Showing papers in "Physics of Fluids in 1986"
TL;DR: In this paper, a general picture of magnetic reconnection in the framework of 2D incompressible resistive magnetohydrodynamic theory is presented, and a theory of the solution in the external and in the diffusion region is developed and analytical expressions in agreement with the simulation results are obtained by means of a variational principle.
Abstract: A general picture of magnetic reconnection in the framework of 2‐D incompressible resistive magnetohydrodynamic theory is presented. Numerical studies of (quasi‐) steady‐state driven reconnection reveal current sheet formation for Mach numbers M=u/vA exceeding the Sweet–Parker reconnection rate MSP=(η/LvA)1/2. Since the thickness δ of the current sheet is found to be invariant to a change of the resistivity η, its length Δ increases rapidly with decreasing η or increasing M, which can be written in the form Δ∼(M/MSP)4, so that Δ reaches the global system size L within a short range of the parameter M/MSP. The results are rather insensitive to the particular choice of boundary conditions. Because of the presence of a current sheet, the overall reconnection process is quite slow. This picture essentially agrees with Syrovatsky’s [Sov. Phys. JETP 33, 933 (1971)] theory and disproves Petschek’s [AAS/NASA Symposium on the Physics of Solar Flares, (NASA, Washington, DC, 1964) p. 425] mechanism of fast magnetic reconnection. A theory of the solution in the external and in the diffusion region is developed and analytical expressions in agreement with the simulation results are obtained by means of a variational principle. For sufficiently long current sheets the tearing mode becomes unstable in spite of the stabilizing effect of the inhomogeneous flow. The tearing mode contributes to the overall reconnection process, but a general assessment of this effect in the asymptotic regime of almost vanishing η is difficult.
752 citations
TL;DR: In this paper, a statistical theory for compressible turbulent shear flows subject to buoyancy effects is developed, and important correlation functions in compressible shear flow are calculated with the aid of a multiscale direct-interaction approximation.
Abstract: A statistical theory for compressible turbulent shear flows subject to buoyancy effects is developed. Important correlation functions in compressible shear flows are calculated with the aid of a multiscale direct‐interaction approximation. They are expressed in the gradient‐diffusion form similar to the eddy‐viscosity representation for the Reynolds stress in incompressible flows. The results obtained are applicable to subgrid modeling, and a Smagorinsky‐type model in compressible flows is constructed.
625 citations
TL;DR: In this article, the conditions for solidification in a laboratory plasma are discussed and conditions for the formation of a coulomb lattice are given for small particles in plasmas.
Abstract: Small particles in plasmas can form a coulomb lattice. The conditions for solidification in a laboratory plasma are discussed.
517 citations
TL;DR: In this article, it was shown both analytically and numerically that the toroidicity not only breaks up the shear Alfven continuous spectrum, but also creates new discrete toroidic induced shear eigenmodes with frequencies inside the continuum gaps.
Abstract: In toroidal plasmas, the toroidal magnetic field is nonuniform over a magnetic surface and causes coupling of different poloidal harmonics. It is shown both analytically and numerically that the toroidicity not only breaks up the shear Alfven continuous spectrum, but also creates new, discrete, toroidicity‐induced shear Alfven eigenmodes with frequencies inside the continuum gaps. Potential applications of the low‐n toroidicity‐induced shear Alfven eigenmodes on plasma heating and instabilities are addressed.
509 citations
TL;DR: In this article, the effects of turbulence on magnetic reconnection were investigated by two-dimensional spectral method magnetohydrodynamic computations and the nonlinear evolution of the periodic sheet pinch configuration was studied as an initial value problem.
Abstract: The effects of turbulence on magnetic reconnection are investigated by two‐dimensional spectral method magnetohydrodynamic computations. The nonlinear evolution of the periodic sheet pinch configuration is studied as an initial value problem. Turbulence is initiated by including a low level of broadband fluctuations in the initial data. Nonlinear features of the evolution, appropriately described as turbulence, are seen early in the solutions and persist throughout the runs. Small‐scale, unsteady coherent electric current and vorticity structures develop in the reconnection zone, resulting in enhanced viscous and resistive dissipation. Unsteady and often spatially asymmetric fluid flow develops. Large‐scale magnetic islands produced by reconnection activity, undergo internal pulsations. Small‐scale magnetic islands, or ‘‘bubbles’’ develop near the reconnection zone, prodcing multiple X points. Large‐amplitude electric field fluctuations, often several times larger than the reconnection electric field, are produced by large island pulsations and by motion of magnetic bubbles. Spectral analysis of the fluctuations show development of broad band excitations, reminiscent of inertial and dissipation range spectra in homogeneous turbulence. Two‐dimensional spectra indicate that the turbulence is broadband in both spatial directions. It is suggested that the turbulence that develops from the randomly perturbed sheet pinchbears a strong resemblence to homogeneous magnetohydrodynamic turbulence, and that analytical theories of reconnection must incorporate these effects.
456 citations
TL;DR: In this article, a theoretical treatment of the stability of miscible displacement in a porous medium is presented, where the base state of uniform velocity and a dispersive concentration profile is time dependent, leading to predictions of the growth rate.
Abstract: A theoretical treatment of the stability of miscible displacement in a porous medium is presented. For a rectilinear displacement process, since the base state of uniform velocity and a dispersive concentration profile is time dependent, we make the quasi‐steady‐state approximation that the base state evolves slowly with respect to the growth of disturbances, leading to predictions of the growth rate. Comparison of results with initial value solutions of the partial differential equations shows that, excluding short times, there is good agreement between the two theories. Comparison of the theory with several experiments in the literature indicates that the theory gives a good prediction of the most dangerous wavelength of unstable fingers. An approximate analysis for transversely anisotropic media has elucidated the role of transverse dispersion in controlling the length scale of fingers.
383 citations
TL;DR: In this article, a strong correlation has been found between the bulk energy per AMU for fast ions measured by charge cups and the x-ray-determined hot electron temperature.
Abstract: Data on the emission of energetic ions produced in laser–matter interactions have been analyzed for a wide variety of laser wavelengths, energies, and pulse lengths. Strong correlation has been found between the bulk energy per AMU for fast ions measured by charge cups and the x‐ray‐determined hot electron temperature. Five theoretical models have been used to explain this correlation. The models include (1) a steady‐state spherically symmetric fluid model with classical electron heat conduction, (2) a steady‐state spherically symmetric fluid model with flux limited electron heat conduction, (3) a simple analytic model of an isothermal rarefaction followed by a free expansion, (4) the lasnex hydrodynamics code [Comments Plasma Phys. Controlled Fusion 2, 85 (1975)], calculations employing a spherical expansion and simple initial conditions, and (5) the lasnex code with its full array of absorption, transport, and emission physics. The results obtained with these models are in good agreement with the experiments and indicate that the detailed shape of the correlation curve between mean fast ion energy and hot electron temperature is due to target surface impurities at the higher temperatures (higher laser intensities) and to the expansion of bulk target material at the lower temperatures (lower laser intensities).
342 citations
TL;DR: In this article, a Langevin model appropriate to constant property turbulent flows is developed from the general equation for the fluid particle velocity increment proposed by Pope in an earlier paper [Phys.fluids 26, 404 (1983)].
Abstract: A Langevin model appropriate to constant property turbulent flows is developed from the general equation for the fluid particle velocity increment proposed by Pope in an earlier paper [Phys. Fluids 26, 404 (1983)]. This model can be viewed as an analogy between the turbulent velocity of a fluid particle and the velocity of a particle undergoing Brownian motion. It is consistent with Kolmogorov’s inertial range scaling, satisfies realizability, and is consistent with second‐order closure models. The objective of the present work is to determine the form of a second‐order tensor appearing in the general model equation as a function of local mean quantities. While the model is not restricted to homogeneous turbulence, the second‐order tensor is evaluated by considering the evolution of the Reynolds stresses in homogeneous flows. A functional form for the tensor is chosen that is linear in the normalized anisotropy tensor and in the mean velocity gradients. The resulting coefficients are evaluated by matching the modeled Reynolds stress evolution to experimental data in homogeneous flows. Constraints are applied to ensure consistency with rapid distortion theory and to satisfy a consistency condition in the limit of two‐dimensional turbulence. A set of coefficients is presented for which the model yields good agreement with available data in homogeneous flows.
341 citations
TL;DR: A kinetic theory for the nonlinear evolution of a magnetic island in a collisionless plasma confined in a toroidal magnetic system is presented in this paper, where an asymptotic analysis of a Grad-Shafranov equation including neoclassical effects such as island bootstrap current defines an equation for the time dependence of the island width.
Abstract: A kinetic theory for the nonlinear evolution of a magnetic island in a collisionless plasma confined in a toroidal magnetic system is presented. An asymptotic analysis of a Grad–Shafranov equation including neoclassical effects such as island bootstrap current defines an equation for the time dependence of the island width. Initially, the island bootstrap current strongly influences the island evolution. As the island surpasses a certain critical width the effect of the island bootstrap current diminishes and the island grows at the Rutherford rate. For current profiles such that Δ’<0 the island bootstrap current saturates the island.
316 citations
TL;DR: In this paper, a study of advection in a two-dimensional Stokes flow slowly modulated in time is presented, showing that chaotic particle motion is readily realizable even for a very low Reynolds number.
Abstract: Chaotic advection can be produced whenever the kinematic equations of motion for passively advected particles give rise to a nonintegrable dynamical system. Although this interpretation of the phenomenon immediately shows that it is possible for flows at any value of Reynolds number, the notion of stochastic particle motion within laminar flows runs counter to common intuition to such a degree that the range of applicability of early model results has been questioned. To dispel lingering doubts of this type a study of advection in a two‐dimensional Stokes flow slowly modulated in time is presented. Even for this very low Reynolds number, manifestly ‘‘laminar’’ flow chaotic particle motion is readily realizable. Standard diagnostics of chaos are computed for various methods of time modulation. Relations to the general ideas of parametric resonance and adiabatic invariance are pointed out.
263 citations
TL;DR: In this paper, a variational principle is developed for computing accurate values of local plasma transport coefficients in nonsymmetric toroidal confinement configurations, and numerical solutions of the linearized drift Fokker-Planck equation are used to obtain the thermodynamic fluxes as functions of collision frequency and the radial electric field.
Abstract: A variational principle is developed for computing accurate values of local plasma transport coefficients in nonsymmetric toroidal confinement configurations. Numerical solutions of the linearized drift Fokker–Planck equation are used to obtain the thermodynamic fluxes as functions of collision frequency and the radial electric field. Effects resulting from the variation of the longitudinal adiabatic invariant J along an orbit (resulting from particle transitions from helically trapped to toroidally trapped orbits) are treated. The velocity‐space distribution resulting from trapped, circulating, and transition particle orbits is well represented by a Legendre polynomial expansion in the pitch angle coordinate. The computational effort is significantly reduced from that required with Monte Carlo methods through use of an efficient treatment of the disparity between the time scales of collisionless and collisional particle dynamics. Numerical computations for a stellarator configuration are presented.
TL;DR: In this paper, two-stream supersonic turbulent mixing layers with a free-stream Mach number of 2.3 on the high-velocity side were experimentally investigated and the spreading rate of the layer is correlated with a velocity ratio of the free streams and a Mach number based on the velocity difference across the layer.
Abstract: Two‐stream supersonic turbulent mixing layers with a free‐stream Mach number of 2.3 on the high‐velocity side are experimentally investigated. A large‐scale vortical structure, which has been believed to dominate the development of incompressible mixing layers, is also observed in the present supersonic layers. The spreading rate of the layer is correlated with a velocity ratio of the free streams and a Mach number based on the velocity difference across the layer.
TL;DR: In this article, a Cartesian tensor expansion of the distribution function of the Fokker-Planck equation was used to find significant departures from standard transport coefficients for the electron current and heat flux in a fully ionized plasma.
Abstract: Significant departures from standard transport coefficients [e.g., Sov. Phys. JETP 6, 338 (1958)] have been found for the electron current and heat flux in a fully ionized plasma. These have been discovered by carrying out a direct and accurate numerical solution of the linearized Fokker–Planck equation using a Cartesian tensor expansion of the distribution function. The results, which were carried out for plasmas of various atomic numbers, show the presence of major inaccuracies (errors of up to 65%) in Braginskii coefficients β■, κ⊥, and κ■ (as conventionally defined) for Hall parameters ωτ in the range 0.3≲ωτ≲30. Surprisingly, α■ and β⊥ are found to depend on τ/(ωτ)2/3 and 1/(ωτ)5/3, and not on τ/(ωτ) and 1/(ωτ)2, respectively, as ωτ→∞. An analytic expansion for large ωτ verifies this result, showing that the relatively cold unmagnetized electrons in the distribution function play a dominant role in the cross‐field transport. The numerical results are fitted to within 15% to a polynomial in ωτ for vari...
TL;DR: In this article, the nonlinear efficiency for a gyrotron oscillator operating at harmonics of the cyclotron frequency has been calculated and is presented as a function of generalized parameters for the second through fifth harmonics.
Abstract: The nonlinear efficiency for a gyrotron oscillator operating at harmonics of the cyclotron frequency has been calculated and is presented as a function of generalized parameters for the second through fifth harmonics. The numerical results are valid for a wide range of operating conditions, including voltage, current, beam radius, cavity dimensions, and operating mode. Relatively high efficiencies are found even at high harmonics; the maximum transverse efficiencies for harmonics 2, 3, 4, and 5 are 0.72, 0.57, 0.45, and 0.36, respectively. The calculation of the efficiency in terms of generalized parameters allows the straightforward design and optimization of harmonic gyrotrons. The influence of the axial profile of the rf field in the gyrotron cavity on the efficiency is also investigated. Improved efficiency can be achieved with asymmetric field profiles. The implications of these results for the generation of millimeter and submillimeter wave radiation by harmonic emission are discussed.
TL;DR: In this paper, a simulation model for steady-state, lower hybrid current drive is described which incorporates a relativistic, one-dimensional Fokker-Planck calculation and a toroidal ray tracing code.
Abstract: A simulation model for steady‐state, lower hybrid current drive is described which incorporates a relativistic, one‐dimensional Fokker–Planck calculation and a toroidal ray tracing code. Two‐dimensional (v⊥) effects are included in the Fokker–Planck analysis in the form of a large perpendicular electron temperature that results from pitch angle scattering. An increase in the parallel refractive index of the lower hybrid waves, arising from toroidal geometry effects, is proposed as a physical mechanism whereby injected rf waves at high phase velocity (ve≪v∥ ≲c) can interact via the Landau resonance with electrons at low phase velocity (v∥ ≲3ve). Numerical results relevant to the Alcator C [Phys. Rev. Lett. 53, 450 (1984)] and PLT [Phys. Rev. Lett. 49, 1255 (1982)] experiments are presented which demonstrate the dependency of the current drive efficiency on various plasma parameters.
TL;DR: In this paper, a redefinition of the Leonard term, the subgrid scale cross term, and the sub-grid scale Reynolds stress is proposed, consistent with a general statement concerning turbulent stresses.
Abstract: A redefinition of the Leonard term, the subgrid scale cross term, and the subgrid scale Reynolds stress is proposed, consistent with a general statement concerning turbulent stresses. The main advantage of the new redefined stresses consists of their term by term Galilean invariance.
TL;DR: In this article, the stability of finite amplitude, two-dimensional solitary waves of permanent form in water of uniform depth with respect to 2D infinitesimal disturbances is investigated and numerically confirmed that the results obtained by Saffman (submitted to J. Fluid Mech.) for the superharmonic instability of periodic waves hold also in the case of solitary waves.
Abstract: The stability of finite amplitude, two‐dimensional solitary waves of permanent form in water of uniform depth with respect to two‐dimensional infinitesimal disturbances is investigated. It is numerically confirmed that the recent analytical results obtained by Saffman (submitted to J. Fluid Mech.) for the ‘‘superharmonic’’ instability of periodic waves hold also in the case of solitary waves.
TL;DR: Differential filters for the large eddy numerical simulation of turbulent flows are defined and their properties are discussed in this article, where the correlations can be expressed exactly in the resolvable scale and the attenuation of the filtered function in the Fourier space can be carefully controlled.
Abstract: Differential filters for the large eddy numerical simulation of turbulent flows are defined and their properties are discussed Their main advantages are that the correlations can be expressed exactly in the so‐called resolvable scale and the attenuation of the filtered function in the Fourier space can be carefully controlled
Abstract: Competing processes previously called ‘‘selective decay’’ and ‘‘dynamic alignment’’ are studied numerically for two‐dimensional magnetohydrodynamic turbulence. In selective decay, the energy decays relatively to mean‐square vector potential, and in dynamic alignment, the energy decays relatively to cross helicity. In the former case, the kinetic (fluid) energy decays to zero and the magnetic energy occupies the largest scales allowed by the boundary conditions. In the latter case, the velocity field and magnetic field become aligned and energetically equipartitioned. An extensive study of the initial value problem, with viscous and resistive dissipation, indicates that four distinct regimes of behavior are possible: a magnetically dominated regime, a velocity dominated regime, a dynamic alignment dominated regime, and a transition regime separating the others in parameter space. An analytical variational problem predicts several features seen in the computations, including geometrical alignment of velocity and magnetic fields, constraints on global quantities in the long‐time limit, and in some cases, condensation to long wavelength modes.
TL;DR: In this paper, an analytic theory of ion-temperature-gradient-driven turbulence in tokamaks is presented, where energyconserving, renormalized spectrum equations are derived and solved in order to obtain the spectra of stationary iontemperature−gradient−driven turbulence.
Abstract: An analytic theory of ion‐temperature‐gradient‐driven turbulence in tokamaks is presented. Energy‐conserving, renormalized spectrum equations are derived and solved in order to obtain the spectra of stationary ion‐temperature‐gradient‐driven turbulence. Corrections to mixing‐length estimates are calculated explicitly. The resulting anomalous ion thermal diffusivity χi=0.4[(π/2)ln(1+ηi)]2[(1+ηi)/τ]2 ρ2pcs/Ls is derived and is found to be consistent with experimentally deduced thermal diffusivities. The associated electron thermal and particle diffusivity, and particle and heat‐pinch velocities are also calculated. The effect of impurity gradients on saturated ion‐temperature‐gradient‐driven turbulence is discussed and a related explanation of density profile steepening during Z‐mode operation is proposed.
TL;DR: In this article, the null collision concept is introduced into the direct simulation Monte Carlo method in the rarefied gas dynamics and it is concluded that the null-collision technique is superior to any other existing techniques in the direct simulating Monte Carlo (DMMC) method.
Abstract: The null‐collision concept is introduced into the direct‐simulation Monte Carlo method in the rarefied gas dynamics. The null‐collision technique overcomes the principle fault in the time‐counter technique and the difficulties in the collision‐frequency technique. The computation time required for the null‐collision technique is comparable to that for the time‐counter technique. Therefore, it is concluded that the null‐collision technique is superior to any other existing techniques in the direct‐simulation Monte Carlo method.
TL;DR: In this article, the effect of viscosity on the capillary instability of a liquid jet is examined and the critical Weber number for convective instability is determined as a function of Reynolds number.
Abstract: The effect of viscosity on the capillary instability of a liquid jet is examined. The critical Weber number for convective instability is determined as a function of Reynolds number and comparison is made with the inviscid limit. It is shown that certain waves that are neutral in the inviscid case exhibit growth for finite Reynolds numbers.
TL;DR: The physical mechanisms for the hydrothermal wave instability found in dynamic thermocapillary liquid layers by Smith and Davis [J. Fluid Mech. 132, 119 (1983)] are described as discussed by the authors.
Abstract: The physical mechanisms for the hydrothermal wave instability found in dynamic thermocapillary liquid layers by Smith and Davis [J. Fluid Mech. 132, 119 (1983)] are described.
TL;DR: In this article, a finite amplitude theory for an ion-acoustic double layer (DL) in a plasma with cold ions and two distinct groups of Boltzmann-Maxwellian distributed hot electrons is presented.
Abstract: A finite amplitude theory for an ion‐acoustic double layer (DL) in a plasma with cold ions and two distinct groups of Boltzmann–Maxwellian distributed hot electrons is presented. Conditions are obtained under which large amplitude stationary double layers can exist. For physical parameters of interest, the DL profiles and the relationship between the maximum DL amplitude and the Mach number are found. A perturbation technique is introduced to derive a modified Korteweg–de Vries (MKdV) equation which governs the dynamics of a weak double layer. It is found that the parameter regimes for the existence of small and finite amplitude DL are not complementary to each other. The relevance of this investigation to space and laboratory plasmas is pointed out.
TL;DR: In this article, it is shown that periodic, axisymmetric oscillations arise spontaneously within the cylindrical sheet emerging from the nozzle and grow with such rapidity along the axial dimension that a sealingoff and encapsulation of the core gas occurs within a few jet diameters.
Abstract: An annular jet flow of liquid surrounding a flow of gas at its core is extremely unstable. Experiments are described in which such a flow is generated by an annular nozzle operated at fairly specific conditions. It is shown that periodic, axisymmetric oscillations arise spontaneously within the cylindrical sheet emerging from the nozzle and grow with such rapidity along the axial dimension that a sealing‐off and encapsulation of the core gas occurs within a few jet diameters. This is closely followed by a pinchoff of the liquid between adjacent bubbles. The liquid shells set free thereby assume spherically symmetric form under capillary forces, and each contains a precisely uniform measure of gas and of liquid on account of the extremely high frequency‐stability of the process. Description is given of the fluid dynamic processes by which the shells are formed, and mention is made of exploiting the instability for the production of rigid shells for technological applications.
TL;DR: The results of direct numerical simulations of chemically reacting, turbulent mixing layers are presented in this article, and the simulation results are shown to be consistent with similarity theory, and are found to be in approximate agreement with laboratory data, even though there are no adjustable parameters in the method.
Abstract: The results of direct numerical simulations of chemically reacting, turbulent mixing layers are presented. The reaction considered is a binary, irreversible reaction with no heat release, so that only the effect of the turbulence on the chemical reaction is investigated. The simulation results are shown to be consistent with similarity theory, and are found to be in approximate agreement with laboratory data, even though there are no adjustable parameters in the method.
TL;DR: In this article, the effect of finite aspect ratio on the impurity contribution to ion thermal conductivity was studied and a simple modification to the pure ion case is obtained with the assumption that the single heavy impurity species is in the Pfirsch-Schluter regime.
Abstract: The effect of finite‐aspect ratio on the impurity contribution to neoclassical ion thermal conductivity is studied A simple modification to the pure‐ion case is obtained with the assumption that the single heavy impurity species is in the Pfirsch–Schluter regime It is found that the impurity contribution is larger than the usual approximation: Zeff times the pure ion thermal conductivity
TL;DR: In this article, linear differential filters, i.e., filters in which the filtered function f and the original function f are connected by a linear differential equation, are studied on a general basis concerning the elliptic operators of second order.
Abstract: Linear differential filters, i.e., filters in which the filtered function f and the original function f are connected by a linear differential equation, are studied on a general basis concerning the elliptic operators of second order. In addition, a particular example of a parabolic filter depending on space and extended to past times is given, and its interest in the context of the large eddy simulation of turbulence is discussed.
TL;DR: In this paper, the transport equations for the Reynolds stresses are closed by modeling the turbulence and mean strain parts of the pressure-strain-rate correlation, and the model constants are determined from simple relationships deduced from measurements in rectilinear and longitudinally curved shear flows.
Abstract: The transport equations for the Reynolds stresses are closed by modeling the turbulence and mean‐strain parts of the pressure‐strain‐rate correlation. The model constants are determined from simple relationships deduced from measurements in rectilinear and longitudinally curved shear flows. It is found that the effects of complex strain fields are more correctly predicted when the influence of the mean‐strain part is reduced from levels indicated by rapid distortion theory, and the turbulence part is adjusted to conform approximately with the measured rates of return to isotropy. The case of the swirling jet is used to illustrate the improved performance of the model.
TL;DR: In this article, the deformation of a hairpin-shaped vortex filament under self-induction and in the presence of shear is studied numerically using the Biot-Savart law.
Abstract: The deformation of a hairpin-shaped vortex filament under self-induction and in the presence of shear is studied numerically using the Biot-Savart law. It is shown that the tip region of an elongated hairpin vortex evolves into a vortex ring and that the presence of mean shear impedes the process. Evolution of a finite-thickness vortex sheet under self-induction is also investigated using the Navier-Stokes equations. The layer evolves into a hairpin vortex which in turn produces a vortex ring of high Reynolds stress content. These results indicate a mechanism for the generation of ring vortices in turbulent shear flows, and a link between the experimental and numerical observation of hairpin vortices and the observation of ring vortices in the outer regions of turbulent boundary layers.