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3.2 Guiding-Center Drifts and Drift-Kinetic Equation

As shown in previous section, for axisymmetric plasmas, the electron drift kinetic equation may be expressed in the general form

∂f- + vgc ⋅∇f = C(f) + Q (f)+ E (f) ∂t
(3.31)

where f = f(p,ξ,ψ,θ,t) is the guiding-center distribution function.

In tokamaks, it can be shown that the guiding center velocity vgc may be decomposed into a fast parallel motion along the field lines, and a vertical drift velocity vD across the magnetic flux surfaces

^ vgc = v∥b + vD
(3.32)

From the general expression (2.36) of the magnetic field B,

B = I (ψ )∇ ϕ+ ∇ψ × ∇ ϕ
(3.33)

one obtains in the (ψ,s,ϕ) coordinates system,

I (ψ)-^ |∇-ψ| B = R ϕ - R ^s
(3.34)

As shown in Appendix A, the gradient in (ψ,s,ϕ ) coordinates is

^ ∇ = ∇ ψ-∂- + ∇s ∂--+ ∇ ϕ-∂-= ∇ ψ -∂-+ ^s-∂-+ ϕ-∂-- ∂ ψ ∂s ∂ϕ ∂ψ ∂s R ∂ϕ
(3.35)

and recalling that the constants of the motion are the total energy (or momentum p) as defined in (2.16) and the magnetic moment μ as given by relation (2.17), following conservations laws

∂-μ ∂s = 0
(3.36)

∂ [ 2 ] ---p ∥ + 2μBme = 0 ∂s
(3.37)

are satisfied.

3.2.1 Drift Velocity from the Conservation of Canonical Momentum

The toroidal canonical momentum is also a constant of the motion because of axisymmetry. It is expressed as

Pϕ = R [γmevϕ + qeAϕ]
(3.38)

where Aϕ is the toroidal component of the vector potential. From the relation

B = ∇ × A
(3.39)

and the expression (A.171) of a rotational in (ψ,s,ϕ) coordinates, we get

[ ] B = -1-∂-(RA ) - 1-∂--(A ) ψ^ R ∂s ϕ R ∂ϕ s [ 1 ∂ |∇ ψ| ∂ ] + -----(A ψ)- ------- (RA ϕ) ^s [R ∂ ϕ R ∂ ψ ( )] + |∇ψ |-∂-(A ) - |∇ ψ | ∂- -A-ψ- ^ϕ (3.40) ∂ψ s ∂s |∇ ψ|
with
^ A ψ = A ⋅ ψ (3.41) As = A ⋅ ^s (3.42) A ϕ = A ⋅ ^ϕ (3.43)

In axisymmetric plasma, this reduces to

-1-∂- ^ B = R ∂s (RA ϕ)ψ |∇ ψ |∂ - --------(RA ϕ)^s [R ∂ψ ( )] + |∇ψ |-∂-(As) - |∇ ψ | ∂- -A-ψ- ^ϕ (3.44) ∂ψ ∂s |∇ ψ|
so that
B = - |∇ψ-|-∂-(RA ) s R ∂ψ ϕ
(3.45)

In addition, we know from expression (3.34) that

|∇ ψ| Bs = - -R---
(3.46)

so that be obtain

∂RA ϕ ------= 1 ∂ψ
(3.47)

Because the toroidal canonical momentum is a constant of the motion, we have

vgc ⋅∇P ϕ = 0
(3.48)

which can be decomposed into

vgc ⋅∇ (Rγmev ϕ) + vgc ⋅∇ (qeRA ϕ) = 0
(3.49)

Using relation (A.169), we get

[ ^ ] vgc ⋅∇ (qeRA ϕ) = vgc ⋅ ∇ ψ ∂-+ ^s-∂-+-ϕ-∂- (qeRA ϕ ) ∂ψ ∂s R ∂ ϕ
(3.50)

which in axisymmetric systems gives

[ ] v ⋅∇ (qA ) = qv ⋅ ∇ ψ∂RA--ϕ+ ^s∂RA-ϕ- gc e ϕ e gc ∂ψ ∂s
(3.51)

Since Bψ = 0, we have from relation (3.44) ∂(RA ϕ)∕∂s = 0 and therefore, using expression (3.47),

vgc ⋅∇ (qeA ϕ) = qevgc ⋅∇ψ
(3.52)

The only velocity accross the flux-surfaces is the drift velocity we are looking for, so that we get, using relation (3.32)

vgc ⋅∇ (qeA ϕ) = qevD ⋅ ∇ψ
(3.53)

and the equation (3.49) becomes

qevD ⋅∇ ψ = - vgc ⋅∇ (Rγmev ϕ)
(3.54)

Assuming a priori that | | |v∥|≫|vD |, a condition that holds in tokamaks, this equation reduces to

-1v∥ vD ⋅∇ ψ = - qe B B ⋅∇ (Rγmev ϕ) v∥ = - Ω--B ⋅∇ (Rv ϕ) (3.55) e
where we used that ∂γ∕∂s = 0 because of the conservation of energy.

The toroidal velocity is related to the parallel velocity by

Bϕ I (ψ) vϕ = ---v∥ = ----v∥ B RB
(3.56)

so that

I (ψ) Rvϕ = ----v∥ B
(3.57)

Since I(ψ ) is a flux function, it can be taken out of the gradient, so that

( ) v∥- v∥ vD ⋅∇ ψ = - ΩeI (ψ)B ⋅∇ B
(3.58)

3.2.2 Drift Velocity from the Expression of Single Particle Drift

The guiding-center drift velocity due to the magnetic field gradient and curvature is

1 ( 2 v2⊥) B × ∇B vD = Ω-- v∥ + 2-- ---B2--- e
(3.59)

and its component perpendicular to the flux-surface can be written as

1 ( v2 ) 1 vD ⋅∇ ψ = --- v2∥ +-⊥- --2∇ ψ × B ⋅∇B Ωe 2 B
(3.60)

Inserting the expression (3.34) of the magnetic field, we find

1 ( v2) |∇ ψ|[ ] vD ⋅∇ψ = ---- v2∥ + -⊥- --2-- I (ψ)^s + |∇ ψ|ϕ^ ⋅∇B Ωe 2 B R
(3.61)

Using (3.35), the equation (3.61) becomes

( 2) [ ] vD ⋅∇ψ = - 1-- v2+ v⊥- |∇-ψ| I (ψ) ∂B-+ |∇-ψ|∂B- Ωe ∥ 2 B2R ∂s R ∂ ϕ
(3.62)

Under the assumption of axisymmetry, we are left with

( ) 1 2 v2⊥ |∇ψ |I (ψ) ∂B vD ⋅∇ ψ = - Ω-- v∥ +-2- ---B2R----∂s- e
(3.63)

With the definition (2.17) of the magnetic moment μ, we rewrite

1 |∇ ψ|I (ψ)( μB ) ∂B vD ⋅∇ ψ = - -------2----- v2∥ + --- --- Ωe B R me ∂s
(3.64)

We have, using the conservation of magnetic momentum (3.36),

( ) ( ) v2∥ + μB ∂B-= v2∥∂B- + B ∂-- μB- me ∂s ∂s ∂s me
(3.65)

Using the conservation of energy (3.37), we get

( μB ) ∂B ∂B ∂ ( v2) v2∥ + --- --- = v2∥--- - B --- -∥ (3.66) me ∂s ∂s ∂s 2 [ ∂v∥ ∂B ] = - v∥ B ----- v∥--- (3.67) ∂s( ) ∂s = - v B2-∂- v∥ (3.68) ∥ ∂s B
and finally, the equation (3.64) becomes
v (v ) vD ⋅∇ ψ = -∥I (ψ ) |∇-ψ|-∂ -∥ Ωe R ∂s B
(3.69)

In addition,

B ⋅∇ = I (ψ )∇ϕ ⋅∇ + ∇ ψ × ∇ ϕ ⋅∇ I (ψ ) ∂ |∇ ψ| ∂ = --------- ------- (3.70) R ∂ϕ R ∂s
and, using axisymmetry,
|∇ ψ| ∂ B ⋅∇ = - -R--∂s-
(3.71)

so that we can rewrite (3.69) as

v∥- ( v∥) vD ⋅∇ ψ = - ΩeI (ψ)B ⋅∇ B
(3.72)

expression which is the same as (3.58).

3.2.3 Case of Circular concentric flux-surfaces

In this case, ψ = ψ(r) and therefore

′ ∇ ψ = ψ (r) ^r
(3.73)

and

vDr = vD ⋅^r = vD-⋅∇-ψ- ψ′
(3.74)

which gives

v∥I-(ψ) ( v∥) vDr = - Ωe ψ′ B ⋅∇ B
(3.75)

In addition,

|∇ ψ| ∂ |ψ′|∂ B ⋅∇ = - --------= - ------- R ∂s R ∂s
(3.76)

and, because

^ ^ ^s⋅θ = ψ ⋅^r = σψ
(3.77)

we find

-∂- σψ-∂- ∂s = r ∂ θ
(3.78)

and

ψ′ ∂ B ⋅∇ = ---- --- Rr ∂θ
(3.79)

so that finally

v∥ I (ψ) ∂ ( v∥) v∥ I (ψ) ∂ ( v∥) vDr = Ω---Rr--∂θ- B- = -r RB---∂θ- Ω- e
(3.80)

When the toroidal field dominates, B ≃ I(ψ)∕R and

v ∂ (v ) vDr ≃ -∥--- -∥ r ∂θ Ω
(3.81)

3.2.4 Steady-State Drift-Kinetic Equation

Using expressions (3.31), (3.32) and (3.58) or (3.72), we obtain in steady-state

v (v ) ∂f v∥^b⋅∇f - -∥-I (ψ) B ⋅∇ -∥ --- = C (f )+ Q (f)+ E (f) Ωe B ∂ ψ
(3.82)

which can be rewritten as

∂f- v∥- |∇-ψ|-∂-(v∥) ∂f- vs∂s + ΩeI (ψ ) R ∂s B ∂ψ = C (f )+ Q (f)+ E (f)
(3.83)

with

(^ ) vs = v∥ b ⋅^s
(3.84)

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