Finite Amplitude Effects Research Articles

Simulating Solar Near-surface Rossby Waves by Inverse Cascade from Supergranule Energy

Rossby waves are found at several levels in the Sun, most recently in its supergranule layer. We show that Rossby waves in the supergranule layer can be excited by an inverse cascade of kinetic energy from the nearly horizontal motions in supergranules. We illustrate how this excitation occurs using a hydrodynamic shallow-water model for a 3D thin rotating spherical shell. We find that initial kinetic energy at small spatial scales inverse cascades quickly to global scales, exciting Rossby waves whose phase velocities are similar to linear Rossby waves on the sphere originally derived by Haurwitz. Modest departures from the Haurwitz formula originate from nonlinear finite amplitude effects and/or the presence of differential rotation. Like supergranules, the initial small-scale motions in our model contain very little vorticity compared to their horizontal divergence, but the resulting Rossby waves are almost all vortical motions. Supergranule kinetic energy could have mainly gone into gravity waves, but we find that most energy inverse cascades to global Rossby waves. Since kinetic energy in supergranules is three or four orders of magnitude larger than that of the observed Rossby waves in the supergranule layer, there is plenty of energy available to drive the inverse-cascade mechanism. Tachocline Rossby waves have previously been shown to play crucial roles in causing seasons of space weather through their nonlinear interactions with global flows and magnetic fields. We briefly discuss how various Rossby waves in the tachocline, convection zone, supergranule layer, and corona can be reconciled in a unified framework.

Finite-amplitude sound propagation effects in volume backscattering measurements for fish abundance estimation

Scientific advice for governmental management of marine resources relies on acoustical observation methods. Quantification and identification of fish and plankton species are often achieved using multi-frequency acoustic data. Accurate measurements of backscattering cross-sections and volume backscattering coefficients are essential. Systematic errors from finite-amplitude sound propagation are demonstrated in experimental survey measurements on Atlantic mackerel using 120 kHz and 200 kHz echosounders and high power settings. Finite-amplitude signal distortion causes excess transmission loss that is not accounted for in fisheries acoustics today, other than by fixed limits on the maximum transmitted power. The demonstrated errors are of a magnitude that can seriously bias abundance estimation and species identification. It is shown how the finite-amplitude effects can be modelled and predicted quantitatively, within a framework of electroacoustic power budget equations. A method is provided to calculate related errors in echosounder calibration and oceanic measurement of acoustic volume backscattering. When accounting for finite-amplitude effects in echosounder signal processing, higher transmit powers can be used when needed to improve signal-to-noise ratio or extend measurement range. The results indicate that historical survey data can be adjusted for such errors using numerical simulations. The echosounder characteristics relevant to finite-amplitude effects can be determined by laboratory measurements.

Could Switchbacks Originate in the Lower Solar Atmosphere? II. Propagation of Switchbacks in the Solar Corona

Abstract The magnetic switchbacks observed recently by the Parker Solar Probe have raised the question about their nature and origin. One of the competing theories of their origin is the interchange reconnection in the solar corona. In this scenario, switchbacks are generated at the reconnection site between open and closed magnetic fields, and are either advected by an upflow or propagate as waves into the solar wind. In this paper we test the wave hypothesis, numerically modeling the propagation of a switchback, modeled as an embedded Alfvén wave packet of constant magnetic field magnitude, through the gravitationally stratified solar corona with different degrees of background magnetic field expansion. While switchbacks propagating in a uniform medium with no gravity are relatively stable, as reported previously, we find that gravitational stratification together with the expansion of the magnetic field act in multiple ways to deform the switchbacks. These include WKB effects, which depend on the degree of magnetic field expansion, and also finite-amplitude effects, such as the symmetry breaking between nonlinear advection and the Lorentz force. In a straight or radially expanding magnetic field the propagating switchbacks unfold into waves that cause minimal magnetic field deflections, while a super-radially expanding magnetic field aids in maintaining strong deflections. Other important effects are the mass uplift the propagating switchbacks induce and the reconnection and drainage of plasmoids contained within the switchbacks. In the Appendix, we examine a series of setups with different switchback configurations and parameters, which broaden the scope of our study.

Characteristics of thermospheric infrasound predicted using ray tracing and weakly non-linear waveform analyses.

Infrasonic signals refracted by thermal gradients in the rarefied upper atmosphere are modeled using a combination of ray tracing and weak shock theory to develop an understanding of thermospheric infrasound signals produced by energetic, transient sources. Canonical arrival structures in the form of u-wave signatures are identified for returns refracted at lower altitudes within the thermosphere, and possible multi-pathing produced by effective sound speed inflections are investigated to elucidate more complex arrival structures, which are found to be spatially localized. Variability in the source characteristics is investigated and it is found that whereas some waveform phase information is lost due to finite amplitude effects, arrival characteristics are strongly dependent on the peak overpressure near the source. Variability in the propagation path is considered using archived atmospheric specifications and implies that despite uncertainties related to the dynamic and sparsely sampled nature of the atmosphere, thermospheric signatures might be useful in estimating the yield for explosive sources. Last, thermospheric arrivals from a failed rocket launch, as well as those from several large chemical explosions, are analyzed and it is found that qualitative trends match those predicted, and analyses here provide additional insight into such signatures.

Two-dimensional spectroscopy beyond the perturbative limit: The influence of finite pulses and detection modes.

Ultra-fast and multi-dimensional spectroscopy gives a powerful looking glass into the dynamics of molecular systems. In particular, two-dimensional electronic spectroscopy (2DES) provides a probe of coherence and the flow of energy within quantum systems, which is not possible with more conventional techniques. While heterodyne-detected (HD) 2DES is increasingly common, more recently fluorescence-detected (FD) 2DES offers new opportunities, including single-molecule experiments. However, in both techniques, it can be difficult to unambiguously identify the pathways that dominate the signal. Therefore, the use of numerically modeling of 2DES is vitally important, which, in turn, requires approximating the pulsing scheme to some degree. Here, we employ non-perturbative time evolution to investigate the effects of finite pulse width and amplitude on 2DES signals. In doing so, we identify key differences in the response of HD and FD detection schemes, as well as the regions of parameter space where the signal is obscured by unwanted artifacts in either technique. Mapping out parameter space in this way provides a guide to choosing experimental conditions and also shows in which limits the usual theoretical approximations work well and in which limits more sophisticated approaches are required.

Transient time-delay focusing of shock waves in thin liquids

Abstract

Spatial characterization of high intensity focused ultrasound fields in the brain

High intensity focused ultrasound (HIFU) has seen widespread clinical adoption as a therapeutic tool, as it may be targeted noninvasively and without ionizing radiation. The converging sound waves induce thermal (e.g., ablation) and mechanical (e.g., radiation force) effects that may be localized to manipulate or destroy tissue. In addition to these primary effects, finite amplitude acoustic influences (e.g., second harmonic generation) become relevant due to the high pressure levels near the focal region. Specifically, improved localization of energy at sonic frequencies might be enabled by exploiting these effects. Here, through experimentally validated simulations, we demonstrate the relative importance of acoustical and material properties on the spatial and temporal properties of the resulting focal field, and the feasibility of attaining finite amplitude effects transcranially. While the relative amplitude of the second-order effect was found to scale with the source pressure and nonlinearity parameter, absorption and pulse modulation frequencies influenced both the distribution and spectral content of the field. These effects were observed for a water reference medium, and thus are likely to be even more prominent in tissue. Recent advancements in trans-skull focusing and understanding of the mechanical properties of neurons warrant further investigations, which will aim to explore these higher-order acoustic effects for noninvasive treatment of central nervous system diseases.

Nonlinear torsional wave propagation in a thin circular rod

Coupled nonlinear wave equations are obtained for the special case of wave motion in a thin rod with circular cross-section resulting from a large-amplitude torsional source. Lagrangian mechanics are employed to obtain the wave equations from expressions for the kinetic and potential energy densities in the rod due to an assumed displacement field. Longitudinal waves are generated at quadratic order in strain by the torsional wave through inertial (centrifugal) and elastic forces. Nonlinear effects on the torsional wave occur at cubic order in strain, due to both interaction with the longitudinal wave and strain-hardening shear nonlinearity. While the torsional wave propagates without dispersion, the longitudinal wave is subject to leading-order effects of geometrical dispersion when its wavelength is comparable to the rod radius. The case of weak nonlinearity is analyzed, for which the torsional wave propagates without nonlinear distortion while finite-amplitude effects generate a longitudinal wave. An analytical solution reveals that inertial and elastic forces resulting from a harmonic travelling torsional wave work against each other in the generation of the longitudinal second harmonic. Numerical solutions for transient source motion highlight interaction between the torsional and longitudinal modes near the source, and effects of geometrical dispersion on the longitudinal wave.

Imprint of local environment on fast radio burst observations

ABSTRACT When fast radio burst (FRB) waves propagate through the local (${\lesssim}\rm 1\, pc$) environment of the FRB source, electrons in the plasma undergo large-amplitude oscillations. The finite-amplitude effects cause the effective plasma frequency and cyclotron frequency to be dependent on the wave strength. The dispersion measure and rotation measure should therefore vary slightly from burst to burst for a repeating source, depending on the luminosity and frequency of the individual burst. Furthermore, free–free absorption of strong waves is suppressed due to the accelerated electrons’ reduced energy exchange in Coulomb collisions. This allows bright low-frequency bursts to propagate through an environment that would be optically thick to low-amplitude waves. Given a large sample of bursts from a repeating source, it would be possible to use the deficit of low-frequency and low-luminosity bursts to infer the emission measure of the local intervening plasma and its distance from the source. Information about the local environment will shed light on the nature of FRB sources.

Effective acoustic metamaterial homogenization based on Hamilton's principle with a multiple scales approximation.

This paper derives and demonstrates a one-dimensional acoustic metamaterial homogenization method. The homogenization method uses a multiple-scales approximation with Hamilton's principle, a weak-form representation of the dynamic equation. While the multiple-scales approximation makes the predicted effective material properties of this method inexact, the method is shown to be highly versatile. Analytical and numerical examples are given showing the ability of the homogenization method to account for viscosity and finite-amplitude effects.

Instability of finite-amplitude gravity–capillary progressive ring waves by an oscillating surface-piercing body

We investigate the instability of finite-amplitude progressive ring waves in deep water, which are radiated by the time-periodic oscillation of a half-submerged sphere (with radius ), under the influence of gravity and surface tension. We use direct numerical simulations of fully nonlinear wave–body interactions to quantify the temporal–spatial evolution of the base and perturbed outgoing ring wave fields, from which the stability of ring waves is analysed. The numerical simulation is based on a mixed Euler–Lagrangian quadratic boundary-element method and accounts for fully nonlinear wave–wave and wave–body interactions in the context of potential flow. We find that the progressive gravity–capillary ring waves (with frequency ) become unstable to (small-amplitude) radial cross-wave disturbances when the body-motion parameter exceeds the threshold value , where is the amplitude of body oscillation and is the wavenumber of the ring wave at subharmonic frequency . The predicted from nonlinear simulations under the assumption of ideal fluid, which decreases with increasing , is generally smaller than the experimental measurement of Tatsuno et al. (Rep. Res. Inst. Appl. Mech. Kyushu University, vol. 17, 1969, pp. 195–215) by approximately 50 %. When the viscous effects in body-surface and free-surface boundary layers are taken into account, the predicted matches the experimental data excellently. The unstable modes are characterized as the progressive radial cross-waves at the subharmonic frequency ( ) with the growth rates generally increasing with . The maximum growth rate is achieved for the cross-wave mode with the azimuthal wavenumber . These distinctive features of instability obtained in numerical simulations are consistent with the experimental observations. From the comparison with the weakly nonlinear analysis of Shen & Liu (J. Fluid Mech., vol. 869, 2019, pp. 439–467), it is found that inclusion of finite-amplitude ring wave effects generally reduces the growth rate of unstable modes but has an insignificant influence on the shape of unstable modes and the value of . Moreover, for moderately steep ring waves, nonlinear interactions of a few unstable modes can excite broadbanded unstable subharmonic cross-wave modes, leading to the formation/observation of distinctive non-axisymmetric wave patterns during long-time evolutions.

On the convergence of the normal form transformation in discrete Rossby and drift wave turbulence

We study numerically the region of convergence of the normal form transformation for the case of the Charney-Hasagawa-Mima (CHM) equation to investigate whether certain finite amplitude effects can be described in normal coordinates. We do this by taking a Galerkin truncation of four Fourier modes making part of two triads: one resonant and one non-resonant, joined together by two common modes. We calculate the normal form transformation directly from the equations of motion of our reduced model, successively applying the algorithm to calculate the transformation up to $7^\textrm{th}$ order to eliminate all non-resonant terms, and keeping up to $8$-wave resonances. We find that the amplitudes at which the normal form transformation diverge very closely match with the amplitudes at which a finite-amplitude phenomenon called $precession$ $resonance$ (Bustamante $et$ $al.$ 2014) occurs, characterised by strong energy transfers. This implies that the precession resonance mechanism cannot be explained using the usual methods of normal forms in wave turbulence theory, so a more general theory for intermediate nonlinearity is required.

Effect of finite amplitude of bottom corrugations on Fabry-Perot resonance of water waves.

Recently, the mechanism of Fabry-Perot (F-P) resonance in optics was extended to monochromatic water waves propagating in a domain with two patches of sinusoidal corrugations on an otherwise flat bottom. Assuming small-amplitude surface waves, an asymptotic linear analytical solution (ALAS) was derived by L. A. Couston etal. Phys. Rev. E 92, 043015 (2015).PLEEE81539-375510.1103/PhysRevE.92.043015 When resonance conditions are met, the ALAS predicts large amplification of the incident waves in the resonator area between the two patches of corrugations. Based on the ALAS, the amplitude of these standing waves is expected to increase exponentially with the relative amplitude of bottom corrugations (δ=d/h, where d is the corrugation amplitude and h the still water depth). In the present work, we examine the effects associated with the assumptions made in deriving the ALAS regarding the effect of a finite amplitude of bottom corrugation (i.e., finite value of δ), still in the linear wave framework. F-P resonance is studied by means of highly accurate numerical simulations, considering either the exact linear water wave problem (system A) or an approximate problem with a first-order expansion of the bottom boundary condition (system B). The numerical model is first validated on a Bragg resonance case, through comparisons with the ALAS, experimental measurements, and existing numerical simulations, showing its ability to represent well the so-called wave-number downshift of Bragg resonance (i.e., the slight decrease in the incident wave number where maximum resonance is reached in comparison with the value predicted by the ALAS). We then analyze how this downshift affects the F-P resonance, especially when the corrugations are of finite amplitude, i.e., δ varying from 0.05 to 0.4. The wave-number downshift appears to have a strong effect on the F-P resonance for δ>0.1: very low wave amplification manifests for the wave number predicted by the ALAS. However, when the incident wave number is slightly decreased (by an amount increasing with δ) the F-P resonance case can be recovered, and the maximum amplification values are found to be close to the predictions from the ALAS (e.g., up to a factor of about 27 for δ=0.4). The variations of the reflection coefficient and enhancement factor obtained from systems A and B as a function of the incident wave number are discussed and compared to ALAS predictions. In particular, it is found that the resonance peak is extremely narrow when δ=0.2 and 0.4.

Subharmonic generation by shear waves in a 1D resonator formed with a relaxing material

The low shear moduli of soft elastic media such as rubber and tissue facilitate the excitation of shear waves that exhibit significant finite-amplitude effects. Measurements performed by Andreev et al. [Acoust. Phys. 57, 779 (2011)] of the amplitude-dependent response of a 1D shear-wave resonator formed with a rubber-like material revealed reasonable agreement with numerical simulations based on a monorelaxing material model. At the previous ASA meeting the present authors developed an augmented Duffing equation as a model for plane shear waves in a resonator formed with a monorelaxing material that is shaken at one end and free at the other [J. Acoust. Soc. Am. 142, 2723 (2017)]. The focus of the previous work was on the response at drive frequencies near the lowest resonance. Here the augmented Duffing model is used to investigate subharmonic generation associated with a drive frequency of approximately three times that of the lowest mode. Conditions on the amplitude and frequency of excitation for whic...

Nonlinear shear wave resonator consisting of a relaxing material

Soft materials such as rubbers, polymers, and tissue exhibit low shear wave speeds, facilitating the generation of shear waves with large acoustic Mach numbers. In addition to finite-amplitude effects that result from cubic nonlinearity, plane shear wave propagation in these materials is subject to frequency-dependent attenuation and dispersion that result from viscoelastic effects. A wave equation for plane shear waves in a relaxing material is obtained from a nonlinear Zener constitutive model that accounts for cubic nonlinearity as well as the attenuation and dispersion associated with relaxation. The wave equation is used to analyze a one-dimensional shear wave resonator comprised of a nonlinear relaxing material that is shaken at one end and free at the other. For excitation of the lowest mode the wave equation is approximated by an augmented Duffing equation, and the resulting frequency-response equation is compared with numerical finite-difference solutions of the original wave equation. Frequency ...

Finite amplitude effects on drop levitation for material properties measurement

The method of exciting shape oscillation of drops to extract material properties has a long history, which is most often coupled with the technique of acoustic levitation to achieve non-contact manipulation of the drop sample. We revisit this method with application to the inference of bulk shear viscosity and surface tension. The literature is replete with references to a “10% oscillation amplitude” as a sufficient condition for the application of Lamb's analytical expressions for the shape oscillations of viscous liquids. Our results show that even a 10% oscillation amplitude leads to dynamic effects which render Lamb's results inapplicable. By comparison with samples of known viscosity and surface tension, we illustrate the complicating finite-amplitude effects (mode-splitting and excess dissipation associated with vorticity) that can occur and then show that sufficiently small oscillations allow us to recover the correct material properties using Lamb's formula.

Dependence of kink oscillation damping on the amplitude

Context. Kink oscillations of coronal loop are one of the most ntensively studied oscillatory phenomena in the solar corona. In the large-amplitude rapidly damped regime these oscillations are observed to have a low quality-factor, with only a few cycles of oscillation detected before they are damped. The specific mechanism responsible for the rapid damping is commonly accepted to be associated with the linear coupling between collective kink oscillations and localised torsional oscillations, the phenomenon of resonant absorption of the kink mode. However, the role of finite amplitude effects is still not clear. Aims. We investigated the empirical dependence of the kink oscillation damping time and its quality factor, defined as the ratio of the damping time to the oscillation period, on the oscillation amplitude. Methods. We analysed decaying kink oscillation events detected previously with TRACE, SDO/AIA and and STEREO/EUVI in the EUV 171A band. Results. We found that the ratio of the kink oscillation damping time to the oscillation period systematically decreases with the oscillation amplitude. The quality factor dependence on the oscillation isplacement amplitude has been approximated by the powerlaw dependence with the exponent of −1/2, however we stress that this is a “by eye”estimate, and a more rigorous estimation of the scaling law requires more accurate measurements and increased statistics. We conclude that damping of kink oscillations of coronal loops depends on the oscillation amplitude, indicating the possible role of nonlinear mechanisms for damping.

Equations describing finite-amplitude effects in acoustic fish abundance estimation

Species identification of fish and abundance estimation of zooplankton employ frequencies typically in the range of 20–400 kHz. Estimates are based on measurement of (a) the single-target backscattering cross section, σbs, used in echo sounder calibration and fish target strength measurements; and (b) the volume backscattering coefficient, sv, used in oceanic surveying. The power budget equations used today for σbs and sv pre-assume small-amplitude signals (i.e., linear sound propagation). For operating frequencies about 100 kHz and higher, finite-amplitude (nonlinear) sound propagation effects in seawater may cause measurements errors at accessible echo sounder transmit power levels. In the present work, power budget equations are derived for σbs and sv that account for finite-amplitude sound propagation effects. The expressions derived can be used to establish upper limits for echo sounder transmit power levels, in order to reduce finite-amplitude errors in calibration and surveying. They can alternatively be used to develop correction factors for calibration and/or survey data already subjected to finite-amplitude errors. The expressions derived are fully consistent with the small-signal expressions used in fisheries acoustics today.

The Impact of Finite-Amplitude Bottom Topography on Internal Wave Generation in the Southern Ocean

Abstract Direct observations in the Southern Ocean report enhanced internal wave activity and turbulence in a kilometer-thick layer above rough bottom topography collocated with the deep-reaching fronts of the Antarctic Circumpolar Current. Linear theory, corrected for finite-amplitude topography based on idealized, two-dimensional numerical simulations, has been recently used to estimate the global distribution of internal wave generation by oceanic currents and eddies. The global estimate shows that the topographic wave generation is a significant sink of energy for geostrophic flows and a source of energy for turbulent mixing in the deep ocean. However, comparison with recent observations from the Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean shows that the linear theory predictions and idealized two-dimensional simulations grossly overestimate the observed levels of turbulent energy dissipation. This study presents two- and three-dimensional, realistic topography simulations of internal lee-wave generation from a steady flow interacting with topography with parameters typical of Drake Passage. The results demonstrate that internal wave generation at three-dimensional, finite bottom topography is reduced compared to the two-dimensional case. The reduction is primarily associated with finite-amplitude bottom topography effects that suppress vertical motions and thus reduce the amplitude of the internal waves radiated from topography. The implication of these results for the global lee-wave generation is discussed.

Finite Amplitude Effects on Landau Damping and Diminished Transportation of Trapped Electrons

We studied the effect of finite amplitude, which was ignored in Landau’s linearized theory and also studied a transition criterion between linear and nonlinear Landau damping. To adjust the finite amplitude of wave and the initial trapped electron flux JxT0, we used particle-in-cell simulation that can uniquely modulate these parameters. We predicted on the finite amplitude of wave potential and kinetic energies; these matched the simulation exactly. Oscillation and damping rates in Landau damping agreed remarkably well with those of theory and differed substantially from theory beyond certain range of amplitude. Either large amplitude or large JxT0 increased degree of damping; this increase diminished the transportation of trapped electrons. More accurate description of linear Landau damping coupled with the finite amplitude effect was possible. Although previous theories predict nonlinear waves in large amplitude, we did not observe the expected nonlinear wave because of diminished JxT0 at some conditions.