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Streamer discharges play a central role in electric breakdown of matter in pulsed electric fields, both in nature and in technology. Reliable and fast computations of the minimal model for negative streamers in simple gases like nitrogen have recently been developed. However, photoionization was not included; it is important in air and poses a major numerical challenge. We here introduce a fast and reliable method to include photoionization into our numerical scheme with adaptice grids, and we discuss its importance for negative streamers. In particular, we identify different propagation regimes where photoionization does or does not play a role. More precisely, when a high voltage pulse is applied to a gap of insulating matter, conducting streamer channels grow through the gap. Streamer propagation is characterized by a strong field enhancement at the channel tip. This field enhancement is created by a thin curved space charge layer around the streamer tip as many computations show. Such computations are quite challenging due to the multiple inherent scales of the process.Recent streamer research largely concentrates on positive streamers in air or other complex gases for industrial applications [2]. This is because positive streamers emerge from needle or wire electrodes at lower voltages than negative ones [1]. Natural discharges such as sprites [3], on the other hand, occur in both polarities [4], in particular, when they are not attached to an electrode and therefore double ended. Photoionization (or alternatively background ionization) is essential for positive streamers: as their tips propagate several orders of magnitude faster than positive ions drift in the local field, a nonlocal photon-mediated ionization reaction is thought to cause the fast propagation of the positive ionization front. Negative streamers, on the other hand, have velocities comparable to the drift velocity of electrons in the local field, therefore a local impact ionization reaction can be sufficient to explain their propagation. This is why photoionization in negative streamers has received much less attention, most recent work concentrating on sprite conditions with relatively low electric fields [5].The nonlocal photoionization reaction depends strongly on gas composition and pressure [6], in particular, it is much more efficient in air than in pure gases. Furthermore, in air its relative importance saturates for pressures well below 60 Torr (≈ 0.1 bar), while it is suppressed like ≈ 60 Torr/p at atmospheric pressure and above. In this paper we study the effects of photoionization on the propagation of negative streamers by means of efficient computations with adaptive grids.Streamer model. Streamer models always contain electron drift and diffusion, space charge effects and the generation of electron ion pairs by essentially local impact ionization. We will use a fluid model in local field approximation as described, e.g., in Refs. [7,8]. A numerical code with adaptive grid refinement was introduced in [8] to investigate negativ...

Streamers are a generic mode of electric breakdown of large gas volumes. They play a role in the initial stages of sparks and lightning, in technical corona reactors and in high altitude sprite discharges above thunderclouds. Streamers are characterized by a self-generated field enhancement at the head of the growing discharge channel. We briefly review recent streamer experiments and sprite observations. Then we sketch our recent work on computations of growing and branching streamers, we discuss concepts and solutions of analytical model reductions, we review different branching concepts and outline a hierarchy of model reductions.

Non-ionized media subject to strong fields can become locally ionized by penetration of fingershaped streamers. We study negative streamers between planar electrodes in a simple deterministic continuum approximation. We observe that for sufficiently large fields, the streamer tip can split. This happens close to the limit of "ideal conductivity". Qualitatively the tip splitting is due to a Laplacian instability quite like in viscous fingering. For future quantitative analytical progress, our stability analysis of planar fronts identifies the screening length as a regularization mechanism.Streamers commonly appear in dielectric breakdown when a sufficiently high voltage is suddenly applied to a medium with low or vanishing conductivity. They consist of extending fingers of ionized matter and are ubiquitous in nature and technology [1,2]. The degree of ionization inside a streamer is low, hence thermal or convection effects are negligible. However, streamers are nonlinear phenomena due to the space charges inside the ionized body that modify the externally applied electric field. While in many applications, streamers by a strongly nonuniform background electric field are forced to propagate towards the cathode through complex mixtures of gases [2-4], we here investigate the basic phenomenon of the primary anode-directed streamer in a simple nonattaching and non-ionized gas and in a uniform background field as in the pioneering experiments of Raether [5]. In previous theoretical work, it is implicitly assumed that streamers in a uniform background field propagate in a stationary manner [6][7][8]. This view seems to be supported by previous simulations [9,10].In this paper we present the first numerical evidence that anode directed (or negative) streamers do branch even in a uniform background field and without initial background ionization in the minimal fully deterministic "fluid model" [1,[6][7][8][9][10], if the field is sufficiently strong. We argue that this happens when the streamer approaches what we suggest to call the Lozansky-Firsov limit of "ideal conductivity" [6]. The streamer then can be understood as an interfacial pattern with a Laplacian instability [11], qualitatively similar to other Laplacian growth problems [12]. For future quantitative analytical progress, we identify the electric screening length as a relevant regularization mechanism. Our finding casts doubts on the existence of a stationary mode of streamer propagation with a fixed head radius. FIG. 1. Evolution of spontaneous branching of anode directed streamers in a strong homogeneous background field at times t = 300, 365, 420 and 450. Model, initial and boundary conditions are discussed in the text. The planar cathode is located at z = 0 and the planar anode at z = 2000 (shown is 0 ≤ z ≤ 1400). The radial coordinate extends from the origin up to r = 2000 (shown is 0 ≤ r ≤ 600). The thin lines denote levels of equal electron density σ with increments of 0.1 or 0.2 as indicated by the labels. The thick lines denote the higher electron de...

This paper examines a class of explicit finite-difference advection schemes derived along the method of lines. An important application field is large-scale atmospheric transport. The paper therefore focuses on the demand of positivity. For the spatial discretization, attention is confined to conservative schemes using five points per direction. The fourth-order central scheme and the family of K-schemes, comprising the second-order central, the second-order upwind, and the third-order upwind biased, are studied. Positivity is enforced through flux limiting. It is concluded that the limited third-order upwind discretization is the best candidate from the four examined. For the time integration attention is confined to a number of explicit Runge-Kutta methods of orders two up to four. With regard to the demand of positivity, these integration methods turn out to behave almost equally and no best method could be identified. '~' 1995 Academic Press, Inc. l. INTRODUCTIONThe subject of this paper is the numerical solution of the partial differential equation for linear advection of a scalar quantity w in an arbitrary velocity field u, given byLinear advection is an important (classical) problem in computational fluid dynamics and has been the subject of numerous investigations. The central theme is how to approximate the advection term \7 · (uw), such that the resulting errors in both phase and amplitude are minimized and the computational cost is still affordable. An important application we have in mind concerns atmospheric transport of chemical species. Then w represents a concentration or density and u a wind field. In addition to the usual accuracy and efficiency requirements, here the main consideration is that the transported concentrations must remain positive, because in actual applications also chemical reactions are modeled for which positivity is a prerequisite for avoiding non-physical chemical instabilities. We emphasize *The research reported helongs to the projects EUSMOG and CIRK which are carried out in cooperation with the Air Laboratory of the RIVM-Tbe Dutch National Institute of Public Health and Environmental Protection. The RIVM is acknowledged for financial support. 35that the demand of positivity is important and that it severely restricts the choice of method, as it is essentially equivalent to the demand of avoiding numerical under-and overshoots in regions of strong variation.The research objective of this paper is to ex.amine a class of positive, finite-difference advection schemes which we consider promising for atmospheric transport applications and to select from this class the best possible candidate. We hereby follow the method-of-lines approach which means that the spatial discretization and temporal integration are considered separately.For the spatial discretization we confine ourselves to stencils using five points per (spatial) direction. We consider this a good starting point since a 5-point stencil is computationally attractive for the following reasons. First, a 5-point stenc...

The evolution of negative streamers during electric breakdown of a non-attaching gas can be described by a two-fluid model for electrons and positive ions. It consists of continuity equations for the charged particles including drift, diffusion and reaction in the local electric field, coupled to the Poisson equation for the electric potential. The model generates field enhancement and steep propagating ionization fronts at the tip of growing ionized filaments. An adaptive grid refinement method for the simulation of these structures is presented. It uses finite volume spatial discretizations and explicit time stepping, which allows the decoupling of the grids for the continuity equations from those for the Poisson equation. Standard refinement methods in which the refinement criterion is based on local error monitors fail due to the pulled character of the streamer front that propagates into a linearly unstable state. We present a refinement method which deals with all these features. Tests on one-dimensional streamer fronts as well as on three-dimensional streamers with cylindrical symmetry (hence effectively 2D for numerical purposes) are carried out successfully. Results on fine grids are presented, they show that such an adaptive grid method is needed to capture the streamer characteristics well. This refinement strategy enables us to adequately compute negative streamers in pure gases in the parameter regime where a physical instability appears: branching streamers.

Abstract. A second-order, L-stable Rosenbrock method from the field of stiff ordinary differential equations is studied for application to atmospheric dispersion problems describing photochemistry, advective, and turbulent diffusive transport. Partial differential equation problems of this type occur in the field of air pollution modeling. The focal point of the paper is to examine the Rosenbrock method for reliable and efficient use as an atmospheric chemical kinetics box-model solver within Strang-type operator splitting. In addition, two W-method versions of the Rosenbrock method are discussed. These versions use an inexact J1tcobian matrix and are meant to provide alternatives for Strang-splitting. Another alternative for Strang-splitting is a technique based on so-called source-splitting. This technique is briefly discussed.

Dedicated to Peter van der Houwen for his numerous contributions in the field of numerical integration of differential equations Summary. The Runge-Kutta-Chebyshev method is ans-stage Runge-Kutta method designed for the explicit integration of stiff systems of ordinary differential equations originating from spatial discretization of parabolic partial differential equations (method of lines). The method possesses an extended real stability interval with a length f3 proportional to s 2 . The method can be applied with s arbitrarily large, which is an attractive feature due to the proportionality of f3 with s 2 . The involved stability property here is internal stability. Internal stability has to do with the propagation of errors over the stages within one single integration step. This internal stability property plays an important role in our examination of full convergence properties of a class of lst and 2nd order schemes. Full convergence means convergence of the fully discrete solution to the solution of the partial differential equation upon simultaneous space-time grid refinement. For a model class of linear problems we prove convergence under the sole condition that the necessary time-step restriction for stability is satisfied. These error bounds are valid for any s and independent of the stiffness of the problem.Numerical examples are given to illustrate the theoretical results.