arXiv subjects

Alejandro Luque

Publications and source records attributed to Alejandro Luque.

At least 19 recordsLinked to original sources

arXiv2026

Partial ionisation cross sections for the binary-encounter Bethe model

The original binary-encounter Bethe model of Kim and Eugene Rudd (1994 Phys. Rev. A 50 3954-67) has proven to be an accurate analytical representation of total impact ionisation cross sections of electrons colliding with atoms and molecules. It is based on a decomposition into partial ionisation cross sections from electrons in bound orbitals. Despite the model's accuracy for total ionisation, its individual partial cross sections for ionisation rely on thresholds calculated theoretically which systematically overestimate the experimental orbital binding energies. Here, we examine the BEB model's performance when based on experimental ionisation thresholds. The resulting partial cross sections of the various final (excited) ionic states produced could help to prefigure subsequent optical radiations and non-radiative transitions in models of plasma physics.

physics.atom-ph↗

arXiv2022

Renormalization and existence of the finite-time blow up solutions for a one-dimensional analogue of the Navier-Stokes equations

The one-dimensional quasi-geostrophic equation is the one-dimensional Fourier-space analogue of the famous Navier-Stokes equations. In their work Li and Sinai have proposed a renormalization approach to the problem of existence of finite-time blow up solutions of this equation. In this setting, existence of finite time blow ups is a consequence of existence of a fixed point for a certain renormalization operator on an appropriate functional space. They have provided a proof of existence of complex-valued finite time blow up solutions of the quasi-geostrophic equation. In this paper we revisit the renormalization problem for the quasi-geostrophic blow ups, prove existence of a family of renormalization fixed points, and deduce existence of real $C^\infty([0,T),C^\infty(\mathbb{R}) \cap L^2(\mathbb{R}))$ solutions to the quasi-geostrophic equation whose energy and enstrophy become unbounded in finite time, different from those found in the previous work of Li and Sinai.

math.AP↗

arXiv2021

MHD equilibria with nonconstant pressure in nondegenerate toroidal domains

We prove the existence of piecewise smooth MHD equilibria in three-dimensional toroidal domains of $\mathbf{R}^3$ where the pressure is constant on the boundary but not in the interior. The pressure is piecewise constant and the plasma current exhibits an arbitrary number of current sheets. We also establish the existence of free boundary steady states surrounded by vacuum with an external surface current. The toroidal domains where these equilibria are shown to exist do not need to be small perturbations of an axisymmetric domain, and in fact they can have any knotted topology. The building blocks we use in our construction are analytic toroidal domains satisfying a certain nondegeneracy condition, which roughly states that there exists a force-free field that is ergodic on the surface of the domain. The proof involves three main ingredients: a gluing construction of piecewise smooth MHD equilibria, a Hamilton-Jacobi equation on the two-dimensional torus that can be understood as a nonlinear deformation of the cohomological equation (so the nondegeneracy assumption plays a major role in the corresponding analysis), and a new KAM theorem tailored for the study of divergence-free fields in three dimensions whose Poincar\'e map cannot be computed explicitly.

math.AP↗

arXiv2019

Beltrami fields with hyperbolic periodic orbits enclosed by knotted invariant tori

We prove that there exist Beltrami fields in Euclidean space, with sharp decay at infinity, which have a prescribed set of invariant tori (possibly knotted or linked) that enclose an arbitrarily large number of hyperbolic periodic orbits. These hyperbolic orbits are cablings over the core curve of each torus. Moreover, the domain bounded by each invariant torus is covered by an almost full measure set of invariant tori. We show that an analogous result holds for high-frequency Beltrami fields on the flat 3-torus.

math.DS↗

arXiv2019

Spontaneous emergence of space stems ahead of negative leaders in lightning and long sparks

We investigate the emergence of space stems ahead of negative leaders. These are luminous spots that appear ahead of an advancing leader mediating the leader's stepped propagation. We show that space stems start as regions of locally depleted conductivity that form in the streamers of the corona around the leader. An attachment instability enhances the electric field leading to strongly inhomogeneous, bright and locally warmer regions ahead of the leader that explain the existing observations. Since the attachment instability is only triggered by fields above 10 kV/cm and internal electric fields are lower in positive than in negative streamers, our results explain why, although common in negative leaders, space stems and stepping are hardly observed if not absent in positive leaders. Further work is required to fully explain the streamer to leader transition, which requires an electric current persisting for timescales longer than the typical attachment time of electrons, around 100 ns.

physics.ao-ph↗

arXiv2018

Effective bounds for the measure of rotations

A fundamental question in Dynamical Systems is to identify regions of phase/parameter space satisfying a given property (stability, linearization, etc). Given a family of analytic circle diffeomorphisms depending on a parameter, we obtain effective (almost optimal) lower bounds of the Lebesgue measure of the set of parameters that are conjugated to a rigid rotation. We estimate this measure using an a-posteriori KAM scheme that relies on quantitative conditions that are checkable using computer-assistance. We carefully describe how the hypotheses in our theorems are reduced to a finite number of computations, and apply our methodology to the case of the Arnold family. Hence we show that obtaining non-asymptotic lower bounds for the applicability of KAM theorems is a feasible task provided one has an a-posteriori theorem to characterize the problem. Finally, as a direct corollary, we produce explicit asymptotic estimates in the so called local reduction setting (\`a la Arnold) which are valid for a global set of rotations.

math.DS↗

arXiv2018

Stationary phase methods and the splitting of separatrices

Using stationary phase methods, we provide an explicit formula for the Melnikov function of the one and a half degrees of freedom system given by a Hamiltonian system subject to a rapidly oscillating perturbation. Remarkably, the Melnikov function turns out to be computable without an explicit knowledge of the separatrix and in the case of non-analytic systems. This is related to a priori stable systems coupled with low regularity perturbations. It also applies to perturbations controlled by wave-type equations, so in particular we also illustrate this result with the motion of charged particles in a rapidly oscillating electromagnetic field. Quasiperiodic perturbations are discussed too.

math.DS↗

arXiv2017

A-posteriori KAM theory with optimal estimates for partially integrable systems

In this paper we present a-posteriori KAM results for existence of $d$-dimensional isotropic invariant tori for $n$-DOF Hamiltonian systems with additional $n-d$ independent first integrals in involution. We carry out a covariant formulation that does not require the use of action-angle variables nor symplectic reduction techniques. The main advantage is that we overcome the curse of dimensionality avoiding the practical shortcomings produced by the use of reduced coordinates, which may cause difficulties and underperformance when quantifying the hypotheses of the KAM theorem in such reduced coordinates. The results include ordinary and (generalized) iso-energetic KAM theorems. The approach is suitable to perform numerical computations and computer assisted proofs.

math.DS↗

arXiv2017

A domain-decomposition method to implement electrostatic free boundary conditions in the radial direction for electric discharges

At high pressure electric discharges typically grow as thin, elongated filaments. In a numerical simulation this large aspect ratio should ideally translate into a narrow, cylindrical computational domain that envelops the discharge as closely as possible. However, the development of the discharge is driven by electrostatic interactions and, if the computational domain is not wide enough, the boundary conditions imposed to the electrostatic potential on the external boundary have a strong effect on the discharge. Most numerical codes for electric discharges circumvent this problem by either using a wide computational domain or by calculating the boundary conditions by integrating the Green's function of an infinite domain. Here we describe an accurate and efficient method to impose free boundary conditions for an elongated electric discharge. To facilitate the use of our method we provide a sample implementation.

physics.plasm-ph↗

arXiv2017

On the sharpness of the R\"ussmann estimates

Estimating the norm of the solution of the linear difference equation $u(\theta)-u(\theta+\omega)=v(\theta)$ plays a fundamental role in KAM theory. Optimal (in certain sense) estimates for the solution of this equation were provided by R\"ussmann in the mid 70's. The aim of this paper is to compare the sharpness of these classic estimates with more specific estimates obtained with the help of the computer. We perform several experiments to quantify the improvement obtained when using computer assisted estimates. By comparing these estimates with the actual norm of the solution, we can analyze the different sources of overestimation, thus encouraging future improvements.

math.DS↗

arXiv2016

Rigorous computer assisted application of KAM theory: a modern approach

In this paper we present and illustrate a general methodology to apply KAM theory in particular problems, based on an {\em a posteriori} approach. We focus on the existence of real-analytic quasi-periodic Lagrangian invariant tori for symplectic maps. The purpose is to verify the hypotheses of a KAM theorem in an a posteriori format: given a parameterization of an approximately invariant torus, we have to check non-resonance (Diophantine) conditions, non-degeneracy conditions and certain inequalities to hold. To check such inequalities we require to control the analytic norm of some functions that depend on the map, the ambient structure and the parameterization. To this end, we propose an efficient computer assisted methodology, using fast Fourier transform, having the same asymptotic cost of using the parameterization method for obtaining numerical approximations of invariant tori. We illustrate our methodology by proving the existence of invariant curves for the standard map (up to $\eps=0.9716$), meandering curves for the non-twist standard map and 2-dimensional tori for the Froeschl\'e map.

math.DS↗

arXiv2015

Arnold diffusion of charged particles in ABC magnetic fields

We prove the existence of diffusing solutions in the motion of a charged particle in the presence of an ABC magnetic field. The equations of motion are modeled by a 3DOF Hamiltonian system depending on two parameters. For small values of these parameters, we obtain a normally hyperbolic invariant manifold and we apply the so-called geometric methods for a priori unstable systems developed by A. Delshams, R. de la Llave, and T.M. Seara. We characterize explicitly sufficient conditions for the existence of a transition chain of invariant tori having heteroclinic connections, thus obtaining global instability (Arnold diffusion). We also check the obtained conditions in a computer assisted proof. ABC magnetic fields are the simplest force-free type solutions of the magnetohydrodynamics equations with periodic boundary conditions, so our results are of potential interest in the study of the motion of plasma charged particles in a tokamak.

nlin.CD↗

arXiv2013

Growing discharge trees with self-consistent charge transport: the collective dynamics of streamers

We introduce the generic structure of a growth model for branched discharge trees that consistently combines a finite channel conductivity with the physical law of charge conservation. It is applicable, e.g., to streamer coronas near tip or wire electrodes and ahead of lightning leaders, to leaders themselves and to the complex breakdown structures of sprite discharges high above thunderclouds. Then we implement and solve the simplest model for positive streamers in ambient air with self-consistent charge transport. We demonstrate that charge conservation contradicts the common assumption of dielectric breakdown models that the electric fields inside all streamers are equal to the so-called stability field and we even find cases of local field inversion. We also discuss the charge distribution inside discharge trees, which provides a natural explanation for the observed reconnections of streamers in laboratory experiments and in sprites. Our simulations show the structure of an overall "streamer of streamers" that we name collective streamer front, and predict effective streamer branching angles, the charge structure within streamer trees, and streamer reconnection.

physics.plasm-ph↗

arXiv2010

Density models for streamer discharges: beyond cylindrical symmetry and homogeneous media

Streamer electrical discharges are often investigated with computer simulations of density models (also called drift-diffusion-reaction models). We review these models, detailing their physical foundations, their range of validity and the most relevant numerical algorithms employed in solving them. We focus particularly on schemes of adaptative refinement, used to resolve the multiple length scales in a streamer discharge without a high computational cost. We then report recent results from these models, emphasizing developments that go beyond cylindrically symmetrical streamers propagating in homogeneous media.

physics.plasm-ph↗

arXiv2010

Probing photo-ionization: simulations of positive streamers in varying N2:O2 mixtures

Photo-ionization is the accepted mechanism for the propagation of positive streamers in air though the parameters are not very well known; the efficiency of this mechanism largely depends on the presence of both nitrogen and oxygen. But experiments show that streamer propagation is amazingly robust against changes of the gas composition; even for pure nitrogen with impurity levels below 1 ppm streamers propagate essentially with the same velocity as in air, but their minimal diameter is smaller, and they branch more frequently. Additionally, they move more in a zigzag fashion and sometimes exhibit a feathery structure. In our simulations, we test the relative importance of photo-ionization and of the background ionization from pulsed repetitive discharges, in air as well as in nitrogen with 1 ppm O2 . We also test reasonable parameter changes of the photo-ionization model. We find that photo- ionization dominates streamer propagation in air for repetition frequencies of at least 1 kHz, while in nitrogen with 1 ppm O2 the effect of the repetition frequency has to be included above 1 Hz. Finally, we explain the feather-like structures around streamer channels that are observed in experiments in nitrogen with high purity, but not in air.

physics.plasm-ph↗

arXiv2010

Review of recent results on streamer discharges and discussion of their relevance for sprites and lightning

It is by now well understood that large sprite discharges at the low air densities of the mesosphere are physically similar to small streamer discharges in air at standard temperature and pressure. This similarity is based on Townsend scaling with air density. First the theoretical basis of Townsend scaling and a list of six possible corrections to scaling are discussed; then the experimental evidence for the similarity between streamers and sprites is reviewed. We then discuss how far present sprite and streamer theory has been developed, and we show how streamer experiments can be interpreted as sprite simulations. We review those results of recent streamer research that are relevant for sprites and other forms of atmospheric electricity and discuss their implications for sprite understanding. These include the large range of streamer diameters and velocities and the overall 3D morphology with branching, interaction and reconnection, the dependence on voltage and polarity, the electron energies in the streamer head and the consecutive chemical efficiency and hard radiation. New theoretical and experimental results concern measurements of streamer spectra in air, the density dependence of streamer heating (hot leaders are unlikely at 80 km altitude and cold streamers are unlikely in liquids), and a discussion of the influence of magnetic fields on thermal electrons or on energetic electrons in streamers or sprites.

physics.ao-ph↗

arXiv2009

Moving boundary approximation for curved streamer ionization fronts: Numerical tests

Recently a moving boundary approximation for the minimal model for negative streamer ionization fronts was extended with effects due to front curvature; this was done through a systematic solvability analysis. A central prediction of this analysis is the existence of a nonvanishing electric field in the streamer interior, whose value is proportional to the front curvature. In this paper we compare this result and other predictions of the solvability analysis with numerical simulations of the minimal model.

physics.plasm-ph↗

arXiv2008

Saffman-Taylor streamers: mutual finger interaction in an electric breakdown

Bunches of streamers form the early stages of sparks and lightning but theory presently concentrates on single streamers or on coarse approximations of whole breakdown trees. Here a periodic array of interacting streamer discharges in a strong homogeneous electric field is studied in PDE approximation in two dimensions. If the period of the streamer array is small enough, the streamers do not branch, but approach uniform translation. When the streamers are close to the branching regime, the enhanced field at the tip of the streamer is close to $2 E_{\infty}$, where $E_{\infty}$ is the homogeneous field applied between the electrodes. We discuss a moving boundary approximation to the set of PDEs. This moving boundary model turns out to be essentially the same as the one for two-fluid Hele-Shaw flows. In two dimensions, this model possesses a known analytical solution. The shape of the 2D interacting streamers in uniform motion obtained from the PDE simulations is actually well fitted by the analytically known ``selected Saffman-Taylor finger''. This finding helps to understand streamer interactions and raises new questions on the general theory of finger selection in moving boundary problems.

physics.plasm-ph↗