Scaling structure of the velocity statistics in atmospheric boundary layers

Abstract

The statistical objects characterizing turbulence in real turbulent flows differ from those of the ideal homogeneous isotropic model. They contain contributions from various two- and three-dimensional aspects, and from the superposition of inhomogeneous and anisotropic contributions. We employ the recently introduced decomposition of statistical tensor objects into irreducible representations of the SO(3) symmetry group (characterized by j and m indices, where j=0ellipsisinfinity,-j</=m</=j) to disentangle some of these contributions, separating the universal and the asymptotic from the specific aspects of the flow. The different j contributions transform differently under rotations, and so form a complete basis in which to represent the tensor objects under study. The experimental data are recorded with hot-wire probes placed at various heights in the atmospheric surface layer. Time series data from single probes and from pairs of probes are analyzed to compute the amplitudes and exponents of different contributions to the second order statistical objects characterized by j=0, 1, and 2. The analysis shows the need to make a careful distinction between long-lived quasi-two-dimensional turbulent motions (close to the ground) and relatively short-lived three-dimensional motions. We demonstrate that the leading scaling exponents in the three leading sectors (j=0, 1, and 2) appear to be different but universal, independent of the positions of the probe, the tensorial component considered, and the large scale properties. The measured values of the scaling exponent are zeta((j=0))(2)=0.68+/-0.01, zeta((j=1))(2)=1.0+/-0.15, and zeta((j=2))(2)=1.38+/-0.10. We present theoretical arguments for the values of these exponents using the Clebsch representation of the Euler equations; neglecting anomalous corrections, the values obtained are 2/3, 1, and 4/3, respectively. Some enigmas and questions for the future are sketched.

Turbulence with pressure: Anomalous scaling of a passive vector field

“…It was shown that the leading terms of the inertial-range behavior are the same for isotropic and anisotropic forcing [57,58]. In the papers [59][60][61][62], the velocity correlation functions were decomposed in the irreducible representations of the rotation group. It was argued that in each sector of the decomposition, scaling behavior can be found with apparently universal exponents.…”

“…The picture outlined above for passively advected fields (a superposition of power laws with universal exponents and nonuniversal amplitudes) seems rather general, being compatible with that established recently in the field of NS turbulence, on the basis of numerical simulations of channel flows and experiments in the atmospheric surface layer; see Refs. [57][58][59][60][61][62] and references therein. It was shown that the leading terms of the inertial-range behavior are the same for isotropic and anisotropic forcing [57,58].…”

“…in Refs. [57][58][59][60][61][62] for description of the NS turbulence. This becomes especially clear if the left-hand side of Eq.…”

Spectrum of anisotropic exponents in hydrodynamic systems with pressure

“…Experiments and simulations on Navier-Stokes (NS) turbulence also indicate that anisotropic sectors possess larger scaling exponents than the isotropic sector [2][3][4]. However, to date, the exponents for ℓ > 2 were not determined with sufficient accuracy.…”

“…Instead, components of the structure functions that belong to different irreducible representations (sectors) of the SO(3) group possess different scaling exponents. Each of these sectors is characterized by the angular momentum indices ℓ and m. By projecting the structure function onto the different sectors, we could measure [2][3][4] the universal scaling exponents in each sector separately.…”