CONTENTS

                                               № 4 (6), 2001

                                                The following articles were originally presented orally at the International Workshop

                                              “Innovative Concepts and Theory of Stellarators”. The manuscripts were refereed as

                                                normal after submission.

1. ON INTERNATIONAL WORKSHOP  “INNOVATIVE CONCEPTS AND THEORY OF STELLARATORS”

 Ya. I. Kolesnichenko

            The international workshop “Innovative Concepts and Theory of Stellarators” (ICTS) was held at Kyiv, 28 - 31 May 2001. It was organized by the Scientific Centre “Institute for Nuclear Research” (KINR), with support of the Ministry of Education and Science of Ukraine and the Science and Technology Center in Ukraine.

The idea to hold the workshop was first discussed by representatives of Max-Planck-Institut für Plasmaphysik (Garching, Germany), National Institute for Fusion Science (Japan), Princeton Plasma Physics Laboratory (USA), and KINR (Ukraine) during the 18th IAEA Fusion Energy Conference in Sorrento, Italy, October 2000. Later, the Programme and Organizing Committee of the workshop was formed, which consisted of Ya. Kolesnichenko (chairman, Ukraine), C. Alejaldre (Spain), S. Hirshman (USA), R. Komirenko (Ukraine), V. Lutsenko (scientific secretary, Ukraine), J. Nührenberg (Germany), V. Shafranov (Russia), K. Stepanov (Ukraine), I. Vyshnevskyj (Ukraine), M. Wakatani (Japan), R. White (USA), H. Wobig (Germany), Yu. Yakovenko (Ukraine), K. Yamazaki (Japan), M. Zarnstorff (USA).

The major objective of the workshop was to discuss and compare the prospects of various non-axisymmetric toroidal fusion devices (optimized stellarators of Wendelstein line, quasi-axisymmetric and quasi-helically-symmetric stellarators etc.) as fusion reactors. Other objectives were to discuss current state of art in theory of plasmas relevant to these devices; to present features of constructed, projected and proposed new systems; to single out most important physics issues to be studied in the nearest future.

The workshop was also intended to foster East-West scientific collaboration, in particular, facilitate contacts of Ukrainian scientists working on stellarator physics with the world fusion community.

The workshop was attended by scientists from Max-Planck-Institut für Plasmaphysik (Garching and Greifswald, Germany), National Institute for Fusion Science (Japan), Kyoto University (Japan), Kurchatov Institute (Russia), Keldysh Institute of Applied Mathematics (Russia), Centre de Reserches en Physique de Plasmas (Switzerland), Kharkiv Institute of Physics and Technology (Ukraine), Kharkiv University (Ukraine), KINR (Ukraine), Université Henry Poincaré (France) and International Centre of Physics in Ukraine, Princeton Plasma Physics Laboratory (USA). They have shown many new interesting results.

So it happened that the workshop was held in the year of anniversaries: 50 years of stellarator research in the world (started in Princeton) and 40 years of stellarator research in Germany and Japan. Therefore, it was worth to present not only new ideas but also recollect a history of stellarators. In particular, in his invited talk, Dr. H. Wobig reported on the development of Wendelstein-line stellarators in Garching from the beginning till nowadays and also showed results of studies of a Helias reactor. New trends and ways to optimize stellarators were presented in an invited talk by Prof. Nührenberg. The other invited talks dealt with national projects developing in USA and Japan. Dr. H. Neilson reported on physics design of NCSX (National Compact Stellarator Experiment) - a quasi-axisymmetric stellarator developing in USA. Prof. K. Matsuoka presented the physics design of a quasi-axisymmetric stellarator developing in Japan. Prof. K. Yamazaki (Japan) showed the results of studies of the Modular Heliotron Reactor (MHD), which extends the biggest machine of stellarator type in the world - Large Helical Device (LHD). There were also many informal discussions on both modern physics of stellarators and history of fusion research. In particular, Prof. V. Shafranov told interesting stories of the initial stage of fusion research in the USSR.

The ICTS workshop was finished with a general discussion where key issues of stellarator physics and possible ways of international collaboration were identified. The workshop participants supported the suggestion of Prof. J. Nührenberg to hold the next workshop of this kind (which will have a status of an IAEA TC Meeting) in Greifswald in 2002.

            The workshop participants were invited to prepare papers based on their presentations for submission to Scientific Papers of the Institute for Nuclear Research. As is seen from the papers here, most participants have chosen this option. We are quite pleased with this response. In addition, all the collection of the workshop materials (including copies of transparencies and materials presenting through a computer projector) is published on Compact Disks, which became possible due to efforts of Dr. V. Lutsenko.

2. STELLARATOR  RESEARCH  AT  THE  IPP  IN  GARCHING

H. Wobig,  W7-AS-Team,  SSS-Group

            Stellarator experiments at the IPP Garching started in 1965 with small devices until in 1976 the first large device Wendelstein 7-A (major radius 2 m, av. plasma radius 10 cm, magnetic field 3,4 T) went into operation. In 1980 this experiment produced the first plasma without toroidal current. Its successor, Wendelstein 7-AS, abandoned the concept of helical windings and generated the stellarator field by a set of modular coils. The same concept is applied to Wendelstein 7-X, (major radius 5,5 m, av. plasma radius 0,5 m, magnetic field 2,5 T), which in addition will be equipped with superconducting modular coils. The paper describes the main experimental results of the Garching stellarators, its future planning and the reactor prospects of the advanced stellarator [1].

3. REACTOR  PLASMA  DESIGN  BASED  ON  LHD

K. Yamazaki and LHD Experimental Group

            The overview of helical reactor concepts and related plasma designs based on the Large Helical Device (LHD) experimental database are described. Firstly, design requirements for helical reactors are clarified with respect to plasma confinement improvement, density limit and beta limit. Several new confinement scaling laws are derived using LHD database in addition to the previous medium-sized helical confinement database. In the previous LHD-type reactor designs two times better plasma confinement time than the conventional LHD scaling law was assumed, which has been already achieved experimentally as “New LHD” scaling laws. One and half times higher plasma density than the conventional helical density limit scaling law has been achieved. This condition is required at the start-up phase of reactors. Higher than half of beta value required in reactors is also achieved in the inward-shifted configuration in LHD experiment, which beta value is beyond the theoretical Mercier stability limit. This inward-shifted magnetic configuration satisfies high beta and low effective helical ripple operations required for reactors. Almost all these normalized requisites have been achieved in the LHD experiment. The present LHD experiment can justify the future prospect of the LHD-type helical devices towards a steady state, efficient and reliable reactor.

4. PHYSICS  DESIGN  OF  CHS-QA  BASED ON  CHS EXPERIMENTS

K. Matsuoka,  S. Okamura,  S. Nishimura,  M. Isobe,  C. Suzuki,  A. Shimizu,  A. Fujisawa,  K. Ida,  T. Minami,  H. Iguchi,  Y. Yoshimura,  M. Osakabe,  I. Nomura,  S. Murakami,  M. Yokoyama,  N. Nakajima,  T. Hayashi,  K. Itoh,  P. Merkel, M. Drevlak,  C. Nührenberg,  S. Gori,  R. Zille, 

 J. Nührenberg

            CHS-qa, a quasi-axisymmetric helical device, has been designed as a post-CHS device in NIFS, the main purpose being to improve the neoclassical and anomalous transports of a helical plasma. In CHS, a variety of improved modes have been observed. On the basis of these improved modes further improvement of transports is to be pursued in CHS-qa by taking into consideration methods verified in other toroidal magnetic configurations, e.g. large velocity / radial electric field shear, and maximum J criterion. The toroidal viscosity is shown to be smaller by two orders than that of CHS and the poloidal viscosity, which is mainly determined by the aspect ratio with additional contribution from residual ripples, is also smaller by roughly l order than that of conventional helical system.  The maximum J criterion is satisfied at the edge region in the vacuum configuration by the presence of small residual ripples and the region where the criterion is satisfied is extended to the core region due to the stellarator shear produced by the bootstrap current in a finite beta plasma. Design priority is put on the low aspect ratio (A_p = 3.2) because of a large plasma volume: R = 1.5 m, a = 47 cm, B = 1.5 T, toroidal period number N = 2, 10 modular coils per period, 8 additional modular toroidal coils, 3 pairs of poloidal coils. Residual ripples can be controlled with these coils to keep flexibility in the experiments.

5. INITIAL  PLASMAS  AND  DIAGNOSTIC  ENERGY  CONTENT  IN  HELIOTRON  J

S. Besshou,  K. Aizawa,  K. Hanatani,  K. Kondo,  T. Mizuuchi,  K. Nagasaki, Y. Nakamura, 

M. Nakasuga,  T. Obiki,  H. Okada,  F. Sano,  M. Wakatani,  W. L. Ang,  T. Hamada,  K. Ichikawa,  Y. Ikeda,  H. Kawazome,  T. Kobayashi,  Y. Liu,  S. Maeno,  Y. Manabe,  H. Shidara,  Y. Suzuki,  M. Takeda,  T. Takamiya, K. Tomiyama,  O. Yamagishi,  Y. Ijiri,  K. Sakamoto,  T. Senju,  M. Shibano,  K. Toshi,  K. Yaguchi

            This report describes the behavior of the initial plasmas in Heliotron J since July 2000. Hydrogen has been successfully produced by electron cyclotron resonance heating (ECH) (53 GHz, P_ECH £ 400 kW, D t £  £ 50 ms). The stored energy was measured with the diamagnetic double loops as a function of magnetic field strength (0.61 T< <B> axis < 1.44 T). The value of Wp » 0.7 kJ, which corresponds to <b> » 0.2 %, was obtained by the second harmonic ECH at <B> axis » 0.95 T with the input power 400 kW, l/2p = 0.18 m and <R> axis = 1.20 m. The values of Wp » 0.8 kJ and <b> » 0.1 % were obtained at <B> axis » 1.44 T by the off-axis fundamental ECH. Preliminary magnetic configuration scan with the vertical field coils controls the plasma position (1.1 m < <R> axis < 1.3 m), the rotational transform, etc. The configuration effects on the energy content are discussed.

6. APPROACHES  TO  DESCRIPTION  OF  THE  TOROIDAL  PLASMA  EQUILIBRIUM  WITH  ISLANDS 

V. D. Shafranov,  M. I. Mikhailov,  A. A. Subbotin

            Two approaches to descriptions of islands with small number of periods in toroidal systems are discussed. One of these uses the helicoidal poloidal magnetic flux and tokamak-like representation of the magnetic field. Another approach suggests the generalization of the parameterisation of the magnetic surfaces used for example, in VMEC equilibrium code, on the configuration with the magnetic islands.

7. ALFVÉN  CONTINUUM  IN  STELLARATORS:  GENERAL  ANALYSIS  AND  SPECIFIC  EXAMPLES

Ya. I. Kolesnichenko,  H. Wobig,  Yu. V. Yakovenko,  J. Kißlinger

            The Alfvén continuous spectrum in three-dimensional toroidal magnetic configurations is analyzed. Alfvén continua in Wendelstein-line stellarators (Wendelstein 7-AS and the designed Helias reactor HSR4/18) are calculated with the code COBRA, and the principal gaps in the Alfvén continua in these devices are found. It is shown that the shape of the plasma cross section strongly affects the Alfvén spectrum. Peculiarities of the Alfvén continuum of low-shear configurations are discussed. The frequencies of the calculated gaps in the Alfvén continuum in W7-AS are compared with the frequencies of the experimentally observed plasma oscillations.

8. ALFVÉN  INSTABILITIES  CAUSED  BY  CIRCULATING  ENERGETIC  IONS  IN  OPTIMIZED  STELLARATORS

Ya. I. Kolesnichenko,  V. V. Lutsenko,  H. Wobig,  Yu. V. Yakovenko

            The work investigates instabilities of Alfvén eigenmodes that can be driven by circulating energetic ions in optimized stellarators of Wendelstein line (Helias). It is shown for the first time that several sideband resonances rather than the only one associated with toroidicity (and known from a theory relevant to tokamaks) may essentially contribute to the instability growth rate. New resonances enhance the instabilities and, moreover, they may result in instabilities in those cases when the conventional resonance is not efficient. Destabilization of the toroidicity-induced Alfvén eigenmodes and the eigenmodes existing due to both specific plasma shaping and Fourier harmonics of the magnetic field of a Helias is considered.

9. CORE-LOCALIZED  ALFVÉN  EIGENMODES  IN  STELLARATORS

Ya. I. Kolesnichenko,  V. V. Lutsenko,  H. Wobig,  Yu. V. Yakovenko

            The work deals with discrete Alfvén eigenmodes in optimized stellarators of the Wendelstein line. It is shown that core-localized Alfvén eigenmodes do exist in these systems. In particular, mirror-induced Alfvén eigenmodes (MAE) and helicity-induced Alfvén eigenmodes (HAEsi) localized in the plasma core of the four-period Helias reactor are found. The results are obtained by solving numerically an eigenmode equation derived in Ya. Kolesnichenko et al., Phys. Plasmas, 8, 491 (2001).

10. EFFECT  OF  DESTABILIZED  ALFVÉN  EIGENMODES  ON  ALPHA  PARTICLES  IN  A  HELIAS  REACTOR

Ya. I. Kolesnichenko,  V. V. Lutsenko,  H. Wobig,  Yu. V. Yakovenko

            The upper limits of the local energy losses of circulating a-particles caused by the various instabilities of Alfvén eigenmodes in a four-period Helias reactor are evaluated. It is found that certain destabilized Alfvén eigenmodes will affect only alphas with the energy well below 3,5 MeV, which seems to open a possibility to remove the helium ash by exciting the corresponding Alfvén eigenmodes by either energetic particles or an antenna system.

11. METRIC  COEFFICIENTS  FOR  A STELLARATOR  CONFIGURATION

Zh. N. Andrushchenko,  Ya. I. Kolesnichenko,  O. A. Silivra,  Yu. V. Yakovenko

            The metric coefficients are analytically calculated for a non-axisymmetric stellarator configuration with a complex shape of the magnetic axis and the magnetic field varying along the axis. Calculations are performed in a working coordinate system, which is a flux coordinate system with “straightened magnetic field lines”, assuming that the magnetic axis is a closed spatial curve with given curvature and torsion. The cross sections of the magnetic surfaces are approximated by ellipses with both elongation and orientation of the main axes varying along the magnetic axis. Effects of triangularity and displacement of magnetic surfaces with respect to the magnetic axis are included in the analysis. Transition from the working coordinates to Boozer ones is carried out, and the metric coefficients are obtained in Boozer coordinates.

12. REDUCED  MHD  EQUATIONS  FOR  ALFVÉN  EIGENMODES  IN  STELLARATORS

O. P. Fesenyuk,  Ya. I. Kolesnichenko,  H. Wobig,  Yu. V. Yakovenko

            Reduced magnetohydrodynamic (MHD) equations are derived, in which the plasma compressibility is taken into account, but fast magnetoacoustic waves are excluded. For the sake of simplicity all terms associated with the pressure gradient and the plasma current are disregarded. However, the continuous spectrum of the obtained equations is shown to coincide exactly with the continuous spectrum of the full MHD equations. First results of the code COBRAS (COntinuum BRanches of Alfvén and Sound waves) intended for calculation of coupled Alfvén and slow continua, are presented. Effect of the compressibility on the Alfvén continuum in Wendelstein-line stellarators is studied.

13. MECHANISMS  OF  STOCHASTIC  DIFFUSION  IN  OPTIMIZED  HELICAL  SYSTEMS

Ya. I. Kolesnichenko,  V. V. Lutsenko,  V. S. Marchenko,  H. Wobig

            Various mechanisms of stochastic diffusion of energetic ions in optimized stellarators are reviewed and analyzed. It follows from the carried out analysis that alpha losses caused by stochastic diffusion in the Wendelstein-line stellarators (Helias configurations) result mainly from successive orbit transformations of the transitioning particles.

14. ENERGY  CONFINEMENT  IN  W  7-AS  AND EXTRAPOLATION  TO  A  HELIAS  REACTOR

H. Wobig,  SSS-Group

            The energy confinement time in Wendelstein 7-AS shows a strong dependence on the rotational transform, which makes it difficult to establish a universal scaling laws for energy confinement. Under optimum conditions the scaling of confinement follows Lackner-Gottardi scaling with an improvement factor of 1.2. This holds in the very neighbourhood of low order rational surfaces (i = 0.34 and i = 0.52). The scaling law together with other ones published in the literature are applied to the Helias reactor HSR4/18    (R = 18 m, a = 2 m, B = 5 T) showing that the conditions of self-sustained burn can be satisfied. In a second part the one-dimensional heat conduction equation is solved for the temperature profile taking into account alpha-particle heating and bremsstrahlung losses. The non-linearity of the equation leads to multiple solutions, the scaling of the stable solution is studied in detail.

15. STELLARATOR  FIELDS  WITH  SMALL  PS  CURRENT AT  SMALL  ROTATIONAL  TRANSFORM

F. Herrnegger

            One aspect of the optimization concept of stellarators is the reduction of the normalized Pfirsch-Schlüter current density  to a reasonable level but obeying other side conditions, e.g., concerning small bootstrap currents, good stability properties, reasonable aspect ratio, etc. This problem is addressed in the present work. Various stellarator vacuum field are given analytically for M = 2, 3, 5, 10, 12 (M is the number of field periods around the torus) where the PS-current density is reduced by more than a factor of ten to rather small values around 0.3 even at small į-values.

16. MODULAR  COIL  SYSTEMS  OF  ADVANCED  STELLARATOR  REACTORS

J. Kisslinger,  C. D. Beidler,  E. Harmeyer,  F. Herrnegger,  H. Wobig

            Only one single super-conducting non-planar coil set is necessary to generate the magnetic field of an advanced stellarator reactor. This concept of modular coils offers a wide range for field optimization with respect to plasma performance. Two HELIAS configurations are considered; one similar to Wendelstein 7-X with 5 field periods and a major radius of 22 m and a more compact configuration with 4 field periods and a major radius of 18m. Both configurations uses 10 coils per period with 5 different coil shapes. The shapes depend on the magnetic field structure and on the distance between plasma and coils. The minimum distance is given by the thickness of blanket and shield and is an essential parameter for the size of the device. The winding packs with trapezoidal cross-sections are split in double pancakes which are wound on steel shells. The trapezoidal cross-section is used in order to reduce the maximum field strength at the conductor on the high-field side. The favoured choice is a Nb-Ti super-conductor because of the established industrial technology and good mechanical properties. Super-fluid helium at 1.8K is used as coolant to ensure safe operation at 10 T. More advanced conductors like Nb-Sn or Nb-Al offer higher magnetic fields at higher coolant temperatures. Their drawbacks are the lower technological development and brittleness. The magnetic force distribution in the coils is inhomogeneous and has radial and lateral components of about the same value. The coils tend to become more circular and planar under the magnetic load. Stiff coil housings with local reinforcements and a system of inter-coil support elements keep the resulting mechanical stress values within technical limits.

17. THE  a-PARTICLES  CONFINEMENT  IN  ZERO  AND FINITE  b  MIRROR-TYPE  STELLARATORS

M. I. Mikhailov,  M. Yu. Isaev,  J. Nührenberg,  A. A. Subbotin,  W. A. Cooper,  M. F. Heyn,  V. N. Kalyuzhnyj,  S. V. Kasilov,  W. Kernbichler, 

V. V. Nemov, M. A. Samitov,  V. D. Shafranov

            Helical magnetic systems with poloidal direction of the lines B = constant on the magnetic surfaces are investigated to clarify in more detail the connection between the shape of the magnetic surfaces and the topology of the B = constant surfaces on the one hand and particle confinement on the other. The possibilities to fulfil the pseudosymmetry condition as well as the condition that the second adiabatic invariant J_êê forms closed contours are investigated numerically for almost zero and finite b values.

18. AVERAGED  DESCRIPTION  OF 3D MHD  EQUILIBRIUM ^*

S. Yu. Medvedev,  V. V. Drozdov,  A. A. Ivanov,  A. A. Martynov, Yu. Yu. Poshekhonov,  M. I. Mikhailov

            A general approach by S. A. Galkin et al. in 1991 to 2D description of MHD equilibrium and stability in 3D systems was proposed. The method requires a background 3D equilibrium with nested flux surfaces to generate the metric of a Riemannian space in which the background equilibrium is described by the 2D equation of Grad-Shafranov type. The equation can be solved then varying plasma profiles and shape to get approximate 3D equilibria. In the framework of the method both planar axis conventional stellarators and configurations with spatial magnetic axis can be studied. In the present report the formulation and numerical realization of the equilibrium problem for stellarators with planar axis is reviewed. The input background equilibria with nested flux surfaces are taken from vacuum magnetic field approximately described by analytic scalar potential.

19. OPTIMISATION  OF  THE  PARTICLES  CONFINEMENT  IN  STELLARATOR WITH  HELICAL  DIRECTION  OF  LINES  B=CONST

A. A. Subbotin,  J. Nührenberg,  M I. Mikhailov,  W. A. Cooper,  M. Yu. Isaev, M. F. Heyn,  V. N. Kalyuzhnyj,  S. V. Kasilov,  W. Kernbichler, 

V. V. Nemov, V. D. Shafranov

            Collisionless particle confinement in stellarator configurations with helical direction of the lines B=const on the magnetic surfaces is investigated numerically for a six-period system. The optimisation is performed with different penalty functions that are connected with the pseudosymmetry condition and the condition that the second adiabatic invariant J_êê forms closed contours. In addition, the effect of b on the particle confinement is studied.

20. ASYMPTOTIC  DESCRIPTION  OF  PLASMA  TURBULENCE: KRYLOV – BOHOLIUBOV  METHODS  AND  QUASI-PARTICLES

P. P. Sosenko,  P. Bertrand,  V. K. Decyk

            The asymptotic theory of charged particle motion in electromagnetic fields is developed for the general case of finite Larmor-radius effects by means of Krylov-Boholiubov averaging method. The correspondence between the general asymptotic methods, elaborated by M. Krylov and M. Boholiubov, the quasi-particle description and gyrokinetics is established. Such a comparison is used to shed more light on the physical sense of the reduced Poisson equation, introduced in gyrokinetics, and the particle polarization drift. It is shown that the modification of the Poisson equation in the asymptotic theory is due to the non-conservation of the magnetic moment and gyrophase tremblings. It is shown that the second-order modification of the adiabatic invariant can determine the conditions of global plasma stability and introduces new nonlinear terms into the reduced Poisson equation. Such a modification is important for several plasma orderings, e.g. MHD type ordering. The feasability of numerical simulation schemes in which the polarization drift is included into the quasi-particle equations of motion, and the Poisson equation remains unchanged is analyzed. A consistent asymptotic model is proposed in which the polarization drift is included into the quasi-particle equations of motion and the particle and quasi-particle velocities are equal. It is shown that in such models there are additional modifications of the reduced Poisson equation. The latter becomes even more complicated in contrast to earlier suggestions.

21. THE  ASYMMETRY  CURRENT  IN  STELLARATORS

Yu. V. Gott,  E. I. Yurchenko

            An analysis of drift trajectories of charged particles in tokamak leads us to the conclusion on the possibility of a new longitudinal electric current which appears, in contrast to the bootstrap current, to be proportional to plasma pressure. The qualitative difference in drift trajectories of particles, which move in co- and counter- direction with respect to magnetic field produces velocity-space asymmetry of the trapping boundary of charged particles. As a result the new electric current generates. We named this current an asymmetry current. The approximate formula for the asymmetry current in tokamaks and stellarators, which is valid for the entire plasma column, including the near-axis region, is obtained. It is shown that the density of asymmetry current is maximal near magnetic axis and decreases at plasma periphery. The possibility of the experimental detecting of the asymmetry current in stellarators is discussed.

22. COMPACT  STELLARATOR-LIKE  CONFIGURATIONS,  CREATED BY  SYSTEM  OF  PLANE  CIRCULAR  CURRENT  COILS

A. V. Georgievskiy,  W. T. Reiersen,  V. A. Rudakov

            Stellarator magnetic configurations with different combinations of plane circular tilted current coils were studied. Two types of coil combination used for forming of the magnetic configurations: cohesion tilted coils (Villarso coils) and uncoupling tilted coils. As torsatron type, so stellarator type systems with multipolarities l = 2, l = 4 and their combinations were studied. For the first time magnetic configurations with good confinement properties created by system of uncoupling coils stellarator type current combinations were obtained. Splitting of l = 2 coil system allows to obtain l = 4 harmonic, that goes to improvement of confinement properties of the configuration. Variation of value corrective field allowed to obtain quasisymmetric properties of the stellarator configuration.

23. EVALUATION  OF  AN  EFFECTIVE  RIPPLE  IN  STELLARATORS

V. V. Nemov,  S. V. Kasilov,  W. Kernbichler,  M. F. Heyn

            Diffusion coefficients and peculiarities of the different representations of Є_ff (effective ripple) are discussed.

24. SMOOTH  TRAJECTORIES  ON  TOROIDAL  MANIFOLDS

S. S. Romanov

            Great advance in plasma magnetic confinement theory is attained by means of trajectories investigation on toroidal manifolds. Since local and global aspects of qualitative trajectories flow are important at that, it is natural to consider such trajectories on smooth manifolds. It is succeeded by means of fixing the local metrics of a manifold to integrate equation for smooth trajectories, to find out connection between infinitesimal and topology trajectory properties, to write down equations of marked curves in evident form. Conditions are given, on which smooth trajectories will be either closed or compact on a manifold. Restrictions of topological invariances are found for loxodromies, for which a trajectory will be a plane curve.