Gyrokinetic internal kink simulations
- Getting linear kink-type structure is easy using GYGLES, EUTERPE, ORB5
- Target: coupling kink with another physics (turbulence, AEs, EPs)
- Challenge: getting kink right (δB|| etc)
- Nonlinear kink evolution
Examples
Internal kink in large aspect ratio tokamak
This case has been considered using GYGLES, EUTERPE, and ORB5 (see Section 4.2).
We consider an “ad-hoc” tokamak with the minor radius ra=1 m, the major radius R0=10 m, the magnetic field at the axis B0=1 T, flat ion and electron temperatures Te=Ti defined by Lx=2ra/ρs=360 with ρs=miTe−−−−√/(eB) the sound gyroradius. The safety factor is q(s)=0.8(1+s2), the flux surface label s=ψ/ψa−−−−√ with ψa the poloidal flux at the plasma edge. The ambient plasma density profile n0i(s)=n0e(s) is given by
n0e(s)=n0exp[−Δnκntanh(s−s0Δn)]
with κn=3.0, s0=0.5, Δn=0.2, and n0 corresponding
to βe=μ0n0eTe/B2=0.0052.
We use Ne=64×106 electron markers, Ni=16×106 ion markers, Ns=200 radial grid points, Nθ=16 poloidal grid points, and Nφ=8 toroidal grid points. The time step is ωciΔt=10. The Fourier filter includes the poloidal modes m∈[−2,2] and the toroidal mode n=1.
The equilibrium was not consistent (no Grad-Shafranov solved), δB∥=0 was used.
Internal kink in a straight tokamak (GYGLES)
Cosidered for “usual” safety factor profiles in PoP2012
For “W7-X like” rotational transform profiles in NF2021
Internal kink (linear) in a tokamak (GYGLES)
Internal kink in nonlinear turbulent environment (tokamak, ORB5)
Temperature profiles
Electrostatic potential evolution including turbulence (poloidal plane)
Electrostatic potential evolution, toroidal mode number n = 1
Electrostatic potential evolution, toroidal spectrum
Reduced model for delta B_||
Recipe: replace on each occurance
Ω∇B=b×∇BB⟹Ωκ=(∇×b)⊥
Proof via vorticilty equation (GK momentum equation) where δB∥ appears as
K=δB˙∥+ΩB⋅∇δψ ,δA˙∥=−∇∥δψ
The following conditions are used:
-
Perpendicular force balance:
δB∥=ξ⊥B⋅∇P⟺δB˙∥=∇ϕ×BB3⋅∇P
-
Ideal Ohm’s law
E∥=b⋅∇ϕ+δA˙∥=0
-
MHD equilibrium
κ=(b⋅∇)b=μ0∇PB2+∇BB
If these condition are satisfied, one can show
K=δB˙∥+ΩB⋅∇ψ=Ωκ⋅∇ϕ
Unperturbed orbits
Original equation (Frieman-Chen, Littlejohn, Hahm, Brizard, etc):
R˙(0)=v∥b+1qsB∗∥(msμBb×∇BB+msv2∥∇×b)=v∥b+vdvd≈1qsB⎛⎝⎜⎜⎜msμBb×∇BBΩ∇B+msv2∥(∇×b)⊥Ωκ⎞⎠⎟⎟⎟ ,(∇×b)⊥=b×(b⋅∇)b
Compressional force balance: replace Ω∇B with Ωκ, see [Graves2019]
R˙(0)=v∥b+msqsB(μB+v2∥)(∇×b)⊥
No need to change unperturbed velocity equation
v˙(0)∥=−μ(b+msv∥qsB∗∥∇×b)⋅∇B ,μ=v2⊥2B
Perturbed equations of motion
No need to change perturbed part of particle orbits
R˙(1)=bB∗∥×∇⟨ϕ−v∥A(s)∥−v∥A(h)∥⟩−qsms⟨A(h)∥⟩(b+msv∥qsB∗∥∇×b)
Changes in energy equation needed. Original equation
ε˙(1)=v∥v˙(1)∥+μR˙(1)⋅∇B=−1B[μBb×∇BB+v2∥(∇×b)⊥]⋅∇⟨ϕ⟩+v∥B[qsBmsv∥b+μBb×∇BB+v2∥(∇×b)⊥]⋅∇⟨A(h)∥⟩+qsmsμB[∇⋅b−msv∥qsB∗∥∇×BB2⋅∇B]⟨A(h)∥⟩
transforms to
ε˙(1)=−1B(μB+v2∥)(∇×b)⊥⋅∇⟨ϕ⟩+v∥B[qsBmsv∥b+(μB+v2∥)(∇×b)⊥]⋅∇⟨A(h)∥⟩+qsmsμB[∇⋅b−msv∥qsB∗∥∇×BB2⋅∇B]⟨A(h)∥⟩
Solving for delta B_|| (EUTERPE)
Equations of motion solved by EUTERPE (note that μ is the magnetic moment per mass):
R⃗ ˙=(v∥−qm⟨A∥⟩)b⃗ ∗+1B∗∥b⃗ ×(mqμ∇B+∇⟨Ψ⟩+mqμ∇δB∥)v˙∥=−qmb⃗ ∗(mqμ∇B+∇⟨Ψ⟩+mqμ∇δB∥)
closed by the pressure balance:
δB∥=−μ0B∑s=i,ems∫μBδfsδ(R⃗ −x⃗ )d6z
Quasineutrality equations and parallel Ampere’s law are not modified. Definitions:
μ=v2⊥2B,B⃗ ∗=B⃗ +mqv∥∇×b⃗
d6z=B∗∥d3Rdv∥dμdα,Ψ=Φ−v∥A∥,⟨Φ⟩=12π∫Φ(R⃗ +ρ⃗ )dα
Background current (driving instability) in GK equations
Which distribution function is appropriate? (shifted Maxwellian is normally used for electrons)
Shift is given by flux-surface averaged ⟨b⋅∇×B⟩
Pfirsch-Schlueter current vs. Shafranov shift
Cancellation of destabilization (PS current) and stabilization (Shafranov shift). Poloidal variation of ambient current has to be included.