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
