We perform numerical simulations of forced homogeneous isotropic turbulence over a range of bulk viscosities, Reynolds numbers and Mach numbers to investigate the scaling of key flow statistics. Using the Helmholtz decomposition, we analyse the scalings of Favre-averaged turbulent kinetic energy (TKE), root-mean-square (r.m.s.) pressure, pressure dilatation, dilatational dissipation and higher-order velocity-gradient moments. Additionally, new models are proposed for the pressure-dilatation term and the bulk-viscosity dependence of dilatational dissipation. Although the solenoidal and dilatational components of the Favre-averaged TKE are not strictly orthogonal, our numerical results demonstrate that their ratio is well approximated by the squared ratio of the corresponding r.m.s. velocities. The r.m.s. pressure approaches the pseudo-sound scaling as bulk viscosity increases. Within the Donzis r.m.s. pressure model (Donzis & John 2020 Phys. Rev. Fluids 5 (8), 084609), we find that the solenoidal contribution becomes dominant for large bulk viscosity. Pressure dilatation is found to depart systematically from pseudo-sound predictions: without bulk viscosity it favours transfer from kinetic to internal energy, while finite bulk viscosity can reverse this transfer at high Mach numbers. The scaling exponent of dilatational dissipation is shown to vary with bulk viscosity, enabling a corrected model for its exponent and prefactor. Velocity-gradient skewness and flatness reveal that the onset of shocklet-induced divergence is delayed with increasing bulk viscosity and may be suppressed entirely. The results extend recent velocity-ratio-based scaling frameworks and provide modelling insights into compressible turbulence.

Based on wave theory for unsaturated porous media, this study develops a thermoviscoelastic wave equation incorporating Kelvin-Voigt viscoelasticity and generalized thermoelasticity. An analytical solution for plane P-wave incidence is derived via Helmholtz decomposition. The findings indicate that displacement amplification factor differs markedly between thermoviscoelastic and thermoelastic models; thermal conductivity has limited influence; horizontal and vertical displacement amplification factors respond distinctly to variations in saturation, relaxation time, and thermal expansion coefficient. Both factors exhibit temperature sensitivity with a 30 K period, reaching a minimum at 293.2 K.

Abstract A linear decomposition of states underpins many classical systems. This is the case of the Helmholtz decomposition, used to split vector fields into divergence-free and potential components, and of the dry Boussinesq system in atmospheric dynamics, where identifying the slow and fast components of the flow can be viewed as a decomposition. The dry Boussinesq system incorporates two leading ingredients of mid-latitude atmospheric motion: rotation and stratification. In both cases the leading order dynamics are linear so we can rely on an eigendecomposition to decompose states. Here we study the extension of dry Boussinesq to incorporate another important ingredient in the atmosphere: moisture and clouds. The key challenge with this system is that nonlinearities are present at leading order due to phase boundaries at cloud edge. Therefore standard tools of linear algebra, relying on eigenvalues and eigenvectors, are not applicable. The question we address in this paper is this: in spite of the nonlinearities, can we find a decomposition for this moist Boussinesq system? We identify such a decomposition adapted to the nonlinear balances arising from water phase boundaries. This decomposition combines perspectives from partial differential equations (PDEs), the geometry, and the conserved energy. Moreover it sheds light on two aspects of previous work. First, this decomposition shows that the nonlinear elliptic PDE used for potential vorticity and moisture inversion can be used outside the limiting system where it was first derived. Second, we are able to rigorously justify, and interpret geometrically, an existing numerical method for this elliptic PDE. This decomposition may be important in applications because, like its linear counterparts, it may be used to analyse observational data. Moreover, by contrast with previous decompositions, its formulation includes the nonlinearity from the presence of clouds and latent heating.

This paper advances the hypothesis that stellar plasma constitutes a candidate form of inorganic living matter when evaluated against a substrate-independent framework for life derived from first principles. The central epistemological argument is that every existing mainstream definition of life is contingently derived from a single biological data point — terrestrial carbon-based life — and therefore cannot legitimately function as a universal criterion. Building from this critique, five substrate-independent criteria for life are formalised using dynamical systems theory, nonequilibrium thermodynamics, Lyapunov stability analysis, Helmholtz decomposition, and information-theoretic transfer entropy. Each criterion is stated as a precise mathematical condition, connected to the organisational feature of living systems it captures, and assessed against the known physics of stellar plasma. The five criteria are formulated as jointly necessary for a physical system to qualify as living. They are not asserted as sufficient. A sufficiency claim is presented only as a testable conjecture. The application to plasma therefore demonstrates compliance with necessary conditions, while the broader question of sufficiency remains open and outside the scope of the results. Empirical support is drawn from Parker Solar Probe observational data, nucleosynthetic inheritance transmitted through stellar supernovae, and laboratory complex plasma experiments. A formal conjecture of joint sufficiency is stated. The framework is situated relative to Assembly Theory through a derived formal bridge between assembly index and transfer entropy. A continuum model of life is proposed, including resolution of the individuation problem through nested temporal scales. Falsifiable predictions are derived and the primary empirical gap identified.

Abstract We are concerned with the existence and uniqueness of an axisymmetric supersonic solution with non-zero vorticity and non-zero angular momentum density for the steady Euler–Poisson system in three-dimensional divergent nozzles when prescribing the velocity, strength of electric field, and the entropy at the entrance. We first reformulate the problem via the method of the Helmholtz decomposition for three-dimensional axisymmetric flows and obtain a solution to the reformulated problem by the iteration method. Furthermore, we deal carefully with singularity issues related to the polar angle on the axis of the divergent nozzle.

Abstract The seasonal evolution of tropical meridional circulation varies across regions and plays a key role in monsoon onset and rainfall migration, with important impacts on agriculture and disaster preparedness. Observations show that these transitions often occur abruptly within days. We develop a new framework to diagnose zonally asymmetric abrupt seasonal changes (ASC) in tropical circulation, using a pseudo‐streamfunction (Ψ pseudo ) and a dual‐component vector index (ASCI). Ψ pseudo recovers the traditional overturning streamfunction when zonally averaged and is decomposed into rotational and divergent parts via Helmholtz decomposition. Strong ASC is found over land regions containing low‐heat‐inertia land area and intense deep convection, notably the Maritime Continent and South America. Unlike the zonally symmetric Hadley cell, rotational flow dominates ASC in these regions, with divergent flow contributing in regions with strong convection. These results highlight the essential role of tropical and extratropical eddies in shaping abrupt seasonal transitions in regional tropical circulation.

ABSTRACT This work focuses on a posteriori error estimation within a non‐conforming framework for the stabilizer‐free immersed weak Galerkin finite element method (IWG‐FEM) applied to elliptic and parabolic problems with non‐smooth coefficients. With an appropriate adaptation of the Helmholtz decomposition of the error, an optimal a posteriori error bound is derived in the weighted ‐seminorm to address elliptic problems with non‐smooth coefficients. For parabolic problems, we analyze a fully discrete scheme that combines the implicit backward Euler time‐stepping method with the non‐conforming IWG‐FEM. A reliable a posteriori error indicator is derived using the time‐dependent Helmholtz decomposition in the ‐norm in time and the weighted ‐seminorm in space without directly relying on the energy argument. Finally, numerical experiments on various test problems confirm the convergence behavior of the proposed error indicators.

This study systematically investigates the coupling between acoustic sources and radiated sound in the acoustic perturbation equations (APE1 and APE2), where the acoustic source terms are constructed from incompressible flow simulations, as well as the influence of source truncation and mesh resolution on the accuracy of acoustic predictions. Both formulations are implemented in the open-source platform OpenFOAM and validated using a canonical two-dimensional circular–cylinder flow at Re=150. The flow field is obtained by solving the incompressible Navier–Stokes equations, while the acoustic field is computed through APE1 and APE2. Modal analysis, Helmholtz decomposition, and beamforming are employed to separate the acoustic and hydrodynamic pressure components and to identify the radiation characteristics of spurious sources. To suppress artificial radiation caused by abrupt source truncation, a smooth Hanning-based truncation approach is proposed and quantitatively assessed. In addition, a systematic mesh resolution study is conducted, leading to the establishment of non-dimensional accuracy criteria for both formulations. Results indicate that APE1 provides stable and accurate acoustic predictions even on moderately refined grids when smooth truncation is applied, whereas APE2 effectively suppresses hydrodynamic contamination, exhibits compact source localization near the wall, and therefore requires no truncation but finer spatial resolution. The proposed truncation and resolution strategies are further validated through simulations of tandem square cylinders, demonstrating good generality for multi-bluff-body configurations. These findings provide practical guidelines for balancing computational cost and acoustic accuracy in incompressible hybrid aeroacoustic simulations.

Mode conversion in nonhomogeneous elastic media makes it challenging to interpret physical properties accurately. Decomposing these modes correctly is crucial across various scientific areas. Recent machine learning approaches have been proposed to address this problem, utilizing the Helmholtz decomposition technique. In this paper, we investigate the capabilities of a physics-informed neural network (PINN) in separating P and S modes by solving a scalar Poisson equation. This scalar formulation offers a dimensionally scalable reduction in computational cost compared to the traditional vector formulation. We verify the proposed method in both homogeneous and realistic nonhomogeneous elastic models as showcases. The obtained separated modes closely match those from conventional numerical techniques, while exhibiting reduced transverse wave leakage.

Accurately modeling immiscible fluid flow in disordered media remains a significant challenge due to the interference of spurious currents. Using the multiple-relaxation-time (MRT) multicomponent pseudopotential lattice Boltzmann method as an exemplar, we perform a Helmholtz decomposition on the anisotropic residual of discrete interaction forces to elucidate its coupling to viscosity pathways. The solenoidal component dissipates through shear viscosity (μ), while the irrotational component engages bulk viscosity (ζ), establishing distinct routes for suppressing spurious currents. Numerical experiments show that employing a 10th-order interaction force markedly reduces the solenoidal share of the residual—by three to four orders of magnitude compared to fourth-order schemes—thereby activating bulk-viscosity control via the energy-mode relaxation parameter (se, sϵ). This approach attains spurious capillary numbers as low as 10−5 and maintains stability at high viscosity ratios, representing up to two orders of magnitude improvement over MRT color-gradient models. The methodology is validated through Laplace pressure tests, two-component Poiseuille flow, Taylor–Bretherton bubble dynamics, and applications in digitized porous media under challenging wettability conditions and high viscosity ratios. From analysis to implementation, the present framework advances high-fidelity multiphase simulations in complex geometries at high viscosity ratios.

Abstract We present a method to quantify total, horizontal kinetic and available potential energies of linear internal waves (IWs) and vortical mode (VM) using only two-dimensional (2D) (depth, along-track distance) measurements of horizontal velocity, such as those commonly taken by oceanic shipboard ADCP (SADCP). Previous IW and VM energy decomposition methods (Bühler et al., and their extensions) require both velocity and buoyancy measurements. Applying Helmholtz decomposition, 2D horizontal kinetic energy wavenumber ( k x , k z ) spectra are projected onto divergent K div and rotational K rot components. IW total energy spectrum is E IW = 2 K div . VM total energy is E VM = 1/Bu[(1 + Bu) K rot − K div ], where is Burger number with N and f the buoyancy and inertial frequencies, k h the horizontal wavenumber magnitude, and k z the vertical wavenumber. IW and VM horizontal kinetic energy ( K ) and available potential energy ( P ) can be inferred from E VM and E IW as functions of Bu. The proposed method, derived directly from IW and VM theoretical polarization relations, is demonstrated using two sets of velocity and density data. The E VM derived by this new method agrees with results computed using the 2014 and 2017 Bühler et al. methods at Bu ∼ O (1) and within a factor of ∼2–3 elsewhere, confirming that IW and VM energy can be separated using only velocity data. At Bu ≪ O (0.1), E VM is dominated by P VM , with K VM / P VM = Bu, and using K alone to extract E VM through the proposed method is challenging due to inherent uncertainty in spectral measurements. This method could be applied to global SADCP datasets to separate upper-ocean IW and VM energy contributions in different dynamical regions at horizontal scales O (100) m– O (100) km and vertical scales O (10)– O (100) m.

We describe a newly developed method for recovering high-resolution images of seismic discontinuities, such as subducting slabs, in 3D. Our method makes use of converted S→P or P→S waves observed by dense arrays of seismometers to infer the locations and relative strengths of seismic discontinuities at depth in a target region. Observed direct and converted waves are backpropagated to their times of origin. The time-reversed wavefield is then separated into its constituent P and S components via the Helmholtz decomposition, and those separated wavefields are used to compute imaging functions that characterize the locations and relative strengths of seismic discontinuities. Imaging functions may be designed to use either S→P or P→S waves, so that users can target those arrivals expected to be most dominant in a given dataset. We have previously demonstrated the efficacy of our method in two dimensions, and we now present a 3D implementation of our technique which addresses the significant computational challenges posed by the size of volumetric wavefield data in three dimensions. Through a series of synthetic examples, we demonstrate that our method is capable of recovering the fine scale structure of a subducting slab given realistic station coverage and earthquake sources. We investigate optimal seismic station geometries for our technique and explore image interpretability in regions with poor data coverage. We find that linear station geometries yield more optimal, interpretable imaging functions than collections of small arrays can. We also show that our method can successfully recover bothS→P or P→S images when realistic shear earthquake sources are used, and we explore the additional computational challenges presented by the high frequency content of S waves. Our results demonstrate the potential for our technique to recover high-resolution information about subducting slabs in real-world regions, given that relatively sparse seismic arrays with only approximately 100 stations are capable of recovering interpretable imaging functions from just a few realistic earthquake sources for multiple discontinuities at significant depth in an area of approximately 400~sq~km.

Abstract We examine the size of and outer surface boundary integrals in estimating the surface magnetic field from magnetohydrodynamic (MHD) simulations. Maxwell's equations tell us , which may be violated due to numerical error. MHD models such as the Space Weather Modeling Framework (SWMF) and the Open Geospace General Circulation Model (OpenGGCM) use different techniques to limit . Analyses of MHD simulations typically assume errors are small. Similarly, analyses commonly use the Biot–Savart Law and magnetospheric current density estimates from MHD simulations to determine the magnetic field at a specific point on Earth. This calculation frequently omits the surface integral over the outer boundary of the simulation volume that the Helmholtz decomposition theorem requires. This paper uses SWMF and OpenGGCM simulations to estimate the magnitudes of the and outer boundary integrals compared to Biot–Savart estimates of the magnetic field on Earth. In the simulations considered, the and outer surface integrals are up to 30% of Biot–Savart estimates when the Biot–Savart estimates are large. We conclude rather than using the Biot–Savart Law to estimate the magnetic field from the magnetosphere, it is better and computationally more efficient to use the integral over the inner boundary of the magnetosphere. The conclusions are the same for a simulation involving a simple change in the interplanetary magnetic field and a more complex superstorm simulation.

In this study, we revisit the pressure–strain interaction in kinetic plasma turbulence. We reexamine the decomposition of pressure–strain interaction into compressive and incompressive parts using Helmholtz theorem. The pressure dilatation ingredient is clearly due to plasma compressions, but here, using 2.5 dimensional kinetic particle-in-cell (PIC) simulations of plasma turbulence, it is demonstrated that the remaining anisotropic part, often called Pi-D, also contains contributions due to compressive, non-solenoidal velocities of the particle species. The compressive Pi-D can play a significant role in systems with low plasma β even if the system starts with small density variations. In addition, the compressive ingredient of Pi-D is found to be strongly anticorrelated with both incompressive Pi-D and pressure dilatation along the current sheets.

This study employs three-dimensional particle-resolved simulations of planar shocks passing through a suspension of stationary solid particles to study wake-induced gas-phase velocity fluctuations, termed pseudo-turbulence. Strong coupling through interphase momentum and energy exchange generates unsteady wakes and shocklets in the interstitial space between particles. A Helmholtz decomposition of the velocity field shows that the majority of pseudo-turbulence is contained in the solenoidal component from particle wakes, whereas the dilatational component corresponds to the downstream edge of the particle curtain where the flow chokes. One-dimensional phase-averaged statistics of pseudo-turbulent kinetic energy (PTKE) are quantified at various stages of flow development. Reduction in PTKE is observed with increasing shock Mach number due to decreased production, consistent with single-phase compressible turbulence. The anisotropy in Reynolds stresses is found to be relatively constant through the curtain and consistent over all the conditions simulated. Analysis of the budget of PTKE shows that the majority of turbulence is produced through drag and balanced by viscous dissipation. The energy spectra of the streamwise gas-phase velocity fluctuations reveal an inertial subrange that begins at the mean interparticle spacing and decays with a power law of $-5/3$ and steepens to $-3$ at scales much smaller than the particle diameter. A two-equation model is proposed for PTKE and its dissipation. The model is implemented within a hyperbolic Eulerian-based two-fluid model and shows excellent agreement with the particle-resolved simulations.

An Investigation of Wave Characteristics by Analyzing the Potential Fields in Peridynamic Media using Nonlocal Helmholtz Decomposition

Abstract We decomposed complex synthetic wavefields in an inhomogeneous sedimentary basin into P-, SV-, and SH-wavefields, and quantitatively evaluated the amplitude, propagation velocity, and propagation direction of coherent waves in each decomposed wavefield within the 0.125–1 Hz frequency band. In sedimentary basins with irregular subsurface structures, P-, SV-, and SH-waves can coexist at the same location and time, propagating as either body waves or surface waves, where SH-waves manifest as Love waves and P- and SV-waves combine to form Rayleigh waves. The relative amplitudes of these wave types depend on both the source radiation pattern and the subsurface geometry. To accurately evaluate the propagation characteristics, such as amplitude and directional variation, of each wave type, it is necessary to first decompose the wavefield by wave type. To date, no studies have addressed this issue from such a perspective. We fully decomposed the reproduced strong-motion waveforms from the 2018 Mw 5.6 earthquake beneath the margin of the Osaka sedimentary basin in Japan—the target event of this study—into P-, SV-, and SH‑wave components using Helmholtz decomposition. By applying semblance analysis to the decomposed wavefields, we quantitatively evaluated the propagation processes of each wave type in the three-dimensional sedimentary basin. Using the derived propagation characteristics, we conducted pseudo-trajectory analysis (PTA) to visualize wave propagation paths, analogous to streamlines in fluid dynamics. We noted spatial differences in the SH‑ and SV‑wavefields. For example, during an early time window, ground motions were oriented northwest–southeast on both sides of the north–south fault zone in the Osaka Plain. These motions result from southwestward-propagating SH-waves in the western region and southeastward-propagating SV-waves in the eastern region. Later, in the western region, Love waves dominated in the 0.125–0.25 Hz band, while Rayleigh waves dominated in the 0.25–0.5 Hz band. The spatiotemporal amplitude variations of these wave types depend on the combined effects of the source radiation pattern and the subsurface structure as noted above. The proposed method can also be applied to identify, where and what types of waves are likely to be generated. Graphical Abstract

Combustion dynamics in aero-engines exhibit complex thermoacoustic behavior involving multiple interacting modes. Recent research on the so-called intrinsic thermoacoustic modes has provided new impetus for exploring the physical origins of these modes and their mutual interactions. The present study concerns the interplay between the intrinsic thermoacoustic modes and duct acoustic modes in a fundamental combustion configuration. The biglobal linear stability analysis of a ducted premixed flame is carried out by exploiting the linearized compressible reactive flow equations. The pure intrinsic thermoacoustic mode is first identified and characterized. Parameter variations are conducted to study the behaviors of eigenmode trajectories in the proximity of an exceptional point. Near the exceptional point, trajectory veering and mode switching are observed. We find that different parameter variation paths can lead to inconsistent results in mode origin identification. Hence, we propose to characterize the thermoacoustic modes based on the flow structures. Specifically, the Helmholtz decomposition is employed to extract the potential and solenoidal components of the thermoacoustic modes and we find the intrinsic thermoacoustic modes and duct acoustic modes exhibit distinct decomposed flow structures.

The current research is concerned with the analysis of the reflected-refracted bulk waves at the interface between elastic isotropic substrate and a liquid overlaying layer. It is shown that at a certain angle of incidence, the S wave in a substrate generates a “geometric” SP head wave, which leads to the appearance of an induced head wave in the liquid layer. This induced head wave may propagate in the liquid layer at a supersonic velocity exceeding the bulk wave velocity in the liquid. The analysis is based on the Helmholtz decomposition of the displacement field, along with the decompositions of stress and strain tensors into spherical and deviatoric parts. It is anticipated that the possible appearance of induced supersonic head waves in the liquid layer can find applications in theoretical and applied geophysics.

The Lamb vector appears in the rotational form of the Navier-Stokes equations and is the source of vortical flow nonlinearity that results in the cascade of energy from large to small scales. We investigate the three-dimensional spatial structure of this vector in the near wake of a NACA0018 wing with square wingtip profile at 10∘ angle of attack and chord-based Reynolds number of 5000. At these conditions, a large recirculation zone forms in the wake and the flow transitions in the separating shear layer. We analyze the spatial footprint of the three components of the Lamb vector and its divergence (the latter for both the mean and the fluctuating velocity fields). By integrating the vertical component of the Lamb vector across the cross-stream direction (which is equal to the vorticity flux for two-dimensional flows) we obtain the local aerodynamic loading along the wing. The vorticity flux is equal to zero at the trailing edge, thereby confirming the Taylor-Sears condition away from the wing tip. We also apply Helmholtz decomposition to extract the potential and solenoidal parts of the Lamb vector. The former part is dominant in the attached and separated shear layers, while the latter is present downstream of the recirculation zone where the flow is turbulent. We decompose the Euclidean norms of the Lamb vector and its potential part into terms that are determined by the time-average field only, the fluctuating field only, as well as mixed terms. We find that the mixed terms play a dominant role and are the slowest to decay. Published by the American Physical Society 2025