Equilibrium Field Research Articles

Layered materials

We propose a continuum model for bodies made of thin and thick layers of different elastic constituents. Assuming the thickness of subsequent layers are very different (as, e.g., in sedimentary rocks), we take a special assumption on the kinematic of thin beds and partly constrain their deformations to those of the neighboring thick courses. The kinematic of thin beds is thus described through the displacements of their middle surface (three fields depending on two coordinates) plus six parameters giving some average information on the displacement field within the bed (bending, torsion, shear, and elongation of normal fibers). The search for minimal potential energy, under imposed boundary traction, gives then the set of equilibrium field equations and boundary conditions for the displacements of the internally constrained body. Under homogeneous stress conditions, it is possible to integrate these equations and find the explicit expression of a global compliance tensor, which gives a Reuss approximation of the homogeneous behavior of the layered continuum.

A Thermodynamically Consistent Relaxation Model for Turbulent Binary Mixture Undergoing Phase Transition

Non-equilibrium phenomena within a turbulent binary mixture undergoing a phase transition are modelled based on the one fluid, homogeneous approach. The set of governing evolution equations is given and the averaged basic balance laws are considered. The continuum under consideration consists of two constituents of which the first is undergoing a phase transition (vaporisation, cavitation, flashing, condensation) during the flow with turbulent transport of mass, heat and momentum. Starting from the Clausius-Duhem inequality for the recoverable specific entropy, several thermodynamical relationships for equilibrium and non-equilibrium fields are developed. The relaxation-retardation constitutive model for turbulent, phasic and diffusive fluxes are described. It is shown that the resulting model uncovers a few known two-phase models of turbulence and also the classical relaxation models as limiting cases.

MHD field line resonances and global modes in three‐dimensional magnetic fields

By assuming a general isotropic pressure distribution P = P(ψ, α), where ψ and α are three‐dimensional scalar functions labeling the field lines with B = ∇ψ × ∇α, we have derived a set of MHD eigenmode equations for both global MHD modes and field line resonances (FLR). Past MHD theories are restricted to isotropic pressures with P = P(ψ) only. The present formulation also allows the plasma mass density to vary along the field line. The linearized ideal MHD equations are cast into a set of global differential equations from which the field line resonance equations of the shear Alfvén waves and slow magnetosonic modes are naturally obtained for general three‐dimensional magnetic field geometries with flux surfaces. Several new terms associated with ∂P/∂α are obtained. In the FLR equations a new term is found in the shear Alfvén FLR equation due to the geodesic curvature and the pressure gradient in the poloidal flux surface. The coupling between the shear Alfvén waves and the magnetosonic waves is through the combined effects of geodesic magnetic field curvature (κs = 2κ · B × ∇ψ/B2) and plasma pressure as previously derived. The properties of the FLR eigenfunctions at the resonance field lines are investigated, and the behavior of the FLR wave solutions near the FLR surface are derived. Numerical solutions of the FLR equations for three‐dimensional magnetospheric fields in equilibrium with high plasma pressure will be presented in a future publication.

Self-consistent modeling of turbulence and transport

We describe an efficient procedure for coupling a turbulent system to a transport equation which evolves the equilibrium fields that drive and are driven by the turbulence. As an example, we apply the procedure to the coupling of turbulence simulations of the two-dimensional Hasegawa–Wakatani equations to a one-dimensional transport equation for the density n. Our coupling scheme uses implicit temporal discretization of the transport equation, rendering it stable for arbitrarily large time steps. This allows the computation of turbulence with self-consistent steady-state equilibrium profiles in a single time step of the transport equations and with a total computational time comparable to that required for the turbulence code alone to reach a statistical steady state with fixed equilibrium profiles. Results are presented for both local and non-local turbulence simulations. In the former, which requires running a separate turbulence simulation for each transport grid cell, the transport flux Γ( x) depends on only local values of n( x) and n ′( x); for this case, Γ is expressed using Fick’s law, Γ=− Dn ′, with D>0 prescribed by the turbulence. In the non-local simulations, Γ( x) depends on the form of n over the entire domain. For such simulations we present two methods for representing Γ that allow for anomalous flux transport, i.e., regions where the flux flows up the local gradient of n.

Least square methods in structural analysis

A new approach to structural analysis is presented. The method uses equilibrium and deformation geometry field relations directly without resorting to an energy formulation. A matrix vector of errors in the field relations is minimized first with respect to the internal quantities, e.g., the moments. Subsequently the error is minimized with respect to the external forces and displacements. In the latter formulation, the internal quantities have been eliminated by back substitution. The least-square method for minimizing functional forms is the precedent for this work. The method is tested on beam and plate problems. The development of the pertinent matrices is interesting and depends on the trial functions used to represent the internal quantities.

Magnetic Flux Emergence into the Solar Corona. II. Global Magnetic Fields with Current Sheets

Hydromagnetic structural and stability properties of global magnetic fields with current sheets are discussed in this paper. These fields describe solar coronal magnetic structures that form when a fresh magnetic field of opposite polarity has emerged into the corona containing a preexisting magnetic field. Three classes of axisymmetric fields are treated. The first is characterized by a continuous normal field distribution at the boundary and an infinitesimally thin current sheet in the field. As ideal hydromagnetic equilibria, these fields are globally stable, but they are resistively unstable within the current sheet. Magnetic reconnection is unavoidable no matter how large the electrical conductivity is. The other two classes of fields may be stable without this kind of resistive nonequilibrium. Of particular interest among them are equilibrium fields with finite-thickness current sheets in force balance with pressure gradients and gravity. These fields are shown to have enough free magnetic energy to let some parts of its field to become open during magnetic reconnection, an effect important for the dynamics of coronal mass ejections.

Self-similarity of near-breaking short gravity wind waves

We show that the asymmetric profiles of the near-breaking wind waves observed in a large wind-wave tank do not depend on the wave scale. The experiments were performed for wind-generated waves of wavelength increasing with fetch. The wave crests just prior to breaking exhibit steep slopes exceeding 0.6 at the front and much smaller but remarkably constant slope at the rear. The statistics of wave geometric properties, such as the along-wind crest-slope distribution or the wavefront steepness, is found to be scale invariant. This implies that the shape of the near-breaking wind waves is self-similar. Similar experiments with paddle-generated waves amplified by wind reveal the same scale-independent features. These results suggest a certain universality of the underlying physical mechanisms leading to breaking for wave fields in equilibrium with wind.

The dynamics of current carriers in standing Alfvén waves: Parallel electric fields in the auroral acceleration region

The acceleration of current carriers in an Alfvén wave current system is considered. The model incorporates a dipole magnetic field geometry, and we present an analytical solution of the two‐fluid equations by successive approximations. The leading solution corresponds to the familiar single‐fluid toroidal oscillations. The next order describes the nonlinear dynamics of electrons responsible for carrying a few μAm−2 field aligned current into the ionosphere. The solution shows how most of the electron acceleration in the magnetosphere occurs within 1 RE of the ionosphere, and that a parallel electric field of the order of 1 mVm−1 is responsible for energising the electrons to 1 keV. The limitations of the electron fluid approximation are considered, and a qualitative solution including electron beams and a modified E∥ is developed in accord with observations. We find that the electron acceleration can be nonlinear, (ve∥∇∥)ve∥ > ωve∥, as a result of our nonuniform equilibrium field geometry even when ve∥ is less than the Alfvén speed. Our calculation also elucidates the processes through which E∥ is generated and supported.

Inflation from tsunami-waves

We investigate inflation driven by the evolution of highly excited quantum states within the framework of out of equilibrium field dynamics. These states are characterized by a non-perturbatively large number of quanta in a band of momenta but with vanishing expectation value of the scalar field. They represent the situation in which initially a non-perturbatively large energy density is localized in a band of high energy quantum modes and are coined tsunami-waves. The self-consistent evolution of this quantum state and the scale factor is studied analytically and numerically. It is shown that the time evolution of these quantum states lead to two consecutive stages of inflation under conditions that are the quantum analogue of slow-roll. The evolution of the scale factor during the first stage has new features that are characteristic of the quantum state. During this initial stage the quantum fluctuations in the highly excited band build up an effective homogeneous condensate with a non-perturbatively large amplitude as a consequence of the large number of quanta. The second stage of inflation is similar to the usual classical chaotic scenario but driven by this effective condensate. The excited quantum modes are already superhorizon in the first stage and do not affect the power spectrum of scalar perturbations. Thus, this tsunami quantum state provides a field theoretical justification for chaotic scenarios driven by a classical homogeneous scalar field of large amplitude.

Entropy principle for the moment systems of degree $\alpha$ associated to the Boltzmann equation. Critical derivatives and non controllable boundary data

moments and of degree \(\alpha (ET_m^\alpha)\). For such theories the entropy principle is still valid only, if the non equilibrium field variables and their derivatives are sufficiently small with respect to the required approximation order. In this paper we prove through simple examples of stationary problems that the entropy principle fails in general, if all the non-equilibrium variables are of the same order of magnitude. This is due to the fact that there exist some derivatives of non equilibrium variables (critical derivatives) that are not small along all the solutions. This property can be used to fix the non controllable boundary data in such a manner that the critical derivatives are kept small for the solution that we may choose. Thus, for the stationary unidimensional case we propose the requirement that the critical derivatives vanish on the boundary, eventually with some successive derivatives. This is a sufficient condition for the validity of the entropy principle at least in a neighborhood of the boundary and makes it possible to assign the non controllable data in a simple manner when the number of moments is greater than 13. We have tested this procedure in several cases of \(ET_m^\alpha\) theories, showing that the criterion implies continuity with respect to the change of the moment number and to the truncation order. In particular for the planar unidimensional heat conduction problem we have obtained a behavior for the temperature that is always the same as the one predicted by the classical Fourier law. This result is in evident contrast with the minimax principle expectation. However we have qualitative differences between the temperature behavior described by Extended Thermodynamics and the one by Fourier-Navier-Stokes theory for heat conduction in radial symmetry.

Design of superconducting coil system for remodeling JT-60

At JAERI a plan of remodeling JT-60 tokamak fusion device by using superconducting coil system is being considered to investigate performance of break-even plasma in long pulse operation. The coil system are composed of Toroidal Field (TF), Central Solenoid (CS) and Equilibrium Field (EF) coils. The TF coil is designed to have a maximum field of 7.4 T at winding with a nominal current of 19.4 kA and a magnetic stored energy of 1.7 GJ which is larger than that of every superconducting coil constructed so far. The coil uses a cable-in-conduit conductor that a Nb/sub 3/Al (or Nb/sub 3/Sn) strand cable is inserted into a square stainless steel conduit with a circular hole and is wound with a react-and-wind technique. The design operating temperature is 4.6 K. The CS which consists of 4 modules is operated with a maximum field of 7.4 T and a maximum ramp up rate of 1.0 T/s. For the conductor, a square Nb/sub 3/Sn cable-in-conduit conductor with stainless steel conduit is used. The EF coil consists of 6 coils that the maximum size coil has an outer diameter of 11 in. The EF4 coil which has a maximum field of 7.4 T and a ramp up rate of 2.2 T/s uses the same conductor as CS and other field coils use a NbTi cable-in-conduit conductor.

Ergodic magnetic limiter for the TCABR

In this work it is considered the effect of an ergodic magnetic limiter on the plasma confined in the TCABR tokamak. The poloidal distribution of the limiter currents is determined taking into account the toroidal geometry of this tokamak. The plasma equilibrium field is analytically obtained by solving the Grad-Schlüter-Shafranov equation in polar toroidal coordinates. The perturbing limiter field is obtained as a vacuum field by solving the Laplace equation. Supposing the limiter action as a sequence of delta-function pulses, a symplectic map is derived by using a Hamiltonian formulation for the perturbed field line. Examples are given to illustrate the influence of this design on the topology of the perturbed field line structure.

Temperature field of radiative equilibrium in a semitransparent slab with a linear refractive index and gray walls

The temperature field in a semitransparent slab of absorbing–emitting gray medium at radiative equilibrium is solved in this paper. The medium has a linear refractive index and the two boundaries are diffuse gray walls. A curved ray tracing technique is combined with a pseudo-source adding method to deduce the radiative intensities on the gray walls. And on the basis of the previous work done by Ben Abdallah and Le Dez, the discrete temperature field in the slab is deduced. The influences of refractive index distribution, boundary wall emissivities and optical thickness on the radiative equilibrium temperature field are examined. The results display the significant influences of the refractive index distribution and the boundary wall emissivities.

Magnetic effects on the viscous boundary layer damping of ther-modes in neutron stars

This paper explores the effects that magnetic fields have on the viscous boundary layers (VBLs) that can form in neutron stars at the crust-core interface, and it investigates the VBL damping of the gravitational-radiation driven r-mode instability. Approximate solutions to the magnetohydrodynamic equations valid in the VBL are found for ordinary-fluid neutron stars. It is shown that magnetic fields above ${10}^{9}{(10}^{10} \mathrm{K}/T)\mathrm{G}$ significantly change the structure of the VBL, and that magnetic fields decrease the VBL damping time. Furthermore, VBL damping completely suppresses the r-mode instability for $B\ensuremath{\gtrsim}{10}^{12} \mathrm{G}$ (independent of the temperature). These bounds refer to the strength of the radial component of the equilibrium field at the location of the core radius. Magnetic effects on the VBL vanish wherever the radial component of the equilibrium field vanishes. Thus, magnetic fields will profoundly affect the VBL damping of the r-mode instability in hot young pulsars (that are cool enough to have formed a solid crust). One can speculate that magnetic fields can affect the VBL damping of this instability in low-mass x-ray binary systems and other cold old pulsars (if they have sufficiently large internal fields).

Reduction of islands in full-pressure stellarator equilibria

The control of magnetic islands is a crucial issue in designing stellarators. Islands are associated with resonant radial magnetic fields at rational rotational-transform surfaces and can lead to chaos and poor plasma confinement. In this article it is shown that variations in the resonant fields of a full pressure stellarator equilibrium can be related to variations in the boundary via a coupling matrix, and that inversion of this matrix determines a boundary modification for which the island content is significantly reduced. The numerical procedure is described and the results of island optimization are presented. Equilibria with islands are computed using the Princeton Iterative Equilibrium Solver, and resonant radial fields are calculated via construction of quadratic-flux-minimizing surfaces. A design candidate for the National Compact Stellarator Experiment [A. Reimann, L. Ku, D. Monticello et al., Phys. Plasmas 8, 2083 (2001)], which has a large island, is used to illustrate the technique. Small variations in the boundary shape are used to reduce island size and to reverse the phase of a major island chain.

The toroidicity-induced Alfvén eigenmode structure in DIII-D: Implications of soft x-ray and beam-ion loss data

The internal structure of the toroidicity-induced Alfvén eigenmode (TAE) is studied by comparing soft x-ray profile and beam ion loss data taken during TAE activity in the DIII-D tokamak [W. W. Heidbrink et al., Nucl. Fusion 37, 1411 (1997)] with predictions from theories based on ideal magnetohydrodynamic (MHD), gyrofluid, and gyrokinetic models. The soft x-ray measurements indicate a centrally peaked eigenfunction, a feature which is closest to the gyrokinetic model’s prediction. The beam ion losses are simulated using a guiding center code. In the simulations, the TAE eigenfunction calculated using the ideal MHD model acts as a perturbation to the equilibrium field. The predicted beam ion losses are an order of magnitude less than the observed ∼6%–8% losses at the peak experimental amplitude of δBr/B0≃2–5×10−4.

Magnetic equilibria resulting from a helically symmetric kink instability

This paper explores magnetic equilibria which could result from the kink instability in a cylindrical magnetic flux tube. We examine a variety of cylindrical magnetic equilibria which are susceptible to the kink, and simulate its evolution in a frictional fluid. We assume that the evolution takes place under conditions of helical symmetry, so the problem becomes effectively two-dimensional. The initial cylindrical equilibrium field is specified in terms of its twist function k(r) = B θ/(rBz ) and for a variety of k(r) functions we calculate linear growth rates for the kink instability, assuming that it develops under helical symmetry with pitch τ. We find that the growth rate is sensitive to the value of τ. We simulate nonlinear evolution of the kink using a Lagrangian frictional code which constrains the field to have helical symmetry of a given pitch τ. Ideal MHD is assumed and the plasma pressure is taken to be small in order to mimic conditions in the solar corona. In some cases the flux tube evolves to a new smooth helically symmetric equilibrium which involves a relatively small change in the maximum electric current. In other cases there is evidence of current-sheet formation.

Role of the Ridley-Watkins-Hilsum mechanism in the stabilization of surface plasma waves

The stabilization of unstable harmonic oscillations of the surface charge density and electromagnetic field that are excited by a low-density electron beam propagating parallel to the plane boundary of a half-space occupied by gallium arsenide (GaAs) is studied in the hydrodynamic approximation. It is shown that, for gallium arsenide semiconductors (and, generally, for other so-called AIIIBV compound semiconductors), these oscillations are stabilized primarily by the Ridley-Watkins-Hilsum mechanism. The equilibrium field amplitudes and equilibrium surface charge densities of both the semiconductor and the beam are obtained for a slightly overcritical level, i.e., for electrons with energies slightly greater than the energy gap between the lower and higher valleys.

Alternative dimensional reduction via the density matrix

We give graphical rules, based on earlier work for the functional Schr\"odinger equation, for constructing the density matrix for scalar and gauge fields in equilibrium at finite temperature T. More useful is a dimensionally reduced effective action (DREA) constructed from the density matrix by further functional integration over the arguments of the density matrix coupled to a source. The DREA is an effective action in one less dimension which may be computed order by order in perturbation theory or by dressed-loop expansions; it encodes all thermal matrix elements. We term the DREA procedure alternative dimensional reduction, to distinguish it from the conventional dimensionally reduced field theory (DRFT) which applies at infinite T. The DREA is useful because it gives a dimensionally reduced theory usable at any T including infinity, where it yields the DRFT, and because it does not and cannot have certain spurious infinities which sometimes occur in the density matrix itself or the conventional DRFT; these come from $\mathrm{ln}T$ factors at infinite temperature. The DREA can be constructed to all orders (in principle) and the only regularizations needed are those which control the ultraviolet behavior of the zero-T theory. An example of spurious divergences in the DRFT occurs in $d=2+1{\ensuremath{\varphi}}^{4}$ theory dimensionally reduced to $d=2.$ We study this theory and show that the rules for the DREA replace these ``wrong'' divergences in physical parameters by calculable powers of $\mathrm{ln}T;$ we also compute the phase transition temperature of this ${\ensuremath{\varphi}}^{4}$ theory in one-loop order. Our density-matrix construction is equivalent to a construction of the Landau-Ginzburg ``coarse-grained free energy'' from a microscopic Hamiltonian.

Intersonic Crack Propagation—Part II: Suddenly Stopping Crack

In Part I of this series, we have obtained the fundamental solution for a mode II intersonic crack which involves a crack moving uniformly at a velocity between the shear and longitudinal wave speeds while subjected to a pair of concentrated forces suddenly appearing at the crack tip and subsequently acting on the crack faces. The fundamental solution can be used to generate solutions for intersonic crack propagation under arbitrary initial equilibrium fields. In this paper, Part II of this series, we study a mode II crack suddenly stopping after propagating intersonically for a short time. The solution is obtained by superposing the fundamental solution and the auxiliary problem of a static crack emitting dynamic dislocations such that the relative crack face displacement in the fundamental solution is negated ahead of where the crack tip has stopped. We find that, after the crack stops moving, the stress intensity factor rapidly rises to a finite value and then starts to change gradually toward the equilibrium value for a static crack. A most interesting feature is that the static value of stress intensity is reached neither instantaneously like a suddenly stopping subsonic crack nor asymptotically like a suddenly stopping edge dislocation. Rather, the dynamic stress intensity factor changes continuously as the shear and Rayleigh waves catch up with the stopped crack tip from behind, approaches negative infinity when the Rayleigh wave arrives, and then suddenly assumes the equilibrium static value when all the waves have passed by. This study is an important step toward the study of intersonic crack propagation with arbitrary, nonuniform velocities.