Linear JOREK n-scan of the ITER hybrid EQDSK case
The first JOREK stability scan uses case (a): the ITER hybrid EQDSK as given, with the TORAX density used only to split its pressure into density and temperature. Each toroidal mode number n runs separately from round-off amplitude on the accepted 80 × 128 equilibrium (entry 67). The growth rate γ is the slope of ½ ln E, where E is the mode’s energy. We give γ in JOREK’s time unit √(μ₀ρ₀), about half a microsecond here.
What happened
The first job was lost. At Δt = 3 the axisymmetric start went numerically unstable, and every linear run then failed on its first step. The rerun used Δt = 1, with a health gate: the linear runs start only if the axisymmetric state keeps its equilibrium flux on axis and positive density and temperature. It passed.
Every linear run then grew over about 16 decades and stopped on JOREK’s own GMRES abort (GMRES is the iterative linear solver). JOREK exits with code 0 when this happens, so our run wrapper recorded all eight runs as completed (8 of 8 case (a) runs ended on the GMRES abort with exit code 0; a test now checks that such a run is recorded as solver_aborted and not retriedC-230). The cause: in this JOREK version a “linear” run freezes the n = 0 part, but the n ≠ 0 equations stay nonlinear, so the matrix drifts as the mode grows. The abort comes after the linear phase, so the growth data are usable. The wrapper now classifies the abort as solver_aborted, keeps the energies and does not retry, and a unit test covers it.
Growth rates
γ is fitted over the energy band 1e-15 to 1e-7, a band we chose after seeing the traces. A fit counts only if its two half-windows, and separate fits to the magnetic and kinetic energy, agree. All eight runs pass (half-window and magnetic/kinetic fits agree within 1.1% (tolerance 2%) in all 8 runs; moving the energy band one decade either way changes γ by at most 0.6%C-231).
| n | where the mode peaks | γ√(μ₀ρ₀) |
|---|---|---|
| 1, 2, 3, 5 | core, at the q minimum | γ√(μ₀ρ₀) 0.012–0.028, peak at psi_N 0.12–0.18, where q has its minimum of about 1.11C-225 |
| 8, 20 | magnetic axis | γ√(μ₀ρ₀) 0.028 and 0.033, confined to psi_N below 0.02C-226 |
| 12 | pedestal | γ√(μ₀ρ₀) 0.0440 (γ 8.3 × 104 s⁻¹), peak at psi_N 0.975, γ/ω* 0.23C-223 |
| 16 | pedestal | γ√(μ₀ρ₀) 0.0374 (γ 7.1 × 104 s⁻¹), peak at psi_N 0.975, γ/ω* 0.14C-224 |
Every n is unstable with diamagnetic terms included, but only n = 12 and 16 are pedestal modes. Their ratio γ/ω* to the ion diamagnetic frequency is below the EPED-like threshold of ¼, but that reading assumes an ideal growth rate, and these include diamagnetic stabilisation.
- Core modes. n = 1–5 sit where the reversed-shear q profile of this EQDSK has its minimum, just above 1. Case (a) also overstates the core pressure (66.6 keV in case (a) (EQDSK pressure split by TORAX density) against 14.4 keV in case (b) (TORAX at 80 s)C-239), which favours such modes.
- Axis modes. n = 8 and 20 are confined to the magnetic axis. A high-n mode there is probably numerical.
- Hidden modes. Where a core or axis mode grows fastest, a slower pedestal mode of the same n is hidden, because the fastest mode dominates the energy.
None of these values meets the convergence rule yet: γ is quoted only after halving Δt and doubling the grid. That check is entry 69.
Where this stands: how to get pedestal growth rates at the six n where a core or axis mode dominates went to the owner (OQ-41).
Technical details → JOREK. Decision → OQ-41.
JOREK acknowledgement. JOREK is developed by the JOREK team and is open source under the GNU LGPL v3. Results here use JOREK at commit 01a52be3. Citations: M. Hoelzl et al., Nuclear Fusion 61, 065001 (2021); M. Hoelzl et al., Nuclear Fusion 66, 116006 (2026). For a list of JOREK team members, refer to the author list of M. Hoelzl, G.T.A. Huijsmans, F.J. Artola, E. Nardon, M. Becoulet, N. Schwarz, A. Cathey, S.J.P. Pamela et al., Nuclear Fusion 64, 112016 (2024).
Technical details → JOREK › case a n scan · JOREK › runner
Decisions → OQ-39 · OQ-41