Non-zero Wavenumber Research Articles

Field line resonances in a cylindrical plasma

An experimental study of the response to an impulsive driver in a low beta helium plasma is presented. Field line resonance (FLR) spectra are recorded and compared to an ideal magnetohydrodynamic (MHD) theory with finite ion cyclotron frequency corrections. The agreement between observed and predicted values is generally within the frequency resolution of the measurements except for the lowest frequency harmonic. The spectrum of the lowest harmonic is compared to various ultra low-frequency (ULF) FLR dispersion relations. Dispersion effects due to nonzero perpendicular wave number are found to be important for the lowest frequency harmonic. Quality factors are measured and compared to theoretical estimates from a two-fluid theory with finite parallel electron temperature. Reflection coefficient values are obtained using measured and estimated Q values.

Low-Frequency Dispersion-Function Factorization and Classification of P-SV Modes by Wavenumber Limits

The dispersion function for P-SV wave propagation in a laterally homogeneous fluid-solid medium is a characteristic function for a multi-point boundary-value problem, and it can be expressed as a product of compound-matrix propagators. It is studied asymptotically when the frequency tends to zero, as a function of the horizontal wavenumber. An entire function is obtained, whose zeros determine the limits as the frequency tends to zero for the horizontal wavenumbers of the wave-guide modes. The entire function can be factorized with scalar factors, which can be evaluated by solution of ordinary differential equation (ODE) systems. The order of its zero at the origin determines the number of modes with vanishing low-frequency wavenumber limits. Interface waves and different types of bending waves are particular examples of such modes. In addition, there will be one factor from each fluid or solid region. Hence, the modes with nonzero wavenumber limits decouple into region-dependent classes at low frequency, and they can be classified according to low-frequency connection to the different regions. The decoupling is elucidated by an asymptotic analysis of the mode shapes.

Linear Stability of Thermal Convection in Rotating Systems with Fixed Heat Flux Boundaries

Linear stability of rotating thermal convection in a horizontal layer of Boussinesq fluid under the fixed heat flux boundary condition is examined by the use of a vertically truncated system up to wavenumber one. When the rotation axis is in the vertical direction, the asymptotic behavior of the critical convection for large rotation rates is almost the same as that under the fixed temperature boundary condition. However, when the rotation axis is horizontal and the lateral boundaries are inclined, the mode with zero horizontal wavenumber remains as the critical mode regardless of the rotation rate. The neutral curve has another local minimum at a nonzero horizontal wavenumber, whose asymptotic behavior coincides with the critical mode under the fixed temperature condition. The difference of the critical horizontal wavenumber between those two geometries is qualitatively understood by the difference of wave characteristics; inertial waves and Rossby waves, respectively.

Electron States in Quasi-One-Dimensional Structures; Azimuthal Applied Magnetic Field

The energy spectrum of a single electron confined in a cylindrical quantum wire is determined within the effective-mass approximation as a function of the azimuthal applied magnetic field. The magnetic field varies linearly with the radial distance across the wire thickness. Unlike in the case of a cylinder in a parallel magnetic field, the azimuthally directed magnetic field preserves the double degeneracy of the non-zero azithumal quantum number (m) states. In this case, the azimuthal magnetic field instead lifts the double degeneracy of the non-zero axial wavenumber (kz) states. A further analysis is the evaluations of the oscillator strengths for optical transitions involving the lowest-order energy subbands of the electron, within the dipole approximation. The radiation field with clear-cut selection rules for optical transitions is that of circularly polarized, incident along the axis of the cylinder. It emerges that the triple degeneracy of the oscillator strengths, when kz ≠ 0, is lifted by the magnetic field. In terms of the arising branches for the oscillator strengths, the transitions for which kz < 0 are enhanced the most by the azimuthal applied magnetic field.

Feeding and dissipative waves in fracture and phase transition. III. Triangular-cell lattice

Wave configurations for modes I and II of crack propagation in an elastic triangular-cell lattice are studied. [Mode III was considered in Part I of the paper: Slepyan, L.I. Feeding and dissipative waves in fracture and phase transition. I. Some 1D structures and a square-cell lattice. J. Mech. Phys. Solids 49 (2001) 469.] A general solution incorporates a complete set of the feeding and dissipative waves. The solution is based on the wave dispersion dependences obtained in an explicit form. Also some general properties and the long-wave asymptotes of the corresponding Green function are found. This results in the determination of the wavenumbers and modes. The macrolevel-associated solutions exist as the sub-Rayleigh crack speed regime for both modes and as a shear-longitudinal wave-speed intersonic regime for mode II only. In particular, it is shown that any intersonic crack speed is possible, whereas only the speed (shear wave speed multiplied by 2 ) corresponds to a positive energy release in the cohesive-zone-free homogeneous-material model. This is a manifestation of the fact that the local energy release in the lattice is not connected with the singularity of the macrolevel field. Microlevel solutions, corresponding to a nonzero feeding wavenumber, exist for both modes, at least from the energy point of view, for any, sub- and super-Rayleigh, intersonic and supersonic crack speed regimes. In particular, in the super-Rayleigh regime, a high-frequency wave delivers energy to the crack, while the macrolevel wave carries energy away from the crack.

Examination of large-scale structures in a turbulent plane mixing layer. Part 2. Dynamical systems model

The temporal dynamics of large-scale structures in a plane turbulent mixing layer are studied through the development of a low-order dynamical system of ordinary differential equations (ODEs). This model is derived by projecting Navier–Stokes equations onto an empirical basis set from the proper orthogonal decomposition (POD) using a Galerkin method. To obtain this low-dimensional set of equations, a truncation is performed that only includes the first POD mode for selected streamwise/spanwise (k1/k3) modes. The initial truncations are for k3 = 0; however, once these truncations are evaluated, non-zero spanwise wavenumbers are added. These truncated systems of equations are then examined in the pseudo-Fourier space in which they are solved and by reconstructing the velocity field. Two different methods for closing the mean streamwise velocity are evaluated that show the importance of introducing, into the low-order dynamical system, a term allowing feedback between the turbulent and mean flows. The results of the numerical simulations show a strongly periodic flow indicative of the spanwise vorticity. The simulated flow had the correct energy distributions in the cross-stream direction. These models also indicated that the events associated with the centre of the mixing layer lead the temporal dynamics. For truncations involving both spanwise and streamwise wavenumbers, the reconstructed velocity field exhibits the main spanwise and streamwise vortical structures known to exist in this flow. The streamwise aligned vorticity is shown to connect spanwise vortex tubes.

Signature of turbulent zonal flows observed in the DIII-D tokamak.

The spectrum of turbulent density fluctuations at long poloidal wavelengths in the edge plasma of the DIII-D tokamak peaks at nonzero radial wave number. The associated electric-potential fluctuations cause sheared E x B flows primarily in the poloidal direction. These zonal flows have been predicted by theory and are believed to regulate the overall level of turbulence and anomalous transport. This study provides the first indirect experimental identification of zonal flows.

Feeding and dissipative waves in fracture and phase transition: II. Phase-transition waves

Discrete and homogeneous models of a structured material are considered to resolve difficulties in the analysis of dynamic phase transition. The discrete model is a chain consisting of particles connected by massless bonds, while the continuous model is represented by a partial differential equation with higher than the second order of coordinate derivatives. The macrolevel constitutive law is represented by a bi-linear stress–strain relation, such that the transition from the first, stiffer phase to the second one is irreversible. Solutions of two types, macrolevel-associated and microlevel, are derived. The first type of solution is characterized by a macrolevel feeding wave (the wave delivering energy to the phase-transition front is of a zero wave number), while the microlevel solutions correspond to a nonzero feeding wave number. Subsonic, intersonic and supersonic phase-transition waves are described. For the homogeneous model it is shown that the contradiction between the limiting stress and energy criteria, inherent for the macrolevel formulation of the problem, is eliminated if and only if the phase transition does not concern the highest-order modulus. Total structure- and speed-dependent dissipation, as the energy carried by microlevel waves away from the phase-transition front, as well as parameters of each dissipative wave are determined. For the fourth-order partial differential equation, the existence of the Maxwell type, dissipation-free, subsonic phase-transition wave is shown. In this case, the microstructure plays the role of a catalyst. Common and distinctive properties of the discrete and homogeneous models are discussed.

Feeding and dissipative waves in fracture and phase transition: I. Some 1D structures and a square-cell lattice

In the lattice structure considered here, crack propagation is caused by feeding waves, carrying energy to the crack front, and accompanied by dissipative waves carrying a part of this energy away from the front (the difference is spent on the bond disintegration). The feeding waves differ by their wavenumber. A zero feeding wavenumber corresponds to a macrolevel-associated solution with the classical homogeneous-material solution as its long-wave approximation. A non-zero wavenumber corresponds to a genuine microlevel solution which has no analogue on the macrolevel. In the latter case, on the crack surfaces and their continuation, the feeding wave is located behind (ahead) the crack front if its group velocity is greater (less) than the phase velocity. Dissipative waves, which appear in both macrolevel-associated and microlevel solutions, are located in accordance with the opposite rule. (Wave dispersion is the underlying phenomenon which allows such a wave configuration to exist.) In contrast to a homogeneous material model, both these solutions permit supersonic crack propagation. Such feeding and dissipative waves and other lattice phenomena are characteristic of dynamic phase transformation as well. In the present paper, mode III crack propagation in a square-cell elastic lattice is studied. Along with the lattice model, some simplified one-dimensional structures are considered allowing one to retrace qualitatively (with no technical difficulties) the main lattice phenomena.

Nonlinear Growth of the Three‐dimensional Undular Instability of a Horizontal Magnetic Layer and the Formation of Arching Flux Tubes

We use an anelastic MHD code to simulate the nonlinear evolution of the three-dimensional undular instability of a horizontal magnetic layer with a fixed field line direction, embedded in an adiabatically stratified atmosphere. We consider the limit of very high plasma β, representing the condition at the base of the solar convection zone. We show that, in the limit of high plasma β and nearly adiabatic stratification, the anelastic formulation gives an accurate description of the magnetic buoyancy instabilities. We specify the thermodynamic conditions of the magnetic layer such that it is stable against pure interchange modes (with zero wavenumber in the direction of the magnetic field) and is unstable only to three-dimensional undular modes (with nonzero wavenumbers in both horizontal directions parallel and perpendicular to the field). Our simulations show that distinct arching flux tubes form as a result of the growth of the three-dimensional undular instability. The apices of the arching tubes become increasingly buoyant because of the diverging mass flow from the apices to the troughs. The field strength at each loop apex decreases with height at a significantly smaller rate in comparison with that for the rise of a horizontal flux tube, because of the stretching of the loop field lines. Even though the initial magnetic field is untwisted, it is found that the upward moving tube cross sections of the arching tubes maintain their cohesion as they rise through the distance of about 1 density scale height included in the simulation domain. The difference in motion between the apices and the troughs causes bending and braiding of the longitudinal field lines, whose restoring tension force improves the cohesion of the rising flux tubes in comparison with previous two-dimensional simulations of the buoyant rise of horizontal flux tubes with no initial twist. In addition, the fact that both the buoyancy and the tension forces grow self-consistently from zero as the tubes arch is also a crucial factor for the cohesion of the rising tubes. The result of our simulations suggests that the minimum value for the ratio of poloidal field strength over toroidal field strength (i.e., twist) at the base of the solar convection zone, necessary to ensure a cohesive rise of magnetic flux through the solar convection zone, may be far less than that suggested by the two-dimensional calculations of the buoyant rise of infinitely long horizontal tubes.

FDTD algorithm for computing the off-plane band structure in a two-dimensional photonic crystal with dielectric or metallic inclusions

An effective numerical method based on the finite-difference time-domain scheme for computing the off-plane band structure of a two-dimensional photonic crystal is presented. The method is an order N method, and requires only a two-dimensional discretization mesh for a given off-plane wave number k z although the off-plane propagation is a three-dimensional problem. The computation time and memory required is thus reduced significantly. The present method can be used for any type of inclusions and no additional effort is needed for metallic inclusions. The off-plane band structures of a square lattice of metallic rods in the air are studied, and a complete bandgap for some nonzero off-plane wave number k z is found.

Early Stage Process of Pattern Formation Phenomena

Pattern formation phenomena have been observed universally in various systems, such as metal alloys and polymer blends systems. 2) Entropy dominates at high temperatures, leading to a homogeneous, disordered phase, while energy becomes crucial with decreasing temperature, and this leads to the formation of the ordered phases. One of the most successful models is known as the Cahn-Hilliard (CH) equations. The familiar spinodal decomposition (SD) processes under a deep quench to an unstable state have been investigated by numerical simulations based on the above time-dependent Ginzburg-Landau ansatze. 4) Under a shallow quench, on the other hand, the fluctuations change the nature of phase transitions from the second to the first order. By using the Hartree approximation, Brazovskii has firstly shown about this effect under the long-range interaction caused by the nonzero wave number. 6) One of the important essences for this Brazovskii effect is implied here that the Landau free-energy fL(ψ) = −τLψ/2 + uLψ/4 is transformed as the effects of the fluctuations into fH(ψ) = −τHψ/2+uHψ/4+wHψ/36, where τL, uL, τH, uH and wH are phenomenological parameters. It leads to that the fluctuation-induced first order transition changes the early stages of ordering to the nucleation and growth (NCG) manner. In spite of that there exist many analytical predictions of the dynamics about fluctuationinduced pattern formation far from equilibrium [see the reference], this is not the case for numerical studies on the morphology transitions under a shallow quench. The purpose of this short note is to propose the simple numerical model, in which the Landau free-energy is extended to the Hartree free-energy so that the freeenergy can have another minimum between the local minima of the double well free-energy. In terms of the cell dynamical system (CDS) approach, Oono and Puri have proposed the hyperbolic tangent form of one-toone map, ∂fL/∂ψ = −α tanh(ψ) + ψ, by taking the Landau free-energy functional fL into account. 3) On the basis of the Hartree free-energy functional fH, in the present study, the CDS map is modified to ∂fH/∂ψ = −α tanh(βψ) + ψ. Notations α and β are adjustable parameters which are determined such that the modified map has the five intersection points with ∂f/∂ψ = 0; i.e.,

Stability of fluid flow in a flexible tube to non-axisymmetric disturbances

The stability of fluid flow in a flexible tube to non-axisymmetric perturbations is analysed in this paper. In the first part of the paper, the equivalents of classical theorems of hydrodynamic stability are derived for inviscid flow in a flexible tube subjected to arbitrary non-axisymmetric disturbances. Perturbations of the form vi = v˜i exp [ik(x − ct) + inθ] are imposed on a steady axisymmetric mean flow U(r) in a flexible tube, and the stability of mean flow velocity profiles and bounds for the phase velocity of the unstable modes are determined for arbitrary values of azimuthal wavenumber n. Here r, θ and x are respectively the radial, azimuthal and axial coordinates, and k and c are the axial wavenumber and phase velocity of disturbances. The flexible wall is represented by a standard constitutive relation which contains inertial, elastic and dissipative terms. The general results indicate that the fluid flow in a flexible tube is stable in the inviscid limit if the quantity Ud[Gscr ]/dr [ges ] 0, and could be unstable for Ud[Gscr ]/dr &lt; 0, where [Gscr ] ≡ rU′/(n2 + k2r2). For the case of Hagen–Poiseuille flow, the general result implies that the flow is stable to axisymmetric disturbances (n = 0), but could be unstable to non-axisymmetric disturbances with any non-zero azimuthal wavenumber (n ≠ 0). This is in marked contrast to plane parallel flows where two-dimensional disturbances are always more unstable than three-dimensional ones (Squire theorem). Some new bounds are derived which place restrictions on the real and imaginary parts of the phase velocity for arbitrary non-axisymmetric disturbances.In the second part of this paper, the stability of the Hagen–Poiseuille flow in a flexible tube to non-axisymmetric disturbances is analysed in the high Reynolds number regime. An asymptotic analysis reveals that the Hagen–Poiseuille flow in a flexible tube is unstable to non-axisymmetric disturbances even in the inviscid limit, and this agrees with the general results derived in this paper. The asymptotic results are extended numerically to the moderate Reynolds number regime. The numerical results reveal that the critical Reynolds number obtained for inviscid instability to non-axisymmetric disturbances is much lower than the critical Reynolds numbers obtained in the previous studies for viscous instability to axisymmetric disturbances when the dimensionless parameter Σ = ρGR2/η2 is large. Here G is the shear modulus of the elastic medium, ρ is the density of the fluid, R is the radius of the tube and η is the viscosity of the fluid. The viscosity of the wall medium is found to have a stabilizing effect on this instability.

Local transit-time dissipation and Landau damping

A generalization of local transit-time dissipation theory to coherent electrostatic wave packets with nonzero mean wave number is presented. The connection between Landau damping and transit time damping is derived in a concise, mathematically rigorous manner, settling a longstanding controversy. It is shown that transit time dissipation involves both Landau-type resonant damping and nonresonant damping. For small wave packets with nonzero mean wave number or asymmetric incident particle distributions, the nonresonant damping can dominate over Landaudamping. In the opposite extreme of infinitely large, constant-amplitude wave packets, the nonresonant part of transit time dissipation vanishes, and only Landau damping remains. All the analytical results presented are verified independently by numerical test-particle calculations.

A Necessary and Sufficient Instability Condition for Inviscid Shear Flow

We derive a condition that is necessary and sufficient for the instability of inviscid, two‐dimensional, plane‐parallel, shear flow with equilibrium velocity profiles that are monotonic, real analytic, functions of the cross‐stream coordinate. The analysis, which is based upon the Nyquist method, includes a means for delineating the possible kinds of bifurcations that involve the presence of the continuous spectrum, including those that occur at nonzero wave number. Several examples are given.

Local transit-time damping of electrostatic wave packets

The theory of local transit-time damping is generalized to coherent electrostatic wave packets with nonzero mean wave number in an unmagnetized plasma. A general analytic formula for the phase-averaged power dissipated locally within an arbitrary three-dimensional wave packet is derived to second order in the fields. This expression is evaluated explicitly for a representative one-dimensional field structure. The result agrees with independent numerical test-particle calculations to within numerical rounding errors for small to moderate field amplitudes, which justify the perturbation expansions. The resulting damping involves both Landau (resonant) and non-Landau (nonresonant) terms, the latter having been omitted in previous works. It is found that the dissipated power depends sensitively on the ratio of the particle velocity to the phase velocity of the packet, the ratio of the wavelength to the size of the packet, and the form of the particle distribution. In general, particles remove energy from some parts of the packet and deposit it in others, thus reshaping it.

Steady-state bifurcation with Euclidean symmetry

We consider systems of partial differential equations equivariant under the Euclidean group E ( n ) \mathbf {E}(n) and undergoing steady-state bifurcation (with nonzero critical wavenumber) from a fully symmetric equilibrium. A rigorous reduction procedure is presented that leads locally to an optimally small system of equations. In particular, when n = 1 n=1 and n = 2 n=2 and for reaction-diffusion equations with general n n , reduction leads to a single equation. (Our results are valid generically, with perturbations consisting of relatively bounded partial differential operators.) In analogy with equivariant bifurcation theory for compact groups, we give a classification of the different types of reduced systems in terms of the absolutely irreducible unitary representations of E ( n ) \mathbf {E}(n) . The representation theory of E ( n ) \mathbf {E}(n) is driven by the irreducible representations of O ( n − 1 ) \mathbf {O}(n-1) . For n = 1 n=1 , this constitutes a mathematical statement of the ‘universality’ of the Ginzburg-Landau equation on the line. (In recent work, we addressed the validity of this equation using related techniques.) When n = 2 n=2 , there are precisely two significantly different types of reduced equation: scalar and pseudoscalar, corresponding to the trivial and nontrivial one-dimensional representations of O ( 1 ) \mathbf {O}(1) . There are infinitely many possibilities for each n ≥ 3 n\ge 3 .

Multiple resonances involving magnetoexcitons in aGaAs/Al0.30Ga0.70Asquantum well

The excitonic states in an undoped ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}}_{0.30}{\mathrm{Ga}}_{0.70}\mathrm{As}$ single quantum well have been studied by magneto-photoluminescence excitation spectroscopy at $T=4.2$ K. A series of regularly spaced steplike features is observed in the magnetic field dispersions of the transition energies of the excited heavy-hole excitons $\mathrm{ns}\mathrm{hh}, n=2,3,4.$ Correspondingly, the splitting between the two circular polarizations of the $2s\mathrm{hh}$ exciton shows pronounced oscillations as a function of magnetic field. We suggest that these features originate from the coupling of the transition between the ground state $1s\mathrm{hh}$ exciton and each of the excited states of the exciton with another excitation in the system. The energy difference between the features, which is about 7.5 meV, suggests that the excitation may be a transverse acoustic phonon with nonzero wave number near the zone boundary.

Morphological instability of similarity solution to the Stefan problem with undercooling and surface tension

This paper concerns morphological stability of a similarity solution to the Stefan problem with surface tension and initial supercooling. The linear stability analysis shows that for a nonzero surface tension each perturbation mode with a nonzero wave number is stable. However, the solution is unstable with respect to perturbations with a zero wave number limit point in their Fourier spectrum.

Theory of magnetostatic waves for in-plane magnetized anisotropic films

A modified formulation of the Damon-Eshbach theory of magnetostatic waves for in-plane magnetized anisotropic films is presented. New results, relative to the usual backward volume wave, nonreciprocal surface wave, and surface mode critical angle properties for isotropic films, are obtained for films with various anisotropies. These include: (i) Surface mode suppression. (ii) A second critical propagation angle for which the backward volume band of dispersion branches becomes inverted and the dispersion curves change to a forward volume mode character. (iii) Mode conversion and a reentrant mode character for the lowest order dispersion branch; a volume mode character at low wave number which converts to a surface character at high wave number. At the conversion point, this mode corresponds to a propagating plane wave with a uniform mode profile across the film thickness but at nonzero propagation wave number. (iv) A third critical in-plane propagation angle at which the surface mode dispersion branch vanishes and the lowest order branch reverts to a pure volume mode character.