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Whistler wave-particle interactions play an important role in the Earth inner magnetospheric dynamics and have been the subject of numerous investigations. By running a global kinetic ring current model (RAM-SCB) in a storm event occurred on Oct 23–24 2002, we obtain the ring current electron distribution at a selected location at MLT of 9 and L of 6 where the electron distribution is composed of a warm population in the form of a partial ring in the velocity space (with energy around 15 keV) in addition to a cool population with a Maxwellian-like distribution. The warm population is likely from the injected plasma sheet electrons during substorm injections that supply fresh source to the inner magnetosphere. These electron distributions are then used as input in an implicit particle-in-cell code (iPIC3D) to study whistler-wave generation and the subsequent wave-particle interactions. We find that whistler waves are excited and propagate in the quasi-parallel direction along the background magnetic field. Several different wave modes are instantaneously generated with different growth rates and frequencies. The wave mode at the maximum growth rate has a frequency around 0.62 ω c e , which corresponds to a parallel resonant energy of 2.5 keV. Linear theory analysis of wave growth is in excellent agreement with the simulation results. These waves grow initially due to the injected warm electrons and are later damped due to cyclotron absorption by electrons whose energy is close to the resonant energy and can effectively attenuate waves. The warm electron population overall experiences net energy loss and anisotropy drop while moving along the diffusion surfaces towards regions of lower phase space density, while the cool electron population undergoes heating when the waves grow, suggesting the cross-population interactions.

A review of the impact of the cold-ion and cold-electron populations in the Earth’s magnetosphere is presented. The cold populations are defined by total energy less than approximately 100 eV, i.e. in the energy range which is strongly affected by spacecraft charging and that often dominates the total plasma density. We also include the warm plasma cloak in the review, since it overlaps partially with the cold energy range and is a population that is still not well understood. The known impacts of cold ions and cold electrons that are discussed are: the source of hot magnetospheric plasma, solar-wind/magnetosphere coupling, magnetotail reconnection and substorms, Kelvin–Helmholtz instabilities on the magnetopause, chorus, hiss, electromagnetic-ion-cyclotron and ultra-low-frequency wave–particle interactions, aurora structuring and spacecraft charging. Other possible impacts are associated with refilling on open-drift trajectories, the remnant layer and plasmapause disruption. A discussion of the difficulty of cold-plasma measurements and the need for new measurement techniques that measure the full cold-ion and cold-electron distribution functions is also presented. There remain a lot of unknowns about the cold-ion and cold-electron populations, associated with their origin, properties, drivers and impacts. These populations will need to be fully understood before the magnetosphere–ionosphere system can be fully understood.

We present an electrostatic Particle-In-Cell (PIC) code on multi-block, locally structured, curvilinear meshes called Curvilinear PIC (CPIC). Multi-block meshes are essential to capture complex geometries accurately and with good mesh quality, something that would not be possible with single-block structured meshes that are often used in PIC and for which CPIC was initially developed. Despite the structured nature of the individual blocks, multi-block meshes resemble unstructured meshes in a global sense and introduce several new challenges, such as the presence of discontinuities in the mesh properties and coordinate orientation changes across adjacent blocks, and polyjunction points where an arbitrary number of blocks meet. In CPIC, these challenges have been met by an approach that features: (1) a curvilinear formulation of the PIC method: each mesh block is mapped from the physical space, where the mesh is curvilinear and arbitrarily distorted, to the logical space, where the mesh is uniform and Cartesian on the unit cube; (2) a mimetic discretization of Poisson's equation suitable for multi-block meshes; and (3) a hybrid (logical-space position/physical-space velocity), asynchronous particle mover that mitigates the performance degradation created by the necessity to track particles as they move across blocks. The numerical accuracy of CPIC was verified using two standard plasma–material interaction tests, which demonstrate good agreement with the corresponding analytic solutions. Compared to PIC codes on unstructured meshes, which have also been used for their flexibility in handling complex geometries but whose performance suffers from issues associated with data locality and indirect data access patterns, PIC codes on multi-block structured meshes may offer the best compromise for capturing complex geometries while also maintaining solution accuracy and computational efficiency.

Based on discussions held at a workshop in Bra-Pollenzo, Italy, this paper elaborates upon 19 outstanding questions of plasma physics in the Earth's magnetosphere. The questions are grouped according to (a) driving processes, (b) radiation belt and ring current issues, (c) auroral physics, (d) internal plasma processes, and (e) magnetosphere-ionosphere mapping issues. Future needs for magnetospheric plasma physics (measurements, techniques, simulations, theories, studies) are outlined.

An active mapping mission is described that unambiguously connects measurements in the Earth's magnetosphere to visible aurora in the atmosphere. The core of the mission is an electron-beam source operated on a spacecraft in the equatorial magnetosphere, with the electron beam traveling along the Earth's magnetic-field lines to the atmosphere, depositing its energy to create an optical beam-spot in the atmosphere at the footpoint of the spacecraft's magnetic-field line. This optical spot can be imaged by ground-based cameras, putting the location of the spacecraft's magnetic footpoint into the context of the optical aurora. Scientific instruments carried on the spacecraft make critical measurements of the properties of the magnetosphere at the locations where the magnetosphere powers the aurora, allowing the determination of the plasma-physics mechanisms by which the magnetosphere drives the aurora, in particular answering the outstanding question of how the magnetosphere drives low-latitude auroral arcs. Long-standing questions in magnetosphere-ionosphere coupling that have not been answered because we could not unambiguously connect locations in the magnetosphere with their image in the ionosphere will finally be addressed. In this paper the properties of a “standard” growth-phase auroral arc are collected, theories of the magnetospheric generation of auroral arcs are reviewed, and critical magnetospheric measurements to discern the mechanisms that drive auroral arcs are determined. Further, the plasma physics of the experiment is investigated, including spacecraft-charging mitigation, beam stability, beam scattering, and electron orbit theory. Tradeoffs (keV versus MeV) concerning the energy of the electron beam are enumerated.

A spectral method for the numerical solution of the multi-dimensional Vlasov-Maxwell equations is presented. The plasma distribution function is expanded in Fourier (for the spatial part) and Hermite (for the velocity part) basis functions, leading to a truncated system of ordinary differential equations for the expansion coefficients (moments) that is discretized with an implicit, second order accurate Crank-Nicolson time discretization. The discrete non-linear system is solved with a preconditioned Jacobian-Free Newton-Krylov method.It is shown analytically that the Fourier-Hermite method features exact conservation laws for total mass, momentum and energy in discrete form. Standard tests involving plasma waves and the whistler instability confirm the validity of the conservation laws numerically. The whistler instability test also shows that we can step over the fastest time scale in the system without incurring in numerical instabilities. Some preconditioning strategies are presented, showing that the number of linear iterations of the Krylov solver can be drastically reduced and a significant gain in performance can be obtained.

A spectral method for kinetic plasma simulations based on the expansion of the velocity dis- tribution function in a variable number of Hermite polynomials is presented. The method is based on a set of non-linear equations that is solved to determine the coefficients of the Hermite expansion satisfying the Vlasov and Poisson equations. In this paper, we first show that this technique combines the fluid and kinetic approaches into one framework. Second, we present an adaptive strategy to increase and decrease the number of Hermite functions dynamically during the simulation. The technique is applied to the Landau damping and two-stream instability test problems. Performance results show 21% and 47% saving of total simulation time in the Landau and two-stream instability test cases, respectively.

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 leap-frog 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 counter-intuitive 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 leap-frog 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.

The equidistribution methods based on L p Monge–Kantorovich optimization and on the deformation method are analyzed primarily in the context of grid adaptation. The first class of methods can be obtained from a variational principle leading to a fluid dynamic formulation based on time-dependent equations for the mass density and the momentum density. In this context, deformation methods arise from a similar fluid formulation by making a specific assumption on the time evolution of the density (but with some degree of freedom for the momentum density). In general, deformation methods do not arise from a variational principle. However, it is possible to prescribe an optimal deformation method, related to L 1 Monge–Kantorovich optimization, by making a further assumption on the momentum density. Thus, the fluid dynamic formulation provides a unified description of equidistribution methods. Some numerical examples using the L p fluid dynamic formulation are also explored.

Mesh-motion (r-refinement) grid adaptivity schemes are attractive due to their potential to minimize the numerical error for a prescribed number of degrees of freedom. However, a key roadblock to a widespread deployment of this class of techniques has been the formulation of robust, reliable mesh-motion governing principles, which (1) guarantee a solution in multiple dimensions (2D and 3D), (2) avoid grid tangling (or folding of the mesh, whereby edges of a grid cell cross somewhere in the domain), and (3) can be solved effectively and efficiently. In this study, we formulate such a mesh-motion governing principle, based on volume equidistribution via Monge-Kantorovich optimization (MK). In earlier publications [1,2], the advantages of this approach with regard to these points have been demonstrated for the time-independent case. In this study, we demonstrate that Monge-Kantorovich equidistribution can in fact be used effectively in a time-stepping context, and delivers an elegant solution to the otherwise pervasive problem of grid tangling in mesh-motion approaches, without resorting to ad hoc time-dependent terms (as in moving-mesh PDEs, or MMPDEs [3,4]). We explore two distinct r-refinement implementations of MK: the direct method, where the current mesh relates to an initial, unchanging mesh, and the sequential method, where the current mesh is related to the previous one in time. We demonstrate that the direct approach is superior with regard to mesh distortion and robustness. The properties of the approach are illustrated with a hyperbolic PDE, the advection of a passive scalar, in 2D and 3D. Velocity flow fields with and without flow shear are considered. Three-dimensional grid, time-step, and nonlinear tolerance convergence studies are presented which demonstrate the optimality of the approach.

A new cell-area equidistribution method for two-dimensional grid adaptation, based on Monge-Kantorovich optimization (or Monge-Kantorovich optimal transport), is presented. The method is based on a rigorous variational principle, in which the L"2 norm of the grid displacement is minimized, constrained locally to produce a prescribed positive-definite cell volume distribution. The procedure involves solving the Monge-Ampere equation: A single, nonlinear, elliptic scalar equation with no free parameters, and with proved existence and uniqueness theorems. We show that, for sufficiently small grid displacement, this method also minimizes the mean grid-cell distortion, measured by the trace of the metric tensor. We solve the Monge-Ampere equation numerically with a Jacobian-Free Newton-Krylov method. The ellipticity property of the Monge-Ampere equation allows multigrid preconditioning techniques to be used effectively, delivering a scalable algorithm under grid refinement. Several challenging test cases demonstrate that this method produces optimal grids in which the constraint is satisfied numerically to truncation error. We also compare this method to the well known deformation method [G. Liao, D. Anderson, Appl. Anal. 44 (1992) 285]. We show that the new method achieves the desired equidistributed grid using comparable computational time, but with considerably better grid quality than the deformation method.