Moments of the distribution function

3.6 Moments of the distribution function

3.6.1 Flux-surface Averaging

Surface densities

We consider the flux-surface averaging of a surface quantity, such as a flux of a current, generally noted Γ (ψ,θ) . It is defined as the averaged flux of Γ through the infinitesimal poloidal surface dS (ψ)

∫ dS(ψ)dS ⋅Γ (ψ,θ) ⟨Γ ⟩S (ψ) =----∫----------- dS(ψ)dS

(3.227)

In the (ψ,θ,ϕ) system, the differential poloidal surface element is given by (A.201) as introduced in Appendix A

-----r----- ^ dS = ||^ ||d ψdθϕ |∇ ψ||ψ ⋅^r|

(3.228)

so that the infinitesimal poloidal surface element dS_p (ψ) is

∫ ∫ ∫ 2π dS = -----r|---|dψd θ = dψ dθ-----r|----| dSp(ψ) dSp(ψ)|∇ψ |||^ψ ⋅^r|| 0 |∇ ψ|||^ψ ⋅^r||

(3.229)

and the flux-surface averaged flux in the toroidal direction is

( ) -1∫ 2π [ ] ⟨Γ ⟩ (ψ) = dSp- dθ----r|---|- ^ϕ⋅Γ (ψ,θ) ϕ dψ 0 |∇ψ |||ψ^⋅^r||

(3.230)

with

	= ∫ ₀^2πdθ	(3.231)
	= ∫ ₀^2πdθ	(3.232)

Defining the new pseudo saftey factor q as

∫ -- 2π dθ-|-1-|-r-B0(ψ-) q (ψ ) ≡ 0 2π |ψ^⋅^r| R BP | |

(3.233)

we get

-- dSp-(ψ) = 2πq-(ψ) dψ B0 (ψ )

(3.234)

and

∫ [ ] ⟨Γ ⟩ (ψ) = --1-- 2π dθ|--1-|r-B0-(ψ) ^ϕ⋅Γ (ψ,θ) ϕ q(ψ ) 0 2π||^ψ ⋅^r||R BP

(3.235)

Volume densities

We consider the flux-surface averaging of a volume quantity, such as a power density, generally noted Φ (ψ,θ) . It is defined as the average value of Φ within the infinitesimal volume dV (ψ )

∫∫ --dV∫(ψ∫)Φ-(ψ,-θ)dV- ⟨Φ ⟩V (ψ ) = dV dV(ψ)

(3.236)

In the (ψ,θ,ϕ) system, the differential volume element is given by (A.202) as introduced in Appendix A

---Rr------ dV = || ^ ||dψd θdϕ |∇ ψ ||ψ ⋅^r|

(3.237)

so that the infinitesimal volume element dV (ψ) of a flux-surface is

∫∫ ∫∫ ∫ 2π ∫ 2π dV = ----Rr|----|dψdθdϕ = d ψ dθ d ϕ----Rr|---|- dV(ψ) dV(ψ)|∇ ψ|||^ψ ⋅^r|| 0 0 |∇ψ |||ψ^⋅^r||

(3.238)

and the flux-surface averaged quantity in the toroidal direction is

( )- 1∫ 2π ∫ 2π ⟨Φ ⟩ (ψ) = dV- dθ dϕ ---Rr|----|Φ (ψ,θ) V dψ 0 0 |∇ ψ|||^ψ ⋅^r||

(3.239)

with

	= ∫ ₀^2πdθ∫ ₀^2πdϕ	(3.240)
	= ∫ ₀^2πdθ∫ ₀^2πdϕ	(3.241)

Under the assumption of axisymmetry, we get

	= 4π² ∫ ₀^2π	(3.242)
	= ∫ ₀^2π	(3.243)

Defining the new pseudo saftey factor as

∫ 2π dθ 1 r B0 (ψ) ^q(ψ) ≡ 2π||----||R----B--- 0 |^ψ ⋅^r| p P

(3.244)

we get

dV-(ψ)- 4π2Rp-^q(ψ)- dψ = B0 (ψ)

(3.245)

and finally

1 ∫ 2π dθ 1 r B (ψ) ⟨Φ⟩V (ψ) = ----- ---||----||-----0---Φ (ψ,θ) ^q(ψ ) 0 2 π|^ψ ⋅^r|Rp BP

(3.246)

3.6.2 Density

Definition

The electron density n_e (ψ,θ) is given by the relation

∫ ∫ +1 ∞ 2 ne (ψ, θ) = 2π -1 dξ 0 p dpf (p,ξ,ψ,θ)

(3.247)

Using the general expression (3.246) of the flux-surface averaging of a volumic quantity

_V	= ∫ ₀^2πn_e
	= ∫ ₀^∞p²dp∫ ₀^2π ∫ _-1⁺¹dξf
	= ∫ ₀^∞p²dp∫ ₀^2π ∫ _-1⁺¹_Tdξf
		(3.248)

where the trapping condition evaluated at the location θ is given by

∘ ------------ B-(ψ,θ)- |ξ| < ξT = 1- B0 (ψ)

(3.249)

Using ξdξ = Ψξ₀dξ₀ with the condition (3.270) on ξ₀

∘ ------------ |ξ0| ≥ 1- ---1--- Ψ (ψ, θ)

(3.250)

one get

∫ [ ] ∫ [ ] ( ∘ -----------) +1 1-∑ +1 1- ∑ ξ0 ---1--- 2 dξ = 2 Ψ (ψ,θ) ξ H |ξ0|- 1 - Ψ (ψ,θ) dξ0 -1 σ= ±1 T -1 σ=±1 T

(3.251)

where H is the usual Heaviside function which is defined as H (x) = 1 for x > 0, and H (x) = 0 elsewhere.

Note that the condition (3.250) is equivalent to

θmin(ψ,ξ0) ≤ θ ≤ θmax(ψ,ξ0)

(3.252)

so that, the integrals over θ and ξ₀ may be permuted,

_V	= ∫ ₀^∞p²dp∫ _-1⁺¹dξ₀×
	_T ∫ _{θ_min}^θ_maxf	(3.253)

where the bounce-averaging of the distribution appears naturally. Therefore, expression (3.371) can be rewriten in the simple form

∫ ∞ ∫ +1 ⟨ne⟩V (ψ) = 2π^q- p2dp dξ0λ{f (ψ,θ,p,ξ0)} ^q 0 - 1

(3.254)

Fokker-Planck Equation

For the zero order distribution function, since f₀ is constant along a field line,

(0) f0(ψ,θ,p,ξ) = f0 (ψ, p,ξ0)

(3.255)

one obtains

^q ∫ ∞ ∫ +1 (0) ⟨ne⟩0V (ψ) = 2π-- p2dp dξ0λf0 (ψ,p,ξ0) ^q 0 -1

(3.256)

Drift Kinetic Equation

When we consider the first order distribution function, we have f₁ = + g, where g is constant along a field line, and therefore its contribution ⟨ne⟩ _V¹ (ψ) has the same expression as for f₀. However, has an explicit dependence upon θ, which is given by (3.206)

^f(ψ,θ,p,ξ) = ξ(ψ,θ,ξ0)^f(0)(ψ, p,ξ0) Ψ (ψ,θ)ξ0

(3.257)

Therefore, the flux-surface averaged density contribution of ^
f is

_V¹	= 2π∫ ₀^∞p²dp∫ _-1⁺¹dξ₀λ ⁽⁰⁾(ψ,p,ξ₀)	(3.258)
	= 2π∫ ₀^∞p²dp∫ _-1⁺¹dξ₀λ_1,-1,0 ⁽⁰⁾(ψ,p,ξ₀)	(3.259)

where

{ } λ- = σ σ----ξ---- λ 1,- 1,0 Ψ (ψ,θ)ξ0

(3.260)

according to the notation in Sec. 2.2.1, since ^
f ⁽⁰⁾ is antisymmetric in the trapped region.

Since ⁽⁰⁾ and g have no definite symmetry properties, both can contribute to the density and

0 1 1 ⟨ne⟩V (ψ) = ⟨ne⟩V (ψ )+ ⟨ne⟩V (ψ )+ ⟨^ne⟩V (ψ )

(3.261)

3.6.3 Current Density

Definition

The density of current carried by electrons is given by

∫∫ ∫ J(x) = q d3pvf (x,p ) e

(3.262)

so that the parallel current density is

∫ ∫∫ 3 J∥(x) = qe d pv∥f (x,p)

(3.263)

which becomes in (ψ,θ,p,ξ) phase space

∫ ∫ J (ψ, θ) = 2πq ∞ p2dp 1 dξ-pξf (ψ,θ,p,ξ) ∥ e 0 - 1 γm

(3.264)

Flux-Surface Averaging

We are usually interested in the flux-surface averaged current density in the toroidal direction. It is generally given by (3.230)

_ϕ	= ∫ ₀^2πJ_∥
	= ∫ ₀^2πJ_∥	(3.265)

and finally, using (2.23)

⟨ ⟩ 1 ∫ 2π dθ 1 r BT J ∥(ψ,θ) J∥ ϕ(ψ) = q- 2-π||----||R-B---Ψ-(ψ,-θ) 0 |^ψ ⋅^r| P

(3.266)

Fokker-Planck Equation

When we consider only the zero order distribution function, we have that f₀ is constant along a field line, so that

f0(ψ,θ,p,ξ) = f(0)(ψ, p,ξ0) 0

(3.267)

where

∘ ------------------- 1 ξ0 = σ 1 - Ψ-(ψ,θ) (1 - ξ2)

(3.268)

Consequently, we find

J_∥⁰	= 2πq_e ∫ ₀^∞p²dp∫ _-1¹dξf₀
	= 2πq_e ∫ ₀^∞p²dp∫ _-1¹dξf₀⁽⁰⁾
	= 2πq_e ∫ ₀^∞p²dp∫ _-1¹dξ₀Ψ×
	Hf₀⁽⁰⁾	(3.269)

where the condition

∘ ------------ 1 |ξ0| ≥ 1- ------- Ψ (ψ, θ)

(3.270)

results from the equation (3.268) and means that only the particle who reach the position θ must be considered. Note that the integrand in the equation (3.269) is odd in ξ₀ for trapped electrons, since f₀⁽⁰⁾ is symmetric in the trapped region. As a consequence, the contribution from trapped electrons vanishes, and (3.269) can be rewritten as

∫ ∞ ∫ 1 J0(ψ,θ) = 2πqe p2dp dξ0Ψ(ψ, θ)H (|ξ0|- ξ0T) pξ0-f(0)(ψ, p,ξ0) ∥ 0 -1 γme 0

(3.271)

Therefore, the flux-surface averaged current density

⟨ ⟩0 1 ∫ 2πd θ 1 r BT J 0∥ (ψ,θ) J∥ ϕ(ψ) = -- ---||----||------------- q 0 2π |^ψ ⋅^r|R BP Ψ (ψ, θ)

(3.272)

becomes

_ϕ⁰	= ∫ ₀^∞dp∫ ₀^2π×
	∫ _-1¹dξ₀ΨHξ₀f₀⁽⁰⁾	(3.273)

The integrals over θ and ξ₀ can be permuted

_ϕ⁰	= ∫ ₀^∞dp∫ _-1¹dξ₀Hξ₀f₀⁽⁰⁾×
	∫ ₀^2π	(3.274)

We recognize the expression of the safety factor (2.51) so that

⟨ ⟩0 2πqe q∫ ∞ p3∫ 1 (0) J∥ ϕ(ψ) = ------- dp -- dξ0H (|ξ0|- ξ0T) ξ0f0 (ψ,p,ξ0) me q 0 γ - 1

(3.275)

Case of circular concentric flux-surfaces In that case, we showed in (2.83) that the safety factor is

ϵ BT q(r) = √-----2B-- 1 - ϵ P

(3.276)

with ϵ = r∕R_p the inverse aspect ratio.

In addition, q (r) becomes

q	= ∫ ₀^2π
	= ∫ ₀^2π
	= ϵ
	=	(3.277)

since R = R_p + r cosθ, and B₀∕B = R∕R₀. We have then

q(r) ∘ 1-+-ϵBT ---- = -------- q(r) 1 - ϵ B

(3.278)

In the case when B_T ≫ B_P, we retrieve the bounce-averaged coefficient s^* and in the large aspect ratio limit ϵ ≪ 1,

q(r) BT lϵi→m0 q(r) = (1+ ϵ) B--

(3.279)

Drift Kinetic Equation

When we consider the first order distribution function, we have f₁ = + g, where g is constant along a field line, and therefore its contribution has the same expression as for f₀. However, has an explicit dependence upon θ, which is given by (3.206)

^f(ψ,θ,p,ξ) = ξ(ψ,θ,ξ0)^f(0)(ψ, p,ξ0) Ψ (ψ,θ)ξ0

(3.280)

where

∘ ------------------- ---1--- 2 ξ0 = σ 1 - Ψ (ψ,θ) (1 - ξ )

(3.281)

Consequently, we find

_∥¹	= 2πq_e ∫ ₀^∞p²dp∫ _-1¹dξ
	= 2πq_e ∫ ₀^∞p²dp∫ _-1¹dξ ⁽⁰⁾(ψ,p,ξ₀)
	= 2πq_e ∫ ₀^∞p²dp∫ _-1¹dξ₀×
	H ⁽⁰⁾	(3.282)

where again the condition

∘ ------------ |ξ0| ≥ 1- ---1--- Ψ (ψ, θ)

(3.283)

results from the equation (3.268) and means that only the particle who reach the poloidal position θ must be considered.

Therefore, the flux-surface averaged current density contribution from ^
f

⟨ ⟩1 ∫ 2π ^J1(ψ,θ) ^J∥ (ψ ) = 1- dθ|--1-|-rBT- -∥------ ϕ q 0 2π||ψ^⋅^r||R BP Ψ (ψ,θ)

(3.284)

becomes

_ϕ¹	= ∫ ₀^∞dp∫ ₀^2π×
	∫ _-1¹dξ₀Hξ₀ ⁽⁰⁾	(3.285)

Note that the condition (3.283) is equivalent to

θmin(ψ,ξ0) ≤ θ ≤ θmax(ψ,ξ0)

(3.286)

so that, permuting the integrals over θ and ξ₀, we find

_ϕ¹	= ∫ ₀^∞dp∫ _-1¹dξ₀ξ₀ ⁽⁰⁾
	∫ _{θ_min}^θ_max	(3.287)

We have then

2 r-BT----1--- = RpI2-(ψ-)-r--B-R-0Ψ -2(ψ,θ) R BP Ψ (ψ, θ) R0B0 Rp BP R2

(3.288)

Then, noting the the integrand in (3.287) is independent of σ , so that the sum over σ for trapped particles can be added, we obtain

_ϕ¹	= ∫ ₀^∞dp∫ _-1¹dξ₀ξ₀ ⁽⁰⁾×
	_T ∫ _{θ_min}^θ_max²Ψ^-2²	(3.289)

We recognize the expression of a bounce coefficients defined by the general relation (2.66) in Sec. 2.2.1, so that we get finally

⟨ ⟩1 2 πqe^qRp BT 0 ∫ ∞ p3 ∫ 1 (0) ^J∥ ϕ(ψ ) =-m---qR---B-- dp-γ dξ0λ2,-2,2ξ0^f (ψ,p,ξ0) e 0 0 0 -1

(3.290)

with

{ ( )2 ( )2} λ2,- 2,2 = λ ξ- Ψ -2 R0- ξ0 R

(3.291)

Case of circular concentric flux-surfaces In that case, we showed in (2.99) that is

B ^q(r) = ϵ--- BP

(3.292)

with ϵ = r∕R_p the inverse aspect ratio.

In addition, q (r) is

q(r) = --ϵ--B-- 1+ ϵBP

(3.293)

and since

R0 = Rp (1 + ϵ)

(3.294)

we have then

^q-RpBT-0 BT-0 q-R0 B0 = B0 ≃ 1

(3.295)

in the limit B_P ≪ B.

Also, in this case,

R Ψ (ψ,θ) = --0 R

(3.296)

so that

{ ξ2} * λ2,-2,2 = λ2,0,0 = λ ξ2 = ^s 0

(3.297)

using notations used in previous publications. The exact expression of ^* in terms of a series expansion is given in relation (4.148).

3.6.4 Power Density Associated with a Flux

Definition

The kinetic energy associated with a relativistic electron of momentum p is

E = m c2(γ - 1) c e

(3.298)

Then, the local energy density of electrons is

∫ 3 2 ε (x ) = d p mec (γ - 1)f (x, p)

(3.299)

The density of power absorbed through the process , P_abs, is

|| ∫ || P Oabs(x) = ∂-ε|| = d3p mec2 (γ - 1) ∂f(x,-p-)|| ∂t O ∂t O

(3.300)

When the operator is described in conservative form, as the divergence of a flux

| ∂f|| O 1--∂-( 2 O) 1-∂-(∘ -----2 O ) ∂t|O = - ∇ p ⋅Sp = - p2∂p p Sp + p∂ξ 1 - ξ Sξ

(3.301)

then the power density becomes

∫ ∞ ∫ +1 [ 1 ∂ ( ) 1 ∂ ( ∘ ------ )] P Oabs = - 2πmec2 p2dp(γ - 1) dξ -2 --- p2SOp - ----- 1- ξ2SOξ 0 -1 p ∂p p ∂ξ

(3.302)

The integration of the S_ξ term gives no contribution, since the particle energy is function of p only

∫ +1 ( ) [ ] dξ ∂-- ∘1---ξ2SO = ∘1----ξ2SO +1 = 0 -1 ∂ξ ξ ξ - 1

(3.303)

and the equation (3.302) reduces to

O 2∫ +1 ∫ ∞ ∂ ( 2 O ) Pabs = - 2πmec dξ (γ - 1)∂p- p Sp dp -1 0

(3.304)

Integrating by parts, we get

O 2∫ +1 ( [ 2 O]∞ ∫ ∞ dγ 2 O ) Pabs = - 2πmec dξ (γ - 1) p Sp 0 - dpp S p dp -1 0

(3.305)

Assuming that lim_p→∞p²S_p = 0, and using

dγ-= ---p-- dp γm2ec2

(3.306)

the equation (3.305) reduces to

∫ ∫ P O (ψ,θ) = 2π +1dξ ∞ dp-p3-SO abs -1 0 γme p

(3.307)

Flux-Surface Averaging

Starting from the general expression of the flux-surface averaging of a volume quantity (3.246), the flux-surface averaged power density ⟨ ⟩
P Oabs _V (ψ) is

⟨ ⟩ 1 ∫ 2π dθ 1 r B POabsV (ψ) = -- ---||----||-----0PaObs(ψ,θ) ^q 0 2π |^ψ ⋅^r|Rp BP

(3.308)

which becomes

⟨ O ⟩ ∫ ∞ p3 1∫ 2π dθ 1 r B0 ∫ +1 O PabsV (ψ) = 2π dpγm-- ^q- 2π||----||R--B-- dξS p 0 e 0 |ψ^⋅^r| p P -1

(3.309)

The sum over σ for trapped electrons can be added, using

∫ _-1¹dξ_TS_p	= ∫ _-1^-ξ_TdξS_p + ∫ _{ξ_T}¹dξS_p + ∫ _{-ξ_T}^ξ_Tdξ ∑ _σ=±1S_p
	= ∫ _-1^-ξ_TdξS_p + ∫ _{ξ_T}¹dξS_p + ∫ _{-ξ_T}^ξ_Tdξ
	= ∫ _-1^-ξ_TdξS_p + ∫ _{ξ_T}¹dξS_p + ∫ _{-ξ_T}^ξ_TdξS_p(ξ)
	= ∫ _-1¹dξS_p	(3.310)

where the trapping condition evaluated at the poloidal location θ is

∘ ------------- B (ψ,θ) |ξ| < ξT = 1 - B----(ψ-) max

(3.311)

Using ξdξ = Ψξ₀dξ₀ with the condition (3.270) on ξ₀

∘ ------------ |ξ0| ≥ 1- ---1--- Ψ (ψ, θ)

(3.312)

we get that

( ∘ -----------) ∫ +1 ∫ +1 Ψ (ψ,θ) ξ0 1 dξ = dξ0----ξ-----H |ξ0|- 1 - Ψ-(ψ,θ) - 1 -1

(3.313)

Note that the condition (3.312) is equivalent to

θmin(ψ,ξ0) ≤ θ ≤ θmax(ψ,ξ0)

(3.314)

so that, permuting the integrals over θ and ξ₀, we find

_V	= 2π ∫ ₀^∞dp ∫ _-1⁺¹dξ₀	(3.315)
	_T ∫ _{θ_min}^θ_maxS_p	(3.316)

We see that the bounce-averaging of the fluxes appears naturally, so that we can rewrite

∫ ∞ 3 ∫ +1 ⟨P O ⟩ (ψ ) = 2π ^q dp-p-- dξ0λ {SO } abs V ^q 0 γme -1 p

(3.317)

Using the definition (3.167), we observe that the flux-surface averaged power density is calculated using the momentum flux component of the bounce-averaged kinetic equation:

⟨ ⟩ ^q ∫ ∞ p3 ∫ +1 P Oabs V (ψ) = 2π-- dp---- d ξ0λS (p0)O ^q 0 γme -1

(3.318)

Case of circular concentric flux-surfaces In that case, we showed in (3.292) that the coefficient is

^q(r) = ϵ B- BP

(3.319)

with ϵ = r∕R_p.

In addition, (r) becomes

	= ∫ ₀^2π
	= ϵ ∫ ₀^2π
	= ϵ ∫ ₀^2π
	=	(3.320)

using the simple relation B∕B₀ = R₀∕R and R₀ = R_p (1+ ϵ) .

We have then

^q(ψ) -----= 1 + ϵ ^q(ψ)

(3.321)

Fokker-Planck Equation

The Fokker-Planck equation (3.107) solves for the zero-order distribution function f₀. The density of power transfered to f₀ through the momentum-space mechanism is then

∫ ∞ 3 ∫ +1 ⟨PO ⟩0 (ψ) = 2π^q- dp-p-- dξ0λS (0)O (f0) abs V ^q 0 γme -1 p

(3.322)

where S_p (f )
0 is given by (3.187)

∂f (0) ∘1----ξ2 ∂f (0) S(p0)O (f0) = - D (0pp)O--0- + -------0D (0pξ)O---0- + F(p0)O f(00) ∂p p ∂ξ0

(3.323)

The momentum-space diffusion and convection elements D_pp⁽⁰⁾, D_pξ⁽⁰⁾ and F_p⁽⁰⁾ associated with a particular mechanism are calculated in chapter 4.

Drift Kinetic Equation

The Fokker-Planck equation (6.1) solves for the first-order distribution function f₁ = + g (3.117). The densities of power transfered to and g through the momentum-space mechanism are then respectively

_V¹	= 2π∫ ₀^∞dp ∫ _-1⁺¹dξ₀λ _p	(3.324)
_V¹	= 2π∫ ₀^∞dp ∫ _-1⁺¹dξ₀λS_p	(3.325)

where _p ( )
^f and S_p (g) are given by (3.187) and (3.216)

_p⁽⁰⁾	= -_pp⁽⁰⁾ + _pξ⁽⁰⁾ + _p⁽⁰⁾ ⁽⁰⁾	(3.326)
S_p⁽⁰⁾	= -D_pp⁽⁰⁾ + D_pξ⁽⁰⁾ + F_p⁽⁰⁾g⁽⁰⁾	(3.327)

The momentum-space diffusion and convection elements D_pp⁽⁰⁾, D_pξ⁽⁰⁾ , F_p⁽⁰⁾, _pp⁽⁰⁾, _pξ⁽⁰⁾ and _p⁽⁰⁾ associated with a particular mechanism are calculated in chapter 4.

3.6.5 Stream Function for Momentum Space fluxes

When transport in configuration space is ignored, and a steady-state regime is assumed to be reached, the Fokker-Planck equation reduces to the conservative equation (3.146)

∇p ⋅ Sp = 0

(3.328)

Because S_p is a divergence-free field vector, it can be expressed as the curl of a stream function

Sp = ∇ × Tp

(3.329)

The expression of a curl in momentum space (p,ξ,φ ) is given by relation (A.279) in Appendix A

S_p	= +	(3.330)
S_ξ	= -	(3.331)
S_φ	= - -	(3.332)

Because S_φ = 0, we can choose T_ξ = T_p = 0, which leads to

S_p	=	(3.333)
S_ξ	=	(3.334)

and we can rewrite

Sp = ∇ × T φ^φ

(3.335)

In order to give a physical meaning to T_φ (p,ξ,ψ) , we define formally

Tφ (ψ,p,ξ) = K (ψ,p,ξ)A (ψ,p,ξ)

(3.336)

where the function A (p,ξ) is such that the flux of electrons between two contours A₁ and A₂ is equal to n_e (ψ ) (A2 - A1) . Lets consider a path γ₁₂ between the contours A₁ and A₂. The total flux of electrons through this path, which is in fact a surface, given the rotational symmetry in φ, is given by

Γ₁₂	= _S₁₂dSS_p ⋅
	= _S₁₂dS ⋅∇× T_φ
	= ∮ _₁₂T_φdl ⋅	(3.337)

By rotational symmetry in φ, and using (A.272), we get

∘ ------ ∘ ------ Γ 12 = 2πp2 1 - ξ22Tφ2 - 2πp1 1 - ξ21Tφ1

(3.338)

If we define

ne(ψ) K (ψ,p,ξ) ≡ ---∘------2 2πp 1 - ξ

(3.339)

we obtain

Γ 12 = ne(ψ )(A2 - A1)

(3.340)

and therefore the total flux between the contours A₁ and A₂ is equal to n_e (ψ ) (A2 - A1 ) . We call A (ψ,p,ξ) the stream function, and we get finally

S_p	=	(3.341)
S_ξ	=	(3.342)

Since there are no fluxes across the internal boundaries in the momentum space, this boundary coincide with a contour A, and therefore we can arbitrarily set this value to 0:

A (0,ξ) = A (p,±1) = 0

(3.343)

Then A can be calculated by any of the integrals

2 ∫ ξ 2 ∫ ξ A (ψ,p,ξ) = 2πp--- dξSp = 2-πp-- dξ′Sp ne(ψ) -1 ne(ψ ) 1

(3.344)

∘ ------∫ 2π---1--ξ2 p ′ ′ A (ψ, p,ξ) = ne(ψ) 0 p dpS ξ

(3.345)

However, A (ψ, p,ξ) remains a function of ξ, which depends upon θ. Starting from the bounce-averaged fluxes, it is interesting to compute a function A (ψ, p,ξ0) , such that

A	= A = 0
S_p	=	(3.346)
S_ξ	=

We first need to demonstrate the existence of such a function. Starting from S_p,

A	= ∫ _-1^ξ₀dξ₀^′
	= ∫ _-1^ξ₀dξ₀^′ _T ∫ _{θ_min}^θ_maxS_p
	= ∫ _-1^ξ₀dξ₀^′ _T ∫ ₀^2πH
	= _T ∫ ₀^2π
	∫ _-1^ξ₀dξ₀^′Hσ
	= _T ∫ ₀^2π∫ _-1^ξdξ^′
	= _T ∫ ₀^2πσA
	= σ _T_V	(3.347)

where we used

∘ ------ θ ≤ θ ≤ θ ⇔ B ≤ B ⇔ 1- 1-≤ |ξ | min max b Ψ 0

(3.348)

Now, starting from S_ξ, we have

A	= ∫ ₀^pp^′dp^′σ
	= ∫ ₀^pp^′dp^′σ _T ∫ _{θ_min}^θ_maxS_ξ
	= ∫ ₀^pdp^′σ _T ∫ _{θ_min}^θ_max
	= _T ∫ _{θ_min}^θ_max∫ ₀^pdp^′
	= _T ∫ _{θ_min}^θ_maxσA
	= σ _T_V	(3.349)

and we find the same function A. The existence of a function A verifying (3.346) is therefore demonstrated. We need now to demonstrate that A verifying (3.346) leads to the bounce-averaged Fokker-Planck equation (3.166):

	= -
	= -
	= -
	= 0	(3.350)

In conclusion, a stream function verifying

-- A (0)(0,ξ0) = A (p,±1 ) = 0

(3.351)

has been found which leads to the bounce-averaged Fokker-Planck equation and which can be calculated from the bounce-averaged fluxes by either

∫ ∫ A (0)(ψ,p,ξ ) = 2πp2-- ξ0dξ′S(0)= -2πp2- ξ0 dξ′S (0) 0 ne(ψ ) -1 0 p ne (ψ) 1 0 p

(3.352)

2π∘1----ξ2 ∫ p (0) A (0)(ψ,p,ξ0) = ---------0 p′dp ′Sξ ne (ψ) 0

(3.353)

relations.

3.6.6 Ohmic electric field

The electrical conductivity of the plasma σ_e is defined as the ratio of the flux averaged current density ⟨ ⟩
J∥ _ϕ⁰ to the flux surface averaged parallel Ohmic electric field ⟨ ⟩
E∥ _ϕ,

⟨ ⟩0 -J∥-ϕ- σe = ⟨E ⟩ ∥ ϕ

(3.354)

By definition,

_ϕ	= ∫ ₀^2πE_∥
	= ∫ ₀^2πE_∥
	= ∫ ₀^2π	(3.355)

Using

1 R20 E∥(ψ, θ) = Ψ-(ψ,-θ)R2-E∥0(ψ)

(3.356)

where E_∥0 (ψ) is the value at the minimum magnetic field B₀, one obtains

_ϕ	= ∫ ₀^2π
	= E_∥0∫ ₀^2π	(3.357)

_ϕ	= E_∥0 ∫ ₀^2π
	= E_∥0 ∫ ₀^2π×

	= E_∥0 ∫ ₀^2π
	= E_∥0λσ
	= E_∥0λ_1,-3,4	(3.358)

Case of circular concentric flux-surfaces In that case,

^q(r) = 1+ ϵ q(r)

(3.359)

and since R_p∕R₀ = 1∕ (1 + ϵ) ,

_ϕ	= λ_1,-1,2E_∥0
	= λ_1,1,0E_∥0	(3.360)

using relation Ψ (r,θ) = R∕R₀. Therefore,

----- ⟨ ⟩ BT ∘ 1 + ϵ E∥ ϕ(r) = --- -----E ∥0 (r) B 1 - ϵ

(3.361)

as λ_1,-1,2 = ∘ ----
11+-ϵϵ for circular concentric flux-surfaces. Moreover, in this limit,

⟨ ⟩(0) (0) -J-∥ϕ-- J∥-- σe = ⟨E ⟩ = E ∥ ϕ ∥0

(3.362)

since

∘ ----- ⟨J ∥⟩(0)(ψ,θ) = BT- 1-+-ϵJ(0) ϕ B 1 - ϵ ∥

(3.363)

with

(0) 2πqe ∫ ∞ 2 ∫ 1 pξ0 (0) J ∥ = -m--- p dp dξ0H (|ξ0|- ξ0T) γm-f0 (ψ,p,ξ0) e 0 - 1

(3.364)

In that case, the neo-classical conductivity can be either calculated from flux surface averaged quantity, or local values at B = B₀.

3.6.7 Fraction of trapped electrons

The ratio between the number of trapped and passing electrons is an important quantity in the neoclassical transport theory, since the parallel viscosity responsible for reduction of the Ohmic conductivity and the bootstrap current level are both roughly proportional to this parameter. Therefore, under the influence of RF waves, its large variation will indicate unambiguously that significant macroscopic changes are to be expected on the current generation and the power absorption due to neoclassical effects. We could expect to encounter such circomstances especially when wave-particle interaction takes place in the near vicinity of the trapped-passing boundary.

The starting point of the calculations is the determination of the flux averaged density ⟨ne⟩ . According to the definition of the electron momentum distribution function f, the local electron density n_e (ψ,θ) is given by the relation

∫ ∫ +1 ∞ 2 ne (ψ, θ) = 2π -1 dξ 0 p dpf (ψ,θ,p,ξ)

(3.365)

Using the general expression (3.246) of the flux-surface averaging of a volumic quantity

_V	= ∫ ₀^2πn_e
	= ∫ ₀^∞p²dp∫ ₀^2π ∫ _-1⁺¹dξf
	= ∫ ₀^∞p²dp∫ ₀^2π×
	∫ _-1⁺¹_Tdξf	(3.366)

where the trapping condition evaluated at the location θ is given by

∘ ------------- -B-(ψ,θ)- |ξ| < ξT = 1 - Bmax (ψ )

(3.367)

Using ξdξ = Ψξ₀dξ₀ with the condition (3.270) on ξ₀

------------ ∘ |ξ0| ≥ 1- ---1--- Ψ (ψ, θ)

(3.368)

one get

∫ [ ] ∫ [ ] ( ∘ -----------) +1 1-∑ +1 1- ∑ ξ0 ---1--- -1 2 dξ = -1 2 Ψ (ψ,θ) ξ H |ξ0|- 1 - Ψ (ψ,θ) dξ0 σ= ±1 T σ=±1 T

(3.369)

Note that the condition (3.368) is equivalent to

θmin(ψ,ξ0) ≤ θ ≤ θmax(ψ,ξ0)

(3.370)

so that, the integrals over θ and ξ₀ may be permuted,

_V	= ∫ ₀^∞p²dp∫ _-1⁺¹dξ₀×
	_T ∫ _{θ_min}^θ_maxf	(3.371)

where the bounce-averaging of the distribution appears naturally. Therefore, expression (3.371) can be rewriten in the simple form

∫ ∫ ^q- ∞ 2 +1 ⟨ne⟩V (ψ) = 2π^q 0 p dp - 1 dξ0λ{f (ψ,θ,p,ξ0)}

(3.372)

and the exact trapped fraction _t is given by the ratio

∫∞ p2dp ∫+ξ0Tλ {f} dξ Ft (ψ) = -0∫--------∫ξ0T---------0 ∞0 p2dp +-11 λ {f} dξ0

(3.373)

where λ is the normalized bounce time 2.11.

Since, {f} ≃ f₀ + ^
f +g ,

∫∞ 2 ∫+ξ0T [ (0) (0)] F (ψ) = ---0-p-dp---ξ0T[-λ--f0--+-g---d]ξ0--- t ∫∞ p2dp ∫+1 λ f(0)+ ^f(0) + g(0) dξ 0 -1 0 0

(3.374)

taking into account that is an odd function of ξ₀ in the trapped region.

When f₀ = f_0M = f_M is a Maxwellian distribution on the magnetic flux surface ψ,

∫ ∫ ∞0 p2dp +-ξξ0Tλf0(0M)dξ0 F Mt (ψ ) = ∫-∞----∫-+1--[0T(0)----(0)]---- 0 p2dp -1 λ f0M + gM d ξ0

(3.375)

taking into account that g_M = 0 for trapped electrons. Neglecting the contribution of g_M, the zero order trapped fraction _t0^M is given by

∫∞ p2dp ∫+ξ0T(ψ) λf(0) (p,ψ)dξ FM (ψ ) = -0∫-------ξ0∫T(ψ)---0M--------0- t0 ∞0 p2dp +-11 λf (00)M (p,ψ )dξ0

(3.376)

which reduces to

∫+ξ0T(ψ )λdξ ∫+ ξ0T(ψ) FMt0 (ψ ) = --ξ∫0T(ψ)----0-= -0∫-----λdξ0- +-11 λdξ0 +01 λdξ0

(3.377)

In this limit, _t0^M is only a function of the geometrical magnetic configuration, while is the general case, _t is a fully kinetic quantity.

Case of circular concentric flux-surfaces In that case, the normalized bounce time is simply

∫ [ ] θmaxdθ-ξ0 2- 1-2 λ (ξ0) = 2π ξ ≃ π J0(ξ0,ξ0T)- 2ξ0TJ2(ξ0,ξ0T ) θmin

(3.378)

which may be expanded up to the second order with an excellent accuracy as shown in Appendix ??. Here,

∘ -2-ϵ- ξ0T = ----- 1 + ϵ

(3.379)

with ϵ = r∕R_p the usual inverse aspect ratio.

It is interesting to estimate the parametric dependence of _t0^M for ϵ ≪ 1. For trapped particles,

[ ( ) [ ( ) ( )]] 2|ξ0| -ξ02 1-2 -ξ20 -ξ20 λ(ξ0) ≃ πξ0T K ξ2 - 2ξ0T K ξ2 - E ξ2 0T 0T 0T

(3.380)

where K (x ) and E (x) are complete elliptic integrals of the first and second kind. Hence,

∫ + ξ0T(ψ) 2√2-√ -∫ 1 λ(ξ0)dξ0 = ---- ϵ x[K (x)- ϵ(K (x )- E (x))]dx 0 π 0

(3.381)

Using the recurrence relation

∫ ∫ n2 1 xnK (x)dx = (n - 1)2 1xn -2K (x)dx + 1 0 0

(3.382)

and since

∫ 1 xE (x)dx = 2∕3 0

(3.383)

according to formulaes (6.147) and (6.132 ) in Ref. [?],

∫ +ξ0T (ψ) 2√2-√- lim λ (ξ0)dξ0 ≃ ---- ϵ (1 - ϵ∕3) ϵ→0 0 π

(3.384)

For circulating electrons,

[ ( ) [ ( ) ( ) ]] λ(ξ ) ≃ 2- K ξ20T - 1-ξ2 K ξ20T - E ξ20T 0 π ξ20 2 0 ξ20 ξ20

(3.385)

and

√-- ∫ 1 2 2 √- ∫ 1 ( K (x) K (x)- E (x)) λ (ξ0) dξ0 = -π-- ϵ √-- -x2--+ ϵ-----x4------ dx +ξ0T(ψ) 2ϵ

(3.386)

From the relation

∫ K (x) E (x) ---2-dx = ------ x x

(3.387)

which is given by formula (5.112.9) of Ref. [?],

∫ _{+ξ_0T}¹λdξ₀	= + ϵ∫ ¹dx
	≃ 1 - + ϵ∫ ¹dx	(3.388)

and using the indefinite integrals

∫ K-(x)--E-(x)dx = - E (x) x

(3.389)

and

∫ E (x ) 1 [ 2 ( 2) ] -x4--dx = 9x3-2(x - 2)E (x)+ 1 - x K (x)

(3.390)

according to formulaes (5.113.1) and (5.112.12) in Ref. [?],

∫ 1 K (x)- E (x ) δ3 K (δ) - E (δ)( ) δ2 π lim δ3 ------4------dx = --- + ------------- 1 - δ2 + --E (δ) ≃ --δ2 δ→0 δ x 3 3 3 4

(3.391)

so that up to the first order term,

∫ 1 λ (ξ0)d ξ0 ≃ 1 + ϵ- 0 2

(3.392)

Consequently,

-- 2√ 2√ - √ - ϵli→m0 FtM0 (ψ) ≃ ---- ϵ ≃ 0.9 ϵ π

(3.393)

and the √ -
ϵ dependence in the limit ϵ ≪ 1 is well recovered, as expected from an intuitive explanation.

It is worth noting that this result is well recovered by a simple Monte-Carlo technique, where the poloidal angle θ is taken to be a uniform random variable between 0 and 2π, as well as ξ between -1 and 1. Using the relation (2.22) which translates ξ to ξ₀ at the minimum B value, and considering that the particle is trapped when |ξ0| ≤ ξ_0T, the fraction of trapped particle found numerically is exactly _t0^M (ψ ) , while the distribution scales like λ (ξ0) .

It is important to precise that _t0^M is not the “ effective” trapped fraction _t^eff. given by the well known relation

∫ 1 F eff.(ψ ) = 1 - 3-⟨h2⟩ ⟨√-xdx--⟩- t 4 0 1 - xh

(3.394)

found repeatedly in the litterature for the bootstrap current or the neoclassical conductivity, where h = B∕B_max and B_max is the maximum value of the magnetic field B along the particle trajectory. This quantity results from the reduction of the conductivity due to trapped particles, or the onset of the bootstrap current. Its expression with notations used in the text is determined from the bootstrap current calculations with the Lorentz collision operator, as shown Sec.5.6.2. It is important to notice that _t^eff. is in principle not a fraction of trapped electrons, and in addition there is no demonstration that _t^eff. ≤ 1 is always satisfied for all magnetic configurations, as mentioned clearly in Ref. [?]. In fact the denomination “ effective” trapped fraction _t^eff. is quite confusing, since it applies only for Maxwellian regime, and is not established as a kinetic quantity like _t0^M. This point is especially important when non-Maxwellian distributions are considered for evaluating the bootstrap current. Consequently, _t^eff. must not be used in such regimes, but only _t as an true physical sense for comparisons between different regimes.

3.6.8 Runaway loss rate

Primary generation When the Ohmic electric field E_∥ exceeds the Dreicer level E_D, a fraction of the electron population run away. The total number of electrons is therefore no more conserved, since the flux S_p≠0 at p = p_max, on the boundary of the integration domain ϒ. The runaway loss rate Γ_R is given by the relation

∫∫ Γ R(ψ,θ) = ϒS p (ψ,pξ)⋅dS

(3.395)

where the element of surface dS (p) = p²dξdφ according to the Appendix A. Therefore, since ∫ dφ = 2π by symmetry, one obtains

∫ +1 Γ R (ψ,θ) = 2πp2c Sp (ψ, pc,ξ)dξ -1

(3.396)

where p_c is the critical momentum above which electrons runaway. The value of p_c corresponds to the threshold where collision drag may not counter-balance electric field acceleration, and its value is therefore dependent of the model used for collisions. Since electrons with p ≥ p_c will never cross back this threshold, they leave rapidly the domain of integration, and Γ_R (ψ,θ) is weakly dependent of p when p ≥ p_c. In particular, Γ_R (ψ, θ) ≈ Γ_R^max, where

∫ +1 Γ max (ψ,θ) = 2πp2 S (ψ, p ,ξ)dξ R max -1 p c

(3.397)

Consequently, p_c is considered as a free parameter in the code. The flux-surface averaged runaway rate ⟨Γ R ⟩ _V is given by the relation

2π ∫ 2πd θ 1 r B0 (ψ) ⟨Γ R⟩V (ψ) =---p2c ---||----||---------Γ R (ψ,θ) ^q 0 2π |^ψ ⋅^r|Rp BP

(3.398)

2π ∫ 2π dθ 1 r B (ψ) ∫ +1 ⟨Γ R ⟩V (ψ) =---p2c --|----|-----0--- Sp (ψ, pc,ξ)dξ ^q 0 2π||^ψ ⋅^r||Rp BP -1

(3.399)

Using the usual condition on the trapped electrons,

∫ +1 ∫ +1 ( ∘ -----------) dξ = Ψ (ψ,θ) ξ0-H |ξ0|- 1 - ---1--- dξ0 -1 -1 ξ Ψ (ψ,θ)

(3.400)

one obtains

_V	= p_c² ∫ _-1⁺¹dξ₀_T ∫ _{θ_min}^θ_maxS_p
	= p_c² ∫ _-1⁺¹λ dξ₀
	= 2πp_c² ∫ _-1⁺¹λS_pdξ₀	(3.401)

since the sum [1∑ ]
2 σ _T may be added, because S_p (ξ) does not depend of the sign of ξ. in the trapped region. The density Δ ⟨nR⟩ _V of runaway electrons lossed per time step Δt is then simply ⟨Γ R ⟩ _V (ψ ) Δt, and in order to preserve the code conservative, a source of thermal electrons corresponding to a similar amount of electrons lost must be added in the set of equations to solve.

Secondary generation The Fokker-Planck approximation for collisions is valid so far electrons suffer only weak deflections. Knock-on processes are consequently neglected. However, for electrons with a high kinetic energy p ≳ lnΛ^†, such an approximation is no more justified, and collisions with large deflections must be considered. The effect may change significantly the picture of the fast tail build-up by leading to an avalanche process that could modify the runaway electron growth rate

--1---d⟨nR⟩V-- --1---Δ-⟨nR⟩V- ⟨Γ R⟩V γR = ⟨nR ⟩ dt ≃ ⟨nR⟩ Δt ≃ ⟨nR⟩ V V V

(3.402)

Here ⟨nR ⟩ _V is the flux surface averaged runaway density.

The source _R of secondary runaway electrons is given by a Krook term of the form [?]

( ) SR (ψ, p,ξ) = --nR-νe- 1-∂--( ---∘--1-------) δ (ξ - ξ*(p)) 4π lnΛee p2∂p 1- 1+ β†2p2 th = S*R δ(ξ - ξ*R (p)) (3.403)

where

∘ ------ * ----σRpβ-†th----- γ---1 ξ (p) = ∘ -----†2-2 = σR γ + 1 1 + βth p + 1

(3.404)

is the pitch-angle of the secondary electron produced by collisions, which is here deduced from momentum and energy conservation of strong elastic collisions between relativistic particles (See Appendix ?? )¹, and

( ) nR νe 1 ∂ 1 nR νe 1 γ + 1 S*R (p) =-------- -2---( ----∘---------) = -------------3-------- 4 πlnΛee p ∂p 1 - 1+ βt†2hp2 4π ln Λee(p)p γ(γ - 1)

(3.405)

Here, γ = ∘ -----†2-2-
1+ β thp and the normalized density of runaway electrons n_R is assumed to be uniform over a magnetic flux surface ψ, so that n_R = n_R (ψ ) = ⟨n ⟩
R _V, lnΛ_ee is the usual Coulomb logarithm for electron-electron collisions, which is a function of p for high energy electrons, and σ_R = v_∥∕v indicates the direction of acceleration for the runaway electrons.

The bounce-averaged expression of _R is given by the usual relation

[ ∑ ] ∫ θmax {SR } = -1- 1- dθ-|-1-|-r--B- ξ0SR (ψ,p,ξ) λ^q 2 σ θmin 2π ||ψ^⋅^r||Rp BP ξ T

(3.406)

[ ∑ ] ∫ θmax {SR} = S*R (p) 1-- 1- dθ|--1-|-r--B- ξ0-δ(ξ - ξ*) λ^q 2 σ T θmin 2π||ψ^⋅^r||Rp BP ξ * * = SR (p) {δ(ξ - ξ )} (3.407)

In order to calculate {δ(ξ - ξ*)} , it is important to recall that ξ is a function of θ, according to the relation

∘ -----------(-----)- ξ = σ 1- Ψ (ψ, θ) 1- ξ20

(3.408)

where Ψ (ψ,θ) = B (ψ,θ) ∕B₀ (ψ ) . Using the general relation δ (g (x)) = ∑ _kδ (x - xk) ∕ |g′(xk)| for the Delta function δ (x) , where x_k are the zeros of function g (x) and g^′ (x) = dg∕dx, one obtains

∑ * * δ (ξ - ξ*) = |-2δ((θ--)θ|k()|ξ|--) R k=1,2 |Ψ ′ ψ,θ*k | 1 - ξ20

(3.409)

where θ_k^* are the poloidal angle values where the secondary electron emerges, which verify the relation

∘ ------(----)(------) * σ 1 - Ψ ψ,θ*k 1 - ξ20 - ξ = 0

(3.410)

* Bθ*k 1 - ξ*2 2 Ψ (ψ,θk) = B---= -1--ξ2-= (----2)-------- 0 0 1- ξ0 (γ + 1)

(3.411)

Indeed, the equation Ψ (ψ, θ*k) = C has in general two distinct solutions in a tokamak magnetic configuration, except at θ^* = {0,π} . Therefore,

[ ] ∫ θ * {δ (ξ - ξ*)} =-1- 1-∑ max dθ-|-1-|-r--B- ξ0 ∑ |--(---2|)ξ|(|-----)δ(θ - θ*) λ^q 2 σ θmin 2π ||ψ^⋅^r||Rp BP ξ |Ψ′ ψ,θ*k | 1 - ξ20 k T k=1,2

(3.412)

∫ 2π ∑ * {δ(ξ - ξ*)} =-1- dθ|-1--|-r--B-ξ0 |-′ (-2*|ξ)||(-----2)δ(θ - θ*k) λ^q 0 2π||^ψ ⋅^r||Rp BP ξ k=1,2 |Ψ ψ,θk | 1 - ξ0

(3.413)

since the integrand is an even function of ξ (or ξ₀). Consequently

* 1 ∑ 1 rθ*k Bθ*k ξ0 2 |ξ*| {δ (ξ - ξ )} = λ^q ||----||--R--B---*ξ-*||Ψ-′ (ψ,θ*)||(1--ξ2) k=1,2|ψ^⋅^r|θ* p P,θk θk k 0 k

(3.414)

1 ∑ 1 rθ* B0 2 |ξ0|(1- ξ*2) {δ(ξ - ξ *)} =--- ||---||----k----- ||-′ (--*)||--(-----2)2-- λq^k=1,2 |^ψ ⋅r^| *Rp BP, θ*k Ψ ψ,θk 1 - ξ0 θk

(3.415)

since

* ξθk*= ξ

(3.416)

Finally,

( ) (0) * 1 ∑ 1 rθ*k B0 2 |ξ0| 1 - ξ*2 {SR } = SR (ψ,p,ξ0) = S R(p) λ^q ||----||--R--B---*||Ψ′ (ψ,-θ*)||-(----2)2-- k=1,2|^ψ ⋅^r|θ* p P,θk k 1 - ξ0 k

(3.417)

and using relations 3.405 and 3.404,

⌊ ⌋ ∑ S(0)(ψ,p,ξ0) = ⟨nR-⟩V νe-1 ---1----B0-(--|ξ0|-)- ⌈|-1--|-r--1-′⌉ R πlnΛeeRp p3 γ(γ - 1)λq^ 1 - ξ202 k=1,2 ||^ψ ⋅^r||BP |Ψ | θ*k

(3.418)

where ⟨nR ⟩ _V = ∫ _-∞^t ⟨Γ R⟩ _Vdt corresponds to the total number of runaway electrons produced at the normalized time t on a flux surface ψ.

The flux-surface averaged source term ⟨SR ⟩ _V is given by the general relation

∫ ∫ ∫ 2π- 2πd-θ|-1--|-r-B0-(ψ) pmax 2 +1 ⟨SR⟩V (ψ) = ^q 0 2π |^ |Rp BP p 2πp dp - 1 SR (ψ, p,ξ)dξ |ψ ⋅^r| min

(3.419)

which becomes

∫ ∫ ∫ 4π2- pmax * 2 2π-dθ--1----r-B0- +1 * ⟨SR ⟩V (ψ ) = ^q SR(p)p dp 2π ||^ ||Rp BP δ(ξ - ξR)dξ pmin 0 |ψ ⋅^r| -1

(3.420)

∫ ∫ ⟨S ⟩ (ψ) = 4π2- pmax S* (p) p2dp 2π dθ|-1--|-r-B0- R V ^q pmin R 0 2π||^ψ ⋅^r||Rp BP

(3.421)

Since by definition ∫ _-1⁺¹δ (ξ - ξ*)
R dξ = 1, and using the relation

∫ 2πdθ 1 r B0 ^q = ---||----||------ 0 2π |^ψ ⋅^r|Rp BP

(3.422)

one finds

∫ 2 pmax * 2 ⟨SR⟩V (ψ) = 4π p SR (p)p dp min

(3.423)

( ) ⟨S ⟩ (ψ ) = πnRνe-( ----∘---1--------- ----∘---1--------) R V ln Λee †2 2 †2 2 1- 1+ βthpmax 1 - 1+ βthp min

(3.424)

assuming a weak dependence of lnΛ_ee with p in the interval of integration.

Recalling that n_R = ⟨nR⟩ _V, the relation between the normalized secondary runaway source and the runaway rate is then simply

( ) ∫ t ⟨SR⟩ (ψ ) = ⟨Γ R⟩ dt-πνe--( ---∘---1---------- ----∘---1-------) V -∞ V lnΛee 1- 1 + β†2p2 1 - 1+ β †2p2 th max th min

(3.425)

where values p_min and p_max correspond respectively to the runaway threshold p_c and the upper momentum boundary of the integration domain of the Fokker-Planck equation. An estimate of p_min is given in Appendix ??. In the limit β_th^†2p_max² ≫ 1,

∫ t ⟨SR ⟩∞ (ψ ) ≃ ⟨Γ R⟩ dt-πνe-∘-------1-------- V - ∞ V ln Λee 1+ β†2p2 - 1 th min

(3.426)

and using the relation β_th^†2p_min² ≈ 2E_c∕E_∥ when E_∥≫ E_*, where E_* is the critical electric field for relativistic electrons,

∞ ∫ t πνe 1 ⟨SR⟩V (ψ) ≃ ⟨Γ R ⟩V dtlnΛ--∘---------------- -∞ ee 1+ 2Ec ∕E∥ - 1 ∫ t E ≃ ⟨Γ R ⟩V dt-πνe--∥- (3.427) -∞ lnΛee Ec

Using relation

E-*-= β†2 ED th

(3.428)

between Dreicer field E_D and the critical electric field E_c, one obtains finaly when E_∥∕E_D ≫ β_th^†2,

∞ ∫ t πνe †2 E∥ ⟨SR ⟩V (ψ ) ≃ ⟨Γ R⟩V dtlnΛ-βthE--- -∞ ee D

(3.429)

A similar expression may be derived in the opposite limit, i.e. when E_∥≳ E_c or E_∥∕E_D ≳ β_th^†2. In that case,

†2 2 -----1---- βth pmin ≈ 1 - E *∕E∥

(3.430)

Λ_eeand

∫ t ∘ -------†2-- ⟨S ⟩∞ (ψ ) ≃ ⟨Γ ⟩ dt-πνe-- 1 - --βth-- R V - ∞ R V ln Λee E ∥∕ED

(3.431)

Since the secondary source term will enhance the fast electron density above p_min, for a given electric field, the effect of strong Coulomb collisions will greatly increase the loss of runaway electrons. The fact that ⟨SR⟩ _V^∞ is not proportional to ⟨Γ R ⟩ _V but to ∫ _-∞^t ⟨Γ R⟩ _Vdt clearly indicates that all the history of the fast tail build-up plays a crucial role on the dynamics at time t, and consequently, only a time evolution is meaningful for this studying this problem which has basically no stationnary regime. A stationnary regime may only be found if E_∥ evolves so that the plasma current is kept time-independent.

3.6.9 Magnetic ripple losses

Though magnetic ripple losses is a full 4 -D problem, it can be considered in a simple manner by defining a super-trapped volume V _ST^p in momentum space,

p ∫ ∞ 2 ∫ +1 VST (ψ) = 2π p dp H (p- pc)(1- H (|ξ0|- ξ0ST))dξ0 0 -1

(3.432)

in which particle escape the plasma. A low energy, it is bounded by the collision detrapping when p ≤ p_c, while the pitch-angle dependence results from the condition that only electrons whose banana tip enter the bad confinement region characterized by the well known criterion α^*≤ 1 are trapped in the magnetic well, in an irreversible manner. As shown in Fig. 3.1, even if this is a rather crude modeling, it captures most of the salient features of the physics. Therefore, all trapped electrons which in addition fullfils the condition p_∥∕p ≤ ξ_0ST are super-trapped. Here, ξ_0ST is deduced from the intersection between the poloidal extend of the banana and the good confinement domain α^* ≥ 1 on a given flux surface [?]. The pitch-angle threshold ξ_0ST depends therefore of the radial position ψ and close to the edge,

lim ξ0ST = ξ0T ψ→ ψa

(3.433)

which indicates that all trapped electrons are expected to escape the magnetic configuration. Furthermore, it is assumed that electrons, once in this magnetic well, do not contribute anymore to the overall momentum dynamics, which is obviously a very crude approximation.

cmcmcm

Figure 3.1: Domain in configuration space where magnetic ripple well takes place for Tore Supra tokamak

An heuristic description of this process may be obtained by introducing a Krook term restricted to the volume V _ST^p in the Fokker-Planck equation

| ∂f (0)| ---0-|| = νdSTf(00) (ψ, p,ξ0) H (p- pc)(1 - H (|ξ0|- ξ0ST)) ∂t |

(3.434)

where ν_{d_ST}^-1 is the drifting time taken by super-trapped electrons for leaving the plasma. In order to reproduce the fact that super-trapped electrons are decoupled from the momentum dynamics, a simple method is to force ν_{d_ST}^-1 ≪ τ_b. Without detailed knowledge of the local dynamics, ν_{d_ST} is taken constant in V _ST^p, which is obviously a coarse approximation. However, in the limit ν_{d_ST}^-1 ≪ τ_b, the shape of the distribution function becomes independent of ν_{d_ST}, since by definition f₀ (ψ,p,ξ0) ≃ 0 in the super-trapped domain.

Obviously, when a Krook term is introduced, the Fokker-Planck equation is no more conservative, since a fraction of fast electrons is definitively leaving the plasma . Assuming the particle loss rate is small, a steady-state solution may be found, provided some external source of electron is added, in order to keep the density locally constant. This important point is discussed in Sec.5.7.1. The new form of the bounce-averaged Fokker-Planck equation is

(0) ∂f0--+ ∇ (ψ,p,ξ ) ⋅S(0) + νd f(0)H (p - pc)(1- H (|ξ0|- ξ0ST)) = 0 ∂t 0 ST 0

(3.435)

and in this stationnary limit lim_t→∞∂f₀∕∂t = 0,

∇ (ψ,p,ξ0) ⋅S(0) = - νdSTf(00)H (p - pc)(1- H (|ξ0|- ξ0ST))

(3.436)

Losses are assumed to be mainly local, since they occur on a very short time scale as compared to the fast electron transport one. Therefore, only the momentum dynamics is considered, and integrating equation (3.436 ), one obtains

	_{V _ST^p}∇ ⋅ S_pJ_pJ_{_ξ₀}dpdξ₀dφ
	= -_{V _ST^p}ν_{d_ST}f₀HJ_pJ_{_ξ₀}dpdξ₀dφ	(3.437)

where J_p and J_{_ξ₀}are the Jacobians as defined in Sec. 3.5.1. The magnetic ripple loss rate Γ_ST (ψ ) on the B₀ axis is simply given by

∫ ∫∫ (0) (0) Γ ST (ψ ) = p νdST f0 H (p - pc)(1 - H (|ξ0|- ξ0ST ))Jdpdξ0dφ VST

(3.438)

∫ pmax ∫ + ξ0ST Γ (0S)T (ψ ) = 2π p2dp νdST f(00)λ (ψ, ξ0)dξ0 pc - ξ0ST

(3.439)

since ∫ dφ = 2π.

An equivalent form can be deduced from the flux of particle leaving the integration domain,

Γ_ST	= _{V _ST^p}∇ ⋅ S_pJ_pJ_{_ξ₀}dpdξ₀dφ
	= _{S_ST^p}S_p ⋅ dS	(3.440)

using the Green-Ostrogradsky theorem, where S_ST^p is the surface that encloses volume V _ST^p. as shown in Fig. 3.1, S_ST^p may be split into two terms corresponding to coordinate surfaces

p p p S ST = SST,p + SST,ξ

(3.441)

where S_ST,p^p is the surface at constant p, while S_ST,ξ₀^p is the surface at constant ξ₀ . Therefore, for the surface S_ST,p^p,

(0) || Sp ⋅dS |p =S (0p)p2λ (ψ, ξ0)dξ0dφ SST,p

(3.442)

(0) || p (0) ∘ ----2- S p ⋅dS |SST,ξ0= - S ξ0 pλ (ψ, ξ0) 1- ξ0dpd φ

(3.443)

according to the differential relations in Appendix A. One obtains finaly

∫ + ξ0ST ∘ --------∫ pmax Γ (S0T)(ψ) = 2πp2c λ(ψ,ξ0)S (0p)dξ0 + 4π λ(ψ,+ ξ0ST) 1 - ξ20ST pS (0ξ0)dp -ξ0ST pc

(3.444)

since the flux S_ξ₀ is a symmetric function of ξ₀.

3.6.10 RF Wave induced cross-field transport

According to adjoint formalism developped by P. Helander [?], the flux surface averaged cross-field particle flux may be expressed as

⟨ I (ψ) ∫ ( ) ⟩ ⟨Γ ⋅∇ ψ⟩ = ----- p∥ ∇p ⋅SRpF - νeDif0 d3p qeB

(3.445)

where |I (ψ)| = RB_T, and the Coulomb deflection frequency ν_D^ei is given by the relation

ei erf(x)- G (x) νD (ψ, p) = νe (ψ )Zeff (ψ )-----x3-----

(3.446)

with x = p∕ √ --
2 and

′ G (x) = erf(x)---xerf(x)- 2x2

(3.447)

Expression (3.445) is only valid in the non-relativistic limit, i.e. when β_th^†≪ 1.

From the expression of the flux divergence ∇_p ⋅ S_p

( ) ∇ ⋅SRF = 1--∂-(p2SRF ) - 1-∂- ∘1---ξ2SRF p p p2∂p p p∂ ξ ξ

(3.448)

one obtains

[ ] ∫ RF 3 ∫ ∞ ∫ +1 ∂ ( 2 RF) 2 ∂ (∘ ------ RF ) p∥∇p ⋅Sp d p = 2π dp ξ p ∂p-p S p - p ∂ξ- 1 - ξ2Sξ dξ 0 - 1

(3.449)

where

∫ ∞ ∫ +1 ∂ (∘ ------ ) - 2π dp ξp2--- 1- ξ2SRFξ dξ ∫ 0 -1( ∂ξ ∫ ) ∞ 2 [ ∘ ----2-RF ]+1 +1 ∘ ----2- RF = - 2π 0 pdp ξ 1- ξ Sξ -1 - -1 1- ξ Sξ dξ ∫ ∞ ∫ +1 ∘ ------ = 2π p2dp 1 - ξ2SRξF dξ 0 - 1 ∫ +1∘ ------ ∫ ∞ 2 RF = 2π 1 - ξ2dξ pS ξ dp (3.450) -1 0

and

∫ ∞ ∫ +1 ∂ ( ) ∫ +1 ∫ ∞ ∂ ( ) 2π dp ξp--- p2SRpF dξ = 2π ξdξ p --- p2SRFp dp 0 -1 ∂p ∫- 1 (0 ∂p ∫ ) +1 [ 3 RF ]∞ ∞ 2 RF = 2π - 1 ξdξ p Sp 0 - 0 p Sp dp ∫ +1 ∫ ∞ = - 2π ξd ξ p2SRpF dp (3.451) -1 0

assuming that lim_p→∞p³S_p = 0. It is important to recall that this condition is more stringent than the equivalent one for RF power calculation, where the condition lim_p→∞p²S_p = 0 must hold.

Therefore

∫ (∫ ∫ ∫ ∫ ) RF 3 +1 ∘ ----2- ∞ 2 RF +1 ∞ 2 RF p∥∇p ⋅Sp d p = 2π - 1 1- ξ dξ 0 p Sξ dp - -1 ξdξ 0 p Sp dp

(3.452)

and finally

∫ ( RF ei ) 3 p∥ ∇p ⋅S p - νD (p)f0 d p ∫ +1∘ ------ ∫ ∞ ∫ +1 ∫ ∞ [ ] = 2π 1- ξ2dξ p2SRξF dp- 2 π ξdξ p2 SRpF + νeDipf0 dp(3.453) -1 0 - 1 0

Since ⟨Γ ⋅∇ ψ⟩ is a volume quantity, the flux surface averaged expression (3.445) may be then expressed as

⟨Γ ⋅∇ψ ⟩ = ⟨Γ ⋅∇ ψ⟩ - ⟨ Γ ⋅∇ ψ⟩ 1 2

(3.454)

where

∫ ∫ ∫ 2πI-(ψ) 2π dθ---1---r-B0-(ψ) 1- +1 ∘ ----2- ∞ 2 RF ⟨Γ ⋅∇ψ ⟩1 = ^q qe 0 2π ||^ ||Rp BP B -1 1- ξ dξ 0 p Sξ dp |ψ ⋅^r|

(3.455)

∫ 2π ∫ +1 ∫ ∞ ⟨Γ ⋅∇ ψ⟩ = 2π-I (ψ-) dθ|--1-|-r-B0-(ψ)-1 ξdξ p2[SRF + νei(p)pf]dp 2 ^q qe 0 2π||^ψ ⋅^r||Rp BP B -1 0 p D

(3.456)

Expressions ⟨Γ ⋅∇ ψ⟩ ₁ and ₂ must be expressed in terms of bounce-averaged quantities so that calculations may be performed numerically in the code. By permuting integrals,

∫ ∞ ∫ 2π ∫ +1∘ ------ ⟨Γ ⋅∇ψ ⟩1 = 2πI-(ψ) p2dp -dθ|-1--|-r-B0-(ψ)-1 1 - ξ2SRξF dξ ^q qe 0 0 2 π||^ψ ⋅^r||Rp BP B -1

(3.457)

and since ∘ ------
1- ξ2 S_ξ is an even function of ξ for trapped electrons, it is equivalent to

∫ ∞ ∫ 2π [ ∑ ] ∫ +1∘ ------ ⟨Γ ⋅∇ψ ⟩ = 2-πI-(ψ-) p2dp -dθ|-1--|-r-B0-(ψ)-1 1- 1 - ξ2SRF dξ 1 ^q qe 0 0 2π ||^ψ ⋅^r||Rp BP B 2 σ=±1 -1 ξ T

(3.458)

Using ξdξ = Ψξ₀dξ₀ with the condition (3.270) on ξ₀

------------ ∘ |ξ0| ≥ 1- ---1--- Ψ (ψ, θ)

(3.459)

we get that

∫ ∫ ( ∘ -----------) +1 +1 Ψ-(ψ,θ)-ξ0 ---1--- dξ = dξ0 ξ H |ξ0|- 1 - Ψ (ψ,θ) - 1 -1

(3.460)

which is equivalent to

θmin(ψ,ξ0) ≤ θ ≤ θmax(ψ,ξ0)

(3.461)

Therefore,

2πI (ψ)∫ ∞ 2 ∫ 2π dθ 1 r B0 (ψ) 1 ⟨Γ ⋅∇ ψ⟩1 = ^q---q-- pdp 2-π||^---||R---B----B- e 0 0 |ψ ⋅^r| p P [ ∑ ] ∫ +1∘ ------ ( ∘ -----------) × 1- 1 - ξ2Ψ3∕2ξ0H |ξ0|- 1- ---1--- SRF(d3ξ.4062) 2σ= ±1 -1 0 ξ Ψ (ψ, θ) ξ T

so that, permuting the integrals over θ and ξ₀, we find

∫ ∫ 2π---I (ψ)- ∞ 2 +1 ∘ ----2- ⟨Γ ⋅∇ ψ⟩1 = ^q qeB0 (ψ) p dp dξ0 1- ξ0 [ ] ∫0 -1 ( )2 1-∑ θmax dθ|-1--|-r--B-ξ0 B0(ψ-) 3∕2 RF × 2 θmin 2π|^ψ ⋅^r|Rp BP ξ B Ψ Sξ (3.463) σ=±1 T | |

∫ ∞ ∫ +1 ∘ ------[ ] ∫ θ ⟨Γ ⋅∇ψ ⟩ = 2-π--I (ψ-) p2dp dξ 1- ξ2 1-∑ max dθ-|-1-|-r--B- ξ0Ψ- 1∕2SRF 1 ^q qeB0 (ψ) 0 -1 0 0 2 θmin 2π ||ψ^⋅^r||Rp BP ξ ξ σ=±1 T

(3.464)

which is equivalent to

^q I (ψ) ∫ ∞ 2 ∫ +1 ∘ ----2-{ -1∕2 RF } ⟨Γ ⋅∇ ψ ⟩1 = 2π ^qq-B--(ψ) p dp λ 1- ξ0 Ψ S ξ dξ0 e 0 0 -1

(3.465)

where we use the bounce-averaged definition

{ } [ ∑ ] ∫ θmax Ψ -1∕2SRF = -1- 1- -dθ|-1--|-r--B-ξ0Ψ -1∕2SRF ξ λ ^q 2σ=±1 θmin 2 π||^ψ ⋅^r||Rp BP ξ ξ T

(3.466)

Much in the same way,

⟨Γ ⋅∇ ψ⟩2 = ⟨Γ ⋅∇ ψ⟩21 + ⟨Γ ⋅∇ ψ⟩22

(3.467)

where

2π I (ψ )∫ ∞ 2 ∫ 2πdθ 1 r B0 (ψ )1 ∫ +1 RF ⟨Γ ⋅ ∇ψ ⟩21 =-^q--q--- p dp 2π-||^---||R---B----B- ξSp dξ e 0 0 |ψ ⋅^r| p P - 1

(3.468)

and

∫ ∫ ∫ 2π-I (ψ-) ∞ ei 3 2π dθ|--1-|-r-B0-(ψ)-1 +1 ⟨Γ ⋅∇ ψ ⟩22 = ^q qe 0 νD p dp 0 2π|^ |Rp BP B -1 ξf0dξ |ψ ⋅^r|

(3.469)

Applying the same technique as for ⟨Γ ⋅∇ ψ ⟩ ₁, using the fact that S_p (ξ) = S_p (- ξ) in the trapped region, one obtains

2π I (ψ )∫ ∞ ∫ 2πdθ 1 r B0 (ψ )1 ⟨Γ ⋅∇ ψ⟩21 = -------- p2dp ---||----||----------- ^q qe 0 0 2π |^ψ ⋅^r|Rp BP B [ ] ∫ +1 ( ∘ -----------) × 1- ∑ Ψξ H |ξ |- 1- ---1--- SRF dξ (3.470) 2 - 1 0 0 Ψ (ψ, θ) p 0 σ=±1 T

^q I (ψ ) ∫ ∞ ∫ +1 { ξ } ⟨Γ ⋅∇ ψ⟩21 = 2π---------- p2dp λξ0σ σ --Ψ- 1SRpF dξ0 ^q qeB0 (ψ) 0 -1 ξ0

(3.471)

since the integrand is odd in ξ₀, and

{ } [ ∑ ] ∫ θmax ( ) σ ξ-Ψ- 1SRF = -1- 1- dθ-|-1--|r--B--ξ0 σ-ξΨ -1SRF ξ0 p λ^q 2 σ=±1 θmin 2π ||^ψ ⋅ ^r||Rp BP ξ ξ0 p [ ]T 1 1 ∑ ∫ θmaxdθ 1 r B0 RF = λ^q- 2- 2π-||^---||R--B--σSp (3.472) σ=±1 T θmin |ψ ⋅ ^r| p P

In a similar way,

2π I (ψ )∫ ∞ ei 3 ∫ 2π dθ 1 r B0 (ψ) 1 ⟨Γ ⋅∇ψ ⟩22 = -^q--q--- νD p dp 2π||^---||R----B---B- e 0 0 |ψ ⋅^r| p P [ ∑ ] ∫ +1 ( ∘ -----------) × 1- Ψξ0H |ξ0|- 1- ---1--- fdξ (3.473) 2 σ=±1 -1 Ψ (ψ, θ) T

and since f₀ is constant on a flux surface ψ,

(0) f0(ψ,θ,p,ξ) = f0 (ψ, p,ξ0)

(3.474)

and an even function in ξ₀ for trapped electrons, one obtains

^q I (ψ) ∫ ∞ ei 3 ∫ +1 (0) { ξ - 1} ⟨Γ ⋅∇ ψ⟩22 = 2π^q-qB--(ψ-) νD p dp ξ0f0 λσ σ ξ-Ψ dξ0 e 0 0 - 1 0

(3.475)

^q I (ψ ) ∫ ∞ ei 3 ∫ +1 (0)-- ⟨Γ ⋅∇ ψ⟩22 = 2π^q-qB--(ψ)- νD pdp ξ0f0 λ1,- 1,0dξ0 e 0 0 -1

(3.476)

Finally, it is necessary to evaluate σ { }
σξξΨ -1SRpF
0 and { }
Ψ -1∕2SRFξ from the quasilinear diffusion coefficients. Starting from the conservative form of the wave-induced fluxes in momentum space

∘ ------ RF RF ∂f0- RF --1---ξ2∂f0- S p = - Dpp ∂p + D pξ p ∂ξ (3.477) ∘ -----2 SRF = - DRF ∂f0-+ DRF --1---ξ-∂f0- (3.478) ξ ξp ∂p ξξ p ∂ξ

and using the fact that f₀ is constant on a flux surface ψ,

{ } { } { ∘ -----2 } σ σ ξ-Ψ-1SRF = - σ σ ξ-Ψ -1DRF ∂f0- + σ σ ξ-Ψ- 1DRF --1---ξ-∂f0- ξ0 p ξ0 pp ∂p ξ0 pξ p ∂ξ { } (0) ∘ -----2{ 2 } (0) = - σ σ ξ-Ψ -1DRF ∂f-0- + --1---ξ0 ξ-Ψ -3∕2DRF ∂f(03.479) ξ0 pp ∂p p ξ20 pξ ∂ξ0

and

{ } { ∘ ------ } { -1∕2 RF } RF -1∕2∂f0- RF -1∕2--1---ξ2∂f0- Ψ Sξ = - D ξp Ψ ∂p + D ξξ Ψ p ∂ξ ∘ ------ { } { RF -1∕2} ∂f-(00) --1---ξ02 -ξ -1 RF ∂f(00) = - D ξp Ψ ∂p + p σ σξ0Ψ D ξξ ∂ξ0(3.480)

Using diffusion coefficients (4.233-4.236) defined in Sec. 4.3,

D_pp	= ∑ _n=-∞^+∞∑ _b(1 - ξ²)D_b,n(p,ξ)	(3.481)
D_pξ	= ∑ _n=-∞^+∞∑ _b -D_b,n(p,ξ)	(3.482)
D_ξp	= ∑ _n=-∞^+∞∑ _b -D_b,n(p,ξ)	(3.483)
D_ξξ	= ∑ _n=-∞^+∞∑ _b²D_b,n(p,ξ)	(3.484)

one obtains

{ ξ } +∑∞ ∑ { ξ } σ σ--Ψ -1DRFpp = σ (1- ξ2)σ--Ψ -1DRFb,n (p,ξ) ξ0 n=-∞ b ξ0 +∑∞ ∑ { } = (1 - ξ20)σ σ ξ-DRFb,n (p,ξ) (3.485) n=-∞ b ξ0

{ } { ∘ ------ [ ] } ξ2 -3∕2 RF +∑ ∞ ∑ --1--ξ2-ξ2 - 3∕2 2 nΩ- RF ξ2Ψ Dpξ = - ξ ξ2Ψ 1- ξ - ωb D b,n (p,ξ) 0 n= -∞ b ∘ ------ 0 +∑ ∞ ∑ 1 - ξ2[ n Ω0] { ξ } = - -------0 1 - ξ20 ----- σ σ --DRbF,n (p,ξ) (3.486) n= -∞ b ξ0 ωb ξ0

{ } +∑ ∞ ∑ { ∘1---ξ2-[ nΩ ] } DRFξp Ψ -1∕2 = - -------- 1- ξ2 - --- Ψ-1∕2DRFb,n (p,ξ) n=- ∞ b ξ ωb +∑ ∞ ∑ ∘ -----2[ ] { } = - --1---ξ0 1 - ξ2- n-Ω0 σ σξ0 ΨDRF (p,ξ)(3.487) n=- ∞ b ξ0 0 ωb ξ b,n

and finally

{ } { ξ -1 RF } +∑ ∞ ∑ 1 [ 2 nΩ ]2 ξ -1 RF σ σξ-Ψ D ξξ = σ σ ξ2 1 - ξ - ω-- ξ-Ψ D b,n (p,ξ) 0 n=- ∞ b b 0 +∑ ∞ ∑ 1 [ nΩ ]2 { ξ } = -2 1- ξ20 - ---0 σ σ-0ΨDRFb,n (p,ξ) (3.488) n=- ∞ b ξ0 ωb ξ

where Ω₀ = Ω∕Ψ is the cyclotron frequency taken at the minimum B value. An interesting result is that σ { -ξ -1 RF }
σξ0Ψ Sp

depends only from ^
D

_b,n^RF(0)D(p,ξ₀) already defined for the wave-induced bootstrap current in Sec.4.3, while { }
Ψ- 1∕2SRF
ξ

depends from a quantity that is very similar to

_b,n^RF(0)F(p,ξ₀). Using definitions

_b,n^RF(0)D(p,ξ₀)	= σ	(3.489)
_b,n^RF(0)F(p,ξ₀)	= σ	(3.490)

one obtains finally

{ } + ∞ ξ- -1 RF ∑ ∑ 2 ^ RF(0)D ^RF (0) σ σ ξ0 Ψ D pp = (1- ξ0)D b,n (p,ξ0) = D pp n=- ∞ b

(3.491)

{ 2 } +∑∞ ∑ ∘ -----2 [ ] ξ2Ψ -3∕2DRpFξ = ---1---ξ0 1- ξ20 - nΩ0- D^RFb,n(0)D (p,ξ0) = ^DRFpξ (0) ξ0 n=-∞ b ξ0 ωb

(3.492)

and

{ } +∑∞ ∑ ∘1---ξ2-[ nΩ ] ( { ξ }) DRFξp Ψ- 1∕2 = - ------0- 1- ξ02- ---0 D^RFb,n(0)F(p,ξ0)+ σ σ-0DRFb,n(p,ξ) n=- ∞ b ξ0 ωb ξ ^ RF(0) ^RF,1(0) = D ξp + D ξp (3.493)

{ } +∑ ∞ ∑ [ ]2( { }) σ σ ξ-Ψ-1D ξξRF = -1 1- ξ02- nΩ0- ^DRF (0)F(p,ξ0)+ σ σ ξ0DRFb,n(p,ξ) ξ0 n= -∞ b ξ20 ωb b,n ξ RF (0) RF,1(0) = D^ξξ + D^ξξ (3.494)

where

∘ ------[ ] { } RF,1(0) +∑∞ ∑ --1---ξ20 2 nΩ0- ξ0 RF ^D ξp = - ξ 1 - ξ0 - ω σ σ ξ D b,n(p,ξ) n=-∞ b 0 b

(3.495)

and

+∑∞ ∑ 1 [ nΩ ]2 { ξ } ^DRFξ,ξ1 (0) = -2 1 - ξ20 - --0- σ σ 0-DRbF,n (p,ξ) n=-∞ b ξ0 ωb ξ

(3.496)

The bounce-averaged quasilinear diffusion coefficient σ { ξ0- RF }
σ ξ D b,n (p,ξ) may be easily deduced from calculations of the RF wave-induced bootstrap current

{ ξ0 RF } γpTe 1 rθbB θbξ30--RF,θb σ σ ξ-Db,n (p,ξ) = p|ξ|λ-^qR----θbξ3D b,n,0 × 0 pB P θb [ ] 1 ∑ ( θ ) || b,(n)||2 H (θb - θmin)H (θmax - θb)σ 2- σδ Nb∥ - N ∥bres |Θ(k3,θ.4b9|7) σ T

using notations defined in Sec. 4.3.

3.6.11 Non-thermal bremsstrahlung

Several other moments of the electron distribution function may be calculated, mainly for diagnosing purposes of the plasma performances. In most cases, the local value of the distribution function f must be determined not only at different plasma radius, but also at various poloidal positions. In that case, the problem is 4 - D, since its shape is function also of the poloidal position on a given flux surface ψ. A good example is the calculation of the non-thermal bremsstrahlung [?], which requires the exact shape of the distribution function f at each plasma position along the lines-of-sight, as well as the local angle between the magnetic field line direction and the direction of observation .

The number of counts N_E₀ that is recorded by a photon detection system in the energy range E₀ ± ΔE between times t_min and t_max is given by the integral

∫ ∫ tmax E0+ ΔE dNE-(t,E)- NE0 = tmin dt E0- ΔE dtdE dE

(3.498)

where dN_E (t) ∕dtdE is the measured photon energy spectrum. Its relation to the effective photon energy spectrum dN_k (t) ∕dkdt emitted by the plasma in the direction of the detector may be expressed as

dN (t,E ) ∫ ∞ dN (t,k ) ---E------ = ηA (k )(1- ηD (k))G (k,E) ---k-----dk dtdE 0 dtdk

(3.499)

where G (k,E ) is the normalized instrumental response function,

∫ ∞ G (k, E)dk = 1 0

(3.500)

which gives the overall broadening of the energy spectrum, η_A (k) the fraction of photons that transmitted rather than being absorbed by various objects along the line-of-sight between the plasma and the detector, and finally, 1 - η_D (k) the fraction that are effectively stopped inside the active part of the photon detector. For most detection systems, G (k,E ) is a complicated function, that is usualy determined experimentaly with monoenergetic photon sources. It incorporates the photoelectric conversion process that may be usualy modeled by a Gaussian shape around the photon energy k whose half-width depends of the type of detector, and the Compton scattering by electrons, which can be approximately described by a Fermi-like function².

Since the plasma is an extended source of photons, all contributions inside the volume ΔV viewing the detector with a solid angle ΔΩ must be added

( ) ∫ ∫ dNk t,k,X,^b⋅ ^d dNk-(t,k)= dV ----------------d Ω dtdk ΔV(k) Δ Ω(k) dtdkdV dΩ

(3.501)

taking into account that photon plasma emissivity depends not only of the plasma position X (inhomogeneity) but also of the angle ⋅between the directions of the magnetic field line _X and the line-of-sight ^
d at X (anisotropy that results from relativistic effects). In principle, both ΔV (k) and ΔΩ (k) are functions of the photon energy, because of the partial transparency of the collimating aperture with k. However, the design of the diaphragm is usualy optimized so that this effect can be neglected.

In the limit where the aperture of the diaphragm is small, so that variation of the photon emissivity transverse to the line-of-sight may be neglected in the field of observation, dN_k (t,k) ∕dtdk may be approximated by the simple sum

( ) dNk (t,k) ∫ lmax dnk t,k,X, ^b⋅d^ ---dtdk-- ≃ GD ----dtdkdΩ------dl lmin

(3.502)

where L_c = l_{c max} - l_{c min} is the chord length in the plasma, and is a geometrical factor that is independent of the position l_c along the line-of-sight³. Here, n_k = dN_k∕dV is the photon density. By definition, the determination of dN_k (t,k) ∕dtdk requires to evaluate X and _X ⋅as a function of l for a given magnetic equilibrium. Since magnetic flux surfaces are nested in tokamaks inside the separatrix, the calculation requires the determination of ψ (lc) , θ (lc) and ⋅= cosθ_d (lc) .

In the appropriate range of energy, the photon density energy spectrum results from the bremsstrahlung process only⁴. It is the sum of two contributions, one arising from electron-ion interactions, the other resulting from electron self-collisions

( ) ( ) ( ) dnk t,k,X, ^b⋅ ^d ∑ dneki t,k,X, ^b⋅d^,Zs dneke t,k,X, ^b⋅d^ -----dtdkdΩ------= -------dtdkd-Ω------- + -----dtdkdΩ------ s

(3.503)

which are related to the respective bremsstrahlung differential cross-sections dσ_ei/dtdkdΩ and dσ_ee∕dtdkdΩ by the relations

ei( ^ ^ ) ∫ ∫∫ ( ^ ) dnk--t,k,X,-b⋅d,Zs-- 3 dσei-k,p,k-⋅ ^p,Zs- dtdkd Ω = ns(t,ψ) d p dtdkd Ω vf (t,X, P)

(3.504)

( ) ( ) ee ^ ^ ∫ ∫∫ ^ dnk---t,k,X,-b⋅d-- = n (t,ψ) d3pdσee--k,p,k ⋅ ^p-vf (t,X, P) dtdkdΩ e dtdkdΩ

(3.505)

where Z_s is the number of protons for the impurity of type s⁵, whose density on the flux surface ψ at time t is n_s (t,ψ) . The velovity v is the velocity of test partcles, in accordance with the definition of the cross-sections. Here cosχ = ⋅ is the cosine of the angle between directions of the incident electron of momentum p and the emitted photon of energy k. If one defines the angles ξ = cosθ_e = ^
b ⋅ and cosθ_d = ^
b ⋅ ^
d , the angle relation between χ, θ_e and θ_d is

cosχ = cosθe cos θd + sinθesinθdcos φ

(3.506)

as shown in Fig. 3.2.

cmcmcm

Figure 3.2: Directions of incident electron and emitted photon with respect to the local magnetic field direction

It is possible to take advantage of the azimuthal symmetry of the distribution function around the field line direction as well as the relations between angles χ, θ_e and θ_d using projection on Legendre polynomials, in order to reduce the required number of integrations. The numerical accuracy for the determination of dn_k∕dtdkdΩ may be then greatly enhanced, while the computational time strongly reduced. Let define the series for a function h (x)

∞ ( ) ∑ 1- (m ) h (x) = m + 2 h Pm (x) m=0

(3.507)

where coefficients h

∫ +1 h(m) = h (x)Pm (x) dx -1

(3.508)

and P_m is the Legendre polynomial of degree m.

Applying the Legendre polynomial series to differential cross-sections dσ^ei∕dtdkdΩ and dσ^ee∕dtdkdΩ and to f (t,X, p,ξ) ,

	= n_s∫ ₀^∞vp²dp∫ ₀^2πdφ∫ _-1⁺¹dξ×
	∑ _m=0^∞∑ _m^′=0^∞×
	fP_mP_m^′	(3.509)

where

( ) (m) ∫ +1 dσ(m) k,p,^k ⋅p^,Zs ( ) dσei-(k,p,Zs)-= --ei---------------Pm ^k ⋅ ^p d(cosχ ) dtdkdΩ - 1 dtdkdΩ

(3.510)

and

∫ (m′) +1 f (t,X, p) = -1 f (t,X, p,ξ)Pm ′ (ξ)dξ

(3.511)

one obtains

	= n_s∫ ₀^∞vp²dp∑ _m=0^∞∑ _m^′=0^∞×
	f×
	∫ ₀^2πdφ∫ _-1⁺¹dξP_mP_m^′	(3.512)

Using the well known sum relation for the Legendre polynomials that holds for angle relation between χ, θ_e and θ_d,

n=∑m (m---n)! n n Pm (cos χ) = Pm (cos θd)Pm (ξ)+ 2 (m + n)!Pm (cosθd)Pm (ξ)cos(nφ ) n=1

(3.513)

where P_mⁿ (x) is the associated Legendre function of degree m and order n, expression (3.512) becomes

	= n_s∫ ₀^∞vp²dp∑ _m=0^∞∑ _m^′=0^∞×
	f∫ ₀^2πdφ×
	∫ _-1⁺¹P_mP_mP_m^′dξ	(3.514)

since

∫ 2π ∫ +1 n∑=m (m - n )! 2 d φ --------Pmn(cosθd)P nm (ξ) cos(nφ)dξ = 0 0 - 1 n=1 (m + n )!

(3.515)

after permutation of integrals over ξ and φ. Using finally the orthogonality relation

∫ +1 δ ′ Pm (x)Pm ′ (x)dx =---mm--- -1 m + 1∕2

(3.516)

where δ_mm^′ is the Kronecker symbol, one obtains the simple relation

	= 2πn_s∫ ₀^∞vp²dp∑ _m=0^∞×
	fP_m	(3.517)

	= 2πn_s∑ _m=0^∞×
	P_m∫ ₀^∞vp²fdp	(3.518)

A similar expression may be obtained for the e-e bremsstrahlung, and the total bremsstrahlung is then

( ) dnk t,k,X,^b ⋅ ^d ∑∞ ( 1) (m) ( ) -----------------= m + -- IB (t,X, p,k)Pm ^b ⋅ ^d dtdkdΩ m=0 2

(3.519)

where the bremsstrahlung function I_B (t,X, p,k) is

I_B	= 2π ∫ ₀^∞vp²fdp
	dp	(3.520)

and the densitites n_s (t,X ) = n_s (t,ψ) and n_e (t,X ) = n_e (t,ψ) are considered to be uniform on a magnetic flux surface ψ.

With this formulation, bremsstrahlung emission may be determined for any direction of observation with the same numerical accuracy. Indeed, the projection of the distribution function and the differential cross-sections over the Legendre polynomial basis is equivalent to determine their value for all azimuthal directions. It is then only necessary to select the interesting direction that is given by the local ⋅value, which depends of the local instrumental arrangement, but also of the magnetic equilibrium. This formulation is particularly convenient when the instrument is made of different chords with different orientations. It is not only important for tangential observation of the plasma, but also for perpendicular ones, since ^
b ⋅ ^
d evolves with ψ as a consequence of the local magnetic shear. Moreover, this method offer the advantage to evaluate dσ_ei (k, p,Zs) ∕dtdkdΩ and dσ_ee (k,p,Zs ) ∕dtdkdΩ only once for various distribution functions, a procedure which may save considerably computer time consumption when the distribution function and the plasma equilibrium, i.e. ⋅evolves with the time t.

From expression (3.519), it is also possible to extract interesting local quantities about the bremsstrahlung, like the mean radiation level in all directions of the configuration space dn_k^4π (t,k,X ) ∕dtdkdΩ

	= ∫ dΩ
	= ∫ ₀^πdθ_d ∫ _-π^π sinθ_ddφ_d	(3.521)
	= ∫ _-1¹dξ_d	(3.522)
	= ∑ _m=0^∞I_B∫ _-1¹dξ_dP_m	(3.523)

and since P₀ (x ) = 1, using the orthogonality relation (3.516), one obtains

dn4π(t,k,X ) 1 --k---------= -I(Bm=0)(t,X, p,k ) dtdkdΩ 2

(3.524)

Much in the same way, the local anisotropy of the photon emission R_B (t,k,X ) may be evaluated from the ratio between the forward emission corresponding to cosθ_d = 1 and the perpendicular one corresponding to cosθ_d = 0.

The determination of I_B (t,X, p,k) requires to evaluate the projection of the electron distribution function given by the electron drift kinetic equation, at all X positions.⁶ Since the magnetic configuration is a toroidaly symmetric, only the radial ψ and poloidal θ positions are necessary, and therefore f (t,X, p,ξ) = f (t,ψ,θ,p,ξ) . Since f (t,X, p) is a linear function of f, it may be split into the three contributions, namely

f	= f₀ + f₁
	= f₀ + + g	(3.525)

where f₀ (t,X, p) are the Legendre coefficients for the zero order distribution function f₀, while f₁ correspond to the first order distribution function f₁.

Like for other moments of the distribution function, starting from the angular relation

∘ ------------------- ξ0 = σ 1 - ---1--- (1 - ξ2) Ψ (ψ,θ)

(3.526)

and using the relation ξdξ = Ψ (ψ,θ) ξ₀dξ₀, one obtains for the zero order distribution function f₀

f₀	= ∫ _-1⁺¹f₀P_mdξ
	= Ψ∫ _-1⁺¹f₀×
	HP_mdξ₀	(3.527)

since f₀ is constant along a magnetic field line, i.e. f₀ (t,ψ, θ,p,ξ) = f₀ (t,ψ, p,ξ0) . Here the Heaviside function H indicates that only electrons who reach the poloidal position θ must be considered. By expanding part of the integrand in (3.527) as a series of Legendre polynomials, according to the relation

( ∘ -----------) ∞∑ ( ) ξ0H |ξ0|- 1 - ---1--- Pl (ξ ) = m ′ + 1 c(mm′)(ψ,θ)Pm ′ (ξ0) ξ Ψ (ψ,θ) m ′=0 2

(3.528)

with

------------ ∫ +1 ( ∘ ) c(mm′)(ψ,θ) = ξ0 H |ξ0|- 1 - ---1--- Pm (ξ)Pm ′ (ξ0)dξ0 -1 ξ Ψ (ψ,θ)

(3.529)

one obtains finaly

∞ ( ) ∫ (m ) ∑ ′ 1- (m′) +1 (0) f0 (t,X, p) = Ψ (ψ, θ) ′ m + 2 cm (ψ, θ) -1 f0 (t,ψ,p,ξ0)Pm ′ (ξ0)dξ0 m =0

(3.530)

∑∞ ( ) ′ f(0m)(t,X,p) = Ψ (ψ,θ) m ′ + 1 c(mm ′)(ψ,θ)f (00)(m )(t,ψ,p) m ′=0 2

(3.531)

where

(0)(m′) ∫ +1 (0) f0 (t,ψ, p) = f0 (t,ψ, p,ξ0) Pm′ (ξ0)dξ0 -1

(3.532)

For the first order distribution function, f₁ = + g, since g is constant is constant along a field line, its contribution is the same as for f₀. Because has an explicit dependence upon θ, which is given by relation (3.280),

	= ∫ _-1⁺¹ P_mdξ
	= ∫ _-1⁺¹ HP_mdξ₀
		(3.533)

------------ ( ∘ ) ∞∑ ( ) H |ξ0|- 1- ---1--- Pm (ξ) = m′ + 1∕2 ^c(mm ′)(ψ, θ)Pm ′(ξ0) Ψ (ψ,θ ) m ′=0

(3.534)

with

( ∘ -----------) (m′) ∫ +1 1 ^cm (ψ,θ) = H |ξ0|- 1- Ψ-(ψ,-θ) Pm (ξ)Pm ′ (ξ0) dξ0 -1

(3.535)

then expression (3.533) becomes

∞∑ ( ) ∫ +1 ^f(m)(t,X, p) = m ′ + 1 ^c(m′)(ψ,θ) ^f(0)(t,ψ, p,ξ0) Pm′ (ξ0)dξ0 m ′=0 2 m -1

(3.536)

Since

(0)(m′) ∫ +1 (0) ^f (t,ψ, p) = ^f (t,ψ, p,ξ0) Pm′ (ξ0)dξ0 -1

(3.537)

one obtains finaly

∑∞ ( ) f^(m )(t,X, p) = m′ + 1- ^c(m ′)(ψ, θ) ^f(0)(m′)(t,ψ, p) m′=0 2 m

(3.538)

It is interesting to notice that the determination of the f (t,X, p) does not require the explicit evaluation of the distribution function f (t,X, p,ξ) at all poloidal positions, and only its value at B_min is needed for the 4 - D problem that is represented by the bremsstrahlung. This result which is a direct consequence of the weak collisional or “banana” regime, is very important for the numerical evaluation. Indeed, all the physics of the trapped-passing electrons is incorporated in the coefficients f (t,X, p) , while the contribution arising from magnetic field line helicity is independently described by cosθ_d = ^
b ⋅ ^
d .

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