Surface-Layer Wind and Turbulence profiling from LIDAR: Theory ...

More documents

Recommendations

Info

Altitude [m] Length of the beam, p’[m] Diameter of scanning cone [m] 40m 43.88m 44.77m 60m 66.97m 68.18m 80m 90.07m 91.70m 100m 113.16m 115.21m 200m 228.64m 232.77m against the direction of the wind, maximum Doppler shifts occur, see. Figure 3. Tabel 1: Geometrical characteristics of the conical scans The Doppler shift results from the difference of the wave vector k r e for the emission and k r r for the reflection. Let B r be the propagation vector of the transmitted Laser beam: Fig. 3:30minute of raw wind measurements obtained from Doppler shifts as function of conical scan angle [0; 2π]. In Fig. 3 the observed Doppler shifts have been transformed into velocities by the following equation: st ( , ) = ut ( )cos( − )sin( Φ ) + wt ( )cos( Φ) (3) θ θ θ d Fig. 2: Details of the interaction between the laser radiation and the horizontal component of the wind The mean wind direction is aligned along the xaxis: r µ ≡ u r , r r v = w= 0 r . To calculate the resulting Doppler shift, we define two frequencies, one representing the emission of the signal, and one representing its reception: r r ⎛ u ⎞ ω' e = ω 0 + ke. µ ⇔ ν ' e = ν ⎜1− cos( θ)sin( Φ) ⎟ ⎝ c ⎠ (1) r r ⎛ u ⎞ ω' r = ω 0 + kr. µ ⇔ ν ' r = ν ⎜1+ cos( θ)sin( Φ) ⎟ ⎝ c ⎠ where ν is the frequency [Hz]. The modulus of wave vector is r 2πν defined as k = . The Doppler shift of an elastic c scattered photon results as the difference between the outgoing and the backscattered radiation: u ∆ ν = ν ' r− ν ' e = 2 ν cos( θ)sin( Φ ) (2) c This represents the Doppler shift detected by the Lidar. The quantity is the angle between the aerosol transported by the wind and the beam. When /2 or 3/2, the Doppler shift is zero. The ideal cases are =0 or ; but due to the conical scanning at fixed angle to vertical there will always be nonzero Doppler shifts wherever the wind comes from. The Doppler shift frequency signal that appears on the Lidar’s detector will give the following typical picture if we sample complete rotations. When the Laser beam is pointing into or and then shown as function of scan angle . A curve has been fitted through the data points. The fitted curve allows calculation of u, w and if the Lidar has been properly oriented (to the North) a curve fit also yields the direction d of the mean wind speed. In normal operation mode the horizontal wind vector is determined from 3 full conical rotations of 1second duration each. During each revolution a total of 25 Doppler spectra are obtained with the present instrument. An average wind speed from the conical scans is then obtained by a curve fit to 3 full rotations of such Doppler measurements and encompasses therefore up to 75 raw spectra as obtained over three revolutions of 2. u(t), w(t) and d are consequently representing averages over 3 seconds in time. Geometry of the Gaussian beam: The radius and the focal width of the Laser beam changing as function of the measurement height and the effective measurement volume is consequently determined by the system optics. The laser beam is characterized by its Rayleigh length z R . 2 πW0 z R = 50.6µ m λ At the output of the fibre this quantity is = , where W 0 is the radius of the beam in the optical fiber. This Rayleigh length represents the halfwidth where most of the power of the laser beam is gathered, according to the Lorentzian distribution of energy among the beam. By adjusting the fiber end near the focal point of the lens, the beam can be focused to take wind measurements at preset heights. The characteristics of the laser beam at the fiber end is via the lens projected to the external focal volume of length z (Rayleigh length) and of focal width W. At a distance p’from the lens the beams crosssection is given by 1 : R
2 ⎛ 2 2 ⎛ p' ⎞ ⎞ W ( p') = W 0 1 ⎜ +⎜ ⎟ z ⎟ ⎝ ⎝ R ⎠ ⎠ The outgoing beam diameter of the ZephIR lidar is ~ 4 cm so that the maximum power density at the instrument is of order of ~ 2 10 2 [ Wcm 2 ]. The Laser beam is focused at the preset distance corresponding to the decided measurement height. In the following the position of the output of the external fibre before the lens is donoted p; and the position of the waist after the lens by p’. When the Laser beam is focused, the characteristic (radius and focal width) will change, as shown in Fig.4. The radius and the focal width, W out (p’) and 2Z’ R , respectively evolve as function of measurement height. (4) radius increases linearly with distance while the focal width increases with distance squared. Wout [mm] 6 5 4 3 2 1 0 Wout(p') 2 Z'r 0 50 100 150 200 250 H [m] 140 120 100 80 60 40 20 0 2 Z'r [m] Fig. 6: Overview of the radius (W out , in blue) and the focal width (2Z’ R, in red) for different altitudes of scanning (H) Fig. 4: Beam propagation Fig.5 shows details at the waist position (focal point) for the focal width (2Z R ’), the radius W out (p’), and the angle of deflection of the Laser beam ’. Fig. 5: Details of the characteristics of the focalized Laser beam As p moves closer (within a fraction of a millimeter) to the focal point of the lens (F=20cm), the further away will the external Laser beam be focused (the waist location, p’) : The volume of the beam, from where plane coherent radiation backscatters, can to a first approximation be considered a doubletruncated cone (See figure 5). The radius of the beam is very small; the focus volume is always very small (0.153 litre at 60m height, 1.174 litre at 100m height). As an approximation we will use only the quantity +/ Z’ R to express the length where the measure will be done. The radius of the beam (blue) increases linearly with the altitude, and is the order of the millimetres within our range of altitude of measurements (40m100m); but the focal width 2Z’ R (red curve) on the other hand grows quadratic about 1 orders of magnitude (from 4m at 40m until 32m at 100m). The focal width +/ “Z R ’represents the range where about 50% of the focussed radiation is gathered, and represents the range at which wind measurements will be taken. During an experiment at the test site Høvsøre in northwestern part of Denmark, the Lidar have been evaluate with respect to precision [2] and [3]. The Lidar data have been compared with calibrated Risø cup anemometers and Metek Sonics (Model USA1) all located on the met tower, the Lidar was installed 200m away from the tower, an example of the comparison are showen in figure 7 (Here measured at 80m height). This data set is obtained over a period of four days of measurements. The data have been filtered because of rainy days, or direction changes resulting in wake from the turbines at the test site and tower effects. H [m] p’[m] W out (p’) [mm] 2Z’ R [m] 40m 43.88m 1mm 4.8m 60m 66.97m 1.7mm 11m 80m 90.07m 2mm 20.2m 100m 113.42m 2.8mm 32.4m 200m 228.64m 5.6mm 128.7m Tab. 2: Main data of the conical scan for different altitudes of scanning Figure 6 shows the characteristics of the external beam as function of measurement distance H. The focussed beams
Page 1: SurfaceLayer Wind and Turbulence
Page 5 and 6: specific points, then squaring and

Surface-Layer Wind and Turbulence profiling from LIDAR: Theory ...

Create successful ePaper yourself

Delete template?

Save as template?