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NASA NTRS2016

Applied Time Domain Stability Margin Assessment for Nonlinear Time-Varying Systems

The baseline stability margins for NASA's Space Launch System (SLS) launch vehicle were generated via the classical approach of linearizing the system equations of motion and determining the gain and phase margins from the resulting frequency domain model. To improve the fidelity of the classical methods, the linear frequency domain approach can be extended by replacing static, memoryless nonlinearities with describing functions. This technique, however, does not address the time varying nature of the dynamics of a launch vehicle in flight. An alternative technique for the evaluation of the stability of the nonlinear launch vehicle dynamics along its trajectory is to incrementally adjust the gain and/or time delay in the time domain simulation until the system exhibits unstable behavior. This technique has the added benefit of providing a direct comparison between the time domain and frequency domain tools in support of simulation validation. This technique was implemented by using the Stability Aerospace Vehicle Analysis Tool (SAVANT) computer simulation to evaluate the stability of the SLS system with the Adaptive Augmenting Control (AAC) active and inactive along its ascent trajectory. The gains for which the vehicle maintains apparent time-domain stability defines the gain margins, and the time delay similarly defines the phase margin. This method of extracting the control stability margins from the time-domain simulation is relatively straightforward and the resultant margins can be compared to the linearized system results. The sections herein describe the techniques employed to extract the time-domain margins, compare the results between these nonlinear and the linear methods, and provide explanations for observed discrepancies. The SLS ascent trajectory was simulated with SAVANT and the classical linear stability margins were evaluated at one second intervals. The linear analysis was performed with the AAC algorithm disabled to attain baseline stability margins. At each time point, the system was linearized about the current operating point using Simulink's built-in solver. Each linearized system in time was evaluated for its rigid-body gain margin (high frequency gain margin), rigid-body phase margin, and aero gain margin (low frequency gain margin) for each control axis. Using the stability margins derived from the baseline linearization approach, the time domain derived stability margins were determined by executing time domain simulations in which axis-specific incremental gain and phase adjustments were made to the nominal system about the expected neutral stability point at specific flight times. The baseline stability margin time histories were used to shift the system gain to various values around the zero margin point such that a precise amount of expected gain margin was maintained throughout flight. When assessing the gain margins, the gain was applied starting at the time point under consideration, thereafter following the variation in the margin found in the linear analysis. When assessing the rigid-body phase margin, a constant time delay was applied to the system starting at the time point under consideration. If the baseline stability margins were correctly determined via the linear analysis, the time domain simulation results should contain unstable behavior at certain gain and phase values. Examples will be shown from repeated simulations with variable added gain and phase lag. Faithfulness of margins calculated from the linear analysis to the nonlinear system will be demonstrated.

Kiefer, J. M.↗

DOE OSTI2020

Results of the International Energy Agency Bioenergy Round Robin on the Analysis of Heteroatoms in Biomass Liquefaction Oils

A round robin study evaluating the analysis of biomass liquefaction oils (BLOs) from fast pyrolysis and hydrothermal liquefaction (HTL) was performed, involving fifteen laboratories in seven countries in order to assess the current status of analytical techniques for the determination of nitrogen, sulfur, and chlorine content in BLOs and evaluate potential differences in origin (i.e. fast pyrolysis vs HTL). The BLOs were produced from a range of feedstocks including pine, mixed softwoods, forest residues, micro-algae, miscanthus, and wheat straw to cover a variety in nitrogen, sulfur, and chlorine content and speciation. Nine samples were distributed, comprised of eight separate BLOs and one blind duplicate produced by five producers. The samples were analyzed for water, carbon, hydrogen, nitrogen, sulfur, and chlorine content. No analytical test method was mandated; laboratories were encouraged to utilize whichever method they determined would be most applicable, relying on the existing body of BLO literature as a guide. The results of this round robin study are presented in this paper. The results of the carbon, hydrogen, and water measurements as reference analyses had relative standard deviations (2.9%, 3.5, and 5.6%, respectively) that were comparable to those found in past round robin studies on fast pyrolysis bio-oil. The analysis of nitrogen, sulfur, and chlorine showed higher levels of variability. Laboratories mostly chose the same method for water, carbon, hydrogen and nitrogen determination whereas there were a variety of methods chosen for sulfur and chlorine determination. The results suggest that specific analytical methods for the determination of nitrogen, sulfur, and chlorine should be further refined to ensure reproducible and accurate results for BLO analysis due to their importance on emissions, material selection, and catalyst activity.

Biomass, IEA Bioenergy, pyrolysis, hydrothermal li↗

NASA NTRS2012

Variable-Speed Simulation of a Dual-Clutch Gearbox Tiltrotor Driveline

This investigation explores the variable-speed operation and shift response of a prototypical two-speed dual-clutch transmission tiltrotor driveline in forward flight. Here, a Comprehensive Variable-Speed Rotorcraft Propulsion System Modeling (CVSRPM) tool developed under a NASA funded NRA program is utilized to simulate the drive system dynamics. In this study, a sequential shifting control strategy is analyzed under a steady forward cruise condition. This investigation attempts to build upon previous variable-speed rotorcraft propulsion studies by 1) including a fully nonlinear transient gas-turbine engine model, 2) including clutch stick-slip friction effects, 3) including shaft flexibility, 4) incorporating a basic flight dynamics model to account for interactions with the flight control system. Through exploring the interactions between the various subsystems, this analysis provides important insights into the continuing development of variable-speed rotorcraft propulsion systems.

DeSmidt, Hans↗

NASA NTRS2024

Critical Load Exceedances for North America and Europe using an Ensemble of Models and an Investigation of Causes for Environmental Impact Estimate Variability: An AQMEII4 Study

Exceedances of critical loads for deposition of sulphur (S) and nitrogen (N) to different ecosystems were estimated using European and North American ensembles of air quality models, under Phase 4 of the Air Quality Model Evaluation International Initiative (AQMEII4), to identify where risk of ecosystem harm is expected to occur based on model deposition estimates. The ensembles were driven by common emissions and lateral boundary condition inputs. Model output was regridded to common North American and Europe 0.125° resolution domains, which were then used to calculate critical load exceedances. New, targeted deposition diagnostics implemented in AQMEII4 allowed an unprecedented level of post-simulation analysis to be carried out and facilitated the identification of specific causes of model-to-model variability in critical load exceedance estimates. New datasets for North American critical loads for acidity for forest soil water and aquatic ecosystems were combined with the ensemble deposition predictions to show a substantial decrease in the area and number of locations in exceedance between 2010 and 2016 (forest soils: 13.2 % to 6.1 %; aquatic ecosystems: 21.2 % to 11.4 %). All models agreed in the direction of the ensemble exceedance change between 2010 and 2016. The North American ensemble also predicted a decrease in both severity and total area in exceedance between the years 2010 and 2016 for eutrophication-impacted ecosystems in the USA (sensitive epiphytic lichen: 81.5 % to 75.8 %). The exceedances for herbaceous community richness also decreased between 2010 and 2016, from 13.9 % to 3.9 %. The uncertainty associated with the North American eutrophication results is high; there were sharp differences between the models in both predictions of total N deposition and the change in N deposition, and hence in the predicted eutrophication exceedances between the two years. The European ensemble was used to predict relatively static exceedances of critical loads with respect to acidification (4.48 % to 4.32 % from 2009 to 2010) while eutrophication exceedance increased slightly (60.2 % to 62.2 %). While most models showed the same changes in critical load exceedances as the ensemble between the two years, the spatial extent and magnitude of exceedances varied significantly between the models. The reasons for this variation were examined in detail by first ranking the relative contribution of different sources of sulphur and nitrogen deposition in terms of deposited mass and model-to-model variability in that deposited mass, followed by their analysis using AQMEII4 diagnostics, along with evaluation of the most recent literature. All models in both the North American and European ensembles had net annual negative biases with respect to observed wet deposition of sulphate, nitrate and ammonium. Diagnostics and recent literature suggest that this bias may stem from insufficient cloud scavenging of aerosols and gases, and may be improved through the incorporation of multiphase hydrometeor scavenging within the modelling frameworks. The inability of North American models to predict the timing of the seasonal peak in wet ammonium ion deposition (observed maximum was in April, while all models predicted a June maximum) may also relate to the need for multiphase hydrometeor scavenging (absence of snow scavenging in all models employed here). High variability in the relative importance of particulate sulphate, nitrate and ammonium deposition fluxes between models was linked to the use of updated particle dry deposition parameterizations in some models. However, recent literature and further development of some of the models within the ensemble suggests these particulate biases may also be ameliorated via the incorporation of multiphase hydrometeor scavenging. Annual sulphur and nitrogen deposition prediction variability was linked to SO 2 and HNO 3 dry deposition parameterizations, and diagnostic analysis showed that the cuticle and soil deposition pathways dominate the deposition mass flux of these species. Further work improving parameterizations for these deposition pathways should reduce variability in model acidifying gas deposition estimates. The absence of base cation chemistry in some models was shown to be a major factor in positive biases in fine mode particulate ammonium and particle nitrate concentrations. Models employing ammonia bidirectional fluxes had both the largest and the smallest magnitude biases, depending on the model and bidirectional flux algorithm employed. A careful analysis of bidirectional flux models suggests that those with poor NH 3 performance may underestimate the extent of NH 3 emissions fluxes from forested areas. Based on these results, an increased process-research focus is therefore recommended for the following model processes and on observations which may assist in model evaluation and improvement: multiphase hydrometeor scavenging combined with updated particle dry deposition, cuticle and soil deposition pathway algorithms for acidifying gases, base cation chemistry and emissions, and NH 3 bidirectional fluxes. Comparisons with satellite observations suggest that oceanic NH 3 emissions sources should be included in regional chemical transport models. The choice of land use database employed within any given model was shown to significantly influence deposition totals in several instances, and employing a common land use database across chemical transport models and critical load calculations is recommended for future work.

critical loads↗

NASA NTRS2001

Interannual Variability of Tropical Precipitation: How Well Do Climate Models Agree With Current Satellite Estimates?

Since the beginning of the World Climate Research Program's Global Precipitation Climatology Project (GPCP) satellite remote sensing of precipitation has made dramatic improvements, particularly for tropical regions. Data from microwave and infrared sensors now form the most critical input to precipitation data sets and can be calibrated with surface gauges to so that the strengths of each data source can be maximized in some statistically optimal sense. Recent availability of the TRMM (Tropical Rainfall Measuring Mission) has further aided in narrowing uncertainties in rainfall over die tropics and subtropics. Although climate modeling efforts have long relied on space-based precipitation estimates for validation, we now are in a position to make more quantitative assessments of model performance, particularly in tropical regions. An integration of the CCM3 using observed SSTs as a lower boundary condition is used to examine how well this model responds to ENSO forcing in terms of anomalous precipitation. An integration of the NCEP spectral model used for the Reanalysis-H effort is also examined. This integration is run with specified SSTs, but with no data assimilation. Our analysis focuses on two aspects of inter-annual variability. First are the spatial anomalies that are indicative of dislocations in Hadley and Walker circulations. Second, we consider the ability of models to replicate observed increases in oceanic precipitation that are noted in satellite observations for large ENSO events. Finally, we consider a slab ocean version of the CCM3 model with prescribed ocean beat transports that mimic upwelling anomalies, but which still allows the surface energy balance to be predicted. This less restrictive experiment is used to understand why model experiments with specified SSTs seem to have noticeably less interannual variability in precipitation than do the satellite observations.

Robertson, Franklin R.↗

NASA NTRS2012

Variability of Black Carbon Deposition to the East Antarctic Plateau, 1800-2000 AD

Refractory black carbon aerosols (rBC) from biomass burning and fossil fuel combustion are deposited to the Antarctic ice sheet and preserve a history of emissions and long-range transport from low- and mid-latitudes. Antarctic ice core rBC records may thus provide information with respect to past combustion aerosol emissions and atmospheric circulation. Here, we present six East Antarctic ice core records of rBC concentrations and fluxes covering the last two centuries with approximately annual resolution (cal. yr. 1800 to 2000). The ice cores were drilled in disparate regions of the high East Antarctic ice sheet, at different elevations and net snow accumulation rates. Annual rBC concentrations were log-normally distributed and geometric means of annual concentrations ranged from 0.10 to 0.18 m cro-g/kg. Average rBC fluxes were determined over the time periods 1800 to 2000 and 1963 to 2000 and ranged from 3.4 to 15.5 m /a and 3.6 to 21.8 micro-g/sq m/a, respectively. Geometric mean concentrations spanning 1800 to 2000 increased linearly with elevation at a rate of 0.025 micro-g/kg/500 m. Spectral analysis of the records revealed significant decadal-scale variability, which at several sites was comparable to decadal ENSO variability.

Bisiaux, M. M.↗

DOE OSTI2022

Kinematic Variables and Feature Engineering for Particle Phenomenology

Kinematic variables have been playing an important role in collider phenomenology, as they expedite discoveries of new particles by separating signal events from unwanted background events and allow for measurements of particle properties such as masses, couplings, spins, etc. For the past 10 years, an enormous number of kinematic variables have been designed and proposed, primarily for the experiments at the Large Hadron Collider, allowing for a drastic reduction of high-dimensional experimental data to lower-dimensional observables, from which one can readily extract underlying features of phase space and develop better-optimized data-analysis strategies. We review these recent developments in the area of phase space kinematics, summarizing the new kinematic variables with important phenomenological implications and physics applications. We also review recently proposed analysis methods and techniques specifically designed to leverage the new kinematic variables. As machine learning is nowadays percolating through many fields of particle physics including collider phenomenology, we discuss the interconnection and mutual complementarity of kinematic variables and machine learning techniques. We finally discuss how the utilization of kinematic variables originally developed for colliders can be extended to other high-energy physics experiments including neutrino experiments.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS2000

Uncertainty Analysis of Composite Structures

A two-phase approach and a computational procedure are presented for predicting the variability in the nonlinear response of composite structures associated with variations in the geometric and material parameters of the structure. In the first phase, hierarchical sensitivity analysis is used to identify the major parameters, which have the most effect on the response quantities of interest. In the second phase, the major parameters are taken to be fuzzy parameters, and a fuzzy set analysis is used to determine the range of variation of the response, associated with preselected variations in the major parameters. The effectiveness of the procedure is demonstrated by means of a numerical example of a cylindrical panel with four T-shaped stiffeners and a circular cutout.

Noor, Ahmed K.↗

NASA NTRS1989

Efficient sensitivity analysis and optimization of a helicopter rotor

Aeroelastic optimization of a system essentially consists of the determination of the optimum values of design variables which minimize the objective function and satisfy certain aeroelastic and geometric constraints. The process of aeroelastic optimization analysis is illustrated. To carry out aeroelastic optimization effectively, one needs a reliable analysis procedure to determine steady response and stability of a rotor system in forward flight. The rotor dynamic analysis used in the present study developed inhouse at the University of Maryland is based on finite elements in space and time. The analysis consists of two major phases: vehicle trim and rotor steady response (coupled trim analysis), and aeroelastic stability of the blade. For a reduction of helicopter vibration, the optimization process requires the sensitivity derivatives of the objective function and aeroelastic stability constraints. For this, the derivatives of steady response, hub loads and blade stability roots are calculated using a direct analytical approach. An automated optimization procedure is developed by coupling the rotor dynamic analysis, design sensitivity analysis and constrained optimization code CONMIN.

Lim, Joon W.↗

NASA NTRS1980

Study of component technologies for fuel cell on-site integrated energy system. Volume 2: Appendices

This data base catalogue was compiled in order to facilitate the analysis of various on site integrated energy system with fuel cell power plants. The catalogue is divided into two sections. The first characterizes individual components in terms of their performance profiles as a function of design parameters. The second characterizes total heating and cooling systems in terms of energy output as a function of input and control variables. The integrated fuel cell systems diagrams and the computer analysis of systems are included as well as the cash flows series for baseline systems.

Lee, W. D.↗

NASA NTRS2008

Atmospheric Parameter Climatologies from AIRS: Monitoring Short-, and Longer-Term Climate Variabilities and 'Trends'

The AIRS instrument is currently the best space-based tool to simultaneously monitor the vertical distribution of key climatically important atmospheric parameters as well as surface properties, and has provided high quality data for more than 5 years. AIRS analysis results produced at the GODDARD/DAAC, based on Versions 4 & 5 of the AIRS retrieval algorithm, are currently available for public use. Here, first we present an assessment of interrelationships of anomalies (proxies of climate variability based on 5 full years, since Sept. 2002) of various climate parameters at different spatial scales. We also present AIRS-retrievals-based global, regional and 1x1 degree grid-scale "trend"-analyses of important atmospheric parameters for this 5-year period. Note that here "trend" simply means the linear fit to the anomaly (relative the mean seasonal cycle) time series of various parameters at the above-mentioned spatial scales, and we present these to illustrate the usefulness of continuing AIRS-based climate observations. Preliminary validation efforts, in terms of intercomparisons of interannual variabilities with other available satellite data analysis results, will also be addressed. For example, we show that the outgoing longwave radiation (OLR) interannual spatial variabilities from the available state-of-the-art CERES measurements and from the AIRS computations are in remarkably good agreement. Version 6 of the AIRS retrieval scheme (currently under development) promises to further improve bias agreements for the absolute values by implementing a more accurate radiative transfer model for the OLR computations and by improving surface emissivity retrievals.

Molnar, Gyula↗

NASA NTRS1995

Automatic differentiation evaluated as a tool for rotorcraft design and optimization

This paper investigates the use of automatic differentiation (AD) as a means for generating sensitivity analyses in rotorcraft design and optimization. This technique transforms an existing computer program into a new program that performs sensitivity analysis in addition to the original analysis. The original FORTRAN program calculates a set of dependent (output) variables from a set of independent (input) variables, the new FORTRAN program calculates the partial derivatives of the dependent variables with respect to the independent variables. The AD technique is a systematic implementation of the chain rule of differentiation, this method produces derivatives to machine accuracy at a cost that is comparable with that of finite-differencing methods. For this study, an analysis code that consists of the Langley-developed hover analysis HOVT, the comprehensive rotor analysis CAMRAD/JA, and associated preprocessors is processed through the AD preprocessor ADIFOR 2.0. The resulting derivatives are compared with derivatives obtained from finite-differencing techniques. The derivatives obtained with ADIFOR 2.0 are exact within machine accuracy and do not depend on the selection of step-size, as are the derivatives obtained with finite-differencing techniques.

Walsh, Joanne L.↗

DOE OSTI2024

Physics and chemistry from parsimonious representations: image analysis via invariant variational autoencoders

Electron, optical, and scanning probe microscopy methods are generating ever increasing volume of image data containing information on atomic and mesoscale structures and functionalities. This necessitates the development of the machine learning methods for discovery of physical and chemical phenomena from the data, such as manifestations of symmetry breaking phenomena in electron and scanning tunneling microscopy images, or variability of the nanoparticles. Variational autoencoders (VAEs) are emerging as a powerful paradigm for the unsupervised data analysis, allowing to disentangle the factors of variability and discover optimal parsimonious representation. Here, we summarize recent developments in VAEs, covering the basic principles and intuition behind the VAEs. The invariant VAEs are introduced as an approach to accommodate scale and translation invariances present in imaging data and separate known factors of variations from the ones to be discovered. We further describe the opportunities enabled by the control over VAE architecture, including conditional, semi-supervised, and joint VAEs. Several case studies of VAE applications for toy models and experimental datasets in Scanning Transmission Electron Microscopy are discussed, emphasizing the deep connection between VAE and basic physical principles. Python codes and datasets discussed in this article are available at https://github.com/saimani5/VAE-tutorials and can be used by researchers as an application guide when applying these to their own datasets.

36 MATERIALS SCIENCE↗

NASA NTRS2013

Quantitative Comparison of the Variability in Observed and Simulated Shortwave Reflectance

The Climate Absolute Radiance and Refractivity Observatory (CLARREO) is a climate observation system that has been designed to monitor the Earth's climate with unprecedented absolute radiometric accuracy and SI traceability. Climate Observation System Simulation Experiments (OSSEs) have been generated to simulate CLARREO hyperspectral shortwave imager measurements to help define the measurement characteristics needed for CLARREO to achieve its objectives. To evaluate how well the OSSE-simulated reflectance spectra reproduce the Earth s climate variability at the beginning of the 21st century, we compared the variability of the OSSE reflectance spectra to that of the reflectance spectra measured by the Scanning Imaging Absorption Spectrometer for Atmospheric Cartography (SCIAMACHY). Principal component analysis (PCA) is a multivariate decomposition technique used to represent and study the variability of hyperspectral radiation measurements. Using PCA, between 99.7%and 99.9%of the total variance the OSSE and SCIAMACHY data sets can be explained by subspaces defined by six principal components (PCs). To quantify how much information is shared between the simulated and observed data sets, we spectrally decomposed the intersection of the two data set subspaces. The results from four cases in 2004 showed that the two data sets share eight (January and October) and seven (April and July) dimensions, which correspond to about 99.9% of the total SCIAMACHY variance for each month. The spectral nature of these shared spaces, understood by examining the transformed eigenvectors calculated from the subspace intersections, exhibit similar physical characteristics to the original PCs calculated from each data set, such as water vapor absorption, vegetation reflectance, and cloud reflectance.

Roberts, Yolanda, L.↗

DOE OSTI2021

Decoding the shift-invariant data: applications for band-excitation scanning probe microscopy *

A shift-invariant variational autoencoder (shift-VAE) is developed as an unsupervised method for the analysis of spectral data in the presence of shifts along the parameter axis, disentangling the physically-relevant shifts from other latent variables. Using synthetic data sets, we show that the shift-VAE latent variables closely match the ground truth parameters. The shift VAE is extended towards the analysis of band-excitation piezoresponse force microscopy data, disentangling the resonance frequency shifts from the peak shape parameters in a model-free unsupervised manner. The extensions of this approach towards denoising of data and model-free dimensionality reduction in imaging and spectroscopic data are further demonstrated. This approach is universal and can also be extended to analysis of x-ray diffraction, photoluminescence, Raman spectra, and other data sets.

36 MATERIALS SCIENCE↗

NASA NTRS2022

Helheim Glacier Ice Velocity Variability Responds to Runoff and Terminus Position, but at Different Timescales

The Greenland Ice Sheet discharges ice to the ocean through hundreds of outlet glaciers. Recent acceleration of Greenland outlet glaciers has been linked to both oceanic and atmospheric drivers. Here, we leverage temporally dense observations, regional climate model output, and newly developed time series analysis tools to assess the most important forcings causing ice flow variability at one of the largest Greenland outlet glaciers, Helheim Glacier, from 2009 to 2017. We find that ice speed correlates most strongly with catchment-integrated runoff at seasonal to interannual scales, while multi-annual flow variability correlates most strongly with multi-annual terminus variability. The disparate time scales and the influence of subglacial topography on Helheim Glacier’s dynamics highlight different regimes that can inform modeling and forecasting of its future. Notably, our results suggest that the recent terminus history observed at Helheim is a response to, rather than the cause of, upstream changes.

Cryospheric science↗

NASA NTRSDate not supplied

Understanding Variability in Mechanical Behavior of Adhesively Bonded Sandwich Joints due to Stochastic Material Damage

A mesoscale progressive damage analysis was applied to the analysis of recently tested adhesively bonded sandwich panels, in which ten nominally identical tension loaded panels failed in three different failure modes, with peakloads ranging from 13.0 –16.1 kips. A parametric finite element model for progressive damage analysis of an adhesively bonded joint (ABJ) considering matrix and fiber damage, and delamination cracks in unidirectional and fabric composite sections of the joint, as well as core crushing, and adhesive fracture was developed. The model was developed using Abaqus/Explicit and utilized continuum damage mechanics material models for intralaminar damage in the facesheet and doubler plies, cohesive elements for delamination and adhesive debonding, and discretized cells considering plasticity to model crushing of the aluminum honeycomb core. Analysis predictions obtained with a pristine model agreed well with the average experimental peak load, strain in the joint, and post-test damage states. Then, various methods to incorporate stochastic material variability into ABJ model were evaluated. These methods included simulation of pre-existing matrix damage, pre-existing adhesive porosity, and uniform modification of adhesive properties. These material and manufacturing defects changed the damage mode predictions in some cases. The analysis of stochastic strength due to the considered defects revealed unintuitive damage mode interaction in the ABJ, explaining some structure-property relationships observed experimentally.

Richard Larson↗

NASA NTRS1991

Stability of propagating modons for small-amplitude perturbations

A stability analysis based on Arnol'd-Liapunov arguments in a gauge-variable formalism is applied to a generalized form of propagating modons. The method was previously applied to the special case of Stern's modon in a quiescent background flow. The analysis presented here shows that propagating-modon solutions in the shallow-water equations are stable to small-amplitude perturbations, regardless of the sign of their translational speed, as long as this is much smaller in magnitude than the solid-rotation speed of the earth. Sufficient conditions for stability are that the given flow be quasigeostrophic, i.e., that the Rossby number be small, and that the associated Froude number be one order of magnitude smaller than the Rossby number.

Sakuma, H.↗