Mean and Fluctuating Quantities
Ocean Surface
u 
3D Turbulence
u  <u   u '
u'
Current Meter
Mean Flow
Fluctuating Flow
One Dimensional Measurement
u'
<     average
 '  Fluctuating
Note:   ' >=0
u
u
time
Three Types of Averages
Ensemble
    
 ( x, t) 
j
average over ensemble members j
Time
T
1
    
dt ( x , t )  average over time for period T

T 0
Space
L
1
    
dx ( x , t )  average over space of length L (1D)

L0
1
3
d
x ( x, t )  average over space of Volume V (3D)

V
Ergodic Hypothesis: Replace ensemble average by either a space or time average
Concept of Correlation Function
Auto Correlation Function
R   '( x2 ) '( x1 )  1 D space
R   '(t2 ) '(t1 ) 
1 D time
R   '( x2 ) '( x1 ) 
3 D space
R   '( x2 , t2 ) '( x1 , t1 ) 
3 D space,time
Cross Correlation Function
R ij  u 'i ( x2 )u ' j ( x1 ) 
R ij  u 'i (t2 )u ' j (t1 ) 
R ij  u 'i ( x2 )u ' j ( x1 ) 
1 D space
1 D time
3 D space
R ij  u 'i ( x2 , t2 )u ' j ( x1 , t1 ) 
3 D space,time
Concept of Spatial Homogeneity and Temporal Stationarity
'


<     average
 '  Fluctuating
Note:   ' >=0
Time or Space
 ( ')  Variance of   (   ) 
2
2
Independent of space and time
Correlation Function R
Space: R(x2  x1 )   '( x2 ) '( x1 )  R(x), x  x2  x1
Time: R(t 2  t1 )   '(t2 ) '(t1 )  R(t), t  t 2  t1
Homogeneous/Stationary I D Correlation Function
R   '( x2 , t ) '( x1 , t )  R(x2  x1 )  R(x)
x  x2  x1
R   '( x , t2 ) '( x, t 1 )  R(t 2  t1 )  R(t)
t  t2  t1 )
Note: R(x) = R(  x)
Proof : R(x)   ' x1  x) ' x1 ) 
Let x'= x1 +x then
R(x)   ' x ') ' x ' x1 )   ' x ' x) ' x ')  R(  x)
Similarly for R(t)   't2 ) 't1 )  R(  t)
Note : R(0)   ' x, t ) ' x, t )  ( ') 2 
Velocity Cross Correlation Function
R uv  u '( x2 , t )v'( x1 , t )  R uv ( x)  R uv (  x)
but R uv  R vu ( x)
Can you show this?
Auto Covariance Function
  '( x1 ) '( x1  x) 

  ( x)
2
 ( ') 
  '(t1 ) '(t1   ) 

  ( )
2
 ( ') 
  '(t1 ) '(t1   ) 

  ( )
2
 ( ') 
1
  , 

x
0
Time or Space Axis


1
  =  d ( )=  d ( )
2 -
0

Temporal Intergal Scale

1
 =  dx  ( x) =  dx  ( x)
2 -
0
Spatial Intergal Scale
Concept of Structure Function
 [u '(t   )  u '(t )]2 
S( ) 
 1    
2
2  (u ') 
S( )

Microscale
I

Integral scale
S( )  0 at   0
S( )  1 at    I
Taylor’s Microscale
Temporal case
S( )  1    

 2 2
3
but       0  
   | 0 




|

O
(

)

0
2

2 
 2 2
2
 1 0 
   | 0  1  2
2  2

where
thus

2
1 2

   | 0
2
2 
2
S( )  2 for    

Spatial case
S(x)  1    x 
S(x) 
x2

2
for x   where

2
1 2

  x  |x 0
2
2 x
How to Calculate Correlation Functions from Data
R ( x )   '( x1 ) '( x1  x ) 
Use Ergodic hypothesis
1
R( x) 
L
1
R ( ) 
T
L
| x|
2

dx1 '( x1 ) '( x1  x)
Space series
L
 | x|
2
T
| |
2

T
 | 
2
dt '(t ) (t   )
x, 
L,T
Time Series
Concept of Spectrum
Temporal Spectrum
 '(   Two Sided Spectrum
(   One Sided Spectrum
1
 '(  
2

 d R(  exp(i )   '(    '( 

R (   u '(t )u '(t   ) 

 d  '(  exp(i )



0
0
  d  2 '  cos( )   d  ) cos( )
where  )  2 ' )

Note:R (0   u '2   d  ) &
0
1
 '(  
2

1

 d R(  cos( )    d R(  cos( )

0
Spatial Spectra
1
 '(k  
2
R( x  


dxR ( x  exp( ikx) 




0
1
(k 
2
 dk  '(k  exp(ikx)   dk (k  cos(kx)
Terminology
  angular frquency ( typical units of rad/sec)
2
T = period ( typical units of secs) =

1 
cyc
f = cyclical frequency = 
(units of
 Hz )
T 2
sec
k  wavenumber ( typical units of rad/m)
 = wavelength ( typical units of meters) =
k = cyclical wavenumber =
1


2
k
k
cyc
(units of
)
2
m
Normalized Correlation Function and Spectra
R( x)
R( x)


x0
2
R (0)  ( ') 
 (0)  1
 ( )  0


1
 =  dx  ( x ) =  dx  ( x )
2 -
0
Integral Scale = Area under curve
You can show that:

 0
 =
2
  ( ') 

x
3D + Time Spectra
R(x , )  u '( x ', t )u '( x ' x , t   ) 

=  d 3kd  k ,   exp[i (k  x   )]

4 k ,  
1
(2 


d 3 xd R(x , ) exp[i (k  x   )]

1D Sepctra:
R(x)  R(x , ) | y 0, z 0, 0   u '( x ', y ', z ', t )u '( x ' x, y ', z ', t ) 
R( )  R(x , ) |x 0, y 0, z 0   u '( x ', y ', z ', t )u '( x ', y ', z ', t   ) 

   2  d 3k  4  k ,  
0

 k1   2  dk2 dk3 d  4  k ,  
0
Gradient Spectra



 u '(t ) u '(t   )  u '(t ) u '(t   ) 
R( )
t



but R( )  R( )  at   0 
R( ) | 0  0

R ( )


 u (t )
u (t )  0
t
u '(t ) u '(t   )
Rg (  
 Spectra of gradient of u
t
t

u '(t   )
 2u '(t   )

 u '(t )
   u '(t )

2
t
t
t
 2u '(t   )

0 (by stationarity)   u '(t )

2

2

R ( 



1
but  ' g (  
 d Rg (  expi ) &
2 
Rg (  

 d  '
g
(  expi )

2
but Rg (   
R (  


  'g (    2  '( 
 g (    2  ( 



d   2  '(  expi )
Spatial Spectra  k 
 ' k   Fourier Transform of R(x)

1
 ' k  
dx R(x) exp[i (kx)]

(2  
 k   2 ' k 

R(x)   dk  ' k  cos(kx)]
0
Gradient Spectra  g  k 
2R
 2 '
 ' 2
|


'

|



(
) |x 0
x 0
x 0
x 2
x 2
x

Using R(x)   dk  k  cos(kx)] 
0


 ' 2
(
) |x 0 =  dk  g  k  =  dk k 2 k 
x
0
0
 g  k   k 2 k 
Use of the Log-Log plot
Example:  k   Ak p
Linear Plot
Log-Log Plot
p=-2
100
10 2
ln( k )
 k 
o
2
( C)
rad / m
o
p=-2
p=2
20
10
2
( C)
ln(
)
rad / m
p=2
101
100
101
0
1 2
20
k
10
101 100
101
ln(k )
10 2
Spectra

R(0)  ( ') 2   dk  k 
 (k)
0
 ( ')  Area under curve
2
k
1
Interpret k as eddy of size k
Gradient Spectra
g (k) =

 ' 2
(
) |x 0 =  dk k 2  k 
x
0
 ' 2
(
) |x 0 
x
k 2 (k)
k
Area under curve
 ( 
 u '2 
is the area under

Consider Model Correlation Function
R    u '  e
2
    e
 2  02
 2  02
 ( )  2 0
u' 
2

e
 2 02

4
 ( 
From  we create a simulation of u ' and
u'
u '
t
u '
t
    e
 2  02
 0  2 Minutes
 ( )
 2( )
Calculation of Spectra
Spectra = Decomposition of Variance into contributions by sines/cosines
f
L
Fourier Series
2 mx
f ( x)   am exp(
)  f ( x  L)
L
m0

1
am 
T
L
2
2 mx
L dtf ( x) exp( L )

2
exp(i )  cos( )  i sin( )
Lim (T  )Fourier Series  Fourier Transform

i 2 mx
f ( x)  LIM  am exp(
)   dkg (k  exp(ikx)
L
L m  0


2
where am  kg (k  k 
L
1
g (k   LIM
L 2
2
m( )  k
L
L
2
i 2 mx
1
L dxf ( x) exp( L )  2

2
Concept of Delta Function
1
 (x)=
2

f ( x) 

 dk exp(ikx)

 dx ' f ( x ') (x-x')


 dxf ( x) exp(ikx)

f ( x)
1
Note: g (k   Lim
L  2
L
2

 dx{  dk ' g (k ' exp(ikx)}exp(ik ' x)

L
2



 dk ' g (k ' (k ' k )

 (k ' k ) = Lim  L (k ' k )
L 
1
 L (k ' k ) 
2
L
2
 dx exp[i(k ' k ) x]

L
2
L
sin[(k ' k ) ]
2

(k ' k )
2 (
)
2
L
sin(k )
2
 L (k ) 
k
2 ( )
2
Calculation of Spectra of u’
x
L
1. Choose Sampling x (digitizing)
2. Calculate  the DFT (Discrete Fourier Transform) of sections of u’
 k )    k  *  k   k
2
1 n
where k 
&    k  *  k     i  k i *  k  
L
n i 1
3. Estimate Spectra by
1
2
3
n
The Uncertainty Principle
Q = Fourier Transform of q
1
Q=
2

 dx exp(ikx)q

If q=exp{(-x/x) 2 } 
with k 
2
x
Q=
1
exp(-k/k ) 2
 k
xk  2 !
x 
x 
x 
k 
k 
k 
Developing the Concept of an Eddy
u=Uexp(ik 0 x)  exp{(-x/x) 2 }
=
k
exp[-{(k-k 0 )/k}2 ]
4
Example k 0  1
Real(u)
x 
x 
x 
exp(ik 0 x)=cos(k 0 x)+isin(k 0 x)
k 
rad
m
2
x

k 
k 
k 
The Eddy
k  k0
u=Uexp(ik 0 x)  exp{(-x/x) 2 }
Q=
x 
2
k0
k
exp{-[( k - k0 ) / k ]2 ]}
4
x
k
k0  1
k  k
k0
k0
k0
k0




k  k0
k0
k0
k0
k0
Q




k  k
kQ
k0
k0
k0
k0




k  1
kQ
k0
k0
k0
k0



