Small-amplitude Analysis Research Articles

Ion-acoustic waves in dense magneto-rotating quantum plasma

A theoretical study is presented for the oblique propagation of linear and nonlinear ion acoustic waves in a dense electron-ion quantum plasma, as that found in dense astrophysical objects like white dwarfs, rotating around an axis at an angle θ with the direction of the constant magnetic field B→=B0ẑ. In the absence of exact analytical solutions, we look for approximate ones by applying different approximation techniques like linearization, reductive perturbation, phase portraits, etc. The linear dispersion relation, obtained as a quadratic equation in the plasma frequency ω2, reveals interesting features. The small amplitude analysis for the nonlinear waves, using the reductive perturbation technique, yields the Korteweg–de Vries equation, whose solutions are solitary waves. The effects of various physical parameters like speed and angle of rotation, strength of the magnetic field, the quantum diffraction term, etc., on the shape of the nonlinear structures, are investigated numerically. It is observed that the different plasma parameters have similar effects on both small and arbitrary amplitude waves—stronger magnetic field, larger quantum effects, and higher speed of rotation decrease their width. Furthermore, as the angle between the rotation axis and magnetic axis decreases, i.e., the rotation is aligned with the direction of the magnetic field, the waves get sharper. Additionally, the energy of the small amplitude solitary wave decreases with an increase in the speed of rotation and stronger quantum effects.

Electron acoustic waves in a 2-electron, dissipative, quantum magneto plasma

A theoretical study is presented for the oblique propagation of linear and nonlinear electron acoustic waves (EAWs) in a two-electron population, dissipative, quantum magnetoplasma. The linear dispersion relation yields a complex plasma wave frequency $\omega$ , denoting a decaying wave, with the amount of damping directly related to the strength of the dissipative force. The small amplitude analysis for the nonlinear structures using the reductive perturbation technique, gives the Korteweg-de Vries-Burgers equation. The numerical solutions depict a soliton for negligible dissipation and a monotonic shock for relatively large dissipation. For intermediate values, the solutions are damped oscillations. The dispersion of the EAWs gets enhanced with increase in strength of the magnetic field and a higher value of the ratio of hot to cold electron density. The stability analysis is also carried out, and the corresponding phase portraits are plotted to observe the trajectories. Of the two critical points, one is a saddle point, hence always unstable, while the other is a stable focus or a stable node, depending on the relative strengths of the plasma parameters.

Dissipative nonlinear waves in a gravitating quantum fluid

Nonlinear wave propagation is studied analytically in a dissipative, self-gravitating Bose Einstein condensate, in the framework of Gross-Pitaevskii model. The linear dispersion relation shows that the effect of dissipation is to suppress dynamical instabilities that destabilize the system. The small amplitude analysis using reductive perturbation technique is found to yield a modified form of KdV equation. The soliton energy, amplitude and velocity are found to decay with time, whereas the soliton width increases, such that the soliton exists for a finite time only

Diffusive Stability of Spatially Periodic Solutions of the Brusselator Model

Applying the Lyapunov–Schmidt reduction approach introduced by Mielke and Schneider in their analysis of the fourth-order scalar Swift–Hohenberg equation, we carry out a rigorous small-amplitude stability analysis of Turing patterns for the canonical second-order system of reaction–diffusion equations given by the Brusselator model. Our results confirm that stability is accurately predicted in the small-amplitude limit by the formal Ginzburg–Landau amplitude equations, rigorously validating the standard weakly unstable approximation and the Eckhaus criterion.

A fast Chebyshev method for simulating flexible-wing propulsion

We develop a highly efficient numerical method to simulate small-amplitude flapping propulsion by a flexible wing in a nearly inviscid fluid. We allow the wing's elastic modulus and mass density to vary arbitrarily, with an eye towards optimizing these distributions for propulsive performance. The method to determine the wing kinematics is based on Chebyshev collocation of the 1D beam equation as coupled to the surrounding 2D fluid flow. Through small-amplitude analysis of the Euler equations (with trailing-edge vortex shedding), the complete hydrodynamics can be represented by a nonlocal operator that acts on the 1D wing kinematics. A class of semi-analytical solutions permits fast evaluation of this operator with O(Nlog⁡N) operations, where N is the number of collocation points on the wing. This is in contrast to the minimum O(N2) cost of a direct 2D fluid solver. The coupled wing–fluid problem is thus recast as a PDE with nonlocal operator, which we solve using a preconditioned iterative method. These techniques yield a solver of near-optimal complexity, O(Nlog⁡N), allowing one to rapidly search the infinite-dimensional parameter space of all possible material distributions and even perform optimization over this space.

Optimization and small-amplitude analysis of Purcell's three-link microswimmer model.

This work studies the motion of Purcell's three-link microswimmer in viscous flow, by using perturbation expansion of its dynamics under small-amplitude strokes. Explicit leading-order expressions and next-order correction terms for the displacement of the swimmer are obtained for the cases of a square or circular gait in the plane of joint angles. The correction terms demonstrate the reversal in movement direction for large stroke amplitudes, which has previously only been shown numerically. In addition, asymptotic expressions for Lighthill's energetic efficiency are obtained for both gaits. These approximations enable calculating optimal stroke amplitudes and swimmer's geometry (i.e. ratio of links' lengths) for maximizing either net displacement or Lighthill's efficiency.

Effects of ionization and ion loss on dust ion-acoustic solitary waves in a collisional dusty plasma with suprathermal electrons

The combined effects of ionization, ion loss, and electron suprathermality on dust ion-acoustic solitary waves in a collisional dusty plasma are examined. Carrying out a small but finite amplitude analysis, a damped Korteweg-de Vries (dK–dV) equation is derived. The damping term decreases with the increase of the spectral index and saturates for Maxwellian electrons. Choosing typical plasma parameters, the analytical approximate solution of the dK-dV equation is numerically analyzed. We first neglect the ionization and ion loss effects and account only for collisions to estimate the relative importance between these damping terms which can act concurrently. Interestingly, we found that as the suprathermal character of the electrons becomes important, the strength of the collisions related dissipation becomes more important and causes the dust ion-acoustic solitary wave amplitude to decay more rapidly. Moreover, the collisional damping may largely prevail over the ionization and ion loss related damping. The latter becomes more effective as the electrons evolve far away from their thermal equilibrium. Our results complement and provide new insights into previously published work on this problem.

Ca2+-Sensitivity and Elementary Steps of the Cross-Bridge Cycle in Papillary Muscle Fibers from the Troponin C (TnC)-A8V Knock-In Mouse, which Exhibits Hypertrophic Cardiomyopathy (HCM)

Mutations of TnC alter its function by two mechanisms: changing Ca2+-affinity, and/or changing the TnC-TnI interaction. A mutation, A8V, exhibits HCM and diastolic dysfunction in both mice and humans as demonstrated by echocardiography. Here we have used homozygous knock-in TnC mutant A8V mice (7-8 months-old) in which the myofilaments carry only mutant protein. These mice develop HCM at 3 months (Circ Cardiovasc Genetic 2015, in press). This mutation is located in the N-helix of TnC, a region known to affect Ca2+-affinity. Here we studied the effects of TnC-A8V mutation in small fiber bundles (90-110µm diameter) from skinned papillary muscles. These were activated at 25°C under the intracellular ionic conditions of working cardiomyocytes (5mM ATP, 1mM Mg2+, 8mM phosphate, 200mM ionic strength, and pH 7.0). With the pCa-tension study, Ca2+-sensitivity increased (3x) and cooperativity decreased (0.8x) compared to the wild-type. The maximum active tension decreased slightly (0.95x). The elementary steps of the cross-bridge cycle were investigated by small-amplitude sinusoidal analysis. With the ATP study, there was no significant change in the ATP affinity (K1) to the myosin head, but cross-bridge detachment rate (k2) and its equilibrium constant (K2) increased to 1.5x, which causes slightly fewer force-generating cross-bridges. With the phosphate study, no effect of the mutation on the rate constants of the force-generation step (k4, k-4) or Pi-release step (1/K5) was observed. When the rigor state was induced, there was no effect of the mutation on stiffness, implying that force/cross-bridge is not altered. We infer from these observations that TnC-A8V mutation recruits more cycling cross-bridges during systole as a result of increased myofilament Ca2+-sensitivity, but unable to relax adequately resulting in diastolic dysfunction, followed by HCM development.

Rarefaction solitons initiated by sheath instability

The instability of the cathode sheath initiated by the cold energetic electron beam is studied by the one-dimensional fluid model. Numerical simulations show the generation of travelling rarefaction solitons at the cathode. It is obtained that the parameters of these solitons strongly depend on the parameters of electron beam. The “stretched” variables are derived using the small-amplitude analysis. These variables are used in order to obtain the Korteweg-de Vries equation describing the propagation of the rarefaction solitons through the plasma with cold energetic electron beam.

Theory of self-resonance after inflation. II. Quantum mechanics and particle-antiparticle asymmetry

We further develop a theory of self-resonance after inflation in a large class of models involving multiple scalar fields. We concentrate on inflaton potentials that carry an internal symmetry, but also analyze weak breaking of this symmetry. This is the second part of a two-part series of papers. Here in part 2 we develop an understanding of the resonance structure from the underlying many-particle quantum mechanics. We begin with a small-amplitude analysis, which obtains the central resonant wave numbers, and relate it to perturbative processes. We show that the dominant resonance structure is determined by (i) the nonrelativistic scattering of many quantum particles and (ii) the application of Bose-Einstein statistics to the adiabatic and isocurvature modes, as introduced in part 1 [M. P. Hertzberg et al., Phys. Rev. D 90, 123528 (2014)]. Other resonance structures are understood in terms of annihilations and decays. We set up Bunch-Davies vacuum initial conditions during inflation and track the evolution of modes including Hubble expansion. In the case of a complex inflaton carrying an internal U(1) symmetry, we show that when the isocurvature instability is active, the inflaton fragments into separate regions of $\ensuremath{\phi}$-particles and anti-$\ensuremath{\phi}$-particles. We then introduce a weak breaking of the U(1) symmetry; this can lead to baryogenesis, as shown by some of us recently [M. P. Hertzberg and J. Karouby, Phys. Lett. B 737, 34 (2014); Phys. Rev. D 89, 063523 (2014)]. Then using our results, we compute corrections to the particle-antiparticle asymmetry from this preheating era.

Nonlinear stable steady solutions to the Ostroumov problem

We investigate the linear and weakly nonlinear solutions to a convection problem that was first studied by Ostroumov in 1947. The problem pertains to the stability of the equations governing convective motion in an infinite vertical fluid layer that is heated from below. Ostroumov’s linear stability analysis yields instability threshold conditions that are characterized by zero wavenumber for the Fourier mode in the vertical direction and by eigenfunctions that are independent of the vertical coordinate. Thus, any undertaking at determining the supercritical nonlinear solutions and their stability through a small amplitude expansion fails. This failure is attributed to the fact that the nonlinear interaction of the linear modes vanish identically. In this paper, we put forth exact and stable similarity type solutions to the Ostroumov problem. These solutions are characterized by the same linear threshold conditions as Ostroumov’s solutions. Moreover, we are able to extend the analysis to the supercritical regime through a small amplitude analysis to obtain steady two-dimensional solutions for a small range of Prandtl numbers. These solutions are found to be stable to general two-dimensional, time-dependent disturbances. Furthermore, when the analysis is extended to the case where the fluid layer thickness is also allowed to be infinite, we find that the infinite two-dimensional fluid region becomes linearly unstable through a Batchelor–Nitsche (BN) instability mechanism. Thus, the nonlinear solutions are obtained through a long wavelength expansion, and consequently our analysis provides the nonlinear development of the BN instability for the case of an unstable and uniform density profile. Finally, numerical solutions of the nonlinear problem are also presented which shed light on the flow patterns and temperature distributions for both the vertical channel and the infinitely extended stratified region problems.

Turbulent mixing driven by spherical implosions. Part 1. Flow description and mixing-layer growth

AbstractWe present large-eddy simulations (LES) of turbulent mixing at a perturbed, spherical interface separating two fluids of differing densities and subsequently impacted by a spherically imploding shock wave. This paper focuses on the differences between two fundamental configurations, keeping fixed the initial shock Mach number ${\approx }1.2$, the density ratio (precisely $|A_0|\approx 0.67$) and the perturbation shape (dominant spherical wavenumber $\ell _0=40$ and amplitude-to-initial radius of $3\, \%$): the incident shock travels from the lighter fluid to the heavy fluid or, inversely, from the heavy to the light fluid. After describing the computational problem we present results on the radially symmetric flow, the mean flow, and the growth of the mixing layer. Turbulent statistics are developed in Part 2 (Lombardini, M., Pullin, D. I. & Meiron, D. I. J. Fluid Mech., vol. 748, 2014, pp. 113–142). A wave-diagram analysis of the radially symmetric flow highlights that the light–heavy mixing layer is processed by consecutive reshocks, and not by reverberating rarefaction waves as is usually observed in planar geometry. Less surprisingly, reshocks process the heavy–light mixing layer as in the planar case. In both configurations, the incident imploding shock and the reshocks induce Richtmyer–Meshkov (RM) instabilities at the density layer. However, we observe differences in the mixing-layer growth because the RM instability occurrences, Rayleigh–Taylor (RT) unstable scenarios (due to the radially accelerated motion of the layer) and phase inversion events are different. A small-amplitude stability analysis along the lines of Bell (Los Alamos Scientific Laboratory Report, LA-1321, 1951) and Plesset (J. Appl. Phys., vol. 25, 1954, pp. 96–98) helps quantify the effects of the mean flow on the mixing-layer growth by decoupling the effects of RT/RM instabilities from Bell–Plesset effects associated with geometric convergence and compressibility for arbitrary convergence ratios. The analysis indicates that baroclinic instabilities are the dominant effect, considering the low convergence ratio (${\approx } 2$) and rather high ($\ell >10$) mode numbers considered.

Longwave convection in a layer of binary mixture with modulated heat flux: weakly nonlinear analysis

We consider dynamics of a binary mixture layer subject to a modulated heat flux at the bottom. Nonlinear evolution for longwave synchronous mode is shown to be governed by a set of nonlocal amplitude equations, solvability conditions of a certain linear nonhomogeneous problem. For the superlattice combining two hexagonal lattices, the set of nonlocal equations can be reduced to the set of Landau equations with cubic and quadratic nonlinear terms. Although this set is conventional for a small-amplitude analysis, in the present work it is valid even for finite-amplitude regimes; the perturbations of both temperature and solute concentration are of order unity, only their gradients are small. Nontrivial matching with known limiting cases is found.

Chronic heart failure decreases cross-bridge kinetics in single skeletal muscle fibres from humans

Skeletal muscle function is impaired in heart failure patients due, in part, to loss of myofibrillar protein content, in particular myosin. In the present study, we utilized small-amplitude sinusoidal analysis for the first time in single human skeletal muscle fibres to measure muscle mechanics, including cross-bridge kinetics, to determine if heart failure further impairs contractile performance by altering myofibrillar protein function. Patients with chronic heart failure (n = 9) and controls (n = 6) were recruited of similar age and physical activity to diminish the potentially confounding effects of ageing and muscle disuse. Patients showed decreased cross-bridge kinetics in myosin heavy chain (MHC) I and IIA fibres, partially due to increased myosin attachment time (t(on)). The increased t(on) compensated for myosin protein loss previously found in heart failure patients by increasing the fraction of the total cycle time myosin is bound to actin, resulting in a similar number of strongly bound cross-bridges in patients and controls. Accordingly, isometric tension did not differ between patients and controls in MHC I or IIA fibres. Patients also had decreased calcium sensitivity in MHC IIA fibres and alterations in the viscoelastic properties of the lattice structure of MHC I and IIA fibres. Collectively, these results show that heart failure alters skeletal muscle contraction at the level of the myosin-actin cross-bridge, leading to changes in muscle mechanics which could contribute to impaired muscle function. Additionally, we uncovered a unique kinetic property of MHC I fibres, a potential indication of two distinct populations of cross-bridges, which may have important physiological consequences.

Flat-top oscillons in an expanding universe

Oscillons are extremely long lived, oscillatory, spatially localized field configurations that arise from generic initial conditions in a large number of non-linear field theories. With an eye towards their cosmological implications, we investigate their properties in an expanding universe. We (1) provide an analytic solution for one dimensional oscillons (for the models under consideration) and discuss their generalization to 3 dimensions, (2) discuss their stability against long wavelength perturbations and (3) estimate the effects of expansion on their shapes and life-times. In particular, we discuss a new, extended class of oscillons with surprisingly flat tops. We show that these flat topped oscillons are more robust against collapse instabilities in (3+1) dimensions than their usual counterparts. Unlike the solutions found in the small amplitude analysis, the width of these configurations is a non-monotonic function of their amplitudes.

A Coaxial Tube Model of the Cerebrospinal Fluid Pulse Propagation in the Spinal Column

The dynamics of the movement of the cerebrospinal fluid (CSF) may play an important role in the genesis of pathological neurological conditions such as syringomyelia, which is characterized by the presence of a cyst (syrinx) in the spinal cord. In order to provide sound theoretical grounds for the hypotheses that attribute the formation and growth of the syrinx to impediments to the normal movement of the CSF, it is necessary to understand various modes through which CSF pulse in the spinal column propagates. Analytical models of small-amplitude wave propagation in fluid-filled coaxial tubes, where the outer tube represents dura, the inner tube represents the spinal cord, and the fluid is the CSF, have been used to that end. However, so far, the tendency was to model one of the two tubes as rigid and to neglect the effect of finite thickness of the tube walls. The aim of this study is to extend the analysis in order to address these two potentially important issues. To that end, classical linear small-amplitude analysis of wave propagation was applied to a system consisting of coaxial tubes of finite thickness filled with inviscid incompressible fluid. General solutions to the governing equations for the case of harmonic waves in the long wave limit were replaced with the boundary conditions to yield the characteristic (dispersion) equation for the system. The four roots of the characteristic equation correspond to four modes of wave propagation, of which the first three are associated with significant motion of the CSF. For the normal range of parameters the speeds of the four modes are c(1)=13 ms, c(2)=14.7 ms, c(3)=30.3 ms, and c(4)=124.5 ms, which are well within the range of values previously reported in experimental and theoretical studies. The modes with the highest and the lowest speeds of propagation can be attributed to the dura and the spinal cord, respectively, whereas the remaining two modes involve some degree of coupling between the two. When the thickness of the spinal cord, is reduced below its normal value, the first mode becomes dominant in terms of the movement of the CSF, and its speed drops significantly. This suggests that the syrinx may be characterized by an abnormally low speed of the CSF pulse.

Altered crossbridge kinetics in the alphaMHC403/+ mouse model of familial hypertrophic cardiomyopathy.

A mutation in the cardiac beta-myosin heavy chain, Arg403Gln (R403Q), causes a severe form of familial hypertrophic cardiomyopathy (FHC) in humans. We used small-amplitude (0.25%) length-perturbation analysis to examine the mechanical properties of skinned left ventricular papillary muscle strips from mouse hearts bearing the R403Q mutation in the alpha-myosin heavy chain (alphaMHC403/+). Myofibrillar disarray with variable penetrance occurred in the left ventricular free wall of the alphaMHC403/+ hearts. In resting strips (pCa 8), dynamic stiffness was approximately 40% greater than in wild-type strips, consistent with elevated diastolic stiffness reported for murine hearts with FHC. At pCa 6 (submaximal activation), strip isometric tension was approximately 3 times higher than for wild-type strips, whereas at pCa 5 (maximal activation), tension was marginally lower. At submaximal calcium activation the characteristic frequencies of the work-producing (b) and work-absorbing (c) steps of the crossbridge were less in alphaMHC403/+ strips than in wild-type strips (b=11+/-1 versus 15+/-1 Hz; c= 58+/-3 versus 66+/-3 Hz; 27 degrees C). At maximal calcium activation, strip oscillatory power was reduced (0. 53+/-0.25 versus 1.03+/-0.18 mW/mm3; 27 degrees C), which is partly attributable to the reduced frequency b, at which crossbridge work is maximum. The results are consistent with the hypothesis that the R403Q mutation reduces the strong binding affinity of myosin for actin. Myosin heads may accumulate in a preforce state that promotes cooperative activation of the thin filament at submaximal calcium but blunts maximal tension and oscillatory power output at maximal calcium. The calcium-dependent effect of the mutation (whether facilitating or debilitating), together with a variable degree of fibrosis and myofibrillar disorder, may contribute to the diversity of clinical symptoms observed in murine FHC.

Mean Flow Evolution of a Baroclinically Unstable Potential Vorticity Front

Quasi-linear theory and numerical models are used to study the mean flow modification of a two-layer shallow water baroclinically unstable flow as a function of Rossby number. This flow has an upper-layer potential vorticity front overlying a quiescent lower layer and is used as a simple representation of the Gulf Stream. Quantities derived from an analytical expansion in the small meander amplitude limit of the (quasi-linear) equations are found to compare quantitatively well with numerical model simulations of the flow in small amplitude and to pertain qualitatively even beyond the instability equilibration, where the meander amplitude is as large as the meander wavelength. The baroclinic evolution is similar for all Rossby numbers, with differences arising from increased asymmetry of the flow with increasing Rossby number. The equilibration of the instability is similar for all Rossby numbers and is due to the acceleration of a strong barotropic shear. This acceleration is predicted from the small amplitude analysis. Quasigeostrophic diagnostics are shown to be useful even for large Rossby number flows such as the Gulf Stream. One qualitative difference that appears is that as the mean flow is modified, a lateral separation of the zonal mean potential vorticity front and the jet maximum appears, consistent with Gulf Stream observations. This feature is found only for finite Rossby number flows.

Rayleigh-Taylor Instability Experiments Examining Feedthrough Growth in an Incompressible, Convergent Geometry

We have performed Rayleigh-Taylor instability and feedthrough growth experiments in a 2D cylindrically convergent geometry by imploding gelatin cylinders with high-pressure gases. Instability growth was seeded by fabricating sinusoidal undulations on the outer surfaces of the cylinders, which were then observed during implosion with high-speed framing cameras. These experiments are the first to study feedthrough perturbation growth in an incompressible, convergent geometry. We compare the results to small-amplitude perturbation analysis and to a recent radiation-drive feedthrough experiment.

The instability of rotating fluid columns subjected to a weak external Coriolis force

Instabilities in rotating fluid columns subjected to a weak external Coriolis force are studied. External Coriolis force alters the base flow distorting circular streamlines of the unperturbed columns. The inviscid part of the modified flow (0,r,−2εr sin φ) is an exact solution of Euler equations. Here ε is the strength (nondimensional) of imposed Coriolis force. It is shown that this distortion leads to three-dimensional instabilities. The instability mechanism is generic. It occurs in many flows having circular streamlines distorted by a spectrum of modes cos mφ, sin mφ (m=2 corresponds to elliptical instability). The instabilities occur for wavelengths and frequencies at the intersection points of dispersion curves for the unperturbed columns. Moreover, the instabilities occur at the points of intersection for which modes are coupled with the external Coriolis mode. Here a case is studied where axisymmetric and helical modes are involved in interactions leading to fully three-dimensional flows. It is shown that rotating fluid columns are unstable to disturbances whose axial wavelengths lie in a band, whose width is proportional to the strength ε of imposed Coriolis force. Parametric resonance between a pair of inertial waves (natural modes of oscillation) and the external Coriolis mode is the physical explanation for the instabilities. Numerical results are presented on growth rates of mixed modes and on widths of unstable regions. The growth rates depend linearly on the strength of external Coriolis force. It is shown that viscosity shifts the instability tongues to positive values of ‖ε‖. The results of small-amplitude perturbation analysis are compared with full numerical simulations of the Navier–Stokes equations. Comments are made on competition between centrifugal and parametric instabilities in Taylor–Couette systems subjected to an external Coriolis force studied in Wiener et al. [J. Stat. Phys. 64, 913 (1991)] and Ning et al. [J. Stat. Phys. 64, 927 (1991)].