J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)107–111
000–000

Contributions of Masahiro Wakatani to the Study of MHD
Properties of Magnetically Confined Plasmas and to theUnderstanding of Helical Systems
CARRERAS Benjamin A.
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, U. S. A.
(Received: 9 December 2003 / Accepted: 3 March 2004)

Abstract
During his professional life, Masahiro Wakatani made multiple contributions to understanding the physics ofmagnetically confined plasmas. A major area of research was nonlinear magnetohydrodynamics. He, directly andthrough many of his students, made significant contributions to this area, particularly in the dynamics of resistive
interchange, tearing, and internal kink instabilities in helical systems. Modeling of the sawtooth oscillations instellarator/heliotron devices and the study of the beta limits may be some of his main achievements. Prof. Wakatani
spent a great deal of time on the study of helical systems, and he played an important role in the physics design of theLarge Helical Device and Heliotron J.
Keywords:
magnetohydrodynamic, helical system, internal disruption, sawtooth oscillation

1. Introduction

toward the physical design of the next generation of helicalsystems, the Large Helical Device (LHD) and Heliotron J.
These studies are described in Sec. 4. In Sec. 5, I consider
one of the most interesting contributions, the flow selforganization in plasmas and subsequent work on the effect offlows on MHD instabilities. Finally, in Sec. 6, I present theconclusions of this paper.

Professor Masahiro Wakatani’s contributions to plasma
physics spanned a period of about 30 years. With more than
230 publications, he has left his mark on many different areas
research. His interest in helical systems was central to mostof his work, but his research was not limited to thoseconfinement systems. To follow his work over those years is
to follow the history of plasma physics during that period oftime. He has been involved with most of the dominant topicsof this field, and his contributions have been extensive.
In this paper, I will concentrate on two aspects of his
research, magnetohydrodynamics (MHD) studies and thephysics of helical systems. His contributions to transport andkinetics will be reviewed in another paper presented at thismeeting [1]. Even by focusing on these two topics, it would
not be possible for me to cover all his work in a short paper.
Therefore, I will select a few aspects of his research that I
feel more significant and will provide just a brief glance at
his work.
The rest of the paper is organized as follows. In Sec. 2, I
look at the historical background of MHD studies during the1970s, when professor Wakatani started his research, and I
review some of his work on nonlinear MHD applied totokamaks. In Sec. 3, I describe his research in the 80s, when
his MHD studies were centered on the currentless operationof helical systems. In the late 80s, Wakatani’s research moved

2. MHD activity in tokamaks: late 70s
In the 70s, two main experimental issues dominated
MHD research: sawtooth oscillations and major disruptions.
The results of the soft X-ray measurements made in 1974 in
the Princeton Large Torus (PLT) [2] showed the existence ofsawtooth-like relaxation oscillations at the tokamak core,
which were linked to an m = 1 precursor oscillation. This
experiment started intense experimental and theoreticalresearch activities that cover more than a decade.
The concern over major disruptions started earlier. From
the first tokamak experiments [3], major disruptions wereperceived as a serious handicap for the long-term performanceof tokamaks. Therefore, understanding disruptions became ahigh priority in the fusion program. In the 70s, research
activity in disruptions was high. Numerous papers on this
topic were published in the proceedings of the International
Atomic Energy Agency (IAEA) meetings in 1974 and 1976.
The soft X-ray diagnostic in the PLT offered one of the first©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: carrerasba@ornl.gov
1107

Carreras B.A. Contributions of Masahiro Wakatani to the Study of MHD Properties of Magnetically Confined Plasmaschange the stability properties of the m = 1 instabilities. Fordιext /dr < 0, the internal kink mode can be unstable and thereconnection mode is stable. However, for dιext /dr > 0, theinternal kink mode is stable and the reconnection mode canbe unstable. In both cases, the tearing mode can be unstable.
Wakatani used stellarator expansion to derive a nonlinear
reduced set of MHD equations for low-β plasmas. He used
these equations and varied the external rotational transformto carry out numerical calculations in the different regimes.
In this way, he was able to study the nonlinear properties ofthese different m = 1 instabilities. In his work, Wakatani
confirmed the analytical results of Rosenbluth, Dagazian, andRutherford [7] on the saturation of the internal kink mode.
He also studied the nonlinear evolution of the m = 1 tearing
mode when both the ideal kink and the reconnection modeare stable. First, the m = 1 tearing mode induces a magnetic
island in the plasma that grows at a fast rate. This evolution
causes a reconnection through the magnetic axis, and themagnetic configuration goes back to being axisymmetric. Thissecond reconnection causes the loss of energy responsible forthe relation oscillation. This mechanism had already beenshown [12-14] and was the standard interpretation of thesawtooth oscillations. He was also able to study the nonlinear
evolution of the reconnection mode. There were no nonlinear
calculations of this instability at that time. Wakatani showedthat this mode has an initial slow growth (Rutherford regime)
followed by a fast evolution, similar to the nonlinear evolutionof the resistive kink mode.

detailed views of a major disruption process [4]. Here, I am
more interested in picking up the role of helical systems in
the control of a major disruption. In an interesting paper by
the Wendelstein VII A group [5], the control of the m = 2
mode by helical winding was discussed in detail. Previous
results from Pulsator indicated the suppression of the m = 2
mode by helical winding, but the W-VII A results had detailed
profile documentation. These types of studies were probably
the most significant contributions to the understanding ofhelical systems at that time.
In trying to identify the mechanism for the sawtoothoscillations, theoretical researchers started identifying possiblem = 1 instabilities for a tokamak. In 1970, Shafranov [6]classified the ideal current-driven instabilities in a straightcylindrical system. In principle, the internal kink mode was agood candidate to explain the sawtooth oscillations. However,the linear growth rate of this instability in cylindricalgeometry was too high compared with the experimental
growth, and the internal kink nonlinearly saturates at finite
amplitude [7]. The saturation is caused by a shift of themagnetic surfaces and by the formation of a sharp current
sheet in the region of compressed flux surfaces. This process
saturates the instability without causing any magnetic
reconnection. Furthermore, it was later shown [8] that theinternal kink is stable in a toroidal geometry. Therefore, it
was necessary to look for other explanations for the precursor
oscillations.
At that time, other possible candidates for the precursor
oscillations were the m = 1 tearing mode [9], with the linear
growth rate scaling as γ ≈ S1/3 and the m = 1 reconnectionmode, with γ ≈ S3/5 [10]. Here, S is the Linquist number. In
1976, Kadomtsev [11] suggested some possible reconnection
mechanisms that would explain the relaxation process
independent of the particular instability.
Numerical calculations offered an approach to resolve
some of the pending issues in the interpretation of theexperimental results [12-14]. However, the limited computercapabilities did not allow making high-resolution calculations
in a toroidal geometry, and calculations done for a cylindricalgeometry did not allow an independent understanding of thedifferent m = 1 instabilities. In particular, at low beta and in acylindrical geometry, the internal kink is marginally stable andthe tearing mode has a faster growth than the reconnection
mode.
This was the research background when Professor
Wakatani started his own research. Wakatani took aninteresting approach to the numerical study of the m = 1
instabilities [15,16]. In a cylindrical geometry, he used
external windings to change the properties of these
instabilities. Given the stellarator expansion [17] for low-β
plasmas, the effect of helical windings comes essentiallythrough the rotational transform, ι(r) = ισ (r) + ιext (r), whereισ (r) is the rotational transform due to the current and ιext (r)is the one induced by the helical windings. Therefore, the useof the external magnetic field decouples the rotationaltransform from the current profile. By varying ιext (r) one can

3. Stellarator currentless operation: the80s
Helical systems took a significant step forward in theearly 80s. The first results of currentless operation wereobtained in both W-VII-A [18] and Heliotron E [19]. Thisability to operate with zero net current changed the potential
role of helical systems as confinement systems. Therefore,
researchers quickly moved to explore their confinement
capabilities. This change had a strong impact in the MHD
studies; they turn to calculation of beta limits for helicalsystems, nonlinear behavior of pressure-driven instabilities,
and analysis of the new experimental results. Wakatani wasdeeply involved in all these activities.
In the early 80s, Wakatani used the stellarator expansion
to derive a linear set of MHD equations incorporating finite
β effects [20]. He used these equations to study the idealinterchange and ballooning stability properties of Heliotron
plasmas. He showed that the critical β effects depend
sensitively on the pressure profiles. In particular, for the n =1 mode, the pressure gradient at the ι = 1 surface is a criticalparameter. He found values of βc (0) in the range of 5.2 to
6.8%, depending on the profile. These results were similar toother calculations carried out at this time [21, 22].
Investigation of ideal and resistive instabilities continued
over the years of operation of Heliotron E. New codes weredeveloped or adapted by Wakatani and collaborators for thesestudies. For instance, the 3-D BETA code [22] developed at2108Carreras B.A. Contributions of Masahiro Wakatani to the Study of MHD Properties of Magnetically Confined Plasmas

the Courant Institute was used in the study of the m = 1/n = 1
instability [23] and in the study of helical axis configurations[24]. The new STEP code developed at Princeton and basedon stellarator expansion was applied to study the effect of the
profiles on stability [25]. From all this research, a better
understanding was gained of the Heliotron E β limits and of
the parameters affecting plasma stability. Also, newinstabilities were discovered, such as the nonresonant resistive
instabilities near the magnetic axis [26].
Because of the renewed interest in helical systems, there
was a great push forward in the development of newcomputational tools for MHD studies. Comparative studiesof the different codes were carried out, and Wakatani and his
collaborators were at the center of these activities [27].
As the high-β regime was explored, helical systems
started to develop a rich phenomenology of MHD activity.
Interpretation of the experimental results became one of
Wakatani’s main research activities. When internal disruptions
were observed in Heliotron E, he revisited some of thenonlinear calculations of internal kink instability but used an
external rotational transform that corresponded to Heliotron
E. The internal kink evolution followed the basic saturation
pattern obtained by Rosenbluth et al. [7], but once the shiftedsurfaces were compressed, finite resistive effects became
important and a full reconnection took place [28].
In 1984, measurements of relaxation oscillations madeat Heliotron E were reported [29] that were clearly affectedby the increased injection power in the machine. Mirnov loops
also identified m = 1 precursor oscillations. These sawtoothingdischarges led to a soft β limit for Heliotron E. However, byusing a gas puff during the neutral beam injection, a quiet
mode was produced with β values up to 〈β〉 ~ 2%. This quiet
mode offered a stable path to high β. Clearly, the experimental
results indicated the importance of the pressure profile, here
controlled with gas puff, in reaching high β. This profile
dependence was expected because of results from previous
MHD studies. However, it was necessary to do more detailed
calculations in order to understand the sawtoothingphenomenon and the stable path to high β.
Wakatani et al. [30] carried out a series of calculationsof the nonlinear evolution of the m = 1/n = 1 resistivepressure-driven instability with pressure profile evolution(heating and transport). They used a reduced set of nonlinear
MHD equations that incorporated finite β effects. In those
calculations, the evolution of the m = 1/n = 1 resistivepressure-driven instability led to a magnetic island formation,
followed by a reconnection leading to two magnetic islands.
When the pressure profile was evolved, it led to relaxation
oscillations with peak beta oscillating between 2.3% and3.3%. This model explained the experimental observations in
Heliotron E.
Professor Wakatani did more than interpret experimental
results; he also helped to plan experiments in order to test
some of the basic ideas resulting from the MHD calculations.
He designed experiments for Heliotron E to test stability
properties of interchange modes by modifying some of thebasic parameters controlling their stability properties. Thiswas done by changing the magnetic field induced by verticalfield coils and toroidal coils. The vertical field coils, by
shifting the magnetic axis, change the magnetic well, while
the toroidal field coils change the rotational transform andthe plasma size. These tests are not simple because both MHD
properties and confinement properties are changed, and theywork often against each other. The moderate β results were
found to be consistent with the MHD calculations [31].

4. Design and operation of new helicalexperiments: late 80’s and 90’s
The experience gained from the W-VII, Heliotron E,
Advanced Toroidal Facility (ATF), the Compact Helical
System (CHS), and other devices gave credibility to the MHD
stability calculations for helical systems. When the decision
about the next generation of stellarators emerged, MHD tools
were available for detailed physics studies and for
optimization of the new experiments. In Japan, the nextgeneration device was LHD, now in operation at the National
Institute for Fusion Science (NIFS). Wakatani and his
collaborators actively participated in the physics evaluationof the LHD [32] and in the development of the physics basis[33] to decide the basic parameters of the device.
In the early 90s, the planning process began for a
successor experiment to Heliotron E. Wakatani and his
collaborators [34] developed a low-aspect-ratio four-fieldperiod concept, which was a hybrid of a Helias and a Heliac.
In the design, they used an l = 1 pitch-modulated continuous
coil to create the helical field. As characteristic of heliotrons,a set of toroidal field coils was added to control the rotationaltransform. These coils also provided a bumpy field that canbe used to study improvement of the neoclassical transport.
Optimization studies followed, and the concept evolved
through a series of physics studies [35-39]. The final form of
this concept [40] led to the Heliotron J device, the goal ofwhich is the study of helical axis configurations. Heliotron J
is now in operation at Kyoto University.
In the 90s, the CHS was in operation, and later on, theLHD was started. Wakatani was involved in the interpretationand modeling of some of the experimental results from thesedevices. One of the questions to which he dedicated time wasthe apparent violation of the Mercier criterion [41]. This was
not a new issue, but it is always difficult to test because of
the high accuracy required in the determination of the plasma
equilibrium. One way of improving the Mercier stability is
by including the effect of net currents [42]. Because thebootstrap current may play a role in the high β plasmas, it
has been pursued to understand the experimental
measurements. An alternative approach is to consider theeffect of local flattening of the pressure profile at a rationalsurface [43]. The research continued investigating the stabilityof pressure profiles with flat spots at several low rationalsurfaces for plasma parameters close to LHD parameters [44]and the self-organization of profiles by resistive interchange
dynamics that leads to such staircase-like pressure profiles.
3109

Carreras B.A. Contributions of Masahiro Wakatani to the Study of MHD Properties of Magnetically Confined Plasmas

many of whom are now strong contributors to plasma physics
research. These students and all other researchers who had
the pleasure of working with him will carry on his legacy. He
was an example for all and a good friend.
For me, the main lesson from his life is that one can bea good researcher and a successful professional and still
remain a very nice and good person.

Another topic of research that was motivated by the LHD
experimental results is the effect of collisionality and β on
the size of magnetic islands [45]. A model was developed
based on magnetic islands induced by resistive interchange
dynamics. Initial results [46] indicate that this model isconsistent with experimental results.

5. Flow effects on MHD stability
Acknowledgments

In this quick review of the MHD research activity ofMasahiro Wakatani, let us go back in time to pick up a very
interesting topic that overlaps with his transport andturbulence studies. In 1987, Hasegawa and Wakatani
published a paper [47] based on the 3-D turbulencecalculations of resistive-interchange instabilities that showed
the self-organization of flows and turbulence. In that paper,
they show that electrostatic turbulence self-organizes to forma macroscopic potential φ (flow stream function), which isonly a function of the radial coordinate. A feature of thispotential is the existence of a surface with φ = 0. This surface
inhibits radial particle transport. The paper anticipated muchof the work that was done in the 90s on flow generation andturbulence suppression, which are the basic mechanisms formodels of transition to enhanced confinement regimes.
This result also has implications for MHD stability.
Some MHD instabilities can lead to generation of global
flows by Reynolds stress, in a very similar way as theelectrostatic turbulence did. These flows can affect thestability of the same MHD modes and can lead to selforganization. Wakatani and collaborators carried out an
investigation of some of these effects during the 90s. In somecases, instabilities are suppressed by sheared flows. This is
the case for resistive interchange modes [48]. Flows withmoderate shear cause a stabilization of these modes. However,for highly sheared flow levels, the Kelvin-Helmholtz
instability can be destabilized. In Heliotron E, these modes
can be destabilized if the poloidal flow shear is large enough.
The main saturation mechanism is the quasi-linear
modification of the poloidal shear-flow profile [49]. There istransfer of energy from low-m to high-m modes, where energy
is dissipated. This transfer of energy contributes to thesaturation of the instability, but its overall effect is small. The
saturation level depends only weakly on resistivity andviscosity.
Flows may also have a destabilizing effect on very
narrow modes as the ideal modes interchange [50]. A
modified version of the Suydam criterion shows the generaldestabilizing character of the sheared flow for these
instabilities.

I’d like to acknowledge the help of S. Hamaguci and K.
Ichiguchi in gathering some of the publications of M.
Wakatani. I also am grateful to the 13th International Toki
Conference program committee for giving me the opportunity
to honor the memory of M. Wakatani, an excellent
collaborator and a good friend for many years. This researchhas been carried out at Oak Ridge National Laboratory,
managed by UT-Battelle, LLC, for the U.S. Department ofEnergy under contract number DE-AC05-00OR22725.

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6. Conclusions
Masahiro Wakatani had a productive professional life. He
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He educated and influenced a large number of students,
4110

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