
Ensemble Modeling of Radiation Belt Electron Flux Decay Following a Geomagnetic Storm: Dependence on Key Input Parameters
M. Hua, J. Bortnik, A. Kellerman, E. Camporeale, Q. Ma (2022)
Space Weather, in press
We perform an ensemble of quasilinear diffusion simulations of the radiation belt electron flux decay for ~6 days at L = 3.5 during the recovery phase of the storm on 7 November 2015, where plasmaspheric hiss dominantly drives the electron flux decay process. Based on Van Allen Probes measurements, we use percentiles to sample the distributions of the four key input parameters, which are the hiss wave amplitude Bw, hiss wave peak frequency fm, background magnetic field B0 and electron density Ne, with 11 points representing that range of each input, leading to 11^4 (~14,600) ensemble members.
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By developing a Lookup Table method to rapidly calculate the timedependent diffusion coefficients, the changing wave environment at every time step is incorporated in our ensemble simulations. The comparison between the ensemble simulations and observation reveals the influence of uncertainties in the input parameters on the simulated electron fluxes. Our results demonstrate that the perturbations in Bw are the primary contributor for discrepancies between modeled and observed electron fluxes, while the simulation errors caused by variations in fm and Ne are strongly energydependent. The simulated electron flux using the wave parameters observed at the 50th percentile agrees with observations, and most of the simulation errors increase with decreasing observational probability density of the parameters, with the largest log accuracy ratio of ~14. Our physicsbased ensemble modeling provides the essential information about the robustness of radiation belt simulation and forecast considering the uncertainties in the plasma wave measurement or parameterization.

Probabilistic prediction of Dst storms onedayahead using FullDisk SoHO Images
A. Hu, C. Shneider, A. Tiwari, E. Camporeale (2022)
Space Weather, in press
We present a new model for the probability that the Disturbance storm time (Dst) index exceeds 100 nT, with a lead time between 1 and 3 days. Dst provides essential information about the strength of the ring current around the Earth caused by the protons and electrons from the solar wind, and it is routinely used as a proxy for geomagnetic storms. The model is developed using an ensemble of Convolutional Neural Networks (CNNs) that are trained using SoHO images (MDI, EIT and LASCO). The relationship between the SoHO images and the solar wind has been investigated by many researchers, but these studies have not explicitly considered using SoHO images to predict the Dst index.
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This work presents a novel methodology to train the individual models and to learn the optimal ensemble weights iteratively, by using a customized classbalanced mean square error (CBMSE) loss function tied to a leastsquares (LS) based ensemble.
The proposed model can predict the probability that Dst < − 100nT 24 hours ahead with a True Skill Statistic (TSS) of 0.62 and Matthews Correlation Coefficient (MCC) of 0.37. The weighted TSS and MCC from Guastavino et al. (2021) is 0.68 and 0.47, respectively. An additional validation during nonEarthdirected CME periods is also conducted which yields a good TSS and MCC score.

New Findings from Explainable SYMH Forecasting using Gradient Boosting Machines
D. Iong, Y. Chen, G. Toth, S. Zou, T. Pulkkinen, J. Ren, E. Camporeale, T. Gombosi (2022)
Space Weather, in press
In this work, we develop gradient boosting machines (GBMs) for forecasting the SYMH index multiple hours ahead using different combinations of solar wind and interplanetary magnetic field (IMF) parameters, derived parameters, and past SYMH values. Using Shapley Additive Explanation (SHAP) values to quantify the contributions from each input to predictions of the SYMH index from GBMs, we show that our predictions are consistent with physical understanding while also providing insight into the complex relationship between the solar wind and Earth's ring current.
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We also perform a direct comparison between GBMs and neural networks presented in prior publications for forecasting the SYMH index by training, validating, and testing them on the same data. We find that the GBMs have a comparable root mean squared error as the best published blackbox neural network schemes and GBMs have better Heidke Skill Scores at predicting strong storms.

Machine Learning Methods Applied to the Global Modeling of EventDriven Pitch Angle Diffusion Coefficients During High Speed Streams
G. Kluth, J.F. Ripoll, S. Has, A. Fischer, M. Mougeot and E. Camporeale (2022) Frontiers in Physics
Whistlermode waves in the inner magnetosphere cause electron precipitation in the atmosphere through the physical process of pitchangle diffusion. The computation of pitchangle diffusion relies on quasilinear theory and becomes timeconsuming as soon as it is performed at high temporal resolution from satellite measurements of ambient wave and plasma properties. Such an effort is nevertheless required to capture accurately the variability and complexity of atmospheric electron precipitation, which are involved in various Earth’s ionospheremagnetosphere coupled problems.
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In this work, we build a global machinelearning model of eventdriven pitchangle diffusion coefficients for storm conditions based on the data of a variety of storms observed by the NASA Van Allen Probes. We first proceed stepbystep by testing 8 nonparametric machine learning methods. With them, we derive machine learning based models of eventdriven diffusion coefficients for the storm of March 2013 associated with highspeed streams. We define 3 diagnostics that allow highlighting of the properties of the selected model and selection of the best method. Three methods are retained for their accuracy/efficiency: spline interpolation, the radial basis method, and neural networks (DNN), the latter being selected for the second step of the study. We then use eventdriven diffusion coefficients computed from 32 highspeed stream storms in order to build for the first time a statistical eventdriven diffusion coefficient that is embedded within the retained DNN model. We achieve a global mean eventdriven model in which we introduce a twoparameter dependence, with both the Kpindex and time kept as in an epoch analysis following the storm evolution. The DNN model does not entail any issue to reproduce quite perfectly its target, i.e., averaged diffusion coefficients, with rare exception in the Landau resonance region. The DNN mean model is then used to analyze how mean diffusion coefficients behave compared with individual ones. We find a poor performance of any mean models compared with individual events, with mean diffusion coefficients computing the general trend at best, due to their large variability. The DNNbased model allows simple and fast data exploration of pitchangle diffusion among its multiple variables. We finally discuss how to conduct uncertainty quantification of FokkerPlanck simulations of storm conditions for space weather nowcasting and forecasting.

Parameter Distributions for the DragBased Modeling of CME Propagation
G. Napoletano, R. Foldes, F. Berrilli, E. Camporeale, at al., (2022) Space Weather, in press
In recent years, ensemble modelling has been widely employed in space weather to estimate uncertainties in forecasts. We here focus on the ensemble modelling of CME arrival times and arrival velocities using a dragbased model, which is wellsuited for this purpose due to its simplicity and low computational cost. Although ensemble techniques have previously been applied to the dragbased model, it is still not clear how to best determine distributions for its input parameters, namely the drag parameter and the solar wind speed.
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The aim of this work is to evaluate statistical distributions for these model parameters starting from a list of past CMEICME events. We employ LASCO coronagraph observations to measure initial CME position and speed, and in situ data to associate them to arrival date and impact speed. For each event we ran a statistical procedure to invert the model equations producing the parameters distributions as the output.
Our results indicate that the distributions employed in previous works were appropriately selected, even though based on restricted samples and heuristic considerations.

On the propagation of uncertainties in radiation belt simulations
Camporeale, E., Y. Shprits, M. Chandorkar, A. Drozdov, S. Wing (2016), Space Weather
We present the first study of the uncertainties associated with radiation belt
simulations, performed in the standard quasilinear diffusion framework. In particular,
we estimate how uncertainties of some input parameters propagate through the nonlinear simulation, producing a distribution of outputs that can be quite broad.
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Here, we restrict our focus on twodimensional simulations (in energy and pitch angle space) of
parallelpropagating chorus waves only, and we study as stochastic input parameters the
geomagnetic index K p (that characterizes the time dependency of an idealized storm),
the latitudinal extent of waves, and the average electron density. We employ a colloca
tion method, thus performing an ensemble of simulations. The results of this work point
to the necessity of shifting to a probabilistic interpretation of radiation belt simulation
results, and suggest that an accurate specification of a timedependent density model
is crucial for modeling the radiation environment.

Turbulence Heating ObserveR – satellite mission proposal
A. Vaivads et al. (2016), J. Plasma Phys., 82
The Universe is permeated by hot, turbulent, magnetized plasmas. Turbulent plasma is
a major constituent of active galactic nuclei, supernova remnants, the intergalactic and
interstellar medium, the solar corona, the solar wind and the Earth’s magnetosphere,
just to mention a few examples. Energy dissipation of turbulent fluctuations plays
a key role in plasma heating and energization, yet we still do not understand the
underlying physical mechanisms involved.
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THOR is a mission designed to answer
the questions of how turbulent plasma is heated and particles accelerated, how the
dissipated energy is partitioned and how dissipation operates in different regimes of
turbulence. THOR is a singlespacecraft mission with an orbit tuned to maximize
data return from regions in nearEarth space – magnetosheath, shock, foreshock and
pristine solar wind – featuring different kinds of turbulence. Here we summarize the
THOR proposal submitted on 15 January 2015 to the ‘Call for a Mediumsize mission
opportunity in ESAs Science Programme for a launch in 2025 (M4)’. THOR has been
selected by European Space Agency (ESA) for the study phase.

Information theoretical approach to discovering solar wind drivers of the outer radiation belt
Wing, S., Johnson, J., Camporeale, E., Reeves, G. (2016), J. Geophys. Res., 121
The solar windmagnetosphere system is nonlinear. The solar wind drivers of geosynchronous electrons with energy range of 1.8–3.5 MeV are investigated using mutual information, conditional mutual information (CMI), and transfer entropy (TE). These information theoretical tools can establish linear and nonlinear relationships as well as information transfer.
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The information transfer from solar wind velocity (Vsw) to geosynchronous MeV electron flux (Je) peaks with a lag time of 2 days. As previously reported, Je is anticorrelated with solar wind density (nsw) with a lag of 1 day. However, this lag time and anticorrelation can be attributed at least partly to the Je(t + 2 days) correlation with Vsw(t) and nsw(t + 1 day) anticorrelation with Vsw(t). Analyses of solar wind driving of the magnetosphere need to consider the large lag times, up to 3 days, in the (Vsw, nsw) anticorrelation. Using CMI to remove the effects of Vsw, the response of Je to nsw is 30% smaller and has a lag time < 24 h, suggesting that the MeV electron loss mechanism due to nsw or solar wind dynamic pressure has to start operating in < 24 h. nsw transfers about 36% as much information as Vsw (the primary driver) to Je. Nonstationarity in the system dynamics is investigated using windowed TE. When the data are ordered according to transfer entropy value, it is possible to understand details of the triangle distribution that has been identified between Je(t + 2 days) versus Vsw(t).

Collisional effects on the numerical recurrence in VlasovPoisson simulations
Pezzi, O., E. Camporeale, F. Valentini (2016), Phys. Plasmas, 23, 022103
The initial state recurrence in numerical simulations of the VlasovPoisson system is a wellknown phenomenon. Here we study the effect on recurrence of artificial collisions modeled through the LenardBernstein operator [A. Lenard and I. B. Bernstein, Phys. Rev. {\bf 112}, 14561459 (1958)]. By decomposing the linear
VlasovPoisson system in the FourierHermite space, the recurrence problem is investigated in the linear regime of the damping of a Langmuir wave and of the onset of the bumpontail instability.
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The analysis is then confirmed and extended to the nonlinear regime through a Eulerian collisional VlasovPoisson code. It is found that, despite being routinely used, an artificial collisionality is not a viable way of preventing recurrence in numerical simulations without compromising the kinetic nature of the solution. Moreover, it is shown how numerical effects associated to the generation of fine velocity scales, can modify the physical features of the system evolution even in nonlinear regime. This means that filamentationlike phenomena, usually associated with low amplitude fluctuations contexts, can play a role even in nonlinear regime.

On the velocity space discretization for the
VlasovPoisson system: comparison between implicit
Hermite spectral and ParticleinCell methods
Camporeale, E., G.L. Delzanno, B.K Bergen, J.D. Moulton (2016), Comp. Phys. Comm., 198, 4758
We describe a spectral method for the numerical solution of the Vlasov
Poisson system where the velocity space is decomposed by means of an Hermite
basis, and the configuration space is discretized via a Fourier decomposition.
The novelty of our approach is an implicit time discretization that
allows exact conservation of charge, momentum and energy.
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The computational
efficiency and the costeffectiveness of this method are compared to the
fullyimplicit PIC method recently introduced by Markidis and Lapenta [52]
and Chen et al. [15]. The following examples are discussed: Langmuir wave,
Landau damping, ionacoustic wave, twostream instability. The Fourier
Hermite spectral method can achieve solutions that are several orders of
magnitude more accurate at a fraction of the cost with respect to PIC.

Waveparticle interactions with parallel whistler waves: nonlinear and timedependent effects revealed by ParticleinCell simulations
Camporeale, E. & Zimbardo, G. (2015), Phys. Plasmas 22, 092104
We present selfconsistent ParticleinCell simulations of the resonant interactions between anisotropic energetic electrons and a population of whistler waves, with parameters relevant to the Earth's radiation belt.
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By tracking PIC particles, and comparing with testparticles simulations we emphasize the importance of including nonlinear effects and time evolution in the modeling of waveparticle interactions, which are excluded in the resonant limit of quasilinear theory routinely used in radiation belt studies. In particular we show that pitch angle diffusion is enhanced during the linear growth phase, and it rapidly saturates. We discuss how the saturation is related to the fact that the domain in which the particles' pitch angle diffuse is bounded, and to the wellknown problem of 90∘ diffusion barrier.

Resonant and nonresonant whistlersparticle interaction in the radiation belts
Camporeale, E. (2015), Geophys. Res. Lett., 42, 31143121,
We study the waveparticle interactions between lower band chorus whistlers and an anisotropic tenuous population of
relativistic electrons.
We present the first direct comparison of firstprinciple ParticleinCell (PIC) simulations with a quasilinear diffusion code.
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In the PIC approach, the waves are selfconsistently generated by the temperature anisotropy instability that quickly
saturates and relaxes the system towards marginal stability.
We show that the quasilinear diffusion and PIC results have significant quantitative mismatch
in regions of energy/pitch angle where the resonance condition is not satisfied.
Moreover, for pitch angles close to the loss cone the diffusion code overestimates the scattering, particularly
at low energies.
This suggests that higher order nonlinear theories should be taken in consideration
in order to capture nonresonant interactions, resonance broadening, and to account for scattering at angles close to 90 degress.

Neutral Vlasov kinetic theory of magnetized plasmas
Tronci, C. & Camporeale, E. (2015), Phys. Plasmas., 22, 020704
The lowfrequency limit of Maxwell equations is considered in the MaxwellVlasov system. This limit produces a neutral Vlasov system that captures essential features of plasma dynamics, while neglecting radiation effects.
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EulerPoincare' reduction theory is used to show that the neutral Vlasov kinetic theory possesses a variational formulation in both Lagrangian and Eulerian coordinates. By construction, the model recovers all collisionless neutral models employed in plasma simulations. Then, comparisons between the neutral Vlasov system and hybrid kineticfluid models are presented in the linear regime.

Approximate semianalytical solutions for the steadystate expansion of a contactor plasma
Camporeale, E., E. Hogan, E. MacDonald (2015) Plasma Sources Sci. Technol., 24 , 025014
We study the steadystate expansion of a collisionless, electrostatic, quasi
neutral plasma plume into vacuum, with a fluid model. We analyze approximate
semianalytical solutions, that can be used in lieu of much more expensive numerical
solutions.
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In particular, we focus on the earlier studies presented in Parks and Katz
(1979) [1], Korsun and Tverdokhlebova (1997) [2], and Ashkenazy and Fruchtman
(2001) [3]. By calculating the error with respect to the numerical solution, we can
judge the range of validity for each solution. Moreover, we introduce a generalization
of earlier models that has a wider range of applicability, in terms of plasma injection
profiles. We conclude by showing a straightforward way to extend the discussed
solutions to the case of a plasma plume injected with nonnull azimuthal velocity.

Electron vortex magnetic holes: a nonlinear coherent plasma structure
Haynes, C. T., Burgess, D., Camporeale, E., & Sundberg, T. (2015), Phys. Plasmas 22, 012309
We report the properties of a novel type of subproton scale magnetic hole found in two dimensional PIC simulations of decaying turbulence with a guide field. The simulations were performed with a realistic value for ion to electron mass ratio. These structures, electron vortex magnetic holes (EVMHs), have circular crosssection. The magnetic field depression is associated with a diamagnetic azimuthal current provided by a population of trapped electrons in petallike orbits.
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The trapped electron population provides a mean azimuthal velocity and since trapping preferentially selects high pitch angles, a perpendicular temperature anisotropy. The structures arise out of initial perturbations in the course of the turbulent evolution of the plasma, and are stable over at least 100 electron gyroperiods. We have verified the model for the EVMH by carrying out test particle and PIC simulations of isolated structures in a uniform plasma. It is found that (quasi)stable structures can be formed provided that there is some initial perpendicular temperature anisotropy at the structure location. The properties of these structures (scale size, trapped population etc.) are able to explain the observed properties of magnetic holes in the terrestrial plasma sheet. EVMHs may also contribute to turbulence properties, such as intermittency, at short scale lengths in other astrophysical plasmas.
 Delzanno, G. L., Camporeale, E., Moulton, J. D., Borovsky, J. E., MacDonald, E. A., & Thomsen, M. F. (2013)
CPIC: A Curvilinear ParticleinCell Code for PlasmaMaterial Interaction Studies.
IEEE Transactions on Plasma Science, 41, 12, 3577
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We describe a new electrostatic particleincell (PIC) code in curvilinear geometry called curvilinear PIC (CPIC). The code models the microscopic (kinetic) evolution of a plasma with the PIC method, coupled with an adaptive computational grid that can conform to arbitrarily shaped domains. CPIC is particularly suited for multiscale problems associated with the interaction of complex objects with plasmas. A map is introduced between the physical space and the logical space, where the grid is uniform and Cartesian. In CPIC, most of the operations are performed in logical space. CPIC was designed following criteria of versatility, robustness, and performance. Its main features are the use of structured meshes, a scalable field solver based on the black box multigrid algorithm and a hybrid mover, where particles' position is in logical space while the velocity is in physical space. Test examples involving the interaction of a plasma with material boundaries are presented.
 Tu, W., Cunningham, G. S., Chen, Y., Henderson, M. G., Camporeale, E., & Reeves, G. D. (2013).
Modeling radiation belt electron dynamics during GEM challenge intervals with the DREAM3D diffusion model.
J. Geophys. Res., 118(10), 61976211.
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As a response to the Geospace Environment Modeling (GEM) “Global Radiation Belt Modeling Challenge,” a 3D diffusion model is used to simulate the radiation belt electron dynamics during two intervals of the Combined Release and Radiation Effects Satellite (CRRES) mission, 15 August to 15 October 1990 and 1 February to 31 July 1991. The 3D diffusion model, developed as part of the Dynamic Radiation Environment Assimilation Model (DREAM) project, includes radial, pitch angle, and momentum diffusion and mixed pitch anglemomentum diffusion, which are driven by dynamic wave databases from the statistical CRRES wave data, including plasmaspheric hiss, lowerband, and upperband chorus. By comparing the DREAM3D model outputs to the CRRES electron phase space density (PSD) data, we find that, with a datadriven boundary condition at Lmax = 5.5, the electron enhancements can generally be explained by radial diffusion, though additional local heating from chorus waves is required. Because the PSD reductions are included in the boundary condition at Lmax = 5.5, our model captures the fast electron dropouts over a large L range, producing better model performance compared to previous published results. Plasmaspheric hiss produces electron losses inside the plasmasphere, but the model still sometimes overestimates the PSD there. Test simulations using reduced radial diffusion coefficients or increased pitch angle diffusion coefficients inside the plasmasphere suggest that better wave models and more realistic radial diffusion coefficients, both inside and outside the plasmasphere, are needed to improve the model performance. Statistically, the results show that, with the datadriven outer boundary condition, including radial diffusion and plasmaspheric hiss is sufficient to model the electrons during geomagnetically quiet times, but to best capture the radiation belt variations during active times, pitch angle and momentum diffusion from chorus waves are required.
 Camporeale, E., Delzanno, G. L., Zaharia, S., & Koller, J. (2013).
On the numerical simulation of particle dynamics in the radiation belt: 1. Implicit and semiimplicit schemes.
J. Geophys. Res., 118(6), 34633475.
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The particle dynamics in the Earth's radiation belt is generally modeled by means of a twodimensional diffusion equation for the particle distribution function in energy and pitch angle. The goal of this paper is to survey and compare different numerical schemes for the solution of the diffusion equation, and to outline the optimal strategy from a numerical point of view. We focus on the general (and more computationally challenging) case where the mixed terms in the diffusion tensor are retained. In Part 1, we compare fully implicit and semiimplicit schemes. For the former, we have analyzed a direct solver based on a LU decomposition routine for sparse matrices, and an iterative incomplete LU preconditioned Generalized Minimal REsidual solver. For the semiimplicit scheme, we have studied an alternating direction implicit scheme. We present a convergence study for a realistic case that shows that the time step and grid size are strongly constrained by the desired accuracy of the solution. We show that the fully implicit scheme is to be preferred in most cases as the more computationally efficient.
 Camporeale, E., Delzanno, G. L., Zaharia, S., & Koller, J. (2013).
On the numerical simulation of particle dynamics in the radiation belt: 2. Procedure based on the diagonalization of the diffusion tensor.
J. Geophys. Res., 118(6), 34763484.
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In this paper, we conclude the survey and comparison of different numerical methods used to solve the diffusion equation for particle dynamics in the Earth's radiation belt, initiated in Camporeale et al. (2013). Here we focus on the diagonalization procedure introduced by Albert and Young (2005) that, by performing a change of coordinates, solves the diffusion equation in a space where the mixed diffusion terms are null. We describe the diagonalization procedure and its numerical implementation, which is not as straightforward as the implementation of a traditional solver in a rectangular domain. We compare the computing times with and without the diagonalization procedure, and we conclude that this procedure is generally not advantageous from the point of view of computational efficiency.
 Delzanno, G. L., & Camporeale, E. (2013).
On particle movers in cylindrical geometry for ParticleInCell simulations.
J. Comput. Physics, 253, 259277.
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Three movers for the orbit integration of charged particles in a given electromagnetic field are analyzed and compared in cylindrical geometry. The classic Boris mover, which is of leapfrog type with position and velocities staggered by half time step, is connected to a second order Strang operator splitting integrator. In general the Boris mover is about 20% faster than the Strang splitting mover without sacrificing much in terms of accuracy. Furthermore, the Boris mover is second order accurate only for a very specific choice of the initial half step needed by the algorithm to get started. Unlike the case in Cartesian geometry, where any initial half step which is at least first order accurate does not compromise the second order accuracy of the method, in cylindrical geometry any attempt to use a more accurate initial half step does in fact decrease the accuracy of the scheme to first order. Through the connection with the Strang operator splitting integrator, this counterintuitive behavior is explained by the fact that the error in the half step velocities of the Boris mover is proportional to the time step of the simulation.
For the case of a uniform and static magnetic field, we discuss the leapfrog cyclotronic mover, cylindrical analogue of the cyclotronic mover of Ref. [L. Patacchini and I. Hutchinson, J. Comput. Phys. 228 (7), 2009], where the step involving acceleration due to inertial forces is combined with the acceleration due to the magnetic part of the Lorentz force. The advantage of a cyclotronic mover is that the gyration of a charged particle in a magnetic field is treated analytically and therefore only the dynamics associated with the electric field needs to be resolved.
 Welling, D. T., Koller, J., & Camporeale, E. (2012).
Verification of SpacePy's radial diffusion radiation belt model.
Geoscientific Model Development, 5(2), 277287.
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Model verification, or the process of ensuring that the prescribed equations are properly solved, is a necessary step in code development. Careful, quantitative verification guides users when selecting grid resolution and time step and gives confidence to code developers that existing code is properly instituted. This work introduces the RadBelt radiation belt model, a new, opensource version of the Dynamic Radiation Environment Assimilation Model (DREAM) and uses the Method of Manufactured Solutions (MMS) to quantitatively verify it. Order of convergence is investigated for a plethora of code configurations and source terms. The ability to apply many different diffusion coefficients, including time constant and time varying, is thoroughly investigated. The model passes all of the tests, demonstrating correct implementation of the numerical solver. The importance of DLL and source term dynamics on the selection of time step and grid size is also explored. Finally, an alternative method to apply the source term is examined to illustrate additional considerations required when nonlinear source terms are used.
 Camporeale, E., Delzanno, G. L., & Colestock, P. (2012)
Lower hybrid to whistler mode conversion on a density striation.
J. Geophys. Res., 117(A10).
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When a wave packet composed of short wavelength lower hybrid modes traveling in an homogeneous plasma region encounters an inhomogeneity, it can resonantly excite long wavelength whistler waves via a linear mechanism known as mode conversion. An enhancement of lower hybrid/whistler activity has been often observed by sounding rockets and satellites in the presence of density depletions (striations) in the upper ionosphere. We address here the process of linear mode conversion of lower hybrid to whistler waves, mediated by a density striation, using a scalarfield formalism (in the limit of cold plasma linear theory) which we solve numerically. We show that the mode conversion can effectively transfer a large amount of energy from the short to the long wavelength modes. We also study how the efficiency scales by changing the properties (width and amplitude) of the density striation. We present a general criterion for the width of the striation that, if fulfilled, maximizes the conversion efficiency. Such a criterion could provide an interpretation of recent laboratory experiments carried out on the Large Plasma Device at UCLA.
 Camporeale, E. (2012).
Nonmodal linear theory for space plasmas.
Space science reviews, 172, 397409.
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The nonmodal approach is a linear theory formalism that emphasizes the transient evolution of a perturbed equilibrium. It differs from the normalmode analysis by not assuming an exponential behavior of physical perturbations.
We discuss works that have applied the nonmodal formalism to the problem of solar wind heating and acceleration. We briefly review the methodology of the Kelvin formalism and of the Generalized Stability Theory, and discuss the cases of both sheared and nonsheared plasmas.
The results and methodology reviewed in this paper could form the basis for a trend of research in solar wind dynamics that has not been yet systematically explored.
 Camporeale, E., D. Burgess (2011)
The dissipation of solar wind turbulent fluctuations at electron scales
Astrophys. J. , 730 114
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We present twodimensional fully kinetic particleincell simulations of decaying electromagnetic fluctuations. The computational box is such that wavelengths ranging from electron to ion gyroradii are resolved. The parameters used are realistic for the solar wind, and the iontoelectron mass ratio is physical. The dissipation of turbulent fluctuations at small scales is thought to be a crucial mechanism for solar wind acceleration and coronal heating. The computational results suggest that a powerlaw cascade of magnetic fluctuations could be sustained up to scales of the electron Larmor radius and smaller. We analyze the simulation results in light of the Vlasov linear theory, and we comment on the particle heating. The dispersion curves of lightly damped modes in this regime suggest that a linear mechanism could be responsible for the observed steepening of power spectra at electron scales, but a straightforward identification of turbulent fluctuations as an ensemble of linear modes is not possible.
 Camporeale, E., T. Passot, D. Burgess (2010)
Implications of a NonModal Linear Theory for the Marginal
Stability State and the Dissipation of Fluctuations in the Solar Wind
Astrophys. J. 715, 260270
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A magnetized plasma with anisotropic particle distributions may be unstable to a number of different kinetic instabilities. The solar wind is often observed in a state which is close to that implying instability, i.e., in a marginal stability state. Normalmode linear theory predicts that fluctuations in a stable plasma damp exponentially. The nonmodal approach for a linearized system differs from a normalmode analysis by following the temporal evolution of some perturbed equilibria, and therefore includes transient effects. We employ a nonmodal approach for studying the stability of a biMaxwellian magnetized plasma using the Landau fluid model, which we briefly describe. We show that biMaxwellian stable equilibria can support transient growth of some physical quantities, and we study how these transients behave when an equilibrium approaches its marginally stable condition. The results obtained with a nonmodal approach are relevant to a reexamination of the concept of linear marginal stability. Moreover, we highlight some aspects of the dissipation of turbulent fluctuations, which suggest that the nonmodal approach should be included in future studies
 Camporeale, E., D. Burgess (2010)
Electron temperature anisotropy in an expanding plasma: Particle in Cell simulations
Astrophys. J. 710, 18481856
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We perform fully kinetic particleincell simulations of a hot plasma that expands radially in a cylindrical geometry. The aim of the paper is to study the consequent development of the electron temperature anisotropy in an expanding plasma flow as found in a collisionless stellar wind. Kinetic plasma theory and simulations have shown that the electron temperature anisotropy is controlled by fluctuations driven by electromagnetic kinetic instabilities. In this study, the temperature anisotropy is driven selfconsistently by the expansion. While the expansion favors an increase of parallel anisotropy (T_par > T_perp ), the onset of the firehose instability will tend to decrease it. We show the results for supersonic, subsonic, and static expansion flows and suggest possible applications of the results for the solar wind and other stellar winds.
 Camporeale, E., D. Burgess and T. Passot (2009)
Transient growth in stable collisionless plasma
Phys. Plasmas. 16, 030703
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The first kinetic study of transient growth for a collisionless homogeneous Maxwellian plasma in a uniform magnetic field is presented. A system which is linearly stable may display transient growth if the linear operator describing its evolution is nonnormal so that its eigenvectors are nonorthogonal. In order to include plasma kinetic effects, a Landau fluid model is employed. The linear operator of the model is shown to be nonnormal and the results suggest that the nonnormality of a collisionless plasma is intrinsically related to its kinetic nature, with the transient growth being more accentuated for smaller scales and higher plasma beta. The results based on linear spectral theory have been confirmed with nonlinear simulations.
 Markidis, S., Camporeale, E., Burgess, D., Rizwanuddin, Lapenta, G. (2009)
Parsek2D: An Implicit Parallel ParticleinCell Code
Numerical Modeling of Space Plasma Flows: ASTRONUM  2008 406, 237
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A ParticleinCell (PIC) code, called Parsek2D, for the simulation of astrophysical and space plasmas is presented. Parsek2D enables simulations with large grid spacing and long time step, by means of implicit time differentiation and parallel computing. An implicit formulation of the PIC algorithm removes the severe numerical stability constraints of explicitly timedifferenced PIC. Parallel supercomputers are needed to meet the computational and memory storage requirements of large scale problems The implicit PIC algorithm, and its implementation on parallel computers are described. Simulations of magnetic reconnection using physical mass ratio are shown.
 Camporeale, E. and D. Burgess (2008)
Electron firehose instability: Kinetic linear theory and twodimensional particleincell simulations
J. Geophys. Res., 113, A07107, doi:10.1029/2008JA013043
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The kinetic electron firehose instability (EFI) is thought to be a crucial mechanism for constraining the observed electron anisotropy in expanding astrophysical plasmas, such as the solar wind. The EFI arises in a biMaxwellian plasma when the parallel temperature is greater than the perpendicular one, and its effect is to reduce anisotropy. We study this mechanism via kinetic linear theory, extending and refining previous results, and by new twodimensional particleincell (PIC) simulations with physical mass ratio. The results of PIC simulations show under which conditions the EFI can indeed be regarded as a constraint for electron distribution function. The detailed electron physics near marginal stability condition is discussed, with emphasis on the competition between growing and damping modes and on wave patterns formed at the nonlinear stage. The results also suggest an observational signature that the EFI has operated, namely the appearance of lowfrequency, quasiperpendicular whistler/electron?cyclotron waves.
 Camporeale, E., G.L. Delzanno, W. Daughton, and G. Lapenta (2006)
New approach for the study of linear Vlasov stability of inhomogeneous systems
Phys. Plasmas. 13, 092110
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This paper presents an alternative technique for solving the linearized VlasovMaxwell set of equations, in which the velocity dependence of the perturbed distribution function is described by means of an infinite series of orthogonal functions, chosen as Hermite polynomials. The orthogonality properties of such functions allow us to decompose the Vlasov equation into a set of infinite coupled linear equations. With a suitable truncation relation, the problem is transformed in an eigenvalue problem. This technique is based on solid but easy concepts, not attempting to evaluate the integration over the unperturbed trajectories and can be applied to any equilibrium. Although the solutions are approximate, because they neglect contributions of higher order coefficients of the series, the physical meaning of the loworder coefficients is clear. Furthermore the accuracy of the solution, which depends on the number of terms taken into account in the Hermite series, appears to be merely a problem of computational power. The method has been tested for a 1D Harris equilibrium, known to give rise to several instabilities like tearing, drift kink, and lower hybrid. The results are shown in agreement with those obtained by Daughton with a traditional technique based on the integration over unperturbed orbits.
 Camporeale E. and G. Lapenta (2005)
Model of bifurcated current sheets in the Earth's magnetotail: Equilibrium and stability
J. Geophys. Res. 110, A07206, doi:10.1029/2004JA010779
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Recent observations of CLUSTER have reconfirmed the recurring presence of bifurcated current sheets. The present paper revisits the model of Schindler and Birn (2002) to describe analytically bifurcated current sheets. Our contributions are in order. First, we describe a number of analytical velocity distribution functions that lead to bifurcated current sheets. Second, we derive necessary and sufficient conditions that determine when current sheets can be produced within the mathematical model of Schindler and Birn (2002). Third, we present a class of bifurcated current sheets, and we describe their properties. Fourth, we study the stability of bifurcated current sheets to the tearing instability finding that bifurcated current sheets tend to be more stable. Finally, we investigate the stability of bifurcated current sheets to the lower hybrid drift instability (LHDI) and kinking instability proving their presence. The work reported here is intended to extend and investigate the properties of instabilities from the typical but academic case of the Harris current sheet to current sheet equilibria that are more realisticly representative of the real magnetotail.
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