Lagrangian Trajectories Research Articles

Quartic polynomial approximation for fluctuations of separation of trajectories in chaos and correlation dimension

We consider the cumulant generating function of the logarithm of the distance between two infinitesimally close trajectories of a chaotic system. Its long-time behavior is given by the generalized Lyapunov exponent providing the logarithmic growth rate of the kth moment of the distance. The Legendre transform of is a large deviations function that gives the probability of rare fluctuations where the logarithmic rate of change of the distance is much larger or much smaller than the mean rate defining the first Lyapunov exponent. The only non-trivial zero of is at minus the correlation dimension of the attractor which for incompressible flows reduces to the space dimension. We describe here general properties constraining the form of and the Gallavotti–Cohen type relations that hold when there is symmetry under time-reversal. This demands studying joint growth rates of infinitesimal distances and volumes. We demonstrate that quartic polynomial approximation for does not violate the Marcinkiewicz theorem on invalidity of polynomial form for the generating function. We propose that this quartic approximation will fit many experimental situations, not having the effective time-reversibility and the short correlation time properties of the quadratic Grassberger–Procaccia estimates. We take the existing for turbulent channel flow and demonstrate that the quartic fit is nearly perfect. The violation of time-reversibility for the Lagrangian trajectories of the incompressible Navier–Stokes (NS) turbulence below the viscous scale is considered. We demonstrate how the fit can be used for finding the correlation dimensions of strange attractors via easily measurable quantities. We provide a simple formula via the Lyapunov exponents, holding in quadratic approximation, and describe the construction of the quartic approximation. A different approximation scheme for finding the correlation dimension from expansion in the flow compressibility is also provided.

Stochastic Closures for Wave\u2013Current Interaction Dynamics

Wave–current interaction (WCI) dynamics energizes and mixes the ocean thermocline by producing a combination of Langmuir circulation, internal waves and turbulent shear flows, which interact over a wide range of time scales. Two complementary approaches exist for approximating different aspects of WCI dynamics. These are the Generalized Lagrangian Mean (GLM) approach and the Gent–McWilliams (GM) approach. Their complementarity is evident in their Kelvin circulation theorems. GLM introduces a wave pseudomomentum per unit mass into its Kelvin circulation integrand, while GM introduces an additional ‘bolus velocity’ to transport its Kelvin circulation loop. The GLM approach models Eulerian momentum, while the GM approach models Lagrangian transport. In principle, both GLM and GM are based on the Euler–Boussinesq (EB) equations for an incompressible, stratified, rotating flow. The differences in their Kelvin theorems arise from differences in how they model the flow map in the Lagrangian for the Hamilton variational principle underlying the EB equations. A recently developed approach for uncertainty quantification in fluid dynamics constrains fluid variational principles to require that Lagrangian trajectories undergo Stochastic Advection by Lie Transport (SALT). Here, we introduce stochastic closure strategies for quantifying uncertainty in WCI by adapting the SALT approach to both the GLM and GM approximations of the EB variational principle. In the GLM framework, we introduce a stochastic group velocity for transport of wave properties, relative to the frame of motion of the Lagrangian mean flow velocity and a stochastic pressure contribution from the fluctuating kinetic energy. In the GM framework, we introduce a stochastic bolus velocity in addition to the mean drift velocity by imposing the SALT constraint in the GM variational principle.

Oil pollution in the Eastern Arabian Sea from invisible sources: A multi-technique approach

The Eastern Arabian Sea (EAS) is affected by oil pollution, as often evidenced by the presence of tarballs along the West Coast of India (WCI). Tarball samples collected during May 2017 along the Goa coast were subject to biomarker fingerprints, and the results matched with Bombay High (BH) oil fingerprints. The present study primarily aims at identifying the potential minor spill areas using Sentinel-imagery. Interestingly, repeated occurrence of oil spills detected at two locations, perfectly matched with BH platforms. The simulated Lagrangian trajectories also depict that tarball particles have originated from those detected locations. In 2017 alone, the quantity of spilled oil was estimated to be 129,392 l. However, spills detected offshore regions of Kachchh and Mangalore were found to be caused by ships. This is the first comprehensive study, tracking the oil pollution sources in the EAS through multi-technique approach - chemical, remote sensing and numerical modeling.

Topology and stochasticity of turbulent magnetic fields.

We present a mathematical formalism for the topology and stochasticity of vector fields based on renormalization group methodology. The concept of a scale-split energy density, ψ_{l,L}=B_{l}·B_{L}/2 for vector field B(x,t) renormalized at scales l and L, is introduced in order to quantify the notion of the field topological deformation, topology change, and stochasticity level. In particular, for magnetic fields, it is shown that the evolution of the field topology is directly related to the field-fluid slippage, which has already been linked to magnetic reconnection in previous work. The magnitude and direction of stochastic magnetic fields, shown to be governed, respectively, by the parallel and vertical components of the renormalized induction equation with respect to the magnetic field, can be studied separately by dividing ψ_{l,L} into two (3+1)-dimensional scalar fields. The velocity field can be approached in a similar way. Magnetic reconnection can then be defined in terms of the extrema of the L_{p} norms of these scalar fields. This formulation in fact clarifies different definitions of magnetic reconnection, which vaguely rely on the magnetic field topology, stochasticity, and energy conversion. Our results support the well-founded yet partly overlooked picture in which magnetic reconnection in turbulent fluids occurs on a wide range of scales as a result of nonlinearities at large scales (turbulence inertial range) and nonidealities at small scales (dissipative range). Lagrangian particle trajectories, as well as magnetic field lines, are stochastic in turbulent magnetized media in the limit of small resistivity and viscosity. The magnetic field tends to reduce its stochasticity induced by the turbulent flow by slipping through the fluid, which may accelerate fluid particles. This suggests that reconnection is a relaxation process by which the magnetic field lowers both its topological entanglements induced by turbulence and its energy level.

A high‐order parallel Eulerian‐Lagrangian algorithm for advection‐diffusion problems on unstructured meshes

SummaryIn this paper, we present a high‐order discontinuous Galerkin Eulerian‐Lagrangian method for the solution of advection‐diffusion problems onstaggered unstructured meshesin two and three space dimensions. The particle trajectories are tracked backward in time by means of ahigh‐orderrepresentation of the velocity field and a linear mapping from the physical to a reference system, hence obtaining a very simple and efficient strategy that permits to follow the Lagrangian trajectories throughout the computational domain. The use of an Eulerian‐Lagrangian discretization increases the overall computational efficiency of the scheme because it is the only explicit method for the discretization of convective terms that admits large time steps without imposing a Courant‐Friedrichs‐Lewy–type stability condition. This property is fully exploited in this work by relying on asemi‐implicitdiscretization of the incompressible Navier‐Stokes equations, in which the pressure is discretized implicitly; thus, the sound speed does not play any role in the restriction of the maximum admissible time step. The resulting mild Courant‐Friedrichs‐Lewy stability condition, which is based only on the fluid velocity, is here overcome by the adoption of the Eulerian‐Lagrangian method for the advection terms and an implicit scheme for the diffusive part of the governing equations. As a consequence, the novel algorithm is able to run simulation with a time step that is defined by the user, depending on the desired efficiency and time scale of the physical phenomena under consideration. Finally, a completeMessage Passing Interface parallelizationof the code is presented, showing that our approach can reach up to 96% of scaling efficiency.

The utility of near-surface water vapor deuterium excess as an indicator of atmospheric moisture source

This study explores the utility of observations of deuterium excess (dx) of water vapor for attribution of vapor to remote source regions and local influences. A Lagrangian back trajectory model was combined with various parameterizations of the dx of ocean evaporation and land evapotranspiration to simulate daily vapor dx at a continental site, a marine site and a coastal site. The model reproduced reasonably well the observed variabilities in the vapor dx at the coastal and the marine site when water vapor at these sites were primarily influenced by large-scale advection from ocean sources. The simple parameterization relating dx of the ocean evaporation to relative humidity is a robust representation of the ocean isotopic evaporation. On the other hand, the model did poorly for the continental site and during the land evapotranspiration dominated months at the coastal and the marine site, confirming the published findings that the water vapor dx near the Earth’s surface can be significantly altered by land evapotranspiration and therefore is not a conserved tracer of humidity from the marine moisture source region. Several parameterizations for the dx of land evapotranspiration suggested by previous studies were tested with the trajectory model, but none brought improvement to the simulation of the dx at the continental site. Our study emphasizes that a fundamental challenge in isotopy hydrology is the lack of understanding of the different fractionation processes of 18O and D associated with land evapotranspiration.

New Insights of the Sicily Channel and Southern Tyrrhenian Sea Variability

The dynamics of the Sicily Channel and the southern Tyrrhenian Sea are highly influenced by the seasonal variability of the Mediterranean basin-wide circulation, by the interannual variability of the numerous mesoscale structures present in the Channel, and by the decadal variability of the adjacent Ionian Sea. In the present study, all these aspects are investigated using in-situ (Lagrangian drifter trajectories and Argo float profiles) and satellite data (Absolute Dynamic Topography, Sea Level Anomaly, Sea Surface Temperature, wind products) over the period from 1993 to 2018. The availability of long time series of data and high-resolution multi-sensor surface currents allow us to add new details on the circulation features and on their driving mechanisms and to detect new permanent eddies not yet described in literature. The structures prevailing in winter are mainly driven by wind, whereas those prevailing in summer are regulated by topographical forcing on surface currents. The strength of the surface structures located at the western entrance of the Ionian Sea and of the mesoscale activity along the northern Sicily coast is modulated by the large-scale internal variability. The vertical hydrological characteristics of these mesoscale eddies are delineated using the Argo float profiles inside these structures.

Bayesian inference of ocean diffusivity from Lagrangian trajectory data

A Bayesian approach is developed for the inference of an eddy-diffusivity field from Lagrangian trajectory data. The motion of Lagrangian particles is modelled by a stochastic differential equation associated with the advection–diffusion equation. An inference scheme is constructed for the unknown parameters that appear in this equation, namely the mean velocity, velocity gradient, and diffusivity tensor. The scheme provides a posterior probability distribution for these parameters, which is sampled using the Metropolis–Hastings algorithm. The approach is applied first to a simple periodic flow, for which the results are compared with the prediction from homogenisation theory, and then to trajectories in a three-layer quasigeostrophic double-gyre simulation. The statistics of the inferred diffusivity tensor are examined for varying sampling interval and compared with a standard diagnostic of ocean diffusivity. The Bayesian approach proves capable of estimating spatially-variable anisotropic diffusivity fields from a relatively modest amount of data while providing a measure of the uncertainty of the estimates.

The transport and decay of radioactive aerosols in a wall-bounded turbulent flow

This paper aims at radiation monitoring and activity inversion when radionuclides are transported in duct or a pipe. The Lagrangian trajectory model was established, and a quick numerical method was used to simulate radioactive aerosols transport and radioactive decay in a wall-bounded flow. Total 12 cases were studied in the simulation according to the particle diameter and density. The dimensionless deposition velocity has been validated by comparing with Wood’s predictions and the experienced gravitational settling velocity. The results show that turbophoresis has a significant effect on particle transport in the wall-bounded flow, leading to particles migration and concentrating near the wall or channel center. It will cause particles to move with different velocities in the streamwise direction, so that their residence times in channel will be complexly distributed. In addition, the gravity settling will enhance the disequilibrium of particle distribution. The activity losses are obtained in detail including particle deposition ratio and radioactive decay. One radionuclide with half-life 7 s was employed in simulation, and the activity error between the estimated and initial values is up to about 40%. The activity error decreases if the half-life becomes large. When the half-life is 7.5 and 0.5 times larger than the mean time spent by airflow through the channel if the gravity is considered or not respectively, the final activity error will be less than ±5% in the present simulation setup.

Sensitivity of vortex pairing and mixing to initial perturbations in stratified shear flows

The effects of different initial perturbations on the evolution of stratified shear flows that are subject to Kelvin-Helmholtz instability and vortex pairing have been investigated through Direct Numerical Simulation (DNS). The effects of purely random perturbations of the background flow are sensitive to the phase of the subharmonic component of the perturbation that has a wavelength double that of the Kelvin-Helmholtz instability. If the phase relationship between the Kelvin-Helmholtz mode and its subharmonic mode is optimal, or close to it, vortex pairing occurs. Vortex paring is delayed when there is a phase difference, and this delay increases with increasing phase difference. In three dimensional simulations vortex pairing is suppressed if the phase difference is sufficiently large, reducing the amount of mixing and mixing efficiency. For a given phase difference close enough to the optimal phase, the response of the flow to eigenvalues perturbations is very similar to the response to random perturbations. In addition to traditional diagnostics, we show quantitatively that a non-modal Fourier component in a random perturbation quickly evolves to be modal and describe the process of vortex pairing using Lagrangian trajectories.

Inertial impaction of particles on a circular cylinder for a wide range of Reynolds and P numbers: A comparative study

In this paper, particulate gas flows over a circular cylinder were studied numerically for wide ranges of flow Reynolds numbers (5 < Re < 5 × 106) and P numbers (0.025 < P < 5 × 104). P number is defined as 9Re/S and S is the particle-fluid density ratio. For Re ≤ 100, the laminar flow simulations were performed while the turbulent flow simulations were done for Re ≥ 103 using unsteady Reynolds-averaged Navier-Stokes (URANS) equations. The Reynolds stress model (RSM) was used to account for the anisotropic turbulent stresses around the cylinder. The computed drag was compared with the available experimental data and earlier numerical results. It was shown that the RSM could properly predict the drag-drop at the critical flow regime. Deposition of particles on the cylinder surface was also investigated using the Lagrangian trajectory analysis. Particle capture efficiencies were evaluated and compared with the available experimental data and theoretical models. The simulation results indicated that for P ≥ 102, the computed capture efficiencies obtained by the viscous flow analysis coincided with those obtained from the inviscid flows. For 1 ≤ P ≤ 10, differences between viscous and inviscid flow collection efficiencies, however, were more than 20% when Stk≤1. However, for P ≤ 0.1, differences between the collection efficiencies obtained by viscous and inviscid flows exceeded 20% when Stk≤2.2. In the case of Re = 103 and P = 10, the highest capture efficiency was obtained by impaction for Stk≥0.5.

Combining domain filling with a self-organizing map to analyze multi-species hydrocarbon signatures on a regional scale.

For the period of the Barnett Coordinated Campaign, October 16-31, 2013, hourly concentrations for 46 volatile organic compounds (VOCs) were recorded at 14 air monitoring stations within the Barnett Shale of North Texas. These measurements are used to identify and analyze multi-species hydrocarbon signatures on a regional scale through the novel combination of two techniques: domain filling with Lagrangian trajectories and the machine learning unsupervised classification algorithm called a self-organizing map (SOM). This combination of techniques is shown to accurately identify concentration enhancements in the lightest measured alkane species at and downwind of the locations of active-permit oil and gas facilities, despite the model having no a priori knowledge of these source locations. Site comparisons further identify the SOM's ability to distinguish between signatures with differing influences from oil- and gas-related processes and from urban processes. A random forest (a machine learning supervised classification) analysis is conducted to further probe the sensitivities of the SOM classification in response to changes in any hydrocarbon species' concentration values. The random forest analysis of four representative classes finds that the SOM classification is appropriately more sensitive to changes in certain urban-related species for urban-related classes, and to changes in oil- and gas-related species for oil- and gas-related classes.

Network measures of mixing.

Transport and mixing processes in fluid flows can be studied directly from Lagrangian trajectory data, such as those obtained from particle tracking experiments. Recent work in this context highlights the application of graph-based approaches, where trajectories serve as nodes and some similarity or distance measure between them is employed to build a (possibly weighted) network, which is then analyzed using spectral methods. Here, we consider the simplest case of an unweighted, undirected network and analytically relate local network measures such as node degree or clustering coefficient to flow structures. In particular, we use these local measures to divide the family of trajectories into groups of similar dynamical behavior via manifold learning methods.

Assessing Lagrangian inverse modelling of urban anthropogenic CO2 fluxes using in situ aircraft and ground-based measurements in the Tokyo area

BackgroundIn order to use in situ measurements to constrain urban anthropogenic emissions of carbon dioxide (CO2), we use a Lagrangian methodology based on diffusive backward trajectory tracer reconstructions and Bayesian inversion. The observations of atmospheric CO2 were collected within the Tokyo Bay Area during the Comprehensive Observation Network for TRace gases by AIrLiner (CONTRAIL) flights, from the Tsukuba tall tower of the Meteorological Research Institute (MRI) of the Japan Meteorological Agency and at two surface sites (Dodaira and Kisai) from the World Data Center for Greenhouse Gases (WDCGG).ResultsWe produce gridded estimates of the CO2 emissions and calculate the averages for different areas within the Kanto plain where Tokyo is located. Using these inversions as reference we investigate the impact of perturbing different elements in the inversion system. We modified the observations amount and location (surface only sparse vs. including aircraft CO2 observations), the background representation, the wind data used to drive the transport model, the prior emissions magnitude and time resolution and error parameters of the inverse model.ConclusionsOptimized fluxes were consistent with other estimates for the unperturbed simulations. Inclusion of CONTRAIL measurements resulted in significant differences in the magnitude of the retrieved fluxes, 13% on average for the whole domain and of up to 21% for the spatiotemporal cells with the highest fluxes. Changes in the background yielded differences in the retrieved fluxes of up to 50% and more. Simulated biases in the modelled transport cause differences in the retrieved fluxes of up to 30% similar to those obtained using different meteorological winds to advect the Lagrangian trajectories. Perturbations to the prior inventory can impact the fluxes by ~ 10% or more depending on the assumptions on the error covariances. All of these factors can cause significant differences in the estimated flux, and highlight the challenges in estimating regional CO2 fluxes from atmospheric observations.

Cloud droplet growth in shallow cumulus clouds considering 1-D and 3-D thermal radiative effects

Abstract. The effect of 1-D and 3-D thermal radiation on cloud droplet growth in shallow cumulus clouds is investigated using large eddy simulations with size-resolved cloud microphysics. A two-step approach is used for separating microphysical effects from dynamical feedbacks. In step one, an offline parcel model is used to describe the onset of rain. The growth of cloud droplets to raindrops is simulated with bin-resolved microphysics along previously recorded Lagrangian trajectories. It is shown that thermal heating and cooling rates can enhance droplet growth and raindrop production. Droplets grow to larger size bins in the 10–30 µm radius range. The main effect in terms of raindrop production arises from recirculating parcels, where a small number of droplets are exposed to strong thermal cooling at cloud edge. These recirculating parcels, comprising about 6 %–7 % of all parcels investigated, make up 45 % of the rain for the no-radiation simulation and up to 60 % when 3-D radiative effects are considered. The effect of 3-D thermal radiation on rain production is stronger than that of 1-D thermal radiation. Three-dimensional thermal radiation can enhance the rain amount up to 40 % compared to standard droplet growth without radiative effects in this idealized framework. In the second stage, fully coupled large eddy simulations show that dynamical effects are stronger than microphysical effects, as far as the production of rain is concerned. Three-dimensional thermal radiative effects again exceed one-dimensional thermal radiative effects. Small amounts of rain are produced in more clouds (over a larger area of the domain) when thermal radiation is applied to microphysics. The dynamical feedback is shown to be an enhanced cloud circulation with stronger subsiding shells at the cloud edges due to thermal cooling and stronger updraft velocities in the cloud center. It is shown that an evaporation–circulation feedback reduces the amount of rain produced in simulations where 3-D thermal radiation is applied to microphysics and dynamics, in comparison to where 3-D thermal radiation is only applied to dynamics.

Statistical Properties of Small Particle Trajectories in a Fully Developed Turbulent State in He-II

Lagrangian trajectories of small particles in a fully developed turbulent state are studied in a rectangular duct. A plate heater is attached on the bottom to generate the thermal counter flow. The bath temperature is changed from 1.7 to 2.1 K and is controlled within 0.1 mK. Small particles made of solid hydrogen are visualized by high-speed camera, and their trajectories are recorded. Their motions indicate complex features depending not only on bath temperature and heater power, but also on particle size. In this study, the Hurst exponent is defined by $$|\mathbf x (t+\tau )-\mathbf x (t)| \propto \tau ^H$$ , where $$\mathbf x (t)$$ denotes the particle position at time t. It was found that there is a characteristic timescale $$\tau _0$$ . For small time separation, $$\tau \le \tau _0$$ , the exponent H is small. However, for large time separation, $$\tau _0 \ll \tau $$ , H is nearly 1.

Climatology and dynamics of the link between dry intrusions and cold fronts during winter. Part I: global climatology

Cold fronts are a primary feature of the day-to-day variability of weather in the midlatitudes, and feature in conceptual extratropical cyclone models alongside the dry intrusion airstream. Here the climatological frequency and spatial distribution of the co-occurrence of these two features are quantified, and the differences in cold front characteristics (intensity, size, and precipitation) when a dry intrusion is present or not are calculated. Fronts are objectively identified in the ECMWF ERA-Interim dataset for the winter seasons in each hemisphere and split into three sub-types: central fronts (within a cyclone area); trailing fronts (outwith the cyclone area but connected to a central front); and isolated fronts (not connected to a cyclone). These are then associated with dry intrusions identified using Lagrangian trajectory analysis. Trailing fronts are most likely to be associated with a DI in both hemispheres, and this occurs more frequently in the western parts of the major storm track regions. Isolated fronts are linked to DIs more frequently on the eastern ends of the storm tracks, and in the subtropics. All front types, when co-occurring with a DI, are stronger in terms of their temperature gradient, are much larger in area, and typically have higher average precipitation. Therefore, climatologically the link with DIs increases the impact of cold fronts. There are some differences in the statistics of the precipitation for trailing and isolated fronts that are further investigated in Part II of this study from the front-centred perspective.

The Euler Equations in Planar Domains with Corners

When the velocity field is not a priori known to be globally almost Lipschitz, global uniqueness of solutions to the two-dimensional Euler equations has been established only in some special cases, and the solutions to which these results apply share the property that the diffuse part of the vorticity is constant near the points where the velocity is insufficiently regular. Assuming that the latter holds initially, the challenge is then to propagate this property along the Euler dynamic via an appropriate control of the Lagrangian trajectories. In domains with obtuse corners and sufficiently smooth elsewhere, Yudovich solutions fail to be almost Lipschitz only near these corners, and we investigate necessary and sufficient conditions for the vorticity to remain constant there. We show that if the vorticity is initially constant near the whole boundary, then it remains such forever (and global weak solutions are unique), provided no corner has angle greater than $\pi$. We also show that this fails in general for domains that do have such corners.

Assessing the Transport of Volcanic Aerosol in the Stratosphere over Tomsk and Vladivostok from Lidar Data

The transport of volcanic aerosol in the atmosphere after the eruptions of the Grimsvotn and Nabro volcanoes in 2011 is analyzed using the method of Lagrangian particle trajectories. It was impossible to identify volcanic aerosol after the Grimsvotn eruption using data of lidar observations over Tomsk and Vladivostok against the existing background aerosol. At that time there was strong horizontal mixing in the Northern Hemisphere atmosphere. Volcanic aerosol formed after the Nabro eruption was clearly manifested in the form of aerosol scattering peaks over Vladivostok and Tomsk. This is proved by data of the CALIPSO space lidar and by the satellite observations of sulfur dioxide with GOME-2. The dynamics of the eruptive aerosol cloud formation over the Northern Hemisphere is traced.

Formation, Development, and Propagation of a Rare Coastal Coccolithophore Bloom

AbstractThis study examines an unprecedented bloom of Emiliania huxleyi along the California coast during the NE Pacific warm anomaly of 2014–2015. Observations of coccolithophore populations from microscopy and flow cytometry, surface current data derived from high‐frequency radar, and satellite ocean color imagery were used to track the population dynamics of the bloom in the Santa Barbara Channel. Results show a coastal bloom of mostly E. huxleyi that reached cell concentrations up to 5.7 × 106 cells per liter and a maximum spatial extent of 1,220 km2. We speculate that the rare cooccurrence of warm water, high water column stability, and an extensive preceding diatom bloom during the anomaly contributed to the development of this bloom. Flow cytometry measurements provided insight on the phases of bloom development (e.g., growth versus senescence) with calcified cells comprising up to 64% of particles containing chlorophyll a and detached‐coccolith:cell ratios ranging from 10 to &gt;100. Lagrangian particle trajectories estimated during two nonoverlapping 48‐ and 72‐hr periods showed the changes in the surface structure of the bloom due to advection by surface currents and nonconservative biological and physical processes. Time rates of change of particulate inorganic carbon were estimated along particle trajectories, with rates ranging from −4 to 6 μmol·L−1·day−1. The approach presented here is likely to be useful for understanding the evolution of coastal phytoplankton bloom events in a general setting.