Delta_mn does not generally exist in favorable-curvature, nonzero pressure gradient plasmas

This is relevant to EF physics and is human written so I am requesting that you read it thoughtfully @logan-nc @parkjk @ebursch

${}\Delta_{mn}{}$ is the jump in the derivative of the pitch-resonant harmonic of the area-weighted surface-normal magnetic field,

$$
\Delta_{mn}\equiv A(\psi) \left[ \frac{ \partial }{ \partial \psi } \bar{b}^\psi \right]{mn} = \left[ \frac{ \partial }{ \partial \psi } \mathcal{J} \mathbf{b}\cdot\nabla\psi \right]{mn}.
$$

According to Glasser 2016 ${}^1{}$, Frobenius analysis of the singular points in the ideal MHD force-balance problem solved by DCON / STRIDE / RDCON / ForceFreeStates reveals that the resonant harmonics reduce in the vicinity of the rational surface with which they are resonant to small and large power series solutions whose leading order terms are

$$
\xi \propto z^{-\frac{1}{2}-\sqrt{ -D_{I} }}, \hspace{0.3cm} z^{-\frac{1}{2} + \sqrt{ -D_{I} }}
$$

with the solution on the left being the "large" and the right being the "small" resonant solutions, where ${}z = \psi-\psi_{res}{}$ and ${}D_{I}{}$ being the toroidal Mercier criterion quantity

\begin{aligned}
D_I = {}& -\frac{1}{4} + \frac{2\pi f P' V'}{q' \chi'^3}
\left\langle \frac{1}{|\nabla\psi|^2} \right\rangle
\left[ 1 - \frac{2\pi f P' V'}{q' \chi'^3}
\left\langle \frac{1}{|\nabla\psi|^2} \right\rangle \right] \\
& + \left\langle \frac{B^2}{|\nabla\psi|^2} \right\rangle
\frac{P' V'^2}{q'^2 \chi'^4}
\left\{ P' \left[ \left\langle \frac{1}{B^2} \right\rangle
+ \left( \frac{2\pi f}{\chi'} \right)^2
\left\langle \frac{1}{B^2 |\nabla\psi|^2} \right\rangle \right]
+ \frac{\chi''}{\chi'} - \frac{V''}{V'} \right\}.
\end{aligned}

The magnetic field solutions are raised by a power from the displacement solutions, with the large solution being dropped by FFS as a finite perturbed energy constraint (Fitzpatrick 2026 ${}^2{}$ showed this is equivalent to no-reconnected-flux). So perturbed magnetic field on either side of the rational surface is in general:

$$
\bar{b}^\psi (z) \propto \begin{cases}
-C_{L} \hspace{2mm} \left| z \right| ^{\frac{1}{2}+\sqrt{ -D_{I} }}, & z < 0, \
C_{R} \hspace{2mm} \left| z \right| ^{\frac{1}{2}+\sqrt{ -D_{I} }}, & z > 0.
\end{cases}
$$

This has a finite derivative jump proportional to

$$
\lim_{ \epsilon \to 0^+ } C_{R}\left| \epsilon \right|^{-\frac{1}{2}+\sqrt{ -D_{I} }}-C_{L}\left| -\epsilon \right|^{-\frac{1}{2}+\sqrt{ -D_{I} }} = \left( C_{R}-C_{L} \right) \left| \epsilon \right|^{-\frac{1}{2}+\sqrt{ -D_{I} }}
$$

We can see that this quantity goes to zero whenever the exponent is greater than zero, blows up whenever the exponent is less than zero, and has a finite nonzero value only when ${}\sqrt{ -D_{I} }{}$ is exactly equal to ${}\frac{1}{2}{}$.

This is only the case when either ${}P'=0{}$, the case assessed in many places where this pfirsch-schluter annoyance is problematic, or where the two RHS terms in ${}D_{I}{}$ are otherwise equal which is determined by curvature and shear. This means Boozer's $\Delta_{mn}$ does not generally exist as a delta function current in tokamaks unless ${}P'=0{}$.

This is solved in Fitzpatrick 2026 ${}^2{}$ by arguing that a finite quantity proportional to ${}C_{R}-C_{L}{}$ is instead the tearing drive at this surface, not the divergent or zero small solution quantity.

Previously, in GPEC, this quantity has been used even though it is divergent by arguing that ideal MHD should not be interpreted below the scales at which it is relevant, and choosing a finite width where the derivative jump is also finite and is seen in some equilibria to plateau. However, the width of and existence of this relative plateau region where this works out well is determined by the beta of the equilibrium. Using high beta equilibria and scanning jump width for this quantity shows no plateau region, as seen here:

This can be scanned infinitely far in using the galerkin integrator which has the exact frobenius form explicit thanks to its finite element method.

My proposal is that we redefine Delta_mn as a quantity proportional to ${}C_{R}-C_{L}{}$, calculable presently by Riccati and Galerkin solvers (known as Delta_coil or Delta_EF, see Wang et al 2020 ${}^3{}$), and examine thoroughly where the finite jump width is an acceptable approximation, which may have significant impact on the predictions of previous error field work. My thought is that it will be ok for core surfaces where P' is lower, but any surfaces that exist in the pedestal will be significantly impacted. This could be a neat PoP that I am interested in writing. Jong-Kyu's thesis says this issue "needs to be resolved in the future" and hopefully this could do that.

[1] A. H. Glasser, The direct criterion of Newcomb for the ideal MHD stability of an axisymmetric toroidal plasma, Physics of Plasmas 23, 072505 (2016).
[2] R. Fitzpatrick, Response of a magnetically diverted tokamak plasma to a resonant magnetic perturbation, Nucl. Fusion 66, 046025 (2026).
[3] Z. Wang, A. H. Glasser, D. Brennan, Y. Liu, and J. Park, Modeling of resistive plasma response in toroidal geometry using an asymptotic matching approach, Physics of Plasmas 27, (2020).