Two-dimensional Wave Number Space Research Articles

Asymptotic analysis of breather modes in a two-dimensional mechanical lattice

We consider a two-dimensional square lattice in which each node is restricted to the plane of the lattice, but is permitted to move in both directions of the lattice. We assume nodes are connected to nearest neighbours along the lattice directions with nonlinear springs, and to diagonal neighbours with linear springs. We consider a generalised Klein–Gordon system, that is, where there is an onsite potential at each node in addition to the (nonlinear) nearest-neighbour interactions. We derive the equations of motion for the displacements from the Hamiltonian. We use asymptotic techniques to derive the form of small amplitude breather solutions, and find necessary conditions required for their existence. We find two types of mode, which we term ‘optical’ and ‘acoustic’, based on the analysis of other lattices which support dispersion relations with multiple branches. In addition to the usual inequality on the sign of the nonlinearity in order for the NLS to be of the focusing type, we obtain an additional ellipticity constraint, that is a restriction in the two-dimensional wavenumber space, required for the spatial differential operator to be elliptic.

Angular distribution of energy spectrum in two-dimensional β-plane turbulence in the long-wave limit

The time-evolution of two-dimensional decaying turbulence governed by the long-wave limit, in which LD/L → 0, of the quasi-geostrophic equation is investigated numerically. Here, LD is the Rossby radius of deformation, and L is the characteristic length scale of the flow. In this system, the ratio of the linear term that originates from the β-term to the nonlinear terms is estimated by a dimensionless number, \documentclass[12pt]{minimal}\begin{document}$\gamma =\beta L_{\rm D}^2/U$\end{document}γ=βLD2/U, where β is the latitudinal gradient of the Coriolis parameter, and U is the characteristic velocity scale. As the value of γ increases, the inverse energy cascade becomes more anisotropic. When γ ⩾ 1, the anisotropy becomes significant and energy accumulates in a wedge-shaped region where \documentclass[12pt]{minimal}\begin{document}$|l|>\sqrt{3}|k|$\end{document}|l|>3|k| in the two-dimensional wavenumber space. Here, k and l are the longitudinal and latitudinal wavenumbers, respectively. When γ is increased further, the energy concentration on the lines of \documentclass[12pt]{minimal}\begin{document}$l=\pm \sqrt{3}k$\end{document}l=±3k is clearly observed. These results are interpreted based on the conservation of zonostrophy, which is an extra invariant other than energy and enstrophy and was determined in a previous study. Considerations concerning the appropriate form of zonostrophy for the long-wave limit and a discussion of the possible relevance to Rossby waves in the ocean are also presented.

Gyrokinetic turbulent transport simulation of a high ion temperature plasma in large helical device experiment

Ion temperature gradient turbulent transport in the large helical device (LHD) is investigated by means of gyrokinetic simulations in comparison with the experimental density fluctuation measurements of ion-scale turbulence. The local gyrokinetic Vlasov simulations are carried out incorporating full geometrical effects of the LHD configuration, and reproduce the turbulent transport levels comparable to the experimental results. Reasonable agreements are also found in the poloidal wavenumber spectra of the density fluctuations obtained from the simulation and the experiment. Numerical analysis of the spectra of the turbulent potential fluctuations on the two-dimensional wavenumber space perpendicular to the magnetic field clarifies the spectral transfer into a high radial wavenumber region which correlates with the regulation of the turbulent transport due to the zonal flows. The resultant transport levels at different flux surfaces are expressed in terms of a simple linear relation between the transport coefficient and the ratio of the squared turbulent potential fluctuation to the averaged zonal flow amplitude.

Physical Mechanism behind Zonal-Flow Generation in Drift-Wave Turbulence

The energetic interaction between drift-wave turbulence and zonal flows is studied experimentally in two-dimensional wave number space. The kinetic energy is found to be transferred nonlocally from the drift waves to the zonal flow. This confirms the theoretical prediction that the parametric-modulational instability is the driving mechanism of zonal flows. The physical mechanism of this nonlocal energetic interaction between and zonal flows and turbulent drift-wave eddies in relation to the suppression of turbulent transport is discussed.

Experimental estimation of the dual cascade in two-dimensional drift-wave turbulence

The dual turbulent cascade is studied experimentally in two-dimensional wavenumber space. A two-field model is used for the analysis and tested on simulated data first. It is found that the energy of the turbulent fluctuations follows an inverse cascade to larger scales while enstrophy is transported in a direct cascade to smaller scales. This provides experimental evidence for the turbulent dual cascade in magnetized plasmas. The analysis also reveals the importance of non-local transfer to the cascading process.

Wave-number-based assessment of the doubly asymptotic approximation. II. Frequency and time domain response

In part I, the surface variational principle (SVP) and the doubly asymptotic approximation (DAA) were used to generate alternative descriptions of the frequency domain wet surface impedance. These are used here to obtain alternative descriptions of the displacement field in slender hemicapped cylindrical shells. This field is represented as a decomposition into a two-dimensional wave-number space. The responses that are obtained are frequency domain transfer functions for a point force at the midplane, evaluated for a discrete set of frequencies. Solutions obtained from first- and second-order versions of DAA are compared to the SVP result. Then temporal responses are obtained by using an inverse fast Fourier transform to evaluate the convolution of the transfer functions with a prototypical temporal excitation. The results reveal that DAA and SVP differ most in the vicinity of fluid-loaded resonances. The largest disagreement is encountered for the axisymmetric portion of the response. It is shown that DAA often greatly underpredicts peak displacements and accelerations, but a few cases lead to overprediction.

Monitoring the surface electronic states of crystals in vapor-phase growth processes under magnetic field

We present a useful technique to monitor the electronic states of the surface atomic layer in magnetic field. The system employs a filament to emit thermal electrons, a grid to control the energy distribution function of incident electrons and a titanium substrate to be investigated. The surface electronic structure of the oxide layer of the titanium substrate was derived by analyzing the transmission current which was measured as a function of magnetic field and electron energy. The results are expressed as contour plots of the quantities related to the transmission probability T in the two-dimensional wave number space; perpendicular and parallel components to the surface. The surface oxidized in air at 200°C for 2 h contained a lot of adsorbed oxygen atoms. The adsrobed oxygen atoms, which showed the non-directional electronic structures, disappeared after the evacuation for two hours at 10 −7 Torr, and then three-dimensional order of surface atoms appeared.

Electronic structures of adsorbed buffer layer on metallic electrodes

We present electronic and structural features of adsorbed nitrogen buffer layers on nickel electrodes covered with a thin surface oxide layer. The electrical structures of adsorbed buffer layer were obtained by a new in situ monitoring technique, which is named as angle-resolved transmission current spectroscopy in magnetic field. The probe is electron beam, whose momentum is controlled by external magnetic field. The measure quantity is transmission probability T of the incident electrons into the surface as functions of electron energy and magnetic field. The results are expressed as contour plots of d T/d k ‖ in two-dimensional wave number space k=(k ‖, k ⊥) . The information on atomic arrangement on the surface was derived at grid potential V g=−1 V for controlling the incident electrons. Significant pressure dependence of the surface electronic structures were derived in 10 −7 Torr< P N2<1×10 −3 Torr. At P<10 −7 Torr, the NiO surface showed a superlattice structure with 2D lattice constant, a=1.894 nm, b=0.936 nm which were 6 times and 3 times of a modified α-NiO structure, whose lattice constant is 6% larger than that of the equilibrium phase. Adsorbed nitrogen atoms formed extra spots in contour plots and a phase transition of surface electronic structure occurred at P N2=6×10 −5 Torr. At P N2=10 −3 Torr, the adsorbed nitrogen molecules showed a clear three-dimensional structure which were related to the sites and energies of adsorbates.

A new technique for monitoring the microscopic electronic surface structures of sputtered thin films

We present a new convenient technique to investigate the microscopic electronic states of condensed atoms or adsorbed atoms on metal substrates in a cylindrical magnetron sputtering system. The substrates are Ni and Ti. The analysis system for surface electronic states is constructed with a tungsten filament for electron emission, a grid for controlling the electron density on the substrate and a cylindrical plate which is used as a metallic substrate. The absorbed electron current to the substrate is measured as functions of the applied voltage (0<40<V) between the filament and the substrate, and the magnetic field (0<100<G) along the axis of the cylinder. By using the free electron model and energy conservation rule, contour plots of electronic states in two-dimensional wave number space are derived. In this experiment, we observed the change of surface potential as a function of incident-electron-density and found some peaks of absorbed current due to the resonant tunneling effect of incident electrons. These peaks strongly depend on the surface conditions of the substrate.

Statistical formulation for an inhomogeneous random water surface: a basis for optical remote sensing of oceans

A statistical formulation for a stationary ocean surface has been fully developed in the angular spectral domain. The angular spectral formulation can be found in the theory of radiometry and optical coherence. The present theory is a rigorous generalization of the conventional framework of ocean-wave statistics, which represent the surface displacement in two-dimensional wave-number space. The present theory is applicable to an inhomogeneous (spatially nonstationary) surface as well as a homogeneous surface. A number of formulas for the statistical functions, such as the cross spectrum, the correlation function for the surface displacement, and the correlation function for the surface slopes, have been derived in connection with the spectrum that characterizes the ocean surface. By taking an adequate spectral model for the surface wave and using the present formalism, it is possible to treat statistically the light emitted from and scattered by the ocean surface, which is characterized by its spectrum. Optical statistical understanding of random ocean surface is essential for optical remote sensing of water and oceans.

Transition in circular Couette flow

Two distinct kinds of transition have been identified in Couette flow between concentric rotating cylinders. The first, which will be called transition by spectral evolution, is characteristic of the motion when the inner cylinder has a larger angular velocity than the outer one. As the speed increases, a succession of secondary modes is excited; the first is the Taylor motion (periodic in the axial direction), and the second is a pattern of travelling waves (periodic in the circumferential direction). Higher modes correspond to harmonics of the two fundamental frequencies of the doubly-periodic flow. This kind of transition may be viewed as a cascade process in which energy is transferred by non-linear interactions through a discrete spectrum to progressively higher frequencies in a two-dimensional wave-number space. At sufficiently large Reynolds numbers the discrete spectrum changes gradually and reversibly to a continuous one by broadening of the initially sharp spectral lines.These periodic flows are not uniquely determined by the Reynolds number. For the case of the inner cylinder rotating and the outer cylinder at rest, as many as 20 or 25 different states (each state being defined by the number of Taylor cells and the number of tangential waves) have been observed at a given speed. As the speed changes, theso states replace each other in a repeatable but irreversible pattern of transitions; vortices appear or disappear in pairs, and waves are added or subtracted. More than 70 such transitions have been found in the speed range up to about 10 times the first critical speed. Regardless of the state, however, the angular velocity of the tangential waves is nearly constant at 0.34 times the angular velocity of the inner cylinder.The second kind of transition, which will be called catastrophic transition, is characteristic of the motion when the outer cylinder has a larger angular velocity than the inner one. At a fixed Reynolds number, the fluid is divided into distinct regions of laminar and turbulent flow, and these regions are separated by interfacial surfaces which may be propagating in either direction. Under some conditions the turbulent regions may appear and disappear in a random way; under other conditions they may form quite regular patterns. One common pattern of particular interest is a spiral band of turbulence which rotates at very nearly the mean angular velocity of the two walls without any change in shape except possibly an occasional shift from a right-hand to a left-hand pattern. One example of this spiral turbulence is being studied in some detail in an attempt to clarify the role played in transition by interfaces and intermittency.