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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

The dependence of the gradients of oxygen and nitrogen-to-oxygen on stellar age in MaNGA galaxies

For this work, we derived the oxygen abundance (O/H), the nitrogen-to-oxygen (N/O) abundance ratio, and their corresponding radial gradients for a sample of 1431 galaxies from the MaNGA DR15 survey using two different realisations of the strong line method: empirical R calibration and the Bayesian model-based HII-CHI-MISTRY (HCM) code. We find that both abundance calculation methods reveal a correlation between the O/H gradient and the stellar mass of a galaxy. This relation is non-linear, with the steepest average gradients in the intermediate mass range and flatter average gradients for high- and low-mass galaxies. The relation between the N/O gradient and the stellar mass is, on average, non-linear with the steepest gradients in the intermediate mass range (log(M/M ⊙ )~10), flatter gradients for high-mass galaxies, and the flattest gradients for low-mass galaxies. However, the general trend of steepening N/O gradients for higher masses, as reported in previous studies, remains evident. We find a dependence between the O/H and N/O gradients and the galaxy mean stellar age traced by the D(4000) index. For galaxies of lower masses, both gradients are, generally, steeper for intermediate values of D(4000) and flatter for low and high values of D(4000). Only the most massive galaxies do not show this correlation. We interpret this behaviour as an evolution of the metallicity gradients with the age of stellar population. Though the galaxies with a positive slope of the D(4000) radial gradient tend to have flatter O/H and N/O gradients, as compared to those with a negative D(4000) gradient.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Colour gradients of low-redshift galaxies in the DESI Legacy Imaging Survey

ABSTRACT Radial colour gradients within galaxies arise from gradients of stellar age, metallicity, and dust reddening. Large samples of colour gradients from wide-area imaging surveys can complement smaller integral-field spectroscopy data sets and can be used to constrain galaxy formation models. Here, we measure colour gradients for low-redshift galaxies (z < 0.1) using photometry from the DESI Legacy Imaging Survey DR9. Our sample comprises ∼93 000 galaxies with spectroscopic redshifts and ∼574 000 galaxies with photometric redshifts. We focus on gradients across a radial range 0.5Reff to Reff, which corresponds to the inner disc of typical late-type systems at low redshift. This region has been the focus of previous statistical studies of colour gradients and has recently been explored by spectroscopic surveys such as MaNGA. We find that the colour gradients of most galaxies in our sample are negative (redder towards the centre), consistent with the literature. We investigate empirical relationships between colour gradient, average g − r and r − z colour, Mr, M⋆, and sSFR. Trends of gradient strength with Mr (M⋆) show an inflection around Mr ∼ −21 ($\log _{10} \, M_\star /\mathrm{M_\odot }\sim 10.5$). Below this mass, colour gradients become steeper with increasing M⋆, whereas colour gradients in more massive galaxies become shallower. We find that positive gradients (bluer stars at smaller radii) are typical for galaxies of $M_{\star }\sim 10^{8}\, \mathrm{M_\odot }$. We compare our results to age and metallicity gradients in two data sets derived from fits of different stellar population libraries to MaNGA spectra, but find no clear consensus explanation for the trends we observe. Both MaNGA data sets seem to imply a significant contribution from dust reddening, in particular, to explain the flatness of colour gradients along the red sequence.

Astronomy & Astrophysics↗
DOE OSTI2022

The half-mass radius of MaNGA galaxies: effect of IMF gradients

Gradients in the stellar populations (SP) of galaxies – e.g. in age, metallicity, stellar initial mass function (IMF) – can result in gradients in the stellar-mass-to-light ratio, M * / L . Such gradients imply that the distribution of the stellar mass and light is different. For old SPs, e.g. in early-type galaxies at z ∼ 0, the M * / L gradients are weak if driven by variations in age and metallicity, but significantly larger if driven by the IMF. A gradient which has larger M * / L in the centre increases the estimated total stellar mass ( M * ) and reduces the scale which contains half this mass ( R e,* ), compared to when the gradient is ignored. For the IMF gradients inferred from fitting MILES simple SP models to the H β, 〈Fe〉, [MgFe], and TiO 2SDSS absorption lines measured in spatially resolved spectra of early-type galaxies in the MaNGA survey, the fractional change in R e,* can be significantly larger than that in M * , especially when the light is more centrally concentrated. The R e,* – M * correlation which results from accounting for IMF gradients is offset to smaller sizes by 0.3 dex compared to when these gradients are ignored. Comparisons with ‘quiescent’ galaxies at higher z must account for evolution in SP gradients (especially age and IMF) and in the light profile before drawing conclusions about how R e,* and M * evolve. Here, the implied merging between higher z and the present is less contrived if R e,* / R e at z ∼ 0 is closer to our IMF-driven gradient calibration than to unity.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Gradient-based surface nuclear magnetic resonance for groundwater investigation

In medical magnetic resonance imaging, spatial localization (imaging) is based upon the application of controlled magnetic field gradients on top of the main magnetic field to spatially modulate the frequency and/or phase of the nuclear magnetic resonance (NMR) signal across the volume of investigation. In this work, we have applied similar physical principles to produce controlled magnetic field gradients during surface NMR-based groundwater investigations. In this approach, a gradient pulse of variable amplitude or duration is applied immediately after the excitation pulse to cause predictable phase encoding of the NMR signal as a function of depth. This approach is also applicable to emerging surface NMR detection methods that use a prepolarization field with fast nonadiabatic turn-off to generate detectable NMR signals from the shallow subsurface. In this case, the gradient pulse is applied after terminating the prepolarization field and provides a heretofore unavailable means of localizing the NMR response as a function of depth. The application of gradients can also be combined with tip-angle-based modulation to yield higher imaging resolution than can be achieved through either gradient- or tip-angle-based imaging alone. We implemented this new gradient-based capability into a surface NMR gradient generation accessory that is compatible with the GMR-Flex instrument and developed surface NMR-specific forward modeling and linear inverse models. We validated the accuracy of this novel gradient-based sNMR technology using computer simulations, experiments using a small pool filled with a discrete layer of bulk water, and field experiments at well-characterized groundwater test sites along Ebey Island, WA, and Larned, KS. The gradient-based sNMR imaging observations were compared with high-resolution direct push NMR results observed at these sites. The results of computer simulations and field experiments indicate improvements in both the detection (signal-to-noise ratio) and spatial resolution of shallow subsurface water content using gradient-based surface NMR, compared with traditional surface NMR imaging methods.

Geochemistry & Geophysics↗
DOE OSTI2026

Strong gradient neoclassical transport in the plateau regime

Strong gradient regions in tokamaks such as the pedestal or internal transport barriers are regions of reduced turbulence where neoclassical transport can play a dominant role. In pedestals, gradient lengths comparable to the ion poloidal gyroradius have been measured. Standard neoclassical theory can miss important strong gradient effects in these regions because it assumes that the gradient length scales of density, temperature and potential are larger than the ion poloidal gyroradius. We extend plateau regime neoclassical theory into regions of gradients of the order of the ion poloidal gyroradius to capture strong gradient effects on transport processes in the pedestal and internal transport barriers. The fundamental idea behind our new framework is to keep a scale separation between the orbit widths and the gradient length scales by performing a large aspect ratio expansion. In the plateau regime, strong gradients cause poloidal variation that is in–out as well as up–down asymmetric. We study two different test cases assuming either radial force balance or the absence of turbulence and show that strong gradient effects can enhance or reduce standard neoclassical theory predictions in the plateau regime in strong gradient regions.

fusion plasma↗
DOE OSTI2021

Gradient-Informed Design Optimization of Select Nuclear Systems

In this work, we present a gradient-informed design optimization of nuclear reactor core components based on neutronics objectives with both continuous and discrete materials. The main argument in favor of using gradient-informed design optimization is that it scales well with increasing dimensionality of the design space. First, a challenge problem with 121 free parameters is solved with a gradient-informed method and then with a genetic algorithm. Then, a challenge problem to optimize the flux profile of a simplified assembly with eight axial zones is solved. Both challenge problems are solved using directly calculated derivatives from Tools for Sensitivity and Uncertainty Analysis Methodology Implementation (TSUNAMI) in the SCALE package. Furthermore, we demonstrate how a discrete optimization problem—selection of materials for 121 voxels—can be lifted into a continuous problem with mixed materials. In the continuous space, adjoint-based gradients are well-defined, and gradient descent is applicable. Then, a forcing function is introduced that with the selection of an appropriately sized hyperparameter can be used to guide the optimized continuous solution back into a discrete solution. This paper presents an account of the challenges that were faced when applying a gradient-informed optimization algorithm using a Monte Carlo calculation to estimate the gradient information and compares a gradient descent optimization method to a genetic algorithm optimization of the same geometry. Overall, this work demonstrates the potential use of adjoint-based gradient calculations in design optimization of nuclear systems.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2022

Examination of stiff ion temperature gradient mode physics in simulations of DIII-D H-mode transport

A systematic evaluation of gyrokinetic and gyrofluid model predictions of ion temperature gradient (ITG) stability and transport using parameters from DIII-D high confinement mode (H-mode) plasmas has been performed. The nonlinear CGYRO code is used to make the gyrokinetic predictions, and the quasilinear TGLF model for the corresponding gyrofluid predictions. The assessments are made at three radii (normalized toroidal flux ρtor = 0.4, 0.55, and 0.7) in three different plasma scenarios with varying levels of neutral beam heating and torque. For each of the nine cases (3 radii × 3 scenarios) considered, ITG turbulence is found to be the dominant long-wavelength instability and transport mechanism. The inclusions of both transverse magnetic fluctuations and dynamic fast beam ions are stabilizing for all cases considered, with strongest effects seen at ρor = 0.4 where the fast ion population and normalized plasma pressure β = 2μ0nT/B2 are highest. The further inclusion of parallel magnetic fluctuations does not have a meaningful impact on the ITG turbulence in these scenarios, but does destabilize (in combination with fast ions) new high-frequency instabilities at ρtor = 0.4 in the high power scenarios. In each case the linear and nonlinear ITG critical gradients are predicted to be lower than the measured ITG scale lengths and their associated uncertainties. Inclusion of equilibrium flow shear in the transport predictions generally leads to an upshift in effective critical gradient rather than a qualitative change in the predicted stiffness, with stronger responses typically seen in the gyrokinetic predictions than in the gyrofluid results. However, in most cases these upshifted gradients still remain below the measured values and their uncertainties. Although the predicted critical gradients are below the measured gradients, both models predicted flux-matching gradients consistent with measured values in six of the nine cases considered, with no clear systematic over- or underprediction. Thus, while the experimental ion temperature profiles do not appear to be closely pinned to the ITG critical gradient, both gyrokinetic and gyrofluid models are able to accurately match the measured gradients reasonably well in most cases.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Impact of Different Thermal Gradients on the Dynamics of Cylindrical Lithium-ion Cells Subject to Accelerated Aging and on Module Performance

This study investigates the impacts of applying different thermal gradient patterns to cylindrical lithium-ion cells in a module on cell dynamics (temperatures, current flows, state of charge), module performance (evolution of resistance, capacity, and energy versus cycle number), and module lifetime. The thermal gradients were generated using cooling plates (CPs) with three different flow-field designs, namely, straight, perpendicular, and U-turn. The study uses computational fluid dynamics (CFD), the pseudo-two-dimensional (P2D) battery model, capacity loss and increased impedance due to the growth of a solid-electrolyte-interphase, and the electric current distribution from module terminals to cells that depends on the series-parallel electrical connections among the cells. The impact of the thermal gradient (resulting from the CP designs) on the variability in resistance, current, state of charge, and voltage among the cells was analyzed and linked to differences in the module's performance. Applying a thermal gradient to parallel-connected strings of series-connected cells led to variation in the current through each parallel string and an imbalance in the voltage of series-connected cells. Module performance is poorer when the thermal gradient causes a voltage imbalance than when it causes a current imbalance. Module performance becomes the worst when both current variation and voltage imbalance happen together. For instance, the module's lifetime (estimated as reaching 80% of its initial capacity) varied by 5% to 17.5%, depending on the magnitude and pattern of the imposed thermal gradient. As the relative orientation between thermal gradients and cells' electrical connectivity influences the module's performance, appropriate consideration should be given to the choice of the CP, especially if large thermal gradients are allowed.

Battery thermal management↗
DOE OSTI2021

Single Molecule Spectroscopy Studies of Acid–Base Chemical Gradients Using Nile Red as a Probe of Local Surface Acidity

Single molecule spectroscopy studies of local acidity along bifunctional acid-base gradients are reported. Gradients are prepared by directional vapor phase diffusion and subsequent reaction of 3-aminopropyl-trimethoxysilane with a uniform silica film. Gradient formation is confirmed by spectroscopic ellipsometry and by static water contact angle measurements. X-ray photoelectron spectroscopy is used to characterize the nitrogen content and degree of nitrogen protonation along the gradient. Nile Red is employed as the probe dye in single molecule spectroscopy studies of these gradients. While Nile Red is well-known for its solvent sensitivity, it is used here, for the first time, to sense the acid/base properties of the film in two-color wide-field fluorescence imaging experiments. The data reveal broad bimodal distributions of Nile Red emission spectra that vary along the gradient direction. The single-molecule results are consistent with solution-phase ensemble acid/base studies of the dye. The former reveal a gradual transition from a surface dominated by basic aminosilane sites at the high-amine end of the gradient to one dominated by acidic silanol sites at the low-amine end. The sub-diffraction-limited spatial resolution afforded by superlocalization of the single molecules reveals spatial correlations in the acid/base properties of the gradient over ~ 200 nm distances. Furthermore, these studies provide data relevant to the use of aminosilane-modified silica in bifunctional, cooperative chemical catalysis.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2026

Ground and excited state gradients with end-to-end differentiable semiempirical quantum chemistry

Accurate and efficient gradients of molecular energy with respect to nuclear degrees of freedom are essential for geometry optimization and molecular dynamics, including simulations that go beyond the Born–Oppenheimer regime. A common approach involves deriving analytical formulas for new electronic structure methods, which is often conceptually difficult and requires tedious coding. Here, we implement analytical, semi-numerical, and automatic differentiation (AD)-based gradient pathways for semiempirical Hamiltonian models in the PYSEQM software package, leveraging both graphics processing unit (GPU) and central processing unit (CPU) architectures. We further extend these capabilities to excited states calculated using the configuration interaction singles and time-dependent Hartree–Fock ansätze. We benchmark wall time, peak memory usage, and accuracy across three molecular families of varying chemical complexity, including systems of up to a thousand atoms. For ground-state simulations, analytical and AD gradients achieve near-identical GPU runtimes, while semi-numerical gradients are slower on GPU but remain competitive on CPU. For excited states, both analytical and custom AD approaches using implicit differentiation show similar performance and low memory requirements, whereas gradients with full AD are memory-limited. AD gradients match analytical ones in accuracy across all tested systems, aided by a quaternion-based diatomic frame rotation for two-center quantities that ensures smooth energy surfaces. Overall, automatic differentiation emerges as a practical alternative to analytical gradients in semiempirical quantum chemistry, offering high accuracy while allowing seamless integration in AI-driven workflows and popular packages, such as PyTorch and JAX. Our results provide actionable guidance for selecting optimal gradient strategies in large-scale ground- and excited-state molecular dynamics simulations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Monsoon Depression Amplification by Horizontal Shear and Humidity Gradients: A Shallow Water Perspective

Transient, synoptic-scale vortices produce a large fraction of total rainfall in most monsoon regions and are often associated with extreme precipitation. However, the mechanism of their amplification remains a topic of active research. For monsoon depressions, which are the most prominent synoptic-scale vortex in the Asian–Australian monsoon, recent work has suggested that meridional gradients in zonal wind in the vortex environment may produce growth through barotropic instability, while meridional gradients in environmental humidity have also been proposed to cause amplification through coupling with precipitating convection. Here, a two-dimensional shallow water model on a sphere with parameterized precipitation is used to examine the relative role played by these two environmental gradients. By systematically varying the meridional moisture gradient and meridional wind shear for both weak, quasi-linear waves and finite-amplitude isolated vortices, we show that rotational winds in the initial vortex are amplified most strongly by meridional shear of the environmental zonal wind, while vortex precipitation rates are most sensitive to environmental moisture gradients. The growth rate in the presence of both gradients is less than the sum of growth rates in the presence of isolated gradients, as the phase relation between moisture and vorticity anomalies becomes distorted with increasing shear. In conclusion, these results suggest that background meridional gradients in both zonal wind and environmental humidity can contribute to the amplification of vortices to monsoon depression strength, but with some degree of decoupling of the dry rotational flow and the moist convection.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Effect of barren plateaus on gradient-free optimization

Barren plateau landscapes correspond to gradients that vanish exponentially in the number of qubits. Such landscapes have been demonstrated for variational quantum algorithms and quantum neural networks with either deep circuits or global cost functions. For obvious reasons, it is expected that gradient-based optimizers will be significantly affected by barren plateaus. However, whether or not gradient-free optimizers are impacted is a topic of debate, with some arguing that gradient-free approaches are unaffected by barren plateaus. Here we show that, indeed, gradient-free optimizers do not solve the barren plateau problem. Our main result proves that cost function differences, which are the basis for making decisions in a gradient-free optimization, are exponentially suppressed in a barren plateau. Hence, without exponential precision, gradient-free optimizers will not make progress in the optimization. We numerically confirm this by training in a barren plateau with several gradient-free optimizers (Nelder-Mead, Powell, and COBYLA algorithms), and show that the numbers of shots required in the optimization grows exponentially with the number of qubits.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Extracting energy from ocean thermal and salinity gradients to power unmanned underwater vehicles: State of the art, current limitations, and future outlook

Thermal gradient energy-generation technologies for powering unmanned underwater vehicles (UUVs) or autonomous sensing systems in the ocean are mainly in the research development phase or commercially available at a limited scale, and salinity-gradient energy-generation technologies have not been adequately researched yet. The demand for self-powered UUVs suitable for long-term deployments has been growing, and further research related to small-scale ocean gradient energy systems is needed. In this study, we conducted a comprehensive review about harvesting energy from ocean thermal or salinity gradients for powering UUVs, focusing on gliders and profiling floats. Thermal gradient energy systems for UUVs based on phase change materials (PCM) cannot provide the energy required for powering autonomous sensing systems because of the systems' low energy conversion efficiency. Besides reducing energy consumption by developing more efficient electrical-mechanical systems, enhancing the thermal conductivity of the PCMs may help address this challenge by increasing the power generation rate of the UUVs. Several other emerging technologies, such as thermoelectric generators, shape memory alloys, and small-scale thermodynamic cycle systems, have shown potential for powering UUVs, but they are still only at the laboratory testing or conceptual design phase. The most advanced power generation technologies based on salinity gradients, reverse electrodialysis and pressure-retarded osmosis, are still not economically viable for large-scale deployment, mainly because of the high cost of the components required to operate in harsh saline environments. Our feasibility evaluation showed that existing salinity gradient power generation technologies are not directly feasible for powering UUVs in the open ocean.

16 TIDAL AND WAVE POWER↗
DOE OSTI2025

Operando Neutron Imaging of Lithium Flux and Gradient Cathode Design for Enhanced Kinetics in High‐Loading All‐Solid‐State Li─S Batteries

All-solid-state Li–sulfur batteries (ASSLSBs) are considered promising candidates for next-generation energy storage owing to their inherent safety, high energy density, and abundant sulfur resources. However, slow redox kinetics greatly limit sulfur utilization during solid-solid sulfur reactions, leading to significant challenges to achieve efficient performance in high-mass-loading ASSLSBs. Here, operando neutron image is employed to directly visualize, for the first time, that sluggish Li + transport kinetics and the uneven distribution of Li + during cathodic reactions are critical factors limiting sulfur conversion. To address this issue, gradient cathode architectures comprising three and five layers are designed, in which catholyte concentrations are strategically varied to optimize Li-ion flux and enhance ionic conductivity of the whole composite cathode electrode. Operando neutron imaging distinctly visualizes and confirms that three-layer gradient approach significantly enhances Li-ion mobility, resulting in more uniform redox reactions and greatly improved sulfur utilization compared to traditional non-gradient structures. Consequently, the three-layer gradient cathode achieves superior rate performance and reduced electrode polarization at high sulfur mass loadings of 4.5 and 6.0 mg cm −2 . Furthermore, the applicability and scalability of this design are demonstrated in a five-layer gradient cathode architecture, achieving an impressive discharge specific capacity increase from 656 mAh g −1 (three-layer gradient) to 1232 mAh g −1 at 1/20 C for ultra-high sulfur loading of 7.5 mg cm −2 . In conclusion, this innovative gradient cathode design offers substantial advancements in understanding and overcoming Li-ion transport limitations, paving the way toward practical, high-energy-density ASSLSBs.

25 ENERGY STORAGE↗
DOE OSTI2024

What is the gradient of a scalar function defined on a subspace of square matrices?

We illustrate a technique to calculate the gradient of scalar functions that are defined on any arbitrary matrix subspace. It generalizes our earlier work titled “What is the gradient of a scalar function of a symmetric matrix ?”(Indian Journal of Pure and Applied Mathematics (2022), https://doi.org/10.1007/s13226-022-00313-x), in which we considered the special case of the subspace of symmetric matrices. Extant methods to calculate the gradient in such cases have an inherent flaw which leads to spurious results that populate several publications, as well as respected textbooks and handbooks on matrix calculus. Here, we examine these sources and results in a rigorous and concrete mathematical setting of a finite-dimensional inner-product space and discover the inherent flaw and also a remedy. We demonstrate two ways to calculate the derivative/gradient and second derivative for scalar functions of matrices defined over an arbitrary matrix subspace; the first method is by considering any (differentiable) extension to the space of square matrices and projection of its gradient onto the given subspace. The second method utilizes an ordered basis and computes each component of the gradient through evaluation of the directional derivative. All the ideas presented are illustrated by non-trivial examples, namely, considering the subspace of 3 x 3 circulant and Toeplitz matrices and presenting the results of gradient-descent with both the spurious and correct gradients. Moreover, our bibliography makes it clear that a rigorous approach to matrix calculus is not common in practice, and our presentation of matrix calculus in the language of inner-product spaces will be significant and meaningful for applied mathematicians, engineers and researchers working in inter-disciplinary fields to avoid the conceptual pitfalls that exist.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Fast model-based scenario optimization in NSTX-U enabled by analytic gradient computation

Model-based optimization offers a systematic approach to advanced scenario planning. In this case, the feedforward-control inputs (actuator trajectories) that are needed to attain and sustain a desired scenario are obtained by solving a nonlinear constrained optimization problem. This class of problems generally minimize a cost function that measures the difference between desired and actual plasma states. Several numerical optimization algorithms, such as sequential quadratic programming, require repeated calculation of the cost function gradients with respect to the input trajectories. Calculating these gradients numerically can be computationally intensive, increasing the time needed to solve the feedforward-control optimization problem. Here, this work introduces a method to analytically calculate these cost function gradients from the current profile evolution model. This can significantly reduce the computational time and allow for fast feedforward-control optimization, which would eventually enable optimal scenario planning between discharges. The performance of the feedforward optimizer with analytical gradients is compared to a traditional optimization algorithm based on numerical gradients for different NSTX-U scenarios. The plasma dynamics in the optimization algorithm are simulated using the Control Oriented Transport SIMulator (COTSIM). Results of the work show that analytical gradients consistently reduce the computation time while achieving trajectories that are comparable to those obtained by traditional optimization algorithms based on numerical gradients.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

The effect of the vertical gradients of photosynthetic parameters on the CO 2 assimilation and transpiration of a Panamanian tropical forest

Terrestrial biosphere models (TBMs) include the representation of vertical gradients in leaf traits associated with modeling photosynthesis, respiration, and stomatal conductance. However, model assumptions associated with these gradients have not been tested in complex tropical forest canopies. Here we compared TBM representation of the vertical gradients of key leaf traits with measurements made in a tropical forest in Panama, and then quantified the impact of the observed gradients on simulated canopy scale CO 2 and water fluxes. Comparison between observed and TBM trait gradients showed divergence that impacted canopy scale simulations of water vapor and CO 2 exchange. Notably, the ratio between the dark respiration rate and the maximum carboxylation rate was lower near the ground than at the top-of-canopy, leaf-level water-use efficiency was markedly higher at the top-of-canopy, and the decrease in maximum carboxylation rate from the top-of-canopy to the ground was less than TBM assumptions. The representation of the gradients of leaf traits in TBMs is typically derived from measurements made within-individual plants, or, for some traits, assumed constant due to a lack of experimental data. Our work shows that these assumptions are not representative of the trait gradients observed in species-rich, complex tropical forests.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

What is the gradient of a scalar function defined on a subspace of square matrices?

We illustrate a technique to calculate the gradient of scalar functions that are defined on any arbitrary matrix subspace. It generalizes our earlier work titled “What is the gradient of a scalar function of a symmetric matrix ?”(Indian Journal of Pure and Applied Mathematics (2022), doi:10.1007/s13226-022-00313-x), in which we considered the special case of the subspace of symmetric matrices. Extant methods to calculate the gradient in such cases have an inherent flaw which leads to spurious results that populate several publications, as well as respected textbooks and handbooks on matrix calculus. We examine these sources and results in a rigorous and concrete mathematical setting of a finite-dimensional inner-product space and discover the inherent flaw and also a remedy. We demonstrate two ways to calculate the derivative/gradient and second derivative for scalar functions of matrices defined over an arbitrary matrix subspace; the first method is by considering any (differentiable) extension to the space of square matrices and projection of its gradient onto the given subspace. The second method utilizes an ordered basis and computes each component of the gradient through evaluation of the directional derivative. All the ideas presented are illustrated by non-trivial examples, namely, considering the subspace of 3 × 3 circulant and Toeplitz matrices and presenting the results of gradient-descent with both the spurious and correct gradients. Moreover, our bibliography makes it clear that a rigorous approach to matrix calculus is not common in practice, and our presentation of matrix calculus in the language of inner-product spaces will be significant and meaningful for applied mathematicians, engineers and researchers working in inter-disciplinary fields to avoid the conceptual pitfalls that exist.

97 MATHEMATICS AND COMPUTING↗