Normalization and definitions

6.3 Normalization and definitions

6.3.1 Temperature, density and effective charge

In order to use parameters without dimensions in the calculations, all of them are normalized to reference values. Here temperature and densities correspond to flux surface-averaged values, though the usual bracket notation is not introduce, for sake on simplicity.

Normalized temperatures, are defined by the relations

-- † T-e(ψ) = Te(ψ )∕Te T ss′ (ψ ) = Tss′ (ψ)∕Te†

(6.36)

where the subscripts e stands for electrons and ss^′ for ion species s in the ionization state s^′, whose charge is Z_ss^′q_e. Here, both T_e (ψ) and T_e^† have defined units, while T_e^† is the reference value for the normalization. According to this rule, the normalized densities are defined in the same way

{ -- † ne (ψ) = ne(ψ)∕n e † nss′ (ψ) = nss′ (ψ )∕ne

(6.37)

and in order to ensure that all quantities are less than unity from the plasma center to the edge, references values are given by the condition

{ T†e = max (Te (ψ ),Tss′ (ψ )) † ne = max (ne(ψ ),nss′ (ψ ))

(6.38)

for 0 ≤ ψ ≤ ψ_max.

In principle, ion densities n_ss^′ (ψ) are obtained from a particle transport code transport code. However, when the plasma is made of two types of ions which are fully ionized (low Z limit), a simple model may be used in order to determine densities n_s (ψ ) with s = {1,2} , knowing electron density n_e (ψ) and effective charge Z_eff (ψ ) profiles. Using the standard definition of Z_eff as defined in Ref. [?]⁸,

2 Z (ψ ) = ∑ ∑ nss′ (ψ)Z-ss′ (6.39) eff s ′ nss′ (ψ)Zss′ s ∑ ∑ = ---1-- nss′ (ψ)Z2ss′ (6.40) ne (ψ) s s′

taking into account of the local electro-neutrality, n_e (ψ )

= ∑ _s ∑ _s^′n_ss^′ (ψ )

Z_ss^′, one obtains in this simple case a system of two equations with two unknowns,

{ n1 (ψ)Z21 + n2 (ψ)Z22 = ne (ψ)Zeff (ψ) n1 (ψ)Z1 + n2 (ψ )Z2 = ne (ψ )

(6.41)

Therefore,

(Z (ψ)- Z ) n1 (ψ ) = ne (ψ)--eff--------2-- Z1(Z1 - Z2 )

(6.42)

(Z1 - Zeff (ψ)) n2 (ψ ) = ne (ψ)-Z-(Z----Z--)-- 2 1 2

(6.43)

and using the normalization rule,

(Z (ψ)- Z ) n1 (ψ ) = ne (ψ)--eff--------2-- Z1(Z1 - Z2 )

(6.44)

n2 (ψ ) = ne (ψ) (Z1---Zeff-(ψ))- Z2(Z1 - Z2 )

(6.45)

since by definition Z_eff (ψ) has no units.

In the calculations, one may also consider several isotopes, since ion inertia play a significant role for the collision operator. Hence,

∑ m nss′ (ψ ) = nss′ (ψ) m

(6.46)

-- ∑ -m nss′ (ψ ) = nss′ (ψ) m

(6.47)

where the masses of isotope m of ion species ss^′ are m_ss^′^m. In the calculations

--m m m ss′ = m ss′∕me

(6.48)

where m_e is the electron rest mass.

6.3.2 Coulomb Logarithm

The electron-electron Coulomb logarithm lnΛ_ee is defined as the logarithm of the ratio of the maximum to the minimum impact parameters of the Coulomb collision, Λ_ee = b_max∕b_min. Due to screening effect at long distance, b_max = λ_D, where λ_D is the Debye length, while b_min is of the order of the Landau scattering length, as discussed in Ref. [?]. Therefore

[ -3] lnΛee(ψ ) = 31.3- 0.5 log(ne (ψ ) m )+ log(Te(ψ)[eV ])

(6.49)

and

† †[ -3] † lnΛee = 31.3- 0.5log(ne m )+ log(Te [eV ])

(6.50)

6.3.3 Time

Regarding time ordering which is considered in the derivation of the drift kinetic equation and the dominant role played by collisions for the electron momentum distribution function build-up, it is natural to scale time evolution to a reference electron-electron collision frequency ν_e^† according to the relation

- t = t∕τc†= tν†e

(6.51)

There is arbitrariness in the choice of ν_e^†, since one can use different definition. The relativistic electron-electron collision frequency is used in the code as given in Ref. [?], instead of the usual thermal one discussed in Ref.[?] and Ref.[?] , since its expression is more simple⁹,

4 † † 2 ν† = -qeneln-Λee-= 4πcren †ln Λ† e 4πε20m2ec3β†t3h β†t3h e ee

(6.52)

Here ε₀ = 8.854187818 × 10^-12 (F∕m ) is the free space permeability, c = 299792458 (km ∕s) the speed of light, q_e = -1.60217733 × 10^-19 (C ) the electric charge of the electron, m_e = 9.1093897 × 10^-31 (Kg ) the electron rest mass, and lnΛ_ee^† is the usual reference electron-electron thermal Coulomb logarithm [?] taken at T_e^† and n_e^† with

∘ ----- † T†e √ --- βth = m-c2-= Θ† e

(6.53)

where m_ec² = 510.99905 (keV ) is the electron rest mass energy. In expression 6.52, the classical electron radius r_e has been introduced, which is given by

2 re =----qe--- 4π ε0mec2

(6.54)

6.3.4 Momentum, velocity, and kinetic energy

In the calculations, the momentum p in relativistic units (mec) is normalized to the thermal reference value p_th^† = m_ev_th^†,

-- † p = p∕pth

(6.55)

and consequently

† † † pth∕mec ≈ vth∕c = βth

(6.56)

since thermal electrons are only weakly relativistic. The well known relativistic Lorentz correction factor γ is then simply given by the relation

∘ ------ ∘ ----†2---- γ = p2 + 1 = p2β th + 1

(6.57)

and in the non-relativistic limit, i.e. when p²β_th^†2 ≪ 1, γ ≈ 1.

Since in relativistic units, the total energy is linked to the relativistic momentum according to the expression,

2 (Ec + 1) = γ

(6.58)

it is straightforward to expression the kinetic energy E_c as a function of p in units of electron rest mass energy m_ec²

(∘ --------- ) Ec = mec2 p2-β †2th + 1- 1

(6.59)

Finally, concerning the normalization of the electron velocity v, one has

v∕c = p∕(γmec ) = pp†th∕ (γmec) --† = pβth∕γ (6.60)

and using v = v∕v_th^†, it comes

--† --† vvth∕c = pβth∕ γ

(6.61)

-- -- v = p∕γ

(6.62)

with

† † vth∕c = βth

(6.63)

6.3.5 Maxwellian electron momentum distribution

The relativistic Maxwellian f_M used in the calculations has the following dependence in p

[ p2 ] [ γ - 1] fM (ψ, p) ∝ α exp - --------- = α exp - ----- (1 + γ)Θ Θ

(6.64)

where Θ = T_e (ψ ) ∕m_ec² and γ² = p² + 1. By definition, f_M is normalized to the local density n_e (ψ) , and the parameter α is then given by the integral

∫ ∞ [ ] 4πα p2exp - γ --1 dp = ne(ψ ) 0 Θ

(6.65)

Recalling that γdγ = pdp,

ne(ψ) α = 4πexp-(Θ-1)Ξ--(Θ--1)- p

(6.66)

where

( ) ∫ ∞ ∘ ------ [ ] Ξp Θ-1 = γ γ2 - 1 exp - γΘ- 1 dγ 1

(6.67)

By integrating by parts,

( ) - 1∫ ∞ ( ) [ ] Ξp Θ -1 = Θ--- γ2 - 1 3∕2 exp - γΘ -1 dγ 3 1 ( ) 4(3∕2)!K2 Θ -1 = 3-√-π----Θ--1--- (6.68)

and using relations (n - 1∕2)

! =

!!∕2ⁿ and n!! = 1.3.....n for positive odd n values, one obtains readily

( ) ( ) K2 Θ- 1 Ξp Θ -1 = -----1--- Θ

(6.69)

and finally

ne(ψ ) [ γ - 1 ] fM (ψ,p) = -----------1--------1-exp - ----- 4πΘ exp(Θ )K2 ([Θ )] Θ = ---ne-(ψ-)----exp - γ- (6.70) 4πΘK2 (Θ -1) Θ

For Θ ≪ 1, using the large argument asymptotic development of K₂ ( -1)
Θ ≈ ∘ ----
π∕2 √ --
Θ exp(-Θ^-1) + O ( - 3∕2)
Θ , the usual expression

[ ] -ne-(ψ)- γ---1 fM (ψ, p) ≃ [2πΘ ]3∕2 exp - Θ [ 2 ] = -ne-(ψ)-exp - ---p----- (6.71) [2πΘ ]3∕2 (1+ γ) Θ

is well recovered. This expression is only valid providing that γ - 1 ≪ 1. Therefore, the condition

p2β†2≪ 1 th

(6.72)

must be fulfiled. Since p may be as large as 30 in calculations, in order to correctly describe momentum dynamics of the fastest electrons, it results that

β † = Θ †2 ≪ 1∕30 th

(6.73)

----- T †≪ 511∘ 1∕30 ≈ 100keV e

(6.74)

Usually, this condition is well satisfied even in the core of tokamak plasmas, where T_e^† never exceeds 20keV. In normalized units,

[ ] ne(ψ)n †e p2β†2 fM (ψ, p) = ---------3∕2-exp - ---------th----†2 [2πΘ (ψ)] (1 + γ)T e(ψ)βth ne(ψ )n†e [ p2 ] = [-----------]3∕2 exp - ------------- (6.75) 2πT e(ψ)β †2th (1+ γ )Te(ψ )

since β_th^† = ∘ --------
T†e∕mec2

. Then, it turns out that

†-- † -- fM (ψ,p) = -nef M (ψ, p) = ne-fM (ψ,p) β†t3h p†3th

(6.76)

since p_th^† = β_th^†, with

-- ne(ψ ) [ p2 ] fM (ψ, p) ≈ [--------]3∕2 exp - ------------- 2π Te(ψ ) (1+ γ )Te(ψ )

(6.77)

which may be expressed in an alternative form, useful for calculating interpolation between full and half-grids,

[ ] -- ne(ψ) γ - 1 fM (ψ,p) ≈ [--------]3∕2 exp - -------†2- 2πT e(ψ) Te(ψ )βth

(6.78)

One can then cross-check that

∫ ∫ † ∫ ∫ 2 π +1 dξ +∞ p2f (ψ, p)dp = 2πp†3n-e +1dξ + ∞ p2f- (ψ,p)dp-= n †n-= n - 1 0 0 M thβ†3 -1 0 0 M e e e th

(6.79)

the local density is well recovered.

6.3.6 Poloidal flux coordinate

For the poloidal flux coordinate, the normalized value is

-- ψ = ψ ∕ψa

(6.80)

where ψ_a is the value at the plasma edge, corresponding usually to the last closed magnetic flux surface, as given by the equilibrium code. Usually, it is taken so that 0 ≤ψ ≤ 1 from the center to the plasma edge. For

6.3.7 Drift kinetic coefficient

The first order drift kinetic equation requires to calculate defined as

v I (ψ) (0) ^f(0) = -∥0-0---∂f-0- Ωe0 ∂ ψ

(6.81)

where by definition B_T0 (ψ ) = |I0(ψ )| ∕R₀ and Ω_e0 = - |e|B0
γme is the electron Larmor frequency, all quantities being determined at the poloidal location where the magnetic field B is minimum, as discussed in Sec. 3.5.5. Since

--† --† vvth∕c = pβth∕ γ

(6.82)

one obtains in normalized units

-(0) γme B ∂f(0) ^f = vξ0----R0 -T-0--0-- qe B0 ∂ψ -- --† γme BT 0R0 ∂f (00) = vvthξ0-q---B--ψ---∂ψ-- e 0 a -- -- cme- † BT-0R0-∂f(00) = pξ0 qe βthB0 ψa ∂ψ- (6.83)

Therefore,

-(0) -(0) f^ = pξ0C^0 ∂f0-- ∂ψ

(6.84)

where

^ cme- † BT-0R0- C0 = qe βth B0 ψa

(6.85)

For circular concentric flux surfaces, since ∂∕∂ψ = (1∕∇ ψ) ∂∕∂r and ∇ψ = RB_P, it turns out that

(0) ^f(0) = vξ γme-BT-0-1-∂f0- qe BP 0B0 ∂r

(6.86)

and naturaly

^ cme- † -q- 1-- C(0) ≈ qe βthapϵ B0

(6.87)

where the safety factor q is approximated by its cylindrical expression q = rBT
RBP- , a_p the plasma minor radius, and ϵ = r∕R_p the inverse aspect ratio.

6.3.8 Momentum convection and diffusion

From the conservative form of the Fokker-Planck equation in momentum space

∂f (0) ∂ ( ) --0-- ∝ --- Sp(0) ∂t ∂p

(6.88)

where

(0) (0)∂f0(0) (0)(0) S p ∝ - D p ∂p + Fp f0

(6.89)

Introducing normalized coordinates, it turns out that

-(0) ( (0) -(0) -- ) ∂-f0- ∝ --1--∂-- - D-p-∂-f0- + F(p0)f (00) ∂t ν†ep†th ∂p p†th ∂p ( (0) -(0) (0) ) = ∂-- - -Dp--∂-f0- + F-p--f(0) ∂p ν†ep†t2h ∂ p ν†ep†th 0 ( -(0) ) = ∂-- - D(0)∂f0--+ F-(0)f(0) (6.90) ∂p- p ∂p- p 0

where normalized diffusive and convective coefficients are

D-(0) = D (0p)∕D † (6.91) --p(0) p F p = F (0)p∕F †p (6.92)

with

D †p = ν†ep†t2h (6.93) † † † F p = νepth (6.94)

For the case of the Ohmic electric field, this normalization leads to introduce naturaly the Dreicer field

† †† E = νepth∕qe

(6.95)

the electric field being normalized according to the relation

E- = E ∕E † ∥0 ∥0

(6.96)

6.3.9 Radial convection and diffusion

From the conservative form of the Fokker-Planck equation in configuration space

∂f0(0) ∂ ( (0)) --∂t- ∝ ∂ψ- |∇ ψ|0Sψ

(6.97)

where

(0) (0) ∂f0(0) (0) (0) S ψ ∝ - D ψ |∇ ψ|0 ∂ψ + Fψ f0

(6.98)

Introducing normalized coordinates, it turns out that

--(0) ( ) ∂f-0- ∝ -1---∂-- |∇ ψ |0S (ψ0) ∂ t ν†eψa∂ ψ ( (0) -(0) ) = -1---∂--( Dψ--|∇ψ |2 ∂f0--+ |∇ ψ| F(0)f-(0)) ν†eψa∂ ψ ψa 0 ∂ψ 0 ψ 0 ( ) ∂ D (0) ∂f(0) F (0)--(0) = ---( -†ψ--|∇ ψ |20 --0--+ |∇ψ |0 -†ψ-f 0 ) (6.99) ∂ψ νeψ2a ∂ψ νeψa

with

--(0) (0) † D ψ = D ψ ∕D ψ (6.100) F-(0) = F (0)∕F † (6.101) ψ ψ ψ

with

D † = ν †ψ2 (6.102) ψ e a F †ψ = νe†ψa (6.103)

However, since it is more convenient to handle diffusion and convection coefficients that scale in m²∕s and m∕s respectively, one must introduce new reference values for the diffusion and convection coefficients. Let

†* † 2 D ψ = νeap (6.104) F †* = ν†ap (6.105) ψ e

the reference diffusion and convection, where a_p is the plasma minor radius as defined in Sec.2.1. The relation is

D(0) = α D (0)∕D †* (6.106) -ψ D ψ ψ F(ψ0) = αFF (0)∕F †* (6.107) ψ ψ

where

α = D†*∕D †= a2∕ψ2 (6.108) D ψ ψ p a αF = F†ψ*∕F †ψ = ap∕ψa (6.109)

6.3.10 Fluxes

In momentum space,

∂f(0) S (0p) ∝ - D (0p)--0--+ F (p0)f(00) ∂p -- --(0) n†e ∂f(00) --(0) n †e-(0) = - D p D†p -†3--†---+ F p F †p-†3f0 ( pth p)th∂ p p(th ) --(0) ν†en†e ∂f-(0) -(0) ν†en†e -(0) = - D p --†2-- ---0- + Fp --†2-- f0 pth ∂p pth ( † †) ( -- -(0) -- -- ) = νen†2e - D(p0)∂f0--+ F (0p)f(00) (6.110) pth ∂p

and

-- S (0p)= S(p0)S †p

(6.111)

where

† ν†en†e- Sp = p†2 th

(6.112)

and with these definitions,

-- -- --(0) -- -- S (0p)∝ -D (0p)∂f-0-+ F(p0)f(00) ∂ p

A similar procedure is applied for dynamics in configuration space,

∂f (0) S(ψ0) ∝ - D (0ψ)ψ|∇ ψ|0--0- + F(ψ0)f0(0) ∂ψ -- --(0) † n †e ∂ f(0) (0) † n†e -(0) = -D ψψD ψψ-†3|∇ ψ|0---0--+ F ψ Fψ -†3-f0 ( pth ) ψa ∂ψ pth( ) --(0) n†e ∂f-(0) n†e -(0) = -D ψψ -†3ν†eψa |∇ ψ|0---0- + F(ψ0) -†3-ν†eψa f0 pth ∂ ψ pth ( † ) ( -- -(0) -- ) = neνe†ψa - D (0ψ)ψ|∇ ψ|0 ∂f0-+ F (ψ0)f(00) (6.113) p†3th ∂ψ

and

-- S (0)= S(ψ0)S † ψ ψ

(6.114)

with

n† Sψ†= -e†3-ν†eψa pth

(6.115)

and with these definitions,

--(0) --(0) --(0) ∂f-0- -(0)-(0) S ψ ∝ -D ψψ|∇ ψ|0 ∂ψ- + Fψ f 0

(6.116)

6.3.11 Current density

The local current density at the minimum B value on the flux surface is given by the relation

∫ ∫ +1 J(0) ∝ 2πqe p2dp ξ0vf(00)dξ0 ∫ - 1∫ -p3- +1 (0) = 2πqe γme dp -1 ξ0f0 dξ0 † ∫ -3 ∫ +1 = 2π qep†4-ne p-dp- ξ f(0)dξ (6.117) me thβ †3 γ - 1 0 0 0 th

Therefore,

J(0) = J (0)∕J†

(6.118)

where

J† = -qep† n† me th e

(6.119)

With this definition

-(0) ∫ p3 -∫ +1 -(0) J ∝ 2π γ-dp ξ0f0 dξ0 - 1

(6.120)

6.3.12 Power density

The power density is deduced from the integral

(0) ∫ +1 ∫ ∞ p3 (0) P ∝ 2π dξ0 γm--Sp dp -1 ∫ 0 e∫ -- p†t4h ν†en†e- +1 ∞ p3 -(0) -- = 2πme p†2 - 1 dξ0 0 γ Sp dp (6.121) th

and defining

-(0) (0) † P = P ∕P

(6.122)

one has

ν†ep†3n †e P † = ---th--- me

(6.123)

With this definition

∫ +1 ∫ ∞ -3 P(0) ∝ 2π dξ p- S-(0)dp -1 0 0 γ p

(6.124)

6.3.13 Electron runaway rate

The electron runaway rate Γ_R is given by the relation

(0) ∫ +1 (0) ΓR ∝ 2 πp2 Sp dξ0 - 1 ∫ -2 †2ν†en†e- +1 -(0) = 2 πp pth p†2 - 1 Sp d ξ0 (6.125) th

Therefore, normalized expression is

-(0) (0) † ΓR = ΓR ∕ ΓR

(6.126)

with

† † † ΓR = νene

(6.127)

so that

-- ∫ +1 -- Γ (0R) ∝ 2πp2 S(p0)dξ0 -1

(6.128)

The consistency of the normalized units may be benchmarked by the approximate relation between the total current and the bulk current, the difference coming from the electron runaway tail. For small Γ_R, the runaway distribution is independent of v_∥, so that it may be approximated by the simple expression f₀ ( )
v∥ ≫ vth ≈ (γ∕E ) δ (p⊥) . By integrating the current integral v_∥f₀ out to p_max, one obtains

( (0)) (0) (0) --1- Γ-R- 2 J (vmax ) ≈ Jbulk + 2me E ∥0 pmax

(6.129)

and in normalized units,

-- -- ( -(0) †) J(0) (vmax) J† ≈ J(0) J† + -1-- ΓR--ΓR- p †2p2max bulk 2me E ∥0E† th

(6.130)

so that

-- -- 1 p†2Γ †( Γ-(0)) J(0)(vmax ) ≈ J (0bu)lk +--- -th†-R†- --R- p2max 2me E J E ∥0

(6.131)

According to their respective expression,

--p†2thΓ †R- ------p†th2νe†n-†e----- m E †J † = m -qe-p†n †ν †p† ∕q = 1 e th eme th ee th e

(6.132)

which demonstrates the overall consistency of the normalization of the different quantities calculated by the code.

The bounce-averaged avalanche source term _R given by relation 3.418, is simply proportional to ⟨n ⟩
R _Vν_e∕lnΛ_ee. Therefore

(0) ⟨nR⟩V-νe-n†eν†e- S R ∝ lnΛ--- lnΛ † ee ee

(6.133)

where lnΛ_ee = lnΛ_ee∕lnΛ_ee^†, and the ratio is

n†eν†e n†e2cr2 ----†-= --†3-e ln Λee βth

(6.134)

6.3.14 Electron magnetic ripple loss rate

By definition, the electron magnetic ripple loss rate Γ_ST is given by a relation relation similar to Γ_R, except that integrals of fluxes are calculated for different boundaries. Therefore, the normalized expression is simply given by the relation

-- Γ (S0T)= Γ (0S)T∕Γ †ST

(6.135)

where

Γ †ST = Γ †R = νe†n †e

(6.136)

6.3.15 Units

In the code, the following units are used

Quantity Units Temperature keV Density m -3 Time s -1 Frequency s

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