Kotenko, V.G.

Volkov, E.D.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

THE INFLUENCE OF THE HELICAL-COIL ANGULAR SIZE ON THE YAMATOR MAGNETIC SURFACE CHARACTERISTICSV.G.Kotenko, E.D.VolkovInstitute of Plasma Physics,   NSC KIPT, 61108 Kharkov, UkrainePACS: 52.55.HcINTRODUCTIONNew  stellarator-type  magnetic  systems,  called  as Yamator,  have  been  considered  in  papers  [1-3]. Numerical calculations were carried out there for the ideal toroidal  models  of  Yamator  magnetic  systems  with filament-like  (i.e.,  vanishing  transverse  dimensions) helical  conductors.  The  calculations  have  demonstrated that  the  closed  magnetic  surface  region  with  a  high magnetic well value -U~0.1-0.5 can exist. The practical realization of those magnetic systems will evidently meet the necessity to determine the influence of real finite size of helical conductors/coils on the basic characteristics of the magnetic surfaces, first of all, on the magnetic well value. In this paper the magnetic surface characteristics are numerically studied for the Yamator with helical coils in the form of finite-angular-size thin strip. The main task of  the  study  has  been  to  get  a  general  idea  about  the tendencies  in  variations  of  Yamator  magnetic  surface characteristics when passing from ideal magnetic systems closer to the real ones.CALCULATION MODELAs  an  initial  ideal  model,  the  l=2,  m=3  Yamator magnetic  system [1]  was  chosen  for  calculations.  This system consists of two (l=2) 2-wire lines with equal-in-magnitude and opposite-in-direction wire currents. The 2-wire lines  are wound round the torus  along the helical lines θ=mϕ and θ=mϕ+pi, θ is the poloidal angle, ϕ is the toroidal angle,  m=3 is the number of helical line pitches on  the  length  of  the  torus.  The  wires  (helical conductors/coils) of each 2- wire line lie on the coaxial tori of the same major radius Ro and different minor radii a1=0.3  and  a2=0.45,  a2-a1=0.15  being  the  distance between the wires of the 2-wire line. Here and in what follows the lengths are given in Ro units. The amplitude of the circular-axis magnetic field generated by the helical current traversing the torus of the minor radius a1 is bo. In the ideal model the magnetic surface configuration forms if the superimposed toroidal circular-axis magnetic field strength is Bo/bo=2.5. The magnetic surface characteristics are presented in Table 1 (System 1, see below).Fig.1  shows  the  scheme  of  the  l=2  m=3  Yamator magnetic  system with  helical  coils  in  the  form of  thin strips. The strip angular sizes are ∆θh=45o (poloidal angle) and ∆ϕh=15o (toroidal angle). The strip was approximated by  11  filament-like  conductors  with  a  toroidal  angular distance of  δϕ=1.5o between them. Each of 11 filament-like conductors obeys the winding law θ=mϕ (cylindrical law).For comparison to the ideal model, the regime of the magnetic  axis  geometry  similarity  was  taken.  The Yamator magnetic system belongs to the type of helical magnetic systems, where the undistorted magnetic axis is Roa1a2Fig.1. A scheme of l=2, m=3 Yamator (top view) with real-angular-size helical coils: there are 11 filamentary helical conductors spaced 1.5о apart along the toroidal  angle in each helical coil. The longitudinal magnetic field  coils are not shown.a helical line, wound round the imaginary torus almost of a circular cross-section and closed on itself after one go-round along the torus.  At the upper right  of Fig.2a the major radius of this torus is designated as Roax (magnetic axis major radius), the minor radius–as rax (magnetic axis minor radius, index 1 concerns the ideal magnetic system, index  11-  the  one  with  a  strip).The  magnetic  axis geometry  similarity  means  fixing  the  value  of  the magnetic-axis  major  radius,  (Roax)1=(Roax)11=Roax=const. For  the  ideal  Yamator  magnetic  system  under consideration,  (Roax)1=1.1372  for  (Bo/bo)1=2.5.  For  the system  with  a  helical  strip  the  calculations  give (Roax)11=1.1372  provided  that  (Bo/bo)11=1.986.  The (Bo/bo)1/(Bo/bo)11 ratio  for  a  given  magnetic  field  Bo strength  points  to  the  necessity  of  raising  the  helical current by a factor of ~1.26 to form the similar (in above-given  sense)  magnetic  surface  configuration  in  the Yamator  with a helical strip.RESULTSFig.2  shows  three  poloidal  magnetic-surface  cross-sections within ½ magnetic field period (a) ϕ=0, b) ϕ=15o, c) ϕ=30o ) for the Yamator magnetic system with a helical strip (dotted lines) and the cross-section of the last closed magnetic  surface (LCMS) for  the ideal  Yamator  model (thin  solid  line).  It  is  seen  from  the  figures  that  the magnetic  surface  existence  region  has  appreciably decreased as  compared  to  that  in  the  ideal  model.  The LCMSs  are  not  equidistant,  i.e.,  in  each  cross-section there exists the θ azimuth region, where LCMS contours Problems of Atomic Science and Technology. 2002. № 4. Series: Plasma Physics (7). P. 65-66                            65coincide.  At  the  upper  right  of  Fig.  2a  the  toroidal projection of the magnetic axis trajectory on an enlarged scale  is  shown.  It  can  be  seen  that  the  magnetic  axis minor  radius  increases,  (rax)11>(rax)1,  as  the  helical-coil angular size increases.0.60 1.00 1.400.80 1.20Roax(r   )ax 1(r   )ax 11Ra)0.60 1.00 1.400.80 1.20 Rb)0.60 1.00 1.400.80 1.20 Rc)Fig.2. The magnetic surface poloidal cross-sections for Yamator magnetic system with helical strip (dotted lines)  and the LCMS for ideal Yamator magnetic system  (thin solid line). The poloidal cross-sections of the coaxial tori  and the helical strips are shown, the longitudinal magnetic field coils are not shown.The geometrical parameters of magnetic systems and magnetic  axes  under  comparison,  and  the  magnetic surface characteristics are summarized in Table 1. Here, –Ulc is the magnetic well value on the LCMS, iax, ilc are the rotational  transform  angles  (in  units  of  2pi)  near  the magnetic axis and on the LCMS, γax,  γlc are the magnetic field  ripples  on  the  magnetic  axis  and  on  the  LCMS, respectively; rlc is the average LCMS radius.Table 1System      1     11    l      2      2    m      3      3    a1     0.3     0.3    a2     0.45     0.45   θ     mϕ     mϕ   θh     0o     45o   ϕh     0o     15o  Bo/bo     2.5    1.986   rk     0.27    0.24-Ulc     0.25    0.184   iax     0.26    0.33   ilc     0.29    0.33   γax     1.15    1.17   γlc     2.28    2.08   Bz/bo     0.0     0.0From the analysis of  Table 1 it becomes clear that in the  Yamators  with  a  thin  helical  strip  of  the  poloidal angular size  ∆θh≤45o the magnetic well is reduced by  ≤20%, the LCMS average radius is cut down by ≤ 11%, the magnetic field ripple value on the LCMS falls off by  ≤11%.  The  magnetic  surface  configuration  still  remains shearless,  the rotational  transform angle increases  by  ≤20%. The calculations were performed for the controlling transverse  magnetic  field  Bz/bo=0,  z  is  the  principal (straight) axis of the systemSUMMARYThe  calculations  performed  have  demonstrated  the possibility  of  existence  of  the  inner  closed  magnetic surface region in  the Yamator  with helical  coils  in  the form of the finite angular size thin strip. In the l=2, m=3 Yamator  with  ∆θh=45o angular  size  helical  coils,  the magnetic surface characteristics, including  the magnetic well  value,  differ  by  no  more  than  ~10-25% from the ideal level.A new method of comparison between the magnetic surface characteristics in ideal and real helical magnetic systems  is  suggested.  It  is  based  on  searching  for  the magnetic-axis  geometry  similarity  regime,  i.e.,  equal values of magnetic-axis major radii.REFERENCES1.  V.G.Kotenko, G.G.Lesnyakov, S.S.Romanov,  Probl. of Atomic  .Science  and  Techn.  Ser.  Plasma  Phys. NSC”KIPT”, Kharkov, 1999, Iss. 1(1), 2(2), p.49-51. 2.  V.G.Kotenko,  G.G.Lesnyakov,  S.S.Romanov. 7th Ukraine  Conf.  on  Contr.  Fus.  and Plasma Phys.,  Kiev, September  20-21,  1999,  Book  of  Abstracts,  p.48  (in Ukrainian.).3.  V.G.Kotenko,  G.G.Lesnyakov,  S.S.Romanov.  Plasma Fus. Res. SERIES: Vol 3(2000), p.154-157.

The influence of the helical-coil angular size on the Yamator magnetic surface characteristics

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The influence of the helical-coil angular size on the Yamator magnetic surface characteristics

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)