A whistle-stop tutorial on
observational turbulence studies
Background and motivation for a statistical
treatment
Khurom Kiyani
motivation
motivation
$
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outline
What do you
think?
outline
Why, energy
transfer & scaling of
course, you silly man!
outline
•Equations of motion and phenomenology of energy
transfer in r and k-space for iHI turbulence.
•Richardson energy cascade and the 5/3rd energy
spectrum (power spectral density)
•Measurement and ensembles
•Higher order two-point statistics
•Some real data from the solar wind
•4/5th (third order) law
•fractal models (if time permits)
outline
•Equations of motion and phenomenology of energy
transfer in r and k-space for iHI turbulence.
•Richardson energy cascade and the 5/3rd energy
spectrum (power spectral density)
•Measurement and ensembles
•Higher order two-point statistics
•Some real data from the solar wind
•4/5th (third order) law
•fractal models (if time permits)
statistical theory of turbulence
∂u
1
2
+ u · ∇u = − ∇p + ν∇ u
∂t
ρ
∇·u=0
Incompressible fluid Navier-Stokes equations
statistical theory of turbulence
∂u
1
1
�
� �
�
�2 �
+
u
·
∇
u
=
−
∇p
+
∇
u
∂t�
ρ
Re
�
∇ ·u =0
�
�
LV
Re =
ν
Reynolds
number
Dimensionless Navier-Stokes equations
the turbulence problem (amongst others)
To understand better the phenomenology,
and thus dynamics, of turbulence
statistical theory of turbulence
∂u
1
2
+ u · ∇u = − ∇p + ν∇ u
∂t
ρ
u = Ūi + ũi
∇·u=0
Reynolds decomposition
statistical theory of turbulence (RANS)
∂ Ūi
∂
1
2
+
Ūi Ūj = − ∂i P̄ + ν∂ Ūi
∂t
∂xj
ρ
∂
�ũi ũj �
+
∂xj
Reynolds stress tensor
(two point spatial correlation/covariance)
(will stay with 2-point statistics)
statistical theory of turbulence
FT
�ũi ũj � ←→ P (k)
Wiener-Khinchin theorem
∂
E(t) = εw (t) − εd(t),
∂t
energy transfer in real space (Karman-Howarth eq)
εw = εd = εT
�
∂ � ��
∂ �
∂ � � ��
ui uk + � ui uj uk + � ui uj uk
∂t
∂xj
∂xj
�
�
1 � � ∂p
∂p� �
�2
�
�
2
= − uk
+ ν ui ∇ uk + uk ∇ ui
+ ui
ρ
∂xi
∂xk
2
∂
E(t) = εw (t) − εd(t),
∂t
energy transfer in real space (Karman-Howarth eq)
εw = εd = εT
�
∂ � ��
∂ �
∂ � � ��
ui uk + � ui uj uk + � ui uj uk
∂t
∂xj
∂xj
�
�
1 � � ∂p
∂p� �
�2
�
�
2
= − uk
+ ν ui ∇ uk + uk ∇ ui
+ ui
ρ
∂xi
∂xk
2


2 ∂E
2
1 ∂S2
1 ∂ 4
ν ∂  4 ∂S2 

=− ε=
+ 4 (r S3) − 4 r

3 ∂t
3
2 ∂t
6r ∂r
r ∂r
∂r
∂ 4
4
−4εr � (r S3)
∂r
4
S3(r) � − εr
5
∂
E(t) = εw (t) − εd(t),
∂t
energy transfer in real space (Karman-Howarth eq)
εw = εd = εT
�
∂ � ��
∂ �
∂ � � ��
ui uk + � ui uj uk + � ui uj uk
∂t
∂xj
∂xj
�
�
1 � � ∂p
∂p� �
�2
�
�
2
= − uk
+ ν ui ∇ uk + uk ∇ ui
+ ui
ρ
∂xi
∂xk
2


2 ∂E
2
1 ∂S2
1 ∂ 4
ν ∂  4 ∂S2 

=− ε=
+ 4 (r S3) − 4 r

3 ∂t
3
2 ∂t
6r ∂r
r ∂r
∂r
∂ 4
4
−4εr � (r S3)
the closure∂rproblem
4
S3(r) � − εr
5
Fourier (k-space) -- mode-coupling






�
∂

2

+ νk  uα(k, t) = Mαβγ (k) d3juβ (j, t)uγ (k − j, t) + fα(k, t)
∂t
kαuα(k, t) = 0
∂
1
2
u(x, t) − ν∇ u(x, t) = −u(x, t) · ∇u(x, t) − ∇p(x, t) + f (x, t)
∂t
ρ
∇ · u(x, t) = 0
energy transfer in Fourier space (Lin equation)






�
�
∂

2

+ 2νk  �u−k uk � = Mk u−k uj uk−j
∂t
�
�
+M−k uk u−j u−k+j + �u−k fk � + �f−k uk �






∂

2
+ 2νk  E(k, t) = T (k, t) + W (k, t)
∂t
��
�
�
T (k, t) = Mk u−k uj uk−j − uk u−j u−k+j
1
��






�
�
∂

2

+ 2νk  �u−k uk � = Mk u−k uj uk−j
∂t
�
�
+M−k uk u−j u−k+j + �u−k fk � + �f−k uk �
energy transfer in Fourier space (Lin equation)


�
�
 ∂

2



+ 2νk ∂ �u−k uk � = Mk u−k uj uk−j

2 �u u � = M u u u
∂t
+
2νk
∂
j k−j
k
∂t�
t) = kT� (k,−k
t) +
W (k, t)
+ 2νk 2−kE(k,
∂t u u u
u−k+j ++
�u−k
fk−k
� + �f
� �f−k uk �
+M−k u+M
�u
f−k
k −k−jk u−j−k+j
k �uk+















�

�




�
�
��
�
�
T (k, t) = Mk u−k uj uk−j − uk u−j u−k+j










��

∂

2
+ 2νk  E(k, t) = T (k, t) + W (k, t)
∂t

∂

2

 E(k, t) = T (k, t) + W (k, t)
+ 2νk
T (k, t) =
 Mk u−k uj uk−j − uk u−j u−k+j
∂t
��
�
�
��
1
��
�
�
T (k, t) = Mk u−k uj uk−j − uk u−j u−k+j
1
��
∂t
�
−k k
−k j k−j
k
�
+M−k uk u−j u−k+j + �u−k fk � + �f−k uk �
energy transfer in Fourier space (Lin equation)






∂

2
+ 2νk  E(k, t) = T (k, t) + W (k, t)
∂t
E(k)
��
-5/3
�
�
T (k, t) = Mk u−k~ukj uk−j − uk u−j u−k+j
��
Energy Spectrum of
Turbulence Velocity
Fluctuations
k
1
outline
•Equations of motion and phenomenology of energy
transfer in r and k-space for iHI turbulence.
•Richardson energy cascade and the 5/3rd energy
spectrum (power spectral density)
•Measurement and ensembles
•Higher order two-point statistics
•Some real data from the solar wind
•4/5th (third order) law
•fractal models (if time permits)
Kolmogorov scaling and Richardson cascade
Energy input
at rate εw
l
bl
Energy
transferred
through
scales
Energy
dissipated at
rate εd
kd-1
Kolmogorov scaling and Richardson cascade
ε
m2s3
E(k)
Energy input
at rate εw
m3s2
k
m-1
Energy
transferred
through
scales
Energy
dissipated at
rate εd
Kolmogorov scaling and Richardson cascade
ε
m2s3
E(k)
Energy input
at rate εw
m3s2
k
m-1
Energy
transferred
through
scales
E(k) ∼ ε
2/3 −5/3
k
Energy
dissipated at
rate εd
more dimensionless numbers
� ε �1/4
kd =
3
ν
ld = 1/kd
Re = (L/ld )
4/3
Kolmogorov scaling and Richardson cascade
outline
•Equations of motion and phenomenology of energy
transfer in r and k-space for iHI turbulence.
•Richardson energy cascade and the 5/3rd energy
spectrum (power spectral density)
•Measurement and ensembles
•Higher order two-point statistics
•Some real data from the solar wind
•4/5th (third order) law
•fractal models (if time permits)
NBR
NBR
HBR
HBR
overflow
overflow
40
40
35
35
-1/2
dB above 10-7 Vrms.Hz-1/2
dB above 10-7 Vrms
.Hz
Frequency (kHz)
(kHz)
Frequency
30
30
30
30
ensembles, ergodicity & statistical stationarity
25
25
20
20
20
20
15
15
10
10
10
10
22
00:10:00
00:10:00
00:25:00
00:25:00
00:40:00
00:40:00
00:55:00
00:55:00
01:10:00
01:10:00
R(Re)
R(Re)
20.20
20.20
20.21
20.21
20.22
20.22
20.22
20.22
20.23
20.23
at_gse(deg)
Lat_gse(deg)
-25.39
-25.39
-25.71
-25.71
-26.03
-26.03
-26.35
-26.35
-26.66
-26.66
LT_gse(h)
LT_gse(h)
13.92
13.92
13.91
13.91
13.89
13.89
13.88
13.88
13.86
13.86
E
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,*
—Bx —By —Bz —|B|
!DD
-*
!DE
G*
!DF
F*
!DG
!D-
E*
!D,
—Vx —Vy —Vz
D*
!DH
%
D***
E***
F***
G***
!"#$%&'$(%)*+,-.
-***
,***
0
6
5
ne ~ 4 cm-3
VA ~ 50 km s-1
Tiㅗ ~ 24 eV
βp ~ 1
|B|~4 nT
Te ~ 22 eV
ρi ~ 111 km
ρe ~ 2 km
Tp ~ 15 eV
5.5
10
5
15
4.5
20
4
25
3.5
100
stationary plasma parameters
200
300
400
500
time (secs x 4)
600
700
!D
;:K%&LE0#'%# %67 .
%
V-GSE
(km/s)
869:;<=%<>$(!0"(%/"$>?%'@$(!0AB0C#
/0$12$3(4%&567.
B-GSE (nT)
UT
UT
55
Ion energy KeV
C4
C4
Frequency (kHz)
40
40
800
Frozen turbulence & Taylor’s hypothesis
∂u
= −V · ∇u
∂t
∼k·V
angles of measurement w.r.t. B
2πfsc
k=
VSW
Taylor frozen-in approx,
for low-frequency
phenomena
-VSW
VA
<1
MHD V
SW
else
Vφ
<1
VSW
MHD inertial range (spectral) scaling exponents
!
2
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MHD inertial range (spectral) scaling exponents
!
2
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MHD inertial range (spectral) scaling exponents
!
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MHD inertial range (spectral) scaling exponents
!
2
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#
Re = (L/ld )
$
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5
Re~10
&
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MHD inertial range (spectral) scaling exponents
!
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no phase
no coherence
no structure
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outline
•Equations of motion and phenomenology of energy
transfer in r and k-space for iHI turbulence.
•Richardson energy cascade and the 5/3rd energy
spectrum (power spectral density)
•Measurement and ensembles
•Higher order two-point statistics
•Some real data from the solar wind
•4/5th (third order) law
•fractal models (if time permits)
Scaling
(beyond spectra)
scaling, fractals and all that jazz ...
&!
Change scale from t to bt
AND
scale x to bHx
!
!&!
x
!"!!
!"&!
!#!!
!#&!
!$!!
!$&!
!
"
#
$
t
%
&
'
'
()"!
#)
)
!#)
bHx
!())
!(#)
!,))
!,#)
!+))
!+#)
!"#
$
$"#
bt
%
%"#
*
&'()
scaling, fractals and all that jazz ...
&!
Change scale from t to bt
AND
scale x to bHx
!
!&!
x
!"!!
!"&!
!#!!
!#&!
!$!!
!$&!
!
"
#
$
t
%
&
'
'
()"!
#)
)
!#)
bHx
!())
If the statistics of
bHx is the same as x then
process is
statistically self-similar
!(#)
!,))
!,#)
!+))
Hurst exponent H
!+#)
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$
$"#
bt
%
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*
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self-similarity
increments
y(t, τ ) = x(t + τ ) − x(t)
)!!
'!!
%!!
τ
12-3
#!!
!
!#!!
!%!!
!'!!
!
"
#
$
%
&
-./0
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&
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self-similarity
increments
y(t, τ ) = x(t + τ ) − x(t)
)!!
'!!
%!!
τ
12-3
#!!
!
!#!!
!%!!
!'!!
!
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self-similarity
)!!
'!!
%!!
τ
12-3
#!!
!
!#!!
probability density function
!%!!
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!
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pdf scaling
P (y, τ ) = τ
−H
Ps (yτ
−H
&"
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)
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)*(+
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pdf scaling
P (y, τ ) = τ
−H
Ps (yτ
−H
)
$%"&
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)!!'
#
"
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Test statistic and its scaling
increments
y(t, τ ) = x(t + τ ) − x(t)
)!!
'!!
%!!
τ
12-3
#!!
!
!#!!
!%!!
!'!!
!
"
#
$
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&
-./0
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%
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#
!
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!'
!
Test statistic and its scaling
increments
y(t, τ ) = x(t + τ ) − x(t)
)!!
'!!
%!!
τ
12-3
#!!
!
!#!!
!%!!
!'!!
!
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#
!
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!
pth order moment
N
�
1
p
p
M (τ ) =
yj
N j=1
Test statistic and its scaling
increments
y(t, τ ) = x(t + τ ) − x(t)
)!!
'!!
%!!
τ
12-3
#!!
!
!#!!
!%!!
!'!!
!
"
#
$
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#
pth order moment
N
�
1
p
p
M (τ ) =
yj
N j=1
moment scaling
!
!#
M (τ ) = M (1)τ
p
!%
!'
!
"
#
$
%
&
-./0
'
(
)
*
p
ζ(p)
"!
&
+,"!
log M (τ ) = log M (1) + ζ(p) log τ
p
p
via ordinary least-squares regression
Heavy-tails and intermittency
probability density
N
�
1
p
p
S (τ ) =
|yi |
N i=1
S1
S2
S3
Sn
Test statistic and its scaling
log M (τ ) = log M (1) + ζ(p) log τ
log10Mp(τ)
p
ζ(p) = pH
single exponent scaling
p
Test statistic and its scaling
log M (τ ) = log M (1) + ζ(p) log τ
p
log10Mp(τ)
p
ζ(p) = pH
single exponent scaling
if ζ(p) non − linear
then multi exponent
scaling
Limit theorems and the origin of scaling
‘All epistemological value of the theory of probability is based
on this: that large scale random phenomena in their collective
action create strict, non-random regularity.’
(Gnedenko and Kolmogorov, Limit Distributions for Sums of Independent
Random Variables)
Limit theorems
Central Limit Theorem (De Moivre, Laplace, Lyapunov)
SN
N
�
1
yi
=√
N i=1
lim SN → Gaussian
N →∞
Limit theorems
Central Limit Theorem (De Moivre, Laplace, Lyapunov)
SN
N
�
1
yi
=√
N i=1
lim SN → Gaussian
N →∞
Generalized Central Limit Theorem (Lévy)
SN =
1
N 1/α
N
�
i=1
yi
lim SN → Lévy
N →∞
Limit theorems
Central Limit Theorem (De Moivre, Laplace, Lyapunov)
SN
N
�
1
yi
=√
N i=1
lim SN → Gaussian
N →∞
Generalized Central Limit Theorem (Lévy)
SN =
1
N 1/α
N
�
yi
i=1
and many others!
lim SN → Lévy
N →∞
Stable processes
Limit theorems and self-similar processes
SN
N
�
1
yi
=√
N i=1
SN =
1
N 1/α
N
�
i=1
yi
Coarse-graining/averaging
Scaling
study of limit theorems and
stable processes have a very
profound link to self-similar
processes
outline
•Equations of motion and phenomenology of energy
transfer in r and k-space for iHI turbulence.
•Richardson energy cascade and the 5/3rd energy
spectrum (power spectral density)
•Measurement and ensembles
•Higher order two-point statistics
•Some real data from the solar wind
•4/5th (third order) law
•fractal models (if time permits)
Test statistic and its scaling
log M (τ ) = log M (1) + ζ(p) log τ
p
p
")%
1
")!
"
!324
#)*
#)'
#)%
#)!
#1
!!
ζ(p) = pH
monoscaling
!"
#
"
!
$
+,-./012
%
&
'
ζ(p) non − linear
multiscaling
(
moment scaling
moment scaling
1.2
Inertial range
1
!(m)
0.8
0.6
0.4
0.2
0
0
1
2
3
Moment m
4
5
non-Gaussian pdfs
log10 (l) PDF(z+,l)
0
a.) case A
−1
−4
−5
−6
0
DF(z+,l)
0
l= 0.0208 L0
l= 0.0052 L0
−3
−2
l= 0.0104 L0
−2
−1
l = 0.0417 L
b.) case B
outline
•Equations of motion and phenomenology of energy
transfer in r and k-space for iHI turbulence.
•Richardson energy cascade and the 5/3rd energy
spectrum (power spectral density)
•Measurement and ensembles
•Higher order two-point statistics
•Some real data from the solar wind
•4/5th (third order) law
•fractal models (if time permits)
4/5ths law (Kolmogorov 1941)


2 ∂E
2
1 ∂S2
1 ∂ 4
ν ∂  4 ∂S2 

=− ε=
+ 4 (r S3) − 4 r

3 ∂t
3
2 ∂t
6r ∂r
r ∂r
∂r
∂ 4
4
−4εr � (r S3)
∂r
4
S3(r) � − εr
5
3
4/5ths law (Kolmogorov 1941)


2 ∂E
2
1 ∂S2
1 ∂ 4
ν ∂  4 ∂S2 

=− ε=
+ 4 (r S3) − 4 r

3 ∂t
3
2 ∂t
6r ∂r
r ∂r
∂r
∂ 4
4
−4εr � (r S3)
∂r
4
S3(r) � − εr
5
3
4/5ths law (Kolmogorov 1941)


2 ∂E
2
1 ∂S2
1 ∂ 4
ν ∂  4 ∂S2 

=− ε=
+ 4 (r S3) − 4 r

3 ∂t
3
2 ∂t
6r ∂r
r ∂r
∂r
∂ 4
4
−4εr � (r S3)
∂r
4
S3(r) � − εr
5
3
4/5ths law (Kolmogorov 1941)


2 ∂E
2
1 ∂S2
1 ∂ 4
ν ∂  4 ∂S2 

=− ε=
+ 4 (r S3) − 4 r

3 ∂t
3
2 ∂t
6r ∂r
r ∂r
∂r
∂ 4
4
−4εr � (r S3)
∂r
4
S3(r) � − εr
5
3
3
2 ∂t
6r ∂r
r ∂r
4/5ths law (Kolmogorov 1941)
∂
4
4
−4εr � (r S3)
∂r


2 ∂E
2
1 ∂S2
1 ∂ 4
ν ∂  4 ∂S2 

=− ε=
+ 4 (r S3) − 4 r

3 ∂t
3
2 ∂t
6r ∂r
r ∂r
∂r
∂ 4
4
−4εr � (r S3)
∂r
4
4
S3(r) � − εr
5
why is the 4/5th law important?
• One of the few exact results from Navier Stokes Equations -- de facto exact closure
of Karman Howarth equations
• Make a direct measurement of the energy transfer rate from simple structure
functions i.e. ‘straight-forward’ moment calculations
• Energy in = Energy out => direct measurement of total energy going into
dissipation and heating (thermodynamics of the system)
• Can be shown valid for each realisation – not dependent on an ensemble average
• Free from intermittency corrections
outline
•Equations of motion and phenomenology of energy
transfer in r and k-space for iHI turbulence.
•Richardson energy cascade and the 5/3rd energy
spectrum (power spectral density)
•Measurement and ensembles
•Higher order two-point statistics
•Some real data from the solar wind
•4/5th (third order) law
•Fractal models (if time permits)
what does all of this mean and why does it matter?
So what?
Power-laws, exponents -- why should we care?
• Many theories and models
‣ main prediction is scaling behaviour and thus scaling exponents
‣ directly measurable from observations
• Information on the scaling of statistical quantities and thus help in prediction
of bulk properties of turbulent flows e.g. ability to calculate Reynolds stresses
in Reynolds averaged equations (RANS).
Power-laws, exponents -- why should we care?
• Many theories and models
‣ main prediction is scaling behaviour and thus scaling exponents
‣ directly measurable from observations
• Information on the scaling of statistical quantities and thus help in prediction
of bulk properties of turbulent flows e.g. ability to calculate Reynolds stresses
in Reynolds averaged equations (RANS).
However
• Need to take a sober attitude to such things -- avoid the temptation to just fit
straight lines to log-log plots. Statistics and errors need to be handled well.
fractal dissipation
Single exponent H
living on a fractal set of
dimension D where
dissipation occurs
Global Scale Invariance
Cantor ‘dust’ courtesy of Andrew Top: http://www.andrewtop.com/IFS3d/
IFS3d.html
Multifractal dissipation
Courtesy of R. Roemer (Warwick): multifractal electronic
wavefunction at metal insulator transition in 3D Anderson model
Multifractal dissipation
multiple exponents h
living on a fractal sets of
dimension D(h) where
dissipation occurs
Local Scale Invariance
the end
Questions?