4 Model Model 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.
5 Main modeling techniques Direct Numeric Simulation (DNS) Reynolds Averaged Navier-Stokes (RANS) Large Eddy Simulation (LES)
6 DNS Model Numerical solution of the Navier-Stokes equation system. All scales of motion are solved. Does not have the closure problem.
7 Kolmogorov micro scale. l length scale of the most energetic eddies. Scales of turbulence
8 DNS model grid dilemma Number of grid points required for all length scales in a turbulent flow: PBL: Re ~ 10 7 DNS requires huge computational effort even for small Re flow (~1000).
9 DNS Model First 3-D turbulence simulations (NCAR) First published DNS work was for isotropic turbulence Re = 35 in a grid of 32 3 (Orszag and Patterson, 1972) Nowadays: grid
10 Small resolved scale in the DNS model Smallest length scale does not need to be the Kolmogorov microscale.
11 Reynolds Number How high should Re be? There are situations where to increase Re means only to increase the sub-inertial interval.
12 DNS Model – Final remarks It has been useful to simulate properties of less complex non-geophysical turbulent flows It 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
13 RANS Model 1. Diagnostic Model 2. Prognostic Model
15 Closure Problem Closure problem occurs when Reynolds average is applied to the equations of motion (Navier-Stoke). The number of unknown is larger than the number of equations.
16 Diagnostic RANS Model Diagnostic RANS model are a set of the empirical expressions derived from the similarity theory valid for the PBL. Zero order closure model
20 Advantages Simplicity Yields variances and characteristic length scales required for air pollution dispersion modeling applications
21 Does not provide height of PBL Valid only for PBL in equilibrium Valid only for PBL over horizontally homogeneous surfaces Restrict to PBL layers and turbulence regimen of the similarity theories Disadvantages
22 Prognostic RANS model Mixing Layer Model (1/2 Order Closure) First Order Closure Model Second Order Closure Model 1.5 Order Closure Model
23 Mixing Layer Model (1/2 Order Closure)
24 Mixing Layer Model Hypothesis: turbulent mixing is strong enough to eliminate vertical gradients of mean thermodynamic (θ = Potential temperature) and dynamic properties in most of the PBL.
26 Advantages Computational simplicity Yields a direct estimate of PBL height
27 Disadvantages Restrict to convective conditions (Stable PBL very strong winds) Does not give information about variance of velocity or characteristic length scales Can only be applied to dispersion of pollutants in the cases when the pollutant is also well mixed in the PBL
First Order Closure Model
29 First Order Closure Model Are based on the analogy between turbulent and molecular diffusion. λ is a characteristic length scale and u is a characteristic velocity scale. Vertical flux Diffusion coefficient
30 First order closure model
31 Advantage Computational simple Works fine for simple flow
33 Disadvantage Requires the determination of the characteristic length and velocity scales It 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 components It does not provide PBL height.
Second Order Closure Model
35 Second 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.
36 Reynolds Stress Tensor Equation Molecular dissipation Transport Tendency to isotropy
37 Parametrization 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 experiments Based on LES simulations
38 TKE balance in the PBL Stable Convective Destruição Térmica Produção Térmica
39 Advantages Provide a direct estimate of the PBL height. Provide a direct estimate of wind components variance.
40 Disadvantages High computational cost Does not provide a direct estimate of the characteristic length scale
1.5 Order Closure Model
Order closure model They are also based on the analogy between molecular and turbulent diffusion where the Turbulent diffusion coefficients are estimated in terms of the characteristic length and velocity scales Characteristic velocity scale is determined by resolving the TKE equation numerically
43 Turbulent kinetic energy (e) equation. 1.5 Order closure model
45 Example 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 direction Iperó Source: Pereira (2003)
46 Advantages Moderate computational cost (mesoscale model) Provides a direct estimate of the PBL height
47 One more equation to solve Extra length scales to estimate Does not provide a direct estimate of wind component variances Disadvantages
49 LES Model The motion equation are filtered in order to describe only motions with a length scale larger than a given threshold.