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

This lecture explores the physical complexity of turbulence, discussing vorticity and helicity dynamics, tensor kinematics, energy spectra, and beyond Kolmogorov turbulence. Topics also include MHD and rotation modeling, adaptive mesh refinement, and applications to the dynamo problem. The lecture emphasizes the need for a deeper understanding of fundamental processes and the use of added complexity in models, along with adequate resolution in space and time.

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

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  1. 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

  2. GENERAL OUTLINE for LECTURES Physical complexity of flows on Earth and beyond Vorticity 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 how II - The Lagrangian averaging model, for MHD and perhaps for fluids II - Adaptive mesh refinement with spectral accuracy II - Application to the dynamo problem at low magnetic Prandtl number

  3. Energy dissipation (Celani) • Eulerian velocity • Acoustic intensity (vorticity fluctuations) From Baudet, Cargèse Summer school

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

  5. Hurricane Francis from Space

  6. * The Sun, and other stars • * The Earth, and other planets - • including extra-solar planets • The solar-terrestrial interactions • (space weather), the magnetospheres, Many parameters and dynamical regimes Many scales, eddies and waves interacting Cluster  MMS, …

  7. Weather, climate and all that … 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)

  8. Seemingly simple questions … • 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

  9. Surface-Atmosphere Interactions • Influences of atmospheric stability, orography, and plants all matter on ecosystem exchanges From Peter Sullivan, NCAR

  10. Surface-Atmosphere Interactions • Mathematical model: Trees viewed as fractal •  ensuing evaluation of transport (drag) coefficientsthrough Random Numerical Simulations (RNS) Meneveau et al. (JHU) Other possible model: corrugation?

  11. Seamless predictions across scales, from hourly to decadal Greenland ice cover By how much is the sea level going to rise?

  12. Seamless predictions across scales, from hourly to decadal + chemistry, biology, economy, policies, society, …

  13. Slide after Mark Rast, TOY Workshop (NCAR)

  14. Modeled SST, West coast Observed SST, East coast Sea Surface Temperatures (SST)

  15. Slide after Krueger, TOY Workshop (NCAR)

  16. Slide after Ian Foster (Argonne), National Energy Modeling system One modeling example of societal complexity: wiring diagrams

  17. Linkingacrossscales Models are advancing to tackle different issues Continental Scale → Focus of modelers: Different Scales (space/time) Different Issues & Problems Different Stakeholders Different Decisions Human/watershed Scale → Slide after Roy Rasmussen, NCAR Center for Hydrometeorology and Remote Sensing, University of California, Irvine

  18. Complex interactions • Rotation, stratification (waves & eddies), radiation, … • Compressibility, moisture, … • Chemistry, biology, hydrology • Math & algorithms • Boundaries, geometry, … • Socio-politico-economical processes

  19. 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, …)

  20. Vorticity equation: ∂tω = curl (v xω) + νΔω + curlF Also: Dtω =∂tω + v. grad ω = ω.Grad v + νΔω + curl F advection stretching by velocity gradients + dissipation + forcing Model: w ~ |ω| Dtw = w . grad v, grad v ~ O(1): exponential growth of vorticity at early times But You can also view w ~ grad v so Dtw = w2 :explosive growth w(t)~ (t-t*)-1 What is really happening? What can get us out of this explosive growth? Vorticity dynamics

  21. Vorticity equation: ∂tω = curl (v xω) + νΔω + curlF Also: Dtω =∂tω + v. grad ω = ω.Grad v + νΔω + curl F advection stretching by velocity gradients + dissipation + forcing Model: w ~ |ω| Dtw = w . grad v, grad v ~ O(1): exponential growth of vorticity at early times But You can also view w ~ grad v so Dtw = 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

  22. 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

  23. 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 Vorticity dynamics and dissipation

  24. 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)

  25. x After Sreenivasan (1998) and Kaneda (Cargèse, 2007) X: Kaneda et al. (2002, 2009), Rλ ~ 1500

  26. Nth-order scaling exponents of structure functions (velocity and temperature differences) n/3 Slide from Lanotte, Cargèse Summer school

  27. 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)

  28. 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, …)

  29. ω Helicity dynamics H is a pseudo (axial) scalar u

  30. ω Helicity dynamics H is a pseudo (axial) scalar u <ui(k)uj*(-k)>= UE(|k|) Pij(|k|)

  31. ω Helicity dynamics H is a pseudo (axial) scalar u <ui(k)uj*(-k)>= UE(|k|) Pij(|k|) + εijlkl UH(|k|)

  32. ω Helicity dynamics H is a pseudo (axial) scalar u <ui(k)uj*(-k)>= UE(|k|) Pij(|k|) + εijlkl UH(|k|) L R 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, …

  33. ω Helicity dynamics H is a pseudo (axial) scalar u <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 …

  34. ω Helicity dynamics H is a pseudo (axial) scalar u <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’’)

  35. ω Helicity dynamics H is a pseudo (axial) scalar u A wild knot L R 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, …

  36. Helicity in tropical cyclones versus shear Molinari & Vollaro, 2010 Shear & helicityin the atmosphere Helicity spectrum in the Planetary Boundary Layer: K41 Koprov, 2005

  37. 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, …)

  38. Helicity dynamics hr=cos(v, ω), non-helical TG flow Blue, hr>0.95; Red, hr<-.95 • 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

  39. Helicity dynamics hr=cos(v, ω), non-helical TG flow Blue, hr>0.95; Red, hr<-.95 • 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

  40. Helicity dynamics hr=cos(v, ω), non-helical TG flow Blue, hr>0.95; Red, hr<-.95 • 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

  41. Vorticity ω=curl v & Relative helicity intensity h=cos(v, ω) Local v-ωalignment (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 Blue, h> 0.95 ; Red, h<-0.95

  42. Exact laws in turbulence With 5 hypotheses: Homogeneity Isotropy Incompressibility Stationarity High Reynolds number Kolmogorov 1941, …

  43. Exercise: The 12th law for 1D Burgers equation x’=x+r u’=u(x’)

  44. The magnetohydrodynamics (MHD) equations • P is the pressure, B is the induction (in Alfvén velocity units), j = ∇ × B is the current, ηisthe resistivity, and div B = 0 (not an assumption). ______ Lorentz force Elsässer: z± = v ± B  ... What is different from the Navier-Stokes eqs.?

  45. 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?

  46. 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 for δF(r) = F(x+r) - F(x) : structure function for field F ; longitudinal component δFL(r) = δF . r/ |r| < δvLδvi2 >+ <δvLδbi2 > -2 < δbLδviδbi > =-(4/D)εTr - <δbLδbi2 > - < δbLδvi2 >+2 < δvLδviδbi > =-(4/D) εcr with εT = - dt(EV+EM) andεc = - dt<v.b> ; D is the space dimension

  47. 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 for δF(r) = F(x+r) - F(x) : structure function for field F ; longitudinal component δFL(r) = δF . r/ |r| < δvLδvi2 >+ <δvLδbi2 > -2 < δbLδviδbi > =-(4/D)εTr - <δbLδbi2 > - < δbLδvi2 >+2 < δvLδviδbi > =-(4/D) εcr with εT = - dt(EV+EM) andεc = - dt<v.b> ; D is the space dimension

  48. 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 for δF(r) = F(x+r) - F(x) : structure function for field F ; longitudinal component δFL(r) = δF . r/ |r| • < δvLδvi2 >+ <δvLδbi2 > -2 < δbLδviδbi> =-(4/D)εTr • - <δbLδbi2 > - < δbLδvi2 >+2 < δvLδviδbi> =-(4/D) εcr • with εT = - dt(EV+EM) andεc = - dt<v.b> ; D is the space dimension • Strong V regime, or strong B regime or Alfvénic (v~B) regime(Ting et al ‘86); in the latter case, v-B correlations play a dynamical role • Where such laws apply, the energy input can be measured, e.g. in the solar wind

  49. 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 whereθ 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)

  50. 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(ℓ) ~ ℓi and E(ℓ) ~ ℓf(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, …

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