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Transport Upscaling under Flow Heterogeneity and Matrix-Diffusion in Three-Dimensional Discrete Fracture Networks
For this work, we investigate the combined effects of network scale flow variability and retention due to matrix-diffusion on the scaling behavior of transport through fractured media. Two of the principal mechanisms controlling the transport of solutes through fractured low-permeability media are broad distributions of flow velocities and retention times in the solid matrix. We study the relative impact of these two processes under different initial conditions using a set of three-dimensional discrete fracture network simulations. We use these simulations to develop and calibrate an upscaled continuous time random walk (CTRW) approach for advective transport based on an Ornstein-Uhlenbeck model for the particle velocities that accounts for the fracture-matrix coupling using a compound Poisson process. This CTRW model can be conditioned on the initial solute distribution and allows to observe late-time scaling behavior at distances beyond what is feasible to observe using high-fidelity direct numerical simulations. We determine that the initial distribution of particles leads to marked differences in the persistent long-term scale behavior in the solute travel time distributions, even those undergoing retention due to matrix diffusion through implementation and analysis of the model.
Aging power spectrum of membrane protein transport and other subordinated random walks
Single-particle tracking offers detailed information about the motion of molecules in complex environments such as those encountered in live cells, but the interpretation of experimental data is challenging. One of the most powerful tools in the characterization of random processes is the power spectral density. However, because anomalous diffusion processes in complex systems are usually not stationary, the traditional Wiener-Khinchin theorem for the analysis of power spectral densities is invalid. Here, we employ a recently developed tool named aging Wiener-Khinchin theorem to derive the power spectral density of fractional Brownian motion coexisting with a scale-free continuous time random walk, the two most typical anomalous diffusion processes. Using this analysis, we characterize the motion of voltage-gated sodium channels on the surface of hippocampal neurons. Our results show aging where the power spectral density can either increase or decrease with observation time depending on the specific parameters of both underlying processes.
A review of spatial Markov models for predicting pre-asymptotic and anomalous transport in porous and fractured media
Heterogeneity across a broad range of scales in geologic porous media often manifests in observations of non-Fickian or anomalous transport. While traditional anomalous transport models can successfully make predictions in certain geological systems, increasing evidence suggests that assumptions relating to independent and identically distributed increments constrain where and when they can be reliably applied. A relatively novel model, the Spatial Markov model (SMM), relaxes the assumption of independence. The SMM belongs to the family of correlated continuous time random walks and has shown promise across a wide range of transport problems relevant to natural porous media. It has been successfully used to model conservative as well as more recently reactive transport in highly complex flows ranging from pore scales to much larger scales of interest in geology and subsurface hydrology. In this review paper we summarize its original development and provide a comprehensive review of its advances and applications as well as lay out a vision for its future development.
Anomalous losses of energetic particles in the presence of an oscillating radial electric field in fusion plasmas
The confinement of energetic particles in nuclear fusion devices is studied in the presence of an oscillating radial electric field and an axisymmetric magnetic equilibrium. Here, it is shown that, despite the poloidal and toroidal symmetries, initially integrable orbits turn into chaotic regions that can potentially intercept the wall of the tokamak, leading to particle losses. It is observed that the losses exhibit algebraic time decay different from the expected exponential decay characteristic of radial diffusive transport. A dynamical explanation of this behaviour is presented, within the continuous time random walk theory. The central point of the analysis is based on the fact that, contrary to the radial displacement, the poloidal angle is not bounded and a proper statistical analysis can therefore be made, showing for the first time that energetic particle transport can be super-diffusive in the poloidal direction and characterised by asymmetric poloidal displacement. The connection between poloidal and radial positions ensured by the conservation of the toroidal canonical momentum, implies that energetic particles spend statistically more time in the inner region of the tokamak than in the outer one, which explains the observed algebraic decay. This indicates that energetic particles might be efficiently slowed down by the thermal population before leaving the system. Also, the asymmetric transport reveals a new possible mechanism of self-generation of momentum.
Geminate exciton fusion fluorescence as a probe of triplet exciton transport after singlet fission
The geminate annihilation of two triplet excitons created by singlet exciton fission is affected by the dimensionality of transport as determined by typically anisotropic triplet exciton mobilities in organic molecular crystals. We analyze this process using a random-walk model where the time dynamics of the geminate annihilation probability is determined by the average exciton hopping times along the crystallographic directions. Here, the model is then applied to the geminate fluorescence dynamics in rubrene, where the main channel for triplet-triplet annihilation is via triplet fusion and subsequent photon emission, and we identify the transitions between transport in one, two, and three dimensions.
Markovian transport processes in a heterogeneous, variably saturated watershed: A multi-domain spatial Markov model
Transport processes in watersheds remain challenging to describe accurately with upscaled models because of their intricate complexities at multiple scales that can lead to multi-modal, non-Fickian breakthrough curves. One of the recent advances in solute transport modeling has been the Lagrangian spatial Markov model (SMM), which describes transport using velocity increments and models the correlated transitions between them. The method has been applied previously to saturated porous media and fractured media, but these successful applications suggest it may be useful for describing watershed scale processes as well. An existing, 3-D, heterogeneous variably saturated model of a headwaters catchment and classical random walk particle tracking were used to generate a forward simulation and the resulting transport was analyzed with a spatial Markov framework. Existing SMM methods were used to analyze the transport paths through the complex, unconfined domain and we found that the vadose zone plays an important role in the correlation structure. As such, a new kind of SMM is proposed, termed a multi-domain SMM (MD-SMM), that uses separate correlation models for auto-transitions in the vadose zone and the saturated zone, and for the cross-transitions between them. The MD-SMM improves representation of transport relative to the single domain model and can naturally delineate times when a particle is in the vadose zone or the saturated zone, adding an additional degree of realism to the upscaled model. Lastly, we show that the transition probability matrix for transport in the saturated portion of this unconfined system is approximately Markovian, which is an important validation of the proposed framework.
A Differentiable Hydrology Approach for Modeling With Time-Varying Parameters
In the analysis of observational data with mathematical models, it is often desirable to let one or more model parameters vary with time to account for changing environmental conditions or to offer more flexibility. A standard approach for dynamic models is to allow for the number of parameters to grow as a function of time, leading to a high-dimensional inference problem for long data records. We propose using Hamiltonian Monte Carlo, a gradient-based Markov chain Monte Carlo (MCMC) method, to perform Bayesian inference for time-varying parameters (TVP) in hydrology models. As derivatives of model error functions with regard to parameters are not available in closed form, we implemented the GR4J rainfall-runoff models in Theano/PyMC3, and Jax/NumPyro, allowing for the application of automatic differentiation techniques. A simulation experiment assessing the viability of these methods for recovering underlying temporal variation in parameters indicates that a discrete Gaussian random walk-based prior appears to be best suited across a range of scenarios while the continuous-time Gaussian process compares relatively poorly. In an analysis of 20 years of daily real-world streamflow records from the Model Parameter Estimation Experiment (MOPEX) hydrology data set with increasing data sparsity and at multiple temporal resolutions, we find that TVP inference with gradient-based MCMC is a flexible and reliable approach for analyzing hydrology models with dynamic parameter sets.
Examining AGN UV/Optical Variability beyond the Simple Damped Random Walk
We present damped harmonic oscillator (DHO) light-curve modeling for a sample of 12,714 spectroscopically confirmed quasars in the Sloan Digital Sky Survey Stripe 82 region. DHO is a second-order continuous-time autoregressive moving-average process, which can be fully described using four independent parameters: a natural oscillation frequency (ω 0 ), a damping ratio (ξ), a characteristic perturbation timescale (τ perturb ), and an amplitude for the perturbing white noise (σ ϵ ). The asymptotic variability amplitude of a DHO process is quantified by σ DHO —a function of ω 0 , ξ, τ perturb , and σ ϵ . We find that both τ perturb and σ ϵ follow different dependencies with rest-frame wavelength (λ RF ) on either side of 2500 Å, whereas σ DHO follows a single power-law relation with λ RF . After correcting for wavelength dependence, σ DHO exhibits anticorrelations with both the Eddington ratio and the black hole mass, while τ perturb —with a typical value of days in the rest frame—shows an anticorrelation with the bolometric luminosity. Modeling active galactic nuclei (AGN) variability as a DHO offers more insight into the workings of accretion disks close to the supermassive black holes at the center of AGN. The newly discovered short-term variability (characterized by τ perturb and σ ϵ ) and its correlation with bolometric luminosity pave the way for new algorithms that will derive fundamental properties (e.g., Eddington ratio) of AGN using photometric data alone.
Optical variability of quasars with 20-yr photometric light curves
ABSTRACT We study the optical gri photometric variability of a sample of 190 quasars within the SDSS Stripe 82 region that have long-term photometric coverage during ∼1998−2020 with SDSS, PanSTARRS-1, the Dark Energy Survey, and dedicated follow-up monitoring with Blanco 4m/DECam. With on average ∼200 nightly epochs per quasar per filter band, we improve the parameter constraints from a Damped Random Walk (DRW) model fit to the light curves over previous studies with 10–15 yr baselines and ≲ 100 epochs. We find that the average damping time-scale τDRW continues to rise with increased baseline, reaching a median value of ∼750 d (g band) in the rest frame of these quasars using the 20-yr light curves. Some quasars may have gradual, long-term trends in their light curves, suggesting that either the DRW fit requires very long baselines to converge, or that the underlying variability is more complex than a single DRW process for these quasars. Using a subset of quasars with better-constrained τDRW (less than 20 per cent of the baseline), we confirm a weak wavelength dependence of τDRW∝λ0.51 ± 0.20. We further quantify optical variability of these quasars over days to decades time-scales using structure function (SF) and power spectrum density (PSD) analyses. The SF and PSD measurements qualitatively confirm the measured (hundreds of days) damping time-scales from the DRW fits. However, the ensemble PSD is steeper than that of a DRW on time-scales less than ∼ a month for these luminous quasars, and this second break point correlates with the longer DRW damping time-scale.
Polarization Angle Swings in Blazars Detected in the Millimeter-wave with the South Pole Telescope
We present the first systematic search for electric vector position angle (EVPA) swings in the millimeter-wave (mm-wave) emission of blazars, using five years of observations from the South Pole Telescope SPT-3G camera at 95 and 150 GHz, and investigate their connection to gamma-ray flares. Of the 168 bright sources in the ~1500 square degrees SPT-3G Main Field, eight have sufficient polarization signal-to-noise for reliable EVPA measurement, four of which have continuous Fermi LAT gamma-ray detections. We detect EVPA swings in all four gamma-ray-active blazars and in none of the remaining four, consistent with the established connection between EVPA swings and high-energy emission seen at optical wavelengths. The observed swing amplitudes and rotation rates are smaller than those found in optical studies, consistent with mm-wave EVPA variability being slower than at shorter wavelengths. Random walk simulations of the polarization angle using a multi-cell model fail to reproduce both the number, amplitude, and duration of the observed swings, suggesting that this mechanism is insufficient to explain the observed swing population. Analysis of EVPA variability on one-week timescales is consistent with the anti-correlation between polarization degree and EVPA rotation rate previously observed at optical wavelengths. Of the 14 detected swings, eight are within 30 days of a gamma-ray flare. While many individual swing-flare associations are found to have a very low probability of happening by chance, the full ensembles of 95 and 150 GHz swing time lags with respect to gamma-ray flares are found to be consistent with random coincidence.
Transient cycling of nitrogen, organic carbon and oxygen within the free-flowing Columbia River corridor: Linking exposure time dependent biogeochemical reactions to river stage fluctuations (Final project report)
The objective of this project was to develop next-generation techniques for representing the transformations of complex reaction networks like those impacting transient river-corridors, then apply them to develop upscaling tools. The project contributed significantly to the development of Lagrangian “mass transfer particle tracking” (MTPT) tools that are the first numerical framework that explicitly separates mixing and spreading processes; a crucial distinction for accurate representations of reaction rates across scales. The MTPT approach uses an operator splitting scheme where physical transport processes are simulated using classical random walk methods and each particle is treated as a “container” that carries with it any number of chemical species. Mixing between particles is simulated using a colocation-probability based mass transfer kernel and reactions are evaluated on each particle after mixing. The method is stable, accurate, and also allows the explicit modeling of residence time distributions of the individual containers of mass. The MTPT scheme allowed us to explore several research questions related to reactive transport in river corridors and complex reaction networks. We found that it is possible to quantify the exposure time distributions (ETDs) of reactants and that this does suggest some pathways to upscaling. However, comparable approximations can be obtained using simplified (reduced dimensionality) MTPT simulations in less time with similar uncertainty, so development of extensive ETD-based methods was not productive. The major findings are that i) the MTPT schemes are robust and highly accurate across a range of arbitrarily complex reactions, ii) the methods can be efficiently parallelized and the parallel performance characteristics of MTPT are predictable, and iii) the MTPT tools allow simultaneous tracking of residence time. The development of these research tools into efficient software packages continues. The methods are already available to the community because open-source, working examples have been included with all publications.