Effects of Driving Scales on Turbulence
Jungyeon Cho
Chungnam National University, South Korea
Related paper:
Cho & Yoo (2016)
Yoo & Cho (2014)
Yoon, Cho, & Kim (2016)
Introduction:
Turbulence is everywhere
Without turbulence:
molecular diffusion coefficient D ~10-5 cm2/sec
( It’s for small molecules in water.)
 Mixing time ~ (size of the cup)2/D ~ 107 sec ~ 0.3 year !
Spectrum of turbulence
v( x ), ( x ), …
Fourier analysis
=
+
+
….

Theories : E(k) ~ k-m
e.g.) E(k)
~ k-5/3 (Kolmogorov spectrum)
E(k)
k~1/l
Measured spectrum (on the Earth)
Energy injection
dissipation
Inertial
range
E(k)  k-5/3
Various types of turbulence in space
Intracluster medium
Turbulence!
Coma cluster
Interstellar medium
The solar wind
3C 465 -- Abell 2634
Driving is required to maintain turbulence
* Without driving, turbulenc decays on a
time scale of order ~L/v
Conventional wisdom for driving:
Physics of inertial range is independent of forcing
 Driving has not been an important issue in turbulence
Driving
E(k)
Inertial range
k
Topic 1. Effects of driving scale
A conventional way of studying turbulence…
=Driving on the largest scale
E(k)
k
2
Maximization of the
inertial range
 Reserchers use numerical simulations to test
theoretical models.
 If we don’t take into account the driving scale,
it is possible to make mistakes
Example: Chandrasekhar-Fermi method
Observation of dust polarization at FIR/sub-mm
 Strength of the mean field on the plane of the sky (B0,sky)
Grains are aligned  long axis  B
B
Radiation from elongated grains
B
B
Aligned grains
Stronger FIR/sub-mm radiation for
this direction
Observations of polarized FIR/sub-mm emission:
B
Aligned grains
Direction of polarization
(Direction of pol  B)
Review: Crutcher 2012, Heiles & Haverkorn (2012)
See also Gonatas+ 1990; Di Francesco+ 2001; Lai+ 2001;
Crutcher+ 2004; Girart+ 2006; Curran & Chrysostomou 2007;
Heyer+ 2008; Mao+ 2008; Tang+ 2009; Sugitani+ 2011,
Kwon+ 2011, Li, McKee & Klein 2012, ...
Turbulence produces a small-scale
magnetic field B
B0: large-scale field
Turbulence
produces a
random field
B on scales
below the
driving scale
Magnetic tension
can creat waves
Alfven waves
Alfven waves
Suppose that we have a mean magnetic field B0
Let’s consider an Alfven wave..
B
B0
Then, what does B look like?
B
B0
B
B0
B0
B
We can also obtain V:
B
V
Summary) For an Alfven wave moving to the right,
B  -V
In fact,
The CF method is based on the following fact:
If the mean field B0 is weak:
If the mean field B0 is strong:
CF method
B0
For Alfven waves,
B0
If we multiply B0, sky on both sides,
B0, sky
To observer
How can we get V and B?
For simplicity, let’s assume that B0  LOS
LOS Velocity
V ~ Vlos
Optically thin line
Vlos
We assume
waves are
independent
To observer
B0
V
Vlos
?
y
by
Suppose that we have
coherent waves
LOS
B0,sky
x
z
Observed Bobs  3D by (=B)
Observed B0,obs  3D B0,sky
 B/B0,sky = Bobs/B0,obs
?
 CF assumed that B/B0,sky = Bobs/B0,obs
Bobs
Plane of the sky

B0,obs
In this case we have Bobs/B0,obs  tan   
Original CF method (C & F 1953):
~
Numerical Simulations (e.g. Ostriker et al. 2001):
CF method has been used by many researchers
Crutcher+ 2004
Q) Is B/B0,sky = Bobs/B0,obs?
 No!
LOS
y
z
B0,sky
x
by
B
Bobs: Random walk byN1/2
(=B*N1/2)
B0,obs: Just adds up  B0,skyN
 The conventional CF-method
overestimates B0,sky by ~N1/2 !
Bobs
B0,obs
See earlier discussions in
Myers & Goodman 1991, Zweibel 1996, Houde et al 2009.
Estimates of B0,sky from
the conventional CF
Conventional CF indeed overestimates B0,sky
* N = 1/kf
* In our simulations, B0,sky = 1
Cho & Yoo (2016)
Then, how to fix it?
We need to know N!
(N=number of independent eddies along the LOS)
 We can get N from the standard deviation of
centroid velocities
Optically thin line
V
Centroid velocity = average velocity
VC has something to do with N1/2
v1
v2
v3
v4
V shows a random walk  Let’s consider (V1+V2+V3+…)
St Dev of (V1+V2+V3+…) ~ N1/2 |V|
V
 St Dev of (V1+V2+V3+…)/N ~ N-1/2 |V|
 VC ~ N-1/2 |V|
 VC /|V| ~ 1/N1/2
VC /Vlos  1/N1/2
Standard deviation
of cent. vel.
Average width of an
optically thin line
Our new CF-method
= conventional CF-method * VC /Vlos
B0,sky

VC /Vlos
Our correction
factor
Conventional CF

VC /Vlos

VC /Vlos
Estimates of B0,sky from
our CF-method
Our simulations show that ’1
* In our simulations B0,sky = 1
Topic 2. Effects of two-scale driving
In astrophysical fluids, there can be many different
driving mechanisms which act on different scales
s=energy injection rate
Ev(k)
ks
k
Suppose we have a driving on a small scale
See Yoo & Cho (2014)
Effects of two-scale driving
Now, let’s add a large-scale driving
L
s
Ev(k)
kL
ks
k
As L increases, the large-scale spectrum will go up.
Increasing s
Fixed s
Yoo & Cho (2014)
Effects of two-scale driving
Q) When do two spectra have similar heights?
L
s
Ev(k)
kL
k
ks
~
Effects of two-scale driving
Example) Suppose that kL/kS =10.
L
s
Ev(k)
kL
ks=10kL
~
k
~ 10-2.5 !
So, even a tiny energy injection on a large scale
can change the large-scale spectrum significantly!
Q) Is it important?
 Answer) Yes!
Example) Turbulent diffusion
We can show that, if
, turbulent diffusion
is dominated by the large-scale driving.
Another example: velocity dispersion
If energy injection rates are same for small and
large-scale driving, the large-scale driving can
produce velocity dispersion
times larger
than the small-scale one.
 The following question may have two answers:
What is the dominant driving mechanism in the ISM?
In terms of energy injection rate, it can be …
But, in terms of velocity dispersion, it can be …
Summary
• If there are N independent eddies along the
line of sight, the traditional CF method
overestimates the B0,sky by a factor of N1/2
• We found that standard deviation of centroid
velocity is proportional to 1/N1/2
• Our modified CF-method performs better
when the driving scale of turbulence is small
• A large-scale driving is more efficient for
turbulence diffusion and velocity dispersion
Topic 3. Sometime it’s useful to drive on small scales:
Diffusion on scales larger than the driving scale?
Lsys
Passive
scalar or B
Diffusion rate?
Small-scale driving
Diffusion rate?
t=0
t=12
t=28
Size  t1/2
Is magnetic diffusion as fast as that of a passive scalar?
Size  t1
Sometimes small-scale driving reveals a new physics!
Cho (2013; 2014)