Lagrangian Scale Research Articles

Eulerian and Lagrangian properties of biophysical intermittency in the ocean

Eulerian and Lagrangian characteristics of simultaneously recorded temperature, salinity and phytoplankton biomass time series are investigated over a seasonal cycle. Time series are analyzed in the spectral and multifractal scaling frameworks. We show that passive scalar time series (temperature and salinity) are characterized by scaling laws representative of Eulerian and Lagrangian turbulence. The two different regimes identified are separated by a transition linked to the characteristic scale of turbulent structures of the same size as the boat. While the scaling behavior exhibited by phytoplankton biomass fluctuations is close to the expected passive scalar behavior over Eulerian scales, it is quite far from it over Lagrangian scales. For these scales, the observed scaling behavior exhibits a significant density‐dependence, indicative of biological activity.

Operational mapping of atmospheric nitrogen deposition to the Baltic Sea

Abstract. A new model system for mapping and forecasting nitrogen deposition to the Baltic Sea has been developed. The system is based on the Lagrangian variable scale transport-chemistry model ACDEP (Atmospheric Chemistry and Deposition model), and aims at delivering deposition estimates to be used as input to marine ecosystem models. The system is tested by comparison of model results to measurements from monitoring stations around the Baltic Sea. The comparison shows that observed annual mean ambient air concentrations and wet depositions are well reproduced by the model. Diurnal mean concentrations of NHx (sum of NH3 and NH4+) and NO2 are fairly well reproduced, whereas concentrations of total nitrate (sum of HNO3 and NO3-) are somewhat overestimated. Wet depositions of nitrate and ammonia are fairly well described for annual mean values, whereas the discrepancy is high for the monthly mean values and the wet depositions are rather poorly described concerning the diurnal mean values. The model calculations show that the annual atmospheric nitrogen deposition has a pronounced south--north gradient with depositions in the range about 1.0 T N km-2 in the south and 0.2 T N km-2 in the north. The results show that in 1999 the maximum diurnal mean deposition to the Danish waters appeared during the summer in the algae growth season. For the northern parts of the Baltic the highest depositions were distributed over most of the year. Total deposition to the Baltic Sea was for the year 1999 estimated to 318 kT N for an area of 464 406 km2 equivalent to an average deposition of 684 kg N/km2.

Lateral diffusivity and Lagrangian scales in the Pacific Ocean as derived from drifter data

The Global Drifter Program/Surface Velocity Program data set is used to derive maps of the lateral diffusivity and Lagrangian scales of velocity, length and time across the whole Pacific. For the lateral diffusivity, the minor principal component of both the Davis' diffusivity tensor and the half growth rate of the single‐particle dispersion tensor were calculated in 5° × 5° bins from drifters for the period 1979–1999. Estimates of the lateral diffusivity, K, obtained by means of the minor principal component approach, are shown to correspond well with other authors' estimates for the Pacific. The highest values of K (2–3 104 m2/s) are found in the eastern equatorial Pacific, while the lowest ones (2–3 103 m2/s) are typical for forty‐odd degrees in latitude northward of the subpolar fronts in both hemispheres and for the subtropics in the eastern part of the South Pacific. High values of K (about 1 × 104 m2/s) are also observed in the Kusoshio/Kuroshio Extension area and in the equatorial Pacific along the whole length of the equator. There are two approximately symmetrical tongues of high K value in the western Pacific centered around 21°N and 24°S, respectively. In general, K has a tendency to increase westward (eastward) in the midlatitudes (equatorial region) and toward the equator in both hemispheres. In the midlatitudes, the Lagrangian length scale L is found to be approximately equal to the first‐mode baroclinic radius of deformation, Ri, while the Rossby number defined as Ro = Tearth/T sinφ, where T is the Lagrangian timescale, Tearth is the period of the Earth's rotation, and φ is the latitude, is less than unity, implying that the quasigeostrophic mesoscale eddies are the main mechanism for lateral mixing. The lateral diffusivity in the midlatitudes is suggested to be parameterized as K = V2Ri, where V2 is the Lagrangian velocity scale inherent to a mesoscale eddy field. In the low latitudes, L becomes less than Ri, and the condition Ro ≤ 1 is no longer valid, which implies the change of the eddy‐controlled mechanism of lateral mixing by another mechanism.

Lagrangian Eddy Scales in the Northern Atlantic Ocean

Abstract Eddy time and length scales are calculated from surface drifter and subsurface float observations in the northern Atlantic Ocean. Outside the energetic Gulf Stream, subsurface timescales are relatively constant at depths from 700 m to 2000 m. Length scale and the characteristic eddy speed decrease with increasing depth below 700 m, but length scale stays relatively constant in the upper several hundred meters of the Gulf Stream. It is suggested that this behavior is due to the Lagrangian sampling of the mesoscale field, in limits set by the Eulerian eddy scales and the eddy kinetic energy. In high-energy regions of the surface and near-surface North Atlantic, the eddy field is in the “frozen field” Lagrangian sampling regime for which the Lagrangian and Eulerian length scales are proportional. However, throughout much of the deep ocean interior, the eddy field may be in the “fixed float” regime for which the Lagrangian and Eulerian timescales are nearly equal. This does not necessarily imply that the deep interior is nearly linear, as fixed-float sampling is possible in a flow field of O(1) nonlinearity.

Lagrangian Eddy Scales in the Northern Atlantic Ocean

Eddy time and length scales are calculated from surface drifter and subsurface float observations in the northern Atlantic Ocean. Outside the energetic Gulf Stream, subsurface timescales are relatively constant at depths from 700 m to 2000 m. Length scale and the characteristic eddy speed decrease with increasing depth below 700 m, but length scale stays relatively constant in the upper several hundred meters of the Gulf Stream. It is suggested that this behavior is due to the Lagrangian sampling of the mesoscale field, in limits set by the Eulerian eddy scales and the eddy kinetic energy. In high-energy regions of the surface and near-surface North Atlantic, the eddy field is in the ‘‘frozen field’’ Lagrangian sampling regime for which the Lagrangian and Eulerian length scales are proportional. However, throughout much of the deep ocean interior, the eddy field may be in the ‘‘fixed float’’ regime for which the Lagrangian and Eulerian timescales are nearly equal. This does not necessarily imply that the deep interior is nearly linear, as fixed-float sampling is possible in a flow field of O(1) nonlinearity.

Numerical simulation of turbulent flow in complex geometries used in power plants

Performance degradations or improvements of coal-fired power stations depend on effective functioning of pulveriser equipment and combustion efficiency of furnaces in boilers. The function of a pulveriser is to grind the lumped coal and transfer the fine coal to the furnace for efficient combustion. However, the presence of several solid objects inside the mill, flow of air and particles takes turn around from inlet to outlet. The flow simulation process involves the geometrical modelling, grid generation and particle trajectories for the given flow conditions, and has been investigated to understand the flow path in grinding chamber, separator and classifier. The behaviour of turbulent air flow motion with fly ash particle paths on Lagrangian scale in computational domain are obtained through CFD/CAD software packages. The understanding developed with reference to recirculation flows in the inlet duct, non-uniform flow over the height of bowl mill and unequal flow at exit, provides valuable insights to designers for optimisation of components for better efficiency.

Lagrangian statistics in the central North Pacific

Lagrangian integral scales, diffusivities, dispersion and velocity spectra are calculated using surface drifter trajectories in the central North Pacific. The meridional integral time scale is relatively homogeneous throughout the region; a large increase in the zonal time and length scales south of Hawaii is attributed to meanders in the North Equatorial Current. Except in this current, the initial dispersion is consistent with Taylor's Theorem. For lags of 20–120 days, the meridional dispersion can be modeled by a constant eddy diffusivity. Shear in the mean zonal currents magnifies the zonal dispersion at long lags. In the Lagrangian spectra, the energetic eddy band is at 3–20 days west of Hawaii, 10–40 days east and north of Hawaii, and 20–60 days in the North Equatorial Current. In the wake of Hawaii, energetic lee vortices produce sharp peaks in the cyclonic and anticyclonic rotary spectra.

Study on horizontal relative diffusion in the troposphere and lower stratosphere

The behaviour of relative diffusion theory and Gifford’s random-force theory for long-range atmospheric diffusion is examined. When a puff scale is smaller than the Lagrangian length scale, √2KTL, an accelerative relative diffusion region exists, i.e., σy∝ t3/2. While the puff diffusion enters a two-dimensional turbulence region, in which the diffusion scale is larger than 500 km, or time scale is larger than 1 day, divergence and convergence are main cause of horizontal diffusion. Between the two above-mentioned regimes, diffusion deviation is given by σy = √2KT. The large-scale horizontal relative diffusion parameters were obtained by analyzing the data of radioactive cloud width collected in air nuclear tests.

Turbulence parameterisation for PBL dispersion models in all stability conditions

Accounting for the current knowledge of the planetary boundary layer (PBL) structure and characteristics, a new set of turbulence parameterisations to be used in atmospheric dispersion models has been derived. That is, expressions for the vertical profiles of the Lagrangian length scale l i and time scale T i and diffusion coefficient K i , i=u, v, w , are proposed. The classical statistical diffusion theory, the observed spectral properties and observed characteristics of energy containing eddies are used to estimate these parameters. The results of this new method are shown to agree with previously determined parameterisations. In addition, these parameterisations give continuous values for the PBL at all elevations (z 0⩽z⩽h, z i) and all stability conditions from unstable to stable, where h and z i are the turbulent heights in stable or neutral and convective PBL, respectively, and L is the Monin–Obukhov length. It is the aim of this work to present the general derivations of these expressions and to show how they compare to previous results. Finally, a validation of the present parameterisation applied in a Lagrangian particle model, will be shown. The Copenhagen data set is simulated. This data set is particularly suited for this validation, since most of the Copenhagen tracer experiments were performed in stability conditions that are the result of the relative combination of wind shear and buoyancy forces. As a consequence, a parameterisation scheme, able to deal contemporary with neutral and slightly convective condition, is to be preferred.

Near-surface circulation of the northeast Pacific Ocean derived from WOCE-SVP satellite-tracked drifters

Statistics of the near-surface circulation in the northeast Pacific Ocean were derived from the trajectories of nearly 100 surface drifters tracked between August 1990 and December 1995 as part of the World Ocean Circulation Experiment's (WOCE) Surface Velocity Program (SVP). Drifters were drogued within the mixed layer (15 m drogue depth) or near the top of the permanent halocline (120 m). All branches of the Alaskan Gyre were well-sampled at both depths, revealing a weak Subarctic Current, a bifurcation of the Subarctic Current near 48°N, 130°W at 15 m depth, and strong, variable flow in the Alaska Current and Alaskan Stream. At 120 m depth, northward flow in the Alaska Current occurred much farther offshore than within the mixed layer. The drifter trajectories revealed interannual variability, with evidence of an intensified Alaskan Gyre during the winters of 1991–92 and 1992–93 and more southerly transport during winter 1994–95. A minimum in eddy kinetic energy was found at both depths within the northern branch of the Subtropical Gyre. Eddy kinetic energies were nearly twice as high in the mixed layer compared to below, and were 2–3 times larger in winter than in summer throughout most of the near-surface Alaskan Gyre. High eddy energies observed near the eastern perimeter of the Alaskan Gyre may be due to the offshore intrusion of eddies formed by coastal current instabilities. Taylor's theory of single-particle dispersion was applied to the drifter ensembles to estimate Lagrangian decorrelation scales and eddy diffusivities. Both the initial dispersion and random walk regimes were identified in the dispersion time series computed for several regions of both ensembles. The integral time scales and eddy diffusivities computed from the dispersion scale linearly with r.m.s. velocity, which is consistent with drifter studies from the Atlantic. An exception is the meridional integral time scales, which were nearly constant throughout the study area and at both drogue depths. The magnitudes of the derived eddy statistics are comparable to those derived from surface drifters in other parts of the world ocean. These are the first Lagrangian estimates of particle dispersion over a broad region of the near-surface North Pacific, and the consistency of the results with previous studies from the Atlantic lends credence to the idea that the simplifying assumptions of Taylor (1921) (Proceedings of the London Mathematical Society Series A 20, 196–221) are reasonably valid throughout the upper ocean. This bodes well for the effective parameterization of near-surface diffusivities in general circulation models. Finally, the drifter-derived velocity statistics were used to speculate on the source regions of waters of possible coastal origin observed at offshore stations during the field studies of the Canadian Joint Global Ocean Flux Study.

The Summer Hydrography and Surface Circulation of the East Siberian Shelf Sea*

During the ice-free summer season in 1995 the authors deployed and subsequently tracked 39 surface drifters to test the hypothesis that the discharge from the Kolyma River forces a buoyancy-driven coastal current from the East Siberian Sea toward Bering Strait. The observed mean flow is statistically significant at the 95% level of confidence, but its direction contradicts their initial hypothesis. Instead of a coastally trapped eastward flow, the authors find a laterally sheared westward flow with maximum velocities offshore that correlate only weakly with the local winds. At a daily, wind-dominated timescale the drifter data reveal spatially coherent flows of up to 0.5 m s21. The Lagrangian autocorrelation scale is about 3 days and the Lagrangian eddy length scale reaches 40 km. This spatial scale exceeds the nearshore internal deformation radius by a factor of 3; however, it more closely corresponds to the internal deformation radius associated with the offshore ice edge. Bulk estimates of the horizontal mixing coefficient resemble typical values of isotropic open ocean dispersion at midlatitudes. Hydrographic observations and oxygen isotope ratios of seawater indicate a low proportion of riverine freshwater relative to sea ice melt in most areas of the East Siberian Sea except close to the Kolyma Delta. The observations require a reevaluation of the conceptual view of the summer surface circulation of the East Siberian Sea. Eastward buoyancy-driven coastal currents do not always form on this shelf despite large river discharge. Instead, ice melt waters of a retreating ice edge act as a line source of buoyancy that in 1995 forced a westward surface flow in the East Siberian Sea.

A case study of longitudinal dispersion in small lowland rivers

Contaminant transport in natural streams is considered on the basis of tracer studies in Moldovian small lowland rivers. Theories describing the spread of conservative and passive pollutants in the context of a longitudinal dispersion model, as well as attempts to estimate the discrepancy between this model and natural processes, are discussed. It is shown that the model agrees with experimental data, with an accuracy ranging from 15 to 20%, at least in the upper [C(t) > 0.5Cmax] concentration distributions. The process behaves in a quasi‐Fickian manner only at distances greater than 80 to 100 times the river width. Primarily, the nonuniformity of the average velocities over cross sections affects the process. An estimate of the Lagrangian spatial scales was proposed, and it was shown that those scales are as large as two to four times the length of alternate bars.

Near‐surface circulation of the Nordic seas as measured by Lagrangian drifters

In the period June 1991 to August 1993, 107 Argos tracked, drifters drogued to 15 m depth, were released in the Nordic seas (or Greenland, Iceland, and Norwegian Seas). The drifter movements revealed the strong and spatially confined current systems along the surface salinity fronts of the Iceland‐Faroe Frontal zone and of the Norwegian coast and along the continental margins and their extensions to the Barents Sea and Spitsbergen. The Norwegian Atlantic Current is composed of three distinct streams (two continental margin and one coastal branches) which join into one single swift mean current west of the Lofoten and Vesterålen Islands, where the strongest measured currents are in excess of 110 cm s−1. In addition to the general cyclonic gyre circulation in the Nordic seas, the drifters indicate smaller cyclonic circulation patterns in all the major subbasins, i.e., the Iceland plateau, the Norwegian, the Lofoten, and the Greenland basins. No surface signature of the East Icelandic Current is disclosed by the drifters. Interpolated and low‐pass‐filtered position data were used to construct maps of 15‐m‐depth ensemble mean velocity, velocity variability, and residence time. Vigorous eddy fields are dominant in the strong currents and in the Lofoten basin. Eulerian correlations indicate that they tend to propagate to the west. In contrast, the Iceland plateau appears quiescent, both in the mean and eddy velocities. Single‐particle diffusivities are computed and are found to be in the range 1–7 × 107 cm2 s−1. The corresponding Lagrangian timescale and space scale are 1–3 days and 10–40 km, respectively. These Lagrangian drifter measurements compose the first basin‐scale, accurate near‐surface velocity data set of the Nordic seas.

Lagrangian Analysis of Nonreactive Pollutant Dispersion in Porous Media by Means of the Particle Image Velocimetry Technique

An experimental technique based on image analysis was used to perform a Lagrangian description of passive pollutant particle motion in a three‐dimensional saturated porous medium. To allow for optical access, the experiment was carried out with Pyrex grains as the solid matrix and glycerol as the liquid phase in order to have two phases with the same refractive index. Statistical analysis of the experimental data allowed for estimation of velocity and displacement probability density functions (pdf), velocity component correlation functions, Lagrangian integral scales, and mechanical dispersion coefficient tensor components. The results obtained suggest that the longitudinal velocity component has a log normal pdf while the transversal component has a symmetrical pdf, which is nevertheless not Gaussian for high values of the kurtosis. Furthermore, the velocity components' autocorrelation functions are well represented by exponential laws, and the integral scale is dependent on filtration velocity and grain size. As foreseen in the theory the total displacement pdf shows the tendency to reach normal distribution after many integral scales. The evaluated dispersion coefficient tensor components are dependent on travel time; the components start from zero and reach an asymptotic value after several integral scales. Furthermore, the tensor is anisotropic, with the longitudinal component greater than the transversal one by about 1 order of magnitude. Comparison with other experimental data shows agreement at least for the longitudinal dispersion component. Dagan's linear theory has been used for comparing the analytical longitudinal component of the dispersion tensor with that obtained by means of the experiments.

A lagrangian study of particle dispersion in the unstable boundary layer

Most of the experimental techniques used to investigate the atmospheric turbulence give an Eulerian description of the velocity field. Nevertheless, pollutant dispersion phenomena are naturally described in the Lagrangian approach as the pollutant acts as tag of the fluid particles. The turbulence in the unstable boundary layer (UBL) of the atmosphere was simulated by means of a laboratory model and investigated from the Lagrangian point of view by use of a measuring technique (the particle tracking velocimetry) based on image analysis. The Lagrangian correlations of the horizontal and vertical velocity components were evaluated. From these, the Lagrangian integral time scales were obtained and compared with the Eulerian ones. The comparison showed that it is not possible to define a single value of the Lagrangian and Eulerian scale ratio for the whole UBL. Moreover, the data set was conditionally sampled in order to describe selectively the behaviour of upward- and downward-moving particles. Finally, assuming the horizontal homogeneity, steadiness, and frozen turbulence, the time histories of the particle locations were used to predict pollutant concentration fields, downwind continuous sources placed at any height.

Investigation on time scales in a low Reynolds number jet flow using particle-tracking velocimetry

The near flow field of an axisymmetric water jet at Reynolds numbers between 2000 and 5000 is investigated using Particle-Tracking Velocimetry. Measurements are taken in the longitudinal section (along the mean flow) and in cross-sections (orthogonal to the mean flow). From the former, correlation coefficients of the two in-plane velocity components in a Lagrangian framework are obtained: thus Lagrangian integral scales can be computed. Those of the streamwise velocity (axial) component increase on moving away from the centreline, whereas the opposite happens for the vertical velocity (radial) component: integral time scales of the two components are almost equal at the interface between jet and ambient fluids. On the other hand, integral scales are almost constant or increase slightly with the axial direction. In cross-sections, fluid ejection and injection from the jet centreline are observed to be connected to counter-rotating vortices (“mushroom”): their number and size change with Reynolds number in agreement with results from other authors. The maximum ejection velocity (orthogonal to the mean jet flow), at 3 nozzle diameters downstream of the outlet, is found to be one half of the mean outlet velocity.

An Analytical Method to Evaluate Mixing Length Scales for the Planetary Boundary Layer

Abstract An analytical method to evaluate the Lagrangian length scales for a turbulent planetary boundary layer (PBL) under stable and convective conditions is described in this paper. The method is based on the Taylor's diffusion theory. Agreement with the mixing lengths found in the literature indicates that the hypothesis of using Lagrangian length scale as a surrogate for mixing length scale in the turbulent PBL is valid.

The circulation and hydrography of conception bay, Newfoundland

Abstract The hydrography and circulation of Conception Bay (Newfoundland) are described based on hydrographic, current‐meter and drifter data collected over four years (1988–1991). The seasonal cycles of temperature (‐1.6 to 13–17°C) and salinity (31–32.5) in the bay closely follow those on the adjacent shelf. Exchange of bottom water was observed in April 1989. Deepwater exchange was observed from late fall to early winter of 1989–90. Tidal currents are weak, 1–2 cm s‐1 for the M2 and K1 constituents. Observed Eulerian mean currents (<3 cm s‐1) are smaller than the standard deviation (1–11 cm s‐1); however, there is a persistent outflowing current of 10 to 20 cm s‐1 within 2 km of the shoreline on the eastern side of the outer bay. The Lagrangian correlation length scale is from 4 to 10 km, in agreement with the weak coherence squared (≤0.4) found between the fixed current‐meter sites separated by greater than 4–5 km. The variable currents (up to 20 cm s‐1) tend to be cyclonic. Cyclonic eddies were obser...

More effective Lagrangians

The ad hoc Skyrme model is generalized to an all-order sum of a class of chiral invariant Lagrangians suggested by Marleau. The imposition of physical constraints leads to relative stable results for the properties of both single baryons and dense matter. In particular, under the assumption that the Lagrangian scales like the non-linear sigma model at low densities and like the free quark gas at high densities, a chiral phase transition must occur.

Summing skyrmions

The Skyrme model has the same high-density behaviour as a free quark gas. However, the inclusion of higher-order terms spoils this agreement. We consider the all-order sum of a class of chiral invariant lagrangians of even order in L μ suggested by Marleau. We prove Marleu's conjecture that these terms are of second order in the derivatives of the chiral angle for the hedgehog case and show that the terms are unique under the additional condition that, for each order, the identity map of the 3-sphere S 3( L) is a solution. The general form of the summation can be restricted by physical constraints leading to stable results. Under the assumption that the lagrangian scales like the non-linear sigma model at low densities and like the free quark gas at high densities, we prove that a chiral phase transition must occur.