Flexibility Coils for Quasi Symmetric Stellarators - Boyd Blackwell
Note: this is a very rough first draft of the technical report describing in detail the work Boyd Blackwell on the NCSX design.
Many sections will be changed, missing section added etc. Please do not distribute this very early version any futther than necessary, and certainly not beyond the group directly involved in the US stellarator desgin effort. Much of the material to be added (20pp) already exists, and can be supplied to anyone interested. Email
Comments on physics are very welcome, although draft 1 will be the first time I ask for comments. General comments on presentation are welcome too, but details including typos should be ignored at this stage.
Additional Rotational Transform......
Method......
Magnetic Field Elements......
Ring Conductor......
Vertical Field......
Toroidal Field Coils......
Classical Helical Windings
Conforming Helical Windings
Tilted Window Coils
Assessment of initial synthesized saddle coil configurations:......
Boozer Spectra and Departures from QAS......
High iota via lowering Bt......
High Iota from Helical Post:......
Spectrum of Saddle Coils (very early model)......
Conclusions:......
Fixed Boundary VMEC......
Field Line Tracing......
Future directions:......
Conclusions (more like “thoughts”)......
Method......
Coil Geometry Considerations......
Other......
AppendiCES
Appendix A: Base Configurations
Modular Coil Set for Vacuum Case 40%, 2 period (ksa7_d9e)......
Incremental effects on transform and surface shape......
The vac2 case......
The QO base case......
Visualizations of various Saddle and Modular Coil sets......
Appendix B: Notes on magnetic characterization and alignment of the PBX/NCSX Coil set......
Appendix C: Basic Vacuum Magnetic Field Properties
Calculation of rotational transform
“Six digit” iota values......
Iota symmetry......
Tokatron
Additional Rotational Transform
The primary flexibility control for magnetic configuration is likely to be the rotational transform and its profile. It will be essential to have good control of total rotational transform, including plasma current (generated by current drive, bootstrap and finite mechanisms). However, for flexibility, control of the external transform, iotaext, is required. Shear can stabilize neoclassical islands (positive shear), and there are indications that is effective in stabilizing the external n=1 (kink) mode, to avoid plasma disruptions.
Method
The ideal (and the only truly correct) way to evaluate alternative designs for flexibility coils would be to calculate magnetic surfaces taking into account the effects of external coil geometry (i.e. a free boundary code), plasma currents and finite beta effects (an MHD code) and magnetic islands (i.e. do not assume the existence of magnetic surfaces). At present, the only code capable of all these is the PIES[1] code, but this code is not convenient to use for detailed parameter scans, because of cpu (and real time) time requirements, and difficulty running without intervention. In this work, maximum advantage was taken of vacuum calculations by magnetic field tracing, and the results were extended where possible to finite beta by the VMEC[2] free boundary code. Conversely, the accuracy of the VMEC code was checked by comparison with the HELIAC code for similar vacuum cases.
Two vacuum configurations were used for most of the work:
- “d9e”[3], a vacuum configuration with the same boundary shape as a finite beta QAS configuration designed for 40% external transform and two field periods, and
- “vac2” a similar QAS configuration actually optimized for vacuum[4], with iotaext ~35%.
In both cases, the configuration was realized by 40 filamentary modular coils, spaced 30cm from the plasma surface. Under these conditions, the filamentary representation was very precise.
The same tests should be repeated with a saddle coil configuration when one with suitably low errors is found. Some results with an early prototype saddle set are presented.
For comparison, some calculations are performed for a four period quasi-omnigenous configuration[5] optimized for zero .
Magnetic Field Elements
Here we examine the effect of the basic elements of the magnetic configuration on rotational transform as a basis for comparison with more specialized windings to be treated in the next section
Ring Conductor
One very simple configuration is a circular conductor. If the geometry of the base configuration is suitable, even a simple ring conductor can produce external transform. The figure shows a small effect from a 0.8m radius horizontal ring located on the midplane ( iotaext ~ 0.012 from 80kA) so the efficiency is only 1/6 of the plasma current efficiency, but the effect may be useful in view of the large number of coils available in PBX-M. The effect on this configuration is ~5x higher than on a 380kA, 0.5m cylindrical plasma, where ~ 0.0028 – 0.0022(edge). When the ring is current is oriented in the sense that increases transform, the surface size is decreased by the usual tendency of currents that add to transform attracting the magnetic surfaces into the conductor (at =90).
It would be interesting to calculate the vector of the transform effect of the existing and proposed equilibrium coils in the NCSX.
Vertical Field
Vertical field normally affects axis position, magnetic well and transform, and is often used to optimize surface size. The effect on transform in this case is relatively weak (< 0.01 for 0.01Tesla, adding for vac2 , but opposite sign (decreasing iota)for D9E) compared to its other effects (e.g. in both cases, positive BV moved the axis out by about 4cm), so it will not be considered further as a primary means of controlling transform. In this case, the axis shift and change in iota are similar to the effect on a current carrying plasma (tokcyl66: 0.005-0.0055, and 3.5cm).
Toroidal Field Coils
An additional toroidal field changes transform very effectively, scaling faster than the additional field (iotaext ~1/Bt2 ) , where Bt is the total toroidal field. For the following, additional toroidal field of 50% was added with idealised circular TFCs (a=1.2 m, R=1.5m). Not much difference is expected with more realistic models, although for studies of the effect on islands, more realistic models should be used.
Classical Helical Windings
Windings of the classical stellarator type for this machine would probably be constructed on a toroidal surface with an elliptical cross-section. The calculations presented are for an ellipticity (ratio of long to short axis) or 1.44. In general, these windings were less efficient than expected, probably because of the mismatch between the winding surface and the outer surface of the plasma.
Conforming Helical Windings
A very efficient configuration (in the sense of minimum current) is a set of helical windings, for example a classical l=2 stellarator winding consisting of 4 interlinked toroidal helices, and wound on a surface that conforms to the surface shape of the plasma, allowing a margin of space for clearance (vacuum vessel, liners etc). These windings at first sight are quite complicated to construct, but as they could share the same (or a nearby) winding surface as the main windings (e.g. NESCOIL optimized saddle coils), simplification could result. The winding law is non-trivial, and is directly affected by the effective poloidal angle of the surface parameterization equations.
Try replotting for 45 degrees? The above examples show badly and better behaved poloidal angle distributions. The badly behaved example on the left was obtained from a simplified VMEC representation using a small number of terms, and the poloidal angle distribution is more crowded into the indentation. Angles in the 5 \degree cross section were offset by 0.5m, and even more crowded around that point. The more regularly spaced example on the right was obtained by optimizing? a fitted surface to (Drevlak?).
Tilted Window Coils
This geometry is inspired (as is the vertical helix) by the presence of large tilted current elements in the computer generated saddle and modular coil sets as illustrated below. The tilted window frame coils are placed so that the diagonal current elements imitate the large current densities in these configurations (e.g.vac2) See also Appendix A for a comparison with sc036, sc009 and sc008.
The n,m spectrum at <r>~ 0.33 m shows a number of higher order components near resonance, e.g. 2,8 ~ 300x smaller than main cpt 0,1. There are also some unidentified components only slightly smaller?
Magnetic Islands and Surface Distortion
Introduction
Given typical transform values just under iota ~ ½, and the fast fall off in the amplitude of magnetic field Fourier components with M, the most likely harmonics to generate islands in the QAS are those with toroidal mode numbers equal to the toroidal periodicity of the configuration. For a 2 period QAS device, with iota ~0.4, this would correspond to members of the N=2 family, the M=4,5 and 6. Ideally, these islands should be studied using a free boundary (to allow for the inclusion of details of conductors) MHD code which does not assume the existence of nested magnetic surfaces and with self consistent bootstrap current, such as the PIES code currently being upgraded for this purpose. Alternatively, analytic estimates of island width may be made from analysis of components of the island-free “weak solutions” to the MHD equilibrium equations produced by the VMEC code, provided that 1/ suitable estimates can be found, and 2/ that the solutions are sufficiently well converged that small details such as islands are meaningful.
Vacuum magnetic field tracing provides an accurate solution, but to a different problem, because the plasma internal currents are ignored. In the following analysis this technique is used, and two approaches are used to bridge the gap between the actual problem, and the one solved. The first approach is to change the rotational transform to a value much closer to that with finite and internal currents, by use of external windings or an artificial plasma current. This produces islands directly, but the additional currents will affect the magnetic field spectrum, and may even introduce resonant components which could be confused with those under study. To minimize this effect, additional windings far way from the magnetic surfaces are used. The second approach is to identify and quantify the spectral components responsible for producing those islands These components are readily detected in spectra of data in the parallel Boozer coordinate ()obtained when integrating numerically along magnetic field lines [Boozer 1980? (Heliac code?)], even if the corresponding island chain is not resonant in the range of transforms present in the configuration. This work is based on the additional assumption that the island size can be at least semi quantitatively related to components in the magnetic spectrum, and that the amplitude of individual components won’t vary too much if the transform changes by a small amount, say 10-20%, and for larger variations in iota, there will still be some consistency in the component amplitudes. (Except for a possible “flattening” of the peak when it actually becomes resonant?)
The likely sources of islands can be interpreted graphically by looking at the vacuum Boozer spectrum.
This is a spectrum of a complex configuration which is simple to interpret because the transform is small, so the multiplet structure in M is clustered closely for each N value (N=0 and N=2 for this example). The configuration is an early saddle configuration with high N=2 contamination and low transform. The dominant term, is 0,1 as in all QAS configurations. Small variations in transform cause the multiplet structure to shrink and grow about the M=0 component, but it is clear that
1/ the amplitude of components falls for rapidly with M (as expected) so that
2/ there are no components of visible amplitude near zero frequency (resonance) which given and reasonable change in transform, could move into resonance
Although used widely for transport and Pfirsch-Schluter current analysis, it is not clear that components m,n of the 1/B2 spectrum are the best predictors of island width. Islands are formed by components perpendicular to the average magnetic surface b , but if instead of an average magnetic surface with an added perturbation we consider exact magnetic field line coordinates, (which exist in this non-resonant situation because the point of calculation is normally chosen outside the island chain) there is no component of B perpendicular to the surface by definition. Perhaps a better measure would be out of plane (or normal) magnetic curvature, but this would contain a resonant denominator term 1/(n- m), which may complicate analysis.
Perhaps the amplitude could be estimated from m,n, and the tendency toward island formation of that component be evaluated by taking the ratio of the normal to geodesic curvature for that component, well away from resonance. While these issues remain unresolved, in the remaining analysis, the amplitude m,n, is used as an estimate of the upper limit of the island driving term.
The following more complex spectrum is more realistic. The configuration is an idealised modular coil set, but but both the transform and the amplitude of undesired components have been increased by subtracting a pure toroidal field from the toroidal field generated by the modular coils. The N=0,2,4 and 6 multiplets are shown, and they overlap in frequency with their nearest neighbor. A small but significant component ( 2,8 , amplitude 1) can be seen near zero frequency, which would be resonant if iota changed slighly (to iota=2/8)
Assessment of initial synthesized saddle coil configurations:
The following configurations were generated in April 98 by Art Brooks using the Drevlak saddle coil generator. Although reasonable errors in Bnorm were obtained, (<3% RMS), the resulting surfaces were not as good as expected. Three different base coil sets are illustrated here, sc005, sc009 and sc023, which shown in more detail in appendix A.
(note: there might be a basic error in the coil generation code here – perhaps in the sense of rotation of the saddle coil itself?
Note also that the =90 section is used, as it is easier to see structure in the more circular shape).
Overall conclusion: A more accurate saddle coil set is necessary before firm conclusions can be reached. However it seems that Bnorm errors less than 1% are needed, as is an outer saddle coil.
Boozer Spectra and Departures from QAS
An important parameter of a flexibility coil is the departure from quasi-Axisymmetry that it induces. The two most likely harmonics are the mirror term (0,N for an N field period machine), and the helical curvature terms N,N. (? Check – is this helical, and does it ever really show up?)
Comparison of nm and Bnm at r/<a>=0.76 for the true vacuum QAS case.
As expected, after normalization, the dominant nm values are roughly twice as large as B nm . 0,1 is about 0.30, consistent with an aspect ratio of 6.6 (assuming an ideal cylindrical plasma), and as the surface of measurement is at s~0.5, this implies the full plasma aspect ratio is about 3.3, which is almost correct. The vertical plasma elongation should decrease the toroidal variation in |B| and hence 0,1 for a given aspect ratio, but this effect is hardly noticeable – why?
The departure in nm is highest for nm, at 2.5% (only 1.3% in Bnm), or 0.08 x 0,1. The general structure is similar ( cf B), but for example, the 0,2 is comparatively much smaller in the mn spectrum. The 2,8 and 2,9 terms are approaching the levels where island generation is noticeable.
High iota via lowering Bt
Adding some opposing pure toroidal field produces a 20% change in transform for a 10% change in total toroidal field (sc039).
The highest non QAS component is 2,2 ~ 2.7%.
Move to island section This is a good example for island amplitude estimation: On surface 4 at r/a=.3/.4, terms from the N=2 family are near resonance with amplitudes approaching 1.0 – by contrast, the surface at r/a = .2/.4 shows the last visible harmonic 3,2 at 0.3.
High Iota from Helical Post:
This is an essentially impractical case (sc041) which adds 1.8MA from a vertical helix of 0.75m radius, similar to a helical post or sawtooth coil.
The additional transform in this case is 0.15, about 2.5x the amount as for the HP vacuum surfaces alone. (The correct comparison is with hp2040, not 2039, because the latter has a TF of 0.5T)
For surface 2 the largest departure is 2,1 ~ -2.4%
This is not a very good surface for analysis, because it is near a low order resonance, and the multiplets overlap. The main departure is the 1,2 sideband just below the 0,2 mirror term. At surface 3, the largest departure is 2,1 ~ -5%, 0.18x 0,1. However this surface is much closer to the edge (than other surface 3s) because of the loss of outer surfaces.
Spectrum of Saddle Coils (very early model)
The N=2 mirror term is very obvious, particularly as it does not decrease near the magnetic axis. It is 1/3rd of the dominant term at <r>=0.3 above (cf <a>=0.4), and 0.4 of the dominant term at r/a=0.5.Conclusions:
(above) The magnetic surfaces of the double helical post alone. The transform is ~0.06.
(In spite of incorrect subtitle) this is a case of reduced transform by additional TF. Iota is down by >2, for a 50% TF increase. This is the lowest aspect ratio (Avs) so far: 0.315.
Appendices