Scalar spectra in the near-dissipation range derived from structure functions

Laser scintillometers (both single and double beam) are sensitive to temperature (and humidity) fluctuations with sizes between inertial range and dissipation range (Hartogensis et al., 2002). Furthermore, large aperture scintillometers become sensitive to fluctuations at those small scales when the signal becomes saturated (Kohsiek et al., 2006). Despite their importance for the interpretation of scintillometer measurements, the exact shapes of the spectra of temperature and humidity are still a point of debate.

In state-of-the-art scintillometry (Hartogensis, 2002), one uses Hill’s (1978) model spectrum to interpret the measured samples. This model is pragmatic and leads to a fair match with experimental data for the scalar spectra. We present an alternative method to estimate the scalar spectrum, relying only on assumptions that can be experimentally verified.

The method, which has been inspired by the recent work of Tatarskii (2005), is based on the simultaneous solution of the differential equations for the second-order structure functions of temperature (D_TT) and longitudinal velocity (D_uu). The third-order structure functions occurring in those differential equations are closed via the second-order structure functions and the validity of these closure relations is checked against experimental evidence.

Following Obukhov and Yaglom (1951) we first assume the closure relations to be independent of separation. Secondly, we use recent DNS results of Watanabe and Gotoh (2004) to incorporate the separation-dependence.

The resulting spectrum exhibits a similar bump as the solution of Hill's model (scalar variance piles up in the spectrum where turbulence does no longer break up the patches of scalar variance into even smaller bits). The exact location and height of the bump depends on the two closure parameters. We assess their values from the inertial subrange in the spectra of field data. Using the natural spread in the two closure parameters and our model for the temperature spectrum, the natural spread in the shape and location of the bump can be assessed. Consequently, it is shown how the variation in the assumed temperature spectrum influences the outcomes of flux calculations of a displaced-beam scintillometer.

Our resulting spectrum is also compared to the same experimental data as done in Hill (1978).

Hartogensis, O.K., DeBruin, H.A.R., Wiel, B.J.H. van de (2002). Displaced-beam small aperture, scintillometer test. Part II: CASES-99 stable boundary layer experiment. Boundary-Layer Meteorol. 105, 149 - 176.

Hill R. (1978). Models of the scalar spectrum for turbulent advection. J. Fluid Mech., 88, 541-562.

Kohsiek W., Meijninger W., De Bruin H. and Beyrich F. (2006). Saturation of the large-aperture scintillometer. Boundary-Layer Meteorol. in press.

Obukhov A. and Yaglom A. (1951). The microstructure of turbulent flow. Prikl. Mat. Mekh., 15, 3. In Russian. Translated into English in NACA Technical Memorandum 1350 (1953).

Tatarskii V. (2005). Use of the 4/5 Kolmogorov equation for describing some characteristics of fully developed turbulence. Phys. Fluids, 14, 035110-1-12.

Watanabe T. and Gotoh T. (2004). Statistics of a passive scalar in homogeneous turbulence. New Journal of Physics, 6, 40.