Rogerio Jorge
@tecnico.ulisboa.pt
Physics Department
Instituto Superior Tecnico
RESEARCH INTERESTS
Plasma Physics
Fusion Energy
Scopus Publications
Scholar Citations
Scholar h-index
Scholar i10-index
Scopus Publications
Nikita Nikulsin, Wrick Sengupta, Rogerio Jorge, and Amitava Bhattacharjee
Cambridge University Press (CUP)
A first-order model is derived for quasisymmetric stellarators where the vacuum field due to coils is dominant, but plasma-current-induced terms are not negligible and can contribute to magnetic differential equations, with $\\beta$ of the order of the ratio induced to vacuum fields. Under these assumptions, it is proven that the aspect ratio must be large and a simple expression can be obtained for the lowest-order vacuum field. The first-order correction, which involves both vacuum and current-driven fields, is governed by a Grad–Shafranov equation and the requirement that flux surfaces exist. These two equations are not always consistent, and so this model is generally overconstrained, but special solutions exist that satisfy both equations simultaneously. One family of such solutions is the set of first-order near-axis solutions. Thus, the first-order near-axis model is a subset of the model presented here. Several other solutions outside the scope of the near-axis model are also found. A case study comparing one such solution to a VMEC-generated solution shows good agreement.
R. Jorge, A. Giuliani, and J. Loizu
AIP Publishing
Single-stage optimization, also known as combined plasma-coil algorithms or direct coil optimization, has recently emerged as a possible method to expedite the design of stellarator devices by including, in a single step, confinement, stability, and engineering constraints. In this work, we show how such frameworks allow us to find new designs in a streamlined manner, yielding a broad range of new configurations. Examples are shown for stellarators with a small number of coils and quasisymmetric stellarators with only one to three coils per half field-period, with external trim coils, helical coils, and a single set of coils generating both a quasi-axisymmetric and a quasi-helical equilibrium.
Wrick Sengupta, Eduardo Rodriguez, Rogerio Jorge, Matt Landreman, and Amitava Bhattacharjee
Cambridge University Press (CUP)
A systematic theory of the asymptotic expansion of the magnetohydrostatics (MHS) equilibrium in the distance from the magnetic axis is developed to include arbitrary smooth currents near the magnetic axis. Compared with the vacuum and the force-free system, an additional magnetic differential equation must be solved to obtain the pressure-driven currents. It is shown that there exist variables in which the rest of the MHS system closely mimics the vacuum system. Thus, a unified treatment of MHS fields is possible. The mathematical structure of the near-axis expansions to arbitrary order is examined carefully to show that the double-periodicity of physical quantities in a toroidal domain can be satisfied order by order. The essential role played by the leading-order Birkhoff–Gustavson normal form in solving the magnetic differential equations is highlighted. Several explicit examples of vacuum, force-free and MHS equilibrium in different geometries are presented.
R. Jorge, W. Dorland, P. Kim, M. Landreman, N. R. Mandell, G. Merlo, and T. Qian
American Physical Society (APS)
Turbulent transport is regarded as one of the key issues in magnetic confinement nuclear fusion, both for tokamaks in stellarators. In this work, we show that a significant decrease in a microstability-based proxy, as opposed to a geometric one, for the turbulent heat flux, namely the quasilinear heat flux, can be obtained in an efficient manner by coupling stellarator optimization with linear gyrokinetic simulations. This is accomplished by computing the quasi-linear heat flux at each step of the optimization process, as well as the deviation from quasisymmetry, and minimizing their sum, leading to a balance between neoclassical and turbulent transport proxy.
M Madeira and R Jorge
IOP Publishing
Abstract With the advances in the optimization of magnetic field equilibria, stellarators have become a serious alternative to the tokamak, bringing this concept to the forefront of the pursuit of fusion energy. In order to be successful in experimentally demonstrating the viability of optimized stellarators, we must overcome any potential hurdles in the construction of its electromagnetic coils. Finding cost-effective ways of increasing the number of operating optimized stellarators could be key in cementing this magnetic confinement concept as a contender for a reactor. In this work, an alternative to modular coils, permanent magnets, are studied and are shown to enable the possibility of converting a tokamak into a stellarator. This is then applied to the case of ISTTOK tokamak, where an engineering design study is conducted.
P. Kim, S. Buller, R. Conlin, W. Dorland, D.W. Dudt, R. Gaur, R. Jorge, E. Kolemen, M. Landreman, N.R. Mandell,et al.
Cambridge University Press (CUP)
We present new stellarator equilibria that have been optimized for reduced turbulent transport using nonlinear gyrokinetic simulations within the optimization loop. The optimization routine involves coupling the pseudo-spectral GPU-native gyrokinetic code GX with the stellarator equilibrium and optimization code DESC. Since using GX allows for fast nonlinear simulations, we directly optimize for reduced nonlinear heat fluxes. To handle the noisy heat flux traces returned by these simulations, we employ the simultaneous perturbation stochastic approximation (SPSA) method that only uses two objective function evaluations for a simple estimate of the gradient. We show several examples that optimize for both reduced heat fluxes and good quasi-symmetry as a proxy for low neoclassical transport. Finally, we run full transport simulations using the T3D stellarator transport code to evaluate the changes in the macroscopic profiles.
P.A. Figueiredo, R. Jorge, J. Ferreira, and P. Rodrigues
Cambridge University Press (CUP)
Recent developments in the design of magnetic confinement fusion devices have allowed the construction of exceptionally optimized stellarator configurations. The near-axis expansion in particular has been proven to enable the construction of magnetic configurations with good confinement properties while taking only a fraction of the usual computation time to generate optimized magnetic equilibria. However, not much is known about the overall features of fast-particle orbits computed in such analytical, yet simplified, equilibria when compared with those originating from accurate equilibrium solutions. This work aims to assess and demonstrate the potential of the near-axis expansion to provide accurate information on particle orbits and to compute loss fractions in moderate to high aspect ratios. The configurations used here are all scaled to fusion-relevant parameters and approximate quasi-symmetry to various degrees. This allows us to understand how deviations from quasi-symmetry affect particle orbits and what are their effects on the estimation of the loss fraction. Guiding-centre trajectories of fusion-born alpha particles are traced using gyronimo and SIMPLE codes under the NEAT framework, showing good numerical agreement. Discrepancies between near-axis and magnetohydrodynamic fields have minor effects on passing particles but significant effects on trapped particles, especially in quasi-helically symmetric magnetic fields. Effective expressions were found for estimating orbit widths and passing–trapped separatrix in quasi-symmetric near-axis fields. Loss fractions agree in the prompt losses regime but diverge afterwards.
A.G. Goodman, K. Camacho Mata, S.A. Henneberg, R. Jorge, M. Landreman, G.G. Plunk, H.M. Smith, R.J.J. Mackenbach, C.D. Beidler, and P. Helander
Cambridge University Press (CUP)
We present a novel method for numerically finding quasi-isodynamic stellarator magnetic fields with excellent fast-particle confinement and extremely small neoclassical transport. The method works particularly well in configurations with only one field period. We examine the properties of these newfound quasi-isodynamic configurations, including their transport coefficients, particle confinement and available energy for trapped-electron-instability-driven turbulence, as well as the degree to which they change when a finite pressure profile is added. We finally discuss the differences between the magnetic axes of the optimized solutions and their respective initial conditions, and conclude with the prospects for future quasi-isodynamic optimization.
R Jorge, A Goodman, M Landreman, J Rodrigues, and F Wechsung
IOP Publishing
Abstract We introduce a novel approach for the simultaneous optimization of plasma physics and coil engineering objectives using fixed-boundary equilibria that is computationally efficient and applicable to a broad range of vacuum and finite plasma pressure scenarios. Our approach treats the plasma boundary and coil shapes as independently optimized variables, penalizing the mismatch between the two using a quadratic flux term in the objective function. Four use cases are presented to demonstrate the effectiveness of the approach, including simple and complex stellarator geometries. As shown here, this method outperforms previous two-stage approaches, achieving smaller plasma objective function values when coils are taken into account.
R. Jorge, G.G. Plunk, M. Drevlak, M. Landreman, J.-F. Lobsien, K. Camacho Mata, and P. Helander
Cambridge University Press (CUP)
A single-field-period quasi-isodynamic stellarator configuration is presented. This configuration, which resembles a twisted strip, is obtained by the method of direct construction, that is, it is found via an expansion in the distance from the magnetic axis. Its discovery, however, relied on an additional step involving numerical optimization, performed within the space of near-axis configurations defined by a set of adjustable magnetic field parameters. This optimization, completed in 30 s on a single CPU core using the SIMSOPT code, yields a solution with excellent confinement, as measured by the conventional figure of merit for neoclassical transport, effective ripple, at a modest aspect ratio of eight. The optimization parameters that led to this configuration are described, its confinement properties are assessed and a set of magnetic field coils is found. The resulting transport at low collisionality is much smaller than that of W7-X, and the device needs significantly fewer coils because of the reduced number of field periods.
Katia Camacho Mata, Gabriel G. Plunk, and Rogerio Jorge
Cambridge University Press (CUP)
We develop the formalism of the first-order near-axis expansion of the magnetohydrodynamic equilibrium equations described by Garren & Boozer (Phys. Fluids B, vol. 3, issue 10, 1991, pp. 2805–2821) and Plunk et al. (J. Plasma Phys., vol. 85, issue 6, 2019; J. Plasma Phys., vol. 87, issue 6, 2021) for the case of a quasi-isodynamic, $N$ -field-period, stellarator-symmetric, single-well magnetic field equilibrium. The importance of the magnetic axis shape is investigated, and we conclude that control of the curvature and torsion is crucial to obtain omnigenous configurations with finite aspect ratio and low effective ripple, especially for a higher number of field periods. For this reason a method is derived to construct classes of axis shapes with favourable curvature and torsion. Solutions are presented, including a three-field-period configuration constructed at an aspect ratio of $A=20$ , with a maximum elongation of $e=3.2$ and an effective ripple under $1\\,\\%$ , which demonstrates that high elongation is not a necessary feature of quasi-isodynamic stellarators.
B.J. Frei, J. Ball, A.C.D. Hoffmann, R. Jorge, P. Ricci, and L. Stenger
Cambridge University Press (CUP)
The derivation and numerical implementation of a linearized version of the gyrokinetic (GK) Coulomb collision operator (Jorge et al., J. Plasma Phys., vol. 85, 2019, 905850604) and of the widely used linearized GK Sugama collision operator (Sugama et al., Phys. Plasmas, vol. 16, 2009, 112503) is reported. An approach based on a Hermite–Laguerre moment expansion of the perturbed gyrocentre distribution function is used, referred to as gyromoment expansion. This approach allows the considering of arbitrary perpendicular wavenumber and expressing the two linearized GK operators as a linear combination of gyromoments where the expansion coefficients are given by closed analytical expressions that depend on the perpendicular wavenumber and on the temperature and mass ratios of the colliding species. The drift-kinetic (DK) limits of the GK linearized Coulomb and Sugama operators are also obtained. Comparisons between the gyromoment approach and the DK Coulomb and GK Sugama operators in the continuum GK code GENE are reported, focusing on the ion-temperature-gradient instability and zonal flow damping, finding an excellent agreement. It is confirmed that stronger collisional damping of the zonal flow residual by the Sugama GK model compared with the GK linearized Coulomb (Pan et al., Phys. Plasmas, vol. 27, 2020, 042307) persists at higher collisionality. Finally, we show that the numerical efficiency of the gyromoment approach increases with collisionality, a desired property for boundary plasma applications.
R Jorge and M Landreman
IOP Publishing
Abstract The stability of the ion-temperature gradient mode in quasisymmetric stellarators is assessed. This is performed using a set of analytical estimates together with linear gyrokinetic simulations. The peak growth rates, their corresponding real frequencies and wave-vectors are identified. A comparison is made between a first-order near-axis expansion model and eleven realistic designs obtained using numerical optimization methods. It is found that while the near-axis expansion is able to replicate the growth rates, real frequencies and perpendicular wave-vector at the inner core (both using simplified dispersion relations and first-principle gyrokinetic simulations), it leads to an overestimation of the growth rate at larger radii. An approximate analytic solution of the ITG dispersion relation for the non-resonant limit suggests growth rates could be systematically higher in quasi-axisymmetric (QA) configurations compared to quasi-helically (QH) symmetric ones. However except for very close to the axis, linear gyrokinetic simulations do not show systematic differences between QA and QH configurations.
B D Dudson, W A Gracias, R Jorge, A H Nielsen, J M B Olsen, P Ricci, C Silva, P Tamain, G Ciraolo, N Fedorczak,et al.
IOP Publishing
Abstract Fluid models used to study the edge plasma region need to be benchmarked against similar conditions given that models can strongly differ in complexity and therefore the results they produce. Via this validation study undertaken through the framework of EUROfusion Enabling Research, four state-of-the art models—GBS, Hermes/BOUT++, hot-edge-sol-electrostatic and TOKAM3X—are compared to experimental plasma turbulence measurements on the ISTTOK tokamak. Statistical comparisons of simulation and experiment data show that fluid models used here can replicate most of the experiment in terms of I sat and V float fluctuations despite their differences. Furthermore, it is shown that without including more complex information (like core turbulence information and domain geometry details and magnetic topological aspects) in fluid models, the results recovered vary from their experimental counterparts. Via the simulations using these codes, it is demonstrated that fluid models continue to be a good cost-effective tool in recovering many global aspects of edge plasma behaviour.
P. Kim, R. Jorge, and W. Dorland
Cambridge University Press (CUP)
A simplified analytical form of the on-axis magnetic well and Mercier's criterion for interchange instabilities for arbitrary three-dimensional magnetic field geometries is derived. For this purpose, a near-axis expansion based on a direct coordinate approach is used by expressing the toroidal magnetic flux in terms of powers of the radial distance to the magnetic axis. For the first time, the magnetic well and Mercier's criterion are then written as a one-dimensional integral with respect to the axis arclength. When compared with the original work of Mercier, the derivation here is presented using modern notation and in a more streamlined manner that highlights essential steps. Finally, these expressions are verified numerically using several quasisymmetric and non-quasisymmetric stellarator configurations including Wendelstein 7-X.
R Jorge and M Landreman
IOP Publishing
Abstract The design of turbulence optimized stellarators has so far relied on three-dimensional equilibrium codes such as VMEC in order to find the minimum of a given objective function. In this work, we propose a complimentary approach based on the near-axis expansion to compute the geometry parameters of neoclassicaly optimized stellarators used in turbulence studies. As shown here, the near-axis expansion can be a reasonable approximation of the geometric parameters relevant for turbulence and stability simulations of the core of existing optimized stellarator designs. In particular, we examine the geometry coefficients that appear in the gyrokinetic equation, the drift-reduced fluid equations and the ideal ballooning equation. This approach may allow for the development of new stellarator optimization techniques significantly faster than conventional methods.
L. M. Perrone, R. Jorge, and P. Ricci
AIP Publishing
A four-dimensional plasma model able to describe the scrape-off layer region of tokamak devices at arbitrary collisionality is derived in the drift-reduced limit. The basis of the model is provided by a drift-kinetic equation that retains the full non-linear Coulomb collision operator and describes arbitrarily far from equilibrium distribution functions. By expanding the dependence of distribution function over the perpendicular velocity in a Laguerre polynomial basis and integrating over the perpendicular velocity, a set of four-dimensional moment equations for the expansion coefficients of the distribution function is obtained. The Coulomb collision operator, as well as Poisson's equation, are evaluated explicitly in terms of perpendicular velocity moments of the distribution function.
R. Jorge, W. Sengupta, and M. Landreman
IOP Publishing
Optimized stellarator configurations and their analytical properties are obtained using a near-axis expansion approach. Such configurations are associated with good confinement as the guiding center particle trajectories and neoclassical transport are isomorphic to those in a tokamak. This makes them appealing as fusion reactor candidates. Using a direct coordinate approach, where the magnetic field and flux surface functions are found explicitly in terms of the position vector at successive orders in the distance to the axis, the set of ordinary differential equations for first and second order quasisymmetry is derived. Examples of quasi-axisymmetric shapes are constructed using a pseudospectral numerical method. Finally, the direct coordinate approach is used to independently verify two hypotheses commonly associated with quasisymmetric magnetic fields, namely that the number of equations exceeds the number of parameters at third order in the expansion and that the near-axis expansion does not prohibit exact quasisymmetry from being achieved on a single flux surface.
Matt Landreman and Rogerio Jorge
Cambridge University Press (CUP)
We have recently demonstrated that by expanding in small distance from the magnetic axis compared with the major radius, stellarator shapes with low neoclassical transport can be generated efficiently. To extend the utility of this new design approach, here we evaluate measures of magnetohydrodynamic interchange stability within the same expansion. In particular, we evaluate the magnetic well, Mercier's criterion, and resistive interchange stability near a magnetic axis of arbitrary shape. In contrast to previous work on interchange stability near the magnetic axis, which used an expansion of the flux coordinates, here we use the ‘inverse expansion’ in which the flux coordinates are the independent variables. Reduced expressions are presented for the magnetic well and stability criterion in the case of quasisymmetry. The analytic results are shown to agree with calculations from the VMEC equilibrium code. Finally, we show that near the axis, Glasser, Greene and Johnson's stability criterion for resistive modes approximately coincides with Mercier's ideal condition.
B. J. Frei, R. Jorge, and P. Ricci
Cambridge University Press (CUP)
A gyrokinetic model is presented that can properly describe large and small amplitude electromagnetic fluctuations occurring on scale lengths ranging from the electron Larmor radius to the equilibrium perpendicular pressure gradient scale length, and the arbitrarily large deviations from thermal equilibrium that are present in the plasma periphery of tokamak devices. The formulation of the gyrokinetic model is based on a second-order accurate description of the single charged particle dynamics, derived from Lie perturbation theory, where the fast particle gyromotion is decoupled from the slow drifts assuming that the ratio of the ion sound Larmor radius to the perpendicular equilibrium pressure scale length is small. The collective behaviour of the plasma is obtained by a gyrokinetic Boltzmann equation that describes the evolution of the gyroaveraged distribution function. The collisional effects are included by a nonlinear gyrokinetic Dougherty collision operator. The gyrokinetic model is then developed into a set of coupled fluid equations referred to as the gyrokinetic moment hierarchy. To obtain this hierarchy, the gyroaveraged distribution function is expanded onto a Hermite–Laguerre velocity-space polynomial basis. Then, the gyrokinetic equation is projected onto the same basis yielding the spatial and temporal evolution of the Hermite–Laguerre expansion coefficients. A closed set of fluid equations for the lowest-order coefficients is presented. The Hermite–Laguerre projection is performed accurately at arbitrary perpendicular wavenumber values. Finally, the self-consistent evolution of the electromagnetic fields is described by a set of gyrokinetic Maxwell equations derived from a variational principle where the velocity integrals are explicitly evaluated.
R. Jorge, W. Sengupta, and M. Landreman
Cambridge University Press (CUP)
A direct construction of equilibrium magnetic fields with toroidal topology at arbitrary order in the distance from the magnetic axis is carried out, yielding an analytical framework able to explore the landscape of possible magnetic flux surfaces in the vicinity of the axis. This framework can provide meaningful analytical insight into the character of high-aspect-ratio stellarator shapes, such as the dependence of the rotational transform and the plasma beta limit on geometrical properties of the resulting flux surfaces. The approach developed here is based on an asymptotic expansion on the inverse aspect ratio of the ideal magnetohydrodynamics equation. The analysis is simplified by using an orthogonal coordinate system relative to the Frenet–Serret frame at the magnetic axis. The magnetic field vector, the toroidal magnetic flux, the current density, the field line label and the rotational transform are derived at arbitrary order in the expansion parameter. Moreover, a comparison with a near-axis expansion formalism employing an inverse coordinate method based on Boozer coordinates (the so-called Garren–Boozer construction) is made, where both methods are shown to agree at lowest order. Finally, as a practical example, a numerical solution using a W7-X equilibrium is presented, and a comparison between the lowest-order solution and the W7-X magnetic field is performed.
R. Jorge, P. Ricci, S. Brunner, S. Gamba, V. Konovets, N. F. Loureiro, L. M. Perrone, and N. Teixeira
Cambridge University Press (CUP)
The dynamics of electron-plasma waves is described at arbitrary collisionality by considering the full Coulomb collision operator. The description is based on a Hermite–Laguerre decomposition of the velocity dependence of the electron distribution function. The damping rate, frequency and eigenmode spectrum of electron-plasma waves are found as functions of the collision frequency and wavelength. A comparison is made between the collisionless Landau damping limit, the Lenard–Bernstein and Dougherty collision operators and the electron–ion collision operator, finding large deviations in the damping rates and eigenmode spectra. A purely damped entropy mode, characteristic of a plasma where pitch-angle scattering effects are dominant with respect to collisionless effects, is shown to emerge numerically, and its dispersion relation is analytically derived. It is shown that such a mode is absent when simplified collision operators are used, and that like-particle collisions strongly influence the damping rate of the entropy mode.
R. Jorge, P. Ricci, and N. F. Loureiro
American Physical Society (APS)
A numerically efficient framework that takes into account the effect of the Coulomb collision operator at arbitrary collisionalities is introduced. Such a model is based on the expansion of the distribution function on a Hermite-Laguerre polynomial basis to study the effects of collisions on magnetized plasma instabilities at arbitrary mean-free path. Focusing on the drift-wave instability, we show that our framework allows retrieving established collisional and collisionless limits. At the intermediate collisionalities relevant for present and future magnetic nuclear fusion devices, deviations with respect to collision operators used in state-of-the-art turbulence simulation codes show the need for retaining the full Coulomb operator in order to obtain both the correct instability growth rate and eigenmode spectrum, which, for example, may significantly impact quantitative predictions of transport. The exponential convergence of the spectral representation that we propose makes the representation of the velocity space dependence, including the full collision operator, more efficient than standard finite difference methods.
João P. S. Bizarro, Hugo Hugon, and Rogério Jorge
American Physical Society (APS)
Ray propagation in weakly turbulent media is described by means of a quasilinear (QL) approach in which the dispersion relation and the ray equations are expanded up to, and including, second-order terms in the medium and ray fluctuations, leading to equations for the ensemble-averaged ray and its root-mean-square (rms) spreading. An important feature of the QL formalism is that the average ray does not coincide with the zero-order, unperturbed ray but may exhibit a drift with respect to the latter that is governed by the mean squared fluctuations. The theory is complete in that equations can be set for all quantities necessary to compute the ray trajectory and the rms spreading along its path, yet they obey an infinite downward recurrence in which equations involving lower-order derivatives of the medium fluctuations are recursively generated by the subsequent higher-order derivative, and which must thus be truncated for practical purposes. Using as examples the propagation of rays in homogeneous media with fluctuations arising from the presence of either a single random mode or a multimode isotropic turbulent spectrum, the QL formalism is validated against Monte Carlo (MC) calculations and, whenever possible, its numerical implementation is verified by comparison with analytical predictions. Choosing $4%$ both for the level of fluctuations and for the maximum ratio between the wavelengths of the propagating ray and of the turbulent modes, so as to remain within the validity of the second-order expansion in the random perturbations and of the eikonal approximation, the overall agreement between QL and MC results is fairly good, particularly for quantities such as the distance traveled by the average ray, its perpendicular rms spread, and the averages of the wave-vector components.
Rogério Jorge, Ednilton S. de Oliveira, and Jorge V. Rocha
WORLD SCIENTIFIC