Ohmic electric field

4.2 Ohmic electric field

The generation of Ohmic current is based on the concept of transformer, where the plasma torus is the secondary circuit. An electric field is induced by the temporal variation ∂ψ∕∂t of the poloidal flux generated by the primary circuit. The induced current density is then calculated by Ohm’s law, J = σ_ΩE, where σ_Ω is the electric conductivity calculated by accounting for the Coulomb interaction between the strongly magnetized components of the plasma. Using Faraday’s law,

∂B- = - ∇ × E ∂t

We consider the surface S (ψ, θ) which is a truncated cone delimited by two circles being C₁, the magnetic axis, and C₂, the toroidal line at position (ψ,θ) . Applying the integral formula to Faraday’s law, we have

∫ ∫ ∫ ∫ ∮ dS ⋅ ∂B-= - dS ⋅∇ × E = - dl ⋅E S(ψ,θ) ∂t S(ψ,θ) C(ψ,θ)

(4.150)

The poloidal flux is given by

∫∫ ∫∫ -1- -1- ψ = 2π dSBP = 2π dS ⋅B S(ψ,θ) S(ψ,θ)

(4.151)

so that we get

∮ ∮ ∂ψ- 2π∂t = C dl⋅E - C (ψ,θ)dl⋅E (4.152) ∫ 21π 2∫ 2π = R dϕE - R (ψ,θ)dϕE (ψ, θ) (4.153) 0 p ϕ0 0 ϕ

where R_p is the major radius on axis and E_ϕp is the electric field on axis. Using axisymmetry, this becomes

∂ψ-= R E - R (ψ,θ)E (ψ,θ) ∂t p ϕp ϕ

(4.154)

and therefore

∂ψ- RE ϕ = RpE ϕp - ∂t

(4.155)

and RE_ϕ is only a function of ψ. We can therefore rewrite

R0 Eϕ (ψ,θ) = Eϕ0-R-

(4.156)

where R₀ is the major radius taken at the the poloidal position θ₀ where the magnetic field B is minimum on a flux-surface.By definition,

E ∥0(ψ) = E∥ (ψ, θ0)

(4.157)

The electric field along the field line can then be obtained by projection, which gives

(^ ^) BT- E ∥ = b ⋅ϕ Eϕ = B E ϕ

(4.158)

so that we get

B B0 R0 B--E ∥ = B---E∥0-R- T T 0

(4.159)

and then

2 E ∥ = BT--B0-R0-E∥0(ψ) = 1-R-0E ∥0 (ψ ) BT0 B R Ψ R2

(4.160)

with Ψ = B∕B₀ as defined in Sec. 2.2.1.

4.2.1 Conservative Form for the Ohmic Electric Field Operator

The effect of the electric fied E_∥ can be expressed in a conservative form as (f₀) = ∇_p ⋅ S^E, where the flux in momentum space is easily expressed in cylindrical coordinates ( )
p ,p⊥
∥ as

( ) E SEp∥ S (p∥,p⊥) = SE p⊥

(4.161)

with

E S p∥ = qeE∥f0 (4.162) SEp = 0 (4.163) ⊥

where q_e is the electronic charge.

The transformation from cylindrical to spherical coordinates is given by

E - 1 E S (p,ξ) = R S(p∥,p⊥)

(4.164)

where is the rotational matrix

( ∘ ------) ξ - 1- ξ2 R = ∘ -----2 1 - ξ ξ

Using ^-1 = ^t we find

SE = ξSE (4.165) p ∘p∥------ SEξ = - 1 - ξ2SEp (4.166) ∥

and S^E countains a convective part only

E E S = Fpf0

(4.167)

with

FE = q ξE p e ∘∥------ FEξ = - qe 1 - ξ2E∥

4.2.2 Bounce Averaged Fokker-Planck Equation

In the Fokker-Planck equation, the diffusion and convection elements are bounce-averaged according to the expressions (3.189)-(3.194), which gives, using (4.167),

DEp(0) = 0

(4.168)

and the convection components

{ } F Ep(0) = qeξE∥ (4.169) { σξ ∘ ------ } F Eξ(0) = - σ √----qe 1 - ξ2E∥ (4.170) Ψξ0

Since the poloidal dependence of the electric field on a flux-surface is given by ()

1 R2 E∥(ψ, θ) =------- -02E∥0(ψ) Ψ (ψ, θ)R

(4.171)

we find

-- F Ep(0) = λ1,--1,2qeξ0E∥0(ψ) (4.172) -λ ∘ ------ E(0) λ1,-1,2 2 F ξ = - λ qe 1 - ξ0E ∥0 (ψ ) (4.173)

where we defined the bounce averaged coefficient (2.66)

{ 2} λ- = λσ σ ξ----1---R-0 1,-1,2 ξ0Ψ (ψ,θ)R2

(4.174)

Expression of λ_1,-1,2

The coefficient λ_1,-1,2, which is known as s^* in the old notation found in the litterature ([?]), is expressed as

[ ] -- σ 1∑ ∫ θmaxdθ 1 r B 1 R2 λ1,-1,2 = -- -- ---||----||------σ------- -02- q^ 2 σ T θmin 2π |^ψ ⋅^r|Rp BP Ψ (ψ, θ)R

(4.175)

Since the integral is odd in σ, the sum over trapped particles vanishes, and we have

-- || λ1,-1,2 = || 0P for trapped particles λ1,- 1,2 for passing particles

(4.176)

P 1∫ 2π dθ 1 r B 1 R20 λ1,-1,2 = ^q- 2π-||^--||R--B-- Ψ-(ψ,-θ)R2- 0 |ψ ⋅^r| p P 1R ∫ 2π dθ 1 r B R = ---0 ---|---|----0 -0- ^qRp 0 2π ||ψ^⋅^r||R BP R ∫ 2π = 1R0--B0- dθ-|-1-|-rBT-0 R0- ^qRp BT 0 0 2π ||ψ^⋅^r||R BP R ∫ 1R0--B0- 2π dθ---1---r-BT = ^qRp BT 0 0 2π ||^ ||R BP |ψ ⋅^r| qR0 B0 = ^qR--B--- (4.177) p T0

Case of circular concentric flux-surfaces

Since in the case of circular concentric flux-surfaces, we have

^q = r--B-- (4.178) Rp BP r B ∫ 2π dθR q = ----T- ----p (4.179) Rp BP 0 2π R

so that

P ∫ 2π dθR0 ∫ 2π dθ 1+ ϵ λ 1,-1,2 = 2π-R- = 2π1-+-ϵcosθ 0 0

(4.180)

This integral can then be performed analytically, as shown in Sec. ??, at formula (??), which gives

∘ ----- P 1+-ϵ- λ1,-1,2 = 1- ϵ

(4.181)

4.2.3 Bounce Averaged Drift Kinetic Equation

In the first order Drift-Kinetic equation, the diffusion and convection flux elements related to ^
f are bounce-averaged according to the expressions (3.218)-(3.223), which gives, using (4.167),