Magnetic confinement apparatus, fusion reactor apparatus and methods of operation thereof

Abstract

A magnetic confinement apparatus (100) comprises an evacuable vessel (10) having a torus shape with an inner torus hole (11) and being configured for accommodating a fusion plasma (1), a magnetic coil device (20) being configured for creating a magnetic confinement field (2) in the vessel (10), wherein the magnetic coil device (20) comprises a plurality of axisymmetric coils (21) being configured for creating the magnetic confinement field (2) with an axisymmetric field equilibrium, and a control device (30) being configured for controlling the magnetic coil device (20), wherein the magnetic coil device (20) further comprises at least two quasi-axisymmetric perturbation (QAP) coils (22) being placed outside the vessel (10) within the torus hole (11) of the torus shape and being arranged for subjecting the axisymmetric field equilibrium of the magnetic confinement field (2) to quasisymmetry-preserving perturbations, so that the magnetic confinement field (2) created by the magnetic coil device (20) has quasi-axisymmetry and a magnetic equilibrium of the magnetic confinement field has a quasi-axisymmetric stellarator configuration of the magnetic confinement apparatus (100). Furthermore, a fusion reactor apparatus (200), including the magnetic confinement apparatus (100) and a method of operating the magnetic confinement apparatus are disclosed.

Description

Magnetic confinement apparatus, fusion reactor apparatus and methods of operation thereofField of the inventionThe invention relates to a magnetic confinement apparatus, being adapted for confining a plasma, in particular a fusion plasma in a fusion reactor apparatus. Furthermore, the invention relates to a fusion reactor apparatus, including the magnetic confinement apparatus, and / or to a method of operating the magnetic confinement apparatus and / or the fusion reactor apparatus. Applications of the invention are available, e. g., in the fields of investigating fusion plasma confinement, executing fusion experiments and / or operating a fusion reactor apparatus.Prior artIn the present specification, reference is made to the following prior art illustrating the technical background of the invention:[1] T. E. Evans et al. "Suppression of Large Edge-Localized Modes in High- Confinement Dlll-D Plasmas with a Stochastic Magnetic Boundary" in "Phys. Rev. Lett." 92 (23 June 2004), p. 235003;[2] T. E. Evans "Resonant magnetic perturbations of edge-plasmas in toroidal confinement devices" in "Plasma Physics and Controlled Fusion" 57.12 (Nov. 2015), p. 123001;[3] W Vll-A Team "Stabilization of the (2, 1) tearing mode and of the current disruption in the W Vll-A stellarator" in "Nuclear Fusion" 20.9 (1980), p. 1093;[4] M. Hirsch et al. "Major results from the stellarator Wendelstein 7-AS" in "Plasma Phys. Contr." F. 50.5 (2008), p. 053001;[5] M. D. Pandya et al. "Low edge safety factor operation and passive disruption avoidance in current carrying plasmas by the addition of stellarator rotational transform" in "Physics of Plasmas" 22.11 (2015), p. 110702;[6] G. J. Hartwell et al. "Design, Construction, and Operation of the Compact Toroidal Hybrid" in "Fusion Science and Technology" 72.1 (2017), pp. 76-90;[7] Paul E. Moroz "Spherical Stellarator Configuration" in "Phys. Rev. Lett." 77 (4 July 1996), pp. 651-654;[8] Paul E. Moroz "Extreme low aspect ratio stellarators" in "Physics Letters" > pp. 60-65;[9] K. Yamazaki et al. "TOKASTAR: a tokamak-stellarator hybrid with possible bean-shaped operation" in "Plasma Physics and Fusion Technology" 17.12 (1985), p. 24;

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[0026] "used SIMSOPT version" in https: / / doi.org / 10.5281 / zenodo.7221578;

[0027] P. Merkel "Solution of stellarator boundary value problems with external currents" in "Nuclear Fusion" 27.5 (1987), p. 867;

[0028] Michael Drevlak "Automated Optimization of Stellarator Coils" in "Fusion Technology" 33.2 (1998), pp. 106-117;

[0029] T. Brown et al. "Engineering Optimization of stellarator coils lead to improvements in device maintenance" in "IEEE 26th Symp. on Fusion Engineering" (2015), pp. 1-6;

[0030] Caoxiang Zhu et al. "New method to design stellarator coils without the winding surface" in "Nuclear Fusion" 58.1 (Nov. 2017), p. 016008;

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[0032] V.D. Shafranov et al. "Use of the virtual-casing principle in calculating the containing magnetic fi in toroidal plasma systems" in "Nuclear Fusion" 12.5 (Sept. 1972), p. 599;

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[0035] C. Mercier "EQUILIBRIUM AND STABILITY OF A TOROIDAL MAGNETO- HYDRODYNAMIC SYSTEM IN THE NEIGHBOURHOOD OF A MAGNETIC AXIS" in "Nucl. Fusion" 4.3 (1964), 213-226;

[0036] P. Helander et al. "GENERALIZED ADIABATIC INVARIANTS IN ONE-DIMENSIONAL HAMIL- TONIAN-SYSTEMS" in "Phys. Rev. Lett." 68.25 (1992), 3659-3662;

[0037] Rogerio Jorge et al. "Single-stage stellarator optimization: Combining coils with fi boundary equilibria" in "Plasma Physics and Controlled Fusion" (2023, arXiv:2302.10622);

[0038] O. Neubauer et al. "Design Features of the Tokamak TEXTOR" in "Fusion Science and Technology" 47.2 (2005), pp. 76-86; and

[0039] A. Weller et al. "Significance of MHD Effects in Stellarator Confinement" in "Fusion Science and Technology" 50.2 (2006), pp. 158-170.Two main types of fusion reactors or fusion experiments are generally known, cor kamak type and the stellarator type. The magnetic confinement field of a tokamak is obtained by a superposition of toroidal fields created with a plurality of toroidal field coils and the poloidal field of a plasma current, typically induced by an additional central solenoid coil. Due to their relatively simple axisymmetric shape and their early success, tokamaks are widely studied in the magnetically confined fusion research. The stellarator creates the magnetic confinement field by a complex, non-rotationally symmetrical magnetic field geometry.There are instances in which three-dimensional (3D) shaping is beneficial for tokamaks, e.g. when resonant magnetic perturbations (RMP) are exploited to mitigate edge-localized modes (ELMs), [1, 2], In addition, it has been found that 3D shaping can help to avoid disruptions, [3, 4, 5, 6], The Compact Toroidal Hybrid (CTH) device studied in [5, 6] is a hybrid of a tokamak and a stellarator with continuous helical coils. Stellarators can produce the necessary rotational transform (which is the inverse of the safety factor) by coils, which is referred to as the external rotation transform. CTH therefore allows the study of how the external rotational transform affects the plasma, e.g. its stability including disruptions.A stellarator with continuous helical coils, such as CTH, is just one of many different types of stellarators. In fact, there are even other hybrid stellarator-tokamak designs, such as the spherical stellarator concept [7, 8] or the tokastar [9], The type of stellarator most similar to tokamaks is the quasi-axisymmetric (QA) stellarator [10, 11], where the magnetic field strength contours close toroidal ly and, when expressed in Boozer coordinates, the magnetic field strength is symmetric in the toroidal angle (as it is in tokamaks). Because of this symmetry, a perfectly quasi-axisymmetric stellarator would share many neoclassical properties with tokamaks. Therefore, a hybrid of a quasi-axisymmetric stellarator and a tokamak would be a very suitable candidate to achieve 3D shaping effects on the plasma while approximately maintaining neoclassical confinement properties.Many quasi-axisymmetric stellarators have been designed in the past, e.g. [10, 12, 14, 16, 15, 13], Designing a hybrid of a tokamak with one of these quasi-axisymmetric stellarators is restricted by the fact that the volume of the tokamak part in such a hybrid would be drastically reduced compared to the stellarator operation and the attainable aspect ratio of the tokamak part would be very large, which would limit its usability and confinement properties.As a method of field shaping, quasisymmetry-preserving perturbations to axisymn have been theoretically explored in a number of works (e.g.,

[0017] ,

[0018] ), and developed theoretically in recent years, starting with the vacuum case

[0019] , followed by the more general case with finite beta

[0020] ,A key finding from these theories was that, even with small deviations from axisymmetry, approximately quasi-axisymmetric equilibria could be found that possess a significant external rotational transform. However, up to now, it was impossible to apply these theoretical concepts in practice because no practical coil configuration was known which would be capable of applying quasisym- metry-preserving perturbations to axisymmetric equilibria.Objective of the inventionThe objective of the invention is to provide an improved magnetic confinement apparatus, being adapted for confining a plasma, in particular a fusion plasma in a fusion reactor apparatus, being capable of avoiding or reducing limitations of conventional techniques. In particular, the objective of the invention is to provide a magnetic confinement apparatus having a practical coil configuration for introducing quasisymmetry-preserving perturbations to axisymmetric equilibria, allowing a tokamak-stellarator-hybrid configuration of magnetic confinement with low complexity, being capable of an improved control of field shaping and / or being capable of retrofitting existing fusion reactors and experiments of the tokamak type for introducing the tokamak-stellarator-hybrid functionality. Furthermore, the objective of the invention is to provide an improved fusion reactor apparatus, including the magnetic confinement apparatus, and to provide an improved method of operating the magnetic confinement apparatus, wherein limitations of conventional techniques are avoided or reduced.Brief summary of the inventionThese objectives are correspondingly solved by a magnetic confinement apparatus, a fusion reactor apparatus and a method of operating the magnetic confinement apparatus, comprising the features of the independent claims, respectively. Preferred embodiments and applications of the invention arise from the dependent claims.According to a first general aspect of the invention, the above objective is solved by a magnetic confinement apparatus, comprising an evacuable vessel having a torus shape with an inner torushole, in particular central torus hole, and being configured for accommodating a fi magnetic coil device being configured for creating a magnetic confinement field in the vessel, wherein the magnetic coil device comprises a plurality of axisymmetric coils being configured for creating the magnetic confinement field with an axisymmetric field equilibrium, and a control device being configured for controlling the magnetic coil device.According to the invention, the magnetic coil device additionally comprises at least two quasi- axisymmetric perturbation (QAP) coils being placed outside the vessel within the torus hole of the torus shape and being arranged for subjecting the axisymmetric field equilibrium of the magnetic confinement field, as created in particular by the axisymmetric coils alone, to quasisymmetry-pre- serving perturbations, so that the magnetic confinement field created by the magnetic coil device has quasi-axisymmetry and a magnetic equilibrium of the magnetic confinement field has a quasi- axisymmetric stellarator configuration of the magnetic confinement apparatus.According to a second general aspect of the invention, the above objective is solved by a fusion reactor apparatus, including the magnetic confinement apparatus according to the first general aspect of the invention or an embodiment thereof. Advantageously, the fusion reactor apparatus may have improved operating conditions in terms of the start-up mechanism and an improved compactness of the coil configuration.According to a third general aspect of the invention, the above objective is solved by a method of operating a magnetic confinement apparatus according to the first general aspect of the invention or an embodiment thereof, comprising the steps of operating the axisymmetric coils for creating the magnetic confinement field having the axisymmetric field equilibrium, and operating the at least two quasi-axisymmetric perturbation coils for subjecting the axisymmetric equilibrium of the magnetic confinement field to the quasisymmetry-preserving perturbations, so that the magnetic confinement field has the quasi-axisymmetric stellarator configuration.Advantageously, the above objective is solved in particular by employing a coil configuration comprising the combination of the axisymmetric coils, e.g. toroidal coils as known from a tokamak, and the quasi-axisymmetric perturbation coils, which are arranged in the torus hole of the vessel torus shape. The inventors have found that the quasi-axisymmetric perturbation coils can be placed in the torus hole which is occupied in a conventional tokamak by the tokamak solenoid coil and the toroidal field coils. Furthermore, the inventors have found that the quasi-axisymmetric perturbation coils are capable of shaping the magnetic confinement field even if being arrangedoutside the vessel, i.e., with a distance from the plasma and separated from the p sei wall. For the first time, a practical coil configuration is proposed which is capable of introducing quasisymmetry-preserving and volume-preserving perturbations to axisymmetric equilibria. The magnetic confinement apparatus has advantages in particular in terms of a large plasma volume (high compactness), a small number of distinctly shaped coil types, and simple and inherently steady-state operation.As a particular advantage, a novel quasi-axisymmetric stellarator-tokamak hybrid design is provided by the inventive magnetic confinement apparatus which would allow a plasma configuration to be varied continuously from an axisymmetric tokamak to a quasi-axisymmetric stellarator. The magnetic confinement apparatus may be operated either as a tokamak or as a low-aspect- ratio, quasi-axisymmetric stellarator, or in an intermediate operation condition. The volumes and aspect ratios of the two distinct tokamak and stellarator modes of operation are kept approximately constant which advantageously means that no volume is lost due to the shaping of the confinement field. Surprisingly, the optimized magnetic confinement apparatus may only employ a number of identically shaped quasi-axisymmetric perturbation coils compared to the tokamak scenario. In other words, the magnetic confinement apparatus may only employ a single type of quasi-axisymmetric perturbation coil.In quasi-axisymmetric (QA) stellarators, the feature of quasi-axisymmetry refers to the characteristic that the magnetic field strength possesses a (hidden) symmetry that is revealed upon transformation to Boozer coordinates (

[0010] ,

[0011] ). It is known that perfectly quasi-axisymmetric stellarators have the same neoclassical properties as tokamaks,

[0010] , Therefore, the inventive magnetic confinement apparatus provides particular advantages in experiments for studying three-dimensional effects on several physical properties, including stability and disruptions, without adversely affecting the neoclassical properties. In addition, the inventive magnetic confinement apparatus allows the evaluation of how much three-dimensional shaping is needed to attain steady-state operation in attractive parameter regimes.A plasma design suitable for the inventive magnetic confinement apparatus can be obtained from the near-axisymmetric solution disclosed in

[0020] , generated by perturbing an initially axisymmetric equilibrium in a manner that preserves quasi-symmetry. Numerical optimization may be applied to find or further improve the degree of quasi-axisymmetry, including single and non-single stage optimization. Designing the quasi-axisymmetric perturbation coils configuration and operation conditions, in particular in terms of shape, size, position, orientation and operation current, maybe obtained by numerical optimization methods. As an example, using the coil opi ture of the SIMSOPT framework, [25,

[0026] ], the inventors have shown that even with only two quasi-axisymmetric perturbation coils a sufficient perturbation of the axisymmetric equilibrium can be obtained. In addition, the inventors have found that none of the plasma volume of the original tokamak equilibrium is lost due to the provision of the quasi-axisymmetric perturbation coils.The term "torus shape" of the evacuable vessel generally refers to the vessel shape equal to or approximated by a torus ring. The term "coil" refers to any conductor, in particular conductor loop, being configured for creating a magnetic field. The conductor loop preferably may be represented by a single closed, non-branched curve.According to a preferred embodiment of the invention, each quasi-axisymmetric perturbation coil may be a vessel-unlinked modular coil extending along the outside the vessel without encircling the vessel.The term "vessel-unlinked modular coil" refers to the particular geometry provided for the QAP coils. These coils are not linked to the plasma vessel (i.e., they do not enclose the plasma vessel), allowing for ease of removal and remounting of the QAP coils and other components of the magnetic confinement apparatus during maintenance, without the need for disassembly of the evacuable vessel. The feature of being unlinked makes them distinct from traditional modular coils, which are linked to the evacuable vessel. As with traditional modular coils, vessel-unlinked modular coils may have multiplicity equal to a multiple of the field period number. Replication of optionally identical coil shapes simplifies the design and manufacturing process. The feature of being unlinked is shared with so-called "saddle" coils, used e.g. for resonant magnetic perturbations in tokamaks. However, unlinked-modular coils may be similar (equal or nearly equal) or greater in size (in particular coil extension along axial direction) to the height of the evacuable vessel (vessel extension in axial direction), making them distinct from saddle coils, which are small relative to the height of the evacuable vessel, and therefore can be oriented to lie everywhere flat to the evacuable vessel. Nearly equal coil and vessel extensions may mean that the axial coil extension differs no more than 10 % from the axial vessel extension. Unlinked modular coils can be partially or fully twisted, and therefore do not need to be oriented in any specific way relative to the surface of the evacuable vessel. The direction along which a QAP coil is most extended is to be referred to as the main coil extension.According to a further preferred embodiment of the invention, the at least two qi, ric perturbation coils may have a twisted arrangement, in particular the shapes of the quasi-ax- isymmetric perturbation coils may conform to each other, wherein inner concave portions of the quasi-axisymmetric perturbation coils along a first main coil extension thereof face each other. In particular in case of providing vessel-unlinked modular coils as the quasi-axisymmetric perturbation coils, each vessel-unlinked modular coil may be arranged such that the first main coil extension is spirally twisted around a central axis of the torus shape and a second coil main extension is perpendicular to the central axis of the torus shape.The twisted arrangement means that the quasi-axisymmetric perturbation coils fit around each other in the form of a twisted bundle within the hole of the torus. The twisted arrangement has advantages for producing rotational transform of the magnetic coil device.With the at least two quasi-axisymmetric perturbation coils being situated within the inner torus hole of the torus shape, the quasi-axisymmetric perturbation coils preferably may be arranged such that they could be included between two co-axial cylinders having their axes parallel to a central axis of the torus shape. In other words, the extension of the quasi-axisymmetric perturbation coils may be restricted to a space defined between two cylinder surfaces with a common axis coinciding with the central axis of the torus shape. The diameter of the smaller, inner cylinder is smaller than the inner diameter of the torus hole. Due to the curved shape of the quasi-axisymmetric perturbation coils, the diameter of the larger, outer cylinder may be larger than the inner diameter of the torus hole. Particularly preferred, a radial extent of the quasi-axisymmetric perturbation coils may be not more than half of the radial extent of the evacuable vessel.The configuration, in particular the size, of the quasi-axisymmetric perturbation coils such that they are arranged between the cylinder surfaces may have particular advantages in terms of compactness of the quasi-axisymmetric perturbation coils and an arrangement close to an inner surface of the vessel.According to a further preferred feature of the invention, the at least two quasi-axisymmetric perturbation coils may be arranged exclusively inside the inner torus hole of the torus shape. Particularly preferred, the radial extension of the quasi-axisymmetric perturbation coils may be equal to or less than an inner diameter of the vessel, which may depend on the vessel height Z. Optionally, the axial extent (extent in the direction aligned with the axis of toroidal symmetry) of the quasi- axisymmetric perturbation coils may be equal to or shorter than the axial extent of the vessel, i. e.coils with a relatively short axial extent may be employed as quasi-axisymmetric p coils. Alternatively, the axial extent of the quasi-axisymmetric perturbation coils may be larger than the axial extent of the vessel. In particular, the height of the quasi-axisymmetric perturbation coils in axial direction may be up to twice of the axial extent of the evacuable vessel.Preferably, all of the at least two quasi-axisymmetric perturbation coils may have an equal shape and size. Advantageously, the inventors have found that only a single type of quasi-axisymmetric perturbation coils may be needed to realize the quasi-axisymmetric stellarator. This is an exceptionally simple coil configuration for a stellarator, which usually requires a large number of distinct three-dimensional coils to produce its field. With this embodiment, manufacturing the quasi- axisymmetric perturbation coils is made easier and costs are reduced. As a further advantage, the configuration of the quasi-axisymmetric perturbation coils may be easily modified by introducing an additional coil.According to a further preferred embodiment of the invention, the at least two quasi-axisymmetric perturbation coils may be non-interlinked coils. The term "non-interlinked coils" refers to coils, which are not engaged with each other, i. e. there is no topological feature of the coils linking, e.g. chaining, them together. Due to employing the topologically non-interlinked coils, taking or introducing single coils from or into the group of quasi-axisymmetric perturbation coils is substantially facilitated, resulting in advantages for manufacturing and assembling the magnetic confinement apparatus.Alternatively, the at least two quasi-axisymmetric perturbation coils may comprise at least one interlinked coil pair. Contrary to the non-interlinked coils, the interlinked coils comprise coils, which are engaged with each other, i. e. which are topologically linked. Interlinked coils can be advantageous when making additional features available for shaping the magnetic field.Further advantages for field shaping may be obtained if, according to a further preferred feature of the invention, the magnetic coil device further comprises at least one resonant magnetic perturbation coil being placed outside the torus shape and being configured for acting in combination with the quasi-axisymmetric perturbation coils so that the magnetic confinement field created by the magnetic coil device has the quasi-axisymmetry.Generally, the at least one resonant magnetic perturbation coil may comprise a coil having a shape and arrangement being optimised in conjunction with the quasi-axisymmetric perturbationcoils, so that the magnetic confinement field created by the magnetic coil device s dition of quasi-axisymmetry in improved manner. The at least one resonant magnetic perturbation coil is provided additionally to the quasi-axisymmetric perturbation coils, and it is configured for contributing to quasi-symmetry of the magnetic confinement field having the quasi-axisymmetric stellarator configuration. To this end, particularly preferred, the at least one resonant magnetic perturbation coil may have a non-planar (curved) shape, extending e.g., above and / or below the evacuable vessel in relation to the axial direction of the torus shape thereof.Another advantage of the invention results from the fact that the number of the quasi-axisymmetric perturbation coils can be selected in a broad range. Preferably, the number of the quasi- axisymmetric perturbation coils may be selected between 2 and 10, for instance between 2 and 8 or 2 and 4. More than ten quasi-axisymmetric perturbation coils are also possible if there is enough space in the inner hole of the torus shape.According to a further, particularly preferred embodiment of the invention, the magnetic confinement apparatus may be configured to be operated without the need of a tokamak solenoid coil. Preferably, the magnetic confinement apparatus, in particular the inner hole thereof, may be free of a tokamak solenoid coil. The inventors have found that advantageously the field generated by the quasi-axisymmetric perturbation coils provides sufficient confinement during plasma startup, so that the tokamak solenoid coil is not required and additional space for arranging the quasi-axisymmetric perturbation coils is obtained.Preferably, the magnetic confinement field having the quasi-axisymmetric stellarator configuration may have an aspect ratio equal to or below 3. A configuration of the quasi-axisymmetric perturbation coils, optionally in combination with the at least one resonant magnetic perturbation coil, such that the aspect ratio is reduced to the above range has the particular advantage that the space in the vessel is substantially better used for magnetic confinement and plasma creation. An aspect ratio within this range has not been achieved in conjunction with quasi-axisymmetry with conventional hybrid approaches.According to a further, particularly preferred embodiment of the invention, the at least two quasi- axisymmetric perturbation coils may be configured such that a quadratic-flux error functional according tois optimized in a numerical optimization framework, wherein BE,nis a normal component of the magnetic field produced by the quasi-axisymmetric perturbation coils with respect to the plasma boundary, Dnis the targeted normal component of the magnetic field and ds represents the surface area element for this surface integral. Preferably, the numerical optimization may start with a geometrically-fixed set of coils which can produce the axisymmetric magnetic field and additionally at least two initial circular coils inside the torus hole whose shape is optimized by minimizing the quadratic-flux error functional such that they become the quasi-axisymmetric perturbation coils.According to a preferred feature of the inventive method, the configuration of the magnetic confinement apparatus may be varied between an axisymmetric tokamak configuration and the quasi-axisymmetric stellarator configuration by controlling the at least two quasi-axisymmetric perturbation coils and optionally the at least one resonant magnetic perturbation coil, in particular controlling a current through each of the controlling the at least two quasi-axisymmetric perturbation coils and optionally the at least one resonant magnetic perturbation coil. Particularly preferred, a continuous variation between both configurations may be provided.Varying the configuration includes setting the tokamak configuration, the stellarator configuration or a configuration therebetween. Variations in both directions are possible, i.e., toward the stellarator configuration or toward the tokamak configuration. For the tokamak configuration, the quasi-axisymmetric perturbation coils may be switched off. For any hybrid or the stellarator configuration, the current needed though each of the at least two quasi-axisymmetric perturbation coils and optionally the at least one resonant magnetic perturbation coil for obtaining the configuration of interest may be calculated, e.g. using the SIMSOPT framework.The configuration of the magnetic confinement apparatus or the fusion reactor apparatus may be monitored by measurements with vessel sensors, e. g. magnetic field sensors. Setting the tokamak configuration, the stellarator configuration or the configuration therebetween may be provided in dependency on an output of the vessel sensors, so that a feedback control of the magnetic confinement apparatus may be provided.Features disclosed in the context of the magnetic confinement apparatus and / or t tor apparatus also represent preferred features of the inventive method of operating the magnetic confinement apparatus and / or the fusion reactor apparatus or embodiments thereof. The aforementioned aspects and inventive and preferred features, in particular with regard to the configuration of the apparatuses as well as the configurations of individual components being described in relation to the apparatuses, also apply for the methods. The preferred embodiments, variants and features of the invention described above are combinable with one another as desired.Brief description of the drawingsFurther details and advantages of the invention are described in the following with reference to the attached drawings, which schematically show in:Figure 1: a side view of an arrangement of quasi-axisymmetric perturbation coils in a magnetic confinement apparatus according to preferred embodiments of the invention;Figure 2: a top view of the arrangement of quasi-axisymmetric perturbation coils as shown in Figure 1, including a schematic illustration of the evacuable vessel;Figure 3: an illustration of an exemplary shape of a single quasi-axisymmetric perturbation coil;Figure 4: illustrations of interlinked (A) or non-interlinked (B) quasi-axisymmetric perturbation coils;Figure 5: a side view of an arrangement of resonant magnetic perturbation coils in a magnetic confinement apparatus according to preferred embodiments of the invention;Figure 6: a top view of the arrangement of resonant magnetic perturbation coils as shown in Figure 5; andFigures 7 to 15: graphical illustrations of features of magnetic confinement obtained with embodiments of the invention.Preferred embodiments of the inventionFeatures of preferred embodiments of the invention are described in the following in particular with reference to the design, arrangement and operation of the quasi-axisymmetric perturbation coils, optional in combination with resonant magnetic perturbation coils. Details in particular of axisymmetric coils, e.g., toroidal coils, providing the tokamak configuration and operation thereof, details of the evacuable vessel and operation thereof, and details of a fusion reactor apparatus and operation thereof are not described as far as they are known per se from prior art.The invention is not restricted to the particular illustrated configuration of the quasi-axisymmetric perturbation coils, but rather can be varied, e. g. with regard to the number and shape of coils. For example, instead of using multiple quasi-axisymmetric perturbation coils of the same type, the quasi-axisymmetric perturbation coils may include at least two different coil types, e.g. different coil shapes and / or dimensions. Furthermore, the working of the invention is not restricted to employing the software tools mentioned below, but rather can be obtained with other software tools providing coil optimization, e.g. using single-stage optimization. The application of the invention is not limited to experiments investigating magnetic confinement in vessels, but can be used also in fusion reactor apparatuses.In the following, features of preferred embodiments are described with reference to the configuration of the magnetic coil device, followed by a description of the theoretical concepts of perturbation of axisymmetric equilibrium and designing in particular the quasi-axisymmetric perturbation coils.Magnetic coil device, including the quasi-axisymmetric perturbation coilsFigures 1 and 2 illustrate features of embodiments of the magnetic confinement apparatus 100 of the invention, comprising the evacuable vessel 10 (for clarity reasons illustrated in Figure 2 only), the magnetic coil device 20 and the control device 30, with a side view (Figure 1) and a top view (Figure 2). Figures 1 and 2 simultaneously represent key features of a fusion reactor apparatus 200 including the magnetic confinement apparatus 100. The axial direction z of the magnetic confinement apparatus is in the drawing plane (Figure 1) or perpendicular to the drawing plane (Figure 2). The radial directions r of the magnetic confinement apparatus extent in the drawing plane of Figure 2.The evacuable vessel 10 is shown in the top view of Figure 2 with dashed lines. Th torus shape with an inner torus hole 11. Details of the vessel 10, that is configured for accommodating a fusion plasma 1, may be provided as known from available fusion reactors or fusion experiments, like the ITER fusion reactor in France or the JT-60SA fusion reactor in Japan or JET in the UK.The magnetic coil device 20 comprises a plurality of axisymmetric coils 21, four quasi-axisymmet- ric perturbation coils 22 and optionally resonant magnetic perturbation coils 23 (see Figures 5 and 6). All coils are coupled with coil current sources (not shown) as known from conventional techniques. The magnetic coil device 20 creates a magnetic confinement field 2 holding the fusion plasma 1 in the vessel 10 with a distance from vessel walls. For clarity reasons, both of the magnetic confinement field 2 and the fusion plasma 1 are schematically illustrated with a grid pattern.The axisymmetric coils 21 comprise e.g. toroidal coils, each with a ring shape surrounding the torus shape. For clarity reasons, Figure 1 shows only 4 examples of the axisymmetric coils 21, while the top view of Figure 2 illustrates the axisymmetric arrangement of the axisymmetric coils 21. Details of the axisymmetric coils 21, in particular the number, shape, arrangement and operation conditions thereof, may be provided as it is known from available tokamak fusion reactors or experiments, e.g., as cited above. In practice, the axisymmetric coils 21 as shown in Figures 1 and 2 with an upright orientation may be combined with poloidal field coils (not shown for clarity reasons). The invention is not restricted to employing the illustrated toroidal coils as axisymmetric coils 21. Alternatively, other coil types, being capable of creating the magnetic confinement field 2 with an axisymmetric field equilibrium, may be employed, like e.g., tilted ring-shaped axisymmetric coils 21 being configured for producing both a toroidal and a poloidal field, so that poloidal field coils may be omitted, or toroidal field coils which are shaped conventionally everywhere but the inboard side of the torus where they are tilted or helically twisting.The quasi-axisymmetric perturbation coils 22 are arranged within the torus hole 11 with a distance from the vessel wall of the vessel 10. Each of the quasi-axisymmetric perturbation coils 22 may have a curved coil shape limited by two cylindrical surfaces pointing in the Z-direction, as shown in Figures 3 and 4. The quasi-axisymmetric perturbation coils 22 may be arranged as pairs 24 of interlinked (engaged) coils 22 (Figure 4A) or as non-interlinked (separate) coils 22 (Figure 4B).As schematically illustrated, the quasi-axisymmetric perturbation coils 22 have a t' ment, wherein, along an azimuthal direction relative to the z axis, for example a first (e.g., lower) end section of a quasi-axisymmetric perturbation coil 22A may overlap with a second (e.g., middle or upper) end section of a neighbouring quasi-axisymmetric perturbation coil 22B (see Figures 2, 3A and 3B). The radial extension of the quasi-axisymmetric perturbation coils 22 is restricted to a space between surfaces of two cylinders 3, 4 shown in cross-sections with dashed lines in Figure 1, wherein the diameter of the inner cylinder 3 is smaller than the diameter of the inner hole 11 and the diameter of the outer cylinder 4 is larger than the diameter of the inner hole 11. The axial extension of the quasi-axisymmetric perturbation coils 22 is larger than the height of the torus shape, in particular the vessel 10 in axial direction. Due to the curved shape of the quasi-axisymmetric perturbation coils 22, end sections thereof, like 22C (see Figure 1) overlap in radius with the vessel 10.The quasi-axisymmetric perturbation coils 22 are configured, i.e., in particular their number, size, position, mutual arrangement and orientation, for subjecting the axisymmetric field equilibrium of the magnetic confinement field (as it would be created if the axisymmetric coils 21 were operated alone), to quasisymmetry-preserving perturbations, so that the magnetic confinement field created by all coils of the magnetic coil device 20 has quasi-axisymmetry and it has a quasi-axisymmetric stellarator configuration.The control device 30 for controlling the magnetic coil device 20 is coupled with all coils 21, 22 and optionally 23, in particular the coil current sources thereof (not shown). The control device 30 may comprise at least one computer circuit creating driving signals supplied to the coil current sources. The driving signals may be created in dependency on the configuration of the quasi-axisymmetric perturbation coils 22 and / or sensor signals of magnetic field sensors representing a current status of the magnetic confinement field 2.Figures 5 and 6 illustrate the arrangement of resonant magnetic perturbation coils 23 in a magnetic confinement apparatus 100 according to preferred embodiments of the invention. For clarity reasons, only the resonant magnetic perturbation coils 23 are shown, while the remaining configuration of the magnetic confinement apparatus 100 may be provided as described with reference to Figures 1 to 4.The resonant magnetic perturbation coils 23 comprise curved loops, preferably having a shape of saddle coils. They are arranged in a space axially adjacent to the torus shape, optionally with anoverlap with the torus shape in axial direction thereof, preferably above and / or be shape, in particular above and / or below the vessel (not shown in Figures 5 and 6)The resonant magnetic perturbation coils 23 are configured, i.e., in particular their number, size, position, mutual arrangement and orientation, for perturbing the axisymmetric field equilibrium of the magnetic confinement field towards quasi-axisymmetry additionally to the effect of the quasi-axisymmetric perturbation coils 22.Creating the quasi-axisymmetric stellarator configuration of the magnetic confinement apparatus by perturbing the axisymmetric field equilibrium is based on the following theoretical considerations applied by the inventors and implemented by numerical iterative procedures, as described below.Perturbed axisymmetric, quasi-axisymmetric equilibriaAs mentioned above, quasisymmetry-preserving perturbations to axisymmetric equilibria have been disclosed e.g., in

[0017] ,

[0018] ,

[0019] and

[0020] , The publication

[0020] is incorporated by reference to the present disclosure, in particular with regard to the construction of quasi-axisymmetric stellarator equilibria via perturbations of given axisymmetric equilibria.The construction of perturbed axisymmetric, quasi-axisymmetric equilibria may be described as follows. To represent a magneto-hydrodynamic (MHD) equilibrium one can use the magnetic flux IJJ (x), and the Boozer poloidal and toroidal angles 0(x) and <p(x) as function of the location, x, which is called "direct formulation". Alternatively, the location x(i , 0, <p) can be treated as the unknown function of the flux and the angles, which is called "inverse formulation". Using the inverse formulation one can perturb x with respect to a small parameter e:The zeroth order term corresponds to the unperturbed, axisymmetric equilibrium. Similarly, other quantities such as the rotational transform L are perturbed. One can now determine the different orders of the force balance equation: j x B = Vp(^) and of the MHD-constraint equation, obtained by setting the contravariant and covariant forms of the magnetic field to be equalThe zeroth order forms of the last two equations is equivalent to the Grad-Shafranov equation which is given in the direct formulation [21, 22],At first order, one obtains a set of four differential equations for the unknowns for Xi and Ki. Assuming the perturbation to be non-axisymmetric (non-zero toroidal mode number), the first order of the rotational transform Li is equal to zero. The external transform therefore may be expected to come in at higher order, which may make it surprising that significant rotational transform can be generated by small perturbations in the field strength.The quasi-symmetry enters the equations by enforcing, at a given order, that dB2 / d<f> = 0 which is equivalent to dJ / d<f> = 0 where J is the Jacobian of the Boozer coordinates.One advantage of using the perturbed axisymmetry method is that quasi-axisymmetry can be achieved with a high number of field periods and / or a small aspect ratio. The alternative approach of using a near axis expansion seems to be limited to either small field period numbers (2 or 3) or high aspect ratios,

[0023] , For the problem of perturbed axisymmetry, the low aspect ratio, high-N limit is accessible but characterized by the curious feature that the perturbations localize on the inboard. As argued in Appendix E of

[0019] , this feature may be traced to solubility of the in-surface part of the MHD constraint equation, which implies the inboard-to-outboard ratio of the amplitude of the perturbation scales roughly asThis factor is clearly large at low aspect ratio (a / R -> 1) or large N. Low aspect ratio, high field period number QA equilibria have some potential advantages, beyond those due to their compactness, including the ability to generate significant external rotational transform with a small deviation from axisymmetry, and the preservation of axisymmetry for deeply trapped particles, which reside on the outboard side of the torus.The perturbed axisymmetric equilibrium may be obtained by solving the set of four coupled first order partial differential equations using the least squares finite element method as described in the publication

[0020] , The plasma boundary shape is obtained from this solution, which can beused along with the plasma profiles, e.g. toroidal current density and plasma pres: tions of radius, as input to a plasma equilibrium solver such as the VMEC code to obtain a standard numerical description of the equilibrium that can be used for the purposes of coil optimization using one of several optimization software frameworks like SIMSOPT or STELLOPT.One example of such a perturbed axisymmetric equilibrium which is implemented with the inventive technique is shown in Figure 7, which shows a poloidal cross section of the perturbed boundary 5, the axisymmetric boundary 6 and an ellipse 7 that would fit within the perturbed boundary 5. The perturbed axisymmetric equilibrium is stellarator symmetric with a number of field periods of 4, a rotational transform profile shown in Figure 8, and with a net toroidal plasma current ltorof about 574 kA. Figure 8 shows the rotational transform, L, of the targeted plasma boundaries versus normalized flux s. The unperturbed case is represented by the dashed line and the perturbed case is represented by the solid line. From a geometrical investigation, it can be seen that the aspect ratio achieved for the axisymmetric boundary fitting within the QA boundary is much smaller than for the conventional QA designs with a value of approximately 3, and therefore the "wasted" volume for the tokamak equilibrium is reduced in advantageous manner.In the following, this presented perturbed boundary, e.g., as shown in Figure 7, is used for configuring the quasi-axisymmetric perturbation coils 22 for the inventive quasi-antisymmetry-stellara- tor-tokamak hybrid.Designing the quasi-axisymmetric perturbation coilsFor providing the configuration of the quasi-axisymmetric perturbation coils 22 and optionally the resonant magnetic perturbation coil 23, the inventors employed the second stage of the known two-stage approach, as described in

[0024] , in particular for providing a near-axisymmetric solution. The inventors have found that the near-axisymmetric solution may be sufficient for obtaining a practical design of the magnetic confinement apparatus 100.For the coil optimization in the two-stage approach, the coil optimization features of the optimization framework SIMSOPT [25, 26] may be used. The optimization process targets, as provided by the two-stage approach, the quadratic-flux error functional, [24, 27, 28, 29, 30, 31]:where BEis the external magnetic field produced by the coils. BEnis the normal co with respect to the plasma boundary. Dnis the targeted normal component of the external magnetic field, and ds is the surface area element for the surface integral. In vacuumDn= 0, since the plasma boundary is a flux surface. If one investigates finite beta plasmas, as it is sufficient for the design of the quasi-axisymmetric perturbation coils 22, the magnetic field produced by plasma currents is determined to be able to calculate Dn. The so-called virtual casing method may be used, as described in

[0032] , In addition, the length of the coils L =Lj is targeted with a scalar penalty functional Q:where Ltis a user specific target value for the sum of the compound coil length. The target functional is kept minimal with only two terms since there is an interest in determining if it is possible to find suitable coils at all. Once feasible coils are established, the targets may be refined more.The axisymmetric coils 21 (tokamak coils) are designed as follows. Axisymmetric equilibria require a net toroidal plasma current to produce rotational transform and flux surfaces. Due to this net toroidal current, the targeted normal component of the external magnetic field Dn, on the plasma boundary has a substantial vertical fraction, which may be cancelled by combining the axisymmetric coils 21 with poloidal field coils or by providing tilted axisymmetric coils 21 (not shown in the figures).It is possible to start with four poloidal field coils where only the radius and the height z are allowed to vary. Next, a plurality of axisymmetric coils 21, e.g., more than 10, like 20 axisymmetric coils 21, are added. The relatively large number of axisymmetric coils 21 allows to reduce coil ripple effects since there is main interest in the geometry of the additional quasi-axisymmetric perturbation coils 22. Once feasible quasi-axisymmetric perturbation coils 22 are found, the number of axisymmetric coils 21 may be reduced. The axisymmetric coils 21 do not need to be optimized to achieve the desired normal component of the magnetic field on the plasma boundary since their total current is determined by the given equilibrium, but they may be optimized with respect to length, toroidal volume filled with magnetic fields, and coil ripple effects. Also additional coils may be added, e.g. for a conventional divertor. This can be done after finding the quasi-axisymmetric coils. Preferably for the quasi-axisymmetric coil optimization, they can be chosen to be relatively large in radius to minimize any coil ripple effects. Alternatively, a current sheet like with NESCOIL

[0027] or REGCOIL

[0031] could be used. This would have eradicated any coil ripple effects.However, in this case, the freedom to use the coil current as an independent degr for the quasi-axisymmetric perturbation coil optimization would be lost.For designing the quasi-axisymmetric perturbation coils 22, the chosen perturbed axisymmetric equilibrium may have e.g., four field periods and is stellarator symmetric. The optimization may be started with either 8 or 4 saddle coils at the inboard side as an initial guess for the additional quasi-axisymmetric perturbation coils 22. While the geometry of the quasi-axisymmetric perturbation coils 22 is optimized, the geometry of the axisymmetric coils 21 and optional poloidal field coils is restricted. In principle the geometry of the axisymmetric coils 21 could be also used to improve the quasi-axisymmetric perturbation coils 22.To evaluate the quality of quasi-symmetry obtained, a couple of measures may be used. The first is the quasi-symmetric error defined as:where the Bm,nare the Fourier coefficients of the magnetic field in Boozer coordinates. Another measure is the effective ripple eeff, which is a proxy for the neoclassical transport

[0033] , To aid comparison the effective ripple may be presented to the power 3 / 2. ROSE / ONSET suites of codes may be used to evaluate the quasi-symmetric error EQA and the effective ripple eeff [34, 28], The third measure has been used to optimize for quasi-symmetry

[0013] and is an alternative quasi-axisymmetric error:The brackets (• • •) indicate flux surface averages. The sum is over a set of flux surfaces sp, N and M determine the helicity of the quasi-symmetry, e.g. B = B(s, MB - Mp), so for quasi-axisymmetry N = 0. 2nG uo is the poloidal current, and 2nl / i0is the toroidal current. This expression for / qs is exactly zero for a quasi-symmetric field.Using free-boundary VMEC (Variational Moments Equilibrium Code) with the magnetic field produced by the coils, the new plasma boundary and field surfaces are obtained. To test if the quasi- axisymmetry properties are maintained with the realization with the coils, the quasi-axisymmetric error may be determined from these VMEC equilibria.It is not guaranteed that flux surfaces exist in vacuum

[0011] , To evaluate the existence, shape, and volume of the flux surfaces, Poincare plots are generated. To generate the Poincare plots, thequasi-axisymmetric error plots and the effective ripple, the ROSE / ONSET suites [3' used.In numerical tests, the inventors found a set of axisymmetric coils 21 and poloidal field coils which can approximately reproduce the axisymmetric equilibrium. In addition to those coils, the inventors found that a single type of quasi-axisymmetric perturbation coils 22 is sufficient to reproduce the perturbed QA equilibrium.The inventors investigated a number of different sets of quasi-axisymmetric perturbation coils 22 found by the coil optimization. All of them are vessel-unlinked modular coils and located on the inboard side. They are not required to be interlinked (with each other; see Figure 4A), but can be. The designs of quasi-axisymmetric perturbation coils 22 provided by the invention comprise in particular embodiments with non-interlinked coils, e.g., 8 identical quasi-axisymmetric perturbation coils 22 without resonant magnetic perturbation coils 23, 8 identical quasi-axisymmetric perturbation coils 22 with resonant magnetic perturbation coils 23, or 4 identical quasi-axisymmetric perturbation coils 22, and with interlinked coils, e.g., 8 pairwise-interlinked quasi-axisymmetric perturbation coils 22, as described below with further details.For all cases, the effective ripple, as shown in Figure 9, and the quasi-axisymmetric errors, as shown in Figure 10 are small. With more details, Figure 9 shows the effective ripple to the power of 3 / 2 vs the normalized toroidal flux s for several coil geometry embodiments. Figure 6 illustrates the quasi-axisymmetry error EQ,A mentioned above, vs normalized toroidal flux s for the same coil geometry embodiments. Curves a, b and c of Figures 9 and 10 share the same interlinked coils set including four pairs of interlinked quasi-axisymmetric perturbation coils 22 distributed as shown in Figure 3B, but differ in plasma profiles: original pressure and current (curve a), vacuum case (curve b), reduced current (curve c). The other cases are non-interlinked coils. Curves d and e of Figures 9 and 10 are non-interlinked coils, without and with the resonant magnetic perturbation coils 23 included (Figures 5 and 6), while the curve f gives an alternative with just four non-interlinked quasi-axisymmetric perturbation coils 22, instead of 8. The inventors found flux surfaces in vacuum for all of the presented coils with Poincare plots.The sets of quasi-axisymmetric perturbation coils 22 with non-interlinked coils comprise e.g., 8 or 4 identical quasi-axisymmetric perturbation coils 2 being arranged as shown in Figure 3B with 4 coils which wrap around at the inboard side without being interlinked with each other. The arrangement of 8 coils is similar to Figure 3B, because the 8 coils comprise 4 pairs of stellarator-symmetric counterparts, wherein the distance between coils of each pair is quite i might open up the possibility to mount them together. Such a pair can be considered as a stellarator-symmetric pair. Accordingly, with the embodiment having 8 non-interlinked quasi-axisymmet- ric perturbation coils 22, each coil 22 of Figure 3B is replaced by a stellarator-symmetric pair.Even though these 8 quasi-axisymmetric perturbation coils 22 achieve small effective ripple and can produce vacuum flux surfaces (see Figures 9, 10), the maximum quadratic flux error can be improved by adding the resonant magnetic perturbation coils 23 on top and / or bottom of the plasma, see Figures 5 and 6.For the finite beta equilibrium, the QA error improves for most of the plasma (Figure 10) whereas the effective ripple eeff only improves for normalized flux values of below 0.25%. The effective ripple varies between 6.4xl0“5and 0.026 (e3 / 2ranges from 5.4 x 10“7and 4.3 x 10“3), see Figure 9. Interestingly, it is in the range of previous optimized quasi-symmetric designs,

[0013] , The case without the additional resonant magnetic perturbation coils 23 also lies in a range of 1.9 x 10“4to 8.0 x 10“3(e3 / 2ranges from 2.5 x 10“6and 7.2 x 10“4) and still fits in a good range of optimized stellarators.With the set of 8 quasi-axisymmetric perturbation coils 22 coils and the additional resonant magnetic perturbation coils 23, the vacuum flux surfaces shown in Figure 11 are obtained. The volume is reduced compared to finite beta equilibrium and the rotational transform ranges from 0.26 to 0.19 which is a large fraction of the total rotational transform at finite beta.If the number of quasi-axisymmetric perturbation coils 22 is reduced to 4, as shown in Figure 3B, advantageously for more space between the coils is required while achieving approximately similar values for the quasi-symmetry, Figure 10, and the effective ripple, Figure 9. The contour lines of the magnetic field strength look nearly straight everywhere but near the inboard midplane and the Fourier harmonics also confirm the quasisymmetry feature. This feature is illustrated in Figure 12, which shows the magnetic field strength in Boozer coordinates for the four coil set of quasi- axisymmetric perturbation coils 22. According to Figure 12A, the contour lines of the magnetic field strength with respect to Boozer angles near the plasma boundary are shown, and in Figure 12B, Fourier harmonics of the magnetic field strength in Boozer coordinates are shown.An embodiment of quasi-axisymmetric perturbation coils 22 with pairwise-interlinked coils (Figure 4A) is illustrated in Figure 3C. This set of quasi-axisymmetric perturbation coils 22 comprises 8pairwise-interlinked coils, which are also located on the inboard side of the torus < beta, both the effective ripple and the quasi-axisymmetric error are comparable with the other quasi-axisymmetric perturbation coil designs, see Figures 9 and 10.As shown in Figure 13, the equilibrium determined with the free-boundary VMEC resembles the feature of the boundary found from the analytic theory (Figure 7): the perturbation compared to the tokamak is dominantly on the inboard side and the outboard side is mostly unaltered. Furthermore, Figure 14 shows that this coil set possesses, as the other coil designs, flux surfaces in vacuum. For the Poincare plots of Figures 14A to 14C, the coil current of the poloidal field coils is set to zero. For the original current of the quasi-axisymmetric perturbation coils 22, the vacuum rotational transform ranges from approximately 0.24 and 0.22. Since it is reaching a low-order rational at the edge (ledge ~ 4 / 18 = 2 / 9) the region becomes stochastic further outside. This explains why the flux surfaces may have a maximum vertical height zmaxof below 0.3. By alternating the ratio of quasi-axisymmetric perturbation coil current to axisymmetric coil current, the rotational transform can be changed. If one increases the quasi-axisymmetric perturbation coil current, the rotational transform also increases but crosses other low-order rationals. If one reduces, on the other hand, the quasi-axisymmetric perturbation coil current in vacuum one can increase the volume with flux surfaces while decreasing the rotational transform, see Figure 14D to 14E. With this reduced quasi-axisymmetric perturbation coil current, the rotational transform stays below 0.1 and the aspect ratio reaches 2.35.Under vacuum conditions with the reduced quasi-axisymmetric perturbation coil current, the effective ripple stays below 1% throughout the plasma (see Figure 9, curve b). At finite beta, a new equilibrium may be found, wherein the effective ripple reaches the smallest values with the reduced quasi-axisymmetric perturbation coil current (see Figure 9). This can be explained by the smaller effect of the perturbation due to a smaller relative quasi-axisymmetric perturbation coil current. The quasi-symmetric error seems, however, to be increased for most of the plasma compared to the other designs (Figure 10), but the magnetic field strength seems to still have mainly quasi-axisymmetric features except for the inboard midplane, as shown in Figure 15. According to Figure 15A, similar to Figure 12, the contour lines of the magnetic field strength with respect to Boozer angles near the plasma boundary are shown, and in Figure 15B, Fourier harmonics of the magnetic field strength in Boozer coordinates are shown.For showing how the neoclassical properties would change going from a tokamak to a quasi- axisymmetry stellarator, the change of the effective ripple may be examined. If a set of equilibriaare generated by slowly increasing the current in the quasi-axisymmetric perturbs fixing the pressure, the plasma current profile and the other coil currents, the rotational transform slowly increases at the edge from 0.46 to 0.54. This is expected, since the 3D shaping adds to the rotational transform, [35, 36], While the rotational transform increases, the quasi-axisymmetric error , given above, monotonically increases from approximately 10“7to 4 * 10“3. As a reference, the equilibrium obtained from the analytic theory has a quasi-axisymmetric error of 2.7 * 10“3without any coil ripple effects. This increase in the quasi-axisymmetric error can be explained from the fact that in a perfectly axisymmetric equilibrium the quasi-axisymmetric error is zero. The added analytical perturbation was only derived to be quasi-symmetric at first order. The quasi-axisymmetric error is therefore expected to be non-zero for any finite perturbation, scaling with the square of its amplitude.In the following, particular further advantages of the invention are summarized. A new compact quasi-axisymmetric stellarator-tokamak hybrid design is presented with a low aspect ratio of below 2.5 for both the tokamak scenario and the quasi-axisymmetric stellarator. Only one type of quasi-axisymmetric stellarator coils is needed in addition to typical tokamak coils to generate the quasi-axisymmetric stellarator. Good quasi-axisymmetry is maintained even in vacuum with good flux surfaces.The inventive magnetic confinement apparatus is a suitable system for studying how 3D shaping affects plasma properties, such as stability, while approximately maintaining neoclassical transport. In addition, it allows investigation of a new start-up concept, whereby one can ramp-up the machine with the vacuum flux surfaces of the quasi-axisymmetric coil currents. This quasi-axisymmetric vacuum start-up provides an alternative scenario compared to other tokamak start-up mechanisms such as the Ohmically-heated one, e. g. in operating a fusion reactor apparatus.Each of the above embodiment sets of quasi-axisymmetric perturbation coils 22 has specific advantages: The set of 8 non-interlinked coils have advantages since they are easier to build compared to the interlinked coils. In addition, they provide more flexibility than the set of 4 non-inter- linked quasi-axisymmetric perturbation coils 22 since one could control the coil currents in the coil pairs individually and therefore doubles the freedom in quasi-axisymmetric perturbation coil currents. The interlinked coils achieve the lowest effective ripple in the center of the plasma and in the vacuum case (see Figure 9). In addition, they allow for more flexibility similarly to the noninterlinked coils. The four identical coils provide the configuration with lowest complexity for two reasons: Firstly, one only has to produce only 4 additional coils compared to a tokamak. Secondly,the spacing between coils is enlarged compared to the other quasi-axisymmetric f coils. In addition, the effective ripple achieves the smallest values for higher values of the normalized toroidal flux while still achieving very small values nearer the axis. Advantageously, since the quasi-axisymmetric perturbation coils are only on the inboard side of the torus shape, it is possible to "upgrade" an existing tokamak. Any aspect ratio of a tokamak can be generated with the inventive technique as long as the QA coils have enough space in the center of the torus shape. The inventive technique is comparable in compactness with previous tokamak-stellarator hybrids including the so-called "spherical stellarator" [7] but with transport opti- mization included via quasi-symmetry.The features of the invention disclosed in the above description, the drawings and the claims can be of significance both individually as well as in combination or sub-combination for the realization of the invention in its various embodiments. The invention is not restricted to the preferred embodiments described above. Rather a plurality of variants and derivatives is possible which also use the inventive concept and therefore fall within the scope of protection. In addition, the invention also claims protection for the subject and features of the sub claims independently of the features and claims to which they refer.

Claims

Claims1. Magnetic confinement apparatus (100), comprising- an evacuable vessel (10) having a torus shape with an inner torus hole (11) and being configured for accommodating a fusion plasma (1),- a magnetic coil device (20) being configured for creating a magnetic confinement field (2) in the vessel (10), wherein the magnetic coil device (20) comprises a plurality of axisymmetric coils (21) being configured for creating the magnetic confinement field (2) with an axisymmetric field equilibrium, and- a control device (30) being configured for controlling the magnetic coil device (20), characterized in that- the magnetic coil device (20) further comprises at least two quasi-axisymmetric perturbation (QAP) coils (22) being placed outside the vessel (10) within the torus hole (11) of the torus shape and being arranged for subjecting the axisymmetric field equilibrium of the magnetic confinement field (2) to quasisymmetry-preserving perturbations, so that the magnetic confinement field (2) created by the magnetic coil device (20) has quasi-axisymmetry and a magnetic equilibrium of the magnetic confinement field has a quasi-axisymmetric stellarator configuration of the magnetic confinement apparatus (100).

2. Magnetic confinement apparatus according to claim 1, wherein- each quasi-axisymmetric perturbation coil (22) is a vessel-unlinked modular coil extending along a curved surface extending outside the vessel (10) without encircling the vessel (10).

3. Magnetic confinement apparatus according to claim 2, wherein- the vessel unlinked-modular coils are similar or greater in size to an axial extent of the vessel (10).

4. Magnetic confinement apparatus according to one of the foregoing claims, wherein- the at least two quasi-axisymmetric perturbation coils (22) have a twisted arrangement, wherein inner concave portions of the quasi-axisymmetric perturbation coils (22) along a first main coil extension thereof face to each other.

5. Magnetic confinement device according to one of the foregoing claims,- the at least two quasi-axisymmetric perturbation coils (22) are situated within the inner torus hole (11) of the torus shape, wherein the quasi-axisymmetric perturbation coils (22) are arranged such that they could be included between two co-axial cylinders having their axes parallel to a central axis (z) of the torus shape.

6. Magnetic confinement apparatus according to one of the foregoing claims, wherein- the at least two quasi-axisymmetric perturbation coils (22) are arranged exclusively inside the inner torus hole (11) of the torus shape.

7. Magnetic confinement apparatus according to one of the foregoing claims, wherein- all of the at least two quasi-axisymmetric perturbation coils (22) have an equal shape and size.

8. Magnetic confinement apparatus according to one of the foregoing claims, wherein- the at least two quasi-axisymmetric perturbation coils (22) are non-interlinked coils.

9. Magnetic confinement apparatus according to one of the claims 1 to 7, wherein- the at least two quasi-axisymmetric perturbation coils (22) comprise at least one interlinked coil pair (24).

10. Magnetic confinement apparatus according to one of the foregoing claims, wherein- the magnetic coil device (20) further comprises at least one resonant magnetic perturbation coil (23) being placed outside the torus shape and being configured for acting in combination with the quasi-axisymmetric perturbation coils (22) so that the magnetic confinement field (2) created by the magnetic coil device (20) has the quasi-axisymmetry.

11. Magnetic confinement apparatus according to one of the foregoing claims, wherein- the number of said quasi-axisymmetric perturbation coils (22) is between 2 and 10.

12. Magnetic confinement apparatus according to one of the foregoing claims, wherein- the magnetic confinement apparatus (100) is configured to be operated without the need of a tokamak solenoid coil.

13. Magnetic confinement apparatus according to one of the foregoing claii- the magnetic confinement field (2) having the quasi-axisymmetric stellarator configuration has an aspect ratio equal to or below 3.

14. Magnetic confinement apparatus according to one of the foregoing claims, wherein- the at least two quasi-axisymmetric perturbation coils (22) are configured such that a quadraticflux error functional according tois optimized in an optimization framework, wherein BE,nis a normal component of the magnetic field produced by the quasi-axisymmetric perturbation coils (22) with respect to the plasma boundary, Dnis the targeted normal component of the magnetic field.

15. Fusion reactor apparatus (200), including a magnetic confinement apparatus (100) according to one of the foregoing claims.

16. Method of operating a magnetic confinement apparatus according to one of the claims 1 to 14, comprising the steps of- operating the axisymmetric coils (21) for creating the magnetic confinement field (2) having the axisymmetric field equilibrium, and- operating the at least two quasi-axisymmetric perturbation coils (22) for subjecting the axisymmetric equilibrium of the magnetic confinement field (2) to the quasisymmetry-preserving perturbations, so that the magnetic confinement field (2) has the quasi-axisymmetric stellarator configuration.

17. Method according to claim 16, wherein- by controlling the at least two quasi-axisymmetric perturbation coils (22), the configuration of the magnetic confinement apparatus is varied between an axisymmetric tokamak configuration and the quasi-axisymmetric stellarator configuration.

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