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