Stationary Electrostatic Waves Research Articles

Nonlinear particle trapping by coherent waves in thermal and nonthermal plasmas

We perform a test particle simulation to study the resonant interaction of electrons with a stationary electrostatic wave. In the simulation, we use a new method to generate nonthermal particle distributions efficiently. This method utilizes inverse cumulative distribution function, and compared with the existing methods in the literature, found to have a negligible deviation from the analytical form of the distribution, particularly in the tail side, where nonthermal particles are situated. The simulation revealed that the trapped particle density varies highly with the width (≈10%–60%) and amplitude (≈13%–19%) of the wave potential. The particle trapping is found to be more efficient in nonthermal plasmas than thermal plasma. It implies that particle transport through waves could be more effective in nonthermal plasmas than thermal plasmas. In addition, the simulation sheds light on the particle trapping time and its relation to the characteristics of the wave potential. Overall, the present study will be helpful to understand the particle trapping by the coherent waves, which are frequently seen in space and astrophysical plasmas.

On the correct implementation of Fermi–Dirac statistics and electron trapping in nonlinear electrostatic plane wave propagation in collisionless plasmas

Quantum statistics and electron trapping have a decisive influence on the propagation characteristics of coherent stationary electrostatic waves. The description of these strictly nonlinear structures, which are of electron hole type and violate linear Vlasov theory due to the particle trapping at any excitation amplitude, is obtained by a correct reduction of the three-dimensional Fermi-Dirac distribution function to one dimension and by a proper incorporation of trapping. For small but finite amplitudes, the holes become of cnoidal wave type and the electron density is shown to be described by a ϕ(x)1/2 rather than a ϕ(x) expansion, where ϕ(x) is the electrostatic potential. The general coefficients are presented for a degenerate plasma as well as the quantum statistical analogue to these steady state coherent structures, including the shape of ϕ(x) and the nonlinear dispersion relation, which describes their phase velocity.

Chaotic particle heating due to an obliquely propagating wave in a magnetized plasma

We study the dynamics of a relativistic charged particle in the presence of a uniform magnetic field and a stationary electrostatic wave that propagates at an arbitrary angle. The wave is considered as a series of periodic pulses which allows us to derive an exact map for the system. In particular, we investigate the heating process of an initially low-energy particle. It is found that abrupt changes in the maximum energy attained by the particle may occur as the angle between the wave propagation and the magnetic field varies. To determine what is the mechanism behind this phenomenon a reduced Hamiltonian that retains the important dynamical features is obtained. Using both Poincaré plots and perturbation theory, we identify that a separatrix reconnection is the key mechanism for the abrupt change in particle response.

Controlling chaos in wave-particle interactions

We analyze the behavior of a relativistic particle moving under the influence of a uniform magnetic field and a stationary electrostatic wave. We work with a set of pulsed waves that allows us to obtain an exact map for the system. We also use a method of control for near-integrable Hamiltonians that consists of the addition of a small and simple control term to the system. This control term creates invariant tori in phase space that prevent chaos from spreading to large regions, making the controlled dynamics more regular. We show numerically that the control term just slightly modifies the system but is able to drastically reduce chaos with a low additional cost of energy. Moreover, we discuss how the control of chaos and the consequent recovery of regular trajectories in phase space are useful to improve regular particle acceleration.

Wavelet representation of plasma particle dynamics in stationary electrostatic waves

The dynamics of plasma particles in the presence of a stationary electrostatic wave is studied by expanding the motion of each particle in powers of the wave amplitude. The coefficients of this expansion are wavelets in velocity space that gradually transform into well-behaved singular functions for t⪢to; to=(m∕k2T)1∕2. The force exerted on the plasma and the rate of kinetic energy transferred to the plasma by the wave are obtained by averaging the amplitude expansion over initial conditions. Higher-order terms show that nonlinear wave-plasma interactions are governed by a universal function M(τ) which is independent of the velocity distribution and depends only on τ=t∕tosc; tosc=(m∕kqEo)1∕2. It is found that the short-time (t≈tosc) behavior of M(τ) describing the transition between linear and nonlinear regimes is given by M(τ)=1−(τ4∕234!)+(41τ8∕278!). Also, using an implicit expansion of the particle motion, a global component Mo(τ)=(4∕τ2)J1(τ2∕2), 0⩽τ<∞, of the universal function is obtained.

Stationary waves produced by the Earth's bow shock

Cyclotron waves should be produced upstream and downstream from the bow shock by current in the shock. Amplitudes and polarizations of waves produced by a steady thin current sheet in the plane of the shock have been calculated by using cold plasma dispersion relations. The steady current leads to waves that are stationary in the shock frame. Waves upstream from the shock are circularly polarized electron waves, the polarization is in the sense of a right-hand screw when the interplanetary field is directed generally toward the shock and of a left-hand screw when the field is directed away. The downstream magnetic wave field has two components, one from the electron cyclotron branch and one from the ion cyclotron branch of the dispersion relation. The second component has a longer wavelength and a larger amplitude (by the ion/electron mass ratio) than the electron wave. It is elliptically rather than circularly polarized and is always polarized in the opposite sense to the upstream electron wave. Therefore, the sense of polarization would be seen to reverse as the shock is traversed in either direction. When the ambient magnetic field is not perpendicular to the shock, there is also a stationary electrostatic wave with its electric field normal to the shock. Both upstream and downstream the ratio of the electrostatic wave to the magnetic wave amplitudes is of the order u/c tan θ, where θ is the angle between the normal to the shock and the magnetic field, and u is the plasma flow velocity.