Inverse Length Scale Research Articles

Metal-insulator transition in an Hubbard-like model with degeneracy in one dimension.

We consider the SU(N) generalization of the one-dimensional Hubbard model with arbitrary degeneracy N (spin and orbital degrees of freedom). This model is integrable and has several unusual properties at low temperatures. The Bethe-Ansatz equations at T=0 are analyzed in the thermodynamic limit in the absence of external fields. In the continuum limit, the effective interaction between the charge degrees of freedom corresponds to a potential of the form [sinh(ax)${]}^{\mathrm{\ensuremath{-}}2}$, where x is the distance between the particles involved and a is an inverse length scale. In the limit N\ensuremath{\rightarrow}\ensuremath{\infty} and in the continuum limit, the charges reduce to a Bose gas interacting via a \ensuremath{\delta}-function potential. We further address here the properties of the charge degrees of freedom for a band filling close to one electron per site. The charge excitations obey Fermi statistics. We find a Mott metal-insulator transition at a critical value ${\mathit{U}}_{\mathit{c}}$ of the Coulomb repulsion. ${\mathit{U}}_{\mathit{c}}$ depends on N (${\mathit{U}}_{\mathit{c}}$=0 for N=2). A qualitative change in the charge-rapidity distribution is found at ${\mathit{U}}_{\mathit{c}}$. The Fermi velocity is finite for U${\mathit{U}}_{\mathit{c}}$, diverges as U\ensuremath{\rightarrow}${\mathit{U}}_{\mathit{c}}$, and vanishes for Ug${\mathit{U}}_{\mathit{c}}$.

Theory of weak shock-like structures

New analytic solutions of weak shock-like structures are presented. One of them slows down as its amplitude and inverse scale length increase, as opposed to the ordinary shock wave and solitons which speed up. Within the assumption of “analyticity” and continuity of entire distribution functions, the amplitudes (ψ +, ψ −) and the inverse scale lengths ( K +, K −) of the non-oscillatory part and the oscillatory part are related by ψ − < ψ + ⩽ 3ψ − and K + ⩾ K −, where equality holds for the lowest order shock-like solutions.

Wave breaking in a compressible atmosphere

Abstract An inviscid, adiabatic, two-dimensional, steady-state model of finite-amplitude gravity waves produced by flow over a ridge is considered. A conservation principle, for motions ihat produce relatively small deviations from a basic adiabatic density distribution, is derived. This principle represents a compressible flow modification of Long's (1953) result for incompressible flow. A nonlinear differential equation for the displacement 6 of a streamline from its undisturbed level is derived under the limitations of the long-wave approximation and uniform upstream flow. The relative importance of compressibility in the present model is measured by the parameter e=(g/c 2)l −1∼10−1, where g is the acceleration of gravity, c is the adiabatic sound speed and l is a characteristic inverse length scale represented by the ratio of the Brunt-Vaisala frequency to the amplitude of the upstream flow. A solution of the differential equation is derived that preserves the first-order (in e) effects of compressibi...

Steady Infiltration From Large Shallow Ponds

Exact quasi‐linear solutions are obtained for steady infiltration from a shallow half plane pond on the surface of an isotropic semi‐infinite porous medium. Three particular boundary value problems are considered, namely, constant Dirichlet boundary conditions in relative permeability and vertical flux, and a mixed boundary value problem. The boundary conditions on the surface of the porous media for these problems correspond physically, first, to fully saturated conditions under the pond and fully unsaturated conditions elsewhere; second, to a constant vertical flux under the pond and zero vertical flux elsewhere; and third, to fully saturated conditions under the pond and zero vertical flux elsewhere. Each problem has essentially the same diffusive boundary layer, or capillary fringe, connecting the fully saturated and unsaturated regions. The total flux per unit length of boundary due to capillarity in the first problem is infinite, and for the latter two problems, finite constants. Strong analogies between our results for a surface source and the corresponding results of Waechter and Philip for buried spherical and cylindrical sources are established. Finally, the exact results for the half plane pond are used in an approximate description of steady infiltration from a large shallow pond of arbitrary shape, subject to the minimum radius of curvature of the pond's perimeter being large relative to 2/α, where α is the inverse length scale introduced in quasi‐linear models. A boundary layer develops from capillarity at the pond's edge, away from which the solution is adequately described by a diffusion equation representing a balance between the vertical flow from gravity and the horizontal flow from capillarity. An approximate expression is obtained analytically for the total flow from large shallow ponds.

Reflection of Hydrostatic Gravity Waves in a Stratified Shear Flow. Part I: Theory

Continuous partial reflection of linear hydrostatic gravity waves that propagate through a stratified shear flow is examined. The complex reflection coefficient R satisfies a Riccati equation, which is a first-order nonlinear differential equation. It is shown that |R|<1 since critical levels and overreflection are not considered. In this case the conservation of wave action flux may be expressed as a relationship between |R| and El−1, where E is the wave energy and l a characteristic inverse vertical length scale of the background state. It is demonstrated that R for a layered model represents a limiting solution of the Riccati equation. A general solution is also derived, under the assumption that the characteristic woe l is directly proportional to the inverse scale height of the characteristic impedance associated with a stratified shear flow. It is shown that the vanishing of |R| at a specific level is analogous to the vanishing of |R| in a three layer model, when the characteristic impedances in the top and the bottom layers satisfy a matching condition. Finally, various properties of the reflection coefficient are displayed for a particular background state. The extension of the theory to encompass other types of wave motion is indicated.

A collisional drift wave description of plasma edge turbulence

Model mode-coupling equations for the resistive drift wave instability are numerically solved for realistic parameters found in tokamak edge plasmas. The Bohm diffusion is found to result if the parallel wavenumber is chosen to maximize the growth rate for a given value of the perpendicular wavenumber. The saturated turbulence energy has a broad frequency spectrum with a large fluctuation level proportional to κ̄ (=ρs/Ln, the normalized inverse scale length of the density gradient) and a wavenumber spectrum of the two-dimensional Kolmogorov–Kraichnan type, ∼k−3.

The velocity evolution of galaxy clustering

We have examined the changing velocity distribution of galaxies as they cluster in computer models of the expanding universe. The models are 4000-body numerical simulations of galaxies with a large range of masses interacting gravitationally. Clustering in velocity space is measured by calculating the residual peculiar velocities around the Hubble expansion. These form ''Hubble streaks as clustering progresses. We distinguish isolated field galaxies from clustered galaxies. In contrast to the usual belief, the velocity dispersion of the most extreme field galaxies does not decrease adiabatically. Rather, it is dominated by the perturbations of distant large clusters as they form and it decreases much more slowly than the inverse expansion length scale, R/sup -1/. The velocity dispersion of extreme field galaxies is a good cosmological indicator of ..cap omega.. = rho/rho/sub crit/. Preliminary comparison of several simulations with observtions shows that our universe agrees better with low density models, ..cap omega..< or =0.1. The velocity dispersion of cluster centers of mass is a good cosmological marker as well. We also suggest another new method for estimating ..cap omega.., based on the history of extreme field galaxies.

Self-Consistent Stationary Density Cavities with Many Bounded Langmuir Waves

Stationary one-dimensional propagation of Langmuir waves trapped in a local density cavity in a plasma is investigated for the case in which the frequency spectrum of the Langmuir waves is broad as compared with the ion acoustic frequency, but in which their wavenumbers are comparable to the inverse scale length of the density cavity. Analytical solutions for the density cavity, which traps many Langmuir wave modes and which is self-consistent with the ponder-omotive potential coming from these eigenmodes, are obtained. The Langmuir wave envelopes exhibit oscillations at the beat frequencies of the different eigenmodes. These solutions are valid when the intensity of the trapped Langmuir waves normalized by the electron thermal energy is much greater than the electron-to-ion mass ratio.

A hierarchy of length scales for weak solutions of the three-dimensional Navier-Stokes equations

Moments of the vorticity are used to define and estimate a hierarchy of time-averaged inverse length scales for weak solutions of the three-dimensional, incompressible Navier-Stokes equations on a periodic box. The estimate for the smallest of these inverse scales coincides with the inverse Kolmogorov length but thereafter the exponents of the Reynolds number rise rapidly for correspondingly higher moments. The implications of these results for the computational resolution of small scale vortical structures are discussed.