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Quasi-helically symmetric toroidal stellarators
Volume 129, number
PHYSICS
2
QUASI-HELICALLY J. NUHRENBERG Max-Planck-Institut
SYMMETRIC
LETTERS
A
9 May 1988
TOROIDAL STELLARATORS
and R. ZILLE
ftir Plasmaphysik,
IPP-EVRATOMAssociation,
Received 30 December 1987; revised manuscript Communicated by R.C. Davidson
received
D-8046 Garching near Miinchen. FRG
19 February
1988; accepted
for publication
9 March 1988
It is computationally shown that there are toroidal stellarators whose magnetic field strength is helically symmetric coordinates. Accordingly, these stellarators, without collisions, strictly confine guiding centre orbits.
The Helias class of stellarators [ 1 ] exhibits a magnetic field strength B whose variation on magnetic surfaces is dominated by a particular Fourier component in magnetic coordinates [ 2,3 ] for which the coordinate jacobian is proportional to l/B’. This Fourier component is B,,_ ,, where the indices m, n indicate Fourier components in these coordinates, s being the flux label and 8, @being the poloidal (index m) and toroidal (index n) variables, respectively. This led us to speculate that toroidal magnetic geometries can be found in which B(s, 8, $) = B( s, 8- 9). If this condition is satisfied, collisionless guiding centre orbits are strictly confined because their motion is governed by B alone, irrespective of the three-dimensional geometry. For convenience the associated constant of motion is given explicitly in the appendix. The conjecture that B(s, 8, @) =B(s, 8- 9) can be achieved was computationally verified by the following procedure. The properties of a stellarator configuration are completely determined by prescribing the shape of an outermost magnetic surface. Parametrizing this shape with a relatively small number of parameters (linked to important physical variables of a stellarator, e.g. rotational transform, magnetic well, parallel current density at finite B), and studying the dependence of physical properties on these parameters yielded the Helias class of stellarators. Optimization of the boundary shape with B=B(s, tY-@) as optimization goal led to the conjectured result. In the following we describe a specific example. 0375-9601/88/S ( North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
Let a Helias boundary surface shape: R=Ro+Ro,, -A,,_,
in magnetic
be given by the following
cos V+R,,2cos2V+(1-Al,o)cosU cos( U- V)-A,,_,
cos( U-2V)
+A,,, cos( u+ V) +A,,2 cos( U+2V) +Az,o cos 2U+A2._,
cos(2U-
V)
+A2,_2 cos(2U-2V), Z=Z,,, +AI,_l
sin V+Ro,* sin 2V+ (1 +A,,o) sin U sin( U- V)+A1,_2 sin( U-2V)
+A,,, sin( U+ V)+A,,* +A2,0 sin 2U+A2,_,
sin( U+2V)
sin(2U-
V)
-A2,_-2 sin(2U-2V), where riods) loidal radius
R, Z, and V/N (with N the number of peare cylindrical coordinates and U is the poparametrization with period 2x. The plasma is then approximately unity; the aspect ratio is given by Bo; BO,i, &,2, Z,,, are three parameters describing the shape of the plasma column as approximated by a closed spatial curve; A, .o, A,, _ , , A,. _ 2 are three ellipticity parameters corresponding to I= 2 fields (axisymmetric, the most slowly turning, and the next faster turning); A,.,, Ai,2 are the two lowest bumpiness parameters corresponding to mirror fields; A2,oand A2._, are the two lowest indentation parameters corresponding to central conductor fields providing indentation; A2,_2 is a triangularity parameter B.V.
113
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129, number
2
PHYSICS
Fig. 1. Flux surface cross-sections of a quasi-helically symmetric d,,,=O.O58, d,,_,=OS, 41,_2=0.046, 4,,,=0.023, 4,,2=-0.003,
LETTERS
A
9 May 1988
stellarator with N=6, d,,,=O.O68, &,=0.248,
R,,=II,Ro,, =0.553, &=0.038;
R0,z=0.027, Z,., =0.395. p=O; the magnetic well is ap-
proximately marginal; the rotational transform and shear are characterized by r(0) 5 1.4, I( 1) z 1.5.
corresponding to an I=3 field. For a fixed number of periods N, fixed aspect ratio Ro, and fixed A,,_, (providing the dominant contribution to the rotational transform for the configurations considered here) the above shape describes a ten-parameter family of stellarators which is only slightly more general than the original Helias family [ 11. Fig. 1 shows the result of an optimization towards B(s, 6- 9) in which the ten variables above were optimized without further constraints for an N=6 period configuration. Fig. 2 and tables, 1, 2 show the structure of B in terms of its Fourier components. Table 1 shows the values at sx f , corresponding to half the plasma radius, table 2 the corresponding results for sz 1 where the optimization B( 1, 8, @) = B( 1, (3-e) was done. Among the dominating Fourier coefficients violating the symmetry are Bz, _ 3 and B3,_*, which are already outside the range of boundary parameters used for the optimization. Approximately one order of magnitude is achieved between the largest remaining Fourier coeffkient and Table 1 Fourier coefftcients n
II,,,,, m 2 0, - 4 Q n d 4 at s = f of the conftguration
in fig. 1
m
-4 -3 -2 -1 0 2 3 4
114
0
1
0.000 0.000 - 0.002 0.002 1.258 0.000 0.000 0.000 0.000
0.000 0.000 0.000 - 0.090 0.001 0.003 0.002 0.000 0.000
2
3 0.000 0.000
0.003 0.002 -0.001 0.001 0.001 0.000 0.000
0.000
-0.001 -0.001 0.000 0.000 0.000 0.000 0.000 0.000
4
5
0.001 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
Volume
129. number
PHYSICS
2
LETTERS
9 May 1988
A
Table 2
Fourier coeffkients B,,,, ma0, -4
1.
m
n
0
-4 -3 -2 -1 0
1
0.001 -0.001 0.001 0.006 1.270 0.000 0.000 0.000 0.000
2 3 4
0.000 0.001
0.006 -0.171 0.002 0.004 0.002 -0.001 0.001
2
3
0.002 - 0.007 0.004 0.008 -0.001 0.001 0.000 0.000 0.000
0.002 - 0.003 -0.014 0.001 0.003 0.001 0.001 0.000 0.000
the dominant B,,_, component. This is to be contrasted with the well-known result pertaining to 1=2 stellarators with circular magnetic axis for which B ,,o m R, ’ , a coefficient which virtually vanishes here. We now discuss some consequences of the existence of quasi-helically symmetric configurations. The size of the parallel current density in a finiteB equilibrium of this type is very small, although it is not explicitly minimized. A convenient measure is given by (ji /j: ), which is 5 4 in the above configuration. Optimization towards j,, = 0 will yield isodynamic equilibria [ 41 if they exist. It turns out that the latter type of optimization is only well defined if the geometry parameters are bounded, which suggests that toroidally closed isodynamic equilibria do not exist at this value of aspect ratio (A z 11). Fig. 3 shows a configuration with (ji /j$ ) S 10-l in which the mirror ratio and the elongation of the surfaces were limited. Small parallel current density leads to a large limit for the equilibrium pvalue. Fig.
4
5 0.000
- 0.005
0.007 -0.003 - 0.002 0.001 0.000 0.000 0.000 0.000
0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000
4 shows magnetic surfaces at (/3) = 0.5 for the quasihelically symmetric stellarator. Avoidance of low-order rational values of rotational transform per period z, is another prerequisite for acceptable equilibria in moderate-shear configurations because a finite pressure gradient drives a diverging parallel current density at rational values of I, so that a necessary condition is flattening of a pressure profile at resonances [ 1,5 1. The divergence of the parallel current density for z,= n/m at &s, m, n results from B,,_, (sres) # 0. Exact quasi-helically symmetric equilibria are thus free of this feature of 3D toroidal equilibria. The computational optimization described above renders B,,,small for m # - n. In addition, in the specific case considered here the unconstrained optimization towards quasi-helical symmetry leads to 1.41 -CI c 1.5, i.e. $ < rP< {, so that resonances up to order 10 do not occur. Furthermore, the reaction of I to /3, with the condition of vanishing net toroidal current imposed, is very small because of the small parallel current density [ 1 I.
surface cross-sections of a Helias stellarator optimized towards min(j: /j: >. N=6, &= 11, &,I =0.435, &,1=0.041, lo, A,,_~=0.5, 4,._2=o.13, 4,,,=o.022,4,.2= -0.003, ~$,=0.036, ~k-~=O.273, kZ=0.045; 8=0; the magnetic hill is characterized by AL”/ V’= 0:02; the rotational transform and shear are characterized by I(0) F=1.3 1,I(1)x 1.36. Fig. 3.
HUX
z,,, = -0.083, &,=o.l
115
Volume 129, number
2
PHYSICS
Fig. 4. Magnetic
LETTERS
A
surfaces at (8) =0.5 of the configuration
A magnetic well at /3=0 is helpful for MHD stability at finite /3. Carrying out the optimization towards quasi-helical symmetry in general leads to configurations without such a well. In the above example the number of periods (N= 6) and A,,_, were chosen such as to produce an approximately marginal magnetic well. Optimization in the vicinity of this configuration with a magnetic well as an additional goal allows a significant magnetic well of approximately 2% to be achieved without significant loss of the approximate quasi-helical symmetry. In keeping with the equilibrium features described above, high &values can be achieved which are stable with respect to resistive interchange modes. The limit occurs at (j3) = 0.1; see fig. 5. It may be seen from the computational realization shown in table 1 that the ripple perturbing the quasihelical symmetry is small, 650.005 at aspect ratio 20. With respect to particle and energy confinement there are two important consequences. Usually the ripple in toroidal stellarators is approximately one order of magnitude larger and leads to an electron
9 May 1988
shown in fig. 1.
heat conduction which is uncomfortably large in the ripple regime and determines the requirements for the circulating power. Alpha particle containment is usually poor in stellarators; a fraction of alpha particles corresponding to $ is usually promptly lost. Both these features are completely changed in quasihelically symmetric configurations. Experimental devices can of course only approximate quasi-helical symmetry. From this point of view the existence of exact quasi-helically symmetric equilibria is of less importance than the demonstration of configurations which approximate this quasi-symmetry sufficiently well. It is also important that a quasihelically symmetric vacuum field configuration retain this property at finite /3. For the configuration shown in fig. 1 we have verified that the changes of the B,,,,p perturbing the quasi-helical symmetry are N 1O-3 in the range 0~ (j3) ~0.05. Summarizing, we think that the physical properties. of quasi-helically symmetric toroidal stellarators, which we have demonstrated to be essentially a sub-class of Helias configurations, encourage fur-
Fig. 5. Flux surface cross-sections of a configuration which is stable with respect to resistive interchange modes and close to that of fig. 1; N=6, I?,= 11, J&,=0.600, R,,2=0.030, Z,,,=O.351, A,.,=O.O99, d,,_,=O.44, 4,,_2=0.053, A,,,=0.022, A,.>= -0.004, dz,,=O.O87. A,,_,=O.295, A2._2=0.063; (p) =0.095; the vacuum magnetic well is given by AV'/V= -0.02;at (p) =0.095, AF’/F’z -0.2 for a bell-shaped pressure profile; at this value of,9 the rotational transform and shear are characterized by I(O) = 1.21, r( 1) = 1.32.
116
PHYSICS LETTERS A
Volume 129. number 2
ther research in this direction. Optimizations (for configurations with a smaller number of periods, N= 4, 5 ) which included such additional goals as a magnetic well and a prescribed range of rotational transform have already been considered [ 61. We are also investigating quasi-axially symmetric toroidal stellarators and will compare them with the class of stellarators considered here in future work.
9 May 1988
Quasi-helical symmetry B = B (s, 8- 9) makes vIl,pII, and j? [ 2,6] also quasi-helically symmetric. Thus, one finds [P,, (z+J) I’ = u+J)PI,,dM) = (I+J)P,,,e -
+ [P,,cz+4 l,si
l -epfl(F;-Fb)
Jg Dj m
,
so that the invariant is given by & -FP -PII (I+J)
Appendix
Here, we will show the constant of motion associated with quasi-helical symmetry (as well as helical symmetry) on the assumption of quasi-helical symmetry. Starting from a form of the guiding centre drift equations which is suitable for the derivation of constants of motion [ 7 1,
.
The work presented here only became possible with the advent of 3D MHD equilibrium codes exploiting the assumption of nested toroidal surfaces. We are thus indebted to F. Bauer, 0. Betancourt, and P. Garabedian as well as S.P. Hirshman for providing us with such codes.
References &Y=
0, m
[B+VX
(p,,B)] ,
Dj= 1+p,, (B-j/B’), and using a representation of B valid in magnetic coordinates [ 2,6],
p,, =mv,,/eB,
B=-
$r.,-
one finds
$~,,=IV@+JW+~VS,
[ 11 J. Ntihrenberg and R. Zille, Phys. Lett. A 114 ( 1986) 129. [2 ] A. Boozer, Phys. Fluids 24 ( 1981) 1999. [ 31 J. Niihrenberg and R. Zille, Proc. 12th Eur. Conf. on Contr. fusion and plasma physics, Budapest, 1985, Vol. 9F (EPS, Budapest, 1985) I, 445. [4] D. Palumbo, Nuovo Cimento XB 53 ( 1968) 507; D. Lortz and J. Ntihrenberg, Proc. Sherwood Meeting, Austin, Texas ( 198 1) 3B48. [ 5 ] S.P. Hirshman, J. Ntihrenberg and R. Zille, 8th Europhysics Conf. on Computational physics, Eibsee, 1986, ECA, Vol. 10D (1986) p. 157. [ 61 J. Ntihrenberg and R. Zille, Proc. of Invited Papers of the Workshop on Theory of fusion plasmas, Varenna, 1987, Nuovo Cimento ( 1987), to be published. [ 71 T.G. Northrop and J.A. Rome, Phys. Fluids 21 (1978) 384; R.G. Littlejohn, Phys. Fluids 24 ( 198 1) 1730; A. Boozer, Phys. Fluids 25 ( 1982) 575.
117
PHYSICS
2
QUASI-HELICALLY J. NUHRENBERG Max-Planck-Institut
SYMMETRIC
LETTERS
A
9 May 1988
TOROIDAL STELLARATORS
and R. ZILLE
ftir Plasmaphysik,
IPP-EVRATOMAssociation,
Received 30 December 1987; revised manuscript Communicated by R.C. Davidson
received
D-8046 Garching near Miinchen. FRG
19 February
1988; accepted
for publication
9 March 1988
It is computationally shown that there are toroidal stellarators whose magnetic field strength is helically symmetric coordinates. Accordingly, these stellarators, without collisions, strictly confine guiding centre orbits.
The Helias class of stellarators [ 1 ] exhibits a magnetic field strength B whose variation on magnetic surfaces is dominated by a particular Fourier component in magnetic coordinates [ 2,3 ] for which the coordinate jacobian is proportional to l/B’. This Fourier component is B,,_ ,, where the indices m, n indicate Fourier components in these coordinates, s being the flux label and 8, @being the poloidal (index m) and toroidal (index n) variables, respectively. This led us to speculate that toroidal magnetic geometries can be found in which B(s, 8, $) = B( s, 8- 9). If this condition is satisfied, collisionless guiding centre orbits are strictly confined because their motion is governed by B alone, irrespective of the three-dimensional geometry. For convenience the associated constant of motion is given explicitly in the appendix. The conjecture that B(s, 8, @) =B(s, 8- 9) can be achieved was computationally verified by the following procedure. The properties of a stellarator configuration are completely determined by prescribing the shape of an outermost magnetic surface. Parametrizing this shape with a relatively small number of parameters (linked to important physical variables of a stellarator, e.g. rotational transform, magnetic well, parallel current density at finite B), and studying the dependence of physical properties on these parameters yielded the Helias class of stellarators. Optimization of the boundary shape with B=B(s, tY-@) as optimization goal led to the conjectured result. In the following we describe a specific example. 0375-9601/88/S ( North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
Let a Helias boundary surface shape: R=Ro+Ro,, -A,,_,
in magnetic
be given by the following
cos V+R,,2cos2V+(1-Al,o)cosU cos( U- V)-A,,_,
cos( U-2V)
+A,,, cos( u+ V) +A,,2 cos( U+2V) +Az,o cos 2U+A2._,
cos(2U-
V)
+A2,_2 cos(2U-2V), Z=Z,,, +AI,_l
sin V+Ro,* sin 2V+ (1 +A,,o) sin U sin( U- V)+A1,_2 sin( U-2V)
+A,,, sin( U+ V)+A,,* +A2,0 sin 2U+A2,_,
sin( U+2V)
sin(2U-
V)
-A2,_-2 sin(2U-2V), where riods) loidal radius
R, Z, and V/N (with N the number of peare cylindrical coordinates and U is the poparametrization with period 2x. The plasma is then approximately unity; the aspect ratio is given by Bo; BO,i, &,2, Z,,, are three parameters describing the shape of the plasma column as approximated by a closed spatial curve; A, .o, A,, _ , , A,. _ 2 are three ellipticity parameters corresponding to I= 2 fields (axisymmetric, the most slowly turning, and the next faster turning); A,.,, Ai,2 are the two lowest bumpiness parameters corresponding to mirror fields; A2,oand A2._, are the two lowest indentation parameters corresponding to central conductor fields providing indentation; A2,_2 is a triangularity parameter B.V.
113
Volume
129, number
2
PHYSICS
Fig. 1. Flux surface cross-sections of a quasi-helically symmetric d,,,=O.O58, d,,_,=OS, 41,_2=0.046, 4,,,=0.023, 4,,2=-0.003,
LETTERS
A
9 May 1988
stellarator with N=6, d,,,=O.O68, &,=0.248,
R,,=II,Ro,, =0.553, &=0.038;
R0,z=0.027, Z,., =0.395. p=O; the magnetic well is ap-
proximately marginal; the rotational transform and shear are characterized by r(0) 5 1.4, I( 1) z 1.5.
corresponding to an I=3 field. For a fixed number of periods N, fixed aspect ratio Ro, and fixed A,,_, (providing the dominant contribution to the rotational transform for the configurations considered here) the above shape describes a ten-parameter family of stellarators which is only slightly more general than the original Helias family [ 11. Fig. 1 shows the result of an optimization towards B(s, 6- 9) in which the ten variables above were optimized without further constraints for an N=6 period configuration. Fig. 2 and tables, 1, 2 show the structure of B in terms of its Fourier components. Table 1 shows the values at sx f , corresponding to half the plasma radius, table 2 the corresponding results for sz 1 where the optimization B( 1, 8, @) = B( 1, (3-e) was done. Among the dominating Fourier coefficients violating the symmetry are Bz, _ 3 and B3,_*, which are already outside the range of boundary parameters used for the optimization. Approximately one order of magnitude is achieved between the largest remaining Fourier coeffkient and Table 1 Fourier coefftcients n
II,,,,, m 2 0, - 4 Q n d 4 at s = f of the conftguration
in fig. 1
m
-4 -3 -2 -1 0 2 3 4
114
0
1
0.000 0.000 - 0.002 0.002 1.258 0.000 0.000 0.000 0.000
0.000 0.000 0.000 - 0.090 0.001 0.003 0.002 0.000 0.000
2
3 0.000 0.000
0.003 0.002 -0.001 0.001 0.001 0.000 0.000
0.000
-0.001 -0.001 0.000 0.000 0.000 0.000 0.000 0.000
4
5
0.001 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
Volume
129. number
PHYSICS
2
LETTERS
9 May 1988
A
Table 2
Fourier coeffkients B,,,, ma0, -4
1.
m
n
0
-4 -3 -2 -1 0
1
0.001 -0.001 0.001 0.006 1.270 0.000 0.000 0.000 0.000
2 3 4
0.000 0.001
0.006 -0.171 0.002 0.004 0.002 -0.001 0.001
2
3
0.002 - 0.007 0.004 0.008 -0.001 0.001 0.000 0.000 0.000
0.002 - 0.003 -0.014 0.001 0.003 0.001 0.001 0.000 0.000
the dominant B,,_, component. This is to be contrasted with the well-known result pertaining to 1=2 stellarators with circular magnetic axis for which B ,,o m R, ’ , a coefficient which virtually vanishes here. We now discuss some consequences of the existence of quasi-helically symmetric configurations. The size of the parallel current density in a finiteB equilibrium of this type is very small, although it is not explicitly minimized. A convenient measure is given by (ji /j: ), which is 5 4 in the above configuration. Optimization towards j,, = 0 will yield isodynamic equilibria [ 41 if they exist. It turns out that the latter type of optimization is only well defined if the geometry parameters are bounded, which suggests that toroidally closed isodynamic equilibria do not exist at this value of aspect ratio (A z 11). Fig. 3 shows a configuration with (ji /j$ ) S 10-l in which the mirror ratio and the elongation of the surfaces were limited. Small parallel current density leads to a large limit for the equilibrium pvalue. Fig.
4
5 0.000
- 0.005
0.007 -0.003 - 0.002 0.001 0.000 0.000 0.000 0.000
0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000
4 shows magnetic surfaces at (/3) = 0.5 for the quasihelically symmetric stellarator. Avoidance of low-order rational values of rotational transform per period z, is another prerequisite for acceptable equilibria in moderate-shear configurations because a finite pressure gradient drives a diverging parallel current density at rational values of I, so that a necessary condition is flattening of a pressure profile at resonances [ 1,5 1. The divergence of the parallel current density for z,= n/m at &s, m, n results from B,,_, (sres) # 0. Exact quasi-helically symmetric equilibria are thus free of this feature of 3D toroidal equilibria. The computational optimization described above renders B,,,small for m # - n. In addition, in the specific case considered here the unconstrained optimization towards quasi-helical symmetry leads to 1.41 -CI c 1.5, i.e. $ < rP< {, so that resonances up to order 10 do not occur. Furthermore, the reaction of I to /3, with the condition of vanishing net toroidal current imposed, is very small because of the small parallel current density [ 1 I.
surface cross-sections of a Helias stellarator optimized towards min(j: /j: >. N=6, &= 11, &,I =0.435, &,1=0.041, lo, A,,_~=0.5, 4,._2=o.13, 4,,,=o.022,4,.2= -0.003, ~$,=0.036, ~k-~=O.273, kZ=0.045; 8=0; the magnetic hill is characterized by AL”/ V’= 0:02; the rotational transform and shear are characterized by I(0) F=1.3 1,I(1)x 1.36. Fig. 3.
HUX
z,,, = -0.083, &,=o.l
115
Volume 129, number
2
PHYSICS
Fig. 4. Magnetic
LETTERS
A
surfaces at (8) =0.5 of the configuration
A magnetic well at /3=0 is helpful for MHD stability at finite /3. Carrying out the optimization towards quasi-helical symmetry in general leads to configurations without such a well. In the above example the number of periods (N= 6) and A,,_, were chosen such as to produce an approximately marginal magnetic well. Optimization in the vicinity of this configuration with a magnetic well as an additional goal allows a significant magnetic well of approximately 2% to be achieved without significant loss of the approximate quasi-helical symmetry. In keeping with the equilibrium features described above, high &values can be achieved which are stable with respect to resistive interchange modes. The limit occurs at (j3) = 0.1; see fig. 5. It may be seen from the computational realization shown in table 1 that the ripple perturbing the quasihelical symmetry is small, 650.005 at aspect ratio 20. With respect to particle and energy confinement there are two important consequences. Usually the ripple in toroidal stellarators is approximately one order of magnitude larger and leads to an electron
9 May 1988
shown in fig. 1.
heat conduction which is uncomfortably large in the ripple regime and determines the requirements for the circulating power. Alpha particle containment is usually poor in stellarators; a fraction of alpha particles corresponding to $ is usually promptly lost. Both these features are completely changed in quasihelically symmetric configurations. Experimental devices can of course only approximate quasi-helical symmetry. From this point of view the existence of exact quasi-helically symmetric equilibria is of less importance than the demonstration of configurations which approximate this quasi-symmetry sufficiently well. It is also important that a quasihelically symmetric vacuum field configuration retain this property at finite /3. For the configuration shown in fig. 1 we have verified that the changes of the B,,,,p perturbing the quasi-helical symmetry are N 1O-3 in the range 0~ (j3) ~0.05. Summarizing, we think that the physical properties. of quasi-helically symmetric toroidal stellarators, which we have demonstrated to be essentially a sub-class of Helias configurations, encourage fur-
Fig. 5. Flux surface cross-sections of a configuration which is stable with respect to resistive interchange modes and close to that of fig. 1; N=6, I?,= 11, J&,=0.600, R,,2=0.030, Z,,,=O.351, A,.,=O.O99, d,,_,=O.44, 4,,_2=0.053, A,,,=0.022, A,.>= -0.004, dz,,=O.O87. A,,_,=O.295, A2._2=0.063; (p) =0.095; the vacuum magnetic well is given by AV'/V= -0.02;at (p) =0.095, AF’/F’z -0.2 for a bell-shaped pressure profile; at this value of,9 the rotational transform and shear are characterized by I(O) = 1.21, r( 1) = 1.32.
116
PHYSICS LETTERS A
Volume 129. number 2
ther research in this direction. Optimizations (for configurations with a smaller number of periods, N= 4, 5 ) which included such additional goals as a magnetic well and a prescribed range of rotational transform have already been considered [ 61. We are also investigating quasi-axially symmetric toroidal stellarators and will compare them with the class of stellarators considered here in future work.
9 May 1988
Quasi-helical symmetry B = B (s, 8- 9) makes vIl,pII, and j? [ 2,6] also quasi-helically symmetric. Thus, one finds [P,, (z+J) I’ = u+J)PI,,dM) = (I+J)P,,,e -
+ [P,,cz+4 l,si
l -epfl(F;-Fb)
Jg Dj m
,
so that the invariant is given by & -FP -PII (I+J)
Appendix
Here, we will show the constant of motion associated with quasi-helical symmetry (as well as helical symmetry) on the assumption of quasi-helical symmetry. Starting from a form of the guiding centre drift equations which is suitable for the derivation of constants of motion [ 7 1,
.
The work presented here only became possible with the advent of 3D MHD equilibrium codes exploiting the assumption of nested toroidal surfaces. We are thus indebted to F. Bauer, 0. Betancourt, and P. Garabedian as well as S.P. Hirshman for providing us with such codes.
References &Y=
0, m
[B+VX
(p,,B)] ,
Dj= 1+p,, (B-j/B’), and using a representation of B valid in magnetic coordinates [ 2,6],
p,, =mv,,/eB,
B=-
$r.,-
one finds
$~,,=IV@+JW+~VS,
[ 11 J. Ntihrenberg and R. Zille, Phys. Lett. A 114 ( 1986) 129. [2 ] A. Boozer, Phys. Fluids 24 ( 1981) 1999. [ 31 J. Niihrenberg and R. Zille, Proc. 12th Eur. Conf. on Contr. fusion and plasma physics, Budapest, 1985, Vol. 9F (EPS, Budapest, 1985) I, 445. [4] D. Palumbo, Nuovo Cimento XB 53 ( 1968) 507; D. Lortz and J. Ntihrenberg, Proc. Sherwood Meeting, Austin, Texas ( 198 1) 3B48. [ 5 ] S.P. Hirshman, J. Ntihrenberg and R. Zille, 8th Europhysics Conf. on Computational physics, Eibsee, 1986, ECA, Vol. 10D (1986) p. 157. [ 61 J. Ntihrenberg and R. Zille, Proc. of Invited Papers of the Workshop on Theory of fusion plasmas, Varenna, 1987, Nuovo Cimento ( 1987), to be published. [ 71 T.G. Northrop and J.A. Rome, Phys. Fluids 21 (1978) 384; R.G. Littlejohn, Phys. Fluids 24 ( 198 1) 1730; A. Boozer, Phys. Fluids 25 ( 1982) 575.
117