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At least 415 records · Page 23
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Performance of high-order Godunov-type methods in simulations of astrophysical low Mach number flows

High-order Godunov methods for gas dynamics have become a standard tool for simulating different classes of astrophysical flows. Their accuracy is mostly determined by the spatial interpolant used to reconstruct the pair of Riemann states at cell interfaces and by the Riemann solver that computes the interface fluxes. In most Godunov-type methods, these two steps can be treated independently, so that many different schemes can in principle be built from the same numerical framework. Because astrophysical simulations often test out the limits of what is feasible with the computational resources available, it is essential to find the scheme that produces the numerical solution with the desired accuracy at the lowest computational cost. However, establishing the best combination of numerical options in a Godunov-type method to be used for simulating a complex hydrodynamic problem is a nontrivial task. In fact, formally more accurate schemes do not always outperform simpler and more diffusive methods, especially if sharp gradients are present in the flow. For this work, we used our fully compressible Seven-League Hydro (SLH) code to test the accuracy of six reconstruction methods and three approximate Riemann solvers on two- and three-dimensional (2D and 3D) problems involving subsonic flows only. We considered Mach numbers in the range from 10 −3 to 10 −1 , which are characteristic of many stellar and geophysical flows. In particular, we considered a well-posed, 2D, Kelvin–Helmholtz instability problem and a 3D turbulent convection zone that excites internal gravity waves in an overlying stable layer. Although the different combinations of numerical methods converge to the same solution with increasing grid resolution for most of the quantities analyzed here, we find that (i) there is a spread of almost four orders of magnitude in computational cost per fixed accuracy between the methods tested in this study, with the most performant method being a combination of a low-dissipation Riemann solver and a sextic reconstruction scheme; (ii) the low-dissipation solver always outperforms conventional Riemann solvers on a fixed grid when the reconstruction scheme is kept the same; (iii) in simulations of turbulent flows, increasing the order of spatial reconstruction reduces the characteristic dissipation length scale achieved on a given grid even if the overall scheme is only second order accurate; (iv) reconstruction methods based on slope-limiting techniques tend to generate artificial, high-frequency acoustic waves during the evolution of the flow; and (v) unlimited reconstruction methods introduce oscillations in the thermal stratification near the convective boundary, where the entropy gradient is steep.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

Automated Fitting of a Semi-empirical Multiphase Equation of State for Carbon

The equation of state (EOS) of carbon is important in high explosive, geophysical, and inertial confinement fusion applications. Within the semi-empirical Sesame framework, the EOS of each phase is represented by a sum of cold, vibrational, and thermal electronic Helmholtz free energy contributions. Each phase has ~5-10 independent parameters that are adjusted to reproduce single-phase data (e.g., thermal expansion, isothermal compression) as well as experimentally- and computationally- derived phase boundaries. Manual calibration of the full multiphase EOS is arduous. We present our progress in development of automated EOS parameterization based on minimization of an objective function. Here, this function encodes deviation of model EOS results from experimental/computational benchmarks. Optimization is implemented as a combination of global (particle swarm) and local optimization techniques.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

High resolution >40 keV x-ray radiography using an edge-on micro-flag backlighter at NIF-ARC

Radiography of low-contrast features in high-density materials evolving on a nanosecond timescale requires a bright photon source in the tens of keV range with high temporal and spatial resolution. One application for sources in this category is the study of dynamic material strength in samples compressed to Mbar pressures at the National Ignition Facility, high-resolution measurements of plastic deformation under conditions relevant to meteor impacts, geophysics, armor development, and inertial confinement fusion. Here, we present radiographic data and the modulation transfer function (MTF) analysis of a multi-component test object probed at ~100 keV effective backlighter energy using a 5 μm-thin dysprosium foil driven by the NIF Advanced Radiographic Capability (ARC) short-pulse laser (~2 kJ, 10 ps). The thin edge of the foil acts as a bright line-projection source of hard x rays, which images the test object at 13.2× magnification into a filtered and shielded image plate detector stack. The system demonstrates a superior contrast of shallow (5 μm amplitude) sinusoidal ripples on gold samples up to 90 μm thick as well as enhanced spatial and temporal resolution using only a small fraction of the laser energy compared to an existing long-pulse-driven backlighter used routinely at the NIF for dynamic strength experiments.

47 OTHER INSTRUMENTATION↗
DOE OSTI2021

Structure of the electrical double layer at the ice–water interface

The surface of ice in contact with water contains sites that undergo deprotonation and protonation and can act as adsorption sites for aqueous ions. Therefore, an electrical double layer should form at this interface and existing models for describing the electrical double layer at metal oxide–water interfaces should be able to be modified to describe the surface charge, surface potential, and ionic occupancy at the ice–water interface. I used a surface complexation model along with literature measurements of the zeta potential of ice in brines of various strength and pH to constrain equilibrium constants. I then made predictions of ion site occupancy, surface charge density, and partitioning of counterions between the Stern and diffuse layers. The equilibrium constant for cation adsorption is more than 5 orders of magnitude larger than the other constants, indicating that this reaction dominates even at low salinity. Deprotonated OH sites are predicted to be slightly more abundant than dangling O sites, consistent with previous work. Surface charge densities are on the order of ±0.001 C/m2 and are always negative at the moderate pH values of interest to atmospheric and geophysical applications (6–9). In this pH range, over 99% of the counterions are contained in the Stern layer. This suggests that diffuse layer polarization will not occur because the ionic concentrations in the diffuse layer are nearly identical to those in the bulk electrolyte and that electrical conduction and polarization in the Stern layer will be negligible due to reduced ion mobility.

Daigle, Hugh (ORCID:0000000260628321)↗
DOE OSTI2022

Dynamic mode decomposition with core sketch

With the increase in collected data volumes, either from experimental measurements or high fidelity simulations, there is an ever-growing need to develop computationally efficient tools to process, analyze, and interpret these datasets. Modal analysis techniques have gained great interest due to their ability to identify patterns in the data and extract valuable information about the system being considered. Dynamic mode decomposition (DMD) relies on elements of the Koopman approximation theory to compute a set of modes, each associated with a fixed oscillation frequency and a decay/growth rate. Extracting these details from large datasets can be computationally expensive due to the need to implement singular value decomposition of the input data matrix. Sketching algorithms have become popular in numerical linear algebra where statistical theoretic approaches are utilized to reduce the cost of major operations. A sketch of a matrix is another matrix, which is significantly smaller, but still sufficiently approximates the original system. We put forth an efficient DMD framework, SketchyDMD, based on a core sketching algorithm that captures information about the range and corange (their mutual relationship) of input data. The proposed sketching-based framework can accelerate various portions of the DMD routines, compared to classical methods that operate directly on the raw input data. We conduct numerical experiments using the spherical shallow water equations as a prototypical model in the context of geophysical flows. In conclusion, we show that the proposed SketchyDMD is superior to existing randomized DMD methods that are based on capturing only the range of the input data.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Graphite resistive heated diamond anvil cell for simultaneous high-pressure and high-temperature diffraction experiments

High-pressure and high-temperature experiments using a resistively heated diamond anvil cell have the advantage of heating samples homogeneously with precise temperature control. Here, we present the design and performance of a graphite resistive heated diamond anvil cell (GRHDAC) setup for powder and single-crystal x-ray diffraction experiments developed at the Extreme Conditions Beamline (P02.2) at PETRA III, Hamburg, Germany. In the GRHDAC, temperatures up to 2000 K can be generated at high pressures by placing it in a water-cooled vacuum chamber. Temperature estimates from thermocouple measurements are within +/–35 K at the sample position up to 800 K and within +90 K between 800 and 1400 K when using a standard seat combination of cBN and WC. Isothermal compression at high temperatures can be achieved by employing a remote membrane control system. Importantly, the advantage of the GRHDAC is demonstrated through the study of geophysical processes in the Earth’s crust and upper mantle region.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2023

Ensemble flow reconstruction in the atmospheric boundary layer from spatially limited measurements through latent diffusion models

Due to costs and practical constraints, field campaigns in the atmospheric boundary layer typically only measure a fraction of the atmospheric volume of interest. Machine learning techniques have previously successfully reconstructed unobserved regions of flow in canonical fluid mechanics problems and two-dimensional geophysical flows, but these techniques have not yet been demonstrated in the three-dimensional atmospheric boundary layer. Here, we conduct a numerical analogue of a field campaign with spatially limited measurements using large-eddy simulation. We pose flow reconstruction as an inpainting problem, and reconstruct realistic samples of turbulent, three-dimensional flow with the use of a latent diffusion model. The diffusion model generates physically plausible turbulent structures on larger spatial scales, even when input observations cover less than 1% of the volume. Through a combination of qualitative visualization and quantitative assessment, we demonstrate that the diffusion model generates meaningfully diverse samples when conditioned on just one observation. These samples successfully serve as initial conditions for a large-eddy simulation code. We find that diffusion models show promise and potential for other applications for other turbulent flow reconstruction problems.

17 WIND ENERGY↗
DOE OSTI2024

Vortex splitting in two-dimensional fluids and non-neutral electron plasmas with smooth vorticity profiles

Initially, elliptical, quasi-two-dimensional (2D) fluid vortices can split into multiple pieces if the aspect ratio is sufficiently large due to the growth and saturation of perturbations known as Love modes on the vortex edge. Presented here are experiments and numerical simulations, showing that the aspect ratio threshold for vortex splitting is significantly higher for vortices with realistic, smooth edges than that predicted by a simple “vortex patch” model, where the vorticity is treated as piecewise constant inside a deformable boundary. The experiments are conducted by exploiting the E × B drift dynamics of collisionless, pure electron plasmas in a Penning–Malmberg trap, which closely model 2D vortex dynamics due to an isomorphism between the Drift–Poisson equations describing the plasmas and the Euler equations describing ideal fluids. The simulations use a particle-in-cell method to model the evolution of a set of point vortices. The aspect ratio splitting threshold ranges up to about twice as large as the vortex patch prediction and depends on the edge vorticity gradient. This is thought to be due to spatial Landau damping, which decreases the vortex aspect ratio over time and, thus, stabilizes the Love modes. Near the threshold, asymmetric splitting events are observed in which one of the split products contains much less circulation than the other. These results are relevant to a wide range of quasi-2D fluid systems, including geophysical fluids, astrophysical disks, and drift-wave eddies in tokamak plasmas.

Physics↗
DOE OSTI2024

The maintenance of coherent vortex topology by Lagrangian chaos in drift-Rossby wave turbulence

Here, this work introduces the “potential vorticity bucket brigade,” a mechanism for explaining the resilience of vortex structures in magnetically confined fusion plasmas and geophysical flows. Drawing parallels with zonal jet formation, we show how inhomogeneous patterns of mixing can reinforce, rather than destroy non-zonal flow structure. We accomplish this through an exact stochastic Lagrangian representation of vorticity transport, together with a near-integrability property, which relates coherent flow topology to fluid relabeling symmetries. We demonstrate these ideas in the context of gradient-driven magnetized plasma turbulence, though the tools we develop here are model-agnostic and applicable beyond the system studied here.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

Measurements of K-edge and L-edge extended x-ray absorption fine structure at the national ignition facility (invited)

High-energy-density laser facilities and advances in dynamic compression techniques have expanded access to material states in the Terapascal regime relevant to inertial confinement fusion, planetary science, and geophysics. However, experimentally determining the material temperature in these extreme conditions has remained a difficult challenge. Extended X-ray Absorption Fine Structure (EXAFS), referring to the modulations in x-ray absorption above an absorption edge from photoelectrons’ interactions with neighboring atoms, has proven to be a versatile and robust technique for probing material temperature and density for mid-to-high Z elements under dynamic compression. The current platform at the National Ignition Facility has developed six configurations for EXAFS measurements between 7 and 18 keV for different absorption edges (Fe K, Co K, Cu K, Ta L3, Pb L3, and Zr K) using a curved-crystal spectrometer and a bright, continuum foil x-ray source. Here, in this work, we describe the platform geometry, x-ray source performance, spectrometer resolution and throughput, design considerations, and data in ambient and dynamic-compression conditions.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Laboratory earthquake forecasting: A machine learning competition

Earthquake prediction, the long-sought holy grail of earthquake science, continues to confound Earth scientists. Could we make advances by crowdsourcing, drawing from the vast knowledge and creativity of the machine learning (ML) community? We used Google’s ML competition platform, Kaggle, to engage the worldwide ML community with a competition to develop and improve data analysis approaches on a forecasting problem that uses laboratory earthquake data. The competitors were tasked with predicting the time remaining before the next earthquake of successive laboratory quake events, based on only a small portion of the laboratory seismic data. The more than 4,500 participating teams created and shared more than 400 computer programs in openly accessible notebooks. Complementing the now well-known features of seismic data that map to fault criticality in the laboratory, the winning teams employed unexpected strategies based on rescaling failure times as a fraction of the seismic cycle and comparing input distribution of training and testing data. In addition to yielding scientific insights into fault processes in the laboratory and their relation with the evolution of the statistical properties of the associated seismic data, the competition serves as a pedagogical tool for teaching ML in geophysics. The approach may provide a model for other competitions in geosciences or other domains of study to help engage the ML community on problems of significance.

58 GEOSCIENCES↗
DOE OSTI2023

Sea-level rise in Southwest Greenland as a contributor to Viking abandonment

The first records of Greenland Vikings date to 985 CE. Archaeological evidence yields insight into how Vikings lived, yet drivers of their disappearance in the 15th century remain enigmatic. Research suggests a combination of environmental and socioeconomic factors, and the climatic shift from the Medieval Warm Period (~900 to 1250 CE) to the Little Ice Age (~1250 to 1900 CE) may have forced them to abandon Greenland. Glacial geomorphology and paleoclimate research suggest that the Southern Greenland Ice Sheet readvanced during Viking occupation, peaking in the Little Ice Age. Counterintuitively, the readvance caused sea-level rise near the ice margin due to increased gravitational attraction toward the ice sheet and crustal subsidence. We estimate ice growth in Southwestern Greenland using geomorphological indicators and lake core data from previous literature. We calculate the effect of ice growth on regional sea level by applying our ice history to a geophysical model of sea level with a resolution of ~1 km across Southwestern Greenland and compare the results to archaeological evidence. The results indicate that sea level rose up to ~3.3 m outside the glaciation zone during Viking settlement, producing shoreline retreat of hundreds of meters. Sea-level rise was progressive and encompassed the entire Eastern Settlement. Moreover, pervasive flooding would have forced abandonment of many coastal sites. These processes likely contributed to the suite of vulnerabilities that led to Viking abandonment of Greenland. Sea-level change thus represents an integral, missing element of the Viking story.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2026

Lunar accelerometer network gravitational observatory (LANGO)

With ground-based interferometers detecting hundreds of gravitational-wave (GW) events, GW astronomy has continued to blossom. U.S. and European scientists are developing plans to construct third-generation ground-based interferometers. ESA has proceeded through the mission formulation phase of space-based interferometer in a lower-frequency band, 10 –4 –0.1 Hz. Despite all these exciting developments, there is still a missing frequency band, 0.1–10 Hz. This mid-frequency band is rich with interesting astrophysical events. Coalescence and merger of intermediate-mass black holes (IMBHs) will occur in this frequency band. Coalescing stellar-mass BHs will pass through this frequency band days before they reach the frequency band of LIGO and Virgo. Detection of such signals would enable a mid-frequency detector to issue an advance notice to the high-frequency GW detectors, as well as to optical, x-ray and γ-ray telescopes. We propose Lunar Accelerometer Network Gravitational Observatory (LANGO) to detect GWs in this frequency band. In the first phase (LANGO 1), we propose to deploy four ambient-temperature (250 K) accelerometers in the tetrahedral or in a square configuration on the hemisphere facing the Earth. After successful operation at 250 K, LANGO would be upgraded to a cryogenic (4 K) version with over two orders of magnitude increased sensitivity (LANGO 2), with coherent rejection of seismic noise implemented. LANGO is a full-tensor detector, capable of determining the source direction and wave polarization. Each test mass (TM) is suspended as a pendulum with resonance frequency ∼ 0.01 Hz, thus is only weakly coupled to the lunar surface horizontally. LANGO is designed to detect the relative motion of globally separated, nearly free, TMs by using the Moon as a large quiet platform. LANGO 1 and 2 accelerometers aim at sensitivities ⩽ 10 –11 m s –2 Hz –1/2 and ⩽ 10 –13 m s –2 Hz –1/2 in the horizontal axes over the frequency band of 1 mHz–10 Hz, which yield GW sensitivities 1.1 x 10 -21 Hz -1/2 and 3.8 x 10 -24 Hz -1/2 at 1 Hz, respectively. The LANGO accelerometers will be 10 3 –10 5 times more sensitive than Apollo seismometers. With such sensitivity, LANGO will also make great contribution to the advancement of lunar geophysics.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Cooperative diffusion in body-centered cubic iron in Earth and super-Earths’ inner core conditions

Abstract The physical chemistry of iron at the inner-core conditions is key to understanding the evolution and habitability of Earth and super-Earth planets. Based on full first-principles simulations, we report cooperative diffusion along the longitudinally fast ⟨ 111 ⟩ directions of body-centered cubic (bcc) iron in temperature ranges of up to 2000–4000 K below melting and pressures of ∼300–4000 GPa. The diffusion is due to the low energy barrier in the corresponding direction and is accompanied by mechanical and dynamical stability, as well as strong elastic anisotropy of bcc iron. These findings provide a possible explanation for seismological signatures of the Earth’s inner core, particularly the positive correlation between P wave velocity and attenuation. The diffusion can also change the detailed mechanism of core convection by increasing the diffusivity and electrical conductivity and lowering the viscosity. The results need to be considered in future geophysical and planetary models and should motivate future studies of materials under extreme conditions.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2023

Local structural investigation of non-crystalline materials at high pressure: the case of GeO 2 glass

Abstract Local structures play a crucial role in the structural polyamorphism and novel electronic properties of amorphous materials, but their accurate measurement at high pressure remains a formidable challenge. In this article, we use the local structure of network-forming GeO 2 glass as an example, to present our recent approaches and advances in high-energy x-ray diffraction, high-pressure x-ray absorption fine structure, and ab initio first-principles density functional theory calculations and simulations. Although GeO 2 glass is one of the best studied materials in the field of high pressure research due to its importance in glass theory and geophysical significance, there are still some long-standing puzzles, such as the existence of appreciable distinct fivefold [5] Ge coordination at low pressure and the sixfold-plus [6+] Ge coordination at ultrahigh pressure. Our work sheds light on the origin of pressure-induced polyamorphism of GeO 2 glass, and the [5] Ge polyhedral units may be the dominant species in the densification mechanism of network-forming glasses from tetrahedral to octahedral amorphous structures.

Physics↗
DOE OSTI2023

Investigating permafrost carbon dynamics in Alaska with artificial intelligence

Abstract Positive feedbacks between permafrost degradation and the release of soil carbon into the atmosphere impact land–atmosphere interactions, disrupt the global carbon cycle, and accelerate climate change. The widespread distribution of thawing permafrost is causing a cascade of geophysical and biochemical disturbances with global impacts. Currently, few earth system models account for permafrost carbon feedback (PCF) mechanisms. This research study integrates artificial intelligence (AI) tools and information derived from field-scale surveys across the tundra and boreal landscapes in Alaska. We identify and interpret the permafrost carbon cycling links and feedback sensitivities with GeoCryoAI, a hybridized multimodal deep learning (DL) architecture of stacked convolutionally layered, memory-encoded recurrent neural networks (NN). This framework integratesin-situmeasurements and flux tower observations for teacher forcing and model training. Preliminary experiments to quantify, validate, and forecast permafrost degradation and carbon efflux across Alaska demonstrate the fidelity of this data-driven architecture. More specifically, GeoCryoAI logs the ecological memory and effectively learns covariate dynamics while demonstrating an aptitude to simulate and forecast PCF dynamics—active layer thickness (ALT), carbon dioxide flux (CO 2 ), and methane flux (CH 4 )—with high precision and minimal loss (i.e. ALT RMSE : 1.327 cm [1969–2022]; CO 2 RMSE : 0.697µmolCO 2 m −2 s −1 [2003–2021]; CH 4 RMSE : 0.715 nmolCH 4 m −2 s −1 [2011–2022]). ALT variability is a sensitive harbinger of change, a unique signal characterizing the PCF, and our model is the first characterization of these dynamics across space and time.

Environmental Sciences & Ecology↗
DOE OSTI2025

Methane emission hotspots in a boreal forest-fen mosaic potentially linked to deep taliks

Permafrost thaw is transforming boreal forests into mosaics of wetlands and drier uplands. Topographic controls on hydrological and ecological conditions impact methane (CH 4 ) fluxes, contributing to uncertainty in local and regional CH 4 budgets and underlying drivers. The objective of this study was to explore CH 4 fluxes and their drivers in a transitioning boreal forest-fen ecosystem (Goldstream Valley, Alaska, USA). This landscape is characterized by thawing discontinuous permafrost and heterogeneous mosaics of fens, collapse-scar channels, and small mounds of permafrost soils. From a survey in July 2021, observed chamber CH4 fluxes included fen areas with intermediate to very high emissions (29.8–635.3 mg CH 4 m −2 d −1 ), clustered locations with CH 4 uptake (−2.11 to −0.7 mg CH 4 m −2 d −1 ), and three anomalous emission hotspots (342.4–772.4 mg CH 4 m −2 d −1 ) that were located near samples with lower emissions. Some surface and near-surface variables partially explained the spatial variation in CH 4 flux. Log-transformed CH 4 flux had a positive linear relationship with soil moisture at 20 cm depth ( R 2 = 0.31, p -value < 1e-5) and negative linear relationships with microtopography ( R 2 = 0.13, p -value < 0.006) and slope ( R 2 = 0.28, p -value < 2e-5). Methane emissions generally occurred in flat, wet, graminoid-dominated fens, whereas CH 4 uptake occurred on permafrost mounds dominated by feather mosses and woody vegetation. However, the CH 4 hotspots occurred on drier, slightly sloped locations with low or undetectable near-surface methanogen abundance, suggesting that CH 4 was produced in deeper soils. When the hotspot samples were omitted, log-transformed CH 4 flux had a positive linear relationship with near-surface methanogen abundance ( R 2 = 0.29, p -value = 0.0023), and stronger linear relationships with soil moisture, slope, and soil macronutrient concentrations. Our findings suggest that some CH 4 emission hotspots could arise from CH 4 in deep taliks. The inference that methanogenesis occurs in deep taliks was strengthened by the identification of intrapermafrost taliks across the study area using low-frequency geophysical induction. This study assesses surface spatial heterogeneity in the context of subsurface permafrost conditions and highlights the complexity of CH 4 flux patterns in transitioning forest-wetland ecosystems. To better inform regional CH 4 budgets, further research is needed to understand the spatial distribution of terrestrial CH 4 hotspots and to resolve their surface, near-surface, and subsurface drivers.

boreal↗
DOE OSTI2021

Thermodynamically consistent semi-compressible fluids: a variational perspective

This paper presents (Lagrangian) variational formulations for single and multicomponent semi-compressible fluids with both reversible (entropy-conserving) and irreversible (entropy-generating) processes. Semi-compressible fluids are useful in describing low-Mach dynamics, since they are soundproof. These models find wide use in many areas of fluid dynamics, including both geophysical and astrophysical fluid dynamics. Specifically, the Boussinesq, anelastic and pseudoincompressible equations are developed through a unified treatment valid for arbitrary Riemannian manifolds, thermodynamic potentials and geopotentials. By design, these formulations obey the 1st and 2nd laws of thermodynamics, ensuring their thermodynamic consistency. This general approach extends and unifies existing work, and helps clarify the thermodynamics of semi-compressible fluids. To further this goal, evolution equations are presented for a wide range of thermodynamic variables: entropy density s, specific entropy η, buoyancy b, temperature T, potential temperature θ and a generic entropic variable χ; along with a general definition of buoyancy valid for all three semicompressible models and arbitrary geopotentials. Finally, the elliptic equation for the pressure perturbation (the Lagrange multiplier that enforces semi-compressibility) is developed for all three equation sets in the case of reversible dynamics, and for the Boussinesq/anelastic equations in the case of irreversible dynamics; and some discussion is given of the difficulty in formulating it for the pseudoincompressible equations with irreversible dynamics.

97 MATHEMATICS AND COMPUTING↗