Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 (2004)
6 (2004)437–441
000–000
J. Plasma Fusion

Bootstrap Current Coefficients in Stellarators
NISHIMURA Shin and SUGAMA Hideo
National Institute for Fusion Science, Toki, 509-5292, Japan(Received: 9 December 2003 / Accepted: 1 December 2004)

Abstract
A method to derive analytical expressions for bootstrap current coefficients is studied. The drift kinetic equation isdivided into two parts corresponding to the effects of local and global structures of magnetic fields. The divided transport coefficients can be approximated by connecting the results of only three types of asymptotic expansions of thedivided equations. The current coefficients obtained by adding these two parts approximately reproduce results of adirect numerical calculation of the drift kinetic equation.
Keywords:
neoclassical transport, bootstrap current, neoclassical viscosity, plasma flow, drift kinetic equation,
non-symmetric toroidal configuration

path. The collision frequency regime boundary determinedby this connecting method corresponds to the bounce frequency of the banana-trapped orbits [6]. Exactly speaking,the parallel viscosity due to the existence of banana trapped
orbits includes also a direct contribution of circulating particles in the banana regime [1-6]. For general collisionality
regimes in stellarators, we have to express the co-existence of
two types of the forces with opposite directions and with different bounce frequencies of the corresponding trapped orbits
by unifying these previously established methods.
On the other hand, various numerical methods to directly solve the DKE without any asymptotic approximations
have been developed to study ripple diffusions in non-symmetric toroidal plasmas. However, these codes often
employed the pitch-angle-scattering collision operator
instead of momentum conserving operators, since the effectof parallel momentum transferring parts in the collision term
was negligible in the calculation of the ripple diffusions [7].
The relation of the numerical solutions given by these codeswith the analytical theories for the flows based on the
moment equations had not been clarified [3,8,9]. Motivatedby recent design activities for advanced stellarators, a method
was developed recently to interpret the direct solution of thelinearized DKE with the pitch-angle-scattering collision
operator within the framework of the moment method [7]. In
this method, the formal expression of the viscosities based on
the numerical solution directly coincides with that in the previous analytical theories. The results given by this methodclarified that the driving forces do not change their dependence on the collisionality (∝ν/υ, ∝(ν/υ)0, ∝(ν/υ)−1) and1. Introduction
The analytical theory of bootstrap currents in non-symmetric toroidal plasmas [1-5] has been constructed based on
the moment equation approach [6]. In this approach, parallel(to the magnetic field) momentum balance equations satisfying particle, momentum, and energy conservation are used
together with neoclassical parallel viscosities given by solutions of the linearized drift kinetic equation (DKE). The viscosity forces include two parts; damping force against theparallel flows and driving force for the flows due to radial
gradient forces. In stellarators, both of contributions due toripple and banana trapped orbits are included in the driving
force. These trapped orbits generate the driving forces in
opposite directions to each other while their contributions to
the damping force have same direction. Here we discuss amethod to express analytically this driving force part of theviscosity under the co-existence of two types driving mechanisms. The treatments of the damping force part in general
non-symmetric toroidal plasmas [7] are substantially identical to those in symmetric plasmas such as tokamaks and thus
details of them are out of scope of this paper. However, weshould note that, even in the symmetric plasmas, the linearized DKE including the complete parts of the Vlasov andcollision operators with the conservative properties could not
be solved analytically. Therefore, for the damping and driving forces in tokamaks [6], a connection formula was used tosmoothly connect results of three types of asymptotic expansions of the DKE; (1) banana regime expansion giving theviscosity of ∝ν/υ, (2) plateau regime expansion giving thatof ∝(ν/υ)0, and (3) Pfirsch-Schlueter regime expansion giving that of ∝(ν/υ)−1, where υ/ν is the collision mean free©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail:shin@nifs.ac.jp
1437

Nishimura S. et al., Bootstrap Current Coefficients in Stellarators

V||GXa − CaL(GXa) = σXa .

their polarity simultaneously. This driving force cannot be
expressed analytically by appropriate connections of only
three types of asymptotic results as the previous tokamak theory used. However, this numerical result also suggests that if
the total driving force can be divided into appropriate twoparts with different polarity and collision frequency dependence, the divided parts can be approximated by combining
previously established analytical methods. This dividing
qualitatively corresponds to the separation of two types driving mechanisms discussed above. Because of difference ofthe pitch-angle variables (banana : the adiabatic invariant of
the drift orbits µ, plateau and Pfirsch-Schlueter : the pitchangle parameter ξ ≡ υ|| /υ [10]) used in analytical calculations,
we have to divide here the linearized DKE itself having a
non-linearity with respect to the magnetic field strength modulation δB on the magnetic flux surface. This separation of
the DKE seems to be uniquely determined by one requirement from the banana regime calculation in which the treatments of trapped and circulating particle distributions must
be separated. Here we extend a dividing method for the DKE
developed in a previous banana regime theory [2-4] to thecollisional regimes. It is shown that the resulting divided
equation set not only includes the previous banana regime
theory, but also is a practically useful separation of the two
types driving mechanisms in general collisionality regimes.
Comparisons with the numerical results to confirm a validityof a derived formula are also shown.

(1)Here, V|| ≡ υξ(B/B)·∇ + VM is the linearized Vlasov operator,
where VM ≡ −(υ/2)[(B·∇lnB)/B](1 − ξ2)∂/∂ξ is the mirror
force operator. The linearized Coulomb collision operator
CaL includes only the pitch-angle-scattering operator [7] andthe energy scattering Krook collision operator for l = 2 spherical harmonic perturbation [6]. The particle velocity υ andthe pitch-angle parameter ξ are used as the velocity spacevariables here. In discussions on the plateau and the PfirschSchlueter asymptotic calculations, we show here the obtained
equations and numerical examples for the Boozer coordinates. The magnetic field strength is assumed to be given by
the Fourier expression B = ΣBmn(Boozer)cos(mθB − nζB), in theflux coordinates (s,θB,ζB), where θB and ζB are the poloidal
and toroidal angles, respectively, and s is an arbitrary label ofa flux surface. By applying the method in refs. [2-4] to subtract the part corresponding to the local structure effect, thesource term σXa corresponding to the radial gradient forces
can be written as σXa = σXa(sym) + σXa(asym) by using

σ (sym)≡ − σUaXa

σ (asym)≡Xa⎡ψ 'Bζ − χ'Bθ⎤V'c+
H2 , (2)2ea χ'ψ ' ⎢⎣4π2 ⎥⎦
〈B2〉

∂B
⎡
B 2
mac
2 υ P2(ξ ) ⎢χ ' (1 − H2)∂θB2ea χ'ψ ' 〈B 〉⎣− ψ ' (1 + H2)

2. Separation of the parallel viscosity force
In the banana regime theory in non-symmetric toroidal
plasmas [2-4], a specific method was used to obtain only theparallel viscosity force directly without solving full part ofthe equation. The distribution function for the circulating particle can be calculated analytically using a Fourier expansion
even in the banana regime where µ is used as the pitch-angle
variable. On the other hand, it was also known that the contribution of the trapped particles to the driving force can be
determined by incompressible particle and heat flow patterns,
which are strongly affected by the local structure of the magnetic field such as the helical ripple in stellarators. Using
these characteristics of distribution functions of circulating
and trapped particles, Shaing, et al. [2-4] subtracted a part
expressing the effect of this local structure from the DKE,
and then calculated the circulating particle distribution basedon the remaining part of the equation. Since this separation
was done on the lowest order equation in the banana regime
expansion, the divided equations did not include Coulomb
collision effects. Here we extend this theory to collisional
regimes by adding a linearized collision operator as following.
To discuss about general collisionality regimes, it isconvenient to consider at first the linearized DKE defined inref. [7]. The solution for the particle species a is determinedby the perturbed distribution functions GXa and GUa. The former function GXa, which is driven by the radial gradient
force, satisfies the equation,∂ B ⎤ mac B 2
υ P2(ξ )+∂ζ B ⎥⎦
ea 〈B2〉∂ G ∂B ⎤⎡ ∂ G ∂B⎢ ∂ζ ∂θ − ∂θ ∂ζ ⎥ ,
B
B
B⎦
⎣ B

(3)

Here, H2 is a constant indicating the flux surface averaged
effect of the local structure of the magnetic field on theincompressible flow. This constant is defined in refs. [2-3]using the Hamada coordinates. Definitions of the other quantities in eqs. (2)-(3) are those in ref. [7] so that χ' and ψ' are
the radial derivatives of the poloidal and toroidal magnetic
fluxes, respectively, and V' is the radial derivative of the volume enclosed by the flux surface, where the derivative isdenoted by ' = d/ds. In eq. (2), Bθ and Bζ are the covariant
poloidal and toroidal components of the magnetic field,
respectively, and P2(ξ) in eq. (3) is the Legendre polynominalof order l = 2. The charge and mass of the particle species aare denoted by ea and ma, respectively, and 〈·〉 represents theflux surface average. The source term in eq. (2) σ Ua ≡−maV||(υξB) = −maυ2P2(ξ)B·∇lnB is that for the other DKE,
V||GUa − CaL(GUa) = σUa defined in ref. [7] to derive the flowdriven part of the perturbation. In eq. (3), G is the generating
function related to the Hamada/Boozer coordinates conversion [7] and can be calculated using the Fourier expansion ofB−2. The equation for GUa is identical to that used to derive
the damping force part in the previous theories for symmetricand non-symmetric toroidal plasmas [1-6]. The analyticalexpressions for this damping force part in non-symmetricplasmas are shown in refs. [5,7]. It should be noted that2438

Nishimura S. et al., Bootstrap Current Coefficients in Stellarators

explicit expression for GUa itself is not always required in
obtaining the asymptotic values of the mono-energetic parallel viscosity coefficients, especially when the 〈B∫ υ||d3 υ〉moments of the DKE is used to obtain the pitch-angle integrated and flux surface averaged viscosity forces [1-6].
As shown in eqs. (24)-(25) in ref. [2] and Appendix D
in ref. [7], σ Xa (asym) vanishes when the magnetic fieldstrength has a complete symmetry such as an axisymmetry, or a helical symmetry, or poloidal symmetry. Only
σXa(sym) remains in these completely symmetric cases. The
superscripts (asym) and (sym) denote that the part exists
only in non-symmetric configurations, and the part exists
even in the symmetric configurations, respectively. The
solution of eq. (1) is given by the sum of two components,
GXa = GXa(sym) + GXa(asym), where GXa(sym) and GXa(asym) are thesolutions of V || G Xa (sym) − C a L [G Xa (sym) ] = σ Xa (sym) and
V||GXa(asym) − CaL[GXa(asym)] = σXa(asym), respectively. The former part GXa(sym) is proportional to GUa and only its absolutevalue is determined by the incompressible particle and heatflows under the asymmetry. Therefore the already known
results on GUa are applicable to this former part. The original purpose of this part in refs. [2-4] was the trapped particles. When we consider the case where the collision operator vanishes, the DKE in ref. [7] using eqs. (2)-(3) coincides
with the lowest order equation in the banana regime expansion in refs. [2-4] and thus eqs. (1)-(3) automatically
include this previous banana regime theory. In this theory,
V||GXa(sym) = σXa(sym) ∝ σUa was used to derive the contribution of the trapped particles. The remaining part
V||GXa(asym) = σXa(asym) was used to derive the remaining effectof the circulating particles without solving the trapped particle part. This contribution of the circulating particles due tothe asymmetry appears in the relatively collisionless limit
where the banana-trapped part of the distribution can betreated by the banana regime expansion. With this separation
for treatments of the circulating and trapped particles, thetotal driving force due to all of the particles was easily
obtained without the complete solution of the equations in
the banana regime theory [2-4]. The moment of the DKE
itself is used also in this step to derive viscosity force without
solving full part of the DKE. Also for general collisionality
regimes, this separation shown in eqs. (2)-(3) is a practicallyuseful separation of the effects of local and global structuresof the magnetic fields which make the ripple and banana
trapped orbits, respectively. In many stellarators with helically twisted structures giving large values for (ψ’/χ’)∂B/∂ζB,
the toroidicity component B10 is the dominant Fourier component of σXa(asym) as shown in ref. [2] while the dominantFourier component of σXa(sym) ∝ B·∇B is the ripple component. In plateau and Pfirsch-Schlueter expansions omitting
the mirror force operator [10], these Fourier compositions of
the source terms σXa(asym) and σXa(sym) directly determine thoseof the response functions GXa(asym) and GXa(sym). The n = 0
component dominating GXa(asym) expresses the banana-trapped
effects while the n ≠ 0 component in GXa(sym) expresses the
ripple-trapped effects. Therefore eqs. (2)-(3) is a practicallyuseful separation of the local and global structure effects
especially when the divided driving forces have different
polarity and collision frequency dependence. As discussed
later on eqs. (4)-(8), this tendency of the separation, in which
the GXa(asym) and GXa(sym) dominate the collisioless and collisional regimes, respectively, is not peculiar to stellaratorswith helically twisted structures only, but fairly universal forvarious types of non-symmetric toroidal plasmas. Hereafter,
we call GXa(asym) “global” part, and call GXa(sym) “local” part,respectively.
Then the transport coefficient Na(K), which expresses
the parallel viscosity as the driving force, defined in ref. [7]is obtained from this solution. We show here the expressionsof “global” part of the normalized mono-energetic coefficient N * ≡ Na(K)/[(c/ea)maυTaK3/2] due to the “global” part ofthe perturbation GXa(asym). Here, υTa is the thermal velocity
and K ≡ (υ/υTa)2. From the relation of σXa(sym) and σUa shownin eq. (2), the remaining part of the coefficient N *(sym) due tothe “local” part of the perturbation is directly obtained fromthe normalized mono-energetic parallel viscosity coefficientM * describing the damping force by the relation
N *(sym) = −M *[(ψ'Bζ − χ'Bθ)/〈B2〉 + H2V'/4π2]/(2χ'ψ'). The
analytical expressions for M * are shown in ref. [7], and M * isalways positive in general toroidal configurations. By combining the analytical expression of the parallel viscosity force
using these coefficients with the parallel momentum balanceeqs. [5,7,11], the bootstrap current coefficients of the neoclassical transport matrix can be obtained.
The detail of the banana regime expansion for the “global” part of the equation is shown in refs. [2-4], and the resulting mono-energetic coefficient N *(asym) is given in the form ofa pitch-angle-integral as

N * (asym)(banana) =〈B2〉V' νDa (K)2 χ'ψ 'fc 4π2 υ≡ NB

νDa (K),υ

1

λW(λ )

0||∫ dλ 〈|υ |/υ〉(4)

where νDa is the pitch-angle-scattering collision frequency[6,7] and fc is the fraction of circulating particles [1-7]. The
integral ∫dλ in the range of 0 ≤ λ ≤ 1 indicates the sum of thecontributions of the circulating particles, and W(λ) is theFourier expanded perturbation of the circulating particle distribution. The explicit expression of this integral in the
Hamada coordinates is given in ref. [3] and that for the
Boozer coordinates is in ref. [4]. In the plateau and PfirschSchlueter expansions neglecting the mirror force operator,
the perturbed distribution functions are obtained by using a
Fourier expansion for the dependence on (θB,ζB) in full rangeof ξ. The connected result given by these expansions is* (asym)
N P-P
=−

1
2 χ'ψ ' 〈B2〉1/2[ χ' (1 − H2)m + ψ ' (1 + H2)n]( χ'm − ψ 'n)[B (Boozer)
]2
mn
a
8
ν (K) 2 1/2 V'm,nχ'm − ψ 'n + 5 T
〈B 〉
υ
4π2
π

∑
3
439

Nishimura S. et al., Bootstrap Current Coefficients in Stellarators

By neglecting the higher order of δB in eq. (5) andby using the definition of H2 [2-3], it can be shown thatN *(asym) (νTa→∞) ≈ 0. This is a result of a characteristic of the“global” part in the banana regime. Although eq. (4) implicitly includes a residual contribution of the trapped particle part
via the incompressible particle and energy conservation [2],the contribution of the deeply trapped particle part almost
vanishes in the pitch-angle integral and the flux surface averaging. The transit mean free path for the plateau/banana transition of the “global” part eq. (8) is usually longer than thatfor the “local” part N *(sym) ∝ M * since the “global” part in thebanana regime expresses the effects of circulating particles
while the transitions of the “local” part is strongly affected by
the ripple trapped effects [7].

2 4π2ν a (K) ⎛ 5π υ Ta ⎞ ⎤
⎡ π
IPS T
+⎢
⎜
⎟
2 1/2 Ip1 +2
υ ⎝ 8 ω Ta ⎠ ⎥⎦5〈B 〉 V'
⎣ 4〈B 〉
2⎛
5π νTa (K) υ Ta ⎞
1
+
⎜
⎟
υ
ω Ta ⎠
8
⎝

−2

(5)

,

where Ipl and IPS are defined by

Ip1 ≡

1
4π2

⎧

χ'm − ψ 'n

∫ dθ ∫ dζ ⎨⎩∑ χ'm − ψ 'n B
B

B

(Boozer)mnm,n⎧ ⎛ ∂ G ∂B
∂ G ∂B ⎞sin(mθB − nζB)⎨ ⎜
−
⎟,
∂
ζ
∂
θ
∂
θB ∂ζ B⎠
⎝
B
B
⎩

(6)

and

IPS ≡ −

1
4π2

⎛

∂B

3. Numerical examples∂B ⎞∫ dθ ∫ dζ ⎜⎝χ' ∂ θ + ψ' ∂ ζ ⎟⎠B

B

B

We assumed the magnetic configurations to be
B = B0 [1 − εt cosθB + εh cos(lθB − nζB)], l = 2, n = 10,
B0 = 1 T, χ' = 0.15 T·m, ψ' = 0.4 T·m, Bθ = 0, Bζ = 4 T·m
and investigated dependence of the driving force on toroidicity εt and helical ripple amplitude εh. In Fig. 1, we illustrate
the procedure to calculate the mono-energetic coefficients N *based on the dividing method in a quasi-helically symmetricconfiguration with εt = 0.01 and εh = 0.05. The “local” partN *(sym) qualitatively corresponds to the driving force due tothe ripple-trapped orbits while the “global” part N *(asym) corresponds to that due to the banana-trapped orbits in thisexample. The N *(sym) dominating the relatively collisional
regimes of ν/υ > 10−4 m−1 is always positive. This polarity
means the driving force making the flow in the direction of
the helical winding. The N *(asym) given by eq. (8) is negativein the collisionality range of ν/υ < 10−1 m−1 but it dominates

B∂G ∂B ⎞⎛ ∂G ∂B−⎜⎟,⎝ ∂ζ B ∂θB ∂θB ∂ζ B ⎠(7)respectively. The transit frequency ω Ta in eq. (5) corresponding to the boundary between the plateau and thePfirsch-Schlueter regimes is defined in refs. [6,11]. The
relation of the anisotropy relaxation collision frequency νTafor the Pfirsch-Schlueter regime with ν D a is given byνTa = 3νDa + νEa using the energy scattering collision frequency νEa [6,7]. In calculations in next section, νTa is replaced by
3νDa, and subscripts and superscripts denoting the types ofcollisions and the particle species are omitted following ref.
[7]. Connecting eq. (4) and eq. (5) at a collision frequency
satisfying |CaLGXa(asym)(banana)| ≈ |VMGXa(asym)(plateau)| gives agood approximation since these terms are treated as higher
order terms in the banana and plateau expansions. Since
dominant functional components in both of these terms are
odd functions of ξ, we use l = 1 Legendre components ofthem: [CaLGXa(asym)(banana)](l = 1) = 3νaDmacfcυξBNB/(2ea〈B2〉)and [VMGXa(asym)(plateau)](l = 1) = 3macυ2ξNP/(2ea〈B2〉1/2), where
NP is an appropriate constant expressing a representativevalue for the mirror force term. The detail of this constant ingeneral multi-helicity stellarator is out of scope here and will
be reported in a separated article. However, in so-called single-helicity stellarator model magnetic fields such as examples in the next section, a method to use the dominant Fouriermode term in Σ of the plateau value of N *(asym) is practicallyvalid. This simplified method is used in the next section.
Using what we have noted above, the “global” part of themono-energetic coefficients is given by a banana/plateau
connection formulaN*(asym)= [ν D(K/υ)][NB + Na*(asym)P-P[1 + |NB/NP|ν aD(K)/υ]2 .

10-2

N*[m-1]

10-3+N*(total)− N*(asym)

10-4
10-5
10-6

− N*
(total)ε t=0.01
εh=0.05+N*(asym)

10-7 -6
10 10-5 10-4 10-3 10-2 10-1 100 101
ν/v [m-1](NB /NP) ν D(K)/υ]/2+N*(sym)a(8)Fig. 1 The procedure to obtain the mono-energetic coefficient N * by using two connection formulas corresponding to two types of driving mechanisms for theflows. The closed and open symbols indicate minus
and plus values, respectively.

Here, the transition collision frequency is multiplied by fc totake the effect of the parallel momentum exchange term(Ha(l = 1) in ref. [7]) into account for the banana regime viscosity force.
4440

Nishimura S. et al., Bootstrap Current Coefficients in Stellarators

over the N *(sym) only at the collisionless limit of ν/υ < 10−4 m−1.
This difference of the dependence of these two parts on thecollisionality qualitatively indicates the difference of thebounce frequencies and fractions of corresponding trapped
orbits. By adding these two components, N * having thepolarity reversal at ν/υ ~ 10−4 m−1 is obtained. Here we callthe change of the sign in N * polarity reversal. This result cannot be obtained by the connection formula in ref. [11] without dividing into two contributions with opposite directions.
Figure 2 shows comparisons of the mono-energetic coefficients N * obtained by using eq. (8), with those obtained
numerically by combining the DKES (Drift Kinetic Equation
Solver) code [12] with our conversion formula [7].
Following ref. [7], we plot the geometrical factor [1-5,7]G(BS) ≡ −〈B2〉N */M * instead of N * in this figure, where M * andconsequently N *(sym) are given by the DKES with the relation[7] M * = (ν/υ)2D*33/[1 −(3/2)(ν/υ)D*33/〈B2〉]. Here, D*33 is an
output of the DKES. Therefore the deviation of an analyticalresult from the numerical one shown here indicates the errorof eq. (8). In the numerical results for the relatively collisionless regimes of ν/υ < 10−3 m−1, N * has a weak dependence on
the radial electric field [3]. This dependence seems to becaused by effects of the trapped/untrapped boundary layer inthe velocity space [13] combined with the bounce-averaged
part of the ripple trapped particle distribution function associated with the ripple diffusions [1,7]. In Fig. 2, we show theresults for the cases with the radial electric field strength of
Er /υ = 1 × 10−3 T, 3 × 10−3 T. The dependence of the numerical results in the collisionless limit of ν/υ < 10−5 m−1 on theradial electric field is saturated in this range of the electricfield strength because of the suppression of the ripple diffusions. Since eq. (4) is derived neglecting this boundary layer
effect [2,3], the connection formula eq. (8) is applicable forthese saturated states in the collisionless regimes and for thecollisional regimes where the dependence on the radial electric field disappears. The suppression of the boundary layer
effect due to the radial electric field is still small around theboundary between the banana and plateau regimes at
ν/υ = 10−3 ~ 10−4 m−1 and thus a deviation of the connection
formula from the numerical results remains fairly large especially for the case with εt = 0.1 and εh = 0.05. However, thepolarity reversal at ν/υ ~ 10−4 m−1 for εt = 0.01, εh = 0.05 andat ν/υ ~ 10−2 m−1 for εt = 0.1, εh = 0.05 can be precisely predicted by the analytical calculation using eq. (8), since it
occurs in the plateau or the Pfirsch-Schlueter regimes of the“asymmetric” parts where the boundary layer effect is small.

4. Concluding remarks
We have presented a method to derive analytical expressions for the bootstrap current coefficients in stellarators.
Since this method separates two types driving mechanisms
for the flows in stellarators, connecting the results of three
types of asymptotic expansions gives a good approximation
for the divided transport coefficients. We have also shown
the numerical results to confirm the validity of derived formulas.

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(1983).
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833 (1992).
[6] S.P. Hirshman and D.J. Sigmar, Nucl. Fusion 21, 1079
(1981).
[7] H. Sugama and S. Nishimura, Phys. Plasmas 9, 4637
(2002).
[8] Y. Ogawa et al., Nucl. Fusion 32, 119 (1992).
[9] H. Maassberg, W. Lotz and J. Nuehrenberg, Phys.
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[10] M. Coronado and H. Wobig, Phys. Fluids 29, 527(1986).
[11] K.Y. Watanabe et al., Nucl. Fusion 35, 335 (1995).
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(1982).

30
axisymmetric2520

εt=0.1G(BS)15εt=0.1105 ε =0.01 ε =0.05
tεh=0.01εh=0.05h

0-5 helical symmetric
Er /v = 1.0e-3, 3.0e-3 [T]-10 -610 10-5 10-4 10-3 10-2 10-1 100 101
ν/v [m-1]
Fig. 2 The geometrical factor G (BS). The connection formula(solid curves), and the numerical results for the caseswith E r / υ = 1 × 10 −3 T(open circles) and 3 × 10 −3T(closed circles) are shown.

5441