Daniel H. E. Dubin

Nonlinear gyrokinetic equations

Nonlinear gyrokinetic equations are derived from a systematic Hamiltonian theory. The derivation employs Lie transforms and a noncanonical perturbation theory first used by Littlejohn for the simpler problem of asymptotically small gyroradius. For definiteness, only electrostatic fluctuations in slab geometry are considered; however, there is a straightforward generalization to arbitrary field geometry and electromagnetic perturbations. An energy invariant for the nonlinear system is derived, and several limiting forms are considered. The weak turbulence theory of the equations is examined. In particular, the wave kinetic equation of Galeev and Sagdeev can ony be derived by an asystematic truncation of the equations, implying that this equation fails to consider all gyrokinetic effects. The equations are simplified for the case of small but finite gyroradius and put in a form suitable for efficient computer simulation. Although it is possible to derive the Terry–Horton and Hasegawa–Mima equations as limit...

Inviscid damping of asymmetries on a two-dimensional vortex

The inviscid damping of an asymmetric perturbation on a two-dimensional circular vortex is examined theoretically, and with an electron plasma experiment. In the experiment, an elliptical perturbation is created by an external impulse. After the impulse, the ellipticity (quadrupole moment) of the vortex exhibits an early stage of exponential decay. The measured decay rate is in good agreement with theory, in which the perturbation is governed by the linearized Euler equations. Often, the exponential decay of ellipticity is slow compared to a vortex rotation period, due to the excitation of a quasimode. A quasimode is a vorticity perturbation that behaves like a single azimuthally propagating wave, which is weakly damped by a resonant interaction with corotating fluid. Analytically, the quasimode appears as a wave packet of undamped continuum modes, with a sharply peaked frequency spectrum, and it decays through interference as the modes disperse. When the exponential decay rate of ellipticity is comparabl...

Vortex crystals from 2D Euler flow: Experiment and simulation

Vortex-in-cell simulations that numerically integrate the 2D Euler equations are compared directly to experiments on magnetized electron columns [K. S. Fine, A. C. Cass, W. G. Flynn, and C. F. Driscoll, “Relaxation of 2D turbulence to vortex crystals,” Phys. Rev. Lett. 75, 3277 (1995)], where turbulent flows relax to metastable vortex crystals. A vortex crystal is a lattice of intense small diameter vortices that rotates rigidly in a lower vorticity background. The simulations and experiments relax at the same rates to vortex crystals with similar vorticity distributions. The relaxation is caused by mixing of the background by the intense vortices: the relaxation rate is peaked when the background circulation is 0.2–0.4 times the total circulation. Close quantitative agreement between experiment and simulation provides strong evidence that vortex crystals can be explained without incorporating physics beyond 2D Euler theory, despite small differences between a magnetized electron column and an ideal 2D fluid.

Non-neutral ion plasmas and crystals, laser cooling, and atomic clocks

Experimental work which uses Penning and Paul traps to confine non‐neutral ion plasmas is discussed. Penning traps use a static uniform magnetic field and a static electric field to confine ions. The Paul trap uses the ponderomotive force from inhomogeneous radio‐frequency fields to confine ions to a region of minimum field strength. In many atomic physics experiments, these traps are designed to produce a harmonic restoring force for small numbers of stored ions (<104). Under these conditions and at low temperatures, both traps produce plasmas with simple shapes whose mode properties can be calculated exactly. Laser cooling has been used to reduce the temperature of trapped ions to less than 10 mK with ion spacings less than 20 μm. At such temperatures and interion spacings, the Coulomb potential energy between nearest neighbor ions is greater than the ion thermal energy and the ions exhibit spatial correlations characteristic of a liquid or crystal. Laser beams also apply a torque which, by changing the...

Excitation of nonlinear electron acoustic waves

A particle in cell (PIC) simulation is used to investigate the excitation of electron acoustic waves (EAWs) and the stability of the EAWs against decay. An EAW is a nonlinear wave with a carefully tailored trapped particle population, and the excitation process must create the trapped particle population. For a collisionless plasma, successful excitation occurs when a relatively low amplitude driver that is spatially and temporally resonant with the EAW is applied for a sufficiently long time (many trapping periods). The excited EAW rings at a nearly constant amplitude long after the driver is turned off, provided the EAW has the largest wavelength that fits in the simulation domain. Otherwise, the excited EAW decays to a longer wavelength EAW. In phase space, this decay to longer wavelength appears as a tendency of the vortex-like trapped particle populations to merge. In a collisional plasma, successful excitation of an EAW requires the driver amplitude to exceed a threshold value. The period for a trap...

Thermal equilibria and thermodynamics of trapped plasmas with a single sign of charge

Plasmas consisting exclusively of particles with a single sign of charge (e.g., pure electron plasmas and pure ion plasmas) can be confined by static electric and magnetic fields (e.g., in a Penning trap) and also be in a state of global thermal equilibrium. This important property distinguishes these totally un-neutralized plasmas from neutral and quasineutral plasmas. This paper reviews the conditions for and structure of the thermal equilibrium states and then develops a thermodynamic theory of the trapped plasmas. Thermodynamics provides hundreds of general relations (Maxwell relations) between partial derivatives of thermodynamic variables with respect to one another. Thermodynamic inequalities place general and useful bounds on various quantities. General and relatively simple expressions are provided for fluctuations of the thermodynamic variables. In practice, trapped plasmas are often made to evolve through a sequence of thermal equilibrium states through the slow addition (or subtraction) of ene...

The phonon wake behind a charge moving relative to a two-dimensional plasma crystal

In a recent experiment a wake was created in a two-dimensional lattice of charged dust grains by a charge moving parallel to the lattice plane. Multiple “Mach cones” were observed in the wake. This paper describes a linear theory of the phonon wake caused by a charge moving relative to a crystalline lattice. The theory predicts multiple structures in the wake that are qualitatively similar to those observed in the experiments. These structures are caused by constructive interference of compressional phonons excited by the moving charge, combined with the strongly dispersive nature of these phonons.

Electron acoustic waves in pure ion plasmas

Standing electron acoustic waves (EAWs) are observed in a pure ion plasma. EAWs are slow nonlinear plasma waves; at small amplitude their phase velocities (vph≃1.4v¯ for small kλD) and their frequencies are in agreement with theory. At moderate amplitude, EAW-type plasma waves can be excited over a broad range of frequencies. This frequency variability comes from the plasma adjusting its velocity distribution so as to make the plasma mode resonant with the drive frequency. Wave-coherent laser-induced fluorescence shows the intimate nature of the wave-particle interaction, and how the particle distribution function is modified by the wave driver until the plasma mode is resonant with the driver.

Vortex motion driven by a background vorticity gradient

The motion of self-trapped vortices on a background vorticity gradient is examined numerically and analytically. The vortices act to level the local background vorticity gradient. Conservation of momentum dictates that positive vortices (“clumps”) and negative vortices (“holes”) react oppositely: clumps move up the gradient whereas holes move down the gradient. A linear analysis gives the trajectory of small clumps and holes that rotate against the local shear. Prograde clumps and holes are always nonlinear, and move along the gradient at a slower rate. This rate vanishes when the background shear is sufficiently large.

Collisional transport in non-neutral plasmas

Several recent experiments have measured collisional transport in non-neutral plasmas (heat conduction, test particle diffusion, and viscosity) that is from 10 to 104 times larger than predicted by classical theory. New guiding center theories of collisional transport have been developed that agree with the measurements. The experiments operate in the guiding center regime rc≪λD, where rc is the cyclotron radius and λD is the Debye length. In this regime, classical transport theory is irrelevant because it implicitly assumes the opposite ordering, λD≪rc, although this ordering is not always satisfied in neutral plasmas.