Inverse Lagrangian Analysis Research Articles

Quantifying uncertainty in analysis of shockless dynamic compression experiments on platinum. I. Inverse Lagrangian analysis

Absolute measurements of solid-material compressibility by magnetically driven shockless dynamic compression experiments to multi-megabar pressures have the potential to greatly improve the accuracy and precision of pressure calibration standards for use in diamond anvil cell experiments. To this end, we apply characteristics-based inverse Lagrangian analysis (ILA) to 11 sets of ramp-compression data on pure platinum (Pt) metal and then reduce the resulting weighted-mean stress–strain curve to the principal isentrope and room-temperature isotherm using simple models for yield stress and Grüneisen parameter. We introduce several improvements to methods for ILA and quasi-isentrope reduction, the latter including calculation of corrections in wave speed instead of stress and pressure to render results largely independent of initial yield stress while enforcing thermodynamic consistency near zero pressure. More importantly, we quantify in detail the propagation of experimental uncertainty through ILA and model uncertainty through quasi-isentrope reduction, considering all potential sources of error except the electrode and window material models used in ILA. Compared to previous approaches, we find larger uncertainty in longitudinal stress. Monte Carlo analysis demonstrates that uncertainty in the yield-stress model constitutes by far the largest contribution to uncertainty in quasi-isentrope reduction corrections. We present a new room-temperature isotherm for Pt up to 444 GPa, with 1-sigma uncertainty at that pressure of just under ±1.2%; the latter is about a factor of three smaller than uncertainty previously reported for multi-megabar ramp-compression experiments on Pt. The result is well represented by a Vinet-form compression curve with (isothermal) bulk modulus K0=270.3±3.8 GPa, pressure derivative K0′=5.66±0.10, and correlation coefficient RK0,K0′=−0.843.

Quantifying uncertainty in analysis of shockless dynamic compression experiments on platinum. II. Bayesian model calibration

Dynamic shockless compression experiments provide the ability to explore material behavior at extreme pressures but relatively low temperatures. Typically, the data from these types of experiments are interpreted through an analytic method called Lagrangian analysis. In this work, alternative analysis methods are explored using modern statistical methods. Specifically, Bayesian model calibration is applied to a new set of platinum data shocklessly compressed to 570 GPa. Several platinum equation-of-state models are evaluated, including traditional parametric forms as well as a novel non-parametric model concept. The results are compared to those in Paper I obtained by inverse Lagrangian analysis. The comparisons suggest that Bayesian calibration is not only a viable framework for precise quantification of the compression path, but also reveals insights pertaining to trade-offs surrounding model form selection, sensitivities of the relevant experimental uncertainties, and assumptions and limitations within Lagrangian analysis. The non-parametric model method, in particular, is found to give precise unbiased results and is expected to be useful over a wide range of applications. The calibration results in estimates of the platinum principal isentrope over the full range of experimental pressures to a standard error of 1.6%, which extends the results from Paper I while maintaining the high precision required for the platinum pressure standard.

Reduced scale stripline platform to extend accessible pressures on the Z machine: Shockless compression of platinum to 650 GPa

Reaching astrophysically relevant high energy density (HED) material states in the laboratory is an ongoing effort at multiple experimental facilities. We have developed a new dynamic compression platform for the Z Pulsed Power Facility that allows for sample sizes 100s of μm in thickness that accommodate multiple grains in order to fully capture bulk properties, such as material strength. A pair of experiments compressed platinum (Pt) to HED conditions and conventional inverse Lagrangian analysis as well as a recent Bayesian calibration technique were used to determine the principal isentrope to 650 GPa with density uncertainties of <2%. These low uncertainties are calculated for single sample experiments, presenting the possibility of even smaller experimental uncertainties with multiple samples the platform allows. Our new platform extends the accessible Pt ramp pressures achievable on the Z machine to over 80% of the pressure recently achieved using the National Ignition Facility planar Hohlraum platform. This new capability, the next generation evolution of the stripline platform, was made possible by advancements in both our understanding of the Z pulsed power driver and our overall magnetohydrodynamic modeling capabilities.

Analysis of shockless dynamic compression data on solids to multi-megabar pressures: Application to tantalum

Magnetically-driven, planar shockless-compression experiments to multi-megabar pressures were performed on tantalum samples using a stripline target geometry. Free-surface velocity waveforms were measured in 15 cases; nine of these in a dual-sample configuration with two samples of different thicknesses on opposing electrodes, and six in a single-sample configuration with a bare electrode opposite the sample. Details are given on the application of inverse Lagrangian analysis (ILA) to these data, including potential sources of error. The most significant source of systematic error, particularly for single-sample experiments, was found to arise from the pulse-shape dependent free-surface reflected wave interactions with the deviatoric-stress response of tantalum. This could cause local, possibly temporary, unloading of material from a ramp compressed state, and thus multi-value response in wave speed that invalidates the free-surface to in-material velocity mapping step of ILA. By averaging all 15 data sets, a final result for the principal quasi-isentrope of tantalum in stress-strain was obtained to a peak longitudinal stress of 330 GPa with conservative uncertainty bounds of ±4.5% in stress. The result agrees well with a tabular equation of state developed at Los Alamos National Laboratory.

Estimation of Scalar Source/Sink Distributions in Plant Canopies Using Lagrangian Dispersion Analysis: Corrections for Atmospheric Stability and Comparison with a Multilayer Canopy Model

Source/sink distributions of heat, water vapour and CO 2 within a rice canopy were inferred using an inverse Lagrangian dispersion analysis and measured mean profiles of temperature, specific humidity and CO 2 mixing ratio. Monin-Obukhov similarity theory was used to account for the effects of atmospheric stability on σ w (z). the standard deviation of vertical velocity and τ L (z), the Lagrangian time scale of the turbulence. Classical surface layer scaling was applied in the inertial sublayer (z > z ruf ) using the similarity parameter ζ = (z - d)/L, where z is height above ground, d is the zero plane displacement height for momentum, L is the Obukhov length, and z ruf 2.3h c , where h c is canopy height. A single length scale h c , was used for the stability parameter 3 = h c /L in the height range 0.25 < z/ h c < 2.5. This choice is justified by mixing layer theory, which shows that within the roughness sublayer there is one dominant turbulence length scale determined by the degree of inflection in the wind profile at the canopy top. In the absence of theoretical or experimental evidence for guidance, standard Monin-Obukhov similarity functions, with ζ = h c /L, were used to calculate the stability dependence of σ w (z) and τ L (z) in the roughness sublayer. For z/h c < 0.25 the turbulence length and time scales are influenced by the presence of the lower surface, and stability effects are minimal. With these assumptions there was excellent agreement between eddy covariance flux measurements and deductions from the inverse Lagrangian analysis. Stability corrections were particularly necessary for night time fluxes when the atmosphere was stably stratified. The inverse Lagrangian analysis provides a useful tool for testing and refining multilayer canopy models used to predict radiation absorption, energy partitioning and CO 2 exchanges within the canopy and at the soil surface. Comparison of model predictions with source strengths deduced from the inverse analysis gave good results. Observed discrepancies may be due to incorrect specification of the turbulent time scales and vertical velocity fluctuations close to the ground. Further investigation of turbulence characteristics within plant canopies is required to resolve these issues.

Source/sink distributions of heat, water vapour, carbon dioxide and methane in a rice canopy estimated using Lagrangian dispersion analysis

Source distributions for heat, water vapour, CO 2 and CH 4 within a rice canopy were derived using measured concentration profiles, a prescribed turbulence field and an inverse Lagrangian analysis of turbulent dispersion of scalars in plant canopies. Measurements were made during IREX96, an international rice experiment in Okayama, Japan. Results for the cumulative fluxes of heat, water vapour and CH 4 at the canopy top were satisfactory once their respective concentration profiles were smoothed using simple analytic functions. According to the inverse analysis, water vapour was emitted relatively uniformly by each of five equi-spaced layers within the canopy, whereas sensible heat fluxes were small (<100 W m −2) and of either sign. Methane fluxes were predicted to be emitted most strongly in the lower 50% of the canopy, as expected from the distribution of micropores along leaves and leaf sheaths, the major pathway for CH 4 loss from the soil–crop system. No smoothing was required for CO 2 concentration profiles and the inverse analysis provided close correspondence between the turning point in the concentration profile is the changeover from respiration by the soil/paddy water and lower canopy to net photosynthesis by the upper canopy. These results could only be obtained by including both the near- and far-field contributions of sources to the total concentration profile. Neglect of the near-field contribution in the inverse analysis led to spurious source distributions. Excellent agreement was obtained between cumulative fluxes of heat, water vapour, CO 2 and CH 4 at the top of the canopy from the inverse analysis and direct eddy covariance measurements when the friction velocity u *>0.1 m s −1, and atmospheric stability was approximately neutral. Nocturnal fluxes of CO 2 and CH 4 from the inverse method exceeded micrometeorological measurements above the canopy by a factor of 2–3 when u *<0.1 m s −1 and stable atmospheric conditions prevailed within and above the canopy. Neglect of these stability effects will lead to an underestimate of the dispersion coefficients (dimension of resistances) in the transport model and hence an overestimate of the fluxes. Further work is required to establish the correct procedure for incorporating stability effects into the inverse analysis.

Inferring Sources and Sinks of Mass and Energy in Plant Canopies through Inverse Lagrangian Analysis

Lagrangian analysis of transport within the canopy space leads to a physical description by which the mean concentration profiles can be predicted for any given source distribution through knowledge of the turbulence statistics. The concentration profiles are related to the source-sink distributions through a matrix of linear equations, a ‘dispersion matrix’, in which the coefficients are functions of σw the standard deviation of the vertical wind velocity, and TL, the Lagrangian time-scale. The Inverse Lagrangian analysis inverts the dispersion matrix to calculate the source/sink distributions from the mean concentration profiles and profiles of σw and TL. Thus the technique allows separate identification of, say, the respiration of CO2 at the soil (or paddy water) surface and the uptake of atmospheric CO2 by plants in the canopy. The observations and the computational scheme required for such applications are relatively simple. The paper describes the use of the technique to infer sources of heat, water vapour and CO2 in a wheat crop.

Measuring and modelling soil evaporation in wheat crops

Soil evaporation is often an important component of the total water loss from plant canopies, but is poorly estimated in many SVAT models. This paper reports the use of micrometeorological and lysimetric methods to measure soil evaporation ( E s ) in the canopy of a wheat crop over a period of 7 weeks as the crop matured and green leaf area index ( L) declined from 3.8 to 0.2. Simultaneous measurements of total canopy evaporation ( E) were made by eddy correlation. The within-canopy micrometeorological methods employed a variance ratio approach and an inverse Lagrangian analysis. The pooled micrometeorological and lysimetric data conformed very well to the relationship E s E = exp(−0.61L) with an r 2 value of 0.95. The relationship should be a useful empirical means for estimating daily or weekly E s in SVAT studies.