arXiv subjects

Ivan Calvo

Publications and source records attributed to Ivan Calvo.

At least 19 recordsLinked to original sources

arXiv2025

Modification of ion-temperature-gradient turbulence by impurities in stellarator plasmas

Recent nonlinear gyrokinetic simulations have shown that impurities can strongly modify the turbulent heat flux in stellarator plasmas. Here, the ion-temperature-gradient (ITG) dispersion relation in a plasma containing impurities is analytically solved in certain limits and an expression for the modification of the ITG growth rate by impurities is derived. The analytical expression is the sum of three terms corresponding to three different physical causes (impurity density gradient, impurity temperature gradient and dilution) of the change in the growth rate. The scalings predicted analytically for the modification of the growth rate are shown to be reproduced by linear gyrokinetic simulations. The conditions for reduction or increase of the ITG growth by impurities are also correctly predicted by the analytical solution to the dispersion relation. Finally, a remarkable correlation is found between the analytical expression for the modification of the growth rate and the modification of the turbulent heat flux obtained from nonlinear gyrokinetic simulations.

physics.plasm-ph↗

arXiv2025

Piecewise omnigenous stellarators with zero bootstrap current

Until now, quasi-isodynamic magnetic fields have been the only known stellarator configurations that, at low collisionality, give small radial neoclassical transport and zero bootstrap current for arbitrary plasma profiles, the latter facilitating control of the magnetic configuration. The recently introduced notion of piecewise omnigenous fields has enormously broadened the space of stellarator configurations with small radial neoclassical transport. In this Letter, the existence of piecewise omnigenous fields that give zero bootstrap current is proven analytically and confirmed numerically. These results establish piecewise omnigenity as an alternative approach to stellarator reactor design.

physics.plasm-ph↗

arXiv2022

Finite orbit width effects in large aspect ratio stellarators

New orbit averaged equations for low collisionality neoclassical fluxes in large aspect ratio stellarators with mirror ratios close to unity are derived. The equations retain finite orbit width effects by employing the second adiabatic invariant $J$ as a velocity space coordinate and they have been implemented in the orbit-averaged neoclassical code KNOSOS. The equations are used to study the $1/\nu$ regime and the lower collisionality regimes. For generic large aspect ratio stellarators with mirror ratios close to unity, as the collision frequency decreases, the $1/\nu$ regime transitions directly into the $\nu$ regime, without passing through a $\sqrt{\nu}$ regime. An explicit formula for the neoclassical fluxes in the $\nu$ regime is obtained. The formula includes the effect of particles that transition between different types of wells. While these transitions produce stochastic scattering independent of the value of the collision frequency in velocity space, the diffusion in real space remains proportional to the collision frequency. The $\sqrt{\nu}$ regime is only recovered in large aspect ratio stellarators close to omnigeneity: large aspect ratio stellarators with large mirror ratios and optimized large aspect ratio stellarators with mirror ratios close to unity. Neoclassical transport in large aspect ratio stellarators with large mirror ratios can be calculated with the orbit-averaged equations derived by \cite{calvo17}. In these stellarators, the $\sqrt{\nu}$ regime exists in the collisionality interval $\epsilon |\ln \epsilon| \ll \nu_{ii} R a/\rho_i v_{ti} \ll 1/\epsilon$. In optimized large aspect ratio stellarators with mirror ratios close to unity, the $\sqrt{\nu}$ regime occurs in an interval of collisionality that depends on the deviation from omnigeneity $\delta$: $\delta^2 |\ln \delta | \ll \nu_{ii} R a/\rho_i v_{ti} \ll 1$.

physics.plasm-ph↗

arXiv2019

Impact of main ion pressure anisotropy on stellarator impurity transport

Main ions influence impurity dynamics through a variety of mechanisms; in particular, via impurity-ion collisions. To lowest order in an expansion in the main ion mass over the impurity mass, the impurity-ion collision operator only depends on the component of the main ion distribution that is odd in the parallel velocity. These lowest order terms give the parallel friction of the impurities with the main ions, which is typically assumed to be the main cause of collisional impurity transport. Next-order terms in the mass ratio expansion of the impurity-ion collision operator, proportional to the component of the main ion distribution that is even in the parallel velocity, are usually neglected. However, in stellarators, the even component of the main ion distribution can be very large. In this article, such next-order terms in the mass ratio expansion of the impurity-ion collision operator are retained, and analytical expressions for the neoclassical radial flux of trace impurities are calculated in the Pfirsch-Schl\"uter, plateau and $1/\nu$ regimes. The new terms provide a drive for impurity transport that is physically very different from parallel friction: they are associated to anisotropy in the pressure of the main ions, which translates into impurity pressure anisotropy. It is argued that main ion pressure anisotropy must be taken into account for a correct description of impurity transport in certain realistic stellarator plasmas. Examples are given by numerically evaluating the analytical expressions for the impurity flux.

physics.plasm-ph↗

arXiv2018

Electrostatic potential variations on stellarator magnetic surfaces in low collisionality regimes

The component of the neoclassical electrostatic potential that is non-constant on the magnetic surface, that we denote by $\tilde\varphi$, can affect radial transport of highly charged impurities, and this has motivated its inclusion in some modern neoclassical codes. The number of neoclassical simulations in which $\tilde\varphi$ is calculated is still scarce, partly because they are usually demanding in terms of computational resources, especially at low collisionality. In this paper the size, the scaling with collisionality and with aspect ratio, and the structure of $\tilde\varphi$ on the magnetic surface are analytically derived in the $1/\nu$, $\sqrt{\nu}$ and superbanana-plateau regimes of stellarators close to omnigeneity; i. e. stellarators that have been optimized for neoclassical transport. It is found that the largest $\tilde\varphi$ that the neoclassical equations admit scales linearly with the inverse aspect ratio and with the size of the deviation from omnigeneity. Using a model for a perturbed omnigeneous configuration, the analytical results are verified and illustrated with calculations by the code KNOSOS. The techniques, results and numerical tools employed in this paper can be applied to neoclassical transport problems in tokamaks with broken axisymmetry.

physics.plasm-ph↗

arXiv2018

Stellarator impurity flux driven by electric fields tangent to magnetic surfaces

The control of impurity accumulation is one of the main challenges for future stellarator fusion reactors. The standard argument to explain this accumulation relies on the, in principle, large inward pinch in the neoclassical impurity flux caused by the typically negative radial electric field in stellarators. This simplified interpretation was proven to be flawed by Helander et al. [Phys. Rev. Lett. 118, 155002 (2017)], who showed that in a relevant regime (low-collisionality main ions and collisional impurities) the radial electric field does not drive impurity transport. In that reference, the effect of the component of the electric field that is tangent to the magnetic surface was not included. In this Letter, an analytical calculation of the neoclassical radial impurity flux incorporating such effect is given, showing that it can be very strong for highly charged impurities and that, once it is taken into account, the dependence of the impurity flux on the radial electric field reappears. Realistic examples are provided in which the inclusion of the tangential electric field leads to impurity expulsion.

physics.plasm-ph↗

arXiv2016

The effect of tangential drifts on neoclassical transport in stellarators close to omnigeneity

In general, the orbit-averaged radial magnetic drift of trapped particles in stellarators is non-zero due to the three-dimensional nature of the magnetic field. Stellarators in which the orbit-averaged radial magnetic drift vanishes are called omnigeneous, and they exhibit neoclassical transport levels comparable to those of axisymmetric tokamaks. However, the effect of deviations from omnigeneity cannot be neglected in practice. For sufficiently low collision frequencies (below the values that define the $1/\nu$ regime), the components of the drifts tangential to the flux surface become relevant. This article focuses on the study of such collisionality regimes in stellarators close to omnigeneity when the gradient of the non-omnigeneous perturbation is small. First, it is proven that closeness to omnigeneity is required to preserve radial locality in the drift-kinetic equation for collisionalities below the $1/\nu$ regime. Then, it is shown that neoclassical transport is determined by two layers in phase space. One of the layers corresponds to the $\sqrt{\nu}$ regime and the other to the superbanana-plateau regime. The importance of the superbanana-plateau layer for the calculation of the tangential electric field is emphasized, as well as the relevance of the latter for neoclassical transport in the collisionality regimes considered in this paper. In particular, the tangential electric field is essential for the emergence of a new subregime of superbanana-plateau transport when the radial electric field is small. A formula for the ion energy flux that includes the $\sqrt{\nu}$ regime and the superbanana-plateau regime is given. The energy flux scales with the square of the size of the deviation from omnigeneity. Finally, it is explained why below a certain collisionality value the formulation presented in this article ceases to be valid.

physics.plasm-ph↗

arXiv2014

Radial transport of toroidal angular momentum in tokamaks

The radial flux of toroidal angular momentum is needed to determine tokamak intrinsic rotation profiles. Its computation requires knowledge of the gyrokinetic distribution functions and turbulent electrostatic potential to second-order in $\epsilon = \rho/L$, where $\rho$ is the ion Larmor radius and $L$ is the variation length of the magnetic field. In this article, a complete set of equations to calculate the radial transport of toroidal angular momentum in any tokamak is presented. In particular, the $O(\epsilon^2)$ equations for the turbulent components of the distribution functions and electrostatic potential are given for the first time without assuming that the poloidal magnetic field over the magnetic field strength is small.

physics.plasm-ph↗

arXiv2014

Less constrained omnigeneous stellarators

A stellarator is said to be omnigeneous if all particles have vanishing average radial drifts. In omnigeneous stellarators, particles are perfectly confined in the absence of turbulence and collisions, whereas in non-omnigeneous configurations, particle can drift large radial distances. One of the consequences of omnigeneity is that the unfavorable inverse scaling with collisionality of the stellarator neoclassical fluxes disappears. In the pioneering and influential article [Cary~J~R and Shasharina~S~G 1997 {\it Phys. Plasmas} {\bf 4} 3323], the conditions that the magnetic field of a stellarator must satisfy to be omnigeneous are derived. However, reference [Cary~J~R and Shasharina~S~G 1997 {\it Phys. Plasmas} {\bf 4} 3323] only considered omnigeneous stellarators in which all the minima of the magnetic field strength on a flux surface must have the same value. The same is assumed for the maxima. We show that omnigenenous magnetic fields can have local minima and maxima with different values. Thus, the parameter space in which omnigeneous stellarators are possible is larger than previously expected. The analysis presented in this article is only valid for orbits with vanishing radial width, and in principle it is not applicable to energetic particles. However, one would expect that improving neoclassical confinement would improve energetic particle confinement.

physics.plasm-ph↗

arXiv2014

Flow damping in stellarators close to quasisymmetry

Quasisymmetric stellarators are a type of optimized stellarators for which flows are undamped to lowest order in an expansion in the normalized Larmor radius. However, perfect quasisymmetry is impossible. Since large flows may be desirable as a means to reduce turbulent transport, it is important to know when a stellarator can be considered to be sufficiently close to quasisymmetry. The answer to this question depends strongly on the size of the spatial gradients of the deviation from quasisymmetry and on the collisionality regime. Recently, formal criteria for closeness to quasisymmetry have been derived in a variety of situations. In particular, the case of deviations with large gradients was solved in the $1/\nu$ regime. Denoting by $\alpha$ a parameter that gives the size of the deviation from quasisymmetry, it was proven that particle fluxes do not scale with $\alpha^{3/2}$, as typically claimed, but with $\alpha$. It was also shown that ripple wells are not necessarily the main cause of transport. This paper reviews those works and presents a new result in another collisionality regime, in which particles trapped in ripple wells are collisional and the rest are collisionless.

physics.plasm-ph↗

arXiv2014

Optimizing stellarators for large flows

Plasma flow is damped in stellarators because they are not intrinsically ambipolar, unlike tokamaks, in which the flux-surface averaged radial electric current vanishes for any value of the radial electric field. Only quasisymmetric stellarators are intrinsically ambipolar, but exact quasisymmetry is impossible to achieve in non-axisymmetric toroidal configurations. By calculating the violation of intrinsic ambipolarity due to deviations from quasisymmetry, one can derive criteria to assess when a stellarator can be considered quasisymmetric in practice, i.e. when the flow damping is weak enough. Let us denote by $\alpha$ a small parameter that controls the size of a perturbation to an exactly quasisymmetric magnetic field. Recently, it has been shown that if the gradient of the perturbation is sufficiently small, the flux-surface averaged radial electric current scales as $\alpha^2$ for any value of the collisionality. It was also argued that when the gradient of the perturbation is large, the quadratic scaling is replaced by a more unfavorable one. In this paper, perturbations with large gradients are rigorously treated. In particular, it is proven that for low collisionality a perturbation with large gradient yields, at best, an $O(|\alpha|)$ deviation from quasisymmetry. Heuristic estimations in the literature incorrectly predicted an $O(|\alpha|^{3/2})$ deviation.

physics.plasm-ph↗

arXiv2014

When omnigeneity fails

A generic non-symmetric magnetic field does not confine magnetized charged particles for long times due to secular magnetic drifts. Stellarator magnetic fields should be omnigeneous (that is, designed such that the secular drifts vanish), but perfect omnigeneity is technically impossible. There always are small deviations from omnigeneity that necessarily have large gradients. The amplification of the energy flux caused by a deviation of size $\epsilon$ is calculated and it is shown that the scaling with $\epsilon$ of the amplification factor can be as large as linear. In opposition to common wisdom, most of the transport is not due to particles trapped in ripple wells, but to the perturbed motion of particles trapped in the omnigeneous magnetic wells around their bounce points.

physics.plasm-ph↗

arXiv2013

Stellarators close to quasisymmetry

Rotation is favorable for confinement, but a stellarator can rotate at high speeds if and only if it is sufficiently close to quasisymmetry. This article investigates how close it needs to be. For a magnetic field $\mathbf{B} = \mathbf{B}_0 + \alpha \mathbf{B}_1$, where $\mathbf{B}_0$ is quasisymmetric, $\alpha\mathbf{B}_1$ is a deviation from quasisymmetry, and $\alpha\ll 1$, the stellarator can rotate at high velocities if $\alpha < \epsilon^{1/2}$, with $\epsilon$ the ion Larmor radius over the characteristic variation length of $\mathbf{B}_0$. The cases in which this result may break down are discussed. If the stellarator is sufficiently quasisymmetric in the above sense, the rotation profile, and equivalently, the long-wavelength radial electric field, are not set neoclassically; instead, they can be affected by turbulent transport. Their computation requires the $O(\epsilon^2)$ pieces of both the turbulent and the long-wavelength components of the distribution function. This article contains the first step towards a formulation to calculate the rotation profile by providing the equations determining the long-wavelength components of the $O(\epsilon^2)$ pieces.

physics.plasm-ph↗

arXiv2012

Long-wavelength limit of gyrokinetics in a turbulent tokamak and its intrinsic ambipolarity

Recently, the electrostatic gyrokinetic Hamiltonian and change of coordinates have been computed to order $\epsilon^2$ in general magnetic geometry. Here $\epsilon$ is the gyrokinetic expansion parameter, the gyroradius over the macroscopic scale length. Starting from these results, the long-wavelength limit of the gyrokinetic Fokker-Planck and quasineutrality equations is taken for tokamak geometry. Employing the set of equations derived in the present article, it is possible to calculate the long-wavelength components of the distribution functions and of the poloidal electric field to order $\epsilon^2$. These higher-order pieces contain both neoclassical and turbulent contributions, and constitute one of the necessary ingredients (the other is given by the short-wavelength components up to second order) that will eventually enter a complete model for the radial transport of toroidal angular momentum in a tokamak in the low flow ordering. Finally, we provide an explicit and detailed proof that the system consisting of second-order gyrokinetic Fokker-Planck and quasineutrality equations leaves the long-wavelength radial electric field undetermined; that is, the turbulent tokamak is intrinsically ambipolar.

physics.plasm-ph↗

arXiv2012

Intrinsic rotation with gyrokinetic models

The generation of intrinsic rotation by turbulence and neoclassical effects in tokamaks is considered. To obtain the complex dependences observed in experiments, it is necessary to have a model of the radial flux of momentum that redistributes the momentum within the tokamak in the absence of a preexisting velocity. When the lowest order gyrokinetic formulation is used, a symmetry of the model precludes this possibility, making small effects in the gyroradius over scale length expansion necessary. These effects that are usually small become important for momentum transport because the symmetry of the lowest order gyrokinetic formulation leads to the cancellation of the lowest order momentum flux. The accuracy to which the gyrokinetic equation needs to be obtained to retain all the physically relevant effects is discussed.

physics.plasm-ph↗

arXiv2010

Phase-space Lagrangian derivation of electrostatic gyrokinetics in general geometry

Gyrokinetic theory is based on an asymptotic expansion in the small parameter $\epsilon$, defined as the ratio of the gyroradius and the characteristic length of variation of the magnetic field. In this article, this ordering is strictly implemented to compute the electrostatic gyrokinetic phase-space Lagrangian in general magnetic geometry to order $\epsilon^2$. In particular, a new expression for the complete second-order gyrokinetic Hamiltonian is provided, showing that in a rigorous treatment of gyrokinetic theory magnetic geometry and turbulence cannot be dealt with independently. The new phase-space gyrokinetic Lagrangian gives a Vlasov equation accurate to order $\epsilon^2$ and a Poisson equation accurate to order $\epsilon$. The final expressions are explicit and can be implemented into any simulation without further computations.

physics.plasm-ph↗

arXiv2009

Zonal flows and long-distance correlations during the formation of the edge shear layer in the TJ-II stellarator

A theoretical interpretation is given for the observed long-distance correlations in potential fluctuations in TJ-II. The value of the correlation increases above the critical point of the transition for the emergence of the plasma edge shear flow layer. Mean (i.e. surface averaged, zero-frequency) sheared flows cannot account for the experimental results. A model consisting of four envelope equations for the fluctuation level, the mean flow shear, the zonal flow amplitude shear, and the averaged pressure gradient is proposed. It is shown that the presence of zonal flows is essential to reproduce the main features of the experimental observations.

physics.plasm-ph↗

arXiv2009

Fractional Levy motion through path integrals

Fractional Levy motion (fLm) is the natural generalization of fractional Brownian motion in the context of self-similar stochastic processes and stable probability distributions. In this paper we give an explicit derivation of the propagator of fLm by using path integral methods. The propagators of Brownian motion and fractional Brownian motion are recovered as particular cases. The fractional diffusion equation corresponding to fLm is also obtained.

cond-mat.stat-mech↗