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3 Equations of motion in cylindrical coordinates

In this section, all quantities are in the normalized form given in Sec. 1.1. For notational simplicity, the over-bars of the notation are omitted. In cylindrical coordinates (R,ϕ,Z), the location vector is written as X = RˆeR(ϕ) + ZˆeZ. Using this, we obtain

dX- = dRˆeR + R dˆeR(ϕ)+ dZ-ˆeZ dt dt dt dt = dRˆeR + R dϕˆeϕ + dZ-ˆeZ. (39) dt dt dt
Substituting this into Eq. (1) gives
dR dϕ dZ B ⋆ μ 1 dt ˆeR + R dtˆeϕ +-dt ˆeZ = B-⋆v∥ + 2πBB-⋆B × ∇B + BB-⋆E × B, ∥ ∥
(40)

from which we obtain the following component equations:

[ ⋆ ] dR-= B--v∥ +---μ---B × ∇B + --1- E× B ⋅ˆeR dt B ⋆∥ 2πBB ⋆∥ BB ⋆∥
(41)

[ ] dZ- B-⋆ ---μ--- --1- dt = B ⋆v∥ + 2πBB ⋆B × ∇B + BB ⋆E × B ⋅ˆeZ, ∥ ∥ ∥
(42)

dϕ 1 [B ⋆ μ 1 ] ---= -- --⋆v∥ +-----⋆-B ×∇B + ---⋆E × B ⋅ˆeϕ, dt R B ∥ 2πBB ∥ BB ∥
(43)

In the cylindrical coordinates, the terms B ×∇B, ∇× b, and b ⋅∇× b are written, respectively, as

|| || || ˆeR ˆeϕ ˆeZ || B × ∇B = ||BR∂B-- 1Bϕ∂B- B∂ZB- ||, ∂R R ∂ϕ ∂Z
(44)

( 1 ∂bZ ∂bϕ ) (∂bR ∂bZ ) ( 1∂(Rbϕ) 1 ∂bR) ∇ ×b = R-∂ϕ-− -∂z ˆeR + -∂Z-− -∂R- ˆeϕ + R---∂R-- − R-∂ϕ-- ˆeZ, (45)
( 1 ∂bZ ∂bϕ) (∂bR ∂bZ ) ( 1∂ (Rb ϕ) 1 ∂bR) b⋅∇ × b = bR R-∂ϕ--− ∂z- + bϕ -∂Z-− -∂R- +bZ R---∂R-- − R-∂-ϕ- .
(46)

Using bR = BR- B, bZ = BZ- B, and bϕ = Bϕ- B, we obtain

( ∂BR- ∂B ( ∂BZ- ∂B ( ∂Bϕ ∂B ||{ ∂∂bRR-= -∂R-BB−2∂RBR- ||{ ∂∂bZR-= -∂R-BB−2∂RBZ- |||{ ∂∂bϕR-= -∂R BB−2∂RBϕ ∂bR-= -∂B∂RZ-B− ∂∂BZBR ∂bZ-= -∂B∂ZZ-B− ∂∂BZBZ ∂bϕ-= ∂B∂ϕZ B-− ∂∂BZBϕ ||( ∂Z ∂BR-BB2− ∂BBR ||( ∂Z ∂BZ-BB2− ∂BBZ ||| ∂Z ∂BϕBB2− ∂BB ∂∂bRϕ-= -∂ϕ-B2∂ϕ--- ∂∂bZϕ-= -∂ϕ-B2∂ϕ--- ( ∂∂bϕϕ-= -∂ϕ-B2∂ϕ-ϕ
(47)

The equation for v∥ is given by Eq. (11), i.e.,

dv∥= − μB-⋆⋅∇B + 2π B⋆-⋅E. dt B∥⋆ B⋆∥
(48)

The first term on the left-hand-side is written

B-⋆ BR⋆∂B- B⋆ϕ-1∂B- B⋆Z-∂B- B ⋆∥ ⋅∇B = B ⋆∥∂R + B⋆∥R ∂ϕ + B∥⋆∂Z ,
(49)

where

v ( 1∂b ∂b ) B⋆R = BR + -∥- ----Z-− --ϕ , 2π R ∂ϕ ∂z
(50)

v ( 1 ∂(Rb ) 1∂b ) B ⋆Z = BZ + -∥- ------ϕ-− ----R- , 2π R ∂R R ∂ϕ
(51)

v ( ∂b ∂b ) B⋆ϕ = Bϕ + -∥- --R-− --Z- , 2π ∂Z ∂R
(52)

and B∥⋆ is given by Eq. (13).

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