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Rotation effect on geodesic and zonal flow modes in tokamak plasmas with isothermal magnetic surfaces

Elfimov, Artour; Galvao, Ricardo Magnus Osorio; Sgalla, Reneé Jordashe Franco
Fonte: IOP PUBLISHING LTD Publicador: IOP PUBLISHING LTD
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
46.29%
The electrostatic geodesic mode oscillations are investigated in rotating large aspect ratio tokamak plasmas with circular isothermal magnetic surfaces. The analysis is carried out within the magnetohydrodynamic model including heat flux to compensate for the non-adiabatic pressure distribution along the magnetic surfaces in plasmas with poloidal rotation. Instead of two standard geodesic modes, three geodesic continua are found. The two higher branches of the geodesic modes have a small frequency up-shift from ordinary geodesic acoustic and sonic modes due to rotation. The lower geodesic continuum is a newzonal flowmode (geodesic Doppler mode) in plasmas with mainly poloidal rotation. Limits to standard geodesic modes are found. Bifurcation of Alfven continuum by geodesic modes at the rational surfaces is also discussed. Due to that, the frequency of combined geodesic continuum extends from the poloidal rotation frequency to the ion-sound band that can have an important role in suppressing plasma turbulence.; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); CNPq (National Council of Brazil for Science and Technology Development); Financiadora de Estudos e Projetos (FINEP); FINEP (Financiadora de Estudos e Projetos)

Investigação cinética de modos geodésicos de baixas frequências em plasmas magnetizados; Kinetic investigation of low frequency geodesic modes in magnetized plasmas

Sgalla, Reneé Jordashe Franco
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Tese de Doutorado Formato: application/pdf
Publicado em 29/07/2014 PT
Relevância na Pesquisa
56.34%
Devido à sua importância em turbulência causada por ondas de deriva e à aplicação com propósitos em diagnósticos de plasma, a investigação de fluxos zonais (ZF) e modos acústicos geodésicos (GAM) tem atraído bastante atenção na literatura em física de plasmas. Nesta tese, primeiramente consideramos efeitos de equilíbrio com rotação poloidal e toroidal nestes modos, posteriormente investigamos efeitos diamagnéticos em GAM a partir de um modelo de dois fluido, no qual incluímos viscosidade paralela de íons e, na parte final, consideramos amortecimento de Landau e efeitos diamagnéticos simultaneamente no estudo de GAM, porém, a partir do modelo girocinético. Efeitos diamagnéticos são causados por termos que envolvem gradientes de densidade e de temperatura provenientes da função Maxwelliana de equilíbrio. O acoplamento entre os harmônicos poloidais, $m = \pm1$, e as derivadas radiais de quantidades macroscópicas do plasma é responsável pelo aumento no valor da frequência no GAM de alta frequência e pela instabilidade no GAM de baixa frequência. Este tipo de instabilidade, que é proporcional à frequência diamagnética de elétrons e à razão entre os gradientes de temperatura e de densidade...

Breakdown of the Debye approximation for the acoustic modes with nanometric wavelengths in glasses

Monaco, Giulio; Giordano, Valentina M.
Fonte: National Academy of Sciences Publicador: National Academy of Sciences
Tipo: Artigo de Revista Científica
EN
Relevância na Pesquisa
46.24%
On the macroscopic scale, the wavelengths of sound waves in glasses are large enough that the details of the disordered microscopic structure are usually irrelevant, and the medium can be considered as a continuum. On decreasing the wavelength this approximation must of course fail at one point. We show here that this takes place unexpectedly on the mesoscopic scale characteristic of the medium range order of glasses, where it still works well for the corresponding crystalline phases. Specifically, we find that the acoustic excitations with nanometric wavelengths show the clear signature of being strongly scattered, indicating the existence of a cross-over between well-defined acoustic modes for larger wavelengths and ill-defined ones for smaller wavelengths. This cross-over region is accompanied by a softening of the sound velocity that quantitatively accounts for the excess observed in the vibrational density of states of glasses over the Debye level at energies of a few milli-electronvolts. These findings thus highlight the acoustic contribution to the well-known universal low-temperature anomalies found in the specific heat of glasses.

Controle ativo de ruído aplicado a dutos com propagação de modos acústicos de alta ordem via particionamento axial; Active noise control applied to ducts with propagation of high- order acoustic modes via partitioning axial

Oliveira, Eider Lúcio de
Fonte: Universidade Federal de Uberlândia Publicador: Universidade Federal de Uberlândia
Tipo: Tese de Doutorado
POR
Relevância na Pesquisa
46.26%
O controle ativo de ruído (CAR) aplicado a dutos onde existe a propagação de ondas planas tem sido alvo de vários estudos. Aplica-se, nesses casos, o controle ativo monocanal. Nos dutos com grandes seções transversais, a propagação acústica se dá com a presença dos modos de alta ordem. Nesses dutos é necessário o controle multicanal, o que leva o sistema CAR a utilizar muitos sensores e atuadores, resultando num projeto bastante complexo. Este trabalho procura aplicar uma metodologia de mudança de geometria dos grandes dutos, para assegurar a propagação de ondas planas no interior dos mesmos e assim possibilitar o uso de um sistema CAR monocanal. A estratégia consiste em dividir a grande seção transversal do duto em áreas menores de forma a garantir ondas planas na banda de frequência de interesse. Várias técnicas e ferramentas foram utilizadas, como simulações do sistema acústico, otimização com algoritmo genético hibridizado e o algoritmo FxLMS para o desenvolvimento do projeto do sistema de controle. Um duto de grande porte foi construído e instrumentado. Além disso, uma placa eletrônica de controle com Processador Digital de Sinais (DSP) foi usada para executar os programas do sistema CAR em tempo real. Foram obtidas atenuações significativas dos níveis de ruído...

Experimental study of active control of high order acoustic modes in ducts

Li, X.; Kestell, C.; Qiu, X.; Zander, A.; Hansen, C.
Fonte: Australian Acoustical Society; C/o Professional Centre, Private Bag 1, Darlinghurst 2010 Publicador: Australian Acoustical Society; C/o Professional Centre, Private Bag 1, Darlinghurst 2010
Tipo: Conference paper
Publicado em //2002 EN
Relevância na Pesquisa
46.26%
Tonal noise radiated from an exhaust stack of a spray dryer system used to make powdered milk has been the source of unwanted community noise.The tone is characterised by a frequency above the first mode cut-on frequency of the exhaust duct and it is not steady in either amplitude or frequency. Passive silencing of the exhaust is a very expensive and inconvenient option and for this reason the feasibility of using active noise control was investigated. Zander and Hansen developed a theoretical model for analysing the effect of active noise control on higher order rectangular duct modes,and used it to evaluate the effects of source size,location and strength. It was shown that the total acoustic power reduction was dependent on the relative locations of the control sources. More recently this work was applied to the practical example of a cylindrical spray dryer exhaust stack to determine the most suitable location and number of control sources and error sensors. Prior to installation in the exhaust stack a half scale model was constructed to test the theory. Here, the effect of the error sensor locations on the active control on higher order acoustic modes propagating in the half scale circular duct is investigated experimentally using real time control with a multi-channel feedforward controller. A tachometer signal at the fan blade pass frequency (BPF) was fed into the controller as the reference signal for the feedforward control. Two configurations of error sensors on the duct wall were tested to evaluate how the error sensor locations affect the control performance...

Enhancement of stimulated Brillouin scattering of higher-order acoustic modes in single-mode optical fiber

Afshar Vahid, S.; Kalosha, V.; Bao, X.; Chen, L.
Fonte: Optical Soc Amer Publicador: Optical Soc Amer
Tipo: Artigo de Revista Científica
Publicado em //2005 EN
Relevância na Pesquisa
46.31%
Solving the elastic wave equation exactly for a GeO2-doped silica fiber with a steplike distribution of the longitudinal and shear velocities and density, we have obtained the dispersion, attenuation, and fields of the leaky acoustic modes supported by the fiber. We have developed a model for stimulated Brillouin scattering of these modes in a pump-probe configuration and provided their Brillouin gains and frequencies for an extended range of core sizes and GeO2 doping. Parameter ranges close to cutoff of the acoustic modes and pump depletion enhance the ratio of higher-order peaks to the main peak in the Brillouin spectrum and are suitable for simultaneous strain-temperature sensing.; Shahraam Afshar V., V. P. Kalosha, Xiaoyi Bao, Liang Chen; This paper was published in Optics Letters and is made available as an electronic reprint with the permission of OSA. The paper can be found at the following URL on the OSA website: http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-20-2685. Systematic or multiple reproduction or distribution to multiple locations via electronic or other means is prohibited and is subject to penalties under law.

Acoustic wave parameter extraction with application to delay line modelling using finite element analysis

Tikka, A.; Al-Sarawi, S.; Abbott, D.
Fonte: International Frequency Sensor Association Publicador: International Frequency Sensor Association
Tipo: Artigo de Revista Científica
Publicado em //2008 EN
Relevância na Pesquisa
46.28%
In this paper we propose a practical approach to develop a SAW parameter extraction technique and simulation method, contrary to the frequently used FEM/BEM numerical simulation technique. The new approach allows for accurate SAW device modelling through a versatile finite element method to automatically include the second order effects in the device by considering the complete set of partial differential equations. The eigenmodes and the harmonic admittance of a periodic structure obtained from the FEM simulations are used to extract the COM/P-matrix parameters based on a fitting technique that is already published in the literature. Comparison between the extracted parameters, using this technique, with currently published parameters for the same device specification was carried out and showed good agreement. As an example, a two dimensional delay line with 10 finger pairs per IDT was modelled using this approach, to outline the mechanical and electrical response of the device for variable acoustic modes and substrate thicknesses.; Ajay Tikka, Said Al-Sarawi and Derek Abbott

Angular Momentum Transport and Variability in Boundary Layers of Accretion Disks Driven by Global Acoustic Modes

Belyaev, Mikhail A.; Rafikov, Roman R.; Stone, James M.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.36%
Disk accretion onto a weakly magnetized central object, e.g. a star, is inevitably accompanied by the formation of a boundary layer near the surface, in which matter slows down from the highly supersonic orbital velocity of the disk to the rotational velocity of the star. We perform high resolution 2D hydrodynamical simulations in the equatorial plane of an astrophysical boundary layer with the goal of exploring the dynamics of non-axisymmetric structures that form there. We generically find that the supersonic shear in the boundary layer excites non-axisymmetric quasi-stationary acoustic modes that are trapped between the surface of the star and a Lindblad resonance in the disk. These modes rotate in a prograde fashion, are stable for hundreds of orbital periods, and have a pattern speed that is less than and of order the rotational velocity at the inner edge of the disk. The origin of these intrinsically global modes is intimately related to the operation of a corotation amplifier in the system. Dissipation of acoustic modes in weak shocks provides a universal mechanism for angular momentum and mass transport even in purely hydrodynamic (i.e. non-magnetized) boundary layers. We discuss the possible implications of these trapped modes for explaining the variability seen in accreting compact objects.; Comment: 41 pages...

Angular Momentum Transport by Acoustic Modes Generated in the Boundary Layer I: Hydrodynamical Theory and Simulations

Belyaev, Mikhail A.; Rafikov, Roman R.; Stone, James M.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 03/12/2012
Relevância na Pesquisa
46.31%
The nature of angular momentum transport in the boundary layers of accretion disks has been one of the central and long-standing issues of accretion disk theory. In this work we demonstrate that acoustic waves excited by supersonic shear in the boundary layer serve as an efficient mechanism of mass, momentum and energy transport at the interface between the disk and the accreting object. We develop the theory of angular momentum transport by acoustic modes in the boundary layer, and support our findings with 3D hydrodynamical simulations, using an isothermal equation of state. Our first major result is the identification of three types of global modes in the boundary layer. We derive dispersion relations for each of these modes that accurately capture the pattern speeds observed in simulations to within a few percent. Second, we show that angular momentum transport in the boundary layer is intrinsically nonlocal, and is driven by radiation of angular momentum away from the boundary layer into both the star and the disk. The picture of angular momentum transport in the boundary layer by waves that can travel large distances before dissipating and redistributing angular momentum and energy to the disk and star is incompatible with the conventional notion of local transport by turbulent stresses. Our results have important implications for semianalytical models that describe the spectral emission from boundary layers.; Comment: 53 pages...

Double adiabatic theory of collisionless geodesic acoustic modes in tokamaks

Hassam, A. B.; Kleva, R. G.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 31/08/2011
Relevância na Pesquisa
46.25%
Collisionless geodesic acoustic modes in tokamaks being supersonic for large "safety factor" q, the CGL (G. Chew, M. Goldberger, F. Low, 1956)1 double-adiabatic fluid closure is applied to formulate a theory for these modes. The basic linear normal mode is obtained. External means to drive these modes at resonance, as has been proposed earlier, are explored. The external drivers considered include external magnetic forces to effect flux surface displacements, as well as non-axisymmetric ion heating. Finally, the damping of these modes from collisional magnetic pumping is investigated using a model set of CGL collision-corrected equations.

Asymptotic analysis of high-frequency acoustic modes in rapidly rotating stars

Lignieres, F.; Georgeot, B.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 10/03/2009
Relevância na Pesquisa
46.38%
The asteroseismology of rapidly rotating pulsating stars is hindered by our poor knowledge of the effect of the rotation on the oscillation properties. Here we present an asymptotic analysis of high-frequency acoustic modes in rapidly rotating stars. We study the Hamiltonian dynamics of acoustic rays in uniformly rotating polytropic stars and show that the phase space structure has a mixed character, regions of chaotic trajectories coexisting with stable structures like island chains or invariant tori. In order to interpret the ray dynamics in terms of acoustic mode properties, we then use tools and concepts developed in the context of quantum physics. Accordingly, the high-frequency acoustic spectrum is a superposition of frequency subsets associated with dynamically independent phase space regions. The sub-spectra associated with stable structures are regular and can be modelled through EBK quantization methods while those associated with chaotic regions are irregular but with generic statistical properties. The results of this asymptotic analysis are successfully confronted with the properties of numerically computed high-frequency acoustic modes. The implications for the asteroseismology of rapidly rotating stars are discussed.; Comment: 21 pages...

A magnetic model for acoustic modes in roAp stars

Zita, E. J.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 16/09/1998
Relevância na Pesquisa
46.25%
The mechanism for excitation of p-modes in rapidly oscillating, peculiar A (roAp, or cool chemically peculiar, CP) stars is unknown. Observations strongly suggest that acoustic modes in roAp stars are causally linked to the stars' magnetic field. We propose that small fluctuations in the shape of the mean magnetic field drive magnetosonic waves, which are observed as p-modes in these stars. The dynamic edge region of roAp stars is a force-free spherical shell. When strongly coupled to the magnetic field, a force-free plasma can oscillate about a minimum in its mean magnetic energy. We describe the stable eigenmodes for this energy minimum in a spherical shell with an open boundary. The wavenumbers, frequencies, and energies of resulting oscillations are consistent with observations of p-modes in roAp stars. Our magnetic model for p-mode oscillations in stars does not invoke convection or opacity mechanisms. We also suggest the possibility of a nonlinear dynamo for such magnetic stars, which lack the convection needed for the usual (-( dynamo.

Electrostatic acoustic modes in a self-gravitating complex plasma with variable charge impurities

Misra, Amar P; Chowdhury, K. Roy; Chowdhury, A. Roy
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 08/01/2005
Relevância na Pesquisa
46.24%
A linear theory of dust acoustic (DA) and dust ion-acoustic (DIA) waves in a self-gravitating collisional dusty plasma contaminated by variable-charge impurities is presented for a self-consistent closed system. The ion-drag force arising from the ion orbital motion as well as momentum transfer from all the ions and self-gravitation are taken into consideration. The physical processes, viz., dust-charge relaxation, ionization, recombination and collisional dissipations are also taken into account self-consistently. The generalized dispersion relation describing the coupling of the acoustic modes with the dust-charge relaxation mode (CRM) and damping of the waves is derived and analyzed numerically.; Comment: 13 pages, 5 figures. Physics Letters A (2005,To be published)

Acoustic modes in spheroidal cavities

Lignières, F.; Rieutord, M.; Valdettaro, L.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 09/10/2001
Relevância na Pesquisa
46.22%
Oscillation modes of rapidly rotating stars have not yet been calculated with precision, rotational effects being generally approximated by perturbation methods. We are developing a numerical method able to account for the deformation of the star by the centrifugal force and, as a first step, we determined the acoustic modes of a uniform density spheroid and studied how the spheroid flatness affects these modes.; Comment: 4 pages, Proceedings of "Semaine de l'Astrophysique Francaise", Edts, SF2A, EdP-Sciences Conf. Series

Optomechanical characterization of acoustic modes in a mirror

Briant, T.; Cohadon, P. F.; Heidmann, A.; Pinard, M.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 21/05/2003
Relevância na Pesquisa
46.31%
We present an experimental study of the internal mechanical vibration modes of a mirror. We determine the frequency repartition of acoustic resonances via a spectral analysis of the Brownian motion of the mirror, and the spatial profile of the acoustic modes by monitoring their mechanical response to a resonant radiation pressure force swept across the mirror surface. We have applied this technique to mirrors with cylindrical and plano-convex geometries, and compared the experimental results to theoretical predictions. We have in particular observed the gaussian modes predicted for plano-convex mirrors.; Comment: 8 pages, 8 figures, RevTeX

Are Solar Acoustic Modes Correlated ?

Foglizzo, T.; Garcia, R. A.; Boumier, P.; Charra, J.; Gabriel, A. H.; Grec, G.; Robillot, J. M.; Cortes, T. Roca; Turck-Chieze, S.; Ulrich, R. K.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 10/09/1997
Relevância na Pesquisa
46.25%
We have studied the statistical properties of the energy of individual acoustic modes, extracted from 310 days of GOLF data near the solar minimum. The exponential distribution of the energy of each mode is clearly seen. The modes are found to be uncorrelated with a +-0.6 % accuracy, thus supporting the hypothesis of stochastic excitation by the solar convection. Nevertheless, the same analysis performed on the same modes just before the solar maximum, using IPHIR data, rejects the hypothesis of no correlation at a 99.3 % confidence level. A simple model suggests that 31.3 +-9.4 % of the energy of each mode is coherent among the modes studied in IPHIR data, correponding to a mean correlation of 10.7 +-5.9 %.; Comment: 10 pages A&A Latex, 13 EPSF figures, to appear in A&A

Corotational Instability of Inertial-Acoustic Modes in Black Hole Accretion Discs and Quasi-Periodic Oscillations

Lai, Dong; Tsang, David
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.25%
We study the global stability of non-axisymmetric p-modes (also called inertial-acoustic modes) trapped in the inner-most regions of accretion discs around black holes. We show that the lowest-order (highest-frequency) p-modes, with frequencies $\omega=(0.5-0.7) m\Omega_{\rm ISCO}$, can be overstable due to general relativistic effects, according to which the radial epicyclic frequency is a non-monotonic function of radius near the black hole. The mode is trapped inside the corotation resonance radius and carries a negative energy. The mode growth arises primarily from wave absorption at the corotation resonance, and the sign of the wave absorption depends on the gradient of the disc vortensity. When the mode frequency is sufficiently high, such that the slope of the vortensity is positive at corotation positive wave energy is absorbed at the resonance, leading to the growth of mode amplitude. We also study how the rapid radial inflow at the inner edge of the disc affects the mode trapping and growth. Our analysis of the behavior of the fluid perturbations in the transonic flow near the ISCO indicates that, while the inflow tends to damp the mode, the damping effect is sufficiently small under some conditions so that net mode growth can still be achieved. We further clarify the role of the Rossby wave instability and show that it does not operate for black hole accretion discs with smooth-varying vortensity profiles. Overstable non-axisymmetric p-modes driven by the corotational instability provide a plausible explanation for the high-frequency (> 100 Hz) quasi-periodic oscillations (HFQPOs) observed from a number of black-hole X-ray binaries in the very high state. The absence of HFQPOs in the soft (thermal) state may result from mode damping due to the radial infall at the ISCO.; Comment: 19 pages...

Nonlinear interactions among solar acoustic modes

Kumar, Pawan; Goldreich, Peter
Fonte: American Astronomical Society Publicador: American Astronomical Society
Tipo: Article; PeerReviewed Formato: application/pdf
Publicado em 01/07/1989
Relevância na Pesquisa
46.52%
We evaluate the rates at which nonlinear interactions transfer energy among the normal modes of a plane-parallel, stratified atmosphere. The atmosphere resembles the outer part of the Sun including the convection zone and the optically thin region above the photosphere up to the temperature minimum. The acoustic modes are assigned energies such that their photospheric velocities match those of the Sun's p-modes. The nonlinearity parameter is the acoustic Mach number, M, the ratio of the total acoustic velocity due to all of the modes to the sound speed. For M^2 ≪ 1 the leading nonlinear interactions are those which couple three-modes. We show that every p-mode in the 5 minute band is involved in many near-resonant triplets. As a consequence, the energy transfer rates are independent of the mode line widths. Because M increases with height, the dominant contributions to the three-mode coupling coefficients occur in the upper part of the convection zone and in the optically thin isothermal layer. Moreover, the coupling coefficients tend to increase with ω and k_h. Nonlinear interactions which couple two trapped modes and one propagating mode drain energy from the trapped modes. They are far more effective than interactions among three trapped modes which drive the modes toward equipartition of energy. Thus...

On the Existence and Stability of Limit Cycles for Longitudinal Acoustic Modes in a Combustion Chamber

Awad, E.; Culick, F. E. C.
Fonte: Taylor & Francis Publicador: Taylor & Francis
Tipo: Article; PeerReviewed Formato: application/pdf
Publicado em //1986
Relevância na Pesquisa
46.43%
Unsteady motions in combustion chambers have previously been treated with an approximate analysis in which an acoustic field is represented as a collection of coupled nonlinear oscillators constructed in one-to-one correspondence to the acoustic modes. Two parameters characterize the linear behavior of each oscillator; a single parameter arises from the nonlinear acoustics carried out to second order in small fluctuations. The formal results are used here as the basis for studying the existence and stability of limit cycles for longitudinal modes. Owing to the special structure of the equations, explicit and precise conclusions can be reached. Existence and stability depend only on the parameters defining the linear motions. The nonlinear gasdynamics influence the amplitudes of motion in the limit cycle. At least one of the acoustic modes must be linearly unstable to produce a nontrivial limit cycle. Generally, energy flows both up and down among the modes, but there are exceptional cases when limit cycles exist only if the fundamental mode is unstable. Explicit results are given for the special cases of two and three modes; the analysis is extendible to any number of modes.

Nonlinear Energy Transport Between Longitudinal Acoustic Modes in Cylindrical Combustion Chambers

Jahnke, Craig C.; Culick, F. E. C.
Fonte: American Institute of Aeronautics and Astronautics Publicador: American Institute of Aeronautics and Astronautics
Tipo: Report or Paper; PeerReviewed Formato: application/pdf
Publicado em //1994
Relevância na Pesquisa
46.4%
The second order nonlinear longitudinal acoustics in a cylindrical combustion chamber are studied for the case of an unstable second mode. A modal analysis is undertaken and a continuation method is used to determine the limit cycle behavior of the time dependent amplitudes of the acoustic modes as functions of the linear stability of the unstable acoustic mode. It is shown that if an insufficient number of modes are included in the truncated system, bifurcations of the primary limit cycle occur. The energy in the limit cycles is analyzed and the bifurcations are shown to occur as a means of increasing the amount of energy transfer out of the unstable acoustic mode and into the stable acoustic modes through the nonlinear terms.