Small drift approximation

3.3 Small drift approximation

We recall the electron drift kinetic equation may be expressed as

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

(3.85)

Each of these terms corresponds to a time evolution, and is therefore associated with a time-scale:

Motion along magnetic field lines. The time scale here is the bounce time τ_b for trapped electrons, or the transit time τ_t for circulating ones, which can be deduced directly from expression (2.10). For circulating electrons
$2πRp-^q 2-πqRp τt = v |ξ0| ~ vTe$ (3.86)

taking λ ~ 1 in that case, and q ≃ that is valid circular plasma cross-sections, and v = v_∥≃ v_Te for thermal electrons. Since v_∥≤v_⊥ ≃v_Te for trapped electrons, as the consequence of the magnetic moment conservation, it turns out that τ_b may be deduced directly from τ_t
$τb = 2πRp^q-~ 2√πqRp- v |ξ0| ϵvTe$ (3.87)

Consequently, τ_b ≥ τ_t, since ϵ ≤ 1, a result which is the consequence of the progressive slowing down of the parallel velocity as far as the electron approaches the turning point.
Vertical drift across magnetic field lines. The time scale here is the drift time τ_d, which corresponds to the time for an electron to drift across the plasma. It is then given by
$∫ ψa Ωe R [ ∂ ( v∥)]- 1 τd = dψ ----------- --- -- 0 v∥I (ψ)|∇ ψ| ∂s B$ (3.88)

using expression 3.69 for the radial component v_D ⋅∇ψ of the drift velocity v_D as defined in 3.59. Using relation I as defined in (2.37), and the fact that dψ∕ = dr for circular concentric plasma cross-sections,
$Ωe ∫ ap dr [ ∂ ( 1) ]-1 τd ≃ -2-- B-- ∂s- B- vTe 0 T$ (3.89)

and since in that magnetic configuration ∂∕∂s ~ 2π∕R_p, ones obtains finally
$∫ ap τd ≃ 2π-Ωe2-Rp -B-dr vT e 0 BT$ (3.90)

or
$Ωe τd ≃ 2πv2--Rpap T e$ (3.91)

Consequently, the small drift parameter δ_d is defined as the ratio
$τt ρL ρL δd ≡ τ- ~ qa--~ a-- d p p$ (3.92)

where the thermal Larmor radius ρ_L = v_Te∕Ω_e has been introduced.
Collisions. The Fokker-Planck theory considers the cumulative effect of many small-angle collisions in calculating the rate of change of a particle distribution. From this theory, the characteristic time scale τ_c corresponds for Coulomb collisons to deflect an electron’s path by a significant angle, on the order of π∕2. This is the electron thermal collision time τ_c whose expression is
$-1 4πε0m2ev3Te τc = νc = q4n lnΛ e e$ (3.93)

where lnΛ is the well known Coulomb logarithm, a slowly varying function of the plasma temperature and density. The collision time scale τ_c holds for circulating electrons.
For trapped ones, it is more physical to consider an alternate collision time scale determined not by the time for deflection of the path by π∕2, but by the time needed for the electron to be deflected so that it is no longer on a trapped orbit. In the limit ϵ ≪ 1, we can approximate the change of the pitch angle necessary to make trapped particles become untrapped
$∘ ----- 2ϵ √ - Δ ξ0 ~ ξ0T ≃ 1+-ϵ-~ ϵ$ (3.94)

Because the small-angle collisions produce a random-walk change in the pitch-angle ξ₀, the effective collision time for detrapping τ_c^eff. or τ_dt is approximately deduced from relation
$ν ν νefcf.~ --c2~ -c Δ ξ0 ϵ$ (3.95)

and
$eff. τdt = τc ~ ϵτc$ (3.96)
Constant electric field acceleration. From the relation (3.30), it is straightforward to estimate that the time scale associated to constant electric field acceleration is
$τ ~ mevTe-~ τ |ED-| e |qe|E∥ c E ∥$ (3.97)

where the well known Dreicer field E_D
$E = ν mevT-e D c qe$ (3.98)

is introduced. Here only circulating particle are concerned.
Quasilinear diffusion. In a similar approach,
$(mevT e)2 D †ql τql ~ ---D-----~ τcD--- ql ql$ (3.99)

where D_ql^† = ν_c².