4ModelModel is a tool used to simulate or forecast the behavior of a dynamic system.Models are based on heuristic methods, statistics description, analytical or numerical solutions, simple physical experiments (analogical model). etc.Dynamic system is a physical process (or set of processes) that evolves in time in which the evolution is governed by a set of physical laws.Atmosphere is a dynamic system.Model hereafter will always implies numerical model.
6DNS Model Numerical solution of the Navier-Stokes equation system. All scales of motion are solved.Does not have the closure problem.
7Scales of turbulence Kolmogorov micro scale. l length scale of the most energetic eddies.
8DNS model “grid dilemma” Number of grid points required for all length scales in a turbulent flow:PBL: Re ~ 107DNS requires huge computational effort even for small Re flow (~1000).
9DNS Model First 3-D turbulence simulations (NCAR) First published DNS work was for isotropic turbulence Re = 35 in a grid of 323 (Orszag and Patterson, 1972)Nowadays: grid 10243
10Small resolved scale in the DNS model Smallest length scale does not need to be the Kolmogorov microscale.
11Reynolds Number How high should Re be? There are situations where to increase Re means only to increase the sub-inertial interval.
12DNS Model – Final remarks It has been useful to simulate properties of less complex non-geophysical turbulent flowsIt is a very powerful tool for research of small Re flows (~ 1000)The application of DNS model for Geophysical flow is is still incipient but very promising
20Advantages Simplicity Yields variances and characteristic length scales required for air pollution dispersion modeling applications
21Disadvantages Does not provide height of PBL Valid only for PBL in equilibriumValid only for PBL over horizontally homogeneous surfacesRestrict to PBL layers and turbulence regimen of the similarity theories
22Prognostic RANS model Mixing Layer Model (1/2 Order Closure) First Order Closure ModelSecond Order Closure Model1.5 Order Closure Model
26Advantages Computational simplicity Yields a direct estimate of PBL height
27DisadvantagesRestrict to convective conditions (Stable PBL very strong winds)Does not give information about variance of velocity or characteristic length scalesCan only be applied to dispersion of pollutants in the cases when the pollutant is also well mixed in the PBL
29First Order Closure Model Are based on the analogy between turbulent and molecular diffusion.Vertical fluxDiffusion coefficientλ is a characteristic length scale and u is a characteristic velocity scale.
33DisadvantageRequires the determination of the characteristic length and velocity scalesIt can not be applied for all regions and stability conditions present in the PBL (turbulence is a properties of the flow)It does not provide variances of the wind speed componentsIt does not provide PBL height.
35Second Order Closure Model SOCM are based on set of equations that describe the first and second order statistic moments and parameterizing the third order terms.
36Reynolds Stress Tensor Equation TransportTendency to isotropyMolecular dissipation
37Parametrization Donaldson (1973) Mellor and Yamada (1974) André et al. (1978)Mellor and Yamada (1982)Therry and Lacarrére (1983)Andrên (1990)Abdella and MacFarlane (1997)Galmarini et al. (1998)Abdella and MacFarlane (2001)Nakanishi (2001)Vu et al. (2002)Nakanishi and Niino (2004)Based on laboratory experimentsBased on LES simulations
38TKE balance in the PBL Convective Stable Destruição Térmica Produção Térmica
39Advantages Provide a direct estimate of the PBL height. Provide a direct estimate of wind components variance.
40Disadvantages High computational cost Does not provide a direct estimate of the characteristic length scale
421.5 Order closure modelThey are also based on the analogy between molecular and turbulent diffusion where theTurbulent diffusion coefficients are estimated in terms of the characteristic length and velocity scalesCharacteristic velocity scale is determined by resolving the TKE equation numerically
431.5 Order closure modelTurbulent kinetic energy (e) equation.
45Example of PBL structure simulated numerically during convective period using mesoscale model with a 1.5 order closure (Iperó, São Paulo, Brazil)Cross section in the East-West directionIperóSource: Pereira (2003)
46Advantages Moderate computational cost (mesoscale model) Provides a direct estimate of the PBL height
47Disadvantages One more equation to solve Extra length scales to estimateDoes not provide a direct estimate of wind component variances