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Slide of the Seminar  !

The spatiotemporal spectrum of turbulent flows!

!

Prof. Patricio Clark di Leoni

ERC Advanced Grant (N. 339032) “NewTURB” (P.I. Prof. Luca Biferale) !

Università degli Studi di Roma Tor Vergata C.F. n. 80213750583 – Partita IVA n. 02133971008 - Via della Ricerca Scientifica, 1 – 00133 ROMA

The spatiotemporal spectrum of turbulent flows

P. Clark di Leoni and P. D. Mininni

Group of Astrophysical Flows

Physics Department

University of Buenos Aires

Roma, ItaliaOctober 8th, 2015

1 / 43

Turbulence under the e�ect of waves

Start with energy equation in Fourier space

12

ˆu2k

ˆt = ≠iÿ

k=p+quú

k · (uq · q)up

Assume uk = U (·)e≠iÊkt

12

ˆU 2k

ˆt = ≠iÿ

k=p+qU ú

k · (Uq · q)Upei(Êk≠Êq≠Êp)t

∆ Êk = Êp + Êq to have interaction!

2 / 43

Turbulence under the e�ect of waves

Start with energy equation in Fourier space

12

ˆu2k

ˆt = ≠iÿ

k=p+quú

k · (uq · q)up

Assume uk = U (·)e≠iÊkt

12

ˆU 2k

ˆt = ≠iÿ

k=p+qU ú

k · (Uq · q)Upei(Êk≠Êq≠Êp)t

∆ Êk = Êp + Êq to have interaction!

3 / 43

Turbulence under the e�ect of waves

Start with energy equation in Fourier space

12

ˆu2k

ˆt = ≠iÿ

k=p+quú

k · (uq · q)up

Assume uk = U (·)e≠iÊkt

12

ˆU 2k

ˆt = ≠iÿ

k=p+qU ú

k · (Uq · q)Upei(Êk≠Êq≠Êp)t

∆ Êk = Êp + Êq to have interaction!

4 / 43

Waves are known to...

I Alter di�usion processes in the ocean Woods, Nature (1980)

I Make the flow anisotropic Cambon and Jacquin, JFM (1989)

I Change the very nature of nonlinear interaction Nazarenko (2011)

5 / 43

Waves are known to...

I Alter di�usion processes in the ocean Woods, Nature (1980)

I Make the flow anisotropic Cambon and Jacquin, JFM (1989)

I Change the very nature of nonlinear interaction Nazarenko (2011)

6 / 43

Waves are known to...

I Alter di�usion processes in the ocean Woods, Nature (1980)

I Make the flow anisotropic Cambon and Jacquin, JFM (1989)

I Change the very nature of nonlinear interaction Nazarenko (2011)

7 / 43

What, how, and why?

I What’s the role of waves in turbulent flows? How do they coexistwith eddies?

I Characterization of the e�ect of waves, and measurements of theamount of energy in wave modes has been done mostly indirectly.

I Space and time resolved spectra (e.g. Yarom and Sharom, NaturePhysics (2014) and Cobelli et al, PRL (2009)) can study the e�ectof waves directly

I Results for rotating, stratified and quantum turbulence.

8 / 43

What, how, and why?

I What’s the role of waves in turbulent flows? How do they coexistwith eddies?

I Characterization of the e�ect of waves, and measurements of theamount of energy in wave modes has been done mostly indirectly.

I Space and time resolved spectra (e.g. Yarom and Sharom, NaturePhysics (2014) and Cobelli et al, PRL (2009)) can study the e�ectof waves directly

I Results for rotating, stratified and quantum turbulence.

9 / 43

What, how, and why?

I What’s the role of waves in turbulent flows? How do they coexistwith eddies?

I Characterization of the e�ect of waves, and measurements of theamount of energy in wave modes has been done mostly indirectly.

I Space and time resolved spectra (e.g. Yarom and Sharom, NaturePhysics (2014) and Cobelli et al, PRL (2009)) can study the e�ectof waves directly

I Results for rotating, stratified and quantum turbulence.

10 / 43

What, how, and why?

I What’s the role of waves in turbulent flows? How do they coexistwith eddies?

I Characterization of the e�ect of waves, and measurements of theamount of energy in wave modes has been done mostly indirectly.

I Space and time resolved spectra (e.g. Yarom and Sharom, NaturePhysics (2014) and Cobelli et al, PRL (2009)) can study the e�ectof waves directly

I Results for rotating, stratified and quantum turbulence.

11 / 43

Simulations

I GHOST: Parallel pseudospectral code with periodic boundaryconditions (Gomez et al 2005, Mininni et al 2011)

I Spatial resolution: 512x512x512

I Rot and Strat cases: Fluid was started from rest and energy wasinjected via forcing terms.

I Simulations were run for 12 turnover times after reaching steadyturbulent state

I Quantum case: No forcing, basically decay run but with a longenough almost steady state.

I High output cadence: over 40 outputs per wave period!

12 / 43

Simulations

I GHOST: Parallel pseudospectral code with periodic boundaryconditions (Gomez et al 2005, Mininni et al 2011)

I Spatial resolution: 512x512x512

I Rot and Strat cases: Fluid was started from rest and energy wasinjected via forcing terms.

I Simulations were run for 12 turnover times after reaching steadyturbulent state

I Quantum case: No forcing, basically decay run but with a longenough almost steady state.

I High output cadence: over 40 outputs per wave period!

13 / 43

Simulations

I GHOST: Parallel pseudospectral code with periodic boundaryconditions (Gomez et al 2005, Mininni et al 2011)

I Spatial resolution: 512x512x512

I Rot and Strat cases: Fluid was started from rest and energy wasinjected via forcing terms.

I Simulations were run for 12 turnover times after reaching steadyturbulent state

I Quantum case: No forcing, basically decay run but with a longenough almost steady state.

I High output cadence: over 40 outputs per wave period!

14 / 43

Simulations

I GHOST: Parallel pseudospectral code with periodic boundaryconditions (Gomez et al 2005, Mininni et al 2011)

I Spatial resolution: 512x512x512

I Rot and Strat cases: Fluid was started from rest and energy wasinjected via forcing terms.

I Simulations were run for 12 turnover times after reaching steadyturbulent state

I Quantum case: No forcing, basically decay run but with a longenough almost steady state.

I High output cadence: over 40 outputs per wave period!

15 / 43

Simulations

I GHOST: Parallel pseudospectral code with periodic boundaryconditions (Gomez et al 2005, Mininni et al 2011)

I Spatial resolution: 512x512x512

I Rot and Strat cases: Fluid was started from rest and energy wasinjected via forcing terms.

I Simulations were run for 12 turnover times after reaching steadyturbulent state

I Quantum case: No forcing, basically decay run but with a longenough almost steady state.

I High output cadence: over 40 outputs per wave period!

16 / 43

Rotating turbulence

Navier Stokes in a rotating frame

ˆuˆt = (Ò ◊ u) ◊ u ≠

Coriolis˙ ˝¸ ˚2� ◊ u ≠ Òp¸˚˙˝

total pressure

+‹Ò2u +forcing˙˝¸˚

F

Rotation axis is along z (parallel direction)

Inertial waves: ÊR = 2�kÎk ∆ Preferential energy transfer towards modes

with small kÎ (Wale�e, PoF 93)

17 / 43

Rotating turbulence

Navier Stokes in a rotating frame

ˆuˆt = (Ò ◊ u) ◊ u ≠

Coriolis˙ ˝¸ ˚2� ◊ u ≠ Òp¸˚˙˝

total pressure

+‹Ò2u +forcing˙˝¸˚

F

Rotation axis is along z (parallel direction)

Inertial waves: ÊR = 2�kÎk

∆ Preferential energy transfer towards modeswith small kÎ (Wale�e, PoF 93)

18 / 43

Rotating turbulence

Navier Stokes in a rotating frame

ˆuˆt = (Ò ◊ u) ◊ u ≠

Coriolis˙ ˝¸ ˚2� ◊ u ≠ Òp¸˚˙˝

total pressure

+‹Ò2u +forcing˙˝¸˚

F

Rotation axis is along z (parallel direction)

Inertial waves: ÊR = 2�kÎk ∆ Preferential energy transfer towards modes

with small kÎ (Wale�e, PoF 93)

19 / 43

Rotating turbulence

e(k‹, kÎ)

Wale�e’s prediction holds! But exactly where are the waves?Clark di Leoni et al, PoF (2014)

20 / 43

Rotating turbulence

E(k, Ê)Only in the larger scales energy accumulates along modes satisfying thedispersion relation of inertial waves!

0 10 20 30 40 50kz (vertical wavenumber)

0

10

20

30

!

!R

10�2

10�1

100

1 80kz0

1

F

ÊR(0, 0, kz) = 2�21 / 43

Rotating turbulence

E(k, Ê)“Loss” of waves is not due to isotropization, but because sweepingmechanisms become faster at those scales

0 10 20 30 40 50kz (vertical wavenumber)

0

10

20

30

!

!R

10�2

10�1

100

ÊR(0, 1, kz) = 2�kzÔ1+k2z

22 / 43

Stratified turbulence

Boussinesq model with no rotation

ˆuˆt = ≠(u · Ò)u ≠

buoyancy˙˝¸˚N◊z ≠Òp + ‹Ò2u +

forcing˙˝¸˚F

ˆ◊

ˆt = u · Ò◊ ≠ Nuz ≠ ŸÒ2◊

Stratification gradient is along z (parallel direction)

Internal waves: ÊS = Nk‹k ∆ Preferential energy transfer towards modes

with small k‹ Waves now travel in the same direction as the mean flow

23 / 43

Stratified turbulence

Boussinesq model with no rotation

ˆuˆt = ≠(u · Ò)u ≠

buoyancy˙˝¸˚N◊z ≠Òp + ‹Ò2u +

forcing˙˝¸˚F

ˆ◊

ˆt = u · Ò◊ ≠ Nuz ≠ ŸÒ2◊

Stratification gradient is along z (parallel direction)

Internal waves: ÊS = Nk‹k

∆ Preferential energy transfer towards modeswith small k‹ Waves now travel in the same direction as the mean flow

24 / 43

Stratified turbulence

Boussinesq model with no rotation

ˆuˆt = ≠(u · Ò)u ≠

buoyancy˙˝¸˚N◊z ≠Òp + ‹Ò2u +

forcing˙˝¸˚F

ˆ◊

ˆt = u · Ò◊ ≠ Nuz ≠ ŸÒ2◊

Stratification gradient is along z (parallel direction)

Internal waves: ÊS = Nk‹k ∆ Preferential energy transfer towards modes

with small k‹ Waves now travel in the same direction as the mean flow

25 / 43

Stratified flows

Previous works

I Develop vertically sheared horizontal winds Smith & Wale�e, JFM(2003)

I These causes Doppler shifting of internal waves Hines, JAS (1991)

I When the phase velocity of a wave matches that of the horizontalwind Critical Layer absorption occurs Hines, JAS (1991) Winters &D’Asaro, JFM (1994)

I This has been observed in the atmosphere Gossard et al, JGR(1970) Kunze et al, JGR (1990)

I Theories of stratified turbulence don’t take this e�ects into account!

26 / 43

Stratified flows

Previous works

I Develop vertically sheared horizontal winds Smith & Wale�e, JFM(2003)

I These causes Doppler shifting of internal waves Hines, JAS (1991)

I When the phase velocity of a wave matches that of the horizontalwind Critical Layer absorption occurs Hines, JAS (1991) Winters &D’Asaro, JFM (1994)

I This has been observed in the atmosphere Gossard et al, JGR(1970) Kunze et al, JGR (1990)

I Theories of stratified turbulence don’t take this e�ects into account!

27 / 43

Stratified flows

Previous works

I Develop vertically sheared horizontal winds Smith & Wale�e, JFM(2003)

I These causes Doppler shifting of internal waves Hines, JAS (1991)

I When the phase velocity of a wave matches that of the horizontalwind Critical Layer absorption occurs Hines, JAS (1991) Winters &D’Asaro, JFM (1994)

I This has been observed in the atmosphere Gossard et al, JGR(1970) Kunze et al, JGR (1990)

I Theories of stratified turbulence don’t take this e�ects into account!

28 / 43

Stratified flows

Previous works

I Develop vertically sheared horizontal winds Smith & Wale�e, JFM(2003)

I These causes Doppler shifting of internal waves Hines, JAS (1991)

I When the phase velocity of a wave matches that of the horizontalwind Critical Layer absorption occurs Hines, JAS (1991) Winters &D’Asaro, JFM (1994)

I This has been observed in the atmosphere Gossard et al, JGR(1970) Kunze et al, JGR (1990)

I Theories of stratified turbulence don’t take this e�ects into account!

29 / 43

Stratified flows

Previous works

I Develop vertically sheared horizontal winds Smith & Wale�e, JFM(2003)

I These causes Doppler shifting of internal waves Hines, JAS (1991)

I When the phase velocity of a wave matches that of the horizontalwind Critical Layer absorption occurs Hines, JAS (1991) Winters &D’Asaro, JFM (1994)

I This has been observed in the atmosphere Gossard et al, JGR(1970) Kunze et al, JGR (1990)

I Theories of stratified turbulence don’t take this e�ects into account!

30 / 43

Stratified turbulence

E(k, Ê)

0 10 20 30 40 50ky (horizontal wavenumber)

0

10

20

30

40

!

!S + Uyky

!S

!S � Uyky CL 10�2

10�1

100

0 80ky0

1

F

Doppler shifting and Critical Layer absorption appear! This indicates anonlocal transfer of energy from the small to the large scales.Clark di Leoni and Mininni, PRE (2015)

31 / 43

Stratified turbulence

E(k, Ê)

0 10 20 30 40 50ky (horizontal wavenumber)

0

10

20

30

40

!

!S + Uyky

!S

!S � Uyky CL 10�2

10�1

100

0 80ky0

1

F

Doppler shifting and Critical Layer absorption appear! This indicates anonlocal transfer of energy from the small to the large scales.Clark di Leoni and Mininni, PRE (2015)

32 / 43

Superfluid turbulence

Gross-Pitaevskii equation

Nonlinear PDE describing a Bose Einstein condensate for wavefunction

i~ˆÂ

ˆt = ≠ ~2

2m Ò2Â + g|Â|2Â

Madelung transformation gives a the Euler equation plus extra term

Â(r, t) =Û

fl(r, t)m ei m

~ „(r,t)

Vorticity is quantized and concentrated along lines with fl = 0

33 / 43

Superfluid turbulence

Gross-Pitaevskii equation

Nonlinear PDE describing a Bose Einstein condensate for wavefunction

i~ˆÂ

ˆt = ≠ ~2

2m Ò2Â + g|Â|2Â

Madelung transformation gives a the Euler equation plus extra term

Â(r, t) =Û

fl(r, t)m ei m

~ „(r,t)

Vorticity is quantized and concentrated along lines with fl = 0

34 / 43

Superfluid turbulence

Gross-Pitaevskii equation

Nonlinear PDE describing a Bose Einstein condensate for wavefunction

i~ˆÂ

ˆt = ≠ ~2

2m Ò2Â + g|Â|2Â

Madelung transformation gives a the Euler equation plus extra term

Â(r, t) =Û

fl(r, t)m ei m

~ „(r,t)

Vorticity is quantized and concentrated along lines with fl = 0

35 / 43

Superfluid turbulence (Gross-Pitaevskii equation)

fl(r)

36 / 43

Kelvin waves

Vortex linex have tension and Kelvin waves can travel through them.Below the inervortex scale we can have Kelvin wave turbulence.Sound waves are also present 37 / 43

Superfluid turbulence (Gross-Pitaevskii equation)

fl(k, Ê)Sound and kelvin waves!

0 20 40 60 80 100 120 140 160k

0

100

200

300

400

500

!

A10�4

10�3

10�2

10�1

100

0 30!0

0.18

⇢(32,!

) C

0 10 20 30 40 50 60k

0

10

20

30

40

50

!

B 10�3

10�2

10�1

38 / 43

Thanks!

39 / 43

Rotating turbulence

Time correlation functions

We studied the time correlation functions for di�erent modes

�ij(k, ·) = Èuúi (k, t)uj(k, t + ·)Ít

È|uúi (k, t)uj(k, t)|Ít

All modes with just parallel components.

40 / 43

Rotating flow

Behaviour of the decorrelation time

·sw : sweeping time; ·NL: nonlinear time; ·Ê: wave period

·t =1·≠2

sw + ·≠2Ê

2≠1/2

Interaction is carried out by the fasted mechanism!For the isotropic case decorrelation is governed by sweeping e�ects as

predicted by Chen & Kraichnan 1989.

41 / 43

Rotating flow

Behaviour of the decorrelation time

·sw : sweeping time; ·NL: nonlinear time; ·Ê: wave period

·t =1·≠2

sw + ·≠2Ê

2≠1/2

Interaction is carried out by the fasted mechanism!For the isotropic case decorrelation is governed by sweeping e�ects as

predicted by Chen & Kraichnan 1989.42 / 43

Rotating flow

Behaviour of the decorrelation time

·sw : sweeping time; ·NL: nonlinear time; ·Ê: wave period

·t =1·≠2

sw + ·≠2Ê

2≠1/2

The point where ·sw = ·Ê does not correspond to the isotropization(Zeman) scales!

43 / 43