Reduced stability diagnostics

VAFT holds many stability-related quantities: beta limits, the Greenwald fraction, $l_i$–$q$ boundaries, the $s$–$\alpha$ ballooning model, Suydam’s and Mercier’s interchange criteria, Bussac’s internal kink, $\Delta’$, and the local stability quantities DCON, RDCON and STRIDE write. They are not interchangeable stable/unstable tests. Some are exact definitions, some are reduced models with strong assumptions, some are empirical correlations of a few machines, and some are heuristics kept only for backward compatibility. This page records which is which.

The relationship to the full solvers is

equilibrium-derived reduced diagnostics
        ├── interpretation
        ├── limiting-case validation
        └── comparison  ──>  DCON / RDCON / STRIDE / GPEC

and not a screening gate. Satisfying Suydam, Mercier, Bussac or $s$–$\alpha$ does not prove MHD stability. For VEST, $a/R_0$ is not small, so where a large-aspect-ratio criterion disagrees with a full-geometry calculation, the disagreement is information about the reduced model.

Logical status

A necessary criterion (Suydam, Mercier) proves instability when violated and proves nothing when satisfied. A marginal boundary ($s$–$\alpha$, Bussac’s $\beta_{p1} \approx 0.3$) separates stable from unstable only within its model.

Inventory

Deprecated heuristics stay for backward compatibility and warn when called: kink_stability_criterion ($\beta_{N,crit} = 2.8\,q_{95}$, unsourced), sawtooth_stability_criterion, beta_stability_boundary and plasma_stability_margins. power_limit_from_beta and power_limit_from_q are misnamed: they are rearranged $\beta$ and $q$ relations, not power limits. They are listed as heuristics and are not yet deprecated.

Suydam and Mercier

Both compare shear stabilisation, $rB^2(q’/q)^2/8\mu_0$, with the pressure drive $p’$. The toroidal average curvature turns the drive into $p’(1 - q^2)$: outside $q = 1$ an outward pressure fall stabilises. Both are necessary conditions for local interchanges.

Sign conventions differ. suydam_criterion and mercier_criterion_circular are positive where the criterion holds. The GGJ index $D_I$, which DCON writes, is negative where it holds, and $D_R = D_I + (H - \tfrac12)^2 \ge D_I$, so a surface can be Mercier-stable and still resistive-interchange unstable.

Bussac internal kink

The taxonomy files Bussac under current-driven modes because it belongs to the $m = 1$ kink family, even though its drive variable is the pressure through $\beta_{p1}$. In a cylinder the $m = n = 1$ internal kink is marginal at leading order, and toroidicity decides it. For a parabolic $q$ inside $r_1$, $\delta\hat W_T \propto (1 - q_0)(13/144 - \beta_{p1}^2)$, with $\beta_{p1} = 2\mu_0(\langle p\rangle_1 - p(r_1))/B_{\theta1}^2$. A core above $\beta_{p1} \approx 0.30$ is unstable. Shaping, other $q$ profiles and the kinetic terms of Porcelli’s model change this. Expose $\epsilon_1 = r_1/R_0$, the elongation and the triangularity alongside the result, so a later study can see how far an equilibrium is from the model.

Where the layers stop

The kernels here take explicit physical quantities and never read an ODS. Evaluating them on a reconstructed equilibrium belongs to vaft.process:

• locating $q = m/n$ surfaces;
• profile derivatives;
• flux-surface averages, the equilibrium-native Mercier $D_I$, and curvature and local shear.

Running and reading solvers belongs to vaft.code.gpec. Those are the next steps of #1635 and are left to the stability workflow. Use MHD stability for the solver side and MHD mode representations across geometries for which mode family each criterion addresses.

References

1. B. R. Suydam, Proc. 2nd UN Int. Conf. Peaceful Uses of Atomic Energy 31 (1958) 157.
2. C. Mercier, Nucl. Fusion 1 (1960) 47.
3. A. H. Glasser, J. M. Greene and J. L. Johnson, Phys. Fluids 18 (1975) 875.
4. M. N. Bussac, R. Pellat, D. Edery and J. L. Soulé, Phys. Rev. Lett. 35 (1975) 1638.
5. J. W. Connor, R. J. Hastie and J. B. Taylor, Phys. Rev. Lett. 40 (1978) 396.
6. C. Z. Cheng, H. P. Furth and A. H. Boozer, Plasma Phys. Control. Fusion 29 (1987) 351.
7. F. Porcelli, D. Boucher and M. N. Rosenbluth, Plasma Phys. Control. Fusion 38 (1996) 2163.
8. J. M. Greene, Comments Plasma Phys. Control. Fusion 17 (1997) 389.
9. J. P. Freidberg, Ideal MHD, Cambridge University Press (2014).