Frank Jenko University of California, Los Angeles www..ucla.edu/~jenko

An introduction to

Turbulence, magnetic fields, and self-organization in laboratory and astrophysical plasmas

Les Houches, France March 23, 2015 Turbulence in lab & natural plasmas Fusion plasmas Space plasmas

Astrophysical plasmas Basic plasma science

What are the fundamental principles of plasma turbulence? Central role of plasma turbulence Particle acceleration & propagation Cross-field transport

Magnetic reconnection Dynamo action

What is the role of plasma turbulence in these processes? Introductory remarks

This lecture is meant to be an introduction; no prior knowledge of turbulence is required

You are invited to ask questions at any time

Lectures may explain, motivate, inspire etc., but they cannot replace individual study... Recommended reading

Also:

P. A. Davidson Turbulence Some turbulence basics Turbulence is ubiquitous…

Active fluids (dense bacterial suspensions) We next summarize the model equations; a detailed motiva- tion is given in SI Appendix. Because our experiments suggest that density fluctuations are negligible (Fig. 2A) we postulate incom- pressibility, ∇ · v 0. The dynamics of v is governed by an incom- ¼ pressible Toner–Tu equation (15–17), supplemented with a Swift– Hohenberg-type fourth-order term (45),

2 2 2 ∂t λ0v · ∇ v −∇p λ1∇v − α β v v Γ0∇ v ð þ Þ ¼ þ ð þ j j Þ þ 2 2 − Γ2 ∇ v; [1] ð Þ where p denotes pressure, and general hydrodynamic considera- tions (52) suggest that λ0 > 1; λ1 > 0 for pusher-swimmers like B. subtilis (see SI Appendix). The α; β -terms in Eq. 1 correspond to ð Þ a quartic Landau-type velocity potential (15–17). For α > 0 and β > 0, the fluid is damped to a globally disordered state with v 0, whereas for α < 0 a global polar ordering is induced. How- ever,¼ such global polar ordering is not observed in suspensions of swimming bacteria, suggesting that other instability mechanisms prevail (53). A detailed stability analysis (SI Appendix) of Eq. 1 implies that the Swift–Hohenberg-type Γ0; Γ2 -terms provide the simplest generic description of self-sustainedð Þ meso-scale turbu- lence in incompressible active flow: For Γ0 < 0 and Γ2 > 0, Fig. 2. Experimental snapshot (A)ofahighlyconcentrated,homogeneous the model exhibits a range of unstable modes, resulting in turbu- quasi-2D bacterial suspension (see also Movie S7 and SI Appendix, Fig. S8). lent states as shown in Fig. 2D. Intuitively, the Γ0; Γ2 -terms de- Flow streamlines v t; r and vorticity fields ω t; r in the turbulent regime, ð Þ ð Þ ð Þ scribe intermediate-range interactions, and their role in Fourier as obtained from (B) quasi-2D bacteria experiments, (C)simulationsofthe space is similar to that of the Landau potential in velocity space deterministic SPR model (a 5, ϕ 0.84), and (D) continuum theory. The ¼ ¼ SI Appendix 1 range of the simulation data in D was adapted to the experimental field ( ). We therefore expect that Eq. describes a wide of view (217 μm × 217 μm) by matching the typical vortex size. (Scale bars, class of quasi-incompressible active fluids. To compare the con- 50 μm.) Simulation parameters are summarized in SI Appendix. tinuum model with experiments and SPR simulations, we next study traditional turbulence measures. minimize oxygen gradients that may cause anisotropic streaming of the oxytactic B. subtilis bacteria (2). To study the effects of Velocity Structure Functions. Building on Kolmogorov’s seminal dimensionality and boundary conditions, experiments were per- work (55), a large part of the classical turbulence literature (27, formed with two different setups: quasi-2D microfluidic chambers 34, 36–38, 40, 41) focuses on identifying the distribution of the flow velocity increments δv t; r; R v t; r R − v t; r . Their with a vertical height H less or equal to the individual body length ð Þ¼ ð þ Þ ð Þ of B. subtilis (approximately 5 μm) and 3D chambers with statistics is commonly characterized in terms of the longitudinal ^ ^ H ≈ 80 μm(SI Appendix, Figs. S6 and S8 and Movies S7–S10). and transverse projections, δv‖ R · δv and δv⊥ T · δv, where ^ ^ ¼ ¼ To focus on the collective dynamics of the microorganisms rather T ϵijRj denotes a unit vector perpendicular to the unit shift than the solvent flow (24, 50), we determined the mean local ¼ð Þ vector R^ R∕ R . The separation-dependent statistical moments motion of B. subtilis directly using particle imaging velocimetry of δv and¼ δvj jdefine the longitudinal and transverse velocity (PIV; see also SI Appendix). A typical snapshot from a quasi-2D ‖ ⊥ structure functions experiment is shown in Fig. 2A. As evident from the inset, local density fluctuations that are important in the swarming/flocking n n S R ≔ δv ; ;n1; 2; …: [2] regime (48, 49, 51) become suppressed at very high filling fractions ‖;⊥ð Þ hð ‖ ⊥Þ i ¼ (SI Appendix, Fig. S5). The corresponding flow fields (Fig. 2B and SI Appendix, Fig. S8) were used for the statistical analysis pre- These functions have been intensely studied in turbulent high-Re sented below. fluids (27, 34, 35, 41) but are unknown for active flow. For isotropic steady-state turbulence, spatial averages · as in Eq. 2 n h i Continuum Theory. The analytical understanding of turbulence become time-independent, and the moments S‖;⊥ reduce to func- phenomena hinges on the availability of simple yet sufficiently tions of the distance R R . ¼ j j accurate continuum models (27). Considerable efforts have been Velocity distributions, increment distributions, and structure made to construct effective field theories for active systems (15– functions for our numerical and experimental data are summar- 17, 19, 31, 32, 52–54), but most of them have yet to be tested ized in Fig. 3. For the SPR model, the velocity statistics can be quantitatively against experiments. Many continuum models dis- calculated either from the raw particle data or from pre-binned tinguish solvent velocity, bacterial velocity and/or orientational flow field data. The two methods produce similar results, order parameter fields, resulting in a prohibitively large number and Fig. 3 shows averages based on individual particle velocities. of phenomenological parameters and making comparison with Generally, we find that both the 2D SPR model and the 2D con- experiments very difficult. Aiming to identify a minimal hydro- tinuum simulations are capable of reproducing the experimen- dynamic model of self-sustained meso-scale turbulence, we study tally measured quasi-2D flow histograms (Fig. 3 A and B) and a simplified continuum theory for incompressible active fluids, structure functions (Fig. 3C). The maxima of the even transverse 2n by focusing solely on the experimentally accessible velocity field structure S⊥ signal a typical vortex size Rv, which is substantially v t; r . By construction, the theory will not be applicable to re- larger in 3D bulk flow than in quasi-2D bacterial flow. Unlike ð Þ gimes where density fluctuations are large (e.g., swarming or their counterparts in high-Re Navier–Stokes flow (27, 34), the flocking), but it can provide a useful basis for quantitative structure functions of active turbulence exhibit only a small re- comparisons with particle simulations and experiments at high gion of power law growth for ℓ ≲ R ≪ Rv and flatten at larger concentrations. distances (Fig. 3C).

14310 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1202032109 Wensink et al. …as well as an important unsolved physics problem

According to a famous statement by …

…and a survey by the British “Institute of Physics” among many of the leading physicists world-wide…

“Millennium Issue” (December 1999)

TURBULENCE: A challenging topic for both basic and applied research

The Turbulence Problem

injection rate ε at large scales is bal- anced by the mean energy dissipation rate at small scales. Richardson and Taylor also appreciated that generic properties of turbulence may be dis- covered in the statistics of velocity dif- ferences between two locations WHAT IS TURBULENCEseparated by a distance r, denoted as THEN? δu(x, x+r) = u(x) – u(x+r). The statis- tics of velocity differences at two loca- tions are an improvement over the statistics of velocity fluctuations at a single location for a number of techni- cal reasons, which we do not discuss here. Longitudinal projections of Turbulence… velocity differences

Figure 4. The Energy Cascade Picture of Turbulence are often measured in modern experi- This figure represents a one-dimensional simplification of the cascade process with ments and are one of the main quanti- β representing the scale factor (usually taken to be 1/2 because of the quadratic ties of interest in the analysis of fluid nonlinearity in the Navier-Stokes equation). The eddies are purposely shown to be turbulence. “space filling” in a lateral sense as they decrease in size. (This figure was modified with permission from Uriel Frisch. 1995. Turbulence: The Legacy of A. N. Kolmogorov. • is an intrinsically nonlinear phenomenonMeasuring velocity differences on Cambridge, UK: Cambridge University Press.) fast time scales and with high preci- sion was a difficult proposition in the early 20th century and required the development of the hot-wire anemometer, in which fluid flowing • occurs only in open systemspast a thin heated wire carries heat away at a rate, proportional to the fluid velocity. Hot-wire anemometry made possible the measurement, on laboratory time scales, of the fluctuat- ing turbulent velocity field at a single point in space (see Figure 5). For a • involves many degrees of freedomflow whose mean velocity is large, velocity differences were inferred from those single-point measurements by Taylor’s “frozen-turbulence” hypothesis.7

• is highly irregular in space and7 If the mean velocity time is large compared with velocity fluctuations, the turbulence can be considered “frozen” in the sense that velocity fluctuations are swept past a single point faster than they would change because of turbulent dynamics. In that case, the spatial separation ∆r is related to Figure 5. Time Series of Velocities in a Turbulent Boundary Layer the time increment ∆t by ∆r = – U∆t, This time series of velocities for an atmospheric turbulent boundary layer with where U is the mean velocity. See also the 7 article “Taylor’s Hypothesis, Hamilton’s Reynolds number Re ~ 2 × 10 was measured at a single location with a hot-wire • often leads to a statistically quasi-stationaryanemometer. The velocity fluctuations are apparently random. (This figure is courtesy of Principle, and the LANS- Model for α Computing Turbulence” on page 152 . Susan Kurien of Los Alamos, who has used data recorded in 1997 by Brindesh Dhruva of Yale University.) state far from thermodynamic equilibrium Number 29 2005 Los Alamos Science 129

Note that these are also the features of LIFE! WHAT IS THE CHALLENGE?

Theories that don’t apply directly:

• nonlinear dynamics

• equilibrium statistical mechanics

• nonequilibrium statistical mechanics near equilibrium

Co-existence of randomness and coherence

Two dangers constantly threaten the world: order and disorder. Paul Valéry VISION BY J. VON NEUMANN

Supercomputers can help to unravel the “mysteries” of turbulence in the spirit of John von Neumann (1949):

„There might be some hope to 'break the deadlock' by extensive, but well-planned, computational efforts. There are strong indications that one could name certain strategic points in this complex, where relevant information must be obtained by direct calculations. This should, in the end, make an attack with analytical methods possible.“ Milestones in turbulence research I

Osborne Reynolds (1842-1912) 1883 Transition from laminar to turbulent flows (Reynolds number) 1895 Reynolds decomposition into mean and fluctuating flows

Ludwig Prandtl (1875-1953) 1904 Recognition of the importance of boundary layers 1925 Mixing length model for turbulent transport

Lewis Fry Richardson (1881-1953) 1922 Book „Weather Prediction by Numerical Process“ Notion of turbulent eddies & (local, direct) energy cascade

Geoffrey Ingram Taylor (1886-1975) 1935 Series of papers on the „Statistical Theory of Turbulence“ Milestones in turbulence research II

Werner Heisenberg (1901-1976) 1923 „Über Stabilität und Turbulenz von Flüssigkeitsströmen“ (PhD Thesis, LMU München, Advisor: Arnold Sommerfeld) 1948 Three papers on the statistical theory of turbulence

Andrey Kolmogorov (1903-1987) 1941 K41 theory: dimensional analysis, -5/3 law (energy spectrum) 1962 K62 theory: scale invariance is broken, problem of intermittency

Robert Kraichnan (1928-2008) 1957- Field-theoretic approach: Direct Interaction Approximation 1967 Inverse energy cascade in two-dimensional fluid turbulence

1973 Field-theoretic approach: Martin-Siggia-Rose formalism Milestones in turbulence research III

Steven Orszag (1943-2011) 1966 Eddy-Damped Quasi-Normal Markovian approximation 1969- Towards Direct Numerical Simulations via spectral methods 1972 First 3D DNS on a 323 grid by S. Orszag & G. Patterson

1948- Numerical weather prediction by J. von Neumann & J. Charney 1963 Large-Eddy Simulation techniques by J. Smagorinsky 1965 Fast Fourier Transform algorithm by J. Cooley & J. Tukey 1977 Cray-1 at the National Center for Atmospheric Research

2002 Largest 3D DNS to date on a 40963 grid by Y. Kaneda et al. Turbulence: The bigger picture

Some grand challenges • Design airplanes, ships, cars etc. • Predict weather & climate • Unravel role of turbulence in space & astrophysics • Predict performance of fusion devices like ITER

Some open problems beyond Homogeneous Isotropic Turbulence • Effects of inhomogeneity, anisotropy, compressibility (role of walls, drive, stratification, rotation etc.) • From fluid to magneto-/multi-fluid to kinetic turbulence

Conceptual approach • Ab initio simulations of complicated problems are not feasible • Seek physics understanding to construct reliable minimal models INSTITUTE FOR PURE AND APPLIED MATHEMATICS Los Angeles, California Turbulence MATHEMATICS OF TURBULENCE Image courtesy of David Goluskin research September 8 – December 12, 2014

ORGANIZING COMMITTEE: Charlie Doering (University of Michigan), Gregory Eyink (Johns HopkinsORGANIZING University), COMMITTEE: Pascale Garaud NAME (UC IN Santa BOLD Cruz), (INSTITUTION), Michael Jolly NAME (Indiana IN University),BOLD Keith (INSTITUTION), NAME IN BOLD (INSTITUTION), NAME IN BOLD (INSTITUTION) today Julien (University of Colorado), Beverley McKeon (Caltech)

Scientific Overview

Turbulence is perhaps the primary paradigm of complex nonlinear multi-scale dynamics. It is ubiquitous in fluid flows and plays a major role in problems ranging from the determination of drag coefficients and heat and mass transfer rates in engineering applications, to important dynamical processes in environmental science, ocean and atmosphere dynamics, geophysics, and astrophysics. Understanding turbulent mixing and transport of heat, mass, and momentum remains an important open challenge for 21st century physics and mathematics.

This IPAM program is centered on fundamental issues in mathematical , scientific computation, and applications including rigorous and reliable mathematical estimates of physically important quantities for solutions of the partial differential equations that are believed, in many situations, to accurately model the essential physical phenomena. Workshop Schedule • Mathematics of Turbulence Tutorials: September 9 - 12, 2014. • Workshop I: Mathematical Analysis of Turbulence. September 29 - October 3, 2014. • Workshop II: Turbulent Transport and Mixing. October 13 - 17, 2014. • Workshop III: Geophysical and Astrophysical Turbulence. October 27 - 31, 2014. • Workshop IV: Turbulence in Engineering Applications. November 17 - 21, 2014. • Culminating Workshop at Lake Arrowhead Conference Center, December 7 – 12, 2014. Participation

This program will bring together physicists, engineers, analysts, and applied mathematicians to share problems, insights, results and solutions. Enhancing communications across these traditional disciplinary boundaries is a central goal of the program.

Full and partial support for long-term participants is available. We are especially interested in applicants who intend to participate in the entire program (September 8 – December 12, 2014), but will consider applications for shorter periods. Funding is available for participants at all academic levels, though recent PhDs, graduate students, and researchers in the early stages of their careers are especially encouraged to apply. Encouraging the careers of women and minority mathematicians and scientists is an important component of IPAM's mission and we welcome their applications. More information and an application is available online. www.ipam.ucla.edu/programs/mt2014 On the physics of 3D fluid turbulence Early observations

Leonardo da Vinci (ca. 1500) The Navier-Stokes equations (1822)

The NSE for incompressible fluids: ν: kinematic viscosity

p: pressure / mass density

Expressed in terms of vorticity :

Claude Navier George Stokes (1785-1836) (1819-1903) moves in the velocity field, and is fittingly called the advective term. It will be present even in amoves steady in state the velocity velocity field, field. and is fittingly called the advective term. It will be present The twoeven terms in on a steady the right state hand velocity side field. of the equation represent the contact forces per volume onThe the twomoves fluid terms particle. in on the the velocity The right first field, hand term, and side the is of fittingly the gradient equation called of th represent theepressure,describesthe advecttheive contact term. It forces will perbe present effect ofvolume forceseven acting on the in a normal fluid steady particle. state on the velocity The surfaces first field. term, of the the fluid gradient parti ofcle. thepressure,describesthe The second term describese theffect shear ofThe forces stress two terms acting forces on normal on the the right on surface the hand surfaces of side the of fluid of the the p equationarticle. fluid parti The representcle. shear Thethe forces second contact are term forces per zero whendescribes the fluidvolume the is shear at on rest. the stress fluid forces particle. on the The surface first term, of the the fluid gradient particle. of The thepressure,describesthe shear forces are zero wheneffect the of fluid forces is at acting rest. normal on the surfaces of the fluid particle. The second term The NSE’s candescribes be non-dimensionalised the shear stress forces by introducing on the surface a chara of thecteristic fluid particle. velocity, TheV ,and shear forces are lenght, L.Definingnew,dimensionlessfields:Thezero NSE’s when can the be fluid non-dimensionalised is at rest. by introducing a characteristic velocity, V ,and lenght, L.Definingnew,dimensionlessfields: Thev NSE’s= V ˜v can, x be= non-dimensionalisedL˜x ,t=(L/V )t,˜ (p/ byρ introducing)=V 2p˜ a characteristic velocity, V ,and lenght, L.Definingnew,dimensionlessfields:v = VSELF-SIMILARITY˜v , x = L˜x ,t=(L/V )t,˜ (p/ &ρ )=PIPEV 2p˜ FLOWS Osborne Reynolds (1883) the NSE’s become the NSE’s become v = V ˜v , x = L˜x ,t=(L/V )t,˜ (p/ρ)=V 2p˜ 1 ∂t˜v + ˜v )˜v = p˜ + ˜v 1 (6) the NSE’s become∂ ·˜v ∇+ ˜v −∇)˜v = Rep˜#+ ˜v Similarity principle (6) t · ∇ −∇ Re# ! ! ˜v =01 (7) ∇ ·∂ ˜v + ˜v ˜v =0)˜v = p˜ + ˜v (7) (6) t ∇ · Where we have introduced a dimensionless number,· ∇ character−∇ isticRe# of the flow, called Where we have introduced a dimensionless! number, characteristic of the flow, called the Reynoldsthe Reynolds number number ˜v =0 (7) increases

∇ · VL VL Where we have introducedRe = aRe dimensionless= number, characteristic of the(8) flow,(8) called

ν speed the Reynolds number ν Reynolds number The non-dimensionalThe non-dimensional NSE’s (6) NSE’s and (6) (7) and reveal (7) an reveal important anFlow importantVL property property of incompressible of incompressible Re = (8) fluids, calledfluids, the called similarity the similarity principle: principle: That flow That on flow large on sc largealesν sc areales governed are governed by the by same the same dynamics as flow on small scales. If the geometry and Reynolds number are the same, dynamicsThe as flow non-dimensional on small scales. NSE’s If the (6) and geometry (7) reveal and an Reynolds importantnumber property are of the incompressible same, then the solutions of the NSE’s are similar, only rescaled. Anexampleofapossible then thefluids, solutions called of the the similarity NSE’s are principle: similar, That only flow rescaled. on large A scnexampleofapossibleales are governed by the same situationsituation is illustrateddynamics is illustrated in asfigure flow in 3. on figure It small shows 3. It scales. a shows cross If a section the cross geometry sectionof a uniformof and a uniform Reynolds flow impending flownumber impending on are the on same, acylinder.Thecharacteristiclengthisthediameteroftheacylinder.Thecharacteristiclengthisthediameterofthethen the solutions of the NSE’s are similar,cylinder, onlycylinder, rescaled. and the and A characteristicnexampleofapossible the characteristic velocity is the uniform velocity of the incoming flow, far awayfromthecylinder.Ifthe velocity is the uniformsituation velocity is illustrated of the in incoming figure 3. It flow, shows far a cross awayfromthecylinder.Ifthe section of a uniform flow impending on diameter of the cylinder and the velocity of the incoming flow are changed, while keeping diameter of theacylinder.Thecharacteristiclengthisthediameterofthe cylinder and the velocity of the incoming flow are changed,cylinder, while and keeping the characteristic the Reynolds number constant, the flow around the cylinder will be the same, only the Reynolds numbervelocity isconstant, the uniform the velocity flow around of the the incoming cylinder flow, wi farll be awa theyfromthecylinder.Ifthe same, only rescaled. rescaled. diameter of the cylinder and the velocity of the incoming flow are changed, while keeping The Reynolds number also serves as a measure of the relative importance of the accel- The Reynolds numberthe Reynolds also serves number as constant, a measure the of flow the relative around theimportance cylinder of wi thell be accel- the same, only eration anderation the viscous and the forces viscous per forces mass. per It mass. is the It ratio is the of ratio the ofcharacteristic the characteristic acceleration acceleration 2 rescaled. 2 V 2/L andV the/L characteristicand the characteristic viscous viscous force per force mass perν massV/L2ν.Turbulentflow,issignifiedbyV/L .Turbulentflow,issignifiedby high ReynoldsThe Reynolds numbers, number as will also be servesdiscussed as a further measure in secti of theon 2.2. relative importance of the accel- high Reynolds numbers,eration and as the will viscous be discussed forces per further mass. in It secti is theon 2.2. ratio of the characteristic acceleration V 2/L and the characteristic viscous force per mass νV/L2.Turbulentflow,issignifiedby high Reynolds numbers, as will be discussed further in section 2.2. 2.1.1 The vorticity equations 2.1.1 The vorticity equations For many purposes it is useful to look at the curl of the velocity field, the so called vorticity, For many purposes it is useful to look at the curl of the velocity field, the so called vorticity, that describes2.1.1 Thethe local vorticity rotation equations of the velocity field. that describes the local rotation of the velocity field. For many purposes it is useful toω look= at thev curl of the velocity field, the so called(9) vorticity, that describes the local rotationω = ofv the∇× velocity field. (9) ∇× The evolution of the vorticity is governed by the vorticity equation, which can be found The evolutionby taking of the the vorticity curl of the is NSE’s governed (6). by the vorticityω = equation,v which can be found (9) by taking the curl of the NSE’s (6). ∇× 1 The evolution of∂ thev vorticity+ v is governed)v = by thep vorticity+ equation,v which can(10) be found ∇× t ∇× · ∇ −∇ × ∇1 Re∇×# by taking∂ thev curl+ of thev NSE’s)v = (6). p + v (10) ∇× t !∇×" · ∇ ! −∇ × ∇ Re∇×# 9 1 ! " ! ∂ v + v )v = p + v (10) ∇× t 9∇× · ∇ −∇ × ∇ Re∇×# ! " ! 9 Conservation laws

Spatial average in a 3D periodic box:

Vorticity:

Kinetic energy Enstrophy

Ideal invariants

Helicity Vortical helicity An enclosed incompressible fluid ˆ ∇ u = k ψ 2 × ρ(ut + u·∇u)+∇p = µ∇ u + f and ∇·u =0 2 ρ(ut + u·∇u)+∇p = µ∇ u + f and and ∇·u =0 2 ρ(ut + u·∇u)+∇p = µ∇ u + f and ∇·u =0 u 2 2 =0 ρ(ut + u·∇ρ(uudt)++ u∇1·p∇=u2 )+µ∇∇up+=fµ∇ u + fand 2 and∇·u =0∇·u =0 2|u| dx = u·f dx − ν |ω| dx d dt V V V 1|u|2 dx = u·f dx −Theν power|ω|2 d xintegral! is: ! ! 2 d 2 2 dt V V V 1|u| dx = u·f dx − ν |ω| dx d ! 1 2 ! !d 2 1 2d 2 2 ∇ψ dxdy = ν ζ dxudy xdt 1V 2u·ωf =x∇ ×νV·u ω x V 2 !! 2 !! 2| | d =!2|u| dx =d −!u f d|x |−dν !|ω| dx dt | | − dt DissipativeV dt V V anomalyV V V ω = ∇ × u ! ! ! ! ! ! lim ν |ω|2 dx $=0∇ with the vorticityν→0 ω = × u Energy dissipationV rate: d 1 2 2 ω = ∇ ×!ωu = ∇ × u ζ dxlimdyν =|ω|2 dνx $=0 ∇ζ dxdy 2 ν→0 V dt !! ! − !! | | lim ν |ω|2 dx $=0 ν→0 2 ∇ × ρ ut + 2u·∇u +V ∇p = µ∇ u lim ν |ω| dx $=0 ! 2 ν→0 V The 2zeroth law" is:! lim ν |ω| dx %=0 % ∇ × ρ ut + u·∇u + ∇p = µ∇ u # ν→0 $ d 2 !! !V 2 " ∇ % ∇ × ρ ut + u·∇u + ∇2p = µ∇ u F (ζ)dxdy = ν ζ F (ζ)dx ·∇ ·∇ 2 dt !! # − $!! | | ∇ ρ uωt +u·u∇"uω =∇ωp uµ + νu∇ ω % × Turbulence∇t + is+ a singular·∇= ∇ limit∇ 2 2 " × ρ u#t + u u +$ p%= µ∇ u ωt + u·∇ω = ω·∇u + ν∇ ω In turbulent flows, the energy dissipation rate ε # (a “dissipative" $ anomaly”). 2 % has a finite ωlimit#t + asu· the∇ω viscosity=$ω·∇ uν tends+ ν∇ toω zero! 2 ω u·∇ω ω·∇u ν 2ωˆ Saturday, March 16, 2013 vx uy = ψ u = uˆıt++vωˆ = =(v+x −∇uy)k 2 ∴ ω·∇u =0 In thisω sense,t + u· ∇theω Euler= ω ·and∇u Navier+ ν∇-Stokesω − ∇ ˆ u = uˆı + vωˆ =(v − u )k ∴ equationsω·∇u =0 are fundamentally different! x y ˆ u = uˆı + vωˆ 10−5 m=(2 s−v1 x − uy)k ∴ ω·∇u =0 ˆ ˆ νair ≈ ˆ u2 = uı + vω=(vx − uy)k ˆ∴ ω·∇u =0 −5 2 −1 ˆ ˆ ∴ ·∇ vxνair ≈u10y =m s ψ u = uı + vω=(vx − uy)k ω u =0 − ∇ −5 2 −1 νair ≈ 10 m s −5 2 −1 νair ≈ 10 m s −5 2 −1 νair 10 m s d ≈1 1 u 2 dx = u·f dx ν ω 2 d 2 1 dt !V | | !V − !V | | 1 1 1 Energy budget scale-by-scale

Low-/High-pass filter Energy budget scale-by-scale (cont’d)

Cumulative kinetic energy between wavenumbers 0 and K:

Scale-by-scale energy budget equation:

Energy flux through wavenumber K:

Cumulative enstrophy:

Cumulative energy injection: Spectral energy transfer

Energy spectrum:

Scale-by-scale energy transfer equation:

Net energy transfer spectrum:

Energy injection spectrum: Spectral energy transfer (cont’d) The Richardson cascade (real space)

„Big whorls have little whorls, little whorls have smaller whorls that feed on their velocity, and so on to viscosity“ The Richardson cascade (Fourier space)

Turbulence as a local cascade in wave number space…

Ek

dissipation inertial Computational range drive range effort range

k f kη k

Much turbulence research addresses the cascade problem. (Important note: In this context, think of an Autobahn, not of a waterfall…) Kolmogorov’s theory from 1941

K41 is based merely on intuition and dimensional analysis – it is not derived rigorously from the Navier-Stokes equation

Key assumptions:

• Scale invariance – like, e.g., in critical phenomena • Central quantity: energy flux ε

1 ∞ E = ⌠ v2 d3x =⌠ E(k) dk 2V ⌡ ⌡ 0 E(k) = C ε2/3 k-5/3

This is the most famous turbulence result: the “-5/3” law.

However, K41 is fundamentally wrong: scale invariance is broken (anomaly)! Intermittency & structure functions Intermittency & non-Gaussian pdf’s

Renner et al. 2001 financial data (exchange rates) Direct numerical simulations Wilczek Wilczek et al . 2008

Structure formation and broken scale invariance Key open issues: Inertial range

• Is the inertial range physics universal (for Re → ∞)?

• If so, can one derive a rigorous IR theory from the NSE?

• How should one, in general, handle the interplay between randomness and coherence? Key issue: Intermittency!

Example: Trapping of tracers in vortex filaments

Note:

The observed deviations from self-similarity can be reproduced qualitatively by relatively simple vortex models.

Biferaleal.2005et Wilczek, Jenko, and Friedrichs 2008 Key open issues: Drive range

• Often, one is interested mainly in the large scales. Here, one encounters an interesting interplay between linear (drive) and nonlinear (damping) physics. – Is it possible to remove the small scales?

• Candidates: LES, dynamical systems approach etc.

Orellano and Wengle 2001 Orellanoand Wengle Turbulence: Further reading Some literature on 3D turbulence