1. 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. Concepci Ó n

2. 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. Concepci Ó n

3. 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. Concepci Ó n

4. PBL Modeling circa 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. Concepci Ó n

5. 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. Concepci Ó n

6. PBL Models 1990s See 1980s R. A. Brown 2003 U. Concepci Ó n

7. Some K-theory Models DateAuthorKRemarks 1902EkmanConstant3-D Spiral for ocean 1905Ekman~ |dV/dz|Equi-angle spiral 1908AkerblomConstantIt works in Atmosphere 1915TaylorConstantV(0) > 0 1930Prandtl, vKL 2 |dV/dz|Mixing length; k 1930Takaya~z 2, e az 1933KohlerZnZn Bessel’s equation 1935RossbyC(z+z o ),KTwo-layer model 1936BlinovaKzBessel’s equation 1940KibelKu*z U* = (  o /  ) ½ 1940 Yudin/Shvets Kz, KhTwo-layer 1950LettauEmpiricalLeipzig profile 1962Blackadar[k|dV/dz] 2 1+kz/C Numerical solution with empirical constants k,C 1973ShirKze -4z Numerical integration 1974Brown K for small eddies only Large eddies explicit as Coherent Structures 1980s – 90s Numerous Troen & Mahrt Ditto, HOC K for all Numerical LES K for OLE mixing R. A. Brown 2001 EGS

8. PBL Models ~ 2000 See 1990s See EGS 2001 OA13 –K-theory –LES models –Complex terrain –Convective Coherent Structures R. A. Brown 2003 U. Concepci Ó n

9. 2001: Chaos in the PBL (Coherent Structures and disorder/confusion) R. A. Brown 2003 U. Concepci Ó n

10. 2000 no yes R.A. Brown 2000 1700 1800 Fluid Mechanics Basics towards a PBL in a GCM R. A. Brown 2003 U. Concepci Ó n

11. fV + K U zz - p z /  = 0 fU - K V zz + p z /  = 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 8 th order instability solution, variable roughness, stratification and baroclinicity, 1996 R. A. Brown 2003 U. Concepci Ó n

12. 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

13. V t +V  V+f(k×V)+  /  +A(z) or U t +(KU z ) z +f V- P y /  = 0 V t +(KV z ) z -f U+ P x /  =A(u 2 w 2 ) Small-scale eddy momentum flux Large-scale eddy = OLE momentum flux Boundary Layer Equations 11-98, 11/99, RAB R. A. Brown 2003 U. Concepci Ó n

14. 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 Scales R. A. Brown 2003 U. Concepci Ó n

15. 1960 –1990 OLE Verification A Typical Cloud-Street Satellite Photo 2-km East Coast USA -- GA Skylab; 1963 R. A. Brown 2003 U. Concepci Ó n