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

2014 KSTAR Workshop 2 / 18

• 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?

Possible fluid models

2014 KSTAR Workshop 3 / 18

• 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 kz simulation).
• They are two-dimensional simulation.

Drift-Alfven Model(DA)

2014 KSTAR Workshop 4 / 18

With cold ion and isothermal electron, dn dt = −n∇·(vE + vpi) , ∇⊥ · nvpi − ∇⊥ · nvde = 1 eµ0 ∇∇2

⊥ ψ ,

Te∇n + n e E = ηJ , In the non-dimensionless form, dn dt = −(1 − ǫn)∂φ ∂y + ǫn ∂n ∂y + ∂ ∂z (j + jext) + β

• j + jext, ψ
• ,

d dtΩ = ǫn ∂n ∂y + ∂ ∂z (j + jext) + β

• j + jext, ψ
• , and

β ∂ψ ∂t = ∂ ∂z (n − φ) − β ∂ ∂y (ψ + ψext) + β

• n − φ, ψ + ψext

− ˜ ηj . where df dt = ∂f ∂t + vE · ∇f = ∂f ∂t + [φ, f] ∇f = ˆ b · ∇f = ˆ b0 · ∇f + 1 B0 [f, ψ]

Reduction of Drift-Alfv´ en to Hasegawa-Wakatani

2014 KSTAR Workshop 5 / 18

The linear parallel electron dynamics can be expressed, ∂j ∂z

• k

= − k2

z/˜

η 1 − i β(ω−ω∗)

k2

⊥ ˜

η

(n − φ) , (1) Here, k2

z/˜

η is the adiabatic parameter in the Hasegawa-Wakatani equation.

• β(ω − ω∗) ≪ k2

⊥˜

η : the current fluctuation leads the non-adiabatic response by π/2, j = i[kz/˜ 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∗) ≫ k2

⊥˜

η j = − kzk2

⊥

β(ω − ω∗)(n − φ). (2) The current is on the opposite phase to the electrostatic response, n − φ. ∇j ∝ −i k2

zk2 ⊥

β(ω − ω∗)(n − φ) . When ω ≫ ω∗, ∇j = −ic1(n − φ) gives the dispersive linear dynamics, it would not directly modify the stability.

Hasegawa-Wakatani Equation(HW)

2014 KSTAR Workshop 6 / 18

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

∂t +

• φ, ∇2

⊥ φ

• = ∇j

(4) where j = j0 + δj , j0 = −D∇0 (φ − n) = −D B0 B0 · ∇ (φ − n) , δj = −D

• ψext, φ − n
• .

The external magnetic field perturbation is given in this work as ψext = ψext sin 2πnext Ly y

• .

(5)

Simulation Parameters

2014 KSTAR Workshop 7 / 18

Device MAST(KirkChapman13) ASDEX(PeerKendl13) Te ( eV) 150 300 ne (1019 m−3) 2 2.5 BT ( T) 0.45 2.0 Cs ( km/s) 84.8 120 ρs ( mm) 1.77 1.25 Ln ( m) 0.1 0.06 ˜ η(˜ η = ρs

Ln me mi νe)

1.072 0.58

• In this work, β = 0.01, ˜

η = 0.01 and kzLn = 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.

On the perturbation amplitude

2014 KSTAR Workshop 8 / 18

• the amplitude of perturbations is calculated,

˜ B B0 = (kyρs) ρs Ln

• βψext
• Therefore, ψext = 100 is estimated

˜ B B0 = (0.1) 1 mm 10 cm

• (0.01)(100) ≃ 10−3

(6)

• In the HW model, the amplitude takes into account the β factor,

ψext(Hw) = βψext(DA)

Linear Eigenvalues in Comparison to HW.

2014 KSTAR Workshop 9 / 18

• Lin. Frequency(c11, γmax= 6.29e-02 at ky = 1.10)

0.5 1.0 1.5 2.0 2.5 3.0 ky

• 1

1 2 3 4 ω

(c11, γmax= 6.29e-02 at ky = 1.10)

0.5 1.0 1.5 2.0 2.5 3.0 ky 0.001 0.010 0.100 1.000 10.000 γ

j/(n-φ)

0.5 1.0 1.5 2.0 2.5 3.0 ky

• 1.0
• 0.5

0.0 0.5 1.0 phase

• The linear frequencies on the unstable branch of the DA is identical to the
• nes of HW.
• The ratio of growth rates to wave frequencies are small.
• At low ky, the current response to the non-adiabatic component (π/2) is

the instability mechanism, however at large ky, it also stabilizes the instability.

• At large ky, even the current response is close to the dissipation.

Standard Runs

2014 KSTAR Workshop 10 / 18

• A pseudo-spectral method with discrete Fourier transform is implemented.
• (∆kx, ∆ky) = (0.1, 0.1) and (Nx, Ny) = (256, 256)
• Given α, after simulations are saturated, the magnetic perturbation is

added.

α = 2.00

200 400 600 800 1000 1200 t 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Eφ

Ψext = 0 Ψext = 05 Ψext = 10 Ψext = 15 Ψext = 20 η = 0.01, β = 0.010(n) 500 1000 1500 2000 2500 t 2 4 6 8 En

ψext = 0.00 ψext = 1.00 ψext = 5.00 ψext = 10.00 ψext = 50.00 ψext = 100.00

Hasegawa Wakatani in action

2014 KSTAR Workshop 11 / 18

As the magnetic field strength changes, turbulence increases.

α = 2.00

5 10 15 20 Ψext 0.8 1.0 1.2 1.4 1.6 1.8 2.0 E/Eref

Eφ Eψ

α = 2.00

5 10 15 20 Ψext 0.30 0.35 0.40 0.45 <Γn>

• 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.

Change in amplitude spectrum

2014 KSTAR Workshop 12 / 18

α = 0.01

0.1 1.0 10.0 ky 10-8 10-6 10-4 10-2 100

|ψ(Ψext = 15)|2 |φ(Ψext = 15)|2 |ψ(Ψext = 0)|2 |φ(Ψext = 0)|2

α = 2.0

0.1 1.0 10.0 ky 10-8 10-6 10-4 10-2

|ψ(Ψext = 20)|2 |φ(Ψext = 20)|2 |ψ(Ψext = 0)|2 |φ(Ψext = 0)|2

α = 0.01

0.1 1.0 10.0 kx 10-8 10-6 10-4 10-2 100

|ψ(Ψext = 15)|2 |φ(Ψext = 15)|2 |ψ(Ψext = 0)|2 |φ(Ψext = 0)|2

α = 2.0

0.1 1.0 10.0 kx 10-8 10-6 10-4 10-2

|ψ(Ψext = 20)|2 |φ(Ψext = 20)|2 |ψ(Ψext = 0)|2 |φ(Ψext = 0)|2

• In the hydrodynamic regime, the decrease in low-k spectra comes with increase

in high-k increase in electrostatic potential, in the kx and ky 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 kx direction.

Fluctuation Energy in DA

2014 KSTAR Workshop 13 / 18

η = 0.01, β = 0.010(n) 1 10 100 1000 ψext 0.1 1.0 10.0 E/Eref

En Eφ Eψ Etot Γn

η = 0.01, β = 0.010(a) 1 10 100 1000 ψext 0.1 1.0 10.0 100.0 E/Eref

En Eφ Eψ Etot Γn

• 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,

ψext = 5 (HW) and ψext = 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.

Change in amplitude spectrum in DA

2014 KSTAR Workshop 14 / 18

η = 0.01, β = 0.01 0.1 1.0 10.0 ky 10-8 10-6 10-4 10-2 100 102

|ψ(Ψext = 500)|2 |φ(Ψext = 500)|2 |n(Ψext = 500)|2 |ψ(Ψext = 0)|2 |φ(Ψext = 0)|2 |n(Ψext = 0)|2

η = 0.01, β = 0.01 0.1 1.0 10.0 ky 10-6 10-4 10-2 100 102 104

|ψ(Ψext = 500)|2 |φ(Ψext = 500)|2 |n(Ψext = 500)|2 |ψ(Ψext = 0)|2 |φ(Ψext = 0)|2 |n(Ψext = 0)|2

η = 0.01, β = 0.01 0.1 1.0 10.0 kx 10-8 10-6 10-4 10-2 100 102

|ψ(Ψext = 500)|2 |φ(Ψext = 500)|2 |n(Ψext = 500)|2 |ψ(Ψext = 0)|2 |φ(Ψext = 0)|2 |n(Ψext = 0)|2

η = 0.01, β = 0.01 0.1 1.0 10.0 kx 10-6 10-4 10-2 100 102 104

|ψ(Ψext = 500)|2 |φ(Ψext = 500)|2 |n(Ψext = 500)|2 |ψ(Ψext = 0)|2 |φ(Ψext = 0)|2 |n(Ψext = 0)|2

• (Left: only nonlinear) All over the spectrum range, the fluctuations increases.
• (Right : full) No change is observed.

How the magnetic perturbation relaxes back?

2014 KSTAR Workshop 15 / 18

200 400 600 800 1000 0.010 1.000 100.000

ψ φ n

• This quickly relaxes to the stationary state.
• The decay rates, 0.007 is close to the linear damping rates of 0.004, 0.005.
• There seems to be very robust“nonlinear equilibrium state”

. And the externally imposed magnetic field is quickly balanced by the plasma response.

Phases of nonlinear current fluctuations

2014 KSTAR Workshop 16 / 18

ky=0.1

• 1.0
• 0.5

0.0 0.5 1.0 θ 0.005 0.010 0.015 P(φ,n-φ)

ψext(a) ψext(n) ψext=0

ky=0.1

• 1.0
• 0.5

0.0 0.5 1.0 θ 0.00 0.01 0.02 0.03 P(φ,n-φ)

ψext(a) ψext(n) ψext=0

• With only nonlinear interaction and no external perturbation, the phase

are randomly distributed.

• The phases between the current and the nonadiabatic components are

giving strong parallel dissipation when the large external perturbation is imposed (left) ψext = 100 and (right) ψext = 500.

Wavenumber-frequency spectrum

2014 KSTAR Workshop 17 / 18

• (Top: no perturbation) The

spectra peak at three linear wave frequencies.

• (Middle: nonlinear only) While

the peaks at the linear wave frequencies remains observable, the spectra becomes noiser.

• (Bottom: full) With a new

peak at ω = 0, the spectra is similar to the spectra without the perturbation.

Conclusion

2014 KSTAR Workshop 18 / 18

• We investigated the effects of the externally short-scale radial magnetic

field in the drift(-Alfv´ en ) wave turbulence in the context of HW and DA models.

• Without no magnetic fluctuation response, the imposed field increases the

fluctuation levels while the transport barely varies.

• With magnetic fluctuation response, nonlinear interaction with the

imposed field increase the transport and fluctuation levels more strongly.

• However, when the imposed field is included even in the linear terms, the

turbulence quickly goes back to the turbulence level as if there is no imposed field. In future,

• large resistivity (hydrodynamics regimes in HW) need to be explored to

see whether this quick response could be observed.