Coulomb collisions

4.1 Coulomb collisions

4.1.1 Small angle scattering

4.1.2 Linearized collision operator

The collision operator used in the calculations may be expressed as¹

∑ ∑ C (f) = C (f,fss′) + C (f, f) s s′

(4.1)

where ∑ _s ∑ _s^′C (f,f ′)
ss describe interactions between electrons and ions of species s in the ionization state s^′ and C (f,f) is the self-collision contribution, as discussed in Ref. [?]. For the electron-ion collisions, it is considered that f_ss^′ is a Maxwellian distribution function, the corresponding temperature being T_ss^′. In the application of the code here foreseen, including RF heating and current drive, collisions dominate thermal particles, and therefore the distribution function f may be expanded about the Maxwellian f_M according to the relation

f ≃ fM + δf

(4.2)

The self-collision operator C (f,f) may be consequently approximated by its linearized form

C (f,f) ≃ C (f,fM )+ C (fM ,f)

(4.3)

where the the relation C (fM ,fM) = 0 has been used, and terms of order δf² have been ignored. It can be shown that the operator C (f,fM ) may be computed as C (f,fss′) and expressed in a conservative form

∑ ∑ C (f,fss′) + C (f,fM ) → ∇p ⋅Sp (f ) s s′

(4.4)

where component S_p and S_ξ of the flux S_p are

∂f0 ∘1----ξ2 ∂f0 Sp = - Dpp----+ --------Dp ξ----+ Fpf0 (4.5) ∂p ∘ --p--- ∂ξ ∂f0- --1---ξ2 ∂f0- Sξ = - D ξp∂p + p D ξξ∂ξ + Fξf0 (4.6)

In the standard notations used in Ref. [?]

(| Dpp = A (ψ, p) |{ D = 0 D pξ= 0 ||( ξp D ξξ = Bt(ψ,p)

(4.7)

and

{ Fp = - F (ψ,p) F ξ = 0

(4.8)

The term C (f ,f )
M requires is specific treatment. By expanding f as a sum of Legendre harmonics according to the relation

∑∞ f (t,X, p,ξ) = (m + 1∕2)f(m)(t,X, p)Pm (ξ) m=0

(4.9)

with

∫ +1 f (m )(t,X, p) = f (t,X, p,ξ)P (ξ)d ξ -1 m

(4.10)

one obtains

∑∞ ( (m) ) C (fM ,f) = (m + 1∕2)C fM ,f (t,X,p )Pm (ξ) m=0

(4.11)

By definition, f (t,X,p) ≃ f_M and, since P₀ (ξ) = 1,

( (m=0) ) C fM ,f (t,X, p)P0 (ξ) ≃ C (fM ,fM ) = 0,

The first non-zero term in the series is then kept, so that

( 3 ) C (fM ,f) ≃ C fM ,--ξf(m=1 )(t,X, p) 2

(4.12)

since P₁ (ξ) = ξ. By construction the linearized electron-electron collision operator conserves momentum, but not energy, so there is no need to introduce an energy loss term in the kinetic equation. Since f is an integral of f, the term C (f ,f)
M introduce a non-linear dependence in the Fokker-Planck or drift kinetic equation. However, even if it is crucial for the current drive problem, including the determination of the boostrap current level, this non-linearity remains weak, so that the rate of convergence towards the solution of the kinetic equation is not significantly affected, even if this term is treated explicitely, regarding the time scheme. For the calculations, the notation used in Ref. [?] is considered, and

( ) ( ) C fM , 3ξf(m=1 )(t,X, p) = - 3ξI fM ,f (m=1 )(t,X, p) 2 2 3- = - 2ξI (t,X, p) (4.13)

In the code, it is possible to choose different collision models for simulations. Most of them have been implemented for benchmarking, the only realistic one being the Belaiev-Budker relativistic collision operator.

4.1.3 Electron-electron collision operators

Belaiev-Budker relativistic collision model

In the calculations, the Belaiev-Budker collision operator is used for weakly relativistic plasmas. This operator ranges from non-relativistic to fully relativistic limits and is therefore very well suited for studying the heating and current drive problems. Its recent formulation in terms of Rosenbluth-like potential has open the possibility to use it in numerical calculations (Ref.[?]). Following the work done in Ref. [?] , normalized coefficients A^ee --
(ψ,p) , F^ee --
(ψ, p) and B_t^ee are

-- -- -- -- ---- Aee(ψ,p) = F-1(ψ,p)+--F2(ψ,-p)Te(ψ ) v

(4.14)

and

-- -- -- Fee(ψ,p) = F 1(ψ,p)+ F 2(ψ, p)

(4.15)

Here,

-- -- 4π--- -- 4π--- -- F 1(ψ,p) = v2 F11 (ψ, p)+ p2F12 (ψ, p)

(4.16)

-- -- 4π ( γζ) -- -- F 2(ψ,p) = --- 1- --- F 21 (ψ, p) v z

(4.17)

-- ∫ p -- ( ) F 11(ψ,p) = p′v′f M ψ,p′ dp′ 0

(4.18)

- -- -- ∫ p -′′( γ ′ζ′) -- ( -′) -′ F12(ψ, p) = pv 1- -z′- fM ψ, p dp 0

(4.19)

-- ∫ ∞ -- ( ) F21(ψ, p) = - p′fM ψ, p′ dp′ p

(4.20)

When relativistic corrections are neglected, 1 - γζ∕z ≈ 0, expressions derived from usual Rosenbluth potentials are recovered [?], and

Aee,nr(ψ, v) = 4πF-11,nr(ψ,v)T-e(ψ) v3

(4.21)

and

-ee,nr -- 4π-- -- F (ψ, v) = v2F 11,nr(ψ,v)

(4.22)

with

-- -- ∫ v ---- ( -) -- F11,nr(ψ, v) = v′2fM ψ, v′ dv′ 0

(4.23)

since p = v in this limit. The expression of F_11,nr for a Maxwellian background is

-- ∫ v ( -′2 ) F11,nr (ψ, v) = ---ne-(ψ-)---- v′2 exp - --v---- dv′ [2πTe (ψ )]3∕2 0 2Te(ψ )

(4.24)

can then be evaluated analyticaly using the coordinate transformation u_ψ = v∕ ∘ -------
2Te(ψ ) ,

-- -- ne(ψ )∫ uψ 2 ( 2) F 11,nr(ψ,v) = -π3∕2- x exp - x dx 0

(4.25)

and by integrating by parts

∫ uψ ( ) [ ( )]u ∫ uψ ( ) exp - x2 dx = x exp - x2 0ψ+ 2 x2 exp - x2 dx 0 0

(4.26)

one obtains

-- -- [ ] F11,nr(ψ, v) = ne(ψ)-Erf (uψ)- u ψErf ′(u ψ) 4π

(4.27)

where

∫ -2-- uψ 2 Erf (u ψ) = √π- exp(- x )dx 0

(4.28)

Consequently

-- n- (ψ) [ ] Fee,nr(ψ, v) =--e2-- Erf (u ψ)- uψErf ′(uψ ) v

(4.29)

and for A^ee,nr (ψ,v) it comes

-- -- Aee,nr(ψ,v-) = ne-(ψ)T-e(ψ) [Erf (u )- u Erf ′(u )] v3 ψ ψ ψ

(4.30)

expressions which correspond to the Maxwellian limit discussed later in this section. The high velocity limit corresponds to the condition u_ψ ≫ 1, and in this case, since lim_{u_ψ→∞}Erf (uψ) = 1, it comes readily

--ee,nr -- ne (ψ) ulψim→ ∞F (ψ,v ) =--v2--

(4.31)

and

-- n- (ψ)T- (ψ) lim Aee,nr(ψ,v) = --e---3-e--- uψ→∞ v

(4.32)

which both are well known relations.

Much in the same way, the expression of coefficient B_t^ee for pitch-angle scattering is

Bee (ψ, p) = Bt1 (ψ,p)+ Bt2 (ψ,p) t

(4.33)

with

-- ∑5 -- Bt1 (ψ,p) = 4π B[tn1](ψ, p) n=1

(4.34)

and

-- -- ∑5 --[n] -- Bt2(ψ,p) = 4π B t2 (ψ, p) n=1t2

(4.35)

where

∫ p B-[1](ψ,p) = -1- p′2f-M (ψ,p′)dp′ t1 2v 0

(4.36)

- --[2] -- 1 ∫ p-′4-- ( -′) -′ B t1 (ψ,p) = - 6vp2 p f M ψ,p dp 0

(4.37)

-- ∫ p -′ -- ( ) B[t31] (ψ, p) = --1--- p′2J1-(p-)fM ψ, p′ dp′ 8γ2z2 0 γ′

(4.38)

∫ - -- --[4] -- --1---- p-′2 J2(p′)-- ( -′) -′ B t1 (ψ,p) = - 4z2 0 p γ′ f M ψ,p dp i+1∕2

(4.39)

- --[5] -- 1 ∫ pp′2( ′ ζ′) -- ( --′) -′ B t1 (ψ,p) = ----2--- γ-′ γ - z′ fM ψ,p dp 4γ i+1∕2 0

(4.40)

and

-- ∫ ∞ -′2 -- B [1t2](ψ,p) = 1- p--fM (ψ,p′)dp′ 2 p v′

(4.41)

∫ -- --[2] -- γ2- ∞ -p′2--- -′ B t2 (ψ,p) = - 6 p γ′2v′fM dp

(4.42)

-[3] -- J1 (p)∫ ∞ p-′2 1 -- ( -′) -′ Bt2 (ψ, p) =-8γz2 - -v′γ-′2fM ψ, p dp p

(4.43)

-- -- ∫ ∞ -′2 -- ( ) B [4t2](ψ,p) = - γJ2(2p)- p′-1′2-fM ψ,p′ dp′ 4z p v γ

(4.44)

( ) ∫ ∞ B[5](ψ, p) = ---1-- γ - ζ- p′2v′f- (ψ, p′) dp′ t2 4γ p2 z p M

(4.45)

Here,

( ) -- 3 J1(p) = - 3 γ + ζ z-+ 2z

(4.46)

-- ζ 2 J2 (p) = γ +--- -γz2 z 3

(4.47)

with

†-- z = β thp

(4.48)

∘ ----2- γ = 1+ z

(4.49)

ζ = sinh- 1z

(4.50)

and f_M --
(ψ,p) is the weakly relativistic normalized Maxwellian distribution function given in Sec.6.3.5.

The first order Legendre correction of the collision operator --
(t,X, p) is expressed as


	= f + ₁ + p₂	(4.51)

where

--(-- -(m=1) --) ∑10 -[n] -- I1 f M,,f0 (ψ,p) = I1 (ψ,p ) n=1

(4.52)

and

7 --(-- -(0)(m=1) --) ∑ -[n] -- I2 f M,,f0 (ψ,p) = I2 (ψ,p ) n=1

(4.53)

The set of coefficients ₁ --
(ψ, p) is

- -[1] -- 1 ∫ p p′3--(m=1 ) -- -′ I1 (ψ, p) = --------- γ′f 0 (ψ,p)dp 3Te,l+1∕2 0

(4.54)

-- 2γ ∫ p -- I [21](ψ,p) = - ---i+1∕2-- p′3f(0m=1)(ψ, p)dp′ 3Te,l+1∕2 0

(4.55)

∫ - -[3] -- -γi+1∕2--- p p′5--(m=1 ) -- -′ I1 (ψ, p) = 5T2 0 γ′f 0 (ψ,p)dp e,l+1∕2

(4.56)

- -[4] -- ∫ p p′( ′ ζ′) -(m=1) -- -′ I1 (ψ,p) = γ′ γ - z′ f0 (ψ,p) dp 0

(4.57)

-- γ ∫ p-′3 -′ -- I[51](ψ,p) = - --i+1∕2- p--J2(p-)f(0m=1)(ψ,p) dp′ Te,l+1∕2 0 γ ′ z ′2

(4.58)

-- -- ∫ --- ( ) -[6] -- γp2 --5T-e(ψ) pp′3 3-- 3γ′ζ′ --(m=1 ) -- -′ I1 (ψ, p) = 6T-2(ψ) 0 γ′ 1+ z′2 - z′3 f 0 (ψ,p)dp e

(4.59)

- -[7] -- γ ∫ p p′3J3 (p′)--(m=1 ) -- -′ I1 (ψ, p) = --†2-2---- γ′--z-′-f 0 (ψ,p)dp 2βth Te (ψ ) 0

(4.60)

∫ p-′3 -′ I[8](ψ,p) = ---γ--- p--J1(p-)f(m=1)(ψ,p) dp′ 1 2T e(ψ) 0 γ ′ z′2 0

(4.61)

-- ∫ --- ( ) -[9] -- --p2-- pp′ γ-′ζ′ --(m=1 ) -- -′ I1 (ψ, p) = T-e(ψ) 0 γ ′ z′ - 1 f 0 (ψ,p)dp

(4.62)

- -[10] -- γ2 ∫ pp′3J4 (ψ )-(m=1) -- -′ I1 (ψ,p) = -----†2-2--- -γ′--z′--f0 (ψ, p)dp 12 βthT e (ψ) 0

(4.63)

and the coefficients ₂ (ψ,p) are

-- ∫ ∞ -- I[12](ψ,p) = ---1--- 1-f(0m=1)(ψ,p) dp′ 3T e(ψ) p γ′

(4.64)

( ) ∫ --[2] -- --2γ--- --p2--- ∞ -(m=1 ) -- -′ I 2 (ψ,p) = - 3T- (ψ) + -2 - f0 (ψ, p)dp e 5Te (ψ) p

(4.65)

( ) -[3] -- ζ 1 ∫ ∞ 1 -(m=1) -- -′ I2 (ψ,p) = γ - z- p2 - γ′f0 (ψ,p) dp p

(4.66)

--[4] -- J (p) ∫ ∞ -(m=1) -- -- I 2 (ψ,p) = - -22----- - f0 (ψ, p)dp′ z Te(ψ) p

(4.67)

-- ( ) ∫ ∞ ( --′2 -- ) -- I[25](ψ, p) = 1+ 3-- 3γζ- ---1--- γ′p----5T-e(ψ)- f (m0=1 )(ψ,p)dp′ z2 z3 6T 2e (ψ) p γ′

(4.68)

( ) -[6] -- J3(p) J1(p) J4(p) ∫ ∞ -(m=1) -- I2 (ψ,p) = ---†2-2----+ --2------- -----†2-2--- - f0 (ψ,p) dp′ 2zβthTe (ψ ) 2z Te(ψ ) 12z βth Te (ψ ) p

(4.69)

( )∫ ∞ -′2 I[7](ψ, p) =----1--- γζ- 1 p- f-(m=1 )(ψ,p)dp′ 2 p2T e(ψ) z p γ′ 0

(4.70)

where

-- 3γζ- 3- 2-3 J3(p) = - z + z + z - 5z

(4.71)

( ) -- 15 15 J4 (p ) = γζ -z2 + 6 - z-+ 11z

(4.72)

Relativistic Maxwellian background

The relativistic Maxwellian limit corresponds to that case where the first order Legendre correction for momentum conservation is neglected, but nevertheless using the Beliaev-Budker formulation for coefficients A^ee --
(ψ, p) , F^ee and B_t^ee. This is an academic case that allows only fruitful comparison with some theorerical works for code benchmarking.

Non-relativistic Maxwellian background

The non-relativistic collision operator with a Maxwellian background is extensively discussed in Ref. [?]. It is an interesting model, since analytical evaluation of the collision integrals may be performed. Its validity is restricted to the limit γ - 1 ≪ 1, where γ is the Lorentz factor defined is Sec.6.3.4. In that case v = p is the unit system here employed. Using the standard notations

-- --ee -- ne(ψ-)-1-[ ′ ] A (ψ,p) = 2v- u2ψ Erf (uψ)- uψErf (uψ)

(4.73)

-ee -- ne (ψ)[ ′ ] F (ψ,p ) =--v2-- Erf (u ψ)- uψErf (uψ )

(4.74)

and

-ee -- ne (ψ) 1 [( ) ] Bt (ψ,p) = -------2- 2u2ψ - 1 Erf (u ψ)+ uψErf ′(uψ ) 4v uψ

(4.75)

where

v- uψ = ∘-------- 2Te(ψ )

(4.76)

∫ -2-- x ( 2) Erf (x) = √ π- exp - y dy 0

(4.77)

is the well know error function defined in Refs [?] and [?] , and its derivative

′ 2 ( 2) Erf (x) = √-π-exp - x

(4.78)

The relation

-ee --2 -- F--(ψ,-p)= 2vuψ-= ---v-- Aee(ψ, p) v2 T e(ψ)

(4.79)

which ensures that the Maxwellian is the correct solution when collisions is the only physical process. In that limit, self-collisions are neglected.

High-velocity limit

Though the high velocity limit u_ψ ≫ 1 corresponds to a restricted range of applications regarding the full electron-electron collision operator, it can contribute usefuly to comparisons with some theoretical calculations. Starting from expressions given in Ref. [?],

-- -- -- Aee(ψ, p) = ne(ψ)T e(ψ) v3

(4.80)

-- Fee(ψ, p) = ne-(ψ) v2

(4.81)

and

-ee -- ne (ψ)[ T-e(ψ)] Bt (ψ,p) = ------ 1 - ---2-- 2v v

(4.82)

With these definitions, the ratio

-ee -- -- F--(ψ,p)-= --v--- Aee(ψ,p) Te(ψ)

(4.83)

is well recovered. In that limit, self-collisions are neglected.

Non-relativistic Lorentz model

This case corresponds to the limit where only pitch-angle scattering of electrons on massive ions with T_ss^′ (ψ) = 0, with large Z_ss^′. Consequently, large simplifications may be performed, and

-ee -- --ee -- -ee -- A (ψ, p) = F (ψ,p) = Bt (ψ,p) = 0

(4.84)

This simple model is very interesting since analytical expressions may be obtained in this limit, which allow accurate code benchmarking, especially for the bootstrap current problem in arbitrary magnetic configuration. Obviously, self-collisions are neglected by definition.

Ultra-relativistic Møller model

The growth rate of runaway electrons by a large constant electric field is often studied using the ultra-relativistic model as derived by C. Møller in the early 1930’s [?]. In this limit, since the ratio of the friction term F^ee --
(ψ,p) to the diffusion one A^ee --
(ψ, p) scales like v as indicated in 4.79 , the contribution of the diffusion A^ee (ψ,p) is simply neglected in the limit v → c. Hence,

-- -- Fee(ψ, p) = ne-(ψ) v2

(4.85)

and

-- Aee (ψ,p) = 0

(4.86)

For the pitch-angle term, the term is simply

-ee -- ne (ψ) Bt (ψ, p) =--2v--

(4.87)

It is important to recall that the term β_th^† arises from the definition of the normalization for p as discussed in Sec.6.3.4. It can be observed that the Møller collision model is equivalent for e-e interactions to the high-velocity limit of the e-e standard collision operator, neglecting all terms of the order of v^-3. Therefore, the model may have analytical solutions since the partial derivative in p of the collision operator is of the first order. However, as -vA^ee --
(ψ, p) ∕F^ee≠T_e (ψ) high numerical instabilities are found in the code. This model is equivalent to T_e (ψ ) = 0 which is obviously not consistent with the initial assumptions. Therefore, in order to ensure a correct numerical stability, the expression 4.80 for A^ee is used with the Møller model. It has been cross-checked that this does not change the final results.

4.1.4 Electron-ion collision operators

Non-relativistic Maxwellian background

Since ions mass is much larger than electron ones, their dynamics is almost non-relativistic. Consequently, electron-ion collisions may be described in this limit considering a Maxwellian ion background. Expressions for arbirary type of ions is also given in Ref. [?] and their validities are also restricted to the limit γ - 1 ≪ 1, where γ is the Lorentz factor defined is Sec.6.3.4. . In that case v = p is the unit system here employed. Using the standard notations

-- ∑ ∑ [ ( ) ( )] ln Λ† Aei(ψ,p) = -1- -1′--Erf usψs′ - uψErf ′ ussψ′ ---e∕ss′Z2ss′nss′ (ψ) 2v s s′ usψs2 lnΛ †e∕e

(4.88)

-- 1 ∑ ∑ [ ( ′) ′ ( ′)] ln Λ† ′ 1 F ei(ψ, p) =--2 Erf usψs - usψs Erf ′ usψs ---e∕†ssZ2ss′nss′ (ψ )-- v s s′ lnΛ e∕e ms

(4.89)

and

--ei -- 1 ∑ ∑ 1 [( ′ ) ( ′) ′ ( ′) ] lnΛ †e∕ss′ -- B t (ψ,p) =--- -ss′2- 2usψs2- 1 Erf usψs + usψsErf ′ usψs ----†---Z2ss′nss′ (ψ) 4v s s′ uψ ln Λe∕e

(4.90)

where

-- ss′ ---v---- --------v------- uψ = √2v-† = √ -∘ ------------ th,ss′ 2 Tss′ (ψ)∕ms

(4.91)

Here, v_th,ss^′^† is the thermal velocity of species s in ionization state s^′, while Erf(x) and Erf^′(x) have the same expressions as for the electron-electron collision term. For a single ion species s fully ionized, the ratio

Fei(ψ,p) v- -ei------= ------ A (ψ,p) T s(ψ)

(4.92)

which means that the electron population is thermalized to the ion temperature T_s (ψ ) . The quantities lnΛ_e∕ss^′^† and lnΛ_e∕e^† are the reference Coulomb logarithms defined in Sec. 6.3.4.

High-velocity limit

For most current drive studies like for the Lower Hybrid wave where the resonance condition is far from the thermal bulk, it is reasonable to consider the high velocity limit of the electron-ion collision operator. Corresponding coefficients A^ei --
(ψ, p) , F^ei --
(ψ,p) and B_t^ei are

-- ∑ ∑ ln Λ† -- Aei (ψ, p) = 1- Z2ss′nss′ (ψ)---e∕ss′T-ss′ (ψ-) v3 s s′ lnΛ †e∕e ms

(4.93)

-- ∑ ∑ lnΛ † ′ F ei(ψ,p) = 12 Z2ss′nss′ (ψ )---e∕ss-1-- v s s′ ln Λ†e∕e ms

(4.94)

and

-- 1 ∑ ∑ lnΛ † ′( T- ′(ψ )) Beti(ψ,p) = --- Z2ss′nss′ (ψ)---e∕s†s- 1- -ss-2--- 2v s s′ lnΛ e∕e msv

(4.95)

where the double sum ∑ _s ∑ _s^′ takes into account of all ions species s in ionization state s^′. Here, n_ss^′ (ψ ) is the normalized ion density at ψ, as introduced in Sec. 6.3.1, and m_s is the ion rest mass normalized to the electron rest mass m_e. Coefficients for the electron-ion collisions given in Ref.[?] are well recovered. Like for the Maxwellian limit,

-- Fei(ψ,p) v- -ei------= ------ A (ψ,p) T s(ψ)

(4.96)

which means that the electron population is thermalized to the ion temperature T_s (ψ ) , when a single ion species s is considered.

Non-relativistic Lorentz model

Since only pitch-angle electron scattering on massive ions with T_ss^′ (ψ) = 0, with large Z_ss^′ is considered in this model, by definition

-ei -- --ei -- A (ψ,p) = F (ψ,p ) = 0

(4.97)

while

--ei -- B t (ψ,p) = 1∕2

(4.98)

The solutions of the Fokker-Planck and the drift kinetic equations is independent in this limit of the B_t^ei value. Here the standard value 1∕2 is chosen as used in several publications for analytical calculations.

Ultra-relativistic Møller model

A simplified term is used where

-ei -- --ei -- A (ψ,p) = F (ψ,p ) = 0

(4.99)

since ion contribution is almost negligible as compared to electrons and

† --ei -- -1-∑ ∑ 2 -- ln-Λe∕ss′- B t (ψ,p ) = 2v Zss′nss′ (ψ) lnΛ † s s′ e∕e

(4.100)

by combining the high-velocity limit expression 4.95, and the momentum dependence given in Ref. [?].

4.1.5 Large angle scattering

4.1.6 Bounce Averaged Fokker-Planck Equation

Flux conservative term

In the Fokker-Planck equation, the diffusion and convection elements are bounce-averaged according to the expressions (3.189)-(3.194), which gives, using (4.7)-(4.8),

DC (0) = {A (ψ, p)} (4.101) pCp(0) D pξ = 0 (4.102) C(0) D ξp = 0 (4.103) C(0) { ξ2 } D ξξ = Ψ-ξ2B (ψ,p) (4.104) 0

and the convection components

FC(0) = - {F (ψ,p)} (4.105) p FCξ(0) = 0 (4.106)

and therefore

DCp(p0) = A (0)(ψ,p) (4.107) C(0) D pξ = 0 (4.108) DC (0) = 0 (4.109) ξp DC (0) = λ2,-1,0B (0) (ψ, p) (4.110) ξξ λ t

and

C (0) (0) Fp = - F (ψ,p) (4.111) FC (0) = 0 (4.112) ξ

Here, coefficients A (p,ψ ) , B_t (p,ψ) and F (p,ψ) are determined at the location where B = B₀ on the magnetic flux surface. However, since A, B_t and F are only functions of the density and temperature that are flux surface quantities as shown in Sec. 4.1.1, their respective values are consequently independent of the poloidal position and therefore, A = A, B_t = B_t and F = F. The bounce coefficient λ_2,-1,0 is defined as (2.66)

{ ξ2 } λ2,- 1,0 = λ Ψξ2- 0

(4.113)

For reference to the litterature ([?]), note that we could also perform the following transformation

{ } { ( ) } -ξ2- 1---Ψ--1---ξ02 Ψξ20 = Ψ ξ20 { ( )} = 1 - 1- 1- 1- ξ20 Ψ Δb = 1 - -ξ2 (4.114) 0

with

{ 1 } Δb ≡ 1 - -- Ψ

(4.115)

The evaluation of Δ_b for circular concentric flux surfaces in given in Appendix ??.

First order Legendre correction

Concerning the term that ensures momentum conservation in the collision operator, one must evaluate

{ ( 3 (m=1 ))} 3 { ( (m=1))} C fM ,--ξf0 = --- ξI fM ,f0 2 2

(4.116)

Making the substitution Ψξ₀dξ₀ = ξdξ in the integral f₀ = ∫ _-1⁺¹ξf₀ (p,ξ,ψ,θ) dξ, one obtains

∫ √ ----- ∫ (m=1 ) - 1-1∕Ψ (0) +1 (0) f0 (p,ξ,ψ,θ) = -1 Ψ ξ0f0 (p,ξ0,ψ)d ξ0 + √1--1∕ΨΨ ξ0f0 (p,ξ0,ψ) dξ0

(4.117)

where the limits of integration come from the relation ξ (ψ, θ,ξ0) = σ ∘ -----------(-----2)
1 - Ψ (ψ,θ) 1 - ξ0 . Since f₀ is symmetric in the region of the phase space ξ₀ ∈ ( )
- ∘1----1∕Ψ,∘1----1∕Ψ- which corresponds to trapped orbits,

----- ∫ + √1-1∕Ψ √----- Ψξ0f(00)(p,ξ0,ψ )dξ0 = 0 - 1-1∕Ψ

(4.118)

one get

∫ +1 f (m=1 )(p,ξ,ψ,θ) = Ψ ξ0f (0)(p,ξ0,ψ)dξ0 0 - 1 0 ∫ +1 (0) = Ψ ξ0f0 (p,ξ0,ψ) dξ0 - 1 = Ψf (00)(m=1)(p,ξ0,ψ) (4.119)

where f₀

is the Legendre integral evaluated at B₀ = B (ψ,θ )
0

, independent of θ. Since the operator

is linear,

{ ( ) } ( ) 3- ξI f0M,f(m=1 ) = 3{ξΨ }I fM ,f(0)(m=1) 2 0 2{ } 0 -ξ 3- ( (0)(m=1)) = σ σξ0Ψ 2ξ0I fM ,f0 -- ( ) = λ1,1,0-3ξ0I fM ,f (0)(m=1 ) (4.120) λ 2 0

and consequently

{ ( )} ( ) C f , 3ξf(m=1) = C (0) f ,f(0)(m=1) M 2 0 M 0 λ- 3 ( ) = - -1,1,0--ξ0I fM ,f(00)(m=1 ) (4.121) λ 2

Expression of λ_1,1,0 This coefficient is expressed as

[ ] -- σ 1 ∑ ∫ θmax dθ 1 r B λ1,1,0 = -- -- ---||---||------ σΨ ^q 2 σ T θmin 2π |ψ^⋅^r| RpBP

(4.122)

Since the integral is odd in σ, the sum over trapped particles vanishes, λ_1,1,0 = 0 for trapped electrons, and λ_1,1,0 = λ_1,1,0^P≠0 for circulating ones. Hence

-P 1 ∫ 2π dθ 1 r B λ1,1,0 = -- ---||----||------Ψ ^q 0 2π |^ψ ⋅ ^r|Rp BP ∫ 2π 2 = 1- dθ-|-1--|r---B---- (4.123) ^q 0 2π ||^ψ ⋅ ^r||Rp BP B0

Case of circular concentric flux-surfaces In this case, ϵ = r∕R_p, ^
ψ ⋅ = 1 and since the ratio B∕B_P is a function of r only

∫ -P 1--B- 2π dθ---1+-ϵ--- λ1,1,0 = q^BP ϵ 2π 1+ ϵcos θ ∫ 2π 0 = dθ---1+-ϵ--- 0 2π 1+ ϵcosθ = s* (4.124)

using the relation

B_P∕B = ϵ. The integral s^*, according to the old notations found in the litterature ([?]),

* ∫ 2π dθ 1 + ϵ s = 2π-1+-ϵ-cosθ 0

(4.125)

can be performed analytically, as shown in Appendix ??, and

-P ∘ 1-+-ϵ λ1,1,0 = ----- 1 - ϵ

(4.126)

Moreover, in this limit, λ_1,1,0^P = λ_1,-1,2^P, as shown in Sec.4.2.2.

4.1.7 Bounce Averaged Drift Kinetic Equation

Flux conservative term

In the first order drift kinetic equation, the diffusion and convection flux elements related to are bounce-averaged according to the expressions (3.218)-(3.223), which gives, using (4.7)-(4.8),

C (0) { ξ } ^Dpp = σ σ Ψξ-A (ψ, p) (4.127) C (0) 0 ^Dpξ = 0 ^C (0) Dξp = 0 C (0) { σξ3 } ^Dξξ = σ Ψ2ξ3Bt (ψ,p) 0

and the convection components

{ } C(0) ξ F^p = - σ σ Ψξ-F (ψ, p) (4.128) ∘ ----2-0{ } F^C(0) = --1--ξ0-σ σξ-(Ψ----1)B (ψ,p) ξ pξ30 ξ0Ψ2 t

and therefore

-- D^C (0) = λ1,--1,0A (ψ,p) (4.129) pp λ D^C (0) = 0 pξ D^Cξ(p0) = 0 -- D^C (0) = λ3,--2,0Bt (ψ,p) ξξ λ

and

-- C(0) λ1,-1 ^Fp = - --λ--F (p) (4.130) ∘ ----2-(-- -- ) ^F C(0) = --1--ξ0--λ1,-1,0 --λ1,-2,0-Bt (ψ,p) ξ pξ30 λ

The following bounce coefficients are defined (2.66)

{ } λ1,-1,0 = λσ -σξ- (4.131) Ψ ξ0 -- { σξ3 } λ3,-2,0 = λσ --2-3 (4.132) { Ψ ξ0} λ- = λσ -σξ-- (4.133) 1,-2,0 Ψ2 ξ0

We also have the following relation, by expanding ξ²

-- ( )-- -- λ1,-2,0 - 1 - ξ20 λ1,- 1,0 λ3,- 2,0 = ----------ξ2----------- 0

(4.134)

First order Legendre correction

Concerning the term that ensures momentum conservation in the collision operator, one must evaluate

{ ( ) } 3-^ 3-{ ( ^(m=1 ))} C fM ,2ξf = - 2 ξI fM ,f

(4.135)

Making like for the Fokker-Planck term the substitution Ψξ₀dξ₀ = ξdξ in the integral = ∫ _-1⁺¹ξ (p,ξ,ψ,θ) dξ, one obtains

	=	∫ _-1^-Ψξ₀ dξ₀
		+ ∫ ⁺¹Ψξ₀ dξ₀		(4.136)

which becomes

∫ ^(m=1 ) 1 ^(0) f (p,ξ,ψ, θ) = -1H (|ξ0|- ξ0T )ξf (p,ξ0,ψ )dξ0

(4.137)

Then,

3{ ( )} 3 ( { }) -- ξI fM , ^f(m=1 ) = --I fM , ξ^f(m=1 ) 2 2

(4.138)

since f_M and are independent of θ. It is therefore necessary to evaluate

{ } 1 [ 1∑ ] ∫ θmaxdθ 1 r B ξ ξf^(m=1 ) = --- -- ---|----|-------0ξf^(m=1) λ ^q 2 σ T θmin 2π ||^ψ ⋅^r||Rp BP ξ [ ] ∫ -1- 1∑ θmaxdθ-|-1--|r--B-- ^(m=1) = λ ^q 2 θmin 2π |^ψ ⋅^r|Rp BP ξ0f σ T | | ξ0 ∫ 2π dθ 1 r B (m=1) = λ-^qH (|ξ0|- ξ0T) 2π-||---||-R--B--^f (4.139) 0 |ψ^⋅^r| p P

since

is independent of σ because of the integration over ξ₀, while [1∑ ]
2 σ

_Tξ₀ = 0 for trapped orbits. Hence,

{ } ξ^f(m=1) ∫ [ ∫ ] ξ0- 2π dθ|--1-|-r--B- 3- 1 ^(0) = λ^qH (|ξ0|- ξ0T) 0 2π|^ |Rp BP 2 -1H (|ξ0|- ξ0T)ξf dξ0 |ψ ⋅^r| ⌊ ⌋ ∫ 1 ∫ 2π = ξ0H (|ξ0|- ξ0T) H (|ξ0|- ξ0T) ^f(0)d ξ0 ⌈1 -dθ|-1--|-r--B-ξ⌉ λ -1 ^q 0 2 π||^ψ ⋅^r||Rp BP ⌊ ⌋ ξ ∫ 1 1 ∫ 2π dθ 1 r B ξ ξ2 = -0H (|ξ0|- ξ0T) H (|ξ0|- ξ0T) ^f(0)ξ0λd ξ0 ⌈-- --||----||------ -0-2⌉ λ -1 λ ^q 0 2π|ψ^⋅^r|Rp BP ξ ξ0 ∫ 1 { 2} = ξ0H (|ξ0|- ξ0T) ξ0H (|ξ0|- ξ0T ) ^f(0)λ ξ- dξ0 λ -1 ξ20 ξ0 ∫ 1 = --H (|ξ0|- ξ0T) ξ0H (|ξ0|- ξ0T ) ^f(0)λ2,0,0dξ0 (4.140) λ -1

Defining

∫ 1 ^f(0)(m=1 ) ≡ ξ0H (|ξ0|- ξ0T ) ^f(0)λ2,0,0dξ0 -1 ∫ 1 (0)-- = ξ0f^ λ2,0,0dξ0 (4.141) -1

one obtains

{ ( ) } ( ) 3- ^(m=1 ) ^(0) 3- ^(0)(m=1 ) C fM ,2ξf = C fM ,2 ξ0f ( ) = 3-ξ0-H (|ξ0|- ξ0T)I f0M , ^f(0)(m=1) (4.142) 2 λ

Case of circular concentric flux-surfaces

Note that in the case of circular concentric flux-surfaces, we can find analytical expressions for the bounce coefficients

-- ∫ 2π dθ 1 λ1,-1,0 = H (|ξ0|- ξ0T) ----- 0 2π∫Ψπ = ---1---H(|ξ0|- ξ0T ) dθ(1 + ϵcosθ) (1 + ϵ) 0 π ---1--- = (1 + ϵ) H(|ξ0|- ξ0T ) (4.143)

∫ -- 2π dθ-1- λ1,-2,0 = H(|ξ0|- ξ0T) 0 2πΨ2 H (|ξ |- ξ )∫ πdθ = ----0---20T-- ---(1+ ϵcos θ)2 (1 + ϵ) ( 0 π ) H (|ξ0|- ξ0T) ϵ2 = --------2---- 1+ 2- (4.144) (1 + ϵ)

so that

-- H (|ξ0|- ξ0T) ( ϵ(1- ϵ∕2)) λ3,-2,0 = ---------2--- ξ20 - ---------- (1+ ϵ)ξ0 (1 + ϵ)

(4.145)

Furthermore,

-- ∫ 2π dθ ξ λ2,0,0 = H (|ξ0|- ξ0T) 2π-ξ- ∫02π 0 2 = H (|ξ0|- ξ0T) dθ-ξ0ξ- 0 2π ξ ξ20 ∫ 2π dθ ξ 1- Ψ (1 - ξ2) = H (|ξ0|- ξ0T) ----0------2----0- 0 2π ξ ξ0 -λ [ ( 2) ] = H (|ξ0|- ξ0T)ξ20 1- 1 - ξ0 {Ψ } (4.146)

and using the expression of {Ψ }

given in Appendix ??,

1 2 ∑∞ 2m {Ψ} = λ-π- ^χm ξ0T J2m m=0

(4.147)

where J_2m is expressed in terms of complete elliptic integrals of the first and second kind, and _m is given by the recurrence relation _m = 2m--1
2m _m-1 with ₀ = 1,

∞ -- 2- ∑ [ ( 2)] ξ20Tm- λ2,0,0 = πH (|ξ0|- ξ0T) χm - ^χm 1- ξ0 ξ20 J2m m=0

(4.148)

Here χ_m is defined in Appendix ??. The series expansion is converging less rapidly than for λ_0,0,0 = λ, therefore, at least first three terms have to be kept for accurate calculations, so that the truncated expression is

-- [ ( ) ( ) ] λ2,0,0 ≃ 2-H (|ξ0|- ξ0T) J0 + 1- 1- ξ20T J2 + 3-- -1- ξ40TJ4 π 2 ξ20 8 2ξ20

(4.149)

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