Amauri Pereira de Oliveira

1. Summer SchoolRio de JaneiroMarch 2009 3. PBL MODELING Amauri Pereira de Oliveira Group of Micrometeorology

2. Topics • Micrometeorology • PBL properties • PBL modeling • Modeling surface-biosphere interaction • Modeling Maritime PBL • Modeling Convective PBL

3. Part 3 PBL MODELLING

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. Scales of turbulence  Kolmogorov micro scale. l length scale of the most energetic eddies.

8. DNS model “grid dilemma” • Number of grid points required for all length scales in a turbulent flow: • PBL: Re ~ 107 • 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 323 (Orszag and Patterson, 1972) • Nowadays: grid 10243

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 Diagnostic Model 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

17. PBL Similarity Theory • Monin-Obukhov: Surface Layer (-1 < z/L < 1) • Free Convection: Surface Layer ( z/L < -1) • Mixing Layer Similarity: Convective PBL • Local Similarity: Stable PBL

20. Advantages • Simplicity • Yields variances and characteristic length scales required for air pollution dispersion modeling applications

21. Disadvantages • 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

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

28. First Order Closure Model

29. Vertical flux Diffusion coefficient 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.

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.

34. 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 Transport Tendency to isotropy Molecular dissipation

37. Based on laboratory experiments 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 LES simulations

38. TKE balance in the PBL Convective Destruição Térmica Stable 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

41. 1.5 Order Closure Model

42. 1.5 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. 1.5 Order closure model Turbulent kinetic energy (e) equation.

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. Disadvantages • One more equation to solve • Extra length scales to estimate • Does not provide a direct estimate of wind component variances

48. LES Model

49. LES Model The motion equation are filtered in order to describe only motions with a length scale larger than a given threshold.

50. Reynolds Average f