Three-dimensional Density Field Research Articles

The dependence of galaxy properties on the underlying three-dimensional matter density field at 2.0 < z < 2.5

The dependence of galaxy properties on the underlying three-dimensional matter density field at 2.0 &amp;lt; <i>z</i> &amp;lt; 2.5

Imprint of massive neutrinos on Persistent Homology of large-scale structure

ABSTRACT Exploiting the Persistent Homology technique and its complementary representations, we examine the footprint of summed neutrino mass ($M_{\nu }$) in the various density fields simulated by the publicly available Quijote suite. The evolution of topological features by utilizing the superlevel filtration on three-dimensional density fields at zero redshift, reveals a remarkable benchmark for constraining the cosmological parameters, particularly $M_{\nu }$ and $\sigma _8$. The abundance of independent closed surfaces (voids) compared to the connected components (clusters) and independent loops (filaments), is more sensitive to the presence of $M_{\nu }$ for $R=5$ Mpc $h^{-1}$ irrespective of whether using the total matter density field (m) or cold dark matter + baryons field ($\mathrm{ \mathrm{cb}}$). Reducing the degeneracy between $M_{\nu }$ and $\sigma _8$ is achieved via Persistent Homology for the m field but not for the $\mathrm{cb}$ field. The uncertainty of $M_{\nu }$ at $1\sigma$ confidenc interval from the joint analysis of Persistent Homology vectorization for the m and $\mathrm{cb}$ fields smoothed by $R=5$ Mpc $h^{-1}$ at $z=0$ reaches 0.0152 and 0.1242 eV, respectively. Noticing the use of the three-dimensional underlying density field at $z=0$, the mentioned uncertainties can be treated as the theoretical lower limits.

Three-Dimensional Density And Flight Direction Measurement In The Ballistic Range

Simultaneous measurement of the three-dimensional density field around the Hayabusa-type re-entry capsule model and the angle of attack using the 3D-BOS (Background Oriented Schlieren) technique was carried out in the ballistic range at the Institute of Fluid Science (IFS), Tohoku University. A Hayabusa-type re-entry capsule model with a diameter of 25 mm was used for the test model, and the free-flight Mach number was 1.20. A total of 22 digital cameras were installed for the two set of 3D-BOS measurement system. Each of 11 digital cameras are synchronized and captures the flow field around the test model at two different moments. In this paper, the model position at the different moments were measured using two sets of 3D-BOS measuring systems based on image processing, and the model flight direction and angle of attack were measured. The results showed that the stagnation point predicted from the angle of attack and the maximum density point showed good agreement. Therefore, we conclude that the density distribution around the reentry capsule model obtained by the 3D-BOS technique can be directly compared with the CFD results.

Algorithm to produce a density field with given two-, three-, and four-point correlation functions

ABSTRACT Here we show how to produce a three-dimensional density field with a given set of higher order correlation functions. Our algorithm enables producing any desired two-, three-, and four-point functions, including odd parity for the last ones. We note that this algorithm produces the desired correlations around a set of ‘primary’ points, matched to how the spherical-harmonic-based algorithms ENCORE and CADENZA measure them. These ‘primary points’ must be used as those around which the correlation functions are measured. We also generalize the algorithm to (i) N-point correlations with $N\ \gt\ 4$, (ii) dimensions other than three, and (iii) beyond scalar quantities. This algorithm should find use in verifying analysis pipelines for higher order statistics in upcoming galaxy redshift surveys, such as Dark Energy Spectroscopic Instrument (DESI), Euclid, Roman, and Spherex, as well as intensity mapping. In particular, it may be helpful in searches for parity violation in the four-point correlation function of these samples, for which producing initial conditions for N-body simulations is both costly and highly model dependent at present, and so alternative methods, such as that developed here, are desirable.

Deep electron cloud‐activity and field‐activity relationships

AbstractChemists have been pursuing general mathematical laws to explain and predict molecular properties for a long time. However, most of the traditional quantitative structure‐activity relationship (QSAR) models have limited application domains; for example, they tend to have poor generalization performance when applied to molecules with parent structures different from those of the trained molecules. This paper attempts to develop a new QSAR method that is theoretically possible to predict various properties of molecules with diverse structures. The proposed deep electron cloud‐activity relationships (DECAR) and deep field‐activity relationships (DFAR) methods consist of three essentials: (1) a large number of molecule entities with activity data as training objects and responses; (2) three‐dimensional electron cloud density (ECD) or related field data by the accurate density functional theory methods as input descriptors; and (3) a deep learning model that is sufficiently flexible and powerful to learn the large data described above. DECAR and DFAR are used to distinguish 977 sweet and 1965 non‐sweet molecules (with 6‐fold data augmentation), and the classification performance is demonstrated to be significantly better than the traditional least squares support vector machine (LS‐SVM) models using traditional descriptors. DECAR and DFAR would provide a possible way to establish a widely applicable, cumulative, and shareable artificial intelligence‐driven QSAR system. They are likely to promote the development of an interactive platform to collect and share the accurate ECD and field data of millions of molecules with annotated activities. With enough input data, we envision the appearance of several deep networks trained for various molecular activities. Finally, we could anticipate a single DECAR or DFAR network to learn and infer various properties of interest for chemical molecules, which will become an open and shared learning and inference tool for chemists.

Super-resolution of time-resolved three-dimensional density fields of the B mode in an underexpanded screeching jet

The estimation of time-resolved three-dimensional (3D) density fields of an underexpanded jet at the nozzle pressure ratio of 2.42, a so-called “spatiotemporal super-resolution” was conducted using non-time-resolved three-dimensional background-oriented schlieren (3D-BOS) and time-resolved microphone measurements. This approach aims to reconstruct three-dimensional density fields associated with the intermittent and switching behavior of the B mode of a screeching jet from the microphone data by constructing a linear regression model. An azimuthal Fourier decomposition is applied to the 3D-BOS and microphone data, and the proper orthogonal decomposition (POD) is performed for each of their azimuthal Fourier modes. The m=1 azimuthal Fourier mode is dominant in both cases, and the leading two POD modes in the m=1 azimuthal mode of the microphone data are associated with the B mode. The linear regression model is constructed from the POD modes of the m=1 azimuthal 3D-BOS data and the first two microphone POD modes of the m=1 azimuthal mode of the microphone data. The three-dimensional density fields reconstructed from each POD mode of the m=1 azimuthal mode of the microphone data have helical structures with opposite rotation directions. The amplitudes of those POD modes change with time, and the azimuthal structure associated with the B mode is determined depending on those amplitudes. The present result showed that intermittency in the flapping to helical structures and their strength can be interpreted by the temporal changes in the strengths of two rotating helical structures with opposite rotation directions.

Bayesian field-level inference of primordial non-Gaussianity using next-generation galaxy surveys

ABSTRACT Detecting and measuring a non-Gaussian signature of primordial origin in the density field is a major science goal of next-generation galaxy surveys. The signal will permit us to determine primordial-physics processes and constrain models of cosmic inflation. While traditional approaches use a limited set of statistical summaries of the galaxy distribution to constrain primordial non-Gaussianity, we present a field-level approach by Bayesian forward modelling the entire three-dimensional galaxy survey. Since our method includes the entire cosmic field in the analysis, it can naturally and fully self-consistently exploit all available information in the large-scale structure, to extract information on the local non-Gaussianity parameter, fnl. Examples include higher order statistics through correlation functions, peculiar velocity fields through redshift-space distortions, and scale-dependent galaxy bias. To illustrate the feasibility of field-level primordial non-Gaussianity inference, we present our approach using a first-order Lagrangian perturbation theory model, approximating structure growth at sufficiently large scales. We demonstrate the performance of our approach through various tests with self-consistent mock galaxy data emulating relevant features of the SDSS-III/BOSS-like survey, and additional tests with a Stage IV mock data set. These tests reveal that the method infers unbiased values of fnl by accurately handling survey geometries, noise, and unknown galaxy biases. We demonstrate that our method can achieve constraints of $\sigma _{{f_\mathrm{nl}}} \approx 8.78$ for SDSS-III/BOSS-like data, indicating potential improvements of a factor ∼2.5 over current published constraints. We perform resolution studies on scales larger than ∼16h−1 Mpc showing the promise of significant constraints with next-generation surveys. Furthermore, the results demonstrate that our method can consistently marginalize all nuisance parameters of the data model. The method further provides an inference of the three-dimensional primordial density field, providing opportunities to explore additional signatures of primordial physics. This first demonstration of a field-level inference pipeline demonstrates a promising complementary path forward for analysing next-generation surveys.

Three-dimensional density measurements of a heated jet using laser-speckle tomographic background-oriented schlieren

Tomographic background-oriented schlieren (TBOS) is a promising optical technique for 3D density field measurement in turbulent flows. Path-integrated information on refractive index gradients is captured by cameras looking through the flow at textured background patterns in the form of apparent displacements of the patterns. A tomographic reconstruction of the three-dimensional refractive index gradients is performed, from which the instantaneous three-dimensional density field is obtained. Most TBOS measurements to date have been limited by large temporal blurring and/or large defocus blurring. Our 15-camera experimental set-up uses a pulsed laser for both illumination and creation of a laser-speckle background pattern. The high-power laser ensures adequate exposures in nanoseconds, so temporal integration is minimised in each measurement. The beam is spread to illuminate a surface that is observed by fifteen cameras that are evenly spaced circumferentially around the flow. Laser speckle patterns are observed by the cameras due to the diffuse reflection and interference of coherent light from roughness on the surface. We develop a methodology for selecting the optimal focal length, focus distance, and lens aperture by considering the compromise between displacement sensitivity, defocus blurring and speckle size. Low temporal- and spatial-blurring 3D temperature field measurements with excellent resolution of turbulent flow features are obtained and good agreement with thermocouple. Classical behaviours of the potential core instability such as vortex rollup and radial ejections are identified. The behaviour of the potential core is examined further, which reveals that the potential core undergoes considerable deformation and fragmentation during the transition to turbulence.

Spatiotemporal vertical velocity variation in the western tropical Pacific and its relation to decadal ocean variability

Prior to the cold phase of the Pacific decadal oscillation (PDO), sea surface water in the western tropical North Pacific is heated. To evaluate the impact of the subsurface water on the upper-ocean (<100 m depth) temperature variation, we examined the vertical velocity variation. We applied altimetry-based gravest empirical mode (AGEM) method to Argo hydrographic profile (2001–2017) and altimetric sea surface height (1993–2017) data in the western tropical Pacific and obtained a time-varying three-dimensional (3D) potential density field in the layer from 100 to 1000 m depth. From the AGEM-derived potential density, we derived a 3D field of vertical velocity below 100 m depth by the P-vector inverse method, which is based on potential density and vorticity conservations, and examined the spatiotemporal characteristics, focusing on the seasonal to quasi-decadal (QD) variations. In the mean state, substantial upward current (>10^{-6},{text {m,s}}^{-1}) is observed in the region of 10^{circ }–17^{circ }N, 140^{circ }E–180^{circ } in the North Equatorial Current, southern part of the North Pacific subtropical gyre. Possibly besides the Ekman upwelling, the upward current contributes to the cooling of the upper ocean in the western tropical North Pacific through the upward advection of the deep cold water. The upward current was found to be intensified (attenuated) by the strengthening (weakening) of the subtropical gyre interior southward flow on the QD timescale in association with frequent occurrences of La Niña (El Niño) events. The weakening of the upward current is responsible for the QD increase in upper-ocean temperature in the western tropical North Pacific, which is equivalent to or larger than 0.2~^{circ }C, several years after the warm phase of the PDO. The QD temperature variation may affect the phase reversal of the PDO through the heat transport of the western boundary current of the subtropical gyre, i.e., the Kuroshio.

Three-dimensional density field of a screeching under-expanded jet in helical mode using multi-view digital holographic interferometry

A synchronised multi-axis digital holographic interferometry set-up is presented for the study of 3-D flow fields with large density gradients. This optical configuration provides instantaneous interferograms with fine spatial resolution in six directions of projection. A regularised tomographic approach taking into account the presence of possible shock waves is furthermore considered to reconstruct 3-D density fields. Applied to a screeching under-expanded supersonic jet with helical dynamics, this set-up is used to provide dense optical phase measurements in the initial region of the jet. The jet mean density field is shown to be satisfactorily estimated with sharply resolved density gradients. In addition, an approach based on azimuthal Fourier transform and snapshot proper orthogonal decomposition (POD) applied to the instantaneous flow observations is proposed to study the main coherent dynamics of the jet. Relying on a cluster analysis of the azimuthal POD mode coefficients, a reduced dynamical model in the POD mode phase space is used as an approximation of the two observed limit cycles. A clear 3-D representation of the density field of a helical instability associated with screech mode C is then evidenced, with two equally probable directions of rotation. Switching between the two directions is reported, highlighting intermittency in the feedback loop. This helical structure is particularly seen to extend to the jet core, driving its internal dynamics and inducing out-of-phase density fluctuations between the outer and inner shear layers. These out-of-phase motions are related to the non-uniform radial distribution of fluctuation phase associated with the outer-layer Kelvin–Helmholtz instability wave.

Minkowski functionals and the nonlinear perturbation theory in the large-scale structure: Second-order effects

The second-order formula of Minkowski functionals in weakly non-Gaussian fields is compared with the numerical $N$-body simulations. Recently, the weakly non-Gaussian formula of Minkowski functionals is extended to include the second-order effects of non-Gaussianity in general dimensions. We apply this formula to the three-dimensional density field in the large-scale structure of the Universe. The parameters of the second-order formula include several kinds of skewness and kurtosis parameters. We apply tree level of nonlinear perturbation theory to estimate these parameters, including novel calculations of quartic cumulants. First we compare the theoretical values with those of numerical simulations on the basis of parameter values, and next we test the performance of the analytic formula combined with the perturbation theory. The second-order formula outperforms the first-order formula in general. The performance of the perturbation theory depends on the smoothing radius applied in defining the Minkowski functionals. The quantitative comparisons are presented in detail.

Three-dimensional density measurements of a heated jet using laser-speckle tomographic background-oriented schlieren

Three-dimensional density field measurement techniques can be used to understand the complex heat transfer and mixing processes that occur in turbulent flows. Tomographic background-oriented schlieren (BOS) is an optical technique that can be used to measure the instantaneous three-dimensional density field in turbulent flows. Light rays propagating through the flow are deflected from their ambient path due to variations in refractive index related to the spatial density gradients. In BOS, a camera is placed looking through the flow at a reference image, which captures path-integrated information on the refractive index gradients in the form of apparent image displacements Richard and Raffel (2001). The displacements recorded simultaneously from many cameras placed around the flow form the basis of a tomographic reconstruction of the three-dimensional refractive index gradients Goldhahn and Seume (2007), from which the density field is obtained through integration of the gradients and application of the Gladstone-Dale relation.

A Case Study of the 3D Water Vapor Tomography Model Based on a Fast Voxel Traversal Algorithm for Ray Tracing

A fast voxel traversal algorithm for ray tracing was applied to build a 4 × 4 × 20 tomography model using the observation data of 11 ground-based Global Navigation Satellite System (GNSS) meteorology (GNSS/MET) stations in Hebei Province, China. The precipitation water vapor (PWV) observed at 05 a.m. (Universal Time Coordinated: UTC) on 10 December 2019, was used to reconstruct three-dimensional (3D) water vapor density fields over the test area. The tomographic results (GNSS_T) show that the water vapor density above this area is mainly below 25 g/m3 and is concentrated between the first to the fourth layers. The vertical distribution conforms to the exponential characteristics, while the horizontal distribution shows a decreasing trend from southwest to northeast. In addition, the results of the 0.25° grid dataset generated by the Global Forecast System (GFS) of the National Center for Environmental Forecasting (NCEP) (GFS_L) were interpolated to the height of the tomographic grid, which is in good agreement with the tomographic results. GFS_L is larger than GNSS_T on the first floor at the surface, with an average deviation of 0.19 g/m3. In contrast, GFS_L from the second floor to the top of the model is smaller than GNSS_T, with the average deviations distributed between −0.08 and −0.15 g/m3.

Cosmic Velocity Field Reconstruction Using AI

Abstract We develop a deep-learning technique to infer the nonlinear velocity field from the dark matter density field. The deep-learning architecture we use is a “U-net” style convolutional neural network, which consists of 15 convolution layers and 2 deconvolution layers. This setup maps the three-dimensional density field of 323 voxels to the three-dimensional velocity or momentum fields of 203 voxels. Through the analysis of the dark matter simulation with a resolution of 2h −1 Mpc, we find that the network can predict the the nonlinearity, complexity, and vorticity of the velocity and momentum fields, as well as the power spectra of their value, divergence, and vorticity and its prediction accuracy reaches the range of k ≃ 1.4 h Mpc−1 with a relative error ranging from 1% to ≲10%. A simple comparison shows that neural networks may have an overwhelming advantage over perturbation theory in the reconstruction of velocity or momentum fields.

The Influence of Mesh Resolution on 3D RANS Flow Simulations in Turbomachinery Flow Parts

The question of the difference mesh refinement degree influence on the results of calculation of the three-dimensional viscous gas flows in the flow parts of turbomachines using the RANS flow models and second order numerical methods is considered. Calculations of flows for a number of turbine and compressor grids on successively refining grids have been performed. We used H-type grids with approximate orthogonalization of cells in the boundary layer. The calculations were carried out using a CFD solver F with the use of an implicit ENO scheme of the second order, a local time step, and a simplified multigrid algorithm. When calculating the flow on fine grids, the following were used: convergence acceleration tools implemented in the solver; truncation of the computational domain with subsequent distribution of the results based on the symmetry property; the computational domain splitting into parts and computations parallelizing. Comparison of the obtained results is carried out, both in terms of qualitative resolution of the complex structure of three-dimensional flows, and in terms of quantitative assessment of losses. Grid convergence was estimated in two ways. In the first, the characteristic two-dimensional distributions of parameters obtained on different grids were visually compared. The purpose of such comparisons was to evaluate the sufficient degree of solution of both the general structure of the flow in grids and its features, namely, shock waves, contact discontinuities, separation zones, wakes, etc. The second estimation method is based on the grid convergence index (GCI). The GCI calculated from the three-dimensional density field was considered in this paper. It is concluded that for scientific research requiring high accuracy of calculations and detailing of the structure of a three-dimensional flow, very fine difference meshes with the number of cells from 106 to 108 in one blade-to-blade channel are needed, while for engineering calculations, under certain conditions, it is sufficient to use meshes with the number of cells less than 1 million in one blade-to-blade channel.

Ray-tracing log-normal simulation for weak gravitational lensing: applicationto the cross-correlation with galaxies

We present an algorithm to self-consistently generate mock weak gravitational lensing convergence fields and galaxy distributions in redshift space. We generate three-dimensional cosmic density fields that follow a log-normal distribution, and ray-trace them to produce convergence maps. As we generate the galaxy distribution from the same density fields in a manner consistent with ray-tracing, the galaxy-convergence cross-power spectrum measured from the mock agrees with the theoretical expectation with high precision.We use this simulation to forecast the quality of galaxy-shear cross-correlation measurements from the SubaruHyper Suprime-Cam (HSC) and Prime Focus Spectrograph (PFS) surveys.We find that the nominal HSC and PFS surveys would detect the cross power spectra with signal-to-noise ratios of 20 and 5 at the lowest (z = 0.7) and highest (z = 2.2) redshift bins, respectively.

Assessment of three-dimensional density measurements from tomographic background-oriented schlieren (BOS)

Tomographic background-oriented schlieren (BOS) is used to measure three-dimensional instantaneous density fields in turbulent flows. Density gradients are related to the refractive index through the Gladstone-Dale relation. Based on path-integrated apparent displacements in a background pattern from path variations in light rays travelling through the flow, a tomographic reconstruction of the three-dimensional refractive index gradients can be obtained. A Poisson equation of the reconstructed gradients is solved to obtain the refractive index field. We examine four sources of measurement error: (i) defocus blurring; (ii) spatial averaging in the finite difference scheme for solving the Poisson equation; (iii) limited-view tomographic reconstruction; and (iv) displacement field noise. Synthetic BOS displacements are generated from the instantaneous density fields of a heated jet at a Reynolds number of 10 000 and jet exit centreline-to-ambient density ratio of 0.83 computed via DNS. Synthetic background displacements are produced by raytracing through the refractive index field obtained using the Gladstone-Dale relation. The virtual BOS setup consists of 15 cameras placed circumferentially around the jet axis, based on a proposed experimental setup. We show that defocus blurring has the greatest impact on the measurement accuracy, using the RMS and peak errors between the measured and true DNS fields as metrics; its impact is greatest in the transitional region and decreases steadily downstream. Sources (ii) and (iv) have marginal impact (for noise 15% of mean displacements). Three reconstruction methods are tested: (i) filtered back-projection (FBP); (ii) an algebraic reconstruction technique (ART); and (iii) sequential FBP-ART. ART and FBP-ART are similarly accurate, producing under half the errors of FBP, which suffers from significant reconstruction artefacts. The reconstruction error is modest compared to the impact of defocus blurring.

Quantitative Flow Visualization of Slightly Underexpanded Microjets by Mach\u2013Zehnder Interferometers

The Mach–Zehnder interferometer with the finite fringe setting is applied for a shock-containing microjet issued from an axisymmetric convergent nozzle with an inner diameter of 1.0 mm at the exit. Experiments are performed at a nozzle pressure ratio of 3.0 to produce a slightly underexpanded sonic jet where the Reynolds number, based upon the diameter and flow properties at the nozzle exit, is 4.45 times 10^4. The reconstruction of the jet density field is performed using the Abel inversion method under the assumption of an axisymmetric flow as well as the Fourier transform method for the phase shift analyses of interferograms. The three-dimensional density field of a shock-containing microjet can be captured with a spatial resolution of 4 upmum, and the near-field shock structure inside the jet plume is shown in the density contour plot at the cross-section including the jet centerline. In addition, the density field of the microjet is illustrated with various techniques, including representation such as the vertical-knife-edge schlieren, the horizontal-knife-edge schlieren, the bright-field schlieren, and the shadowgraphy. In addition to experiments, the Reynolds averaged Navier–Stokes (RANS) simulation with the SST k–omega turbulence model is carried out to model the microjets, and a quantitative comparison with the experiments is performed. The jet centerline density profile obtained by the present experiment is quantitatively compared with those from the previous Mach–Zehnder interferometer, the background oriented schlieren, as well as the RANS simulation.

The highly variable time evolution of star-forming cores identified with dendrograms

ABSTRACT We investigate the time evolution of dense cores identified in molecular cloud simulations using dendrograms, which are a common tool to identify hierarchical structure in simulations and observations of star formation. We develop an algorithm to link dendrogram structures through time using the three-dimensional density field from magnetohydrodynamical simulations, thus creating histories for all dense cores in the domain. We find that the population-wide distributions of core properties are relatively invariant in time, and quantities like the core mass function match with observations. Despite this consistency, an individual core may undergo large (&amp;gt;40 per cent), stochastic variations due to the redefinition of the dendrogram structure between time-steps. This variation occurs independent of environment and stellar content. We identify a population of short-lived (&amp;lt;200 kyr) overdensities masquerading as dense cores that may comprise $\sim\!20$ per cent of any time snapshot. Finally, we note the importance of considering the full history of cores when interpreting the origin of the initial mass function; we find that, especially for systems containing multiple stars, the core mass defined by a dendrogram leaf in a snapshot is typically less than the final system stellar mass. This work reinforces that there is no time-stable density contour that defines a star-forming core. The dendrogram itself can induce significant structure variation between time-steps due to small changes in the density field. Thus, one must use caution when comparing dendrograms of regions with different ages or environment properties because differences in dendrogram structure may not come solely from the physical evolution of dense cores.

Three-dimensional reconstruction of a microjet with a Mach disk by Mach–Zehnder interferometers

The Mach–Zehnder interferometer with the finite fringe method is applied for the first time to diagnose the three-dimensional density field of a shock-containing microjet issued from a convergent nozzle with an inner diameter of 1 mm at the exit. Experiments are performed at a nozzle pressure ratio of 4.0 to produce an underexpanded free jet with a Mach disk in the first shock-cell where the Reynolds number, based upon the diameter and flow properties at the nozzle exit, is . Interferogram analyses for reconstructing the jet density fields are performed using the Abel inversion method, in which the analysis of the phase shift of the deformed fringe relative to the background fringe is carried out by the Fourier transform method. In addition to experiments, the flow field of the shock-containing microjet is simulated by solving the Reynolds-averaged Navier–Stokes equations with the SST – turbulence model for a quantitative mutual comparison between the simulation and the experiment. The detailed variation of the density field associated with the Mach disk, the slip streams and the outer shear layers near the jet boundaries are successfully captured. Furthermore, the density profile along the jet centreline obtained by the present experiment is quantitatively compared with those from prior quantitative visualization studies, such as rainbow schlieren deflectometry, background oriented schlieren, moire schlieren and Mach–Zehnder interferometer. The three-dimensional contour map coloured by the magnitude of the density gradient vector inside the microjet reveals the flow topology of the near-field shock systems and elucidates the spatial variation of the shock strength.