1 به نام خدا

2 Turbulence Turbulence is is beautiful beautiful beautiful beautiful

3 Turbulence? Every day phenomena: nature --- administration Leonardo da Vinci : 1452 –1519van Gogh Turbolenza

4 Generation of the random motions from a smooth flow: The dynamic systems approach) chaos) The development of singular solutions of the ideal fuid equations The excitation of instabilities and their effects

5 Two body system (earth and sun) Two body system (earth and sun) Sun an attractor of the Earth Three body system as a nonlinear interaction. 1)The trajectory is chaotic 2)Sensitive Dependence on Initial Conditions. 3) the trajectory never repeats. 4) All chaotic systems are nonlinear. s E classical chaotic systems deterministic. Quantum chaotic systems non-deterministic. Play 1

6 What is turbulence ? Whorls and eddies of all sizes Chaotic and unpredictable Unsteady and irregular Diffusivity, rapid mixing The motion is nonlinear Play 2

7 HD and MHD equations HD and MHD equations Viscosity Thermal pressure Speed of light Using Faraday’s law we have Which NL Vis NLRes Viscose-stress tensor Navier Stokes equation:Magnetohydrodynamics:

8 For large Re non-linear regime For small Re dissipative regime When non-linear terms dominate, energy is transfer from large to small scales along a turbulence cascade. We observe Universality only at high Re.

9 Turbulence in a magnetized plasma ITER Good confinement Magnetized plasma is usually unstable and turbulent

10 High magnetic Reynolds number ~ 10 5

11 Conservation equations: 1.The momentum-conservation law 2. The energy-conservation law 3. The cross-helicity-conservation law 4. The magnetic-helicity-conservation law Where A is the vector potential and Natural phenomenon is dissipative and conservation equations are broken

12 We introduce two dimensionless variables which were first introduced by Elsasser, now: Which is total pressure In the presence of a background magnetic field, and neglecting the dissipation terms the fields describe Alfven waves propagating in the two opposite direction of The interesting property of the Elsasser field is that there is no self-coupling in the nonlinear term but only cross- coupling

13 Turbulence? E(k) ~ k -5/3 Energy cascade, input at large scales – dissipation at small scales A.N. Kolmogorov

14 What is different between the laminar flow and turbulence? Osborne Reynolds showed that as increases, less than a threshold value for example in Reynolds ‘s experimental it was 2300, the flow is laminar. Re the fluid velocity dose not change with time and all of streamline are parallel with the one of the axial system And in high Re number turbulence state is obtained. streamlines become non-distinction because of mixing with the surrounding flow Play 3

15 High Reynolds numbers in MHD (Re, Rm) lead to global scale of the order of the system size down to very small scales, where dissipation occurs.

16 Spectral properties of MHD equations One of the practical way to study MHD turbulence is direct numerical simulation (DNS) The nonlinear terms in MHD equations become numerically expensive convolution sums. Based on Pseudospectral method the nonlinear terms are compute by applying Fast Fourier transforms (FFTs) and shuttle between real-and Fourier-space.

17 velocity and magnetic fields or Elsasser fields, can be written in Fourier representation Where It is evident that the dissipative terms becomes very simple in k space while the nonlinear terms obviously are more complicated In the high Reynolds number, the nonlinear term be dominant. Since turbulence dynamics contain many transfer process of certain quantities, such as, energy or helicity, between different scale, the Fourier representation which directly yields the corresponding spectral densities, is very convenient. The transfer process of a conserved quantity is called a cascade.

18 Phenomenological models in MHD turbulence According to the kolmogorov’s theory three range of lengths can be identifed Injection range inertial range where energy is transferred towards smaller and smaller lengths, without any production of energy or dissipation Dissipation range

19 Production range Inertial rangeDissipation range Viscous dissipation Energy production Transfer of energy to successively smaller scale injection energy Flux of energy Dissipation of energy the spectrum exhibits a power law behavior

20 Direct cascade Inertial range Dissipation range Derived range Small scale structure Large scale structure Energy K k spectral slope is “UNIVERSAL”

21 spectral-transfer process is local, i.e., the mode interactions are dominated by wavenumbers of the same order of magnitude. Divide the inertial range into a discrete number of scales. A typical turbulence eddy of scale can be represented by the average difference in velocity between two points a distance apart or by the Fourier component The time taken for transfer of energy between two neighboring scales and or turnover, time of the eddy is given

22 The energy flux is constant across the inertial range Then energy spectrum is obtained The well-known Kolmogorov spectrum

23 Iroshnikov-Kraichnan (IK) spectrum: The process is based on the interaction of eddies of size with the magnetic field Alfven waves For two colliding Alfven waves of extent the interaction time to exchange energy nonlinearly is given by which is much shorter than the nonmagnetic eddy-distortion time

24 inertial range scaling of the Elsasser fields And energy spectrum Iroshnikov-Kraichnan (IK) spectrum

25 Compensated Energies with Total energy Magnetic energy Kinetic energy Universality

26 Turbulence Structures : Self-similarity which was proposed by Kolmogorov related to the lack of any characteristic length in the inertial range ??? Intermittent character monofractality or multifractality behavior I will discuss in the next seminar……

27 Applications: DNA sequences (C.K Peng,PRE,1994- Z. Chen,PRE 2002) heart rate dynamics (A. Bunde,PRL,2000-Y. Ashkenazy,PRL,2001) Neuron spiking(S. Bahar,Europhys. Lett. 2001) human gait (Hausdorff J.M,J.Appl.Physiology 1997) long-time weather records(E.Bounde,PRL,1998) cloud structure( K. Ivanova,Europhys. Lett.2000) geology ) B.D Malamud,J. Stat. Plan. Infer,1999) ethnology (C.L. Alados, Ethnology,2000) economical time series (R.N. Mantegna, Nature 1998, PRE,2004)) Stock Markets, M.R. Rahimitabar PRE 2007, oil price, M.Momeni,PRE, Submitted) solid state physics, Rough Surface of graphene )morphology( Geophysics(Earth’s liquid core, Seismic ( Space plasma ( MHD, M.Momeni,PRE 2008, sunspot time series and the solar wind( Discharge Current DNA sequences [30,34], heart rate dynamics [35–37], neuron spiking [38], human gait [39],

28 The end