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3.4 Low collision limit and bounce averaging

3.4.1 Fokker-Planck Equation

The Fokker-Planck Equation is

∂f0- vs∂s = C (f0) + Q (f0)+ E (f0)
(3.103)

In the low collision regime, which is characterized by the condition

* τb ν ≡ τ--≪ 1 dt
(3.104)

where τdt = ϵτc is the collision detrapping time, as defined in the previous section, it is assumed that electrons circulating or trapped are able to complete their orbit in a time too short for collisions to deflect them from their orbit. As a consequence, the dominant term in the Fokker-Planck equation is simply

vs∂f0-= 0 ∂s
(3.105)

so that f0 is constant along the field lines.

Then, performing a bounce-averaging, we have

{ ∂f0} vs-∂s- = 1 τ- b[ ] 1∑ 2- σT ∫ sminsmax ds |v-|- svs∂f0 -∂s-
= 1 -- τb[ ] 1∑ -- 2 σT σ[f0]sminsmax (3.106)

For passing particles, the positions smin and smax coincide, so that [f ] 0sminsmax = 0 and the term vanishes. For trapped particles, the term also vanishes because of the sum over σ = ±1, since, by definition, v∥ = 0 at the turning points smin and smax, and consequently f0 is independent of the sign of σ.

Therefore, the bounce-averaged Fokker-Planck equation becomes

{C (f0)} + {Q (f0)} + {E (f0)} = 0
(3.107)

with f0 constant along the field lines.

3.4.2 Drift-Kinetic Equation

The drift kinetic equation is

∂f1- v∥- |∇-ψ|-∂-(v∥ ) ∂f0 vs ∂s + ΩeI (ψ) R ∂s B ∂ψ = C(f1)+ Q (f1)+ E (f1)
(3.108)

In the low collisiona regime ν*≪ 1, the dominant term is

∂f1 v∥ |∇ ψ| ∂ (v∥) ∂f0 vs ---+ --I (ψ )-------- -- ----= 0 ∂s Ωe R ∂s B ∂ψ
(3.109)

which gives

f = f^+ g 1
(3.110)

where

∫ ^ -v∥-- -∂-( v∥) ∂f0- f = - dsvsΩeI (ψ)BP ∂s B ∂ψ
(3.111)

and g is a constant function along the field lines. Noting that

^ BP- b⋅^s = - B
(3.112)

we get

^ f = ∫ ds1-- ΩeI(ψ )B∂-- ∂s(v∥) B∂f0- ∂ ψ
= γmeI-(ψ-) qe∂f0- ∂ψ∫ ds-∂- ∂s( v ) -∥ B
= v∥- ΩeI(ψ)∂f0- ∂ψ (3.113)

where we used the fact that ∂f0∕∂s = 0.

Then, performing a bounce-averaging, we find again, using the same argument as in (3.106), that

{ ∂f } vs--1- = 0 ∂s
(3.114)

In addition, we have

{ ( ) } v∥I (ψ) |∇-ψ|-∂ v∥ ∂f0- Ωe R ∂s B ∂ψ
= 1- τb[ ∑ ] 1- 2 σT ∫ sminsmax ds-- |vs|v -∥- ΩeI(ψ)|∇-ψ-| R∂-- ∂s( v ) -∥ B∂f0- ∂ψ
= 1- τbγmeI-(ψ)- qe∂f0- ∂ψ[ ∑ ] 1- 2 σT σ ∫ sminsmax ds-∂- ∂s( ) v∥ B
= 1- τbγmeI-(ψ)- qe∂f0- ∂ψ[ ∑ ] 1- 2 σT σ[ ] v∥ Bsminsmax (3.115)

Again, for passing particles, the positions smin and smax coincide, so that [ ] v∥∕Bsminsmax = 0 and the term vanishes. For trapped particles, the term also vanishes because v∥→ 0 at the turning points smin and smax.

Consequently, we find that the bounce-averaged drift kinetic equation becomes

{C (f1)} + {Q (f1)} + {E (f1)} = 0
(3.116)

where .

^ f1 = f + g
(3.117)

v∥ ∂f0 ^f = ---I (ψ)---- Ωe ∂ψ
(3.118)

and g is then given by

{ ( )} { ( )} { ( )} {C(g)} + {Q (g)} + {E (g)} = - C f^ - Q ^f - E f^
(3.119)

using the fact that all operators are linear.

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