sion, we establish the main property of the transformation of magnetic ..... H. * . 25 with a poloidal flux H as a Hamiltonian function. For the magnetic perturbation 16 the latter has a form. H d. q r. R , * A x * ...... Here ¯ means averaging over a set of N initial conditions ...... tional Atomic Energy Agency, Vienna, 1989, Vol. 1, p. 9.
PHYSICS OF PLASMAS
VOLUME 6, NUMBER 1
JANUARY 1999
Asymptotical and mapping methods in study of ergodic divertor magnetic field in a toroidal system S. S. Abdullaev and K. H. Finken Institut fu¨r Plasmaphysik, Forschungszentrum Ju¨lich GmbH, EURATOM Association, Trilateral Euregio Cluster, D-52425 Ju¨lich, Germany
K. H. Spatschek Institut fu¨r Theoretische Physik I, Heinrich-Heine-Universita¨t Du¨sseldorf, Universita¨tsstrasse 1, D-40225 Du¨sseldorf, Germany
~Received 14 May 1998; accepted 29 September 1998! Asymptotical and mapping methods to study the structure of magnetic field perturbations and magnetic field line dynamics in a tokamak ergodic divertor in toroidal geometry are developed. The investigation is applied to the Dynamic Ergodic Divertor under construction for the Torus Experiment for the Technology Oriented Research ~TEXTOR-94! Tokamak at Ju¨lich @Fusion Eng. Design 37, 337 ~1997!#. An ideal coil configuration designed to create resonant magnetic perturbations at the plasma edge is considered. In cylindrical geometry, the analytical expressions for the vacuum magnetic field perturbations of such a coil system are derived, and its properties are studied. Corrections to the magnetic field due to the toroidicity are presented. The asymptotical analysis of transformation of magnetic perturbation into the Hamiltonian perturbation in toroidal geometry is carried out, and the asymptotic formulas for the spectrum of the Hamiltonian perturbations are found. A new method of integration of Hamiltonian equations is developed. It is based on a canonical transformation of variables that replaces the dynamics of a continuous Hamiltonian system by a symplectic mapping. The form of the mapping is established in the first order of perturbation theory. It is shown that the mapping well reproduces Poincare´ sections of field lines, as well as their statistical properties in an ergodic zone obtained by the numerical integration of field line equations. The mapping is applied to study, in particular, the formation of a stochastic layer and the statistical properties of field lines at the plasma edge. © 1999 American Institute of Physics. @S1070-664X~99!02501-X#
I. INTRODUCTION
ties at the plasma edge in the ergodic divertor operation have been extensively studied in several recent works, Refs. 10– 18. Particularly, an approximate formula for the spectrum of the magnetic perturbations has been proposed in Ref. 11. These works were mainly based on numerical codes. The statistical properties of field lines in the ergodic zone were studied using qualitative criteria of overlapping of resonances, e.g., the Chirikov criteria ~Ref. 19!. So far, the study of field line dynamics and the formation of an ergodic zone at the plasma edge in ergodic divertors has mainly been based on field line tracing codes ~Refs. 13, 14, 16–18!, which require huge computational times. Moreover, as was noted in Ref. 16, the field line tracing codes lose the accuracy in the stochastic zone in a few poloidal turns. Recently, in Refs. 20, 21, the computationally more efficient mapping method has been proposed for this purpose. The method is based on the description of field lines by a fluxpreserving perturbed twist mapping ~see, e.g., Refs. 22, 23!. Application of the area-preserving maps for description of field lines in magnetic confinement devices has been discussed for a long time. Particularly, the Chirikov–Taylor standard map ~Ref. 19! and twist maps have been extensively studied to calculate the magnetic field line diffusivity in Refs. 24–29. However, as was recently noted in Ref. 30, these maps do not take into account the main features of
Control of the plasma edge is one of the important problems in nuclear fusion research. Particularly, the concept of an ergodic divertor ~Refs. 1–3! has been developed for tokamaks to exhaust particles from the plasma edge and to reduce the penetration of wall-released impurities into the plasma core. The idea of the ergodic divertor is based on the creation of a perturbed magnetic field at the plasma edge by a special coil arrangement. This creates a stochastic zone of magnetic field lines that guides the ionized particles through the laminar zone to special divertor plates. A novel element to the concept of the ergodic divertor has been introduced by the dynamic nature of the Dynamic Ergodic Divertor ~DED!, under construction for the Ju¨lich Tokamak: Torus Experiment for Technology Oriented Research ~TEXTOR-94! ~Ref. 4!. In addition to the conventional concept of the ergodic divertor as implemented in Tore Supra ~see, e.g., Refs. 5–7!, the DED also permits the operation with a rotating magnetic perturbation field. This allows us, in particular, to broaden the heating footprints to avoid hot spots on the divertor plates. The energy and particle transports in the stochastic zone depends significantly on the topology of the perturbed magnetic field and the structure of the field lines ~Refs. 8–12!. The structure of the magnetic field and its statistical proper1070-664X/99/6(1)/153/22/$15.00
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© 1999 American Institute of Physics
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Phys. Plasmas, Vol. 6, No. 1, January 1999
field line dynamics and magnetic perturbations in tokamaks, particularly in ergodic divertors, where there is a strong radial dependence of the perturbation field. A global mapping model of magnetic field lines in a tokamak that takes into account these features has been recently proposed in Refs. 30 and 31. One should note that specific algebraic maps have been proposed in Refs. 32–33 for a description of field lines in a poloidal divertor tokamak. Generalized separatrix maps have been introduced in Refs. 34 and 35 to study perturbed field lines near the magnetic separatrix. They were applied to control magnetic footprints on the divertor plates in Ref. 36. The relation between continuous Hamiltonian systems and the corresponding maps was mentioned in Ref. 37. It was emphasized that the perturbed twist mapping in the form presented in Refs. 22, 23 does not really describe the corresponding continuous Hamiltonian system. One should note that the perturbed twist mapping has not been derived directly from the Hamiltonian equations, and therefore it is not clear how the variables in the mapping are related to the ones in the Hamiltonian system. In the present work we develop asymptotical and new mapping methods to study the topology of perturbed magnetic fields, the spectrum of magnetic perturbations, and magnetic field lines dynamics in an ergodic divertor tokamak. The study is carried out for the geometry of the DED of the TEXTOR-94. There are three important steps in the study. First, we shall derive analytical expressions for the magnetic field perturbations generated by the ideal coil configuration and study their asymptotics at the different radial distances. A secondary important step in the study of the field line dynamics is the representation of field line equations in Hamiltonian form, particularly the transformation of magnetic field perturbations into Hamiltonian terms in toroidal geometry. The behavior of field lines at the plasma edge is mainly determined by the spectrum of Hamiltonian perturbations. We shall study this transformation using asymptotic methods. The final step is a study of field lines dynamics by solving the corresponding Hamiltonian equations. The direct integration of the Hamiltonian system usually takes a long computational time. We shall develop a new method of integration of Hamiltonian equations that reduces the dynamics of continuous Hamiltonian system to a symplectic mapping. It is a new version of the perturbation theory for integrable systems. Unlike the averaging procedure in classical mechanics ~see, e.g., Refs. 38, 39! consisting of the change of variables in the perturbed motion equations that eliminates fast phases in equations, the new procedure consists of canonical transformation of variables that adds fast phases in equations and transforms the Hamiltonian to the new one with periodic d-kick perturbations in time. The solution of the new system between ‘‘kicks’’ is determined by the unperturbed Hamiltonian. The relation between solutions after crossing the ‘‘kick’’ is given by the canonical transformation. Using this relation, one can easily establish the mapping describing the system evolution in one period of perturba-
Abdullaev, Finken, and Spatschek
tion. The developed mapping method will be compared with the results of direct numerical integration of the Hamiltonian system, and it will be applied to study the stochastic zone of field lines at the plasma edge. One should note that unlike Ref. 20, where the perturbed twist mapping was proposed to model field line dynamics in an ergodic divertor tokamak, the mapping in the present work appears as a result of the rigorous and regular procedure of the symplectic transformation of variables in Hamiltonian field line equations. This procedure exactly replaces the dynamics of a Hamiltonian system by the symplectic mapping of a given form. Only generating functions in the mapping should be found as solutions of certain equations, which may be easily solved using perturbation theory. The contents of the work is the following. The analytical description of the ideal divertor coil configuration and the magnetic field generated by this system are presented in Sec. II. We first study the perturbed magnetic field in cylindrical geometry and find their asymptotic forms at the plasma edge. Then we take into account the corrections to the magnetic field due to toroidicity of the system. Section III is devoted to the Hamiltonian description of field line equations in an ergodic divertor. Using the methods of asymptotic expansion, we establish the main property of the transformation of magnetic perturbations in geometrical space into Hamiltonian perturbations in toroidal geometry. Based on the specific features of this transformation, we propose an asymptotic formula for the spectrum of magnetic perturbations in Hamiltonian form. A new integration method of Hamiltonian field line equations is presented in Sec. IV. Particularly, the derivation of the so-called symmetric map for field lines, and the comparison of the results of the mapping and the direct integration of Hamiltonian field line equations are given. The formation of the ergodic zone of field lines at the plasma edge, and some statistical properties of them, are studied in Sec. V. The results are summarized in Sec. VI. In order to make the whole paper easier to read, some long analytical computations are summarized in Appendices A, B, C.
II. COIL SYSTEM AND MAGNETIC FIELD PERTURBATIONS
The coil system for the TEXTOR-DED consists of a set of 16 helical coils, which will be installed on the inboard side of the vacuum vessel of TEXTOR-94 ~Ref. 40!. Here we consider the ideal coil configuration designed to create resonant magnetic perturbations at the plasma edge. A. System of helical coils
Consider cylindrical coordinates x5(r, u ,z), and related to them coordinates (r, u , w ) on the torus, with the major radius R, and the minor radius r. We have z5R w ; u is the poloidal angle, and w is the toroidal angle. Suppose that 16 identical helical coils on the inboard circumference of radius r c are starting at toroidal angles w s j 5( j21) p /8 and poloidal angles u s j 5 p 2 u c , and ending after one toroidal turn at
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Phys. Plasmas, Vol. 6, No. 1, January 1999
Abdullaev, Finken, and Spatschek
u 1~ w ! 5 u 1~ 0 ! 1
155
uc w. p
Because the helical current system is periodic in toroidal direction w, the product m 0 u c / p 5n 0 should be an integer number. Without loss of the generality, one can put u 1 (0) 50. Therefore, the current density J( u , f ) may be written in the symmetric form `
J ~ u , w ! 5J 0 g ~ u !
(
sin@~ 21 ! k ~ 2k11 !~ m 0 u 2n 0 w ! 1 v t # .
k50
~6! FIG. 1. Sketch of the positions of the divertor coils in the ~w,u! plane.
The Fourier decomposition of the current density ~6! is `
J~ u,w !5
u e j 5 p 1 u c ( j51,2,...,16). The schematic positions of coils in the ~w,u! plane are shown in Fig. 1. The poloidal extension of the coil system is D u 52 u c . Let u j 5 u 1~ w ! 1
j21 2 p 4 m0
~1!
I j 5I d sin@ p ~ j21 ! /21 v t # ,
j51,2,...,16,
~2!
where I d is the current amplitude and v 52 p f is the current frequency. Introduce a linear density J( u , w ) of currents as 16
(
j51
I j r 21 c d~ u2u j !.
~3!
Using ~1! and ~2!, Eq. ~3! may be written as `
J ~ u , w ! 5I d r 21 c g~ u !
S
(
j52`
3 d u 2 u 12
sin~ p j/21 v t !
D
j 2p , 4 m0
H
1, 0,
for p 2 u c , u , p 1 u c , otherwise.
Using the Poisson summation rule, `
(
j52`
`
d ~ j2n ! 5112 ( cos~ 2 p pn ! , p51
Eq. ~4! may be reduced to `
J ~ u , w ! 5J 0 g ~ u !
(
k50
sin@~ 21 ! k ~ 2k11 !
3m 0 ~ u 2 u 1 ! 1 v t # ,
J ~mk ! sin~ m u 2 ~ 2k11 ! n 0 w
1 ~ 21 ! k v t ! ,
J ~mk ! 5 ~ 21 ! k J 0 g ~mk ! ,
~7!
g (k) m
is a Fourier transformation of the product where g( u )sin@(2k11)m0u#: g ~mk ! 5 ~ 21 ! m1m 0
sin~ m2m ~0k ! ! u c ~ m2m ~0k ! ! p
,
m ~0k ! 5 ~ 2k11 ! m 0 .
~8!
As one can see from ~7! for the ideal coil configuration, the linear current density J( u , w ) has only toroidal Fourier components n proportional to n 0 . It does not have the n 50 components that may generate a net poloidal field. The latter would affect the plasma equilibrium. However, in the technical implementation of the DED the coils are bundled in quartets ~Ref. 40!. Such a coil configuration creates nonoscillating components of the current density in w. They generate n50 magnetic perturbations that may disturb the plasma equilibrium. These components should be compensated by introducing additional, so-called compensation coils positioned near these areas ~Refs. 16, 40!.
~4!
where g( u ) is a stepwise function with nonzero values in the area covered by coils: g~ u !5
( (
m52` k50
~k!
be the poloidal positions of coils at a given poloidal section w 5const, where m 0 is a certain number; u 1 ( w ) is the position of the first coil at the section w 5const. The distribution of current on coils is the following:
J~ u,w !5
`
J 0 5I d
2m 0 . prc
~5!
The poloidal position of the first helical coil u 1 ( w ) is a linear function of w:
B. Magnetic field perturbations
In this section we derive the formulas for the magnetic field created by the helical coil system in a cylindrical approximation. The toroidal corrections will be taken into account in the next section. The magnetic field generated by the current density J( u ) ~7! may be found by the Fourier method. It is presented in Appendix A. Here we summarize the main formulas for the magnetic field and their asymptotics for r,r c . The magnetic field B(r, u , w ) may be expressed via the scalar potential F(r, u , w ): B5“F. The scalar potential F(r, u , w ) of the magnetic field created by the system of helical currents ~7! has the following general form inside the cylinder r,r c : `
F ~ r, u , w ! 5
`
( (
k50 m52`
F m, ~ 2k11 ! n 0 ~ r !
3cos@ m u 2 ~ 2k11 ! n 0 w 1 ~ 21 ! k v t # ,
~9!
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Phys. Plasmas, Vol. 6, No. 1, January 1999
Abdullaev, Finken, and Spatschek
with the Fourier components F m,n ~ r ! 5 ~ 21 ! k11
2m 0 m 0 I d nr c pumu Rc
3cos a 0 g ~mk ! K u8m u
S D S D
nr c nr I umu . Rc Rc
Here R c is the major radius of torus at the high field side, a 0 5 u 0 r c / p R c is the angle between the current direction and the toroidal axis w, and I m (z) and K m (z) are the modified Bessel functions. At the plasma edge r,r c , all terms in ~9! with the higher toroidal harmonics n5(2k11)n 0 , and (k.1) are negligibly small because of the strong radial decay factor (r/r c ) (2k11)m 0 of the corresponding poloidal harmonics. Therefore, at the plasma edge the perturbation magnetic field can be approximated by the main terms for k50: `
F ~ r, u , w ! 5
(
m52`
F m ~ r ! cos~ m u 2n 0 w 1 v t ! ,
~10!
with the spectrum F m ~ r ! [F mn 0 ~ r ! 52 m 0 I d cos a 0 3g ~m0 ! K u8m u
2m 0 n 0 r c pumu Rc
S D S D
n 0r c n 0r I umu . Rc Rc
~11!
At the plasma edge r,r c this formula may be simplified using the asymptotics of the Bessel functions for small arguments: F m~ r ! 5
SD
m 0 m 0 I d cos a 0 ~ 0 ! r gm pumu rc
umu
@ 11 f m ~ r,r c !# ,
~12!
for mÞ0,
where f m (r,r c ) are small corrections of order O(1/m) ~see Appendix A!. Since the poloidal modes m are localized near the central mode m 0 520, these corrections are negligibly small. Consider the particular case of a straight current system with coils parallel to the toroidal axis w, i.e., a 0 50. The magnetic field created by this coil system is described by Eq. ~12! with f m [0, which is exact for this case. The radial component of the perturbed magnetic field B r (r, u , w ) inside the torus r,r c is defined by the expression
(
with B m ~ r ! 5B 0 g ~m0 !
SD r rc
~13!
for mÞ0,
~14!
where B 0 is the amplitude of the magnetic perturbation,
m 0 I d n 0 cos a 0 m 0 I d n 0 ' , B 05 u cr c u cr c and g (0) m is the spectrum of poloidal harmonics:
sin~ m2m 0 ! u c . p ~ m2m 0 !
~15!
Figure 2 shows the radial component of the perturbed magnetic field in the ~u,w! plane at the plasma edge r 50.46 m for the TEXTOR-DED parameters: the mean major radius R 0 51.75 m, r c 50.5325 m, R c 5R 0 2r c 51.22 m. The poloidal extension of the coils is D u 52 p /5, n 0 54, m 0 520. The magnetic field perturbation B may be also presented by the vector potential A of the electromagnetic field. In the case of helical currents considered above, the toroidal component A w of the vector potential A gives the main contribution, and the other components may be neglected. Indeed, one can notice that the toroidal component of the magnetic field perturbation B w is much smaller than both the radial and the poloidal ones: the perturbation field is mainly localized in a finite poloidal interval, and it has the main poloidal harmonics m 0 . Then, according to ~10!, one obtains B w5
1 ]F n0 5 F. R c ]w R c
On the other hand, we have u B ru ' u B uu ;
m0 F. r
From these estimations it follows that n 0r !1. m 0R c
Neglecting the poloidal component B w , the divergence-free perturbed magnetic field may be presented as
u m u 21
@ 11 f m ~ r,r c !# ,
g ~m0 ! 5 ~ 21 ! m1m 0
u B wu / u B ru '
`
]F 5 B r ~ r, u , w ! 5 B ~ r ! cos~ m u 2n 0 w 1 v t ! , ] r m52` m
FIG. 2. The radial component of the perturbed magnetic field in the ~w,u! plane at r50.46 m.
S
B~ r, u , w ! 5 er
D
1 ] ] 2e A ~ r, u , w ! , r ]u u ]r w
~16!
where A w (r, u , w ) is a toroidal component of the vector potential A w ~ r, u , w ! 52
(m m 21 rB m~ r ! cos~ m u 2n 0 w 1 v t ! .
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Phys. Plasmas, Vol. 6, No. 1, January 1999
Abdullaev, Finken, and Spatschek
C. Toroidal corrections
Up to now we have considered the cylindrical approximation for the magnetic field. The effect of toroidicity may be taken into account, multiplying the scalar potential F(r, u , w ) by AR 0 /R, if the small corrections (n 0 r c /2R c ) m13 are neglected for each poloidal component m ~see, e.g., Ref. 41!: `
F ~ r, u , w ! 5 ˜ m ~ r, u ! 5 F
(
m52`
˜ m ~ r, u ! cos~ m u 2n 0 w 1 v t ! , F ~17!
A
R0 F ~ r !, R m
where F mn (r) are the Fourier components of the scalar potential in a cylindrical approximation ~11!, R5R 0 1r cos u is a major radius, and R 0 is a mean major radius. Then the toroidal component A w of the vector potential is replaced by A w ~ r, u , w ! 52
(m m 21 rB˜ m~ r, u ! cos~ m u 2n 0 w 1 v t ! ,
˜ m ~ r, u ! ]F ] ˜B m ~ r, u ! 5 5 ]r ]r
A
R0 F ~ r !. R m
~18!
Using Eq. ~12!, one can obtain the following estimation for the radial components of the magnetic field perturbations at the plasma edge r,r c : ˜B m ~ r, u ! 5B 0 g ~ 0 ! m
S
SD A
3 12
r rc
u m u 21
R0 R 0 1r cos u
D
r cos u . 2m ~ R 0 1r cos u !
~19!
The spectrum of poloidal harmonics of the perturbation magnetic field is localized near the central poloidal mode m 0 520. Then, the second term on the right-hand side of the last expression is much smaller than the first one, and it may be neglected. One can estimate the radial magnetic field B r at the plasma edge r,r c . Since the field is localized at the highfield side u 5 p and its poloidal spectrum is localized around m 0 520, one can obtain for the amplitude of the magnetic field, maxu B r u '
A
SD
R 0 m 0I dn 0 r R 0 2r r c u c rc
m 0 21
.
At the plasma boundary r50.46 m we have obtained maxuBru5162.7 G for the following TEXTOR-DED parameters: R 0 51.75 m, r c 50.5325 m, the current I d 515 kA, u c 5 p /5, and n 0 54. This estimation is in a good agreement with the result (maxuBru5160.5 G) obtained in Ref. 16 by the direct calculations of the helical coil system in toroidal geometry using the Gourdon code. III. FIELD LINE EQUATIONS
magnetic field lines dx/d t 5B. The independent variable t is related to the length element dl via d t 5 u Bu 21 dl, dl5 u dxu . Suppose that the axisymmetric magnetic field, B~ 0 ! ~ r, u ! 5 ~ 0,B ~u0 ! ~ r, u ! ,B ~w0 ! ~ r, u !! ,
~20!
describes the equilibrium configuration. Then the field lines are helical curves on the cylindrical ~toroidal! surfaces with r5const. Let us now consider the effect of magnetic perturbations on field lines. As was shown in Sec. II, the toroidal component of the magnetic perturbations B(1) (x) ~created by a set of helical coils! is much smaller that its radial and poloidal components, and may be presented in the form ~16!. In a tokamak, the toroidal component of the magnetic field B w(0) (r, u ) is much stronger than the magnetic perturbations B(1) (x), and it never vanishes. That allows one to parametrize the field lines x~t! in the form r5r( w ), u 5 u ( w ), choosing the toroidal angle w as an independent variable. Then the equation for the perturbed field lines may be written as B ~r1 ! ~ x! dr 5R ~ 0 ! , dw B w ~ r, u !
d u R B ~u0 ! ~ r, u ! R B ~u1 ! ~ x! 5 1 . d w r B ~w0 ! ~ r, u ! r B ~w0 ! ~ r, u !
~21! In the absence of perturbations, field lines lie on the nested magnetic surfaces r5const. Introducing the new variable u * 5 u * ( u ,r) ~intrinsic coordinate! with the property D u * 5 u * ( u 12 p ,r)2 u * ( u ,r)52 p , i.e., 1 u * ~ u ,r ! 5 q~ r !
E
r B ~w0 ! ~ r, u 8 ! d u 8, R B ~u0 ! ~ r, u 8 !
u
0
~22!
where q~ r !5
E
1 p
p
0
r B ~w0 ! ~ r, u ! du, R B ~u0 ! ~ r, u !
~23!
field lines may be presented as
u * 5 w /q ~ r ! 1const.
r5const,
~24!
Introducing a normalized toroidal flux c canonically conjugated to u * , the perturbed field line equations may be presented in a Hamiltonian form ~Refs. 42–44!: du* ]H , 5 dw ]C
dC ]H 52 . dw ]u*
~25!
with a poloidal flux H as a Hamiltonian function. For the magnetic perturbation ~16! the latter has a form H5
E
dc R ~ c , u * ! A w @ x~ u * , c , w !# 2 . q @ r ~ c !# B t R 20 ~ a !
~26!
The variable c for the equilibrium field ~20! is equal to
c5
A. Hamiltonian description
In this section we recall the field line equations and their Hamiltonian representation. Let us start with the equation for
157
5
1 2 p B t R 20 ~ a ! 1 2 p B t R 20 ~ a !
E E
dSB~ 0 !
Cr r
0
dr 8 r 8
E
2p
0
d u B ~w0 ! ~ r 8 , u ! ,
~27!
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Phys. Plasmas, Vol. 6, No. 1, January 1999
Abdullaev, Finken, and Spatschek
where B t is a toroidal field at the mean major radius R 0 (a). Equations ~22! and ~27! determine the relationship between the new ~intrinsic! coordinates ( c , u * ) and the old ~geometrical! coordinates (r, u ). The transformation (r, u )→( c , u * ) depends on the equilibrium magnetic configuration B(0) (r, u ). For the large aspect ratio R 0 /a of plasma with a circular cross section, the analytical relationship between the geometrical coordinates (r, u ) and the intrinsic coordinates (C, u * ) is given in Ref. 13. Below we recall these formulas and derive some new useful relationships between these coordinates. B. Equilibrium configuration of the magnetic field
Consider a magnetic equilibrium of the tokamak plasma with nested, circular magnetic surfaces. The effects of plasma pressure and electric current leads to an outward shift of the magnetic surfaces. Let R 0 (r) be the position of the center of the magnetic surface of radius r. The shift of R 0 (r) with respect to the center of the plasma ~of radius a! is given by ~see Ref. 13! D ~ r ! 5R 0 ~ r ! 2R 0 ~ a ! 5 @ R 20 ~ a ! 1 ~ L11 !~ a 2 2r 2 !# 1/22R 0 ~ a ! ,
~28!
where L5 b pol1l i /221. Here, b pol is the ratio of the plasma pressure to the magnetic pressure of the poloidal field B (0) u ; l i is the internal inductance. The quantity D(r) is known as the Shafranov shift of magnetic surfaces. The poloidal field B (0) u on each magnetic surface may be represented as B ~u0 ! ~ r, u ! 5
S
D
m 0I p r 11L cos u , 2pr R0
~29!
m 0I w R 0~ a ! 5B t 2pR R 0~ r !
1 r 11 cos u R0
~30!
F S
r 5a 2
R 20 ~ r ! 2R 20 ~ a ! L11
D
r'R 0 ~ a !@ 12 ~ 12 c ! 2 # 1/2.
,
~33!
The relation between the intrinsic poloidal angle u * and the geometrical one u may be found from Eq. ~22! or by integrating the equation 1 r B ~w0 ! ~ r, u ! du* 5 du q ~ r ! R B ~u0 ! ~ r, u ! 5
r2 Iw 1 1 , 2 q ~ r ! R 0 ~ r ! I p ~ 11 e cos u ! 2 ~ 11L e cos u ! ~34!
where «5r/R 0 (r) is the inverse aspect ratio. The result of integration is given in Ref. 13. However, the obtained relation u * 5 u * ( u ,r) is rather complex and does not allow one to obtain the inverse relationship u 5 u ( u * ,r). Below we derive new formulas for these relationships using expansions in powers of the inverse aspect ratio «. The details of the calculations is given in Appendix B. Expanding the right side of Eq. ~34! in a series of powers of « and integrating the obtained expression, the relation u * 5 u * ( u ,r) may be presented in the form M
u * ~ u ,r ! 5 u 1 ( a m sin m u 1O ~ « M 11 ! , m51
~35!
where the expansion coefficients a m are series in powers of «: M
a m 5 ( a ~mk ! « m1k 1O ~ « M 11 ! . k50
a 1 5a 1 «1 ~ 43 a 3 2 12 a 1 a 2 ! « 3 1O ~ « 5 ! ,
a 4 5 321 a 4 « 4 1O ~ « 5 ! .
The coefficients a m are polynomial functions of the plasma parameter L: m
R 0~ r ! r2 12 12 2 R 0~ a ! R 0~ r ! 2
R 20 ~ a !
1/2
a 3 5 121 a 3 « 3 1O ~ « 5 ! ,
According to ~27! one can obtain the following relationships between the normalized toroidal flux c and the radius r of a magnetic surface:
2
r2
a 2 5 14 « 2 $ a 2 1 @~ a 4 2 21 a 22 ! « 2 # % 1O ~ « 5 ! ,
,
m 0I w B t5 . 2 p R 0~ a !
c5
S
c '12 12
We have derived the coefficients a m for the case M 54:
where I p is the plasma current. The toroidal magnetic field is determined by the current I w , i.e., B ~w0 ! ~ r, u ! 5
For small Shafranov shifts, D(r)!R 0 (a), these formulas may be simplified to
DG
a m 5 ~ 21 !
q~ r !5 ~31!
,
R 0 ~ r ! 5R 0 ~ a ! $ 2 ~ L11 ! c 1 @ 11 ~ L11 !~ L12 ! c 2 1 ~ L11 ! a 2 /R 20 ~ a !# 1/2% .
(
k50
~ m2k11 ! L k .
The safety factor q(r) may be also presented as a series of powers «:
1/2
,
m
r2 R 20 ~ r !
S
D
1 3 Iw 11 a 2 « 2 1 a 4 « 4 1O ~ « 8 ! , ~36! Ip 2 8
Equation ~35!, expressing the intrinsic poloidal angle u * via the geometrical poloidal angle u, allows one to invert it and to find u in terms of u * , M
~32!
u ~ u * ,r ! 5 u * 1 ( a m* sin m u * 1O ~ « M 11 ! . m51
~37!
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Phys. Plasmas, Vol. 6, No. 1, January 1999
Abdullaev, Finken, and Spatschek
H 0~ c ! 5
E
159
dc q @ r ~ c !#
is an unperturbed Hamiltonian, and
e H 1 ~ u * , c , w ! 52
R ~ r, u ! A w @ x~ u * , c , w !# B t R 20 ~ a !
is its perturbed part. In ~38! we have introduced a small dimensionless parameter of magnetic perturbation e 5B 0 /B t . According to ~17!, ~18!, and ~38!, the perturbed Hamiltonian e H 1 at the plasma edge for the magnetic field perturbation created by helical coils may be approximated as `
FIG. 3. The dependence of the intrinsic coordinate u * on the poloidal angle u at the distance r50.46 m. The plasma parameter L50.6 and the major radius R 0 51.75 m have been used.
e H 15 e
(
m52`
H m ~ r, u ! 5
* can be expressed in Here, the expansion coefficients a m terms of the coefficients a m . For the case M 54 they are presented in Appendix B. Figure 3 shows the dependence u * on u at the distance r50.46 m for the plasma parameter L50.6, and R 0 (a) 51.75 m. The radial dependence of the safety factor q(r) is shown in Fig. 4 for the same parameters. For the current I w of the magnetic system and the plasma current I p , we have chosen such values that the q53 magnetic surface is located at r50.43 m. Note that for M 54 the accuracy of the formulas ~35! and ~37! are sufficiently high, and they deviate from the exact formulas by less than 1%. However, when calculating the higher-order derivatives of type d k u * /d u k (k.2) using these formulas, strong deviations from their exact values occur because the series becomes divergent. C. Hamiltonian perturbations
Consider Hamiltonian field line equations ~25! with Hamiltonian, H5H ~ u * , c , w ! 5H 0 ~ c ! 1 e H 1 ~ u * , c , w ! ,
~38!
where
H m ~ r, u ! cos~ m u 2n 0 w 1 v t ! ,
r c AR 0 ~ a !@ R 0 ~ a ! 1r cos u # mR 20 ~ a !
g ~m0 !
SD r rc
~39!
m
,
where r stands for a geometrical radial coordinate, which, in contrast to r, here stands for the radius of magnetic surfaces. The relation between these coordinates is determined by the Shafranov shift D(r):
r 5 Ar 2 1D 2 ~ r ! 12rD ~ r ! cos u . The perturbed part of the poloidal flux e H 1 ~39! determines the behavior of field lines under external magnetic perturbations. Particularly, a Fourier decomposition of the perturbation, `
e H 1~ u *, c , w ! 5 e
(
m52`
* ~ c ! cos~ m u * 2n 0 w 1 v t ! , Hm ~40!
allows one to study the resonant perturbation on field lines. * ( c ) strongly affects The term (m,n 0 ) with the amplitude H m the field lines near the resonant magnetic surface q @ r mn 0 ( c ) # 5m/n 0 , destroying them and changing the topology of the field lines. * ( c ) may be determined from ~39! The amplitudes H m and ~40! by amplitudes H m of perturbations in geometrical coordinates: `
*~ c ! 5 Hm
(
m 8 52`
S mm 8 ~ c ! ,
~41!
where S mm 8 ~ c ! 5
1 2p
E
2p
0
H m 8 @ r ~ c ! , u ~ c , u * !#
3e i ~ m 8 u ~ u * , c ! 2m u * ! d u * .
FIG. 4. The radial profile of the safety factor q(r) for the same parameters as in Fig. 3.
~42!
One cannot obtain analytically a simple expression for the integrals ~42!. However, the main features of the integral allow one to estimate the integral using asymptotic methods. It reveals the main physical peculiarities of the transformation of the perturbation field spectrum in geometrical space into the one in intrinsic coordinates. The detailed asymptotic analysis of the integral ~42! is given in Appendix C. Here we present the main results.
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160
Phys. Plasmas, Vol. 6, No. 1, January 1999
Abdullaev, Finken, and Spatschek
The asymptotic form of the integral ~42! is mainly determined by the behavior of the geometrical poloidal angle u as a function of the intrinsic angle u * near the two values: u * 50 and u * 5 p , i.e., by the derivatives
g 15 b 15
U U
du du*
du du*
,
g 35
u 50
b 35
,
U U
d 3u d u *3
u5p
d 3u d u *3
~43!
,
u 50
~44!
, u5p
taken at the low-field side ( u * 50) and at the high-field side ( u * 5 p ). One should note the inequalities g 1 .1, g 3 ,0, 0 , b 1 ,1, and b 3 .0. The coefficients g 1 and b 1 determine the tangents of the curves u vs u * at the low-field side and at the high-field side, respectively ~see Fig. 3!. It is shown in Appendix C that for fixed values of m, the main contribution to the integral ~42! comes from two intervals of m 8 . For small values of m 8 less than @ m/ g 1 1x c (m 8 u g 3 u /2) 1/3(x c '2) # , the integral S mm 8 ( c ) may be expressed as a product of the field taken on the low-field side u 50 on the transformation matrix A mm 8 ( c ): S mm 8 ~ c ! 5A mm 8 ~ c ! H m 8 ~ r ~ c ! ,0! .
~45!
The transformation matrix A mm 8 ( c ) is well described by the Airy function Ai(z): A mm 8 ~ c ! 5 '
1 2p
S
E
2p
0
2 u g 3u m 8
e i ~ m 8 u ~ u * , c ! 2m u * ! d u *
D S 1/3
Ai 2
D
g 1 m 8 2m . ~ u g 3 u m 8 /2! 1/3
~46!
Similarly, for the values of m 8 .m/ b 1 2x c (m 8 b 3 /2) 1/3 (x c '3) the quantity S mm 8 ( c ) may be approximated by the product S mm 8 ~ c ! 5A mm 8 ~ c ! H m 8 @ r ~ c ! , p #
~47!
of the transformation matrix A mm 8 ( c ) on the field taken on the high-field side u 5 p . The transformation matrix in this case is equal to A mm 8 ~ c ! ' ~ 21 ! m1m 8
S D S 2 b 3m 8
1/3
Ai
D
b 1 m 8 2m . ~ b 3 m 8 /2! 1/3
~48!
For the intermediate values of m 8 @ m/ g 1 1x c (m 8 u g 3 u /2) 1/3,m 8 ,m/ b 1 1x c (m 8 b 3 /2) 1/3# unlike ~46! and ~48! the integral ~42! is proportional to 1/Am 8 . That * may be neglected due to the rapid oscilcontribution to H m lations of A mm 8 with m 8 . An example of the dependence of the transformation matrix A mm 8 on m 8 at the fixed value m512 is shown in Fig. 5 for the plasma parameter L50.5 and the minor radius r 50.46 m. The corresponding coefficients are b 1 50.5489 and g 1 51.8218. Figure 5~a! shows A mm 8 itself, and Fig. 5~b! shows it after multiplication by (21) m 8 . The solid curves correspond to the exact numerical evaluation the integral ~42!, and the dashed curves describe their asymptotics by
FIG. 5. The dependence of the transformation integral ~46! A mm 8 on m 8 for the fixed m512: ~a! A mm 8 ; ~b! (21) m 8 A mm 8 . The solid curve describes the exact numerical calculation, the dashed curve corresponds to the asymptotics by the Airy function.
Airy functions. As can be seen from Fig. 5, the asymptotic formulas well describe the transformation matrix A mm 8 near the values m 8 'm/ g 1 and m 8 'm/ b 1 . The perturbation field created by the set of helical coils is localized in the finite interval of poloidal angles: p 2 u c , u , p 1 u c at the high-field side. The spectrum g (0) m of the perturbation ~39! of such a field contains the factor (21) m . Then one can see from ~45!–~48! that the contribution to the * ( c ) for fixed m originates from the spectral amplitude H m amplitudes H m 8 ( c ) with m 8 larger than m and located near the number m/ b 1 . In the example shown in Fig. 5~b! it corresponds to m 8 '20. If the perturbation field localized on the low-field side ( u 50), the spectrum g (0) does not contain the factor m * ( c ) comes from the (21) m . Then the contribution to H m harmonics m 8 lower than m, being located near the number m/ g 1 @see also Fig. 5~a!#.
D. Asymptotic formula for the spectrum of * „c… perturbation H m
We have established the main properties of the transformation of the magnetic field perturbation into Hamiltonian
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Phys. Plasmas, Vol. 6, No. 1, January 1999
Abdullaev, Finken, and Spatschek
161
form. The specific features of this transformation allow one to obtain the asymptotic formula for the spatial spectrum * ( c ). Hm Consider Eq. ~41! for the perturbation spectrum. Using the Poisson summation rule, one can rewrite it as an integral,
*~ c ! 5 Hm
E
`
2`
S
D
`
S mm 8 ~ c ! 112
(
k51
cos~ 2 p km 8 ! dm 8 . ~49!
Neglecting the rapid oscillating terms containing trigonometric functions cos(2pkm8), taking into account that the terms * ( c ), and m 8 near m/ b 1 give the main contributions to H mn using Eqs. ~47! and ~48!, the integral ~49! may be approximated as
* ~ c ! 5 ~ 21 ! m Hm 3Ai
S
E S b 8D `
2 3m
2`
b 1 m 8 2m ~ b 3 m 8 /2! 1/3
D
1/3
˜ ~ m 8 ! dm 8 , H
~50!
FIG. 6. A comparison of the exact and the asymptotic formulas for the spectrum of perturbation at the minor radius r50.43 m. The plasma parameter is b pol51.0. The solid line corresponds to the numerical calculation of ~41!, and the dashed line describes the asymptotic spectrum ~53!.
where ˜ ~ m 8 ! [ ~ 21 ! m 8 H m @ r ~ c ! , p # H 8 is a slowly varying function of m 8 since the factor (21) m 8 in H m 8 (r( c ), p ) cancels. Introducing the new integration variable, x5
b 1 m 8 2m b 1 m 8 2m , 1/3 ' ~ b 3 m 8 /2! ~ b 3 m/2b 1 ! 1/3
the integral ~50! may be reduced to
* ~ c ! 5 ~ 21 ! m Hm
1 b1
E
`
2`
˜ Ai~ x ! H
S
D
m1x ~ m b 3 /2b 1 ! 1/3 dx. b1 ~51!
The Airy function Ai(x) oscillates for x,0 and exponentially decays for x.0. It has a local maximum at x c '21. ˜ in ~51! is a smooth funcOn the other hand, the spectrum H tion of x. Then the integral ~51! can be estimated by the Laplace method. The integration gives
*~ c ! ' Hm
~ 21 ! m C
b1
˜ H
S
D
m1x c ~ m b 3 /2b 1 ! 1/3 , b1
C5 A2 p / u x c u Ai~ x c ! .
~52!
The formula ~52! approximately represents the spectrum of perturbations in intrinsic coordinates. The agreement of the asymptotic formula ~52! with the exact numerical calculations of the sum ~41!, ~42! may be improved by choosing x c and C as adjustable parameters. According to ~39!, ~50!, and ~52!, the spectrum of perturba* ( c ) can be presented in the form tion H m
* ~ c ! 5 ~ 21 ! m1m 0 C Hm
sin~ m * 2m 0 ! u c 3 , p b 1 m * ~ m * 2m 0 ! with
S
r c AR 0 ~ a !~ R 0 ~ a ! 2r ! r2D ~ r ! rc R 20 ~ a !
D
m*
m *5
m1x c ~ m b 3 /2b 1 ! 1/3 , b1
C and x c are now considered as fitting parameters whose values should be adjusted to obtain the good agreement with * ( c ) via ~41!, ~42!. the direct numerical calculation of H m The parameters C and x c were found for the TEXTORDED parameters (R 0 51.75 m, a50.46 m) and for the different plasma parameters b pol ~or L!. The fitting was performed for an interval of poloidal modes m:8
