Large Aspect Ratio Limit Research Articles

On multiple solutions of the Grad–Shafranov equation

The Grad–Shafranov equation (GSE) for axisymmetric MHD equilibria is a nonlinear, scalar PDE which in principle can have zero, one or more non-trivial solutions. The conditions for the existence of multiple solutions has been little explored in the literature so far. We develop a simple analytic model to calculate multiple solutions in the large aspect ratio limit. We compare the results to the recently developed deflated continuation method to find multiple solutions in a realistic geometry and right-hand side of the GSE using the finite element method. The analytic model is surprisingly accurate in calculating multiple solutions of the GSE for given boundary conditions and the two methods agree well in limiting cases. We examine the effect of plasma shaping and aspect ratio on the multiple solutions and show that shaping generally does not alter the number of solutions. We discuss implications for predictive modelling, equilibrium reconstruction, plasma stability and disruptions.

Rotation of a fibre in simple shear flow of a dilute polymer solution

The motion of a freely rotating prolate spheroid in a simple shear flow of a dilute polymeric solution is examined in the limit of large particle aspect ratio, $\kappa$ . A regular perturbation expansion in the polymer concentration, $c$ , a generalized reciprocal theorem, and slender body theory to represent the velocity field of a Newtonian fluid around the spheroid are used to obtain the $O(c)$ correction to the particle's orientational dynamics. The resulting dynamical system predicts a range of orientational behaviours qualitatively dependent upon $c\, De$ ( $De$ is the imposed shear rate times the polymer relaxation time) and $\kappa$ and quantitatively on $c$ . At a small but finite $c\, De$ , the particle spirals towards a limit cycle near the vorticity axis for all initial conditions. Upon increasing $\kappa$ , the limit cycle becomes smaller. Thus, ultimately the particle undergoes a periodic motion around and at a small angle from the vorticity axis. At moderate $c\, De$ , a particle starting near the flow–gradient plane departs it monotonically instead of spirally, as this plane (a limit cycle at smaller $c\, De$ ) obtains a saddle and an unstable node. The former is close to the flow direction. Upon further increasing $c\, De$ , the saddle node changes to a stable node. Therefore, depending upon the initial condition, a particle may either approach a periodic orbit near the vorticity axis or obtain a stable orientation near the flow direction. Upon further increasing $c\, De$ , the limit cycle near the vorticity axis vanishes, and the particle aligns with the flow direction for all starting orientations.

Frequency response of cantilevered plates of small aspect ratio immersed in viscous fluids

Comprehensive theoretical models for the dynamic response of slender cantilevered beams immersed in fluid have been widely reported, while the distinct behavior of wide cantilevered plates has received comparatively little attention. In this article, we develop an exact analytical theory for the resonant response of rectangular cantilevered plates of zero length-to-width aspect ratio that are immersed in unbounded viscous fluids. Unlike the opposite slender limit of large aspect ratio, the hydrodynamic load experienced by zero-aspect-ratio cantilevered plates is inherently non-local, which can strongly affect the individual mode shapes of the plate. In addition, finite-element-method simulations are reported for two- and three-dimensional cases of zero and finite aspect ratio, respectively. Accuracy of the present theory and that of Atkinson and Manrique de Lara [J. Sound Vib. 300, 352 (2007)] for small viscosity and zero aspect ratio is assessed using the former simulations. The latter simulations are used to clarify the regime of validity of the present theory as a function of aspect ratio, along with that of existing theory for slender (large aspect ratio) beams. The results of this study are expected to be of practical importance to micro- and nano-electromechanical system design and their applications.

Merging of the superbanana plateau and transport regimes in nearly quasisymmetric stellarators

Alpha particle confinement is one of the most demanding issues for stellarators. It now seems clear that it is possible to design optimized stellarators that confine the background plasma at near tokamak radial transport levels. Moreover, adequate collisionless alpha particle confinement is possible in the core of a highly optimized stellarator. Here, the collisional confinement of barely trapped alphas in an optimized stellarator is considered by accounting for the resonance due to the reversal in direction of the drift within a flux surface and investigating the sensitive role of magnetic shear in keeping this resonance close to the passing boundary in some nearly quasisymmetric stellarator configurations. The treatment relies on a narrow collisional boundary layer formulation that combines the responses of both these resonant pitch angle alphas and the remaining barely trapped alphas. A novel merged regime treatment leads to explicit expressions for the energy diffusivity for both superbanana plateau (or resonant plateau) and $\sqrt \nu$ transport in the large aspect ratio limit for a slowing down tail alpha distribution function, where $\nu$ is the effective pitch angle scattering collision frequency of the trapped alphas off the background ions. Depending on the details of the optimization scheme and the sign of the magnetic shear, modest magnetic shear can be used to reduce superbanana (or resonant) plateau transport to below the $\sqrt \nu$ transport level. In addition, a quasilinear equation retaining spatial diffusion is derived for a general alpha distribution function that allows the radial alpha transport to modify the distribution so it is no longer isotropic in velocity space.

Improved neoclassical plasma confinement and turbulence suppression in a magnetic well in tokamak reactors

It has been known that the width of the trapped particles, i.e. bananas, is reduced or squeezed in a magnetic well in tokamaks. The magnetic squeezing factor S B depends on the energy and pitch angle of the particles. When ρ p/a ∼ 1, S B becomes appreciably larger than unity; it can be of the order of 2 for a parabolic well with B −1 d2 B/dr 2 ∼ 2a −2. Here, ρ p is the poloidal gyro-radius, B is the magnetic field strength, r is the local minor radius, and a is the minor radius. However, the real orbit width measured in terms of the poloidal magnetic flux is still less than a because of the magnetic squeezing; for S B = 2, it is a 30% reduction. The transport consequences of the squeezed bananas in a magnetic well are calculated by solving the drift kinetic equation utilizing the constants of motion in the large aspect ratio limit. Consequently, neoclassical ion heat conductivity χ h in the reactor relevant collisionality regime (i.e. banana regime) is reduced by a factor of ; for S B = 2, it is almost a factor of three reduction for the same temperature. The ion temperature scaling of χ h improves from the conventional, and weakly favorable scaling to the strongly favorable scaling for high temperature fusion grade tokamak plasmas; the scaling however does not take the temperature dependence of the magnetic well itself into account. Here, T i is the ion temperature. In addition, it is shown that plasma turbulence is suppressed in a magnetic well using the reduced decorrelation time argument and anomalous plasma confinement is improved. The combined improvements result in a favorable temperature scaling in the ion energy confinement time, and make burning aneutronic fuels feasible. The implications on a tokamak with a magnetic well as a fusion reactor are discussed.

Magnetic reconnection in 3D fusion devices: non-linear reduced equations and linear current-driven instabilities

Magnetic reconnection in 3D fusion devices is investigated. With the use of Boozer co-ordinates, we reduce the non-linear resistive magnetohydrodynamic equations in the limit of large aspect ratio and finite pressure fluctuations, to obtain a set of non-linear equations suitable for magnetic reconnection studies in stellarators. Magnetic flux unfreezing due to a finite electron mass is also considered. Equations that govern the linear regime and some of their general properties are given. We emphasise the role of magnetic geometry and identify how some aspects of stellarator optimisation could have an impact on reconnecting instabilities, in particular by exacerbating those enabled by electron inertia. The effect of 3D coupling on the linear reconnection rates and the mode structure is quantitatively addressed in the case in which the equilibrium rotational transform has one specific resonant location for which one mode can reconnect while coupled to an arbitrary number of non-resonant harmonics. The full problem is rigorously reduced to an equivalent cylindrical one, by introducing some geometrically modified plasma inertial and dissipative scales. The 3D scalings for the growth rates of reconnection instabilities and their destabilisation criteria are given.

Schwarz–Christoffel accessory parameter for quadrilaterals via isomonodromy

We develop the recent proposal by the authors to exploit the isomonodromic tau function defined by Jimbo, Miwa and Ueno (JMU) to solve the accessory parameter problem in conformal mapping theory. We focus here on mappings of Schwarz–Christoffel type: in particular, the mapping from the upper half plane to a four-sided polygon where the sides are all straight lines. We show that one can obtain the relevant accessory parameters—the pre-image of the polygonal vertices—via a special ‘zero curvature limit’ in which the radius of curvature of some of the edges tends to zero. We apply the procedure to rectangular domains where the JMU tau function is given by a ratio of Riemann theta functions, known as the Picard solution, and take the zero curvature limit to recover the accessory parameter obtained by Nehari using quite different methods. We then turn to trapezoids, deriving new asymptotic formulas for the accessory parameters in the limit of large and small aspect ratios. Our work lends a new geometrical perspective to problems of isomonodromy that we believe provides theoretical insight, while also showing how classical problems in conformal mapping can benefit from new ideas emerging from isomonodromic deformation theory.

Edge corrections for parallel-plate capacitors

As a result of field fringing, the capacitance of a parallel-plate capacitor differs from that predicted by the textbook formula. Using singular perturbations and conformal mapping techniques, we calculate the leading-order correction to the capacitance in the limit of large aspect ratio. We additionally obtain a comparable approximation for the electrostatic attraction between the plates.

Influence of Toroidal Flow on Stationary Density of Collisionless Plasmas

Starting from the given passive particle equilibrium particle cylindrical profiles, we built self-consistent stationary conditions of the Maxwell-Vlasov equation at thermodynamic equilibrium with non-flat density profiles. The solutions to the obtained equations are then discussed. It appears that the presence of an azimuthal (poloidal) flow in the plasma can ensure radial confinement, while the presence of a longitudinal (toroidal) flow can enhance greatly the confinement. Moreover in the global physically reasonable situation, we find that no unstable point can emerge in the effective integrable Hamiltonian of the individual particles, hinting at some stability of the confinement when considering a toroidal geometry in the large aspect ratio limit.

Effective magnetic susceptibility of suspensions of ferromagnetic particles

The effective susceptibility χeff of suspensions of ferromagnetic particles in a liquid was measured using inductance measurements. These measurements were used to test a model that predicts how χeff varies due to demagnetization, as a function of sample aspect ratio, particle packing fraction, and particle aspect ratio [R. Skomski, G. C. Hadjipanayis, and D. J. Sellmyer, IEEE Trans. Magn. 43, 2956–2958 (2007)]. For spherical particles or cylindrical particles forcibly aligned with an external magnetic field, the model can be fitted to the measured data with agreement within 13%. This model predicts suspensions of aligned, large-aspect-ratio particles should have the largest χeff, approaching the particle material susceptibility in the limit of large particle aspect ratio. However, χeff was found to be no larger than about 4 for cylindrical iron particles of various aspect ratios, close to the value obtained for spheres. This results from the random alignment of non-spherical particles relative to the magnetic field naturally found in suspensions, which increases the demagnetization effect and limits χeff. The contribution of random particle alignments to the demagnetization effect and χeff remains to be accounted for in models.

Gyrokinetic particle simulation of electrostatic microturbulence with impurity ions

Impurity is an important factor that can affect significantly turbulent transport in tokamaks. In order to study the impurity physics, we implement a new impurity module in the gyrokinetic particle simulation code GTC (Gyrokinetic Toroidal Code). With an improved numerical scheme, we expand the validity of gyrokinetic Poisson equation in the GTC to the short wavelength region, for both non-zonal and zonal parts of the perturbed Poisson equation. Verifications of this new scheme are carried out on the linear instability and zonal flow response. The linear simulation of the ion temperature gradient (ITG) instability including the impurity ions shows that the new Poisson solver can obtain the correct linear growth rate and frequency at the thermal ion gyro-radius scale. The residual zonal flow with impurities obtained via the new zonal flow solver is consistent with the numerical and analytical predictions in the large aspect-ratio limit. The nonlinear simulation of the ITG turbulence shows that the turbulent transport is significantly reduced by the impurity ions through decreasing the linear growth rate of the instability.

Turbulent thermal superstructures in Rayleigh-Bénard convection

We report the observation of superstructures, i.e.\ very large-scale and long living coherent structures in highly turbulent Rayleigh-B\'enard convection up to Rayleigh $Ra=10^9$. We perform direct numerical simulations in horizontally periodic domains with aspect ratios up to $\Gamma=128$. In the considered $Ra$ number regime the thermal superstructures have a horizontal extend of six to seven times the height of the domain and their size is independent of $Ra$. Many laboratory experiments and numerical simulations have focused on small aspect ratio cells in order to achieve the highest possible $Ra$. However, here we show that for very high $Ra$ integral quantities such as the Nusselt number and volume averaged Reynolds number only converge to the large aspect ratio limit around $\Gamma \approx 4$, while horizontally averaged statistics such as standard deviation and kurtosis converge around $\Gamma \approx 8$, and the integral scale converges around $\Gamma \approx 32$, and the peak position of the temperature variance and turbulent kinetic energy spectra only around $\Gamma \approx 64$.

Efficiency of wave-driven rigid body rotation toroidal confinement

The compensation of vertical drifts in toroidal magnetic fields through a wave-driven poloidal rotation is compared with compensation through the wave driven toroidal current generation to support the classical magnetic rotational transform. The advantages and drawbacks associated with the sustainment of a radial electric field are compared with those associated with the sustainment of a poloidal magnetic field both in terms of energy content and power dissipation. The energy content of a radial electric field is found to be smaller than the energy content of a poloidal magnetic field for a similar set of orbits. The wave driven radial electric field generation efficiency is similarly shown, at least in the limit of large aspect ratio, to be larger than the efficiency of wave-driven toroidal current generation.

Revisiting the Instability and Bifurcation Behavior of Soft Dielectrics

Development of soft electromechanical materials is critical for several tantalizing applications such as human-like robots, stretchable electronics, actuators, energy harvesting, among others. Soft dielectrics can be easily deformed by an electric field through the so-called electrostatic Maxwell stress. The highly nonlinear coupling between the mechanical and electrical effects in soft dielectrics gives rise to a rich variety of instability and bifurcation behavior. Depending upon the context, instabilities can either be detrimental, or more intriguingly, exploited for enhanced multifunctional behavior. In this work, we revisit the instability and bifurcation behavior of a finite block made of a soft dielectric material that is simultaneously subjected to both mechanical and electrical stimuli. An excellent literature already exists that has addressed the same topic. However, barring a few exceptions, most works have focused on the consideration of homogeneous deformation and accordingly, relatively fewer insights are at hand regarding the compressive stress state. In our work, we allow for fairly general and inhomogeneous deformation modes and, in the case of a neo-Hookean material, present closed-form solutions to the instability and bifurcation behavior of soft dielectrics. Our results, in the asymptotic limit of large aspect ratio, agree well with Euler's prediction for the buckling of a slender block and, furthermore, in the limit of zero aspect ratio are the same as Biot's critical strain of surface instability of a compressed homogeneous half-space of a neo-Hookean material. A key physical insight that emerges from our analysis is that soft dielectrics can be used as actuators within an expanded range of electric field than hitherto believed.

Effect of the Shafranov shift and the gradient of β on intrinsic momentum transport in up–down asymmetric tokamaks

Tokamaks with up–down asymmetric poloidal cross-sections spontaneously rotate due to turbulent transport of momentum. In this work, we investigate the effect of the Shafranov shift on this intrinsic rotation, primarily by analyzing tokamaks with tilted elliptical flux surfaces. By expanding the Grad–Shafranov equation in the large aspect ratio limit we calculate the magnitude and direction of the Shafranov shift in tilted elliptical tokamaks. The results show that, while the Shafranov shift becomes up–down asymmetric and depends strongly on the tilt angle of the flux surfaces, it is insensitive to the shape of the current and pressure profiles (when the geometry, total plasma current, and average pressure gradient are kept fixed). Next, local nonlinear gyrokinetic simulations of these MHD equilibria are performed with GS2, which reveal that the Shafranov shift can significantly enhance the momentum transport. However, to be consistent, the effect of (i.e. the radial gradient of β) on the magnetic equilibrium was also included, which was found to significantly reduce momentum transport. Including these two competing effects broadens the rotation profile, but leaves the on-axis value of the rotation roughly unchanged. Consequently, the shape of the β profile has a significant effect on the rotation profile of an up–down asymmetric tokamak.

Evaluation of toroidal torque by non-resonant magnetic perturbations in tokamaks for resonant transport regimes using a Hamiltonian approach

Toroidal torque generated by neoclassical viscosity caused by external non-resonant, non-axisymmetric perturbations has a significant influence on toroidal plasma rotation in tokamaks. In this article, a derivation for the expressions of toroidal torque and radial transport in resonant regimes is provided within quasilinear theory in canonical action-angle variables. The proposed approach treats all low-collisional quasilinear resonant neoclassical toroidal viscosity regimes including superbanana-plateau and drift-orbit resonances in a unified way and allows for magnetic drift in all regimes. It is valid for perturbations on toroidally symmetric flux surfaces of the unperturbed equilibrium without specific assumptions on geometry or aspect ratio. The resulting expressions are shown to match the existing analytical results in the large aspect ratio limit. Numerical results from the newly developed code NEO-RT are compared to calculations by the quasilinear version of the code NEO-2 at low collisionalities. The importance of the magnetic shear term in the magnetic drift frequency and a significant effect of the magnetic drift on drift-orbit resonances are demonstrated.

Neoclassical toroidal plasma viscosity with effects of finite banana width for finite aspect ratio tokamaks

Theory for neoclassical toroidal plasma viscosity has been developed to model transport phenomena, especially, toroidal plasma rotation for tokamaks with broken symmetry. Theoretical predictions are in agreement with the results of the numerical codes in the large aspect ratio limit. The theory has since been extended to include effects of finite aspect ratio and finite plasma β. Here, β is the ratio of the plasma thermal pressure to the magnetic field pressure. However, there are cases where the radial wavelength of the self-consistent perturbed magnetic field strength B on the perturbed magnetic surface is comparable to the width of the trapped particles, i.e., bananas. To accommodate those cases, the theory for neoclassical toroidal plasma viscosity is further extended here to include the effects of the finite banana width. The extended theory is developed using the orbit averaged drift kinetic equation in the low collisionality regimes. The results of the theory can now be used to model plasma transport, including toroidal plasma rotation, in real finite aspect ratio, and finite plasma β tokamaks with the radial wavelength of the perturbed symmetry breaking magnetic field strength comparable to or longer than the banana width.

Random geometric graph description of connectedness percolation in rod systems.

The problem of continuum percolation in dispersions of rods is reformulated in terms of weighted random geometric graphs. Nodes (or sites or vertices) in the graph represent spatial locations occupied by the centers of the rods. The probability that an edge (or link) connects any randomly selected pair of nodes depends upon the rod volume fraction as well as the distribution over their sizes and shapes, and also upon quantities that characterize their state of dispersion (such as the orientational distribution function). We employ the observation that contributions from closed loops of connected rods are negligible in the limit of large aspect ratios to obtain percolation thresholds that are fully equivalent to those calculated within the second-virial approximation of the connectedness Ornstein-Zernike equation. Our formulation can account for effects due to interactions between the rods, and many-body features can be partially addressed by suitable choices for the edge probabilities.

Spatial shape of avalanches in the Brownian force model

.We study the Brownian force model (BFM), a solvable model of avalanche statistics for an interface, in a general discrete setting. The BFM describes the overdamped motion of elastically coupled particles driven by a parabolic well in independent Brownian force landscapes. Avalanches are defined as the collective jump of the particles in response to an arbitrary monotonous change in the well position (i.e. in the applied force). We derive an exact formula for the joint probability distribution of these jumps. From it we obtain the joint density of local avalanche sizes for stationary driving in the quasi-static limit near the depinning threshold. A saddle-point analysis predicts the spatial shape of avalanches in the limit of large aspect ratios for the continuum version of the model. We then study fluctuations around this saddle point, and obtain the leading corrections to the mean shape, the fluctuations around the mean shape and the shape asymmetry, for finite aspect ratios. Our results are finally confronted to numerical simulations.

Purcell effect for finite-length metal-coated and metal nanowires

We investigate the modification (enhancement and suppression) of the spontaneous emission rate of a dipole emitter in two configurations: inside a finite-length semiconductor nanowire surrounded by bulk metal and in the vicinity of a finite metal nanowire. Our analysis is based on a first-principle approach, which is reduced to a seminumeric one in the limit of large nanowire aspect ratios. The numerical calculations are carried out for an emitter in a GaAs nanowire embedded in Ag or Au and for that nearby an Ag or Au nanowire in vacuum or dielectric. We consider in detail the Purcell and β factors as functions of the cylinder radius, the emitter position, and the transition frequency for both configurations. We contrast the results for finite-length nanowires with those obtained in the infinite-length approximation and find considerable differences in the Purcell factor magnitude.