25F06
A new solution method for longstanding problemsof resistive MHD stability analysis
抵抗性 MHD 安定性解析の未解決問題に対する新しい近似解法
Masaru Furukawa, Shinji Tokuda1)
古川 勝，徳田 伸二 1)
Grad. Sch. Frontier Sci., Univ. Tokyo, Kashiwa-shi, Chiba 277-8561, Japan
International Fusion Energy Research Center, Rokkasho-mura, Aomori 039-3212, Japan東京大学 大学院新領域創成科学研究科 〒 277-8561 千葉県柏市柏の葉 5 − 1 − 51)
国際核融合エネルギー研究センター 〒 039-3212 青森県六ヶ所村大字尾駮字表舘２番１６６1)

We have developed a new solution method for longstanding problems of resistive magnetohydrodynamics (MHD) stability analysis. Since this is a conference proceeding of an invited talk, the main
body of the study has been published elsewhere. In this proceeding, we point out the diﬃcultiesof the classical method based on the matched asymptotic expansion, and brieﬂy explain notion of
our new method.

1. Introduction
One of the important pieces in nuclear fusion
development is magnetohydrodynamics (MHD)
stability analysis of conﬁned plasmas. Finite
plasma resistivity introduces several resistive
MHD instabilities such as tearing modes. Since
large-scale magnetic islands degrade the plasma
conﬁnement signiﬁcantly and even cause disruption, control and suppression of the islands are
one of the urgent issues. To understand physics
of them, accurate calculation of the resistive
MHD stability is indispensable.
A standard method of the stability analysis
has been the asymptotic matching method[1,2].
The method relies on assumptions such as smallness of the resistivity and the slow dynamics.
Then the resistivity and the inertia terms canbe neglected in most of the region in the plasma,that we call outer region. There we solve theinertia-less, ideal MHD equation or the so-called
Newcomb equation[3]. In the outer region, thedominant eﬀect is magnetic tension. The magnetic tension however vanishes at a resonant surface inside the plasma if it exists, and thus resistivity and the inertia need to be retained in athin layer around the resonant surface, which wecall an inner layer. Since the inner layer is thin,
we can simplify the governing equation. To focus
on the inside of the thin layer and on the slow
dynamics, we may re-scale the radial coordinate
and the frequency by using resistivity as a small
parameter, leading to the inner equation.
The resistivity term has the highest-order spa-cial derivative; it is singular perturbation. Since
that term is dropped in the outer region, thenumber of independent solutions becomes two,instead of four in the inner layer. Two of the four
independent solutions in the inner layer have thesame asymptotic forms which can be matched
onto the outer solution. The two solutions can beobtained as the Frobenius series around the resonant surface, and are called small and large solutions. The ratio of the small solution to the largesolution, called matching data, plays crucial rolein the asymptotic matching method. The matching then gives us the dispersion relation.
The asymptotic matching method is well established mathematically. However, it has somediﬃculties in practice. (i) Although the resistivity and the inertia are neglected in the outerregion, they can be important there if the plasma
is close to marginal stability against ideal MHD.
A plasma close to its ideal MHD stability limit
may be preferable to improve eﬃciency of a fusion reactor. Such a situation may be simulated
by a cylindrical plasma with q = 1 surface inside
the plasma, where q is the safety factor. The
m/n = 1/1 internal kink mode is then marginallystable against ideal MHD, where m and n are
the poloidal and toroidal mode numbers, respectively. (ii) The method cannot be applied in theﬁrst place if the resonant surface becomes an irregular singular point of the Newcomb equation.
An important type of discharge in fusion development has non-monotonic q proﬁle, and theminimum-q position can be such irregular sin-gularity. (iii) Accurate numerical computationof the matching data is still diﬃcult in toroidal
plasmas even though some sophisticated theory
have been developed[4,5]. (iv) In the plasma
close to marginal stability against ideal MHD,
the matching data diverges. Although a numerical scheme to calculate huge matching data wasdeveloped[6], it is reported that the accuracy ofthe matching data strongly depend on the local
equilibrium accuracy and grid arrangement[7].
(v) Careful treatment is required in solving theinner equation numerically since the radial coordinate is re-scaled into unbounded space[8].
We pointed out these diﬃculties, and resolved
them by our new matching method in Ref. [9,10].
The key ingredients of the new method are anutilization of an inner region with a ﬁnite width,and an ordering scheme for the outer region. We
will brieﬂy explain notion of the new method below. Readers interested in details of the formulation and applications may refer to Ref. [9,10].
2. Notion of new matching method
In our new method, we adopt a ﬁnite-width
inner region, instead of inﬁnitely thin inner layer.
Neither the radial coordinate nor the frequency
are re-scaled via small resistivity. The solutionsin the inner and outer regions are then matcheddirectly, not asymptotically, by imposing continuity of perturbed magnetic ﬁeld at the boundaries, which are reasonably apart from the resonant surface. Since the boundaries or the matching points are apart from the singularity, we canfully avoid the diﬃculties in the numerical computation. Accurate computation of the matching
data, including the divergent case, and the careful treatment of the unbounded space are unnecessary. Furthermore, our method is applicable tothe irregular singularity case, since the method
does not rely on the Frobenius series solution.
Note that the ﬁnite-width inner region wasoriginally introduced for ideal MHD modes[11],and extended to resistive wall modes in rotating cylindrical plasmas[12]. The important difference from these studies is that the resistivity term increases the order of spacial derivative.
We then need to select two of the four independent solutions in the inner region, that can match
onto the outer solution. A natural idea for theselection may be to impose smooth disappearance of parallel electric ﬁeld Ek as approaching
the matching points from the inner side[9]. The
application results were mostly satisfactory.

What was exceptional is the m/n = 1/1 internal kink mode. This mode is marginally stableagainst ideal MHD. In the q < 1 region, which
is treated as ideal MHD region in the asymptotic matching method, Ek does not disappear
smoothly. The remaining Ek indicate the relative importance of the resistivity and the inertia.
Thus we have developed an ordering scheme forthe outer region to include those eﬀects perturbatively. The ordering is similar to the one adopted
for the inner layer in the asymptotic matching
method, but is diﬀerent in the estimation of radial derivative. Since the boundary of the outerregion is reasonably far from the singularity, wecan assume that the radial derivative is of orderunity. The ordering scheme then gives us a hierarchy of generalized Newcomb equations. The
lowest- and the ﬁrst-order equations agree withthe conventional one, and the second-order equation includes the resistivity and the inertia as inhomogeneous terms. Note that the homogeneouspart has the same diﬀerential operator at eachorder. By including the second-order correction,
we have achieved suﬃciently accurate stability
calculations also for the concerned m/n = 1/1
internal kink modes.
Acknowledgments
This work was supported by KAKENHI
19760595 and KAKENHI 23760805.
References[1] H. P. Furth, J. Killeen and M. N. Rosenbluth:
Phys. Fluids 6 (1963) 459.
[2] B. Coppi, J. M. Greene, and J. L. Johnson: Nucl.
Fusion 6 (1966) 101.
[3] W. A. Newcomb: Ann. Phys. 10 (1960) 232.
[4] A. Pletzer, A. Bondeson, and R. L. Dewar: J.
Comput. Phys. 115 (1994) 530.
[5] S. Tokuda and T. Watanabe: Phys. Plasmas 6(1999) 3012.
[6] R. L. Dewar and M. Persson: Phys. Fluids B 5(1993) 4273.
[7] D. P. Brennan, E. J. Strait, A. D. Turnbull, M.
S. Chu, R. J. La Haye, T. C. Luce, T. S. Taylor,S. Kruger, and A. Pletzer: Phys. Plasmas 9 (2002)
2998.
[8] S. Tokuda: Nucl. Fusion 41 (2001) 1037.
[9] M. Furukawa, S. Tokuda and L. -J. Zheng: Phys.
Plasmas 17 (2010) 052502.
[10] M. Furukawa and S. Tokuda: Phys. Plasmas 18(2011) 062502.
[11] Y. Kagei and S. Tokuda: Plasma Fusion Res. 3(2008) 039.
[12] J. Shiraishi, S. Tokuda, and N. Aiba: Phys. Plasmas 17 (2010) 012504.