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2.2 Particle motion

2.2.1 Arbitrary configuration

Transit or Bounce Time

Normalized Expression The transit, or bounce time, is defined as the time for a passing particle to complete a full orbit in the poloidal plane, and for a trapped particle to complete half a bounce period. Note that this is possible only in the approximation of zero banana width. Otherwise, the bounce motion would be no longer symmetric in the forth and back motions, and both would need to be accounted for. We define then

∫ ∫ τ (ψ) = smax ds--= smax|ds|-B- b smin |vs| smin |v∥|BP
(2.4)

where vs is the guiding center velocity along the poloidal field lines, and v∥ is its velocity parallel to the magnetic field. B is the magnitude of the magnetic field, while BP is the magnitude of its poloidal component as shown in Fig. 2.5. The limits smin and smax are defined in (2.2) for passing electrons, and are the positions, along the field lines, of turning points for trapped electrons.

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Figure 2.5: Guiding center velocity definition

The differential arc length ds along the poloidal field line is generally expressed in curvilinear coordinates ( 1 2 3) u ,u ,u as (A.13)

∘ ----i--j- ds = gijdu du
(2.5)

where the gij are the metric coefficients, defined in (A.12). In the (ψ,θ,ϕ) coordinates, the variations dψ and dϕ are essentially zero along the poloidal field line. As a consequence, (2.5) becomes

√--- ds = g22dθ
(2.6)

The velocity and momentum are related through the relativistic factor γ(p) introduced in Sec. 6.3.4, and therefore, we have

v∥ p∥ v ≃ p = ξ
(2.7)

in the weak relativistic regime of tokamak plasmas, where the pitch-angle cosine ξ is defined in (A.247)

We get

∫ θmax τb (ψ ) = -2π-- dθ√g22-ξ0-B- v|ξ0| θmin 2π ξ BP
(2.8)

where ξ0 is the pitch angle cosine at the position θ0 of minimum B-field

θ0 ≡ θ (B = B0 (ψ ))
(2.9)

and the limits θmin and θmax will be calculated in the next subsection.

The bounce time can be normalized as such:

2πR ^q (ψ ) τb(ψ,ξ0) = ---p-----λ (ψ, ξ0) v|ξ0|
(2.10)

with

∫ θmax √ --- λ (ψ,ξ0) = -1--- dθ--g22ξ0 B-- ^q(ψ) θmin 2π Rp ξ BP
(2.11)

and

∫ √--- 2π dθ--g22-B-- ^q (ψ ) ≡ 2π Rp BP 0
(2.12)

The bounce time is normalized to the transit time of particles with parallel momentum only, such that λ(ψ,±1 ) = 1.

The covariant metric element g22 is given by (A.10)-(A.12), which is in the (ψ, θ,ϕ) system becomes (A.192)

√ --- |--r-| g22 = |J∇ ψ × ∇ ϕ| = |^ψ ⋅^r| | |
(2.13)

Consequently, the normalized bounce time takes the form

∫ λ (ψ,ξ ) = -1--- θmax dθ-|-1-|-r--B- ξ0 0 ^q(ψ) θmin 2π ||ψ^⋅^r||Rp BP ξ
(2.14)

with

∫ 2π ^q (ψ ) = dθ-|-1-|-r--B-- 0 2π ||ψ^⋅^r|| Rp BP
(2.15)

Particle Motion in the Magnetic Field The particle motion along the magnetic field lines exhibits one constant of the motion, the energy (or the total momentum p), and an adiabatic invariant, the magnetic moment μ. They are given by the equations

2 2 2 p = p⊥ + p∥
(2.16)

p2 μ = --⊥--- 2meB
(2.17)

such that, as a function of the moment component ( ) p∥0,p⊥0 at the location θ0 of minimum B-field, we have

p⊥2 + p ∥2 = p ⊥02 + p ∥02 (2.18)
2 --p⊥---- B (ψ,θ) = 2 --p⊥0- B0 (ψ ) (2.19)

Using the transformation (A.250-A.251) from (p ,p ) ∥ ⊥ to (p,ξ), the system (2.18-2.19) becomes

p2 = p 02 (2.20)
2 -1--ξ--- B (ψ,θ) = 2 -1--ξ0 B0 (ψ ) (2.21)

We get an expression for ξ as a function of ξ0:

∘ -----------(----2)- ξ(ψ,θ,ξ0) = σ 1- Ψ(ψ, θ) 1- ξ0
(2.22)

where σ = sign(ξ0) = sign( ) v∥, and Ψ(ψ,θ) is the ratio of the total magnetic field B to its minimum value B0

B (ψ,θ) Ψ (ψ,θ) ≡ -------- B0 (ψ)
(2.23)

The trapping condition is given by |ξ| 0 < ξ0T (ψ) , where ξ0T (ψ ) is the pitch angle, defined at the minimum B0(ψ ) on a given flux-surface, such that the parallel velocity of the particle vanishes at the maximum Bmax(ψ ). An expression for ξ0T (ψ ) can then be obtained from (2.22): setting ξ(ξ0T,B = Bmax (ψ )) = 0, we get

2 B0 (ψ) ξ0T (ψ) = 1- B----(ψ)- max
(2.24)

The turning points are

| | - π for passing particles θmin(ψ,ξ0) = ||θ for trapped particles T min
(2.25)

|| θmax (ψ,ξ0) = || π for passing particles θT max for trapped particles
(2.26)

We can determine the turning angles θT min(ψ, ξ) 0 and θT max(ψ,ξ ) 0 as the position where ξ(ψ, θ,ξ0) = 0. At this position, we have B = Bb(ψ,ξ0), where Bb(ψ,ξ0) is then given by (2.22)

B0-(ψ) Bb(ψ, ξ0) = 1 - ξ2 0
(2.27)

so that

θT min(ψ,ξ ) 0 = θ(B = B |θ < θ ) b 0[2π] (2.28)
θT max(ψ, ξ0) = θ(B = Bb|θ > θ0)[2π] (2.29)

where θ0 is given by (2.9).

Calculation of λ(ψ,ξ0) From the Output Data of Equilibrium Codes The numerical calculation of λ(ψ,ξ0) can be carried from the output of any magnetic equilibrium code. In the kinetic code here considered, we use HELENA for magnetic flux surface calculations [?], since it is used in the the CRONOS tokamak simulation package [?].

Data are assumed to be the parametrization of the flux-surfaces R(ψ, θ) and Z(ψ,θ), and the three components of the magnetic field BR(ψ, θ), BZ(ψ,θ) and Bϕ(ψ,θ). From these components we derive directly the toroidal and poloidal components of the field, as well as the total field:

BT (ψ,θ) = |B ϕ(ψ, θ)|
BP (ψ,θ) = ∘ -------------------- B2R (ψ,θ)+ B2Z (ψ, θ)
B(ψ,θ) = ∘ -------------------- B2T (ψ,θ)+ B2P (ψ, θ) (2.30)

and also

Rp = R(0,θ) (2.31)
Zp = Z(0,θ) (2.32)

We also have an expression for r

∘ --------------2----------------2 r(ψ, θ) = (R (ψ,θ) - Rp) + (Z (ψ,θ) - Zp)
(2.33)

and, using relation

^r = ( ^) ^r ⋅R^ R + ( ^) ^r ⋅Z^ Z
= (R (ψ,θ )- Rp ) ------r------R^ + ( Z (ψ, θ)- Zp ) -----r------^Z (2.34)

that can be easily deduced from vector relation in Fig. 2.2, we get an expression for the scalar product

( ) ( ) ∇ ψ ⋅ ^R (R (ψ,θ)- Rp )+ ∇ ψ ⋅Z^ (Z (ψ,θ) - Zp) ^ψ ⋅^r = ------------------------------------------------ r|∇ ψ|
(2.35)

In a toroidal axisymmetric geometry, the magnetic field can be expressed generally as

B = I (ψ )∇ ϕ+ ∇ψ × ∇ ϕ
(2.36)

so that

BT = |I (ψ)| |∇ ϕ| = |I (ψ-)| R (2.37)
BP = |∇ψ | |∇ϕ | = |∇ψ-| R (2.38)

We also have

BT = I(ψ)∇ϕ = Bϕ^ϕ (2.39)
BP = ∇ψ ×∇ϕ = -BP ^s (2.40)

and therefore

∇ ϕ × BP = ∇ϕ × (∇ ψ × ∇ϕ ) = |∇ ϕ|2 ∇ψ
(2.41)

so that

∇ ψ = ∇ϕ-×-B2P--= R ^ϕ× BP |∇ ϕ|
(2.42)

and we have the projections

( ) ∇ ψ ⋅R^ = R^R ⋅^ϕ× BP = -RBZ (2.43)
( ) ∇ ψ ⋅Z^ = R^Z ⋅^ϕ× BP = RBR (2.44)

Finally, the expressions for the normalized bounce time λ and q^ that are used in numerical calculations are

[ ] ∫ θmax B (R - Rp)2 + (Z - Zp)2 λ (ψ,ξ0) = -1--- dθ-------------------------------ξ0- ^q(ψ) θmin 2π Rp |BR (Z - Zp )- BZ (R - Rp )| ξ
(2.45)

with

∫ [ 2 2] 2πdθ---B--(R---Rp-)-+-(Z---Zp)----- q^(ψ) = 0 2π Rp |BR (Z - Zp)- BZ (R - Rp )|
(2.46)

where R,Z,BR,BZ and B are functions of (ψ, θ), and ξ is a function of (ψ,θ,ξ0) given by (2.22).

Safety Factor q(ψ ) The (averaged) safety factor q is defined in Ref. [?] in a general way as

I-(ψ-)δV-⟨ -2⟩ q(ψ) = 4π2 δψ R
(2.47)

where V is the volume enclosed by a flux-surface and ⟨⟩ denotes the flux-surface average.

It can be expressed as

∫ 2π ∫ 2π q(ψ) = I (ψ ) dθ- dϕ-J-- 0 2π 0 2π R2
(2.48)

where the Jacobian J is given by (A.195)

J = |∇ ψ × ∇ θ ⋅∇ ϕ|-1
= ----Rr|----| |∇ ψ|||^ψ ⋅^r||
= -r- BP|-1-|- ||^ψ ⋅r^|| (2.49)

where (2.40) is used

We obtain

∫ 2π dθ|--1-|--r--- q(ψ ) = I (ψ) 0 2π|^ |BP R2 |ψ ⋅^r|
(2.50)

and, using (2.36), we finally have

∫ 2π dθ 1 r BT q(ψ) = 2-π||----||R-B-- 0 |^ψ ⋅^r| P
(2.51)

The expression of q(ψ ) and its relation to ^q (ψ ) in the simplified case of circular concentric flux-surfaces will be addressed in sub-section 2.2.2.

Using (2.33) and (2.35), we find the expression

∫ [ 2 2] 2π dθ---(R----Rp)-+-(Z---Zp-)--BT-- q(ψ) = 2πR |BR (Z - Zp) - BZ (R - Rp)| 0
(2.52)

that is convenient for the numerical evaluation.

Toroidal Extent of Banana Orbits

We are interested in calculating the toroidal extent of banana orbits, that is, the toroidal angle corresponding to the path done by a trapped particle between two turning points. It is given by

∫ ϕmax Δ ϕ = ϕmax - ϕmin = dϕ ϕmin
(2.53)

and can be expressed as a function of the length element dl along the path, using (A.198)

∫ ∫ l(ϕmax) --dϕ- l(ϕmax) dl(ϕ-) Δ ϕ = l(ϕ ) dl(ϕ)dl (ϕ ) = l(ϕ ) R min min
(2.54)

The poloidal and toroidal elements are related through the local angle of the magnetic field,

dl(ϕ) = BT- dl(θ) BP
(2.55)

so that

∫ l(ϕmax) d ϕ ∫ l(θmax)1 BT Δ ϕ = dl(ϕ)----- = -----dl(θ) l(ϕmin) dl(ϕ) l(θmin) R BP
(2.56)

Using (A.197), we get

∫ θ Δ ϕ = max dθ|-1--|r-BT- θmin ||^ψ ⋅^r||R BP
(2.57)

Defining the integral

∫ θ q (ψ, ξ) = maxdθ-|-1--|r-BT- T 0 θmin 2π ||^ψ ⋅^r||R BP
(2.58)

we find that the toroidal extent of banana orbits is

Δ-ϕ 2π = qT (ψ,ξ0)
(2.59)

Note that at the trapped/passing limit, we have

Δ ϕ lim --- = qT (ψ,ξ0T) = q(ψ) ξ0→ξ0T 2π
(2.60)

Therefore, we retrieve the interpretation of the safety factors, which is the number of toroidal rotations Δϕ∕2π for one poloidal rotation.

Bounce Average

In order to reduce the dimension of kinetic equations, it is important to define an average over the poloidal motion, which anihilates the term that accounts for the time evolution of the variations of the distribution function along the field lines. The natural average is

[ ∑ ] ∫ smax {A } = 1- 1- -dsA τb 2 σ T smin |vs|
(2.61)

where the sum over σ applies to trapped particles only.

It can be rewritten in terms of the normalized bounce time λ using expression (2.11)

1 [ 1∑ ] ∫ θmax dθ√g--- B ξ {A} = --- -- ------22----0 A λ^q 2 σ T θmin 2π Rp BP ξ
(2.62)

or

[ ] ∫ 1-- 1∑ θmax dθ--1----r--B-ξ0 {A } = λ^q 2 2π||^ ||Rp BP ξ A σ T θmin |ψ ⋅^r|
(2.63)

using relation (2.13).

Another expression uses the output data from equilibrium codes. Following the work in the previous section, we find

[ ] [ ] ∫ θ 2 2 {A } = 1-- 1-∑ max dθ---B--(R---Rp)-+-(Z---Zp-)----ξ0A λ^q 2 θmin 2πRp |BR (Z - Zp) - BZ (R - Rp)| ξ σ T
(2.64)

or explicitely

{A } = [ ] ⌊ ∫ θmax B (R - R )2 + (Z - Z )2 ⌋ ⌈ dθ-------------p----------p-----ξ0⌉ θmin 2π Rp|BR (Z - Zp )- BZ (R - Rp )| ξ-1×
[ 1 ∑ ] -- 2 σT ∫ θminθmax dθ --- 2π [ ] B (R - Rp )2 + (Z - Zp)2 ------------------------------ Rp |BR (Z - Zp)- BZ (R - Rp)|ξ -0 ξA (2.65)

The bounce averaging of momentum-space operators in the kinetic equations leads to a set of coefficients that all have a similar structure, denoted λk,l,m and λk,l,m, which are define as

{( )k ( )m } ξ(ψ,-θ,ξ0)- Ψl (ψ,θ) -R0(ψ-) = λk,l,m-(ψ,ξ0)- ξ0 R (ψ,θ) λ(ψ, ξ0)
(2.66)

and

{ } -- ( ξ(ψ,θ,ξ0))k ( R0 (ψ) )m λk,l,m (ψ,ξ0) σ σ ---------- Ψl(ψ, θ) ------- = ------------ ξ0 R (ψ, θ) λ (ψ,ξ0)
(2.67)

where

R0 (ψ) ≡ R (ψ,θ0)
(2.68)

Note that by definition, λ0,0,0 = λ. In addition,

-- ||λ for passing particles λk,l,m = || k,l,m 0 for trapped particles
(2.69)

2.2.2 Circular configuration

Parametrization of the Flux-Surfaces

In this case, we have ψ = ψ(r) and therefore it is easier to work in the (r,θ,ϕ) coordinate to account for the symmetry in the problem. The normalized radius is

-r- ρ(ψ) = ap
(2.70)

We have now

^ψ = ^r
(2.71)

so that (A.113)

√g---= r 22
(2.72)

Magnetic Field

The toroidal field is

B (r,θ) =-|I (r)| T R (r,θ)
(2.73)

and the poloidal field is

|∇ψ (r)| BP (r,θ) = -R(r,θ)-
(2.74)

where

||dψ (r)|| 1 ||dψ (ρ)|| |∇ ψ(r)| = ||-----|| = --||------|| dr ap dρ
(2.75)

is now only a function of r or ρ.

The total field is then

∘ ---------------- I2 (r) + |∇ ψ (r)|2 B (r,θ) = ------------------ R (r,θ )
(2.76)

and can be written as

R0 B (r,θ) = B0 (r) R-(r,θ)-
(2.77)

with

∘ ---------------- I2(r)+ |∇ ψ(r)|2 B0 (r) = -------R---------- 0
(2.78)

Consequently, we ratio of magnetic fields Ψ as defined in (2.23) becomes

R Ψ (r,θ) = ---0--- R (r,θ)
(2.79)

and

∘ ---------------2 ∘ ------------- -B- --I2-(r)-+-|∇-ψ-(r)|- -I2-(r)-- BP = |∇ ψ(r)| = 1+ |∇ ψ (r)|2
(2.80)

is a function of r only.

Safety factor

The safety factor given by expression (2.51) becomes

∫ ∫ 2πd-θr-BT- -r-BT- 2πd-θRp- q(r) = 2π R BP = Rp BP 2π R 0 0
(2.81)

The averaged value of Rp∕R is evaluated in (??). It gives

∫ 2π dθ-Rp-= ∘----1------- 0 2π R 1- (r∕Rp )2
(2.82)

so that

-----1-------r--BT- q(ψ ) = ∘ ---------2-Rp BP 1- (r∕Rp )
(2.83)

Note that in the factor ∘ ----------2 1 - (r∕Rp) is usually neglected, which is valid only in the large aspect ratio approximation, i.e. when the inverse aspect ratio ϵ defined as

-r- ϵ = Rp
(2.84)

is much less than unity.

Particle Motion

Using relation (A.95),

R (r,θ) = Rp + rcosθ
(2.85)

and recalling that the minimum B-field B0 corresponds to the poloidal angle value in that case

θ0 = 0
(2.86)

we find

Rmin(r) = Rp - r = Rp(1 - ϵ) (2.87)
Rmax(r) = Rp + r = Rp(1 + ϵ) = R0(r) (2.88)

Therefore, the expression (2.79) becomes

--1+--ϵ-- Ψ (ρ,θ) = 1+ ϵcos θ
(2.89)

and using relation (2.77)

1 + ϵ Bmax (r) = B0 (r)1---ϵ
(2.90)

expression (2.24) is

2ϵ ξ20T (r) = ----- 1+ ϵ
(2.91)

The pitch-angle cosine ξ is then given by combining relations (2.22 ) and (2.89)

--------------------- ∘ 1+ ϵ ( ) ξ(r,θ,ξ0) = σ 1 - --------- 1- ξ20 1 + ϵcosθ
(2.92)

and the the turning angles are obtained from expression (2.27), or in the present notation

B (r,θT) = Bb (r,ξ0) = B0-(r) 1 - ξ20
(2.93)

Using relation (2.89), one obtains

---1+-ϵ---- B0-(r) B0 (r)1+ ϵcosθT = 1 - ξ2 0
(2.94)

and then

2 1 + ϵcosθT ϵ (1 - cosθT) ξ0 = 1 - ---1+-ϵ----= ----1-+-ϵ---
(2.95)

so that

[ 2ξ2] θT = arccos 1 - -20 ξ0T
(2.96)

and finally by symmetry

θT min = -θT (2.97)
θT max = θT

Bounce Time

Using (2.72), the normalized bounce time reduces to

∫ θmax λ(r,ξ0) = ϵ dθ-ξ0-B- q^ θmin 2π ξ BP
(2.98)

with, using definition (2.15)

∫ 2π dθ-B-- ^q (r) = ϵ 0 2π BP
(2.99)

Because B∕BP only a function of r, as seen in (2.80), and can be taken out of the integrals, we get finally

∫ θmax dθ ξ λ (r,ξ0) = ----0 θmin 2π ξ
(2.100)

This integral can be performed analytically in a series expansion whose coefficients are calculated in (??). Note that in the case where BT ≫ BP and in the large aspect ratio approximation ϵ ≪ 1, we have ^q (r) → q(r), which explains the notations, and the introduction of pseudo safety factor like ^q . Other new definitions of pseudo safety factors will be introduced throughout the next sections, based on similar arguments.

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