Driven, dissipative, energy-conserving magnetohydrodynamic equilibria. Part 2. The screw pinch

A cylindrically symmetric, electrically driven, dissipative, energy-conserving magnetohydrodynamic equilibrium model is considered. The high-magneticfield Braginskii ion thermal conductivity perpendicular to the local magnetic field and the complete electron resistivity tensor are included in an energy equation and in Ohm's law. The expressions for the resistivity tensor and thermal conductivity depend on number density, temperature, and the poloidal and axial (z-component) magnetic field, which are functions of radius that are obtained as part of the equilibrium solution. The model has plasma-confining solutions, by which is meant solutions characterized by the separation of the plasma into two concentric regions separated by a transition region that is an internal boundary layer. The inner region is the region of confined plasma, and the outer region is the region of unconfined plasma. The inner region has average values of temperature, pressure, and axial and poloidal current densities that are orders of magnitude larger than in the outer region. The temperature, axial current density and pressure gradient vary rapidly by orders of magnitude in the transition region. The number density, thermal conductivity and Dreicer electric field have a global minimum in the transition region, while the Hall resistivity, Alfvén speed, normalized charge separation, Debye length, (ωλ)ion and the radial electric field have global maxima in the transition region. As a result of the Hall and electron-pressure-gradient effects, the transition region is an electric dipole layer in which the normalized charge separation is localized and in which the radial electric field can be large. The model has an intrinsic value of β, about 13·3%, which must be exceeded in order that a plasma-confining solution exist. The model has an intrinsic length scale that, for plasma-confining solutions, is a measure of the thickness of the boundary-layer transition region. If appropriate boundary conditions are given at R = 0 then the equilibrium is uniquely determined. If appropriate boundary conditions are given at any outer boundary R = a then the equilibrium exhibits a bifurcation into two states, one of which exhibits plasma confinement and carries a larger axial current than the other, which is almost homogeneous and cannot confine a plasma. Exact expressions for the two values of the axial current in the bifurcation are derived. If the boundary conditions are given at R = a then a solution exists if and only if the constant driving electric field exceeds a critical value. An exact expression for this critical electric field is derived. It is conjectured that the bifurcation is associated with an electric-field-driven transition in a real plasma, between states with different rotation rates, energy dissipation rates and confinement properties. Such a transition may serve as a relatively simple example of the L—H mode transition observed in tokamaks.

Download Full-text

On driven, dissipative, energy-conserving magnetohydrodynamic equilibria

Journal of Plasma Physics ◽

10.1017/s0022377800016482 ◽

1992 ◽

Vol 48 (2) ◽

pp. 177-207 ◽

Cited By ~ 4

Author(s):

Michael L. Goodman†

Keyword(s):

Magnetic Field ◽

Thermal Conductivity ◽

Critical Point ◽

Transition Region ◽

Current Density ◽

Outer Region ◽

Internal Boundary ◽

Cylinder Wall ◽

Number Density ◽

Energy Conserving

A cylindrically symmetric, electrically driven, dissipative, energy-conserving magnetohydrodynamic equilibrium model is considered. The high-magnetic-field Braginskii electron electrical resistivity η parallel to a constant axial magnetic field B and ion thermal conductivity ĸ perpendicular to B are included in an energy equation and in Ohm's law. The expressions for η and ĸ depend on number density and temperature, which are functions of radius that are obtained as part of the equilibrium solution. The model has plasma-confining solutions, by which are meant solutions characterized by the separation of the plasma into two regions separated by a relatively thin transition region that is an internal boundary layer across which temperature and current density vary rapidly. The inner region has a temperature, pressure and current density that are much larger than in the outer region. The number density and thermal conductivity attain their minimum values in the transition region. The model has an intrinsic value of β, about 6.6%, which must be exceeded in order that a plasma-confining solution exist. The model has an intrinsic length scale, which, for plasma-confining solutions, is a measure of the thickness of the transition region separating the inner and outer regions of plasma. A larger class of transport coefficients is modelled by artificially changing η and ĸ by changing the constant coefficients ηO and ĸO that occur in their expressions. Increasing ĸO transforms a state that does not exhibit confinement into one that does, improves the confinement in a state that already exhibits it, and leads to an increase in ĸ in the confined region of plasma. The improvement in confinement consists in a decrease in the thickness of the transition region. Decreasing ηO subject to certain constraints, also transforms a state that does not exhibit confinement into one that does, improves the confinement in a state that already exhibits it, and leads to a decrease in η in the confined region of plasma. Increasing ηO up to a critical point increases the current, temperature, and volume of the confined region of plasma, and causes the thickness of the transition region to increase. If ηO is increased beyond the critical point, a plasma-confining state cannot exist. In all cases it is found that an increase in ĸ and a decrease in η in the confined region of plasma are associated with an improvement in the confinement properties of the equilibrium state. If the pressure and temperature are given on the cylinder wall, the equilibrium bifurcates when the electric field decreases below a critical value. The equilibrium can bifurcate into a state that exhibits confinement and a state that does not.

Download Full-text

A class of driven, dissipative, energy-conserving magnetohydrodynamic equilibria with flow

Journal of Plasma Physics ◽

10.1017/s0022377898006928 ◽

1998 ◽

Vol 60 (3) ◽

pp. 587-625 ◽

Cited By ~ 3

Author(s):

MICHAEL L. GOODMAN

Keyword(s):

Heat Flux ◽

Electric Field ◽

Boundary Conditions ◽

Current Density ◽

Hydrogen Plasma ◽

Transport Coefficients ◽

Outer Region ◽

Radial Electric Field ◽

Axial Current ◽

Transport Processes

The classical transport coefficients provide an accurate description of transport processes in collision-dominated plasmas. These transport coefficients are used in a cylindrically symmetric, electrically driven, steady-state magnetohydrodynamic (MHD) model with flow and an energy equation to study the effects of transport processes on MHD equilibria. The transport coefficients, which are functions of number density, temperature and magnetic field strength, are computed self-consistently as functions of radius R. The model has plasma-confining solutions characterized by the existence of an inner region of plasma with values of temperature, pressure and current density that are orders of magnitude larger than in the surrounding, outer region of plasma that extends outward to the boundary of the cylinder at R=a. The inner and outer regions are separated by a boundary layer that is an electric-dipole layer in which the relative charge separation is localized, and in which the radial electric field, temperature, pressure and axial current density vary rapidly. By analogy with laboratory fusion plasmas in confinement devices, the plasma in the inner region is confined plasma, and the plasma in the outer region is unconfined plasma. The solutions studied demonstrate that the thermoelectric current density, driven by the temperature gradient, can make the main contribution to the current density, and that the thermoelectric component of the electron heat flux, driven by an effective electric field, can make a large contribution to the total heat flux. These solutions also demonstrate that the electron pressure gradient and Hall terms in Ohm's law can make dominant contributions to the radial electric field. These results indicate that the common practice of neglecting thermoelectric effects and the Hall and electron pressure-gradient terms in Ohm's law is not always justified, and can lead to large errors. The model has three, intrinsic, universal values of β at which qualitative changes in the solutions occur. These values are universal in that they only depend on the ion charge number and the electron-to-ion mass ratio. The first such value of β (about 3.2% for a hydrogen plasma), when crossed, signals a change in sign of the radial gradient of the number density, and must be exceeded in order that a plasma-confining solution exist for a plasma with no flow. The second such value of β (about 10.4% for a hydrogen plasma), when crossed, signals a change in sign of the poloidal current density. Some of the solutions presented exhibit this current reversal. The third such value of β is about 2.67 for a hydrogen plasma. When β is greater than or equal to this value, the thermoelectric, effective electric-field-driven component of the electron heat flux cancels 50% or more of the temperature-gradient-driven ion heat flux. If appropriate boundary conditions are given on the axis R=0 of the cylinder, the equilibrium is uniquely determined. Analytical evidence is presented that, together with earlier work, strongly suggests that if appropriate boundary conditions are enforced at the outer boundary R=a then the equilibrium exhibits a bifurcation into two states, one of which exhibits plasma confinement and carries a larger axial current than the other state, which is close to global thermodynamic equilibrium, and so is not plasma-confining. Exact expressions for the two values of the axial current in the bifurcation are presented. Whether or not a bifurcation can occur is determined by the values of a critical electric field determined by the boundary conditions at R=a, and the constant driving electric field, which is specified. An exact expression for the critical electric field is presented. Although the ranges of the physical quantities computed by the model are a subset of those describing fusion plasmas in tokamaks, the model may be applied to any two-component, electron–ion, collision-dominated plasma for which the ion cyclotron frequency is much larger than the ion–ion Coulomb collision frequency, such as the plasma in magnetic flux tubes in the solar interior, photosphere, lower transition region, and possibly the upper transition region and lower corona.

Download Full-text

Anomalous losses of energetic particles in the presence of an oscillating radial electric field in fusion plasmas

Journal of Plasma Physics ◽

10.1017/s002237782000029x ◽

2020 ◽

Vol 86 (2) ◽

Author(s):

David Zarzoso ◽

Diego del-Castillo-Negrete

Keyword(s):

Electric Field ◽

Radial Electric Field ◽

Energetic Particles ◽

Fusion Plasmas ◽

Canonical Momentum ◽

Algebraic Decay ◽

Energetic Particle Transport ◽

Inner Region ◽

First Time ◽

Magnetic Equilibrium

The confinement of energetic particles in nuclear fusion devices is studied in the presence of an oscillating radial electric field and an axisymmetric magnetic equilibrium. It is shown that, despite the poloidal and toroidal symmetries, initially integrable orbits turn into chaotic regions that can potentially intercept the wall of the tokamak, leading to particle losses. It is observed that the losses exhibit algebraic time decay different from the expected exponential decay characteristic of radial diffusive transport. A dynamical explanation of this behaviour is presented, within the continuous time random walk theory. The central point of the analysis is based on the fact that, contrary to the radial displacement, the poloidal angle is not bounded and a proper statistical analysis can therefore be made, showing for the first time that energetic particle transport can be super-diffusive in the poloidal direction and characterised by asymmetric poloidal displacement. The connection between poloidal and radial positions ensured by the conservation of the toroidal canonical momentum, implies that energetic particles spend statistically more time in the inner region of the tokamak than in the outer one, which explains the observed algebraic decay. This indicates that energetic particles might be efficiently slowed down by the thermal population before leaving the system. Also, the asymmetric transport reveals a new possible mechanism of self-generation of momentum.

Download Full-text

Calculation of the radial electric field with RF sheath boundary conditions in divertor geometry

Nuclear Fusion ◽

10.1088/1741-4326/aa9e9a ◽

2018 ◽

Vol 58 (2) ◽

pp. 026027 ◽

Cited By ~ 2

Author(s):

B. Gui ◽

T. Y. Xia ◽

X. Q. Xu ◽

J.R. Myra ◽

X. T. Xiao

Keyword(s):

Electric Field ◽

Boundary Conditions ◽

Radial Electric Field

Download Full-text

The root of a comet tail: Rosetta ion observations at comet 67P/Churyumov–Gerasimenko

Astronomy and Astrophysics ◽

10.1051/0004-6361/201832842 ◽

2018 ◽

Vol 616 ◽

pp. A21 ◽

Cited By ~ 10

Author(s):

E. Behar ◽

H. Nilsson ◽

P. Henri ◽

L. Berčič ◽

G. Nicolaou ◽

...

Keyword(s):

Magnetic Field ◽

Solar Wind ◽

Electric Field ◽

Outer Region ◽

Field Measurements ◽

Night Side ◽

Combined Action ◽

Single Night ◽

Inner Region ◽

Comet 67P

Context. The first 1000 km of the ion tail of comet 67P/Churyumov–Gerasimenko were explored by the European Rosetta spacecraft, 2.7 au away from the Sun. Aims. We characterised the dynamics of both the solar wind and the cometary ions on the night-side of the comet’s atmosphere. Methods. We analysed in situ ion and magnetic field measurements and compared the data to a semi-analytical model. Results. The cometary ions are observed flowing close to radially away from the nucleus during the entire excursion. The solar wind is deflected by its interaction with the new-born cometary ions. Two concentric regions appear, an inner region dominated by the expanding cometary ions and an outer region dominated by the solar wind particles. Conclusions. The single night-side excursion operated by Rosetta revealed that the near radial flow of the cometary ions can be explained by the combined action of three different electric field components, resulting from the ion motion, the electron pressure gradients, and the magnetic field draping. The observed solar wind deflection is governed mostly by the motional electric field −uion × B.

Download Full-text

Numerical Simulation on Transient Electrophoretic Motion of a Circular Particle in a T-Shaped Slit Microchannel

Fluids Engineering ◽

10.1115/imece2003-42791 ◽

2003 ◽

Author(s):

Chunzhen Ye ◽

Dongqing Li

Keyword(s):

Numerical Simulation ◽

Liquid Phase ◽

Flow Field ◽

Electric Field ◽

Particle Motion ◽

Outer Region ◽

Double Layers ◽

Circular Particle ◽

Electric Potentials ◽

Inner Region

This paper considers the electrophoretic motion of a circular particle in a T-shaped slit microchannel, where the size of the channel is close to that of the particle. During the process, the electric field (i.e., the gradient of the electric potential) changes with the particle motion, which in return influences the flow field and the particle motion. Therefore, the electric field, the flow field and the particle motion are coupled together, and this is an unsteady process. The objective is to obtain a fundamental understanding of the characteristics of the particle motion in the complicated T-shaped junction region. Such influences on the electric field and the particle motion are investigated as the applied electric potentials, the geometry of the channel and the size of the particle. In the theoretical analysis, the liquid phase is divided into the inner region and the outer region. The inner region consists of the electrical double layers and the outer region consists of the remainder of the liquid. Under the assumption of thin electrical double layer, a mathematical model governing the inner region, the outer region and the particle motion is developed. A direct numerical simulation method using the finite element method is employed. In this method, a continuous hydrodynamic model is adopted. By this model, both the liquid phase in the outer region and the particle phase are governed by the same momentum equations. ALE method is used to track the surface of the particle at each time step. The numerical results show that the electric field is influenced by the applied electric potentials, the geometry of the channel and the particle suspension, and that the particle motion is mainly dominated by the local electric field. It is also found that the magnitude of the particle motion is dependent on its own size in the same channel.

Download Full-text

Electron shock waves moving into an ionized medium

Laser and Particle Beams ◽

10.1017/s0263034600009502 ◽

1995 ◽

Vol 13 (3) ◽

pp. 377-382

Author(s):

Mostafa Hemmati

Keyword(s):

Shock Waves ◽

Electric Field ◽

Boundary Conditions ◽

Transition Region ◽

Fluid Model ◽

Class Ii ◽

Poisson’S Equation ◽

Electron Velocity ◽

Poisson's Equation ◽

Dynamical Transition

The propagation of electron driven shock waves has been investigated by employing a one-dimensional, three-component fluid model. In the fluid model, the basic set of equations consists of equations of conservation of mass, momentum, and energy, plus Poisson's equation. The wave is assumed to be a shock front followed by a dynamical transition region. Following Fowler's (1976) categorization of breakdown waves, the waves propagating into a preionized medium will be referred to as Class II Waves. To describe the breakdown waves, Shelton and Fowler (1968) used the terms proforce and antiforce waves, depending on whether the applied electric field force on electrons was with or against the direction of wave propagation. Breakdown waves, i.e., return strokes of lightning flashes, therefore, will be referred to as Antiforce Class II waves. The shock boundary conditions and Poisson's equation for Antiforce Class II waves are different from those for Antiforce Waves. The use of a newly derived set of boundary conditions and Poisson's equation for Antiforce Class II waves allows for a successful integration of the set of fluid equations through the dynamical transition region. The wave structure, i.e., electric field, electron concentration, electron temperature, and electron velocity, are very sensitive to the ion concentration ahead of the wave.

Download Full-text

Boundary-induced electrophoresis of uncharged conducting particles: near-contact approximation

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2008.0533 ◽

2009 ◽

Vol 465 (2106) ◽

pp. 1939-1948 ◽

Cited By ~ 6

Author(s):

Mohammad Abu Hamed ◽

Ehud Yariv

Keyword(s):

Electric Field ◽

Outer Region ◽

Uniform Electric Field ◽

Net Charge ◽

Polarizable Particle ◽

Hydrodynamic Stresses ◽

Inner Region ◽

Correction Terms ◽

Gap Width ◽

Maxwell Stresses

A zero net charge ideally polarizable particle is suspended within an electrolyte solution, nearly in contact with an uncharged non-polarizable wall. This system is exposed to a uniform electric field that is applied parallel to the wall. Assuming a thin Debye thickness, the induced-charge electro-osmotic flow is investigated with the goal of obtaining an approximation for the force experienced by the particle. Singular perturbations in terms of the dimensionless gap width δ are used to represent the small-gap singular limit δ ≪1. The fluid is decomposed into two asymptotic regions: an inner gap region, where the electric field and strain rate are large, and an outer region, where they are moderate. The leading contribution to the force arises from hydrodynamic stresses in the inner region, while contributions from both hydrodynamic stresses at the outer region and Maxwell stresses in both regions appear in higher order correction terms.

Download Full-text