Radial Wavenumber Research Articles

Radial propagation of geodesic acoustic modes in up-down asymmetric magnetic geometries

The energy approach to the analysis of the propagation of geodesic acoustic modes presented in the work of Hager and Hallatschek [Phys. Plasmas 16, 072503 (2009)] is generalized to up-down asymmetric magnetic geometries including a model for single-null configuration. By removing the neoclassical cancellation effects, up-down asymmetry can trigger a nonvanishing group velocity at zero radial wavenumber. A theoretical insight in this effect is provided by analytical calculations combined with numerical gyrokinetic and two-fluid studies. Thereby, an useful estimate of the group velocity at zero radial wavenumber is derived within a two-fluid framework.

Spontaneous‐adjustment emission of inertia‐gravity waves by unsteady vortical motion in the hurricane core

AbstractSpontaneous‐adjustment emission of inertia‐gravity waves (IGWs) from unsteady vortical motion in the hurricane core is further examined. An idealized divergent barotropic model simulation is conducted in an unforced framework using a highly asymmetric potential vorticity (PV) field, consisting of an offset monopole embedded in an elliptically shaped ring, which is motivated by concentric eyewalls that were observed in the inner core of hurricane Ivan (2004). The simulation proceeds in two phases. The first phase is marked primarily by inner‐core potential vorticity (PV) mixing and rearrangement as the monopole interacts with the ring, however axisymmetrization does not occur. This leads to a second phase characterized by a prolonged (lasting at least 36 h) period of spontaneous‐adjustment emission of IGWs to the environment. During the spontaneous‐adjustment phase, unsteady vortical motion acts as an IGW source, constraining the frequency and radial wavenumber of the waves. Diagnostics of this simulation reveal two important conclusions. First, the spontaneous radiation of IGWs is insignificant for vortex intensification or decay; rather, PV mixing in the early portion was more important. Secondly, using a non‐divergent barotropic model simulation of the same configuration and idealized simulations, we demonstrate that non‐axisymmetrization in this case‐study is caused primarily by PV (Rossby wave) dynamics, with the Rossby‐IGW instability only playing a minor role. These results support the existence of a slow quasi‐manifold with exponentially small fuzziness at small Froude and large Rossby numbers. Copyright © 2010 Royal Meteorological Society

The kink-type instability of toroidal stellar magnetic fields with thermal diffusion

The stability of toroidal magnetic fields in rotating radiative stellar zones is studied for realistic values of both the Prandtl numbers. The two considered models for the magnetic geometry represent fields with odd and even symmetry with respect to the equator. In the linear theory in Boussinesq approximation the resulting complex eigenfrequency (including growth rate and drift rate) are calculated for a given radial wave number of a nonaxisymmetric perturbation with m = 1. The ratio of the Alfvén frequency, ΩA, to the rate of the basic rotation, Ω, controls the eigenfrequency of the solution. For strong fields with ΩA > Ω the solutions do not feel the thermal diffusion. The growth rate runs with ΩA and the drift rate is close to −Ω so that the magnetic pattern will rest in the laboratory system. For weaker fields with ΩA < Ω the growth rate strongly depends on the thermal conductivity. For fields with dipolar parity and for typical values of the heat conductivity, the resulting very small growth rates are almost identical with those for vanishing gravity. For fields with dipolar symmetry, the differential rotation of any stellar radiative zone (like the solar tachocline) is shown as basically stabilizing the instability independent of the sign of the shear. Finally, the current-driven kink-type instability of a toroidal background field is proposed as a model for the magnetism of Ap stars. The recent observation of a lower magnetic field threshold of about 300 Gauss for Ap stars is understood as corresponding to the minimum magnetic field producing the instability. This article is dedicated to Andrew Soward on the occasion of his 65th birthday.

Determination of Tectonic and Crustal Structure of the Eastern Pontide Orogenic Belt (NE Turkey) Using Gravity and Magnetic Data

The Eastern Pontide Orogenic Belt is one of the most complex geodynamic settings within the Alpine belt. Subduction polarity, which is responsible for the formation of the Eastern Pontide Magmatic Arc, is still under debate because of limited geological, geophysical and geochemical data. This orogenic belt is mainly divided into three subzones depending on lithological characteristics and facies changes as in Northern, Southern and Axial Zones from North to South. These zones are separated from each other by near-vertical faults that display the block-faulting tectonic style of this belt. In this study, the tectonic and crustal structure of the Eastern Pontides, which as yet have not been prospected by using geophysical data, has been investigated with potential field data. The horizontal gradient map obtained from gravity data shows a number of steep and gentle lineaments. It seems that these lineaments E-W, NE and NW-trending correspond to major structural zones of continental crust. Additionally, The Moho depth and Curie point depth variation maps of the Eastern Pontide Orogenic Belt have been computed with the power spectral method of the radial wavenumber carried out by using the fast Fourier transform method. As a result of this method, we estimated that the depths of the Moho and Curie point varied between 29.0 ± 1.2-47.2 ± 1.9 km and 14.3 ± 0.7-27.9 ± 1.4 km, respectively. Our findings indicate that the Moho depth generally increases from north to south in the region. However, the Curie point depth level within the crust has an undulating surface, not a horizontal one.

Theory of fine-scale zonal flow generation from trapped electron mode turbulence

Most existing zonal flow generation theory has been developed with a usual assumption of qrρθi⪡1 (qr is the radial wave number of zonal flow and ρθi is the ion poloidal gyroradius). However, recent nonlinear gyrokinetic simulations of trapped electron mode turbulence exhibit a relatively short radial scale of the zonal flows with qrρθi∼1 [Z. Lin et al., Proceedings of the 21st International Conference on Plasma Physics and Controlled Nuclear Fusion Research, Chengdu, China, 2006 (International Atomic Energy Agency, Vienna, 2006); D. Ernst et al., Phys. Plasmas 16, 055906 (2009)]. This work reports an extension of zonal flow growth calculation to this short wavelength regime via the wave kinetics approach. A generalized expression for the polarization shielding for arbitrary radial wavelength [L. Wang and T. S. Hahm, Phys. Plasmas 16, 062309 (2009)] which extends the Rosenbluth–Hinton formula in the long wavelength limit is applied.

Poloidal eigenmode of the geodesic acoustic mode in the limit of high safety factor

AbstractThe poloidal eigenmode of the geodesic acoustic mode (GAM) is studied in the limit of high safety factor. In this limit, the poloidal gyroradius cannot be treated as a perturbation or as an expansion parameter. Analytical expressions for the poloidal structure of the GAM potential, the radial wavenumber dependence of the frequency, the phase velocity, and the group velocity are obtained. The spatial structure of the poloidal eigenmode including the higher-order gyroradius effect is revealed theoretically.

Observation of the parametric-modulational instability between the drift-wave fluctuation and azimuthally symmetric sheared radial electric field oscillation in a cylindrical laboratory plasma

Observation of the parametric-modulational interaction between the drift-wave fluctuation (7–8 kHz) and azimuthally symmetric sheared radial electric field structure (∼0.4 kHz) in a cylindrical laboratory plasma is presented. Oscillation of the sheared radial electric field is synchronized at modulations of the radial wave number and Reynolds stress per mass density of the drift-wave spectrum. Bispectral analysis at the location where the sheared radial electric field has finite radial wave numbers shows that nonlinear energy transfers from the drift wave to the sheared radial electric field occur. Nonlocal energy transfers of fluctuations via “channel of the azimuthally symmetric sheared radial electric field” in spectral space as well as real space are discovered.

Two-and-Three-Dimensional Crustal Thickness of the Eastern Pontides (NE Turkey)

The Eastern Pontide orogenic belt is divided into three subzones (northern, southern and axial zones) based on present lithologies and facies associations. NE-SW-, NW-SE- and E-W-trending fault systems, which play an important role in palaeotectonics and neotectonics of the Eastern Pontides, separate these zones. Three different methods were used to estimate the Moho depth from observed gravity value, namely (i) empirical relationship between Moho depth and Bouguer anomaly; (ii) spectral analysis of the radial wave number and; (iii) by the gravity inversion method. Power spectrum, a statistical approach, is a widely used technique to determine the depth of geological sources successfully. The empirical linear relations between the Bouguer anomalies and seismically determined crustal thicknesses have been used to compute depths to the Conrad and Moho discontinuities, which are consistent with the average depths obtained from the power spectrum method. The crustal structure was also determined from the inversion of a Bouguer anomaly profile along longitude 39°. Gravity inversion results are consistent with those obtained from the empirical relations and power spectrum methods. We calculated the maximum crustal thickness of 43.8 km in the studied region by using the gravity inversion method, which also showed that crustal thickness increases from north to south.

On Active Noise Reduction in a Cylindrical Duct with Flow

An analytical approach to active-noise reduction is presented in the case of a line-source in a cylindrical enclosure, minimizing the noise in a sector, corresponding to (i) the passenger head area of an aircraft cabin for a cylindrical fuselage, in the absence of flow, and (ii) a sector or an annulus of noise reduction outside of a cylindrical duct carrying a uniform axial flow of an arbitrary Mach number. In both cases, the noise is assumed to consist of the superposition of modes, and the anti-noise is used to cancel the fundamental mode and/or specified harmonics. The total acoustic energy in the region of interest is calculated for the residual and original sound field, and their ratio specifies the noise-reduction function. The latter is minimized by adjusting the source position, and the noise reduction achieved is plotted versus the dimensionless radial wave number, taking into account mean flow effects. The case of the original noise field, consisting of the fundamental and the anti-noise source set to cancel this, is taken as the baseline for further comparison. Cases with several anti-noise line-sources set to cancel various combinations of the fundamental and harmonics are also considered; for example n anti-noise sources are used to cancel the fundamental and first n-1 harmonics. For a given noise field, the improvement in noise-reduction performance with the number of anti-noise sources is demonstrated, both for cylindrical and planar enclosures. The addition of anti-noise sources while reducing noise at low frequencies can cause an increase in noise at high frequencies; the latter may be countered by passive means. All results obtained follow from the calculation of the noise-reduction function in terms of Bessel functions, which can be evaluated with their zeros, using asymptotic methods, which are shown to be reasonably accurate.

Characterizing the gravitational instability in cooling accretion discs

In this paper we perform a systematic analysis of the structure induced by the onset of gravitational instabilities in cooling gaseous accretion discs. It is well known that for low enough cooling rates the disc reaches a quasi-steady configuration, with the instability saturating at a finite amplitude such that the disc is kept close to marginal stability. We analyse the dependence of the saturation amplitude on the imposed cooling rate, and we find that it scales with the inverse square root of the cooling parameter β = t cool /t dyn . This indicates that the heating rate induced by the instability is proportional to the energy density of the density waves excited by the disc self-gravity. In particular, we find that at saturation the energy dissipated per dynamical time by weak shocks due to the gravitational perturbations is of the order of 20 per cent of the wave energy. We further perform a Fourier analysis of the disc structure, and subsequently determine the dominant radial and azimuthal wavenumbers of the density waves. While the number of spiral arms (corresponding to the azimuthal wavenumber) is fairly constant with radius, we find that the disc displays a range of radial wavenumbers whose mean increases with increasing radius. The dominant modes closely match the locally most unstable wavelength as predicted by linear perturbation analysis. As a consequence, we numerically demonstrate that the density waves excited in relatively low-mass discs M disc /M * ∼ 0.1 are always close to corotation, deviating from it by approximately 10 per cent. This result can be understood in terms of the constancy of the Doppler-shifted phase Mach number of the flow - the pattern speed self-adjusts so that the flow into spiral arms is always sonic. This has profound effects on the degree to which the extraction of energy and angular momentum from the mean flow through density waves can be modelled as a viscous process. Our results thus provide (i) a detailed description of how the self-regulation mechanism is established for low cooling rates, (ii) a clarification of the conditions required for describing the transport induced by self-gravity through an effective viscosity, (iii) an estimate of the maximum amplitude of the density perturbation before fragmentation takes place and finally (iv) a simple recipe to estimate the density perturbation in different thermal regimes.

Coexistence of Zonal Flows and Drift-Waves in a Cylindrical Magnetized Plasma

Spatiotemporal structures of fluctuations with frequencies lower than the ion cyclotron frequency in a cylindrical magnetized plasma are investigated. Drift-wave and low-frequency zonal flow coexist. Electrostatic potentials of the zonal flow and the drift-wave are distributed widely in radius. The radial wave number profile of the zonal flow has a shear structure at the radial location where the drift-wave has a maximal normalized fluctuation amplitude. On the other hand, the radial wave number profile of the drift-wave shows vortex tilting, resulting in the generation of stationary turbulence Reynolds stress gradient per mass density. The envelope and bispectral analyses indicate significant nonlinear interactions between the zonal flow and the drift-wave.

Physical interpretation of spiralling-columnar convection in a rapidly rotating annulus with radial propagation properties of Rossby waves

To aid the physical understanding of spiralling-columnar convection emerging in rapidly rotating spheres and spherical shells, two-dimensional thermal convection in a rapidly rotating annulus is investigated through the radial propagation properties of topographic Rossby waves. Two kinds of the boundaries containing the fluid in the axial direction are considered: a convex type modelling a spherical geometry and a concave type for comparison. The linear stability of a basic state with no motion and uniformly unstable stratification is examined and spirally elongated structures of critical convection are obtained for small Prandtl numbers. An analysis of the energy budget shows that a part of the kinetic energy generated in the region with slightly inclined boundaries is dynamically transferred and dissipates through viscosity in the region with strongly inclined boundaries. This indicates that the Rossby waves propagate from the region with slightly inclined boundaries to the region with strongly inclined boundaries. It is presented that the appearance of a spiral structure corresponds to an increase of the local radial wavenumber of the Rossby waves propagating in the radial direction. The flow patterns obtained using the dispersion relation of the Rossby waves coincide with those of the tailing part of the spiral structure obtained numerically. As the Prandtl number increases, the Rossby waves barely propagate because of strong viscous dissipation, and the flow pattern is localized in the region with slightly inclined boundaries. For convex boundaries with unstable stratification concentrating near the outer boundary and concave boundaries with unstable stratification confined near the inner boundary, the flow patterns tilt in the direction inverse to the case of uniform unstable stratification. The tilting direction of the flow pattern is not determined by the curvature of the boundaries considered but instead by the radial propagation direction of the Rossby waves excited by thermal convection.

Measurements of spatial structure of plasma potential and density fluctuations by multichannel heavy ion beam probe on large helical device

Heavy ion beam probe (HIBP) on large helical device is currently equipped with three channel detectors, which can observe three spatial points simultaneously inside the plasma with resolution of approximately 10 mm. The beam trajectories and observation point location are calculated numerically and optimized allowing for the identification of the mode structure in multichannel (up to 9) HIBP measurements. The calculations show that the radial and poloidal wavenumbers can be identified by proper changing and choosing of the beam energy and trajectory.

Scattering of electrons via bulk LO-phonon modes near a cylindrical core under the influence of an external azimuthal magnetic field

An investigation is carried out of the interactions of electrons with longitudinal optical (LO) phonons under the influence of an azimuthally directed external magnetic field. The magnetic field is thought to be induced by a steady current passed along the axis of an infinitely long cylindrical core of radius R . Confinement of electrons to regions near the core is assumed to be due to a parabolically varying electric potential of the heterojunction. As anticipated, the azimuthal magnetic field is found to lift the double degeneracy of the non-zero electron's axial wave number ( k z ) states while that of the non-zero azimuthal quantum number ( m ) states is preserved. Furthermore, the azimuthal magnetic field breaks the parabolicity symmetry in k z of the electron's energy subbands. Notably, as the field is increased, the k z < 0 branch of the electron's energy subband is systematically lowered and this is accompanied by a shift of the electron's subband energy minimum to higher values of | k z | . The scattering rates of electrons via bulk LO-phonon modes are obtained employing the Fröhlich interaction Hamiltonian. The intrasubband scattering integrals are found to be characterized by strong oscillations in their variations with the phonon radial wave number, nevertheless, enveloped by a rapid decay of their amplitudes. The intrasubband emission scattering rates exhibit a divergent behaviour for electron energies very near the LO zone frequency, reminiscent of the character of quasi-one-dimensional electronic states. In contrast to the axial applied magnetic field configuration, again, the azimuthal magnetic field breaks the symmetry of the ± k z branches of the intrasubband scattering rates. In general, the k z < 0 branch of the scattering rates is found to be systematically more depressed compared to that of the corresponding positive electron's wave number.

Modal-Analyse im Bogen- oder Ring-Kanal. Teil II: Ring-Kanal

Sound fields in ducts with ring-shaped cross-sections or in flat ducts with a circular bend of their axis can be represented as mode sums of modes in the cylindrical co-ordinate system in which the duct wall surfaces are co-ordinate surfcaces. The modal analysis of the sound fields in both types of ducts proceeds in parallel over wide parts; the numerical solutions, however, reveal significant differences. The case of the duct bend was treated in Part I of this paper; this part will be concerned with the ring-shaped duct. Several forms of developments of the characteristic equation for the radial wave numbers as mode solutions will be described; all of them intend to approximate the characteristic equation by a polynomial equation so that its solutions can be used as starters for a numerical solution of the complete charateristic equation, thus leading to sets of mode solutions. Finally, a set of modes from a flat duct with un-symmetrical lining, which are easily and safely determined by a continued-fraction expansion, are used as starters for modes of a ring-shaped channel with same width and lining. This works quite well under certain parameter conditions.

Analysis of Munk waveguide normal modes based on the Rayleigh‐Schrödinger perturbation and adiabatic invariant theories

This work concerns to obtain accurate explicit formulas related to discrete spectrum of the Munk deepwater acoustic wave guide problem. To this end, based on the Rayleigh‐Schrödinger perturbation theory, we developed analytic results related to the discrete values of the radial wave numbers as well as to the eigen pressure felds. We compare these previous results with JWKB ones. In addition, based on invariant adiabatic theory to wave equation [1,2], we tried to improve the accuracy of the JWKB calculation applying Olver's uniform asymptotic expansion (UAE) theory for the solution of the second‐order differential equation with two turning points [3‐5]. Comparing these three above particular frameworks, we conclude that UAE theory is accurate and it well describes the Munk wave guide normal modes. [1] L. M. Brekhovskikh and Yu. P. Lysanov, Fundamentals of Ocean Acoustics, Spinger, NY, 2001. [2] B. G. Katsnelson and V. G. Petnikov, Shallow Water Acoustics. Spinger, UK, 2002. [3] F. W. J. Olver, Asymptotics and special functions, Academic Press, London, 1974. [4] L. G. Guimarães and H. M. Nussenzveig, J. Mod. Optic., 41, 625 (1994). [5] P. C. G. de Moraes and L. G. Guimarães, JQSRT, 74 757 (2002).

Effect of Wave-Type Mean Flow on the Modulational Process of Zonal Flow Instability

The effects of an external mean flow on the modulational instability process of zonal flow in drift wave turbulence are theoretically studied based on Hasegawa–Mima (HM) turbulence model. Based on the coherent mode coupling approach, a dispersion relation of the zonal flow instability involving the external mean flow with a wave-type characterized by the amplitude |φ f |, radial wave number k f , and frequency ω f is derived. As an example, the zonal flow driven by ion temperature gradient (ITG) turbulence is sampled as the mean flow acting on the modulational process of zonal flow instability in electron temperature gradient (ETG) turbulence. It is shown that the growth rate of the zonal flow, γ q , is suppressed by the mean flow with a fitting relation γ q ≃γ q 0 -α|φ f | 2 k 2 f , where γ q 0 is the growth rate of the zonal flow in the absence of mean flow and α is a positive numerical constant. This formula is applicable to a strong shearing regime where the zonal flow instability is stabilized. The suppression mechanism is investigated and found to originate from frequency mismatch due to an increase of the real frequency of zonal flow |Ω q |.

Mode-Analysis in Curved and Ring-Shaped Ducts. Part I: Curved Ducts

Sound fields in ducts with ring-shaped cross-sections or in flat ducts with a circular bend of their axis can be represented as mode sums of modes in the cylindrical co-ordinate system in which the duct wall surfaces are co-ordinate surfcaces. Ring-shaped lined ducts are a common form of silencers; bent ducts with absorbing walls are favorite constructions as turning vanes for the flow in the corners of wind tunnels. The modal analysis of the sound fields in both types of ducts proceeds in parallel over wide parts. The most significant differences are found in the determination of the mode wave numbers. Whereas in the ring-shaped duct the arguments of the cylindrical functions are sought as radial wave numbers for given (mostly integer) orders of these functions, their (complex) orders are to be determined as azimuthal wave numbers in the curved duct, in which the arguments of the cylindrical functions can be supposed as known quantities, because their determination then is a rather simple standard task. If more than just the first-order approximation is required to the attenuation in a duct system containing a ring-shaped or a bent duct section (for that approximation the knowledge of the least attenuated mode would be sufficient) a set of mode wave numbers must be evaluated. This difficult sub-task of the modal analysis in both types of ducts is the principal topic of the present paper. The present part I of the paper is mainly concerned with curved ducts.

Spectral characteristics of geodesic acoustic mode in the HL-2A tokamak

The spectral characteristics of the geodesic acoustic mode (GAM) are investigated systematically by applying the two-point correlation technique and bispectral analysis to electric field fluctuations measured by electrostatic probe arrays on the HuanLiuqi-2A (HL-2A) tokamak. The three-dimensional wavenumber and frequency spectrum for the GAM has been measured for the first time. The spectrum provides definite evidence for the GAM which is characterized by kθ = kϕ = 0 and krρi ≈ 0.04–0.09 with the full width at half maximum Δkrρi ≈ 0.03–0.07. The radial wavenumber spectrum shows that the localized GAM packet propagates in the outward direction with approximately the same phase and group velocity. The cross-bicoherences involving the Reynolds stress and auto-bicoherences of potential (), radial electric field () and density () fluctuations have been estimated for comparisons. Strong nonlinear coupling between the GAM and broadband turbulence is observed in all summed bicoherences, except for the summed auto-bicoherence of density fluctuations. All cross- and auto-bicoherences, except for the auto-bicoherence of density fluctuations, for interactions satisfying f1 + f2 = fGAM are found to have a peaked feature in the frequency range f1 < 100 kHz. This peaked feature might reflect the resonance property in the nonlinear coupling between the GAM and ambient turbulence.

Cylindrical Hall – MHD Waves: A Nonlinear Solution

The exact nonlinear cylindrical solution for incompressible Hall – magnetohydrodynamic (HMHD) waves, including dissipation, essentially from electron – neutral collisions, is obtained in a uniformly rotating, weakly ionized plasma such as exists in photospheric flux tubes. The ω – k relation of the waves, called here Hall – MHD waves, demonstrates the dispersive nature of the waves, introduced by the Hall effect, at large axial and radial wavenumbers. The Hall – MHD waves are in general elliptically polarized. The partially ionized plasma supports lower frequency modes, lowered by the factor δ≡ratio of the ion mass density to the neutral particle mass density, as compared to the fully ionized plasma (δ=1). The relation between the velocity and the magnetic field fluctuations departs significantly from the equipartition found in Alfven waves. These short-wavelength and arbitrarily large amplitude waves could contribute toward the heating of the solar atmosphere.