Turbulence anisotropy carried by streaks in the neutral atmospheric surface layer

Summary

We investigate the relationships between coherent structures and turbulence anisotropy in the neutral planetary boundary (PBL) layer by an empirical orthogonal function (EOF) analysis of large-eddy simulation (LES) data. The simulated flow contains near-surface streaks similar to those revealed by recent observations. The EOF analysis aims at extracting recurrent patterns from the signal based on its spatial correlations. Not all horizontal wave-vectors present recurrent patterns : we identify a band in Fourier space where the first EOF carries significantly more energy than the next ones. The EOF streamlines bear more resemblance with optimal perturbations of an Ekman boundary layer than with unstable normal modes. The scale and direction of streaks estimated from an examination of the LES flow field correspond to that of the EOFs. Thus these patterns are indeed produced by structures with a spatial coherence on the vertical. The corresponding structure of the turbulence is strongly altered. For horizontal wave vectors belonging to this band, the turbulence extends on a vertical range much smaller than for wave vectors with comparable magnitude and unspecified direction. This observation is particularly true for the fluctuations of horizontal velocity and less for vertical velocity. The velocity fluctuations are also distributed in a less isotropic way, as made apparent by the depleted ratio of vertical and horizontal turbulent kinetic energies (TKEs).

1. Introduction

Important features of the near-surface turbulence in the neutrally-stratified PBL are a strong anisotropy and the presence of coherent structures. The near-surface flow is characterized by streaks, alternating bands of higher and lower streamwise velocity. (Foster, 1997) pointed out that optimal perturbations of the Ekman flow present large transient amplifications and that their scale and orientation are in broad agreement with those of near-surface streaks. Linear analyzes lack however all of the nonlinear mechanisms that support turbulence. Especially, the decay of streaks appears to be a nonlinear process (Drobinski and Foster, 2003). Concomitant with the presence of streaks, turbulence displays anisotropic characteristics: the variances of the wind velocity fluctuations differ for the three components ; close to the ground, the wind velocity spectra deviate from the -5/3 spectral slope expected for isotropic turbulence with the existence of a -1 power law at intermediate spectral subrange (Katul and Chu, 1998 ; Hunt and Carlotti, 2001).

We consider two aspects of anisotropy : the dependence of turbulent quantities on the orientation of the horizontal wave vector relatively to the geostrophic wind, and the vertical/horizontal ratio of the velocity components and of the extracted extracted flow patterns.

2. Significant EOFs

The simulation runs the non-hydrostatic LES model Méso-NH and models a neutrally-stratified atmospheric boundary layer in a box with size (L,l,H)=(3km,3km,750m). The mesh cell is a cube of side 6.25m. The lateral boundary conditions are periodic while the ground surface is rough with a roughness length z0=10cm.

The EOFs are periodic, similar in structure to the normal modes of the linear theory. For a given horizontal wave vector, indexed by two integers m and n, the vertical structure of the EOFs is obtained by diagonalizing the matrix made of velocity cross-correlations at different altitudes. We consider that when the energy fraction explained by the first EOF exceeds a threshold of 50% the wave vector is "strongly structured". Such wave vectors lie in a narrow range of the Fourier space close the line m+n=0, in agreement with qualitative observation of streaks.

3. Anisotropy

3a. Vertical extent

Since the EOFs represent the correlations between the different altitudes, the latter are never considered independently. Therefore we characterize the vertical extent of the turbulent structures through their TKE center, the column average of altitude weighted by the TKE. The height Z(m,n) can then be interpreted as a penetration height up to which a given Fourier mode or EOF contributes significantly to its part of the more commonly used horizontal Fourier spectrum.

We compute Z(m,n) as a function of the horizontal wavenumber, using as weigh the horizontal energy of either the corresponding Fourier mode or the dominant EOF. Short structures have a small vertical extent and the TKE center is much lower for wave vectors on the line m+n=0 than for wave vectors of comparable magnitude and different direction. This anisotropy is more pronounced for the TKE center based on first EOFs. We compute and discuss the TKE center obtained from the vertical energy as well.

3b. Vertical vs horizontal turbulent kinetic energy

If the velocity fluctuations had isotropically distributed directions, as in locally isotropic turbulence, the ratio of the vertical TKE over the horizontal TKE would be equal to 1/2. We display this ratio as a function of the horizontal wavenumber. It approaches the isotropic value of 1/2 only for very small scales. Elsewhere the TKE ratio is in favor of the horizontal TKE, especially at large horizontal scales but also for ``strongly structured'' wave vectors. We display the ratio of the contribution of the first EOF to the vertical and horizontal TKEs as well. The features discussed above are clearly visible in an even more pronounced way.

One expects isotropy to be violated by large scale structures since their vertical extent is limited by the PBL depth, resulting in a low geometric aspect ratio. Because of incompressibility, such structures have a low vertical velocity. Conversely, horizontally short structures are free to achieve an aspect ratio of 1 or more and the corresponding TKE ratio. This is clearly not the case at ``strongly structured'' wave vectors for which a typical ratio is about 0.2 even at large wave vectors. This is consistent with the observation that the corresponding flow patterns are closer to the ground that those of equivalent horizontal scale and different direction, resulting in a lower vertical/horizontal aspect ratio. The TKE ratio for the total signal is about 0.17 which is remarkably similar to the ratio found for the ``strongly structured'' wave vectors. Hence the ``strongly structured'' wave vectors carry a turbulence anisotropy representative of the anisotropy present in the total signal.