Meteorology and Climate Centre, UCD, July 2006
Spectra and Vortices of
Two-Dimensional Turbulence
Colm Connaughton
Center for Nonlinear Studies
Los Alamos National Laboratory
U.S.A.
Joint work with:
M. Chertkov (LANL)
I. Kolokolov, V. Lebedev (Landau Institute)
University College Dublin, July 2006 – p. 1
3 Dimensional Turbulence
Generation of small scale structure from large scale motion.
Described by Navier-Stokes equations :
∂~u
+ ~u · ∇~u = −∇p + ν∇2 ~u + f~
∂t
∇ · ~u = 0
University College Dublin, July 2006 – p. 2
Phenomenology of 3 Dimensional Turbulence in a Nutshell
Reynolds number R =
LU
ν .
Energy injected into large
eddies.
Energy removed from small
eddies at viscous scale.
Transfer by interaction
between eddies.
Concept of inertial range
K41 : In the limit of ∞ R, all small scale statistical properties
depend only on the local scale, k, and the energy dissipation rate, ǫ.
Dimensional analysis :
2
3
E(k) = cǫ k
− 53
Kolmogorov spectrum
University College Dublin, July 2006 – p. 3
Two dimensional turbulence
Generation of large scale structure from small scale motion
Notion of an inverse cascade
Two dimensional turbulence looks really different to three
dimensions.
Why?
University College Dublin, July 2006 – p. 4
Special Role of Vorticity in Two Dimensions
Vorticity, ω = ∇ × ~u, is a scalar in 2D:
∂ω
+ (~u · ∇) ω = forcing + dissipation
∂t
where
~u = (ux , uy ) =
∂ψ ∂ψ
,−
∂y
∂x
with stream-function, ψ obtained from solving
ω = −∇2 ψ.
R
In 2D, unlike 3D, both total energy, E = |~u|2 d~x and total
R 2
enstrophy, H = ω d~x are conserved. This fact profoundly alters
the physics.
University College Dublin, July 2006 – p. 5
Kraichnan-Leith-Batchelor Phenomenology
A stationary state requires two cascades with two independent
dissipation mechanisms with two fluxes :
Direct cascade of enstrophy from the forcing scale to small
scales. K41 Hypothesis gives
2
3
E(k) ∼ ζ k −3 .
Inverse cascade of energy from the forcing scale to large
scales.
2
− 53
3
E(k) ∼ ǫ k .
This picture assumes infinite inertial ranges for both cascades.
In reality there are finite size effects which are more potent in
2D than in 3D.
University College Dublin, July 2006 – p. 6
Are the KLB spectra realised?
Clean numerical or experimental observations of the KLB dual
cascade are difficult but possible (recent large simulations of
Boffetta, Celani et al.)
A problem occurs when the inverse cascade is blocked by the
largest scale - a fact which Kraichnan himself was aware of.
What is the true stationary spectrum of two dimensional
turbulence in a finite box?
Several contradictory scenarios in the literature.
Smith and Yakhot ("condensation" at large scales)
Borue (k −3 spectrum)
Tran and Bowman (k −3 and k −5/3 )
University College Dublin, July 2006 – p. 7
k −5/3 spectrum at early times
L. Smith and V. Yakhot Phys. Rev. Lett. 1993:
Everyone(?) agrees that the k −5/3 scaling is correct during the
establishment of the inverse cascade.
For an infinite system this is the full story.
University College Dublin, July 2006 – p. 8
"Condensation" of energy at the large scales
Later times
For a finite system the front
cannot continue indefinitely.
No (or very little) large scale
dissipation
Inverse cascade eventually
reaches the largest scale, k0 .
Compensated spectra show
energy pile-up at largest scale
Emergence of “condensate”
e
E(k) ∼ Ec (t) δ(k − k0 ) + E(k).
University College Dublin, July 2006 – p. 9
Alternatively : k −3 at later times
V. Borue Phys. Rev. Lett. 1994
Late times
Hypo-viscosity at large scales.
Inverse cascade probably
reaches the largest scale, k0 ,
but weakly.
Compensated stationary
spectra show k −3 scaling
(almost) everywhere.
No sign of condensation in the
sense of Smith and Yakhot.
University College Dublin, July 2006 – p. 10
Or both spectra can co-exist?
Tran and Bowman Phys. Rev. E 2004 :
Late times
Almost no dissipation at large
scales.
Inverse cascade reaches the
largest scale, k0 .
Observe a clean cross-over
− 53
−3
from k to k .
Non-stationary.
No "condensate".
University College Dublin, July 2006 – p. 11
Forcing and dissipation
We force and dissipate the vorticity, ω, directly. The forcing and
dissipation are modeled in ~k-space as follows :
ω̇~k + NL ω~k , ψ~k = f~k − γi (~k)ω~k − γu (~k)ω~k
with
p
f~k = F (k)ζ(~k, t) Intermediate scale forcing
γu (~k) = νu k αu
γi (~k) = νi k −αi
Small scale dissipation
Large scale dissipation
< ζ(~k1 , t1 )ζ(~k2 , t2 ) >= δ(~k1 − ~k2 )δ(t1 − t2 ) Physical values are
αi = 0 (friction) and αu = 2 (viscous dissipation)
University College Dublin, July 2006 – p. 12
Realisation of the "normal" inverse cascade
If the large scale dissipation is strong enough we should be able to
arrest the k −5/3 before it reaches k0 .
Late times
0.1
E(k)
k^-5/3
Input energy flux ≈ 0.004.
E(k)
0.01
Kolmogorov constant ≈ 5.5.
Very little direct cascade.
0.001
About 50% of energy goes
upscale and 50% downscale
1e-04
1e-05
1
10
k
100
Clean, stationary k −5/3 spectrum.
University College Dublin, July 2006 – p. 13
Flow field in physical space
6
5
y
4
3
2
1
Velocity field
F4>6
0
0
1
2
3
4
x
5
6
University College Dublin, July 2006 – p. 14
Energy and Enstrophy Fluxes
0.0025
10
energy flux
Energy growth rate
enstrophy flux
0.002
8
0.0015
0.001
Enstrophy Flux
Energy Flux
6
0.0005
0
4
-0.0005
2
-0.001
-0.0015
0
-0.002
10
20
30
40
50
k
60
70
80
90
100
10
20
30
40
50
k
60
70
80
90
100
Fluxes are computed using standard expressions (eg
Kraichnan, 1967)
Behave as expected
University College Dublin, July 2006 – p. 15
Critical value the large scale dissipation
Characteristic velocity fluctuation at scale r can be related to
the exponent of the energy spectrum :
(δv)r ∼ (ǫr)
x−1
2
E(k) ∼ k −x (1 < x < 3)
K-L-B spectra : x = 5/3 gives (δv)r ∼ (ǫr)1/3 , x = 3 gives
(δv)r ∼ (ǫr)
Dissipation (large) scale, ξ, can be estimated by balancing
terms :
−αi
~v · ∇~v ∼ νi ∇
~v ⇒ ξ = (νi ǫ
3
− 31 − 2+3αi
)
Inverse cascade is blocked when ξ ∼ L. Occurs when the
dissipation is too weak. Critical value :
νi∗
1
3
=ǫ L
−
2+3αi
3
.
University College Dublin, July 2006 – p. 16
Stationary states for decreasing νi
0.25
nu_i=30000
nu_i=3000
nu_i=300
nu_i=30
nu_i=3
nu_i=0.3
0.2
total energy
0.15
0.1
0.05
0
0
100
200
300
400
500
600
700
800
900
1000
time
As νi decreases it takes longer for the system to reach a
stationary state.
Relative size of the fluctuations increases as the energy gets
concentrated at larger scales.
Click here to view the movie
University College Dublin, July 2006 – p. 17
Spectra with decreasing νi
1
nu_i=30000
nu_i=3000
nu_i=300
nu=30
nu_i=3
nu_i=0.3
nu_i=0.03
0.1
E(k)
0.01
0.001
1e-04
1e-05
1
10
k
100
For large dissipation you see a stationary 5/3 spectrum as
expected (range is small)
Deviations occur when the dissipation weakens.
Reasonable agreement with estimated value of νi∗ .
University College Dublin, July 2006 – p. 18
Flow field in physical space
6
5
y
4
3
2
1
Velocity field
F4>6
0
0
1
2
3
4
x
5
6
University College Dublin, July 2006 – p. 19
Emergence of a k −3 regime
1
Stationary state nu_i=0.03
Stationary state nu_i=30000
k^-3
k^-5/3
Comparison of stationary
states for νi = 30000 and
νi = 0.03.
0.1
E(k)
0.01
Both −5/3 and −3 (stationary)
spectra can be observed
depending on the efficiency of
large scale dissipation.
0.001
1e-04
1e-05
1
10
k
100
Evidence is qualitative.
“Strength” of the coherent component of the flow depends on the
relative efficiencies of the forcing and large scale damping.
University College Dublin, July 2006 – p. 20
Experimental measurements by Shats et al.
University College Dublin, July 2006 – p. 21
Experimental measurements by Shats et al.
Experimental measurements
of spectra show k −5/3 at early
times.
Replaced by k −3.3 after the
emergence of the large scale
coherent part.
This k −3.3 is more robust than
the vortices themselves.
Physical space vorticity configuration is dependent on the
boundary conditions.
University College Dublin, July 2006 – p. 22
Is there a cross-over?
1
Stationary state nu_i=0.03
k^-5/3
k^-3
0.1
Plot compensated spectra.
0.01
E(k)
Answer : probably. Also
consistent with previous work.
0.001
1e-04
Cross-over regime requires
tuning of parameters.
1e-05
1
10
k
100
0.8
50
k^5/3 E(k) : stationary
k^3 E(k) : stationary
45
0.7
40
0.6
35
30
k^3 E(k)
k^5/3 E(k)
0.5
0.4
25
20
0.3
15
0.2
10
0.1
5
0
0
10
20
30
40
k
50
60
70
80
10
20
30
40
50
60
70
80
k
University College Dublin, July 2006 – p. 23
Zero large scale dissipation
10
16
Total energy
E(k) @ t=1000
E(k) @ t=2000
E(k) @ t=5000
E(k) @ t=10000
k^-5/3
k^-3
1
14
12
0.1
Total energy
10
E(k)
0.01
0.001
8
6
1e-04
4
1e-05
2
1e-06
0
1
10
k
100
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
time
k −3 seems to be established throughout the inertial range.
Spectrum is completely non-stationary.
Very large fluctuations in the energy - nonlocality?
University College Dublin, July 2006 – p. 24
Growth of large scale mean flow
1000
Maximum omega
t^0.5
Omega
Large scale flow fluctuates but
is not really turbulent.
100
Amplitude of vortex dipole
√
grows like t − t∗ .
Flow profile within vortices
evolves in a self-similar way.
10
10
100
1000
t
12
10
v_azim t=10000
v_azim t=12000
v_azim t=14000
v_azim t=16000
11
v_azim t=10000
v_azim t=12000
v_azim t=14000
v_azim t=16000
9
10
8
7
8
v_azimuthal(r)
Radial velocity profiles :
v_azimuthal(r)
9
7
6
6
5
5
4
4
3
3
2
2
0
0.5
1
1.5
2
r
2.5
3
3.5
0
0.5
1
1.5
2
2.5
3
3.5
r
University College Dublin, July 2006 – p. 25
Anatomy of the large scale vortices
Vorticity profile within each
vortex is approximately
1000
k_f=0.12
k_f=0.03
r^-1.25
Vorticity
100
Ω(r) ≈ r−1.25
10
This exponent lacks an
obvious theoretical
explanation.
1
0.1
0.01
0.01
0.1
r
1
Independent of the forcing
scale. Persists when pumping
is turned off.
University College Dublin, July 2006 – p. 26
Statistics of velocity increments
Intermittency for nu_i=0.03
t=100
t=500
t=900
t=1000
Gaussian value
14
t=100
t=500
t=900
t=1000
Gaussian Value
14
12
12
10
8
F4(r)
F4(r)
10
6
8
6
4
4
2
2
0
0
0
0.5
1
1.5
r
2
2.5
3
0.5
1
1.5
2
2.5
3
r
Usual inverse cascade
Blocked inverse cascade
Moments of velocity increments (Yakhot et al.):
Sn (r) = h||(~u(~x + ~r) − ~u(~x)) · ~r/r||n i
“Flatness” F4 (r) =
S4 (r)
S2 (r)2
Measures deviation from Guassian statistics.
“Small scale intermittency” is entirely vortex dominated.
University College Dublin, July 2006 – p. 27
Preliminary conclusions
There is still some work to be done to understand exactly what
is the final stationary spectrum of 2-D turbulence in a finite box
in the inverse cascade regime.
Considering finite size effects and tunable (large scale)
dissipation together allows a consistent picture at a qualitative
level.
Configurational details are probably not universal.
However, the emergence of a smooth coherent flow at large
scales might be a common signature of finite size effects.
Does this have anything to do with atmospheric science?
University College Dublin, July 2006 – p. 28