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Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Rende (CS) – Italy Models for turbulence.

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Presentation on theme: "Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Rende (CS) – Italy Models for turbulence."— Presentation transcript:

1 Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Rende (CS) – Italy Models for turbulence

2 Outline of talk 1)Why we need a model to describe turbulence? 2)Two kind of models introduced here: (a) shell models; (b) low-dimensional Galerkin approximation. 3)We are interested not just to investigate properties of simplified models per se, rather we are interested to understand to what extend simplified models can mimic the gross features of REAL turbulent flows. Biological or social complex phenomena can be described by simplified toy models which are just caricature of reality, derived from turbulence models First approach Write equations (if any exists!) of the phenomena and simplifies that equations to toy model Second approach Cannot write equations, just collect experimental data and try to write toy models which can reproduce observations

3 Acknowledgments Pierluigi Veltri, Annick Pouquet, Angelo Vulpiani, Guido Boffetta, Helène Politano Roberto Bruno, Vanni Antoni and the whole crew in Padova for experiments on laboratory and solar wind plasmas Paolo Giuliani (PhD thesis on MHD shell model) Fabio Lepreti (PhD thesis on solar flares) Luca Sorriso (PhD thesis on solar wind turbulence)

4 Turbulence: Solar wind as a wind tunnel In situ measurements of high amplitude fluctuations for all fields (velocity, magnetic, temperature…) A unique possibility to measure low-frequency turbulence in plasmas over a wide range of scales. Results from Helios 2

5 Turbulence in plasmas: laboratory Plasma generated for nuclear fusion, confined in a reversed field pinch configuration. High amplitude fluctuations of magnetic field, measurements (time series) at the edge of plasma column, where the toroidal field changes sign. Data from RFX (Padua) Italy

6 Turbulence: numerical simulations High resolution direct numerical simulations of MHD equations. Mainly in 2D configurations. R 1600 Space collocation points Fluctuations BOTH in space and time

7 Turbulence: Solar atmosphere Solar flares: dissipative bursts within turbulent environment ? Turbulent convection observed on the photosphere (granular dynamics), superimposed to global oscillations acoustic modes

8 Turbulence: different examples Strong defect turbulence in Nematic Liquid Crystal films Density fluctuations in the early universe originate massive objects The Jupiters atmosphere

9 Main features of turbulent flows 1) Randomness in space and time 2) Turbulent structures on all scales 3) Unpredictability and instability to very small perturbations

10 Whats the problem Nonlinear Dissipative Incompressible Navier-Stokes equation u velocity field P pressure kinematic viscosity Turbulence is the result of nonlinear dynamics z+z+ z-z- Hydromagnetic flows: the same structure of NS equations Nonlinear interactions happens only between fluctuations propagating in opposite direction with respect to the magnetic field. Elsasser variables

11 Fourier analysis Consider a periodic box of size L, Fourier analysis Divergenceless condition 3D 2D

12 Equation for Fourier modes The evolution of the field for a single wave vector is related to fields of ALL other wave vectors (convolution term) for which k = p + q. Infinite number of modes involved in nonlinear interactions for inviscid flows

13 In the limit of high R, assuming a Kolmogorov spectrum E(k) ~ k -5/3 dissipation takes place at scale: Why models for turbulence? Typical values at present reached by high resolution direct simulations R ~ Input Output Transfer the # of equations to be solved is proportional to For space plasmas: R ~ At these values it is not possible to have an inertial range extended for more than one decade. No possibility to verify asymptotic scaling laws, statistics...

14 Two kind of approximations 1) To investigate dynamics of large-scales and dynamics due to invariants of the motion: 2) To investigate scaling laws, statistical properties and dynamics related to the energy cascade:

15 Fluid flows become turbulent as Re Osborne Reynolds noted that as Re increases a fluid flow bifurcates toward a turbulent regime Flow past a cylinder viscosity. U is the inflow speed, L is the size of flow U L Look here

16 Landau vs. Ruelle & Takens 2) Ruelle & Takens: incommensurable frequencies cannot coexist, the motion becomes rapidly aperiodic and turbulence suddenly will appear, just after three (or four) bifurcations. The system lies on a subspace of the phase space: a strange attractor. 1)Landau: turbulence appears at the end of an infinite serie of Hopf bifurcations, each adding an incommensurable frequency to the flow The more frequencies The more stochasticity We can understand what attractor means, but what about strangeness?

17 The realm of experiments PRL, 1975

18 Gollub & Swinney, 1975 Incommensurable frequencies cannot coexist

19 E.N. Lorenz (1963) The presence of a strange attractor simplifies the description of turbulence Even if the phase space has infinite dimensions, the system lies on a subspace (strange attractor). THE SYSTEM CAN BE DESCRIBED BY ONLY A SMALL SET OF VARIABLES Edward Lorenz in 1963: a Galerkin approximation with only three modes to get a simplified model of convective rolls in the atmosphere. The trajectories of this system, for certain settings, never settle down to a fixed point, never approach a stable limit cycle, yet never diverge to infinity. Butterfly effect: Extreme sensitivity to every small fluctuations in the initial conditions.

20 The idea was that, although a hydrodynamical system has a very large number of degree of freedom, technically speaking infinitely many, most of them will be inactive at the onset of turbulence, leaving only few interacting active modes, which nevertheless can generate a complex and unpredictable evolution. Bohr, Jensen, Paladin & Vulpiani, Dynamical system approach to turbulence Cambridge Univ. Press. Simplified models Dissipation in a complex system, is responsible for the elimination of many degree of freedoms, reducing the system to very few dimensions Coullet, Eckmann & Koch, J. Stat. Phys. 25, 1 (1981).

21 Chaotic dynamics from Navier-Stokes equations Let us add an external forcing term to restore turbulence Chaotic dynamics in a deterministic system 1) Stochastic behaviour (randomness) 2) No predictability poor mans NS equation U. Frisch

22 Sensitivity to initial conditions A transformation leads to the tent map A small uncertainty surely will grows in time ! No predictability in finite times Sensitivity of flow to every small perturbations Numbers written in binary format Iterates of the tent map lead to the Bernoulli shift

23 Chaotic dynamic leads to stochasticity Apply the map n times Ergodic theorem: Let f(x) an integrable function, and let f(T n (x 0 )) calculated over all iterates of the map. Then for almost all x 0 As a consequence of the chaoticity, the trajectory of a SINGLE orbit covers ALL the allowed phase space The ensemble is generated by the dynamics, from the uniform measure in [0,1].

24 Dynamics vs. statistics While the details of turbulent motions are extremely sensitive to triggering disturbances, statistical properties are not (otherwise there would be little significance in the averages!) 1) Stochastic behaviour: the dynamics is unpredictable both in space and time. 2) Predictability is introduced at a statistical level (via the ergodic theorem and the properties of chaos !). The measured velocity field is a stochastic field with gaussian statistics. 3) On every scale details of the plots are different but statistical properties seems to be the same (apparent self-similarity). Atmospheric flow

25 How to build up shell models (1) 1) Introduce a logarithmic spacing of the wave vectors space (shells); In this way we can investigate properties of turbulence at very high Reynolds numbers. We are not interested in the dynamics of each wave vector mode of Fourier expansion, rather in the gross properties of dynamics at small scales. The intershell ratio in general is set equal to = 2.

26 How to build up shell models (2) 2) Assign to each shell ONLY two dynamical variables; These fields take into account the averaged effects of velocity modes between k n and k n+1, that is fluctuations across eddies at the scale r n ~ k n -1 To compare with properties of real flows remember that shell fields represent usual increments at a given scale In this way we ruled out the possibility to investigate BOTH spatial and temporal properties of turbulence. For example the 2-th order moment is related to the usual spectrum u n (t) u(x+r) – u(x) spacetime

27 Measurements In situ satellite measurements of velocity and magnetic field, the sample is transported with the solar wind velocity SW frame Taylors hypothesis: The time dependence of u(x,t) comes from the spatial argument of u Satellite frame The time variation of u at a fixed spatial location (supersonic V SW ), are reinterpreted as being a spatial variation of u.

28 How to build up shell models (3) 3) Write a nonlinear equations with couplings among variables belonging to local shells; Different shell models have been built up with different coupling terms 4) Fix the coupling coefficients M ij imposing the conservation of ideal invariants.

29 Invariants Invariants of the dynamics in absence of dissipation and forcing: 1) total energy 2) cross-helicity 3) magnetic helicity 2D 3D In absence of magnetic field only two invariants: kinetic energy and kinetic helicity. H k (t) disappears in presence of magnetic field 2D 3D

30 GOY shell model The model conserves also a surrogate of magnetic helicity Conserved quantities Positive definite: 2D caseNon positive definite: 3D case There is the possibility to introduce 2D and 3D shell models. Gledzer, Ohkitamni & Yamada (1973, 1989) for the hydrodynamic case.

31 Phase invariance A phase invariance is present in shell models, and this constraints the possible set of stationary correlation functions with a nonzero mean value GOY shell model is invariant under Other constrants exists for high order correlations Modified shell model Owing to this phase invariance the only quadratic form with a mean value different from zero is Constraint With the constraint This simplifies the spectrum of correlations.

32 Old MHD shell model -1 Gloaguen, Leorat, Pouquet, & Grappin (1986) Real variables, only nearest shells involved, one free parameter. Conserved quantities Main investigations: 1) Transition to chaos in N-mode models (Gloaguen et al., 1986) 2) Time intermittency (Carbone, 1994) Desnyansky & Novikov (1974) for the hydrodynamic analog

33 Old MHD shell model -2 NOT dynamical models. Introduced in order to investigate spectral properties of turbulence, and competitions between the nonlinear energy cascade and some linear instabilities (reconnection,..) Main investigations: 1) The first model of development of turbulence in solar surges 2) Spectral properties of anisotropic MHD turbulence Obtained in the framework of closure approximations EDQNM, Direct Interaction Approximation Anticipated results of high resolution numerical simulations

34 Properties 3D model: dynamo action Numerical simulations with: N = 24 shells; viscosity = Constant forcing acting on large-scale: f 4 + = f 4 - = (1 + i) ONLY velocity field is injected Time evolution of magnetic energy K -2/3 time The Kolmogorov spectrum is a fixed point of the system Starting from a seed the magnetic energy increases towards a kind of equipartition with kinetic energy. E(k n ) = / k n

35 Properties 2D: anti-dynamo K -4/3 The 2D model shows a kind of anti-dynamo action: A seed of magnetic field cannot increase. The spectrum expected for 2D kinetic situation due to a cascade of 2D hydrodynamical invariant H(t) cannot decreases H(t) – H(0) is bounded Convergence for large t only when the magnetic energy is zero. From the shell model we have:

36 Turbulent dynamo and anti-dynamo? What turbulent dynamo action means in the shell model There exists some invariant subspaces which can act like attractors for all solutions (stable subspaces). The fluid subspace is stable (in 2D case) or unstable (in 3D case). We will come back to this point in the following Magnetic energy 3D Magnetic energy 2D

37 Dynamical alignment Alfvènic state: fixed point of MHD shell model. Strong correlations between velocity and magnetic fields for each shell. Alfvènic state is a strong attractor for the model. The system falls on it, for different kind of constant forcing. The fixed point is destabilized when we use a Langevin equation for the external forcing term, with a correlation time τ (eddy-turnover time) Time evolution of velocity and magnetic field for the mode n = 7, with constant forcing terms.

38 Properties: spectrum and flux Kolmogorov fixed point of the system. Inertial and dissipative ranges + intermediate range visible in shell models Numerical simulations with: N = 26 shells; viscosity = Flux: an exact relationship which takes the role of the Kolmogorovs 4/5-law

39 Properties: spectrum and flux Kolmogorov fixed point of the system Inertial and dissipative ranges + intermediate range visible in shell models Numerical simulations with: N = 26 shells; viscosity = Flux: an exact relationship which takes the role of the Kolmogorovs 4/5-law

40 Evolution of magnetic field spectrum 0.9AU 0.7AU 0.3AU trace of magnetic field spectral matrix power density frequency the spectral break moves to lower frequency with increasing distance from the sun This was interpreted as an evidence that non-linear interactions are at work producing a turbulent cascade process 1/f 1/f 5/3

41 Observations of the Kraichnans scaling Old observations of magnetic turbulence in the solar wind seems to show that a Kraichnans scaling law is visible at intermediate scales. k -3/2

42 Properties: Time intermittency Velocity field Magnetic field n = 1 n = 9

43 Fluctuations in plasmas Increasing scales Velocity increments at 3 different scales in the solar wind: Δu r = u(t + r) – u(t) Small scale: STRUCTURES Large scale: random signal

44 Phenomenology: fluid-like Let us consider the dissipation rate for both pseudo-energies (stochastic quantities equality in law!) The energy transfer rate is scaling invariant only when h = 1/3 The characteristic time (eddy-turnover time) is the time of life of turbulent eddies Kolmogorov scaling q-th order moments r 1/k

45 Phenomenology: magnetically dominated In this case there is a physical time, the Alfvèn time, which represents the sweeping of Alfvenic fluctuations due to the large-scale magnetic field The energy transfer rate is scaling invariant only when h = 1/4 Kraichnan scaling q-th order moments r 1/k Since the Alfvèn time in some case is LESSER than the eddy- turnover time, the cascade is effectively realized in a time T:

46 Why high-order moments? Let x a stochastic variable distributed according to a Probability Density Function (pdf) p(x), the n-th order moment is Through the inverse transform the pdf can be written in terms of moments, and moments can be obtained through the knowledge of pdf Gaussian process: the 2-th order moment suffices to fully determine pdf. High-order moments are uniquely defined from the 2-th order (in this sense energy spectra are interesting!) Characteristic function

47 Anomalous scaling laws A departure from the Kolmogorov law must be attributed to time intermittency in the shell model. The structure functions in the model Scaling exponents obtained in the range where the flux scales as k n -1 Fields play the same role the same amount of intermittency The departure from the Kolmogorov law measures the amount of intermittency Δu r u n k n ~ 1/r ζ q = q/3 Kolmogorov scaling

48 Inertial range in real experiments? A linear range is visible only in the slow solar wind Magnetic field in the solar wind. Helios data.

49 Extended self-similarity The m-th order structure function (m = 3 or m = 4) plays the role of a generalized scale In this case we can measure only the RELATIVE scaling exponents The range of self-similarity extends over all the range covered by the measurements, BEYOND the inertial range Just a way to get scaling exponents For fluid flows, scaling exponents obtained through ESS coincides with scaling exponents measured in the inertial range.

50 Departure from the Kolmogorovs laws Solar wind: Intermittency is stronger for magnetic field than for velocity field. Scaling for velocity field coincide with fluid flows Fluid flows: Intermittency is stronger for passive scalar Sharp variations of magnetic field

51 Magnetic turbulence in laboratory plasma The departure from the linear scale increases going towards the wall Turbulence more intermittent near the external wall r/a normalized distance Similar to edge turbulence in fluid flows

52 Numerical simulations Intermittency is different for different fields. In particular magnetic field more intermittent than velocity field Incompressible MHD equations in 2D configurations High resolutions points. Averages in both space and time. No Taylor hypothesis when we are dealing with simulations

53 Comparison with velocity in fluid flows A collection of data from laboratory fluid flows (black symbols) and solar wind velocity (white symbols). Differences only for high order moments. Not fully reliable !

54 Probability distribution functions Fluctuations are stochastic variables, so the structure functions are defined in terms of pdfs: For a gaussian pdf Anomalous scaling exponents implies that pdfs have also anomalous scalings The kurtosis increases as the scale becomes smaller Fluctuations at small scales increasingly depart from a GAUSSIAN

55 i.e. the pdfs of normalized fields increments at different scales collapse on the same shape (self-similarity) About self-similarity And let us consider the normalized variables Let the scaling law holds for differences Then by changing the scale r r, it can be shown that, if h = cost. pdfs at two scales are related

56 Experimental evidences in atmospheric fluid flows No global self-similarity! PDFs are not Gaussians PDFs changes with scale Large scales Inertial range Small scales Departure from self-similarity

57 Plasmas and shell model: the same property The full line corresponds to a fit made by using a multifractal model to describe the scaling of Pdfs. In the following I describe this model.

58 A multifractal model for pdfs each describing the statistics in different regions of volume S(h) each of different variance (h) each weighted by the occurrence of S(h) the sum of gaussians of different width (blue) gives the resulting stretched PDF (red) This is achieved introducing the distribution L( ) and computing the convolution with a Gaussian G According to the multifractal model (scaling exponents h(x) depend on the position) the PDF of a field u at scale r can be described as a superposition of Gaussians

59 Evidence: conditioned pdfs In 2D numerical simulations, we have calculated the pdfs of fluctuations, CONDITIONED to a given value of the energy flux (x,r) At each scale they collapse to a GAUSSIAN with different values of. –0.1 < + < 0.1, = < + < 1.0, = 0.9

60 A model for the weight of each gaussian The parameter ² can be used to characterize the scaling of the shape of the PDFs, that is the intermittency of the field! As ² increases, L( ) is wider then more and more Gaussians of different width are summed and the tails of P( u) become higher Width (variance) of the Log- normal distribution When ²= 0, L( ) is a -function centered in 0 so that: Gaussian P( u)

61 Scaling of ² and relevant parameters The parameter ² is found to behave as a power-law of the scale 2d MHD To characterize intermittency, only two parameters are needed, namely: ² max, the maximum value of the parameter ² within its scaling range, represents the strength of intermittency (the intermittency level at the bottom of the energy cascade), the slope of the power-law, representing the efficiency of the non-linear cascade (measures how fast energy is concentrated on structures at smaller and smaller scales)

62 Scaling of ² for solar wind turbulence solid symbols: fast streams Magnetic field Velocity field open symbols: slow streams magnetic field is more intermittent than velocity

63 Scaling of ² for numerical simulations ² max (v) = 0.8 ² max (b) = 1.1 (b) = 0.8 (v) = 0.5

64 Turbulence: structures+background ? A description of turbulence: coherent structures present on ALL scales within the sea of a gaussian background. They contain most of the energy of the flow and play an important dynamical role. Examples from Jupiters atmosphere Need for space AND scale analysis

65 Orthogonal Wavelets decomposition Let us consider a signal f(x) made by N = 2 m samples (being x = 1), and build up a set of functions starting from a mother wavelet Then we generates from this a set of analysing wavelets by DILATIONS and TRANSLATIONS Scale Position

66 Local Intermittency Measure The energy content, at each scale, is not uniformly distributed in space L L.i.m. greater than a threshold means that at a given scale and position the energy content is greater than the average at that scale L Gaussian backgroundStructures Complete signal l.i.m. smaller than threshold l.i.m. larger than threshold

67 In the solar wind The sequence of intermittent events generates a point process. Statistical properties of the process gives information on the underlying physics which generated the point process. Point process

68 Waiting times between structures Interesting! the underlying cascade process is NOT POISSONIAN, that is the intermittent (more energetic) bursts are NOT INDEPENDENT (memory) Solar wind The times between events are distributed according to a power law Pdf(Δt) ~ Δ t -β The turbulent energy cascade generates intermittent coherent events at small scales.

69 Power law distribution for waiting times Turbulent flows share this characteristic. Power law is generated through the chaotic dynamics and must be reproduced by models for turbulence. Fluid flow Laboratory plasma

70 Waiting times in the MHD shell model Time intermittency in the shell model is able to capture also that property of real turbulence Chaotic dynamics generates non poissonian events

71 What kind of intermittent structures ? Solar wind: tangential discontinuity (current sheet) Minimum variance analysis around isolated structure allows to identify them Solar Wind: shock (compressive structures)

72 Magnetic structures in laboratory plasmas RFX edge magnetic turbulence: current sheets Current sheets are naturally produced as coherent, intermittent structures by nonlinear interactions

73 Intermittent structures in laboratory plasmas RFX edge turbulence of electrical potential Structures are potential holes

74 Dynamics of intermittent structures Relationship between intermittent structures of edge turbulence and disruptions of the plasma columns at the center of RFX Time evolution of floating potential Minima are related to disruptions Appearence of intermittent structures in the electrostatic turbulence at the edge of the plasma columns (vertical lines) We dont have explanation for this!

75 Statistical flares Dissipation of (turbulent?) magnetic energy Ratio of EIT full Sun images in Fe XII 195A to Fe IX/X 171A. Temperature distribution in the Sun's corona: - dark areas cooler regions - bright areas hotter regions

76 Solar flares are impulsive events Time series of flare events Hard X-ray ( > 20 keV): Intermittent spikes Duration 1-2 s, E max ~ erg Numerous smaller spikes down to erg (detection limit) X-ray corona: superposition of a very large number of flares Nano flares

77 Power law statistics of bursts Total energy, peak energy and (more or less!) lifetime of individual bursts seems to be distributed according to power laws.

78 The Parkers conjecture (1988) Nanoflares correspond to dissipation of many small current sheets, forming in the bipolar regions as a consequence of the continous shuffling and intermixing of the footpoints of the field in the photospheric convection. Current sheets: tangential discontinuity which become increasingly severe with the continuing winding and interweaving eventually producing intense magnetic dissipation in association with magnetic reconnection.

79 Self-Organized Criticality (P. Bak et al., 1987) A paradigm for complex dissipative systems exibiting bursts, is invoked as a model to describe ALSO turbulence. Self-organized state critical state (at the border line of chaos) reached by the system apparently without tuning parameters. Critical state attractor, robust with respect to variations of parameters and with respect to randomness.

80 Sandpile model Size, lifetimes and number of sand grains in each avalanche are power law distributed. Lack of any typical length Avalanches of all size i.e. FRACTAL PROCESS SANDPILE IS THE PROTOTIPE OF SOC Sandpile profile is the critical state. Perturbed with one single sand grain added at a random position. When the local slope exceedes a critical value the sand in excess is redistributed to nearest sites generating an avalanche whose dimension L is that of the marginally stable region.

81 Sand Pile Model for Solar Flares Power peak, total energy and duration are power law distributed. Cellular Automata model for reconnection : Vector field B i on a 3D lattice Local slope dB i = B i - j w j B i+j When | dB i | > some treshold: instability at position i : Field readjusted in the nearby positions so that the grid point i becomes stable The readjustment can destabilize nearby points producing an avalanche (flare) The coronal magnetic field spontaneously evolves in a self-organized state (critical profile). Perturbations: convective random motion at footpoints of magnetic loops. Avalanche: reconnection event

82 Waiting times between solar flares Sand piles cannot describe all observed features of solar flares (Boffetta, Carbone, Giuliani, Veltri, Vulpiani, 1999) Intermittency in sand piles is produced by isolated and completely random singularities Poisson process pdf of waiting times must be exponential (see inset in figure) On the contrary flares from the GOES dataset show asymptotic POWER LAW DISTRIBUTION P(Δt) Δt

83 The origin of power law distribution for waiting times This suggests to try to fit the WTD with a Lèvy distribution whose characteristic function is The parameter 0 < 2, for = 2 one recovers the definition of a Gaussian. A rescaling gives the same statistical properties The waiting time sequence forms a temporal point process, statistically self-similar

84 The Lèvy function For large t this function behaves like a power law P( t) t -(1 + ) WTD is a Lèvy function. A fit on the GOES flares gives the non trivial value Stable distribution, obtained through the Central Limit Theorem by relaxing the hypothesis of finite variance The underlying process has long (infinite) correlation, and is a non Poissonian point process.

85 Parkers conjecture modified Nanoflares correspond to dissipation of many small current sheets, forming in the nonlinear cascade occuring inside coronal magnetic structure as a consequence of the power input in the form of Alfven waves due to footpoint motion. Current sheets: coherent intermittent small scale structures of MHD turbulence

86 Dissipative bursts in shell model The energy dissipation rate is intermittent in time. Energy is dissipated through impulsive isolated events (bursts).

87 Multifractal structure of dissipation Coarse-grained dissipation has been generated from simulations Moments of dissipation have a scaling law

88 Singularity spectrum Spectrum of singularities described by the function f( ), which represents the fractal dimension of the space where dissipation related to a singularity. As p is varied we select different singularities from an entire (continuous) spectrum From saddle-point we get Inverse transform

89 Inside bursts Through a threshold process we can identify and isolate each dissipative bursts to make statistics

90 Some different statistics Let us define some statistics on impulsive events 1) Total energy of bursts 2) Time duration 3) Energy of peak In all cases we found power laws, the scaling exponents depend on threshold.

91 The waiting times The time between two bursts is, and let us calculate the pdf p( WE FOUND A POWER LAW Even dissipative bursts are NOT INDEPENDENT

92 Statistics of dissipative events Pdfs of normalized fluctuations of energy released in the MHD shell model, are the same as normalized fluctuations of solar flares energy flux. flares shell model

93 Could SOC describes turbulent cascade?* Kadanoff sand pileDissipative Kadanoff sand pile 1.PDFs are non gaussian and collapse to a single PDF (fractal) 2.Esponential distribution for waiting times: 3. avalanches are INDEPENDENT events * Apart for the 4/5-law Simulations of Kadanoff SOC model: Rescaled energy fluctuations at different scales and waiting times at the smallest scale

94 Need for a correct definition of time scale (not often discussed in literature). In the sand pile no way to define a timescale (no time series). Avalanches must be considered as a collection of instantaneous events. The Running sandpile: in each temporal step (properly defined in this model) the system is continuously fed with a finite deposition rate J in and the unstable sites are simultaneously updated the energy dissipated can be properly followed step by step, so that time series are obtained. The Running sandpile

95 Running sandpile Simulations of running sandpile: Rescaled energy fluctuations at different scales and waiting times at the smallest scale Running sand pile with two different deposition rates. Low J in non gaussian pdfs and exponential distribution for waiting times High J in gaussian pdfs and power law for waiting times

96 1/f spectra in the Running Sandpile From the running sandpile model we can get continuous time series. From time series obtained in this way (for example of total dissipated energy) we can easily get power spectra. Unless in the classical SOC model, 1/f spectra are visible, at large scales, but only for high values of J in. f -1 This is an interesting property, with profound consequences. The 1/f spectrum is ubiquitous in nature

97 Anomalous transport in laboratory plasmas Diffusion causes loss of particles, energy, … In general turbulent fluctuations of electric field enhance loss, the transport is called anomaloues since it is due to turbulence. Perhaps the main cause of disruption of magnetic confinement needed to achieve nuclear fusion. Anomalous transport A problem with language: Plasma physics: Transport driven by turbulent fluctuations Physics of fluids: Transport with non-Gaussian features

98 Fluxes of particles in Tokamak The generation of BARRIERS for transport is a way to enhance confinement in plasmas. We need models of turbulent fluctuations

99 SOC-Paradigm for Turbulent Transport Plasma confined in toroidal devices is dominated by anomalous transport (on machine scales) driven by fluctuations (on microscopic scales). SOC apparently solves the paradox. The marginally unstable profile of plasma is continuously perturbed by driving gradients (sand grains microscopic level). 1/f spectrum obtained for the floating potential at the edge of RFX (Padua). Note: The SOC mechanism continuously can sustain active bursty transport (avalanches macroscopic level), and relaxes back to the linearly least unstable profile. The dominant scale for the transport is the system scale.

100 Bursts of density fluctuations at the edge of plasma revealed through both microwave reflectometry and electrostatic probes. Power laws for waiting times: The SOC-PARADIGM does not describe all features of observations. Waiting times between transport events

101 A modifications of sandpile model Sanchez, Newman, Carreras, PRL 88, (2002) Note 2 : Correlated input (necessarily!) correlated output (SOC is a linear model) Note 1: Power laws with scaling exponents greater than 3 corresponds to gaussian processes NOT to Poisson processes (the central limit theorem is actually not broken) Introduces correlated input to reproduce power laws in waiting times. Quite trivial!

102 A different sandpile model Gruzinov, Diamond, Rosenbluth, PRL 89, (2002) Modification of output Two unstable ranges with different rules for grains toppling. When the second range is unstable the height of the pile is lowered at a level of the first range (coupling between internal and external structures). Formation of pedestal region with bursty transport

103 Charged particles diffusion Anomalous diffusion is not a trivial problem! Diffusion is anomalous (non-Gaussian) when the central limit theorem is broken. This leads to very restrictive conditions Typical problem: Lagrangian evolution of particles in a given fluid flow. Chaotic behaviour is assured by non-integrability. Anomalous transport ALSO in very simple laminar fluid flows!

104 3D velocity field from shell model Using a shell model (in the wave vectors space) it is possible to build up a model for a turbulent field (in the physical space) Introduce a wave vector with a given amplitude k n = k 0 2 n and random directions. Use an inverse transform on a shell model (with random coefficients C n ) to get a velocity and magnetic field. (e.g. P. Kalliopi & L.V.)

105 A different approach A simple model for turbulence with coherent structures at all dynamical scales: Perhaps there is no need to run a shell model + the equations of motion for a test particle. Amplitudes a n and b n are related to energy spectrum. Wave vectors have random directions and amplitudes k n = 2 n k 0 Time evolution is related to the eddy-turnover time. Reproduce characteristics of pair diffusion, …

106 Barrier for transport in plasmas Since turbulent fluctuations causes losses, barriers are tentatively generated with a simple equation in mind: No turbulent fluctuations No anomalous transport For example: Shear flows are able to decorrelate turbulent eddies and to kill fluctuations. Mechanism: stretching and distortion of eddies because different points inside an eddy have different speeds. The eddy loses coherence, the eddy turnover time decreases turbulent intensity decreases. Low High mode confinement transition have been observed in real experiments (a lot of money to generate a shear flow in a tokamak!!).

107 Confining turbulence ? In astrophysics, turbulent fluctuations are useful since they CONFINE cosmic rays within the galaxy Test-particle simulations in electrostatic turbulence 2D slab geometry B 0 = (0,0,B) A simple model for electrostatic turbulence with coherent structures at all dynamical scales E X B drift

108 A barrier for the transport A barrier has been generated by randomizing the phases of the field ONLY within a narrow strip at the border of the integration domain. Q(x,y) = strain 2 – vorticity 2

109 Random phases Correlated phases (weak superdiffusion) Diffusive properties D e ~ D e ~ ~ 0.68 = 0.5

110 Reduction of particle flux When the barrier is active we observe a reduction of the flux of particles Cumulative number of particles as a function of time which escape from the integration Region. Different curves refers to different values of the amplitude of the barrier. No barrier

111 Symmetrization of particle flux-1 When the barrier is active we observe a symmetrization of the flux. barrier Particle flux through the line N+N+ N-N- N # of particles which cross the line

112 Simmetrization of particle flux-2 # of crossings for each particle of a line near the border Without barrier particles leave the integration region after some few crossings. The flux is mainly directed from the center towards the border. With the barrier active, particles are trapped and make a standard diffusive motion inside the integration region. The flux is symmetric, each particle makes multiple crossings of the line in both directions.

113 Experiments at Castor tokamak (Prague) A barrier have been generated by biasing the electric field with a weak perturbation on the border (low amount of money!!) Principle of control: perturbate rather than kill turbulence!. Control Ring in Castor Time [µs]poloidal mode number Poloidal Angle (°) before during Pascal Devynck et al., 2003 Perhaps crazy people taking more seriously than ourself our continuous playing in the realm of tokamak plasma physicists

114 Fluxes are reduced and symmetrized Particle Flux during « open loop »PDF of the Particle Flux The positive bursts (towards the wall) still exist but a backward flux (towards the plasma) is created.

115 Galerkin approximation Models can be obtained by retaining only a finite number of interacting modes in the convolution sum. For example in 2D MHD The convolution sum involves an infinite set of wave vectors Rugged invariants of motion: they remain invariant in time for each triad of interacting wave vectors which satisfy the condition k = p + q

116 Simplified N-modes models Simplified models can be obtained by retaining only a finite number of interacting modes in the convolution sum. Among the infinite modes which satisfy k = p + q, retain only wave vectors which lye within a region of width N The result is a Pandoras box of different N-modes models whose dynamics exactly conserve the rugged invariants.

117 Models vs. Simulations Main advantages : rugged invariants are conserved in absence of dissipation, true dissipationless runs. Example: N = 25 Main disadvantages : higher computational times (N 2 vs. N log N)

118 Note: the occurrence of an inverse cascade of magnetic helicity in shell models is yet controversial 2D example: inverse cascade t = 0 Coarse-grained energy averaged over circular shells of amplitudes m = (k x 2 + k y 2 ) 1/2 Equipartition between kinetic and magnetic energy at small scales and dominance of magnetic energy at largest scale

119 2D example: self-similarity in the decay In the inviscid limit, constant quantities Kinetic and magnetic enstrophy decay in time, but their ratio tends to a fixed value. In the limit μ 0 and N, we found Δ 1. Equipartition between kinetic and magnetic energy on small scales in the inviscid case.

120 Do your own model ! If you have some free time to spent, and you want see chaotic trajectories on your screen, you could investigate time behaviour of some N-mode models (N 5). You can find nice sequences of bifurcations, transitions to chaos, very beautiful attractors, etc… (for fluid flows see e.g Franceschini & Tebaldi, 1979; J. Lee, 1987; …) k 1 = (1,1) k 2 = (2,-1) k 3 = (3,0) k 4 = (1,2) k 5 = (0,1) k 1 = (1,1) k 2 = (2,-1) k 3 = (3,0) k 4 = (1,2) k 5 = (0,1) k 6 = (1,0) k 7 = (1,-2) Etc.. N = 5 N = 7 Some triads which satisfy k i = k j + k m k 1 = (1,1) k 2 = (2,-1) k 3 = (3,0) N = 3 No chaos here

121 A triad-interaction model The most basic model to investigate nonlinear interactions in 2D MHD k 1 = (1,1) k 2 = (2,-1) k 3 = (3,0) V i (t) = Re[v(k i,t)] B i (t) = Re[b(k i,t)] Only real fields How a simple model can be interesting without chaos? No chaos here!

122 Free decay: asymptotic states E : energy H c : cross-helicity A : magnetic helicity Starting from any initial condition, the system evolves towards a curve in the parameter space (A/E, 2Hc/E) Analysis of a wide serie of different numerical simulations on free decay 2D MHD reported by Ting, Mattheus and Montgomery (1986). The 3-modes real model seems to reproduce these results. μ = 0.01

123 Selective decay and dynamical alignment Selective decay (SD) due to inverse cascade (large-scale magnetic field) Variational principle Extreme points of the curve represents decay of rugged invariants with respect to total energy. Dynamical alignment (DA) due to approximately equal Decay of energies of alfvènic fluctuations (alignment between velocity and magnetic field) DA SD The curve … does not represent the locus of the extrema of anything over its entire range of variation. (Ting et al., 1986) Laboratory experiments Astrophysics

124 Time-invariant subspaces Fluid equations are characterized by the presence of time-invariant subspaces, which are interesting for the dynamics of the system. A point in the phase space S, evolves according to a time-translation operator Let I S a subspace of S, and let Φ(0) I a vector of I. The subspace I is invariant in time if, for each vector Φ(0), the time evolution is able to maintain the vector Φ(t) on I. Example: the fluid subspace of MHD Φ(0)={v(k,0),b(k,0)} such that b(k,0) = 0. From MHD equations b(k,t) = 0 for each time.

125 Subspaces in the 3-modes model Fluid Alfvènic (fixed point) Cross-helicity = 0 (B 1,V 2,B 3 ) (B 1,B 2,V 3 ) (V 1,B 2,B 3 ) Subspaces Subspaces due to symmetries can be generalized to the true MHD equation to any N-order truncation

126 Example (B 1,V 2,B 3 ) Cross-helicity = 0 Example of invariant subspace When μ = 0 two invariants the motion is bounded on a line given by the intersection of the circle E with the cylinder A. The system reduces to a Duffing equation without forcing term. Solution in terms of elliptic function dn

127 Stable and unstable subspaces Stability of subspaces are investigated according to time evolution of distance from a given subspace. Let Φ(0) and Γ(0) such that E ext « E in at t = 0. Let us investigate the time evolution of both E in and E ext Example: D= E ext is the distance of the point from a given subspace.

128 Subspace (V 1,V 2,V 3 ) dissipation = 0.0 dissipation = 0.01 Stable (no dynamo effect) Selective dissipation Attractor

129 Subspace (B 1,B 2,V 3 ) dissipation = 0.0 dissipation = 0.01 Stable (Magnetic field on the largest scales) Selective dissipation Attractor

130 Subspace (B 1,V 2,B 3 ) dissipation = 0.0 dissipation = 0.01 Unstable (inverse cascade at work from k 3 ) The subspace repels all nearest trajectories.

131 Subspace (V 1,B 2,B 3 ) dissipation = 0.0 dissipation = 0.01 Unstable (inverse cascade at work from k 2 and k 3 ) The subspace repels all trajectories

132 Attractors and repellers V i = B i V i = - B i (B 1,B 2,V 3 ) (V 1,V 2,V 3 )(V 1,B 2,B 3 ) (B 1,V 2,B 3 ) Only one wave vector survive Attractors drive the system towards Repellers drive the system towards the whole

133 Do you remember? What turbulent dynamo action means in the shell model There exists some invariant subspaces which can act like attractors for all solutions (stable subspaces). The fluid subspace is stable (in 2D case) or unstable (in 3D case). The structure of stable and unstable time-invariant subspaces of real MHD are reproduced in the GOY Shell model Magnetic energy 3D Magnetic energy 2D

134 Models for low-β plasmas y x z L a a/L << 1 β << 1 When Laboratory plasmas Coronal loops

135 Reduced MHD equations Incompressible 2D MHD in perpendicular variables Alfven wave propagation along background magnetic field Total energy and cross-helicity survive. Only two time invariants in ideal RMHD

136 Simplified models A Galerkin approximation with N-modes of 2D MHD on each plane, and a finite difference scheme to solve the propagation in the perpendicular direction. Periodic boundaries conditions at z = 0 and z = L to simulate toroidal situations. Simulations with N sez = 256 and N = 18. The cylinder has been divided in N sez planes at fixed z n.

137 The Galerkin truncature model Actually A is quasi-invariant in the model No inverse cascade, but a kind of self-organization due to the fact that ΔA/A « 1 ? Both magnetic and kinetic energies accumulates at m = 1. for all z. Equipartition between energies. Inverse cascade without conservation of A ?

138 Self-organization in RMHD Magnetic energy on the wave vectors plane (m,n) R = 14R = 21 A kind of self-organization also in the vertical direction. Depending on the aspect ratio the spectrum is dominated by some few modes (the higher R the more modes are present in the spectrum).

139 Quasi-single helicity states in RFX Quasi-single helicity states observed in laboratory plasmas in some situation (example RFX). Characterized by: a) the mode m = 1 in the transverse plane; b) a few dominant modes in the toroidal direction, depending on the aspect ratio (the higher R the more modes are present in the spectrum). Spectrum for m = 1 Time evolution of some modes

140 A Hybrid Shell Model RMHD equations in the wave vector space perpendicular to B 0 : A shell model in the wave vector space perpendicular to B 0 can be derived: (Hybrid : the space dependence along B 0 is kept)

141 Boundary Conditions Space dependence along B 0 allows to chose boundary conditions: Total reflection is imposed at the upper boundary A random gaussian motion with autocorrelation time t c = 300 s is imposed at the lower boundary only on the largest scales The level of velocity fluctuations at lower boundary is of the order of photospheric motions v ~ c A ~ 1 Km/s Model parameters: L ~ Km, R ~ 6, c A ~ Km/s

142 Energy balance After a transient a statistical equilibrium is reached between incoming flux, outcoming flux and dissipation Stored Energy Energy flux Dissipated Power The level of fluctuations inside the loop is considerably higher than that imposed at the lower loop boundary Dissipated power displays a sequence of spikes

143 Energy spectra A Kolmogorov spectrum is formed mainly on magnetic energy Magnetic energy dominates with respect to kinetic energy Magnetic Energy Kinetic Energy

144 Statistical analysis of dissipated power Power laws are recovered on Power peak, burst duration, burst energy and waiting time distributions The obtained energy range correspond to nanoflare energy range Power PeakBurst duration Burst EnergyWaiting time

145 Low-dimensional models for coherent structures In many turbulent flows one observes coherent structures on large-scales. In these cases the basic features of the system can be described by few variables Proper Orthogonal Decomposition (POD) is a tool that allows one to build up, from numerical simulations or direct spatio-temporal experiments, a low-dimensional system which models the spatially coherent structures.

146 Proper Orthogonal Decomposition The field is decomposed as: The functions which describe the base are NOT GIVEN A PRIORI (empirical eigenfunctions). We want to find a basis that is OPTIMAL for the data set in the sense that a finite dimensional representation of the field u(r,t) describes typical members of the ensemble better than representations in ANY other base This is achieved through a maximization of the average of the proiection of u on An inner product is defined

147 Empirical eigenfunctions whose kernel is the averaged autocorrelation function. Very huge computational efforts ! The maximum is reached through a variational method thus obtaining the integral equation In the framework of POD, j represents the energy associated to j -th mode. They are ordered as j > j+1 lower modes contain more energy.

148 Low-dimensional models Through empirical eigenfunctions, we can reconstruct the field using only a finite number N of modes In this way we capture the maximum allowed for energy with respect to any other truncature with N modes. Low-dimensional models can be build up through a Galerkin approximation of equations which governes the flow The coupling coefficients depend on the empirical eigenfunctions

149 Turbulent convection – Time behavior We analysed line of sight velocity field of solar photosphere from telescope THEMIS (on July, 1, 1999). 32 images of width 30 x 30 (1 = 725 km) sampled every 1.25 minute) j = 0,1 aperiodic behaviour convective overshooting j = 2,3 oscillatory behaviour T about 5 min 5 minutes oscillations The behaviour of other modes is not well defined both behaviors

150 Turbulent convection – Spatial behavior 0 1 spatial pattern similar to granulation pattern Spatial scale about 700 km. Modes j = 0, 1 are mainly due to a granular contribution. 2, 3 largest structures and low contrasts (with exceptions of definite and isolate regions). These eigenfunctions are associated to oscillatory phenomena characterized by a period of 5 minutes.

151 Reconstruction of velocity field The velocity field has been reconstructed using only J = 0, 1 J = 2, 3

152 Playing with POD POD have been used to describe spatio-temporal behaviour of the 11-years solar cycle Daily observations ( ) of green coronal emission line nm. For every day 72 values of intensities from 0 to 355 degrees of position angle Time Angle

153 3 POD modes Original Reconstruction with 3 POD modes Reconstruction with 2 POD modes Reconstruction with 1 POD mode periodicities + migration + stochasticity

154 Conclusions Became a Plasma Physicist Deadline for applications: September 28, 2003 Acknowledge Loukas Vlahos and the local organizing committee

155 Let sand piles evolve …

156 Dont care about… Avalanchesor Avalanches or

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