amauri pereira de oliveira group of micrometeorology summer school rio de janeiro march 2009 3. pbl...
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Amauri Pereira de OliveiraGroup of
Micrometeorology
Summer SchoolRio de JaneiroMarch 2009
3. PBL MODELING
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Topics
1. Micrometeorology
2. PBL properties
3. PBL modeling
4. Modeling surface-biosphere interaction
5. Modeling Maritime PBL
6. Modeling Convective PBL
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Part 3
PBL MODELLING
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ModelModel 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.
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Main modeling techniques
• Direct Numeric Simulation (DNS)
• Reynolds Averaged Navier-Stokes (RANS)
• Large Eddy Simulation (LES)
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DNS Model
• Numerical solution of the Navier-Stokes equation system.
• All scales of motion are solved.
• Does not have the closure problem.
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Kolmogorov micro scale.
l length scale of the most energetic eddies.
Scales of turbulence
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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).
493 Re
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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
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Small resolved scale in the DNS model
Smallest length scale does not need to be the Kolmogorov microscale.
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Reynolds Number
How high should Re be?
There are situations where to increase Re means only to increase the sub-inertial interval.
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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
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RANS Model
1. Diagnostic Model
2. Prognostic Model
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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.
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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
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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
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Advantages
• Simplicity
• Yields variances and characteristic length scales required for air pollution dispersion modeling applications
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• 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
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Prognostic RANS model
• Mixing Layer Model (1/2 Order Closure)
• First Order Closure Model
• Second Order Closure Model
• 1.5 Order Closure Model
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Mixing Layer Model (1/2 Order Closure)
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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.
0z
M
bzaw
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Advantages
• Computational simplicity
• Yields a direct estimate of PBL height
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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
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First Order Closure Model
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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.
z
uKwu M
Vertical flux
uK 1M
Diffusion coefficient
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First order closure model
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Advantage
• Computational simple
• Works fine for simple flow
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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.
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Second Order Closure Model
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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.
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Reynolds Stress Tensor Equation
Molecular dissipation
Transport Tendency to isotropy
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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
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TKE balance in the PBL
Stable
Convective
Destruição Térmica
Produção Térmica
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Advantages
•Provide a direct estimate of the PBL height.
•Provide a direct estimate of wind components variance.
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Disadvantages
•High computational cost
•Does not provide a direct estimate of the characteristic length scale
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1.5 Order Closure Model
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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
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z
eK
ze
ee
zK
g
z
v
z
uK
t
eM1
23
2H0
22
M
Turbulent kinetic energy (e) equation.
1.5 Order closure model
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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)
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Advantages
• Moderate computational cost (mesoscale model)
• Provides a direct estimate of the PBL height
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One more equation to solve
Extra length scales to estimate
Does not provide a direct estimate of wind component variances
Disadvantages
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LES Model
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LES Model
The motion equation are filtered in order
to describe only motions with a length
scale larger than a given threshold.
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Reynolds Average
f
)x('f)x(f)x(f
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LES Filter
)x(f)x(f~
)x(f
f
large eddies
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Convective Boundary Layer
Updraft
Source: Marques Filho (2004)
Cross section
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Convective PBL – LES Simulation
Source: Marques Filho (2004)
( zi /L ~ - 800)
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Spectral Properties – LES Simulation
Fonte: Marques Filho (2004)
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Advantages
Large scale turbulence is simulated
directly and sub grid (less dependent on
geometry flow) is parameterized.
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Disadvantages
Computational cost is high