Amauri Pereira de Oliveira Summer School Rio de Janeiro March 2009 3. PBL MODELING Amauri Pereira de Oliveira Group of Micrometeorology

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

Part 3 PBL MODELLING

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.

Main modeling techniques Direct Numeric Simulation (DNS) Reynolds Averaged Navier-Stokes (RANS) Large Eddy Simulation (LES)

DNS Model Numerical solution of the Navier-Stokes equation system. All scales of motion are solved. Does not have the closure problem.

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

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).

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

Small resolved scale in the DNS model Smallest length scale does not need to be the Kolmogorov microscale.

Reynolds Number How high should Re be? There are situations where to increase Re means only to increase the sub-inertial interval.

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

RANS Model Diagnostic Model Prognostic Model

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.

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

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

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

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

Prognostic RANS model Mixing Layer Model (1/2 Order Closure) First Order Closure Model Second Order Closure Model 1.5 Order Closure Model

Mixing Layer Model (1/2 Order Closure)

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.

Advantages Computational simplicity Yields a direct estimate of PBL height

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

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.

First order closure model

Advantage Computational simple Works fine for simple flow

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

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.

Reynolds Stress Tensor Equation Transport Tendency to isotropy Molecular dissipation

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

TKE balance in the PBL Convective Stable Destruição Térmica Produção Térmica

Advantages Provide a direct estimate of the PBL height. Provide a direct estimate of wind components variance.

Disadvantages High computational cost Does not provide a direct estimate of the characteristic length scale

1.5 Order Closure Model

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

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

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)

Advantages Moderate computational cost (mesoscale model) Provides a direct estimate of the PBL height

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

LES Model

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

Reynolds Average f

LES Filter f large eddies

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

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

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

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

Disadvantages Computational cost is high