Buoyant Current Research Articles

Vertical convection regimes in a two-dimensional rectangular cavity: Prandtl and aspect ratio dependence

Vertical convection is the fluid motion that is induced by the heating and cooling of two opposed vertical boundaries of a rectangular cavity (see e.g. Wang et al., J. Fluid Mech., vol. 917, 2021, A6). We consider the linear stability of the steady two-dimensional flow reached at Rayleigh numbers of O( $10^8$ ). As a function of the Prandtl number, $Pr$ , and the height-to-width aspect ratio of the domain, $A$ , the base flow of each case is computed numerically and linear simulations are used to obtain the properties of the leading linear instability mode. Flow regimes depend on the presence of a circulation in the entire cavity, detachment of the thermal layer from the boundary or the corner regions and on the oscillation frequency relative to the natural frequency of oscillation in the stably temperature-stratified interior, allowing for the presence of internal waves or not. Accordingly, the regime is called slow or fast, respectively. Either the global circulation or internal waves in the interior may couple the top and bottom buoyancy currents, while their absence implies asymmetry in their perturbation amplitude. Six flow regimes are found in the range of $0.1 \leq Pr \leq 4$ and $0.5 \leq A \leq 2$ . For $Pr \lessapprox 0.4$ and $A>1$ , the base flow is driven by a large circulation in the entire cavity. For $Pr \gtrapprox 0.7$ , the thermal boundary layers are thin and the instability is driven by the motion along the wall and the detached boundary layer. A transition between these regimes is marked by a dramatic change in oscillation frequency at $Pr = 0.55 \pm 0.15$ and $A <2$ .

Automated FerryBox monitoring reveals the first recorded river induced crude oil seep transport to the Strait of Magellan in southern Patagonia

This study presents the first documented occurrence of a natural crude oil seep plume associated with river discharge along the Strait of Magellan in southern Patagonia in modern times. Between September and December 2022, hydrocarbon signals were detected using a crude oil sensor integrated into a FerryBox system that traversed the Strait of Magellan and several channels of southern Patagonia, covering approximately 510 km. The highest levels of crude oil signals were observed in the mid-basin of the Strait of Magellan. These signals exhibited a strong negative correlation with sea surface salinity, coinciding with the water discharge from the San Juan River. Notably, during periods of high river discharge, typically exceeding 15 m3 s−1, a distinct crude oil plume was detected moving towards the Magellan Strait. Conversely, when river discharge fell below this threshold, no noticeable crude oil signal was observed. As river discharge decreased and winds intensified during the austral summer, the crude oil signal gradually dissipated. This observation suggests that the dispersion of crude oil becomes limited during periods of low river discharge, as buoyant currents remain confined close to the coast. Historical records indicate that this seep has been releasing hydrocarbons into the Strait of Magellan for at least the past 120 years, implying a long history of chronic crude oil input into this relatively isolated region of the world. This finding shows the potential contribution to the understanding of marine ecosystems dynamics and potential pollutants in poorly studied regions through the use of automated monitoring FerryBox system, enabling both spatial and temporal high-resolution surveys.

Analysis of the first and second laws of thermodynamics for MHD two-phase natural convection of water/Fe2O3 ferro-nanofluids in a 3D baffled enclosure

The study of the flow of nanofluids (NFs) in the presence of external fields and the effect of these fields on the heat transfer rate of NFs is the subject of engineering and medical science. Magnetic fields are among the external fields applied to fluids. These fields receive a lot of attention in recent years due to their specific features and applications. This study conducts a numerical investigation of the natural convection of water/ Fe2O3 NFs in nanoparticle volume fractions of φ =0, 2, and 4% =0, 2, and 4% in a baffled l-shaped enclosure under a magnetic field. Calculations with different magnetic field intensities are examined. The natural convection is considered laminar and examined at different Rayleigh numbers. The effects of geometric shape changes (baffle displacement) on the heat transfer and entropy generation rates (S˙g) are also investigated. This study is mainly conducted to investigate the effect of the proposed parameters on the NF flow, heat transfer, average and local Nusselt numbers, the curves of velocity distribution and dimensionless temperature distribution, and S˙g. According to the results, increasing the Rayleigh number causes an increase in the temperature difference between hot and cold surfaces and intensification of the buoyancy currents. This increases local and average heat transfer and S˙g. Nuave increases by 59.20 and 38.168%, respectively, with an increase in the Rayleigh number from 104 to 105 and 106 in φ=4%. The results also showed that increasing the φ and magnetic field intensity decreases heat transfer and increases S˙g. So that in Ra=104 and 105, the S˙g increases from about 40 to 75 with increasing the φ and in Ra=106 the S˙g increases from about 8000 to 15,000. This can be attributed mainly to the reduction in the intensity of buoyancy currents. In general, any factor that increases heat transfer causes an increase in S˙g. The Nuave increases 4.64 and 9.29% 4.64 and 9.29% with increasing the φ from φ=0 to 2.4% in Ha= 40.

ВЛИЯНИЕ ВЕТРОВОГО АПВЕЛЛИНГА И СТОКА РЕКИ АМУР НА ТЕРМОХАЛИННУЮ СТРУКТУРУ ВОД У СЕВЕРО-ВОСТОЧНОГО ПОБЕРЕЖЬЯ ОСТРОВА САХАЛИН

The impact of the wind-driven upwelling and the Amur River discharge on the thermohaline water structure off the northeastern coast of Sakhalin was analyzed using hydrographic data аnd satellite information on the sea surface temperature distribution. The interaction of upwelling water and the coastal current depends on wind conditions and the Amur river runoff. Coastal upwelling develops at relatively low river runoff values (summer low water). The increase in the Amur River discharge during the summer flood enhances the interaction of the upwelling and the coastal buoyancy current. The Amur river runoff flows directly along the shore, which tears cold upwelling water off the coast. The advection of the cold upwelling water from the coast towards the open sea increases with the consecutive alternation of upwelling and downwelling events.

Mechanisms of Estuarine Salt‐Plug Formation by an Along‐Shelf Buoyant Current: A Numerical Model Approach

AbstractWhile estuarine salt plugs can develop worldwide in basins adjacent to buoyant coastal currents, their formation has been scarcely documented. This study aims to investigate a mechanism for salt‐plug formation that disregards evaporation processes but involves a buoyant coastal current modified by wind stresses. A numerical model, Delft3D, is used to study the salt‐plug formation in an idealized bay connected to the ocean by a single inlet. Inspired by recent observations, the numerical experiments simulate eight different scenarios of tidal and wind forcing under the influence of an along‐shelf buoyant current. Results show salt‐plug formation that induces inverse exchange at the inlet, with inflow at the surface and outflow underneath. This exchange circulation is modified by wind action. The persistence of the salt plug depends on tidal flushing and on wind forcing. Two numerical experiments with nonstationary buoyant currents and nonstationary winds indicate that: (a) a salt plug forms when a buoyant current is active on the shelf and traps salty water that enters the bay during times of buoyant current relaxation; (b) the presence of the buoyant current induces an inverse circulation at the inlet, modified by the wind action; and (c) onshore and then downwelling winds enhance the inverse circulation at the inlet, while offshore and then upwelling winds stall it. Bay flushing times increase due to the presence of the salt plug. This study represents an initial attempt to identify the role of wind and buoyant coastal currents on salt‐plug formation.

Influence of reservoir properties on the dynamics of a migrating current of carbon dioxide

Storage of carbon dioxide (CO2) in saline aquifers is a promising tool to stabilize the anthropogenic CO2 emissions. At the reservoir conditions, injected CO2 is buoyant with respect to the ambient fluid (brine) and spreads as a current laterally and toward the top cap rock of the aquifer, with the potential risk of a leakage into the upper aquifer layers. However, CO2 is partially soluble in brine and the resulting mixture (CO2 + brine) is denser than both starting fluids. This heavy mixture makes the configuration unstable, producing a convective flow that enhances the dissolution of CO2. Motivated by this geophysical problem, we analyze the influence of the porous medium properties on the evolution of a buoyant current that is weakly soluble with the ambient fluid. A time-dependent large-scale model [C. W. MacMinn et al., “Spreading and convective dissolution of carbon dioxide in vertically confined, horizontal aquifers,” Water Resour. Res. 48, W11516 (2012)] is used to analyze the evolution of the flow. In this work, we include additional physical effects to this model, and we investigate the role of horizontal confinement, anisotropy, and dispersion of the porous layer in the dynamics of the fluid injected. The effect of anisotropy and dispersion is accounted by changing the dissolution rate of CO2 in brine, which is obtained from experiments and Darcy simulations and represents a parameter for the model. Our results reveal that while the confinement has a remarkable effect on the long-term dynamics, i.e., on the lifetime of the current, anisotropic permeability and dispersion of the medium influence mainly the short-term behavior of the flow. Finally, we outline possible implications for the CO2 sequestration process.

Исследование формирования и распространения речного плюма Дуная на основе численного моделирования

The relevance of the studied circulation caused by the water river runoff deals with the anthropological impact on the ecological state of the shelf. River waters, entering into the sea, form mesoscale structures in the delta’s area, characterized by low salinity with a high level of suspended matter and dissolved organic matter. Such structures are called “plume” in modern literature. In this case, the inertial motion of the plume is free to form a rounded area or “bulge” like anticyclonic circulation type of flow. The purpose of this work is to investigate the propagation of freshened waters, formed by the river inflow, the formation of the waters hydrological structure regularities, dynamics of the buoyancy current, and thermohaline front formation on the base of numerical modeling. Numerical modeling is used to study the formation of a river plume and downward propagation of buoyancy current on the shelf without taking into account the tides forcing. A three-dimensional σ-coordinate numerical model was used, adapted for the shelf and estuaries. The calculations were carried out for a rectangular box area. The influence of changes in the main parameters of the runoff, the mouth geometry and wind forcing are considered to development of the plume and the alongshore propagation of buoyancy current. The obtained modeling results for the conditions of the Northwestern Black Sea shelf and the Danube discharge are consistent with the estimates of plume characteristics based on the archival hydrological observations data of water temperature and salinity. The results of this work can be used for further study of hydrological processes in the region of river mouths, the peculiarities of the plumes formation and evolution, assessment of the suspended matter, biogenic elements and microplastics transport in the sea and ocean coastal zones.

Internal tides as a major process in Amazon continental shelf fine sediment transport

The description of hydrodynamics associated with the extensive reef system on the shelf break adjacent to the Amazon River is still a challenge for ocean sciences. Despite the discharge of more than one billion tons of cohesive sediment per year, the outer continental shelf of the world's largest river presents very low concentrations of suspended sediment near the bottom and an absence of modern fine sediment deposits nearly one hundred kilometers before the shelf break. The offshore limit of the subaqueous delta consists of a sigmoidal clinoform standing between 40 and 70 m in depth, a depositional feature that cannot be explained solely by estuarine-like gravitational circulation. This paper aims to test the hypothesis that internal tides have a major role in the control of offshore fine sediment transport. For that, we implement a set of tridimensional, non-hydrostatic, and high-resolution (up to 2 m, vertical, and 2 km, horizontal) Delft3D models. The experiments showed that even disregarding river plume buoyancy, wind drag, superficial waves, and ocean currents, the exclusive interaction between barotropic tidal currents, bathymetry, and the stratification structure of the ocean is capable of generating asymmetrical current patterns compatible with modern deposition. The maximum shelf slope and the relative depth between the outer shelf and the pycnocline represent the main factors influencing the generation and shoreward propagation of internal tides. Over time, spring-neap cycles are eventually capable of reverting cross-shore subtidal transport tendencies, while seasonal variability in ocean stratification modulates the intensity of baroclinic processes.

Modeling the Dynamics of Small‐Scale River and Creek Plumes in Tidal Waters

AbstractThe fate of discharges from small rivers and mountainous streams are little studied relative to the dynamics of large river plumes. Flows from small watersheds are episodic, forming transient low‐salinity surface plumes that vary tidally due to interaction of outflow inertia with buoyancy and ambient tidal currents. An implementation of the Regional Ocean Modeling System (ROMS v3.0), a three‐dimensional, free‐surface, terrain‐following numerical model, is applied to small river outflows (≤10 m3 s−1) entering a tidal ocean, where they form small plumes of scale ~103 m or smaller. Analysis of the momentum balance points to three distinct zones: (i) an inertia‐driven near field, where advection, pressure gradient, and vertical stress divergence control the plume dynamics, (ii) a buoyancy‐driven midfield, where pressure gradient, lateral stress divergence, and rotational accelerations are dominant, and (iii) an advective far field, where local accelerations induced by ambient tidal currents determine the fate of the river/estuarine discharge. The response of these small tidal plumes to different buoyancy forcing and outflow rate is explored. With increasing buoyancy, plumes change from narrow/elongated, bottom‐attached flows to radially expanding, surface‐layer flows. Weak outflow promotes stratification within the plume layer, and with stronger outflow, the plume layer becomes thinner and well‐mixed. When compared with prototypical large‐river plumes in which Coriolis effects are important, (i) in these small plumes, there is no bulge and no coastal buoyancy current, that is, shore contact is mostly absent, and (ii) the plume is strongly influenced by ambient tidal currents, forming a tidal plume that is deflected upcoast/downcoast from the river mouth.

Laboratory experiments modeling the transport and deposition of sediments by glacial plumes rising under an ice shelf

Particles that descend from a buoyant current are carried back toward the source resulting in a sediment deposit whose depth decreases linearly with distance from the source beyond a recirculation zone. The experimental results are used to predict the deposit of sediments from glacial plumes occurring around Antarctica.

Heating of a water droplet on inclined transparent polydimethylsiloxane (PDMS) surface

Heat transfer from replicated hydrophobic and transparent PDMS surface to a water droplet is considered. The laser texturing of alumina surface is carried out to obtain hydrophobic surface and later the textured surface is replicated by polydimethylsiloxane (PDMS) via a solvent casting method. An experiment is carried out to assess the water droplet pinning on the replicated inclined PDMS surface while keeping the replicated surface at uniform temperature (308 K). The flow and temperature fields inside the inclined water droplet are simulated in line with the experimental conditions. The flow velocities inside the droplet are validated through the data obtained from the particle imaging velocimetry (PIV). It is found that the velocity predictions agree well with the PIV data. The droplet heating results in a circulation cell inside the droplet due to the Marangoni and the buoyancy currents. Increasing inclination angle of the PDMS replicated surface enhances the maximum velocity inside the droplet, which is more pronounced for the large size droplets. The Nusselt and the Bond numbers increase with the inclination angle of the PDMS surface.

The influence of runoff and wind on the dispersion patterns of suspended sediment in the Zhujiang (Pearl) River Estuary based on MODIS data

Cloud-free moderate-resolution imaging spectroradiometer (MODIS) images of the Zhujiang (Pearl) River Estuary (ZRE) taken between 2002 and 2012 are retrieved and used to study the spatial and temporal patterns of suspended sediment concentrations (SSCs) across the estuary under runoff, wind, and tropical storm conditions. Five typical dispersal patterns of suspended sediments in the estuary are defined: Case I shows generally low SSCs under low dynamics; Case II shows a river-dominant dispersal pattern of suspended sediments from the outlets, particularly from Modaomen, Jiaomen, Hengmen, and others; Case III shows wind-dominant dispersal of high SSCs derived from the west shoal and southwesterly transport under a strong NE wind; Case IV is the combination of relatively large runoff and wind; and Case V is caused by a strong tropical storm with high river discharge and wind, which is characterized by the high SSCs across the entire estuary that are transported eastward by winddriven and buoyancy currents outside the estuary. Runoff is a dominant factor that controls seasonal and annual SSC variations in the ZRE, with the area of high SSCs being largest in the summer and smallest in the spring. The correlation coefficients between the monthly averaged river-suspended sediment discharge and the area of the high SSCs are approximately 0.6. The wind power over the west shoal increases with a wind speed, which induces more sediment resuspension and shows a close relationship between the wind speed and high SSC area.

Anisotropic distribution of scalar in stably stratified turbulence

We derive the anisotropic mean-square scalar (or temperature variance) spectrum and the buoyancy (or heat) flux for a stably stratified turbulent fluid by assuming the buoyancy terms in the governing dynamical equations as perturbations of the otherwise isotropic dynamics. This allows us to investigate their anisotropic distribution across scales larger and smaller than the Ozmidov scale. The ensuing 1D vertical wavenumber temperature spectrum at large scales is found to be in good agreement with stratospheric observations. The polar plot of the angle-dependent expression for temperature variance signifies the enhanced anisotropy with decreasing Froude number. It also indicates that with increasing stratification, the anisotropy persists down to smaller and smaller scales. In addition, we also display various polar plots for the components of buoyancy current, showing their anisotropic distribution.

Experimental and computational investigation of a latent heat energy storage system with a staggered heat exchanger for various phase change materials

This work reports the operation of a Latent Heat Thermal Energy Storage system (LHTES) utilizing a staggered heat exchanger (HE) and using various organic Phase Change Materials (PCMs). In a LHTES test rig set measurements regarding energy storage and release were performed in the working temperature range of each Phase Change Material. Nominal melting temperatures of the PCMs used were 40–53 °C. Computational Fluid Dynamics (CFD) simulation was applied to follow the operation of the test rig. The test rig consisted of a compact insulated tank, filled with PCM, a staggered heat exchanger to supply or extract thermal energy by the PCM and a water pump to circulate water as a Heat Transfer Fluid (HTF). Different HTF flow rates affect charging (melting) and discharging (solidification) processes but more significant was the effect of heat transfer mechanisms occurring. The latter was confirmed by inserting buoyancy currents created due to convection in a CFD simulation program where melting time was reduced compared to the same conditions with only conduction occurring. The suggested LHTES configuration is a promising compact unit despite the PCMs thermal resistance and solidification hysteresis phenomena, as well as the heat transfer mechanism strongly affecting the energy storage process.

Meanders and eddy formation by a buoyant coastal current flowing over a sloping topography

Abstract. This study investigates the linear and non-linear instability of a buoyant coastal current flowing along a sloping topography. In fact, the bathymetry strongly impacts the formation of meanders or eddies and leads to different dynamical regimes that can both enhance or prevent the cross-shore transport. We use the Regional Ocean Modeling System (ROMS) to run simulations in an idealized channel configuration, using a fixed coastal current structure and testing its unstable evolution for various depths and topographic slopes. The experiments are integrated beyond the linear stage of the instability, since our focus is on the non-linear end state, namely the formation of coastal eddies or meanders, to classify the dynamical regimes. We find three non-linear end states, whose properties cannot be deduced solely from the linear instability analysis. They correspond to a quasi-stable coastal current, the propagation of coastal meanders, and the formation of coherent eddies. We show that the topographic parameter Tp, defined as the ratio of the topographic Rossby wave speed over the current speed, plays a key role in controlling the amplitude of the unstable cross-shore perturbations. This result emphasizes the limitations of linear stability analysis to predict the formation of coastal eddies, because it does not account for the non-linear saturation of the cross-shore perturbations, which is predominant for large negative Tp values. We show that a second dimensionless parameter, the vertical aspect ratio γ, controls the transition from meanders to coherent eddies. We suggest the use of the parameter space (Tp, γ) to describe the emergence of coastal eddies or meanders from an unstable buoyant current. By knowing the values of Tp and γ for an observed flow, which can be calculated from hydrological sections, we can identify which non-linear end state characterizes that flow – namely if it is quasi-stable, meanders, or forms eddies.

Water droplet mobility on a hydrophobic surface under a thermal radiative heating

Side heating of a water droplet on the hydrophobic surface is considered and the droplet internal fluidity is examined. An experiment is carried out to assess the droplet mobility on the hydrophobic surface when subjected to the heating load. Temperature and flow fields are simulated in line with the experimental conditions and fluid acceleration inside the droplet is predicted for various droplet sizes. Predictions of the flow field are validated via particle image velocimetry (PIV) measurements. The adhesion, inertia, and shear forces are determined for possible rolling off the droplet on the hydrophobic surface. The Bond and the Nusselt numbers are predicted and the influence of the droplet size on the Bond and Nusselt numbers are presented. It is found that the velocity predictions agree well with the PIV data. Two counter rotating circulation cells are formed inside the droplet due to the combination of the Marangoni and buoyancy currents. The Bond number remains less than unity (in the range 0.03–0.2) for all the droplet sizes incorporated in the study (15–100μL). This indicates that the Marangoni current dominates over the buoyancy current inside the droplet. The droplet fluid inertia force becomes greater than the adhesion and the fluid shear forces during the heating period; consequently, the droplet rolls off on the hydrophobic surface, which is also observed from the experiments. The Nusselt and the Bond Numbers increase with increasing droplet volume and reach maximum of 151 and 0.2 for the droplet size of 100μL.

Heat Transfer and Fluid Flow Characteristics in a Sessile Droplet on Oil-Impregnated Surface Under Thermal Disturbance

The present study examines the flow field and heat transfer inside a sessile droplet on oil-impregnated surface when subjected to a small temperature difference at the droplet–oil interface. Temperature and flow fields inside the droplet are predicted and the flow velocities predicted are validated through the data obtained from a particle image velocimetry (PIV). Several images of droplets in varying sizes are analyzed and the droplet geometric features and experimental conditions are incorporated in the simulations. A polycarbonate wafer is used to texture the surface via incorporating a solvent-induced crystallization method. Silicon oil is used for impregnation of the textured surfaces. It is found that two counter-rotating circulation cells are formed in the droplet because of the combined effect of the Marangoni and buoyant currents on the flow field. A new dimensionless number (Merve number (MN)) is introduced to assess the behavior of the Nusselt and the Bond numbers with the droplet size. The Merve number represents the ratio of the gravitational force over the surface tension force associated with the sessile droplet and it differs from the Weber number. The Nusselt number demonstrates three distinct behaviors with the Merve number; in which case, the Nusselt number increases sharply for the range 0.8 ≤ MN ≤ 1. The Bond number increases with increasing the Merve number, provided that its values remain less than unity, which indicates the Marangoni current is dominant in the flow field.

Effect of Hall Current on MHD Oscillatory Slip Flow with Varying Temperature and Concentration

An investigation of the effect of Hall current on an unsteady MHD mixed convective oscillatory flow of an electrically conducting fluid through a planar channel filled with saturated porous medium is carried out in this paper. The effect of buoyancy, heat source, thermal radiation, chemical reaction and Hall current are taken into account with slip velocity, varying temperature and concentration at the lower boundary. A series solution is found using perturbation techniques. The effects of various parameters on the main and cross flow velocity, temperature, Skin friction, rate of heat and mass transfer are discussed numerically.Â

Modeling the tidal and sub-tidal hydrodynamics in a shallow, micro-tidal estuary

The three-dimensional hydrodynamics of Galveston Bay were simulated in two periods of several month duration. The physical setting of Galveston Bay is described by synthesis of long-term observations. Several processes in addition to tidal hydrodynamics and baroclinic circulation processes contribute substantially to the observed variability of currents, water level and salinity. The model was therefore forced with realistic water levels, river discharges, winds, coastal buoyancy currents (due to the Mississippi River plume) and surface heat fluxes. Quantitative metrics were used to evaluate model performance against observations and both spatial and temporal variability in tidal and sub-tidal hydrodynamics were generally well represented by the model. Three different unstructured meshes were tested, a triangular mesh that under-resolved the shipping channel, a triangular mesh that resolved it, and a mixed quadrilateral-triangular grid with approximately equivalent resolution. It is shown that salinity and sub-tidal velocity are better predicted when the important topographic features, such as the shipping channel, are resolved. It was necessary to increase the seabed drag roughness in the mixed quadrilateral-triangular grid simulation to attain similar performance to the equivalent triangular mesh.

Simulating surface oil transport during the Deepwater Horizon oil spill: Experiments with the BioCast system

The U.S. Naval Research Laboratory (NRL) is developing nowcast/forecast software systems designed to combine satellite ocean color data streams with physical circulation models in order to produce prognostic fields of ocean surface materials. The Deepwater Horizon oil spill in the Gulf of Mexico provided a test case for the Bio-Optical Forecasting (BioCast) system to rapidly combine the latest satellite imagery of the oil slick distribution with surface circulation fields in order to produce oil slick transport scenarios and forecasts. In one such sequence of experiments, MODIS satellite true color images were combined with high-resolution ocean circulation forecasts from the Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS®) to produce 96-h oil transport simulations. These oil forecasts predicted a major oil slick landfall at Grand Isle, Louisiana, USA that was subsequently observed. A key driver of the landfall scenario was the development of a coastal buoyancy current associated with Mississippi River Delta freshwater outflow. In another series of experiments, longer-term regional circulation model results were combined with oil slick source/sink scenarios to simulate the observed containment of surface oil within the Gulf of Mexico. Both sets of experiments underscore the importance of identifying and simulating potential hydrodynamic conduits of surface oil transport. The addition of explicit sources and sinks of surface oil concentrations provides a framework for increasingly complex oil spill modeling efforts that extend beyond horizontal trajectory analysis.