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**Vincenzo Carbone Models for turbulence**

Dipartimento di Fisica, Università della Calabria Models for turbulence Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Rende (CS) – Italy Khalkidiki, Grece 2003

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**Outline of talk Why we need a model to describe turbulence?**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Outline of talk Why we need a model to describe turbulence? Two kind of models introduced here: (a) shell models; (b) low-dimensional Galerkin approximation. 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 Second approach First approach Cannot write equations, just collect experimental data and try to write toy models which can reproduce observations Write equations (if any exists!) of the phenomena and simplifies that equations to toy model Khalkidiki, Grece 2003

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**Acknowledgments Pierluigi Veltri, Annick Pouquet, Angelo Vulpiani,**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Acknowledgments Pierluigi Veltri, Annick Pouquet, Angelo Vulpiani, Guido Boffetta, Helène Politano Paolo Giuliani (PhD thesis on MHD shell model) Fabio Lepreti (PhD thesis on solar flares) Luca Sorriso (PhD thesis on solar wind turbulence) Roberto Bruno, Vanni Antoni and the whole crew in Padova for experiments on laboratory and solar wind plasmas Khalkidiki, Grece 2003

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**Turbulence: Solar wind as a wind tunnel**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Turbulence: Solar wind as a wind tunnel Results from Helios 2 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. Khalkidiki, Grece 2003

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**Turbulence in plasmas: laboratory**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Turbulence in plasmas: laboratory Data from RFX (Padua) Italy 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. Khalkidiki, Grece 2003

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**Turbulence: numerical simulations**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 Khalkidiki, Grece 2003

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**Turbulence: Solar atmosphere**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Turbulence: Solar atmosphere Solar flares: dissipative bursts within turbulent environment ? Turbulent convection observed on the photosphere (granular dynamics), superimposed to global oscillations acoustic modes Khalkidiki, Grece 2003

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**“Turbulence”: different examples**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria “Turbulence”: different examples Strong defect turbulence in Nematic Liquid Crystal films Density fluctuations in the early universe originate massive objects The Jupiter’s atmosphere Khalkidiki, Grece 2003

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**Main features of turbulent flows**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 Khalkidiki, Grece 2003

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**What’s the problem Turbulence is the result of nonlinear dynamics**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria What’s the problem Turbulence is the result of nonlinear dynamics Incompressible Navier-Stokes equation u velocity field P pressure n kinematic viscosity Nonlinear Dissipative Hydromagnetic flows: the same “structure” of NS equations z+ z- Elsasser variables Nonlinear interactions happens only between fluctuations propagating in opposite direction with respect to the magnetic field. Khalkidiki, Grece 2003

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**Fourier analysis 3D 2D Consider a periodic Divergenceless condition**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Fourier analysis Consider a periodic box of size L, Fourier analysis Divergenceless condition 3D 2D Khalkidiki, Grece 2003

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**Equation for Fourier modes**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 Khalkidiki, Grece 2003

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**Why models for turbulence?**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Why models for turbulence? Input In the limit of high R, assuming a Kolmogorov spectrum E(k) ~ k-5/3 dissipation takes place at scale: Transfer Output 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... Typical values at present reached by high resolution direct simulations R ~ Khalkidiki, Grece 2003

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**Two kind of approximations**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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: Khalkidiki, Grece 2003

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**Fluid flows become turbulent as Re **

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 n. U is the inflow speed, L is the size of flow Look here L U Khalkidiki, Grece 2003

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**Landau vs. Ruelle & Takens**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Landau vs. Ruelle & Takens 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 2) Ruelle & Takens: incommensurable frequencies cannot coexist, the motion becomes rapidly aperiodic and turbulence suddenly will appear, just after three (or four) bifurcations. We can understand what “attractor” means, but what about strangeness? The system lies on a subspace of the phase space: a “strange attractor”. Khalkidiki, Grece 2003

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**The realm of experiments**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria The realm of experiments PRL, 1975 Khalkidiki, Grece 2003

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**Gollub & Swinney, 1975 Incommensurable frequencies cannot coexist**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Gollub & Swinney, 1975 Incommensurable frequencies cannot coexist Khalkidiki, Grece 2003

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**E.N. Lorenz (1963) Even if the phase space has infinite**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria E.N. Lorenz (1963) 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 The presence of a strange attractor simplifies the description of turbulence 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. Khalkidiki, Grece 2003

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Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Simplified models “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. 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). Khalkidiki, Grece 2003

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**Chaotic dynamics from Navier-Stokes equations**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Chaotic dynamics from Navier-Stokes equations Let us add an external forcing term to restore turbulence 1) Stochastic behaviour (randomness) 2) No predictability Chaotic dynamics in a deterministic system poor man’s NS equation U. Frisch Khalkidiki, Grece 2003

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**Sensitivity to initial conditions**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Sensitivity to initial conditions A transformation leads to the tent map Numbers written in binary format Iterates of the tent map lead to the “Bernoulli shift” A small uncertainty surely will grows in time ! No predictability in finite times Sensitivity of flow to every small perturbations Khalkidiki, Grece 2003

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**Chaotic dynamic leads to stochasticity**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Chaotic dynamic leads to stochasticity Apply the map n times As a consequence of the chaoticity, the trajectory of a SINGLE orbit covers ALL the allowed phase space Ergodic theorem: Let f(x) an integrable function, and let f(Tn(x0)) calculated over all iterates of the map. Then for almost all x0 The ensemble is generated by the dynamics, from the uniform measure in [0,1]. Khalkidiki, Grece 2003

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**Dynamics vs. statistics**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Dynamics vs. statistics 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 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!) Khalkidiki, Grece 2003

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**How to build up shell models (1)**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria How to build up shell models (1) 1) Introduce a logarithmic spacing of the wave vectors space (shells); The intershell ratio in general is set equal to = 2. 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. Khalkidiki, Grece 2003

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**How to build up shell models (2)**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria How to build up shell models (2) 2) Assign to each shell ONLY two dynamical variables; In this way we ruled out the possibility to investigate BOTH spatial and temporal properties of turbulence. These fields take into account the averaged effects of velocity modes between kn and kn+1, that is fluctuations across eddies at the scale rn ~ kn-1 To compare with properties of real flows remember that shell fields represent usual increments at a given scale un(t) u(x+r) – u(x) For example the 2-th order moment is related to the usual spectrum time space Khalkidiki, Grece 2003

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Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Measurements In situ satellite measurements of velocity and magnetic field, the sample is transported with the solar wind velocity Satellite frame SW frame Taylor’s hypothesis: The time dependence of u(x,t) comes from the spatial argument of u’ The time variation of u at a fixed spatial location (supersonic VSW), are reinterpreted as being a spatial variation of u’. Khalkidiki, Grece 2003

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**How to build up shell models (3)**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 Mij imposing the conservation of ideal invariants. Khalkidiki, Grece 2003

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**Invariants Invariants of the dynamics in absence of dissipation and**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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. Hk(t) disappears in presence of magnetic field 3D 2D Khalkidiki, Grece 2003

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Vincenzo Carbone Dipartimento di Fisica, Università della Calabria GOY shell model The model conserves also a “surrogate” of magnetic helicity Conserved quantities There is the possibility to introduce “2D” and “3D” shell models. Gledzer, Ohkitamni & Yamada (1973, 1989) for the hydrodynamic case. Positive definite: 2D case Non positive definite: 3D case Khalkidiki, Grece 2003

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Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 With the constraint Owing to this phase invariance the only quadratic form with a mean value different from zero is Other constrants exists for high order correlations Modified shell model This simplifies the spectrum of correlations. Constraint Khalkidiki, Grece 2003

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**“Old” MHD shell model -1 Gloaguen, Leorat, Pouquet, & Grappin (1986)**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria “Old” MHD shell model -1 Gloaguen, Leorat, Pouquet, & Grappin (1986) Real variables, only nearest shells involved, one free parameter. Conserved quantities Desnyansky & Novikov (1974) for the hydrodynamic analog Main investigations: 1) Transition to chaos in N-mode models (Gloaguen et al., 1986) 2) Time intermittency (Carbone, 1994) Khalkidiki, Grece 2003

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**“Old” MHD shell model -2 NOT dynamical models. Introduced in order to**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria “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,..) Obtained in the framework of closure approximations EDQNM, Direct Interaction Approximation Main investigations: 1) The first model of development of turbulence in solar surges 2) Spectral properties of anisotropic MHD turbulence Anticipated results of high resolution numerical simulations Khalkidiki, Grece 2003

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**Properties 3D model: “dynamo action”**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Properties 3D model: “dynamo action” Constant forcing acting on large-scale: f4+ = f4- = (1 + i) 10-3 ONLY velocity field is injected Numerical simulations with: N = 24 shells; viscosity = 10-8 Time evolution of magnetic energy The Kolmogorov spectrum is a fixed point of the system K-2/3 E(kn) = <|un|2> / kn Starting from a seed the magnetic energy increases towards a kind of equipartition with kinetic energy. time Khalkidiki, Grece 2003

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**Properties 2D: “anti-dynamo”**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Properties 2D: “anti-dynamo” 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 K-4/3 From the shell model we have: H(t) cannot decreases H(t) – H(0) is bounded Convergence for large t only when the magnetic energy is zero. Khalkidiki, Grece 2003

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**“Turbulent dynamo” and “anti-dynamo”?**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria “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). Magnetic energy 3D Magnetic energy 2D We will come back to this point in the following Khalkidiki, Grece 2003

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**Dynamical alignment Alfvènic state: fixed point of MHD shell model.**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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. Time evolution of velocity and magnetic field for the mode n = 7, with constant forcing terms. The fixed point is destabilized when we use a Langevin equation for the external forcing term, with a correlation time τ (eddy-turnover time) Khalkidiki, Grece 2003

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**Properties: spectrum and flux**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Properties: spectrum and flux Numerical simulations with: N = 26 shells; viscosity = 0.5 ∙ 10-9 Kolmogorov fixed point of the system. Inertial and dissipative ranges + intermediate range visible in shell models Flux: an exact relationship which takes the role of the Kolmogorov’s “4/5”-law Khalkidiki, Grece 2003

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**Properties: spectrum and flux**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Properties: spectrum and flux Numerical simulations with: N = 26 shells; viscosity = 0.5 ∙ 10-9 Kolmogorov fixed point of the system Inertial and dissipative ranges + intermediate range visible in shell models Flux: an exact relationship which takes the role of the Kolmogorov’s “4/5”-law Khalkidiki, Grece 2003

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**Evolution of magnetic field spectrum**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Evolution of magnetic field spectrum trace of magnetic field spectral matrix 10 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 1/f the spectral break moves to lower frequency with increasing distance from the sun -0.89 1/f5/3 -1.06 -1.07 This was interpreted as an evidence that non-linear interactions are at work producing a turbulent cascade process -1.72 power density -1.67 0.3AU -1.70 0.7AU 0.9AU frequency Khalkidiki, Grece 2003

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**Observations of the Kraichnan’s scaling**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Observations of the Kraichnan’s scaling Old observations of magnetic turbulence in the solar wind seems to show that a Kraichnan’s scaling law is visible at intermediate scales. k-3/2 Khalkidiki, Grece 2003

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**Properties: Time intermittency**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Properties: Time intermittency Velocity field Magnetic field n = 1 n = 9 Khalkidiki, Grece 2003

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**Fluctuations in plasmas**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Fluctuations in plasmas Small scale: STRUCTURES Increasing scales Large scale: random signal Velocity increments at 3 different scales in the solar wind: Δur = u(t + r) – u(t) Khalkidiki, Grece 2003

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**Phenomenology: fluid-like**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Phenomenology: fluid-like Let us consider the dissipation rate for both pseudo-energies (stochastic quantities equality in law!) The characteristic time (eddy-turnover time) is the time of life of turbulent eddies The energy transfer rate is scaling invariant only when h = 1/3 Kolmogorov scaling q-th order moments r 1/k Khalkidiki, Grece 2003

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**Phenomenology: magnetically dominated**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 Since the Alfvèn time in some case is LESSER than the eddy- turnover time, the cascade is effectively realized in a time T: The energy transfer rate is scaling invariant only when h = 1/4 Kraichnan scaling q-th order moments r 1/k Khalkidiki, Grece 2003

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**Why high-order moments?**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 Characteristic function 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!) Khalkidiki, Grece 2003

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**Anomalous scaling laws**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Anomalous scaling laws Δur un The “structure functions” in the model ζq = q/3 Kolmogorov scaling kn ~ 1/r Scaling exponents obtained in the range where the flux scales as kn-1 A departure from the Kolmogorov law must be attributed to time intermittency in the shell model. The departure from the Kolmogorov law measures the “amount” of intermittency Fields play the same role the same “amount” of intermittency Khalkidiki, Grece 2003

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**Inertial range in real experiments?**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Inertial range in real experiments? A linear range is visible only in the slow solar wind Magnetic field in the solar wind. Helios data. Khalkidiki, Grece 2003

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**Extended self-similarity**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 For fluid flows, scaling exponents obtained through ESS coincides with scaling exponents measured in the inertial range. Just a way to get scaling exponents Khalkidiki, Grece 2003

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**Departure from the Kolmogorov’s laws**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Departure from the Kolmogorov’s 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 Khalkidiki, Grece 2003

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**Magnetic turbulence in laboratory plasma**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Magnetic turbulence in laboratory plasma The departure from the linear scale increases going towards the wall Turbulence more intermittent near the external wall Similar to edge turbulence in fluid flows r/a normalized distance Khalkidiki, Grece 2003

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**Numerical simulations**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Numerical simulations Incompressible MHD equations in 2D configurations High resolutions points. Averages in both space and time. No Taylor hypothesis when we are dealing with simulations Intermittency is different for different fields. In particular magnetic field more intermittent than velocity field Khalkidiki, Grece 2003

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**Comparison with velocity in fluid flows**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 ! Khalkidiki, Grece 2003

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**Probability distribution functions**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Probability distribution functions Fluctuations are stochastic variables, so the structure functions are defined in terms of pdfs: For a gaussian pdf The kurtosis increases as the scale becomes smaller Fluctuations at small scales increasingly depart from a GAUSSIAN Anomalous scaling exponents implies that pdfs have also anomalous scalings Khalkidiki, Grece 2003

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**About self-similarity**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria About self-similarity Let the scaling law holds for differences And let us consider the normalized variables Then by changing the scale r r, it can be shown that, if h = cost. pdfs at two scales are related i.e. the pdfs of normalized fields increments at different scales collapse on the same shape (self-similarity) Khalkidiki, Grece 2003

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**Departure from self-similarity**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Departure from self-similarity Experimental evidences in atmospheric fluid flows PDFs are not Gaussians PDFs changes with scale Small scales No global self-similarity! Inertial range Large scales Khalkidiki, Grece 2003

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**Plasmas and shell model: the same property**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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. Khalkidiki, Grece 2003

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**A multifractal model for pdfs**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria A multifractal model for pdfs According to the multifractal model (scaling exponents h(x) depend on the position) the PDF of a field du at scale r can be described as a superposition of Gaussians each describing the statistics in different regions of volume S(h) each of different variance (h) each weighted by the occurrence of S(h) This is achieved introducing the distribution L(s) and computing the convolution with a Gaussian G the sum of gaussians of different width (blue) gives the resulting “stretched” PDF (red) Khalkidiki, Grece 2003

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**Evidence: conditioned pdfs**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Evidence: conditioned pdfs In 2D numerical simulations, we have calculated the pdfs of fluctuations, CONDITIONED to a given value of the energy flux e(x,r) –0.1 < e+ < 0.1, s = 0.4 0.9 < e+ < 1.0, s = 0.9 At each scale they collapse to a GAUSSIAN with different values of s. Khalkidiki, Grece 2003

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**A model for the weight of each gaussian**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria A model for the weight of each gaussian Width (variance) of the Log-normal distribution When l²= 0, L(s) is a d-function centered in s0 so that: Gaussian P(du) As l² increases, L(s) is wider then more and more Gaussians of different width are summed and the tails of P(du) become higher The parameter ² can be used to characterize the scaling of the shape of the PDFs, that is the intermittency of the field! Khalkidiki, Grece 2003

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**Scaling of ² and relevant parameters**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Scaling of ² and relevant parameters The parameter l² is found to behave as a power-law of the scale To characterize intermittency, only two parameters are needed, namely: l²max, the maximum value of the parameter l² within its scaling range, represents the strength of intermittency (the intermittency level at the bottom of the energy cascade) b, 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) 2d MHD Khalkidiki, Grece 2003

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**Scaling of ² for solar wind turbulence**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Scaling of ² for solar wind turbulence Magnetic field solid symbols: fast streams open symbols: slow streams Velocity field magnetic field is more intermittent than velocity Khalkidiki, Grece 2003

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**Scaling of ² for numerical simulations**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Scaling of ² for numerical simulations l²max (b) = 1.1 l²max (v) = 0.8 b (v) = 0.5 b (b) = 0.8 Khalkidiki, Grece 2003

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**Turbulence: “structures”+background ?**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 Jupiter’s atmosphere Need for space AND scale analysis Khalkidiki, Grece 2003

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**Orthogonal Wavelets decomposition**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Orthogonal Wavelets decomposition Let us consider a signal f(x) made by N = 2m samples (being Dx = 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 Khalkidiki, Grece 2003

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**Local Intermittency Measure**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Local Intermittency Measure The energy content, at each scale, is not uniformly distributed in space 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 L l.i.m. larger than threshold l.i.m. smaller than threshold Complete signal Gaussian background Structures Khalkidiki, Grece 2003

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**In the solar wind Point process The sequence of intermittent**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 Khalkidiki, Grece 2003

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**Waiting times between “structures”**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Waiting times between “structures” The times between events are distributed according to a power law Pdf(Δt) ~ Δt -β The turbulent energy cascade generates intermittent “coherent” events at small scales. Solar wind Interesting! the underlying cascade process is NOT POISSONIAN, that is the intermittent (more energetic) bursts are NOT INDEPENDENT (memory) Khalkidiki, Grece 2003

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**Power law distribution for waiting times**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Power law distribution for waiting times Fluid flow Laboratory plasma Turbulent flows share this characteristic. Power law is generated through the chaotic dynamics and must be reproduced by models for turbulence. Khalkidiki, Grece 2003

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**Waiting times in the MHD shell model**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 Khalkidiki, Grece 2003

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**What kind of intermittent structures ?**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria What kind of intermittent structures ? Minimum variance analysis around isolated structure allows to identify them Solar Wind: shock (compressive structures) Solar wind: tangential discontinuity (current sheet) Khalkidiki, Grece 2003

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**Magnetic structures in laboratory plasmas**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Magnetic structures in laboratory plasmas RFX edge magnetic turbulence: current sheets Current sheets are naturally produced as coherent, intermittent structures by nonlinear interactions Khalkidiki, Grece 2003

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**Intermittent structures in laboratory plasmas**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Intermittent structures in laboratory plasmas RFX edge turbulence of electrical potential Structures are potential holes Khalkidiki, Grece 2003

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**Dynamics of intermittent structures**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 don’t have explanation for this! Khalkidiki, Grece 2003

75
Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Statistical flares 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 Dissipation of (turbulent?) magnetic energy Khalkidiki, Grece 2003

76
**Solar flares are impulsive events**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Solar flares are impulsive events Hard X-ray ( > 20 keV): Intermittent spikes Duration 1-2 s, Emax ~ 1027 erg Numerous smaller spikes down to 1024 erg (detection limit) Time series of flare events X-ray corona: superposition of a very large number of flares Nano flares Khalkidiki, Grece 2003

77
**Power law statistics of bursts**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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. Khalkidiki, Grece 2003

78
**The Parker’s conjecture (1988)**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria The Parker’s 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. Khalkidiki, Grece 2003

79
**Self-Organized Criticality (P. Bak et al., 1987)**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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. Khalkidiki, Grece 2003

80
**Sandpile model Avalanches of all size i.e. FRACTAL PROCESS**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Sandpile model 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. Lack of any typical length Avalanches of all size i.e. FRACTAL PROCESS Size, lifetimes and number of sand grains in each avalanche are power law distributed. Khalkidiki, Grece 2003

81
**Sand Pile Model for Solar Flares**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Sand Pile Model for Solar Flares 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 Cellular Automata model for reconnection : Vector field Bi on a 3D lattice Local slope dBi = Bi -Sj wj Bi+j When | dBi| > 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) Power peak, total energy and duration are power law distributed. Khalkidiki, Grece 2003

82
**Waiting times between solar flares**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 -b b 2.38 0.03 Khalkidiki, Grece 2003

83
**The origin of power law distribution for waiting times**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria The origin of power law distribution for waiting times The waiting time sequence forms a “temporal point process”, statistically self-similar A rescaling gives the same statistical properties 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. Khalkidiki, Grece 2003

84
**The Lèvy function For large t this function behaves like a power law**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 1.38 0.06 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. Khalkidiki, Grece 2003

85
**Parker’s conjecture modified**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Parker’s 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 Khalkidiki, Grece 2003

86
**Dissipative bursts in shell model**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Dissipative bursts in shell model The energy dissipation rate is intermittent in time. Energy is dissipated through impulsive isolated events (bursts). Khalkidiki, Grece 2003

87
**Multifractal structure of dissipation**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Multifractal structure of dissipation Coarse-grained dissipation has been generated from simulations Moments of dissipation have a scaling law Khalkidiki, Grece 2003

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Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Singularity spectrum Spectrum of singularities described by the function f(), which represents the fractal dimension of the space where dissipation related to a singularity . From saddle-point we get Inverse transform As p is varied we select different singularities from an entire (continuous) spectrum Khalkidiki, Grece 2003

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Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Inside bursts Through a threshold process we can identify and isolate each dissipative bursts to make statistics Khalkidiki, Grece 2003

90
**Some different statistics**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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. Khalkidiki, Grece 2003

91
**The waiting times The time between two bursts is t, and**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria The waiting times The time between two bursts is t, and let us calculate the pdf p(t). WE FOUND A POWER LAW Even dissipative bursts are NOT INDEPENDENT Khalkidiki, Grece 2003

92
**Statistics of dissipative events**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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. shell model flares Khalkidiki, Grece 2003

93
**Could SOC describes turbulent cascade?***

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Could SOC describes turbulent cascade?* Simulations of Kadanoff SOC model: Rescaled energy fluctuations at different scales and waiting times at the smallest scale Kadanoff sand pile Dissipative Kadanoff sand pile PDFs are non gaussian and collapse to a single PDF (fractal) Esponential distribution for waiting times: avalanches are INDEPENDENT events * Apart for the 4/5-law Khalkidiki, Grece 2003

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Vincenzo Carbone Dipartimento di Fisica, Università della Calabria The Running sandpile 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 Jin and the unstable sites are simultaneously updated the energy dissipated can be properly followed step by step, so that time series are obtained. Khalkidiki, Grece 2003

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Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 Jin non gaussian pdfs and exponential distribution for waiting times High Jin gaussian pdfs and power law for waiting times Khalkidiki, Grece 2003

96
**1/f spectra in the Running Sandpile**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 Jin. f-1 This is an interesting property, with profound consequences. The 1/f spectrum is ubiquitous in nature Khalkidiki, Grece 2003

97
**Anomalous transport in laboratory plasmas**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Anomalous transport in laboratory plasmas Diffusion causes loss of particles, energy, … Perhaps the main cause of disruption of magnetic confinement needed to achieve nuclear fusion. In general turbulent fluctuations of electric field enhance loss, the transport is called “anomaloues” since it is due to turbulence. Anomalous transport A problem with language: Plasma physics: Transport driven by turbulent fluctuations Physics of fluids: Transport with non-Gaussian features Khalkidiki, Grece 2003

98
**Fluxes of particles in Tokamak**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 Khalkidiki, Grece 2003

99
**“SOC-Paradigm” for Turbulent Transport**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria “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. Khalkidiki, Grece 2003

100
**Waiting times between transport events**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Waiting times between transport events 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. Khalkidiki, Grece 2003

101
**A modifications of sandpile model**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria A modifications of sandpile model Sanchez, Newman, Carreras, PRL 88, (2002) Introduces correlated input to reproduce power laws in waiting times. Quite trivial! 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) Note 2: Correlated input (necessarily!) correlated output (SOC is a linear model) Khalkidiki, Grece 2003

102
**A “different” sandpile model**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria A “different” sandpile model Gruzinov, Diamond, Rosenbluth, PRL 89, (2002) 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 Modification of output Khalkidiki, Grece 2003

103
**Charged particles diffusion**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Charged particles diffusion 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! 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 Khalkidiki, Grece 2003

104
**3D velocity field from shell model**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 kn = k0 2n and random directions. Use an “inverse transform” on a shell model (with random coefficients Cn) to get a velocity and magnetic field. (e.g. P. Kalliopi & L.V.) Khalkidiki, Grece 2003

105
**A different approach Perhaps there is no need to run a shell model**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria A different approach Perhaps there is no need to run a shell model + the equations of motion for a test particle. A simple model for turbulence with coherent structures at all dynamical scales: Amplitudes an and bn are related to energy spectrum. Wave vectors have random directions and amplitudes kn = 2n k0 Time evolution is related to the eddy-turnover time. Reproduce characteristics of pair diffusion, … Khalkidiki, Grece 2003

106
**Barrier for transport in plasmas**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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!!). Khalkidiki, Grece 2003

107
**Confining turbulence ? E X B drift**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Confining turbulence ? In astrophysics, turbulent fluctuations are useful since they CONFINE cosmic rays within the galaxy E X B drift Test-particle simulations in electrostatic turbulence 2D slab geometry B0 = (0,0,B) A simple model for electrostatic turbulence with coherent structures at all dynamical scales Khalkidiki, Grece 2003

108
**A barrier for the transport**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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) = strain2 – vorticity2 Khalkidiki, Grece 2003

109
**Diffusive properties De ~ 10-3 = 0.5 De ~ 1 0.1 ~ 0.68**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Diffusive properties De ~ 10-3 Correlated phases (weak superdiffusion) = 0.5 De ~ 1 0.1 ~ 0.68 Random phases Khalkidiki, Grece 2003

110
**Reduction of particle flux**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Reduction of particle flux When the barrier is active we observe a reduction of the flux of particles No barrier 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. Khalkidiki, Grece 2003

111
**Symmetrization of particle flux-1**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 # of particles which cross the line Khalkidiki, Grece 2003

112
**Simmetrization of particle flux-2**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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. Khalkidiki, Grece 2003

113
**Experiments at Castor tokamak (Prague)**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 before Poloidal Angle (°) during Pascal Devynck et al., 2003 Perhaps crazy people taking more seriously than ourself our “continuous playing” in the realm of tokamak plasma physicists Time [µs] poloidal mode number Khalkidiki, Grece 2003

114
**Fluxes are reduced and symmetrized**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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. Khalkidiki, Grece 2003

115
**Galerkin approximation**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 Khalkidiki, Grece 2003

116
**Simplified N-modes models**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 “Pandora’s box” of different N-modes models whose dynamics exactly conserve the rugged invariants. Khalkidiki, Grece 2003

117
**Models vs. Simulations Main advantages :**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Models vs. Simulations Main advantages : rugged invariants are conserved in absence of dissipation, true dissipationless runs. Main disadvantages : higher computational times (N2 vs. N log N) Example: N = 25 Khalkidiki, Grece 2003

118
**2D example: inverse cascade**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 2D example: inverse cascade Coarse-grained energy averaged over circular shells of amplitudes m = (kx2 + ky2)1/2 t = 0 Equipartition between kinetic and magnetic energy at small scales and dominance of magnetic energy at largest scale Note: the occurrence of an inverse cascade of magnetic helicity in shell models is yet controversial Khalkidiki, Grece 2003

119
**2D example: self-similarity in the decay**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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. Khalkidiki, Grece 2003

120
Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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; …) Some triads which satisfy ki= kj + km N = 7 N = 3 N = 5 Etc.. k1 = (1,1) k2 = (2,-1) k3 = (3,0) k4 = (1,2) k5 = (0,1) k6 = (1,0) k7 = (1,-2) k1 = (1,1) k2 = (2,-1) k3 = (3,0) k1 = (1,1) k2 = (2,-1) k3 = (3,0) k4 = (1,2) k5 = (0,1) No chaos here Khalkidiki, Grece 2003

121
**A triad-interaction model**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria A triad-interaction model The most basic model to investigate nonlinear interactions in 2D MHD k1 = (1,1) k2 = (2,-1) k3 = (3,0) Vi(t) = Re[v(ki,t)] Bi(t) = Re[b(ki,t)] Only real fields No chaos here! How a simple model can be interesting without chaos? Khalkidiki, Grece 2003

122
**Free decay: asymptotic states**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Free decay: asymptotic states Analysis of a wide serie of different numerical simulations on free decay 2D MHD reported by Ting, Mattheus and Montgomery (1986). Starting from any initial condition, the system evolves towards a curve in the parameter space (A/E, 2Hc/E) μ = 0.01 The 3-modes real model seems to reproduce these results. E : energy Hc : cross-helicity A : magnetic helicity Khalkidiki, Grece 2003

123
**Selective decay and dynamical alignment**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Selective decay and dynamical alignment Extreme points of the curve represents decay of rugged invariants with respect to total energy. Variational principle Selective decay (SD) due to inverse cascade (large-scale magnetic field) Laboratory experiments Dynamical alignment (DA) due to approximately equal Decay of energies of alfvènic fluctuations (alignment between velocity and magnetic field) DA Astrophysics SD The curve “… does not represent the locus of the extrema of anything over its entire range of variation”. (Ting et al., 1986) DA Khalkidiki, Grece 2003

124
**Time-invariant subspaces**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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. Khalkidiki, Grece 2003

125
**Subspaces in the 3-modes model**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Subspaces in the 3-modes model Fluid Alfvènic (fixed point) Cross-helicity = 0 (B1,B2,V3) Subspaces due to symmetries can be generalized to the true MHD equation to any N-order truncation Subspaces (B1,V2,B3) (V1,B2,B3) Khalkidiki, Grece 2003

126
**Example (B1,V2,B3) Cross-helicity = 0**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Example (B1,V2,B3) Example of invariant subspace Cross-helicity = 0 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 Duffin’g equation without forcing term. Solution in terms of elliptic function dn Khalkidiki, Grece 2003

127
**Stable and unstable subspaces**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Stable and unstable subspaces Stability of subspaces are investigated according to time evolution of distance from a given subspace. Example: Let Φ(0) and Γ(0) such that Eext « Ein at t = 0. D= √Eext is the distance of the point from a given subspace. Let us investigate the time evolution of both Ein and Eext Khalkidiki, Grece 2003

128
**Subspace (V1,V2,V3) dissipation = 0.0 dissipation = 0.01 Stable**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Subspace (V1,V2,V3) dissipation = 0.0 dissipation = 0.01 Stable (no dynamo effect) Selective dissipation Attractor Khalkidiki, Grece 2003

129
**Subspace (B1,B2,V3) dissipation = 0.0 dissipation = 0.01 Stable**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Subspace (B1,B2,V3) dissipation = 0.0 dissipation = 0.01 Stable (Magnetic field on the largest scales) Selective dissipation Attractor Khalkidiki, Grece 2003

130
**Subspace (B1,V2,B3) dissipation = 0.0 dissipation = 0.01**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Subspace (B1,V2,B3) dissipation = 0.0 dissipation = 0.01 Unstable (inverse cascade at work from k3) The subspace repels all nearest trajectories. Khalkidiki, Grece 2003

131
**Subspace (V1,B2,B3) dissipation = 0.0 dissipation = 0.01**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Subspace (V1,B2,B3) dissipation = 0.0 dissipation = 0.01 Unstable (inverse cascade at work from k2 and k3) The subspace repels all trajectories Khalkidiki, Grece 2003

132
**Attractors and “repellers”**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Attractors and “repellers” Vi = Bi Only one wave vector survive (V1,V2,V3) (V1,B2,B3) (B1,B2,V3) (B1,V2,B3) Attractors drive the system towards “Repellers” drive the system towards the whole Vi = - Bi Khalkidiki, Grece 2003

133
Do you remember? Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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). Magnetic energy 3D Magnetic energy 2D The structure of stable and unstable time-invariant subspaces of real MHD are reproduced in the GOY Shell model Khalkidiki, Grece 2003

134
**Models for low-β plasmas**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Models for low-β plasmas When L z a/L << 1 β << 1 y a x Laboratory plasmas Coronal loops Khalkidiki, Grece 2003

135
**Reduced MHD equations Incompressible 2D MHD in perpendicular variables**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 Khalkidiki, Grece 2003

136
**Simplified models The cylinder has been divided in**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Simplified models The cylinder has been divided in Nsez “planes” at fixed zn. 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 Nsez = 256 and N = 18. Khalkidiki, Grece 2003

137
**The Galerkin truncature model**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria The Galerkin truncature model Both magnetic and kinetic energies accumulates at m = 1. for all z. Equipartition between energies. Inverse cascade without conservation of A ? 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 ? Khalkidiki, Grece 2003

138
**Self-organization in RMHD**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Self-organization in RMHD R = 14 R = 21 Magnetic energy on the wave vectors plane (m,n) 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). Khalkidiki, Grece 2003

139
**Quasi-single helicity states in RFX**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Quasi-single helicity states in RFX Quasi-single helicity states observed in laboratory plasmas in some situation (example RFX). Time evolution of some modes Spectrum for m = 1 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). Khalkidiki, Grece 2003

140
Vincenzo Carbone Dipartimento di Fisica, Università della Calabria A Hybrid Shell Model RMHD equations in the wave vector space perpendicular to B0 : A shell model in the wave vector space perpendicular to B0 can be derived: (Hybrid : the space dependence along B0 is kept) Khalkidiki, Grece 2003

141
Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Boundary Conditions Space dependence along B0 allows to chose boundary conditions: Total reflection is imposed at the upper boundary A random gaussian motion with autocorrelation time tc = 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 dv ~ cA ~ 1 Km/s Model parameters: L ~ Km, R ~ 6, cA ~ Km/s Khalkidiki, Grece 2003

142
Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Energy balance Stored Energy After a transient a statistical equilibrium is reached between incoming flux, outcoming flux and dissipation Energy flux The level of fluctuations inside the loop is considerably higher than that imposed at the lower loop boundary Dissipated Power Dissipated power displays a sequence of spikes Khalkidiki, Grece 2003

143
Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Energy spectra Magnetic Energy Kinetic Energy A Kolmogorov spectrum is formed mainly on magnetic energy Magnetic energy dominates with respect to kinetic energy Khalkidiki, Grece 2003

144
**Statistical analysis of dissipated power**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Statistical analysis of dissipated power Power Peak Burst duration Burst Energy Waiting time Power laws are recovered on Power peak, burst duration, burst energy and waiting time distributions The obtained energy range correspond to nanoflare energy range Khalkidiki, Grece 2003

145
**Low-dimensional models for coherent structures**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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. Khalkidiki, Grece 2003

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**Proper Orthogonal Decomposition**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 Khalkidiki, Grece 2003

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**Empirical eigenfunctions**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Empirical eigenfunctions The maximum is reached through a variational method thus obtaining the integral equation whose kernel is the averaged autocorrelation function. Very huge computational efforts ! 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. Khalkidiki, Grece 2003

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**Low-dimensional models**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 Khalkidiki, Grece 2003

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**Turbulent convection – Time behavior**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 Khalkidiki, Grece 2003

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**Turbulent convection – Spatial behavior**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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. Khalkidiki, Grece 2003

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**Reconstruction of velocity field**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Reconstruction of velocity field The velocity field has been reconstructed using only J = 0, 1 J = 2, 3 Khalkidiki, Grece 2003

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**Playing with POD POD have been used to describe spatio-temporal**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 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 Angle Time Khalkidiki, Grece 2003

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**3 POD modes Original Reconstruction with 1 POD mode periodicities**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria 3 POD modes Original Reconstruction with 1 POD mode periodicities Reconstruction with 2 POD modes + migration Reconstruction with 3 POD modes + stochasticity Khalkidiki, Grece 2003

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**Conclusions Became a “Plasma Physicist” Acknowledge Loukas**

Vincenzo Carbone Dipartimento di Fisica, Università della Calabria Conclusions Became a “Plasma Physicist” Acknowledge Loukas Vlahos and the local organizing committee Deadline for applications: September 28, 2003 Khalkidiki, Grece 2003

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**Let sand piles evolve … Vincenzo Carbone Dipartimento di Fisica,**

Università della Calabria Let sand piles evolve … Khalkidiki, Grece 2003

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**Don’t care about… Avalanches or Vincenzo Carbone**

Dipartimento di Fisica, Università della Calabria Don’t care about… Avalanches or Khalkidiki, Grece 2003

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