High-resolution Three-dimensional Numerical Simulations Research Articles

Hydrodynamic turbulence in disks with embedded planets

The vertical shear instability (VSI) is a source of hydrodynamic turbulence that can drive vigorous vertical mixing and moderate levels of accretion in protoplanetary disks, and it could be observable in the near future. With high-resolution three-dimensional numerical hydrodynamics simulations, we modeled the behavior of the VSI in protoplanetary disks with and without embedded planets. We then measured its accretion and mixing capabilities by comparing the full Reynolds stress, which includes the contribution of nonaxisymmetric features, such as spiral arms and vortices, to the Reynolds stress due to the azimuthally averaged velocity field, which can be attributed to good approximation to the VSI. We verified that the VSI can contribute to the accretion stress and showed that, depending on disk conditions, an embedded planet can coexist with or suppress VSI turbulent stress. Specifically, the presence of spiral shocks launched by a planet or planet-generated vortices can interfere with the VSI near the planet’s vicinity, with the instability recovering at large enough distances from the planet or vortex. Our results suggest that observations of VSI signatures are unlikely in disks that contain massive, nonaxisymmetric features.

Transition to turbulence in nonuniform coronal loops driven by torsional Alfvén waves

It has been shown in a previous work that torsional Alfvén waves can drive turbulence in nonuniform coronal loops with a purely axial magnetic field. Here we explore the role of the magnetic twist. We modeled a coronal loop as a transversely nonuniform straight flux tube, anchored in the photosphere, and embedded in a uniform coronal environment. We considered that the magnetic field is twisted and control the strength of magnetic twist by a free parameter of the model. We excited the longitudinally fundamental mode of standing torsional Alfvén waves, whose temporal evolution was obtained by means of high-resolution three-dimensional ideal magnetohydrodynamic numerical simulations. We find that phase mixing of torsional Alfvén waves creates velocity shear in the direction perpendicular to the magnetic field lines. The velocity shear eventually triggers the Kelvin–Helmholtz instability (KHi). In weakly twisted magnetic tubes, the KHi is able to grow nonlinearly, and subsequently, turbulence is driven in the coronal loop in a similar manner as in the untwisted case. When the magnetic twist remains weak, it delays the onset of the KHi and slows the development of turbulence down. In contrast, magnetic tension can suppress the nonlinear growth of the KHi when the magnetic twist is strong enough, even when the KHi has locally been excited by the phase-mixing shear. Thus, turbulence is not generated in strongly twisted loops.

High-resolution direct simulation of deep water breaking waves: transition to turbulence, bubbles and droplets production

We present high-resolution three-dimensional (3-D) direct numerical simulations of breaking waves solving for the two-phase Navier–Stokes equations. We investigate the role of the Reynolds number (Re, wave inertia relative to viscous effects) and Bond number (Bo, wave scale over the capillary length) on the energy, bubble and droplet statistics of strong plunging breakers. We explore the asymptotic regimes at highReandBo, and compare with laboratory breaking waves. Energetically, the breaking wave transitions from laminar to 3-D turbulent flow on a time scale that depends on the turbulentReup to a limiting value$Re_\lambda \sim 100$, consistent with the mixing transition in other canonical turbulent flows. We characterize the role of capillary effects on the impacting jet and ingested main cavity shape and subsequent fragmentation process, and extend the buoyant-energetic scaling from Deikeet al.(J. Fluid Mech., vol. 801, 2016, pp. 91–129) to account for the cavity shape and its scale separation from the Hinze scale,$r_H$. We confirm two regimes in the bubble size distribution,$N(r/r_H)\propto (r/r_H)^{-10/3}$for$r>r_H$, and$\propto (r/r_H)^{-3/2}$for$r<r_H$. Bubbles are resolved up to one order of magnitude below$r_H$, and we observe a good collapse of the numerical data compared to laboratory breaking waves (Deane & Stokes,Nature, vol. 418 (6900), 2002, pp. 839–844). We resolve droplet statistics at highBoin good agreement with recent experiments (Erininet al.,Geophys. Res. Lett., vol. 46 (14), 2019, pp. 8244–8251), with a distribution shape close to$N_d(r_d)\propto r_d^{-2}$. The evolution of the droplet statistics appears controlled by the details of the impact process and subsequent splash-up. We discuss velocity distributions for the droplets, finding ejection velocities up to four times the phase speed of the wave, which are produced during the most intense splashing events of the breaking process.

Spatial asymmetry of nonlinear internal waves in the Lombok Strait

Nonlinear internal waves (NLIWs) are an energetic oceanic phenomenon near the Lombok Strait; however, their generation and three-dimensional characteristics remain poorly understood. To identify the generation and evolution processes of local NLIWs, we implement high-resolution (Δx=100 m) three-dimensional numerical simulations with realistic topography and tidal forcing, and a set of two-dimensional idealized numerical simulations. Barotropic semidiurnal tides (D2 in short) are verified to predominantly determine the local NLIW generation, but barotropic diurnal tides (D1 in short) can significantly modulate it. Particularly, D1 would enhance (weaken) the amplitudes of NLIWs during the propagation processes when they are in phase (out of phase) with D2. In addition, the three-dimensional model results also suggest that internal wave energy fluxes attenuate dramatically when NLIWs are radiating away from the generation site, due to the radial spreading effects of wave energy along the crest line. However, as NLIWs propagate over the complex bottom topography, large energy fluxes mainly occur in the east-central area of the crest line, resulting in a west-east asymmetry of internal wave energy field. The three-dimensional model results indicate that the northward NLIWs are more numerous and better identified than the southward ones, which are in accordance with the satellite image (MODIS). By a series of two-dimensional idealized numerical simulations, we conclude that the northward NLIWs is 43% larger than the southward ones, due to the joint effect of north–south asymmetric slope steepness on two flanks of the sill in the Lombok Strait and a southward deeper Indian Ocean, between them, the role of asymmetric water depth is more significant (38%) than the asymmetric slope steepness (13%).

Discs and outflows in the early phases of massive star formation: Influence of magnetic fields and ambipolar diffusion

Context. Massive star formation remains one of the most challenging problems in astrophysics, as illustrated by the fundamental issues of the radiative pressure barrier and the initial fragmentation. The wide variety of physical processes involved, in particular the protostellar radiative feedback, increase the complexity of massive star formation in comparison with its low-mass counterpart. Aims. We aim to study the details of mass accretion and ejection in the vicinity of massive star forming cores using high-resolution (5 au) three-dimensional numerical simulations. We investigated the mechanisms at the origin of outflows (radiative force versus magnetic acceleration). We characterised the properties of the disc forming around massive protostars depending on the physics included: hydrodynamics, magnetic fields, and ambipolar diffusion. Methods. We used state-of-the-art three-dimensional adaptive-mesh-refinement models of massive dense core collapse, which integrate the equations of (resistive) grey radiation magnetohydrodynamics, and include sink particle evolution. For the first time, we include both protostellar radiative feedback via pre-main-sequence evolutionary tracks and magnetic ambipolar diffusion. To determine the role of magnetic fields and ambipolar diffusion play in the formation of outflows and discs, we studied three different cases: a purely hydrodynamical run, a magnetised simulation under the ideal approximation (perfect coupling), and a calculation with ambipolar diffusion (resistive case). In the most micro-physically complex model (resistive MHD), we also investigated the effect the initial amplitude of both magnetic field and solid body rotation have on the final properties of the massive protostellar system. We used simple criteria to identify the outflow and disc material and follow their evolution as the central star accretes mass up to 20 M⊙ in most of our models. The radiative, magnetic, and hydrodynamical properties of the outflows and discs are quantitatively measured and cross-compared between models. Results. Massive stars form in all our models, together with outflows and discs. The outflow is completely different when magnetic fields are introduced, so magneto-centrifugal processes are the main driver of the outflow up to stellar masses of 20 M⊙. Then, the disc properties heavily depend on the physics included. In particular, the disc formed in the ideal and resistive runs show opposite properties in terms of plasma beta; that is, the ratio of thermal-to-magnetic pressures and of magnetic field topology. While the disc in the ideal case is dominated by the magnetic pressure and the toroidal magnetic fields, the one formed in the resistive runs is dominated by the thermal pressure and essentially has a vertical magnetic field in the inner regions (R < 100−200 au). Conclusions. We find that magnetic processes dominate the early evolution of massive protostellar systems (M⋆ < 20 M⊙) and shapes the accretion and ejection as well as the disc formation. Ambipolar diffusion is mainly at work at disc scales and regulates its properties. We predict magnetic field’s topology within the disc and outflows, as well as disc masses and radii to be compared with observations. Lastly, our finding for the outflow and disc properties are reminiscent of the low-mass star formation framework, suggesting that accretion and ejection in young massive and low-mass protostars are regulated by the same physical processes in the early stages.

Numerical study of the Kelvin-Helmholtz instability and its effect on synthetic emission from magnetized jets

Context.Nonthermal emission from active galactic nucleus (AGN) jets extends up to large scales, even though they are prone to many magnetohydrodynamic instabilities.Aims.The main focus of this study is to understand the effect of magnetohydrodynamic instabilities on the nonthermal emission from large-scale AGN jets.Methods.We performed high-resolution three-dimensional numerical magnetohydrodynamic simulations of a plasma column to investigate the dynamical and emission properties of jet configurations at kiloparsec scales with different magnetic field profiles, jet speeds, and density contrast. We also obtained synthetic nonthermal emission signatures for different viewing angles using an approach that assumes static particle spectra and that is obtained by evolving the particle spectra using Lagrangian macroparticles incorporating the effects of shock acceleration and radiative losses.Results.We find that shocks due to the Kelvin-Helmholtz (KH) instability in the axial magnetic field configurations can strongly affect the jet dynamics. Additionally, we also find weak biconical shocks in the underdense jet columns. The helical magnetic field hinders the vortex growth at the shear surface and thereby stabilizes the jet column. With the evolving particle spectra approach, the synthetic spectral energy distributions obtained for cases with strong KH instability show multiple humps ranging from the radio to the TeV gamma-ray band.Conclusions.We conclude that high-energy electrons that are accelerated in the vicinity of freshly formed shocks due to KH instability result in high X-ray emission.

Linking dissipation, anisotropy, and intermittency in rotating stratified turbulence at the threshold of linear shear instabilities

Analyzing a large database of high-resolution three-dimensional direct numerical simulations of decaying rotating stratified flows, we show that anomalous mixing and dissipation, marked anisotropy, and strong intermittency are all observed simultaneously in an intermediate regime of parameters in which both waves and eddies interact nonlinearly. A critical behavior governed by the stratification occurs at Richardson numbers of order unity and with the flow close to being in a state of instability. This confirms the central dynamical role, in rotating stratified turbulence, of large-scale intermittency, which occurs in the vertical velocity and temperature fluctuations, as an adjustment mechanism of the energy transfer in the presence of strong waves.

Three-dimensional simulations of solar wind turbulence with the hybrid code CAMELIA

We investigate the spectral properties of plasma turbulence from fluid to sub-ion scales by means of high-resolution three-dimensional (3D) numerical simulations performed with the hybrid particle- in-cell (HPIC) code CAMELIA. We produce extended turbulent spectra with well-defined power laws for the magnetic, ion bulk velocity, density, and electric fluctuations. The present results are in good agreement with previous two-dimensional (2D) HPIC simulations, especially in the kinetic range of scales, and reproduce several features observed in solar wind spectra. By providing scaling tests on many different architectures and convergence studies, we prove CAMELIA to represent a very efficient, accurate and reliable tool for investigating the development of the turbulent cascade in the solar wind, being able to cover simultaneously several decades in wavenumber, also in 3D.

Power spectra of outflow-driven turbulence

We investigate the power spectra of outflow-driven turbulence through high-resolution three-dimensional isothermal numerical simulations where the turbulence is driven locally in real-space by a simple spherical outflow model. The resulting turbulent flow saturates at an average Mach number of ~2.5 and is analysed through density and velocity power spectra, including an investigation of the evolution of the solenoidal and compressional components. We obtain a shallow density power spectrum with a slope of ~-1.2 attributed to the presence of a network of localised dense filamentary structures formed by strong shock interactions. The total velocity power spectrum slope is found to be ~-2.0, representative of Burgers shock dominated turbulence model. The density weighted velocity power spectrum slope is measured as ~-1.6, slightly less than the expected Kolmogorov scaling value (slope of -5/3) found in previous works. The discrepancy may be caused by the nature of our real space driving model and we suggest there is no universal scaling law for supersonic compressible turbulence. We find that on average, solenoidal modes slightly dominate in our turbulence model as the interaction between strong curved compressible shocks generates solenoidal modes, and compressible modes decay faster.

Linear and non-linear evolution of current-carrying highly magnetized jets

We investigate the linear and nonlinear evolution of current-carrying jets in a periodic configuration by means of high resolution three-dimensional numerical simulations. The jets under consideration are strongly magnetized with a variable pitch profile and initially in equilibrium under the action of a force-free magnetic field. The growth of current-driven (CDI) and Kelvin-Helmholtz (KHI) instabilities is quantified using three selected cases corresponding to static, Alfvenic and super-Alfvenic jets. During the early stages, we observe large-scale helical deformations of the jet corresponding to the growth of the initially excited CDI mode. A direct comparison between our simulation results and the analytical growth rates obtained from linear theory reveals good agreement on condition that high-resolution and accurate discretization algorithms are employed. After the initial linear phase, the jet structure is significantly altered and, while slowly-moving jets show increasing helical deformations, larger velocity shear are violently disrupted on a few Alfven crossing time leaving a turbulent flow structure. Overall, kinetic and magnetic energies are quickly dissipated into heat and during the saturated regime the jet momentum is redistributed on a larger surface area with most of the jet mass travelling at smaller velocities. The effectiveness of this process is regulated by the onset of KHI instabilities taking place at the jet/ambient interface and can be held responsible for vigorous jet braking and entrainment.

Topology of hydrothermal waves in liquid bridges and dissipative structures of transported particles

High-resolution three-dimensional numerical simulations are carried out for hydrothermal waves in a thermocapillary liquid bridge with Prandtl number Pr=4 and length-to-radius aspect ratio Γ=0.66. The flow topology is analyzed using Poincaré sections in a frame of reference co-rotating with the phase velocity of the wave. We find regions of regular and chaotic motion. The regular regions are shown to be of key importance for dissipative structures of transported particles. Suspended particles which are passively advected in the bulk, but experience dissipation in a thin layer below the free surface, can rapidly form dissipative structures, also called particle accumulation structures. The shape and the formation time of the particulate structures are determined by the location of the invariant tori of the flow field with respect to the sub-surface layer in which the dissipation of the particle motion acts. The results from a hard-wall particle-free-surface interaction model are in good agreement with experimental observations.

High-resolution large eddy simulations of cavitating gasoline–ethanol blends

Cavitation plays an important role in the formation of sprays in fuel injection systems. With the increasing use of gasoline–ethanol blends, there is a need to understand how changes in fluid properties due to the use of these fuels can alter cavitation behavior. Gasoline–ethanol blends are azeotropic mixtures whose properties are difficult to model. We have tabulated the thermodynamic properties of gasoline–ethanol blends using a method developed for flash-boiling simulations. The properties of neat gasoline and ethanol were obtained from National Institute of Standards and Technology REFPROP data, and blends from 0% to 85% ethanol by mass have been tabulated. We have undertaken high-resolution three-dimensional numerical simulations of cavitating flow in a 500-µm-diameter submerged nozzle using the in-house HRMFoam homogeneous relaxation model constructed from the OpenFOAM toolkit. The simulations are conducted at 1 MPa inlet pressure and atmospheric outlet pressure, corresponding to a cavitation number range of 1.066–1.084 and a Reynolds number range of 15,000–40,000. For the pure gasoline case, the numerical simulations are compared with synchrotron X-ray radiography measurements. Despite significant variation in the fluid properties, the distribution of cavitation vapor in the nozzle is relatively unaffected by the gasoline–ethanol ratio. The vapor remains attached to the nozzle wall, resulting in an unstable annular two-phase jet in the outlet. Including turbulence at the conditions studied does not significantly change mixing behavior, because the thermal nonequilibrium at the vapor–liquid interfaces acts to low-pass filter the turbulent fluctuations in both the nozzle boundary layer and jet mixing layer.

The Generation of Turbulence below Midlevel Cloud Bases: The Effect of Cooling due to Sublimation of Snow

AbstractIn the author’s experience as a forecaster, commercial aircraft sometimes report turbulence beneath midlevel clouds that extend above upper frontal zones. Turbulence caused by Kelvin–Helmholtz instability occurs in upper frontal zones with strong vertical shear of horizontal winds. However, the turbulence seems to occur not only in the cloud bases (where upper frontal zones are) but also below the cloud bases where the vertical shear is not strong. Because those clouds are usually accompanied by precipitation that does not reach the ground, cooling by evaporation or sublimation seems to contribute to the generation of turbulence. In this paper, the mechanisms generating turbulence below midlevel cloud bases are examined by using observations and high-resolution three-dimensional numerical simulations with idealized initial conditions. The numerical simulations showed that the following sequence of events led to turbulence. Falling snow sublimated below cloud bases and cooled the air, which created absolute instability. This generated Rayleigh–Bénard convection cells. The vertical motion caused turbulence. The horizontal scale of the convection was about 800–1000 m, and the variations of vertical wind velocity were up to about 7 m s−1. The cloud base was accompanied by a virga-like distribution of snow. Sensitivity experiments showed the appropriate conditions to cause the turbulence: 1) the cloud-base temperature was between about 0° and −15°C, 2) the relative humidity in subcloud layers was sufficiently low, and 3) the stability in subcloud layers was weak. The results of the numerical simulations agreed well with the observations.

BURNING THERMALS IN TYPE Ia SUPERNOVAE

We develop a one-dimensional theoretical model for thermals burning in Type Ia supernovae based on the entrainment assumption of Morton, Taylor and Turner. Extensions of the standard model are required to account for the burning and for the expansion of the thermal due to changes in the background stratification found in the full star. The model is compared with high-resolution three-dimensional numerical simulations, both in a uniform environment, and in a full-star setting. The simulations in a uniform environment present compelling agreement with the predicted power-laws and provide model constants for the full-star model, which then provides excellent agreement with the full-star simulation. The importance of the different components in the model are compared, and are all shown to be relevant. An examination of the effect of initial conditions was then conducted using the one-dimensional model, which would have been infeasible in three dimensions. More mass was burned when the ignition kernel was larger and closer to the center of the star. The turbulent flame speed was found to be important during the early-time evolution of the thermal, but played a diminished role at later times when the evolution is dominated by the large-scale hydrodynamics responsible for entrainment. However, a higher flame speed effectively gave a larger initial ignition kernel and so resulted in more mass burned. This suggests that future studies should focus on the early-time behavior of these thermals (in particular, the transition to turbulence), and that the choice of turbulent flame speed does not play a significant role in the dynamics once the thermal has become established.

Self-aggregation of clouds in conditionally unstable moist convection

The behavior of moist Rayleigh-Bénard convection is investigated using a Boussinesq model with a simplified thermodynamics for phase transitions. This idealized configuration makes the problem accessible to high-resolution three-dimensional direct numerical simulations without small-scale parameterizations of the turbulence for extended layers with aspect ratios up to 64. Our study is focused on the frequently observed conditionally unstable environment that is stably stratified for unsaturated air, but is unstable for cloudy air. We find that no sharp threshold for the transition to convective turbulence exists, a situation similar to wall-bounded shear flows. Rather, the transition depends on the amplitude of the initial perturbation of the quiescent equilibrium and on the aspect ratio of the convective domain. In contrast to the classical dry Rayleigh-Bénard case, convection is highly asymmetric with respect to the vertical direction. Moist upwelling air inside turbulent cloud aggregates is surrounded by ambient regions of slowly descending unsaturated air. It is also found that conditionally unstable moist convection is inefficient at transporting energy. Our study suggests that there is an upper bound on the Nusselt number in moist convection that is lower than that of the classical dry case.

High-resolution 3D relativistic MHD simulations of jets

Relativistic magnetized jets are key elements in active galactic nuclei and in other astrophysical environments. Their structure and evolution involve a complex non-linear physics that can be approached by numerical studies only. Still, owing to a number of challenging computational aspects, only a few numerical investigations have been undertaken so far. In this paper, we present high-resolution three-dimensional numerical simulations of relativistic magnetized jets carrying an initially toroidal magnetic field. The presence of a substantial toroidal component of the field is nowadays commonly invoked and held responsible for the process of jet acceleration and collimation. We find that the typical nose cone structures, commonly observed in axisymmetric two-dimensional simulations, are not produced in the three-dimensional case. Rather, the toroidal field gives rise to strong current-driven kink instabilities leading to jet wiggling. However, it appears to be able to maintain a highly relativistic spine along its full length. By comparing low- and high-resolution simulations, we emphasize the impact of resolution on the jet dynamical properties.

Structure of Convective Circulation in the Atmospheric Boundary Layer over the Northwestern Pacific Ocean under a Subtropical High

To confirm the structure of convective circulation in the atmospheric boundary layer (ABL) over the northwestern Pacific Ocean around the Southwest Islands of Japan under a subtropical high, we perform intensive observations using radiosonde and Aerosonde in August 2002 and high-resolution three-dimensional numerical simulations. A well-mixed subcloud layer exists below 0.7 km during the observation period. The absolute values of the correlation coefficient between anomalies in potential temperature and mixing ratio of water vapor are mainly small (less than 0.3) in the lower subcloud layer. The frequency of a positive correlation coefficient at a height of 0.1 km is only 46%. This suggests that fewer warm moist thermals exist in the lower subcloud layer.The simulation results using 100-m horizontal grid resolution show a moist air mass in updraft regions. These thermals are driven by positive buoyant flux, and this positive buoyancy is contributed by moisture, i.e., density anomalies in water vapor and dry air, instead of heat. This should be attributed to the supply of small sensible and abundant latent heat fluxes from the sea surface under the small air-sea temperature difference.

Relativistic Hydrodynamic Flows Using Spatial and Temporal Adaptive Structured Mesh Refinement

Astrophysical relativistic flow problems require high-resolution three-dimensional (3D) numerical simulations. In this paper, we describe a new parallel 3D code for simulations of special relativistic hydrodynamics (SRHD) using both spatially and temporally structured adaptive mesh refinement (AMR). We used the method of lines to discretize the SRHD equations spatially and a total variation diminishing (TVD) Runge-Kutta scheme for time integration. For spatial reconstruction, we have implemented piecewise linear method (PLM), piecewise parabolic method (PPM), third-order convex essentially nonoscillatory (CENO) and third- and fifth-order weighted essentially nonoscillatory (WENO) schemes. Flux is computed using either direct flux reconstruction or approximate Riemann solvers including HLL, modified Marquina flux, local Lax-Friedrichs flux formulas, and HLLC. The AMR part of the code is built on top of the cosmological Eulerian AMR code enzo. We discuss the coupling of the AMR framework with the relativistic solvers. Via various test problems, we emphasize the importance of resolution studies in relativistic flow simulations because extremely high resolution is required especially when shear flows are present in the problem. We also present the results of two 3D simulations of astrophysical jets: AGN jets and GRB jets. Resolution study of those two cases further highlights the need of high resolutions to calculate accurately relativistic flow problems.

The origin of the stationary frontal wave packet spontaneously generated in rotating stratified vortex dipoles

The origin of the stationary frontal wave packet spontaneously generated in rotating and stably stratified vortex dipoles is investigated through high-resolution three-dimensional numerical simulations of non-hydrostatic volume-preserving flow under the f-plane and Boussinesq approximations. The wave packet is rendered better at mid-depths using ageostrophic quantities like the vertical velocity or the vertical shear of the ageostrophic vertical vorticity. The analysis of the origin of vertical velocity anomalies in shallow layers using the generalized omega-equation reveals that these anomalies are related to the material rate of change of the ageostrophic differential vorticity, which in shallow layers are themselves related to the large-scale ageostrophic flow along the dipole axis, and in particular, to the advective acceleration. It is found that on the anticyclonic side of the dipole axis the combined effect of the speed and centripetal accelerations causes an anticyclonic rotation of the horizontal ageostrophic vorticity vector in a time scale of about one inertial period. These facts support the hypothesis that the origin of the stationary and spontaneously generated frontal wave packet at mid-depths is the large acceleration of the fluid particles as they move along the anticyclonic side of the dipole axis in shallow layers.

A comparative study of the turbulent Rayleigh–Taylor instability using high-resolution three-dimensional numerical simulations: The Alpha-Group collaboration

The turbulent Rayleigh–Taylor instability is investigated in the limit of strong mode-coupling using a variety of high-resolution, multimode, three dimensional numerical simulations (NS). The perturbations are initialized with only short wavelength modes so that the self-similar evolution (i.e., bubble diameter Db∝amplitude hb) occurs solely by the nonlinear coupling (merger) of saturated modes. After an initial transient, it is found that hb∼αbAgt2, where A=Atwood number, g=acceleration, and t=time. The NS yield Db∼hb/3 in agreement with experiment but the simulation value αb∼0.025±0.003 is smaller than the experimental value αb∼0.057±0.008. By analyzing the dominant bubbles, it is found that the small value of αb can be attributed to a density dilution due to fine-scale mixing in our NS without interface reconstruction (IR) or an equivalent entrainment in our NS with IR. This may be characteristic of the mode coupling limit studied here and the associated αb may represent a lower bound that is insensitive to the initial amplitude. Larger values of αb can be obtained in the presence of additional long wavelength perturbations and this may be more characteristic of experiments. Here, the simulation data are also analyzed in terms of bubble dynamics, energy balance and the density fluctuation spectra.