Initial Perturbation Amplitude Research Articles

A Model of a Source of Shock Wave Metal Ejection Based on Richtmyer–Meshkov Instability Theory

This paper provides a solution for the problem about effect of a shock wave profile on space–time distribution of velocity, density and mass of particles ejected from material free surface. It is shown that in liquid phase state of material a shock wave amplitude does not affect the ejected material mass, but is determined by the ratio β = k 2 × a 0 × Δx (a 0 is an initial amplitude of perturbation, k = 2π/λ is a wave number, λ is perturbation wavelength, Δx is a triangular shock wave pulse width).

Strength and Viscosity Effects on Perturbed Shock Front Stability in Metals

Computational modeling and experimental measurements on metal samples subject to a laser-driven, ablative Richtmyer-Meshkov instability showed differences between viscosity and strength effects. In particular, numerical and analytical solutions, coupled with measurements of fed-through perturbations, generated by perturbed shock fronts onto initially flat surfaces, show promise as a validation method for models of deviatoric response in the postshocked material. Analysis shows that measurements of shock perturbation amplitudes at low sample thickness-to-wavelength ratios are not enough to differentiate between strength and viscosity effects, but that surface displacement data of the fed-through perturbations appears to resolve the ambiguity. Additionally, analytical and numerical results show shock front perturbation evolution dependence on initial perturbation amplitude and wavelength is significantly different in viscous and materials with strength, suggesting simple experimental geometry changes should provide data supporting one model or the other.

Dynamics of pulsed-ultrasound driven gas-liquid interfaces

Pulsed ultrasound has been shown to be capable of inducing lung hemorrhage in mammals, though the physical mechanism remains unknown. We model ultrasound-alveolar interactions as a compressible fluid system and perform numerical experiments to investigate the relevant dynamics. Previously we demonstrated that the interaction between trapezoidal acoustic waves and perturbed gas-liquid interfaces, such as those in the lung, can generate sufficient baroclinic vorticity to appreciably deform the interface under certain conditions. In this work we study the dynamics of perturbed liquid-gas interfaces driven by a single ultrasound pulse in and out of the clinically relevant range with frequencies 1.5-3 MHz and amplitudes 1-15 MPa. We calculate theoretical stresses and strains and compare to established failure criteria. We observe that for ultrasound pulses that significantly deform the interface during the wave-interface interaction, there is residual vorticity left at the interface which can deform the interface long after the passage of the wave. This effect increases with wave amplitude, pulse duration, and initial interface perturbation amplitude. Previously presented scaling laws for the interface perturbation amplitude and arc length growth are tested. While we find that nominal interface strains greater than 1 occur we recognize that the applicability of this study to diagnostic lung ultrasound is limited by the accuracy of the model.

The impact of non-ideal magnetohydrodynamics on binary star formation

We investigate the effect of non-ideal magnetohydrodynamics (MHD) on the formation of binary stars using a suite of three-dimensional smoothed particle magnetohydrodynamics simulations of the gravitational collapse of one solar mass, rotating, perturbed molecular cloud cores. Alongside the role of Ohmic resistivity, ambipolar diffusion and the Hall effect, we also examine the effects of magnetic field strength, orientation and amplitude of the density perturbation. When modelling sub-critical cores, ideal MHD models do not collapse whereas non-ideal MHD models collapse to form single protostars. In super-critical ideal MHD models, increasing the magnetic field strength or decreasing the initial density perturbation amplitude decreases the initial binary separation. Strong magnetic fields initially perpendicular to the rotation axis suppress the formation of binaries and yield discs with magnetic fields ~10 times stronger than if the magnetic field was initially aligned with the rotation axis. When non-ideal MHD is included, the resulting discs are larger and more massive, and the binary forms on a wider orbit. Small differences in the super-critical cores caused by non-ideal MHD effects are amplified by the binary interaction near periastron. Overall, the non-ideal effects have only a small impact on binary formation and early evolution, with the initial conditions playing the dominant role.

General theory of the plasmoid instability

A general theory of the onset and development of the plasmoid instability is formulated by means of a principle of least time. The scaling relations for the final aspect ratio, transition time to rapid onset, growth rate, and number of plasmoids are derived and shown to depend on the initial perturbation amplitude (ŵ0), the characteristic rate of current sheet evolution (1/τ), and the Lundquist number (S). They are not simple power laws, and are proportional to Sατβ[ln f(S,τ,ŵ0)]σ. The detailed dynamics of the instability is also elucidated, and shown to comprise of a period of quiescence followed by sudden growth over a short time scale.

The Linear Stability of a Wavetrain Propagating on Water of Variable Depth

A narrow-banded spectrum of gravity waves propagating in one horizontal direction at the surface of water with finite depth may be governed by a nonlinear Schroźdinger equation (NLSE). If the depth is nonuniform, then the coefficients in the NLSE are variable and an additional linear term due to conservation of wave action flux is present, which may act as a dissipative term that causes the wave envelope to broaden and decrease in amplitude or which may act as a growth term that causes the wave envelope to focus and increase in amplitude. The addition of linear, viscous dissipation either enhances the dissipative process or competes with the growth effect. Here we consider such a variable-coefficient, dissipative NLSE. We find a solution that has uniform amplitude in the appropriate reference frame and examine its linear stability. We find that for waves propagating into shallower water, both bathymetry and viscous dissipation act to stabilize the modulational instability. For waves propagating into deeper water, bathymetry acts to enhance instability, but viscous dissipation eventually causes stabilization for small enough initial perturbation amplitudes.

Primordial black holes as a novel probe of primordial gravitational waves

We propose a novel method to probe primordial gravitational waves by means of primordial black holes (PBHs). When the amplitude of primordial tensor perturbations on comoving scales much smaller than those relevant to Cosmic Microwave Background is very large, it induces scalar perturbations due to second-order effects substantially. If the amplitude of resultant scalar perturbations becomes too large, then PBHs are overproduced to a level that is inconsistent with a variety of existing observations constraining their abundance. This leads to upper bounds on the amplitude of initial tensor perturbations on super-horizon scales. These upper bounds from PBHs are compared with other existing bounds.

Prediction and predictability of a catastrophic local extreme precipitation event through cloud-resolving ensemble analysis and forecasting with Doppler radar observations

Local extreme rain usually resulted in disasters such as flash floods and landslides. Upon today, it is still one of the most difficult tasks for operational weather forecast centers to predict those events accurately. In this paper, we simulate an extreme precipitation event with ensemble Kalman filter (EnKF) assimilation of Doppler radial-velocity observations, and analyze the uncertainties of the assimilation. The results demonstrate that, without assimilation radar data, neither a single initialization of deterministic forecast nor an ensemble forecast with adding perturbations or multiple physical parameterizations can predict the location of strong precipitation. However, forecast was significantly improved with assimilation of radar data, especially the location of the precipitation. The direct cause of the improvement is the buildup of a deep mesoscale convection system with EnKF assimilation of radar data. Under a large scale background favorable for mesoscale convection, efficient perturbations of upstream mid-low level meridional wind and moisture are key factors for the assimilation and forecast. Uncertainty still exists for the forecast of this case due to its limited predictability. Both the difference of large scale initial fields and the difference of analysis obtained from EnKF assimilation due to small amplitude of initial perturbations could have critical influences to the event's prediction. Forecast could be improved through more cycles of EnKF assimilation. Sensitivity tests also support that more accurate forecasts are expected through improving numerical models and observations.

Effect of initial perturbation amplitude on Richtmyer-Meshkov flows induced by strong shocks

We systematically study the effect of the initial perturbation on Richtmyer-Meshkov (RM) flows induced by strong shocks in fluids with contrasting densities. Smooth Particle Hydrodynamics simulations are employed. A broad range of shock strengths and density ratios is considered. The amplitude of the initial single mode sinusoidal perturbation of the interface varies from 0% to 100% of its wavelength. The simulations results are compared, wherever possible, with four rigorous theories, and with other experiments and simulations, achieving good quantitative and qualitative agreement. Our study is focused on early time dynamics of the Richtmyer-Meshkov instability (RMI). We analyze the initial growth-rate of RMI immediately after the shock passage, when the perturbation amplitude increases linearly with time. For the first time, to the authors' knowledge, we find that the initial growth-rate of RMI is a non-monotone function of the initial perturbation amplitude, thus restraining the amount of energy that can be deposited by the shock at the interface. The maximum value of the initial growth-rate depends on the shock strength and the density ratio, whereas the corresponding value of the initial perturbation amplitude depends only slightly on the shock strength and density ratio.

On the thermal sensitivity of binary formation in collapsing molecular clouds

We report the results of a numerical study on the initial formation stages of low-mass protostellar binary systems. We determine the separation of protostellar binaries formed as a function of the initial thermal state by varying the initial temperature in a slightly modified version of the Burkert and Bodenheimer collapse test. We find that the outcome is highly sensitive to both the initial temperature of the cloud and the initial amplitude of azimuthal density perturbation A. For A=10 %, variations of only 1 unit Kelvin below 10 K lead to changes of up to 100 AU ( i.e. of order 30 %) in the instantaneous separation, whereas for this small A the initial temperatures above 10 K yield, instead of a binary, a single low-mass fragment that never reaches protostellar densities. Protostellar binaries, however, do emerge when the perturbation amplitude is increased from 10 % to 25 %. We also investigate the impact of the critical density which governs the transition from isothermal to adiabatic thermodynamic behaviour of the collapsing gas. We find that the critical density not only affects the overall structural evolution of the gas envelope, but also the size of the rotating disk structures formed during collapse as well as the number of protostellar fragments resulting from the final fragmentation of the disks. This mechanism can give rise to young protostellar objects constituting bound multiple stellar systems.

Magneto-Rayleigh-Taylor instability in solid media

A linear analysis of the magneto-Rayleigh-Taylor instability at the interface between a Newtonian fluid and an elastic-plastic solid is performed by considering a uniform magnetic B→, parallel to the interface, which has diffused into the fluid but not into the solid. It is found that the magnetic field attributes elastic properties to the viscous fluid which enhance the stability region by stabilizing all the perturbation wavelengths shorter than λ0∝B2 for any initial perturbation amplitude. Longer wavelengths are stabilized by the mechanical properties of the solid provided that the initial perturbation wavelength is smaller than a threshold value determined by the yield strength and the shear modulus of the solid. Beyond this threshold, the amplitude grows initially with a growth rate reduced by the solid strength properties. However, such properties do not affect the asymptotic growth rate which is only determined by the magnetic field and the fluid viscosity. The described physical situation intends to resemble some of the features present in recent experiments involving the magnetic shockless acceleration of flyers plates.

Cross-code comparisons of mixing during the implosion of dense cylindrical and spherical shells

We present simulations of the implosion of a dense shell in two-dimensional (2D) spherical and cylindrical geometry performed with four different compressible, Eulerian codes: RAGE, FLASH, CASTRO, and PPM. We follow the growth of instabilities on the inner face of the dense shell. Three codes employed Cartesian grid geometry, and one (FLASH) employed polar grid geometry. While the codes are similar, they employ different advection algorithms, limiters, adaptive mesh refinement (AMR) schemes, and interface-preservation techniques. We find that the growth rate of the instability is largely insensitive to the choice of grid geometry or other implementation details specific to an individual code, provided the grid resolution is sufficiently fine. Overall, all simulations from different codes compare very well on the fine grids for which we tested them, though they show slight differences in small-scale mixing. Simulations produced by codes that explicitly limit numerical diffusion show a smaller amount of small-scale mixing than codes that do not. This difference is most prominent for low-mode perturbations where little instability finger interaction takes place, and less prominent for high- or multi-mode simulations where a great deal of interaction takes place, though it is still present. We present RAGE and FLASH simulations to quantify the initial perturbation amplitude to wavelength ratio at which metrics of mixing agree across codes, and find that bubble/spike amplitudes are converged for low-mode and high-mode simulations in which the perturbation amplitude is more than 1% and 5% of the wavelength of the perturbation, respectively. Other metrics of small-scale mixing depend on details of multi-fluid advection and do not converge between codes for the resolutions that were accessible.

Stabilizing effect of optimally amplified streaks in parallel wakes

AbstractWe show that optimal perturbations artificially forced in parallel wakes can be used to completely suppress the absolute instability and to reduce the maximum temporal growth rate of the inflectional instability. To this end we compute optimal transient energy growths of stable streamwise uniform perturbations supported by a parallel wake for a set of Reynolds numbers and spanwise wavenumbers. The maximum growth rates are shown to be proportional to the square of the Reynolds number and to increase with spanwise wavelengths with sinuous perturbations slightly more amplified than varicose ones. Optimal initial conditions consist of streamwise vortices and the optimally amplified perturbations are streamwise streaks. Families of nonlinear streaky wakes are then computed by direct numerical simulation using optimal initial vortices of increasing amplitude as initial conditions. The stabilizing effect of nonlinear streaks on temporal and spatiotemporal growth rates is then determined by analysing the linear impulse response supported by the maximum amplitude streaky wakes profiles. This analysis reveals that at $\mathit{Re}= 50$, streaks of spanwise amplitude ${A}_{s} \approx 8\hspace{0.167em} \% {U}_{\infty } $ can completely suppress the absolute instability, converting it into a convective instability. The sensitivity of the absolute and maximum temporal growth rates to streak amplitudes is found to be quadratic, as has been recently predicted. As the sensitivity to two-dimensional (2D, spanwise uniform) perturbations is linear, three-dimensional (3D) perturbations become more effective than the 2D ones only at finite amplitudes. Concerning the investigated cases, 3D perturbations become more effective than the 2D ones for streak amplitudes ${A}_{s} \gtrsim 3\hspace{0.167em} \% {U}_{\infty } $ in reducing the maximum temporal amplification and ${A}_{s} \gtrsim 12\hspace{0.167em} \% {U}_{\infty } $ in reducing the absolute growth rate. However, due to the large optimal energy growths they experience, 3D optimal perturbations are found to be much more efficient than 2D perturbations in terms of initial perturbation amplitudes. Despite their lower maximum transient amplification, varicose streaks are found to be always more effective than sinuous ones in stabilizing the wakes, in accordance with previous findings.

New directions for Rayleigh–Taylor mixing

We study the Rayleigh-Taylor (RT) mixing layer, presenting simulations in agreement with experimental data. This problem is an idealized subproblem of important scientific and engineering problems, such as gravitationally induced mixing in oceanography and performance assessment for inertial confinement fusion. Engineering codes commonly achieve correct simulations through the calibration of adjustable parameters. In this sense, they are interpolative and not predictive. As computational science moves from the interpolative to the predictive and reduces the reliance on experiment, the quality of decision making improves. The diagnosis of errors in a multi-parameter, multi-physics setting is daunting, so we address this issue in the proposed idealized setting. The validation tests presented are thus a test for engineering codes, when used for complex problems containing RT features. The RT growth rate, characterized by a dimensionless but non-universal parameter α, describes the outer edge of the mixing zone. Increasingly accurate front tracking/large eddy simulations reveal the non-universality of the growth rate and agreement with experimental data. Increased mesh resolution allows reduction in the role of key subgrid models. We study the effect of long-wavelength perturbations on the mixing growth rate. A self-similar power law for the initial perturbation amplitudes is here inferred from experimental data. We show a maximum ±5% effect on the growth rate. Large (factors of 2) effects, as predicted in some models and many simulations, are inconsistent with the experimental data of Youngs and co-authors. The inconsistency of the model lies in the treatment of the dynamics of bubbles, which are the shortest-wavelength modes for this problem. An alternative theory for this shortest wavelength, based on the bubble merger model, was previously shown to be consistent with experimental data.

Numerical simulation of equatorial plasma bubbles over Cachimbo: COPEX campaign

The problem of day-to-day variability in onset of equatorial spread F (ESF) is addressed using data from the 2002 COPEX observational campaign in Brazil and numerical modeling. The observational results show that for values of virtual height of the F layer base less than 355km at around 18:35 LT, and for the prereversal peak enhancement of the vertical plasma drift (Vp) less than 30m/s, the spread-F (ESF) was absent on four nights over Cachimbo (9.5°S, 54.8°W, dip latitude=−2.1°). In this work we analyze the geophysical conditions for the generation of the irregularities by comparing the nights with and without the ESF. In the comparison a numerical code is used to simulate plasma irregularity development in an extended altitude range from the bottom of the equatorial F layer. The code uses the flux corrected transport method with Boris–Book’s flux limiter for the spatial integration and a predictor–corrector method for the direct time integration of the continuity equation for O+ and the SOR (Successive-Over-Relaxation) method for electric potential equation. The code is tested with different evening eastward electric fields (or vertical drifts Vp<30m/s and Vp>30m/s) in order to study the influence of the prereversal enhancement in the zonal electric field on plasma bubble formation and development. The code also takes into account the zonal wind, the vertical electric field and the collision frequency of ions with neutrals and the amplitude of initial perturbation. The simulation shows a good agreement with the observational results of the ESF. The results of the code suggest that the instability can grow at the F layer bottomside by the Rayleigh–Taylor mechanism only when the Vp>30m/s. In the analyzed cases we have considered the competition of other geophysical parameters in the generation of plasma structures.

Richtmyer–Meshkov instability of a liquid–gas interface driven by a cylindrical imploding pressure wave

The compression of a cylindrical gas bubble by an imploding molten lead (Pb) shell may be accompanied by the development of the Richtmyer–Meshkov (RM) instability at the liquid–gas interface due to the initial imperfection of the interface. A converging pressure wave impinging upon the interface causes a shell of liquid to detach and continue to travel inwards, compressing the gas bubble. The efficiency of compression and collapse evolution can be affected by development of the RM instability. Investigations have been performed in the regime of extreme Atwood number A=-1 with the additional complexity of modeling liquid cavitation in the working fluid. Simulations have been carried out using the open source CFD software OpenFOAM on a set of parameters relevant to the prototype compression system under development at General Fusion Inc. for use as a Magnetized Target Fusion (MTF) driver.After validating the numerical setup in planar geometry, simulations have been carried out in 2D cylindrical geometry for both initially smooth and perturbed interfaces. Where possible, results have been validated against existing theoretical models and very good agreement has been found. While our main focus is on the effects of initial perturbation amplitude and azimuthal mode number, we also address differences between this problem and those usually considered, such as RM instability at an interface between two gases with a moderate density ratio. One important difference is the formation of narrow molten lead jets rapidly propagating inwards during the final stages of the collapse. Jet behavior has been observed for a range of azimuthal mode numbers and perturbation amplitudes.

Revisiting linear dynamics of non-axisymmetric perturbations in weakly magnetized accretion discs

We investigate linear dynamics of non-axisymmetric perturbations in incompressible, vertically stratified Keplerian discs with a weak vertical magnetic field in the shearing box approximation. Perturbations are decomposed into shearing waves whose evolution is followed via numerical integration of the linearized ideal MHD equations. There are two basic modes in the system -- inertia-gravity waves and magnetic mode, which displays the magnetorotational instability (MRI). As distinct from previous studies, we introduce `eigenvariables' characterizing each (counter-propagating) component of the inertia-gravity and magnetic modes, which are governed by a set of four first order coupled ordinary differential equations. This allowed us to identify a new process of linear coupling of the two above non-axisymmetric modes due to the disc's differential rotation. We did a comparative analysis of the dynamics of non-axisymmetric and axisymmetric magnetic mode perturbations. It is shown that the growth of optimal and close-to-optimal non-axisymmetric harmonics of this mode, having transient nature, can prevail over the exponential growth of axisymmetric ones (i.e., over the axisymmetric MRI) during dynamical time. A possible implication of this result for axisymmetric channel solutions is discussed. Specifically, the formation of the channel may be affected/impeded by non-axisymmetric modes already at the early linear stage leading to its untimely disruption -- the outcome strongly depends on the amplitude and spectrum of initial perturbation. So, this competition may result in an uncertainty in the magnetic mode's non-linear dynamics. It is also shown that a maximum non-axisymmetric growth is at vertical wavelengths close to the scale-height for which compressibility effects are important. This indirectly suggests that compressibility plays a role in the dynamics of the non-axisymmetric MRI.

Optimal nonlinear excitation of decadal variability of the North Atlantic thermohaline circulation

Nonlinear development of salinity perturbations in the Atlantic thermohaline circulation (THC) is investigated with a three-dimensional ocean circulation model, using the conditional nonlinear optimal perturbation method. The results show two types of optimal initial perturbations of sea surface salinity, one associated with freshwater and the other with salinity. Both types of perturbations excite decadal variability of the THC. Under the same amplitude of initial perturbation, the decadal variation induced by the freshwater perturbation is much stronger than that by the salinity perturbation, suggesting that the THC is more sensitive to freshwater than salinity perturbation. As the amplitude of initial perturbation increases, the decadal variations become stronger for both perturbations. For salinity perturbations, recovery time of the THC to return to steady state gradually saturates with increasing amplitude, whereas this recovery time increases remarkably for freshwater perturbations. A nonlinear (advective) feedback between density and velocity anomalies is proposed to explain these characteristics of decadal variability excitation. The results are consistent with previous ones from simple box models, and highlight the importance of nonlinear feedback in decadal THC variability.

Rayleigh-Taylor stability boundary at solid-liquid interfaces

A previous model for the Rayleigh-Taylor instability [A. R. Piriz, J. J. López Cela, and N. A. Tahir, Phys. Rev. E 80, 046305 (2009)] has been extended in order to study an interface between an elastic-plastic solid and a Newtonian liquid and determine the stability region given by the initial perturbation amplitude ξ(0) and wavelength λ. The stability region is found to be enhanced by the effect of the liquid viscosity, but it reaches an asymptote for a sufficiently high viscosity. In addition, it is also found that the boundary for the transition from the elastic to the plastic regime get closer to the stability boundary up to both boundaries coincide for a high enough liquid viscosity, thus making the onset of plastic flow a sufficient condition for instability.

The early phase of multiple proto-stellar system emerging from collapse of molecular cloud under various initial thermal states

An attempt is made here to revisit structure formation in a proto-stellar cloud during the early phase of evolution. Molecular cloud subjected to a set of various initial conditions in terms of initial temperature and amplitude of azimuthal density perturbation is investigated numerically. Special emphasis remained on the analysis of ring and spiral type instabilities that have shown dependence on certain initial conditions chosen for a rotating solar mass cloud of molecular hydrogen. Generally, a star forming hydrogen gas is considered to be initially at 10K. We have found that a possible oscillation around this typical value can affect the fate of a collapsing cloud in terms of its evolving structural properties leading to proto-star formation. We explored the initial temperature range of cloud between 8K to 12K and compared physical properties of each within the first phase of proto-star formation. We suggest that the spiral structures are more likely to form in strongly perturbed molecular cores that initiate their phase of collapse from temperatures below 10K. Whereas, cores with initial temperatures above 10K develop, instead of spiral, a ring type structure which subsequently experiences the fragmentation. A transition from spiral to ring instability can be observed at a typical initial temperature of 10K.