1. Boundary Layer Equations
Vt + V•V + f(k×V) + / + A(z) = 0 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
R. A. Brown 2003 U. ConcepciÓn

2. VG =  (u*,HT, Ta – Ts) = n(P, , f)
There exists a solution for the PBL boundary layer (Brown, 1974, Brown and Liu, 1982),
U/VG = ei  - e –z[e-iz + ieiz]sin  + U2
where VG is the geostrophic wind vector, the angle between U10 and VG is [u*, HT, (Ta – Ts,)PBL] and the effect of the organized large eddies (OLE) in the PBL is represented by U2(u*, Ta – Ts, HT)
This may be written:
U/VG ={(u*), U2(u*), u*, zo(u*), VT(HT), (Ta – Ts), }
Or U/VG = [u*, VT(HT), (Ta – Ts), , k, a] =  {u*, HT, Ta – Ts},
for  = 0.15, k= 0.4 and a = 1
Or simply,
VG =  (u*,HT, Ta – Ts) = n(P, , f)
Hence VG/u*(k, a, ) and P (u*), = fn(o)
R. A. Brown 2003 U. ConcepciÓn

3. Geostrophic Wind direction
VG = P / ( f )
Mid- PBL
V = P / ( f) - Fviscous
Geostrophic Wind direction
25° (Stable strat.)
18° (neutral stratification)
Surface wind
5° (Unstable strat.)
-10° to 40° (Thermal Wind Effect)
R. A. Brown 2003 U. ConcepciÓn

4. Published Questions asked of PBL Modellers
 Why are you publishing papers where the Navier-Stokes equations are being used for turbulence, where they are invalid? Bound.-Layer Meteor. 1976; Fluid Mechanics of the Atmosphere, 1991
 Why are you using a procedure (Higher Order Closure) that failed observationally and mathematically in classical fluid mechanics? Turbulence & Diffusion Meeting, 1974; The Global Atmos. & Ocean Systems, 1994
 Why are you using K-theory when it is invalid in a large-eddy environment? Turbulence & Diffusion Meeting, 1974; Bound.-Layer Meteor. 1976; Fluid Mechanics of the Atmosphere, 1991; The Global Atmos. & Ocean Systems, 1994.
R. A. Brown 2003 U. ConcepciÓn

5. Well, for one thing, they have a lot of names…
Questions asked of OLE Modellers
So large-eddies muck up the PBL models used in GCMs. How often are they there? AMS T & D, 1978; NMC 1994.
Well, for one thing, they have a lot of names…
Large-eddies
finite perturbation secondary flows
Rolls
Coherent Structures
Counter-rotating helical roll vortices
Organized large eddies (OLE)
Random and organized secondary flows
Cloud streets, windrows, slicks & seif sand dunes
Turbulence ?
R. A. Brown 2003 U. ConcepciÓn

6. What does Theory say? The analytic solution for a PBL
fV + K Uzz - pz / = 0
fU - K Vzz + pz /  = 0
Solution, U (f, K,p ) was found by Ekman in 1904.
Unfortunately, this was almost never observed
fV + K Uzz - pz/ = 0
fU - K Vzz + pz/ = A(v2w2)
Solution, U (f, K,p ) found in OLE are part of solution for 80% of observed conditions (near-neutral to convection).
Unfortunately, difficult to observe
The complete nonlinear solution for OLE exists, including 8th order instability solution, variable roughness, stratification and baroclinicity, 1996
R. A. Brown 2003 U. ConcepciÓn

7. K-Theory Vs Similarity
1970: OLE is just a complicated theory for cloud streets
1980: OLE do occur. Analytically, Numerically, and some Observations --- and they all look similar. But….
1974 “Single parameter Similarity Model w/OLE is just a theory”
1980 “OK. They also occur in numerical models and a few experiments. And, if the theory is correct, diffusion models (used in all GCMs) have wrong physics. But, how often are OLE present globally? Need data not just theory” 1990
1998: SAR (RADARSAT) shows OLE produce signature on ocean surface
R. A. Brown 2003 U. ConcepciÓn

8. HOC vs Analytic PBL Models
Current GCMs all have versions of HOC PBL models (Modified K-theory)
Large-eddies cannot appear in HOC models GCM resolution
Large-eddies appear in PBL routinely (from SAR pictures)
Current GCMs have wrong PBL physics, errors in Surface Winds
R. A. Brown 2003 U. ConcepciÓn

9. R. A. Brown 2003 U. ConcepciÓn

10. Why must we consider a PBL (planetary boundary layer) model?
The satellite measures the mean density of the capillaries and short gravity waves on the ocean surface. There is no good theory relating this to anything geophysically worthwhile.
There exists a raw empirical parameterization between surface roughness and near surface winds (for over flat, smooth land surface).
There is a nonlinear analytic solution of the PBL in a rotating frame of reference (but it contains OLE).
R. A. Brown 2003 U. ConcepciÓn

11. Geostrophic Flow  Ocean surface Ekman Layer with OLE Stratification
VG(u*) effects
Ekman Layer with OLE
Thermal Wind
Nonlinear OLE
Non steady-state
Advection,centrifugal terms
U10(u*) effects
Stratification
U10
Surface Layer
Surface Stress, u* 
Variable Surface Roughness
Ocean surface
R. A. Brown 2003 U. ConcepciÓn

12. Taking measurements in the Rolls with Tower & Sondes
1-km
Station A
2 - 5 km
Station B
R. A. Brown 2003 U. ConcepciÓn

13. OLE U Station A V Hodograph Counter-rotating Helical Roll Vortices
1 km 
OLE U 
Station A
1- 3 km 
Hodograph
from convergent zone V
R. A. Brown 2003 U. ConcepciÓn

14. OLE U Station B V Z/ Hodograph    from center zone
Lateral Motion of OLE
1-2 m/s near neutral
0 convective   U 
Z/
Hodograph
from center zone
Station B V
R. A. Brown 2003 U. ConcepciÓn

15. Langmuir Circulations = Rolls
windrows U
10-50 m
Averaged hodograph V
R. A. Brown 2003 U. ConcepciÓn

16. TURBULENCE & Approximations in the boundary layer
Required
MODEL # Layers Comment
Analytic PBL U = f(UG, , Ta - Ts, Ta)
K-Theory for small eddies Equator, Inversion,
Explicit Large eddies limitations
Numerical PBL LES Equator, Inversion, limited
K-Theory for small eddies Re #, Domain Limited Explicit Large eddies
GCM PBL Bad Physics (no LES)
K-Theory for all eddies
All Fluxes (u*) depend on
Bulk Coefficients for Flux land empirical formulas
Surface Layer Approx.
R. A. Brown 2003 U. ConcepciÓn

17. The Problems with getting ‘Surface Truth’ in the PBL
Satellite sensor wind model functions need surface truth for empirical formulas
Coherent Structures adversely impact the PBL measurements
Buoy averaging and coverage is inadequate
Rawinsonde & dropsonde point measurements differ by 50% from mean
R. A. Brown 2003 U. ConcepciÓn

18. The OLE consequences
The winds are non-homogeneous at the surface over a 1-5 km horizontal distance. Upper high velocity wind is advected to the surface in lines. OLE must be taken into account in surface truth measurements (In the average and point values) (Brown, Canadian Jn. Remote Sensing, 28, , 2002)
The average wind profile is different from the Ekman solution --- nonlinear winds (and hence fluxes of momentum, heat, CO2 etc.) are 10-50% different, depending on stratification and thermal wind. In a satellite’s 25-km footprint there will be 10-OLE so the periodic effect will be the average. Not true at 6-km or less --- the SAR resolution. (Brown & Foster, The Global Atmos.-Ocean System, 2, , 1994; ), 2, , 1994; ), 2, , 1994.)
3. The PBL contains advecting flow not amenable to diffusion modeling. Numerical models cannot portray correct physics of mean flow without extreme increase in resolution.
R. A. Brown 2003 U. ConcepciÓn

19. Proof of Rolls
Or, the mathematical solution for the PBL flow that includes coherent structures (OLE) is correct
R. A. Brown 2003 U. ConcepciÓn

20. Status of organized large eddies (OLE) verification
Radar from tetroon balloons show Rolls (OLE) circulation (1971)
Occasional airplane campaigns in cold air outbreaks show Rolls ( ).
Ground based Lidar detects OLE (1996-).
Satellite derived surface pressures [better with enhanced OLE mixing; correct  (1996)]
Satellite SAR data statistics of ocean surface roll signature (1978; 1986; 1997-).
R. A. Brown 2003 U. ConcepciÓn

21. R. A. Brown 2003 U. ConcepciÓn

22. Analysis of frequency of SAR PBL Roll Signature by RADARSAT
Of a random sampling of X 100 km SAR images of the North Pacific (1997 – 1999)
Minimum Occurrence of Roll Vortices by Month %
Jan Feb Mar Apr May Sep Oct Nov Dec
no no
data data
Note: This indicates a minimum for occurrence of Rolls
Gad Levy, R.A. Brown 2001
R. A. Brown 2003 U. ConcepciÓn

23. CONCLUSIONS Climatological u* and heat fluxes need to be revised.
The surface layer relation, hence U10 {u*(o )} works well. Global, smooth U10 (x,y,t) are available†.
The PBL relation with OLE, hence VG(o ) or P(o ), works well. Global P(x,y,t) are available†.
There is almost no surface truth --- buoy or GCM surface winds with U10 > 25 m/s.
The U10 model function can be extrapolated to about 35 m/s. There are indications that o responds to the sea state up to U10 ~ 40 m/s., H-pol ~ 60 m/s?
Climatological u* and heat fluxes need to be revised.
The nonlinear equilibrium PBL model is right. †
R. A. Brown 2003 U. ConcepciÓn

24. The OLE consequences
The winds are non-homogeneous at the surface over 1-5 km horizontal distance. High velocity winds can be advected to the surface in lines. Must be taken into account in surface truth measurements (averaging).
The average wind profile is DIFFERENT from the Ekman solution --- nonlinear winds are 10-50% different, depending on stratification and thermal wind. In a 25-km footprint there will be 10-OLE so the periodic effect will be averaged. Not true at 6-km or less --- SAR resolution
3. The PBL contains ADVECTING flow not amenable to diffusion modeling. Numerical models cannot portray correct physics of mean flow without extreme increase in resolution.
R. A. Brown 2003 U. ConcepciÓn