5
0.0
0.5
1.0
x10−2
-6
-4
-2
0
2
4
6
100 140 180 220 260
B r [T
]
θ [deg]
x10-3
1
2
(a) (b)
Fig. A.2 a Contour plot of the radial magnetic field perturbation Br (r, θ,ϕ) in (θ, r ) plane at thecross section ϕ = 204.4◦; b Poloidal dependence of the magnetic field perturbation Br (r, θ,ϕ)
at the given radial coordinate r = 43cm and at the two cross sections: curve 1—ϕ = 0◦, curve2—ϕ = 204.4◦. The toroidal mode n = 4 and the perturbation current is Id = 15 kA
one expects that the perturbation field, B(n)r (r, θ,ϕ) of the nth toroidal mode has the
following radial dependence,
B(n)r (r, θ,ϕ) ∝
(r
rci
)γn
, γn = nm0
n0− 1. (A.35)
The examples of spatial dependence of the perturbation field are shown inFigs. A.2a, b; (a) the contour plots of Br (r, θ,ϕ) in the (θ, r ) plane at the fixedtoroidal section ϕ = constant; (b) the poloidal variation Br (r, θ,ϕ) at the fixedradius r and toroidal angle ϕ. The perturbation DED current is taken Id = 15 kA.
A.1.7 Summary of Formulas
Below we give a summary of formulas for the perturbation magnetic field of theDED in the DC operation. The toroidal component of the vector potential is givenby
Aϕ(r, θ,ϕ) = −Bcιn
√R0
R
∑
m
rgmn
|m|(
r
rc
)|m|−1
cos (mθ + nϕ + χn) , (A.36)
where the coefficients
gmn = (−1)mCmnsin[(m + nm0/4)θc]
(m + nm0/4)π,
Cmn = sin[(m + nm0/4)Δθ/2](m + nm0/4)Δθ/2
, (A.37)
344 Appendix A: Magnetic Field in Ergodic Divertors
Table A.1 Coefficients φhand ιn for the different n
n χn ιn
1 3π/16 sin(π/4)/[4 sin(π/16)]2 3π/8 1/[2 sin(π/8)]4 5π/4
√2
describes the poloidal mode spectrum at the given toroidal mode n. In Eq. (A.36)the quantity Bc = μ0m0 Id/πrc is the characteristic magnitude of the DEDmagneticfield, Id is the DED current, the constant m0 determines the central poloidal modenumber nm0/4, rc is the minor radius of the DED coils, χn ≡ χn0. The geometricalpoloidal angle θ is related to the cylindrical coordinates R, Z as θ = arctan(Z/[R −R0]), and the parameter θc is the half of the poloidal section covered by the DEDcoils, Δθ is a geometrical parameter of the coil configuration.
The toroidal mode number n takes the value n = 4 for the 12/4 DED modeconfiguration, n = 2 for the 6/2 mode, and n = 1 for the 3/1 mode, respectively. Thephases χn and the factor ιn in Eq. (A.36) are determined by the coil configuration.For the particular configuration they given by
χn = m0n
4(π − θ0) − χn + π
2, (A.38)
where θ0 is a poloidal angle of the first coil at the section ϕ = 0. The coefficientsχn and ιn for the different values of n are given in Table A.1. The parameters rc, θc,θ0, Δθ, and m0 are determined by the geometry of coil configuration, and take fixedvalues, rc = 0.5325 m, θc = 35.49◦, θ0 = 169.35◦, Δθ = 17.745◦, and m0 ≈ 20.
A.2 Magnetic Field of Tore Supra ED
As it was noted in Sect. 9.1.1 and shown in Fig. 9.2 that the ED coil configuration ofthe Tore Supra consists of six identical modules located on outer board the torus. Wemodel the each module by the coil windings shown in Fig. A.3 where arrows indicatethe current direction. The current flows from the feeder located at the beginning ofthe first section j = 1 of the inner side of the winding shown in Fig. A.3a and returnsthrough the outer side of the winding shown in Fig. A.3b. The minor radii of theinner and outer sides are rc1 = 84 cm rc2 = 86 cm, respectively Below we calculatethe magnetic field, created by the current flowing in this coil system by consideringthe inner and outer parts of the winding separately, and by summing those partsafterwards.
Suppose that the modules are centered near the toroidal angles ϕk = (k − 1)Δϕ,Δϕ = 2π/6. The poloidal angle as a function of the toroidal angle of a point locatedin the j th section in the kth module (k = 1, 2, . . . , 6) is given by
Appendix A: Magnetic Field in Ergodic Divertors 345
(a) (b)
ϕ
θ
δθ
Δϕ
Δθ1
j=1
j=8
θ1(0)ϕkrc1=84 cm
ϕ
θ
δθ
Δϕ
Δθ2
ϕkrc2=86 cm
Fig. A.3 Model scheme of one module of the ED coils: a the inner winding at rc1 = 84 cm; b theouter winding at rc2 = 86 cm
θ(k)j (ϕ) = θ
(k)1 (ϕ) + ( j − 1)δθ, j = 1, 2, . . . , N , (A.39)
where N = 8 for the inner winding and N = 6 for the outer winding, and δθ is thepoloidal spacing shown in Fig. A.3. The coordinates of the first coils on each moduleare described by
θ(k)1 (ϕ) = θ1(0) + α (ϕ − ϕk) ,
for ϕk − Δϕ/2 < ϕ < ϕk + Δϕ/2, (A.40)
where α is the slope of a coil creating a helical magnetic perturbation.The poloidal extension, Δθ, of a module shown in Fig. A.3 can be expressed as
a function of the poloidal position of the first coils, θ1(0), at the toroidal sectionϕ = ϕk :
Δθ = 2|θ1(0)| + αΔϕ. (A.41)
One should note that this model of coils is not fully equivalent to the Tore Supracoils. In the latter case the distance between sections of coils in each module are notequidistant along the poloidal angle θ. It slightly decreases with the distance fromthe equatorial plane θ = 0 (Ghendrih 1995).
A.2.1 Current Density
We describe the current, I j , which flows in a coil section by
I (i)j = Id cos (π j) = (−1) j+1, j = 1, 2, . . . , N ,
346 Appendix A: Magnetic Field in Ergodic Divertors
where Id is the current flowing in the coil, i = 1 for the inner part of the windingand i = 2 for its outer part.
Below we shall consider only the long helical section coils since they create themagnetic field perturbations that are resonant with the magnetic field lines of theplasma. The vertical short sections of coils do not contribute to the resonant field,therefore they will not be taken into account.
One can introduce the current density vector j(r, θ,ϕ) of the coil system as
ji (r, θ,ϕ) = e(i) δ(r − rci )
rci
6∑
k=1
g(k)ϕ (ϕ)
N∑
j=1
I (i)j δ
(θ − θ
(k)j (ϕ)
), (A.42)
where e(i) = (er , eθ, eϕ) = (0, sinα0i , cosα0i ) is a unit vector along the helical
section of the coils, α0i = αrci/Rc, Rc = R0 + rci , (i = 1, 2). Here g(k)ϕ (ϕ) is a step
function of the toroidal angle ϕ which takes a non-zero value in the areas coveredby coils, i.e.,
g(k)ϕ (ϕ) =
{1, for ϕk − Δϕ/2 < ϕ < ϕk + Δϕ/2,
0, elsewhere,
Introducing the step function gi (θ) solely depending on the poloidal angle
gi (θ) ={1, for − Δθi/2 < θ < Δθi/2,
0, elsewhere,
the current density (A.42) can after some transformations be reduced to
ji (r, θ,ϕ) = e(i)δ(r − rci )J (i)0 gi (θ)
6∑
k=1
g(k)ϕ (ϕ)
×∞∑
s=−∞cos {m0(2s − 1) [(θ − θ0) − α(ϕ − ϕk)]} . (A.43)
where
m0 = π
δθ, J (i)
0 = m0 Id
πrci, θ0 = θ1(0) − δθ.
One can show that the current density (A.43) can be expanded into a Fourierseries,
ji (r, θ,ϕ) = 2e(i)∞∑
m=−∞
∞∑
n=−∞
∞∑
s=1
j (si)mn (r) cos
(mθ − nϕ + χ(s)
mn
), (A.44)
Appendix A: Magnetic Field in Ergodic Divertors 347
with the Fourier coefficients
j (si)mn (r) = (−1)qδ (r − rci ) J (i)
0 C (s)n g
(s)mi ,
χs = m0(2s − 1)θ0, (A.45)
where
J (i)0 = 6J (i)
0 ΔϕΔθi
(2π)2= m0 Id
πrci
6ΔϕΔθi
(2π)2, (A.46)
g(s)mi = sin ([m − m0(2s − 1)]Δθi/2)
[m − m0(2s − 1)]Δθi/2, (A.47)
C (s)n = sin ([n − m0(2s − 1)α]Δϕ/2)
[n − m0(2s − 1)α]Δϕ/2. (A.48)
The toroidal mode number n takes values n = 6q, q = 0,±1,±2, . . .. As onecan see from Eqs. (A.45) and (A.48), the biggest effect occurs when the ED coils aredesigned in such a way that the product m0α is close to the toroidal mode n0 = 6,i.e., |m0α − n0| � Δϕ/2. Then in the sum (A.44), the main contribution comesfrom the terms with the toroidal numbers n = (2s −1)n0, (s = 1, 2, . . .). Leaving inEq. (A.44) only these terms we have
ji (r, θ,ϕ) = 2e(i)∞∑
m=−∞
∞∑
s=1
j (si)mn (r) × cos (mθ − (2s − 1)n0ϕ + χs) , (A.49)
where n0 = 6, i = 1, 2.
A.2.2 Magnetic Field
The magnetic field created by the helical currents can be found using the proceduresimilar in Sect. A.1.6.
Φ(r, θ,ϕ) =√
R0
R0 + r cos θ[Φ1(r, θ,ϕ) + Φ2(r, θ,ϕ)] , (A.50)
where
Φi (r, θ,ϕ) =∞∑
m=−∞
∑
n
Φ(i)mn(r) sin (mθ − nϕ + χs) . (A.51)
The toroidal modes takes n = (2s − 1)n0, (s = 1, 2, . . .) and
348 Appendix A: Magnetic Field in Ergodic Divertors
(a) (b)
0
0.2
0.4
0.6
0.8
1
0.5 0.6 0.7 0.8 0.9 1
f mn(
r)
r/rc
m=68
1012
14
0
0.01
0.02
0.03
0.04
0.05
0.06
0.5 0.6 0.7 0.8 0.9 1
Δfm
n(r)
r/rc
m=68
1012
14
Fig. A.4 a Radial dependence of the function fmn(r) for different poloidal modes m. b Relativedeviation of the fmn(r) from the power law (r/rc)
m : Δ fmn(r) = | fmn(r) − (r/rc)m |/ fmn(r).
Parameters are rc = 85 cm, R0 = 238 cm, n = 6
Φ(i)mn(r) = −B(i)
c C (s)n g(si)
m f (i)mn(r)
rci
m,
f (i)mn(r) = −2nrci
RciK ′
m
(nrci
Rci
)Im
(nr
Rci
),
B(i)c = 2μom0 Id cos(α0i )
πrci
6ΔϕΔθi
(2π)2, (A.52)
where Im(z) and Km(z) are the modified Bessel functions (K ′m(z) ≡ d Km(z)/dz).
Here B(i)c is the characteristic amplitude of the strength of the perturbation mag-
netic field.1 For the typical parameters of the ED of Tore Supra (rc = 0.85 m, Id =22.5 kA) we have Bc ≈ 425 G.
The radial dependences of the perturbation field are described by functions f (i)mn(r)
which are shown in Fig. A.4 for a several mode numbers m. For large mode numberm (m ≥ 4) the radial dependence is well described by the following asymptoticalformula f (i)
mn(r) ≈ (r/rci )m . Then the radial component of the magnetic field Br can
be represented as
Br (r, θ,ϕ) = ∂Φ
∂r=∑
n
B(n)r (r, θ,ϕ), (A.53)
B(n)r (r, θ,ϕ) =
∞∑
m=−∞Bmn(r, θ) sin (mθ − nϕ + χs) ,
where
1 We should note that the definition of B(i)c as well as coefficients g
(si)m are slightly different from
the corresponding ones given by Eqs. (A.29) and (A.18).
Appendix A: Magnetic Field in Ergodic Divertors 349
(a) (b)
−4
−3
−2
−1
0
1
2
3
4x10
−2
−0.5−0.4−0.3−0.2−0.1 0 0.1 0.2 0.3 0.4 0.5
62
64
66
68
70
72
74
76
78
80
θ/2π
r [
cm]
-6
-4
-2
0
2
4
6
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Br
[T]
θ/2π
1
2
3
ϕ=01 - r=80 cm2 - r=70 cm3 - r=60 cm
x10-2
Fig. A.5 Radial component of the perturbation field Br : a contour plot in the (θ, r ) plane (ϕ = 0); bpoloidal dependence at different radial positions r : 1—r = 80 cm, 2—r = 70 cm, 3—r = 60 cm.Id = 22.5 kA
Bmn(r, θ) =[
B(1)c g
(s)m1
(r
rc1
)m−1
+ B(2)c g
(s)m2
(r
rc2
)m−1]
×√
R0
R0 + r cos θ
(1 − r cos θ
2m (R0 + r cos θ)
). (A.54)
where B(i)mns = −B(i)
c C (s)n g
(si)m . The toroidal component of the vector potential of the
perturbation field is given by
Aϕ(r, θ,ϕ) = εB0R0
∞∑
m=−∞
∑
n
amn(r, θ) cos (mθ − nϕ + χmn) ,
amn(r, θ) = m−1r Bmn(r, θ), (A.55)
where ε stands for the dimensionless perturbation parameter, defined as ε = B(1)c /B0,
B0 is the strength of the toroidal field. The dimensionless Fourier coefficients,amn(r, θ), are then given by
amn(r, θ) = a(1)mn(r, θ) + δa(2)
mn(r, θ),
where
a(i)mn(r, θ) = g(i)
mrci
m R0
r
m
d
dr
(√R0
R0 + r cos θ
(r
rci
)m)
. (A.56)
350 Appendix A: Magnetic Field in Ergodic Divertors
Here (i = 1, 2), δ = B(2)c /B(1)
c = rc1Δθ2/ (rc2Δθ1). The phase χmn = χs andtoroidal mode number n = (2s − 1)n0.
The angular dependencies of the perturbation field Br (r, θ,ϕ) at the fixed valuesof radial coordinate r and the toroidal angle ϕ = 0 are plotted in Fig. A.5. The radialdependence of the perturbation field is determined by (A.35). The power law of theradial decay of perturbation field has the lowest exponent, γn=6 = m0 − 1, for thetoroidalmode n = 6 . For the value δθ = 18◦, one hasm0 = π/δθ = 10 the exponentγn=6 = 9. For the next toroidal mode n = 18 we have γn=18 = 3m0 − 1 = 29.
Appendix BMagnetic Field of a Set of Saddle Coils
In this Appendix we give some details of calculations of the magnetic field createdby a set of saddle coils in a tokamak geometry described in Sect. 3.4.1. The method issimilar to the one in the classicalmagnetostatic to the calculation of themagnetic fieldcreated by the circular current loop (see, e.g., Jackson (1998)). But in our case theproblem is reduced to the new integrals which can be considered as the generalizedelliptic integrals.
As was shown in Sect. 3.4.1 the magnetic field created by the set of saddle coilscan be composed a sum of magnetic fields from the horizontal segments lying on thesurfaces Z = const and the ones lying the vertical surfaces (R, Z) (see also Fig. 3.8).We consider the calculations of these magnetic fields separately.
B.1 The magnetic Field of the Current Loop
Let (R, Z ,ϕ) be the cylindrical coordinate system. Consider a current–carryingfilament coil given by curve G in the 3D space. Let dl be an element of this curve,and J(R, Z ,ϕ) be a current flowing in the coil in the form of the circular loop asshown in Fig. 3.8. For the circular loop of radius R j lying in the horizontal planeZ j = constant the current density j(R, Z ,ϕ) has only a component in theϕ direction,
jϕ(R, Z ,ϕ) = Ic(ϕ)δ(R − R j )δ(Z − Z j ),
jx (R, Z ,ϕ) = − jϕ(R, Z ,ϕ) sinϕ,
jy(R, Z ,ϕ) = jϕ(R, Z ,ϕ) cosϕ, (B.1)
where Ic(ϕ) is a current depending on the toroidal angleϕ. Then, the vector potentialA at the point P(R, Z ,ϕ) is determined by the Biot–Savart law,
S. Abdullaev, Magnetic Stochasticity in Magnetically Confined Fusion Plasmas, 351Springer Series on Atomic, Optical, and Plasma Physics 78,DOI: 10.1007/978-3-319-01890-4, © Springer International Publishing Switzerland 2014
352 Appendix B: Magnetic Field of a Set of Saddle Coils
A(R, Z ,ϕ) = μo
4π
∫
G
j(R′, Z ′,ϕ′)dV ′
|r − r j | ,
Ax (R, Z ,ϕ) = μo
4π
∫
G
jx (R′, Z ′,ϕ′)dV ′
|r − r j | = −μo R j
4π
2π∫
0
Iϕ(ϕ j ) sinϕ j dϕ j
|r − r j | ,
Ay(R, Z ,ϕ) = μo
4π
∫
G
jy(R′, Z ′,ϕ j )dV ′
|r − r j | = μo R j
4π
2π∫
0
Iϕ(ϕ j ) cosϕ j dϕ j
|r − r j | , (B.2)
where dV = Rd Rd Zdϕ is a volume element, r j = (R j , Z j ,ϕ j ) are point coordi-nates at the curve G,
|r − r j | =√
R2 + R2j + (Z − Z j )2 − 2R R j cos(ϕ − ϕ j ).
The toroidal component of the vector potential Aϕ is
Aϕ(R, Z ,ϕ) = −Ax (R, Z ,ϕ) sinϕ + Ay(R, Z ,ϕ) cosϕ
= μo R j
4π
∫ 2π
0
Iϕ(ϕ j ) cos(ϕ j − ϕ)dϕ j
|r − r j | . (B.3)
Introducing the notations
D j =√
(R + R j )2 + (Z − Z j )2, k2 = 4R R j/D2j ,
and replacing the integration variable from ϕ j to φ,
φ = (ϕ − ϕ j + π)/2,
0 ≤ ϕ j ≤ 2π, (ϕ + π)/2 ≥ φ ≥ (ϕ − π)/2,
we obtain
Aϕ(R, Z ,ϕ) = −2μo R j
4πD
π/2∫
−π/2
Ic(ϕ − 2φ + π) cos(2φ)dφ√1 − k2 sin2 φ
. (B.4)
Suppose, that the current Ic(ϕ) can be presented as a Fourier series
Ic(ϕ) = I (0)c +
∞∑
n=1
I (n)c cos(nϕ + χn) = I (0)
c + 1
2
∞∑
n=−∞I (n)c einϕ+iχn , (B.5)
Appendix B: Magnetic Field of a Set of Saddle Coils 353
where
I (n)c eiχn = 1
π
2π∫
0
Ic(ϕ)e−inϕdϕ.
Consider a coil system consisting of N pairs of current loops as shown in Figs. 3.7and 3.8. The distribution of current Ic(ϕ) corresponding to this system is
Ic(ϕ) = I0
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
0 for 2kπ/N ≤ ϕ ≤ 2kπ/N + ϕd ,
1 for 2kπ/N + ϕd ≤ ϕ ≤ (2k + 1)π/N − ϕd ,
0 for (2k + 1)π/N − ϕd ≤ ϕ ≤ (2k + 1)π/N + ϕd ,
−1 for (2k + 1)π/N + ϕd ≤ ϕ ≤ 2(k + 1)π/N − ϕd ,
0 for 2(k + 1)π/N − ϕd ≤ ϕ ≤ 2(k + 1)π/N ,
(B.6)
where k = 0, 1, . . . , n − 1, 2kπ/N ≤ ϕ ≤ 2(k + 1)π/N . After some lengthycalculations one can obtain the following formula for the Fourier components I (n)
cfor this current distribution,
π I (n)c eiχn /I0 = 4i
ne−inπ sin
( nπ
2N
)sin
( nπ
2N− nϕd
) sin(nπ)
sin(nπ/N ).
This expression does not vanish only for non-integer numbers n = (2s + 1)N ,(s = 0, 1, 2, . . .):
I (n)c eiχn = Ic
4
(2s + 1)πeiπ/2 cos [(2s + 1)Nϕd ] . (B.7)
Finally, we have
Ic(ϕ) = 4I0π
∞∑
s=0
cos [(2s + 1)Nϕd ]
2s + 1sin [(2s + 1)Nϕ] . (B.8)
Similar formulas can be obtained when the set of coils consists of odd numbers ofsaddle coils as the case shown in Fig. 3.6.
B.1.1 Vector Potential
Using (B.5) the vector potential (B.4) can be reduced to
Aϕ(R, Z ,ϕ) = −μo R j
πD j
{I (0)c a0(R, Z) +
∞∑
n=1
I (n)c a(s)
n (R, Z) sin(nϕ)
}, (B.9)
354 Appendix B: Magnetic Field of a Set of Saddle Coils
where
a0(R, Z) =π/2∫
0
cos(2φ)dφ√1 − k2 sin2 φ
= 1
k2
[(k2 − 2)K (k) + 2E(k)
], (B.10)
a(s)n (R, Z) = (−1)n
π/2∫
0
cos(2nφ) cos(2φ)dφ√1 − k2 sin2 φ
= 1
k2
[(k2 − 2)Kn(k) + 2En(k)
].
(B.11)Here K (k) and E(k) are the complete elliptic integrals with argument k, and theintegral Kn(k) and En(k) are defined by
Kn(k) =π/2∫
0
cos(2nφ)dφ√1 − k2 sin2 φ
,
En(k) =π/2∫
0
cos(2nφ)
√1 − k2 sin2 φdφ. (B.12)
They can be called the generalized elliptic integrals. Note, that K (k) = Kn=0(k) andE(k) = En=0(k).
Therefore the vector potential of the magnetic field created by the circular loopand the corresponding normalized perturbation poloidal flux can be presented as
Aϕ(R, Z ,ϕ) =∞∑
n=0
A(n)ϕ (R, Z) sin(nϕ),
ψ(pert)(R, Z ,ϕ) = − R Aϕ(R, Z)
B0R20
= ε
∞∑
n=0
ψn(R, Z) sin(nϕ), (B.13)
where
A(n)ϕ (R, Z) = μo I0in D j
4πRLn(k),
ψn(R, Z) = − in D j
R0Ln(k), (B.14)
Here the following notations are introduced
Appendix B: Magnetic Field of a Set of Saddle Coils 355
Ln(k) =(1 − k2
2
)Kn(k) − En(k),
in = I (n)c
I0= (−1)n 4
π
cos [(2s + 1)Nϕd ]
2s + 1, n = (2s + 1)N . (B.15)
The non-dimensional perturbation parameter ε is defined as
ε = μo I04πB0R0
. (B.16)
The (R, Z ) components of the magnetic field are given by
(BR, BZ ) =∞∑
n=0
(B(n)
R (R, Z), B(n)Z (R, Z)
)sin(nϕ), (B.17)
where
B(n)Z (R, Z) = 1
R
∂[
R A(n)ϕ (R, Z)
]
∂R= B0εin
{∂D j
∂RLn(k) + D j
∂k2j∂R
d Ln(k)
dk2j
},
(B.18)
B(n)R (R, Z) = −∂ A(n)
ϕ (R, Z)
∂Z= −B0εin
{∂D j
∂ZLn(k) + D j
∂k2j∂Z
d Ln(k)
dk2j
}.
(B.19)
B.1.2 Approximation of the Integrals Ln(k)
One can establish the approximation of the integrals Ln(k), En(k) (B.15) and itsderivative d Ln(k)/dk2 by the series of an expansion in m = k2:
Ln(k) =M∑
k=1
akmk +M∑
k=0
bkmk1 ln
1
m1+ RM , (B.20)
d Ln(k)
dm=
M∑
k=1
ckmk − d
dm1
M∑
k=0
dkmk1 ln
1
m1+ RM , (B.21)
where m1 = 1 − m. The coefficients ak, bk , ck, dk , (k = 1, 2, . . . , M) found by thefitting with the numerically calculated ones are presented in Tables B.1 and B.2 forthe case M = 3.
The dependencies of the integrals Ln(m) and and its derivative d Ln(m)/dm onthe module m are plotted in Fig. B.1a, b.
356 Appendix B: Magnetic Field of a Set of Saddle Coils
Table B.1 The coefficients ak , bk in Eq. (B.20) for the case M = 3
n = 1 n = 2 n = 3 n = 4 n = 5 n = 6
a1 −15.1318 12.614 6.44849 1.05883 8.68808 −20.2141a2 24.7153 −22.0231 −7.95135 −5.24451 −9.80602 28.0533a3 −9.75115 8.97702 2.30145 3.36486 2.04908 −8.8537b0 −0.227805 0.227776 −0.247866 0.227436 −0.226976 0.226234b1 1.58137 −0.867444 −2.0157 1.67606 −3.26677 4.89141b2 9.47854 −8.3374 −3.3891 −1.38598 −4.72399 12.0068b3 3.9163 −3.64604 −0.796313 −1.58287 −0.466855 3.09053|RM | < 3 × 10−3 3 × 10−3 2 × 10−3 2 × 10−3 10−3 2 × 10−3
Table B.2 The coefficients ck , dk in Eq. (B.21) for the case M = 3
n = 1 n = 2 n = 3 n = 4 n = 5 n = 6
c1 0.750263 1.16925 −6.19823 14.8482 −27.3604 43.1621c2 0.919855 1.28526 −6.18702 14.7706 −27.1634 42.7855c3 −1.93061 −0.149124 5.64138 −16.2511 32.8243 −54.8624d0 0.248363 −0.248326 0.248192 −0.247847 0.24714 −0.24593d1 −0.16046 0.886782 −2.04892 3.56763 −5.3332 7.22269d2 0.661822 0.594552 −3.54806 8.851 −16.6575 26.5599d3 0.392412 −0.063162 −0.849296 2.67554 −5.61181 9.6156|RM | < 2.5 × 10−6 4 × 10−6 ×10−5 2 × 10−5 3 × 10−5 4 × 10−5
(a) (b)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
|Ln(m
)|
m
n=1
n=0 23
4 0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1
|dL
n(m
)/dm
|
m
n=1
n=02
34
Fig. B.1 The integral Ln(m) and and its derivative d Ln(m)/dm as the functions of m = k2 for theseveral numbers n
Appendix B: Magnetic Field of a Set of Saddle Coils 357
B.2 Magnetic Field of the Vertical Segments of a Setof Saddle Coils
Above we have considered the calculation of the magnetic field created by the hori-zontal segments of the set of saddle coils. Now consider the magnetic field from thesegments of the set on the vertical sections (R, Z ).
B.2.1 Magnetic Field from a Single Vertical Wire
First we consider a single straight wire located along the vertical lines located at thesection (Rc = const, ϕ0 = const) between Z1 < Z < Z2. The current density isgiven by
jz = I0R−10 δ(ϕ − ϕ0)δ(R − R0)
{1, for Z1 < Z < Z2,
0, for Z < Z1, Z > Z2.(B.22)
The magnetic field generated by this wire can be described by the vector potential(0, 0, AZ ) with the non-zero z-component:
AZ (R, Z ,ϕ) = μo
4π
∫
G
jz(R′, Z ′,ϕ′)dV ′
|r − r0| . (B.23)
Using (B.22) it can be calculated directly,
AZ (R, Z ,ϕ) = μo I04π
Z2∫
Z1
d Z0√a2 + (Z − Z0)2
= μo I04π
[ln
Z2 − Z +√a2 + (Z2 − Z)2
Z1 − Z +√a2 + (Z − Z1)2
],
a2 = R2 + R20 − 2R R0 cos(ϕ − ϕ0), (B.24)
where we used the integral
x∫
0
dx√a2 + x2
= ln(x +√
a2 + x2) − ln a. (B.25)
358 Appendix B: Magnetic Field of a Set of Saddle Coils
B.2.2 Set of Straight Wires
Now we consider a set of 2N straight wires located at fixed toroidal angles ϕ(±)k =
(π/N )k ± ϕd , (k = 0, 1, . . . , 2N − 1) between circular loops of radii R1 at Z1and radii R2 at Z2. Suppose the directions of even/odd currents are positive/negativewith the respect to Z axis, and have the same magnitude, i.e., I (±)
k = (−1)k I0. (seeFig. 3.8). Then the current density can be presented as
j = ( jR, jZ ) = (sinα, cosα) j (R, Z ,ϕ), (B.26)
where α is an inclination angle of the wire with respect to the vertical axis Z , i.e.,
tanα = R2 − R1
Z2 − Z1,
and the current density
j (R, Z ,ϕ) = I0δ(R − R0(Z))R−1
×2N−1∑
k=0
(−1)k[δ
(ϕ − kπ
N− ϕd
)+ δ
(ϕ − kπ
N+ ϕd
)]. (B.27)
Using the Poisson summation formula
δ(ϕ) = 1
2π
∞∑
n=−∞cos nϕ = 1
2π
∞∑
n=−∞einϕ, (B.28)
Eq. (B.26) can be transformed to
j (R, Z ,ϕ) = I0δ(R − R0)
πR
∞∑
n=−∞einϕ cos(nϕd)e−iπn(1−1/2N )eiπ/2 sin(πn)
cos(πn/2N ).
(B.29)All terms in (B.29) vanish except ns = (2s + 1)N , (s = 0,±1,±2, . . .). The lattergive
j (R, Z ,ϕ) = I0δ (R − R0(Z)) R−1∞∑
s=0
Js cos (nsϕ) , (B.30)
where
Js = 4N
π(−1)N+1+s cos [nsϕd ] . (B.31)
Appendix B: Magnetic Field of a Set of Saddle Coils 359
B.2.3 Vector Potential
Using the current density (B.26) and (B.30) the magnetic field created by the verticalsegments of the saddle coils can be presented by the R- and Z -components of thevector potential,
A(R, Z ,ϕ) = (AR, AZ ) = μo
4π
∫( jR, jZ )(R′, Z ′,ϕ′) sin dV ′
|r − r0|= μo I0
4π
∞∑
s=0
Js
∫(eR, eZ ) cos(nsϕ
′)d Z ′dϕ′
|r − r0|
= μo I04π
∞∑
s=0
Js
[a(s)(R, Z) cos(nsϕ) + b(s)(R, Z) sin(nsϕ)
], (B.32)
where
a(s)(R, Z) =Z2∫
Z1
2π∫
0
(eR, eZ ) cos(nsφ)d Z ′dφ
D(φ, Z ′),
b(s)(R, Z) = −Z2∫
Z1
2π∫
0
(eR, eZ ) sin(nsφ)d Z ′dφ
D(φ, Z ′), (B.33)
and
eR = sinα(Z ′), eZ = cosα(Z ′),
D(φ, Z ′) =√
R2 + R20(Z ′) − 2R R0(Z ′) cosφ + (Z − Z ′)2. (B.34)
If the segment of a wire connecting Z1 and Z2 is a straight the integral overZ ′ can be calculated analytically. If this segment is not straight it can be presentedas a composed by M number of small straight segments. Each straight segment iconnects the points (Ri , Zi ) and (Ri+1, Zi+1), (i = 1, 2, . . . , M − 1). Then thevectors as(R, Z), bs(R, Z) is given by the sum
a(s)(R, Z) =M∑
i=1
(sinαi , cosαi )a(i)s (R, Z),
b(s)(R, Z) =M∑
i=1
(sinαi , cosαi )b(i)s (R, Z), (B.35)
where sinαi = sinα(Zi ), cosαi = cosα(Zi ),
360 Appendix B: Magnetic Field of a Set of Saddle Coils
a(i)s (R, Z) =
Zi+1∫
Zi
2π∫
0
cos(nsφ)d Z ′dφ
D(φ, Z ′),
b(i)s (R, Z) = −
Zi+1∫
Zi
2π∫
0
sin(nsφ)d Z ′dφ
D(φ, Z ′). (B.36)
The function R0(Z) describes the straight segments between points (Ri , Zi ) and(Ri+1, Zi+1), (i = 1, 2, . . . , M − 1), i.e.,
R0(Z) = Ri + Ri+1 − Ri
Zi+1 − Zi(Z − Zi ) = Ri + βi (Z − Zi ),
βi = Ri+1 − Ri
Zi+1 − Zi= tanαi , Zi ≤ Z ≤ Zi+1. (B.37)
The denominator D(φ, Z ′) in (B.36) can be presented in the form,
D(φ, Z ′) =√
Ai
[a2
i + (Zi − Z ′)2],
where Ai , ai and Zi are given by
Ai = 1 + β2i , Zi = 1
1 + β2i
[Z + Rβi cosφ − (Ri − βi Zi )βi
],
a2i = Ci − Z2
i ,
Ci = 1
1 + β2i
[R2 + (Ri − βi Z1)
2 − 2R(Ri − βi Zi ) cosφ + Z2].
It allows to integrate (B.36) over Z ′ thus reducing them to the one-dimensionalintegral
a(i)s (R, Z) = 1√
Ai
2π∫
0
L(i)g (φ) cos(nsφ)dφ,
b(i)s (R, Z) = − 1√
Ai
2π∫
0
L(i)g (φ) sin(nsφ)dφ, (B.38)
where
L(i)g (φ) = ln
⎡
⎣Zi+1 − Zi +
√a2
i + (Zi+1 − Zi )2
Zi − Zi +√
a2 + (Zi − Zi )2
⎤
⎦ .
Appendix B: Magnetic Field of a Set of Saddle Coils 361
The numerical integration shows that the integral b(i)s (R, Z) is negligibly small.
Furthermore, we neglect this integral.
B.2.4 The Perturbation Poloidal Flux
According to (3.4) the perturbation poloidal flux ψ(pert) created by the vertical seg-ments of saddle coils is determined by the function g(R, Z ,ϕ) given by (3.6) and(3.8). Using (B.32) and (3.8) the perturbation flux ψ(pert) on the given magneticsurface ψ =const can be reduced to
ψ(pert)(R, Z ,ϕ) = ∂
∂ϕ
∫G (R, Z ,ϕ) dϑ, (B.39)
where
G (R, Z ,ϕ) = ε
∞∑
ns
Gns (R, Z) cos(nsϕ), (B.40)
Gns (R, Z ,ϕ) = Js R−10
[a(s)
R (R, Z)d R
dϑ+ a(s)
Z (R, Z)d Z
dϑ
].
The perturbation parameter ε is defined by Eq. (B.16). Using the relation dϑ =dϕ/q(ψ) and the equations of magnetic field lines (1.19), the function Gns (R, Z) isreduced to
Gns (R, Z) = Jsq(ψ)R
R0Bϕ
(a(s)
R (R, Z)BR + a(s)Z (R, Z)BZ
), (B.41)
where BR and BZ are the poloidal components of the equilibrium magnetic field.On the given magnetic surface ψ = const the function Gn(R, Z) is the 2π–
periodic function of the angle variable ϑ, and therefore it can be expanded into theFourier series,
Gn(R, Z) =∑
m
Gmn(ψ) cos (mϑ + φmn) . (B.42)
Finally, from (B.40) and (B.42) we obtain the following presentation of the pertur-bation poloidal flux,
ψ(pert)(R, Z ,ϕ) = −ε∑
m,n
n
mGmn(ψ) sin (mϑ + φmn) sin(nϕ), (B.43)
which can be also rewritten in the form given by Eqs. (3.50) and (3.51).
362 Appendix B: Magnetic Field of a Set of Saddle Coils
B.3 Numerical Calculations of Fourier Components
The Fourier expansion of the poloidal fluxes ψn(ψ,ϑ) ≡ ψn (R(ψ,ϑ), Z(ψ,ϑ))
given by Eqs. (3.47) and (3.50) can be presented in the following form
ψn(ψ,ϑ) =∞∑
m=0
Hmn(ψ) cos(mϑ + χmn)
=∞∑
m=0
[H (c)
mn (ψ) cosmϑ + H (s)mn (ψ) sinmϑ
], (B.44)
where Fourier components Hmn(ψ), H (c)mn (ψ), H (s)
mn (ψ) are given by the integrals
Hmn(ψ) =[(
H (c)mn (ψ)
)2 +(
H (s)mn (ψ)
)2]1/2,
H (c)mn (ψ) = Hmn(ψ) cosχmn = 1
π
π∫
−π
ψn(ψ,ϑ) cosmϑdϑ,
H (s)mn (ψ) = −Hmn(ψ) sinχmn = 1
π
π∫
−π
ψn(ψ,ϑ) sinmϑdϑ,
H (c)m=0,n(ψ) = 1
2π
π∫
−π
ψn(ψ,ϑ)dϑ. (B.45)
The phases χmn are found by the following rules
χ(0)mn = arctan
(∣∣∣∣∣H (s)
mn (ψ)
H (c)mn (ψ)
∣∣∣∣∣
),
χmn =
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
χ(0)mn, if H (c)
mn (ψ) > 0 and H (s)mn (ψ) > 0,
π − χ(0)mn, if H (c)
mn (ψ) < 0 and H (s)mn (ψ) > 0,
π + χ(0)mn, if H (c)
mn (ψ) < 0 and H (s)mn (ψ) < 0,
2π − χ(0)mn, if H (c)
mn (ψ) > 0 and H (s)mn (ψ) < 0.
(B.46)
Appendix CCalculations of the Poincaré Integrals
Here we present the detailed calculations of the Poincaré integrals (6.59) for theperturbation magnetic fluxes ψ1(ψ,ϑ,ϕ) of type
ψ1 (ψ,ϑ,ϕ) = ψn (ψ,ϑ) sin (nϕ + χn) ,
ψn (ψ,ϑ) =∞∑
m=1
Hmn(ψ)eimϑ =∞∑
m=1
|Hmn(ψ)| eim(ϑ−ϑ0). (C.1)
We assume that the integration in (6.59) is taken over the unperturbed orbit (ψ =const,ϑ(ϕ) = ϕ/q(ψ)) one poloidal turn starting from and ending at the section Σs
(see Fig. 2.4). Recall that ϑ = ±π at the section Σs . Using (C.1) the integral (6.59)is reduced to
P (ϕ,ψ) =πq(ψ)∫
−πq(ψ)
ψn(ψ,ϑ(ϕ′)
)sin
(n[ϕ + ϕ′]) dϕ′
= Kn(ψ) sin(nϕ + χn) + Ln(ψ) cos(nϕ + χn), (C.2)
where
Rn(ψ) = Kn(ψ) + i Ln(ψ) =πq(ψ)∫
−πq(ψ)
ψn (ψ,ϑ(ϕ)) einϕdϕ. (C.3)
Using a Fourier series of ψn (ψ,ϑ) in ϑ we have
S. Abdullaev, Magnetic Stochasticity in Magnetically Confined Fusion Plasmas, 363Springer Series on Atomic, Optical, and Plasma Physics 78,DOI: 10.1007/978-3-319-01890-4, © Springer International Publishing Switzerland 2014
364 Appendix C: Calculations of the Poincaré Integrals
Kn(ψ) = 1
2
∞∑
m=1
Hmn(ψ)
πq∫
−πq
[ei(m/q+n)ϕ + ei(m/q−n)ϕ
]dϕ
= q(ψ)
∞∑
m=1
Hmn(ψ)
[sin (π[m + nq])
m + nq+ sin (π[m − nq])
m − nq
]. (C.4)
Recalling that Hmn(ψ) = |Hmn| exp(−imϑ0) one has at the resonant surface ψm0,n ,(q(ψm0,n) = m0/n),
Kn(ψm0,n) = πq(ψm0,n)|Hm0n(ψm0,n)| cos (nqϑ0) . (C.5)
One can similarly obtain
Ln(ψ) = 1
2i
∞∑
m=1
Hmn(ψ)
πq∫
−πq
[ei(m/q+n)ϕ − ei(m/q−n)ϕ
]dϕ
= −iq(ψ)
∞∑
m=1
Hmn(ψ)
[sin (π[m + nq])
m + nq− sin (π[m − nq])
m − nq
], (C.6)
andLn(ψm0,n) = πq(ψm0,n)|Hm0n(ψm0,n)| sin (nqϑ0) . (C.7)
At the arbitrary ψ the function Rn(ψ) = Kn(ψ) + i Ln(ψ) can be presented as
Rn(ψ) = R(reg)n (ψ) + R(osc)
n (ψ), (C.8)
where R(reg)n (ψ) is the regular part defined as
R(reg)n (ψ) = πq(ψ)H∗
n (ψ; m = nq), (C.9)
and R(osc)n (ψ) is the oscillatory part of the integral Rn(ψ). According to the definition
of the R(reg)n (ψ) the oscillatory part R(osc)
n (ψ) has zeros at the resonant surfacesψmn ,q(ψmn) = m/n, i.e., R(osc)
n (ψmn) = 0. In Eq. (C.9) the function Hn(ψ; m) is definedby Hmn(ψ) by extending the discrete mode number m to the continuous one.
The relation (C.9) can be also obtained from (C.3). Indeed, by replacing thevariable ϕ to ϑ we have
Rn(ψ) = q(ψ)
π∫
−π
ψn (ψ,ϑ(ϕ)) einqϑdϑ. (C.10)
Appendix C: Calculations of the Poincaré Integrals 365
For the integer values of the product nq, nq = m, the latter up to the constant factorπ coincides with the Fourier coefficients Hmn , i.e., Rn(ψ) = πq(ψ)H∗
mn(ψ).
C.1 Determination of Oscillatory Parts of Rn(ψ)
Below we present the method of calculations of R(osc)n (ψ) based on the contour
integration. It can be applied when there exists an analytical formula of the fluxψn(ψ,ϑ). Specifically we use the poloidal fluxes of the perturbation magnetic fieldfor Model I (9.28) and Model II (9.29).
C.1.1 Model I
For Model I with the flux function ψn(ψ,ϑ) (C.1) the integral (C.3) can be taken bya contour integral. Changing the integration variable ϕ to ϑ = ϕ/q the integral (C.3)is reduced to
Rn(ψ) = e−b
π∫
−π
cos(ϑ − ϑ0) − e−b
1 + e−2b − 2e−b cos(ϑ − ϑ0)einqϑdϑ. (C.11)
Introducing a complex integration variable z = eiϑ, dϑ = dz/ i z, we have
sin(ϑ − ϑ0) = eiϑ0
2zi
(z2e−2iϑ0 − 1
),
cos(ϑ − ϑ0) = eiϑ0
2z
(1 + z2e−2iϑ0
),
1 + e−2b − 2e−b cos(ϑ − ϑ0) = z1z[(z − z1)(z2 − z)] ,
z1 = e−b+iϑ0 , z2 = eb+iϑ0 .
The integral (C.11) can be then rewritten as
Rn(ψ) = e2iϑ0
2i
∫
Cϕ
f (z)dz, (C.12)
where Cϕ is the segment −π < θ < π on the unit circle |z| = 1, and
f (z) = znq−1[1 + z2e−2iϑ0 − 2z∗
1z]
(z − z1)(z2 − z).
366 Appendix C: Calculations of the Poincaré Integrals
Fig. C.1 Integration contourof the integral (C.13)
I
II
z1
z2
Cε
Cϕ
The function f (z) has two poles at z = z1 = e−b+iϑ0 and z = z2 = eb+iϑ0 locatedinside |z1| < 1 and outside |z2| > 1 the circle |z| = 1. Moreover, for the nonintegervalue of the product nq the function f (z) is multivalued because of term znq , i.e.,f(eiθ) �= f
(ei(θ+2π)
). To apply the residue formula to integrate (C.12) we choose
the closed contour C along which the function f (z) is single-valued. The contourC shown in Fig. C.1 contains the segment Cϕ on the unit circle |z| = 1 and cuts Iand I I along the radii z = r exp(±iπ), ε < r < 1, respectively, and the circle Cε ofradius |z| = ε.
The residue formula for the integral over the function f (z) along the contour Cgives,
∮
Cf (z)dz = 2πiRes [ f (z); z = z1] = 2πie−nqbeinqϑ0e−2iϑ0 . (C.13)
The left hand side of (C.13) consists of four integrals,
∮
Cf (z)dz =
(∫
Cϕ
+∫
I+∫
I I+∫
Cε
)f (z)dz. (C.14)
It is not difficult to see that at nq > 0 the integral along the contour Cε vanish atthe limit ε → 0. The integrals along the cuts I and I I , where zI = r exp(iπ) andzI I = r exp(−iπ), respectively, are reduced to
(∫
I+∫
I I
)f (z)dz = 2i sin(πnq)FI (ψ,ϑ0), (C.15)
Appendix C: Calculations of the Poincaré Integrals 367
where
FI (ψ,ϑ0) =1∫
0
φ(r)rnq−1dr,
φ(I )(r) = 1 + r2e−2iϑ0 + 2re−iϑ0e−b
(r + e−b+iϑ0)(r + eb+iϑ0). (C.16)
From Eqs. (C.12), (C.13) and (C.15) it follows that
Rn(ψ) = R(reg)n (ψ) + R(osc)
n (ψ),
R(reg)n (ψ) = πeinqϑ0e−nqb = πe−nqb [cos(nqϑ0) + i sin(nqϑ0)] ,
R(osc)n (ψ) = −e2iϑ0 sin(πnq)FI (ψ,ϑ0). (C.17)
C.1.2 Asymptotical Expansion
For large nq � 1 the function FI (ψ,ϑ0) (C.16) can be estimated by the asymptoticexpansion in powers of 1/nq. It can be done using an integration by part,
1∫
0
φ(I )(r)rnq−1dr =⎡
⎣φ(I )0
nq− 1
nq
1∫
0
dφ(I )(r)
drrnqdr
⎤
⎦
= 1
nq
{φ(I )0 − 1
nq + 1φ(I )1 + · · ·
}, (C.18)
where
φ(I )0 = φ(I )(1) = cosϑ0 + e−b
cosϑ0 + cosh b,
φ1 = dφ(r)
dr
∣∣∣∣r=1
= −isinh b sin ϑ0
(cosϑ0 + cosh b)2. (C.19)
At ϑ0 = 0 and ϑ0 = π, we have
φ(I )0 (0) = 1 + e−b
1 + cosh b, φ
(I )0 (π) = − 1 − e−b
cosh b − 1.
Near the separatrix b → 0, it has the following asymptotics, φ0(π) ≈ −(2/b)
(1 + O(b)).Finally, the leading terms of asymptotical expansion of R(osc)
n (ψ), in 1/nq can bepresented as
368 Appendix C: Calculations of the Poincaré Integrals
K (osc)n (ψ) = − sin(πnq)
nq
cosϑ0 + e−b
cosϑ0 + cosh b,
L(osc)n (ψ) = sin(πnq)
nq(nq + 1)
sin ϑ0 sinh b
(cosϑ0 + cosh b)2. (C.20)
C.1.3 Model II
For Model II (9.29), integrating by part one arrives to
Rn(ψ) = − nq
π∫
−π
arctan
[sin(ϑ − ϑ0)
cos(ϑ − ϑ0) − eb
]einqϑdϑ
= − 2 sin(πnq)F(ϑ0)
+ ie−b
π∫
−π
cos(ϑ − ϑ0) − e−b
1 + e−2b − 2e−b cos(ϑ − ϑ0)einqϑdϑ, (C.21)
where
F(ϑ) = arctan
[sin ϑ
cosϑ + eb
]. (C.22)
The integral in the last term of (C.21) coincides with the integral (C.11). Then using(C.17), we obtain
Rn(ψ) = R(reg)n (ψ) + R(osc)
n (ψ),
R(reg)n (ψ) = πieinqϑ0e−nqb,
R(osc)n (ψ) = − sin(πnq)
[2F(ϑ0) + ie2iϑ0 FI (ψ,ϑ0)
]. (C.23)
Using (C.20) they can be also presented by
K (reg)n (ψ) = −π sin(nqϑ0)e
−nqb,
L(reg)n (ψ) = π cos(nqϑ0)e
−nqb,
K (osc)n (ψ) = − sin(πnq)
[2F(ϑ0) + 1
nq(nq + 1)
sin ϑ0 sinh b
(cosϑ0 + cosh b)2
],
L(osc)n (ψ) = − sin(πnq)
[2F(ϑ0) + 1
nq
cosϑ0 + e−b
cosϑ0 + cosh b
]. (C.24)
Appendix DAdvanced Version of the Symplectic Mappingfor Hamiltonian Systems
Below we construct the alternative form of the mapping (ϑk, Ik) → (ϑk+1, Ik+1)
for the Hamiltonian system given by Eqs. (6.5) and (6.6). Similar to the methodgiven in Sect. 6.2.2 it is based on the canonical change of variables and the classicalperturbation theory in a finite time interval (see Abdullaev (2002, 2006)). In theinterval tk ≤ t ≤ tk+1 we perform a such a canonical transformation of variables(ϑ, I ) → (Θ, J ) that the new HamiltonianH in the new canonical variables (Θ, J )acquires fast oscillating perturbation terms, i.e.,
H = H0(J ) + εH1(Θ, J, t),
H1(Θ, J, t) = −2H1 (Θ, J, t)N∑
s=1
cos (s Mnt) , (D.1)
where M ≥ 1 is an integer number, N � 1 is the number of harmonics. The canoni-cal change of variables (ϑ, I ) → (Θ, J ) is implemented via the generating functionF(J,ϑ, t) = Jϑ + εS(J,ϑ, t), where the generating function S = S(J,ϑ, t) satis-fies the Hamilton-Jacobi equation
H0
(ϑ, J + ε
∂S
∂ϑ, t
)+ ε
∂S
∂t= H(ϑ, J, t, ε). (D.2)
The generating function S(J,ϑ, t) is sought as a series in powers of ε similar to(6.15). Expanding the Hamilton–Jacobi equation (D.2) in powers of of ε one obtainsH0(J ) = H0(J ), and the equations for Si , (i = 1, 2, . . .):
∂S1∂t
+ ∂H0
∂ J
∂S1∂ϑ
= H1(ϑ, J, t) − H1(ϑ, J, t), (D.3)
∂S j
∂t+ ∂H0
∂ J
∂S j
∂ϑ= −Fj (ϑ, J, t), j ≥ 2, (D.4)
where Fj (ϑ, I, t) are thepolynomial functions of derivatives∂S1/∂ϑ, . . ., ∂S j−1/∂ϑ.
S. Abdullaev, Magnetic Stochasticity in Magnetically Confined Fusion Plasmas, 369Springer Series on Atomic, Optical, and Plasma Physics 78,DOI: 10.1007/978-3-319-01890-4, © Springer International Publishing Switzerland 2014
370 Appendix D: Advanced Version of the Symplectic Mapping for Hamiltonian Systems
In the first order of the perturbation parameter ε the generating function is givenby the integral
S1(ϑ, J, t, t0) =t∫
t0
[H1
(ϑ(t ′), J, t ′
)− H1(ϑ(t ′), J, t ′
) ]dt ′, (D.5)
taken along the unperturbed field line ϑ(t ′) = ϑ + ω(J )(t − t ′), J = const. Thevariable t and the free parameter t0 lie in the intervals: tk ≤ t ≤ tk+1, tk < t0 < tk+1.Using Eq. (D.1) we have
S1(J,ϑ, t, t0) = −t∫
t0
H1(ϑ(t ′), J, t ′)N∑
s=−N
cos(sMnt ′
)dt ′
= − 2π
Mn
t∫
t0
H1(ϑ(t ′), I, t ′)∞∑
k=−∞δ(t ′ − tk
)dt ′, at N → ∞,
(D.6)
where tk = k(2π/Mn), (k = 0,±1,±2, . . .). From Eq. (D.6) it follows that thegenerating function S1 vanishes in the time interval tk < t < tk+1. But it takesnon-zero values at t = tk and t = tk+1 given by Eq. (6.47).
Note that the higher order generating functions S j , ( j ≥ 2), vanish in the wholetime interval tk ≤ t ≤ tk+1 (see Abdullaev (2006)).
Suppose that (Θ(t), J (t)) is a solution of the newHamiltonian (D.1) in the intervaltk < t < tk+1. Then the mapping can be presented by the following successivecanonical transformations
Jk = Jk − ε∂Sk
∂ϑk, Θk = ϑk + ε
∂Sk
∂ Jk,
(Θk+1, Jk+1) = M(Θk, Jk),
Jk+1 = Jk+1 + ε∂Sk+1
∂ϑk+1, ϑk+1 = Θk+1 − ε
∂Sk+1
∂ Jk+1, (D.7)
where (Θk, Jk) = (Θ(tk), J (tk)). The mapping (Θk+1, Jk+1) = M(Θk, Jk) can beconstructed using the regular procedure described in Refs. Abdullaev (2002, 2006).For the Hamiltonian system (D.1) it reads as
Appendix D: Advanced Version of the Symplectic Mapping for Hamiltonian Systems 371
Jk = Jk − ε∂Gk
∂Θk, Θk = Θk + ε
∂Gk
∂ Jk,
Θk+1 = Θk + tk+1 − tkq(Jk)
, Jk+1 = Jk,
Jk+1 = Jk+1 + ε∂Gk+1
∂Θk+1, Θk+1 = Θk+1 − ε
∂Gk+1
∂ Jk+1, (D.8)
determined by the generating function G( J ,Θ, t, t0): Gk = G( Jk,Θk, tk, t0),Gk+1 = G( Jk+1,Θk+1, tk+1, t0). In the first order of ε it is given by
G1(J,Θ, t, t0) = −t∫
t0
H1(Θ(t ′), J, t ′)dt ′
= 2
t∫
t0
H1(Θ(t ′), J, t ′)N∑
s=1
cos(s Mnt ′
)dt ′. (D.9)
As seen from Eq. (D.9) the generating function G1 is determined as an integral fromthe fast oscillating functions.One should expect thatG1 decreaseswith increasing thenumber M . Taking the asymptotical expansion of the integral (D.9) one can obtainthe following estimation for the generating function G1 = G1(J,Θ, tk+1, tk):
G1 ≈ 4π
Mn
N∑
s=1
⎡
⎣P∑
p=1
1
(s Mn)2pf (2p)(tk) + O([smΩ]−2P )
⎤
⎦ , (D.10)
where f (t) ≡ H1(Θ(t), J, t), f (p)(t) ≡ d p f (t)/dt p, (p = 1, 2, . . .). Supposingthat n−2p f (2p)(tk) ∼ 1, one can obtain the following estimation for G1 at largevalues N � 1, P � 1:
|G1| � 2π3
M3n= n2
12(Δt)3 , (D.11)
where Δt = tk+1 − tk = 2π/Mn is a step of the mapping. At the moderately largevalues of M ≥ 8÷ 10 one can neglect the generating function G1, and the mapping(D.7) is reduced to the form (6.44) with the generating functions (6.47). For instance,for M = 8 the divergence of the flux coordinate I of a regular orbit from the onecalculated for M = 128 is 3.6× 10−7 per one toroidal turn (for ε = 2× 10−3). Thisis sufficiently accurate to plot Poincare sections. We have chosen M = 16 for thecalculation of diffusion coefficients.
Amore detailed study of the accuracy of the describedmapping procedure requiresa special investigation.
Appendix EEigenvalues of the Jacobi Matrix
E.1 Jacobi Matrix
In this appendix we present the calculations of the Jacobi matrix (7.44), its eigen-values (7.52) and the Lyapunov exponents of the mapping (6.29)–(6.31) [or (7.32)].We present the latter in the form
M = T−T0T+, (E.1)
of three successive mappings, T−T0T+, each of them are given by Eqs. (6.29), (6.30)and (6.31), respectively. Then the Jacobian matrix (7.44) can be written as a productof three Jacobian matrices, corresponding to three successive mappings,
Jk = Mk+1M0Mk, (E.2)
where
Mk =⎛
⎜⎝
∂ Jk
∂ Ik
∂ Jk
∂ϑk∂Θk
∂ Ik
∂Θk
∂ϑk
⎞
⎟⎠ , (E.3)
M0 =
⎛
⎜⎜⎝
∂ Jk+1
∂ Jk
∂ Jk+1
∂Θk∂Θk
∂ Jk
∂Θk
∂Θk
⎞
⎟⎟⎠ =(
1 0w′(Jk) (tk+1 − tk) 1
), (E.4)
Mk+1 =
⎛
⎜⎜⎝
∂ Ik+1
∂ Jk+1
∂ Ik+1
∂Θk∂ϑk+1
∂ Jk+1
∂ϑk+1
∂Θk
⎞
⎟⎟⎠ . (E.5)
S. Abdullaev, Magnetic Stochasticity in Magnetically Confined Fusion Plasmas, 373Springer Series on Atomic, Optical, and Plasma Physics 78,DOI: 10.1007/978-3-319-01890-4, © Springer International Publishing Switzerland 2014
374 Appendix E: Eigenvalues of the Jacobi Matrix
The derivatives in the matrices (E.3) and (E.5) are easily calculated from themappings given by Eqs. (6.29) and (6.31):
∂ Jk
∂ Ik= 1
1 + εG Jϑ(ϕk),
∂ Jk
∂ϑk= − εGϑϑ(ϕk)
1 + εG Jϑ(ϕk),
∂Θk
∂ Ik= εG J J (ϕk)
1 + εG Jϑ(ϕk),
∂Θk
∂ϑk= 1 + εG Jϑ(ϕk) − ε2G J J (ϕk)Gϑϑ(ϕk)
1 + εG Jϑ(ϕk), (E.6)
and
∂ Ik+1
∂ Jk= 1 + εG Jϑ(ϕk+1) − ε2G J J (ϕk+1)Gϑϑ(ϕk+1)
1 + εG Jϑ(ϕk+1),
∂ Ik+1
∂Θk= εGϑϑ(ϕk+1)
1 + εG Jϑ(ϕk+1),
∂ϑk+1
∂ Jk= − εG J J (ϕk+1)
1 + εG Jϑ(ϕk+1),
∂ϑk+1
∂Θk= 1
1 + εG Jϑ(ϕk+1), (E.7)
where
G J J (ϕ) ≡ ∂2G
∂ J 2 , G Jϑ(ϕ) ≡ ∂2G
∂ J∂ϑ,
Gϑϑ(ϕ) ≡ ∂2G
∂ϑ2 , G(ϕ) = G(ϑ, J,ϕ,ϕ0). (E.8)
E.2 Jacobi Matrix at the Fixed Points
Below we calculate the eigenvalues of the Jacobi matrix J of the mapping (7.32) atthe fixed points considered in Sect. 7.2.2. Putting ϕk = ϕ0 − 2πm0, ϕk+1 = ϕ0 wereduce the corresponding Jacobi matrix to
∂ Ik+1
∂ Ik= 1
1 + εG J,ϑ,
∂ Ik+1
∂ϑk= − εGϑϑ
1 + εG Jϑ,
∂ϑk+1
∂ Ik= w′(Ik+1)2πm0
1 + εG J,ϑ+ εG J J
1 + εG Jϑ(ϕk),
∂ϑk+1
∂ϑk= εGϑϑw′(Ik+1)2πm0
1 + εG J,ϑ+ 1 + εG Jϑ − ε2G J J Gϑϑ
1 + εG Jϑ. (E.9)
Appendix E: Eigenvalues of the Jacobi Matrix 375
Using the generating function G (6.34) in the first order of ε we have
G1 = G1(ϑ0, J,ϕ0 − 2πm0,ϕ0)
= 2πm0
∑
m,n
Hmn(J )[a(xmn) sin (mϑ0 − nϕ0 + χmn)
+ b(xmn) cos (mϑ0 − nϕ0 + χmn)]. (E.10)
According to (7.41) the coefficients a(xmn), b(xmn) near the resonant surfacesIm0n0 , q(Im0n0) = m0/n0, are of order
a(xmn) ∝ ε2,
b(xmn) ={1 + C0ε for m/n = m0/n0,
C0ε for m/n �= m0/n0,
C0ε = −q ′mq2 (I0 − Im0n0). (E.11)
Neglecting small terms of order of ε and ε2, The generating function G1 (E.10) atthe fixed point I0 is reduced to
G1 = 2πm0
∑
m/n=m0/n0
Hmn(I0) cos (mϑ0 − nϕ0 + χmn) . (E.12)
The trace TrJ of the matrix can be reduced to
Tr J = ∂ Ik+1
∂ Ik+ ∂ϑk+1
∂ϑk= 1
1 + εG J,ϑ+ εGϑϑw′(Ik+1)2πm0
1 + εG J,ϑ
+ 1 + εG Jϑ − ε2G J J Gϑϑ
1 + εG Jϑ. (E.13)
Neglecting the terms of ε2 we obtain Eq. (7.53).
E.3 Eigenvalues of Jacobi Matrix
Consider the linearized equations (7.51) near the fixed point. We introduce newcoordinates (x, y) by rotating the coordinates (δJ, δϑ) around the point (0, 0), i.e.,
(xy
)= U
(δJδϑ
), U =
(cosφ − sin φsin φ cosφ
), (E.14)
376 Appendix E: Eigenvalues of the Jacobi Matrix
which would reduce the Eq. (7.51) into form
xk+1 = λ1 xk,
yk+1 = λ2 yk, (E.15)
where λi are constants. The angle φ are found from the eigenvalue problem of thematrix Λ,
ΛU = EU, E =(
λ1 00 λ2
). (E.16)
The eigenvalues λ of the 2 × 2 matrix Λ with det Λ = 1 are determined by thesolution of the quadratic equation
λ2 − 2Aλ + 1 = 0, A = 1
2Tr Λ = 1
2
(∂ Ik+1
∂ Ik+ ∂ϑk+1
∂ϑk
). (E.17)
From Eq. (E.14) we have two solutions:
tan φ1 = − tan φ2 = λ1 − Λ11
Λ12= Λ21
Λ22 − λ2. (E.18)
Appendix FFeatures of the Perturbation Hamiltonianof Guiding-Center Motion
In this appendix we analysis the perturbation Hamiltonian of guiding-center motion(5.38) in the action-angle variables (ϑ,ϑϕ, J, Iϕ) for the magnetic perturbationscreated by the set of saddle coils. Furthermore, we consider only time-independentmagnetic perturbations and neglect the perturbations of the electric field.
The magnetic perturbation fluxes ψ(pert)p (R, Z ,ϕ) of the set of saddle coils
were given in Sect. 3.4 by Eqs. (3.43) and (3.49). Using the latter the perturbationHamiltonian (5.34) can be reduced to
εh1 ≡ εh1(z,ϕ, pz, pϕ, pt ) = ε
3∑
j=1
∑
n
H ( j)n (R, Z) sin(nϕ), (F.1)
where
H ( j)n (R, Z) = Zq
uϕψ( j)n (R, Z)
xc. (F.2)
The dimensionless perturbation parameter ε is defined by Eq. (3.46). The functionsψ
( j)n (R, Z), ( j = 1, 2, 3), are given by (3.44) and (3.50). We recall that R = R0x ,
Z = R0z.Using the relations (5.21) and (5.22) between the coordinates (R = Roxc(pz), Z =
R0z,ϕ) and the action-angle variables (ϑ,ϑϕ, J, Iϕ) the perturbation Hamiltonian(F.1) can be written as
εh1 = ε
3∑
j=1
∑
n
V ( j)n (ϑ; J, Iϕ) sin
(nϑϕ + nG(ϑ; J, Iϕ)
), (F.3)
whereV ( j)
n (ϑ; J, Iϕ) ≡ H ( j)n
(R(ϑ; J, Iϕ), Z(ϑ; J, Iϕ)
).
S. Abdullaev, Magnetic Stochasticity in Magnetically Confined Fusion Plasmas, 377Springer Series on Atomic, Optical, and Plasma Physics 78,DOI: 10.1007/978-3-319-01890-4, © Springer International Publishing Switzerland 2014
378 Appendix F: Features of the Perturbation Hamiltonian of Guiding-Center Motion
For brevitywe notate the first function as V (ϑ), i.e., V ( j)(ϑ) ≡ V ( j)n (ϑ; J, Iϕ). Using
the relation
sin(nϑϕ + nG(ϑ; J, Iϕ)
) = sin(nϑϕ)A(ϑ) + cos(nϑϕ)B(ϑ), (F.4)
where A(ϑ) and B(ϑ) are the 2π–periodic functions of ϑ given by the Fourier series,
A(ϑ) = cos[nG(ϑ; J, Iϕ)
], B(ϑ) = sin
[G(ϑ; J, Iϕ)
], (F.5)
the perturbation Hamiltonian (F.3) can be presented as
εh1 = ε
3∑
j=1
∑
n
V ( j)n (ϑ)
[A(ϑ) sin(nϑϕ) + B(ϑ) cos(nϑϕ)
]. (F.6)
To expand the perturbationHamiltonian h1 in a Fourier series inϑ, one can separatelyfind the Fourier expansions of the functions V ( j)
n (ϑ), A(ϑ), and B(ϑ), and thenmultiply them. Below we study the features of these functions and their Fourierexpansions.
F.1 Fourier Asymptotics of V (ϑ)
Suppose that the function V (ϑ) is given by the following Fourier series
V (ϑ) =∑
m
|Vm | cos(mϑ + χm) = 1
2
∞∑
m=1
(Vmeimϑ + V ∗
me−imϑ)
,
Vm = |Vm |eiχm , (F.7)
with the Fourier coefficients Vm .The typical dependencies of the function V (ϑ) on the angle variable ϑ = 2πt/T
for the different types of guiding center orbits of α-particle are plotted in Fig.F.1 forthe magnetic perturbation created by the one horizontal segment of the saddle coilsin the ITER-like plasma. The corresponding orbits were shown in Fig. 5.8a, b. Theenergy of theα-particle is E = 1MeV, and the ratioλI = 0.8. The plasmaparametersare given in Sects. 2.3.1 and 3.4, and Table 2.2 and Fig. 3.6. The correspondingperturbation parameter ε is equal to 3.043 × 10−3 Ic, where the perturbation currentIc is in MA. As seen from Fig. F.1a the perturbation function V (ϑ) has a pulselike behavior along the orbit. Such a dependence is due to the similar behavior ofthe perturbation poloidal flux ψ
( j)n (R, Z) as was shown in Fig. 3.9. For the passing
orbits (curves 1 and 2) V (ϑ) has only one peak, while the trapped orbits (curve 3 and4) it has the two peaks (see Fig. F.1a, b).
Appendix F: Features of the Perturbation Hamiltonian of Guiding-Center Motion 379
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0 0.2 0.4 0.6 0.8 1
V(ϑ
)
ϑ/2π
12
3
4
x10-2
Fig. F.1 Dependencies of the perturbation function V (ϑ) (F.1) on the angle variable ϑ along theseveral guiding–center orbits of α-particle. Curve 1 and 2 correspond co-passing and barely co-passing orbits, respectively; curve 3 and 4 correspond to the barely trapped and the trapped orbits,respectively. The magnetic perturbation created by the one single horizontal segment of the setof saddle coils located at (R2, Z2). The energy Ek = 1 MeV, the ratio λI = 0.8. The plasmaparameters and the RMP coils positions are given in Fig. 3.6. The toroidal mode of the perturbationmagnetic field is n = 3. The perturbation current Ic = 1 kA
Fig. F.2 Fitting of V (ϑ)
with the Lorentzian pulsefunction (F.8) near the oneits peak. Curve 1 (solid line)corresponds to V (ϑ) and curve2 (dashed line) correspondsto VL (ϑ) with the appropriatefitting parameters a, σ, and ϑ0
0
0.1
0.2
0.3
0.4
0.5
0.2 0.4 0.6 0.8
V(ϑ
)
ϑ/2π
x10-3
12
Each of peaks in the dependence V (ϑ) versus ϑ similar to the one for ψ( j)n (R, Z)
can be approximated by the Lorentzian pulse form (3.52),
VL(ϑ) = a
(ϑ − ϑ0)2/σ2 + 1, (F.8)
with the appropriate amplitude a, the pulse width σ, and its location ϑ0. The fittingof the one of peaks by the Lorentzian pulse form is shown in Fig. F.2.
Therefore using the arguments similar to the ones in Sect. 3.4.4 we can obtain theasymptotical formulas for the poloidal spectra Vm in Eq. (F.7) similar to Eq. (3.54).
380 Appendix F: Features of the Perturbation Hamiltonian of Guiding-Center Motion
However, unlike from the latter the spectrum Vm for the trapped motion will containthe contributions from two Lorentzian peaks.
According to the localization principle2 the asymptotic estimation of the Fourierintegral
Vm = 1
2π
2π∫
0
V (ϑ)e−imϑdϑ, (F.9)
for large m is given by the sum of the contributions from each critical points of thefunction V (ϑ) corresponding its peaks. Near these points V (ϑ) is well approximatedby the Lorentzian function (F.8) whose Fourier transform is given by the exponentialform (3.53). Therefore, the asymptotical formula for Vm at large m will be given by
Vm ≈ 1
2T
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
A(o) exp
(−mC (o)
T− imϑ
(o)0
)− A(i) exp
(−mC (i)
T− imϑ
(i)0
),
for trapped orbits,
A(o) exp
(−mC (o)
T− imϑ
(o)0
), for passing orbits.
(F.10)The parameters A(o,i), C (o,i), and ϑ(o,i)
0 depend on the orbit characteristics. The sub-scripts (o) and (i) stand for the outer and inner branches of the orbit. Particularly,A(o,i) and C (o,i) are similar to the parameters A and C in the asymptotical formula(3.54) and take finite values at the separatrix of the banana orbit as well as at the mag-netic separatrix where the period of motion (transit time) T has singularities as wasshown in Figs. 5.12b and 5.13b. One should notice the period T in the asymptoticalformula (F.9) plays the similar role as the safety factor q(ψ) in Eq. (3.54).
In order to find the parameters A(o,i), C (o,i), and ϑ(o,i)0 it is not necessary to calcu-
late the numerical Fourier transform of V (ϑ) (F.9) as it has been used in Sect. 3.4.4.It is sufficient to find the amplitude a, the width σ, and the peak’s location ϑ0 byfitting the function V (ϑ) near its peaks with the Lorentzian function (F.8). Accordingto Eq. (3.53) the relations between (a,σ) and (A, C) are A = aσT , C = σT .
Figure F.3a, b shows the typical dependencies of the exponents C (o,i) and theamplitudes A(o,i) on the toroidal momentum pϕ = Iϕ. Curve 1 corresponds to thepassing orbits at pϕ > ps and to the outer branch of the trapped orbits at pϕ < ps ,while curve 2 corresponds to the inner branch of the trapped orbits.
The phases ϑ(o,i)0 are also similar to the ones in Eq. (3.54). They determine the
angular positions of the horizontal segments of the saddle coils. For the orbits nearthe separatrix their distanceΔϑmeasured from the outer equatorial plane ϑ = 0 goesto zero inversely proportional to the period of motion T . Therefore, the productΔϑThas a finite nonzero value at the separatrix. The typical dependence of the productΔϑT on the toroidal momentum of a particle pϕ = Iϕ is shown in Fig. F.4.
2 The localization principle in the asymptotic expansion of integral can be found in many booksdevoted to asymptotic methods in analysis, for example, by Fedoryuk (1989).
Appendix F: Features of the Perturbation Hamiltonian of Guiding-Center Motion 381
1
2
3
4
0.02 0.04 0.06 0.08
C(p
ϕ)/T
tr
pϕ
ps1
2
0.4
0.5
0.6
0.7
0.8
0.9
0.02 0.04 0.06 0.08
A(p ϕ
)/Ttr
pϕ
ps1
2
x10-2(a) (b)
Fig. F.3 Typical dependencies of the exponents C (o,i)(pϕ) (a) and the amplitudes A(o,i)(pϕ) (b)on the momentum pϕ = Iϕ in the asymptotical formula (F.9) (Ttr = 2πR0/
√2ma Ek is the
characteristic transit time). The plasma parameters and the RMP coils positions are given Fig. F.1.Curve 1 corresponds to the passing orbits at pϕ > ps and to the outer branch of the trapped orbitsat pϕ < ps , curve 2 corresponds to the inner branch of the trapped orbits
Fig. F.4 Typical dependen-cies of the phases ϑ(o,i)
0 on themomentum pϕ = Iϕ in theasymptotical formula (F.9).The plasma parameters andthe RMP coils positions aregiven Fig. F.1. Numberingof curves are the same as inFig. F.3
0.6
0.64
0.68
0.72
0.76
0.02 0.04 0.06 0.08
ΔϑT
/Ttr
pϕ
1- Δϑ=2π−ϑ02- Δϑ=ϑ0−π
ps
1
2
Using Eqs. (F.7) and (F.10) we can obtain the asymptotic formula for the functionV (ϑ) similar to the perturbation poloidal flux ψ
( j)n (3.67),
Vasym(ϑ) ={
V (o)(ϑ) − V (i)(ϑ), for trapped orbits,
V (o)(ϑ), for passing orbits,(F.11)
where
V (l)(ϑ) = A(l)
2T Λ(l)
(1 − e−2α(l)
), l = (o, i),
Λ(l) = 1 + e−2α(l) − 2e−α(l)cos
(ϑ − ϑ(l)
), α(l) = C (l)/T . (F.12)
382 Appendix F: Features of the Perturbation Hamiltonian of Guiding-Center Motion
Fig. F.5 Comparison ofV (ϑ) with the asymptoticpresentation (F.11) for thetwo trapped orbits. Solidcurves correspond to V (ϑ)
and dashed curves correspondto Vasym(ϑ) with the fittingparameters A, C , and ϑ0
-1.5
-1
-0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1
V(ϑ
)
ϑ/2π
x10-2
The comparison of the asymptotic formula (F.11) for the perturbation function V (ϑ)
with its numerically calculated values is shown in Fig. F.5. It shows that the asymp-totic formula well describes the positions, the amplitudes, and the widths of peaksof V (ϑ).
F.2 Fourier Expansions of the Functions G(ϑ), A(ϑ), B(ϑ)
We present the function G(ϑ) in a Fourier series,
G(ϑ; J, Iϕ) =M∑
s=1
gs sin(sϑ), (F.13)
where gs ≡ gs(J, Iϕ) is the Fourier coefficients depending on the action variables(J, Iϕ). Typical dependencies of the first three coefficients gs , s = 1, 2, 3 on thetoroidal momentum pϕ = Iϕ are plotted in Fig. F.6. The coefficient g1 is of a twoorder larger than the ones gs , (s = 2, 3, . . .) for the both trapped, pϕ < ps , andpassing, pϕ > ps , orbits. They reach their maximal values only at pϕ → ps , i.e.,for the barely trapped and passing orbits.
The functions A(ϑ) and B(ϑ) in (F.5) are the 2π-periodic functions of ϑ given bythe Fourier series,
A(ϑ) =∞∑
m=0
am cos(mϑ), B(ϑ) =∞∑
m=1
bm sin(mϑ). (F.14)
Figure F.7 show the dependencies of the functions A(ϑ), B(ϑ), and G(ϑ) on theangle variable ϑ for the four different types of orbits: (a) passing; (b) barely passing;(c) barely trapped; (d) trapped.
Appendix F: Features of the Perturbation Hamiltonian of Guiding-Center Motion 383
Fig. F.6 Fourier coefficientsgs(p) as functions of thetoroidalmomentum pϕ:a E =1.0 MeV, b E = 3.5 MeV:curve 1 s = 1, curve 2 s = 2,curve 3 s = 3
10-3
10-2
10-1
100
101
0.04 0.06 0.08
g s(p
ϕ)
pϕ
ps
1
1
22
3 3
(a) (b)
To find the Fourier coefficients am , bm , in (F.5) one can use the following formulae(Abramowitz and Stegun (1965), page 361)
cos(z cos θ) = J0(z) + 2∞∑
k=1
J2k(z) cos(2kθ),
sin(z cos θ) = 2∞∑
k=0
J2k+1(z) sin[(2k + 1)θ]. (F.15)
where Jk(z) are the Bessel functions. Then presenting the coefficients A(ϑ) and B(ϑ)
in the complex form,
A(ϑ) + i B(ϑ) = exp
(in
M∑
s=1
gs sin(sϑ)
)=
M∏
s=1
exp (ings sin(sϑ)) , (F.16)
and expanding the product in (F.16) in a trigonometric series of sin(kϑ), cos(kϑ),one finds the coefficients am , bm . Particularly, the first three nonzero coefficientsm = 0, 1, 2 are given by
a0 =M∏
s=1
J0(ngs), b1 = 2J1(ng1)
M∏
s=2
J0(ngs),
a2 = 2J2(ng1)
M∏
s=2
J0(ngs), b2 = 2J0(ng1)J1(ng2)
M∏
s=3
J0(ngs),
b3 = 2J3(ng1)
M∏
s=2
J0(ngs) + J0(ng1)J0(ng2)J2(ng3)
M∏
s=4
J0(ngs).
384 Appendix F: Features of the Perturbation Hamiltonian of Guiding-Center Motion
-3
-2
-1
0
1
2
3
0 0.2 0.4 0.6 0.8 1
A(ϑ
), B
(ϑ),
G(ϑ
)
ϑ/2π
1
2
3
(b)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
A(ϑ
), B
(ϑ),
G(ϑ
)
ϑ/2π
1
23
(a)
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1
A(ϑ
), B
(ϑ),
G(ϑ
)
ϑ/2π
1
2
3
(c)
-3
-2
-1
0
1
2
3
0 0.2 0.4 0.6 0.8 1
A(ϑ
), B
(ϑ),
G(ϑ
)
ϑ/2π
1
2
3
(d)
Fig. F.7 Dependence of the coefficeints A(ϑ), B(ϑ), G(ϑ) on the angle variable ϑ for α-particleof energy E = 1.0 MeV: a passing orbit; b barely passing; c barely trapped orbit; d trapped orbit.Curve 1 corresponds to A(ϑ), curve 2 corresponds to B(ϑ), curve 3 corresponds to G(ϑ). Thetoroidal mode n = 3
For deeply passing orbits the largest coefficient g1 � 1. Therefore a0 � 1, but thecoefficients am , bm , (m > 1) are much smaller, and can be neglected. Then we haveA(ϑ) ≈ 1, B(ϑ) ≈ 0 (see Fig. F.7a).
For deeply trapped orbits the largest coefficient g1 ∼ 1 and one can neglect othersgs , (s > 1). Then only the coefficient g1 can be retained giving G(ϑ) ≈ g1 sin ϑ (seeFig. F.7d). Therefore we have A(ϑ) ≈ cos(ng1 sin ϑ), B(ϑ) ≈ sin(ng1 sin ϑ).
For the barely trapped orbits the large number of coefficients gs give the contribu-tions to G(ϑ). The latter becomes a steep function of ϑ at the low field side ϑ = 0,πas shown in Fig. F.7c. The functions A(ϑ) and B(ϑ) become also fast-oscillatingthere.
Appendix GDerivation of the Many-Dimensional Full-TurnTransfer Mapping
Below we present the detailed derivation of the full-turn transfer mapping (6.53).First we rewrite the mapping (6.9)–(6.11) in the following form
Ik+1 = Ik + ε
(∂Sk+1
∂ϑk+1− ∂Sk
∂ϑk
), (G.1)
ϑk+1 = ϑk + w(Jk, ε)(τk+1 − τk) − ε
(∂Sk+1
∂ Jk− ∂Sk
∂ Jk
), (G.2)
where Sk ≡ S(ϑk, Jk, τk, τ0, ε), Sk+1 ≡ S(ϑk+1, Jk, τk+1, τ0, ε) are the values ofthe generating function S(ϑ, J, t, τ0, ε) at the time instants τ = τk and τ = τk+1,respectively. The intermediate action variables J are given by
Jk = Ik − ε∂Sk
∂ϑk= Ik+1 − ε
∂Sk+1
∂ϑk+1. (G.3)
In general the mapping (G.1)–(G.3) is exact and valid for arbitrary magnitude of theperturbation parameter ε.
We intend to derive the simplified form of the Poincaré mapping to the sectionΣ of the phase space where the angle variable ϑ ≡ ϑ1 takes constant value ±π,mod 2π. Furthermore we introduce the following notations for the rest of anglevariables Θ = (ϑ2, . . . ,ϑN ,ϑN+1 = t), and the frequencies ω(J ) = ω1(J ), Ω =(ω2, . . . ,ωN ,ωN+1 = 1). We consider the cases of small perturbation parameterε � 1.
In the first order of the perturbation parameter ε the generating function S(ϑ, J, t,τ0, ε) is determined by the integral (6.21). Without loosing the generality one canput H1(J ) = 0. Then according to (6.26) w(J, ε) = ω(J ) and Eq. (6.21) is reducedto
S(ϑ,Θ, J, τ , τ0) = −τ∫
τ0
H1(ϑ0(τ′),Θ0(τ
′)J )dτ ′, (G.4)
S. Abdullaev, Magnetic Stochasticity in Magnetically Confined Fusion Plasmas, 385Springer Series on Atomic, Optical, and Plasma Physics 78,DOI: 10.1007/978-3-319-01890-4, © Springer International Publishing Switzerland 2014
386 Appendix G: Derivation of the Many-Dimensional Full-Turn Transfer Mapping
where integration is taken along the unperturbed trajectory ϑ0(τ′) = ϑ0(τ ) +
ω(J )(τ ′ − τ ), Θ0(τ′) = Θ0(τ ) + Ω(J )(τ ′ − τ ), [ϑ ≡ ϑ0(τ ),Θ ≡ Θ0(τ )].
Since ϑk+1 = ϑk + 2π for the angle variable ϑ from the mapping (G.2) it followsthat
τk+1 − τk = 2π
ω(Jk)+ O(ε), (G.5)
where the terms of order of O(ε) are much smaller than the period of motion alongthe angle ϑ and they can be neglected. The unperturbed orbits ϑ0(τ
′),Θ0(τ′) in (G.4)
can be presented as
ϑ0(τ′) = ϑk+1 + ω(Jk)(τ
′ − τk+1) = ϑk + ω(Jk)(τ′ − τk),
Θ0(τ′) = Θk+1 + Ω(Jk)(τ
′ − τk+1) = Θk + Ω(Jk)(τ′ − τk).
In (G.4) we introduce the new integration variable τ = τ ′ − τ0, where τ0 is the timeinstant when the angle variable ϑ0(τ ) crosses the section ϑ = 0, i.e.,
τ0 = τk+1 + τk
2= τk + π
ω(Jk)= τk+1 − π
ω(Jk). (G.6)
Then ϑ0(τ′) = ϑ0(τ + τ0) = ϑ0(τ ) = ω(J )τ , ϑ0(τk) = −π, and ϑ0(τk+1) = π.
Thus the generating functions Sk in (G.1)–(G.3) are reduced to
Sk ≡ S(−)(Θk, Jk) =0∫
−π/ω(Jk )
H1(ϑ0(τ ),Θ0(τ + τ0), Jk)dτ ,
Sk+1 ≡ S(+)(Θk+1, Jk) = −π/ω(Jk )∫
0
H1(ϑ0(τ ),Θ0(τ + τ0), Jk)dτ , (G.7)
where Θk ≡ Θ(τk). The argument Θ0(τ + τ0) of the perturbation Hamiltonian H1under the integrals (G.7) is a function of Θk or Θk+1, i.e.,
Θ0(τ + τ0) = Θ0(τ ) + Θ0(τ0) − Θ0, (G.8)
where Θ0 is the constant initial phase, and Θ0(τ0) is given by
Θ0(τ0) = Θk + πΩ(Jk)
ω(Jk)= Θk+1 − πΩ(Jk)
ω(Jk). (G.9)
Equation (G.9) gives the relation between Θk and Θk+1 in the zero-order of ε. Usingthis relation the derivative ∂Sk+1/∂Θk+1 can be presented as
Appendix G: Derivation of the Many-Dimensional Full-Turn Transfer Mapping 387
∂Sk+1
∂Θk+1= ∂Sk+1
∂Θk
∂Θk
∂Θk+1= ∂Sk+1
∂Θk+ O(ε).
Then the mapping (G.1)–(G.3) for the action variables I = (I2, . . . , IN , In+1 = H)
and the angle variables Θ can be reduced to
Ik+1 = Ik − ε∂Pk
∂Θk, (G.10)
Θk+1 = Θk + 2πΩ(Jk)
ω(Jk)+ ε
∂Pk
∂ Jk, (G.11)
where the function Pk = S(−) − S(+), according to (G.7), is given by the integral
Pk ≡ P
(Θk + πΩ(Jk)
ω(Jk), Jk
)= S(−)(Θk, Jk) − S(+)(Θk+1, Jk)
=π/ω(J )∫
−π/ω(J )
H1
(ϑ(τ ),Θ0(τ ) + Θk + πΩ(Jk)
ω(Jk)+ Θ0, Jk
)dτ . (G.12)
We will call the function P(Θ, J ) the Poincaré integral. It depends on the interme-diate action variables J which are related to the action variables Ik and Ik+1:
Jk = Ik − ε∂Sk
∂Θk= Ik+1 − ε
∂Sk+1
∂Θk+1. (G.13)
The system of Eqs. (G.10)–(G.13) with the generating functions (G.7) and (G.12)describe the full–turn transfer mapping (Θk, Ik) → (Θk+1, Ik+1) to the section Σ .This form of the mapping is an invariant with respect to time–reverse τ → −τ , i.e.,k ↔ k + 1. This property of the mapping (G.10)–(G.13) is due to the intermediateaction variables Jk : the generating functions Sk , Sk+1, and Pk are the functions ofthese intermediate action variables.
The mapping (G.10)–(G.13) can be simplified by replacing the intermediate vari-ables J by the action variables (Ik, Ik+1) using the relations (G.13). However, sucha replacement should preserve the two properties of the mapping: (i) the invariancewith respect to time–reverse, and (ii) the area–preserving (or the symplectic) (6.8).These requirements can be satisfied under the following transformations. The ratioof frequencies α(J ) = πΩ(J )/ω(J ) in (G.11) can be transformed to the followingsymmetric form
α(Jk) ≈ 1
2
[α(Ik) + α(Ik+1)
]+ 1
2
∂α(J )
∂ J· (2Jk − Ik − Ik+1)
= 1
2
[α(Ik) + α(Ik+1)
]− ε
2
∂α(J )
∂ J·(
∂Sk
∂Θk+ ∂Sk+1
∂Θk+1
). (G.14)
388 Appendix G: Derivation of the Many-Dimensional Full-Turn Transfer Mapping
Using (G.9) and (G.13) the Poincaré integral P(Θ0, Jk) can be transformed as
P(Θ0, Jk) = P
(Θk + α(Jk), Ik+1 − ε
∂Sk+1
∂Θk+1
)
≈ P (Θk + α(Ik), Ik+1) − ε∂P
∂Θk· ∂α(J )
∂ J· ∂Sk
∂Θk− ε
∂P
∂ Jk· ∂Sk+1
∂Θk+1.
(G.15)
Neglecting the terms of order of ε in (G.14) and (G.15) the mapping (G.10) and(G.11) is reduced to
Ik+1 = Ik − ε∂Pk
∂Θk, (G.16)
Θk+1 = Θk + πΩ(Ik)
ω(Ik)+ πΩ(Ik+1)
ω(Ik+1)+ ε
∂Pk
∂ Ik+1, (G.17)
where
Pk ≡ P
(Θk + πΩ(Ik)
ω(Ik), Ik+1
). (G.18)
It is not difficult to show that the mapping in the form (G.16) and (G.17) is invariantwith respect to the transformation k ↔ k + 1 and the area–preserving, i.e.,
∣∣∣∣∂ (Θk+1, Ik+1)
∂ (Θk, Ik)
∣∣∣∣ = 1.
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Index
AAction, 3, 102Action-angle variables, 6, 12
guiding center motion, 102Adiabatic invariant, 83, 171Airy function, 52Arnold diffusion, 173Aspect ratio, 4
BBanana tip, 96Basin, 250Biot-Savart law, 59Boozer coordinates, 9Bounce frequencies, 103Bounce oscillations, 103Bounce time, 103
CCanonical transformation
guiding center, 80Chaotic scattering system, 250Chirikov parameter, 214Chirikov’s criterion, 214Clebsch form, 9Collisional diffusion
Kadomtsev-Pogutse regime, 284Laval regime, 284Rechester-Rosenbluth regime, 283
Collisional diffusion coefficientparallel, 283perpendicular, 283
Collisionless regime , 283Connection length, 31, 36, 45, 288
Convection coefficient, 267Correlation function, 264, 288
phase, 268Correlation length, 274, 276, 288Correlation time, 266, 268, 288Cylindrical coordinate system, 3
DDensity
rational drift surfaces, 326rational tori, 305
Diffusion coefficient, 267empiric formula, 289field lines, 272fractal, 308particles, 271quasilinear, 268, 269, 273runaway electrons, 322
Divertor tokamak, 28, 257double–null, 29model, 11single–null, 11, 29, 58snowflake, 42, 221
Drift surface, 93Dynamical chaos, 197
EElliptic fixed point, 11, 182Ergodic divertor, viii, 227, 228, 333
dynamic (DED), 230, 260, 333Ergodic limiter, 228, 260Ergodic motion, 264Ergodic zone, 228
S. Abdullaev, Magnetic Stochasticity in Magnetically Confined Fusion Plasmas, 409Springer Series on Atomic, Optical, and Plasma Physics 78,DOI: 10.1007/978-3-319-01890-4, © Springer International Publishing Switzerland 2014
410 Index
FFast ions, 97, 172Fixed points, 108Fractal, 250Fundamental problem of dynamics, 125
GGauge transformation, 2Golden KAM tori, 170Grad-Shafranov equation, 20
analytical solutions, 20, 27Gyrofrequency, 79
normalized, 86radial oscillations, 79, 86reference, 79
Gyrophase, 80, 87, 171, 317
HHamilton-Jacobi equation, 125, 167Hamiltonian equation, 2, 4
autonomous, 3guiding center, 82, 84, 87, 89in flux coordinates, 6, 49non–autonomous, 3relativistic, 77
Hamiltonian flow, 123Hamiltonian function, 3
guiding center, 80, 82, 86, 89in flux coordinates, 6, 49perturbation, 169, 316relativistic, 77
Hamiltonian monodromy, 116Helical coils, 1, 230, 333Helical currents
density, 1, 3, 7, 14, 333, 335, 339, 346Horseshoe map, 198Hyperbolic fixed point, 11, 33, 182
guiding center, 108Hyperbolicity, 198
IIntegrability, 174integrability, 124Integral invariant, 102Internal (I-) coils, 234Internal transport barriers, 223Invariant tori, 167
KKAM theory, 6, 166
Kinetic equation, 267Kolmogorov entropy, 266Kolmogorov length, 280
finite-length, 280Kolmogorov time, 266, 279Kolmogorov’s technique, 168Kolmogorov’s theorem, 166Kolmogorov-Sinai entropy, 266Kubo number, 297, 315
LLaminar plot, 252Laminar zone, 252, 276Liouville’s equation, 267Lobe, 188, 198Lorentzian pulse, 47, 64, 379Lyapunov exponent, 180
finite-time, 280
MMagnetic chaos, viiiMagnetic coordinates, 9Magnetic field, 1
in flux coordinates, 9poloidal, 5, 23reversed shear, 223toroidal, 5, 21, 23
Magnetic field lines, 1equation, 1open, 250
Magnetic field perturbations, 47DED, 7, 339helical, 50model, 55, 232, 248poloidal spectra, 50, 236poloidal spectra asymptotics, 54, 57, 64,65, 229, 231
poloidal spectra model, 232, 245resonant, 227, 229saddle coils, 60, 63
Magnetic field, plasmaequilibrium, 19standard model, 21three–wire model, 42two–wire model, 38wire model, 36
Magnetic footprints, 154, 195, 243, 255Magnetic island, 204Magnetic moment, 83Magnetic perturbations
poloidal spectra, 50toroidal spectra, 50
Index 411
Magnetic separatrix, 16, 19, 28Magnetic shear
measure, 205reversed, 183, 223, 309
Magnetic stochasticity, viiiMagnetic surface, 1, 6
shearless, 205Magnetic turbulence, 313, 314Major radius, 3Manifold
splitting, 187stable, 186, 190unstable, 186, 190
Mappingbackward, 137Chirikov–Taylor, 157, 162field lines, 129forward, 137full–turn transfer, 133, 135–137, 146, 183full-turn transfer, 299, 320interpolated cell, 163models, 161separatrix, 133, 146, 162, 183standard, 157, 162symplectic form, 125Wobig-Mendonc, 158
Mean free path, 283Melnikov integral, 138, 139, 187Mixing, 264Monodromy matrix, 181Multipliers, 181
NNon-twist system, 225Nonlinear resonance, 202
2D–system, 210standard, 204width, 204, 211
Nonlinearity degree, 205Nontwist mappings, 225Nulls of a magnetic field, 11Numerical codes
FLOC, 122Gourdon, 50, 122MASTOC, 50, 122TRIP3D, 50, 122TRIPND, 50, 122
OO-point, 11
PParticle orbits
pinch, 98potato, 98trapped, 96, 101
Pendulum, 203Periodic orbits, 174
stable, 181, 182unstable, 181, 182, 185, 186
Perturbation parameter ε, 10, 49, 61, 342Perturbation series, 126
generating function, 126Hamiltonian function, 126
Phase correlation function, 274phase space, 123Plasma
current, 24density, 283, 291temperature, 283, 291
Poincaré integral, 55, 135, 138, 143, 187, 387Poincaré mapping, 122Poincaré section, 122Poincaré–Birkhoff theorem, 174, 176, 203Poincaré–Melnikov integrals, 139Poisson summation rule, 298Poloidal angle, 9Poloidal flux, 7, 9
perturbation field, 48, 49, 60, 62, 63, 71Primary resonance approximation, 147Private flux zone, 251Probability distribution function (PDF), 266
QQuasilinear theory, 268Quasitoroidal coordinates, 3
RRandom walk model, 285Reference energy, 84Reference momentum, 84Reference time, 84Reference velocity, 84Relativistic factor, 78, 87, 88Rescaling invariance, 225Rotational transform, 10Runaway electrons, 96, 331
diffusion, 322transport, 313
SSaddle coils, 58
412 Index
Safety factor q, 10q95, 16asymptotics, 16, 35, 44cylindrical plasma, 25effective, 105, 119toroidal plasma, 25two-wire model, 39
Shafranov shift, 22Shielding factor, 323Spherical tokamak, 29Stagnation orbit, 96Stochastic layer, 212, 214
rescaling invariance, 214width, 220, 221
Superconvergence, 169Symplectic mapping, 125
TTangles
heteroclinic, 196homoclinic, 196
Test particle model, 263Tokamak
DIII-D, 50, 234EAST, 330ITER, 29, 59, 331JET, 218MAST, 258NSTX, 29, 42, 59TCV, 42
TEXT, 260TEXTOR, 50, 95, 230, 260Tore Supra, 50, 228, 260
Tokamak divertor maps, 161Tokamap, 158
revtokamap, 159symmetric, 159
Toroidal angle, 3, 9Toroidal flux, 7, 9Toroidal procession, 96, 106Torus
non-resonant, 166resonant, 166
Transit frequencies, 103, 105Transit oscillations, 103, 172Transit time, 103, 112, 117, 296Transport barriers, 306
VVariational principle , 2Vector potential, 2
WWada property, 253Winding number, 10
XX-point, 11, 19, 29, 32, 149