33
3
Turbulence and
Dispersion of
Contaminants in the
Planetary Boundary Layer
Gervásio Annes Degrazia, Antonio Gledson
Oliveira Goulart, and Debora Regina Roberti
CONTENTS
3.1 Introduction .................................................................................................... 33
3.1.1 Taylor’s Model .................................................................................... 35
3.1.1.1 Some Considerations about Taylor’s Model ......................... 38
3.2 The Wiener–Khinchin Theorem: Selecting Energy-Containing Eddies ........ 39
3.3 Relations between Lagrangian and Eulerian Statistics ..................................44
3.4 A Heuristic Formulation for the Scale Factor β i .............................................46
3.5 Derivation of an Eddy Diffusivity for Inhomogeneous Turbulence
in a Convective Boundary Layer .................................................................... 50
3.5.1 Turbulent Transport Modeling of Contaminants during the
Decaying of a CBL ............................................................................. 55
3.5.1.1 Energy Density Spectrum Dynamical Equation ................. 56
3.6 Analysis of the Low-Frequency Horizontal Wind Oscillations
Employing the Navier–Stokes Equations ....................................................... 62
3.6.1 Analytical Solution of the Simplifi ed Navier–Stokes Equations ........ 62
References ................................................................................................................66
3.1 INTRODUCTION
We consider here turbulent dispersion in geophysical fl ows, such as the planetary
boundary layer (PBL), where Reynolds numbers are very large (≈10 7 , Wyngaard,
1982), so that all or some of the possible symmetries permitted by the equations (and
the boundary conditions) are restored in a statistical sense and turbulence is known
as fully developed turbulence (FDT) (Frisch, 1995). Furthermore, the FDT in the
PBL shows scale-invariance (small eddies are as space-fi lling as large ones) and selfsimilarity (the general aspect of turbulent signal is independent of where the window
© 2010 by Taylor and Francis Group, LLC
3
Turbulence and
Dispersion of
Contaminants in the
Planetary Boundary Layer
Gervásio Annes Degrazia, Antonio Gledson
Oliveira Goulart, and Debora Regina Roberti
CONTENTS
3.1 Introduction .................................................................................................... 33
3.1.1 Taylor’s Model .................................................................................... 35
3.1.1.1 Some Considerations about Taylor’s Model ......................... 38
3.2 The Wiener–Khinchin Theorem: Selecting Energy-Containing Eddies ........ 39
3.3 Relations between Lagrangian and Eulerian Statistics ..................................44
3.4 A Heuristic Formulation for the Scale Factor β i .............................................46
3.5 Derivation of an Eddy Diffusivity for Inhomogeneous Turbulence
in a Convective Boundary Layer .................................................................... 50
3.5.1 Turbulent Transport Modeling of Contaminants during the
Decaying of a CBL ............................................................................. 55
3.5.1.1 Energy Density Spectrum Dynamical Equation ................. 56
3.6 Analysis of the Low-Frequency Horizontal Wind Oscillations
Employing the Navier–Stokes Equations ....................................................... 62
3.6.1 Analytical Solution of the Simplifi ed Navier–Stokes Equations ........ 62
References ................................................................................................................66
3.1 INTRODUCTION
We consider here turbulent dispersion in geophysical fl ows, such as the planetary
boundary layer (PBL), where Reynolds numbers are very large (≈10 7 , Wyngaard,
1982), so that all or some of the possible symmetries permitted by the equations (and
the boundary conditions) are restored in a statistical sense and turbulence is known
as fully developed turbulence (FDT) (Frisch, 1995). Furthermore, the FDT in the
PBL shows scale-invariance (small eddies are as space-fi lling as large ones) and selfsimilarity (the general aspect of turbulent signal is independent of where the window
© 2010 by Taylor and Francis Group, LLC