Collisional Viscosity Research Articles

Shear reduction of collisional transport: Experiments and theory

Experiments and theory on collisional diffusion and viscosity in quiescent single-species plasmas demonstrate enhanced transport in the two-dimensional (2D) bounce-averaged regime, limited by shear in the plasma rotation. For long plasma columns, the measured diffusion agrees quantitatively with recent theories of three-dimensional long-range E×B drift collisions, and is substantially larger than predicted for classical velocity-scattering collisions. For short plasmas, diffusion is observed to be enhanced by Nb, the number of times a thermal particle bounces axially before being separated by shear. Equivalently, recent theory in the 2D bounce-averaged regime shows how diffusion decreases with increasing shear, generalizing the zero-shear perspective which gives Bohm diffusion. Viscosity is similarly enhanced in the 2D regime, but there is presently only qualitative agreement with theory. These results apply to both non-neutral and neutral plasmas, and provide the first rigorous analysis of shear reduction of transport in a paradigmatic system.

On the relationship between mixing length and strongturbulence estimates for transport due to drift turbulence

For a simple two-dimensional nonlinear model of the collisional driftinstability the relationship between mixing length and strong turbulence fluxesinvestigated. Calculation of the turbulent wavenumber spectrum within astrong turbulence model shows that the transport is dominated by an inertialrange contribution; this yields a similar result for the fluxes to that from animproved mixing length theory. The nature of the spectrum in the dissipationregion, where collisional viscosity dominates, provides an explanation of anapparent paradox that arises when transport fluxes are cast in a formproportional to viscosity (or the diffusion coefficient).

Sawtooth oscillations in a damped/driven cryogenic electron plasma: Experiment and theory

Measurements have been made of nonlinear sawtooth oscillations of the displacement of a magnetized electron column in a cryogenic, cylindrical trap. First reported 7 years ago, these oscillations occur when the displacement is destabilized by a resistive wall and damped by a temperature-dependent collisional viscosity. A typical evolution can last for thousands of seconds. Measurements show that oscillations of the plasma displacement are accompanied by oscillations in the plasma temperature. A simple predator-and-prey model of the temperature and displacement gives rise to a limit cycle solution due to the nonmonotonic dependence of the viscosity on temperature. These limit cycles are in good quantitative agreement with the measured sawtooth oscillations.

Collisional viscosity of FCC particles in a CFB

AbstractParticle velocity distributions were measured for a flow of 75‐μm FCC particles in a CFB using a new video‐digital camera technique. From the spread of particle histograms, a random oscillating particle velocity was determined. This random velocity was used to compute the powder viscosity with the help of dense‐phase kinetic theory of granular flow. There was excellent agreement between this kinetic‐theory measurement and previous macroscopic viscosity measurements. The dimensionless group kinematic viscosity divided by the oscillating velocity and particle diameter correlated all available data as a function of particle volume fraction.

Double hysteresis in L/H transition and compound dithers

To study L/H transition physics, the hysteresis, generated by the ambipolar condition, is examined. Three mechanisms for the bipolar losses, i.e. the loss cone loss, collisional bulk viscosity loss of ions, and the anomalous loss, are simultaneously retained, and the competition between them is investigated. It is found that fivefold multiple branches exist in the gradient - flux relation, when the effective ion collision frequency is close to unity. Multiple bifurcations in the hysteresis curve appear. The new type of the limit cycle oscillation, compound dither, is predicted to occur.

Nonlinear magnetohydrodynamic dynamo

The self-consistent nonlinear evolution and saturation of the dynamo, including the back reaction of the magnetic field on the flow through the Lorentz J×B force, is investigated via simulation of the fully compressible magnetohydrodynamic (MHD) equations. The saturated state is found to be highly turbulent. The energy in the saturated magnetic field is only a small fraction of the kinetic energy in the flow which drives the dynamo. However, as the collision frequency decreases and the Reynolds number R increases, the ratio of magnetic to kinetic energy in the saturated state increases gradually. The nonlinear viscosity generated by the turbulent fluctuations rises rapidly relative to the collisional viscosity as R increases, such that the total transport of momentum remains virtually unchanged as the collisional viscosity is reduced. The scale lengths of the magnetic and velocity fluctuations both decrease as R increases, so that the scale size of the magnetic field remains comparable to the scale size of the flow.

Plasma cloud expansion in the ionosphere: Three‐dimensional simulation

A three‐dimensional time‐dependent model was developed to study the characteristics of a plasma cloud expansion in the ionosphere. The model makes it possible to use realistic ionospheric and cloud parameters. With the model the parallel and perpendicular motion of both the cloud and background ions can be studied. The electrostatic potential is solved in three dimensions taking into account the large parallel‐to‐perpendicular conductivity ratio. The model takes account of ion inertia, pressure gradients, the Lorentz force, the stress tensor (both collisional viscosity and the finite Larmor radius effect), Coulomb collisions and collisions between neutral and ionized species, and the finite parallel conductivity. Three sample simulations are presented: a plasma expansion of a nearly spherical 1 km Ba+ cloud, both with and without a background neutral wind, and a long thin Ba+ cloudlet. With or without the neutral wind the effective potential, which is different from the electrostatic potential if the electron temperature is included, is constant along the magnetic field for typical cloud sizes. The expanding plasma clouds become elongated in the magnetic field direction. The released Ba+ ions push the background O+ ions away along the magnetic field as they expand. Consequently, a hole develops in the background O+ distribution at the cloud location and on the two sides of the cloud O+ bumps form. The entire three‐dimensional structure, composed of the plasma cloud and the background plasma embedded in the cloud, slowly rotates about the magnetic field, with the ions and electrons rotating in opposite directions. The cloud configuration takes the shape of a rotating ellipsoid with a major axis that expands with time. Perpendicular to the magnetic field, in the absence of the neutral wind the motion is insignificant compared to the parallel motion. With a neutral wind the motion along the magnetic field and the rotational motion are qualitatively unchanged, but the cloud and the perturbed background structure move in the direction of the wind, with a speed less than the wind speed. Perpendicular to the magnetic field the deformation of the cloud induced by the wind is characterized by steepening of the backside. For an extended cloudlet along the magnetic field the potential is not constant along B. Also, the parallel expansion of the cloud and the snowplow effect are not important. The sheared rotational motion and diffusion act to smooth initial density perturbations on the cloud surface at early times, but later an instability develops that creates new density structures.

Collisional viscosity and finite ion Larmor radius effects on resistive internal kink modes

The theory of resistive m0 = 1 modes in toroidal experiments is considered for parameters relevant to high temperature regimes of the type predicted for ignited plasmas. A kinetic theory is developed for the ion response within the singular layer, including full finite ion Larmor radius effects and ion-ion collisional viscosity. Analytical and numerical solutions of the linear dispersion equation are presented and this formalism is applied to situations appropriate for planned ignition devices.

A Dissipative Model of Plasma Equilibrium in Toroidal Systems

In order to describe a steady-state plasma equilibrium in tokamaks, stellarators or other non-axisymmetric configurations, the model of ideal MHD with isotropic plasma pressure is widely used: 0 = - ∇p + j x B . The ideal MHD-model of a toroidal plasma equilibrium requires the existence of closed magnetic surfaces. Several numerical codes have been developed in the past to solve the three-dimensional equilibrium problem, but so far no existence theorem for a solution has been proved. Another difficulty is the formation of magnetic islands and field line ergodisation, which can only be described in terms of ideal MHD if the plasma pressure is constant in the ergodic region. In order to describe the formation of magnetic islands and ergodisation of surfaces properly, additional dissipative terms have to be incorporated to allow decoupling of the plasma and magnetic field. In a collisional plasma, viscosity and inelastic collisions introduce such dissipative processes. In the model used a friction term proportional to the velocity v of the plasma is included. Such a term originates from charge exchange interaction of the plasma with a neutral background. With these modifications, the equilibrium problem reduces to a set of quasilinear elliptic equations for the pressure, the electric potential and the magnetic field. The paper deals with an existence theorem based on the Fixed-Point method of Schauder. It can be shown that a self-consistent and unique equilibrium exists if the friction term is large and the plasma pressure is sufficiently low. The essential role of the dissipative terms is to remove the singularities of the ideal MHD model on rational magnetic surfaces. The problem has a strong similarity to Bénard cell convection, and consequently a similar behaviour such as bifurcation and exchange of stability is expected.

Resistive ballooning modes in high-beta tokamaks

Resistive ballooning mode stability is examined using one-fluid, two-fluid, and kinetic theories to determine the domain of validity of the resistive-fluid theories in the high-beta Doublet III tokamak. In general, it is found that the requirements for the validity of the resistive-fluid theories in Doublet III are only satisfied on the cold, outermost edge of the plasma with temperatures T ≤ 25 eV. In the domain of validity, the two-fluid and kinetic theories are qualitatively the same and indicate that the dominant effect is parallel resistivity with small corrections from perpendicular resistivity, fluid compression, thermal conduction, and ion collisional viscosity. The generic role of resistive ballooning modes in tokamaks and the implications for anomalous transport are also discussed.

Ion-electrostatic instabilities in current-carrying magnetized plasmas

A fluid analysis of the stability of obliquely propagating ion-electrostatic waves to current flow along the magnetic field in a fully ionized, collisional plasma is presented. The stabilizing or destabilizing effects of frictional force, collisional energy transfer, electron and ion viscosity, electron and ion thermal conductivity, and thermal force are investigated. Simple analytical expressions for the growth rate arising from each dissipative mechanism are derived for ion-cyclotron, cosine ion-acoustic, ion-acoustic, and cosine ion-cyclotron waves.

Damping of nonlinear magneto-acoustic waves

Nonlinear magneto-acoustic waves in a turbulent plasma are simulated by collisional MHD equations. Damping of these waves due to electrical conductivity arising from micro-instabilities and collisional viscosity are analyzed. Numerical investigation of competing effects due to non-linearity and dissipation has been carried out. It is found that finite amplitude perturbation leads to the formation of a shock in both the slow and the fast mode propagation. Collisional viscosity plays an important role in the damping of nonlinear magneto-acoustic waves in the astrophysical and space plasmas.

Transport processes in tokamak-type installations in two-fluid approximation

Transport processes across a strong magnetic field in axisymmetric magnetic traps are studied in two-fluid magnetohydrodynamics with anisotropic temperature. Account is taken of inertia and collisional viscosity, and of the nonuniformity of the equilibrium temperatures. In the case of low toroidality r/R ≪ 1, expressions are obtained and analysed for the heat and particle fluxes, and equations are obtained for the ambipolar electric field, the mean velocity of the plasma, and the electric current along the magnetic field lines.

Viscosity-dominated convective plasma diffusion

It is shown that, in addition to the classical (D⊥ ∼ B−2), plateau (D⊥ ∼ B0), and Bohm (D⊥ ∼ B−1) diffusions, there is a new region of diffusion between the classical and the plateau regions where the diffusion increases with increasing the magnetic field. In this region, the lifetime of the convective cells is determined by the ion collisional viscosity and, therefore, this region is important for a plasma with large ion mass.

Effects of Parallel Wavelength on the Collisional Drift Instability

An extensive investigation of the dependence of drift wave characteristics upon parallel wavelength in collisional potassium Q-machine plasmas is presented. Results are compared with a two-fluid slab-model theory which includes the effects of parallel resistivity and thermal conductivity for electrons and Larmor radius, collisional viscosity, and motion parallel to the magnetic field for ions. The general stability characteristics of the drift wave are confirmed: stability at low magnetic field (B/k⊥≲0.4 kG·cm) due to ion viscosity, stability at both long- and short-parallel wavelength (λ‖ ≳ 200 and λ‖≲100 cm, respectively) due to the combined effects of ion viscosity and parallel electron fluid expansion and stability of individual modes at high magnetic field (B/k⊥ ≈ 1 kG·cm, λ‖ ≈ 100 cm) due to ion parallel motion. Quantitative agreement between experiment and theory is demonstrated for the detailed parametric dependences of instability onset, complex frequency, and oscillation amplitude on parallel wavelength, magnetic field, and density.

Drift Instabilities in a Uniformly Rotating Plasma Cylinder

The dispersion relation for low-frequency, collisional, electrostatic waves in a uniformly rotating plasma cylinder is obtained in terms of several positive-definite bilinear forms whose magnitude may easily be estimated. The effects of ion transverse collisional viscosity, electron parallel resistivity, ion parallel motion, and an equilibrium radial electric field are included in the theory. The close correspondence between the present dispersion relation and the dispersion relation for localized, slab-model modes justifies the use of the latter in the interpretation of Q-machine experiments. Another result is that ion collisional viscosity stabilizes the density-gradient driven drift wave if the axial wavelength λ‖ satisfies λ‖ < 2πσr0 (M/me)1/4, (σ ∼ 0.7; r0 is the density gradient scale length; M, me are the ion and electron masses, respectively). Overstable waves, which are driven by the centrifugal force and which resemble flutes with a finite parallel wavelength, can exist if λ‖ is sufficiently long.