On a numerical model for extracting TKE dissipation rate from very high frequency (VHF) radar spectral width

Abstract

The standard model (e.g., Hocking in Earth Planets Space 51:525–541, 1999), varepsilon = c_{0} sigma^{2} N, (where sigma is the radar spectral width assumed to be equal to vertical turbulence velocity fluctuation sqrt {overline{{w^{2} }} }, N is the buoyancy frequency, and c0 is a constant), derived from Weinstock (J Atmos Sci 35:1022–1027, 1978; J Atmos Sci 38:880–883, 1981) formulation, has been used extensively for estimating the turbulence kinetic energy (TKE) dissipation rate varepsilon under stable stratification from VHF radar Doppler spectral width sigma. The Weinstock model can be derived by simply integrating the TKE spectrum in the wavenumber space from the buoyancy wavenumber k_{text{B}} = frac{N}{sigma } to infty. However, it ignores the radar volume dimensions and hence its spatial weighting characteristics. Labitt (Some basic relations concerning the radar measurements of air turbulence, MIT Lincoln Laboratory, ATC Working Paper NO 46WP-5001, 1979) and White et al. (J Atmos Ocean Technol 16:1967–1972, 1999) formulations do take into account the radar spatial weighting characteristics, but assume that the wavenumber range in the integration of TKE spectrum extends from 0 to infty. The White et al. model accounts for wind speed effects, whereas the other two do not. More importantly, all three formulations make the assumption that k−5/3 spectral shape of TKE spectrum extends across the entire wavenumber range of integration. It is traditional to use Weinstock formulation for k_{text{B}}^{ - 1} < 2a,2b (where a and b are radar volume dimensions in the horizontal and vertical directions) and White et al. formulation (without wind advection) for k_{text{B}}^{ - 1} > 2a,2b. However, there is no need to invoke these asymptotic limits. We present here a numerical model, which is valid for all values of buoyancy wavenumber k_{text{B}} and transitions from varepsilon simsigma^{2} behavior at lower values of sigma in accordance with Weinstock’s model, to varepsilon simsigma^{3} at higher values of sigma, in agreement with Chen (J Atmos Sci 31:2222–2225, 1974) and Bertin et al. (Radio Sci 32:791–804, 1997). It can also account for the effects of wind speed, as well the beam width and altitude. Following Hocking (J Atmos Terr Phys 48:655–670, 1986, Earth Planets Space 51:525–541, 1999), the model also takes into account contributions of velocity fluctuations beyond the inertial subrange. The model has universal applicability and can also be applied to convective turbulence in the atmospheric column. It can also be used to explore the parameter space and hence the influence of various parameters and assumptions on the extracted varepsilon values. In this note, we demonstrate the utility of the numerical model and make available a MATLAB code of the model for potential use by the radar community. The model results are also compared against in situ turbulence measurements using an unmanned aerial vehicle (UAV) flown in the vicinity of the MU radar in Shigaraki, Japan, during the ShUREX 2016 campaign.