Lecture 41
Ferritic thin wall and the ferromagnetic wall mode
We can extend the same constant-\(\psi \) model to a wall with finite relative permeability
\[\hat \mu \equiv \frac {\mu _w}{\mu _0} > 1.\]
A ferritic wall has two competing effects. First, finite conductivity still allows eddy currents that oppose
the perturbation. Second, the high permeability attracts and compresses the perturbed magnetic flux into
the wall. The latter effect weakens the ideal-wall stabilization and tends to make the mode more
unstable.
A convenient way to organize the matching is to characterize the plasma by the tearing-mode-style
logarithmic derivative at the plasma edge,
\[\boxed { \Delta _a^* \equiv \left .\frac {a}{m\psi _{\rm nw}}\frac {d\psi _{\rm nw}}{dr}\right |_{r=a} = \frac {1}{m}\left [ \left .\frac {a}{\delta B_{r,{\rm nw}}}\frac {d\delta B_{r,{\rm nw}}}{dr}\right |_{r=a} \right ], }\]
where \(\psi _{\rm nw}\) (equivalently \(\delta B_{r,{\rm nw}}\)) is taken from the plasma/no-wall solution and \(\delta B_r = i m\psi /r\). The point of this notation is that
the plasma enters the exterior matching only through this single ratio: once \(\Delta _a^*\) is known at \(r=a\), the vacuum
region, resistive wall, and ferritic wall can all be matched algebraically. For a generic exterior vacuum
solution \[\psi (r)=Ar^m+\frac {C}{r^m},\]
one immediately finds \[\boxed { \frac {\Delta _a^*+1}{\Delta _a^*-1} = -\frac {A}{C}\,a^{2m}, }\]
so the combination \((\Delta _a^*+1)/(\Delta _a^*-1)\) simply measures the relative admixture of the growing and decaying vacuum
harmonics. This is why it is such a convenient quantity for matching all the exterior regions.
For a cylindrical long-wavelength mode with poloidal number \(m\), the standard thin-wall ferromagnetic
dispersion relation can be written as Kurita et al. (2003); Bergerson et al. (2008)
\[\boxed { \frac {\Delta _a^*+1}{\Delta _a^*-1} = \frac {\gamma \tau _f-\dfrac {md}{2b}(\hat \mu -\hat \mu ^{-1})} {1+\gamma \tau _f+\dfrac {md}{2b}(\hat \mu +\hat \mu ^{-1}-2)} \left (\frac {a}{b}\right )^{2m}, }\]
where \[\boxed { \tau _f \equiv \frac {\mu _0 b d}{2m\eta _w}. }\]
Here \(\tau _f\) is written in the standard cylindrical ferromagnetic-wall convention; compared with the \(m=1\) wall time
used above, it differs only by a numerical geometry factor. An important point is that the skin time
entering the thin-wall equation is still proportional to \(\mu _0\), not \(\mu _w\): the local increase in magnetic
diffusion time inside the ferritic material is compensated by the compression of the flux at the
wall.
For later convenience define
\[\alpha \equiv \frac {\Delta _a^*+1}{\Delta _a^*-1},\]
and neglect plasma inertia so that \(\alpha \) is independent of \(\gamma \), the growth rate can be written as \[\boxed { \gamma \tau _f = \Gamma _w + \Gamma _\mu , }\]
with the ordinary resistive-wall contribution \[\boxed { \Gamma _w = \frac {\alpha }{(a/b)^{2m}-\alpha }, }\]
and the ferritic correction \[\boxed { \Gamma _\mu = \left (\frac {md}{2b}\right ) \frac { \alpha (\hat \mu +\hat \mu ^{-1}-2) + (\hat \mu -\hat \mu ^{-1})(a/b)^{2m} }{(a/b)^{2m}-\alpha }. }\]
When \(\hat \mu =1\), the ferritic correction vanishes and the ordinary resistive wall mode is recovered. For \(\hat \mu >1\), the extra
term drives the mode in the unstable direction, so the growth rate is larger than for a non-ferritic
resistive wall and the ideal-wall stability window is reduced. This is the ferromagnetic wall
mode.
For weak ferritic response, \(\hat \mu =1+\epsilon \) with \(|\epsilon |\ll 1\), the leading correction is linear,
\[\Gamma _\mu \simeq \left (\frac {md}{b}\right ) \frac {\epsilon \,(a/b)^{2m}}{(a/b)^{2m}-\alpha },\]
so even modest permeability shifts the resistive-wall growth rate.
Physical interpretation
A nearby ideal conductor stabilizes the external kink by forcing the vacuum perturbation to vanish at the
wall and thereby increasing the magnetic energy of the displacement. A resistive wall only
maintains that constraint for a finite time \(\tau _w\), which produces the slow resistive wall mode. A
ferritic wall, however, attracts the perturbed flux into the material and thereby reduces the
efficacy of that ideal-wall stabilization. In this sense, the ferritic wall makes the physical wall
look magnetically farther away, so the mode is more unstable even though the wall remains
conducting.
Takeaways
Ferritic material does not simply behave like a better conducting wall. Its high
permeability compresses perturbed flux into the wall and weakens the ideal-wall
stabilization that would otherwise oppose the external kink. In the thin-wall model
this appears as the additive correction \(\Gamma _\mu \) to the usual resistive-wall growth rate, so the
ferromagnetic wall mode is the same RWM physics modified by magnetic permeability
rather than by conductivity alone.
Bibliography
G. Kurita, T. Tuda, M. Azumi, S. Ishida, S. Takeji, A. Sakasai, M. Matsukawa, T. Ozeki,
and M. Kikuchi. Ferromagnetic and resistive wall effects on the beta limit in a tokamak. Nuclear
Fusion, 43(9):949–954, 2003. ISSN 0029-5515. doi:10.1088/0029-5515/43/9/319.
W. F. Bergerson, D. A. Hannum, C. C. Hegna, R. D. Kendrick, J. S. Sarff, and C. B. Forest.
Observation of resistive and ferritic wall modes in a line-tied pinch. Physical Review Letters, 101
(23):235005, 2008. doi:10.1103/physrevlett.101.235005.