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Stellarators Present and Future of Stellarators Magnetic Coordinates Further Reading Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading Stellarators Arturo Alonso1 Laboratorio Nacional de Fusión Laboratorio Nacional de Fusi´on,CIEMAT, Madrid 1st Costa Rica Training Workshop in Fusion for Latin-American Region Cartago, Nov. 26th, 2019 1arturo.alonso at ciemat.es, Present address: Max Plank Institut f¨urPlasmaphysik, Greifswald Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading Stellarators 1 Brief history of stellarators 2 Present day stellarators and reactor prospects 3 Working with a 3D geometry: magnetic coordinates 4 Further reading Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading Stellarators 1 Brief history of stellarators 2 Present day stellarators and reactor prospects 3 Working with a 3D geometry: magnetic coordinates 4 Further reading Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading Stellarator: the funny-looking magnetic confinement device Stellarators are plasma devices for Magnetic Confinement Fusion (MCF) in which, similar to Tokamaks and other concepts, the motion of the charged particles (the plasma constituents) is limited by a strong magnetic field. As in a tokamak, they require: I Nested flux surfaces. I Rotational transform or safety factor (i.e. line twist). Different from tokamaks, the confining B-field is created by external coils only. and, consequently, a 3D shaping of the flux surfaces is required to generate rotational transform. Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading Brief history of stellarators The first stellarator (model A) was designed by Lyman Spitzer in 1951 at Princeton University. It had 8-shape and B ∼ 0:1T and 5cm diameter. Spitzer was the first to realise that it was possible to generate rotational transform in ways other than inducing a toroidal plasma current (e.g. by magnetic axis' torsion) and demonstrated that confinement improved w.r.t. a simple torus. Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading Brief history of stellarators Further models were planned (B,C,D). Model C intended to achieve break-even and model D was (in their plans) a D-T reactor with 7.5T and 14GW fusion power2. \Classical" stellarators were developed with helical coils, rotational transform and shear. However, the progress in performance of bigger, higher-B devices was below expectations. Energy diffusion was of Bohm-type, T D ∼ D ∼ Bohm eB i.e. weak scaling with B (from classical diffusion D ∼ B−2) and system size independent ! very bad news for fusion. 2Stix 1998. Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading The Tokamak stampede Controlled fusion energy research was declassified in 1958. In the 3rd IAE Conference in 1968, 1 keV temperatures were reported in the Soviet's Tokamak device (twice larger than before). Received with skepticism. An UK group from Culham set out to measure electron temperature and density with a new diagnostic: the Thomson scattering. The confirmation was presented a year later in the International Symposium on Closed Confinement Systems [Holcomb 1969]. Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading Stellarators 1 Brief history of stellarators 2 Present day stellarators and reactor prospects 3 Working with a 3D geometry: magnetic coordinates 4 Further reading Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading Main lines of present stellarator devices Quasi-symmetric stellarator Mainly developed in the US. Devices: HSX (Wisconsin Univ.), NCSX (Princeton, unfinished and discontinued). Modular coils, compact designs, non-resonant divertor. Heliotron Developed in Japan. Devices: Heliotron-J (Kyoto Univ.), LHD (NIFS). Simple (but helical) coils, large shear, simple divertor implementation. Helias Developed at IPP in Garching/Greifswald. Devices: W7-AS (Garching), W7-X (Greifswald). The only running `optimised' stellarator, modular coils, island (resonant) divertor. Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading Principles of Helias optimization The W7-X stellarator has been optimised for Good flux surfaces of the vacuum magnetic field Low Shafranov shift, good MHD stability and a stiff equilibrium up to a plasma β of 4{5%(very low bootstrap current) Good energy and particle confinement, i.e., small neoclassical transport losses (drift optimisation) A suitable divertor concept for controlled particle and energy exhaust (island divertor) Good confinement of fast particles (α particles in a reactor) Feasibility of the modular coil set. Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading Energy confinement time in modern stellarators (I) Data from these and other devices are collected in international databases to ISS04 2:28 0:64 −0:61 0:54 0:84 0:41 τE / a R P n¯e B ι2=3 : determine the empirical scaling of τE with basic plasma and device parameters. The energy confinement time is defined in steady state as Plasma energy content W τE = = ; Heating power PHeat and measures the quality of the confinement. The latest stellarator scaling is the so-called ISS04 [Yamada et al. 2005]. Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading Energy confinement time in modern stellarators (II) ISS04 scaling implies a gyro-Bohm diffusivity ISS04 2:28 0:64 −0:61 0:54 0:84 0:41 τ ISS04 / a2:72R0:08T −1:56n¯−0:18B2:15ι1:05 : τE / a R P n¯e B ι2=3 : E e 2=3 Energy confinement time τE and the and therefore concludes that characteristic diffusivity D are related as DISS04 / 1=B2:15. Simple transport ordering arguments D τ −1 ∼ : yield a gyro-Bohm scaling, DGB / ρ2v a E a2 ∗ t (with ρ∗ ≡ Larmor radius=a and vt ≡ but note that P = 3nT V and, after thermal velocity). τE GB 3 −1:5 2 substitution in the ISS04 one gets Then τE / a T B . Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading Fusion power plant projections based on Stellarators Extrapolations of the empirical energy scaling allow for projections to prospective fusion power plants. FPP name HSR4/18 FFHR ARIES-CS Based on W7-X LHD NCSX R (m) 18 14 7.75 B (T) 5.0 6.2 5.7 T (keV) 4.96 5.4 6.4 n (1020 m−3) 2.6 2.1 4.0 β (%) 4.2 3.0 6.4 Table: Characteristic parameters of stellarator fusion power Figure: The HELIAS-5B power plants. Note that the FPP minimum size (R) is sometimes plant concept. Shown are the determined by the space needed for the Li breeding blanket plasma vessel, coils, support and coil shielding (∼ 1:5m) rather than by confinement. structure, the magnet support, and the outer cryostat. Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading Pros and cons of the stellarator reactor compared to a tokamak Cons Pros 3D-ty worsens thermal and fast particle No need to induce current allows for collisional transport { careful design of steady-state operation. the B-field required to meet with No disruptions and, therefore, no self-sustained fusion conditions. run-away electrons (two of the most Breeding blanket geometry, coil serious problems tokamaks face) manufacture and remote maintenance ELMs are less characterised, but appear are more challenging. to be more benign (lower H-mode More diverse and less studied (i.e. pedestal). H-mode or divertor physics) {higher risk No (Greenwald) density limit { higher in extrapolating to a FPP. fusion power ∼ n2 and more efficient Theory and codes are less developed α-particle slow-down. {again higher risk. Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading Pros and cons of the stellarator reactor compared to a tokamak Tokamaks are fat, but yet Look at present not so bad. Vitali Shafranov, in a 1980 article on True, just in the steady state Stellarators [Shafranov 1980], quotes these They can hardly operate, \humorous rhymes improvised at one of And, besides, their large disruptions the Soviet-American meetings devoted to May end up with wall destructions. the evaluation of tokamaks and stellarators In the future, it might be better as reactor systems": To use a quiet stellarator. But to say `of course' or `yes', One should build the PLS! Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading Stellarators 1 Brief history of stellarators 2 Present day stellarators and reactor prospects 3 Working with a 3D geometry: magnetic coordinates 4 Further reading Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading Working with a 3D geometry Flux coordinates To work with physics equations in a 3D field with nested flux surfaces one needs to define an appropriate set of coordinates. `Flux coordinates' are set of coordinates ( (x); θ(x); φ(x)) such that I labels the flux surfaces, i.e. (x) = 0 is the equation for a flux surfaces. I Common choices for are toroidal or poloidal magnetic fluxes ΨT ; ΨP , volume enclosed V and p effective radius r = V/πLax. I θ; φ are poloidal and toroidal angles to move within a surface. Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading Working with a 3D geometry The coordinate transformation ( (x); θ(x); φ(x)) induces a natural basis of gradient vectors i fe gi=1;2;3 = fr ; rθ; rφg and its dual basis @x @x @x fe g = ; ; i i=1;2;3 @ @θ @φ with ei · e = δi (1 if i = j, 0 otherwise) so j j where (i; j; k) are cyclic permutations of p j k (1; 2; 3) and g is the Jacobian, e × e p j k ei = = g e × e ; ei · ej × ek p −1 g = jr · rθ × rφj = je · eθ × eφj : Brief history of stellarators Present and future of stellarators Magnetic coordinates Further reading Working with a 3D geometry Magnetic field representation Since r · B = 0 we can write the B-field as The field can also be represented in the B = r × rν ; contra-variant basis as where ν( ; θ; φ) ≡ field stream-function.
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