Proceedings of ITC/ISHW2007

Mercier Stability Improvement in Nonlinear Developmentof LHD Plasma
Katsuji ICHIGUCHI1) and Benjamin A. CARRERAS2)
1)

National Institute for Fusion Science, 322-6 Oroshi-cho, Toki 509-5292, Japan
2)
BACV Solutions Inc., 110 Mohawk, Oak Ridge, Tennessee 37831, USA(Received / Accepted )Improvement of linear stability due to the nonlinear saturation of interchange modes in the increase of thebeta value is studied for the inward-shifted LHD plasma. For this study, a multi-scale numerical scheme isutilized. In this scheme, the beta value is increased by adding small pressure increment to the backgroundpressure. We focus on the dependence of the Mercier stability on the profiles of the pressure increment. It isobtained that the pressure profile approaches to the marginally stable profile when fixed profiles are employed forthe pressure increment.
Keywords: MHD, LHD, multi-scale simulation, Mercier criterion, interchange mode, self-organization
1. Introduction
In the LHD experiments, good confinement of theplasma has been observed in the magnetic configuration
with the vacuum magnetic axis located Rax = 3.6m[1].
However, linear ideal interchange modes or Mercier modes
were predicted to be unstable in this configuration. In order to investigate the stabilizing mechanism of the modes,
we developed a nonlinear MHD code, NORM, based on
the reduced MHD equations[2, 3]. In such investigation, it
is crucial to follow the continuous change of the pressure
profile in the increase of the beta value. For this purpose,
we have also developed a multi-scale simulation scheme[4]by utilizing the NORM code and the VMEC code[5]. Thisscheme treats both the equilibrium change in the long timescale and the nonlinear dynamics of the instability in the
short time scale simultaneously.
In the multi-scale scheme, the beta value is increased
by adding a small increment of pressure to the backgroundpressure obtained as the results of the nonlinear dynamics. In this case, there is a freedom in the determinationof the profile of the pressure increment. One choice forthe profile is to use the shape similar to the backgroundpressure profile obtained by the nonlinear evolution. In
the original study[4], we applied this pressure increment
to the inward shifted configuration of LHD. We found aself-organization of the pressure profile which indicated astable path to high beta regime.
On the other hand, the profiles of the heat deposition
and the particle supply in experiments are usually fixed in
the increase of beta. In order to take this situation into
account, we consider to use a fixed profile for the increment of the pressure in the present study. We employ two
types of increment profile and compare the results withthat of the similar increment profile. Particularly, we foauthor’s e-mail: ichiguch@nifs.ac.jp

cus on how the Mercier stability is improved by the selforganization of the pressure profile due to the nonlinear
saturation of the interchange mode.
2. Multi-scale scheme with fixed pressure increment
The multi-scale scheme used in the present analysis
is explained in Ref.[4] precisely. Here we start from abrief review of the multi-scale scheme, and then, explain
the choice of the pressure increment profile and the conditions in the calculation.
The scheme consists of iterative calculations of nonlinear dynamics of the perturbations by the NORM codeand three-dimensional equilibrium by the VMEC code. In
this case, we divide the whole calculation time into shorttime intervals. At t = ti , the beginning of an interval,
we calculate new equilibrium quantities at the higher beta
value with the VMEC code as the values of t = ti+1 , thebeginning of the next interval. In order to keep a smooth
continuity of the perturbation, we also divide the interval
between ti and ti+1 into some sub-intervals and employ a
linear interpolation of the equilibrium quantities by using
the equilibrium quantities of t = ti and ti+1 . Then, the nonlinear dynamics is calculated for each sub-interval with theinterpolated equilibrium quantities with the NORM code.
When we calculate the equilibrium with the VMEC
code, we incorporate the pressure deformation due to thenonlinear dynamics into the pressure profile. At t = ti , thetotal pressure is obtained asP̃mn ,Pitot = Pi +(1)m  0 or n  0

where the tilde means a perturbed quantity and m and n
are the poloidal and the toroidal mode numbers. Here Pidenotes the average pressure which is given byPi = Pieq + P̃i00 .

(2)

Proceedings of ITC/ISHW2007

0.0

unstable

-6

2/5
3/7
1/2
3/5
2/3-8log EkDI

1.0-10stable-12<β>=0.221%-14
10000 20000 30000 40000 50000 60000<β>=0.444%-1.00t (τA)

0.2 0.4 0.6 0.8
ρ

1Fig. 2 Time evolution of kinetic energy for each case of the pressure increment.

Fig. 1 Profiles of DI of the equilibrium for the pressure profileof Peq = P0 (1 − ρ2 )(1 − ρ8 ) in the inward-shifted LHD
plasma. Each symbol shows the value of DI at the position of the resonant surface.

The average pressure includes the eﬀect of the nonlinear
i+1by using Pi asdynamics through P̃i00 . We calculate Peqi+1Peq= Pi + ΔPi+1 .

(3)Here ΔPi+1 denotes the increment of the pressure, which
gives the increase of beta. In the original study[4], we employed a similar increment profile given by
ΔPi+1 = Pi

βi+1 − βi.
βi

ΔPi+1 = PI (1 − ρ2 )(1 − ρ8 )ΔP= PI (1 − ρ ) ,<P>

<ι>
0.8

1.0t=60000

0.70.6 3/5

0.5(6)

where ρ denotes the square-root of normalized toroidal
magnetic flux. The factor PI is adjusted so as to give agiven beta increment. Hereafter, we call the increments
given by (4)-(6) ‘similar increment’, ‘parabolic increment’and ‘parabola-squared increment’, respectively.
We apply the scheme to the LHD plasma for the three
types of pressure increment under following numericalconditions. We choose the configuration with the vacuum
magnetic axis located at Rax = 3.6m. We assume the resistivity of S = 106 , where S is the magnetic Reynolds
number. We examine the evolution for 0.221% ≤ β ≤0.498%. One time interval is 2500τA , where τA is Alfvéntime. We increase the beta value by Δβ = 0.0138% every time interval. In the equilibrium calculation with theVMEC code, we use the free boundary condition and theno net-current condition. The time interval is divided into
10 sub-intervals for the linear interpolation.
To give the initial state, we start from the equilibriumfor Peq = P0 (1 − ρ2 )(1 − ρ8 ) at β = 0.221%. As is shownt=10000
t=20000
t=30000
t=40000
t=50000
t=60000

0.5 1/2
3/7(5)

and2 2in Fig.1, the core region of ρ ≤ 0.44 of this equilibrium isMercier unstable. The absolute value of DI [6] decreases inthe ρ direction. We follow the nonlinear evolution of the interchange mode for this equilibrium and obtain a saturation
at t = 10000τA . We employ the saturated state as the initialstate of the multi-scale calculation and set t = 10000τA as
the initial time. Then, the beta value reaches β = 0.498%
at t = 60000τA .

(4)

In the present study, we also consider two kinds of fixed
profile for the increment given byi+1

ΔP : similar
ΔP : parabola
ΔP : parabola-squared

0.4 2/5000.3
0.2 0.4 0.6 0.8 1.0
ρFig. 3 Time evolution of average pressure in the case of theparabolic increment.

3. Self-organization of pressure profile
We follow the evolution of the plasma for the three
types of increment profile and compare the resultant pressure profile. Figure 2 shows the time evolution of the total kinetic energy for the three pressure increments. It iscommon that the kinetic energy varies smoothly comparedwith the time scale of sub-interval. This feature indicatesthe multi-scale approach works well also in the fixed increment cases. The evolution of the parabolic incrementcase is close to that of the similar increment case, while
the evolution in the parabola-squared increment case is alittle more active.
Figure 3 shows the time evolution of the average pressure in the case of the parabolic increment. As in the caseof the similar increment case[4], weak excitation and mild

Proceedings of ITC/ISHW2007

Fig. 4 Bird’s-eye view of total pressure at t=60000τA for similar increment (left), parabolic increment (center) and parabola-squared
increment (right).

saturation of the interchange modes occur. The saturation
generates locally flat structure at the resonant surfaces in
the average pressure profile. Since the Mercier quantity
DI is a decreasing function of ρ as shown in Fig.1, the flat
region is generated from inward to outward of the plasma
as the beta increases. Similar tendency is observed in theparabola-squared increment case.
Figure 4 shows the bird’s-eye view of the total pressure at the final time of t = 60000τA . The deformationof the total pressure is almost θ independent for all increment cases. This implies that almost all of the resonantinterchange modes are saturated in a low level without any
significant excitation. In other words, in each increment
profile, the total plasma pressure evolves so that fluctuations are suppressed in the increase of beta.
Remarkable diﬀerence between the three increment
profiles is seen in the average pressure profile at the finalstate. Figure 5 shows the profile of the average pressure att = 60000τA . In the similar increment case, a global flat
structure is generated in the core region of ρ < 0.4. In theparabolic increment case, the gradient of the pressure is recovered in the core region. In the parabola-squared increment case, the gradient becomes larger. These diﬀerences
are attributed to the gradient of the increment profile. In
any case of the increment, the pressure profile is flattenedin the core region once at low beta because (m,n)=(5,2)
and (7,3) modes are saturated in the region. In the similar
increment case, the average pressure is increased so thatthe shape should be maintained. Therefore, the local flatstructure generated at low beta is kept even at high beta.
On the other hand, in the fixed increment cases, thegradient of the increment profile is always added to thetotal pressure. Therefore, the local flat structure of the average pressure tends to be smoothed out. Furthermore, theresonant mode can be excited again at the flattened region
when the local pressure gradient enhanced by the increment exceeds a critical value. Since the driving force ofthe mode should be quite weak, it saturates immediately
to generate a narrower flat region in the average pressure
profile. Thus, the local pressure gradient approaches to thecritical value through this process. The critical value canbe measured in terms of DI as explained in the next section.

In the parabolic increment case, the increment profileis the same as the equilibrium profile used in the initialstate generation. Therefore, this process is limited in thecore region. On the other hand, the more steep gradientis added in the parabola-squared increment case. The region of the process extends to the outer region including
the surfaces resonant with the (5,3) and the (3,2) modes.
4. Mercier stability improvement
The global feature of the DI profile is common in
all cases of the increment. Figure 1 (red line) shows theDI profile for the equilibrium with the pressure profile of
Peq = P0 (1 − ρ2 )(1 − ρ8 ) at β = 0.444%. In this case,the wide region of ρ ≤ 0.60 is Mercier unstable. On theother hand, DI has negative values around the resonantsurfaces as shown in Fig.6, which shows the DI profiles
at t = 60000τA (β = 0.498%) for the three cases of thepressure increment. This comparison shows that the nonlinear saturation of the interchange mode stabilizes itself
through the local pressure flattening.
The diﬀerence in the structure of the pressure profile
is reflected to the precise structure of DI profile. In thesimilar increment case, the improved values of DI are 1.05, -0.49 and -0.21 at resonant surfaces with ι = 2/5,´
3/7 and 1/2. There is a tendency that the absolute value
is a decreasing function of ρ s , where ρ s is the positionof the resonant surfaces. This tendency is related to theMercier stability at the initial equilibrium. The profileof DI at β = 0.221% implies that the driving force of
the interchange mode is also the decreasing function of ρ.
Therefore, the local deformation at inner resonant surface
is larger than that at outer surface. Since such structure isalmost maintained during the beta increase, the resonantsurface is more stabilized beyond the marginal stability.
On the contrary, the absolute value of DI is limited in
the level of -0.32 in the parabolic increment case. In thiscase, even once the pressure profile is locally flattened, theenhancement of the gradient of the pressure degrades theMercier stability. Therefore, the value of DI approaches
to a marginal value in the increase of beta. This tendency
is the same as in the case of the parabola-squared increment. In this case, the local improvement is observed also

Proceedings of ITC/ISHW2007

<ι>

<P>

1.0

ΔP:similar
ΔP:parabolic
ΔP:parabolasquared

0.8
1.0

0.5

0.6

unstable

0.0

0.5
0.4

00DI

0.7
ΔP:similar
ΔP:parabolic
ΔP:parabolasquared

0.3
0.2 0.4 0.6 0.8 1.0
ρ

3/7-1.00

1/2 3/52/3

stable2/5

0.2 0.4 0.6 0.8
ρ1Fig. 6 Profiles of DI at t = 60000τA .

Fig. 5 Average pressure profiles at t = 60000τA .

around the surfaces of ι = 3/5 and 2/3. Including these´surfaces, the absolute value of DI is limited in the level of-0.29. The enhancement of the pressure gradient broughtby the parabola-squared increment is larger than that by
the parabolic increment case. Nevertheless, the maximum
value of DI is in the similar level of DI ∼ −0.3. This result
indicates that this value of DI corresponds to the marginalpressure gradient independent of the increment profile, if a
fixed increment profile is employed. It can be concluded
that the local pressure gradient is determined in the increase of beta so that DI at the resonant surface should
achieve to the marginal value.
5. Conclusions
The local improvement of the Mercier stability in the
nonlinear evolution of the interchange mode is studied in
the inward-shifted LHD plasma. The beta increase effect is incorporated by employing the multi-scale numerical scheme. The plasma is Mercier unstable in a wide
region if there is no deformation of the pressure profile.
However, the nonlinear saturation of the interchange mode
locally improves the Mercier stability around the resonantsurface through the generation of the local flat structure in
the pressure profile.
The absolute value of DI in the stabilized region depends on the pressure increment profile. If we use the similar increment profile, the absolute value of negative DI
becomes much larger in the vicinity of the axis than thatin the outer region. This is attributed to that the locally flat
structure in the pressure profile is maintained in the beta increase. On the other hand, if we use a parabolic increment
profile, the reduction of the pressure gradient is compensated by the increment pressure. Therefore, the absolutevalues of DI at all resonant surfaces are in a small level.
In the case of the parabola-squared increment profile, theimprovement of the Mercier stability extends to the outer
rational surfaces. Even in this case, the absolute values ofDI are also limited in a small level including the outer rational surfaces. The level is almost the same as that in theparabolic increment case. These results indicates that theenhancement and the reduction of the pressure gradient isbalanced so as to give a critical pressure gradient. The former is due to adding the increment pressure and the latteris due to the nonlinear saturation of the mode. In otherwords, in the case of the fixed profile of the pressure increment, the plasma is self-organized so that the pressure
profile approaches to the marginally stable profile at theresonant surfaces with respect to the Mercier stability.
As a future plan, we consider to include an eﬀect of
the equilibrium diﬀusion. In this case, we can expect thatthe positive DI values in the regions between the resonantsurfaces also approach to marginal value.
Acknowledgments
This work is supported by NIFS cooperation programs
NIFS07KLDD012 and NIFS07KNXN092. and also partly
supported by the Grant-in-Aid for Scientific Research (C)17560736 of the Japan Society for the Promotion of Science.
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