Eulerian Time Scales Research Articles

Drifter observations of coastal surface currents during CODE: The statistical and dynamical views

Observations of near‐surface coastal currents were made off the Northern California coast during CODE by using 164 current‐following drifters. These observations are used to describe the two‐dimensional structure of the mean surface flow and the scales of its variability. The mean flow is a broad equatorward current, strongly sheared only within 10 km of the shore, and a mean offshore flow producing an average divergence ∇H·u ≈ 3 × 10−6 s−1. Divergence is uniformly distributed across the shelf, but variation of the alongshore flow causes an upwelling center near Point Arena. The spatial correlation scale is less than 40 km in both the alongshore and across‐shelf directions, even though 55% of the surface kinetic energy is described by a single mode with gradual across‐shelf variation and an alongshore wavelength of the order 200 km. The surface flow is well correlated with flow at 30‐m depth. The Lagrangian time scale (≈1.5 day) is significantly shorter than the Eulerian time scale (≈5 days), indicating that the flow is dominated by highly nonlinear quasi‐stationary eddies. Drifter displacements indicate that the mean lateral eddy transport of passive scalars can be described by an anisotropic and inhomogeneous eddy diffusivity, but this diffusivity cannot be used to relate eddy Reynolds stresses and the mean shear. Analysis of two‐particle separations, which determine the size of dispersing property clouds, shows that dispersal cannot be described by a scale‐dependent diffusivity and indicates the importance of small‐scale convergences in retarding dispersal. Over the entire 100‐km by 50‐km region the surface layer heat budget is dominated by upwelling cooling and surface heating, with onshore eddy heat flux playing a smaller role. Substantial convergence of the alongshore eddy heat flux is apparently required to balance upwelling cooling in the upwelling center. Drifters are found to have a Lagrangian mean acceleration caused by eddy processes. Analysis of this acceleration and of the horizontal flow contributions to the eddy Reynolds stress allows examination of the importance of eddy processes in the mean momentum budget. While the alongshore flow must be in approximate geostrophic balance, there is a clear pattern to the eddy forcing, which appears to be important in the alongshore momentum equation.

Estimates of integral time scales from a 100-M meteorological tower at a plains site

Four distinct types of autocorrelograms were observed using high-frequency vertical velocity data measured at 100 m above a flat terrain. Several types of nonstationary atmospheric motions due to low frequency fluctuations were examined. Under nighttime stable conditions, these phenomena were found to lead to abnormally slow exponential decay of the autocorrelation function. Several different techniques for estimating Eulerian integral time scales were compared in order to select an appropriate method of estimation. When grouped by stability classes, the Eulerian integral time scales decrease slightly with increasing stability, but generally exhibit no significant correlation with other meteorological parameters. Using a postulated relation, estimates of the Lagrangian to Eulerian integral time scale ratio range from 3 to 5 under unstable conditions and 15 to 25 under stable conditions. Under unstable conditions the average Lagrangian integral time scale is on the order of 40 s and exhibits no significant correlation with several pertinent meteorological parameters. Under stable conditions, the Lagrangian integral time scale correlates well with the Monin-Obukhov length and temperature lapse rate.

Two-Dimensional Simulations of Drainage Winds and Diffusion Compared to Observations

A vertically integrated dynamical drainage flow model is developed from conservation equations for momentum and mass in a terrain-following coordinate system. Wind fields from the dynamical model drive a Monte Carlo transport and diffusion model. The model needs only topographic data, an Eulerian or Lagrangian time scale and a surface drag coefficient for input data, and can be started with a motionless atmosphere. Model wind and diffusion predictions are compared to observations from the rugged Geysers, California area. Model winds generally agree with observed surface winds, and in some cases may give better estimates of area-averaged flow than point observations. Tracer gas concentration contours agree qualitatively with observed contours, and point predictions of maximum concentrations were correctly predicted to within factors of 2 to 10. Standard statistical tests of model skill showed that the accuracy of the predictions varied significantly from canyon to canyon in the Geysers area. Model wind predictions are also compared to observations from the South Carolina Savannah River Plant, which has gently rolling terrain. The model correctly simulated the slower development of drainage winds and slower deepening of the drainage layer in the Savannah River Valley, relative to the Geysers, California simulations. The South Carolina simulations and observations suggest that drainage winds are more frequent in the southeastern United States than is generally recognized. They may be responsible for some of the errors in air pollution concentration predictions made by Gaussian models, which assume homogeneous winds and turbulence.

Statistics of atmospheric turbulence within and above a corn canopy

Two three-dimensional split-film anemometers were used to measure turbulence statistics within and above a corn canopy. Normalised profiles of mean windspeed, root-mean-square velocity, momentum flux, and heat flux were constructed from half-hourly averages by dividing within-canopy measurements by the simultaneous canopy-top measurement. With the exception of the heat flux, these profiles showed consistent shape from day to day. Time series of the three velocity components were recorded on magnetic tape and subsequently analysed to obtain Eulerian time and length scales and the power spectrum of each component at several heights. The timescale was found to have a local minimum value at the top of the canopy. However the length scale L wformed from the timescale and the root-mean-square vertical velocity varied with height as L w≃ 0.1 z. The power-spectra were non-dimensionalised to facilitate comparison of spectra at different heights and times. All spectra had -5/3 regions spanning at least two decades in frequency.

Etude de la diffusion de la chaleur en aval d'une source lineaire placee dans une couche limite turbulente

Mean temperatures are studied in a wind tunnel downstream of a line source (heated wire) located in a turbulent boundary layer. Measurements are made successively at four distances from the wall. Longitudinal evolution of mean temperatures are presented. Analysis of experimental results indicates a strong influence of the position of the source on diffusion. Peak value temperatures as a function of downstream distance is fitted with power laws. Good agreement is found with Shlien and Corrsin [38] measurements. The lagrangian turbulent Prandtl number defined by Shlien and Corrsin is in the range from 0.8 to 1.7. Comparison with the usual turbulent Prandtl number shows some difference when the source is located near the wall. The lagrangian integral scale is found of the same order as the eulerian integral time scale of the vertical velocity fluctuation in an ‘optimum’ convected frame.

Lagrangian and Eulerian Time-Scale Relations in the Daytime Boundary Layer

Abstract Lagrangian (neutral balloon) and Eulerian (tower and aircraft) turbulence observations were made in the daytime mixed layer near Boulder, Colorado. Average sampling time was ∼25 min. Average Lagrangian time scale is ∼70 s and average ratio of Lagrangian to Eulerian time scales (β = TL/TE) is about 1.7. The ratio β is inversely proportional to turbulence intensity i. These data support the formula β = 0.7/i. Lagrangian time scale for the vertical component of turbulence at heights above ∼100 m is given by the formula TL = 0.17zi/σμ where zi is mixing depth. This formula is valid for the horizontal components of turbulence at all heights in the mixed layer. Lagrangian spectra in the inertial subrange are best represented by the formula Fr(n) = 0.2ϵn−2.

DISTORTED TURBULENCE IN AXISYMMETRIC FLOW

A solution to the rapid-distortion theory for small-scale turbulence in flow round an axisymmetric obstacle is derived. General formulae for velocity covariances and Eulerian time scales are obtained and are evaluated for the particular case of flow round a sphere. The large-scale limit for this flow is also discussed.

The expected deviation of observed concentrations from predicted ensemble means

This note presents an analysis to estimate the expected deviation of observed concentrations from predicted ensemble means. It is shown that the expected deviation ε is related to the sampling time T by the expression ε 2 = 2( Γ − 1) T i / T In the above equation, Γ is the instantaneous peak to mean ratio at the receptor under consideration and T i is the Eulerian time scale controlling dispersion of the plume. As reasonable estimates of these parameters can be made we can compute ε for various meteorological conditions. For example, for a tall stack emitting pollutants in an unstable boundary layer e can be as large as 100 % at the point of maximum concentration. This points to the importance of knowing ε when comparing predictions with observations.

Turbulence in Lake Huron

In order to improve the understanding of diffusive processes in large bodies of water, the instantaneous speed and direction of the current at a fixed point two meters below the surface of Lake Huron have been measured during the summer of 1967. The root mean square of the current fluctuations, both in the streamwise and lateral directions, was of the order of 0·05 of the mean current but varied substantially from day to day. The spectra of these fluctuations showed large amounts of energy at frequencies below 0·1 cycles min−1.Based on three hour records, the Eulerian time scales were found to be of the order of 10 min while the Lagrangian time scales, calculated from diffusion studies in the same area the previous summer, were of the order of 40 min.

Lagrangian-Eulerian time-scale ratios estimated from constant volume balloon flights past a tall tower

The ratio of Lagrangian and Eulerian time scales of motion in the planetary boundary layer is estimated from radar observations of 35 constant volume balloons (tetroons) flown past the 460 m high BREN tower at the Nevada Test Site. On the average, the peak energies of the Eulerian and tetroon vertical velocity spectra occur at periods near 6 and 18 minutes, yielding a mean Lagrangian-Eulerian scale ratio, β, of about three, β increases with increase in wind speed, decreases with increase in r.m.s. vertical velocity, and is inversely related to vertical turbulence intensity, with β ranging from 4.5 at a turbulence intensity of 0.05 to 2.5 at turbulence intensities exceeding 0.3.

Lagrangian‐Eulerian time‐scale ratios estimated from constant volume balloon flights past a tall tower

AbstractThe ratio of Lagrangian and Eulerian time scales of motion in the planetary boundary layer is estimated from radar observations of 35 constant volume balloons (tetroons) flown past the 460 m high BREN tower at the Nevada Test Site. On the average, the peak energies of the Eulerian and tetroon vertical velocity spectra occur at periods near 6 and 18 minutes, yielding a mean Lagrangian‐Eulerian scale ratio, β, of about three, β increases with increase in wind speed, decreases with increase in r.m.s. vertical velocity, and is inversely related to vertical turbulence intensity, with β ranging from 4.5 at a turbulence intensity of 0.05 to 2.5 at turbulence intensities exceeding 0.3.

An optical study of turbulence

This paper describes theoretical and experimental work carried out at the Cavendish Laboratory of the University of Cambridge. The main object of the work was to develop a new technique for measuring the structure of fluid turbulence.A parallel beam of light is passed through the turbulent region, containing refractive index fluctuations, and analyzed on exit by gratings of periodic transmissivity. Two forms of analysis yield (a) the spatial power spectrum of the refractive index fluctuations in the turbulence, and (b) the velocity distribution within the beam aperture. The method does not disturb the fluid physically, does not depend on the existence of a mean flow velocity, and works well in liquids.One of the limitations of this single-beam method is that it produces information averaged along the path length of the beam in the turbulence, and to overcome this a cross-beam technique, using two beams intersecting at right-angles, has been developed in theory. This method gives the spatial power spectrum of the refractive index fluctuations, as does the single beam method, but the results are characteristic only of the volume of intersection of the beams.The paper first discusses the theory of the single-beam and crossed-beam techniques, and then experimental results obtained with the single-beam method.The turbulent region investigated was a rectangular tank of water, heated from below and cooled from above, producing convective turbulence of high Rayleigh number (4·1 × 108), a system difficult to analyze by conventional methods of measurement, such as the hot-wire anemometer.Spectral density functions (power spectra) of refractive index, and hence in this case temperature fluctuations, have been measured, as have velocity distributions. Statistical analysis of the results also gives useful information about the Eulerian time scale of the turbulent field.

Dispersion Prediction from Current Meters

Two-dimensional dispersion plumes for the near shore area of Nanticoke on Lake Erie are predicted by applying turbulent diffusion concepts to recording current meter data. Eulerian integral time scales are found from autocorrelation coefficients, based on monthly data. Lagrangian integral space scales and one-dimensional diffusion coefficients are then predicted. Average monthly probability distributions are found based on north-south and east-west diffusion coefficients. The prediction equation is an average of long and short time diffusion equations. The results show better dilution nearer the shore and parallel to it. It has been assumed that vertical diffusion is negligible, that the Reynold's number is large, and that the effective diffusion coefficients are constant over long periods.

Estimations from geostrophic trajectories of horizontal diffusivity in the mid‐latitude troposphere and lower stratosphere

AbstractZonal and meridional diffusivities Kxx and Kyy have been found at mid‐latitudes from the spreads of sets of geostrophic trajectories at levels between 700 mb and 30 mb during each season of 1965. Values have been calculated using the relationship R = e−PT cos qr for the form of the corresponding Lagrangian autocorrelograms. In addition a similar set of Eulerian data has been analysed to obtain autocorrelograms and by comparison with the trajectory data Eulerian‐Lagrangian scale relationships have been obtained.The results show values of Kxx ranging from 0·2.106 to about 20.106 and Kyy from 0·2. 106 to about 6.106 m2 sec−1. In both cases the lowest values are found in the summer stratosphere and the highest in the spring stratosphere and in the upper troposphere in winter. Their distributions principally follow that of the wind variances. Integral time scales are mainly between 5 and 30 hours meridionally and 10 and 50 hours zonally and are greatest in the stratosphere and lowest in the mid‐troposphere. The contributions of damping and harmonic effects to these time scales are discussed. Comparison of the Lagrangian and Eulerian time scales indicates a mean value β of about 0·6 for their ratio with the meridional values smaller than the zonal values.

Relation between Eulerian and Lagrangian Statistics

R (x, r, t) is the Eulerian correlation function in the space with zero mean motion. R(t), the Lagrangian correlation function, may be found from the correlation function R'(x, r, t) based on the sub-ensemble of ``Eulerian trials'' for which the fluid particle at (x, r, t) is the same as that at (0, 0, 0). The hypothesis that R may be substituted for R' yields a connection between R(t) and R(x, r, t). The Lagrangian time scale is found to be about one-third the Eulerian time scale. The apparent Eulerian time scale depends on the intensity of turbulence. Data on the ratio of Lagrangian to apparent Eulerian time scales agree farily well with the analysis.

Estimates of the Relations between Eulerian and Lagrangian Scales in Large Reynolds Number Turbulence

With Eulerian and Lagrangian spectra approximated by their “inertial subrange” forms between appropriate wave number and frequency limits, it is found that the Lagrangian integral time scale is roughly equal to the Eulerian integral length scale divided by the root-mean-square velocity. The corresponding estimate for the “microscale” ratio shows fair agreement with the large Reynolds number form of Heisen-berg's result. A similar approach to Eulerian time scales gives values approximately equal to the Lagrangian scales.

Concentration fluctuations in a stirred baffled vessel

AbstractA stirred vessel is maintained in a quasisteady state by continuously introducing water and a conductive tracer solution and continuously removing the resulting mixture by overflow. Detailed properties of the system are elucidated by measuring tracer concentration changes with a conductivity probe in a volume element of the order of 0.3 cu. mm. By the use of suitable electronic equipment the following statistical concentration parameters are measured in experiments with three diameters of flat‐bladed turbines and with a tenfold range of rotational speed: temporal mean, total root‐mean‐square fluctuation, spectral distribution of fluctuations, and Eulerian micro time scale. Results are interpreted in terms of behavior of time‐average and fluctuation measures in three regions, the generation region within the impeller volume, the decay region which starts in the horizontal fluid sheet issuing from the impeller, and the recirculation flow region in the vessel.Generation of concentration fluctuations within the confines of the impeller is described in terms of a model involving gross mixing, without significant decay, in the wake of each flat blade. Distributions of mean and fluctuating concentrations are suggested to provide a measure of uniformity in the vessel as a whole. The former, in particular, was remarkably constant throughout the vessel.