Abstract
If quasiaxisymmetry is preserved, nonaxisymmetric shaping can be used to design tokamaks that do not require current drive, are resilient to disruptions, and have robust plasma stability without feedback. Suggestions for addressing the critical issues of tokamaks can only be validated when presented with sufficient specificity that validating experiments can be designed. The purpose of this paper is to provide that specificity for nonaxisymmetric shaping. Whether nonaxisymmetric shaping is essential, or just an alternative strategy, to the success of tokamakfusion systems can only be assessed after axisymmetric alternatives are suggested and subjected to a similar study. Sequences of threefieldperiod quasiaxisymmetric plasmas are studied. These sequences address the questions: (1) What can be achieved at various levels of nonaxisymmetric shaping? (2) What simplifications to the coils can be achieved by going to a larger aspect ratio? (3) What range of shaping can be achieved in a single experimental facility? The sequences of plasmas found in this study provide a set of interesting and potentially important configurations.
This work was supported by U.S. Department of Energy through the Grant No. ER54333 to Columbia University and the Contract No. DEAC0209CH11466 to Princeton Plasma Physics Laboratory.
I. INTRODUCTION
II. TOKAMAK SHAPING
III. IMPORTANCE OF SHAPING
IV. APPROACH AND METHODS
V. CONFIGURATIONS WITH INCREASING LEVELS OF ROTATIONAL TRANSFORM
VI. CONFIGURATIONS WITH THE INCREASING ASPECT RATIO
VII. DESIGN OF A MULTIFUNCTIONAL STELLARATOR
VIII. SUMMARY AND CONCLUSIONS
Key Topics
 Tokamaks
 31.0
 Toroidal plasma confinement
 23.0
 Stellarators
 18.0
 Magnetic fields
 14.0
 Current drive
 11.0
Figures
QA configurations with the rotational transform provided by the threedimensional shaping in (a) 0:05, (b) 0.10, (c) 0.20, and (d) 0.30, shown in four cross sections with equally spaced toroidal angles over half period. Configuration (a) is passively stable to the vertical mode, (b) removes the need for current drive at , (c) remains in vacuum chamber if plasma pressure and current vanish instantaneously, and (d) is passively stable to the wall mode.
QA configurations with the rotational transform provided by the threedimensional shaping in (a) 0:05, (b) 0.10, (c) 0.20, and (d) 0.30, shown in four cross sections with equally spaced toroidal angles over half period. Configuration (a) is passively stable to the vertical mode, (b) removes the need for current drive at , (c) remains in vacuum chamber if plasma pressure and current vanish instantaneously, and (d) is passively stable to the wall mode.
A modular coil set designed for the configuration shown in Fig. 1(a) with the current winding surface conformal to the boundary of the plasma and displaced outward by a distance equal to , where is the average minor radius of the plasma. The left frame shows the coils, six per field period, viewed from the top. The right frame shows the contours of current potential on the flattened winding surface in one field period with the abscissa being the toroidal angle, , and the ordinate the poloidal angle, . The toroidal angle starts at the crescentshaped cross section and the poloidal angle starts at the outboard midplane.
A modular coil set designed for the configuration shown in Fig. 1(a) with the current winding surface conformal to the boundary of the plasma and displaced outward by a distance equal to , where is the average minor radius of the plasma. The left frame shows the coils, six per field period, viewed from the top. The right frame shows the contours of current potential on the flattened winding surface in one field period with the abscissa being the toroidal angle, , and the ordinate the poloidal angle, . The toroidal angle starts at the crescentshaped cross section and the poloidal angle starts at the outboard midplane.
Rotational transform for the plasma configuration shown in Fig. 1(d) as function of the normalized toroidal flux. The dotted line is the external transform supplied by the shaping. The solid line is the total transform including the internal contribution from the plasmadriven bootstrap current at .
Rotational transform for the plasma configuration shown in Fig. 1(d) as function of the normalized toroidal flux. The dotted line is the external transform supplied by the shaping. The solid line is the total transform including the internal contribution from the plasmadriven bootstrap current at .
Contours of flux surfaces for the configuration shown in Fig. 1(d) at from a PIES calculation showing the configuration has good surface quality despite the existence of rational values in the iota profile.
Contours of flux surfaces for the configuration shown in Fig. 1(d) at from a PIES calculation showing the configuration has good surface quality despite the existence of rational values in the iota profile.
QA configurations with rotational transform provided by shaping in (a) 0.40, (b) 0.50, and (c) 0.60, shown in four cross sections equally spaced in toroidal angles over half period. The vacuum transform accounts for ≈70%, 80%, and 90% of the total transform at .
QA configurations with rotational transform provided by shaping in (a) 0.40, (b) 0.50, and (c) 0.60, shown in four cross sections equally spaced in toroidal angles over half period. The vacuum transform accounts for ≈70%, 80%, and 90% of the total transform at .
QA configurations with three periods, rotational transform from shaping ≈0.3 and are MHD stable to the external kink modes at . The aspect ratios are: (a) , (b) , (c) , (d) , and (e) .
QA configurations with three periods, rotational transform from shaping ≈0.3 and are MHD stable to the external kink modes at . The aspect ratios are: (a) , (b) , (c) , (d) , and (e) .
Contours of flux surfaces for the configuration shown in Fig. 6(e) with at from a VMEC calculation. The Shafranov shift of the magnetic axis is about 32% of the half width at the crescentshaped section.
Contours of flux surfaces for the configuration shown in Fig. 6(e) with at from a VMEC calculation. The Shafranov shift of the magnetic axis is about 32% of the half width at the crescentshaped section.
Top view of modular coils constructed for configurations (b) with , (d) with , and (e) with of Fig. 6. Coil winding surfaces have been constructed such that the inboard midplane is displaced by , whereas the outboard midplane is displaced by , where is the plasma minor radius, with interpolation made for locations in between. In all cases, there are three types of coils for each half period for a total of 18 coils.
Top view of modular coils constructed for configurations (b) with , (d) with , and (e) with of Fig. 6. Coil winding surfaces have been constructed such that the inboard midplane is displaced by , whereas the outboard midplane is displaced by , where is the plasma minor radius, with interpolation made for locations in between. In all cases, there are three types of coils for each half period for a total of 18 coils.
Left two frames: current carrying surface (dotted lines) relative to the plasma configuration shown in Fig. 1(d) in two cross sections. Right frame: the arrangement of the windowpane coils wound on the current carrying surface. Currents in the windowpane coils may be controlled to produce plasma configurations (a)–(d) of Fig. 1. The shading here indicates current levels for configuration (d) in linear scale, where darker shading corresponds to larger current.
Left two frames: current carrying surface (dotted lines) relative to the plasma configuration shown in Fig. 1(d) in two cross sections. Right frame: the arrangement of the windowpane coils wound on the current carrying surface. Currents in the windowpane coils may be controlled to produce plasma configurations (a)–(d) of Fig. 1. The shading here indicates current levels for configuration (d) in linear scale, where darker shading corresponds to larger current.
Comparison of the last closed magnetic surface between the target plasma and the plasma reconstructed using the field and the windowpane coils whose currents are adjusted to minimize the normal field on the target boundary for the configuration with an external transform of ≈0.05 of Fig. 1(a). Left two frames: the right frame shows the comparison of the total rotational transform, target (solid) vs reconstructed (dotted) at .
Comparison of the last closed magnetic surface between the target plasma and the plasma reconstructed using the field and the windowpane coils whose currents are adjusted to minimize the normal field on the target boundary for the configuration with an external transform of ≈0.05 of Fig. 1(a). Left two frames: the right frame shows the comparison of the total rotational transform, target (solid) vs reconstructed (dotted) at .
Comparison of the last closed magnetic surface between the target plasma and the plasma reconstructed using the field and the windowpane coils whose currents are adjusted to minimize the normal field on the target boundary for the configuration with an external transform of ≈0.3 of Fig. 1(d). Left two frames: The right frame shows the comparison of the total rotational transform, target (solid) vs reconstructed (dotted) at .
Comparison of the last closed magnetic surface between the target plasma and the plasma reconstructed using the field and the windowpane coils whose currents are adjusted to minimize the normal field on the target boundary for the configuration with an external transform of ≈0.3 of Fig. 1(d). Left two frames: The right frame shows the comparison of the total rotational transform, target (solid) vs reconstructed (dotted) at .
Tables
The important Fourier coefficients for describing the boundary for the configurations shown in Fig. 1 are given with the normalization that . The coefficient is a measure of the plasma minor radius and the major radius.
The important Fourier coefficients for describing the boundary for the configurations shown in Fig. 1 are given with the normalization that . The coefficient is a measure of the plasma minor radius and the major radius.
Fourier coefficients that describe the boundary for the configurations shown in Fig. 5 are given. The normalization is .
Fourier coefficients that describe the boundary for the configurations shown in Fig. 5 are given. The normalization is .
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