Amauri Pereira de Oliveira

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Summer SchoolRio de JaneiroMarch 2009 3. PBL
MODELING

• Amauri Pereira de Oliveira

Group of Micrometeorology
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Topics

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

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Part 3
PBL MODELLING
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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.
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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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Scales of turbulence
? Kolmogorov micro scale. l length scale of
the most energetic eddies.
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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).

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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

• Diagnostic Model
• 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 lt z/L lt 1)
• Free Convection Surface Layer ( z/L lt -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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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

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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 (? Potential temperature) and
dynamic properties in most of the PBL.
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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.
? is a characteristic length scale and u is a
characteristic velocity scale.
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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
Transport
Tendency to isotropy
Molecular dissipation
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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 LES simulations
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TKE balance in the PBL
Convective
Destruição Térmica
Stable
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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1.5 Order closure model
Turbulent kinetic energy (e) equation.
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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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Disadvantages

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

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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
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LES Filter
f
large eddies
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Convective Boundary Layer
Cross section
Updraft
Source Marques Filho (2004)
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Convective PBL LES Simulation
( zi /L - 800)
Source Marques Filho (2004)
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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