TUBULENT FLUXES OF HEAT, MOISTURE AND
MOMENTUM: MEASUREMENTS AND PARAMETERIZATION
General definitions and dimensions:
(1)
Flux of property c: Fc = wc = wc + w’c’
If the area considered is small and horizontally uniform and during a
given time atmospheric conditions are steady, than
w = 0
(2)
where
Momentum flux
=w’v’ [kg/(ms2)=N/m2]
(3)
Sensible heat flux:
Qh=Cpw’’ [J/(m2s)=W/m2] (4)
Latent heat flux:
Qe=Lw’q’ [J/(m2s)=W/m2]
(5)
Thus, we need to measure
w’ is vertical velocity fluctuation
v’ is horizontal velocity fluctuation
 is air density
’ is potential temperature
fluctuation
Cp is specific heat capacity at
constant pressure
q’ is specific humidity fluctuation
L is specific heat of evaporation
w’v’, w’’, w’q’
The eddy correlation method
Statistical meaning of fluxes:
w’v’, w’’, w’q’ can be considered as the
second mixed moments, i.e.
co-variances of variables
Requirements for the direct measurements
 Covariances should be measured, and
not variances
 Time resolution should be high (10-20 Hz)
 Time of record should be relatively
long (more than 20 min)
Instrumentation:
 Sonic anemometer
 Fast-response thermometer
 Fast-response infrared hydrometer
Problems:
 Technically difficult
 Expensive
ship
Stationary
platform
Sonic anemometer is based on the
estimation of speed of sound
V = [(t1-t2)C2] / 2d
We need to measure
variances of wind
velocity in
3 directions
Description of a typical eddy correlation package:
The 3-D sonic anemometer uses three pairs of orthogonally oriented, ultrasonic
transmit/receive transducers to measure the transit time of sound signals traveling
between the transducer pairs. The wind speed along each transducer axis is
determined from the difference in transit times.
The sonic temperature is computed from the speed of sound which is determined from
the average transit time along the vertical axis. A pair of measurements are made along
each axis normally 100 times per second. 10 measurements are averaged to produce 10
wind measurements along each axis and 10 temperatures each second.
The infrared hygrometer measures the water vapor density by detecting the absorption
of infrared radiation by water vapor in the light path. Two infrared wavelength bands are
used, one centered on a band strongly absorbed by water vapor, and one centered on a
band (the reference band) which is not aborbed. By normalizing the absorption band by
the reference band, instrument drifts caused by light source and photodetector
changes are eliminated. Measurements are made 40 times per second. 4 measurements
are averaged to produce 10 water vapor density measurements each second.
Parameterization of surface turbulent fluxes
massively
measured
SST, Ta, q, V, SLP
parameterization
w’v’, w’’, w’q’
What should be a combination of massively measured parameters to
be compared with direct flux measurements?
From K-theory the relationship between stress
and vertical distribution of wind under stationary
conditions:
Km is eddy diffusivity ~ [m2s-1] ~ [UL].
u/z is velocity gradient ~ [s-1] ~ [U/L]
 zx   K m
u
z
Km(u/z) ~ [U2]
What is this velocity? This is the velocity related to the mean speed of
turbulent eddies u*. Now for the stress:
 zx   u
2
*
(10)
i.e. eddy velocities in neutral conditions
are independent on height.
Velocity u* is called friction velocity and serves as indicator of velocity scale in
the exchange theory. It is proportional to root-mean square vertical velocity:
u* ~ w
We need also the size of the largest eddies. It is related to the height,
because the upper limit of eddy is determined by the distance from
the sea surface. Thus, for Km:
Where  is the von Karman
Km:= u*z
(11)
constant of proportionality (0.4)
 zx   K m
u
non-dimensional wind shear
z
m 
ln z
/u*
u
z  u
u* z

 zx   u*2
 u
u*  ln z
(12)
Measurements of mean velocity at
several levels, plotted on a log-scale
give a straight live with a slope of
/u*
Integration of (12) gives equation for the vertical
distribution of mean velocity:
z0 is the integration constant. It
u*
z
(13) represents the height, on which
u  ln
the mean wind speed calculated
 z0
by (13) goes to zero. This is the
so-called roughness length.
Land data:
We can calculate
friction velocity
and wind speed
at any level step
by step
Von Karman constant
Sea surface:


the shape of the surface and the roughness lengths vary with
wind speed;
marine roughness length may depend on the characteristics of
short capillary waves, but does not depend on longer waves (sea);
wind sea is not what the
roughness length is about!!!

marine roughness elements are not stationary, they move
together with significant waves.
Dimensional analysis under light wind speed when the ocean surface
is smooth shows that: z0 ~ /u*
however, no observational
evidence has been found for that, except for Roll (1961).
Henry Charnock (1955) assumed that roughness lenght depends on
surface stress and gravity, and derived experimental formula:
z0 = 0.0123u*2 / g
(14)
where 0.0123 is the Charnock «constant», which varies in different
studies from 0.011 to 0.018.
On aerodynamically smooth surfaces he stress is exerted by viscosity
and within the laminar sublayer the stress is proportional to wind
speed
z0 ~ /u*
On rough surfaces, the effects of viscosity are negligible and the
transfer of stress to the surface is done by the pressure differences
between the upwind and downwind sides of the obstacles
z0 = 0.0123u*2 / g
under neutral conditions:
neutral drag coefficient:
Stress:
u*
z
u  ln
 z0
 zx   u ,
2
*
Cdn 
u*2
u
2
Cdn 
,
 zx   Cdn u
2

ln z
(15)
z0
Passive mean properties under neutral conditions (,q)
For the potential temperature and humidity with the use of K-theory:
  w' '   K t

z
,
Eddy diffusivity for temperature
  w' q'   K q
q
z
(16)
Eddy diffusivity for humidity
We introduce now (with the analogy to u*) two new scales: q* and *
  w' '   u* * ,
  w' q'  u* *
(17)
For non-dimensional temperature and humidity gradients:
Km
Km
z  
 
z  q   q
t 


, q 


(18)
 * z  *  ln z K t
q* z
q*  ln z K q
Ratio of exchange coefficients for passive property and momentum is
a fundamental question of boundary layer turbulence. In general, it
depends on stability, but for the neutral conditions can be taken as
constant.
lnz
Measurements of mean velocity at
several levels, plotted on a log-scale
give a straight live with a slope of
/q*
/u*
q
Unlike the momentum, the mean values of θ, q do not approach zero at
the surface, but come to the values which depend on th eprocesses at
the surface. Let’s define them as θa,
q*
z
q  qa  ln
 z aq
*
z
   a  ln
 z a
qa and integrate (18):
Integration constants zaθ and zaq are
the heights at which <θ> and <q> are
equal to the surface values θa and qa.
Although these are analogues of the
roughness length, the mechanisms
producing zaθ and zaq are completely
different from those for zaq.
under neutral
conditions:
  w' '   Ct u     a
  w' q'   Cq u  q  qa


Bulk formulas for the transfer of sensible heat and moisture
(still under the neutral conditions)
Ctn 
neutral coefficients
for heat and moisture
transfer:
Cqn 

2
 ln z ln z 


z
z
0
ta 

2
 z

z
 ln z ln z 
0
qa 

Now, if the conditions are not neutral, we can:
1. Measure the mean variables and derive the already known
product u(SST-)
2. Measure the flux directly using eddy correlation method
3. Try to statistically compare these two
(SST-Ta) V
Ct=?
w’’
But!!!! what to do when the conditions are not neutral???
In this case we have to account for the modification of profiles of
wind and passive properties due to surface layer instability.
We have to consider the balance of turbulent kinetic energy
Turbulent kinetic energy per unit mass:
TKE = ½ u’i2 = ½ (u’2+ v’2+ w’2)
(21)
Reynolds averaging of TKE:
ui2
2

ui
2
2

ui'2
2
 EM  TKE
The energy of mean motion
The eddy energy
Equation of TKE transformation can be derived theoretically:
see, e.g. Blackadar, A. “Turbulence an diffusion in the
atmosphere”. We will consider it as a generally assumed
conservation equation of energy.
(22)
Conservation of the total kinetic energy:
Transformation to and
from mechanical energy
u 
d

2

   gw  p d    1   p u 
T
ki i
dt
dt
 xk
2
i
Work done
by stress at
boundary
Transformation
through the
potential energy
Transformation
to and from
internal energy
(23)
External
sources
Steps skipped (Blackadar 1997, Appendix A):
1) Reynolds averaging of the total kinetic energy of conservation:
D
DEM  TKE 
g
 g w 
 w  p
   M 
Dt

Dt


 xk
1
p
ui  pki ui   uk ui ui   uk ui / 2
2
ki
(24)

2) Multiplication of the Reynolds averaged Navier-Stockes equation
term-by-term by ui and summing over i:
D
 ui
DEM
1 
 pki   uk ui  ui
 g w  p
  M  uk ui

Dt
Dt
xk
 xk

(25)

(26)
3) Subtraction (25) from (24) term by term:
 ui
DTKE 
g
1 

 w    uk ui

Dt

xk
 xk
 p u   
ki i
uk ui
2
2
 ui
DTKE 
g
1 

 w    uk ui

Dt

xk
 xk
DTKE 
B  
Dt
Production of
TKE by
buoyancy (+/-)
(reversible)
 M
TKE transform
into internal
energy (+)
irreversible
 p u   
ki i

Mechanical
production of
TKE
(normally
positive)
XE
uk ui
2
2
(27)
External
sources
(with either
sign)

The mechanical production rate (from K-theory):
 u
 u
M
 K m 
 z
 z
2
 u*3
 
 z

(28)
The buoyant production rate (in the dry atmosphere):
B
g

 w 
g

w  
gQh
Cp  T
, Qh  C p 

Kt

z
(29)
Ri – gradient
Richardson
number
The flux Richardson number:

gQh
B
g Kt
z
Rf 


u
M
 Km   u
Cp  T 

z
 z
T



2
(30)
defines the stability: Rf>0 (stable), Rf<0 (unstable, turbulence
is maintained by convection).
Height on which the two rates of TKE production are equal gives the
length scale, known as Monin-Obukhov length:
hL 
C p  T u*3
gQh
,
L
C p  T u*3
gQh
(31)
 independent on height
 has the same sign as the Richardson number, defining the stability
 falls to infinity under neutral conditions
Now we can derive a non-dimensional height:
  z L,
(32)
which gives the length scale and implies similarity of wind profiles
(Monin-Obukhov 1954):
m 
z  u
u* z
  m ( )
Universal function
to be estimated
(33)
Estimation of ():
  16  1
4
m
1. KEYPS – equation:
Sellers
Panofsky
Yamamoto
Ellison
Kazansky
3
m
(34)
Solution exists but it is
not convenient to use
2. Bussinger et al. (1971) from the experiments
in Kansas:   1  4.7 ,   0 ( stable)
m
m  1  15  4 ,   0 (unstable)
1
3. Dyer (1974):
m  1  5 ,   0 ( stable)
m  1  16  4 ,   0 (unstable)
1
4. Large and Pond (1981):
m  1  7 ,   0 ( stable)
m  1  16  4 ,   0 (unstable)
1
ln z
unstable
neutral
stable
U
General formulation:
m  1   ,   0 ( stable)
m  1    ,   0 (unstable)

Similarity functions for temperature and humidity:
t  q  m2 ,   0 ( stable)
t  q   ,   0 (unstable)
2
m
Now we can finally derive the bulk formulae!!!
(35)
Since we need to estimate flux at a given height z, the equations:
z  
z  q
 m ( ),
 t ( ),
 q ( )
z
 * z
q* z
z  u
u*
should be integrated from the surface to this height, that gives the
values of mean variables at height z:

u*   zu 
u z  u0  ln    m m ,
   z0 


 *   zt 
 z   0  ln    t t ,
   z0 t 

Stability correction
functions, which are
the integrals of nondimensional profiles
(Paulson 1970):
(36)

q*   zq 
qz  q0  ln
 q q 


   z0 q 

 1  m1 
 1  m1 

  ln 
  2 tan 1 m1  ,   0 (unstable), m  1  m ,   0 ( stable)
m  2 ln 
2
 2 
 2 
 1  t1 
,   0 (unstable),
t  t  2 ln 
t  t  1  t ,   0 ( stable)
 2 
(37)
Bulk formulae:
 z 

Cd   2 ln  u   m 
  z0 

   Cd u z2 ,
Qh  C p Ct ( z   0 ),
  z
Ct   2 ln  t
  z0t
Qq  L Cq (q z  q0 )
1
1
  z 


  t  ln  u   m  ,

   z0 

(38)
1
  zq 
  z 

2
  q  ln  u   m 
Cq   ln 
 
  z0 q 
   z0 

The problem of transfer
coefficients:
Neutral coefficients at a given
reference height (h=10m):
2
  10 
Cdn   ln   ,
  z0 
2
1
2
1
  10    10 
2
Ctn   ln   ln   ,
  z0t    z0 
Cd
1

,
Cdn 1  Cdn  ln  zu 10  m  zu L 
Cd Cdn
Ct

,
1/ 2
Ctn 1  Ctn Cdn
ln zt 10  t 














10
10
 ln  
Cqn   2 ln 
 
Cd Cdn

Ct
  z0 q    z0 

1/ 2
Ctn 1  Cqn Cdn
 ln zq 10  q 
Thus, we need to know either
roughness length, or neutral
transfer coefficients to determine the fluxes!
1
1
Inertial dissipation method:
TKE – equation:
DTKE 
 B
Dt
XE
 M  XU

XP

for the steady horizontally homogeneous flow:
0

g

2  u
w   u*
Vertical divergence
of turbulent transport
Scaling parameter:
z

L

u2
1 

w

wp  
z z
2
 z
Vertical divergence
of pressure transport
z  u*3
 m  U
 P  
McBean and Elliott (1975):
 
u2
1 


w

wp

2
 z
 z

  XU  X P

is independent on L
 U  P
Large and Pond (1981):
Very important: this does not mean that both U, P are small (a
very mistake of many), simply they balance each other. We do not
neglect terms XU and XP, but we neglect the sum of these!
Now we can assume:
or
B+P=
(39)
z
  m 
L
(40)
Surprise!!!!: if we know , we can find u* !!!!!!!!!!!
How to know  ?
Attribute of turbulence No.5 from the last lecture:
Turbulence is dissipative by nature
Molecular destruction of turbulent motions is largest for the smallest
size eddies. Thus, if we have only “very small eddies”, they will be
destructed by molecular processes and turbulence will decay. But from
where “the smallest eddies” come from?
Only from the lager scale eddies
Thus, there should be a dissipation which allows larger scale
eddies to become smaller eddies. And the more intense small
scale turbulence is, the greater rate of dissipation occurs.
Thus, dissipation, as well as the other terms in the TKE equation
should depend on the size of eddies. This allows us to consider
the TKE equation in a spectral form, where the contributions of
terms will depend on the wave length or the eddy size
Traditional TKE equation, assuming that the vertical divergence of
turbulent transport and pressure transport are neglected and the flow
is steady:
DTKE 
 B  M   0
Dt
Let’s assume that we observed that this
works for a particular environment
during a given time:
Let’s assume that we are able to estimate
all terms for the eddies of different sizes.
Now, let’s plot TKE vs eddy size (wave
number)
?
Is it also valid for the particular
range of the eddy sizes?
TKE
large eddies
small eddies
Spectral representation of TKE equation (Batchelor 1953, Stull 1988):
S t , k 
t

g

 t , k    t , k 
DTKE 
 B
Dt
The local time
tendency of
the k-th
component
of TKE
u
TRt , k 

 2k 2 S t , k 
z
z
 M
Buoyant
production
associated with
the k-th component
of <w’’>

?
Mechanical
production
associated with
the k-th
component
of <w’u’>
 
Viscous
dissipation
of the k-th
component
of TKE
Convergence of
TKE transport
across the spectrum
S
Generation of TKE
Viscosity
Small eddies
Large eddies
Here eddies get the energy inertially from
the larger size eddies and loss it in the
same way to smaller size eddies!
Kolmogorov (1941), Obukhov (1941):
S (k )  K
2/3
k
5 / 3
K is the Kolmogorov constant (0.5510%)
Mid-size eddies feel
neither the effect of
viscosity
nor
the
generation. How do
they get their energy?
How do they loss it?
The cascade rate of energy
down the spectrum must
balance the dissipation
rate at the smallest eddies
sizes.
Spatial structure of turbulence: frozen turbulence (Taylor 1938):
If the mean wind U is directed in x-direction, spatial statistics of U can
be considered, assuming that the U(x) is frozen in time, i.e. these
statistics move along the x-axis with the mean speed U.
k = f / U, f = /2
links time and space for
turbulent motion
Now we can re-write the
Kolmogorov’s hypothesis:
S ( f )  K
2/3
f
5 / 3
U / 2 
2/3
(41)
But now we can measure the time series of, e.g. wind speed at high
frequency and derive the friction velocity from (40), (41):
 fS ( f ) 
u*  

 K 
1/ 2
 2
z

 U m  z L 

f

1/ 3
(42)
S ( f )  K
2/3
f
For the fluxes of passive scalars
you need to determine the spectral
level of fluctuations in the inertial
sub-range and solve the budget
equation for this scalar:
 K  fSq ( f ) 
wq  

   q S u ( f ) 
1/ 2
u*2
Kolmogorov constant pertaining
to q
dissipation function for q
5 / 3
U / 2 
2/3
Spectral
similarity
Summary of inertial dissipation method:
So, you do not need to measure <w’x’> anymore, you now need only:
1. Make fast response measurements of velocity to get the time
series
2. Calculate the spectrum of the time series
3. Plot the spectra on log-log graph
4. Find inertial subrange, i.e. the portion of the spectrum that
exhibits a –5/3 slope
5. Fit a strait line to this part of your graph
6. Pick any point on this line and determine the values of S and k
7. Compute mean wind speed U during your measurements
8. Solve the equation for :
S ( f )  K 2 / 3 f 5 / 3 U / 2 
2/3
9. Find wind sress:
1/ 2
 fS ( f ) 
u*  
 K 
Find sensible and latent fluxes:
 2
z
 
 U m  z L 
1/ 3

f

 K  fSq ( f ) 
wq  



S
(
f
)


q
u
1/ 2
u*2