Effects of (Externally Imposed) Short-Scale Radial Magnetic Field on - PowerPoint PPT Presentation

Effects of (Externally Imposed) Short-Scale Radial Magnetic Field on Drift Alfv´ en Turbulence Juhyung Kim Department of Physics, KAIST 2014 KSTAR Conference, Jeong Seon, Kangwon-Do 02/25/2014

Motivations While the suppression mechanism of edge-localized mode(ELM) by ● resonant magnetic perturbation(RLM) has been thought to be attributed to stochastic magnetic field lines, there have been some evidences that that may not be the case. One alternative suggestion is the route via zonal flow(ZF) modification by ● RMP. While the mechanisms involves ZF, which is beyond the current work, here ● we investigate how turbulence changes within the Hasegawa-Wakatani model and drift-Alfv´ en model, given a short-scale magnetic perturbation. What is the origin of gyro-radius magnetic perturbation? It could arise ● directly from external perturbation or, via MHD or macroscopic plasma responses. How would this oscillatory field affect micro-turbulence? 2014 KSTAR Workshop 2 / 18

Possible fluid models Full gyrofluid models: the best choice with the tokamak geometry! ● Drift-Alfv´ en model is the most minimal model that is equipped with magnetic ● response of plasma to the external magnetic field. Particle transport can be calculated. Electromagnetic ion-temperature-gradient model could be appropriate when ● thermal transport, not particle transport, is of interest. Hasegawa-Wakatani model is the simplest. Since the instability comes from the ● parallel dynamics of electrons, the magnetic perturbation can perturb the field lines along which electrons follow. However, magnetic response could not be modeled directly. In this talk, we implement the Hasegawa-Wakatani and a drift-Alfv´ en model. The caveats are that We don’t deal with ZFs. ● The models are implemented in the slab geometry (fixed k z simulation). ● They are two-dimensional simulation. ● 2014 KSTAR Workshop 3 / 18

Drift-Alfven Model(DA) With cold ion and isothermal electron, d n d t = − n ∇ · ( v E + v pi ) , 1 ∇ � ∇ 2 ∇ ⊥ · n v pi − ∇ ⊥ · n v de = ⊥ ψ , eµ 0 T e ∇ � n + n e E � = ηJ , In the non-dimensionless form, d n d t = − (1 − ǫ n ) ∂φ ∂n ∂y + ∂ ∂z ( j + j ext ) + β j + j ext , ψ � � ∂y + ǫ n , d ∂n ∂y + ∂ ∂z ( j + j ext ) + β j + j ext , ψ � � d t Ω = ǫ n , and β ∂ψ ∂t = ∂ ∂z ( n − φ ) − β ∂ ∂y ( ψ + ψ ext ) + β n − φ, ψ + ψ ext � � − ˜ ηj . where d f d t = ∂f ∂t + v E · ∇ f = ∂f ∂t + [ φ, f ] b 0 · ∇ f + 1 ∇ � f = ˆ b · ∇ f = ˆ [ f, ψ ] 2014 KSTAR Workshop 4 / 18 B 0

Reduction of Drift-Alfv´ en to Hasegawa-Wakatani The linear parallel electron dynamics can be expressed, k 2 � ∂j � z / ˜ η = − ( n − φ ) , (1) 1 − i β ( ω − ω ∗ ) ∂z k k 2 ⊥ ˜ η Here, k 2 z / ˜ η is the adiabatic parameter in the Hasegawa-Wakatani equation. β ( ω − ω ∗ ) ≪ k 2 ⊥ ˜ η : the current fluctuation leads the non-adiabatic response by ● π/ 2 , j = i [ k z / ˜ η )]( n − φ ) . ∇ � j ∝ − α ( n − φ ) . And the term will give the collisional dissipation. This keeps the density and the electrostatic potential close, n ∼ φ when α � O (1) . β ( ω − ω ∗ ) ≫ k 2 ⊥ ˜ η ● k z k 2 ⊥ j = − β ( ω − ω ∗ )( n − φ ) . (2) The current is on the opposite phase to the electrostatic response, n − φ . k 2 z k 2 ⊥ ∇ � j ∝ − i β ( ω − ω ∗ )( n − φ ) . When ω ≫ ω ∗ , ∇ � j = − ic 1 ( n − φ ) gives the dispersive linear dynamics, it would 2014 KSTAR Workshop 5 / 18 not directly modify the stability.

Hasegawa-Wakatani Equation(HW) Leconte (2011, 2012) suggested the use of Hasegawa-Wakatani model in the context of RMP, calculating Maxwell stress on zonal flow. Following his work, ∂n ∂t + [ φ, n ] = − ∂φ ∂y + ∇ � j � (3) ∂ ∇ 2 ⊥ φ φ, ∇ 2 � � + ⊥ φ = ∇ � j � (4) ∂t where j � = j � 0 + δj � , B 0 j � 0 = − D � ∇ � 0 ( φ − n ) = − D � · ∇ ( φ − n ) , B 0 ψ ext , φ − n � � δj � = − D � . The external magnetic field perturbation is given in this work as � 2 πn ext � ψ ext = ψ ext sin y . (5) 0 L y 2014 KSTAR Workshop 6 / 18

Simulation Parameters Device MAST(KirkChapman13) ASDEX(PeerKendl13) T e ( eV) 150 300 n e (10 19 m − 3 ) 2 2.5 B T ( T) 0.45 2.0 C s ( km / s) 84.8 120 ρ s ( mm) 1.77 1.25 L n ( m) 0.1 0.06 η = ρ s m e η ( ˜ ˜ m i ν e ) 1.072 0.58 L n In this work, β = 0 . 01 , ˜ η = 0 . 01 and k z L n = 0 . 1 ● In a parameter range, β = 0 . 01 ∼ 0 . 1 and ˜ η = 0 . 01 ∼ 0 . 1 , the basic ● picture in the following does not change. 2014 KSTAR Workshop 7 / 18

On the perturbation amplitude the amplitude of perturbations is calculated, ● � ρ s ˜ � B βψ ext = ( k y ρ s ) B 0 L n Therefore, ψ ext = 100 is estimated ● � 1 mm ˜ � B (0 . 01)(100) ≃ 10 − 3 = (0 . 1) (6) B 0 10 cm In the HW model, the amplitude takes into account the β factor, ● ψ ext (Hw) = βψ ext (DA) 2014 KSTAR Workshop 8 / 18

Linear Eigenvalues in Comparison to HW. Lin. Frequency(c11, γ max = 6.29e-02 at k y = 1.10) (c11, γ max = 6.29e-02 at k y = 1.10) j/(n- φ ) 1.0 4 10.000 3 0.5 1.000 2 phase 0.0 ω γ 0.100 1 -0.5 0 0.010 -1 -1.0 0.001 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 k y k y k y The linear frequencies on the unstable branch of the DA is identical to the ● ones of HW. The ratio of growth rates to wave frequencies are small. ● At low k y , the current response to the non-adiabatic component ( π/ 2 ) is ● the instability mechanism, however at large k y , it also stabilizes the instability. At large k y , even the current response is close to the dissipation. ● 2014 KSTAR Workshop 9 / 18

Standard Runs A pseudo-spectral method with discrete Fourier transform is implemented. ● (∆ k x , ∆ k y ) = (0 . 1 , 0 . 1) and ( N x , N y ) = (256 , 256) ● Given α , after simulations are saturated, the magnetic perturbation is ● added. α = 2.00 η = 0.01, β = 0.010(n) 8 ψ ext = 0.00 Ψ ext = 0 3.0 ψ ext = 1.00 Ψ ext = 05 ψ ext = 5.00 ψ ext = 10.00 Ψ ext = 10 ψ ext = 50.00 Ψ ext = 15 ψ ext = 100.00 2.5 6 Ψ ext = 20 2.0 E φ 4 E n 1.5 1.0 2 0.5 0.0 0 0 200 400 600 800 1000 1200 0 500 1000 1500 2000 2500 t t 2014 KSTAR Workshop 10 / 18

Hasegawa Wakatani in action As the magnetic field strength changes, turbulence increases. α = 2.00 α = 2.00 E φ 2.0 0.45 E ψ 1.8 0.40 1.6 E/E ref < Γ n > 1.4 0.35 1.2 1.0 0.30 0.8 0 5 10 15 20 0 5 10 15 20 Ψ ext Ψ ext In the adiabatic regime, the total energies increase. ● The particle flux driven by turbulence varies within 10%, (compare with ● the amplitudes). The effect is negligible. ● 2014 KSTAR Workshop 11 / 18

Change in amplitude spectrum α = 0.01 α = 2.0 10 0 10 -2 10 -2 10 -4 10 -4 10 -6 10 -6 | φ ( Ψ ext = 0)| 2 | φ ( Ψ ext = 0)| 2 | ψ ( Ψ ext = 0)| 2 | ψ ( Ψ ext = 0)| 2 | φ ( Ψ ext = 15)| 2 | φ ( Ψ ext = 20)| 2 | ψ ( Ψ ext = 15)| 2 | ψ ( Ψ ext = 20)| 2 10 -8 10 -8 0.1 1.0 10.0 0.1 1.0 10.0 k y k y α = 0.01 α = 2.0 10 0 10 -2 10 -2 10 -4 10 -4 10 -6 10 -6 | φ ( Ψ ext = 0)| 2 | φ ( Ψ ext = 0)| 2 | ψ ( Ψ ext = 0)| 2 | ψ ( Ψ ext = 0)| 2 | φ ( Ψ ext = 15)| 2 | φ ( Ψ ext = 20)| 2 | ψ ( Ψ ext = 15)| 2 | ψ ( Ψ ext = 20)| 2 10 -8 10 -8 0.1 1.0 10.0 0.1 1.0 10.0 k x k x In the hydrodynamic regime, the decrease in low- k spectra comes with increase ● in high- k increase in electrostatic potential, in the k x and k y direction. k In the adiabatic regime, the increase in low k does not change the spectrum ● except high- k spectra of density fluctuation in the k x direction. 2014 KSTAR Workshop 12 / 18

Fluctuation Energy in DA η = 0.01, β = 0.010(n) η = 0.01, β = 0.010(a) E n E n E φ E φ 100.0 E ψ E ψ E tot E tot Γ n Γ n 10.0 10.0 E/E ref E/E ref 1.0 1.0 0.1 0.1 1 10 100 1000 1 10 100 1000 ψ ext ψ ext The left graph shows how turbulent energy changes when the magnetic ● field is imposed only in the nonlinear terms. In comparison to the HW, at the same amplitudes of the magnetic fields, ● ψ ext = 5 (HW) and ψ ext = 500 (DA), the turbulence enhancement is much stronger in the drift-Alfv´ en . However, when the magnetic field is imposed in all the terms, the ● turbulence enhancement disappears. 2014 KSTAR Workshop 13 / 18

Change in amplitude spectrum in DA η = 0.01, β = 0.01 η = 0.01, β = 0.01 10 2 10 4 10 0 10 2 10 -2 10 0 10 -4 10 -2 10 -6 |n( Ψ ext = 0)| 2 |n( Ψ ext = 0)| 2 | φ ( Ψ ext = 0)| 2 | φ ( Ψ ext = 0)| 2 10 -4 | ψ ( Ψ ext = 0)| 2 | ψ ( Ψ ext = 0)| 2 |n( Ψ ext = 500)| 2 |n( Ψ ext = 500)| 2 | φ ( Ψ ext = 500)| 2 | φ ( Ψ ext = 500)| 2 10 -8 | ψ ( Ψ ext = 500)| 2 | ψ ( Ψ ext = 500)| 2 10 -6 0.1 1.0 10.0 0.1 1.0 10.0 k y k y η = 0.01, β = 0.01 η = 0.01, β = 0.01 10 2 10 4 10 0 10 2 10 -2 10 0 10 -4 10 -2 10 -6 |n( Ψ ext = 0)| 2 |n( Ψ ext = 0)| 2 | φ ( Ψ ext = 0)| 2 | φ ( Ψ ext = 0)| 2 10 -4 | ψ ( Ψ ext = 0)| 2 | ψ ( Ψ ext = 0)| 2 |n( Ψ ext = 500)| 2 |n( Ψ ext = 500)| 2 | φ ( Ψ ext = 500)| 2 | φ ( Ψ ext = 500)| 2 10 -8 | ψ ( Ψ ext = 500)| 2 | ψ ( Ψ ext = 500)| 2 10 -6 0.1 1.0 10.0 0.1 1.0 10.0 k x k x (Left: only nonlinear) All over the spectrum range, the fluctuations increases. ● (Right : full) No change is observed. ● 2014 KSTAR Workshop 14 / 18