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Mechanism of the Atmospheric Ball Lightning Using the Triple Beltrami Equation Ana M´arcia Alves Taveira and Paulo Hiroshi Sakanaka Instituto de F´ısica “Gleb Wataghin”, UNICAMP, C.P. 6165 13083-970, Campinas, SP, Brazil November 17, 2004 Abstract Ball lightning, also known as fire ball, is a luminous globe which occurs in the course of a thunderstorm. It has been the object of investigation by figures in science since the early nineteenth century. The difficult and long-standing problem of ball lightning has attracted few meteorologists or atmospheric scientists. Rather, physicists constitute the larger part of the company studying the atmospheric fire balls. Taking as model, the two fluid plasma consisting of electrons and one species ions, for the fire balls physics and considering that the plasma flow is a finite quantity, we can derive the equation of relaxed energy state, maintaining the helicity constant, in the form of triple Beltrami equations for the magnetic field: s B~ + p B~ + q B~ + rB~ =0 ∇×∇×∇× ∇×∇× ∇ × where B~ is the magnetic field, s, p, q, and r are constants to be determined through boundary conditions. This equation is coupled with an equation which describes the hydrodynamic vortex. When s = p = 0, we have the Taylor relaxed state of plasma [1], without the fluid flow. In the case of electron-ion plasma, both s and p are non-zero quantities. In particular, when electron mass is neglected the results is s = 0 and it describes the Double Beltrami relaxed ccsd-00002014, version 1 - 17 Nov 2004 minimum energy state as derived by S. M. Mahajan and Z. Yoshida [2]. Its solution is a spheromak type solution. The problem of the formation of an isolated luminous mass in the sky and the moderate persistence of the resulting form combined with observations of 1 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 1, FEBRUARY 1998 41 Two-Dimensional Modeling of Positive Streamer Propagation in Flue Gases in Sphere-Plane Gaps Natalya Yu. Babaeva and George V. Naidis Abstract— The results of two-dimensional numerical simula- II. SIMULATIONS tion of positive streamer dynamics in flue gases in sphere-plane electrode configuration are presented. The q-values (numbers The 2-D streamer model is analogous to that used earlier of radicals produced per 100 eV of input electrical energy) for for air, and is described in detail in [12] and [13]. Streamer the production of chemically active particles participating in flue propagation in the sphere-plane gap (the sphere radius gas cleaning from toxic components are calculated. Obtained – cm, the ratio of the gap length to the sphere radius q -values are almost independent on the discharge conditions. – in the mixture N O CO H O Simple estimates for q-values, based on analytical streamer the- ory, are shown to agree with the results of numerical simulation. at the molecule number density cm and the gas temperature K Index Terms— Corona discharge, flue gas cleaning, positive has been simulated. As it has been shown in [9], the electron streamer. energy distribution function (EEDF) in flue gas mixtures is close (at the values of reduced electric field typical for streamer propagation) to the EEDF in air. So in our model for I. INTRODUCTION flue gases, the rate constants of excitation and dissociation of ULSED positive corona discharges in flue gases are gas components have been taken calculated with EEDF in air. Pactively studied in connection with their use for gas Approximations of the excitation rate constants of nitrogen cleaning from toxic components NO and CO [1]. Such triplet electronic states and the rate constants of oxygen discharges have a structure of a number of streamers—thin dissociation and dissociative excitation presented in [14] have plasma channels propagating in discharge gap. Active particles been used. The data on the nitrogen dissociation rate constant taking part in the removal of toxic components are produced in have been taken from [15]. Calculation of carbon dioxide the regions of high electric field—in the streamer heads. The dissociation and water dissociative attachment rate constants problem of simulation of the cleaning process includes two has been made with the use of cross sections presented in stages. The first stage is calculation of the rates of primary [16] and [17]. The dependencies of the ionization coefficient active particles production by streamers, the second is the and the drift velocity of electrons on in flue gas have modeling of subsequent chemical transformations in the flue been taken the same as in air, in accordance with [9]. The gas mixture. The chemical part of the problem has been photoionization model derived in [18] for nitrogen-oxygen considered in a number of works [2]–[8]. In most of these mixtures has been used modified by the additional account works, some assumptions are made about initial concentrations of absorption of ionizing photons by H O and CO molecules of radicals, because available information about the efficiency (corresponding absorption coefficients are taken from [19] and of generation of active particles by streamers is rather poor. [20]). The attachment coefficient in flue gas differs from that In previous works [9]–[11], calculations of radical production in air, but the role of attachment is not essential for short by streamers in flue gases have been done with the use of streamers considered here (see [12]). The difference between one-dimensional (1-D) streamer models. In the frame of 1- parameters of streamers in air and flue gases is caused mainly D models, the results depend on the choice of the value of by the shortening of the path length of ionizing photons in flue streamer radius which is an input parameter of the model. A gases in comparison with air due to the strong absorption by more rigorous approach is based on two-dimensional (2-D) H O and CO molecules. streamer modeling. In the present paper, the results of 2-D In Fig. 1, the distributions of the electric field and the simulation of positive streamer propagation in flue gas are electron number density along the streamer axis are given. The -values (numbers of particles produced per 100 compared for streamers propagating in flue gas and in air. eV of input electrical energy) for the production of primary The value of the electric field in the streamer head in chemically active components are obtained. flue gas is greater than in air. Correspondingly, the electron number density in the streamer channel in flue gas is also greater than in air (correlation of and obtained by 2- Manuscript received February 24, 1997; revised September 19, 1997. This D simulation agrees with the results of analytical streamer work was supported by INTAS under Contract 94-4207 and by NWO (Dutch theory; see [12]). Organization for Scientific Research). The dependencies of the velocity of streamer propagation The authors are with the Institute for High Temperatures, Russian Academy of Sciences, Moscow 127412, Russia. and current on streamer length in flue gas and in air Publisher Item Identifier S 0093-3813(98)00695-X. are shown in Fig. 2. The values of and in flue gas are 0093–3813/98$10.00 1998 IEEE 42 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 1, FEBRUARY 1998 (a) (a) (b) Fig. 1. (a) Electric field and (b) electron number density distributions along (b) the streamer axis in air (lines 1) and flue gas (lines 2), for sphere radius sph aHXPcm, gap length d aIcm, applied voltage aIRkV. Fig. 2. (a) Dependencies of streamer velocity and (b) current on its length in air (lines 1, 3) and flue gas (lines 2, 4), for sph aHXHS cm, d aHXS cm, aTkV (lines 1, 2) and for sph aHXPcm, d aIcm, aIR slightly greater than in air (for the same external conditions). kV (lines 3, 4). Streamer radius in flue gas is about 10–20% less than in air. The results of simulation show that, as for streamers in air [12], the values of and are almost independent on the effective width of the radial distribution of electrons [which external conditions of the discharge (the applied voltage, the can be estimated as is smaller than that of sphere radius, and the gap length). Streamer radius, velocity, nitrogen atoms. This fact is caused by stronger dependence and current increase with the applied voltage. of ionization coefficient on in comparison with the Concentrations have been calculated of chemically active dissociation coefficient (the electric field in the streamer head components generated in the flue gas mixture: electronically is maximal at the axis and decreases in radial direction). excited nitrogen molecules N (producing radicals in collisions The results of calculation show that, as in air [21], the with O CO and H O molecules), nitrogen atoms N, oxygen dependence of the -values on external conditions in flue gas atoms in ground state O P , and in excited state O D is weak. As an example, in Fig. 4, the -values for nitrogen CO and OH radicals. In Fig. 3, axial values of concentrations atoms production are given corresponding to various parame- and linear number densities (concentrations ters of the gap and applied voltages. They slowly decrease with integrated in radial direction) of electrons and nitrogen atoms growth of streamer length. Calculated -values for production in streamer channel are presented for two time moments of of chemically active components are given in Table I. Also, streamer propagation in flue gas. It is seen that atoms are estimations of the -values are presented in the table obtained generated mainly in streamer head, their concentration in with the use of analytical streamer theory (see below). the channel does not change during streamer propagation.