In a pair of linked articles (called Papers I and II, respectively), we apply the concept of Lagrangian Coherent Structures (LCSs) borrowed from the study of dynamical systems to magnetic field configurations in order to separate regions where field lines have a different kind of behaviour. In the present article, Paper I, after recalling the definition and the properties of the LCSs, we show how this conceptual framework can be applied to the study of particle transport in a magnetized plasma. Furthermore, we introduce a simplified model that allows us to consider explicitly the case where the magnetic configuration evolves in time on time scales comparable to the particle transit time through the configuration. In contrast with previous works on this topic, this analysis requires that a system that is aperiodic in time be investigated. Published by AIP Publishing.

In a pair of linked articles (called Article I and II respectively) we apply the concept of Lagrangian Coherent Structures borrowed from the study of Dynamical Systems to magnetic field configurations in order to separate regions where field lines have different kind of behavior. In the present article, article II, by means of a numerical procedure we investigate the Lagrangian Coherent Structures in the case of a two-dimensional magnetic configuration with two island chains that are generated by magnetic reconnection and evolve nonlinearly in time. The comparison with previous results, obtained by assuming a fixed magnetic field configuration, allows us to explore the dependence of transport barriers on the particle velocity.

transport in a time dependent magnetic configuration D. Grasso1, G. Di Giannatale2, M.V. Falessi3, F. Pegoraro4,T.J. Schep5 1 ISC CNR and Politecnico di Torino, Dip. Energia C.so Duca degli Abruzzi 24, Torino. Italy 2 IGI CNR, Corso Stati Uniti 4, Padova, Italy 3 ENEA, C. R. Frascati, Via E. Fermi 45, Frascati, Italy 4 Dip. Fisica E. Fermi, Pisa University, largo Pontecorvo 3, Pisa, Italy 5 Dep. Applied Physics, Eindhoven Univ. of Technology, 5600MB Eindhoven, The Netherlands

The concept of Lagrangian Coherent Structures (LCS) is here applied to investigate in a new perspective the particles transport in chaotic magnetic configurations. A paradigmatic and simplified magnetic configuration is revisited before extending the analysis to a more realistic one. Preliminary results concerning the typical dynamics of the Reversed Field Pinch (RFP) device are shown.

The properties of two-dimensional turbulence in a circular domain are investigated within the framework of the punctuated point-vortex model. Vortex dynamics is governed by Hamiltonian equations, and it is interrupted by instantaneous events resulting in vortex merging. Motion of about 100 point vortices is simulated using an accurate, symplectic integration method. Ensembles of like-sign vortices relax to a quasi-lattice state. Vortices with zero total vorticity tend to be randomized. Their motion still does not become fully chaotic. We observe emergence of long lived large dipoles (co-propagating pairs of vortices with opposite signs), which affect the evolution of the whole vortex ensemble. The presence of such dipoles accelerate the vortex decay rate. The decay exponent has been estimated as ξ ≃ 1.7, which is much larger than ξ ≃ 0.7, reported in previous studies of decaying turbulence. Since dipole dynamics depends on specific properties of the point vortex system, our findings suggest that a universal decay exponent in such systems does not exist.

A dynamical system framework is used to describe transport processes in plasmas embedded in a magnetic field. For periodic systems with one degree of freedom, the Poincaré map provides a splitting of the phase space into regions where particles have different kinds of motion: periodic, quasi-periodic or chaotic. The boundaries of these regions are transport barriers, i.e. a trajectory cannot cross such boundaries throughout the evolution of the system. Lagrangian coherent structures generalize this method to systems with the most general time dependence, splitting the phase space into regions with different qualitative behaviours. This leads to the definition of finite-time transport barriers, i.e. trajectories cannot cross the barrier for a finite amount of time. This methodology can be used to identify fast recirculating regions in the dynamical system and to characterize the transport between them.

The transitional phase from local to global chaos in the magnetic field of a reconnecting current layer is investigated. Regions where the magnetic field is stochastic exist next to regions where the field is more regular. In regions between stochastic layers and between a stochastic layer and an island structure, the field of the finite time Lyapunov exponent (FTLE) shows a structure with ridges. These ridges, which are special gradient lines that are transverse to the direction of minimum curvature of this field, are approximate Lagrangian coherent structures (LCS) that act as barriers for the transport of field lines.

The transport properties of magnetized plasma configurations are studied that arise from a one-dimensional, current layer that is unstable to reconnecting modes. These magnetic configurations are partially stochastic. It is shown that ridges in the Finite Time Lyapunov Exponents (FTLE) distribution are aligned with the invariant manifolds related to the lines of uniform hyperbolicity. It is shown that these ridges form approximate Lagrangian Coherent Structures (LCS) and act as barriers to the transport of magnetic field lines.

A numerical contour dynamics code has been employed to calculate the stable and unstable manifolds related to two interacting magnetic island chains. The magnetic configuration is generated by a nonlinear reconnection process described in D. Borgogno et al. (cid:3) Phys. Plasmas. 12 , 032309 (cid:1) 2005 (cid:2)(cid:4) . The appearance of the first homoclinic and heteroclinic intersections of the dominant manifolds are shown and one of the associated uniformly hyperbolic orbits is given. The stickiness of the field lines around the island and the eventual development of global stochasticity are discussed. The basic geometry of the magnetic configuration is periodic so that the structure of the manifolds may be compared with the one obtained with Poincaré plots. © 2008 American Institute of Physics . (cid:3) DOI: 10.1063/1.2999539 (cid:4)

The drift-Alfvén equations for a quasineutral plasma that is permeated by a strong background magnetic field describe the plasma dynamics in the transverse plane. Three scalar fields—which are linear combinations of parallel current, fluid vorticity, and density—are incompressibly advected by three streaming potentials. These three “generalized vorticities” are point-wise conserved. Their dynamics is studied numerically using the method of Contour Dynamics, which for this purpose is generalized from one to three different types of vorticity. In the regime where the electron inertial skin depth de is smaller than the ion sound gyroradius, the interaction dynamics of patches of generalized vorticity shows a preference for the formation of structures with length scale O(de). In this regime quasiperiodic splitting-merging cycles of single vorticity patches occur. These vortex interactions are considered to be the paradigm for the interactions of separated vortical structures in a strongly magnetized turbulent plasma.

The effect of Raman instabilities on the production of fast electrons in laser-plasma interaction has been investigated for laser intensities well above the electron trapping threshold. The results of one-dimensional particle-in-cell Simulations show that in this regime the presence of Raman backscattering (RBS) hampers fast-electron production, and that its suppression increases the yield of high-energy electrons (> 15 MeV). Such suppression has been realized either through deletion of all backscattered radiation from the simulations or through direct stimulation of Raman forward scattering (RFS). An increased high-energy electron yield has been observed for both methods. In addition, the influence of various laser and plasma parameters on the production of highly energetic electrons has been investigated. Specifically, the effects of plasma density ramps, skews in the temporal envelopes of the laser pulses, and laser frequency chirp (both pulse-length preserving and bandwidth preserving) have been examined. For each parameter, its influence on the yield of hi h-energy electrons can be explained from the way it affects the balance between RBS and RFS excitation in laser-plasma interaction.

This Letter deals with scaling laws that describe transport and correlation effects in anisotropic media. The method of multi-scale continuum percolation is used. Multi-scale continuum percolation in 2D random flows is based upon a description in terms of a hierarchy of spatial scales;. In that theory the correlation function of the velocity scales as lambda(-alpha). On the other hand. fractal theory leads to the scaling with time, lambda alpha t(H), where H is the Hurst factor. A closer examination of fractal and percolation concepts allows us to obtain not only the value of the exponents but also the relationship between them. It is shown that super-diffusive, 1/2 < H < 1, and sub-diffusive behavior, 0 < H < 1/2, in an anisotropic medium can be described by a single scaling law obtained from percolation theory. The model of double diffusion (H = 1/4) and the one of Dreizin-Dykhne (H = 3/4) are treated as examples. The connection between these scaling laws and the order of the fractional time derivative in the transport equation is pointed out. (C) 2003 Elsevier B.V. All rights reserved.

Plasmas and magnetic fields are inseparably related in numerous physical circumstances. This is not only the case in natural occurring plasmas like the solar corona and the earth magnetic tail, but also in laboratory plasmas like tokamaks and stellarators.

The effect of Raman backscattering (RBS) on high-energy electron generation in laser-plasma interaction has been investigated for laser intensities well above the wave breaking and electron trapping threshold. One-dimensional particle-in-cell simulations show that suppression of RBS increases the high-energy electron yield in this regime. RBS-induced heating causes heavy beam loading and damping of the laser wake. Its suppression leads to higher wake amplitudes and higher particle energies. RBS suppression through direct stimulation of Raman forward scatter is demonstrated. The implications for high-energy electron production through laser-plasma interaction are discussed.

The problem of the generation of mean magnetic fields by small-scale turbulence within the framework of electron magnetohydrodynamics (EMHD) is considered. Two EMHD models are investigated, a two and one-half dimensional (212D) model in which the magnetic field has all three spatial components but, due to a strong external field, depends only on two coordinates, and a fully three-dimensional (3D) model with an imposed stationary and homogeneous magnetic field. It is shown that in the case of 212D turbulence two possible mechanisms are responsible for the generation of mean magnetic fields. The first one is similar to the α-effect in the MHD dynamo problem and is due to a nonzero helicity of the turbulence. The second one is related to the anisotropy of the turbulence, which can give rise to negative dissipation (resistivity, viscosity) of the mean field. The influence of electron inertia on the above effects is analyzed. Inertia results in a qualitative modification of the helicity effects and may lead to a change in sign of the turbulent viscosity. The criteria for the generation of mean magnetic fields are obtained. In the case of the 3D model, the generation of large-scale helicons by the small-scale helicon turbulence is studied within the framework of the adiabatic approximation. A closed set of equations for the evolution of both the magnetic field of the large-scale helicon and of the generalized action of the small-scale turbulence is obtained. The criterion for the resonant instability of a large-scale helicon due to its interaction with small-scale helicon turbulence is obtained.

The spectral properties of decaying turbulence in 212-dimensional electron magnetohydrodynamics are studied numerically. In the range kde<1 the energy exhibits a direct cascade while mean square momentum exhibits an inverse cascade. Their spectra are characterized by k−7/3 and k−13/3, respectively. The self-similar decay state of the turbulence is reached after an initial phase of fast exchange between the axial and poloidal magnetic energies. The time behavior t−2/3 of the total energy is found to be consistent with that obtained from selective decay. The maximum of the energy spectrum shifts towards low mode numbers and decays in time as t−1, in agreement with the infrared scaling of the turbulence. In the large de limit, both energy and mean square generalized momentum exhibit direct cascades. No stationary turbulent state could be found as long as the axial kinetic energy is large as compared to the poloidal kinetic energy initially. The global physical quantities decay well before turbulent macroscopic quantities have established similar space–time behavior, and the turbulence is infected by the lack of stationarity. The system decouples into a Navier–Stokes equation and a passive scalar equation only if the poloidal kinetic energy is larger than or equal to the axial kinetic energy. In this limit the k−5/3 and k−3 spectra of the poloidal kinetic energy are recovered.

Electron bunches produced in self-modulated laser wakefield experiments usually have a broad energy spectrum, with most electrons at low energy (1-3 MeV) and only a small fraction at high energy. We propose and investigate further acceleration of such bunches in a channel-guided resonant laser wakefield accelerator. Two-dimensional simulations with and without the effects of self-consistent beam loading are performed and compared. These results indicate that it is possible to trap about 40 % of the injected bunch charge and accelerate this fraction to an average energy of about 50 MeV in a plasma channel of a few mm. In recent years, experiments on the self-modulated laser wakefield (SMLW) accelerator [1] have been conducted at various laboratories [2–8]. In these experiments, usually a highpower laser pulse is focused on a gas jet. The laser pulse is sufficiently strong to create a plasma inside the gas jet and to excite a high-amplitude plasma wakefield as a result of selfmodulation. Subsequently, electrons from the plasma are trapped and accelerated in this wakefield. The electron bunches produced in these experiments have been characterized by measurements of the energy spectrum, the amount of charge, and the angular distribution. Typically, the energy spectrum of SMLW-bunches is exponential, ∝ exp(−γmc/Te), with an effective electron temperature Te in the MeV-range, where γ is the relativistic factor of the electrons [2–8]. The bunches contain high charge: up to 8 nC has been reported 1 [6,7]. The opening angles are usually small, and decrease with increasing energy: in several experiments, a tightly focused “hot core” of high energy electrons has been found [3,8]. Numerical simulations of the acceleration process have been performed with particle-incell (PIC) codes [9,11]. These simulation results show that, depending on the laser and plasma parameters involved, there are different mechanisms that lead to electron trapping and acceleration. These include wave breaking of the Raman forward scattered plasma wave [9], coupling to Raman backward scattering [10], and direct laser acceleration [11]. In this paper, we propose and investigate a novel two-stage laser wakefield accelerator (LWFA). The first stage is a SMLW accelerator, in which electrons are self-trapped and accelerated from the gas jet plasma. The SMLW electron bunch, with the bulk of the electrons at low energy, is then injected into the second stage, which is a channel-guided resonant LWFA. This second acceleration stage, which we call the post-acceleration, is studied in detail with fluid and particle simulations, with and without the effects of self-consistent beam loading. The SMLW and resonant LWFA regimes are characterized primarily by the plasma density. For the SMLW, the plasma density is relatively high (> 10 cm−3, but still below the critical density), such that the laser pulse extends over several plasma waves λp and the laser power P exceeds the critical power for relativistic self-focusing Pc [12]. Here λp = 2π/ωp, ωp = (4πnee /m) is the plasma frequency, ne is the plasma density, Pc[GW] 17(λp/λ), P [GW] 21.5(a0r0/λ), r0 is the spot size of the radial laser profile (assumed Gaussian), a0 8.5 × 10−9λ[μm](I [W/cm])1/2 is the laser strength parameter, and λ is the laser wavelength. In this regime, the laser pulse is highly unstable to self-modulation [13–15], which drives a plasma wave to sufficiently high amplitudes such that it traps and accelerates electrons from the background plasma. For the resonant LWFA a lower density (∼ 10 cm−3) is used such that the laser pulse is approximately equal to the plasma wavelength and the plasma wave is driven in a controlled manner, which allows for a controlled acceleration process. However, for the resonant LWFA it is usually not possible to trap background electrons from the plasma and instead the 2 accelerated electron bunch must be externally injected. In this paper we propose to inject the SMLW bunches into a resonant LWFA to exploit the advantages of both schemes. Because the resonant LWFA involves a much larger dephasing distance, and consequently much higher final electron energies, some form of laser pulse guiding [12] is needed. In this report a parabolic plasma channel is assumed to provide this guiding. Such channels have been produced in experiments [16–18]. In the conventional view of resonant LWFA [1], it is assumed that high energy, high quality bunches are produced by first injecting (from an external source) low energy, high quality electron bunches. To maintain low energy spread, the injected bunch must be short compared to the plasma wavelength and it must be injected at the proper wakefield phase [19,20]. In the two-stage LWFA considered here, the injected bunch (produced in the first stage by the SMLW) is non-ideal in the sense that the initial energy spread of the bunch is large and the bunch is not short compared to the plasma wave of the second stage. Nevertheless, during the post-acceleration process, a large fraction (40 %) of the injected bunch is trapped and accelerated in the wakefield, resulting in a high energy (50 MeV) bunch with somewhat reduced energy spread (60 %). Such a high energy bunch is useful for various applications in which a small energy spread is not essential, such as in nuclear activation for novel isotope production. For the post-acceleration simulations, we use a model distribution for the injected SMLWbunch. This model distribution does not directly use data from SMLW experiments or simulations, but it is constructed in such a way that it contains a number of features found in most experiments and simulations. Using this model for the injected bunch, the postacceleration process is studied in detail via simulations based on a 2D code that combines a particle description for the electron bunch with a fluid model for the wakefields, including all beam loading effects. In the Appendix, a brief description of this code is given. This paper is organized as follows: In Sec. I, a description of the injected bunch model is given. In Sec. II, beam loading effects are discussed in detail. Sec. III contains the simulation results of the post-acceleration process. In Sec. IV we mention nuclear activa3 tion experiments as a possible application for accelerated SMLW-bunches. Conclusions are offered in Sec. V. I. DESCRIPTION OF INJECTED BUNCH For the effective electron temperature Te of the energy distribution a value of 3.4 MeV has been chosen. The longitudinal bunch distribution is shown in Fig. 1, which is a plot of energy versus t−z/c. A number of features found in PIC-simulations [9] have been included in this distribution. The simulations typically show a microbunching with a period slightly larger than the plasma wavelength. This is due to both the increase of wavelength in the nonlinear regime and the effect of beam loading (electrons trapped in wakefield buckets modify the wakefield). Also visible in Fig. 1 is a correlation between energy and t−z/c, i.e., electrons from the first few wakefield buckets have gained more energy than those trapped in other buckets. This is another beam loading effect also found in PIC simulations. In Fig. 2, a color contour plot of the bunch density as a function of x, t−z/c is given. The bunch width w is varied in our simulations. The charge of the bunch is varied proportional to w (this will be explained in sec. III) such that the peak density is constant (at about 25 % of the plasma density ne0 on the axis of the plasma channel). In this plot, the microbunching is visible again. A plot of the assumed opening angle versus energy is given in Fig. 3. This distribution reflects the experimentally observed correlation between opening angle and energy (smaller opening angle for high-energy electrons). II. BEAM LOADING EFFECTS To illustrate the importance of beam loading effects, two runs with the same initial electron distribution are compared, one with beam loading for which all bunch wakefields are included and a test particle run without beam loading for which no bunch wakefields are calculated. The width of the electron bunch is w = 28μm, the charge is 2 nC for 4 the case with beam loading (test particle run is effectively the zero bunch charge limit). Other parameters are: plasma density on axis ne0 = 3.4 × 10 cm−3, laser power P = 40 TW, laser spot size r0 = 38 μm, pulse duration 30 fs, which imply a peak laser strength of a0 0.9. The pulse profiles in the axial and transverse directions are assumed to be Gaussian. Furthermore, the laser pulse propagates in a parabolic plasma channel with the channel depth given by the matched beam condition ∆n = 1/πrer 2 0 7.8×10 cm−3, where re is the classical electron radius. Color contours of Fx, Fz after an acceleration distance of 2 mm are shown in Figs. 4 and 5. Here, Fx = −e(Ex − cBy)/mωpc, Fz = −eEz/mωpc denote the (normalized) transverse and longitudinal components of the Lorentz force on a relativistic bunch electron. The solid black lines in these figures indicate the electron bunch density contours. Comparison of the color contours shows the influence of the bunch on the fields. The amplitude of the longitudinal force behind the bunch is considerably lower for the 2 nC bunch (beam loading case, Fig. 4), indicating that the bunch is taking energy from the plasma wave [23,24]. Behind the bunch (t− z/c > 350 fs) the on-axis accelerating field has been completely canceled by the bunch wakefield. Depending on the position of the electrons in the wakefield, the bunch could also increase the wake amplitude by giving energy to the plasma wave, but inside the bunch, wakefields from the bunch are always decelerating, i.e. the bunch always lowers the value of Fz. The amplitude of the transverse force behind the bunch is considerably larger for the 2 nC bunch, indicating that the bunch induces strong focusing forces. It is important to note that these forces are always focusing insid

A new regime of laser wakefield acceleration of an injected electron bunch is described. In this regime, the bunch charge is so high that the bunch wakefields play an important role in the bunch dynamics. In particular, the transverse bunch wakefield induces a strong self-focusing that suppresses the transverse emittance growth arising from misalignment errors. The decelerating longitudinal bunch wakefield, however, is not so strong that it completely cancels the accelerating laser wakefield. In fact, the induced energy spread can be compensated by exploiting phase slippage effects. These features make the new regime interesting for high beam quality laser wakefield acceleration.

Electron bunches produced in self-modulated laser wakefield experiments usually have a broad energy distribution, with most electrons at low energy (1--3 MeV) and only a small fraction at high energy. We propose and investigate further acceleration of such bunches in a channel-guided resonant laser wakefield accelerator. Two-dimensional simulations with and without the effects of self-consistent beam loading are performed and compared. These results indicate that it is possible to trap about $40%$ of the injected bunch charge and accelerate this fraction to an average energy of about 50 MeV in a plasma channel of a few mm.