Direct numerical simulation of fully three-dimensional, compressible and nonlinear magnetohydrodynamic equations in the Large Helical Device is carried out in combination with the passive particle simulation. In the simulation, strong vortical motions are excited by the pressure-driven instability and form the mushroom-like structures of pressure. It is shown by the passive particles analysis that the fluid volumes around the resonant magnetic surfaces experience finite compressibility and toroidal deformation, which are both excited by the strong vortical motions. The passive particles simulation helps us to investigate local structures even for low Fourier wavenumber modes

H. MIURA

M. OKAMOTO

N. NAKAJIMA

T. HAYASHI

National Institute for Fusion Science (NIFS-Repository)

J. Plasma Physics (2006), vol. 72, part 6, pp. 1095–1099. c© 2006 Cambridge University Pressdoi:10.1017/S002237780600571X Printed in the United Kingdom1095Vortical, toroidal and compressible motionsin 3D MHD simulations of LHDH. M IURA1,3, N. NAKAJ IMA1,3, T. HAYASH I2,3†and M. OKAMOTO1,21Theory and Computer Simulation Center,National Institute for Fusion Science, 322-6 Oroshi, Toki, Gifu 509-5292, Japan2Division of Theory and Data Analysis,National Institute for Fusion Science, 322-6 Oroshi, Toki, Gifu 509-5292, Japan3Department of Fusion Science, The Graduate University for Advanced Studies(SOKENDAI),322-6 Oroshi, Toki, Gifu 509-5292, Japan(Received 25 August 2005 and accepted 12 June 2006)Abstract. Direct numerical simulation of fully three-dimensional, compressible andnonlinear magnetohydrodynamic equations in the Large Helical Device is car-ried out in combination with the passive particle simulation. In the simulation,strong vortical motions are excited by the pressure-driven instability and form themushroom-like structures of pressure. It is shown by the passive particles analysisthat the ﬂuid volumes around the resonant magnetic surfaces experience ﬁnitecompressibility and toroidal deformation, which are both excited by the strongvortical motions. The passive particles simulation helps us to investigate localstructures even for low Fourier wavenumber modes.1. IntroductionThe Large Helical Device (LHD) is the world’s largest heliotron/torsatron typehelical device. It has a set of L=2/M =10 helical coils with a major radius of3.9m (Motojima et al. 1999). Recent LHD experiments revealed that plasmascan be conﬁned relatively well even under the magnetic conﬁguration with va-cuum magnetic axis position Rax = 3.6m, which is considered unstable in thesense of Magnetohydrodynamic (MHD) instability. In order to understand howplasma survives MHD instabilities, direct numerical simulation (DNS) of fullythree-dimensional, compressible and nonlinear MHD equations is carried out byour numerical code, the MHD In Non-Orthogonal System (MINOS). Details of theMINOS code and conditions of the DNS are reported in Miura et al. (2004, 2007).In traditional linear analysis and reduced MHD simulations, toroidal ﬂow andcompressibility are often discarded. This is partly because the ﬂuid remains in-compressible if the perturbation is initially incompressible in the framework of thereduced MHD equations. The toroidal velocity can also be omitted if the perturba-tion is incompressible. However, we have shown that the toroidal component of thevelocity grows rapidly from a weakly perturbed state. (Miura et al. 2004, 2007).† Deceased April 20061096 H. Miura et al.Figure 1. Full 3D views of MHD instabilities obtained by DNS. (a) Pressure isosurface withtwo helical coils. (b) Stream lines, pressure isosurfaces and contours on a poloidal section.Furthermore, the compressibility works to reduce the linear growth rate to 1/2–1/5 of the growth rate obtained under the incompressible assumption. This impliesthat the toroidal ﬂow and compressibility should be included in numerical studies tostudy the conﬁnement mechanism of plasmas in LHD. Although we have shown thesigniﬁcance of these effects, the detailed mechanism by which these effects inﬂuencethe nonlinear saturations of the growth still remains unclear. Aiming to clarify themechanism, we propose an analysis with the displacement vector by the combin-ation with the passive scalar simulation. We show below that the new approachworks well and is worth incorporating into further research on this subject.2. DNS with passive scalar simulationsOur simulation starts from an unstable initial equilibrium with the vacuum mag-netic axis position 3.6m and β0 = 4%. As is reported in Miura et al. (2004), asimulation starting from this equilibrium shows strong instability driven by thepressure gradient. In the simulations, a rapid growth of unstable Fourier modesm/n = 2/1 and 1/1 of the pressure are observed where m and n are the poloidaland toroidal wavenumber in the Boozer coordinate. A typical view of them/n = 2/1structure of the pressure is shown in Fig. 1(a). In Fig. 1(a), the isosurface of thepressure is shown with two helical coils. We clearly observe that the isosurfaces aredeformed by the growth of the m/n = 2/1 Fourier mode. In Fig. 1(b), stream lines,pressure isosurfaces and contours on a poloidal section are shown. Spirals of streamrepresent vortical motions associated with the m/n = 2/1 mode. The stream linesare inclined to the toroidal direction due to the active toroidal ﬂows. The strongvortices, which consist of two anti-parallel vortex pairs, advect each other and formthe mushroom-like structures of the pressure, as are observed in the contours ofthe pressure in a poloidal section.The roles of the toroidal ﬂow and compressibility are studied more closely withthe help of the passive particles simulation. In the passive particles simulation, theposition of the ith particle ξi(t) is tracked by solving the equationdξidt= v(ξi, t) (2.1)where v(ξ, t) is the ﬂuid velocity at the position ξ. Since the motions of particlesdepend only on the ﬂuid velocity, the vector δξ = ξ(t) − ξ(t = 0) represents theVortical motions in 3D MHD simulations of LHD 1097Toroidal displacementEntrainmentby vortical motionsFigure 2. Deformations of passive surfaces. Poloidal cross-sections of the pressure contourand the m/n = 2/1 passive surface coincide with each other at the initial time.displacement vector which is often used to study the linear instability of torusplasmas (Friedberg 1987). The particles are initially located on the rational surfacesι/2π = 1/2, 2/3, 3/4, 1 uniformly on the θ–ζ plane of the Boozer coordinate where θand ζ are the poloidal and toroidal angles, respectively, composing the passive sur-faces. We also put particles on the neighboring surfaces to each rotational surfacesfor use later. The passive surfaces coincide with the resonant magnetic surfaces ifthe magnetic ﬁelds are frozen into the ﬂuid, as they are in ideal MHD. Althoughthe passive surfaces depart from the resonant magnetic surfaces because of the non-ideal MHD effects, it is still worth studying the behavior of them because it servesto study the roles of physical elements such as toroidal ﬂows and compressibility.First, we study the deformations of the passive surfaces. In Fig. 2, the passivesurfaces are shown with isosurfaces and contours on a poloidal plane of the pressureat the saturation time of the energy growth. The white meshes represent the passivesurfaces. The vertexes of meshes are the position of passive particles. The inner-mostsurface in Fig. 2 coincides initially with the rotational surface ι/2π = 0.5. In thecourse of the time evolution, the surface is deformed into spirals by the roll-up of thevortices. It can be veriﬁed by studying the toroidal displacement ξ · B/|B|, whereB is the magnetic ﬁeld vector, that the surface is also elongated or compressed intothe toroidal direction by the toroidal ﬂows, although it may not be clearly seen inFig. 2. The toroidal deformation is to the left-hand side of the paper in the inner sideof the torus and to the right-hand side in the outer side. The toroidal deformationis likely anm/n = 1/1 structure. The observation is consistent with the analysis bymeans of the Fourier power spectra of the parallel velocity, reported in Miura et al.(2007). The m/n = 1/1 toroidal ﬂow is excited by the couplings of the m/n = 2/1primarily unstable mode and m/n = 1/0 perturbation added to the system due tonoise in the MHD equilibrium. While the directions of ﬂuid motions and its timehistory are not observed from the streamlines in Fig. 1(b), the passive surfaces inFig. 2 represent the history of the ﬂuid motions. The passive surface representationhelps us to understand the evolutions of ﬂuid motions, especially on the rationalsurface, and helps us to understand the evolution of the MHD instability.Next the signiﬁcance of compressibility is investigated. Although it is possible toestimate the local compressibility by studying the dilatational ﬁeld ∇ · v, it becomeseasier by making use of the passive particles informations especially when we are1098 H. Miura et al.11.11.211.331160 180 200 220 240 260Volume ratiostι/2π = 1/2ι/2π = 2/3ι/2π = 3/4ι/2π = 1outermostFigure 3. Ratio of the volumes of small rectangular boxes to their initial volumes.interested in the compressibility in the neighbourhood of the rational surfaces. Thesix vertices of a rectangular box are composed of the passive particles neighboring-each other. Suppose that a position of one vertex is identiﬁed by the sets of variables(ι/2π, θ, ζ). Then the other ﬁve vertices are given by shifting the position to δι, δθ,δζ in each direction, respectively. The volumes become larger or smaller in theircourse of time evolutions if they experience some compressibility. In Fig. 3, timeevolutions of the volumes of the small rectangular boxes on some typical rotationalsurfaces are shown. The time region is restricted to 160  t/τA  260, that is,from the middle of the linear regime to the beginning of the nonlinear stage, sothat the separation between two neighboring particles does not become too large.The volumes are all averaged over each rotational surface and normalized by theinitial volume. We ﬁnd in Fig. 3 that the volume of the elements which are initiallylocated on the ι/2π = 1/2 surface grows rapidly. The volumes in the plasma cores(that is, ι/2π = 1/2, 2/3 and 3/4 surfaces) experience expansion of the volume,while the volumes in the outermost surface of the magnetic ﬁeld line shrink. Notethat the compressibility is weak in the simulation. The root mean square of thedilatation ∇ · v is roughly 1/10–1/100 of that of the vorticity ∇ × v. However, theanalysis in the above shows that the local contributions of compressibility aroundthe magnetic surfaces is never negligible.We emphasize here again that all rectangular boxes distributed on the rationalsurfaces experience ﬁnite expansion or compression of the volume. Since the passiveparticles essentially represent the displacement vectors adopted in the linear MHDanalysis, the large change of the volumes shows that ﬁnite compressibility worksto the dynamics on the displacement vector. While the volume change could alsobe studied by evaluating the dilatation of the velocity ﬁeld, ∇ · v, the passivevector information can further be used to study the parallel and perpendiculardisplacement and so on. The passive particle information can provide us with richinformation about plasma motion.3. Concluding remarksWe have studied the ﬂuid motions associated with the growth of instability inLHD. The pressure-driven instability excites the strong m/n = 2/1 vortex pairs,which in turn generates considerable strength of toroidal ﬂows. Analysis withthe passive particle simulations have revealed that the volumes which are putVortical motions in 3D MHD simulations of LHD 1099initially on rotational surfaces experience ﬁnite compressibility, even though themean compressibility is weak. Our study in this article shows that the passiveparticle simulations can serve to explore various aspects of local plasma motions.ReferencesMotojima, O. et al. 1999 Initial physics achievements of Large Helical Device experiments.Phys. Plasmas 6, 225–245.Miura, H. et al. 2004 Non-disruptive MHD dynamics in inward-shifted LHD conﬁgurationsIn: Procedings of the 19th IAEA Fusion Energy Conference, IAEA-CSP-25/CD,http://www-naweb.IAEA.org/napc/physics.fec.htm.Miura, H. et al. 2005, to appear in Fusion Science and Technology (2007).Freidberg, J. P. 1987 In: Ideal Magnetohydrodynamics, New York: Plenum Press.

Vortical, toroidal and compressible motions in 3D MHD simulations of LHD

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