1. Amauri Pereira de Oliveira
Summer School Rio de Janeiro March 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. First Order Closure Model
Are based on the analogy between turbulent and molecular diffusion.
Vertical flux
Diffusion coefficient
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. 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 Convective Stable 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

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

51. LES Filter f
large eddies

52. Convective Boundary Layer
Cross section
Updraft
Source: Marques Filho (2004)

53. Convective PBL – LES Simulation
( zi /L ~ - 800)
Source: Marques Filho (2004)

54. Spectral Properties – LES Simulation
Fonte: Marques Filho (2004)

55. Advantages
Large scale turbulence is simulated directly and sub grid (less dependent on geometry flow) is parameterized.

56. Disadvantages
Computational cost is high