Gyrokinetic internal kink simulations


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(ss0Δn)]

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

ΩB=b×BBΩκ=(×b)

Proof via vorticilty equation (GK momentum equation) where δB appears as
K=δB˙+ΩBδψ ,δA˙=δψ

The following conditions are used:

  1. Perpendicular force balance:

    δB=ξBPδB˙=ϕ×BB3P

  2. Ideal Ohm’s law

    E=bϕ+δA˙=0

  3. MHD equilibrium

    κ=(b)b=μ0PB2+BB

If these condition are satisfied, one can show

K=δB˙+ΩBψ=Ωκϕ

Unperturbed orbits

Original equation (Frieman-Chen, Littlejohn, Hahm, Brizard, etc):

R˙(0)=vb+1qsB(msμBb×BB+msv2×b)=vb+vdvd1qsB(msμBb×BBΩB+msv2(×b)Ωκ) ,(×b)=b×(b)b

Compressional force balance: replace ΩB with Ωκ, see [Graves2019]

R˙(0)=vb+msqsB(μB+v2)(×b)

No need to change unperturbed velocity equation

v˙(0)=μ(b+msvqsB×b)B ,μ=v22B


Perturbed equations of motion

No need to change perturbed part of particle orbits

R˙(1)=bB×ϕvA(s)vA(h)qsmsA(h)(b+msvqsB×b)

Changes in energy equation needed. Original equation

ε˙(1)=vv˙(1)+μR˙(1)B=1B[μBb×BB+v2(×b)]ϕ+vB[qsBmsvb+μBb×BB+v2(×b)]A(h)+qsmsμB[bmsvqsB×BB2B]A(h)

transforms to

ε˙(1)=1B(μB+v2)(×b)ϕ+vB[qsBmsvb+(μB+v2)(×b)]A(h)+qsmsμB[bmsvqsB×BB2B]A(h)


Solving for delta B_|| (EUTERPE)

Equations of motion solved by EUTERPE (note that μ is the magnetic moment per mass):

R˙=(vqmA)b+1Bb×(mqμB+Ψ+mqμδB)v˙=qmb(mqμB+Ψ+mqμδB)

closed by the pressure balance:

δB=μ0Bs=i,emsμBδfsδ(Rx)d6z

Quasineutrality equations and parallel Ampere’s law are not modified. Definitions:

μ=v22B,B=B+mqv×b

d6z=Bd3Rdvdμdα,Ψ=ΦvA,Φ=12πΦ(R+ρ)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.