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B.4 Wave scattering by fluctuations

Low frequency fluctuations of the electron density or the magnetic field can strongly affect the propagation of rf waves in a magnetized plasma. This effect has been extensively investigated for the Lower Hybrid wave by solving a wave kinetic equation in the weak turbulence approximation. [14, 15]. In this approach, the flow of rf energy is described by a usual ray-tracing between random scattering events where the initial wave vector k can be modified both in amplitude and direction with the constraint to remain solution of the local dispersion relation. This physical mechanism is a good candidate for explaining a possible strong upshift of the power spectrum as the Lower Hybrid wave propagates in the plasma, even if the toroidal mode coupling for bridging the spectral gap is very weak. Though primarily dedicated to the Lower Hybrid wave, the concept of wave scattering by low frequency density or magnetic field fluctuations can be easily extended to other types of rf waves provided that ^k ∥ ≪ k∥ and ^ω ≪ ω, where k∥ = k ⋅ b and ^ k∥ = ^ k ⋅ b are the projections of the rf and fluctuation wave vectors respectively along the direction of the local magnetic field b = B∕∥B ∥, while ω and ^ω are the rf and fluctuation frequencies respectively. Consequently, the scattering process becomes a two dimensional problem in the plane perpendicular to the local magnetic field direction, a reasonable assumption regarding the large difference between the long wave length of the low frequency fluctuations and the short length of the rf waves along b.4

Calculations are organized in three steps. In the first section, the components of the scattered rf wave vector are calculated, according to the curvilinear coordinate system chosen for describing the magnetic equilibrium in C3PO. The wave kinetic equation corresponding to the wave-wave interaction is then derived from first principles in the second section. Finally, in the third section, the wave kinetic equation is solved using a Monte-Carlo method and the procedure for the numerical implementation in the code is explicited.

Before entering into the details of the calculations, it is possible to estimate the impact of a simple transfer from the toroidal mode number n to the poloidal one m at fixed k∥0 on the ray properties, by changing the initial conditions. The benchmark case presented in Sec.E.1 is considered using Nϕ0 = -1.8. From a scan of m∈{- 300,300} with steps Δm = 50, it is possible to show that a variation of m and n strongly affect the ray propagation, even if k∥0 remains unchanged.

B.4.1 Calculation of the scattered wave vector

When the rf wave vector k is scattered by low density fluctuations, it may change both in module and direction, with the conditions

(135)

where k′ is the scattered wave vector. Its perpendicular component k ⊥′ = ∥k ′⊥∥, where k⊥′ = k′× b, is determined by solving the local dispersion relation (??) at fixed k ∥, and its direction results from a rotation Rβ of an angle β around the magnetic field direction b so that

(136)

For unlike transitions, the scattered wave moves to a different branch of propagation and k⊥′≠k ⊥, otherwise k⊥′ = k ⊥ for like transitions. From the relations (??), (??) and (??), all the components ( ′ ′ ′) kρ,m ,n of k′ can be deduced from those of k.5

The wave scattering may be therefore described as a two steps process, and the mode conversion M and the rotation Rβ.

B.4.2 Derivation of the wave kinetic equation

B.4.3 Solution of the wave kinetic equation

Using the Rodrigues general formula for the rotation Rβ of a vector k,

(137)

it turns out that each components of ^ k is given by the relations

^k ⋅ ^ρ = cosβk ⋅ ^ρ + (1 - cos β)(k ⋅ b )b ⋅ ^ρ + sin β (b × k) ⋅ ^ρ ^k ⋅ ^θ = cos βk ⋅ ^θ + (1 - cosβ )(k ⋅ b)b ⋅θ^+ sinβ (b × k ) ⋅ ^θ ^ ^ ^ ^ ^ k ⋅ϕ = cosβk ⋅ϕ + (1 - cosβ )(k ⋅ b)b ⋅ϕ + sin β (b × k ) ⋅ϕ

Using the vectorial relations

(b × k ) ⋅ ^ρ = (^ρ × b ) ⋅ k ( ) (b × k ) ⋅ ^θ = ^θ × b ⋅ k ( ) (b × k ) ⋅ ^ϕ = ^ϕ × b ⋅ k
and the expression on the magnetic field b = σBT^ϕ+ σIPŝ according to the relation (??), recalling that T = BT ∕B and P = BP ∕B, one obtains
( ) (b × k ) ⋅ ^ρ = σBT ^ρ × ^ϕ ⋅ k + σIP (^ρ × ˆs) ⋅ k ( ) ( ) (b × k ) ⋅ ^θ = σ T ^θ × ^ϕ ⋅ k + σ P ^θ × ˆs ⋅ k B I ( ) ( ) ^ ^ ^ ^ (b × k) ⋅ ϕ = σBT ϕ × ϕ ⋅ k + σI P ϕ × ˆs ⋅ k
which simplifies to
(b × k ) ⋅ ^ρ = - σ Tˆs ⋅ k + σ P ^ϕ ⋅ k B I (b × k) ⋅ ^θ = σBT ^r ⋅ k - σIP sinα ^ϕ ⋅ k (b × k) ⋅ϕ^= - σIP ^ρ ⋅ k
since
^ρ × ^ϕ = - ^s θ^× ϕ^= ^r
and
^ρ × ˆs = ^ϕ ^θ × ˆs = - sinα ^ϕ ^ ϕ × ˆs = - ^ρ
using the relations ˆs = sin α^r + cos αθ^ and ^θ= ^ϕ׈r . Using (??),
m ˆs ⋅ k =- cosα r n ϕ^⋅ k = -- R ^r ⋅ k = kρ ∥∇ ρ∥cos α m- ^ρ ⋅ k = kρ∥ ∇ ρ∥ - sin α r ^ m- θ ⋅ k = - sinαk ρ∥∇ ρ∥ + r
and
m n (b × k ) ⋅ ^ρ = - σBT--cos α + σIP -- r R n (b × k ) ⋅ ^θ = σBT kρ∥∇ ρ∥ cosα - σI P sin α-- R( ) (b × k) ⋅ ^ϕ = - σIP kρ∥∇ ρ∥ - sinα m- r

Finally, since

b ⋅ ^ρ = 0 b ⋅ ^θ = σIP cosα b ⋅ ^ϕ = σBT
one obtains
m^ ( m ) ( m n) ^kρ∥∇ ρ∥ - sin α-- = cos β kρ∥ ∇ρ ∥ - sin α- + sin β - σBT -- cosα + σIP -- r ( r ) r R ( ) - sin α^k ∥∇ ρ∥+ ^m- = cos β - sin αk ∥∇ ρ∥ + m- + (1 - cos β)k σ P cosα+sin β σ T k ∥∇ ρ ∥cosα - σ P sin α n- ρ r ρ r ∥ I B ρ I R ^n n ( m ) R-= cosβ R-+ (1 - cos β)k ∥σBT - sin βσIP kρ∥∇ ρ∥ - sinα -r
or
( ( ) ) ^kρ∥∇ ρ∥ = cosβk ρ∥∇ ρ∥+ (1 - cosβ) k∥σIP tan α+sin β σBT-- kρ∥∇ ρ∥ sin α - m- + σIP n- cosα r R ^m m σIP σBT ( m ) -- = cosβ -- + (1 - cosβ )k∥----- + sin β ----- k ρ∥∇ ρ∥ - -- sin α r r cos α cosα ( r ) ^n- = cos β n + (1 - cosβ) k σ T + sin βσ P - k ∥∇ ρ∥ + sinα m- R R ∥ B I ρ r
(138)
where k ⋅ b = k∥ and the scattered wave vector is
(139)

If β = 0, it is straightforward to demonstrate that ^k = k. In addition, the relation ^k ⋅ b = k ⋅ b or

(140)

is also well satisfied whatever β, as expected from the initial assumption on the invariance of k∥ with the rotation of angle β, which is intrinsically considered in the Rodrigues formula (??).

In the cold limit approximation, the plasma has bi-refringent optical properties which means that for a given k∥, the dispersion relation cannot have more than two different roots in k⊥.

From the definitions k⊥ = k × b and k⊥′ = k′× b, the relation k ⊥⋅ k⊥′ = k ⊥k⊥′ cos β becomes

and

(141)

For like mode scattering, k⊥′ = k ⊥ and

(142)

a relation which can be well verified by replacing ( m ′ n ′) k′ρ∥∇ ρ ∥,---,-- r R from (138) into (??). Therefore, for like mode scattering, ( ) k′ρ,m ′,n ′ have just to be deduced from (138) knowing (kρ,m, n).

For unlike mode scattering, k⊥′≠k ⊥,

Therefore, The coordinates of k′ are therefore determined by relations (??) and (??) and (??),

(143)

By definition, the variation of the poloidal mode number m is made possible by the breaking of the toroidal symmetry resulting from the fluctuations, so that the toroidal mode number n is no more conserved in the scattering process.

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