[next] [prev] [prev-tail] [tail] [up]

5.8 Moments of the Distribution Function

5.8.1 Flux discretization for moment calculations

The calculation of momentum-space moments of the distribution function, such as the density of power absorbed or the stream function, requires to calculate the bounce-averaged momentum space fluxes Sp(0) associated with a distribution function. Considering a given distribution function f(0) (which could be f0(0), ^f (0) or g(0)) and given momentum-space diffusion Dp(0)O and convection Fp(0)Ocoefficients, associated with a particular physical mechanism O (such as collisions, RF waves and DC electric field) and calculated in chapter 4, these fluxes are given by (3.187) or equivalently (3.216) and their components are

Sp(0) = -D pp(0)O∂f(0) -∂p-- + ∘ ------ 1- ξ2 ---p--0-Dpξ(0)O∂f(0) ∂-ξ-- 0 + Fp(0)Of(0) (5.557)
Sξ0(0) = -D ξp(0)O∂f(0) ∂p + ∘ ------ --1--ξ20- pDξξ(0)O∂f(0)- ∂ ξ0 + Fξ(0)Of(0) (5.558)

5.8.2 Numerical integrals for moment calculations

Density

From calculations in Sec.3.6, the flux surface averaged density ⟨ne⟩V,l+1∕2 at ψl+1∕2 may be expressed as a sum of three terms

0 1 1 ⟨ne⟩V,l+1∕2 = ⟨ne⟩V,l+1∕2 + ⟨ne⟩V,l+1∕2 + ⟨^ne⟩V,l+1∕2

the first one corresponding to the zero order Fokker-Planck equation

np∑- 1n∑ξ0-1 ⟨ne⟩0 = 2π ^ql+1∕2- p2 λl+1∕2,j+1∕2f (0) Δp Δξ V,l+1∕2 ^ql+1∕2 i=0 j=0 i+1∕2 0,l+1∕2,i+1∕2,j+1∕2 i+1∕2 0,j+1∕2
(5.573)

while the two others result from the solution of the electron drift kinetic equation

np∑-1nξ∑0-1 ⟨ne⟩1 = 2π^ql+1∕2 p2 λl+1∕2,j+1∕2g(0) Δpi+1∕2Δ ξ0,j+1∕2 V,l+1∕2 ^ql+1∕2 i=0 j=0 i+1∕2 l+1∕2,i+1∕2,j+1∕2
(5.574)

and

np∑-1nξ∑0-1 ⟨^ne⟩1 = 2π^ql+1∕2 p2 λl+1∕2,j+1∕2^f(0) Δp Δ ξ V,l+1∕2 ^ql+1∕2 i=0 j=0 i+1∕2 1,-1,0 l+1∕2,i+1∕2,j+1∕2 i+1∕2 0,j+1∕2
(5.575)

Current Density

In a similar way, the flux surface averaged parallel current ⟨ ⟩ J∥ϕ,l+1∕2 at ψl+1∕2 may be expressed as a sum of three terms

⟨ ⟩ ⟨ ⟩0 ⟨ ⟩1 ⟨ ^⟩1 J∥ ϕ,l+1∕2 = J∥ ϕ,l+1∕2 + J∥ ϕ,l+1∕2 + J∥ ϕ,l+1∕2
(5.576)

where the zero order term is

⟨ ⟩ J∥ϕ,l+1∕20 = 2πq ---e- meq -l+1∕2- ql+1∕2 ∑ i=0np-1 ∑ j=0nξ0-1p3 -i+1∕2 γi+1∕2H(| | ) |ξ0,j+1 ∕2|- ξ0T,l+1∕2
× ξ0,j+1∕2f0,l+1∕2,i+1∕2,j+1∕2(0)Δp i+1∕2Δξ0,j+1∕2 (5.577)

The first order term arising from function g is

⟨ ⟩ J∥ϕ,l+1∕21 = 2πqe ----- meql+1∕2 ------ ql+1∕2 ∑ i=0np-1 ∑ j=0nξ0-1p3i+1∕2 ------ γi+1∕2H(| | ) |ξ0,j+1 ∕2|- ξ0T,l+1∕2
× ξ0,j+1∕2gl+1∕2,i+1∕2,j+1∕2(0)Δp i+1∕2Δξ0,j+1∕2 (5.578)

while the other one which results from f^ is more complex and

⟨ ⟩ J^∥ϕ,l+1∕21 = 2πq ---e- me^q -l+1∕2 ql+1∕2 R ----p--- R0,l+1∕2B -T0,l+1∕2- B0,l+1 ∕2 ∑ i=0np-1 ∑ j=0nξ0-1p3 -i+1∕2- γi+1∕2λ2,-2,2l+1∕2,j+1∕2
× ξ0,j+1∕2f^ l+1∕2,i+1∕2,j+1∕2(0)Δp i+1∕2Δξ0,j+1∕2. (5.579)

Power Density Associated with a Flux

The density of power absorbed by the plasma through a particular mechanism is the sum

⟨ ⟩0 ⟨ ⟩0 ⟨ ⟩1 ⟨ ⟩1 P Oabs V,l+1∕2 = POabs V,l+1∕2 + POabs V,l+1∕2 + ^PaObs V,l+1∕2

where the respective contributions are given by the equations (3.322) and (3.324-3.325) and discretized as

⟨ O ⟩ PabsV,l+1∕20 = 2π- me^ql+1∕2- ^ql+1∕2 ∑ i=1np ∑ j=0nξ0-1p3i γiλl+1∕2,j+1∕2S p,l+1∕2,i,j+1∕2(0)O( (0)) f0ΔpiΔξ0,j+1∕2 (5.580)
⟨ ⟩ PaObsV,l+1∕21 = 2π --- me^ql+1∕2 ------ ^ql+1∕2 ∑ i=1np ∑ j=0nξ0-1p3 -i γiλl+1∕2,j+1∕2S p,l+1∕2,i,j+1∕2(0)O( ) g(0)ΔpiΔξ0,j+1∕2 (5.581)
⟨ ⟩ P^OabsV,l+1∕21 = 2π --- me^q -l+1∕2- ^ql+1∕2 ∑ i=1np ∑ j=0nξ0-1p3 -i γiλl+1∕2,j+1∕2^S p,l+1∕2,i,j+1∕2(0)O( ) ^f(0)ΔpiΔξ0,j+1∕2 (5.582)

The discretization of the momentum-space flux components Sp,l+1∕2,i,j+1∕2(0)O and S^p,l+1∕2,i,j+1∕2(0)O is done in (5.565).

Stream Function for Momentum Space fluxes

The stream function gives the local direction of the momentum-space fluxes, and its gradient is an indication of the flux intensity. Because it is a flux function, it is naturally defined on the momentum-space flux grid. According to the three equivalent expressions for the stream function (3.352-3.353), we have the following discretizations, for 1 ≤ i ≤ np and 1 ≤ j ≤ nξ - 1

Al+1∕2,i,j(0) = 2 -2πpi-- ne,l+1∕2 ∑ m=0j-1Δξ 0,m+1∕2Sp,l+1∕2,i,m+1∕2(0) (5.583)
Al+1∕2,i,j(0) = 2πp2 ----i-- ne,l+1∕2 ∑ m=jnξ-1Δξ 0,m+1∕2Sp,l+1∕2,i,m+1∕2(0) (5.584)

and

∘ ------- 2π 1- ξ20,ji∑-1 A (l+0)1∕2,i,j =----------- pm+1∕2Δpm+1 ∕2S(ξ0,)l+1∕2,m+1 ∕2,j ne,l+1 ∕2 m=0
(5.585)

where the boundary conditions are

A (0l+)1∕2,0,j = A (0l+)1∕2,i,0 = A (0l+)1∕2,i,n = 0 ξ
(5.586)

The discretization of the momentum-space flux components Sp,l+1∕2,i,j+1∕2(0)O and Sξ,l+1∕2,i+1∕2,j(0)O is done in (5.565) and (5.572).

Ohmic electric field

As shown in Sec. 4.2, the flux surface averaged parallel Ohmic electrid field ⟨ ⟩ E∥ϕ(ψ) may be expressed as a function of its local value E∥0(ψ ) taken at the poloidal position where the magnetic field B is minimum, and

⟨ ⟩ ^ql+1∕2 Rp BT 0,l+1∕2-l+1 ∕2,j+1∕2 E∥ ϕ,l+1∕2 = E∥0,l+1∕2----------------------λ1,-3,4 ql+1∕2 R0,l+1∕2 B0,l+1∕2
(5.587)

Exact and effective fractions of trapped electrons

The exact fraction of trapped electrons is given by relation,

Ft,l+1∕2 = ⌊ n∑p-1nξ∑0-1 ⌈ p2 H (||ξ0,j+1∕2||- ξ0T,l+1∕2)λl+1∕2,j+1∕2 i=0 j=0 i+1∕2
×( ) ] ^f(l+0)1∕2,i+1∕2,j+1∕2 + g(l0+)1∕2,i+1∕2,j+1∕2 Δpi+1∕2Δξ0,j+1∕2
×⌊ n∑p- 1nξ0∑- 1 ⌈ p2i+1∕2λl+1∕2,j+1∕2 i=0 j=0
×( (0) ^(0) (0) ) ] f0,l+1∕2,i+1∕2,j+1∕2 + fl+1∕2,i+1∕2,j+1∕2 + gl+1∕2,i+1∕2,j+1∕2 Δpi+1 ∕2Δ ξ0,j+1∕2-1
(5.588)

according to calculations given in Sec. 3.6.

The effective fraction of trapped electrons, as deduced from the Lorentz model in Sec. 5.6.2 is

Ft,l+1∕2eff. = 3- 2BT-0,l+1∕2 B0,l+1∕2×
⌊ nξ -1 ⌈ ^ql+1∕2BT-0,l+1∕2 ∑0 2 l+1∕2,j+1∕2 ql+1∕2 B0,l+1∕2 ξ0,j+1∕2λ2,- 2,2 Δ ξ0,j+1∕2 j=0
q R n∑ξ0-1 (| | ) - -l+1∕2--0,l+1∕2- σj+1∕2H |ξ0,j+1∕2|- ξ0T,l+1∕2 ql+1∕2 Rp j=0
× ( || ||) ] ξ0,j+1∕2IL ψl+1∕2,ξ0,j+1∕2 Δ ξ0,j+1∕2 (5.589)

Runaway loss rate

As shown in Sec. 3.6, the runaway loss rate ⟨Γ R⟩V (ψ) is given

nξ-1 ^ql+1∕2- 2 ∑0 l+1∕2,j+1∕2 (0) ⟨Γ R⟩V,l+1∕2 = ^ql+1∕22πpnp-1 λ Sp,l+1∕2,np-1,j+1∕2Δ ξ0,j+1∕2 j=0
(5.590)

Magnetic ripple losses

In a similar way, the magnetic ripple loss rate ΓST (0)(ψ) is given by

ΓST,l+1∕2(0) = 2πp ic2 ∑ j=0nξ0-1λl+1∕2,j+1∕2(1 - H (||ξ ||- ξ )) 0,j+1 ∕2 0ST,l+1∕2Sp,l+1∕2,ic,j+1∕2(0)Δξ 0,j+1∕2
+ 4πλl+1∕2,jST ∘ --------- 1 - ξ2 0ST,j ∑ i=0np-1p i+1∕2H(p - p ) i+1∕2 ic+1∕2Sξ0,l+1∕2,i+1∕2,jST(0)Δp i+1∕2 (5.591)

and the second formulation given in Sec.3.6 is

ΓST,l+1∕2(0) = 2π ∑ i=0np-1 ∑ j=0nξ0-1( (| | )) 1- H |ξ0,j+1∕2|- ξ0ST,l+1∕2H( ) pi+1∕2 - pic+1∕2
× νdST,l+1∕2,i+1∕2,j+1∕2f0,l+1∕2,i+1∕2,j+1∕2(0)Δξ 0,j+1∕2Δpi+1∕2 (5.592)

where ic is the index number that corresponds to the detrapping threshold by collisions, and ξ0ST,l+1∕2 is the picth-angle boundary value between super-trapped and trapped electrons.

Non-thermal bremsstrahlung

For the bremsstrahlung emission, it is necessary to calculate numericaly all coefficients of the Legendre series. Since Legendre polynomials Pm strongly oscillate between -1 and +1, as their order m increases, it is not possible to evaluate accurately integrals of the type

∫ +1 h(m) = h (x)Pm (x) dx -1
(5.593)

by standard integration techniques, like trapezoidal or Simpson rules. The only possibility is to replace integral (5.593) by a discrete sum, namely a Gaussian quadrature,

∑N h(m ) = wnh (xn)Pm (xn) n=1
(5.594)

where weights wn and abscissas xn are determined independently. Here, the use of Legendre polynomials lead to consider the fast ans accurate Gauss-Legendre algorithm as derived by G. B. Ribicki in Ref. [?], where xn are N zeros of the Legendre polynomial of degree N, the weights being given by the relation

---------2-------- wn = 2 [ ′ ]2 (1 - xn) PN (xn)
(5.595)

Very accurate determination of h(m ) may be obtained by this method, which requires a value of N = 50.

[next] [prev] [prev-tail] [front] [up]