Optimization of

particle confinement time in CNT

Abstract :  Automatic simulations for different angles between coils and ratios IL-current / PF-current in CNT are carried out. SimPIMF  code v2.7 in optimization mode is run in a similar manner as for UST_1. The method to calculate collisionless losses has been included in the automatic optimization in order to obtain the best configurations of coils in relation to the losses of particles. It is a new feature in SimPIMF and it was not implemented for UST_1 optimization. Although provisional and not contrasted some optimums are obtained. Plasma size, magnetic well, Iota, Iota profile, average ripple,  are also automatically calculated as in the previous UST_1 optimizations. Only two parameters are taken (angle and ratio) so most of the variables cannot be optimised altogether.

   CNT, Columbia Nonneutral Torus, is an innovative stellarator in Columbia University mainly dedicated to the study of non-neutral plasmas.

   Some simulations and test recalculations for CNT stellarator are shown in [1] mainly to test SimPIMF for a CNT style stellarator. The contrast with the results from CNT team were satisfactory so the modellisation of CNT in SimPIMF is correct or almost correct.

New modules added to SimPIMF

   The code with the new additions will be the version 2.7

* A module to throw random particles on a magnetic surface. It becomes a necessity in order to calculate confinement of particles in a reasonable computing time. The previous method throws the particles inside a circular torus and  more particles are needed to have a serious estimation of the confinement time.

   The calculations do not need to be accurate but enough to select the best configurations and to obtain similar results in the final accurate manual simulation.

*  D, T, and 3He ions are defined in order to be used in the simulations.

 * An improvement in the code to save results in the file. Now the change from one series of data to be recorded to another is simpler.

General features

  The process to simulate and optimise coils is described in [3] and some additional tips in  [4], applied to UST_1 optimization in both cases, but the essence is the same for CNT. This method is an old method of optimization before the "reverse engineering" method appeared. SimPIMF was the only method for optimization in UST_1 but not very adequate (much better a NESCOIL-like code). In the case of CNT style, with very simple coils, perhaps one configuration of good quality could be found. The problem is that also perhaps a configuration with good confinement could not exist. Also in [2], CNT team, this possibility of optimization is stated.

  The differences with respect the optimization for UST_1 are: 

- The rough pre-calculations include a pre-calculation of the particle confinement time. The computer time for this simulation is about ten times smaller than the definitive calculation. It results in an important reduction of processing time.

- A final more accurate simulation of the particle confinement time is done for the candidates  selected automatically in the rough pre-calculations.

-  The Iota calculations and other are faster here than in [1] mainly because  t_Step is larger. Reasons : a) Iota is not so important here because the focus is particle confinement time. b) It was noticed that longer Steps gave enough accuracy.

- The present process of optimization includes a simulation of a series of protons with drifts. Automatic simulations of protons and automatic simulation under drifts is done for the first time in the present optimization.

 *  All calculations are vacuum calculations without plasma pressure effect. Electric fields and ambipolarity condition is not included in SimPIMF yet. It should be implemented in the future.

   Loops of optimization

  Number of coils = 4 ;  (2 IL and   2 PF)

IL coil current = 170000 A-turn  (a typical value for CNT)

PF current is generated in the simulation. Alpha =  IF current /PF current =  Parameter 2 (named as Pitch 2 in the past)

 Coef. for the magnetic grid = 40 for the rough pre-selection. 80 for the rest (one side of grid = 1/80m)

nº Plas  : Order number of the "plasma"

nº Part : number of particles of the plasma.

t_Step : increment of time per Step, nanosec.

KSteps : number of steps of simulation /1000

Table 1

First test in collisional regime

  Initially a run of optimization composed of 72 structures was tried. The code simulated the plasma nº 7 in collisional regime. A density of 5e19 m^-3 was choosen, T protons =  500 eV monoenergetic. Virtual collisions method  'V'  is used. Simulation time = 1ms

   The simulation gives a table of results. Several best results are obtained, for example taking major weight on the  particle confinement time variable.

   However different particle confinement times are obtained at different densities and T (in relation to low collisionality regime, plateau regime, P-S regime, and the collision frequency).   It requires the establishment of a working condition (a particular density and T) to be optimised, otherwise it would be necessary to optimise an average of conditions. The last would take excessive computing time and is almost unfeasible using a PC.

   Density do not participate in collisionless regime and T only in the increase of Energy and so in the velocity drift but in a more lineal manner (the variations due to change in regime are absolutely no lineal)

   Finally the simulation of particles in absolute collisionless regime was decided. It can be considered a limit of the 1/nu regime. If particles are well confined under no collisions it can be supposed a good confinement under 1/nu regime.

  However

* A) more computing time is necessary to obtain results in collisionless regime because more particles are needed to distribute them in the space and pitch angle (in collision simulation the random essence of collisions distributes the particles in space and direction-angle). Moreover

* B) SimPIMF v2.5 was not prepared to throw particles randomly on a particular magnetic surface.

  Because of this the first test was done in collisional regime.

Optimization of particle confinement time in collisionless regime

  Conditions : Energy of protons for plasma 7 simulation = 100eV, monoenergetic distribution. 200 ions  H+ . Magnetic surface to throw particles defined by x+radius = 0.2 * radius of IL coils. The particles escape when they are out of the magnetic boundaries, not exactly the LCFS. Simulation under drifts. Particles thrown in random direction of velocity.

   Initially a test run with 15 particles for 'Rough Tau particle' and 50 for ' 'Accurate' Tau particle conf.' was tried. The variability in the 'Rough Tau particle' was excessive so the number of particles and t_Step were increased. 1000 particles, or less for estimations, are considered for TJ-II (many fourier modes) [5]. Conditions for the next results are expressed in Table 1. t_Step is at the maximum but the number of particles is more relevant for the accuracy.

The results for a test for Parameter 1 = 0.7679 rad , Par. 2 = 3.0 are

Table 2

The results have small dispersion for 200 particles. Max. error ~10%. The 'rough' is only an estimation and it is enough so far. The threshold is set to Tau =230 so to obtain most of the accurate Tau =250 or superior.

Coarse intervals

  A first run with coarse intervals is carried out to know about the best branches of parameters.

     Coarse intervals of optimization :  Parameter 1 (Pitch 1) = [32º , 44º]  [0,489 rad , 0,768 rad] 4.0 points ; Parameter 2 =[1.0 , 9.0]  4 points ; 

   25 different structures analysed. It lasted  2.5 PC-hours. Only 5 structures passed the threshold of 230 microsec of particle confinement time so they are simulated in detail. Each one of the three detailed simulation of Plasma 7 lasted ~1700s  so most of the time is dedicated to the detailed simulation and not to the rough pre-calculation. It saves much time.

The results are :

Iota_1: Iota at mag. axis + 100mm on x+ 

% Trap :  Proportion of trapped particles = % particles that bounce

A rip : Average ripple for the magnetic surfaces calculated in 0.22 * IL radius.

Plas Size  :  Plasma size in mm.

Iota_2 : Iota at mag. axis +30mm on x+ 

Tau rou =  Tau rough  :  Estimation of Tau particles to decide if the structure is interesting (threshold = 230 in Table 3 )

Tau  :   Final more accurate Tau particles

Pitch 1 = Parameter 1 =  Tilted amgle of IL coils

Pitch 2 = Parameter 1 =  Ratio IL-current / PF-current  = Alpha

Prof  :  Idea of Iota profile  = I(30) / I(100)

Well = Magnetic well depth at axis +100mm in %

NOTE: The field 'Bmin' is not copied to the table. 'Bmin' is the standard deviation of Bmin. It is not necessary if particle confinement time is calculated. Besides it should be tested in CNT where modular rippple might be confused with helical ripple. Bmin was important in the optimization of UST_1.

Results

Table 3

Conclusions and comments

 Configurations are denoted as XXYY : 7730 corresponds to   Parameter 1 = XX = 0.77 radian,  and Parameter 2 =  Alpha = YY = 3.0 .

* 1) 'Tau Rough' and 'Tau' are similar and the difference between them express the degree of accuracy, the error, in the calculation of 'Tau Rough' and indirectly of 'Tau'.

* 2)  The threshold was fixed at 230 microsec for 'Tau Roug' and it is correct because only 5 accurate calculation are executed in order to speed up the process.

* 3)  In all cases the  Pitch 2  = 3.0  gives maximum  Tau so the next finer optimization will be run around 3.0 .

* 4) Magnetic well is negative in all the cases, and relatively high. It is an important weakness.

* 5)  The Tau maximum is at Pitch 1 = 0,628  ,  Pitch 2 = 3.0   ;  Tau =  298 microsec. Configuration 6330

   This value and any of the others are not specially promising because all passing particles live the maximum of 500 microsec (for this simulation) while trapped particles live an average of ~170microsec  and should live near 500 microsec or more.

* 6)  Pitch 2 should be non linear to have a more relevant results from the simulation.

 * 7)  Plasma size is only the dimension on x+ and it has low relevance in CNT  (plasma shape is flat at high angles).

* 8) A correlation between % Trapped and Tau seems to appear but must be contrasted with simulations with more particles. The minimum of trapped particles is at the maximum of Tau.

*  9)  Averege ripple decreases at higher Tilt Angles and reaches 0,286 . The value is almost 3 times higher than in UST_1, formed by 12 modular coils.

* 10) Collisionless transport is excesive in every configuration if radial electric field effect is not included,

Finer intervals

     Finner intervals of optimization :  Parameter 1 (Pitch 1) = [32º , 40º]  [0,5585 rad , 0,6806 rad] 8.0 points ; Parameter 2 = [2.0 , 4.5]  5 points ; 
 

   54 different structures analysed. It lasted  8  PC-hours. 13 structures passed the threshold of 280 microsec of particle confinement time so they are simulated in detail.

Table 4

Only the values accurately calculated are shown. Ordered by Tau

Conclusions and comments

* 1)  The results are similar to the ones from the coarse intervals. No important improvement is observed.

* 2)  In many cases Pitch 2  = 2.0 -  2.5 gives maximum  Tau. 

* 3)  The Tau maximum is at Pitch 1 = 0,663  ,  Pitch 2 = 2.0   ;  Tau =  325 microsec. Configuration 6620

 
*  4)  Averege ripple is specially low. It is expected in some degree (ripple - trapped particles - collisionless confinement)

* 5)  Perhaps the reasoning in [6] for W7-X is applicable to this simulation, "For neoclassical ion transport, configurational details are less important than the presence of a radial electric field, which.... reduces the diffusion coefficient well below the plateau regime. Therefore, electron behaviour is of great importance since it is less affected by the electric field (electrons largely being in the 1/nu regime)".

   So it seems that the simulation for collisionless H+ is not relevant if electric fields are not included.

*  6)  The calculation of the regime for H+ and electrons has not been done in CNT. It was calculated for UST_1 and iIt is necessary also for CNT.

    Possible future research

   Contrast results from SimPIMF with results from other well-known codes and analytical approximations. Simulate collsionless electrons. In a medium term, try to include electric fields in the simulations. However, other priorities, like the development of ECRH heating for UST_1, limits near future improvement of this optimization.

 References

[1] " Simulations  and  recalculations in CNT " ,  Vicente M. Queral . See  “Past R&D" in this web

[2] "The Columbia Nonneutral Torus:  a new experiment to confine
 nonneutral and positron-electron  plasmas in a stellarator"  , Thomas Sunn Pedersen, Allen H. Boozer, Jason Paul Kremer et al.

[3]  "Optimization of Iota, Iota profile, plasma size, Bmin and magnetic well" , Vicente M. Queral . See  “Past R&D" in this web

[4] "Optimization of modular coils by means of SimPIMF v2.1"  Vicente M. Queral . See  “Past R&D" in this web

[5] "Monte Carlo simulation of neoclassical transport for the TJ-II stellarator", Victor Tribaldos   Physics PLasma  April  2001

[6]  "Physics and engineering design for wendelstein 7-X" Craig Beidler, G. Grieger , et al. Fusion Tech. 1989

Date of publication 24-12-2006