A few issues in turbulence and how to cope with them using computers

Annick Pouquet, NCAR

Alex Alexakis!, Julien Baerenzung&, Marc-Etienne Brachet!, Jonathan Pietarila-Graham&&, Aimé Fournier, Darryl Holm@, Giorgio Krstulovic*, Ed Lee#, Bill Matthaeus%, Pablo Mininni^, Jean-François Pinton!!, Hélène Politano*, Yannick Ponty*, Duane Rosenberg, Amrik Sen & Josh Stawarz

! &!! ENS, Paris and Lyon & MPI, Postdam

&& LANL @ Imperial College & LANL

* Observatoire de Nice # U. Leuwen

% Bartol, U. Delaware, ^ and Universidad de Buenos Aires

CU Boulder, July 2011 , pouquet@ucar.edu

Physical complexity of flows on Earth and beyondVorticity and helicity dynamics

Kinematics of tensors and methodology Exact laws, structures and different energy spectra in MHD? Complexity of phenomenology: beyond Kolmogorov

Weak turbulence and beyond, towards strong turbulence with closures

***************

II – Some results for MHD and for rotation

II - Modeling: why and howII - The Lagrangian averaging model, for MHD and perhaps for fluids II - Adaptive mesh refinement with spectral accuracyII - Application to the dynamo problem at low magnetic Prandtl number

GENERAL OUTLINE for LECTURES

• Energy dissipation (Celani)

• Eulerian velocity

• Acoustic intensity (vorticity fluctuations)

From Baudet, Cargèse Summer school

Observations of galactic magnetic fields (after Brandenburg & Subramanian, 2005)

Hurricane Francis from SpaceHurricane Francis from Space

Many parameters and dynamical regimes

Many scales, eddies and waves interacting

Cluster MMS, …

* The Sun, and other stars* The Earth, and other planets -including extra-solar planets

• The solar-terrestrial interactions (space weather), the magnetospheres,

In order to progress, one needs:

^ A deeper understanding of underlying fundamental processes (minimalist approach)

^ An added complexity in models (maximalist approach: Physics, Chemistry, Biology, Socio-economics,

…)

^ An adequate resolution, both in space and time, both observationally, experimentally and numerically (expensive approach)

Weather, climate and all that …

• By how much is the sea-level going to rise by, say, year 2030?

• What does it take to control the global temperature so that it be increasing by

at most two degrees? Inverse problem

Seemingly simple questions …

Surface-Atmosphere Interactions

• Influences of atmospheric stability, orography, and plants all matter on ecosystem exchanges

From Peter Sullivan, NCAR

Surface-Atmosphere Interactions

Mathematical model: Trees viewed as fractal

ensuing evaluation of transport (drag) coefficients through Random Numerical Simulations (RNS)

Meneveau et al. (JHU)

Other possible model: corrugation?

Seamless predictions across scales, from hourly to decadal

By how much is the sea level going to rise?

Greenland ice cover

Seamless predictions across scales, from hourly to decadal

+ chemistry, biology, economy, policies, society, …

Slide after Mark Rast, TOY Workshop (NCAR)

Modeled SST, West coast Observed SST, East coast

Sea Surface Temperatures (SST)

Slide after Krueger, TOY Workshop (NCAR)

Slide after Ian Foster (Argonne), National Energy Modeling system

One modeling example of societal complexity: wiring diagrams

Models are advancing to tackle different issues

Continental Scale →

Human/watershed Scale →

Different Scales (space/time)Different Issues & ProblemsDifferent StakeholdersDifferent Decisions

Focus of modelers:

Slide after Roy Rasmussen, NCAR

Linkingacrossscales

Center for Hydrometeorology and Remote Sensing, University of California, Irvine

Complex interactions

• Rotation, stratification (waves & eddies), radiation, …

• Compressibility, moisture, …

• Chemistry, biology, hydrology• Math & algorithms• Boundaries, geometry, …

• Socio-politico-economical processes

Multi-scale Interactions

• Large small, or climate weather

• Small large, or weather climate: orography / bathymetry circulation pattern and pluviosity

cloud cover climate

^ Butterfly effect^ Eddy-viscosity and anomalous transport coefficients^ Beating of 2 small frequencies (eddy-noise)^ Inverse cascades (Statistical mech. with 2 or more`` temperatures’’)^ The Andes, the Rockies, the Alps, the Pacific coast^ Memory effect (decadal time scales, El Niño, …)

Vorticity dynamics

• Vorticity equation: ∂tω = curl (v x ω) + ν Δ ω + curl F

Also:

Dt ω =∂t ω + v. grad ω = ω. Grad v + ν Δ ω + curl F advection stretching by velocity gradients + dissipation + forcing

Model: w ~ |ω|

Dt w = w . grad v, grad v ~ O(1): exponential growth of vorticity at early timesButYou can also view w ~ grad v so

Dt w = w2 : explosive growth w(t)~ (t-t*)-1

What is really happening?

What can get us out of this explosive growth?

Vorticity dynamics

• Vorticity equation: ∂tω = curl (v x ω) + ν Δ ω + curl F

Also:

Dt ω =∂t ω + v. grad ω = ω. Grad v + ν Δ ω + curl F advection stretching by velocity gradients + dissipation + forcing

Model: w ~ |ω|

Dt w = w . grad v, grad v ~ O(1): exponential growth of vorticity at early timesButYou can also view w ~ grad v so

Dt w = w2 : explosive growth w(t)~ (t-t*)-1

What is really happening? What can get us out of this explosive growth?

What is the role of the geometry of structures?

Vorticity dynamics and dissipation

DE/DT = ε = -2 ν < ω2>

ν 0, ω infinity?, ε = O(1) ~ U3/L with U~1, L~1

E(k)~k-5/3 Ω (k) ~ k2-5/3 ~ k+1/3: the vorticity peaks at the dissipation scale

Could there be other expressions for ε?

Vorticity dynamics and dissipation

DE/DT = ε = -2 ν < ω2>

ν 0, ω infinity?, ε = O(1) ~ U3/L with U~1, L~1

Could there be other expressions for ε?

Should we introduce another time-scale, such as a wave period? For example, for rotating flows:

ε = [U3/L ] * Ro = U4/L2Ω with Ro=U/LΩ the Rossby number

Ro<<1 at high rotation: less nonlinear transfer because of linear waves

Energy conservation ?

• Energy equation:

Dt <v2>/2 = ε = - ν <ω2>

What happens to the energy conservation when

ν 0 ; does <ω2> infinity ?

There are indications that the energy dissipation rate is finite

ε~ Urms3 / L0

i.e. ε ~ O(1) for Urms ~ 1 and L0 ~ 1

After Sreenivasan (1998) and Ishihara and Kaneda (2002)

After Sreenivasan (1998) and Kaneda (Cargèse, 2007)

X: Kaneda et al. (2002, 2009), Rλ ~ 1500

x

Nth-order scaling exponents of structure functions

(velocity and temperature differences)

Slide from Lanotte, Cargèse Summer school

n/3

Invariants of the Euler equations

Invariants in the absence of dissipation & forcing (ν=0=F):

* Kinetic energy EV = <v2>/2 , together with:

• In three dimensions: kinetic helicity HV = <v. ω > (mid 60s, Moreau; Moffatt; after Woltjer for MHD, mid 50s)

Invariants of the Euler equations

Invariants in the absence of dissipation & forcing (ν=0=F):

* Kinetic energy EV = <v2>/2 , together with:

• In three dimensions: kinetic helicity HV = <v. ω > (mid 60s, Moreau; Moffatt; after Woltjer for MHD, mid 50s)

• Statistical equilibria: no indication of inverse cascade (Kraichnan, 1973)

• In two dimensions: < ω2 >, the ``enstrophy’’ (and more, …)

dVH uω

ω

u

Helicity dynamicsH is a pseudo (axial) scalar

dVH uω

ω

u

Helicity dynamics H is a pseudo (axial) scalar

<ui(k)uj*(-k)>= UE(|k|) Pij(|k|)

dVH uω

ω

u

Helicity dynamicsH is a pseudo (axial) scalar

<ui(k)uj*(-k)>= UE(|k|) Pij(|k|) + εijlkl UH(|k|)

dVH uω

ω

u

Helicity dynamicsH is a pseudo (axial) scalar

<ui(k)uj*(-k)>= UE(|k|) Pij(|k|) + εijlkl UH(|k|)

For particles, helicity is S.P where S is spin vector and P is momentum, versus chirality for fluids and knots (Kelvin, 1873, 1904).L,R: important differences: thalidomide, aspartame, …

L R

dVH uω

ω

u

Helicity dynamicsH is a pseudo (axial) scalar

<ui(k)uj*(-k)>= UE(|k|) Pij(|k|) + εijlkl UH(|k|)

Two defining functions: UE & UH, or E(k) and H(k)

A priori two different scaling laws …

dVH uω

ω

u

Helicity dynamicsH is a pseudo (axial) scalar

<ui(k)uj*(-k)>= UE(|k|) Pij(|k|) + εijlkl UH(|k|)

Two defining functions: UE & UH, or E(k) and H(k)

A priori two different scaling laws …And more if anisotropic (``polarization’’)

ω

u

Helicity dynamics H is a pseudo (axial) scalar

For particles, helicity is S.P where S is spin vector and P is momentum, versus chirality for fluids and knots (Kelvin, 1873, 1904).L,R: important differences: thalidomide, aspartame, …

L R

A wild knot

Shear & helicityin the atmosphere

Koprov, 2005

Helicity spectrum in thePlanetary Boundary Layer: K41

Helicity in tropical cyclonesversus shearMolinari & Vollaro, 2010

Helicity in Hurricane Andrew, Xu & Wu, 2003

Strong helicity where magnetic field is active, Komm et al., 2003

Role of small-scale helicity in the dynamo process (Parker, 1950s, …)

Helicity dynamics

hr=cos(v, ω), non-helical TG flow

Blue, Blue, hhrr>0.9>0.95; Red, 5; Red, hhrr<-.9<-.955

• Zoom on 3D NS flow

• Perspective volume rendering of relative helicity i.e. the degree of alignment between velocity and vorticity for the Taylor-Green (TG) vortex, non-helical globally

Helicity dynamics

• Evolution equation for the local helicity density (Matthaeus et al., PRL 2008):

∂t(v. ω) + v. grad(v. ω) = ω.grad(v2/2 - P) + νΔ (v. ω)

+ forcing

v. ω (x) can grow locally on a fast (nonlinear) time-scale

hr=cos(v, ω), non-helical TG flow

Blue, Blue, hhrr>0.9>0.95; Red, 5; Red, hhrr<-.9<-.955

Helicity dynamics

• Evolution equation for the local helicity density (Matthaeus et al., PRL 2008):

∂t(v. ω) + v. grad(v. ω) = ω.grad(v2/2 - P) + νΔ (v. ω)

+ forcing

v. ω (x) can grow locally on a fast (nonlinear) time-scale

The several ways to a 0 solution

hr=cos(v, ω), non-helical TG flow

Blue, Blue, hhrr>0.9>0.95; Red, 5; Red, hhrr<-.9<-.955

Vorticity ω=curl v & Relative helicity intensity h=cos(v, ω) Local v-ω alignment (Beltramization). Tsinober & Levich, Phys. Lett. (1983); Moffatt, J. Fluid

Mech. (1985); Farge, Pellegrino, & Schneider, PRL (2001), Holm & Kerr PRL (2002).

no local mirror symmetry, and weak nonlinearities in the small scales

Blu

e, h

> 0

.95

; R

ed

, h

<-0

.95

Blu

e, h

> 0

.95

; R

ed

, h

<-0

.95

Exact laws in turbulence

With 5 hypotheses:

HomogeneityIsotropyIncompressibilityStationarityHigh Reynolds number

Kolmogorov 1941, …

Exercise:

The 12th law for 1D Burgers equation

x’=x+ru’=u(x’)

The magnetohydrodynamics (MHD) equations

• P is the pressure, B is the induction (in Alfvén velocity units), j = ∇ × B is the current, η is the resistivity, and div B = 0 (not an assumption).

______ Lorentz force

Elsässer: z± = v ± B ... What is different from the Navier-Stokes eqs.?

Invariants of the MHD equations

Invariants (no dissipation, no forcing, ν=0=η=F):

* Total energy ET = <v2+ b2>/2 , together with:

• In three dimensions: cross-helicity HC =<v.b>, and magnetic helicity HM = <A.b >/2 (Woltjer, mid 50s)

• In two dimensions: HM =0; < A2 >, the square magnetic potential, is invariant (and more, …), with b = curl A

Does it matter for the MHD equations when ν≠0, η ≠0? Does conservation hold for vanishing viscosity & resistivity?

Two coupled exact laws in MHD [1]Politano+AP,GRL 25, 1998

In terms of velocity & magnetic field, two+ invariants (<|v|2+|b|2>/2 & <v.b> ), and two scaling laws for MHD forδF(r) = F(x+r) - F(x) : structure function for field F ;

longitudinal component δFL(r) = δF . r/ |r|

< δvLδvi2 >+ <δvLδbi

2 > -2 < δbLδviδbi > = - (4/D) εTr

- <δbLδbi2 > - < δbLδvi

2 >+2 < δvLδviδbi > = - (4/D) εc r with εT = - dt(EV + EM) and εc = - dt<v.b> ; D is the space dimension

Two coupled scaling laws in MHD [2]Politano+AP,GRL 25, 1998

In terms of velocity & magnetic field, two+ invariants (<|v|2+|b|2>/2 & <v.b> ), and two scaling laws for MHD forδF(r) = F(x+r) - F(x) : structure function for field F ;

longitudinal component δFL(r) = δF . r/ |r|

< δvLδvi2 >+ <δvLδbi

2 > -2 < δbLδviδbi > = - (4/D) εTr

- <δbLδbi2 > - < δbLδvi

2 >+2 < δvLδviδbi > = - (4/D) εc r with εT = - dt(EV + EM) and εc = - dt<v.b> ; D is the space dimension

Two coupled scaling laws in MHD [2]Politano+AP,GRL 25, 1998

THE ROLE OF VELOCITY-MAGNETIC FIELD CORRELATIONS

• Exact law (Politano & AP,1998): dimensionally, < δb2(l) δv(l) > ~ l , does it imply δv ~ l 1/3 as for fluid turbulence? No, because of v-b correlations

• Boldyrev, 2006: what if

< δb2(l) δv(l) θ(l) > ~ l , so e.g., δv(l) ~ δb(l) ~ l1/4 , δθ(l)~ l1/4

where θ is the angle (degree of alignment) of V & B

This scaling is compatible with IK, E(k) ~ k-3/2 (but role of anisotropy)

It implies a variation of V-B alignment with scales (observed numerically)

Several time scales in a turbulent flow* Local turn-over time TNL ~ l/ul ~ k-2/3 for a Kolmogorov spectrum

* Integral time Tint ~ Lint/Urms ~ k0

* Global advection time TE ~ l/Urms~ k-1

* Diffusive time Tdiss ~ l2/ν ~ k-2

T(ℓℓ) ~ ℓℓi and E(ℓℓ) ~ ℓℓf(i)

One can define other time scales, e.g., associated with waves TW

Different regimes and different spectral indices for energy spectra?

E.g., rotation, stratification, compressibility, magnetic fields, …

The GReC experimentFlow Reynolds Numbers and Scales

• Reynolds Numbers: Re = 8.105 to 109 • Integral Scale: L ~ 30 cm

• Taylor Reynolds: Rλ = 1300 to 6000 • Taylor Scale: λ ~ 1.7 mm to 0.4 mm

• Kolmogorov Scale: 20 μm to 0.4 μm

• Time series up to 109 samples w/ sampling @ 1.25 MHz

Slide from Baudet, Cargèse Summer school

2nd order experimental data

Slide from Baudet, Cargèse Summer school

+: 1962

Compensatedenergy spectra

Numerical data

• Kaneda & Ishihara, 2006

• Red: 40963 grid for Navier-Stokes turbulence

Exercise: What is the criterium to choose the Reynolds number at a given grid resolution?

K41-Compensated spectraKaneda Ishihara 2006

Kolmogorov constant

Slide from Kaneda, Cargèse Summer school

Slow convergence …

Is equipartition stemming from statistical mechanics equilibrium (ν=0) responsible for the bottleneck effect (Frisch et al., 2008)?

Result from a 2-pt closure of turbulence (Bos et al.)

Slide from Cambon, Cargèse Summer school

Slow convergence …

Compensated energy spectrum, with bottleneck

VAPOR freeware (John Clyne & Alan Norton, CISL) What happens to a flow when the Reynolds number is increased?

Navier-Stokes grids with N3 points, Taylor-Green flow

643 & 2563

10243 & 20483 grids

Pablo Mininni (NCAR, and Pittsburgh for the

highest resolution run)

VAPOR freeware (John Clyne & Alan Norton, CISL) What happens to a flow when the Reynolds number is increased?

Navier-Stokes grids with N3 points, Taylor-Green flow

643 & 2563

10243 & 20483 grids

Pablo Mininni (NCAR, and Pittsburgh for the

highest resolution run)

Answer: more structures, more scales in interaction, more complexity?

SEMI-ZOOM

Vortex filament complex with particle paths in red

Navier-Stokes

Rλ ~ 1200

20483 grid points

Zoom on vortex tube with helical field lines

Navier-Stokes equations in three dimensions

Rλ ~ 1000

The classical picture of turbulence

uℓ,ℓ uℓ/8, ℓ/8uℓ/4, ℓ/4uℓ/2,ℓ/2

E(k) k

є= є= dE/dtdE/dt : : energy dissipation rateE ~ kE(k) E ~ kE(k) (locality)(locality) and and τ τ ~~ℓℓ/u/u ℓ ℓ (turn-over time)(turn-over time),,

So:So: єє~ u~ uℓℓ33//ℓℓ

and E(k) = CE(k) = CKK єє2/32/3 k k -5/3-5/3

F

v

The classical picture of turbulence

uℓ,ℓ uℓ/8, ℓ/8uℓ/4, ℓ/4uℓ/2,ℓ/2

E(k) kє= є= dE/dtdE/dt , E~kE(k) and , E~kE(k) and ττ~~ℓℓ/u/uℓℓ, , єє~ u~ uℓℓ

33//ℓℓ E(k) ~ E(k) ~ єє2/32/3 k k --

5/35/3

• Can we ignore long-range interactions

e.g. climate at large scale weather at small scale• Can we ignore the aspect ratio & the structure of eddies?

F

v

Some history on transfer and locality in fluid turbulence

• Theory: Kolmogorov (1941); Kraichnan (1971, 1976)

• Direct Numerical Simulations – Importance of non local interactions:

Domaradzki PRL (1987), Phys. Fluids (1988), Kerr (1990)

- Local energy transfer with non local interactions: Domaradzki & Rogallo (1990), Yeung & Brasseur (1991), Ohkitani (1992)

• Models and different measures of locality: Herring (70’s), Zhou (1993, 1996)• Locality in wavelet space: Kishida (1999)

• Bounds on locality: Eyink (1994, 2005)

• Nonlocality for NS and MHD fluids, including Hall MHD

Scaling with Reynolds number of the ratio of nonlocal to local energy flux

at the Taylor scale

Scaling laws barely appear for the largest resolution run:A self-similar Kolmogorov-like behavior becomes apparent only for sufficiently large Reynolds numbers (Navier-Stokes run)

20483

Intermittency: higher-order data

Slide from Baudet, Cargèse Summer school

p=2 3

9

Extreme events as a function of resolution, and thus of Reynolds number (MHD example)

• Probability Distribution Function of current density (with Jrms ~ 1):

purple (483 grid points), 963, 1923, 3843, 7683 & 15363 grids

The flow becomes progressively more non-Gaussian, with large wings

Extrema from ~12 to 130 in units of Jrms : strong but rare events, or intermittency

NS intermittency: Anomalous exponents (using Extended Self Similarity, or ESS) for the TG & ABC flows: there are small but measurable differences

between them

TT22(K,Q)(K,Q)

1.9 101.9 10-5-5

Conclusions for Navier-Stokes• Slow return to homogeneity and isotropy at small scales • Kolmogorov spectrum but departures from self-similarity at higher

order• Intermittency: discrepancies viewed on PDFs and anomalous

exponents between different flows & between active / non-active regions of a given flow: is there a weak break of universality?

• Influence of forcing scale, as seen on enstrophy & as measured on energy transfer, leading to quantifiable degree of interactions between disparate scales

Models such as RDT, Lagrangian model leading to advection by smooth flow may increase in value …

• Scaling of such properties with Reynolds number?• What is the role of helicity (2nd invariant of the Euler equations)?

Thank you for your attention!