Leif Kristensen

How Long Is Long Enough When Measuring Fluxes and Other Turbulence Statistics

Abstract It is determined how long a time series must be to estimate covariances and moments up to fourth order with a specified statistical significance. For a given averaging time T there is a systematic difference between the true flux or moment and the ensemble average of the time means of the same quantities. This difference, referred to here as the systematic error, is a decreasing function of T tending to zero for T→∞. The variance of the time mean of the flux or moment, the so-called error variance, represents the random scatter of individual realizations, which, when T is much larger than the integral time scale T of the time series, is also a decreasing function of T. This makes it possible to assess the minimum value of T necessary to obtain systematic and random errors smaller than specified values. Assuming that the time series are either Gaussian processes with exponential correlation functions or a skewed process derived from a Gaussian, we obtain expressions for the systematic and random e...

How Close is Close Enough When Measuring Scalar Fluxes with Displaced Sensors

Abstract To improve the quality of scalar-flux measurements, the two-point covariance between the vertical velocity w and a scalar s, separated in space both horizontally and vertically, is studied. The measurements of such two-point covariances between vertical velocity and temperature with horizontal and vertical separations show good agreement with a symmetric turbulence model when the displacement is horizontal. However, a similar model does not work for vertical displacements because up–down asymmetry exists; that is, there is a lack of reflection symmetry of the covariance function. The second-order equation for conservation of two-point covariance of w and s reveals the reason for this up–down asymmetry and determines its character. On the basis of our measurements, the “loss of flux” for a given lateral displacement decreases with increasing height of the sensors. For example, at a height of z = 10 m with a sensor displacement of D = 0.2 m, less than 1% of the flux is lost, whereas at z = 1 m ...

The spectral velocity tensor for homogeneous boundary layer turbulence

We have postulated a simple model for the spectral tensor Φij(k) of an anisotropic, but homogeneous turbulent velocity field. It is a simple generalization of the spectral tensor Φinfijpiso(k) for isotropic turbulence and we show how in the limit of isotropy, Φij(k) becomes equal to Φinfijpiso(k). Whereas Φinfijpiso(k) is determined entirely by one scalar function of k = ¦k¦, namely the energy spectrum, we need three independent scalar functions of k to specify Φij(k). We show how it is possible by means of the three stream-wise velocity component spectra to determine the three scalar functions in Φij(k) by solving two uncoupled, ordinary linear differential equations of first and second order. The analytic form of the component spectra each has a set of three parameters: the variance and the integral length scale of the velocity component and a dimensionless parameter, which governs the curvature of the spectrum in the transition domain from the inertial subrange towards lower wave numbers. When the three sets of parameters are the same, the three spectra correspond to isotropic turbulence and they are all interrelated and related to the energy spectrum. We show how it is possible to obtain these spectral forms in the neutral surface layer and in the convective boundary layer from data reported in the literature. The spectral tensor is used to predict the lateral coherences for all three velocity components and these predictions are compared with coherences obtained in two experiments, one using three masts at a horizontally homogeneous site in Denmark and one employing two aircraft flying in formation over eastern Colorado. Comparison shows reasonable agreement although with considerable experimental scatter.

Uncorrelated Noise in Turbulence Measurements

Abstract We show that the error variance contributed by random uncorrelated measurement noise can be merged with the error variance contributed by real variations in the atmosphere to obtain a single expression for the total error variance when the sampling time is much less than the integral scale of atmospheric variability. We assume that the measured signal is a representation of a variable that is continuous on the scale of interest in the atmosphere. The characteristics of this noise are similar, but not identical, to quantization noise, whose properties are briefly described. Uncorrelated noise affects the autocovariance function (or, equivalently, the structure function) only between zero and the first lag, while its effect is smeared across the entire power spectrum. For this reason, quantities such as variance dissipation may be more conveniently estimated from the structure function than from the spectrum. The modeling results are confirmed by artificially modifying a test time series with Poiss...

In search of a gust definition

We propose a simple gust definition based on the theory of excursions by Rice (1944 and 1945). We discuss the relation to the distribution of extreme events and demonstrate theoretically and experimentally that the most probable extreme event is very close to being identical to the gust according to our definition. We demonstrate how it is possible to predict the gust on the basis of the measured mean wind and variance rather than rely on actually measured extreme excursions. Our gust definition also allows us to predict the average duration of a gust.

Cup Anemometer Behavior in Turbulent Environments

Abstract The behavior of the cup anemometer rotor in turbulent atmospheric flow is discussed in terms of a general equation of motion. This equates the rate of change s of the rotation rate s of the rotor to a forcing F(s, h, w), which is proportional to the torque and a function of s and of the total horizontal and the vertical wind velocity components, h, and w, respectively. To determine the so-called overspeeding, it is necessary to carry out first-and second-order perturbation calculations around the response curve obtained in a laminar flow. From this curve, which for the purpose of this paper can be considered linear, five constraints are derived between the first and second partial derivatives of F. These constraints provide sufficient information for deriving an expression for the overspeeding to which four distinctly different biases contribute—one for each of the velocity components and one from the covariance between streamwise velocity components ũ and w. A phenomenological model of...

Cup Anemometer Overspeeding

Abstract Statistical considerations are applied to a general equation of motion for cup anemometers in a turbulent wind. It is shown that the relative overspeeding ΔS/S can be expressed as ΔS/S = Ih2 · Js(l0/Λs) + cIw2, where Is and Iw are the horizontal and the vertical turbulence intensifies, respectively. The function Js depends on the shape of the spectrum of horizontal turbulent energy, l0 is the distance constant for the anemometer, and Λs is a characteristic length scale of the horizontal turbulence. The constant c is of order unity. If Λs is suitably chosen as the scale of the energy-containing eddies, then Js is satisfactorily approximated by Js = (1 + Λs/l0)−1 in most atmospheric applications.

Lateral dispersion of pollutants in a very stable atmosphere—the effect of meandering

Abstract A model based on single particle diffusion is introduced to account for the effect of “meandering” on lateral plume dispersion in a very stable atmosphere. It is assumed that small scale atmospheric turbulence is absent, so that only large horizontal eddies are effective. A formula for the lateral standard deviation σ y as function of observation time, distance from source, mean wind speed, lateral turbulence intensity, and scale of the atmospheric motion is derived. Climatological time series of temperature lapse rates, wind speeds, and wind directions can be used as input to calculate σ y . Meteorological data from Riso and the small island Sprogo have been analysed in order to identify all situations in which the atmosphere is so stable that small scale turbulence cannot exist. The purpose is to see in how many of these situations meandering is also absent. The results show that, as a rule, meandering will be present in a strongly stable atmosphere with low wind speeds. If the dispersion by meandering is not taken into account, estimates of mean concentrations can easily be a factor of 4–6 too high.

The Effect of Line Averaging on Scalar Flux Measurements with a Sonic Anemometer near the Surface

Abstract We present a theoretical analysis of the effect of line averaging by a sonic anemometer on scalar fluxes and an observational study of this phenomenon in the atmospheric surface layer. The theoretical analysis rests on an axisymmetric model for the cross-spectral tensor of vertical velocity and scalar fluctuations, limiting the validity of the derived transfer function to a constant-flux layer. Observations of temperature flux in the unstable surface layer confirm the theoretical prediction that line averaging does not significantly affect the flux estimate down to heights only several times the sonic path length. However, the observations exhibit large scatter at small height, indicating that problems with the representativity of the measurement and not with line averaging may become a limiting factor. Based on the analysis of the Kansas data and the characteristics of the transfer function, we infer that temperature flux measurements will, in general, be affected by some line averaging under st...

The perennial cup anemometer

A short version of the history of the cup anemometer precedes a more technical discussion of the special features of this instrument. These include its extremely linear calibration and the non-linearity of its response to wind speed changes. A simple conceptual model by Schrenk is used to demonstrate this and to explain why the cup anemometer is able to start from a zero rotation rate at zero wind to one corresponding to a sudden change in the ambient wind speed to a finite value. The same model is used to show that the cup anemometer should be characterized by a distance constant rather than by a time constant. The bias in the measured mean wind speed due to the random variations in the three velocity components is discussed in terms of standard, semiquantitative turbulence models, and the main thesis is that this bias is overwhelmingly dominated by the fluctuations of the lateral wind velocity component, i.e. the wind component perpendicular to the mean wind direction, and not, as is often assumed, by the longitudinal wind velocity component. It is shown theoretically and tested experimentally that the bias due to lateral wind velocity fluctuations can be significantly reduced by means of a special data processing of the simultaneous signals from a cup anemometer and a wind vane. This means that, with care, the overall overspeeding can be reduced to less than 1%. Copyright