Inertial Modes Research Articles

Hydrodynamic Stability Analysis of the Neutron Star Core

Abstract Hydrodynamic instabilities and turbulence in neutron stars have been suggested to be related to observable spin variations in pulsars, such as spin glitches, timing noise, and precession (nutation). Accounting for the stabilizing effects of the stellar magnetic field, we revisit the issue of whether the inertial modes of a neutron star can become unstable when the neutron and proton condensates flow with respect to one another. The neutron and proton condensates are coupled through the motion of imperfectly pinned vorticity (vortex slippage) and vortex-mediated scattering (mutual friction). Previously-identified two-stream instabilities that occur when the two condensates rotate with respect to one another in the outer core are stabilized by the toroidal component of the magnetic field. This stabilization occurs when the Alfvén speed of the toroidal component of the magnetic field becomes larger than the relative rotational velocity of the condensates, corresponding to toroidal field strengths in excess of ≃1010 G. In contrast with previous studies, we find that spin-down of a neutron star under a steady torque is stable. The Donnelly–Glaberson instability is not stabilized by the magnetic field and could play an important role if neutron stars undergo precession.

Inertia-driven jetting regimes in microfluidic coflows

Inertial breakup modes are identified at microscale for co-flowing jets using a high-pressure mixer.

Nonlinear thermal inertial waves in rotating fluid spheres

ABSTRACTWhen the Prandtl number is sufficiently small, the rotational constraint on convection in rotating systems is largely broken by inertial effects and the corresponding convective instability in leading order is described by non-dissipative thermal inertial waves propagating in either prograde or retrograde direction while buoyancy forces maintain the waves against viscous damping at next order. We investigate, via direct numerical simulation using a finite element method, the nonlinear properties of thermal inertial waves in rapidly rotating, self-gravitating, internally heated Boussinesq fluid spheres, concentrating on the liquid metal gallium with the Prandtl number . We analyze the nonlinear numerical solution by decomposing it into the spectrum of mathematically complete spherical inertial modes. Different forms of nonlinear convection, representing a sequence of bifurcations, are identified with increasing Rayleigh number. Near the threshold of convective instability, we show that the primary bifurcation is supercritical, dominated by a single thermal-inertial-wave mode, and consistent with the prediction of the existing asymptotic theory. At moderately supercritical Rayleigh numbers, we reveal that various thermal-inertial-wave modes with different radial/azimuthal/equatorial symmetries – which are convectively excited and maintained – become nonlinearly interactive, progressively breaking the temporal, radial, azimuthal and equatorial symmetries of the primary solution and, then, leading to a weakly turbulent state.

Mechanisms for magnetic field generation in precessing cubes

ABSTRACTIt is shown that flows in precessing cubes develop at certain parameters large axisymmetric components in the velocity field which are large enough to either generate magnetic fields by themselves, or to contribute to the dynamo effect if inertial modes are already excited and acting as a dynamo. This effect disappears at small Ekman numbers. The critical magnetic Reynolds number also increases at low Ekman numbers because of turbulence and small-scale structures.

Mechanism of bone-conducted hearing: mathematical approach.

For better understanding of bone-conducted (BC) hearing, a mechanical BC model is formulated using the Wentzel-Kramers-Brillouin (WKB) method. The BC hearing can be generally described by three main mechanisms: (1) cochlear fluid inertia, (2) in-phase motion of the outer bony shell, and (3) out-of-phase motion of the outer bony shell. Specifically, the second and third mechanisms can be identically explained by symmetric pressure compression-expansion and anti-symmetric compression-expansion, respectively. In this study, simulation results show that both the symmetric and anti-symmetric compression-expansion modes become significant at frequencies above 7 kHz while the fluid inertial mode is dominant at lower frequencies. The density difference between the scala fluid and soft cells of basilar membrane and the amplitude of the anti-symmetric compression-expansion input are identified as the difference between the air conduction and bone conduction. The natural frequency of the cochlear duct wall determines the magnitudes between the three mechanism and is approximated to be in the order of 10MHz and above.

Properties of conditionally filtered equations: Conservation, normal modes, and variational formulation

Conditionally filtered equations have recently been proposed as a basis for modeling the atmospheric boundary layer and convection. Conditional filtering decomposes the fluid into a number of categories or components, such as convective updraughts and the background environment, and derives governing equations for the dynamics of each component. Because of the novelty and unfamiliarity of these equations, it is important to establish some of their physical and mathematical properties and to examine whether their solutions might behave in counterintuitive or even unphysical ways. It is also important to understand the properties of the equations in order to develop suitable numerical solution methods. The conditionally filtered equations are shown to have conservation laws for mass, entropy, momentum or axial angular momentum, energy, and potential vorticity. The normal modes of the conditionally filtered equations include the usual acoustic, inertio‐gravity, and Rossby modes of the standard compressible Euler equations. In addition, the equations support modes with different perturbations in the different fluid components that resemble gravity modes and inertial modes but with zero pressure perturbation. These modes make no contribution to the total filter‐scale fluid motion, and their amplitude diminishes as the filter scale diminishes. Finally, it is shown that the conditionally filtered equations have a natural variational formulation, which can be used as a basis for systematically deriving consistent approximations.

Convective Excitation of Inertial Modes in Binary Neutron Star Mergers.

We present the first very long-term simulations (extending up to ∼140 ms after merger) of binary neutron star mergers with piecewise polytropic equations of state and in full general relativity. Our simulations reveal that, at a time of 30-50ms after merger, parts of the star become convectively unstable, which triggers the excitation of inertial modes. The excited inertial modes are sustained up to several tens of milliseconds and are potentially observable by the planned third-generation gravitational-wave detectors at frequencies of a few kilohertz. Since inertial modes depend on the rotation rate of the star and they are triggered by a convective instability in the postmerger remnant, their detection in gravitational waves will provide a unique opportunity to probe the rotational and thermal state of the merger remnant. In addition, our findings have implications for the long-term evolution and stability of binary neutron star remnants.

Single-mode nonlinear Langevin emulation of magnetohydrodynamic turbulence.

Based on the Langevin equation of Brownian motion, we present a simple model that emulates a typical mode in incompressible magnetohydrodynamic turbulence, providing a demonstration of several key properties. The model equation is consistent with von Kármán decay law and Kolmogorov's symmetries. We primarily focus on the behavior of inertial range modes, although we also attempt to include some properties of the large-scale modes. Dissipation scales are not considered. Results from the model are compared with results from published direct numerical simulations.

Inertial Waves and Steady Flows in a Liquid Filled Librating Cylinder

The fluid flow in a non-uniformly rotating (librating) cylinder about a horizontal axis is experimentally studied. In the absence of librations the fluid performs a solid-body rotation together with the cavity. Librations lead to the appearance of steady zonal flow in the whole cylinder and the intensive steady toroidal flows near the cavity corners. If the frequency of librations is twice lower than the mean rotation rate the inertial waves are excited. The oscillating motion associated with the propagation of inertial wave in the fluid bulk leads to the appearance of an additional steady flow in the Stokes boundary layers on the cavity side wall. In this case the heavy particles of the visualizer are assembled on the side wall into ring structures. The patterns are determined by the structure of steady flow, which in turn depends on the number of reflections of inertial wave beams from the cavity side wall. For some frequencies, inertial waves experience spatial resonance, resulting in inertial modes, which are eigenmodes of the cavity geometry. The resonance of the inertial modes modifies the steady flow structure close to the boundary layer that is manifested in the direct rebuilding of patterns. It is shown that the intensity of zonal flow, as well as the intensity of steady flows excited by inertial waves, is proportional to the square of the amplitude of librations.

Greenland shark (Somniosus microcephalus) feeding behavior on static fishing gear, effect of SMART (Selective Magnetic and Repellent-Treated) hook deterrent technology, and factors influencing entanglement in bottom longlines.

The Greenland Shark (Somniosus microcephalus) is the most common bycatch in the Greenland halibut (Reinhardtius hippoglossoides) bottom longline fishery in Cumberland Sound, Canada. Historically, this inshore fishery has been prosecuted through the ice during winter but winter storms and unpredictable landfast ice conditions since the mid-1990s have led to interest in developing a summer fishery during the ice-free season. However, bycatch of Greenland shark was found to increase substantially with 570 sharks captured during an experimental Greenland halibut summer fishery (i.e., mean of 6.3 sharks per 1,000 hooks set) and mortality was reported to be about 50% due in part to fishers killing sharks that were severely entangled in longline gear. This study investigated whether the SMART (Selective Magnetic and Repellent-Treated) hook technology is a practical deterrent to Greenland shark predation and subsequent bycatch on bottom longlines. Greenland shark feeding behavior, feeding kinematics, and variables affecting entanglement/disentanglement and release are also described. The SMART hook failed to deter Greenland shark predation, i.e., all sharks were captured on SMART hooks, some with more than one SMART hook in their jaw. Moreover, recently captured Greenland sharks did not exhibit a behavioral response to SMART hooks. In situ observations of Greenland shark feeding show that this species uses a powerful inertial suction mode of feeding and was able to draw bait into the mouth from a distance of 25–35 cm. This method of feeding is suggested to negate the potential deterrent effects of electropositive metal and magnetic alloy substitutions to the SMART hook technology. The number of hooks entangled by a Greenland shark and time to disentangle and live-release a shark was found to increase with body length.

Rotating thermal convection in liquid gallium: multi-modal flow, absent steady columns

Earth’s magnetic field is generated by convective motions in its liquid metal core. In this fluid, the heat diffuses significantly more than momentum, and thus the Prandtl number $Pr$ is well below unity. The thermally driven convective flow dynamics of liquid metals are very different from moderate-$Pr$ fluids, such as water and those used in current dynamo simulations. In order to characterise rapidly rotating thermal convection in low-$Pr$ number fluids, we have performed laboratory experiments in a cylinder of aspect ratio $\unicode[STIX]{x1D6E4}=1.94$ using liquid gallium ($Pr\simeq 0.025$) as the working fluid. The Ekman number varies from $E\simeq 5\times 10^{-6}$ to $5\times 10^{-5}$ and the Rayleigh number varies from $Ra\simeq 2\times 10^{5}$ to $1.5\times 10^{7}$. Using spectral analysis stemming from point-wise temperature measurements within the fluid and measurements of the Nusselt number $Nu$, we characterise the different styles of low-$Pr$ rotating convective flow. The convection threshold is first overcome in the form of container-scale inertial oscillatory modes. At stronger forcing, sidewall-attached modes are identified for the first time in liquid metal laboratory experiments. These wall modes coexist with the bulk oscillatory modes. At $Ra$ well below the values where steady rotating columnar convection occurs, the bulk flow becomes turbulent. Our results imply that rotating convective flows in liquid metals do not develop in the form of quasisteady columns, as in moderate-$Pr$ fluids, but in the form of oscillatory convective motions. Thus, thermally driven flows in low-$Pr$ geophysical and astrophysical fluids can differ substantively from those occurring in $Pr\simeq 1$ models. Furthermore, our experimental results show that relatively low-frequency wall modes are an essential dynamical component of rapidly rotating convection in liquid metals.

Axisymmetric inertial modes in a spherical shell at low Ekman numbers

We investigate the asymptotic properties of axisymmetric inertial modes propagating in a spherical shell when viscosity tends to zero. We identify three kinds of eigenmodes whose eigenvalues follow very different laws as the Ekman number$E$becomes very small. First are modes associated with attractors of characteristics that are made of thin shear layers closely following the periodic orbit traced by the characteristic attractor. Second are modes made of shear layers that connect the critical latitude singularities of the two hemispheres of the inner boundary of the spherical shell. Third are quasi-regular modes associated with the frequency of neutral periodic orbits of characteristics. We thoroughly analyse a subset of attractor modes for which numerical solutions point to an asymptotic law governing the eigenvalues. We show that three length scales proportional to$E^{1/6}$,$E^{1/4}$and$E^{1/3}$control the shape of the shear layers that are associated with these modes. These scales point out the key role of the small parameter$E^{1/12}$in these oscillatory flows. With a simplified model of the viscous Poincaré equation, we can give an approximate analytical formula that reproduces the velocity field in such shear layers. Finally, we also present an analysis of the quasi-regular modes whose frequencies are close to$\sin (\unicode[STIX]{x03C0}/4)$and explain why a fluid inside a spherical shell cannot respond to any periodic forcing at this frequency when viscosity vanishes.

Triadic resonances in the wide-gap spherical Couette system

The spherical Couette system, consisting of a viscous fluid between two differentially rotating concentric spheres, is studied using numerical simulations and compared with experiments performed at BTU Cottbus-Senftenberg, Germany. We concentrate on the case where the outer boundary rotates fast enough for the Coriolis force to play an important role in the force balance, and the inner boundary rotates slower or in the opposite direction as compared to the outer boundary. As the magnitude of differential rotation is increased, the system is found to transition through three distinct hydrodynamic regimes. The first regime consists of the emergence of the first non-axisymmetric instability. Thereafter one finds the onset of ‘fast’ equatorially antisymmetric inertial modes, with pairs of inertial modes forming triadic resonances with the first instability. A further increase in the magnitude of differential rotation leads to the flow transitioning to turbulence. Using an artificial excitation, we study how the background flow modifies the inertial mode frequency and structure, thereby causing departures from the eigenmodes of a full sphere and a spherical shell. We investigate triadic resonances of pairs of inertial modes with the fundamental instability. We explore possible onset mechanisms through numerical experiments.

On triadic resonance as an instability mechanism in precessing cylinder flow

Contained rotating flows subject to precessional forcing are well known to exhibit rapid and energetic transitions to disorder. Triadic resonance of inertial modes has been previously proposed as an instability mechanism in such flows, and that idea was developed into a successful model for predicting instability in a cylindrical container when departures from solid-body rotation are sufficiently small. Using direct numerical simulation and dynamic mode decomposition, we analyse instabilities of precessing cylinder flows whose three-dimensional basic states, steady in the gimbal frame of reference, may depart substantially from solid-body rotation. In the gimbal frame, the instability can be interpreted as resulting from a supercritical Hopf bifurcation that results in a limit-cycle flow. In the cylinder frame of reference, the basic state is a rotating wave with azimuthal wavenumber $m=1$, and the instability satisfies triadic-resonance conditions with the instability mode maintaining a fixed orientation with respect to the basic state. Thus, we are able to demonstrate the existence of two alternative but congruent explanations for the instability. Additionally, we show that basic states may depart substantially from solid-body rotation even with modest cylinder tilt angles, and growth rates for instabilities may be sufficiently large that nonlinear saturation to disordered states can occur within approximately ten cylinder revolutions, in agreement with experimental observations.

Precession effects on a liquid planetary core

Motivated by the desire to understand the rich dynamics of precessionally driven flow in a liquid planetary core, we investigate, through numerical simulations, the precessing fluid motion in a rotating cylindrical annulus, which simultaneously possesses slow precession. The same problemhas been studied extensively in cylinders, where the precessing flow is characterized by three key parameters: the Ekman number E, the Poincaré number Po and the radius-height aspect ratio Γ. While in an annulus, there is another parameter, the inner-radius-height aspect ratio ϒ, which also plays an important role in controlling the structure and evolution of the flow. By decomposing the nonlinear solution into a set of inertial modes, we demonstrate the properties of both weakly and moderately precessing flows. It is found that, when the precessional force is weak, the flow is stable with a constant amplitude of kinetic energy. As the precessional force increases, our simulation suggests that the nonlinear interaction between the boundary effects and the inertial modes can trigger more turbulence, introducing a transitional regime of rich dynamics to disordered flow. The inertial mode u111, followed by u113 or u112, always dominates the precessing flow when 0.001 ≤ Po ≤ 0.05, ranging from weak to moderate precession. Moreover, the precessing flow in an annulus shows more stability than in a cylinder which is likely to be caused by the effect of the inner boundary that restricts the growth of resonant and non-resonant inertial modes. Furthermore, the mechanism of triadic resonance is not found in the transitional regime from a laminar to disordered flow.

Using gravitational-wave data to constrain dynamical tides in neutron star binaries

We discuss the role of dynamical tidal effects for inspiralling neutron star binaries, focussing on features that may be considered "unmodelled" in gravitational-wave searches. In order to cover the range of possibilities, we consider i) individual oscillation modes becoming resonant with the tide, ii) the elliptical instability, where a pair of inertial modes exhibit a nonlinear resonance with the tide, and iii) the non-resonant p-g instability which may arise as high order p- and g-modes in the star couple nonlinearly to the tide. In each case, we estimate the amount of additional energy loss that needs to be associated with the dynamical tide in order for the effect to impact on an observed gravitational-wave signal. We explore to what extent the involved neutron star physics may be considered known and how one may be able to use observational data to constrain theory.

Review on Multi-Degree-of-Freedom Piezoelectric Motion Stage

Multi-degree-of-freedom (multi-DOF) piezoelectric motion stage is an important part of modern manufacturing industry. In order to perform high precision outputs in a large travel range, various multi-DOF piezoelectric motion stages have been proposed and applied for decades. This paper is concentrated on reviewing the research on a piezoelectric motion stage. It involves the major components, operating principles of different actuation modes, design schemes, and structures of the multi-DOF piezoelectric motion stages. In this paper, the summary and classification of the actuation modes used in the multi-DOF piezoelectric motion stages are presented, mainly including the resonance actuation mode, inertial actuation mode, clamping and feeding actuation mode, and direct actuation mode; the multi-DOF piezoelectric motion stage is divided into multi-actuator type and single-actuator type according to the number of actuators used in the motion stage, in which the multi-actuator type includes series, parallel, and series–parallel types from the viewpoint of structure mechanism; in addition, the output performances of the stages are compared, and the state of the arts of the stages are discussed and summarized; the further efforts and future research perspectives are highlighted.

Осреднённые течения, возбуждаемые инерционными модами в либрирующем цилиндре

Steady flow excited by oscillatory motion of the fluid in a non-uniformly rotating (librating) cylinder is experimentally investigated. Librations lead to the propagation of inertial waves, which are born near the junction of the side and end walls of the cavity. As a result of multiple reflections from the cavity walls, the inertial waves experience a spatial resonance, exciting so-called inertial modes. The latter are a system of vortices in which the direction of rotation of the fluid varies during the period of librations. It is found that the oscillatory motion in the fluid bulk leads to the appearance of intense steady flow in the Stokes boundary layer on the side wall of the cylinder. The flow structure has the form of a system of averaged toroidal vortices located along the entire side cavity wall. The number of vortices is determined by the axial wave number of the excited mode and does not depend on the radial wave number. It is shown that the intensity of the steady flow is proportional to the square of the libration amplitude and strongly depends on the number of the mode.

Tilted incompressible Coriolis modes in spheroids

The incompressible flow of a uniform fluid, which fills a rigid spheroid rotating about an arbitrary axis fixed in an inertial frame, is dominated at small Rossby and Ekman numbers by the rotation through the Coriolis force. The effects of rotation on the flow can be found by treating the Coriolis force modified by a pressure gradient as a skew-symmetric bounded linear operator $\boldsymbol{{\mathcal{C}}}$ acting on smooth inviscid incompressible flows in the spheroid. It is shown that the space of incompressible polynomial flows of degree $N$ or less in the spheroid is invariant under $\boldsymbol{{\mathcal{C}}}$ for any $N$. The skew symmetry of $\boldsymbol{{\mathcal{C}}}$ implies the Coriolis operator $\boldsymbol{{\mathcal{C}}}$ is non-defective for such flows with an orthogonal set of eigenmodes (inertial and geostrophic modes) which form a basis for the finite-dimensional space of spheroidal polynomial flows. The eigenmodes are tilted if the rotation axis is not aligned with the symmetry axis of the spheroid. The non-defective property of $\boldsymbol{{\mathcal{C}}}$ enables enumeration of the modes and proof of their completeness using the Weierstrass polynomial approximation theorem. The fundamental tool, which is required to establish invariance of spheroidal polynomial flows under $\boldsymbol{{\mathcal{C}}}$ and completeness of the Coriolis modes, is that the solution of the polynomial Poisson–Neumann problem, i.e. Poisson’s equation with Neumann boundary condition and polynomial data, in a spheroid is a polynomial. The Coriolis modes of degree one and all geostrophic modes are explicitly constructed. Only the modes of degree one have non-zero angular momentum in the boundary frame.

Resonant tidal excitation of oscillation modes in merging binary neutron stars: Inertial-gravity modes

In coalescing neutron star (NS) binaries, tidal force can resonantly excite low-frequency (< 500 Hz) oscillation modes in the NS, transferring energy between the orbit and the NS. This resonant tide can induce phase shift in the gravitational waveforms, and potentially provide a new window of studying NS interior using gravitational waves. Previous works have considered tidal excitations of pure g-modes (due to stable stratification of the star) and pure inertial modes (due to Coriolis force), with the rotational effect treated in an approximate manner. However, for realistic NSs, the buoyancy and rotational effects can be comparable, giving rise to mixed inertial-gravity modes. We develop a non-perturbative numerical spectral code to compute the frequencies and tidal coupling coefficients of these modes. We then calculate the phase shift in the gravitational waveform due to each resonance during binary inspiral. We adopt polytropic NS models with a parameterized stratification. We derive relevant scaling relations and survey how the phase shift depends on various properties of the NS. We find that for canonical NSs (with mass M = 1.4M_sun and radius R = 10 km) and modest rotation rates (< 300 Hz), the gravitational wave phase shift due to a resonance is generally less than 0.01 radian. But the phase shift is a strong function of R and M, and can reach a radian or more for low-mass NSs with larger radii (R > 15 km). Significant phase shift can also be produced when the combination of stratification and rotation gives rise to a very low frequency (< 20 Hz in the inertial frame) modified g-mode. We also find that some inertial modes can be strongly affected by stratification, and that the m = 1 r-mode, previously identified to have a small but finite inertial-frame frequency based on the Cowling approximation, in fact has essentially zero frequency, and therefore cannot be excited.