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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]

In this paper, we extend the multiscale approach developed in Abel et al (2012 Rep. Prog. Phys. submitted) by exploiting the scale separation between ions and the electrons. The gyrokinetic equation is expanded in powers of the electron to ion mass ratio, which provides a rigorous method for deriving the reduced electron model. We prove that ion-scale electromagnetic turbulence cannot change the magnetic topology, and argue that to lowest order the magnetic field lies on fluctuating flux surfaces. These flux surfaces are used to construct magnetic coordinates, and in these coordinates a closed system of equations for the electron response to ion-scale turbulence is derived. All fast electron timescales have been eliminated from these equations. We also use these magnetic surfaces to construct transport equations for electrons and for electron heat in terms of the reduced electron model. [ABSTRACT FROM AUTHOR]

Differential rotation is known to suppress linear instabilities in fusion plasmas. However, numerical experiments show that even in the absence of growing eigenmodes, subcritical fluctuations that grow transiently can lead to sustained turbulence, limiting the ability of the velocity shear to suppress anomalous transport. Here transient growth of electrostatic fluctuations driven by the parallel velocity gradient (PVG) and the ion temperature gradient (ITG) in the presence of a perpendicular (E × B) velocity shear is considered. The maximally simplified (but, as numerical simulations suggest, most promising for transport reduction) case of zero magnetic shear is treated in the framework of a local shearing box approximation. In this case there are no linearly growing eigenmodes, so all excitations are transient. In the PVG-dominated regime, the maximum amplification factor is found to be eN with N &agr; q/∈ (safety factor/inverse aspect ratio), the maximally amplified wavenumbers perpendicular and parallel to the magnetic field are related by kyρi ≈ (∈/q)1/3k∥vth i/S, where ρi is the ion Larmor radius, vth i the ion thermal speed and S the E × B shear. In the ITG-dominated regime, N is independent of wavenumber and N &agr; vth i/(LT S), where LT is the ion-temperature scale length. Intermediate ITG-PVG regimes are also analysed and N is calculated as a function of q/∈, LT and S. Analytical results are corroborated and supplemented by linear gyrokinetic numerical tests. Regimes with N ≲ 1 for all wavenumbers are possible for sufficiently low values of q/∈ (≲7 in our model); ion-scale turbulence is expected to be fully suppressed in such regimes. For cases when it is not suppressed, an elementary heuristic theory of subcritical PVG turbulence leading to a scaling of the associated ion heat flux with q, ∈, S and LT is proposed; it is argued that the transport is much less 'stiff' than in the ITG regime. [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]

ABSTRACT Weakly collisional magnetized cosmic plasmas have a dynamical tendency to develop pressure anisotropies with respect to the local direction of the magnetic field. These anisotropies trigger plasma instabilities at scales just above the ion Larmor radius ρi and much below the mean free path λmfp. They have growth rates of a fraction of the ion cyclotron frequency, which is much faster than either the global dynamics or even local turbulence. Despite their microscopic nature, these instabilities dramatically modify the transport properties and, therefore, the macroscopic dynamics of the plasma. The non-linear evolution of these instabilities is expected to drive pressure anisotropies towards marginal stability values, controlled by the plasma beta βi. Here this non-linear evolution is worked out in an ab initio kinetic calculation for the simplest analytically tractable example - the parallel ( k⊥= 0) firehose instability in a high-beta plasma. An asymptotic theory is constructed, based on a particular physical ordering and leading to a closed non-linear equation for the firehose turbulence. In the non-linear regime, both the analytical theory and the numerical solution predict secular ( ∝ t) growth of magnetic fluctuations. The fluctuations develop a k−3∥ spectrum, extending from scales somewhat larger than ρi to the maximum scale that grows secularly with time ( ∝ t1/2); the relative pressure anisotropy ( p⊥− p∥)/ p∥ tends to the marginal value −2/βi. The marginal state is achieved via changes in the magnetic field, not particle scattering. When a parallel ion heat flux is present, the parallel firehose mutates into the new gyrothermal instability (GTI), which continues to exist up to firehose-stable values of pressure anisotropy, which can be positive and are limited by the magnitude of the ion heat flux. The non-linear evolution of the GTI also features secular growth of magnetic fluctuations, but the fluctuation spectrum is eventually dominated by modes around a maximal scale ∼ρi lT/λmfp, where lT is the scale of the parallel temperature variation. Implications for momentum and heat transport are speculated about. This study is motivated by our interest in the dynamics of galaxy cluster plasmas (which are used as the main astrophysical example), but its relevance to solar wind and accretion flow plasmas is also briefly discussed. [ABSTRACT FROM AUTHOR]

We consider the problem of self-regulated heating and cooling in galaxy clusters and the implications for cluster magnetic fields and turbulence. Viscous heating of a weakly collisional magnetized plasma is regulated by the pressure anisotropy with respect to the local direction of the magnetic field. The intracluster medium is a high-beta plasma, where pressure anisotropies caused by the turbulent stresses and the consequent local changes in the magnetic field will trigger very fast microscale instabilities. We argue that the net effect of these instabilities will be to pin the pressure anisotropies at a marginal level, controlled by the plasma beta parameter. This gives rise to local heating rates that turn out to be comparable to the radiative cooling rates. Furthermore, we show that a balance between this heating and bremsstrahlung cooling is thermally stable, unlike the often conjectured balance between cooling and thermal conduction. Given a sufficient (and probably self-regulating) supply of turbulent power, this provides a physical mechanism for mitigating cooling flows and preventing cluster core collapse. For observed density and temperature profiles, the assumed balance of viscous heating and radiative cooling allows us to predict magnetic field strengths, turbulent velocities and turbulence scales as functions of distance from the centre. Specific predictions and comparisons with observations are given for several different clusters. Our predictions can be further tested by future observations of cluster magnetic fields and turbulent velocities. [ABSTRACT FROM AUTHOR]

Two-dimensional gyrokinetics is a simple paradigm for the study of kinetic magnetised plasma turbulence. In this paper, we present a comprehensive theoretical framework for this turbulence. We study both the inverse and direct cascades (the ‘dual cascade’), driven by a homogeneous and isotropic random forcing. The key characteristic length of gyrokinetics, the Larmor radius, divides scales into two physically distinct ranges. For scales larger than the Larmor radius, we derive the familiar Charney–Hasegawa–Mima equation from the gyrokinetic system, and explain its relationship to gyrokinetics. At scales smaller than the Larmor radius, a dual cascade occurs in phase space (two dimensions in position space plus one dimension in velocity space) via a nonlinear phase-mixing process. We show that at these sub-Larmor scales, the turbulence is self-similar and exhibits power-law spectra in position and velocity space. We propose a Hankel-transform formalism to characterise velocity-space spectra. We derive the exact relations for third-order structure functions, analogous to Kolmogorov's four-fifths and Yaglom's four-thirds laws and valid at both long and short wavelengths. We show how the general gyrokinetic invariants are related to the particular invariants that control the dual cascade in the long- and short-wavelength limits. We describe the full range of cascades from the fluid to the fully kinetic range. [ABSTRACT FROM AUTHOR]

Both global dynamics and turbulence in magnetized weakly collisional cosmic plasmas are described by general magnetofluid equations that contain pressure anisotropies and heat fluxes that must be calculated from microscopic plasma kinetic theory. It is shown that even without a detailed calculation of the pressure anisotropy or the heat fluxes, one finds the macroscale dynamics to be generically unstable to microscale Alfvénically polarized fluctuations. Two instabilities that can be treated this way are considered in detail: the parallel firehose instability (including the finite Larmor radius effects that determine the growth rate and scale of the fastest growing mode) and the gyrothermal instability (GTI). The latter is a new result – it is shown that a parallel ion heat flux destabilizes Alfvénically polarized fluctuations even in the absence of the negative pressure anisotropy required for the firehose. The main physical conclusion is that both pressure anisotropies and heat fluxes associated with the macroscale dynamics trigger plasma microinstabilities and, therefore, their values will likely be set by the non-linear evolution of these instabilities. Ideas for understanding this non-linear evolution are discussed. It is argued that cosmic plasmas will generically be ‘three-scale systems’, comprising global dynamics, mesoscale turbulence and microscale plasma fluctuations. The astrophysical example of cool cores of galaxy clusters is considered quantitatively and it is noted that observations point to turbulence in clusters (velocity, magnetic and temperature fluctuations) being in a marginal state with respect to plasma microinstabilities and so it is the plasma microphysics that is likely to set the heating and conduction properties of the intracluster medium. In particular, a lower bound on the scale of temperature fluctuations implied by the GTI is derived. [ABSTRACT FROM AUTHOR]

Two-dimensional numerical simulations of the effect of background turbulence on two-dimensional resistive magnetic reconnection are presented. For sufficiently small values of the resistivity (η) and moderate values of the turbulent power (ε), the reconnection rate is found to have a much weaker dependence on η than the Sweet–Parker scaling of and is even consistent with an η independent value. For a given value of η, the dependence of the reconnection rate on the turbulent power exhibits a critical threshold in ε above which the reconnection rate is significantly enhanced. [ABSTRACT FROM AUTHOR]

The characteristics of near-unity-β equilibria are investigated with two codes. CUBE is a multigrid Grad–Shafranov solver [Gourdain et al., J. Comput. Phys. 216, 275 (2006)], and Ophidian was written to compute solutions using analytic unity-β equilibria [Cowley et al., Phys. Fluids B 3, 2066 (1991)]. Results from each method are qualitatively and quantitatively compared across a spectrum of mutually relevant parameters. These comparisons corroborate the theoretical results and provide benchmarks for high-resolution numerical results available from CUBE. Both tools facilitate the exploration of the properties of high-β equilibria, such as a highly diamagnetic plasma and its ramifications for stability and transport. [ABSTRACT FROM AUTHOR]

A new analytically and numerically manageable model collision operator is developed specifically for turbulence simulations. The like-particle collision operator includes both pitch-angle scattering and energy diffusion and satisfies the physical constraints required for collision operators: it conserves particles, momentum, and energy, obeys Boltzmann’s H-theorem (collisions cannot decrease entropy), vanishes on a Maxwellian, and efficiently dissipates small-scale structure in the velocity space. The process of transforming this collision operator into the gyroaveraged form for use in gyrokinetic simulations is detailed. The gyroaveraged model operator is shown to have more suitable behavior at small scales in phase space than previously suggested models. Model operators for electron-ion and ion-electron collisions are also presented. [ABSTRACT FROM AUTHOR]

Current sheets formed in magnetic reconnection events are found to be unstable to high-wavenumber perturbations. The instability is very fast: its maximum growth rate scales as S1/4vA/LCS, where LCS is the length of the sheet, vA the Alfvén speed, and S the Lundquist number. As a result, a chain of plasmoids (secondary islands) is formed, whose number scales as S3/8. [ABSTRACT FROM AUTHOR]

This paper is a detailed report on a programme of direct numerical simulations of incompressible nonhelical randomly forced magnetohydrodynamic (MHD) turbulence that are used to settle a long-standing issue in the turbulent dynamo theory and demonstrate that the fluctuation dynamo exists in the limit of large magnetic Reynolds number Rm ⪢ 1 and small magnetic Prandtl number Pm ⪡ 1. The dependence of the critical Rmc for dynamo versus the hydrodynamic Reynolds number Re is obtained for 1 ≲ Re ≲ 6700. In the limit Pm ⪡ 1,Rmc is at most three times larger than for the previously well established dynamo at large and moderate Prandtl numbers: Rmc ≲ 200 for Re ≳ 6000 compared to Rmc ~ 60 for Pm ⩾ 1. The stability curve Rmc(Re) (and, it is argued, the nature of the dynamo) is substantially different from the case of the simulations and liquid-metal experiments with a mean flow. It is not as yet possible to determine numerically whether the growth rate of the magnetic energy is ∞Rm½ in the limit Re ⪢ Rm ⪢ 1, as should be the case if the dynamo is driven by the inertial-range motions at the resistive scale, or tends to an Rm-independent value comparable to the turnover rate of the outer-scale motions. The magnetic-energy spectrum in the low-Pm regime is qualitatively different from the Pm ⩾ 1 case and appears to develop a negative spectral slope, although current resolutions are insufficient to determine its asymptotic form. At Rm ϵ (1,Rmc), the magnetic fluctuations induced via the tangling by turbulence of a weak mean field are investigated and the possibility of a k-1 spectrum above the resistive scale is examined. At low Rm < 1, the induced fluctuations are well described by the quasistatic approximation; the k-11/3 spectrum is confirmed for the first time in direct numerical simulations. Applications of the results on turbulent induction to understanding the nonlocal energy transfer from the dynamo-generated magnetic field to smaller-scale magnetic fluctuations are discussed. The results reported here are of fundamental importance for understanding the genesis of small-scale magnetic fields in cosmic plasmas. [ABSTRACT FROM AUTHOR]

Observations of galaxy clusters show that the intracluster medium (ICM) is likely to be turbulent and is certainly magnetized. The properties of this magnetized turbulence are determined both by fundamental nonlinear magnetohydrodynamic interactions and by the plasma physics of the ICM, which has very low collisionality. Cluster plasma threaded by weak magnetic fields is subject to firehose and mirror instabilities. These saturate and produce fluctuations at the ion gyroscale, which can scatter particles, increasing the effective collision rate and, therefore, the effective Reynolds number of the ICM. A simple way to model this effect is proposed. The model yields a self-accelerating fluctuation dynamo whereby the field grows explosively fast, reaching the observed, dynamically important, field strength in a fraction of the cluster lifetime independent of the exact strength of the seed field. It is suggested that the saturated state of the cluster turbulence is a combination of the conventional isotropic magnetohydrodynamic turbulence, characterized by folded, direction-reversing magnetic fields and an Alfvén-wave cascade at collisionless scales. An argument is proposed to constrain the reversal scale of the folded field. The picture that emerges appears to be in qualitative agreement with observations of magnetic fields in clusters. [ABSTRACT FROM AUTHOR]

One of the outstanding challenges of solar physics is understanding the mechanism for heating the solar corona, a low-β, low-resistivity plasma driven slowly by the shuffling of "footpoints" (ends of flux tubes) anchored in the photosphere. The corona's low resistivity means that enormous current densities must be present for Ohmic dissipation to be effective. Two ways that this can occur in a slowly evolving, "quasiequilibrium" medium are by the formation of current sheets or by the excitation of very low-frequency Alfvén resonances. Both processes lead to singular current densities in the ideal limit, and both may result from the excitation of very low- (or zero-) frequency shear Alfvén resonances. Thus, understanding the conditions under which zero-frequency Alfvén resonances can occur may ultimately be important in understanding coronal heating. This paper examines shear Alfvén resonances in line-tied magnetic geometries, with special attention being paid to geometries with discontinuities in the field line mapping as candidates for having zero-frequency Alfvén resonances. Two such geometries have been examined in detail: a two-dimensional X-point and a three-dimensional magnetic null. Qualitative differences in the two spectra are observed. The X-point geometry has Alfvén resonances of vanishing frequency, whereas the magnetic null geometry has a finite cutoff in resonant frequency for all field lines in the system. The Alfvén continuum for a null depends on its geometry, however, and much lower (though still nonzero) resonant frequencies are found when the flux surfaces are moderately eccentric. Consequently, a subset of magnetic nulls in complex, line tied geometries may have flux surfaces that can be driven resonantly by a driver of much lower frequency than for similar media without nulls. To the extent that nulls are generic features of complex magnetic geometries {and there is some reason to... [ABSTRACT FROM AUTHOR]

Stochastic magnetic fields in a weakly collisional plasma enhance the collisionality of electrons in the medium compared to the classical Spitzer values. The reduction in the thermal conductivity may play an important role in the evolution of galaxy clusters. This work extends prior work by examining the parallel propagation of electrons in plasma media where the magnetic field amplitude sampled along a field line is a stochastic function of the distance. New physics is exhibited when many large-amplitude magnetic mirrors are present; the parallel transport becomes subdiffusive rather than diffusive in this limit. The transition between diffusion and subdiffusion can be obtained from properties of the asymptotic solution to the kinetic equation in the vicinity of a solitary magnetic mirror. This transition has been computed as a function of the power law exponent in the distribution of mirror maxima, and the asymptotic scaling of the mean-squared electron displacement with time has been obtained in these systems as a function of this exponent. Monte Carlo comparisons are in agreement with these results. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The theory of beat wave excitation in a slightly inhomogeneous plasma is presented. While the theory is general it applies directly to the experiments performed in Alaska by the ionospheric heating facilities HIPAS [High Power Auroral Simulation, Radio Sci. 25, 1269 (1990)] and HAARP [High Frequency Active Auroral Research Project, Geophys. Res. Lett. 25, 257 (1998)]. A ray tracing (WKB) formalism appropriate for computations is developed. The computational implementation of this formalism and extensive results will be presented in a follow up to this paper. Also the beat wave excitation of upper-hybrid waves is investigated analytically. The complicated trajectories of these waves in the plasma are described. When this beat wave reaches the plasma wave resonance it can, in the HIPAS-HAARP experiments, attain nonlinear amplitude. The electrostatic upper-hybrid waves are trapped around the density maximum of the ionosphere. This trapping is investigated in detail. Beat wave pumping of the trapped modes is possible using HAARP and HIPAS or with split beams from HAARP. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Through the principal use of the reduced magnetohydrodynamic version of the finite aspect ratio code [L. A. Charlton et al., J. Comput. Phys. 86, 270 (1990)], an m/n=1/1 resistive kink mode was poloidally rotated with the accompanying rotational shear. It was observed that the growth rate of this unstable mode can either decrease or increase as the applied equilibrium rotation is increased to near poloidal sonic speeds. Shear in the poloidal rotation profile is stabilizing, but only if the destabilizing effects of bulk rotation can be overcome. Therefore, the mode's stability was sensitive to the location of the rotation's peak relative to the eigenmode's spatial extent. The destabilizing effects of bulk rotation are apparently a rotationally enhanced beta, and the stabilizing effects appear to be caused by exceeding a critical rotational shear spatially averaged over the eigenmode structure. © 2001 American Institute of Physics. [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]

The ion-temperature-gradient-driven instability is considered in this paper. Physical pictures are presented to clarify the nature of the instability. The saturation of a single eddy is modeled by a simple nonlinear equation. It is shown that eddies that are elongated in the direction of the temperature gradient are the most unstable and have the highest saturation amplitudes. In a sheared magnetic field, such elongated eddies twist with the field lines. This structure is shown to be an alternative to the usual Fourier mode picture in which the mode is localized around the surface where k|| =0. These elongated twisting eddies, which are an integral part of the ‘‘ballooning mode’’ structure, could survive in a torus. The elongated eddies are shown to be unstable to secondary instabilities that are driven by the large gradients in the long eddy. It is argued that the ‘‘mixing length’’ is affected by this nonlinear process, and is unlikely to be a linear eigenmode width. [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 separation of the cylindrical tearing mode stability problem into a resistive resonant layer calculation and an external marginal ideal magnetohydrodynamic (MHD) calculation (Δ’ calculation) is generalized to axisymmetric toroidal geometry. The general structure of this separation is analyzed and the marginal ideal MHD information (the toroidal generalization of Δ’) required to discuss stability is isolated. This can then, in principle, be combined with relevant resonant layer calculations to determine tearing mode growth rates in realistic situations. Two examples are given: the first is an analytic treatment of toroidally coupled (m=1, n=1) and (m=2, n=1) tearing modes in a large aspect ratio torus; the second, a numerical treatment of the toroidal coupling of three tearing modes through finite pressure effects in a large aspect ratio torus. In addition, the use of a coupling integral approach for determining the stability of coupled tearing modes is discussed. Finally, the possibility of using initial value resistive MHD codes in realistic toroidal geometry to determine the necessary information from the ideal MHD marginal solution is discussed. [ABSTRACT FROM AUTHOR]

This paper examines the stability of tearing modes in a sheared slab when the width of the tearing layer is much smaller than the ion Larmor radius. The ion response is nonlocal, and the quasineutrality retains its full integral form. An expansion procedure is introduced to solve the quasineutrality equation in powers of the width of the tearing layer over the ion Larmor radius. The expansion procedure is applied to the collisionless and semicollisional tearing modes. It is found that first-order terms in the expansion are strongly stabilizing. The physics of the mode and of the stabilization are discussed. Tearing modes are observed in experiments even though the slab theory predicts stability. It is proposed that these modes grow from an equilibrium with islands at the rational surfaces. If the equilibrium islands are wider than the ion Larmor radius, the mode is unstable when δ' is positive. [ABSTRACT FROM AUTHOR]

The usual kinetic description of a plasma is extended to include variables to describe the nuclear spin. The distribution function, over phase space and the new spin variables, provides a sufficient description of a spin-polarized plasma. The evolution for the distribution function is given. The equations derived are used to calculate depolarization caused by four processes: inhomogeneous fields, collisions, collisions in inhomogeneous fields, and waves. It is found that depolarization by field inhomogeneity on scales large compared with the gyroradius is totally negligible. The same is true for collisional depolarization. Collisions in inhomogeneous fields yield a depolarization rate of order 10-3 sec-1 for deuterons and a negligible rate for tritons in a typical fusion reactor design. This is still sufficiently small on reactor time scales. However, small-amplitude magnetic fluctuations (of order 1 G) resonant with the spin precession frequency can lead to significant depolarization (depolarizes triton in 10 sec and deuteron in 100 sec). [ABSTRACT FROM AUTHOR]

The scope and detail of physics contained in computational models for fluid (density, momentum, energy) transport in toroidal plasmas have steadily increased during the past two decades. There has been considerable success in the development and verification of models for sources and sinks of particles, energy, momentum, and magnetic flux. Transport codes have collectively become very useful tools in interpreting experimental data and in providing guidance for new experiments. However, a more thorough understanding of the fundamental transport processes of magnetically confined plasmas and development of improved computational models are needed to enhance the predictive capabilities of transport codes. It is argued that fluid transport modeling by itself cannot lead to a complete understanding of transport—there must be a very strong collaboration among theory, experiment, and modeling on both the fluid and kinetic levels. [ABSTRACT FROM AUTHOR]

Using a simple model for the layer of the tearing mode, it is demonstrated that toroidally coupled tearing modes with two rational surfaces are most unstable when the ω*’s of the electrons at the rational surfaces are equal. The onset of instability may then occur because of the tuning of ω* rather than the passage of Δ’-like quantities through zero. This mechanism for the onset of instability is sharp since the resonance is narrow. The effect of toroidal rotation is also discussed. [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 theoretical framework for low-frequency electromagnetic (drift-)kinetic turbulence in a collisionless, multi-species plasma is presented. The result generalises reduced magnetohydrodynamics (RMHD) and kinetic RMHD (Schekochihin et al., Astrophys. J. Suppl. Ser., vol. 182, 2009, pp. 310–377) to the case where the mean distribution function of the plasma is pressure-anisotropic and different ion species are allowed to drift with respect to each other – a situation routinely encountered in the solar wind and presumably ubiquitous in hot dilute astrophysical plasmas such as the intracluster medium. Two main objectives are achieved. First, in a non-Maxwellian plasma, the relationships between fluctuating fields (e.g. the Alfvén ratio) are order-unity modified compared to the more commonly considered Maxwellian case, and so a quantitative theory is developed to support quantitative measurements now possible in the solar wind. Beyond these order-unity corrections, the main physical feature of low-frequency plasma turbulence survives the generalisation to non-Maxwellian distributions: Alfvénic and compressive fluctuations are energetically decoupled, with the latter passively advected by the former; the Alfvénic cascade is fluid, satisfying RMHD equations (with the Alfvén speed modified by pressure anisotropy and species drifts), whereas the compressive cascade is kinetic and subject to collisionless damping (and for a bi-Maxwellian plasma splits into three independent collisionless cascades). Secondly, the organising principle of this turbulence is elucidated in the form of a conservation law for the appropriately generalised kinetic free energy. It is shown that non-Maxwellian features in the distribution function reduce the rate of phase mixing and the efficacy of magnetic stresses, and that these changes influence the partitioning of free energy amongst the various cascade channels. As the firehose or mirror instability thresholds are approached, the dynamics of the plasma are modified so as to reduce the energetic cost of bending magnetic-field lines or of compressing/rarefying them. Finally, it is shown that this theory can be derived as a long-wavelength limit of non-Maxwellian slab gyrokinetics. [ABSTRACT FROM PUBLISHER]

Slow dynamical changes in magnetic-field strength and invariance of the particles’ magnetic moments generate ubiquitous pressure anisotropies in weakly collisional, magnetized astrophysical plasmas. This renders them unstable to fast, small-scale mirror and firehose instabilities, which are capable of exerting feedback on the macroscale dynamics of the system. By way of a new asymptotic theory of the early non-linear evolution of the mirror instability in a plasma subject to slow shearing or compression, we show that the instability does not saturate quasi-linearly at a steady, low-amplitude level. Instead, the trapping of particles in small-scale mirrors leads to non-linear secular growth of magnetic perturbations, δB/B ∝ t2/3. Our theory explains recent collisionless simulation results, provides a prediction of the mirror evolution in weakly collisional plasmas and establishes a foundation for a theory of non-linear mirror dynamics with trapping, valid up to δB/B = O(1). [ABSTRACT FROM PUBLISHER]

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]

Within the single fluid theory for a toroidal, resistive plasma, the favorable average curvature effect [Glasser et al., Phys. Fluids 18, 875 (1975)], which is responsible for the strong stabilization of the classical tearing mode at finite pressure, can also introduce a strong screening effect to the externally applied resonant magnetic field. Contrary to conventional understanding, this screening, occurring at slow plasma rotation, is enhanced when decreasing the plasma flow speed. The plasma rotation frequency, below which this screening effect is observed, depends on the plasma pressure and resistivity. For the simple toroidal case considered here, the toroidal rotation frequency has to be below ∼10-5ωA, with ωA being the Alfvén frequency. In addition, the same curvature effect leads to enhanced toroidal coupling of poloidal Fourier harmonics inside the resistive layer, as well as reversing the sign of the electromagnetic torque at slow plasma flow. [ABSTRACT FROM AUTHOR]

Theoretical approaches to low-frequency magnetized turbulence in collisionless and weakly collisional astrophysical plasmas are reviewed. The proper starting point for an analytical description of these plasmas is kinetic theory, not fluid equations. The anisotropy of the turbulence is used to systematically derive a series of reduced analytical models. Above the ion gyroscale, it is shown rigourously that the Alfvén waves decouple from the electron-density and magnetic-field-strength fluctuations and satisfy the reduced MHD equations. The density and field-strength fluctuations (slow waves and the entropy mode in the fluid limit), determined kinetically, are passively mixed by the Alfvén waves. The resulting hybrid fluid-kinetic description of the low-frequency turbulence is valid independently of collisionality. Below the ion gyroscale, the turbulent cascade is partially converted into a cascade of kinetic Alfvén waves, damped at the electron gyroscale. This cascade is described by a pair of fluid-like equations, which are a reduced version of the electron MHD. The development of these theoretical models is motivated by observations of the turbulence in the solar wind and interstellar medium. In the latter case, the turbulence is spatially inhomogeneous and the anisotropic Alfvénic turbulence in the presence of a strong mean field may coexist with isotropic MHD turbulence that has no mean field. [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]