Stable Surface Waves Research Articles

PIC simulations of stable surface waves on a subcritical fast magnetosonic shock front

We study with particle-in-cell (PIC) simulations the stability of fast magnetosonic shocks. They expand across a collisionless plasma and an orthogonal magnetic field that is aligned with one of the directions resolved by the 2D simulations. The shock speed is 1.6 times the fast magnetosonic speed when it enters a layer with a reduced density of mobile ions, which decreases the shock speed by up to 15% in 1D simulations. In the 2D simulations, the density of mobile ions in the layer varies sinusoidally perpendicularly to the shock normal. We resolve one sine period. This variation only leads to small changes in the shock speed evidencing a restoring force that opposes a shock deformation. As the shock propagates through the layer, the ion density becomes increasingly spatially modulated along the shock front and the magnetic field bulges out where the mobile ion density is lowest. The perturbed shock eventually reaches a steady state. Once it leaves the layer, the perturbations of the ion density and magnetic field oscillate along its front at a frequency close to the lower-hybrid frequency; the shock is mediated by a standing wave composed of obliquely propagating lower-hybrid waves. We perform three 2D simulations with different box lengths along the shock front. The shock front oscillations are aperiodically damped in the smallest box with the fastest variation of the ion density, strongly damped in the intermediate one, and weakly damped in the largest box. The shock front oscillations perturb the magnetic field in a spatial interval that extends by several electron skin depths upstream and downstream of the shock front and could give rise to Whistler waves that propagate along the shock’s magnetic field overshoot. Similar waves were observed in hybrid and PIC simulations and by the MMS satellite mission.

Theory and analysis of acoustic-gravity waves in a free-surface compressible and stratified ocean

Waves propagate in a free-surface ocean due to compressibility and gravity (and surface tension at much smaller scale). Analytical solutions have long been derived independently for acoustic and gravity waves, i.e., acoustic waves or internal-gravity waves in an unbounded ocean, surface-gravity waves in a free-surface-ocean, and acoustic or internal modes in a bounded ocean. In the present study, surface tension and earth-rotation are neglected and a simple, unified model based on inner and boundary dispersion relations is derived for waves propagating in a compressible, stratified, free-surface ocean. Branches of acoustic gravity wave solutions are identified and visually analysed in phase-space. Taylor developments are then carried out with respect to small parameters describing stratification and compressibility and are compared with numerical approximations of the intersection of inner and boundary dispersion surfaces. Finally, the model recovers the known approximations for swell, long-surface waves, internal-gravity waves, internal modes, acoustic waves or acoustic modes, and also provides information on the modifications of these solutions due to stratification and compressibility and on the coupling of acoustic and gravity waves. Two peculiar regions of the acoustic-gravity wave phase-space are more specifically highlighted and studied in detail: one for long waves shedding new light on the distinction between surface waves and low-order internal modes, the other for marginally stable surface waves of intermediate length-scale.

Application of seismic interferometry by multidimensional deconvolution to ambient seismic noise recorded in Malargüe, Argentina

Obtaining new seismic responses from existing recordings is generally referred to as seismic interferometry (SI). Conventionally, the SI responses are retrieved by simple crosscorrelation of recordings made by separate receivers: one of the receivers acts as a ‘virtual source’ whose response is retrieved at the other receivers.When SI is applied to recordings of ambient seismic noise, mostly surface waves are retrieved. The newly retrieved surface wave responses can be used to extract receiver-receiver phase velocities. These phase velocities often serve as input parameters for tomographic inverse problems. Another application of SI exploits the tempo- ral stability of the multiply scattered arrivals of the newly retrieved surface wave responses. Temporal variations in the stability and/or arrival time of these multiply scattered arrivals can often be linked to temporally varying parameters such as hydrocarbon production and precip- itation. For all applications, however, the accuracy of the retrieved responses is paramount. Correct response retrieval relies on a uniform illumination of the receivers: irregularities in the illumination pattern degrade the accuracy of the newly retrieved responses. In practice, the illumination pattern is often far from uniform. In that case, simple crosscorrelation of separate receiver recordings only yields an estimate of the actual, correct virtual-source response. Re- formulating the theory underlying SI by crosscorrelation as a multidimensional deconvolution (MDD) process, allows this estimate to be improved. SI by MDD corrects for the non-uniform illumination pattern by means of a so-called point-spread function (PSF), which captures the irregularities in the illumination pattern. Deconvolution by this PSF removes the imprint of the irregularities on the responses obtained through simple crosscorrelation. We apply SI by MDD to surface wave data recorded by theMalargue seismic array in western Argentina. The aperture of the array is approximately 60 km and it is located on a plateau just east of the Andean mountain range. The array has a T-shape: the receivers along one of the two lines act as virtual sources whose responses are recorded by the receivers along the other (perpendicular) line.We select time windows dominated by surface wave noise travelling in a favourable direction, that is, traversing the line of virtual sources before arriving at the receivers at which we aim to retrieve the virtual-source responses. These time windows are selected through a frequency-dependent slowness analysis along the two receiver lines. From the selected time windows, estimates of virtual-source responses are retrieved by means of crosscorrelations. Similarly, crosscorrelations between the positions of the virtual sources are computed to build the PSF. We use the PSF to deconvolve the effect of illumination irregularities and the source function from the virtual-source responses retrieved by crosscorrelation. The combined effect of time-window selection and MDD results in more accurate and temporally stable surface wave responses.

Damping of vertically excited surface wave in weakly viscous fluid

In a vertically oscillating circular cylindrical container, singular perturbation theory of two-time scale expansions is developed in weakly viscous fluids to investigate the motion of single free surface standing wave by linearizing the Navier-Stokes equation. The fluid field is divided into an outer potential flow region and an inner boundary layer region. The solutions of both two regions are obtained and a linear amplitude equation incorporating damping term and external excitation is derived. The condition to appear stable surface wave is obtained and the critical curve is determined. In addition, an analytical expression of damping coefficient is determined. Finally, the dispersion relation, which has been derived from the inviscid fluid approximation, is modified by adding linear damping. It is found that the modified results are reasonably closer to experimental results than former theory. Result shows that when forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when forcing frequency is high, the surface tension of the fluid is prominent.

Instability analysis of nonlinear surface waves in a circular cylindrical container subjected to a vertical excitation

Singular perturbation theory of two-time scale expansions was developed both in inviscid and weak viscous fluids to investigate the motion of single surface standing wave in a liquid-filled circular cylindrical vessel, which is subject to a vertical periodical oscillation. Firstly, it is assumed that the fluid in the circular cylindrical vessel is inviscid, incompressible and the motion is irrotational, a nonlinear evolution equation of slowly varying complex amplitude, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from solvability condition of high-order approximation. It shows that when forced frequency is low, the effect of surface tension on mode selection of surface wave is not important. However, when forced frequency is high, the influence of surface tension is significant, and can not be neglected. This proved that the surface tension has the function, which causes free surface returning to equilibrium location. Theoretical results much close to experimental results when the surface tension is considered. In fact, the damping will appear in actual physical system due to dissipation of viscosity of fluid. Based upon weakly viscous fluids assumption, the fluid field was divided into an outer potential flow region and an inner boundary layer region. A linear amplitude equation of slowly varying complex amplitude, which incorporates damping term and external excitation, was derived from linearized Navier–Stokes equation. The analytical expression of damping coefficient was determined and the relation between damping and other related parameters (such as viscosity, forced amplitude and depth of fluid) was presented. The nonlinear amplitude equation and a dispersion, which had been derived from the inviscid fluid approximation, were modified by adding linear damping. It was found that the modified results much reasonably close to experimental results. Moreover, the influence both of the surface tension and the weak viscosity on the mode formation was described by comparing theoretical and experimental results. The results show that when the forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when the forcing frequency is high, the surface tension of the fluid is prominent. Finally, instability of the surface wave is analyzed and properties of the solutions of the modified amplitude equation are determined together with phase-plane trajectories. A necessary condition of forming stable surface wave is obtained and unstable regions are illustrated.

Electromigration-induced wave propagation on surfaces of voids in metallic thin films: Hopf bifurcation for high grain symmetry

The propagation of stable surface waves induced by surface electromigration on voids in metallic thin films is analyzed by self-consistent numerical simulations of current-driven surface morphological evolution. Emphasis is placed on single-crystalline films characterized by high symmetry of surface diffusional anisotropy, representative of [1 0 0]-oriented grains in fcc metals. The theoretical analysis predicts the onset of stable time-periodic states for the void surface morphological response as either the applied electric field strength, or the void size, or the strength of the diffusional anisotropy is increased over a critical value. These stable states correspond to waves propagating on surfaces of voids that migrate along the metallic film at constant speeds. For parameter values below the critical ones, morphological evolution leads to stable steady states for the void surface morphology. Therefore, the onset of stable surface wave propagation corresponds to a Hopf bifurcation in the current-induced surface morphological response. The examined nonlinear dynamics is very rich, leading to different sequences of morphological transitions and instabilities as different operating conditions and system parameters are varied.

Instability of the vertically forced surface wave in a circular cylindrical container

The nonlinear amplitude equation, which was derived by Jian Yongjun employing expansion of two-time scales in inviscid fluids in a vertically oscillating circular cylindrical vessel, is modified by introducing a damping term due to the viscous dissipation of this system. Instability of the surface wave is analysed and properties of the solutions of the modified equation are determined together with phase-plane trajectories. A necessary condition of forming a stable surface wave is obtained and unstable regions are illustrated. Research results show that the stable pattern of surface wave will not lose its stability to an infinitesimal disturbance.

Vertically forced surface wave in weakly viscous fluids bounded in a circular cylindrical vessel

Two-time scale perturbation expansions were developed in weakly viscous fluids to investigate surface wave motions by linearizing the Navier-Stokes equation in a circular cylindrical vessel which is subject to a vertical oscillation. The fluid field was divided into an outer potential flow region and an inner boundary layer region. A linear amplitude equation of slowly varying complex amplitude, which incorporates a damping term and external excitation, was derived for the weakly viscid fluids. The condition for the appearance of stable surface waves was obtained and the critical curve was determined. In addition, an analytical expression for the damping coefficient was determined and the relationship between damping and other related parameters (such as viscosity, forced amplitude, forced frequency and the depth of fluid, etc.) was presented. Finally, the influence both of the surface tension and the weak viscosity on the mode formation was described by comparing theoretical and experimental results. The results show that when the forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when the forcing frequency is high, the surface tension of the fluid is prominent.

The shearing instability of ion flow at the high-latitude tail of magnetopause boundary layer

Shearing instability of ion flow in an inhomogeneous plasma background in the magnetopause boundary layer at the high-latitude magnetotail is studied in this paper. By considering tail-aligned currents, we find that the instability excitation strongly depends on the disturbed wavelength. A quasi-critical wave number for instability is obtained. For relatively long perturbations, the instability tends to be excited at the inner edge of the boundary layer. The stable surface waves at the magnetopause and the K-H instability at the inner edge of the boundary layer can exist at the same time. This may contribute to the continuous transfer of momentum toward the magnetosphere.

Data‐dependent non‐double‐couple components of shallow earthquake source mechanisms: Effects of waveform inversion instability

Seismic moment tensors are routinely determined for global earthquakes using waveform inversions, and the resulting double‐couple and non‐double‐couple characteristics are extensively used in tectonic analyses. For shallow events, several moment tensor components are intrinsically poorly resolved, especially for surface wave inversions, which can lead to artificial non‐double‐couple components or can obscure actual non‐double‐couple components in the source representations. The relatively greater instability of surface wave inversions can cause differences between inversions based on only body waves and joint inversions of body and surface waves. The Harvard CMT catalog exhibits systematic variations in non‐double‐couple components with seismic moment, which can be attributed to use of only body waves in inversions for small events and combined body and surface waves for large events. Systematic behavior of non‐double‐couple components with regional strain environment is significantly less dependent on seismic moment when solutions influenced by less stable surface waves are omitted.

Surface waves in laser-produced plasma double layers

The surface-tension pressure for a charged double layer is calculated. This result is used to derive the dispersion relation for surface waves. The stability criterion defines stable surface waves with wavelength lambda /sub sw/ >

A Second-Order JWKB Approximation with One Turning Point and Two Singular Points: Stability of an Accelerating Liquid Sphere

A formal higher-order matching procedure is employed to obtain a second-order asymptotic solution, of the JWKB type, to a Legendre-like differential equation with a large parameter. The equation has two second-order poles and a first-order turning point. In addition to the usual nonuniformity, the second-order JWKB approximation exhibits a divergent integral at these points. Eigenfunctions and eigenvalues, valid to the second order of approximation, are found by simultaneously matching the latter approximation to a turning-point expansion and two boundary-layer expansions. The solutions, which heretofore have not been described, are appropriate to the neutral stable surface waves manifest by an accelerating liquid sphere.

Profile and Speed of Finite Amplitude Waves in a Falling Liquid Layer

The profiles as well as the speeds of supercritically stable surface waves are obtained for various values of surface tension, Reynolds numbers, and the angle of inclination of a falling liquid layer. The calculated values of wave speeds are in good agreement with Kapitza's measurements and the theoretically obtained wave profiles seem to resemble those appearing in his photographs. Profiles of “generalized solitary waves” which propagated at speeds several times that of the kinematic speed are also found.