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This theme issue collects together papers summarising the conceptual design of the Spherical Tokamak for Energy Production (STEP). In 2019, the UK government funded the first design stages of a prototype fusion powerplant based on a compact toroidal geometry, called STEP. The primary technical aims of STEP are to produce net energy, to be self-sufficient in tritium fuel and to demonstrate a maintenance regime that would extrapolate to appropriate availability for commercial powerplants. After 5 years and over 1000 person-years of detailed scientific and engineering conceptual design, this theme issue acts as a compendium of the current design basis for STEP, noting that this is a snapshot in time and that the design will continue to evolve. This article is part of the theme issue 'Delivering Fusion Energy – The Spherical Tokamak for Energy Production (STEP)'. [ABSTRACT FROM AUTHOR]

We have extended our study of the competition between the drive and stabilization of plasma microinstabilities by sheared flow to include electromagnetic effects at low plasma β (the ratio of plasma to magnetic pressure). The extended system of characteristic equations is formulated, for a dissipative fluid model developed from the gyrokinetic equation, using a twisting mode representation in sheared slab geometry and focusing on the ion temperature gradient mode. Perpendicular flow shear convects perturbations along the field at the speed we denote as Mcs (where cs is the sound speed). is required to make the system characteristics unidirectional and inhibit eigenmode formation, leaving only transitory perturbations in the system. This typically represents a much larger flow shear than in the electrostatic case, which only needs M > 1. Numerical investigation of the region shows the driving terms can conflict, as in the electrostatic case, giving low growth rates over a range of parameters. Also, at modest drive strengths and low β values typical of experiments, including electromagnetic effects does not significantly alter the growth rates. For stronger flow shear and higher β, geometry characteristic of the spherical tokamak mitigates the effect of an instability of the shear Alfvén wave, driven by the parallel flow shear. [ABSTRACT FROM AUTHOR]

Calculations of tearing mode stability in tokamaks split conveniently into one in an external region, where marginally stable ideal magnetohydrodynamics (MHD) is applicable, and one in a resonant layer around the rational surface where sophisticated kinetic physics is needed. These two regions are coupled by the stability parameter Δ′. Axisymmetric pressure and current perturbations localized around the rational surface significantly alter Δ′. Equations governing the changes in the external solution and Δ′ are derived for arbitrary perturbations in axisymmetric toroidal geometry. These equations can be used in two ways: (i) the Δ′ can be calculated for a physically occurring perturbation to the pressure or current; (ii) alternatively we can use these equations to calculate Δ′ for profiles with a pressure gradient at the rational surface in terms of the value when the perturbation removes this gradient. It is the second application we focus on here since resistive magnetohydrodynamics (MHD) codes do not contain the appropriate layer physics and therefore cannot predict stability for realistic hot plasma directly. They can, however, be used to calculate Δ′. Existing methods (Ham et al 2012 Plasma Phys. Control. Fusion54 025009) for extracting Δ′ from resistive codes are unsatisfactory when there is a finite pressure gradient at the rational surface and favourable average curvature because of the Glasser stabilizing effect (Glasser et al 1975 Phys. Fluids18 875). To overcome this difficulty we introduce a specific artificial pressure flattening function that allows the earlier approach to be used. The technique is first tested numerically in cylindrical geometry with an artificial favourable curvature. Its application to toroidal geometry is then demonstrated using the toroidal tokamak tearing mode stability code T7 (Fitzpatrick et al 1993 Nucl. Fusion33 1533) which employs an approximate analytic equilibrium. The prospects for applying this approach to resistive MHD codes such as MARS-F (Liu et al 2000 Phys. Plasmas7 3681) which utilize a fully toroidal equilibrium are discussed. [ABSTRACT FROM AUTHOR]

This paper presents a complete theoretical framework for studying turbulence and transport in rapidly rotating tokamak plasmas. The fundamental scale separations present in plasma turbulence are codified as an asymptotic expansion in the ratio ϵ = ρi/a of the gyroradius to the equilibrium scale length. Proceeding order by order in this expansion, a set of coupled multiscale equations is developed. They describe an instantaneous equilibrium, the fluctuations driven by gradients in the equilibrium quantities, and the transport-timescale evolution of mean profiles of these quantities driven by the interplay between the equilibrium and the fluctuations. The equilibrium distribution functions are local Maxwellians with each flux surface rotating toroidally as a rigid body. The magnetic equilibrium is obtained from the generalized Grad–Shafranov equation for a rotating plasma, determining the magnetic flux function from the mean pressure and velocity profiles of the plasma. The slow (resistive-timescale) evolution of the magnetic field is given by an evolution equation for the safety factor q. Large-scale deviations of the distribution function from a Maxwellian are given by neoclassical theory. The fluctuations are determined by the ‘high-flow’ gyrokinetic equation, from which we derive the governing principle for gyrokinetic turbulence in tokamaks: the conservation and local (in space) cascade of the free energy of the fluctuations (i.e. there is no turbulence spreading). Transport equations for the evolution of the mean density, temperature and flow velocity profiles are derived. These transport equations show how the neoclassical and fluctuating corrections to the equilibrium Maxwellian act back upon the mean profiles through fluxes and heating. The energy and entropy conservation laws for the mean profiles are derived from the transport equations. Total energy, thermal, kinetic and magnetic, is conserved and there is no net turbulent heating. Entropy is produced by the action of fluxes flattening gradients, Ohmic heating and the equilibration of interspecies temperature differences. This equilibration is found to include both turbulent and collisional contributions. Finally, this framework is condensed, in the low-Mach-number limit, to a more concise set of equations suitable for numerical implementation. [ABSTRACT FROM AUTHOR]

First-principles numerical simulations are used to describe a transport bifurcation in a differentially rotating tokamak plasma. Such a bifurcation is more probable in a region of zero magnetic shear than one of finite magnetic shear, because in the former case the component of the sheared toroidal flow that is perpendicular to the magnetic field has the strongest suppressing effect on the turbulence. In the zero-magnetic-shear regime, there are no growing linear eigenmodes at any finite value of flow shear. However, subcritical turbulence can be sustained, owing to the existence of modes, driven by the ion temperature gradient and the parallel velocity gradient, which grow transiently. Nonetheless, in a parameter space containing a wide range of temperature gradients and velocity shears, there is a sizeable window where all turbulence is suppressed. Combined with the relatively low transport of momentum by collisional (neoclassical) mechanisms, this produces the conditions for a bifurcation from low to high temperature and velocity gradients. A parametric model is constructed which accurately describes the combined effect of the temperature gradient and the flow gradient over a wide range of their values. Using this parametric model, it is shown that in the reduced-transport state, heat is transported almost neoclassically, while momentum transport is dominated by subcritical parallel-velocity-gradient-driven turbulence. It is further shown that for any given input of torque, there is an optimum input of heat which maximises the temperature gradient. The parametric model describes both the behaviour of the subcritical turbulence (which cannot be modelled by the quasi-linear methods used in current transport codes) and the complicated effect of the flow shear on the transport stiffness. It may prove useful for transport modelling of tokamaks with sheared flows. [ABSTRACT FROM AUTHOR]

High-beta (β>ε/q2) large-aspect-ratio (ε<1) tokamak equilibria are shown to be always stable to ideal magnetohydrodynamics (MHD) modes that are localized about a flux surface. Both the ballooning and interchange modes are shown to be stable. This work uses the analytic high-beta large-aspect-ratio tokamak equilibria developed by Cowley et al. [Phys. Fluids B 3, 2069 (1991)], which are valid for arbitrary pressure and safety factor profiles. The stability results make no assumption about these profiles nor the shape of the boundary. [ABSTRACT FROM AUTHOR]

An analytic solution of the high-beta (∈&βmacr;[sub p] ∼ βq² / ∈ » 1) equilibrium of a large aspect ratio tokamak is presented. Two arbitrary flux functions, the pressure profile p(ψ) and the safety factor profile q(ψ), specify the equilibrium. The solution splits into two asymptotic regions: the core region where ψ is a function of the major radius alone and a narrow boundary layer region adjoining the conducting wall. The solutions in the two regions are asymptotically matched to each other. For monotonic pressure profiles, the Shafranov shift is equal to the minor radius. For beta much bigger than 1, the solution contains a region (in place of the magnetic axis) of zero magnetic field and constant pressure. At high beta the quantity β[sub I] which is essentially proportional to the pressure over the total current squared, is largely independent of pressure. The important ramifications of limited β[sub I] for high-beta reactors are discussed. Generalizations to shaped cross sections and hollow pressure profiles are outlined. The problem of equilibrium reconstruction in the high-beta regime is also considered. [ABSTRACT FROM AUTHOR]

Toroidal calculations on m = 2 magnetic feedback are presented. These calculations are performed using a coupled Δ' formulation and show that the feedback can have a pronounced effect on the sideband (m = 1 and 3) content. [ABSTRACT FROM AUTHOR]

The theory of tokamak stability to nonlinear 'ballooning' displacements of elliptical magnetic flux tubes is presented. Above a critical pressure profile the energy stored in the plasma may be lowered by finite (but not infinitesimal) displacements of such tubes (metastability). Above a higher pressure profile, the linear stability boundary, such tubes are linearly and nonlinearly unstable. The predicted saturated flux tube displacement can be of the order of the pressure gradient scale length. Plasma transport from these displaced flux tubes may explain the rapid loss of confinement in some experiments. [ABSTRACT FROM AUTHOR]

A numerical study is carried out, based on a simple toroidal tokamak equilibrium, to demonstrate the radial re-distribution of the electromagnetic torque density, as a result of a rotating resistive plasma (linear) response to a static resonant magnetic perturbation field. The computed electromagnetic torque peaks at several radial locations even in the presence of a single rational surface, due to resonances between the rotating response, in the plasma frame, and both Alfvén and sound continuum waves. These peaks tend to merge together to form a rather global torque distribution, when the plasma resistivity is large. The continuum resonance induced net electromagnetic torque remains finite even in the limit of an ideal plasma. [ABSTRACT FROM AUTHOR]

Theoretically, high-β (β∼1) tokamaks offer a large fusion power efficiency advantage over low-β devices. However, if high-β tokamaks are inherently unstable then such devices will never be realized. In particular, kink modes are thought to be the most serious obstacle to high-β operations. High-β tokamaks are characterized by a very large Shafranov shift with a thin "boundary layer" on the outboard side of the device and a large "core" region of vertical flux surfaces comprising most of the central volume. In this paper, the energy principle is used to compute the magnetohydrodynamic internal kink stability of such devices in the large aspect ratio limit with a low toroidal mode number. A class of internal kink mode similar to the usual low-β kink is present; the stability against these modes is computed. A set of parameters describing a kink stable high-β equilibrium is given. Stability is shown to be dependent on the shape of the plasma boundary. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Sheared toroidal flows can cause bifurcations to zero-turbulent-transport states in tokamak plasmas. The maximum temperature gradients that can be reached are limited by subcritical turbulence driven by the parallel velocity gradient. Here it is shown that q/є (magnetic field pitch/inverse aspect ratio) is a critical control parameter for sheared tokamak turbulence. By reducing q/є, far higher temperature gradients can be achieved without triggering turbulence, in some instances comparable to those found experimentally in transport barriers. The zero-turbulence manifold is mapped out, in the zero-magnetic-shear limit, over the parameter space (γE, q/є, R/LT), where γE is the perpendicular flow shear and R/LT is the normalized inverse temperature gradient scale. The extent to which it can be constructed from linear theory is discussed. [ABSTRACT FROM AUTHOR]