The Dynamic Ergodic Divertor (DED) of TEXTOR is presently being installed. It consists of sixteen helically wound coils occupying about 30% of the wall at the HFS. The coils follow field lines on a “pre-selected” magnetic surface and are fed individually outside the vessel. A perturbation field is created by the electrical currents in the perturbation coils with Fourier components resonant to the magnetic surfaces. The stochastic boundary layer is generated in the outermost region of the plasma, which due to long and short connection lengths can be divided into ergodic and laminar regions. Field line tracing and mapping techniques were used to analyse properties of the TEXTOR-DED plasma boundary. The DED will operate with several frequencies (DC or AC up to 10 kHz). In the “dynamic” operation the convective heat flux is deposited to a large plasma-facing surface and forces are transferred to the plasma edge, what can introduce a differential rotation of the plasma

Abdullaev, S.S.

Finken, K.H.

Jakubowski, M.W.

Kobayashi, M.

Lehnen, M.

Wolf, R.

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

PHYSICS OF THE DYNAMIC ERGODIC DIVERTORM.W. Jakubowski†, K.H. Finken, S.S. Abdullaev, M. Kobayashi, M. Lehnen, R. WolfInstitut für Plasmaphysik, Forschungszentrum Jülich GmbH, Association EUROATOM–KFA, Trilateral Euregio Cluster, D-52425 Jülich Germany† Also: Institute of Physics, University of Opole, 45 –052 Opole, PolandThe Dynamic Ergodic Divertor (DED) of TEXTOR is presently being installed. It consists of sixteen helically wound coils occupying about 30% of the wall at the HFS. The coils follow field lines on a “pre-selected” magnetic surface and are fed individually outside  the vessel. A perturbation field is created by the electrical currents in the perturbation coils with Fourier components resonant to the magnetic surfaces. The stochastic boundary layer is generated in the outermost region of the plasma, which due to long and short connection lengths can be divided into ergodic and  laminar regions. Field line tracing and mapping techniques were used to analyse properties of the TEXTOR-DED plasma boundary. The DED will operate with several frequencies (DC or AC up to 10 kHz). In the “dynamic” operation the convective heat flux is deposited to a large plasma-facing surface and forces are transferred to the plasma edge, what can introduce a differential rotation of the plasma.PACS: 52.55FaINTRODUCTIONThe  Dynamic  Ergodic  Divertor  was  developed  for spreading  a  convective  heat  flux  over  a  larger  plasma-facing  surface1.  An  ergodicity  of  the  magnetic  field  is created by the resonant  components  of  the perturbation magnetic  field  originating from the  currents  flowing in the  DED  coils.  For  the  low  perturbation  current amplitude,  magnetic  islands  centred  on  the  resonant magnetic  surfaces  are  created.  With  increasing  current, the  islands  start  to  overlap  destroying  magnetic  flux surfaces. This creates a specific topology of the magnetic field, where the field lines are not restricted to the flux surfaces, but “diffuse” over the whole ergodic region. The term  “ergodic”  means  that  the  field  line  will  pass infinitesimally  close  to  any  pre-selected  point  of  the available space. The magnetic field in the edge is an open system; field lines intersect  the wall. A distance between two  intersections  is  called  connection  length.  If  the connection  length  is  large  as  compared  to  the Kolmogorov length the corresponding field lines form a proper ergodic zone; if it is smaller or of the order of the Kolmogorov length, the field lines establish the laminar zone.  These  terms  will  be  explained   in  the  following sectionsThe  magnetic  perturbation  field  can  be  rotated  with different  frequencies  (up  to  10  kHz).  Applied  the magnetic field should:a)Prevent the divertor  plates  from the overheating and excessive erosion;b)Decrease  significantly  an  amount  of  impurities entering the plasma core;c) Create  a  plasma  rotation  (in  the  high  frequency operation),  what  may  improve  a  plasma  confinement and delay disruptions.In  the  following  sections  the  DED  arrangement  is described  and  a  short  introduction  to  the  Hamiltonian formalism  for  the  calculations  of  the  magnetic  field structure is provided. A discussion of this structure and of the dynamic aspects is given.THE EXPERIMENTAL SET-UPThe  DED  (as  shown  on  a  figure  1)  consists  of  four quadruples of coils. Each coil goes helically once around the torus following field lines on a 3≈q  surface. Every coil is supplied individually from the outside of the vessel by the 4-phase, either DC or AC (50 Hz, 1 – 10 kHz) with the current amplitude up to 15 kA. The main perturbation modes  (m =  10,..,14  poloidal  mode  numbers,  n  =  4 toroidal  mode  number)  are  centred  at  the  respective magnetic flux surface with q = 12/4. The 12/4 mode was chosen as a standard operational mode due to technical constrains  (such  as  current  density,  skin  effect  or  heat capacity)  and  physical  requirements.  One  of  the  most important physical requirement is not to disturb a  q = 2 surface (this could lead to interactions with intrinsic MHD modes and affects the plasma stability). The perturbation field for the 12/4 mode scales likercoilBeffmrrδ ∝ ж цз чи ш, where meff ≈ 20. It  decays very rapidly with distance from the coils.  The 4-phase current can be operated with several frequencies : DC, 50 Hz and at seven frequencies in the band 1 kHz — 10 kHz). These frequencies correspond to the rotating perturbation field, where the phase velocity of 42                         Problems of Atomic Science and Technology. 2002. № 4. Series: Plasma Physics (7). P. 42-44 Fig.  1. Sketch of the DED coils arrangement.the  propagating  field  projected  onto  a  poloidal  plane equals 12 m/s, 240 m/s and 2400 m/s respectively. The flexibility of the power supply system allows to connect coils  in several ways creating m = 6, n = 2 and m =3, n = 1 modes. There is also a possibility to mix the 12/4 mode with the 6/2 mode.HAMILTONIAN FORMALISM FOR FIELD LINE TRACINGThe  Hamilton  formalism  is  used  to  investigate  the magnetic  field  of  tokamaks  and  stellarators.  For  the destroyed magnetic flux surfaces, as it is the case for the TEXTOR-DED, the perturbation theory can be used. The field  line  equations  can  be  written  in  the  toroidal coordinates system as:1 1,   Z RB BdZ dRR d B R d Bϕ ϕϕ ϕ= =By replacing the time by the toroidal direction ϕ, one can introduce  the  Hamiltonian-Jacobi  equations.  Supposing that0 0 0 0/ lnZB B R R A B R Rϕ = = −Юand introducing new variables:( )00 0,  / ,  ln(1 )/ zR R R p xz Z R x − = += =one obtains:,  zzdz H dp Hd p d zϕ ϕ∂ ∂= = −∂ ∂where  2 20 0/H RA B Rϕ= ,  R0 is  the  major  radius  of  the tokamak  and  B0 is  the  magnetic  field  at  R0.  One  can introduce the perturbation field originating from the DED coils by dividing the Hamiltonian into two parts:0 1H H Hε= +where H0 is an unperturbed part and1 cos( )mnmH b m n tε ε θ ϕҐ= − Ґ= − + Ωеis a perturbed part written in the form of a Fourier series (ϕ, θ are toroidal and poloidal angles respectively).2 If the spatial  Fourier  spectrum has  more  than  one  frequency, island chains are created. The islands are created at the ( ) /mnq r m n= flux  surfaces  by  the  perturbation  field which  has  a  spatial  Fourier  component  bmn at  rmn.  A standard  parameter  to  describe  the  degree  of  the ergodization is the Chirikov parameter1, ,,1, ,21, ,( )28 ( )         m n m nm nm n m nm n m nsh trr rr b b mqRm L B rσ+++∆ + ∆=−=+ ж цЧз чи шIt is the ratio between the islands width and the distance between  island  chains  (∆m,n is  the  island  width  at  the radius  rm.n,  Lsh – shear characteristic length) A region of the plasma is ergodic if  1σ і ,  what simply means that islands from different island chains are overlapping. The Kolmogorov length is related to the Chirikov parameter.4 / 302KL qRpi σ−= ЧIt  simply  characterises  the  distance  from  where  on initially  neighbouring  field  lines  starts  to  separate exponentially. STRUCTURE OF THE MAGNETIC FIELD IN THE PLASMA BOUNDARY For  the  considerations  of  chaotic  systems  so  called Poincaré plots are used, which allow a two dimensional presentation of the magnetic field  structure. It is created by  marking  intersections  of  field  lines  with  a  chosen poloidal  plane  of  the  torus.  One  example  for  the superposition of the TEXTOR equilibrium field and the perturbation field going from the DED is shown in Fig. 2.The  figure  shows  an  outermost  region  of  the  poloidal plane, which is bent to a rectangular shape. The low field side is at the left and right side of the graph, high field side in the centre of the picture. The DED coils and target plates are presented as a rectangle at the top of the figure. Presented an example is the case of a strong ergodization of the edge magnetic field. The island chains are indicated by the corresponding mode numbers. One can see typical features of the magnetic field structure. The inner zone is the confinement zone with non-perturbed magnetic flux surfaces and some well separated island chains. The flux 43Fig. 2. Poincaré plot for the magnetic field lines in the edge at the full perturbation currentFig. 3. The Chirikov parameter versus a plasma radius for βpol = 0.0 and 1 – rres = 0.43 m, 2 – rres = 0.46 msurface and island chains are bent due to the Shafranov shift. The requirement of the Chirikov parameter less than one for the m/n = 8/4 flux surface is fulfilled  (see Fig. 3)The ergodic regionFor the second case in the Fig. 3 the region 42 cm ≤ r ≤ 46 cm forms the ergodic layer, which shows a high level of stochasticity with some remaining island chains. The Chirikov parameter reaches a value above 4 at the very edge.The flux surfaces are destroyed and field lines “diffuse” over the whole ergodic region. Connection lengths are of order  of  10000 meters.  It  enhances  the  radial  transport coefficient  significantly.  It  is  found that  ergodization is not restricted to a regime close to “design parameters” in the  operational  space  ( pol res1,  r( 3) 42 cmqβ = = = ),  but that it covers large operational space. Key parameters to modify the degree of  ergodization are variations of the perturbation field amplitude, of the plasma current, of βpol, and of  the plasma position3.  The perturbation spectrum depends on the pitch angle of the field lines, which can be adjusted by the value of beta poloidal. By modifying βpol and Iplasma one can shift the ergodic layer either outward or inward in the range 0.9 1.0raЈ Ј  keeping the same level of ergodization. The ergodic region is connected with the walls only via very thin structures called fingers.The laminar zoneThe  magnetic  field  of  TEXTOR-DED  is  an  “open” system.  It  means that  field  lines  can  intersect  tokamak walls,  and  ions  following  the  field  lines  will  be neutralised  there.  The  consideration  of  the  connection lengths  (relative  to  the  Kolmogorov  length)  elucidates specific structure of the plasma very edge. It is found that despite of the strong ergodization, rather large continuous areas  exist,  with  relatively  short  connection  lengths:  4 flux tubes with LC = 1 poloidal turn (because of four fold toroidal symmetry of the system), 4 with LC = 2 poloidal turns continuous areas. There exist areas with even higher value of LC, but their dimensions are small as compared to the Larmor ion orbits and have therefore no meaning for the particle orbits. Due to short connection lengths field lines  in  the  laminar  zone  are  not  visualised  on  the Poincaré plot (white region in the Fig. 2). A laminar plot was  developed  to  investigate  properties  of  the  laminar zone.  The  idea  is  similar  as  for  the  Poincaré  plot,  but additionally connection lengths are marked with different colours or grey tones. One example is shown in Fig. 4, each shade of grey denotes a zone with different number of poloidal turns (1 poloidal turn  ≈ 30 m). The plasma current  is  470 kA and  βpol = 1.   Very short  connection lengths in the very edge (< 2 poloidal turns) suggest to treat  that  region  in  a  similar  way  as  the  SOL  of  the poloidal  divertor.  The  laminar  zone  comparing  to  the ergodic  region  has  relatively  simple  structure  and transport is mostly convective, along the flux tubes, what is reflected in the power deposition pattern onto divertor tiles4.Dynamic aspectsThe dynamic option of the DED has been introduced in order to distribute the heat flow to the divertor target plate to a large area. During the static DED operation, the heat flux  is  guided  towards  the  divertor  target  plate;  the divertor strike points follow helically the DED coils. By the  rotation  of  the  perturbation  field,  the  heat  load  is distributed over the whole target area.The  high  frequency  aspect  of  the  DED-field  has  been analysed in cylindrical approximation5. It has been shown that  the „low frequency“ (relative to  Ωi)  electromagnetic wave of the DED effectively propagates in the area between coils and resonance layer as the compressional Alfvén wave (fast wave)6. At the resonance layer of the plasma, different approximations are described either by   a resistive annulus or by tearing modes. The interaction of the external rotating field with the current driven in the shielding layer results in the transfer of angular momentum between the DED-coils and the plasma6. The maximum poloidal torque applied to the plasma amounts to about 50 Nm; this maximum occurs at a frequency which seems to depend mainly on the width of the current layer. In detail it  depends on the assumed plasma temperature, on the applied frequency and on the assumed  island  or  ergodization  width.  The  toroidal projection of the applied force has about the same value as the one imposed by tangential NBI.  Due to the combined action of the DED and NBI torques, a differential rotation may  be  induced.  It  may  be  speculated  whether  the differential  rotation  suppresses  convective  cells  and  thus improves the confinement.44Fig. 4. The topology of the field line connection lengths of the laminar zone on the HFS of the plasma.0123451 5 5 1 6 0 1 6 5 1 7 0 1 7 5 1 8 0 1 8 5 1 9 0 1 9 5 2 0 0 2 0 54 24 34 44 54 64 74 8θ [ d e g . ]  r [cm] 1Dynamic Ergodic Divertor (special issue), Fusion Engineering and Design  37 (1999) 335-448 (15 contributions)2 S.S.Abdullaev et al., Phys. Plasmas, 6 (1999)3 K.H.Finken et al., Nucl. Fusion, 39 (1999), 6374 Th.Eich et al., Nucl. Fusion, 40 (2000), 1759 5 Finken, K.H., Nucl. Fusion, 30 (1999) 7076  Faulconer, D.W., Koch, R., Fusion Engineering and  Design, 37 (1997) 399

Physics of the dynamic ergodic divertor

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Physics of the dynamic ergodic divertor

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