Spectra and Vortices of Two-Dimensional Turbulence · 2006. 7. 24. · spectra can be observed...
Transcript of Spectra and Vortices of Two-Dimensional Turbulence · 2006. 7. 24. · spectra can be observed...
-
Meteorology and Climate Centre, UCD, July 2006
Spectra and Vortices ofTwo-Dimensional Turbulence
Colm ConnaughtonCenter for Nonlinear Studies
Los Alamos National LaboratoryU.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
∂t+ ~u · ∇~u = −∇p+ ν∇2~u+ ~f
∇ · ~u = 0
University College Dublin, July 2006 – p. 2
-
Phenomenology of 3 Dimensional Turbulence in a Nutshell
Reynolds number R = LUν .
Energy injected into largeeddies.
Energy removed from smalleddies at viscous scale.
Transfer by interactionbetween eddies.
Concept of inertial rangeK41 : In the limit of ∞ R, all small scale statistical propertiesdepend only on the local scale, k, and the energy dissipation rate, ǫ.Dimensional analysis :
E(k) = cǫ2
3k−5
3 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 threedimensions.
Why?
University College Dublin, July 2006 – p. 4
-
Special Role of Vorticity in Two Dimensions
Vorticity, ω = ∇× ~u, is a scalar in 2D:
∂ω
∂t+ (~u · ∇)ω = forcing + dissipation
where
~u = (ux, uy) =
(∂ψ
∂y,−∂ψ
∂x
)
with stream-function, ψ obtained from solving
ω = −∇2ψ.
In 2D, unlike 3D, both total energy, E =∫|~u|2 d~x and total
enstrophy, H =∫ω2d~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 independentdissipation mechanisms with two fluxes :
Direct cascade of enstrophy from the forcing scale to smallscales. K41 Hypothesis gives
E(k) ∼ ζ 23k−3.
Inverse cascade of energy from the forcing scale to largescales.
E(k) ∼ ǫ 23k− 53 .This picture assumes infinite inertial ranges for both cascades.In reality there are finite size effects which are more potent in2D than in 3D.
University College Dublin, July 2006 – p. 6
-
Are the KLB spectra realised?
Clean numerical or experimental observations of the KLB dualcascade are difficult but possible (recent large simulations ofBoffetta, Celani et al.)
A problem occurs when the inverse cascade is blocked by thelargest scale - a fact which Kraichnan himself was aware of.
What is the true stationary spectrum of two dimensionalturbulence 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 theestablishment 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 frontcannot continue indefinitely.
No (or very little) large scaledissipation
Inverse cascade eventuallyreaches the largest scale, k0.
Compensated spectra showenergy pile-up at largest scale
Emergence of “condensate”E(k) ∼ Ec(t) δ(k − k0) + Ẽ(k).
University College Dublin, July 2006 – p. 9
-
Alternatively : k−3 at later times
V. Borue Phys. Rev. Lett. 1994Late times
Hypo-viscosity at large scales.
Inverse cascade probablyreaches the largest scale, k0,but weakly.
Compensated stationaryspectra show k−3 scaling(almost) everywhere.
No sign of condensation in thesense 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 largescales.
Inverse cascade reaches thelargest scale, k0.
Observe a clean cross-overfrom k−3 to k−
5
3 .
Non-stationary.
No "condensate".
University College Dublin, July 2006 – p. 11
-
Forcing and dissipation
We force and dissipate the vorticity, ω, directly. The forcing anddissipation 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
f~k =√F (k)ζ(~k, t) Intermediate scale forcing
γu(~k) = νu kαu Small scale dissipation
γi(~k) = νi k−αi Large scale dissipation
< ζ(~k1, t1)ζ(~k2, t2) >= δ(~k1 − ~k2)δ(t1 − t2) Physical values areαi = 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 toarrest the k−5/3 before it reaches k0.
1e-05
1e-04
0.001
0.01
0.1
1 10 100
E(k
)
k
E(k)k^-5/3
Late times
Input energy flux ≈ 0.004.Kolmogorov constant ≈ 5.5.Very little direct cascade.
About 50% of energy goesupscale and 50% downscale
Clean, stationary k−5/3 spec-trum.
University College Dublin, July 2006 – p. 13
-
Flow field in physical space
0
1
2
3
4
5
6
0 1 2 3 4 5 6
y
x
Velocity fieldF4>6
University College Dublin, July 2006 – p. 14
-
Energy and Enstrophy Fluxes
-0.002
-0.0015
-0.001
-0.0005
0
0.0005
0.001
0.0015
0.002
0.0025
10 20 30 40 50 60 70 80 90 100
Ene
rgy
Flu
x
k
energy fluxEnergy growth rate
0
2
4
6
8
10
10 20 30 40 50 60 70 80 90 100
Ens
trop
hy F
lux
k
enstrophy flux
Fluxes are computed using standard expressions (egKraichnan, 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 tothe 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 balancingterms :
~v · ∇~v ∼ νi∇−αi~v ⇒ ξ = (νiǫ−1
3 )−
3
2+3αi
Inverse cascade is blocked when ξ ∼ L. Occurs when thedissipation is too weak. Critical value :
ν∗i = ǫ1
3L−2+3αi
3 .
University College Dublin, July 2006 – p. 16
-
Stationary states for decreasingνi
0
0.05
0.1
0.15
0.2
0.25
0 100 200 300 400 500 600 700 800 900 1000
tota
l ene
rgy
time
nu_i=30000nu_i=3000
nu_i=300nu_i=30
nu_i=3nu_i=0.3
As νi decreases it takes longer for the system to reach astationary state.
Relative size of the fluctuations increases as the energy getsconcentrated at larger scales.
Click here to view the movie
University College Dublin, July 2006 – p. 17
-
Spectra with decreasingνi
1e-05
1e-04
0.001
0.01
0.1
1
1 10 100
E(k
)
k
nu_i=30000nu_i=3000
nu_i=300nu=30nu_i=3
nu_i=0.3nu_i=0.03
For large dissipation you see a stationary 5/3 spectrum asexpected (range is small)
Deviations occur when the dissipation weakens.
Reasonable agreement with estimated value of ν∗i .
University College Dublin, July 2006 – p. 18
-
Flow field in physical space
0
1
2
3
4
5
6
0 1 2 3 4 5 6
y
x
Velocity fieldF4>6
University College Dublin, July 2006 – p. 19
-
Emergence of ak−3 regime
1e-05
1e-04
0.001
0.01
0.1
1
1 10 100
E(k
)
k
Stationary state nu_i=0.03Stationary state nu_i=30000
k^-3k^-5/3 Comparison of stationary
states for νi = 30000 andνi = 0.03.
Both −5/3 and −3 (stationary)spectra can be observeddepending on the efficiency oflarge scale dissipation.
Evidence is qualitative.“Strength” of the coherent component of the flow depends on therelative 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 measurementsof spectra show k−5/3 at earlytimes.
Replaced by k−3.3 after theemergence of the large scalecoherent part.
This k−3.3 is more robust thanthe vortices themselves.
Physical space vorticity con-figuration is dependent on theboundary conditions.
University College Dublin, July 2006 – p. 22
-
Is there a cross-over?
1e-05
1e-04
0.001
0.01
0.1
1
1 10 100
E(k
)
k
Stationary state nu_i=0.03k^-5/3
k^-3
Plot compensated spectra.
Answer : probably. Alsoconsistent with previous work.
Cross-over regime requirestuning of parameters.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
10 20 30 40 50 60 70 80
k^5/
3 E
(k)
k
k^5/3 E(k) : stationary
0
5
10
15
20
25
30
35
40
45
50
10 20 30 40 50 60 70 80
k^3
E(k
)
k
k^3 E(k) : stationary
University College Dublin, July 2006 – p. 23
-
Zero large scale dissipation
1e-06
1e-05
1e-04
0.001
0.01
0.1
1
10
1 10 100
E(k
)
k
E(k) @ t=1000 E(k) @ t=2000 E(k) @ t=5000
E(k) @ t=10000k^-5/3k^-3
0
2
4
6
8
10
12
14
16
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Tot
al e
nerg
ytime
Total energy
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
10
100
1000
10 100 1000
Om
ega
t
Maximum omegat^0.5
Large scale flow fluctuates butis not really turbulent.
Amplitude of vortex dipolegrows like
√t− t∗.
Flow profile within vorticesevolves in a self-similar way.
Radial velocity profiles :
2
3
4
5
6
7
8
9
10
11
12
0 0.5 1 1.5 2 2.5 3 3.5
v_az
imut
hal(r
)
r
v_azim t=10000v_azim t=12000v_azim t=14000v_azim t=16000
2
3
4
5
6
7
8
9
10
0 0.5 1 1.5 2 2.5 3 3.5
v_az
imut
hal(r
)
r
v_azim t=10000v_azim t=12000v_azim t=14000v_azim t=16000
University College Dublin, July 2006 – p. 25
-
Anatomy of the large scale vortices
0.01
0.1
1
10
100
1000
0.01 0.1 1
Vor
ticity
r
k_f=0.12k_f=0.03
r^-1.25
Vorticity profile within eachvortex is approximately
Ω(r) ≈ r−1.25
This exponent lacks anobvious theoreticalexplanation.
Independent of the forcingscale. Persists when pumpingis turned off.
University College Dublin, July 2006 – p. 26
-
Statistics of velocity increments
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3
F4(
r)
r
t=100t=500t=900
t=1000Gaussian value
0
2
4
6
8
10
12
14
0.5 1 1.5 2 2.5 3
F4(
r)
r
Intermittency for nu_i=0.03
t=100t=500t=900
t=1000Gaussian Value
Usual inverse cascade Blocked inverse cascade
Moments of velocity increments (Yakhot et al.):
Sn(r) = 〈||(~u(~x+ ~r) − ~u(~x)) · ~r/r||n〉
“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 whatis the final stationary spectrum of 2-D turbulence in a finite boxin the inverse cascade regime.
Considering finite size effects and tunable (large scale)dissipation together allows a consistent picture at a qualitativelevel.
Configurational details are probably not universal.
However, the emergence of a smooth coherent flow at largescales 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
Meteorology and Climate Centre, UCD, July 20063 Dimensional TurbulencePhenomenology of 3 Dimensional Turbulence in a NutshellTwo dimensional turbulenceSpecial Role of Vorticity in Two DimensionsKraichnan-Leith-Batchelor PhenomenologyAre the KLB spectra realised?$k^{-5/3}$ spectrum at early times"Condensation" of energy at the large scalesAlternatively : $ k^{-3} $ at later timesOr both spectra can co-exist?Forcing and dissipationRealisation of the "normal" inverse cascadeFlow field in physical spaceEnergy and Enstrophy FluxesCritical value the large scale dissipationStationary states for decreasing $u _i$Spectra with decreasing $u _i$Flow field in physical spaceEmergence of a $k^{-3}$ regimeExperimental measurements by Shats et al. Experimental measurements by Shats et al. Is there a cross-over?Zero large scale dissipationGrowth of large scale mean flowAnatomy of the large scale vorticesStatistics of velocity incrementsPreliminary conclusions