The Helena code solves the stationary equilibrium equations
(the Grad-Shafranov equation) for a general Tokamak geometry.
The code makes use of two dimensional cubic Hermite finite elements.
This allows a continuous representation of both the magnetic
flux and the magnetic field. The elements are isoparametric,
i.e. the poloidal plane is discretised using the same cubic
finite elements as used for the representation of the poloidal
flux.
IAS | 0/1 | up-down symmetric/asymmetric shape |
ISHAPE |
1 | shape given by ellip, tria, quad |
2 | shape given by fourier series fm(i) |
ELLIP
x = cos(t + tria sin(t) + quad sin(2t) )
y = ellip sin(t)
| 1. | ellipticity (b/a) |
TRIA | 0. | triangularity |
QUAD | 0. | quadrangularity |
FM(1:MFM)
for ias=1, ishape= 2 : fm(2*m+1)=cos(m);
fm(2*m+2)=sin(m);
for ias=0, ishape= 2 : fm(m)=cos(m-1);
for ias=0, ishape=-2 : fm(2*m+1)=cos(m);
fm(2*m+2) not used;
radius is normalised to 1 at theta=0, pi
| | fourier series of radius(theta) (b/a) for plasma boundary (for ISHAPE=±2) |
MFM | | number of harmonics in FM |
MHARM | 128 | number of harmonics for grid generation (MFM/2) |
IMESH | 0 | equidistant radial grid |
2 | Gaussian enrichment of radial grid |
XR1 | 99. | position of radial grid accumulation (for IMESH=2) |
SIG1 | 99. | width of radial grid accumulation (for IMESH=2) |
IARC | 0 | equidistant poloidal grid |
1 | equal arclength poloidal grid |
NR | 51 |
number of radial grid points |
NP | 33 |
number of poloidal grid points |
NRMAP | 101 |
number of radial grid points in mapping to flux surface grid (should be 2*NR) |
NPMAP | 65 |
number of poloidal grid points in mapping (should be 2*NP) |
NCHI
for IAS=0 : NCHI = 2n + 1
for IAS=1 : NCHI = 2n
| 129 |
number of poloidal grid points written to output file (should be 2*NMAP) |
NRCUR | 51 |
number of radial grid points for iteration current profile (should be NR) |
NPCUR | 33 |
number of poloidal grid points (should be NP) |
NITER | 50 (ICUR≠0) or 1 (ICUR=0) |
maximum number of iterations over current profile |
AMIX
the iteration of the GS equation is :
psin+1= amix psin-1+ (1-amix)psin
amix =0.5 or larger may be usefull
for 'extreme' equilibria
| 0 |
parameter for control of main iteration of GS equation |
ERRCUR | 10-5 |
error allowed at end of iteration current profile |
ABB | 1. |
parameter for control of current profile iteration |
BBB | 1 |
parameter for control of current profile iteration |
NBB | 4 |
FF' profile updated after every NBB iterations |
NMESH | 1 |
maximum number of iterations of ? |
&SHAPE IAS=0, ISHAPE=1, ELLIP=1.5, TRIA=0.2, QUAD=0., &END
&PROFILE IPAI =1, API=-1., BPI=0.,
IGAM =1, AGA=-1., BGA=0.,
ICUR =0,
&END
&PHYS EPS=0.3, B=0.1 &END
&NUM NR=21, NP=9,
NRMAP=21, NPMAP=17, NCHI=33,
NITER=50,
&END
&PRI NPR1 = 1, NPR2=0 &END
&PLOT NPL1 =1 &END
&BALL NQB =1 &END
*****************************************
* HELENA : GRAD-SHAFRANOV EQUILIBRIUM *
*****************************************
* VERSION 12 (30-7-1999) *
*****************************************
$SHAPE ELLIP = 1.500, TRIA = 0.200, QUAD = 0.000
MHARM =128, ISHAPE = 1, ISOL = 0, IAS = 0
$PROFILE IGAM = 1 AGA = -1.000, BGA = 0.000 CGA = 0.000 DGA = 0.000,
EGA = 0.000, FGA = 0.000, GGA = 0.000, HGA = 0.000
IPAI = 1 API = -1.000, BPI = 0.000 CPI = 0.000 DPI = 0.000,
EPI = 0.000, FPI = 0.000, GPI = 0.000, HPI = 0.000
$PHYS EPS = 0.300, ALFA = 1.000, B = 0.100, C = 1.000,
XIAB = -1.000, BETAP = -1.0000
$NUM NR = 11, NP = 9, NRMAP = 11, NPMAP = 17, NCHI = 33, NITER = 50
$PRI NPR1 = 1
$PLOT NPL1 = 1
***********************************************
* INPUT PROFILES : *
***********************************************
* PSI, dP/dPSI, Gamma *
***********************************************
0.0000E+00 0.1000E+01 0.1000E+01
0.5000E-01 0.9500E+00 0.9500E+00
0.1000E+00 0.9000E+00 0.9000E+00
0.1500E+00 0.8500E+00 0.8500E+00
0.2000E+00 0.8000E+00 0.8000E+00
0.2500E+00 0.7500E+00 0.7500E+00
0.3000E+00 0.7000E+00 0.7000E+00
0.3500E+00 0.6500E+00 0.6500E+00
0.4000E+00 0.6000E+00 0.6000E+00
0.4500E+00 0.5500E+00 0.5500E+00
0.5000E+00 0.5000E+00 0.5000E+00
0.5500E+00 0.4500E+00 0.4500E+00
0.6000E+00 0.4000E+00 0.4000E+00
0.6500E+00 0.3500E+00 0.3500E+00
0.7000E+00 0.3000E+00 0.3000E+00
0.7500E+00 0.2500E+00 0.2500E+00
0.8000E+00 0.2000E+00 0.2000E+00
0.8500E+00 0.1500E+00 0.1500E+00
0.9000E+00 0.1000E+00 0.1000E+00
0.9500E+00 0.5000E-01 0.5000E-01
0.1000E+01 -0.8674E-18 -0.8674E-18
***************************************
* FINAL ITERATION, NR= 11 NP= 9 *
***************************************
REST = 8.543E-09 ITERATIONS : 15
***************************************
MAGNETIC AXIS : 0.05333
POLOIDAL BETA : 0.1506E+00
TOROIDAL BETA : 0.1650E-01
BETA STAR : 0.2435E-01
NORM. BETA : 0.00816
TOTAL CURRENT : 0.2540E+01
TOTAL AREA : 0.4687E+01
TOTAL VOLUME : 0.2901E+02
INT. INDUCTANCE : 0.9227E+00
POL. FLUX : 0.1000E+01
***************************************
*******************************************************
* I PSI S ERROR LENGTH BUSSAC *
*******************************************************
10 0.0100 0.1000 0.9904 9.90E-01 0.6280 0.1694
9 0.0400 0.2000 0.9615 9.62E-01 1.2613 0.1691
8 0.0900 0.3000 0.9133 9.13E-01 1.9059 0.1685
7 0.1600 0.4000 0.8456 8.46E-01 2.5684 0.1678
6 0.2500 0.5000 0.7580 7.58E-01 3.2573 0.1667
5 0.3600 0.6000 0.6504 6.50E-01 3.9842 0.1654
4 0.4900 0.7000 0.5219 5.22E-01 4.7661 0.1639
3 0.6400 0.8000 0.3719 3.72E-01 5.6314 0.1621
2 0.8100 0.9000 0.1989 1.99E-01 6.6364 0.1605
1 1.0000 1.0000 0.0000 -1.04E-17 7.9419 0.1613
*******************************************************
*************************************
* PRESSURE PROFILE BEFORE NORM. : *
*************************************
0.32052761E-01 0.31414911E-01 0.29539824E-01 0.26542891E-01
0.22616428E-01 0.18029678E-01 0.13128811E-01 0.83369231E-02
0.41540378E-02 0.11571047E-02 0.00000000E+00
*************************************
RADIUS : 0.2953E+00
B0 : 0.8841E+00
CPSURF : 0.7709E-01
Q ON AXIS = 0.5471
Q PROFILE :
5.4709E-01( 1) 5.4908E-01( 2) 5.5540E-01( 3) 5.6660E-01( 4)
5.8428E-01( 5) 6.1110E-01( 6) 6.5184E-01( 7) 7.1552E-01( 8)
8.2141E-01( 9) 1.0203E+00(10) 1.5281E+00(11)
***************************************************
MAX. ERROR IN JACOBIAN AFTER MAPPING : 1.874E-03 0.300000 1.767146 8 19
***************************************************
***************************************************
* I, X, PSI, P, Q *
***************************************************
1 -0.1000E+01 0.1000E+01 0.0000E+00 0.1528E+01
2 -0.8500E+00 0.8100E+00 0.1157E-02 0.1020E+01
3 -0.7206E+00 0.6400E+00 0.4154E-02 0.8214E+00
4 -0.6044E+00 0.4900E+00 0.8337E-02 0.7155E+00
5 -0.4972E+00 0.3600E+00 0.1313E-01 0.6518E+00
6 -0.3967E+00 0.2500E+00 0.1803E-01 0.6111E+00
7 -0.3011E+00 0.1600E+00 0.2262E-01 0.5843E+00
8 -0.2092E+00 0.9000E-01 0.2654E-01 0.5666E+00
9 -0.1199E+00 0.4000E-01 0.2954E-01 0.5554E+00
10 -0.3264E-01 0.1000E-01 0.3141E-01 0.5491E+00
11 0.5333E-01 0.0000E+00 0.3205E-01 0.5471E+00
12 0.1386E+00 0.1000E-01 0.3141E-01 0.5491E+00
13 0.2238E+00 0.4000E-01 0.2954E-01 0.5554E+00
14 0.3094E+00 0.9000E-01 0.2654E-01 0.5666E+00
15 0.3961E+00 0.1600E+00 0.2262E-01 0.5843E+00
16 0.4846E+00 0.2500E+00 0.1803E-01 0.6111E+00
17 0.5758E+00 0.3600E+00 0.1313E-01 0.6518E+00
18 0.6708E+00 0.4900E+00 0.8337E-02 0.7155E+00
19 0.7713E+00 0.6400E+00 0.4154E-02 0.8214E+00
20 0.8796E+00 0.8100E+00 0.1157E-02 0.1020E+01
21 0.1000E+01 0.1000E+01 0.0000E+00 0.1528E+01
***************************************************
*******************************************************************
* IDEAL AND RESISTIVE MERCIER CRITERION *
*******************************************************************
*I, SS, D_I, D_R, H, Q, SHEAR, GEONC *
*******************************************************************
1 1.000 -2.50E-01 0.00E+00 0.000 1.528 5.495 2.49E+01
2 0.900 -2.49E-01 -1.85E-03 0.003 1.020 2.419 9.91E+00
3 0.800 -2.31E-01 1.33E-02 0.006 0.821 1.327 7.04E+00
4 0.700 -1.80E-01 6.22E-02 0.008 0.716 0.774 6.30E+00
5 0.600 -6.07E-02 1.80E-01 0.010 0.652 0.457 6.55E+00
6 0.500 1.99E-01 4.37E-01 0.012 0.611 0.266 7.73E+00
7 0.400 7.57E-01 9.94E-01 0.014 0.584 0.149 1.04E+01
8 0.300 2.07E+00 2.30E+00 0.015 0.567 0.076 1.67E+01
9 0.200 6.11E+00 6.35E+00 0.017 0.555 0.031 3.49E+01
10 0.100 3.76E+01 3.78E+01 0.021 0.549 0.006 1.34E+02
*******************************************************************
*******************************
* BALLOONING STABILITY *
* PROGRAM HELBAL VERSION 1 *
*******************************
0.000000000000000E+000 50.0000000000000
11 33
AFTER SCALE: SCALEQ= 1.0000E+00
CPSURF = 7.7086E-02
QS
5.47087651E-01 5.49080627E-01 5.55399632E-01 5.66598322E-01 5.84276019E-01
6.11103901E-01 6.51840228E-01 7.15520350E-01 8.21408650E-01 1.02030021E+00
1.52806320E+00
P0
2.50556128E-02 2.45570061E-02 2.30912528E-02 2.07485530E-02 1.76792404E-02
1.40937822E-02 1.02627790E-02 6.51696490E-03 3.24720742E-03 9.04507623E-04
0.00000000E+00
RBPHI
1.00000000E+00 9.97581944E-01 9.90439402E-01 9.78915339E-01 9.63608423E-01
9.45413650E-01 9.25577670E-01 9.05762854E-01 8.88104996E-01 8.75234549E-01
8.70214384E-01
*********************************************************************
* I, FLUX, Q, SHEAR, ALPHA, FMARG, BALLOONING STAB. *
*********************************************************************
2 0.010 0.549 0.00647 0.01994 0.697 UNSTABLE AT T = 40.743
3 0.040 0.555 0.03143 0.03957 0.500 UNSTABLE AT T = 22.777
4 0.090 0.567 0.07458 0.05856 0.500 UNSTABLE AT T = 19.439
5 0.160 0.584 0.14757 0.07664 0.746 UNSTABLE AT T = 26.114
6 0.250 0.611 0.27169 0.09357 1.301 STABLE
7 0.360 0.652 0.44469 0.10902 1.984 STABLE
8 0.490 0.716 0.84916 0.12212 3.403 STABLE
9 0.640 0.821 1.10247 0.12983 4.463 STABLE
10 0.810 1.020 3.30565 0.11894 14.855 STABLE
11 1.000 1.528 3.32292 0.00000 2047.500 STABLE
*********************************************************************
*************************************************
REAL WORLD OUTPUT :
*************************************************
MAJOR RADIUS : 1.0000 [m]
MAGNETIC FIELD : 1.0000 [T]
CENTRAL DENSITY : 1.0000 10^19 [m^-3]
ZEFF : 1.0000
TE/(TE+TI) : 0.5000
*************************************************
**************************************************
S, P [Pa], Ne [10^19m^-3], Te [eV], Ti [eV],
**************************************************
1.000E-01 2.500E+04 9.900E-01 7.880E+03 7.880E+03 1.344E+05 1.256E+05 1.578E-01 3.771E+06
2.000E-01 2.351E+04 9.600E-01 7.642E+03 7.642E+03 1.858E+05 1.717E+05 2.158E-01 3.665E+06
3.000E-01 2.112E+04 9.100E-01 7.244E+03 7.244E+03 2.134E+05 1.973E+05 2.480E-01 3.488E+06
4.000E-01 1.800E+04 8.400E-01 6.686E+03 6.686E+03 2.232E+05 2.073E+05 2.605E-01 3.237E+06
5.000E-01 1.435E+04 7.500E-01 5.970E+03 5.970E+03 2.177E+05 2.033E+05 2.555E-01 2.909E+06
6.000E-01 1.045E+04 6.400E-01 5.094E+03 5.094E+03 1.982E+05 1.864E+05 2.343E-01 2.501E+06
7.000E-01 6.634E+03 5.100E-01 4.060E+03 4.060E+03 1.662E+05 1.574E+05 1.978E-01 2.010E+06
8.000E-01 3.306E+03 3.600E-01 2.866E+03 2.866E+03 1.224E+05 1.168E+05 1.468E-01 1.433E+06
9.000E-01 9.208E+02 1.900E-01 1.512E+03 1.512E+03 6.710E+04 6.462E+04 8.121E-02 7.663E+05
***********************************************************************
* S, Q, Fcirc, NU_e, NU_i, SIG(Spitz),SIG(neo) *
***********************************************************************
1.000E-01 5.491E-01 7.695E-01 2.180E-03 1.811E-03 7.224E+08 5.211E+08
2.000E-01 5.554E-01 6.737E-01 8.033E-04 6.668E-04 6.904E+08 4.253E+08
3.000E-01 5.666E-01 6.019E-01 4.699E-04 3.896E-04 6.381E+08 3.458E+08
4.000E-01 5.843E-01 5.425E-01 3.402E-04 2.816E-04 5.671E+08 2.742E+08
5.000E-01 6.111E-01 4.913E-01 2.842E-04 2.348E-04 4.799E+08 2.086E+08
6.000E-01 6.518E-01 4.457E-01 2.689E-04 2.214E-04 3.800E+08 1.491E+08
7.000E-01 7.155E-01 4.044E-01 2.917E-04 2.392E-04 2.720E+08 9.661E+07
8.000E-01 8.214E-01 3.663E-01 3.835E-04 3.122E-04 1.629E+08 5.244E+07
9.000E-01 1.020E+00 3.311E-01 7.382E-04 5.928E-04 6.359E+07 1.864E+07
*****************************************
* NeoClassical Stability Condition *
*****************************************
S = 0.100 q = 0.549 J.B= 0.169 D_I =-3.758E+01 D_nc = 1.170E+01 Mercier Unstable
S = 0.200 q = 0.555 J.B= 0.234 D_I =-6.113E+00 D_nc = 3.425E+00 Mercier Unstable
S = 0.300 q = 0.567 J.B= 0.268 D_I =-2.069E+00 D_nc = 1.692E+00 Mercier Unstable
S = 0.400 q = 0.584 J.B= 0.281 D_I =-7.573E-01 D_nc = 9.694E-01 Mercier Unstable
S = 0.500 q = 0.611 J.B= 0.274 D_I =-1.989E-01 D_nc = 5.849E-01 Mercier Unstable
S = 0.600 q = 0.652 J.B= 0.249 D_I = 6.066E-02 D_nc = 3.548E-01 D_nc+D_R/(a-H) = 5.986E-01
S = 0.700 q = 0.716 J.B= 0.209 D_I = 1.801E-01 D_nc = 2.095E-01 D_nc+D_R/(a-H) = 2.774E-01
S = 0.800 q = 0.821 J.B= 0.154 D_I = 2.312E-01 D_nc = 1.144E-01 D_nc+D_R/(a-H) = 1.281E-01
S = 0.900 q = 1.020 J.B= 0.084 D_I = 2.487E-01 D_nc = 4.936E-02 D_nc+D_R/(a-H) = 4.751E-02