Modeling the Atmospheric
Boundary Layer (2)
Review of last lecture
• Reynolds averaging: Separation of mean and turbulent components
u = U + u’, < u’ > = 0
• Intensity of turbulence: turbulent kinetic energy (TKE)
TKE = ‹ u’ 2 + v’ 2 + w’ 2 › /2
• Eddy fluxes
Fx = - <u’w’>/z
The turbulence closure problem
Governing equations for the mean turbulent flow
u
x
vw 0
u
t
 u u  v u  w u   1
v
t
 u v  v v  w v   1
w
t
 u w  v w  w w   1

t
r
t
y
z
x
y
x
u
u
y
z
z
ˆ
p
 x
 (  u u )
 (  u v)
 (  u w )
fv 1
1
1
  vx
ˆ
p
 y
 (  vu )
 (  vv)
 (  vw )
fu  1
1
1
  vy


y
z

x
v 
y

z
 
1  ( Cp u  )
 Cp
x
r
x
w

 v r  w r  
y
z

y

y
z
z
 (  w u )
 (  w v)
 (  w w )
ˆ
g 1
1
1
  vz


x

y

z
x


x
ˆ
p
 z


x
1  (  L v ur)
L v
x


 
1  ( Cp v  )
Cp
y
1  (  L v vr)
L v
y

 
1  (  Cp w  )
Cp
z

1  (  L v w r)
L v
z
,
ˆ
p  p0  p
Where
p  R d  Tv ,
Tv  (1  0.608r),
  T (1000)
p
R d / Cd
 Qr
 Qt
The turbulence closure problem
• For large-scale atmospheric circulation, we have six
fundamental equations (conservation of mass,
momentum, heat and water vapor) and six unknowns
(p, u, v, w, T, q). So we can solve the equations to get
the unknowns.
• When considering turbulent motions, we have five
more unknowns (eddy fluxes of u, v, w, T, q)
• We have fewer fundamental equations than
unknowns when dealing with turbulent motions. The
search for additional laws to match the number of
equations with the number of unknowns is commonly
labeled the turbulence closure problem.
The current status of the turbulent
closure problem
• For surface layer (surface flux): nearly solved except for
some extreme conditions (e.g. huricane’s boundary
layer)
• For mixed layer: not solved. There is a variety of
approaches available:
(1) Local theories
(2) Non-local theories
However, these have not converged towards a
commonly accepted BL theory, and they often show
some biases when comparing with observations.
Surface layer
Eddy flux is assumed to be proportional to the vertical
gradience of the mean state variable
• Sensible heat flux
<w’h’> = Qh =  Cd Cp V (Tsurface - Tair)
• Latent heat flux
L <w’q’> = Qe =  Cd L V (qsurface - qair)
Where  is the air density, Cd is flux transfer coefficient,
Cp is specific heat of air, V is surface wind speed,
Tsurface is surface temperature, Tair is air temperature,
qsurface is surface specific humidity, qair is surface air
specific humidity
Mixed layer theory I: Local theories
• K-theory: In eddy-diffusivity (often called K-theory)
models, the turbulent flux of an adiabatically conserved
quantity a (such as θ in the absence of saturation, but
not temperature T, which decreases when an air parcel
is adiabatically lifted) is related to its gradient:
< w’a’ > = - Ka dA/dz
• The local effect is always down-gradient (i.e. from high
value to low value)
• The key question is how to specify Ka in terms of known
quantities.
Three commonly used approaches:
(1) First-order closure
(2) 1.5-order closure or TKE closure
(3) K-profile
First-order closure
z
w u  
u v
,
z z
 k m u
z
0
k m : eddy viscosity (exchange coefficient,
Turbulent
transport
x
coefficient of turbulent transport)
w v   k m v ,
z
w   
 k h 
z
z
,

z
,
r
z
Turbulent
transport
w r    k e r ,
z
k m : (a) unit
Z
m
s
0
2
x
;
(b) function of motion.
ke  k h  1.35k m
km
(by Prof. Ping Zhu of FIU)
Example: The Ekman layer
• Assumption Km = constant (First order closure)
• Then the boundary layer momentum equations are:
• Vertical boundary conditions:
• Solutions:
General circulation of the oceans
• Ocean surface currents – horizontal water motions
• Transfer energy and influence overlying atmosphere
• Surface currents result from frictional drag caused by wind Ekman Spiral
• Water moves at a 45o angle (right)
in N.H. to prevailing wind direction
• Greater angle at depth
Global surface currents
• Surface currents mainly driven by surface winds
• North/ South Equatorial Currents pile water westward, create the Equatorial
Countercurrent
• western ocean basins –warm poleward moving currents (example: Gulf Stream)
• eastern basins –cold currents, directed equatorward
Pressure grad.
force
Centrifugal
force
Secondary
circulation
Pressure grad.
force
Centrifugal
force
2
V
R
Frictional
force
2
Inflow
V
R
Pressure grad. Centrifugal
force
force
Boundary layer
Ekman pumping
(by Prof. Ping Zhu of FIU)
Hurricane Vortex
Coriolis
force
Centrifugal
force
Diverging
Spin down
Pressure grad.
force
L
Converging
Buoyancy Spin up
X
Ekman
Pumping
Boundary
Layer
It is the convective clouds that generate spin up process to overcome
the spin down process induced by the Ekman pumping
To close the system, first order turbulent closure is to use first-order moments
to parameterize second-order moments.
Second order turbulent closure is to use second-order moments to
parameterize third-order moments.
Third order turbulent closure is to use third-order moments to parameterize
fourth-order moments.
Fourth order turbulent closure is to use fourth-order moments to parameterize
fifth-order moments.
………
Why higher order closure is better than lower order closure?
The advantage of higher-order turbulence closure is that parameterizations
of unclosed higher-order moments, e.g., fourth-order moments, might be
very crude, but the prognosed third-order moments can be precise since
there are enough remaining physics in their budget equations. The
second-order moment equations bring in more physics, making them even
more precise, and so on down to the first-order moments.
Mixed layer theory II: Non-local theories
•
•
•
•
•
Any eddy diffusivity approach will not be entirely accurate if most of the
turbulent fluxes are carried by organized eddies filling the entire boundary
layer.
The non-local effect could be counter-gradient.
Consequently, a variety of ‘nonlocal’ schemes which explicitly model the
effects of these boundary layer filling eddies in some way have been
proposed.
A difficulty with this approach is that the structure of the turbulence
depends on the BL stability, baroclinicity, history, moist processes, etc.,
and no nonlocal parameterization proposed to date has comprehensively
addressed the effects of all these processes on the large-eddy structure.
Nonlocal schemes are most attractive when the vertical structure and
turbulent transports in a specific type of boundary layer (i. e. neutral or
convective) must be known to high accuracy.
Illustration of non-local turbulent mixing concept
1
1
2
2
3
3
4
4
1
1
2
2
3
3
4
4
(by Prof. Ping Zhu of FIU)
Summary
• The turbulent closure problem: Number of unknowns
> Number of equations
• Surface layer: related to gradient
• Mixed layer:
Local theories (K-theory): < w’a’ >= - Ka dA/dz
always down-gradient
Non-local theories: organized eddies filling the
entire BL, could be counter-gradient