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Poster 24 One-dimensional eddy diffusivities for growing turbulence in the convective boundary layer
Developments in Environmental Science, Volume 6 C. Borrego and E. Renner (Editors) Copyright r 2007 Elsevier Ltd. All rights reserved. ISSN: 1474-8177/DOI:10.1016/S1474-8177(07)06824-6
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Poster 24 One-dimensional eddy diffusivities for growing turbulence in the convective boundary layer Antonio Goulart, Umberto Rizza, Davidson Moreira, Marco T. Vilhena, Gerva´sio Degrazia and Jonas Carvalho Abstract In this work a general method to derive eddy diffusivities in a convective growing turbulence in the planetary boundary layer is proposed. The method is based in a model for the budget equation describing the 3-D energy density spectrum and the Taylor statistical diffusion theory. 1. Introduction
Exists a vast literature regarding the issues of parameterization and pollutant dispersion simulation in the Planetary Boundary Layer (PBL), but for Convective Boundary Layer (CBL) growing is scarce. In the work a general method to derive eddy diffusivities in a convective growing turbulence in the planetary boundary layer is proposed. The method is based in a model for the budget equation describing the 3-D energy density spectrum and the Taylor statistical diffusion theory. First, on the basis of a dimensional analysis, the unknown inertial transport term present in dynamical equation for the 3D spectrum is parameterized. The 3-D energy density spectrum equation is resolved. The vertical onedimensional vertical spectrum is derived from the 3-D spectrum decaying, employing a weight function that allows to select the magnitude of the vertical spectral component for the production of the decaying 3-D energy density spectrum. The eddy diffusivity is calculated from expression suggested by Goulart et al. (2004). 2. Turbulent energy equation in the growing CBL
In order to derive the spectral form of the turbulent energy equation we must recall that is possible to derive a spectral form of the turbulent
One-Dimensional Eddy Diffusivities for Growing Turbulence
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energy equation from the momentum conservation law, expressed through the Navier-Stokes equations. Indeed, for a homogeneous turbulent flow, the spectral form of the turbulent energy equation reads like Hinze (1975), @ g Eðk; t; zÞ ¼ W ðk; t; zÞ þ (1) Hðk; t; zÞ 2nk2 Eðk; t; zÞ @t T0 where: (g/T0)H(k, t; z) is the buoyancy term and W(k, t; z) is the energytransfer-spectrum function that represents the contribution due to the inertial transfer of energy among different wave-numbers. Is assumed that H(k, t; z) is pt 2=3 Hðk; t; zÞ ¼ c1 gc 0 k2=3 E 0 ðk; zÞ sin (2) 2tf where c1 is a constant to be determined from experiments or model simulations, and tf is the time at which the height of CBL becomes constant. Pao (1965) parameterized the term W(k, t) on the basis of dimensional analysis, as follows: @ 1 1=3 5=3 @ m2 2=3 1=3 k Eðk; tÞ (3) W ðk; t; zÞ ¼ ða k Eðk; tÞÞ @k @k w h where a is the Kolmogorov constant, e is the rate of molecular dissipation of kinetic energy, w is the velocity scale, h is the height of CBL and m2 is a dimensionless constant determined from initial conditions. Substituting Eqs. (3) and (2) in Eq. (1) yields an expression for the energy spectrum function E(k, t; z). The one-dimensional spectrum components in CLC is calculated as follows: Rt ð1=TÞ 0 F w ðk; t; zÞdt F w ðk; t; zÞ ¼ aðkÞ Eðk; t; zÞ (4) Rt ð1=TÞ 0 Eðk; t; zÞdt where the ratio between the two integrals is a weight function that indicates that the w component takes part in the construction of the 3D spectrum and a(k) is the proportionality constant. The eddy diffusivity is calculated as follows (see Pao, 1965): Z 0:55 1 E w ðk; t; zÞ sw hX sin K z ðt; zÞ ¼ k dk (5) sw 0 k 0:55w
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Antonio Goulart et al.
REFERENCES Goulart, A., Moreira, D., Carvalho, J., Tirabassi, T., 2004. Derivation of eddy diffusivities from an unsteady turbulence spectrum. Atmos. Environ. 38, 6121–6124. Hinze, J.O., 1975. Turbulence. McGraw-Hill, p. 790. Pao, Y.H., 1965. Structure of turbulent velocity and scalar fields at large wavenumbers. Physics Fluids 8, 1063.
808
Poster 24 One-dimensional eddy diffusivities for growing turbulence in the convective boundary layer Antonio Goulart, Umberto Rizza, Davidson Moreira, Marco T. Vilhena, Gerva´sio Degrazia and Jonas Carvalho Abstract In this work a general method to derive eddy diffusivities in a convective growing turbulence in the planetary boundary layer is proposed. The method is based in a model for the budget equation describing the 3-D energy density spectrum and the Taylor statistical diffusion theory. 1. Introduction
Exists a vast literature regarding the issues of parameterization and pollutant dispersion simulation in the Planetary Boundary Layer (PBL), but for Convective Boundary Layer (CBL) growing is scarce. In the work a general method to derive eddy diffusivities in a convective growing turbulence in the planetary boundary layer is proposed. The method is based in a model for the budget equation describing the 3-D energy density spectrum and the Taylor statistical diffusion theory. First, on the basis of a dimensional analysis, the unknown inertial transport term present in dynamical equation for the 3D spectrum is parameterized. The 3-D energy density spectrum equation is resolved. The vertical onedimensional vertical spectrum is derived from the 3-D spectrum decaying, employing a weight function that allows to select the magnitude of the vertical spectral component for the production of the decaying 3-D energy density spectrum. The eddy diffusivity is calculated from expression suggested by Goulart et al. (2004). 2. Turbulent energy equation in the growing CBL
In order to derive the spectral form of the turbulent energy equation we must recall that is possible to derive a spectral form of the turbulent
One-Dimensional Eddy Diffusivities for Growing Turbulence
809
energy equation from the momentum conservation law, expressed through the Navier-Stokes equations. Indeed, for a homogeneous turbulent flow, the spectral form of the turbulent energy equation reads like Hinze (1975), @ g Eðk; t; zÞ ¼ W ðk; t; zÞ þ (1) Hðk; t; zÞ 2nk2 Eðk; t; zÞ @t T0 where: (g/T0)H(k, t; z) is the buoyancy term and W(k, t; z) is the energytransfer-spectrum function that represents the contribution due to the inertial transfer of energy among different wave-numbers. Is assumed that H(k, t; z) is pt 2=3 Hðk; t; zÞ ¼ c1 gc 0 k2=3 E 0 ðk; zÞ sin (2) 2tf where c1 is a constant to be determined from experiments or model simulations, and tf is the time at which the height of CBL becomes constant. Pao (1965) parameterized the term W(k, t) on the basis of dimensional analysis, as follows: @ 1 1=3 5=3 @ m2 2=3 1=3 k Eðk; tÞ (3) W ðk; t; zÞ ¼ ða k Eðk; tÞÞ @k @k w h where a is the Kolmogorov constant, e is the rate of molecular dissipation of kinetic energy, w is the velocity scale, h is the height of CBL and m2 is a dimensionless constant determined from initial conditions. Substituting Eqs. (3) and (2) in Eq. (1) yields an expression for the energy spectrum function E(k, t; z). The one-dimensional spectrum components in CLC is calculated as follows: Rt ð1=TÞ 0 F w ðk; t; zÞdt F w ðk; t; zÞ ¼ aðkÞ Eðk; t; zÞ (4) Rt ð1=TÞ 0 Eðk; t; zÞdt where the ratio between the two integrals is a weight function that indicates that the w component takes part in the construction of the 3D spectrum and a(k) is the proportionality constant. The eddy diffusivity is calculated as follows (see Pao, 1965): Z 0:55 1 E w ðk; t; zÞ sw hX sin K z ðt; zÞ ¼ k dk (5) sw 0 k 0:55w
810
Antonio Goulart et al.
REFERENCES Goulart, A., Moreira, D., Carvalho, J., Tirabassi, T., 2004. Derivation of eddy diffusivities from an unsteady turbulence spectrum. Atmos. Environ. 38, 6121–6124. Hinze, J.O., 1975. Turbulence. McGraw-Hill, p. 790. Pao, Y.H., 1965. Structure of turbulent velocity and scalar fields at large wavenumbers. Physics Fluids 8, 1063.