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

1998 ◽  
Vol 60 (3) ◽  
pp. 587-625 ◽  
Author(s):  
MICHAEL L. GOODMAN

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.

1993 ◽  
Vol 49 (1) ◽  
pp. 125-159 ◽  
Author(s):  
Michael L. Goodman

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.


1999 ◽  
Vol 39 (11Y) ◽  
pp. 2119-2125
Author(s):  
V.S Tsypin ◽  
I.C Nascimento ◽  
R.M.O Galvão ◽  
A.G Elfimov ◽  
M Tendler ◽  
...  

2010 ◽  
Vol 655 ◽  
pp. 105-121 ◽  
Author(s):  
EHUD YARIV

A cation-selective conducting particle is suspended in an electrolyte solution and is exposed to a uniformly applied electric field. The electrokinetic transport processes are described in a closed mathematical model, consisting of differential equations, representing the physical transport in the electrolyte, and boundary conditions, representing the physicochemical conditions on the particle boundary and at large distances away from it. Solving this mathematical problem would in principle provide the electrokinetic flow about the particle and its concomitant velocity relative to the otherwise quiescent fluid.Using matched asymptotic expansions, this problem is analysed in the thin-Debye-layer limit. A macroscale description is extracted, whereby effective boundary conditions represent appropriate asymptotic matching with the Debye-scale fields. This description significantly differs from that corresponding to a chemically inert particle. Thus, ion selectivity on the particle surface results in a macroscale salt concentration polarization, whereby the electric potential is rendered non-harmonic. Moreover, the uniform Dirichlet condition governing this potential on the particle surface is transformed into a non-uniform Dirichlet condition on the macroscale particle boundary. The Dukhin–Derjaguin slip formula still holds, but with a non-uniform zeta potential that depends, through the cation-exchange kinetics, upon the salt concentration and electric field distributions. For weak fields, an approximate solution is obtained as a perturbation to a reference state. The linearized solution corresponds to a uniform zeta potential; it predicts a particle velocity which is proportional to the applied field. The associated electrokinetic flow is driven by two different agents, electric field and salinity gradients, which are of comparable magnitude. Accordingly, this flow differs significantly from that occurring in electrophoresis of chemically inert particles.


2018 ◽  
Vol 58 (2) ◽  
pp. 026027 ◽  
Author(s):  
B. Gui ◽  
T. Y. Xia ◽  
X. Q. Xu ◽  
J.R. Myra ◽  
X. T. Xiao

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Saber Mohammadi ◽  
Akram Khodayari ◽  
Arash Ahmadi

Electrocaloric response of the PMN-10PT is measured experimentally and compared with the numerical results. Based on the compatibility of the experimental and numerical results, feasibility of using ferroelectric materials as an electrothermal transducer has been investigated. In this study, electrocaloric response of three different ferroelectric capacitors (PMN-10PT, PMN-25PT, and PZN-4.5PT) under an applied periodic electric field have been investigated. Alternative switching of the electrocaloric elements with specific boundary conditions generates a directed heat flux. It can be concluded that each ferroelectric material can be used as a transducer in a special temperature range that in which it has good electrocaloric response.


2000 ◽  
Vol 402 ◽  
pp. 225-253 ◽  
Author(s):  
CHRISTOPHER J. ELKINS ◽  
JOHN K. EATON

Measurements in the turbulent momentum and thermal boundary layers on a rotating disk with a uniform heat flux surface are described for Reynolds numbers up to 106. Measurements include mean velocities and temperatures, all six Reynolds stresses, turbulent temperature fluctuations, and three turbulent heat fluxes. The mean velocity profiles have no wake region, but the mean temperature profiles do. The turbulent temperature fluctuations have a large peak in the outer layer, and there is a third turbulent heat flux in the cross-flow direction. Correlation coefficients and structure parameters are not constant across the boundary layer as they are in two-dimensional boundary layers (2DBLs), and their values are lower. The turbulent Prandtl number agrees with 2DBL values in the lower part of the outer region but is reduced from the 2DBL values higher in the boundary layer. In the outer region of the boundary layer, the transport processes differ significantly from what is observed in two-dimensional turbulent boundary layers: ejections dominate the transport of momentum while both ejections and sweeps contribute to the transport of the passive scalar.


2015 ◽  
Vol 58 (1) ◽  
pp. 014007 ◽  
Author(s):  
F Ryter ◽  
M Cavedon ◽  
T Happel ◽  
R M McDermott ◽  
E Viezzer ◽  
...  

Sign in / Sign up

Export Citation Format

Share Document