1 Laboratório Associado de Plasma, Instituto Nacional de Pesquisas Espaciais São José dos Campos, SP, Brazil Fluid model of electron cyclotron current drive G.O. Ludwig 17 th IAEA Technical Meeting on Research Using Small Fusion Devices Centro de Fusão Nuclear, Instituto Superior Técnico 22 nd to 24 th October 2007 Lisbon, Portugal

2 Fluid model of electron cyclotron current drive (ECCD) G.O. Ludwig Abstract A macroscopic model was recently proposed for describing the dynamics of ECCD in tokamak plasma [1]. This model depends on the adoption of a suitable distribution function for the relativistic magnetized stream of current-carrying electrons [2], thus avoiding the use of complex quasi-linear 3-D Fokker-Planck codes. The model is readily applied to examine the current drive efficiency of high- power ECCD experiments [3,4,5,6], reproducing the main experimental features. Fokker-Planck (F-P) equation F-P equation for streaming electrons neglecting trapped particle and radial transport effects The fluxes due to both an inductive electric field and diffusion by RF waves are given by

3 Moments of the Fokker-Planck equation The rates of change of the kinetic energy and momentum give the equations of motion where the collision frequency of energetic electrons with the background plasma is normalized to c The volumetric densities of driven power and force are given in terms of the driving fluxes by These quantities must satisfy the electrodynamic relation which gives the rate at which RF energy is being converted to kinetic form

4 Quasi-linear RF diffusion dyadic In cylindrical coordinates in momentum space the gyro-averaged quasi-linear diffusion dyadic is In the small gyroradius limit for the fundamental resonance of electron cyclotron waves one has where the diffusion coefficient strength is expressed in terms of the perpendicular wave amplitude by For a narrow spectrum centered at n ║, within a small range Δn ║, the function Δ(ξ) is ~ a delta function

5 RF power and force densities The average values are evaluated according to the transformation (ξ, β ║ )→(πp ┴ 2, p ║ ) For Δn ║ →0 the leading terms in the expressions of the power and force densities are reducing the evaluation of average values to single integrals along the resonance lines

6 Resonance lines Resonance lines in velocity space normalized to the speed of light for ω=0.982Ω e The curves correspond to the refractive index n ║ =0, 0.15, 0.3, 0.5 and 0.9 from center to edge The dashed lines represent the trapped particle region for a mirror ratio 2/3, although the present model does not include its effect Trapped particle effects can be included on an average basis in the present zero-dimensional model

7 Distribution function of a magnetized relativistic electron stream (Dory-Guest-Harris type) The distribution function of a weakly collisional, strongly magnetized relativistic electron stream is [2] where the normalization coefficient is determined by the condition n ℓ = ∫ f ℓ d 3 p The intensive thermodynamic quantities satisfy the Lorentz transformation properties Taking ℓ = 0, T→∞ and = 0 the Jüttner distribution for an isotropic relativistic gas is recovered In the warm plasma limit the Chew-Goldberger-Low double adiabatic equations are recovered

8 Macroscopic quantities in the rest frame of the stream (vanishing momentum density) Four parameters:, ℓ, T ║, T → three macroscopic quantities in the rest frame ( =0) Energy density, magnetization and enthalpy density of component ℓ (apply a boost to lab frame) Parallel and perpendicular pressures (P ║, M/n and P ┴ are Lorentz invariants) Take stream distribution function as a sum of ℓ = 0 (bi-Maxwellian) and ℓ = 1 (DGH-like) components Four parameters:, F, T ║, T → three macroscopic quantities in the rest frame

9 Macroscopic model of ECCD Using the above form of the distribution function both the energy and parallel momentum losses by collisions with the background plasma, and the average perpendicular velocity are evaluated with satisfying the electrodynamic relation (conversion of RF to kinetic energy) The electron stream in ECCD does not perform work and there is no heat addition if the RF power input is balanced by power lost through collisions, leading to the thermodynamic relation (Gibb’s law) The equations of motion constrained by the electrodynamic and thermodynamic relations give four equations depending on the parameters, F, T ║ and T of the distribution function

10 ECCD discharges The expression of the parallel force density shows that the RF driven current vanishes when n ║ →0 For a fully RF driven tokamak plasma the inductive electric field is put equal to zero In the steady-state the global figure of merit is defined by with Relative density of high-energy electrons as a function of n ║ for ω = (0.975,0.982,0.985)Ω e

11 Application to the DIII-D electron cyclotron current drive experiment [4, 5] Tokamak parameters: R=1.7m, a=0.6m, κ=1.8, B=2.0T Plasma parameters: n e =2.0  m -3, T e =2.7keV, Z eff =1.4, I p =900kA ECCD experiment: f RF =110GHz (2 nd harmonic, ω=0.982Ω e ), W RF =1.5MW, I RF =75kA, ζ RF =0.05 Assumptions from ray-tracing: φ T =20º, n ║ = sin(φ T )=0.342, Δn ║ =0.02, D 0 =? (reference solution) Equilibrium solutions: < D 0 / (m e 2 c 2 ν c ) < 2.24 → < ΔA p /A p < ~published results [5] Assume normalized diffusion coefficient strength D 0 / (m e 2 c 2 ν c ) =1; scan n ║ for fixed W RF φTφT F<β║><β║>T ║ 0 /(m e c 2 )T 0 /(m e c 2 )T ┴ 0 /(m e c 2 ) 0o0o o φTφT n/n e ΔA p /A p P RF /(nm e c 2 ν c ) F ┴ /(nm e cν c ) <β┴><β┴> 0o0o o

12 Application Electron distribution function truncated at of its maximum value and contour plot of the distribution function for DIII-D (ω=0.982Ω e, Δn ║ =0.02, D 0 /(m e 2 c 2 ν c )=1, W RF =1.5 MW) φ T =0 o φ T =20 o

13 Application Logarithmic plots of the distribution function (continuous line at pitch angle 0 o, dashed line 90 o and dotted line 180 o ) φ T =0 o φ T =20 o φ T =0 o : contribution of high-energy electrons (~128 kV) in the tail of the distribution along the magnetic field lines; much lower energy in the perpendicular direction (<12 kV) φ T =20 o : about the same energy (~50 keV) in the parallel and perpendicular directions; small asymmetry between the forward and backward directions due to drift motion ( ≈ 0.03) The proposed form of the distribution function may be useful for fitting experimental results

14 Application Stream parameters (grey circles ω=0.975Ω e, black circles ω=0.982Ω e, dots ω=0.985Ω e, line φ T =20 o ) increases with n ║ ; cooling in the parallel direction and demagnetization (T→∞ at maximum n ║ )

15 Application Figure of merit (grey circles ω=0.975Ω e, black circles ω=0.982Ω e, dots ω=0.985Ω e, line φ T =20 o ) Decrease in the stream density for large n ║ ; compensated by increase in effective ΔA p (fixed W RF )

16 Application RF power density (grey circles ω=0.975Ω e, black circles ω=0.982Ω e, dots ω=0.985Ω e, line φ T =20 o ) RF power deposition limited by collisions; decrease in F ┴ compensated by increase in ; increase in T ┴ implies demagnetization of the high-energy stream (<12 → 60 keV for 0 o →25 o )

17 Experimental results DIII-D [5] Figures 1 and 2 (left): ΔA p /A p ≈ 0.13; ζ RF ≈ 0.05 at φ T =20 o (experimentalists definition) Figure 3 (right): Fluid model results (W RF =1.5 MW; vertical line n ║ =0.342, φ T =20 o )

18 Experimental results TCV [6] TCV model: similar to DIII-D model, but larger stream density and larger effective area of power deposition (same W RF =1.5 MW in small volume) Figure 1 (right): Experimental results 12 o <φ T <30 o ; 12 keV<T ┴ <50 keV Figure 2 (above): Fluid model results for DIII-D φ T =0 o → T ┴ <12 keV; φ T =20 o → T ┴ =51.1 keV; φ T =25 o → T ┴ ≈ 60 keV

19 Conclusions Macroscopic model based on a suitable distribution function of the high-energy electrons Effective value of the diffusion coefficient strength determined from reference solution: given toroidal launching angle, input power and global figure of merit obtained poloidal area of power deposition consistent with experimental values Main features of ECCD reproduced for fixed input power and diffusion coefficient strength: refractive index scan gives consistent variation of the global figure of merit perpendicular temperature variation consistent with HXR measurements Proposed form of the distribution function may become an useful tool for fitting experimental results Provides simple model for studying the time evolution of ECCD and possible instabilities (Weibel) Future work The model must be tested using results of ray-tracing codes Must include pitch-angle scattering due to Weibel instability (also true for F-P codes) Must include trapped particle effects

20 Acknowledgment Work partially supported by the IAEA under contract BRA/12932 References [1] G.O. Ludwig, “Macroscopic model of electron cyclotron current drive”, Plasma Phys. Control. Fusion (2007) submitted. [2] G.O. Ludwig, “Relativistic distribution functions, fluid equations and equations of state for magnetized electron streams”, Plasma Phys. Control. Fusion 49, (2007 ). [3] O. Sauter et al., “Steady-state fully noninductive current driven by electron cyclotron waves in a magnetically confined plasma”, Phys. Rev. Lett. 84, (2000). [4] R.W. Harvey, O. Sauter, R. Prater and P. Nikkola, “Radial transport and electron-cyclotron-current drive in the TCV and DIII-D Tokamaks”, Phys. Rev. Lett. 88, (2002). [5] C.C. Petty et al., “Detailed measurements of the electron cyclotron current drive efficiency on DIII- D”, Nucl. Fusion 42, (2002). [6] S. Coda et al., “Electron cyclotron current drive and suprathermal electron dynamics in the TCV tokamak”, Nucl. Fusion 43, (2003).