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

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.

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

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

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2H0

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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

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