Engineering topics

Montgomery, D.

Publications and source records attributed to Montgomery, D..

At least 55 records · Page 3

NASA NTRS1979

Most probable states in magnetohydrodynamics

A method is developed for selecting 'most probable' MHD states compatible with certain global constraints (e.g., the energy, the toroidal magnetic flux, the total toroidal electric current, or the magnetic helicity). These states occur as the solutions of nonlinear partial differential equations which are far simpler than the original MHD equations. The stabilized Z pinch is discussed by way of illustration of the theory of probable states.

Montgomery, D.↗

NASA NTRS1979

Possible inverse cascade behavior for drift-wave turbulence

The turbulent spectral properties of the dynamical equation of Hasegawa and Mima (1978) governing the evolution of the electrostatic potential in drift-wave turbulence is investigated for two formulations of the problem: (1) as a nondissipative initial value problem, with the potential represented by a truncated Fourier series with large number of terms, and (2) as a dissipative problem with a small viscous dissipation at very short spatial scales, and a long wavelength forcing term at longer wavelengths. It is found that Hasegawa and Mima's prediction for the nondissipative, truncated initial value modal problem is accurate, but substantial differences exist for the forced dissipative case between computer results and analytical predictions based on a wave kinetic equation of Kadomtsev. Much better agreement is found with a simple dual-cascade model based on Kraichnan's generalization of Kolmogorov's cascade arguments.

Fyfe, D.↗

NASA NTRS1979

Two-dimensional magnetohydrodynamic turbulence - Cylindrical, non-dissipative model

Incompressible magnetohydrodynamic turbulence is treated in the presence of cylindrical boundaries which are perfectly conducting and rigidly smooth. The model treated is non-dissipative and two-dimensional, the variation of all quantities in the axial direction being ignored. Equilibrium Gibbs ensemble predictions are explored assuming the constraint of constant axial current (appropriate to tokamak operation). No small-amplitude approximations are made. The expectation value of the turbulent kinetic energy is found to approach zero for the state of maximum mean-square vector potential to energy ratio. These are the only states for which large velocity fluctuations are not expected.

Montgomery, D.↗

NASA NTRS1978

Three-dimensional magnetohydrodynamic turbulence in cylindrical geometry

A theory is developed in which a perfectly conducting cylindrical wall is used to describe incompressible MHD turbulence. Dynamic equations of incompressible MHD and its associated fields are expanded in a system of vector eigenfunctions of the curl. The total energy, magnetic helicity, and cross helicity have simple quadratic expressions in the expansion coefficients. The invariants are temporally constant for a truncation at a large, but finite, number of coefficients. The invariants are used to construct canonical distributions. It is noted that for all initial conditions, including quiescent ones, the mean-square velocity fields have finite values. The stability problem is transformed into a search for values of the integral variants which minimize the mean-square velocity fields. This leads to an extremal helicity principle.

Montgomery, D.↗

NASA NTRS1978

Statistical formulation of one-dimensional electron fluid turbulence

A one-dimensional electron fluid model is investigated using the mathematical methods of modern fluid turbulence theory. Nondissipative equilibrium canonical distributions are determined in a phase space whose coordinates are the real and imaginary parts of the Fourier coefficients for the field variables. Spectral densities are calculated, yielding a wavenumber electric field energy spectrum proportional to 1/k squared for large wavenumbers. The equations of motion are numerically integrated and the resulting spectra are found to compare well with the theoretical predictions.

Fyfe, D.↗

NASA NTRS1977

Implications of Navier-Stokes turbulence theory for plasma turbulence

The methodology of Navier-Stokes fluid turbulence theory is reviewed, with emphasis placed on the relevance of the Navier-Stokes concepts for understanding plasma turbulence. After a brief consideration of the three-dimensional case, two-dimensional problems are discussed. In addition, MHD turbulence and turbulence in Vlasov plasmas are treated. In particular, the direct interaction approximation developed by Kraichnan (1959) is generalized from Navier-Stokes turbulence theory to produce a computable set of differentio-integral equations for a Vlasov plasma.

Montgomery, D.↗

NASA NTRS1977

A statistical formulation of one-dimensional electron fluid turbulence

A one-dimensional electron fluid model is investigated using the mathematical methods of modern fluid turbulence theory. Non-dissipative equilibrium canonical distributions are determined in a phase space whose co-ordinates are the real and imaginary parts of the Fourier coefficients for the field variables. Spectral densities are calculated, yielding a wavenumber electric field energy spectrum proportional to k to the negative second power for large wavenumbers. The equations of motion are numerically integrated and the resulting spectra are found to compare well with the theoretical predictions.

Fyfe, D.↗

NASA NTRS1977

Dissipative, forced turbulence in two-dimensional magnetohydrodynamics

The equations of motion of a turbulent two-dimensional MHD flow are solved in the presence of finite viscosity and resistivity for the case when external mechanical and/or magnetic forces act on the fluid, the goal of the study being to verify the existence of a MHD dynamo effect which is represented by a substantial back-transfer of mean square vector potential to the longest allowed Fourier wavelengths. The regime explored is that for which the mechanical and magnetic Reynolds numbers are in the range 100 to 1000. It is concluded that mechanical forcing terms alone cannot lead to dynamo action, but that dynamo action can result from either magnetic forcing terms or from both mechanical and magnetic forcing terms simultaneously.

Fyfe, D.↗

NASA NTRS1977

Magnetic dynamo action in two-dimensional turbulent magneto-hydrodynamics

Two-dimensional magnetohydrodynamic turbulence is explored by means of numerical simulation. Previous analytical theory, based on non-dissipative constants of the motion in a truncated Fourier representation, is verified by following the evolution of highly non-equilibrium initial conditions numerically. Dynamo action (conversion of a significant fraction of turbulent kinetic energy into long-wavelength magnetic field energy) is observed. It is conjectured that in the presence of dissipation and external forcing, a dual cascade will be observed for zero-helicity situations. Energy will cascade to higher wavenumbers simultaneously with a cascade of mean square vector potential to lower wavenumbers, leading to an omni-directional magnetic energy spectrum.

Fyfe, D.↗

NASA NTRS1977

Turbulent diffusion from a quasi-kinematical point of view

An estimate for the coefficient of self-diffusion can be derived whenever the Eulerian velocity spectrum is known. The result is more general than those previously obtained. A comparison is made with computed test-particle diffusion in an inviscid two-dimensional Navier-Stokes fluid.

Salu, Y.↗

NASA NTRS1977

Implications of Navier-Stokes turbulence theory for plasma turbulence

Plasma turbulence is considered within the framework of the Navier-Stokes turbulence theory. Two-dimensional turbulence is discussed, noting inverse cascades, and compared to three-dimensional turbulence. MHD turbulence is described with reference to applications of the Navier-Stokes theory and the possibility of inverse magnetic cascades. Turbulence in Vlasov plasmas is outlined on the basis of the direct interaction approximation developed for Navier-Stokes fluids (Kraichnan, 1958-1959).

Montgomery, D.↗

NASA NTRS1976

High-beta turbulence in two-dimensional magnetohydrodynamics

Equations of ideal magnetohydrodynamics are used to study incompressible turbulent flows in a specified geometry where all the field quantities vary with only two spatial dimensions. The procedures adopted are basically those of Kraichnan (1967), in which classical equilibrium ensembles are built around constants of the motion identified from the Fourier-transformed equations of motion. Once the constants of the motion are identified, the statistical formulation of the problem is presented in a phase space whose coordinates are the real and imaginary parts of the Fourier coefficients. Canonical ensembles are constructed in this phase space by classical arguments. The theory developed permits isolation of some qualitatively new gross physical effects which have so far not been calculated. One of the more novel physical effects is the appearance of macroscopic structures involving long-wavelength, self-generated, magnetic fields for a wide range of initial parameters.

Fyfe, D.↗

NASA NTRS1976

Magnetic dynamo action in two-dimensional turbulent magneto-hydrodynamics

Two-dimensional magnetohydrodynamic turbulence is explored by means of numerical simulation. Previous analytical theory, based on non-dissipative constants of the motion in a truncated Fourier representation, is verified by following the evolution of highly non-equilibrium initial conditions numerically. Dynamo action (conversion of a significant fraction of turbulent kinetic energy into long-wavelength magnetic field energy) is observed. It is conjectured that in the presence of dissipation and external forcing, a dual cascade will be observed for zero-helicity situations. Energy will cascade to higher wave numbers simultaneously with a cascade of mean square vector potential to lower wave numbers, leading to an omni-directional magnetic energy spectrum which varies as 1/k 3 at lower wave numbers, simultaneously with a buildup of magnetic excitation at the lowest wave number of the system. Equipartition of kinetic and magnetic energies is expected at the highest wave numbers in the system.

Fyfe, D.↗

NASA NTRS1976

A BBGKY framework for fluid turbulence

A framework is presented for a systematic kinetic theory of turbulence originating from the Liouville equation for the Fourier coefficients of fluid variables. The real and imaginary parts of these Fourier coefficients play the role that particle coordinates (positions and momenta) play in the BBGKY theory. The basic relations of the problem are the incompressible Navier-Stokes equations in two dimensions with zero viscosity, with the probability distributions of Fourier coefficients rather than moments being the basic variables of the theory. A kinetic equation is derived and shown to possess a number of requirements that any reasonable kinetic equation must have: conservation laws, positive-definite spectral densities, and an H-theorem. The major lack in the theory is any reliable information on the relaxation predicted by the complicated linear operator H. Closure of the hierarchy is achieved by the hypothesis that the five-coefficient correlation function is negligible. Problems associated with inclusion of viscosity and external driving forces are discussed.

Montgomery, D.↗

NASA NTRS1976

Plasma kinetic processes in a strong d.c. magnetic field

Recent results in the kinetic theory of a strongly magnetized plasma are surveyed. Emphasis is on the electrostatic guiding-center plasma in two dimensions, in both the fluid and 'charged rod' descriptions. The basic kinetic description of the plasma is in terms of the statistically-distributed Fourier coefficients associated with the velocity and 'enstrophy' (charge density) fields. It is a universal tendency in such media for enstrophy to flow to shorter wavelengths but for energy to flow to longer wavelengths. A consequence of the energy flow to longer wavelengths is the generation of long-range order in the form of macroscopic vortices. These kinds of structure have been called 'convection cells' and can be extraordinarily efficient in transporting particles transverse to a magnetic field. The tendency to vortex formation can be disrupted by collisions between particles. Modifications of the Fokker-Planck equation for a plasma produced by a strong dc magnetic field are considered in both two and three dimensions.

Montgomery, D.↗

NASA NTRS1976

Dissipative, forced turbulence in two-dimensional magnetohydrodynamics

The equations of motion for turbulent two-dimensional magnetohydrodynamic flows are solved in the presence of finite viscosity and resistivity, for the case in which external forces (mechanical and/or magnetic) act on the fluid. The goal is to verify the existence of a magnetohydrodynamic dynamo effect which is represented mathematically by a substantial back-transfer of mean square vector potential to the longest allowed Fourier wavelengths. External forces consisting of a random part plus a fraction of the value at the previous time step are employed, after the manner of Lilly for the Navier-Stokes case. The regime explored is that for which the mechanical and magnetic Reynolds numbers are in the region of 100 to 1000. The conclusions are that mechanical forcing terms alone cannot lead to dynamo action, but that dynamo action can result from either magnetic forcing terms or from both mechanical and magnetic forcing terms simultaneously.

Fyfe, D.↗

NASA NTRS1976

Plasma kinetic processes in a strong d.c. magnetic field

The electrostatic guiding center approximation is used to describe the kinetic processes of a strongly magnetized tenuous plasma in two dimensions, in both the fluid and charged rod descriptions. The basic kinetic description of the plasma is in terms of the statistically-distributed Fourier coefficients associated with the velocity and enstrophy (charge density) fields. It is shown that the system exhibits a tendency to long range order characterized by macroscopic vorticity concentrations comparable in size to that of the system. These types of structure have historically been called convection cells and can be quite efficient at transporting particles transverse to a magnetic field.

Montgomery, D.↗

NASA NTRS1975

High-beta turbulence in two-dimensional magnetohydrodynamics

Incompressible turbulent flows were investigated in the framework of ideal magnetohydrodynamics. Equilibrium canonical distributions are determined in a phase whose coordinates are the real and imaginary parts of the Fourier coefficients for the field variables. The magnetic field and fluid velocity have variable x and y components, and all field quantities are independent of z. Three constants of the motion are found which survive the truncation in Fourier space and permit the construction of canonical distributions with three independent temperatures. Spectral densities are calculated. One of the more novel physical effects is the appearance of macroscopic structures involving long wavelength, self-generated, magnetic fields ("magnetic islands"). In the presence of finite dissipation, energy cascades to higher wave numbers can be accompanied by vector potential cascades to lower wave numbers, in much the same way that in the fluid dynamic case, energy cascades to lower wave numbers accompany entropy cascades to higher wave numbers.

Fyfe, D.↗