Some Definitions

• Turbulence:– Usually includes the terms “random”

and “unpredictable”• Becomes:

– Turns in to; produces as a result

• Chaos– Usually includes the terms “apparently

random behavior”, “nonlinear dynamics”

– Also: A condition or place of great disorder or confusion

• PBL– A place where Turbulence is dominant,

a Chaos (coherent structure) solution reigns; and disorder & confusion exist

R. A. Brown 2003 U. ConcepciÓn

Coherent Structure

• Organization within an otherwise turbulent field

• Nonlinear

• Vortex solution

• Contains same parcels

• In the PBL, Organized Large Eddies (OLE)

• Nomenclature problem with ‘Chaos’

R. A. Brown 2003 U. ConcepciÓn

Status of PBL Modeling< 1960

• Ekman 1904

• Ackerblom 1908

• Taylor 1915

• Boussinesq 1903; K-Theory, Mixing length

• Leipzig et al. K(z) profiles 1950

• Two-layer model, Rossby & Montgomery 1935

R. A. Brown 2003 U. ConcepciÓn

PBL Modelingcirca 1970

• K-Theory continued• Log layer/surface layer

– Monin Similarity, 1971

• Dimensional Analysis Similarity, parameters A, B & C

• Higher Order Closure• Ekman Instability, 1960s• Nonlinear solution with Coherent

Structures, 1970• Large-Eddy Numerical Simulation

1971 (Deardorff)

R. A. Brown 2003 U. ConcepciÓn

PBL Modeling~ 1980

• K-Theory continued

• LES Modeling

• Two-layer models with Surface Layer patched to a nonlinear Ekman layer, 1974– Analytic Similarity , with

A(), B() 1974– Coherent Structures re HOC

• Remote Sensing Applications --- global data

R. A. Brown 2003 U. ConcepciÓn

PBL Models1990s

• See 1980s

R. A. Brown 2003 U. ConcepciÓn

Some K-theory Models

Date Author K Remarks

1902 Ekman Constant 3-D Spiral for ocean

1905 Ekman ~ |dV/dz| Equi-angle spiral

1908 Akerblom Constant It works in Atmosphere

1915 Taylor Constant V(0) > 0

1930 Prandtl, vK L2 |dV/dz| Mixing length; k

1930 Takaya ~z2, eaz

1933 Kohler Zn Bessel’s equation

1935 Rossby C(z+zo),K Two-layer model

1936 Blinova Kz Bessel’s equation

1940 Kibel Ku*z U* = (o/)½

1940 Yudin/Shvets Kz, Kh Two-layer

1950 Lettau Empirical Leipzig profile

1962 Blackadar [k|dV/dz]2

1+kz/C

Numerical solution with empirical constants k,C

1973 Shir Kze-4z Numerical integration

1974 Brown K for small eddies only

Large eddies explicit as Coherent Structures

1980s – 90s

NumerousTroen & Mahrt

Ditto, HOC

K for all

Numerical LES

K for OLE mixing

R. A. Brown 2001 EGS

PBL Models~ 2000

• See 1990s

• See EGS 2001 OA13– K-theory– LES models– Complex terrain– Convective Coherent Structures

R. A. Brown 2003 U. ConcepciÓn

2001: Chaos in the PBL (Coherent Structures and disorder/confusion)

R. A. Brown 2003 U. ConcepciÓn

Navier/ Stokes Equations

Kinetic Theory Maxwell- Boltzmann Equations

Euler Equations

Ekman Equations

Numerical Equations

PBL Equations rotating f.o.r.

K-theory

Thirteen Moment Equations

Newton’s Second Law

Similarity A & B

Two-Layer Similarity Equations

Insufficient Computer Power

HOC Approximation

Surface Layer

Are OLE present?

Nonlinear Solution with OLE

1900

2000

noyes

R.A. Brown 2000

1700

1800

Fluid Mechanics Basics towards a PBL in a GCM

R. A. Brown 2003 U. ConcepciÓn

fV + K Uzz - pz/ = 0

fU - K Vzz + pz/ = 0

Solution, U (f, K,p ) found by Ekman in 1904.

What does Theory say?The analytic solution for a PBL

Unfortunately, this was almost never observed

Fortunately, the complete nonlinear solution for OLE exists including 8th order instability solution, variable roughness, stratification and baroclinicity, 1996

R. A. Brown 2003 U. ConcepciÓn

A certain moth that is born, breeds and dies in July, thinks that it never rains in Seattle

A certain meteorologist measures winds on a tower and/or with sondes in a convective PBL for an hour, thinks that it is a good average

R.A. Brown 2000

Vt+V•V+f(k×V)+/+A(z)

or

Ut+(KUz)z+f V- Py/ = 0

Vt+(KVz)z -f U+ Px/ =A(u2w2)

Small-scale eddy momentum flux

Large-scale eddy = OLE momentum flux

Boundary Layer Equations

11-98, 11/99, RABR. A. Brown 2003 U. ConcepciÓn

Small-Scale Effects

• Scales & Scaling Basics– A. Some Pertinent Questions

– B. Scatterometry Basics

– C. Fluid Mechanics Basics

• The OLE Challenge– A. A Pertinent Question

– B. Theory

• Rolls & Meteorologists

• Local versus Mean

• Langmuir Circulations

• ScalesR. A. Brown 2003 U. ConcepciÓn

1960 –1990 OLE Verification A Typical Cloud-Street Satellite Photo

2-km

East Coast USA -- GA

Skylab; 1963R. A. Brown 2003 U. ConcepciÓn