- Curso Internacional de Difusi'on Turbulenta Vilanova

1

I Stratified rotating turbulence II Turbulence models

Curso Internacional de Difusi’on Turbulenta Vilanova

Claude Cambon ´ Laboratoire de Mecanique des Fluides et d’Acoustique CNRS – ECL – UCB Lyon I September 4-8, 2006 CTRL-L switch

2

3

Part I: Stratified Rotating Turbulence

Claude Cambon L.M.F.A, UMR CNRS n0 5509, Ecole Centrale de Lyon, France

Team “Ondes et turbulence” : Fabien Godeferd, Lukas Liechtenstein


September 4-8, 2006

Claude Cambon

4

Part I: Stratified Rotating Turbulence

• A simplified context: 3D, no physical boundaries • Governing equations • Eigenmodes decomposition • Results from “RDT”, “KS” and DNS • Achievements and open problems -) Pure rotation: 2D or not 2D ? cyclonic/anticyclonic asymmetry. -) Pure stratification: horizontal layering

• Geometric constraints. Discussion Curso Internacional de Difusi’on Turbulenta Vilanova

September 4-8, 2006

Claude Cambon

5

Part II: Turbulence Models

• Brief survey of modelling methods • Linear (so-called “Rapid Distortion”) theory • A one-particle linear diffusion model • KS with linear dynamics • From RDT to EDQNM • Fully anisotropic description : angle-dependent spectra • RDT approach to the baroclinic instability


September 4-8, 2006

Claude Cambon

6

Lagrangian approach

Trajectories

x˙ = u ˜ (X, t, t0 ) x=x Curso Internacional de Difusi’on Turbulenta Vilanova

September 4-8, 2006

Claude Cambon

7

Governing equations

∂p ∂ui ∂ui 2 + f i3j uj − bδi3 + = ν∇ ui − uj , |{z} ∂t ∂xi ∂xj | {z } Coriolis

∂ui =0 ∂xi

buoyancy

∂b + ∂t

2

N u3 | {z }

∂b = Pr ν∇ b − uj ∂xj 2

stratification 2 external parameters N and f (frequencies) Valid for a liquid or a gas. Pr characterizes the diffusivity of the stratifying agent (temperature, salt)


September 4-8, 2006

Claude Cambon

8

Geophysical context Ω

λ

= Ω cosλ

g

2Ω → f = 2Ω cos λ Stable stratification in ocean (under the mixed zone) and in atmosphere (temporary inversion in troposphere, stratosphere)


September 4-8, 2006

Claude Cambon

9

A first cartoon of linear effects

Fluctuating pressure ? incompressibility ?


September 4-8, 2006

Claude Cambon

10

Eigenmodes decomposition

• incompressibility and pressure → 3D Fourier space (u, b)(x, t) =   P ık·x  (0) (1) ıσk t (−1) −ıσk t  + a+1 N e + a−1 N e e   |a0 N {z } | {z } vortex (QG)

wave (AG)

p • Dispersion law σk = N 2 sin2 θ + f 2 cos2 θ

• Linear dynamics: slow amplitudes a0,±1 are constant. General case • Advantages k·ˆ u = 0, five (u1 , u2 , u3 , b, p) → three (a0 , a+1 , a−1 ).


September 4-8, 2006

Claude Cambon

11

Geometric aspects

TORO

VORTEX

WAVE POLO TOTAL

Toroidal/poloidal (standard) and “Vortex/wave” (f /N − depending)


September 4-8, 2006

Claude Cambon

12

Wave aspects n

Cg k Forcing zone

Mc Ewan (1967), Mowbray & Rarity (1967), Godeferd & Lollini, JFM (1999) Curso Internacional de Difusi’on Turbulenta Vilanova

September 4-8, 2006

Claude Cambon

13

One-particle diffusion: linear models

∆ij =< (xi − Xi )(xj − Xj ) > Taylor (1921) ∆ij

=

RtRt 0

0

< vi (t0 )vj (t00 ) > dt0 dt00

• Simplified Corrsin Hypothesis (Kaneda & Ishida 2000) < vi (X, t)vj (X, t0 ) >=< ui (x, t)uj (x, t0 ) > two-time “RDT” with isotropic initial data : (symb. < w(t)w(t0 ) >= G(t, 0)G(t0 , 0)

< w(0)w(0) >)

w ˆi (k, t) = Gij (k, t, 0)w ˆj (k, 0) ou as (k, t) = as (k, 0), s = 0, ±1 • “Kinematic Simulation”: random model for the slow amplitudes, prescribed isotropic spectrum, “true” computation of trajectories Curso Internacional de Difusi’on Turbulenta Vilanova

September 4-8, 2006

Claude Cambon

14

Vertical diffusivity

Relevance of the linear model 2

∆33 (t) = q (0)

Z

1

1 − cos σ(x)t dx (1 − x ) 2 σ (x) 2

0

Vertical diffusivity is a functional of the dispersion law only ! Curso Internacional de Difusi’on Turbulenta Vilanova

September 4-8, 2006

Claude Cambon

15

Horizontal diffusivity

Dominant ballistic (“vortex induced”) except for pure rotation Curso Internacional de Difusi’on Turbulenta Vilanova

September 4-8, 2006

Claude Cambon

16

Pure rotation

Cambon, Godeferd, Nicolleau & Vassilicos, JFM (2004)


September 4-8, 2006

Claude Cambon

17

Linkage with “pancake ” and “cigar” structures ?


September 4-8, 2006

Claude Cambon

18

(a)


(b)

September 4-8, 2006

Claude Cambon

19

Trajectories no instantaneous structures for KS · · ·

·

· · · · a paradox ? (Kimura & Herring, Liechtenstein et al., JoT, 2005 Curso Internacional de Difusi’on Turbulenta Vilanova

September 4-8, 2006

Claude Cambon

19

Trajectories no instantaneous structures for KS · · ·

·

STRATIFIED

ROTATING

DNS: Nonlinear Anisotropic Coherent vort. KS: Linear Isotropic spect. Incoherent · · · · a paradox ? (Kimura & Herring, Liechtenstein et al., JoT, 2005 Curso Internacional de Difusi’on Turbulenta Vilanova

September 4-8, 2006

Claude Cambon

20

Waves and coherent vortices competing ?

• Relevance of linear solution depends on the order and type of statistical correlations -) doubles: 2 point 1 time: eıσt , e−ıσt -) doubles: 2 point 2 time -) triples: 3 point 1 time: eı(±σk ±σp ±σq )t

→ nonlinear · · · ·

• Dynamical role of dispersive waves. Purely ballistic contribution from the vortex mode.

• What about organised vorticity structures ?


September 4-8, 2006

Claude Cambon

21

Statistical nonlinear theory

• From classical EDQNM (isotropic, no rotation, Orszag 1970,Bos & Bertoglio, 2006) ... -1

10

-2

10

K

4

t=2.6 t=2.8 t=3.2 t=4.0

-5/3

K

-3

E(K)

10

-4

10

10-5 -6

10

10-7 0 10

K2 1

10

2

10 K

3

10

10

4

• ... to EDQNM3 → (A) QNM energy equation (Bellet et al., JFM, 2006) Curso Internacional de Difusi’on Turbulenta Vilanova

September 4-8, 2006

Claude Cambon

22

Angle-dependent spectrum

• Isotropy breaking by spectral transfer T (e) (k): directional anisotropy: 100

1

0.01

0.0001

1e-06

2

1e-08

2

4πk e(k, tf ) = 4πk e(k, cos | {z θ}, tf )

0.1

1

10

100

kk /k

• Spherical averaging → E(k, tf ), prefactor E ∼ Curso Internacional de Difusi’on Turbulenta Vilanova

Ω −3 , not 2D ! tk September 4-8, 2006

Claude Cambon

23

AQNM and DNS 0.01

k −2

E(kn , θm )

0.001 0.0001

100

1e−05

1

1e−06 0.01

1e−07

cos θ ≈ 0

0.0001

1e−08

cos θ ≈ 1

1e−09 1e−10

1

10

1e-06

k

100

1e-08 0.1

1

10

100

5123 DNS by Liechtenstein et al., JoT, 2005 Curso Internacional de Difusi’on Turbulenta Vilanova

September 4-8, 2006

Claude Cambon

24

g

(a)

(b)

Ω

Conical region in which energy concentrates

(c)

(d)


September 4-8, 2006

Claude Cambon

25

Perspectives and open issues, pure rotation

• Dynamics of the slow mode ? AQNM, EDQNM3, classical Wave-Turbulence (Galtier 2003, also Waleffe 1993) and under-resolved DNS/LES

• EDQNM3 and AQNM poorly exploited: much more statistics must be extracted • including triple vorticity correlations for cyclonic/ anticyclonic vorticity asymmetry 0

S

ω

10

−1

10

−1

10

0

10

Ωt/2π

1

10

10

2

Morize et al. (2005)


September 4-8, 2006

Claude Cambon

26

Stably-stratified turbulence

• A significant nonpropagating mode : (vertical) vortex, toroidal, QG, linearized   PV ... P ık·x a0 N (0) +a+1 N (1) eıσk t + a−1 N (−1) e−ıσk t  v(x, t) = e | {z } vortex

• v = (u, b), gravity waves with σk = N kk⊥ Curso Internacional de Difusi’on Turbulenta Vilanova

September 4-8, 2006

Claude Cambon

27

0.01

0.01 0.001

0.001 1e−04 1e−05

E(kn , θm )

E(kn , θm )

1e−04

1e−06 1e−07

cos θ ≈ 0

1e−08

1e−06 1e−07

cos θ ≈ 0

1e−08

1e−09 1e−10

1e−05

(a) 1

cos θ ≈ 1 10

1e−09

k

100

1e−10

(b) 1

cos θ ≈ 1 10

k

100

Angle-dependent toroidal and poloidal modes (Liechtenstein, 2006) Curso Internacional de Difusi’on Turbulenta Vilanova

September 4-8, 2006

Claude Cambon

28

Perspectives and open issues: pure stratification

• Motion in a stratified fluid is not quasi-2D but ANTI-2D ! • Toroidal turbulence: statistical approach to horizontal layering (vs. zig-zag instability, Billand & Chomaz, Lindborg ...)

• Very good EDQNM2-DNS comparisons (Godeferd & Cambon 1994, Godeferd & Staquet 2003): improvements ? EDQNM3, ED, very high Reynolds

• Rotation + stratification (DNS by Liechtenstein et al.), revisiting a QG model


September 4-8, 2006

Claude Cambon

29

Confinment. Geometric constraints

• Rotation with confinment 0

S

ω

10

−1

10

−1

10

0

10

1

Ωt/2π

10

2

10

• Stratification with (weaker) rotation: DNS in flattened boxes (Lindborg 2006, Brethouwer et al. 2006) Curso Internacional de Difusi’on Turbulenta Vilanova

September 4-8, 2006

Claude Cambon