Lagrangian Scale Research Articles

Large-scale convective flow sustained by thermally active Lagrangian tracers

Non-isothermal particles suspended in a fluid lead to complex interactions – the particles respond to changes in the fluid flow, which in turn is modified by their temperature anomaly. Here, we perform a novel proof-of-concept numerical study based on tracer particles that are thermally coupled to the fluid. We imagine that particles can adjust their internal temperature reacting to some local fluid properties, and follow simple, hard-wired active control protocols. We study the case where instabilities are induced by switching the particle temperature from hot to cold depending on whether it is ascending or descending in the flow. A macroscopic transition from stable to unstable convective flow is achieved, depending on the number of active particles and their excess negative/positive temperature. The stable state is characterized by a flow with low turbulent kinetic energy, strongly stable temperature gradient, and no large-scale features. The convective state is characterized by higher turbulent kinetic energy, self-sustaining large-scale convection, and weakly stable temperature gradients. Individually, the particles promote the formation of stable temperature gradients, while their aggregated effect induces large-scale convection. When the Lagrangian temperature scale is small, a weakly convective laminar system forms. The Lagrangian approach is also compared to a uniform Eulerian bulk heating with the same mean injection profile, and no such transition is observed. Our empirical approach shows that thermal convection can be controlled by pure Lagrangian forcing, and opens the way for other data-driven particle-based protocols to enhance or deplete large-scale motion in thermal flows.

University of Warsaw Lagrangian Cloud Model (UWLCM) 2.0: adaptation of a mixed Eulerian–Lagrangian numerical model for heterogeneous computing clusters

Abstract. A numerical cloud model with Lagrangian particles coupled to an Eulerian flow is adapted for distributed memory systems. Eulerian and Lagrangian calculations can be done in parallel on CPUs and GPUs, respectively. The fraction of time when CPUs and GPUs work simultaneously is maximized at around 80 % for an optimal ratio of CPU and GPU workloads. The optimal ratio of workloads is different for different systems because it depends on the relation between computing performance of CPUs and GPUs. GPU workload can be adjusted by changing the number of Lagrangian particles, which is limited by device memory. Lagrangian computations scale with the number of nodes better than Eulerian computations because the former do not require collective communications. This means that the ratio of CPU and GPU computation times also depends on the number of nodes. Therefore, for a fixed number of Lagrangian particles, there is an optimal number of nodes, for which the time CPUs and GPUs work simultaneously is maximized. Scaling efficiency up to this optimal number of nodes is close to 100 %. Simulations that use both CPUs and GPUs take between 10 and 120 times less time and use between 10 to 60 times less energy than simulations run on CPUs only. Simulations with Lagrangian microphysics take up to 8 times longer to finish than simulations with Eulerian bulk microphysics, but the difference decreases as more nodes are used. The presented method of adaptation for computing clusters can be used in any numerical model with Lagrangian particles coupled to an Eulerian fluid flow.

Convective rain cell properties and the resulting precipitation scaling in a warm‐temperate climate

AbstractConvective precipitation events have been shown to intensify at rates exceeding the Clausius–Clapeyron rate (CC rate) of ca. 7% K−1 under current climate conditions. In this study, we relate atmospheric variables (low‐level dew point temperature, convective available potential energy, and vertical wind shear), which are regarded as ingredients for severe deep convection, to properties of convective rain cells (cell area, maximum precipitation intensity, lifetime, precipitation sum, and cell speed). The rain cell properties are obtained from a rain gauge‐adjusted radar dataset in a mid‐latitude region, which is characterized by a temperate climate with warm summers (Germany). Different Lagrangian cell properties scale with dew point temperature at varying rates. While the maximum precipitation intensity of cells scales consistently at the CC rate, the area and precipitation sum per cell scale at varying rates above the CC rate. We show that this super‐CC scaling is caused by a covarying increase of convective available potential energy with dew point temperature. Wind shear increases the precipitation sum per cell mainly by increasing the spatial cell extent. From a Eulerian point of view, this increase is partly compensated by a higher cell velocity, which leads to Eulerian precipitation scaling rates close to and slightly above the CC rate. Thus, Eulerian scaling rates of convective precipitation are modulated by convective available potential energy and vertical wind shear, making it unlikely that present scaling rates can be applied to future climate conditions. Furthermore, we show that cells that cause heavy precipitation at fixed locations occur at low vertical wind shear and, thus, move relatively slowly compared to typical cells.

Scale-invariant spins and tumbles in turbulence

A Lagrangian perspective has yielded many new insights in our quest to reveal the intricacies of turbulent flows. Much of this progress has been possible by following the trajectories of idealised, inertialess objects (tracers) traversing through the flow. Their spins and tumbles provide a glimpse into the underlying local velocity gradients of the turbulent field. While it is known that the spinning and tumbling rates of anisotropic particles are modified in turbulence – compared with those in a random flow field – a quantitative explanation for this has remained elusive. Now, Pujara et al. (J. Fluid Mech., vol. 922, 2021, R6) have made an attempt to predict the split between spinning and tumbling rates by accessing the particle's alignment with the local vorticity. Their analysis of filtered turbulent fields reveals a Lagrangian scale invariance, whereby key quantities relating to the particle's rotational statistics are preserved from the dissipative to the integral scale.

Towards a particle trajectory modelling approach in support of South African search and rescue operations at sea

ABSTRACT The ability to provide rapid decision support and more precise search area coordinates for rescuers to conduct search and rescue operations at sea are of high impact value for marine and maritime stakeholders. Search and rescue operations rely on accurate information about metocean conditions to locate objects in the ocean. These include local knowledge, operational ocean and wind forecasts and empirical drift relationships between ocean currents, ocean surface winds and the objects being searched for. To provide more accurate decision support for rescuers looking for persons or objects lost at sea, a virtual particle tracking tool was combined with an empirical Leeway drift model. The Lagrangian Ocean Search Targets (LOST) application builds on a Lagrangian ocean analysis framework which has been adapted to provide real-time estimates of the positions of objects based on operational ocean and wind forecasts. LOST incorporates the impact of ocean currents, surface winds and stochastic motion, the latter being critical in accounting for sub-grid scale processes that are not resolved in the ocean and wind forecasts. This study assesses the accuracy of LOST, demonstrating its feasibility as a decision support tool for search and rescue operations by applying it to three use cases in the South African regional ocean. These use cases are real-life scenarios that highlight the value of combining state-of-the-art ocean and wind forecasting systems with Lagrangian ocean analyses frameworks and sub-grid scale parameterisation to support global operational oceanography.

A Lagrangian-to-Eulerian Metric to Identify Estuarine Pelagic Habitats

Estuaries are among the world’s most productive ecosystems, but recent natural and anthropogenic changes have stressed these ecosystems. Tools to assess estuarine pelagic habitats are important to support and maintain healthy ecosystem function. In this work, we demonstrate that estuarine pelagic habitats can be identified by a simple ratio, termed the LE ratio, that takes into account the tidal excursion along a channel (a Lagrangian length scale) and the distance along that channel (an Eulerian length scale). To develop and assess this concept, numerical simulations of the 1D advection–dispersion equation of a conservative tracer and tidal excursion estimates based on data were used to formulize a conceptual model and to define exchange zones within a tidal channel. This conceptual model was then used to predict the extent of pelagic habitats in a terminal channel network in the Sacramento–San Joaquin Delta. Exchange zones mapped onto these channels were found to be in good agreement with independent estimates of residence time. Sensitivity analyses of the numerical model suggest that productive pelagic habitats can be expanded by a factor of 2 by either increasing dispersion or increasing spring–neap variability in mean tidal velocity. Such changes can also enhance flushing in upper channel reaches. These findings are relevant for tidal marsh restoration projects that aim to expand beneficial aquatic habitats by varying exchange or residence time over the spring–neap cycle, because this variability may interact synergistically with varying rates of phytoplankton growth due to spatiotemporal changes in environmental conditions.

Hydrodynamic Dispersion in Porous Media and the Significance of Lagrangian Time and Space Scales

Transport in porous media is critical for many applications in the environment and in the chemical process industry. A key parameter for modeling this transport is the hydrodynamic dispersion coefficient for particles and scalars in a porous medium, which has been found to depend on properties of the medium structure, on the dispersing compound, and on the flow field characteristics. Previous studies have resulted in suggestions of different equation forms, showing the relationship between the hydrodynamic dispersion coefficient for various types of porous media in various flow regimes and the Peclet number. The Peclet number is calculated based on a Eulerian length scale, such as the diameter of the spheres in packed beds, or the pore diameter. However, the nature of hydrodynamic dispersion is Lagrangian, and it should take the molecular diffusion effects, as well as the convection effects, into account. This work shifts attention to the Lagrangian time and length scales for the definition of the Peclet number. It is focused on the dependence of the longitudinal hydrodynamic dispersion coefficient on the effective Lagrangian Peclet number by using a Lagrangian length scale and the effective molecular diffusivity. The lattice Boltzmann method (LBM) was employed to simulate flow in porous media that were constituted by packed spheres, and Lagrangian particle tracking (LPT) was used to track the movement of individual dispersing particles. It was found that the hydrodynamic dispersion coefficient linearly depends on the effective Lagrangian Peclet number for packed beds with different types of packing. This linear equation describing the dependence of the dispersion coefficient on the effective Lagrangian Peclet number is both simpler and more accurate than the one formed using the effective Eulerian Peclet number. In addition, the slope of the line is a characteristic coefficient for a given medium.

Turbulent Diffusion and Eddy Scales in Rivers

The relationship between turbulent diffusion with Eulerian and Lagrangian scales of turbulence in natural flows is considered. An estimate of the depth-averaged Lagrangian time scale as a function of Eulerian scale is suggested. The vertical turbulent diffusion in open natural flows is studied. The mean velocity profile is described by a power law. On the assumption of flat flow under incomplete self-similarity of the global Reynolds number, a universal expression for the vertical transfer coefficient is obtained. This expression enables one to estimate the time and length of complete mixing using minimum experimental data. The coefficients of longitudinal and vertical diffusion in different rivers are compared with each other and the universal ratio of these coefficients is suggested.

Influence of initial conditions on absolute and relative dispersion in semi-enclosed basins.

Absolute and relative dispersion are fundamental quantities employed in order to assess the mixing strength of a basin. There exists a time scale called Lagrangian Integral Scale associated to absolute dispersion that highlights the occurrence of the transition from a quadratic dependence on time to a linear dependence on time. Such a time scale is commonly adopted as an indicator of the duration needed to lose the influence of the initial conditions. This work aims to show that in a semi-enclosed basin the choice of the formulation in order to calculate the absolute dispersion can lead to different results. Moreover, the influence of initial conditions can persist beyond the Lagrangian Integral Scale. Such an influence can be appreciated by evaluating absolute and relative dispersion recursively by changing the initial conditions. Furthermore, finite-size Lyapunov exponents characterize the different regimes of the basin.

Turbulent structures of buoyant jet in cross-flow studied through large-eddy simulation

In the present paper we study buoyant (plume) and non-buoyant (jet) fluid injection in a neutrally stratified uniform cross-flow. Both cases are of practical importance in environmental fluid mechanics. The study is carried out numerically, using highly resolved large-eddy simulation in conjunction with the Lagrangian dynamic sub-grid scale model for both momentum and scalar transport equations. The velocity ratio is $$\kappa =8$$ . In the plume case, the Froude number is $$F=10$$ , such to allow the use of the Boussinesq approximation. The simulations are successfully validated against experimental data and well established semi-empirical relations. The study shows the existence of three different regions as regards the plume evolution, each of them characterised by different peculiarities: in momentum-buoyancy region the plume exhibits an almost steady cylindrical shape with relative small turbulence structures; in deflection region the plume is deviated horizontally and a high shear rate is detected; in entrainment region the vortex pair develops, along with the sausage-like turbulent structure. The comparison between the plume and the jet case shows that the latter has a higher eccentricity while its trajectory height is sensibly lower. Also, the sausage-like structures are not present. Finally, an empirical formula for the jet trajectory is given, although its full validation will require additional studies.

Single-particle statistics in the southern Gulf of Mexico

We present single-particle statistics of a large data set of surface drifters released in the southern Gulf of Mexico. The study is focused on the probability density functions of velocities, Lagrangian scales and absolute dispersion estimates. The probability distributions display a non-Gaussian shape with extended tails. It is shown that the long tails are mainly associated with the Campeche gyre (an intense cyclonic circulation frequently formed in the region) and, to a lesser extent, with a flow along the western margin of the basin. The results are discussed in light of previous studies on ocean dispersion that have reported Gaussian and non-Gaussian distributions in other regions.

Service system design with immobile servers, stochastic demand and concave-cost capacity selection

The service system design problem is a location-allocation problem with service quality considerations that is often modeled as a network of M/M/1 queues to minimize facility setup, customer access, and waiting costs. Traditionally, capacity decisions are either ignored or modeled as a selection among discrete capacity levels. In this work, we study the general continuous capacity case and account for economies-of-scale in its cost through an increasing concave function. We focus on the special square-root case that has been shown to model capacity in terms of the number of servers needed under Poisson arrivals and exponential service times. The problem is formulated as a mixed-integer nonlinear program with concave and convex terms in the objective function. Two novel resolution approaches are proposed: In the first, the problem is reformulated as a mixed-integer quadratic program with fourth-degree polynomial equality constraints. These constraints and the quadratic objective function are approximated using piecewise-linear segments. In the second, we use Lagrangian relaxation to decompose the problem and reformulate the subproblems as second-order cone programs that are solved at multiple utilization levels. The Lagrangian multipliers are updated using a cutting-plane method and a feasible solution is obtained by solving the corresponding set-covering formulation. The solution approaches are tested and compared. The linearization approach provides high quality solutions within short computational times for small instances and lower accuracy; whereas the Lagrangian approach scales well as size increases.

Fourth- and Higher-order Interface Tracking Via Mapping and Adjusting Regular Semianalytic sets Represented by Cubic Splines

This work is a further development and the culmination along our research line of interface tracking in two dimensions [Zhang and Liu, J. Comput. Phys., 27 (2008), pp. 4063--4088; Zhang, SIAM J. Numer. Anal., 51 (2013), pp. 2822--2850; Zhang and Fogelson, SIAM J. Sci. Comput., 36 (2014), pp. A2369--A2400; Zhang and Fogelson, SIAM J. Numer. Anal., 54 (2016), pp. 530--560]. Previously, we have proposed an analytic framework for explicit interface tracking and a fourth-order interface tracking method. In this paper, we continue our approach to propose a new method, called the cubic MARS method, that features (1) a solid theoretical foundation consisting of well developed concepts from distinct branches of mathematics, (2) a representation of the interface with cubic splines, (3) a novel algorithm for Boolean operations on linear/curvilinear polygons, and (4) fourth-, sixth-, and eighth-order accuracy via a refined control of the relation between the Eulerian grid size $h$ and a Lagrangian length scale $h_L$ of interface markers. Volume conservation of the cubic MARS method also has convergence rates at 4, 6, and 8. A simple analysis and numerical results from several benchmark problems show that the cubic MARS method is superior to existing interface tracking methods in terms of accuracy and efficiency. In particular, for the single-vortex test and the deformation test, errors of an eighth-order cubic MARS method are close to machine precision on a 128-by-128 Eulerian grid! Currently the new method does not tackle topological changes of the interface, but its theoretical foundation provides a solid footing for future treatments of such phenomena.

Lagrangian transport by breaking surface waves

The Lagrangian transport due to non-breaking and breaking focusing wave packets is examined. We present direct numerical simulations of the two-phase air–water Navier–Stokes equations describing focusing wave packets, investigating the Lagrangian drift by tracking tracer particles in the water before, during and after the breaking event. The net horizontal transport for non-breaking focusing packets is well described by the classical Stokes drift, both at the surface and in the bulk of the fluid, where the e-folding scale of the evanescent vertical profile is given by the characteristic wavenumber. For focusing wave packets that lead to breaking, we observe an added drift that can be ten times larger than the classical Stokes drift for a non-breaking packet at the surface, while the initial depth of the broken fluid scales with the wave height at breaking. We find that the breaking induced Lagrangian transport scales with the breaking strength. A simple scaling argument is proposed to describe this added drift and is found to be consistent with the direct numerical simulations. Applications to upper ocean processes are discussed.

Consistency relations for the Lagrangian halo bias and their implications

The protohalo patches from which halos form are defined by a number of constraints imposed on the Lagrangian dark matter density field. Each of these constraints contributes to biasing the spatial distribution of the protohalos relative to the matter. We show how measurements of this spatial distribution -- linear combinations of protohalo bias factors -- can be used to make inferences about the physics of halo formation. Our analysis exploits the fact that halo bias factors satisfy consistency relations which encode this physics, and that these relations are the same even for sub-populations in which assembly bias has played a role. We illustrate our methods using a model in which three parameters matter: a density threshold, the local slope and the curvature of the smoothed density field. The latter two are nearly degenerate; our approach naturally allows one to build an accurate effective two-parameter model for which the consistency relations still apply. This, with an accurate description of the smoothing window, allows one to describe the protohalo-matter cross-correlation very well, both in Fourier and configuration space. We then use our determination of the large scale bias parameters together with the consistency relations, to estimate the enclosed density and mean slope on the Lagrangian radius scale of the protohalos. Direct measurements of these quantities, made on smaller scales than those on which the bias parameters are typically measured, are in good agreement.

Large-eddy simulation of thin film evaporation and condensation from a hot plate in enclosure: First order statistics

Numerical investigation of water evaporation and condensation in buoyancy driven flow, along with the thermal coupling between fluid and surrounding solids, is interesting for many industrial applications. The physical complexity of the evaporation and condensation processes, the mutual thermal influence of water change of phase and solid–fluid heat transfer, the anisotropy of turbulence quantities are challenging problems from numerical and theoretical side. The archetypal case of a vertical hot plate inside a cold square enclosure, filled with humid air, is studied. The solid surfaces are wetted by a thin water film. The plate cooling process due to film evaporation is analysed. Numerical simulation adopts the large-eddy methodology along with the dynamic Lagrangian sub-grid scale model. The conjugate heat transfer technique accounts for the solid–fluid thermal coupling, while the water phase is modelled under the thin film assumption. First, a preliminary case with isothermal solid boundaries and fixed film thickness is reproduced at Ra=5×108. The absence of surface heat transfer leads to analogous distribution of temperature and humidity. The cavity is characterised by strong stratification that confines the motion in the upper part. The same setting is used to simulate the case of dry air. It is found that the presence of vapour increases the velocity by a maximum of 20%. Also, the heat transfer generated by the water change of phase, in case of humid air, overcomes the other heat transfer modes. Successively, conjugate heat transfer and water film model are activated, and three cases are studied changing the plate material. The specific heat ρCp of materials is the parameter controlling the plate cooling process, that is mainly due to evaporation and the evolution of the thermodynamic field within the enclosure. An analysis of the dew-point temperature suggests that recondensation onto the plate surface cannot occur, even for materials that are rapidly cooled.

Comparing isopycnal eddy diffusivities in the Southern Ocean with predictions from linear theory

We show that the potential vorticity diffusivity predicted by linear stability analysis (LSA), is the same as a linearized version of Lagrangian cross-stream isopycnal diffusivity. Both can be written in terms of the same expression − the product of the eddy kinetic energy (EKE) and the integral time scale that involves the Lagrangian decay scale γ or the growth rate ωi of the most unstable wave, and a frequency that is related to the difference of the mean flow speed and real part of the phase speed of the unstable waves.Diffusivities from LSA are compared to Lagrangian isopycnal eddy diffusivities estimated from more than 700,000 numerical particles in the Southern Ocean of an eddying model. They show different spatial dependency. LSA predicts eddy diffusivities that are enhanced at the steering level where the mean flow speed equals the phase speed of the unstable waves. In contrast, Lagrangian diffusivities exhibit no clear steering level maxima, but are instead surface intensified in many places. The differences between the Lagrangian and diffusivities from LSA can be understood because EKE predicted from LSA differs from the simulated one, and because the estimated decay scale γ is on average about 4 times larger than the largest linear growth rate. The diagnosed Lagrangian integral time scale has maxima at the depth where the mean flow speed equals the phase speed of the most unstable wave, but the diffusivity maxima are shifted towards the surface because the simulated EKE decreases rapidly with depth. Possibilities for a simple parameterization for the diffusivity are discussed.

Surface dispersion in the Gulf of California

Surface dispersion is measured in the Gulf of California by means of Argos drifters released along this semi-enclosed, elongated basin. First, basic one-particle statistics (Lagrangian scales, absolute dispersion and diffusion coefficients) are estimated along and across the Gulf. Absolute dispersion shows a nearly ballistic regime during the Lagrangian time scale (<2days) in both directions (it grows as ∼t2, where t is time). During the subsequent 30days, absolute dispersion enters a random-walk regime (∼t) along the Gulf, while being saturated across the basin due to the lateral boundaries. Secondly, the analysis is extended to two-particle statistics (relative dispersion between pairs of drifters and Finite Scale Lyapunov Exponents, FSLE). Relative dispersion is nearly exponential in both directions during the first few days, though evidence is not conclusive. During the subsequent 30days, it grows as ∼t1.5 along the Gulf, while being saturated across the basin again. It is shown that relative dispersion along the Gulf is proportional to t̂3, where t̂ represents a shifted time that depends on the initial separation of the particles. This form of the Richardson regime is consistently measured for particles that are sufficiently separated (30km or more). The Richardson regime is verified with the FSLE for particle separations ranging from 30 to 140km, approximately. The obtained dispersion properties are discussed in terms of the main circulation features within the basin, such as mesoscale vortices that occupy the width of the Gulf. These structures might retain buoys during days or weeks, thus preventing or delaying further displacements and therefore affecting the particle dispersion. The vortices are also an important mechanism to translate particles across the Gulf, between the Peninsula and the continent, thus promoting the saturation of dispersion along this direction.

Characteristics of the Near-Surface Currents in the Indian Ocean as Deduced from Satellite-Tracked Surface Drifters. Part II: Lagrangian Statistics

AbstractLagrangian statistics of the surface circulation in the Indian Ocean (IO) are investigated using drifter observations during 1985–2013. The methodology isolates the influence of low-frequency variations and horizontal shear of mean flow. The estimated Lagrangian statistics are spatially inhomogeneous and anisotropic over the IO basin, with values of ~6–85 × 107 cm2 s−1 for diffusivity, ~2–7 days for integral time scale, and ~33–223 km for length scale. Large diffusivities (&gt;20 × 107 cm2 s−1) occur in the central-eastern equatorial IO and the eastern African coast. Small diffusivities (~6–8 × 107 cm2 s−1) appear in the subtropical gyre of the southern IO and the southeastern Arabian Sea. The equatorial IO has the largest zonal diffusivity (~85 × 107 cm2 s−1), corresponding to the largest time scale (~7 days) and length scale (~223 km), while the eastern coast of Somalia has the largest meridional diffusivity (~31 × 107 cm2 s−1). The minor component of the Lagrangian length scale is approximately equal to the first baroclinic Rossby radius (R1) at midlatitudes (R1 ~ 30–50 km), while the major component equals R1 in the equatorial region (R1 &gt; 80 km). The periods of the energetic eddy-containing bands in the IO in Lagrangian spectra range from several days to a couple of months, where anticyclones dominate. A significant result is that the drifter-derived diffusivities asymptote to constant values in relatively short time lags (~10 days) for some subregions of the IO if they are correctly calculated. This is an important contribution to the ongoing debate regarding drifter-based diffusivity estimates with relatively short Lagrangian velocity time series versus tracer-based estimates.

RBF-vortex methods for the barotropic vorticity equation on a sphere

Vortex blob methods approximate a flow as a sum of many small vortices of Gaussian shape, and adaptively move the vortex centers with the current. Gaussian radial basis functions (RBFs) do exactly the same. However, RBFs solve an exact interpolation problem — expensive but accurate — while vortex methods sacrifice accuracy through quasi-interpolation for the absence of a matrix inversion. We show that vortex-RBF algorithms with spectral accuracy are stable for flows on the sphere. The version in Eulerian coordinates is fast; the fully-Lagrangian variant is much slower for a given basis size N, but is highly adaptive for advection-dominated flows. Both versions are excellent for small-to-medium-N problems — N up to 10,000, say, where N is the number of RBF grid points/vortex blobs. Neither is good for large N problems because the cost of the Eulerian model scales as N2 per timestep while the Lagrangian vortex-RBF method scales as N3. The slow-Lagrangian scheme is unique among vortex methods in being genuinely (like its Eulerian sibling) a spectrally-accurate method: for laminar flows the error falls exponentially fast with N.