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Register a new userHamiltonian fluid reduction of the 1.5D Vlasov-Maxwell equations
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We consider the Vlasov-Maxwell equations with one spatial direction and two momenta, one in the longitudinal direction and one in the transverse direction. By solving the Jacobi identity, we derive reduced Hamiltonian fluid models for the density, the fluid momenta and the second order moments, related to the pressure tensor. We also provide the Casimir invariants of the reduced Poisson bracket. We show that the linearization of the equations of motion around homogeneous equilibria reproduces some essential feature of the kinetic model, the Weibel instability.
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We consider linear stability of steady states of 1(1/2) and 3D Vlasov-Maxwell systems for collisionless plasmas. The linearized systems can be written as separable Hamiltonian systems with constraints. By using a general theory for separable Hamiltonian systems, we recover the sharp linear stability criteria obtained previously by different approaches. Moreover, we obtain the exponential trichotomy estimates for the linearized Vlasov-Maxwell systems in both relativistic and nonrelativistic cases.
We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in Ref. 1. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with a modified Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincar{e} theory. Possible applications of these formulations include continuum geometric integration techniques, large-eddy simulation models and Casimir type stability methods. [1] H. Cendra et. al., Journal of Mathematical Physics 39, 3138 (1998)
In this paper we present energy-conserving, mixed discontinuous Galerkin (DG) and continuous Galerkin (CG) schemes for the solution of a broad class of physical systems described by Hamiltonian evolution equations. These systems often arise in fluid mechanics (incompressible Euler equations) and plasma physics (Vlasov--Poisson equations and gyrokinetic equations), for example. The dynamics is described by a distribution function that evolves given a Hamiltonian and a corresponding Poisson bracket operator, with the Hamiltonian itself computed from field equations. Hamiltonian systems have several conserved quantities, including the quadratic invariants of total energy and the $L_2$ norm of the distribution function. For accurate simulations one must ensure that these quadratic invariants are conserved by the discrete scheme. We show that using a discontinuous Galerkin scheme to evolve the distribution function and ensuring that the Hamiltonian lies in its continuous subspace leads to an energy-conserving scheme in the continuous-time limit. Further, the $L_2$ norm is conserved if central fluxes are used to update the distribution function, but decays monotonically when using upwind fluxes. The conservation of density and $L_2$ norm is then used to show that the entropy is a non-decreasing function of time. The proofs shown here apply to any Hamiltonian system, including ones in which the Poisson bracket operator is non-canonical (for example, the gyrokinetic equations). We demonstrate the ability of the scheme to solve the Vlasov--Poisson and incompressible Euler equations in 2D and provide references where we have applied these schemes to solve the much more complex 5D electrostatic and electromagnetic gyrokinetic equations.
We analyze nonlinear aspects of the self-consistent wave-particle interaction using Hamiltonian dynamics in the single wave model, where the wave is modified due to the particle dynamics. This interaction plays an important role in the emergence of plasma instabilities and turbulence. The simplest case, where one particle (N = 1) is coupled with one wave (M = 1), is completely integrable, and the nonlinear effects reduce to the wave potential pulsating while the particle either remains trapped or circulates forever. On increasing the number of particles (N = 2, M = 1), integrability is lost and chaos develops. Our analyses identify the two standard ways for chaos to appear and grow (the homoclinic tangle born from a separatrix, and the resonance overlap near an elliptic fixed point). Moreover, a strong form of chaos occurs when the energy is high enough for the wave amplitude to vanish occasionally.
Nonlinear solitary solutions to the Vlasov-Poisson set of equations are studied in order to investigate their stability by employing a fully-kinetic simulation approach. The study is carried out in the ion-acoustic regime for a collisionless, electrostatic and Maxwellian electron-ion plasma. The trapped population of electrons is modeled based on well-known Schamel distribution function. Head-on mutual collisions of nonlinear solutions are performed in order to examine their collisional stability. The findings include three major aspects: (I) These nonlinear solutions are found to be divided into three categories based on their Mach numbers, i.e. stable, semi-stable and unstable. Semi-stable solutions indicates a smooth transition from stable to unstable solutions for increasing Mach number. (II) The stability of solutions is traced back to a condition imposed on averaged velocities, i.e. net neutrality. It is shown that a bipolar structure is produced in the flux of electrons,early in the temporal evolution. This bipolar structure acts as the seed of the net-neutrality instability, which tips off the energy balance of nonlinear solution during collisions. As the Mach number increases, the amplitude of bipolar structure grows and results in a stronger instability. (III) It is established that during mutual collisions, a merging process of electron holes can happen to a variety of degrees, based on their velocity characteristics. Specifically, the number of rotations of electron holes around each other (in the merging phase) varies. Furthermore, it is observed that in case of a non-integer number of rotations, two electron holes exchange their phase space cores.
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