lagrangian statistics of turbulent flows in fluids and plasmas · instead, physicists focus on...

LAGRANGIAN STATISTICS

OF TURBULENT FLOWS

IN FLUIDS AND PLASMAS

DISSERTATION

zur

Erlangung des Grades

»Doktor der Naturwissenschaften«an der Fakultät für Physik und Astronomie

der Ruhr-Universität Bochum

von

H H

aus

D

B

. Gutachter: Prof. Dr. Rainer Grauer

. Gutachter: PD Dr. Horst Fichtner

Datum der Disputation: ..

Acknowledgments

I gratefully acknowledge the support of many people who have made the comple-

tion of this work possible.

First I would like to express my gratitude to Rainer Grauer, my thesis adviser.

His diversified knowledge on turbulence and numerical methods has been a great

support. He encouraged my enthusiasm for turbulence research and guided me with

optimism. Rainer Grauer accorded me scientific freedom and time for important

discussions.

Thanks to Wolf-Christian Müller for providing me with his magnetohydro-

dynamic code and lots of practical information on performing direct numerical

simulations. Thanks also to Angela Busse for useful discussions and comparisons

of numerical data. I would like to thank Rudolf Friedrich for his fruitful ideas and

comments on Lagrangian statistics and Oliver Kamps for our discussions on the

differences between two- and three-dimensional turbulence.

Among all the members of the Department of Theoretical Physics I in Bochum

I would especially like to thank Holger Schmitz for teaching me generic ++ code

design, Holger Sebert for the development of the / library and Jürgen Möllenhoff

for his assistance in using the Linux-Opteron cluster.

Many thanks to Heinz Joeres for improving the language of this work and Robert

Memering for revising the layout and typography.

Finally I thank my family and my friends for their unconditioned support. Above

all I want to mention the most important person in my life, Annika.

This work benefited from support through of the Deutsche Forschungsge-

sellschaft. Access to the multiprocessor computer at the Forschungszentrum

Jülich was made available through project .

Contents

List of Figures

List of Tables

Introduction

Basic equations . The Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . The magnetohydrodynamic equations . . . . . . . . . . . . . . . Lagrangian coordinates . . . . . . . . . . . . . . . . . . . . . .

Phenomenological description of turbulence . The inertial range of scales . . . . . . . . . . . . . . . . . . . . . Navier-Stokes phenomenology . . . . . . . . . . . . . . . . . .

.. The Richardson cascade . . . . . . . . . . . . . . . . . . .. theory . . . . . . . . . . . . . . . . . . . . . . . . . .. The energy spectrum . . . . . . . . . . . . . . . . . . .

. phenomenology . . . . . . . . . . . . . . . . . . . . . . .

Eulerian intermittency . Models of intermittency . . . . . . . . . . . . . . . . . . . . . .

.. The Obukhov-Kolmogorov model . . . . . . . . . . . . .. The β-model . . . . . . . . . . . . . . . . . . . . . . . .. The bifractal model . . . . . . . . . . . . . . . . . . . . .. The multifractal model . . . . . . . . . . . . . . . . . . .. The She-Lévêque model . . . . . . . . . . . . . . . . . . .. Probability density functions . . . . . . . . . . . . . . . .. Structure functions . . . . . . . . . . . . . . . . . . . .

... Navier-Stokes . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . .

Lagrangian intermittency . Models of intermittency . . . . . . . . . . . . . . . . . . . . . . . Multifractal Navier-Stokes turbulence . . . . . . . . . . . . . .

.. Acceleration statistics . . . . . . . . . . . . . . . . . . . .. Multifractal turbulence . . . . . . . . . . . . . . .

Contents

. Probability density functions . . . . . . . . . . . . . . . . . . . .. Navier-Stokes . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. Structure functions . . . . . . . . . . . . . . . . . . . . . . . . .. Navier-Stokes . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. Structures and Lagrangian intermittency . . . . . . . . . . . . . . Alternative increments . . . . . . . . . . . . . . . . . . . . . . .

.. The norm increment . . . . . . . . . . . . . . . . . . . .. The equal time increment . . . . . . . . . . . . . . . . .

. Frozen Navier-Stokes turbulence . . . . . . . . . . . . . . . . . . Decorrelated Navier-Stokes . . . . . . . . . . . . . . . . . . . .

Numerical methods . Solving the basic equations . . . . . . . . . . . . . . . . . . . .

.. Accuracy of the spectral method . . . . . . . . . . . . . .. Dealiasing . . . . . . . . . . . . . . . . . . . . . . . . . .. The constraint of incompressibility . . . . . . . . . . . .

. Time stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Runge-Kutta third order . . . . . . . . . . . . . . . . . .. Trapezoidal Leapfrog . . . . . . . . . . . . . . . . . . .

. Need for high resolutions . . . . . . . . . . . . . . . . . . . . . . Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . The interpolation . . . . . . . . . . . . . . . . . . . . . . . . .

.. Tri-linear interpolation . . . . . . . . . . . . . . . . . . .. Tri-cubic interpolation . . . . . . . . . . . . . . . . . .

. Influence of the numerical precision . . . . . . . . . . . . . . . . Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.. Initial conditions . . . . . . . . . . . . . . . . . . . . . .. Forcing . . . . . . . . . . . . . . . . . . . . . . . . . .

. Code design . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Summary

Bibliography

List of Figures

. Drawing of turbulent water by Leonardo da Vinci; Sketch of the

Richardson cascade of eddies . . . . . . . . . . . . . . . . . . . . . . Sketch of the energy spectrum of a turbulent flow . . . . . . . . . . . Energy spectra of Navier-Stokes turbulence with normal- and hyper-

viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discretisation of a spherical volume; Corrected and uncorrected com-

pensated energy spectra . . . . . . . . . . . . . . . . . . . . . . . . . Compensated energy spectra in turbulence . . . . . . . . . . .

. Eulerian s of the velocity field and its spatial increment . . . . . . Energy dissipation in a slice from a Navier-Stokes simulation . . . . . The most dissipative structures in Navier-Stokes turbulence . . . . . . Eulerian scaling exponents in Navier-Stokes and turbulence . . . The most dissipative structures in turbulence . . . . . . . . . . . Eulerian s of the spatial velocity increment . . . . . . . . . . . . . Contributions to structure functions from an intermittent . . . . Eulerian velocity structure functions in Navier-Stokes turbulence . . . Logarithmic derivative of Eulerian structure functions in a Navier-

Stokes flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relative Eulerian structure functions () in Navier-Stokes turbulence . Eulerian structure functions in turbulence . . . . . . . . . . . . Relative structure functions () in a flow . . . . . . . . . . .

. Velocity and acceleration of a tracer in Navier-Stokes turbulence and

corresponding s . . . . . . . . . . . . . . . . . . . . . . . . . . . Lagrangian s in Navier-Stokes turbulence, comparison to multi-

fractal prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tracer trapped in a vortex filament in a Navier-Stokes flow . . . . . . Lagrangian s of velocity and magnetic field increments in

turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tracer in the vicinity of strong dissipative structures in a flow .

List of Figures

. Lagrangian s of the velocity increments and magnetic field incre-

ments in turbulence . . . . . . . . . . . . . . . . . . . . . . . . Lagrangian velocity structure functions in Navier-Stokes turbulence . Logarithmic derivative of Lagrangian velocity structure functions in a

Navier-Stokes flow . . . . . . . . . . . . . . . . . . . . . . . . . . . Relative Lagrangian velocity structure functions () in Navier-Stokes

turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lagrangian structure functions in turbulence . . . . . . . . . . . Relative Lagrangian velocity structure functions () in turbulence . Scaling exponents in Navier-Stokes and turbulence together with

the multifractal prediction . . . . . . . . . . . . . . . . . . . . . . . Structure functions depending on the norm of the velocity increment

in a Navier-Stokes flow . . . . . . . . . . . . . . . . . . . . . . . . . Structure functions depending on the equal time increment in a Navier-

Stokes flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tracer trajectory and vortex filaments in a frozen Navier-Stokes flow . Lagrangian structure functions in frozen Navier-Stokes turbulence . . Trajectories of tracers in a Navier-Stokes and uncorrelated flow . . .

. Comparison of the energy spectrum and Eulerian structure functions

for different numbers of grid points . . . . . . . . . . . . . . . . . . . Partitioning of the parallel direction by the . . . . . . . . . . . . Partitioning of the parallel direction for the physical fields and tracer

particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle together with its surrounding cube of grid cells . . . . . . . . Energy spectra and Eulerian structure functions for different floating-

point configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . Eulerian s and tracer trajectories for different floating-point config-

urations and interpolation schemes . . . . . . . . . . . . . . . . . . . Lagrangian s and structure functions for different floating-point

configurations and interpolation schemes . . . . . . . . . . . . . . . . Initial and evolved tracer density . . . . . . . . . . . . . . . . . . . . Fluctuating energy and enstrophy in statistically stationary Navier-

Stokes turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . .

List of Tables

. Eulerian scaling exponents from the She-Lévêque model . . . . . . . . Eulerian scaling exponents from the -She-Lévêque model . . . . . Measured Eulerian scaling exponents in Navier-Stokes turbulence . . . Measured Eulerian scaling exponents in turbulence . . . . . .

. Multifractal Lagrangian scaling exponents in Navier-Stokes . . . . . . Multifractal Lagrangian scaling exponents in . . . . . . . . . . . Measured Lagrangian scaling exponents in Navier-Stokes . . . . . . . Measured Lagrangian scaling exponents in a flow . . . . . . . . Measured scaling exponents of the norm increment in Navier-Stokes . Measured scaling exponents of the equal time increment in Navier-

Stokes turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measured Lagrangian scaling exponents in frozen Navier-Stokes turbu-

lence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measured Eulerian and Lagrangian scaling exponents in a decorrelated

Navier-Stokes flow . . . . . . . . . . . . . . . . . . . . . . . . . .

. Floating-point precision configurations and interpolation schemes . . Parameters of the numerical simulations . . . . . . . . . . . . . . .

Chapter

Introduction

It is easy to come into contact with turbulence in everyday life. This can be done

by watching the sky and its atmospheric motion visualized by the clouds or by

observing a wake of a turbulent jet of an airplane which becomes visible by its vapor

trail. It can also be recognized during a walk on the shores of a bubbly river. In many

cases turbulence is the generic state of motion. The examples mentioned above

have several eye-catching features in common. On the one hand their motions

have a chaotic character, on the other hand they contain structures in the form

of whirls. These eddies cover a large range of spatial scales. In the atmosphere the

largest whirls can only be seen from a satellite. They are as large as entire countries

while the smallest are hidden because of the finite resolution of the human eye.

The scientific problem of turbulence can roughly be divided into two parts,

namely engineering and physical issues, which are of course strongly related. Engi-

neers are mostly concerned with application-oriented problems like determining

and modeling drag coefficients of specific bodies exposed to turbulent flows. Physi-

cists focus on the intrinsic and universal properties of turbulence.

There are features which all turbulent flows share. Although their generation is

based on different forces and although they are enclosed by specific boundaries,

they have a universal character. The forces and boundaries naturally affect the large

scales of the motion. From these large scales turbulence generates a whole range of

whirls of different sizes down to the smallest scale where the dissipation transforms

the kinetic energy into heat. The universality occurs at scales much smaller than

the boundary or forcing scale. Here the information of the geometry of the flow

is lost and the motion is completely determined by the inertial interaction of the

eddies. This range is called inertial range and extends to whirls larger than the

Chapter Introduction

dissipation scale. Due to its universality and absence of a typical length scale the

flow is expected to show scaling behavior within this range. Physical theories often

deal with these fundamental features of the inertial range of scales.

The Navier-Stokes equations are believed to describe the whole ensemble of fluid

motions accurately and in detail. At first sight this might be amazing because of the

small number of terms involved. The complexity arises from the specific boundaries

and especially from its non-linearity. The Schrödinger equation for example is linear

which makes theoretical handling much easier. Another problem of the Navier-

Stokes equations is their vast content of information. The information contained is

temporally and spatially resolved, i. e. the motion of every individual fluid element

is described exactly. That makes it impossible to find analytical turbulent solutions.

However, because of the chaotic nature of turbulence a temporally and spatially

resolved solution is undesired in many cases. Instead, physicists focus on statistical

quantities such as the energy content or probability density functions (s) of the

acceleration of fluid elements.

The first theoretical model of turbulence dates back to when Kolmogorov

[a,b] published two papers concerned with inertial range properties of turbulent

flows. It is mainly a phenomenological theory, that means it is based on heuristic

arguments derived from physical intuition instead of being rigorously derived

from the Navier-Stokes equations. Some years later it was recognized that the

assumption of self-similarity, the basis of Kolmogorov’s theory, is violated by

turbulence. This manifests itself in non-Gaussian s of velocity increments and

in a subtle, anomalous scaling behavior, called intermittency. In Kolmogorov

[] and Obukhov [] presented a model trying to take this effect into account.

However, measurements deviate from their theory. The only model predicting a

scaling behavior in agreement with experiments originates from She and Lévêque

[] in . This is also a phenomenological model. Up to now only a few

results have been derived starting from the Navier-Stokes equations.

The situation in plasma turbulence, described by the magnetohydrodynamic

() equations, is similar to the neutral case. Indeed, plasmas seldom occur in

everyday life. However, most of the visible matter in space consists of plasmas. It is

suggested that turbulence plays an important role in many astrophysical problems

such as star formation. The first models originates from Iroshnikov [] and

Kraichnan [] in . Further improvements of their ideas were published by

Goldreich and S.Sridhar [] in and recently by Boldyrev []. All these

models predict the scaling law of the energy spectrum based on a phenomenological

basis.

Most of the earlier work on turbulence was done in the so-called Eulerian

coordinates. The temporal changes in the velocity field are considered at fixed

points in space. For example, measurements use hot wire probes at fixed locations

to record the temporal velocity fluctuations and numerical simulations conveniently

solve the Navier-Stokes equations on a fixed spatial grid. The enormous progress

in particle-tracking techniques triggered a new interest in tackling the problem

of turbulence in Lagrangian coordinates. These coordinates follow to motion of

fluid elements, called tracers, and therefore evolve in time. Measurements, which

were done in Cornell using optical techniques by Voth et al. [] and in Lyon

using acoustical techniques by Mordant et al. [] provide the possibility of

following the trajectories of tracers in turbulent flows at high Reynolds numbers

with high precision. It is useful to consider Lagrangian coordinates for several

reasons. First, they provide an additional access to turbulence, which may help in

building up theoretical models. Recently, Friedrich [] presented a closure of a

hierarchy of statistical evolution equations, which makes explicit use of Lagrangian

coordinates. A second reason is that the Lagrangian point of view is naturally

adapted to problems concerning diffusion and dispersion of particles (see for

example Yeung and Borgas []). Turbulence greatly enhances the rate of mixing.

This is sometimes desirable for instance, to reduce the concentration of pollution

from a toxic source or in combustion devices or chemical reactors to enhance the

rate of reaction. Tackling problems of this type is preferably done in Lagrangian

coordinates.

Due to the rising speed of computers direct numerical simulations () solving

the Navier-Stokes and equations have become a valuable tool for studying tur-

bulence. Apart from measurements they provide idealized data which, in some cases,

can better be compared to theories than experiments. Especially for conducting

flows there are only a few astrophysical measurements (see Armstrong et al. []),

which makes a falsification of models difficult. The tracking of individual tracers

for a Lagrangian description is impossible in these environments. High resolution

numerical simulation can fill this gap. To achieve fully developed turbulence at

high Reynolds numbers the use of state of the art super-computers is indispensable.

Because of their huge number of processors parallelized computations have to be

performed. In order to integrate the tracers the velocity field has to be interpolated

Chapter Introduction

accurately at the tracer positions. This work uses a tri-cubic scheme and analyzes

the differences in the statistical results when using a tri-linear scheme, which is

done by Biferale et al. [b].

This work is mainly concerned with the Lagrangian scaling behavior in the

inertial range of scales of three-dimensional Navier-Stokes and turbulence.

While for hydrodynamic flows experimental measurements as well as numerical

data exist, the scaling laws of turbulence are not known, yet. High resolution

of turbulent flows are performed and trajectories of millions of tracers advanced

by the underlying velocity field are recorded.

Experimental measurements by Mordant et al. [] and numerical measure-

ments by Biferale et al. [b] of the scaling exponents of Navier-Stokes flows

differ, due to the range used for the evaluation. The appropriate scaling range is

still under discussion. This work aims at clarifying the controversial points of view.

The motivation for considering turbulence in addition to Navier-Stokes

turbulence is primarily not to relate the measurements to specific astrophysical

problems but to have a second turbulent system for studying Lagrangian turbulence.

The comparison of a neutral and conducting flow is fruitful. Navier-Stokes and

turbulence differ significantly due to the influence of the magnetic field. The

most dissipative and coherent structures which have a deep impact on the scaling

within the inertial range are completely different. These structures are reflected

in the tracer trajectories and the Lagrangian scaling behavior. A crucial point is

that these structures have a different influence on the observed degree of intermit-

tency when measured in the Eulerian or Lagrangian framework. These findings

explain the shortcomings of a model predicting the Lagrangian scaling behavior

of Navier-Stokes turbulence proposed by Biferale et al. [a]. In addition this

work considers frozen turbulence with a static velocity field as a simplified system

to disentangle contributions originating in the dynamical evolution from features

arising purely from the transition to Lagrangian coordinates. The comparison of

Lagrangian hydrodynamic to turbulence reveals features of turbulence, which

are hidden for the Eulerian treatment.

This work is organized as follows: The underlying equations of motions are

introduced in the following chapter. To familiarize with turbulence modeling

and to provide fundamental definitions in chapter some important results of

phenomenological models of Navier-Stokes and turbulence are reviewed.

The problem of intermittency is defined and analyzed in chapter in Eulerian

coordinates. The main results of this work are given in chapter . This chapter deals

with Lagrangian intermittency in Navier-Stokes and turbulence. In chapter

the numerical methods used will be explained. Finally, chapter gives a summary

of this work and its results.

Chapter

Basic equations

A turbulent flow can be described as a continuous fluid. Although it consists of

discrete molecules a kinetic description is not required in most cases. Kinetic effects

are taking place below the viscous scale which is the smallest turbulent scale. All

interesting scales are therefore dominated by collisions and can be treated by a fluid

description.

In this chapter the basic equations used to describe a turbulent flow will be

explained. In the hydrodynamic () case these are the Navier-Stokes equations

presented in Section .. In many cases a conducting fluid can be described by the

magnetohydrodynamic () equations as outlined in Section ..

. The Navier-Stokes equations

An incompressible fluid is believed to be correctly described by the set of the nearly

-year-old Navier-Stokes equations,

∂t~u + (~u · ∇)~u = − 1

ρ0

∇p + ν∆~u, (.)

∇ · ~u = 0, (.)

ρ0 denoting the constant mass density, p the pressure and ν the kinematic viscosity.

These equations are written in Eulerian coordinates. The velocity field ~u = ~u(~x, t)

has to be evaluated at the fixed points ~x in space at a time t.

The Euler equations,

∂t~u + (~u · ∇)~u = − 1

ρ0

∇p, ∇ · ~u = 0, (.)

Chapter Basic equations

are years old and are believed to describe in-viscous flows. They appear to be

the same as the Navier-Stokes equations (.) for ν = 0. Nevertheless they do have

different features as the Navier-Stokes equations with small but finite viscosity. On

the one hand they exhibit singularities. These are damped in Navier-Stokes flows

by the viscosity. On the other hand the boundary conditions differ. While one

has no-slip conditions at the walls due to the viscosity (a thin layer sticks to the

wall) one can have parallel velocities at the boundaries in solutions of the Euler

equations.

Introducing a typical length scale L and velocity U one can rescale (.) using

the following replacements,

~x′ =~x

L, ~u′ =

~u

U, t′ =

U

Lt. (.)

Inserting (.) into (.) and introducing a rescaled pressure p′ = pρ0U2 yields

∂′t~u′ + (~u′ · ∇′)~u′ = −∇′p′ + ν ′∆′~u′. (.)

This dimensionless equation contains the famous parameter

R =1

ν ′=

LU

ν, (.)

called Reynolds number. The important implication is that a rescaled configuration

of a turbulent flow will display the same features as the original flow, provided that

the Reynolds numbers are identical. This is particularly important for engineers

who want to study, for example, a turbulent flow behind an airplane using a

downsized model in their laboratory. Apart from the definition (.) measurements

often use the so-called Taylor-Reynolds number

Rλ =√

15R.

In the following the primes in (.) will be omitted and the set of equations

∂t~u + (~u · ∇)~u = −∇p + ν∆~u, (.)

∇ · ~u = 0, (.)

will simply be called the Navier-Stokes equations. In the case of an incompressible

flow one can re-express (.) as

∂t~u +∇ · (~u~u) = −∇p + ν∆~u. (.)

. The magnetohydrodynamic equations

Sometimes it is convenient to write the Navier-Stokes equations in terms of the

vorticity

~ω = ∇× ~u

Taking the curl of . yields the evolution equation for the vorticity ~ω,

∂t~ω = ∇× (~u× ~ω) + ν∆~ω. (.)

The continuity equation . guarantees the incompressibility and determines

the pressure p. Taking the divergence of . yields

∆p = ∇ · (~u · ∇~u), (.)

which is a Poisson equation and establishes together with appropriate boundary

conditions the pressure p.

In the limit of vanishing viscosity there are two invariants in Navier-Stokes flows.

The total energy

E =1

2

∫|~u|2d3x (.)

and the kinetic helicity

HK =1

2

∫~u · ~ωd3x. (.)

Allowing for a finite viscosity these quantities decay in time. The energy dissipation

rate ε given by

ε = −dE

dt=

∫ν|∇~u|2 (.)

is a central quantity of many phenomenological models of turbulence (see Sec-

tion ..).


Under certain conditions a plasma can be described as a single fluid. The resulting

equations are the magnetohydrodynamic () equations. They can be derived as

velocity moments of the Boltzmann equation. (see e. g. Chen [], Nicholson

[], Boyd and Sanderson [], Sturrock []). The equations for the velocity

field are the Navier-Stokes equations (.) with the Lorentz force added on the

right hand side,

∂t~u = − (~u · ∇) ~u +1

cρ0

~j × ~B + ν∆~u− 1

ρ0

∇p,

∇ · ~u = 0.


As in the Navier-Stokes case the system will be considered to be incompressible,

ρ0 denoting the constant density and ν denoting the kinematic viscosity. The

evolution of the electric and magnetic fields are described by the Maxwell equations,

neglecting the dielectric current. In Ohm’s laws, coupling the electric and magnetic

field of a moving charge, only the resistivity is taken into account. Using Gaussian

units the resulting equations read,

∂t~B = −c∇× ~E, (.)

4π

c~j = ∇× ~B, (.)

∇ · ~B = 0, (.)

~E = −1

c~u× ~B + ηd

~j. (.)

The resistivity ηd is assumed to be constant in space. The equations can

be written dimensionless as the Navier-Stokes equations (.) by introducing the

following normalizations,

~x′ =~x

L, ~u′ =

~u

U, t′ =

U

Lt,

~B′ =~B√

4πρ0U, p′ =

p

ρ0U2.

Inserting Ohm’s law (.) for ~E, Faraday’s law (.) for ~j and omitting the

primes yields

∂t~u = − (~u · ∇) ~u +(∇× ~B

)× ~B −∇p + ν∆~u, (.)

∂t~B = ∇×

(~u× ~B

)+ ηd∆ ~B, (.)

∇ · ~B = ∇ · ~u = 0. (.)

ηd is essentially the reciprocal magnetic Reynolds number,

Rm =UL

ηd

. (.)

Taking the curl of (.) yields the vorticity formulation of the momentum equa-

tion,

∂t~ω = ∇×[~u× ~ω +

(∇× ~B

)× ~B

]+ ν∆~ω. (.)

The ratio of the magnetic Reynolds number and the hydrodynamic Reynolds

number

Prm =ν

ηd

(.)


is called magnetic Prandtl number. It is a measure of the ratio of the magnetic and

kinetic energy of the flow.

The ideal equations (neglecting the viscosity ν and resistivity ηd) have three

invariants. Besides the total energy

E =1

2

∫|~u|2 + | ~B|2d3x, (.)

the cross helicity

Hc =1

2

∫~u · ~Bd3x (.)

and the magnetic helicity

Hm =1

2

∫~A · ~Bd3x, ∇× ~A = ~B, (.)

are conserved. The energy dissipation rate, i. e. the decay of the total energy due to

viscous and resistive dissipation is

ε = −dE

dt=

∫ν|∇~u|2 + ηd|~j|2d3x. (.)

Magnetic helicity decays slower than the total energy. This is because the definition

involves smoother fields ( ~A is the integral of ~B). The dissipation which depends

on the roughness is therefore smaller. The differing decay of these ideally conserved

quantities is called selective decay (see e. g. Biskamp []).

The equations (.) to (.) have a symmetric form when written in

terms of the Elsässer variables

~z± = ~u± ~B.

Adding and subtracting equations (.) to (.) yields

∂t~z± + ~z∓ · ∇~z± = −∇P +

1

2(ν + ηd)∇2~z± +

1

2(ν − ηd)∇2~z∓, (.)

∇ · ~z± = 0, (.)

where P is the total pressure, P = p + 12B2. The formulation (.) is useful

because it points out the occurrence of Alfvén waves in plasmas. Neglecting the

dissipation terms and linearizing these equations yields solutions travelling with

Alfvén speed vA = B0/√

4πρ0, B0 denoting the mean magnetic field strength.

For these linear modes the velocity and magnetic field perturbations are parallel.


Alfvén waves travel along a magnetic field (serving as a guide field) and bend the

field lines into the transverse direction. Magnetic field lines try to resist bending

which accounts for the wave speed vA. The waves corresponding to ~z± move in

opposite directions. A last important feature to mention is that no self-coupling

of the Elsässer fields ~z± occurs in the nonlinear term of (.). Only waves with

opposite velocity directions interact.

The importance of Alfvén waves in incompressible turbulence is a funda-

mental aspect included in several phenomenological models of turbulence. These

models will be reviewed in Section ..

. Lagrangian coordinates

The equations in the previous sections were written in Eulerian coordinates (~x, t).

Instead, one can also use Lagrangian coordinates ( ~X(~y, t), t) = (~y, t) which are the

initial position of a tracer and time t. The Lagrangian coordinates stick to the fluid

elements and follow their trajectories. The equations of motion are

d

dt~X(~y, t) = ~u( ~X(~y, t), t), (.)

where ~u( ~X(~y, t), t) is the Eulerian velocity at ~X(~y, t) given by the underlying

equations of motion presented in the previous sections.

Chapter

Phenomenological description of

turbulence

The equations of motion describing a neutral turbulent fluid, the Navier-Stokes

equations, have been known for nearly years. Nevertheless, a theory starting

from this set of equations and predicting central statistical quantities of importance

in turbulent flows is still missing. The situation in the case of a conducting fluid

is similar. Again the equations of motion have been known for about years,

but a theory starting from the basic equations is also missing. Due to the lack of

precise data from experiments even basic properties of turbulent flows (see

Section .) are still under discussion.

Up to now most of the theories dealing with separated aspects of turbulence have

been based on phenomenology, i. e. dimensional analysis and heuristic arguments.

The originator of this branch of turbulence modeling is A. N. Kolmogorov. In

he published several papers which are still the basis for improved models

of turbulence. Due to its great influence on the research of turbulence and its

importance for more sophisticated models presented in the following sections,

some results of the so-called theory will be presented in Section ...

Some phenomenological models trying to describe turbulence will briefly

be reviewed in Section ..

A lot of the following models deal with the prediction of quantities within a

range of scales between the large scales of forcing and the small scales of dissipation.

In the next section some ideas of this inertial range of scales will be presented.

Chapter Phenomenological description of turbulence

. The inertial range of scales

For the following let us consider the example of a turbulent wake behind a cylinder.

Although this scenario is directly connected to neutral flows it will clarify the

meaning of the inertial range also in the case of conducting fluids. In most cases of

turbulent flows the forcing acts on large scales, like the diameter L of a cylinder,

called integral scale. The dissipation smooths the smallest scales due to the viscosity

of the fluid and transforms kinetic energy into heat. The scale η at which the

viscosity becomes important is called Kolmogorov scale. Between these two limiting

spatial scales the inertial range of scales exists. It extends from large scales small

enough that the information about the geometry like the flow limiting cylinder

is lost down to small scales large enough that the influence of the dissipation is

negligible. Within this range the energy is solely transferred from one scale to

another by the action of the non-linear term. The non-linear terms of the Navier-

Stokes equations (.) and equations (.) and (.) conserve energy. The

energy is only changed by the forcing terms and the dissipation terms.

In the inertial range the explicit information of the turbulence generating force

as well as the information of the underlying dissipative process is unimportant, this

range of scales possesses universal features. The same statistical quantities can be

measured in jets, atmospheric flows, oceanic flows and even in turbulent boundary

layers like a turbulent flow behind a cylinder (see e. g. Frisch []).

Sections . and . deal with phenomenological models describing features of

the inertial range of scales in Navier-Stokes and turbulence, respectively.

. Navier-Stokes phenomenology

Turbulent flows are omnipresent. They can be observed by looking at the atmo-

spheric motion visualized by the clouds in the sky, at a whirling river or at the fume

of a cigarette. This easy accessibility of observations of turbulence provides every

intent observer with some features of turbulence. A whirling flow is constituted

of eddies of different sizes. These interact, merge and break up. Already Leonardo

da Vinci drew a detailed picture of turbulent water generated by a waterfall (see

Figure .).

The generation and decay of eddies of different sizes in a turbulent flow leads to

the cascade picture of turbulence presented in the following section.


Figure .: Left: Drawing of turbulent water by Leonardo da Vinci (th century), right: Sketch of

the Richardson cascade of eddies. Large eddies break up into smaller ones

.. The Richardson cascade

The picture of a turbulent flow consisting of eddies of different sizes with a

cascading energy is the basis for almost every model of turbulence. It was couched

in terms by L. F. Richardson:

Big whorls have little whorls

That feed on their velocity

And little whorls have lesser whorls

And so on to viscosity

— L. F. Richardson ()

An eddy of a given size becomes unstable, breaks up and transfers its energy to

daughter eddies of smaller size. After a period of time these also become unstable

and break up into even smaller ones. This cascade ends at the dissipation scale,

which is the smallest scale of turbulent motion in the flow.

.. theory

Kolmogorov made the first attempt to quantify the picture of a cascade of eddies

breaking up successively. In this section the main features of his theory of will

be reviewed (see e. g. Frisch [], Pope []).

Kolmogorov stated three assumptions on turbulent flows which constitute the

basis of his further derivations:


Kolmogorov’s hypothesis of local isotropy

At sufficiently high Reynolds number, the small-scale turbulent mo-

tions (l L) are statistically isotropic.

This assumption states that the dynamics of the small-scale motions are universal.

During the process of cascading the signature of the flow generating geometry is

lost. The forcing or boundary introduces typically anisotropy. In the example of the

flow behind a cylinder, the motion of the largest eddies directly behind the cylinder

will strongly be affected by the cylinder. In the wake, away from this boundary, the

cascade takes place and statistical isotropy is restored at small scales in a frame of

reference moving with the mean flow velocity.

Kolmogorov’s first similarity hypothesis

In every turbulent flow at sufficiently high Reynolds number, the

statistics of the small-scale motions (l L) have an universal form

that is uniquely determined by ν and ε.

For scales smaller than the largest scale the statistical features of the flow are entirely

determined by two inherent statistical quantities, the viscosity ν and the energy

dissipation rate ε. For stationary turbulence the dissipation rate equals the energy

input rate and energy transfer-rate from scale to scale.

Kolmogorov’s second similarity hypothesis

In every turbulent flow at sufficiently high Reynolds number, the

statistics of the motions of scale l in the inertial range L l η have

a universal form that is determined by ε, independent of ν.

The smaller the considered scale the more important the viscosity becomes. For

turbulent flows with a broad range of scales only the smallest scales are affected

by the dissipation. Therefore the statistics of the scales in between, called inertial

range, depend only on the energy dissipation rate ε. Kolmogorov postulated that

this dissipation rate is a finite non-vanishing quantity even in the limit of an infinite

Reynolds number corresponding to a vanishing viscosity. Recently, the energy dissi-

pation rate was measured numerically in turbulent flows up to a Taylor-Reynolds

number of Rλ = 1201 on the Earth Simulator by Kaneda and Ishihara [].

And indeed, for high Reynolds numbers the energy dissipation rate approaches a

constant while decreasing the viscosity.


An important implication is that the spatial derivatives of the velocities have to

grow when decreasing the viscosity in order to keep the energy dissipation rate

(.) constant. And indeed the flow of the Euler equations (.) tends to form

singularities (see e. g. Grauer et al. []). With increasing Reynolds number the

gradients become larger and larger. Only the viscous term added in the Navier-

Stokes equations prevents the solution from forming singularities. A little bit later

in this section we will see that Kolmogorov was able to predict the roughness of a

turbulent flow from his hypothesis.

The observation that a turbulent flow produces small structures from large

structures is in accordance with the cascade picture of energy transfer. The initial

large and smooth eddies shrink and try to build singularities. Due to this shrinking

the energy is transferred to smaller scales. This cascade only ends at the scale where

the dissipation smooths the motion again.

>From Kolmogorov’s first similarity hypothesis one can estimate the scale where

dissipation takes place. This scale has the dimension of length, m, the viscosity ν

the dimension m2/s and the energy dissipation rate ε the dimension m2/s3. This

implies that the only possible combination of ν and ε to constitute a length is

η =

(ν3

ε

) 14

, (.)

which is called Kolmogorov scale or dissipation scale. With the same arguments one

can estimate the according Kolmogorov speed and Kolmogorov time scale,

uη = (εν)14 , (.)

τη =(ν

ε

) 12. (.)

The latter is sometimes called dissipation time scale.

The quantities (.)–(.) estimate the smallest turbulent scales. In the same

spirit the largest scales can be estimated. The largest spatial scale (usually the energy

injection scale), called integral scale L, should only depend on the energy of the

large scale motion and the energy dissipation rate (which equals the rate of energy

transfer in stationary turbulence). This energy is approximately the total energy

because the contribution of the small scales is negligible. From the equipartition

theorem follows,

E =3

2u2

0,


u0 ≡ urms denoting the root mean square velocity and therefore

L =(2/3E)3/2

ε.

The integral time scale, often called large eddy turn-over time is accordingly the ratio

of L and u0

TL =L

u0

. (.)

After this period of time the current state of the flow is statistically independent of

the former state. It is possible to express the Reynolds number (.) in terms of

this integral scale L and the dissipation scale η,

R = (L/η)4/3. (.)

Kolmogorov assumes that a turbulent flow is self-similar below the energy injec-

tion scale. An important consequence of this self-similarity (and therefore space

filling) assumption is that the turbulent flow possesses a unique scaling exponent h

such that the velocity differences δlu = u(x + l)− u(x) at scale l obey,

δu(λl) = λhδu(l). (.)

This exponent h does not depend on the considered scale l. From this assumption

one can directly compute the value of the unique scaling exponent h. Rescaling

l → λl and the according velocity difference δlu → λhδlu as in (.) yields for

the associated time t → t1−h. If one now also rescales the energy dissipation rate

ε → λ3h−1ε and demands that ε must not depend on the considered scale one gets

the scaling exponent

h =1

3. (.)

Therefore a turbulent flow is not smooth within the inertial range. If it is differen-

tiable, the velocity differences δlu would have a scaling exponent h = 1 for spatial

separations l. Then one would end up with the definition of the derivative of the

velocity field at the point x. However, the velocity differences in a turbulent flow

approach zero more slowly than in a differentiable flow. Below the inertial range the

dissipation smooths the motion and the flow becomes indeed differentiable. These

features will be discussed in detail by looking at measurement in Section ...

The result of the theory predicting the scaling exponent (.) is important

because it implies the scaling properties of the structure functions

Sp(l) = 〈|δu|p〉 ≡ 〈|u(x + l)− u(x)|p〉 ∼ lζp . (.)


Integral scale Inertial range of scales Dissipation scale

Energy input Dissipationless turbulent cascade Viscous dissipation into heat

log

E(k

)

L−10 log kη−1

Figure .: Sketch of the energy spectrum of a turbulent flow

The unique value of h yields

〈|u(x + l)− u(x)|p〉 ∼ lp3 . (.)

Therefore the scaling exponents ζp predicted by the theory are a linear function

of p,

ζp =p

3. (.)

The scaling of the structure function has been an active topic of research for more

than years, since experiments by Anselmet et al. [] disagreed for large orders

p with the Kolmogorov scaling .. This deviation from the dimensional prediction

is attributed to the phenomenon of intermittency. This will be discussed in detail

in chapter .

The scaling of the second order structure function S2(l) has implications for the

scaling of the corresponding energy spectrum. The second order structure function

can be expressed in terms of the correlation function Γ(l) = 〈u(x + l)u(x)〉 as

S2(l) = 2Γ(0)− 2Γ(l). The correlation function itself is the Fourier transform of

the energy spectrum. A short computation (see Frisch []) shows

E(k) ∼ k−n, for S2(l) ∼ ln−1.

>From (.) the energy spectrum has the form

E(k) ∼ k−53 , (.)

within the inertial range.


The cascade picture is summarized in terms of the energy spectrum in Figure ..

The energy is introduced into the system at large scales, the integral scale. From

there the energy is transported from scale to scale by inertial interaction described

by the non-linear terms in the Navier-Stokes equations. As the cascade reaches the

dissipation scale viscosity becomes important, smooths the flow and converts the

kinetic energy of the eddies into heat.

A last but important result of Kolmogorov’s theory which should be men-

tioned is the 4/5th law (see e. g. Frisch []). Up to now this has been one of the

few exact results derived from the Navier-Stokes equations for turbulent flows. This

law predicts a linear scaling law of the third order longitudinal structure function,

S3(l) = 〈((~u(~x +~l)− ~u(~x)) · ~l)3〉 =4

5εl. (.)

Measurements often deal with longitudinal structure functions. However, from

dimensional analysis it is difficult to distinguish between directions. From now on

if the direction is not explicitly mentioned, longitudinal increments are meant.

Since the 4/5th law is derived from the underlying equations, it can be seen as a

restriction and benchmark for all models of turbulence. Furthermore it can be used

to define the inertial range. The inertial energy cascade is spoiled by the forcing at

large scales and by viscous damping as small scales. The inertial range of scales can

be defined as the range of scales in which the 4/5th law (.) holds.

.. The energy spectrum

A central quantity of many theories of turbulence is the scaling law of the energy

spectrum within the inertial range of scales. As we have seen in the last section,

the theory predicts a scaling exponent of −5/3. This law has been observed in

several experiments (see e. g. Sreenivasan and Stolovitzky [], Frisch []). Due

to the limited computational performance, direct numerical simulations display

only a narrow inertial range which complicates the determination of the exact

scaling exponent. Conveniently in numerical simulations one considers angle-

averaged energy spectra,

E(k) =

∫Ω

E(kx, ky, kz)dΩ.

The left part of Figure . shows the angle-averaged energy spectrum compensated

by the prediction taken from simulations with normal and hyper-viscosity.


Here hyper-viscosity means replacing the Laplacian ∆ in the Navier-Stokes equa-

tions (.) by its square. This is sometimes done to enlarge the inertial range.

The dissipation range becomes steeper. Both spectra in . display a bottle-neck

effect in front of the dissipation range, but the hyper-viscous bottle-neck is more

pronounced.

0.001

0.01

0.1

1

1 10 100

E(k

)/k(-

5/3)

k

normal viscosityhyper viscosity

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

1 10 100

E(k

)

0.001

0.01

0.1

1 10 100

E(k

)/k(-

5/3)

k

normal viscosityhyper viscosity

1e-14

1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1

1 10 100

E(k

)

Figure .: Energy spectra taken from stationary Navier-Stokes simulations with normal and hyper-

viscosity and 10243 grid points. Left: Compensated angle-averaged spectra, inset: spectra without

compensation; Right: Compensated one-dimensional spectra, inset: spectra without compensation

The bottle-neck effect is usually explained by the lack of smaller-scale eddies

while approaching the dissipation scale η which makes the cascade less efficient

around η and leads to a pile up of energy at η. Haugen and Brandenburg []

showed that the range where the bottle-neck appears and therefore the inertial

range spoils is of the same order no matter which order of hyper-viscosity is used.

With increasing the order of the hyper-viscosity, the amplitude of the bottle-neck

increases, while the width stays approximately constant.

As this work is mostly concerned with the scaling behavior within the inertial

range and as the structure functions are also spoiled by the use of hyper-viscosity

in the following only data obtained with normal viscosity will be presented.

In experiments one often measures a longitudinal one-dimensional energy spec-

trum

E(k) =

∫|ux(kx, ky, kz)|2dkydkz,


r

dx

0.01

0.1

1 10 100

E(k

)/k(-

5/3)

k

correcteduncorrected

Figure .: Left: Volume of a spherical shell and its discretisation, Right: Corrected and uncorrected

compensated energy spectra, shifted for clarity

instead of an angle-averaged spectrum. Such one-dimensional spectra show a less

pronounced bottle-neck, which is a direct consequence of the relation between the

one-dimensional and its angle-averaged counterpart (see Dobler et al. [] for

details). The right part of Figure . shows the one-dimensional spectrum again for

the same normal and hyper-viscous flows. Indeed, the bottle-neck is weaker than

in the left figure, but nevertheless visible.

The determination of a deviation of the scaling exponent from the prediction

−5/3 of 0.03 and 0.01 as suggested by Haugen and Brandenburg [] and

Kaneda and Ishihara [], respectively, is not possible due to the limited inertial

range.

A remark on the energy spectrum obtained from numerical simulations: As the

computations were done on a discretized grid the angle-averaged spectrum has

to be corrected. The situation is depicted in the left part of Figure . for two

dimensions. Angle-averaged means integration over a spherical shell at the radius

r of a size dx. The numerical integration sums up all cubes which lie within this

shell. Clearly there is a difference in the considered volume especially for the inner

shells. In three dimensions the volume of a spherical shell at the radius r of size dx

is,

Vshell(r) =4

3π

(r +

dx

2

)3

− 4

3π

(r − dx

2

)3

=π

3

(dx3 + 12r2dx

).

. phenomenology

The discretized volume is,

Vnum(r) = n(r)dx3,

where n(r) denotes the number of cubes lying in the shell at radius r. The require-

ment Vshell(r) = Vnum(r) yields

Vnum(i) · π

3

(1 + 3i2

n

)= Vshell(i), (.)

where the discretized radius r = idx has been inserted.

Practically one has to count the cubes n(i) corresponding to each radius i and

multiply the energy of this shell by (.). The right part of Figure . shows the

corrected and uncorrected compensated energy spectrum. The corrected curve is

smoother than the uncorrected, i. e. the spiky character of the uncorrected spectrum

is an artifact of the discretisation.

. phenomenology

The phenomenological models of turbulence in the spirit of Kolmogorov’s

theory date back to Iroshnikov [] and Kraichnan []. In this section the

most important features of phenomenology will be briefly reviewed. For a

presentation in more detail see for example Biskamp [].

The equation written in terms of the Elsässer variable (.) show that only

Alfvén waves traveling in opposite directions interact. Iroshnikov and Kraichnan

proposed the so called Alfvén effect. They assumed that the cascade process is mainly

due to the scattering of Alfvén waves. Two different time scales are of importance:

The Alfvén time

τA =l

vA

(.)

and the nonlinear time scale

τ± =l

δz±l. (.)

δz±l denotes a wave packet of size l, sometime also called eddy of size l. In general

τA τ±. The change of the amplitude ∆δz± during one collision is the time scale

of the collision τA divided by the time scale of the nonlinear interaction τ±l , hence

∆δz±lδz±

=τA

τ±l 1.


As different eddies are not correlated, these small perturbations add up randomly,

resulting in a diffusion process. Thus it needs N ∼ (δzl/∆δz±l )2 collisions to have

a substantial change of the wave packets. The energy-transfer time for the Alfvénic

scattering is

T±l ∼ NτA ∼ (τ±l )2/τA (.)

which is increased by a factor of τ±l /τA compared to the Kolmogorov prediction.

One can define energy quantities E± =∫

(z±)2d3x corresponding to the Alfvén

waves z±. In Section . it was mentioned that both the total energy E = E+ +E−

and Hc = E+ − E− are ideal invariants. Therefore the same is true for E±. The

corresponding energy fluxes of these cascading quantities are

ε± ∼ (δz±l )2

T±l

∼ (δzl)4τA

l2,

where an assumption of weakly correlated velocity and magnetic field (δlz+ = δlz

−)

and (.) and (.) is used. Inserting the Alfvén time (.) yields the Iroshnikov-

Kraichnan scaling behavior

δzl ∼ (εvA)1/4l1/4

and the corresponding energy spectrum

Ek ∼ k−3/2. (.)

The spectrum is less steep than the Kolmogorov spectrum because of the longer

energy-transfer time. The cascade process is therefore less efficient.

The Iroshnikov-Kraichnan spectrum is not found in turbulence simulations (see

Mueller et al. []). The measured spectrum is of Kolmogorov type. An

flow is in general anisotropic at small scales due to the large magnetic field of the

large scales. This anisotropy is not taken into account by the Iroshnikov-Kraichnan

model. The eddy size is the same in the field-parallel as in the field-perpendicular

direction. The model does not distinguish between field parallel and perpendicular

directions. Goldreich and S.Sridhar [] proposed a model which tries to take

the anisotropy into account. The key idea is the following: The parallel dimension

l‖ and perpendicular dimension l⊥ of an eddy differ because the magnetic field

lines resist bending so that the eddies will be elongated along the parallel magnetic

field direction. However, the corresponding time scales have to fulfill

δzl⊥/l⊥ ∼ vA/l‖, (.)

. phenomenology

called critical balance. The spectral cascade takes mainly place in the perpendicular

direction with the Kolmogorov energy flux

ε ∼ δz3l⊥

/l⊥. (.)

Using relation (.) yields

k⊥/k‖ ∼ (Lk⊥)1/3. (.)

The anisotropy increases with k.

In summary the Goldreich-Sridhar model proposes an anisotropic cascade pro-

cess. The perpendicular spectrum due to the Kolmogorov energy flux is

E(k⊥) ∼ k−5/3⊥ ,

while the parallel spectrum is

E(k‖) ∼ k−5/2‖ , (.)

which follows from the former by inserting (.). Unfortunately in simu-

lations with a strong external magnetic field (see e. g. Mueller et al. []) the

perpendicular spectrum is found to be close to the Iroshnikov-Kraichnan spectrum

(.). These findings contradict the Goldreich-Sridhar spectrum (.).

Recently, Boldyrev [] proposed a model which assumes a scale dependent

alignment of the magnetic field fluctuations δ ~Bl and velocity fluctuations δ~vl.

He proposes that the alignment of these fluctuations reach the maximal level

consistent with a constant energy flux through this scale. This is achieved if the

fluctuations align their directions within the angle Φl ∝ l1/4. His derivation leads

to a perpendicular spectrum of type in a strong magnetic background field. He

argues that such a spectrum should also be observable in isotropic turbulence

at very high Reynolds numbers where the local magnetic field of the large eddies is

strong enough to act as an external field for the smaller scale eddies. Up to now the

Reynolds numbers achieved in numerical simulations are insufficient to verify his

proposition.

In Figure . energy spectra of isotropic are shown for two different Reynolds

numbers. It is an angle-averaged spectrum without distinguishing between parallel

and perpendicular contributions, because the precise scaling law is not important

for the following sections. A small inertial range with a scaling law is visible. The


1e-05

1e-04

0.001

0.01

0.1

1

1 10 100

E(k

)/k(-

5/3)

k

Rλ= 234Rλ= 107

IK

1e-09 1e-08 1e-07 1e-06 1e-05 1e-04 0.001

0.01 0.1

1 10 100

E(k

)

Figure .: Compensated energy spectra in turbulence for two different Reynolds numbers,

straight line: Iroshnikov-Kraichnan scaling (.), inset: no compensation

displayed spectra seem to be more of Boldyrev type than of Kolmogorov type.

However, the Reynolds number and therefore the scaling range is too small and

might be spoiled by a bottle-neck effect to evaluate the exact scaling exponent.

Chapter

Eulerian intermittency

An intermittent statistical variable has a non-Gaussian probability density distribu-

tion function (). Measurements of such a variable show large deviations from

the mean much more often than Gaussian distributed variables. Examples are as we

will see later on the local dissipation rates, velocity gradients and the acceleration

of fluid tracers. Since measurements provided the evidence that turbulence displays

non-Gaussian s, intermittency has become an active topic of research. The

theory is based on the assumption that turbulence is self-similar, which does not

allow for intermittency. Therefore the theory had to be improved. A lot of

theories have been proposed to described the findings from the measurements.

This chapter deals with intermittency in the Eulerian framework. Several mod-

els and measurements will be presented. Afterwards the implications from the

Lagrangian point of view are discussed in chapter .

As was mentioned above, intermittency manifests itself in a non-Gaussian shape

of the s of the considered statistical variable. Figure . shows the of the

velocity u and the velocity increment u(x + l)− u(x) = δlu(x) for l = 0.8η. For

comparison a Gaussian is also shown. While the of u is nearly Gaussian,

the of δu exhibits stretched tails. Section .. will comment on this topic in

detail.

Apart from looking at the one can compute the according moments. These

structure functions (introduced in Section ..), are

Sp(l) = 〈|δul|p〉 = 〈|u(x + l)− u(l)|p〉 , (.)

angular brackets denotes spatial averaging. Within the inertial range of scales

these structure functions are assumed to exhibit scaling behavior with scaling

Chapter Eulerian intermittency

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

-15 -10 -5 0 5 10 15

PDF(δlu(x))PDF(u(x))

Gaussian PDF

Figure .: Eulerian s of the velocity field u(x) and its spatial increment δlu(x) for l = 0.8η in

Navier-Stokes turbulence, normalized to unit variance,Rλ = 316

exponent ζp:

〈|δlu|p〉 ∼ lζp . (.)

The theory predicts a linear law for these ζp (see (.)). In Section .. we

will see that there are considerable differences between measured exponents and the

prediction, especially for high p. This is not surprising because the structure

functions of higher order are dominated by the tails of the according , which

are clearly non-Gaussian as seen in Figure ..

A crucial point of the derivation of the theory is the assumption of self-

similarity. Beside the velocity increments, also the energy dissipation rate is assumed

to show scaling behavior with scaling exponent τd,⟨εdl

⟩∼ lτd . (.)

The theory predicts a scale independent energy dissipation rate ε, that means

however one chooses the size of the averaging volume, the energy dissipation rate

stays the same. Contrarily, measurements reveal the spiky character of ε (see Fig-

ure .) and therefore its intermittent character (the averaged energy dissipation

rates do depend on the considered scale l). Kolmogorov [] introduced the re-

fined self-similarity hypothesis, which allows for a scale dependent energy dissipation

Figure .: Energy dissipation in a slice from a Navier-Stokes simulation at Rλ = 316

rate in contrast to the primary theory. Hence the dissipation rate now varies

according to the size l of the averaging volume, ε = εl. Within the inertial range

the fluctuations can only depend on the energy dissipation rate and the scale l. The

viscosity does not matter. Hence from dimensional analysis the fluctuations on the

scale l are

δul ∼ (εll)13 . (.)

There are several models of intermittency, which take into account the violation of

self-similarity of the energy dissipation rate. The first model of intermittency was

introduced by Obukhov and Kolmogorov in (see Section ..). It only agrees

well with measurements in describing the scaling exponents ζp up to p ' 10.

The curved shape of the function of scaling exponents measured by Anselmet et al.

[] led to the multifractal perception of turbulence which will be introduced

in Section ... The β- and bifractal model can be seen as a precursor to the


multifractal model (see Section .. and .., respectively).

Because of its outstanding agreement with experimental and numerical results,

the She-Lévêque Model (see Section ..) is currently the most successful model

in describing the scaling laws of the Eulerian structure functions. Furthermore,

this model is the basis for a multifractal model of the Lagrangian statistics (see

Section .). Due to its relevance both for Eulerian and Lagrangian statistics it will

be introduced in detail in Section ...

. Models of intermittency

.. The Obukhov-Kolmogorov model

In Obukhov and Kolmogorov introduced the first model of intermittency (see

Frisch [], Biskamp []). They assumed a cascade process of self-similar eddy

fragmentation. Going from one scale ln to the next smaller scale ln+1, the large cube

is subdivided into eight equally sized smaller cubes, i. e. ln+1 = ln/2. The mean

dissipation of each sub-cube is generated by multiplying the parent dissipation rate

εn by a positive random variable W with mean 〈W 〉 = 1. This process is continued

until the dissipation scale η is reached. After n iterations one has 23n cubes with

edges of length ln = L2−n. In each sub-cube of size l a dissipation of

εl = εW1W2 · · ·Wn (.)

is assumed. An important assumption is that these Wi are independent and identi-

cally distributed random variables Wi. The dissipation rate εl is therefore generated

by a multiplicative random process. It follows from the central limit theorem that

the of εl reaches a log normal distribution for n 1. The scaling exponents

can be computed to

ζn =n

3− µ

n(n− 3)

18,

µ is called the intermittency parameter. For µ = 0 one ends up with self-similar

scaling. From experiments and numerical simulations one derives a value µ ' 0.2.

The ζn agree well with the experimental results up to n ' 10. Afterwards the

modeled ζn reach a maximum at n = 16 and then fall off. This fact does not agree

with the observations.


.. The β-model

The assumption of the theory and the model described in the previous section,

that the cascade of eddies is self-similar contradicts the observation of intermittency.

The cascade to smaller eddies is not space-filling. One observes active regions,

which are separated from each other by calm regions. A first model including this

lack of self-similarity of the cascade process is the β-model which is described in

the following.

Within the β-model the cascade of eddy fragmentation is assumed not to be

space-filling. Herein β is the factor by which the eddy-occupied volume decreases

going from a parent eddy to its child eddies.

Every parent eddy ln has child eddies of the size ln+1 = rln. The factor r

determines the reduction in size after a breakup of an eddy and does not depend

on n. Starting from the largest eddies of size L and going to smaller eddies one

obtains,

ln = rnL, n =ln(

lnL

)ln r

.

Because β prescribes the active fraction of the original volume after the breakup,

the probability pn of being within an active region is

pn = βn = βln( ln

L )lnr =

(lnL

) ln βln r

The space filling factor β is related to the Hausdorff dimension of the considered

regions. The probability of being within a distance l from a structure of dimension

D can be computed by counting the boxes of size l needed to cover a certain

structure. To cover the entire volume one needs (L/l)3 cubes. To cover a line one

needs L/l boxes and for a plane (L/l)2 boxes. Correspondingly for small l, the

probability of being within a distance of l of a structure of dimension D is the

fraction of the number of boxes covering the structure and the entire space, i. e.

pl ∝ l3−D, l → 0. (.)

The quantity 3−D is called the co-dimension of the structure.

An important question is the scaling behavior of the structure functions (.)

predicted by this model. For homogeneous, isotropic turbulence together with the

abbreviation u(r + l)− u(r) = ul (.) can be estimated as,

Sp(l) ∼ upl pl, (.)


pl denoting the probability of being within a distance l of an active region intro-

duced above. Here ul denotes the typical velocity difference of an eddy of size l and

does not depend on x. Therefore spatial averaging can be replaced by multiplying

ul by the probability pl.

To compute the scale dependence of the velocity ul of the active eddies one has

to rely on dimensional arguments. The energy contained in the scale l is El ∼ u2l pl,

because just the fraction pl of the entire set is active. Inserting (.) yields

El ∼ u2l

(l

L

)3−D

,

and the energy flux πl at the scale l

πl ∼El

τl

=Elul

l=

u3l

l

(l

L

)3−D

. (.)

This flux can be related to the forcing scale L by use of the usual argument that

within the inertial range the energy flux does not depend on the scale l,

πl ∼ ε ∼ E0

TL

=u2

0

TL

=u3

0

L. (.)

Equating (.) and (.) yields

ul ∼ u0

(l

L

) 13− 3−D

3

. (.)

Within the β-model the scaling exponent h of the velocity field is therefore

h = 13− 3−D

3and depends only on the co-dimension of the structures.

Inserting (.) into (.) yields the scaling behavior of the structure functions

Sp(l) ∼ up0

(l

L

)ζp

,

with

ζp =p

3+ (3−D)

(1− p

3

)As a consequence the energy spectrum shows the scaling

E ∼ k−( 53+ 3−D

3 )

For D < 3 this spectrum is steeper than the prediction.


.. The bifractal model

According to the β-model the velocity differences over a separation l scales as (see

.)

ul ∼ u0

(l

L

)h

, h =1

3− 3−D

3. (.)

The β-model allows for a single scaling exponent h depending on the space-filling

factor β. This exponent leads to a linear scaling behavior ζp = ph. However, as was

pointed out in the beginning of this chapter, measurements display a non-linear

scaling behavior.

Within the β-model the set M on which (.) holds is the entire set. A straight-

forward extension to overcome this restriction is done by introducing two sets M1

and M2 with

ul(r) ∼

(

lL

)h1, r ∈ M1, dimM1 = D1(

lL

)h2, r ∈ M2, dimM2 = D2

Using (.) the according structure functions are

Sp(l) = µ1upl p

1l + µ2u

pl p

2l

= µ1

(l

L

)ph1(

l

L

)3−D1

+ µ2

(l

L

)ph2(

l

L

)3−D2

, µ1, µ2 const.

For small l L only the term with the smaller exponent of h1 and h2 will

dominate. Thus the structure functions Sp(l) are

Sp(l) ∼ lζp , ζp = minh

(ph1 + 3−D1, ph2 + 3−D2).

If the two sets M1 and M2 have differing co-dimensions one obtains a piecewise

linear scaling function.

.. The multifractal model

The measured function of ζp is curved, but differentiable (see Figure .), therefore

the bifractal model has to be extended. The β- and bifractal model serve as the basis

for the multifractal interpretation of intermittency. Instead of having just one set

M and therefore one scaling exponent h for the β-model or two sets M1, M2 and

therefore two scaling exponents h1 and h2 the multifractal model allows for a whole


range of sets and according scaling exponents h in an interval I = (hmin, hmax).

The entire volume is now split into several Mh each with scaling behavior,

ul(r)

u0

∼(

l

L

)h

, r ∈ Mh. (.)

The structure functions are in analogy to the bifractal model

Sp(l)

up0

=

∫I

(l

L

)ph+3−D(h)

dµ(h). (.)

As in the β-model (.) the co-dimension of the considered structures 3−D(h)

enters this relation and accounts for the probability of observing a set with the

scaling exponent h. Similarly to the bifractal model µ(h) is the weight of the scaling

exponents. Again, for l → 0 the smallest h dominates and determines the scaling

behaviorSp(l)

up0

∼(

l

L

)ζp

,

with

ζp = infh

(ph + 3−D(h)) . (.)

The equation (.) constitutes a Legendre transformation between the function ζ

depending on the variable p and the so-called singularity spectrum D depending on

scaling exponent h. The infimum of ζ(h(p)) = infh(ph + 3−D(h)) is exactly h∗

with D′(h∗(p)) = p, because ∂hζp|h=h∗ = ∂h(ph + 3−D(h))|h=h∗ = 0.

Within the multifractal framework one can compute the dissipation scale. The

starting point is the turn-over time τl of an eddy of size l,

δlu =l

τl

. (.)

In words: The typical velocity difference of an eddy is given by its size l divided

by its turn-over time τl. Or in other words: If one measures a velocity difference

ul over a distance l, a tracer would need a time τ to pass this distance. At the

dissipation scale the turn-over time (.) equals the viscous time scale l2/ν, which

corresponds to the viscous term of the Navier-Stokes equations (.). Equating

these two time scales yields

η ∼ ν

δηu. (.)

In the multifractal framework the velocity differences δlu obey the scaling relation

(.) and therefore

η =

(νLh

u0

) 11+h

. (.)


With the Reynolds number R (.) this reads (.),

η = LR− 11+h . (.)

The corresponding multifractal dissipation time scale can be found starting again

from relation (.) on dissipation scale η,

τη ∼η

δηu.

Inserting the multifractal scaling relation (.) and afterwards the expression for

the dissipation scale (.) yields

τη = ν1−h1+h L2h(1+h)u

2(1+h)0 .

The multifractal model is a description of intermittency. However, in order to

compute specific scaling exponents one has to know the according singularity

spectrum. This procedure is applied in Section . for the Lagrangian scaling

exponents. The next section presents a model in the spirit of the multifractal ansatz,

which is in good agreement with measurements.

.. The She-Lévêque model

The She and Lévêque [] model is again a phenomenological model like all

models presented above, that means instead of starting from the underlying Navier-

Stokes equations, it is based on physical intuition.

>From the assumption of refined self-similarity (.) follows that the fluctuations

δul show the same scaling behavior as (εl)13 . Inserting the scaling laws (.) and

(.) implies the relationp

3+ τ p

3= ζp

for the scaling exponents. Within the model structures are characterized by their

degree of singularity. The intensity of a structure of p-th order is

ε(p)l =

⟨εp+1l

⟩〈εp

l 〉. (.)

The greater p is, the more the singular structures contribute to ε(p)l .

Now lets look at the scaling behavior of these ε(p)l . The intensity of the lowest

order ε(0) = 〈εl〉 is scale independent. This is the mean energy dissipation rate. In


contrast, ε(∞) is divergent, because the peaks of ε (see Figure .) for l → 0 will not

be smeared out and therefore give huge contributions. ε(∞)l is the intensity of the

most singular structures averaged on the scale l. From dimensional analysis follows

ε(∞)l ∼ δE∞

τl

.

She and Lévêque choose to have normal temporal scaling, that means there is one

time scale for all different intensities,

τl ∼ ε13 l

23 .

Furthermore, they argue that the intermittency shows up through the consumed

energy δE∞ by the nearly singular structures. In Navier-Stokes flows these struc-

tures have filamentary character. If a strong filament acts on the surrounding

macroscopic fluid it dissipates the velocity component along its axis of symmetry.

Hence, it dissipates energy of the order of u20. This implies

ε(∞) ∼ ε

(l

L

)− 23

∼ l−23 . (.)

Together with (.) one finds for p →∞

τp+1 − τp ∼ −2

3.

This is a recursion equation for τp which can be solved by regarding it as a differen-

tiation. It follows

τp = −2

3p + C0 + f(p), with f(∞) = 0. (.)

For p →∞ one picks out the most singular structures. Then (.) simplifies to

τp = −23p + C0. Looking at the Legendre transformation (.) it becomes clear

that the constant C0 is the co-dimension and hmin = 2/3 is the scaling exponent

of the energy dissipation of the most singular structures.

Another important assumption of the model is the cascade picture: Structures

ε(p+1)l arise from weaker structures ε

(p)l and end up finally as singular structures

ε(∞)l . It follows that a structure ε

(p+1)l can be interpolated as

ε(p+1)l = Apε

(p)β

l ε(∞)(1−β)

l , 0 ≤ β ≤ 1.


Inserting (.) and (.) one ends up with a homogeneous difference equation

for f(p) in (.),

f(p + 2)− (1− β)f(p + 1) + βf(p) = 0

With the ansatz f(p) = αrp two solutions r1 = β and r2 = 1 are found. The

boundary condition f(∞) forbids the trivial solution r2. Further boundary condi-

tions on f(p) are

• ζ3 = 1 = 1 + τ1 ⇒ τ1 = 0

• The support of the dissipative structures is compact, 〈ε0l 〉 = const = lτ0 ⇒

τ0 = 0

This two conditions imply α = −C0 and β = C0−2/3C0

.

Finally, one obtains the scaling exponents within the She-Lévêque model

τd = −2

3d + C0

(1−

(C0 − 2

3

C0

)d)

, (.)

ζp =p

9+ C0

(1−

(C0 − 2

3

C0

) p3

). (.)

Now one can insert the co-dimension C0 of the most singular structures to obtain

the specific scaling exponents. It is worth stressing that this model contains no free

parameter once the co-dimension is fixed.

Navier-Stokes In Navier-Stokes turbulence the most singular structures are vortex

filaments (see Figure .). Accordingly the co-dimension is 2 and

ζp =p

9+ 2

(1−

(2

3

) p3

). (.)

The resulting scaling exponents are summarized in Table . up to order 10 and

shown in Figure . together with measurements and the prediction.

First one observes, that the exact linear relation of the third order structure

function is reproduced. The deviations of the low-order exponents from the

prediction are small. Significant differences do not arise until the fifth order. The

prediction of the model is in good agreement with numerically measured exponents

(see Section .. for details) and experiments by She and Lévêque [], Anselmet

et al. [].


Figure .: The most dissipative structures in Navier-Stokes turbulence, isosurfaces of the vorticity,

Reynolds number Rλ = 178

Within the multifractal framework one can compute the singularity spectrum

D(h) corresponding to the scaling relation (.). Performing a Legendre transfor-

mation yields

D(h) = infp

(ph + 3− ζ(p))

= 1 + p∗(h)

(1− 1

9

)+ 2

(2

3

) p∗(h)3

, with

p∗(h) =3

ln(

23

) ln

(1− 9h

6 ln(

23

)) .

(.)


0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7 8

ζ p

p

K41She-Leveque

modified She-Leveque (MHD)Navier-Stokes Rλ=316

MHD Rλ=234

Figure .: Eulerian scaling exponents ζp: (.), She-Lévêque (.), Navier-Stokes measure-

ments () Table ., modified She-Lévêque () (.), measurements () Table .

p

ζp . . . . . . . . . .

Table .: Eulerian scaling exponents ζp obtained from the She and Lévêque [] model

MHD turbulence The She-Lévêque model was translated for magnetohydro-

dynamic flows by Grauer et al. [], Horbury and Balogh [], Mueller and

Biskamp []. While the former based their model on the Iroshnikov-Kraichnan

cascade (see Section .) the latter assumed a cascade of Kolmogorov type. Nu-

merical measurements done by Biskamp and Mueller [] seem to verify the

assumption of a Kolmogorov cascade. In this case the presented She-Lévêque model

only has to take into account the specific co-dimension of the most singular struc-

tures of flows. These structures are current and vortex sheets (see Figure .).

Accordingly, the co-dimension C0 is 1, hence

ζp =p

9+ 1−

(1

3

) p3

. (.)


Figure .: The most dissipative structures in turbulence, isosurfaces of the vorticity (red) and

current density (blue), Reynolds number Rλ = 107

p

ζp . . . . . . . . . .

Table .: Eulerian scaling exponents ζp for flows from the modified She-Lévêque model by

Horbury and Balogh [], Mueller and Biskamp []

The resulting scaling exponents are summarized in Table . up to p = 10 and

shown in Figure . together with the She-Lévêque prediction for Navier-Stokes

flows and experimental data. Comparing these exponents to their hydrodynamic

counterparts (see Table .) one observes that turbulence is more intermittent

than Navier-Stokes turbulence. This is because the singular structures have a higher

dimensionality in and therefore a higher weight.


.. Probability density functions

Intermittency is related to the frequent occurring of extreme events. A measure for

the frequency of the occurrence is the probability density function () of the

considered random variable.

This section only focuses on the s emerging in Navier-Stokes turbulence,

because the situation in turbulence is similar. Therefore all statements made

so far remain true if the underlying hydrodynamic variable is exchanged by the

corresponding variable.

The P (δu, l) ≡ P (δlu) describes the probability of observing a velocity

increment δu over a distance l. It depends on two variables. The central limit

theorem states that if a statistical process is uncorrelated, i. e. every measurement is

independent of the previous measurements, the distribution of the corresponding

random variable displays a Gaussian distribution. Contrarily, if the measurements

are correlated the associated will deviate from a Gaussian distribution.

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

-15 -10 -5 0 5 10 15

P(δ

l u)

δl u/σδl u

l=0.8 ηl=3.2 ηl=40 η

l=400 ηgauss

Figure .: Eulerian s of the spatial velocity increment δτu for several separations l in Navier-

Stokes turbulence, Rλ = 316, shifted for clarity, normalized to unit variance

As we have seen in the beginning of this chapter the velocity field itself has a

Gaussian distribution function, while the of the velocity increments displays


stretched tails. To be more precise Figure . shows s of the velocity increment

in a Navier-Stokes flow for several separations l. From top to bottom the separation

increases. The of the largest spatial distance is a Gaussian distribution. With

decreasing separation the tails get more and more stretched. It should be stressed

that the plot is given in a logarithmic scale. Even comparing quite a small increment

of δlu = 5, i. e. five standard deviations, for the smallest and largest separation

l displayed in the plot, the corresponding probability changes from 1 · 10−3 to

5 · 10−6. The probability of observing such an event decreases by nearly three orders

of magnitude. This difference becomes even larger for larger separations. Extreme

events situated in the tails of the s are negligible for Gaussian processes while

they occur much more often for intermittent variables.

The reason for the intermittent distribution of the velocity increments at small

scales are the coherent structures formed and sustained by the turbulent flow. As

we have seen from Figure . at small scales the flow is highly structured into vortex

filaments. Therefore, nearby points in space are highly correlated. These small

structures preferably enter the s for separations of the order of the diameter of

the structure. Since the vorticity is the curl of the velocity, the velocity changes

rapidly when crossing a vortex filament. The change in the velocity is maximal if

the considered separation l is as large as the diameter of the vortex tube. These

filaments have a diameter of the order of the Kolmogorov length, which is the

smallest turbulent scale of the flow. Therefore, the most intermittent emerges

for small increments.

Increasing the separation scale points entering the increment become more and

more uncorrelated. The probability that they belong to the same coherent structure

decreases. Due to the interaction of vortices the spatial correlation approaches zero

only at the largest scale, the integral scale. Measuring the increment over these

large separations correspond to randomly choosing points from regions in the

flow which are independent. Now the central limit theorem holds and predicts a

Gaussian distribution, which is indeed observed in Figure . for separations l of

the order of the integral scale.

Another measure of intermittency are the moments of the s, called structure

functions. These will be presented in detail in the following section.


.. Structure functions

In this section the scaling behavior of hydrodynamic turbulence will be measured

using numerical simulations and compared to the predictions made in Section ..

These measurements might be seen as a benchmark of the numerics.

As we have seen in the preceding sections a statistical quantity of great interest

for building up a theory describing intermittency are the structure functions of

the velocity increment. In general a structure function of order p of the random

variable f(l) which has a probability density distribution P (f, l) is the pth moment

of the ,

Sp(l) = 〈f(l)p〉 =

∫|f(l)|pP (f, l)df. (.)

In turbulence a random variable is the longitudinal velocity increment

δ~u~l(~x) ≡ δ~u(~x, l) ≡ (~u(~x +~l)− ~u(~x))) · ~l

over a distance . It fluctuates in space. In the case of isotropic turbulence the corre-

sponding P (δu, l) depends on the velocity increment δu and the separation l.

The structure functions (.) of the longitudinal velocity increment are

Sp(l) = 〈δlu(~x)p〉 ≡∫|δlu|pP (δu, l))dδu. (.)

Accordingly, the velocity structure functions are the moments of the s of the

velocity increments presented in the previous section. As we have seen they exhibit

stretched tails for small separations l and therefore express intermittency.

Structure functions are also a measure of intermittency, because increasing the

order p is connected to focus more and more on the extreme events situated in the

tails of the according s. To illustrate this fact, Figure . shows the product of

the pth order of u and the P (a). The chosen has strong stretched tails

and corresponds to the acceleration of a tracer (see Section ..). The product

enters the integral of .. The larger the value of the product is the larger is

the contribution to the structure function. From Figure . it becomes clear that

structure functions of small orders of p are unaffected by the tails of the . The

main contribution stems from the core. The situation changes when considering

larger orders of p, for example p = 6. Now the main contribution to the structure

function stems from the tail of the . Although they occur quite seldom, they

determine the shape of the structure function. From this argumentation it is clear


1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1

0 10 20 30 40 50a/σa

pdfp=2p=4p=6p=8

Figure .: Contributions to structure functions from an intermittent ; solid line: of the

acceleration a of a tracer, normalized to unit variance σa; discontinuous lines: product of the

and ap for several orders p

that understanding intermittency is directly connected to the understanding of the

occurrences of extreme events in turbulent flows.

The structure functions are indeed connected to the coherent structures emerging

in turbulence. As we have seen in the previous section the stretched tails originate

from small scale structures. Contrarily, in self-similar flows the s are identically

Gaussian s for all spatial separations l. Due to its self-similarity structures look

the same on all spatial scales. Therefore the structures contributing mainly to the

structure functions are similar on all spatial scales. Taking high orders of p does

not result in selecting specific structures. This is not true in intermittent flows.

Because of the stretched tails the product entering the structure function is sensitive

to the order p. For a given order p a specific region of the gives the main

contribution. Associated to this region is a structure so that the order p picks out a

specific structure. For high orders only the most intense structures contribute.


... Navier-Stokes

In Navier-Stokes turbulence one conveniently focuses on the scaling behavior of

the velocity increments.⟨|δ~l~u|

p⟩

=⟨|(~u(~x +~l)− ~u(~x)) · ~l|p

⟩∼ lζp . (.)

The corresponding ideal invariant is the kinetic energy which cascades from large

to small spatial scales. The most intense velocity differences at small scales originate

from regions of high vorticity. As can be seen in Figure . these regions have a

filamentary character and are separated from each other. Contrarily, at large scales

the big eddies are space filling. Therefore the velocity field is not self-similar.

1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1

10 100 1000

Sp

l/η

p=1p=2p=3p=4p=5p=6p=7p=8 0

0.002 0.004 0.006 0.008

0.01 0.012 0.014 0.016 0.018

1 10 100 1000

S3/

l

Figure .: Eulerian velocity structure functions for a Navier-Stokes flow at Reynolds number

Rλ = 316, inset: compensated third order structure function

Different structures contribute to compute the structure functions depending

on the considered order p and distance l. Indeed, the structure functions show

intermittent behavior, called anomalous scaling.

Figure . shows structures functions for a Navier-Stokes flow for several orders

of p. At first sight only a tiny region of each structure function appears to obey a

scaling law. The influence of the forcing and dissipation is significant on large and

small scales, respectively. However, in between a scaling region seems to exist.


The 4/5th law prescribes a linear scaling behavior for the third order structure

function. The inset of Figure . shows the compensated function. One clearly

observes a range of scales where this function is flat and therefore follows the 4/5th

law.

To determine the scaling exponents of other orders p it is reasonable to look at

the logarithmic derivative of the structure functions (see Figure .). The charac-

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

10 100 1000

d lo

g(S

p) /

d lo

g(l)

l/η

p=1p=2p=3p=4p=5p=6p=7p=8

Figure .: Logarithmic derivative of Eulerian structure functions in a Navier-Stokes flow, Rλ = 316

teristic shape can be explained as follows: For small spatial separations within the

dissipation range the flow is differentiable (see Section ..). The scaling exponents

are p. Entering the inertial range of scales the flow becomes rough and follows a

different scaling law. The deviation of this law from the prediction expresses

intermittency. While approaching the large scales the scaling range ends because

the turbulent flow is determined by the forcing. The velocity differences saturate

and the according logarithmic derivatives tend to zero.

All plotted functions show a plateau. The y-value of this plateau is the scaling

exponent of the corresponding structure function and can be read from the ordinate

axis. The plateau of the third order function has an exponent of unity in good

agreement with the 4/5th law. The range of this plateau can be called inertial range


of scales as pointed out at the end of Section ... It extends approximately from

60η to 250η (indicated by the vertical bars). This shows that even for the highest

achievable Reynolds number of this work the inertial range of scales is quite narrow.

In Section . the dependence of the size of this range on the Reynolds number

will be discussed in detail.

The absolute scaling exponents listed in Table . are computed by the follow-

ing procedure. As mentioned before, the logarithmic derivatives of the structure

functions (see Figure .) have a characteristic shape. The plateau lies between

local extrema of the second derivative. This interval of the third order function is

assumed to be the scaling range. The corresponding scaling exponent is the mean

value within this interval. The error is given by the maximum deviation from the

mean value.

The computed absolute values are in good agreement with the She-Lévêque

prediction . and experiments (see e. g. Anselmet et al. []). The possible

errors are quite large. There is a way to extract the exponents more precisely. This

procedure is called extended self-similarity ().

Extended self-similarity (ESS) The idea is not to look at the absolute scaling of

the structure functions, but to compute their relative scaling behavior. This idea has

been introduced by Benzi et al. []. And in fact the relative structure functions

display scaling behavior over a much larger range than each of them does individu-

ally. Because of the 4/5th law the third order function is conveniently taken as the

reference function. Figure . shows structure functions of several order plotted

against the third order function. These relative functions are linear functions over a

broad range of scales. The inset shows the corresponding logarithmic derivatives

which are nearly constant. The relative scaling exponent is given by this constant.

The procedure of computing the exponents is the following. The inertial

range is assumed to be the range where the compensated third order structure

function (see inset of .) stays above 90% of its maximum value. Within this

range a straight line is fitted to the functions by the linear regression procedure.

The slope of the straight line gives the scaling exponent. The error is the standard

deviation of several realizations. These correspond to different points in time each

separated by a large eddy turn-over time.

The extracted scaling exponents are listed in Table .. The values are in very

good agreement with the absolute values.


1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1

1e-05 1e-04 0.001 0.01

Sp

S3

p=1p=2p=4p=5p=6p=7p=8 0

0.5

1

1.5

2

2.5

Sp

Figure .: Relative Eulerian structure functions () in a Navier-Stokes flow, inset: logarithmic

derivative

p ζp (absolute values) ζp/ζ3 (using )

1 0.37± 0.04 0.363± 1.4 · 10−3

2 0.71± 0.07 0.696± 1.3 · 10−3

3 1.02± 0.10 1

4 1.30± 0.13 1.276± 2.5 · 10−3

5 1.56± 0.15 1.526± 6.2 · 10−3

6 1.79± 0.17 1.751± 1.2 · 10−2

7 2.00± 0.18 1.954± 2.0 · 10−2

8 2.19± 0.20 2.136± 4.2 · 10−2

Table .: Eulerian scaling exponents ζp of Navier-Stokes turbulence, Rλ = 316

If a clear scaling range exists the absolute scaling exponents can be computed

directly. The advantage of the procedure is significant if an explicit scaling

region is absent due to a small Reynolds number. Then one has to rely on the

assumption of to get any information about the suspected scaling behavior in

an inertial range.


1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

10 100

S(Z

+) p

l/η

0 0.005 0.01

0.015 0.02

0.025 0.03

0.035

1 10 100

S(Z

+) 3

/l

(a) orders p = 1–8 from top to bottom, inset:

compensated third order structure function

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

10 100

d lo

g(S

(Z+) p

) / d

log(

l)

l/η

p=1p=2p=3p=4p=5p=6p=7p=8

(b) logarithmic derivative

Figure .: Eulerian structure functions of z+ for an flow at Reynolds number Rλ = 234

... MHD

In magnetohydrodynamic flows the cascading quantities are the Elsässer variables

~z±, as can be seen from the symmetry of the equations written in terms

of these variables (.). Therefore, Eulerian models of intermittency deal with

structure functions of increments of the Elsässer variables,⟨δ~l~z

±p⟩=⟨|(~z±(~x +~l)− ~z±(~x)) · ~l|p

⟩∼ lζp . (.)

These structure functions are shown in Figure .(a) for an flow. A scaling

region is hardly observable. As is pointed out in sections . and .. one assumes a

linear scaling law for the third order structure function as in the Navier-Stokes case.

The inset of Figure .(a) shows the compensated third order structure function.

A region in which this function stays constant cannot be observed.

To have a closer look at the scaling behavior of the structure functions in Fig-

ure .(b) the logarithmic derivatives are given for several orders of p. As already

indicated by the compensated function, no clear plateau is observable. The fact that

no scaling region is visible can be attributed to the relatively low Reynolds number.

However, this Reynolds number is at present nearly the highest one achievable by

numerical simulations.

Unfortunately, due to the lack of a plateau in Figure .(b) absolute scaling

exponents cannot be extracted. Because of this fact one conveniently applies


to flows to extract relative scaling exponents (see e. g. Biskamp and Mueller

[]). As the third order structure function is expected to scale linearly the

function of order p is plotted against this function. A diagram displaying the

relative structure functions is shown in Figure .. Now the situation gets better.

1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1

1e-05 1e-04 0.001 0.01

S(Z

+) p

S3

p=1p=2p=4p=5p=6p=7p=8

0

0.5

1

1.5

2

2.5

Figure .: Relative structure functions () of ~z± for an flow at Reynolds number Rλ = 234,

shifted for clarity, inset: logarithmic derivative

The straight lines are the scaling laws from the She-Lévêque model for flows

(see Table .). They fit the measured exponents over a large range of scales.

A more precise conclusion can be drawn from the inset of Figure .. There

the relative logarithmic slopes are shown. Although the curves are not constant

but curved especially for high orders of p, a good agreement with the prediction is

observed.

To extract the relative scaling exponents, the same procedure is applied as in the

Navier-Stokes case. Again the inertial range is defined by the assumption that the

third order structure function obeys a linear scaling law. The relative slopes are

extracted from this region. The measured exponents are given in Table .. They

are in good agreement with the prediction (see Table .).


p ζp/ζ3 . . . . . . .

∆(ζp/ζ3) . . . . . . .

Table .: Eulerian scaling exponents ζp of turbulence, Rλ = 234

It is worth stressing that the scaling exponents of turbulence indeed display

a more intermittent behavior than the corresponding exponents in Navier-Stokes

turbulence.

Chapter

Lagrangian intermittency

The previous section investigated the phenomenon of intermittency in Eulerian

coordinates. The velocity differences δlu were taken between points in space sepa-

rated by a distance l. In Section . another set of coordinates have been introduced.

These Lagrangian coordinates follow the trajectories of fluid elements. Apart from

measuring velocity differences with respect to spatial separations is it interesting to

measure velocity increments along a tracer trajectory to have a second access to the

problem of intermittency. Several implications arising from the use of Lagrangian

coordinates will be discussed in this chapter.

Intermittency also shows up in the Lagrangian framework. Figure . shows the

x-component of the acceleration and velocity in Navier-Stokes turbulence. The

strong fluctuations correspond to a helical trajectory (see Figure .(a)). The inset

shows the according s of the acceleration and velocity. The of the velocity

has a Gaussian shape and the acceleration exhibits strong non-Gaussian tails.

Comparing the of the Lagrangian acceleration (see Figure .) and the

Eulerian velocity gradient (see Figure .) one immediately recognizes that the

tails of the former s are more stretched than their Eulerian counterparts. The

temporal intermittency measured in terms of tracer particles is more pronounced

than the spatial intermittency measured on a fixed grid.

This is a first indicator that Lagrangian intermittency differs from Eulerian

intermittency (we will see several others in the following sections).

Building up a theory of Lagrangian turbulence is of great importance for several

reasons. First from the theoretical point of view. Tackling a problem (like intermit-

tency in turbulent flows) from two differing directions (like the Lagrangian and

Hamiltonian formalism in classical mechanics) is often fruitful to obtain a more

Chapter Lagrangian intermittency

-5

-4

-3

-2

-1

0

1

2

3

4

0 0.5 1 1.5 2 2.5 3t/TL

axux

1e-07 1e-06 1e-05 1e-04 0.001

0.01 0.1

1

-30 -20 -10 0 10 20 30a/σa, u/σu

P(u)P(a)

Figure .: x component of the velocity (green) and acceleration (red) in Navier-Stokes turbulence,

inset: of the velocity (green) and acceleration (red)

complete understanding of the problem. A very promising approach based on a

Markovian closure of a hierarchy of n-point mixed Lagrangian-Eulerian probability

density functions was recently introduced by Friedrich []. This closure takes

explicit advantage of a Lagrangian description.

Second, the Lagrangian point of view is the particle point of view. In order to

describe dispersion and diffusion of particles in turbulent flows it is generic to use

Lagrangian coordinates (see e. g. Yeung and Borgas [], Ouellette et al. []).

Furthermore, the Lagrangian point of view can be seen as a limiting case of

the motion of small passive particles exposed to a turbulent flow. Real particles

might have a density different from the surrounding fluid. Then the motion of the

particles and the fluid decouple due to the effect of inertia. Bec et al. [] showed

that the smaller this effect of inertia is the more exactly the particles behave like

Lagrangian tracers. Lagrangian intermittency is connected to extreme events such

as strong accelerations along a particle path. The statistics of small impurities and

therefore intermittency in turbulent flows are of great interest for environmental

problems like the formation of rain droplets (see Falkovich et al. []) in warm

clouds and for engineering problems like spray combustion in Diesel engines (see

Post and Abraham []).


Experimental measurements of the statistical properties of Lagrangian turbulence

are laborious. The trajectory of small passive particles exposed to fully developed

turbulence have to be recorded precisely. Recently, two experiments investigated the

Lagrangian scaling properties. In Cornell measurements were done using optical

techniques by Voth et al. [] and in Lyon measurements using acoustical

techniques by Mordant et al. []. Apart from these experiments numerical

simulations have been performed by Biferale et al. [b]. The results concerning

the scaling behavior differ from the experiments. The Lagrangian scaling behavior

is still under discussion. In this chapter the differences in the scaling exponents

between the experiments and the numerical simulations will be explained in detail.

To describe the anomalous scaling behavior observed in the Lagrangian frame-

work Biferale et al. [a] proposed a model based on the She-Lévêque model

presented in Section ... Unfortunately, the predicted scaling exponents differ

from the measured ones. To understand the deviation it is fruitful to investigate the

scaling behavior of Lagrangian turbulence. The scaling behavior of Lagrangian

turbulence has not been analyzed up to now. The comparison of neutral and

conducting flows yields an explanation of the failure of the model by Biferale et al.

[a]. The failure and the explanation will be discussed in detail in this chapter.

In the following section the model by Biferale et al. [a] will be presented.

After that a translation to the case of a conducting fluid will be proposed.


As in the Eulerian case one can quantify the degree of intermittency by looking

at the scaling behavior of structure functions. The Lagrangian point of view deals

with the statistics of physical quantities along the trajectory of tracer particles. The

structure functions therefore depend on a time lag instead of a spatial separation as

in the Eulerian case. The Lagrangian velocity increment

δτ~v = ~v( ~X(~y, t + τ), t + τ)− ~v(~y, t) (.)

written abbreviated as

δτv = v(t + τ)− v(t)

yields the according Lagrangian structure functions,

Sp(τ) = 〈|δτv|p〉 , (.)


angular brackets denoting temporal averaging. They are believed to show scaling

behavior as their Eulerian counterparts. In an inertial time range, spanning from

times much longer than the dissipation time scale (.) to time scales much shorter

than the integral time scale (.) one would therefore expect

Sp(τ) ∼ τ ζLp . (.)

The Lagrangian scaling exponents are denoted by ζLp .

To build up a theory for this Lagrangian scaling exponents, it seem natural to

translate existing Eulerian models into the Lagrangian description.

A first attempt to relate Eulerian space-based statistics to Lagrangian time-based

statistics can be made by the relation (.). Inserting this relation into the

scaling prediction (.) yields the Lagrangian scaling relation

ζLp =

p

2. (.)

This relation is based on the assumptions, especially on self-similarity. The

linearity of the scaling function is not found in measurements as we will see in

Section ... As in the Eulerian case the function ζLp is not linear but curved due

to intermittency.

In the Eulerian framework the She-Lévêque model (see Section ..) together

with the multifractal interpretation was very successful in predicting the anomalous

scaling exponents observed in measurements. For Navier-Stokes turbulence there

is a model by Biferale et al. [a] predicting the ζLp and taking intermittency

into account by using the She-Lévêque model. This model will be presented in the

following section.

. Multifractal Navier-Stokes turbulence

The basis of the multifractal model of Lagrangian turbulence proposed by Biferale

et al. [a] is the successful scaling relation ζ(p) from the She-Lévêque model

(.) and the according singularity spectrum D(h) (.).

The proposed model assumes that the Eulerian velocity fluctuations δlu are of

the same order of magnitude as the Lagrangian velocity fluctuations δτv,

δlu ∼ δτv, (.)


p

ζLp . . . . . . . . .

Table .: Lagrangian scaling exponents ζLp obtained from the multifractal model for Navier-Stokes

turbulence by Biferale et al. [a]

with l and τ related by .. Now it is possible to set up a multifractal description

as follows. Fromδlu

u0

∼(

l

L

)h

,

together with (.) and (.) one obtains a multifractal time scale

τ ∼ Lh

u0

l1−h.

Solving this for l and inserting the resulting term into (.) yields the according

multifractal structure function within the Lagrangian framework,

Sp(τ)

vp0

=

∫I

(τ

TL

) ph+3−D(h)1−h

dµ(h). (.)

Again, for τ → 0 only the smallest exponent,

ζLp = inf

h

hp + 3−D(h)

1− h,

contributes to the integral in (.). The singularity spectrum of the She-Lévêque

model yields the scaling exponents summarized in Table .. These values will be

compared to numerical measurements in Section ...

The presented multifractal ansatz possesses a certain monotonicity property. This

means that if there are two different sets of structure function exponents and one

of these is more intermittent than the other in the Eulerian picture, then this one

is also more intermittent in the Lagrangian framework. To see this it is sufficient

to look at structure functions of high order p. One observes that the value of h∗

where the infimum of

hp + 3−D(h)

is assumed goes to hmin (the minimal multifractal scaling exponent) for high values

of p. Thus the asymptotic behavior reads

ζLp = hminp + 3−D(hmin) , p 1 .


For the saddle point evaluation of the Lagrangian structure functions (see (.))

one has to find the infimum ofhp + 3−D(h)

1− h

so that the asymptotic behavior is given by

ζLp =

hminp + 3−D(hmin)

1− hmin, p 1 .

Since both in Navier-Stokes and the value of hmin = 1/9 is identical, the

degree of intermittency is determined by D(hmin). This is valid both for the

Eulerian as well as for the Lagrangian model which guarantees the monotonicity

property.

.. Acceleration statistics

In the framework of the previously introduced multifractal model Biferale et al.

[a] derived an expression for the probability density function of the accelera-

tion of fluid tracers.

The acceleration aη at the dissipation scale is given by

aη =δv

τη

,

which is the velocity change after a time lag of the order of the Kolmogorov time

scale τη divided by this time lag. Using (.) yields in terms of spatial velocity

differences

aη ∼δru

τη

.

With the multifractal scaling relation (.) for δru and the expression for the

Kolmogorov time scale (.) one obtains

aη =νu0

Lhηh−2.

After inserting the multifractal dissipation scale (.) one ends up with the

multifractal acceleration on the dissipation scale

aη = ν2h−11+h L− 3h

1+h u3

1+h

0 . (.)

Now we turn to the of P (aη). The probability of measuring an acceleration a

between a1 and a2 is ∫ a2

a1

P (a)da.


Measurements show that the velocity field u has a Gaussian density distribution

function (see e. g. Figure .). This observation is an important ingredient to

derive the acceleration statistics. In general, a variable transformation implies for

differential forms

P (u)du = P (u(a))du

dada. (.)

The acceleration can now be derived by assuming a Gaussian distribution,

P (u0) =1√

2πσ2u0

exp

(− u2

0

2σu20

), σ2

u0=⟨u2

0

⟩,

for the large scale velocity field u0.

In the multifractal framework quantities like the dissipation time scale include

a scaling exponent h, which varies. Therefore, one has to take into account the

probability of observing a certain value of h. This probability is in analogy to (.)

given by

P (h) =

(τη(h, u0)

TL(u0)

) 3−D(h)1−h

in the Lagrangian case. The acceleration is therefore

P (aη)daη =

∫h

P (u0(aη))P (h)du0

daη

dhdaη (.)

In order to use (.) one needs u0(aη). Solving (.) for u0 yields

u0(aη) = ν1−2h

3 Lha1+h3

η ,du0

daη

=1 + h

3ν

1−2h3 Lha

h−23

η

Inserting this into (.) and integration over u0 gives the acceleration ,

P (aη) ∼∫

h

dha(h−5+D(h))/3η ν(7−2h−2D(h))/3LD(h)+h−3σ−1

u0

× exp

(−a

2(1+h)/3η ν2(1−2h)/3L2h

2σ2u0

).

This can be written in terms of the Reynolds number 〈a2〉 ∝ Rχλ, where

χ = suph2(D(h)− 4h− 1)/(1− h),

P (a) ∼∫ hmax

hmin

dh a((h−5+D(h))/3)Ry(h)λ

× exp

(−1

2a2(1+h)/3R

z(h)λ

),


with a = a/σa, σa = 〈a2〉1/2, y(h) = χ(h− 5 + D(h))/6 + 2(2D(h) + 2h− 7)/3

and z(h) = χ(1+h)/3+4(2h−1)/3. These results will be compared to numerical

measurements in Section ...

It is worth mentioning that the multifractal prediction contains three parameters.

The first parameter is hidden in the relation 〈a2〉 ∝ Rχλ when normalizing the

width of the . However, this parameter can be determined from one simulation

and should then be kept fixed for other Reynolds numbers. Accordingly to the

She-Lévêque model the value of hmin is given by hmin = 1/9. Biferale et al. [a]

use a different value to obtain a good agreement to their numerical data. The last

parameter is a free amplitude in the normalization.

.. Multifractal turbulence

One can also apply the model presented in the previous section to turbulence.

A reasonable singularity spectrum in this case is the spectrum resulting from the

scaling exponents (.). Applying the same procedure presented in the previous

section one ends up with the Lagrangian multifractal scaling exponents listed

in Table .. These values and the resulting s will be compared to numerical

p

ζLp . . . . . . . . .

Table .: Lagrangian scaling exponents ζLp for , obtained from the modified multifractal

model

measurements in Section .. and Section .., respectively.

. Probability density functions

As in the Eulerian case it is instructive to look at the s of the random variables. In

the Lagrangian framework temporal changes in variables are under consideration.

.. Navier-Stokes

In Navier-Stokes flows an interesting variable is the temporal velocity increment. In

Eulerian coordinates one conveniently considers longitudinal velocity increments.

The notion of longitudinal becomes less clear when switching to the Lagrangian


point of view. Therefore experimental and numerical measurements often consider

velocity increments projected onto a fixed axis ~e (see e. g. Mordant et al. [],

Voth et al. []),

δτv = (~v(t + τ)− ~v(t)) · ~e. (.)

In this work the increment is projected onto the three coordinate axes. As the flow

is isotropic the statistical result does not depend on the chosen axis. To increase the

data set all projections are incorporated into a single .

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

-40 -20 0 20 40

P(δ

τ v)

δτ v/σδτ v

accelerationτ=0.25 τητ=1.25 τη

τ=5 τητ=75 τη

gauss

(a) time lag τ increases from top to bottom,

curves are shifted for clarity, red solid line: tracer

acceleration, black dashed line: Gaussian distri-

bution

101

10-2

10-5

10-8

0 10 20 30 40 50

P(a

)

a/σa

0 0.2 0.4 0.6 0.8

1 1.2 1.4

0 10 20 30 40 50

a4 P(a

)

a/σa

(b) tracer acceleration for Rλ = 122, hmin =0.175 (x) and Rλ = 178, hmin = 0.16, (+) to-

gether with the multifractal prediction, inset:

multiplied by a4

Figure .: Lagrangian s of the temporal velocity increment δτv, normalized to unit variance,

shifted for clarity

Figure .(a) shows s of the increment (.) in hydrodynamic turbulence.

Similar to the Eulerian coordinates the shape of the s differs accordingly to the

size of the time lag τ . For small time lags of the order of the dissipation time scale the

exhibit strong stretched tails. Considering small temporal velocity increments

is identical to considering the tracer acceleration as can be seen in Figure .(a).

Clearly the acceleration is a highly intermittent variable. The question arises which

type of tracer motion is responsible for these huge accelerations. Figure .(a) shows

how a tracer trajectory looks when undergoing extreme accelerations. Isosurfaces of

the vorticity are shown in red. Several filament-like structure are observable. The

isosurfaces given in this figure correspond to a snapshot of the vorticity field at a

fixed point in time. The points of the trajectory belong to different points in time.


The tracer is strongly influenced by a filament. A movie would display that the

tracer is trapped in this filament while it bends and moves. Figure .(b) shows the

corresponding y-component of the velocity of the tracer. The high accelerations

correspond to helical motion in a vortex filament, where velocity fluctuations of

many urms occur. Accordingly the stretched tails of the s are generated by

trapping events of tracers in strong vortical structures. As in Eulerian coordinates

the coherent structures have a deep impact on the Lagrangian statistics of tracers.

(a) isosurface of the vorticity (red) at a fixed

point in time and a trajectory of a tracer (red =

high acceleration, blue = low acceleration), time

spacing between two spheres = 0.25τη

-4

-3

-2

-1

0

1

2

3

4

0 10 20 30 40 50 60 70 80 90 100

u y/u

rms

t/τη

(b) y-component of the velocity of the tracer

given in terms of the root mean square velocity

Figure .: Tracer trapped in a vortex filament in Navier-Stokes turbulence at Rλ = 122

Increasing the time lag the s become more and more Gaussian. As can be

seen from Figure . the time a tracer spends trapped in a vortex filament can

last for several decades of dissipation times scales. However, after a certain time it

leaves the structure or the structure breaks up. The correlation decreases with time.

Approaching the integral time scale the actual and initial velocity of the tracer are

uncorrelated. This is reflected in the Gaussian shape of long time increments.

The multifractal model for Lagrangian turbulence made a prediction for the

of the acceleration (see Section ..). To compare the prediction with the

simulations, Figure .(b) shows the predicted acceleration together with the

measured one. The shape of the is well recovered. The multifractal model


works for the s. However, one has to keep in mind that there are parameters of

the model which have to be chosen appropriately to get this agreement.

..

Lagrangian statistics of conducting flows are concerned with the s of the

temporal velocity and magnetic field increments. The magnetic field is recorded

along a tracer trajectory, while the tracer is advanced by the underlying velocity

field. Figure .(a) shows the of the velocity increment, while in Figure .(b)

the of the magnetic field is displayed for several time lags τ . Both s show

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

-30 -20 -10 0 10 20 30

P(δ

τ v)

δτ v/σδτ v

τ=0.13 τητ=1.25 τητ=6.25 τη

τ=25 τηgauss

(a) velocity field increments δτv

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

-30 -20 -10 0 10 20 30

P(δ

τ B

)

δτ B/σδτ B

τ=0.13 τητ=1.25 τητ=6.25 τη

τ=25 τηgauss

(b) magnetic field increments δτB

Figure .: Lagrangian s of velocity and magnetic field increments in an flow at Rλ = 234,

curves are shifted for clarity, τ increases from top to bottom, solid line: Gaussian distribution

a transition from a highly intermittent for short time lags to a Gaussian

for large time lags. The important conclusion to draw is that both the velocity and

the magnetic field increments are intermittent variables. The s of the magnetic

field are slightly more stretched due to the higher amount of energy stored in the

magnetic field than in the kinetic motion.

The situation is similar to the Navier-Stokes case. The small scale coherent

structures are responsible for high velocity and magnetic field fluctuations. However,

as was pointed out in sections .. and .. the most dissipative structures are

completely different in Navier-Stokes and turbulence. While the former are

vortex filaments the latter are vortex and current sheets. Figure .(a) shows a

trajectory which undergoes a strong acceleration. The corresponding z-component


of the velocity is given in Figure .(b). The tracer was accelerated along a vortex

sheet (red) and approaches a current sheet where it changes its direction drastically.

The inflection point produces a large acceleration situated in the far tails of the

corresponding . Contrary to Navier-Stokes turbulence tracers do not become

trapped in these dissipative structures. They move along and leave them.

The transition to a Gaussian distribution can be explained as in the Navier-Stokes

case. For increasing time lags the influence of a single structure decreases. The fields

at the current and initial position of the tracer starts to decorrelate.

(a) isosurface of the vorticity (red) and current

density (blue) at a fixed point in time and a

trajectory of a tracer (red = high acceleration,

blue = low acceleration), time spacing between

two spheres = 0.125τη

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12 14

u z/u

rms

t/tη

(b) z-component of the velocity of the tracer given

in terms of the root mean square velocity

Figure .: Tracer in the vicinity of strong dissipative structures in turbulence at Rλ = 270

To compare the multifractal prediction for the tracer acceleration from Sec-

tion .. Figure . displays the of the measured velocity and magnetic field

increments for small time lags τ . Included in this figure is the prediction for the

acceleration. The tails are well recovered by the multifractal modeling, while it

is not possible to fit the prediction to the core of the . This is not a major

drawback of the multifractal modeling because the model is concerned with inertial

range properties and the core corresponds to motions within the dissipation range.

. Structure functions

100

10-4

10-8

0 5 10 15 20 25 30 35δτ v/σδτ v

, δτ B/σδτ B

P(δτ v)P(δτ B)

multifractal prediction

Figure .: Lagrangian s of the velocity increments and magnetic field increments in

turbulence, solid line: multifractal prediction with hmin = 0.16, Rλ = 107


The following sections focus on the Lagrangian structure functions. As we have

seen in the last section the increment s are highly intermittent for small time

lags. This will be reflected in the scaling behavior of the corresponding structure

function. They are computed from the trajectories of the tracers advanced by the

turbulent flow. In the case of the velocity increments . the structure functions

reads

Sp(τ) = 〈|δτv|p〉 =

∫|δτv|pP (δv, τ)dδv. (.)

P (δv, τ) are the s presented in the previous section, the angular brackets

denote averaging over all particles and over several realizations. To be more precise,

the structure functions are computed as follows. The tracers are seeded into a

statistically stationary flow at time t0. The increment δτv, τ = t− t0, is computed

for all tracers and incorporated into the average. After each following integral time

this procedure is repeated. After an integral time the flow is statistically independent

of the previous measurement and can be seen as a new realization. The surrounding


flow is isotropic. Therefore, no preferred direction exists. The increments projected

onto the three coordinate axes obey the same statistics. In order to obtain a larger

data set averages contain all projections.

.. Navier-Stokes

The structure functions under consideration in Navier-Stokes turbulence are the

moments of the velocity increments (.),

Sp = 〈|δτv|p〉 = 〈|v(t + τ)− v(t)|p〉 ∼ τ ζLp . (.)

The angular brackets denote averaging over all tracers and several realizations.

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

1 10

Sp

τ/τη

p=1p=2p=3p=4p=5p=6p=7p=8

1 1.5

2 2.5

3 3.5

4 4.5

5 5.5

6

0.1 1 10 100

S2/

(ετ)

Figure .: Lagrangian velocity structure functions in Navier-Stokes turbulence, Rλ = 316, inset:

compensated second order structure function

The numerically computed Lagrangian structure functions are shown in Fig-

ure .. Qualitatively they look the same as their Eulerian counterparts (see Fig-

ure .). They seem to saturate at time lags beyond approximately dissipation

time scales. The motion approaches the integral scale and begins to feel the forcing

at large scales.


Because of relation (.) one expects the second order structure function to scale

linearly. The inset of Figure . shows the compensated second order structure

function. The expected plateau is not observable.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

1 10

d lo

g(S

p)/d

log

τ

τ/τη

p=1p=2p=3p=4p=5p=6p=7p=8

Figure .: Logarithmic derivative of Lagrangian velocity structure functions in Navier-Stokes

turbulence,Rλ = 316

To have a closer look at an inertial range in Figure . the logarithmic derivatives

of the structure functions are plotted up to eighth order. These functions show the

same characteristic shape as their Eulerian counterparts (see Figure .). The range

where the flow becomes smooth is situated approximately below 0.1τη which is

smaller than the dimensional estimate of τη. The saturation of the structure func-

tions and correspondingly the integral time is approached beyond approximately

50τη. It is assumed to observe a scaling range in between these limits.

The lower order functions show no plateau and therefore no clear scaling range.

The formation of such a plateau with increasing order is probably an artifact of an

insufficient statistic. In order to compute a statistical quantity of increasing order

the necessary amount of data grows. This is because high-order statistics are quite

sensitive to extreme events which are seldom even in intermittent flows. If there is

a real plateau for high orders of p then also low order functions should display it.

It should be mentioned that also experimental measurements do not show a clear


scaling range (see e. g. Mordant et al. []). Even for higher Reynolds numbers

than achieved in this work experimentalist rely on to extract scaling exponents.

An explanation of the fact that the Lagrangian scaling behavior is worse than the

Eulerian one might be the explicit time dependence of the Lagrangian coordinates.

The Eulerian structure functions can be computed using every individual snapshot

of the turbulent field. From this point of view only one moment in time enters an

Eulerian structure function. To get a better data basis, structure functions from

several points in time will be averaged. However, this averaging only smooths

the structure function, the overall shape remains the same. If a single structure

function does not show a clear scaling behavior, the averaged one does not either.

Contrary, the Lagrangian statistic is explicitly time dependent. There is a finite

correlation time of the flow. Even the most coherent structures break up and

formerly correlated points might become uncorrelated. A tracer moving within a

turbulent field spends a certain time in regions having a specific scaling exponent.

The crucial point is now that this region itself evolves in time. It might change its

local scaling exponent due to a break-up of a filament during the passing time of the

tracer. Then the observed Lagrangian scaling law is spoiled by the time evolution

of the flow. That this argument is reasonable shows Section .. and Section ..

The former section investigates a temporal increment along a tracer trajectory but

evaluated at equal times. The latter section is concerned with Lagrangian statistics

in frozen turbulent fields. There tracers are advanced by a frozen turbulent velocity

field. The field is static and does not evolve in time. In both examples the effect

that might spoil the Lagrangian scaling behavior is suppressed. And indeed, the

measured structure functions do show a clear scaling range.

Now we return to the determination of the scaling exponents. Although there

is no clear scaling range, the same procedure for evaluating the absolute scaling

exponent can be applied as for the Eulerian case (see Section ...). Following

this procedure the scaling range would be 4.6τη ≤ τ ≤ 13.5τη. The corresponding

absolute exponents are listed in Table .. It is a bit surprising that the second order

exponent deviates from the prediction of unity. Two remarks have to be given. On

the one hand the prediction is founded only on dimensional grounds. There is no

strict derivation from the underlying Navier-Stokes equations as in the case of the

4/5th law. On the other hand the computed absolute values should not be taken

too seriously because of the lack of a clear scaling range and therefore large error

bars.


p ζLp (absolute values) ζL

p /ζL2 (using ) ζL

p /ζL2 (using )

Rλ 316 316 178

1 0.51± 5.3 · 10−2 0.57± 7.7 · 10−3 0.57± 5 · 10−3

2 0.90± 7.7 · 10−2 1 1

3 1.21± 9.2 · 10−2 1.309± 2.3 · 10−2 1.30± 1.6 · 10−2

4 1.44± 1.1 · 10−1 1.503± 5.5 · 10−2 1.51± 4.1 · 10−2

5 1.61± 1.3 · 10−1 1.698± 9.5 · 10−2 1.65± 7.5 · 10−2

6 1.74± 1.6 · 10−1 1.824± 1.4 · 10−1 1.76± 1.1 · 10−1

7 1.83± 2.0 · 10−1 1.924± 1.8 · 10−1 1.84± 1.6 · 10−1

8 1.90± 2.5 · 10−1 2.006± 2.4 · 10−1 1.90± 2.1 · 10−1

Table .: Lagrangian scaling exponents ζLp for Navier-Stokes turbulence

To obtain more precise scaling exponents one uses the assumption of extended

self-similarity as in the Eulerian framework. In Lagrangian turbulence the reference

structure function is that of the second order because of (.). Figure . shows

the structure functions plotted against the second order structure function. The

functions are clearly more bent than in the Eulerian case (see Figure .). The

computed scaling exponents will therefore be sensitive to the range chosen for

the evaluation. Following the procedure presented in Section ... the scaling

range is assumed to be the region where the second order structure function obeys

a linear scaling law. Again this region is chosen to be the interval in which the

compensated function (see inset of Figure .) stays above % of its maximum

value. The assumed inertial range then extends from approximately . to .

dissipation times. This range is similar to the range used to compute the absolute

scaling exponents (see previous paragraph). Table . lists the computed scaling

exponents for two different Taylor-Reynolds numbers. They are in good agreement.

Due to the large error bars a Reynolds number dependence cannot be detected.

A comparison with the prediction of the multifractal model of Biferale et al.

presented in Section . exhibits a disagreement. The computed values are smaller

and therefore more intermittent.

Attention has to be paid to this contradiction. The scaling exponents measured in

this work are in good agreement with recent experiments by Mordant et al. []

and Porta et al. []. It is worth mentioning that the two groups use completely

different experimental setups but that their results are in good agreement. However,


1e-10

1e-08

1e-06

1e-04

0.01

1e-04 0.001 0.01

Sp

S2

p=1p=3p=4p=5p=6p=7p=8

0

0.5

1

1.5

2

2.5

1e-04 0.001 0.01 0.1

Figure .: Relative Lagrangian velocity structure functions () in Navier-Stokes turbulence, solid

lines: computed scaling behavior, Rλ = 316, inset: logarithmic derivative

the presented scaling exponents differ from simulations done by Biferale et al.

[a]. It is important to stress that this is mostly due to the range chosen for

evaluating the scaling exponents. While this work and the experiments are severely

guided by the assumption of a linear scaling law of the second order structure

function, Biferale et al. choose their inertial range at significantly longer time lags.

Biferale et al. [] argue that for short time lags the statistics are spoiled by

trapping events of tracers in vortex filaments. Indeed such events occur as was

discussed in the previous section. To quantify the influence of trapping events on

the statistics, Biferale et al. filtered out segments of trajectories which show an

acceleration larger than 7arms and last at least for a few dissipation times. With

this procedure they found that the knee in the logarithmic derivatives of the

structure functions (see inset of Figure .) becomes weaker when decreasing

the time window for detecting trapping events. They propose to measure the

scaling behavior starting from time lags much longer than the typical time period

of a trapping event. They measure in the range 10τη ≤ τ ≤ 50τη and find a

good agreement with the prediction of their multifractal model. However, there


are several arguments indicating that the interval chosen in this work is more

appropriate.

First the range proposed by Biferale et al. contradicts the assumption of a linear

scaling law of the second order structure function S2, because the peak of the

compensated function S2/τ lies at approximately 8τη.

Second, considering the logarithmic derivative (see Figure .) the scaling range

proposed by Biferale et al. is situated clearly beyond the flattest interval which

is located at approximately 8τη. This flattest interval is a reasonable choice for

evaluating the scaling exponents, because both frozen Lagrangian turbulence and

structure functions of another time increment show clear scaling behavior in this

range as we will see in Section .. and ., respectively.

Third using too rigorously might be dangerous as it is only an assumption

which even in the Eulerian case is not firmly based on theoretical grounds. The

curvature of the functions is a hint at a lack of extended self-similarity in the

Lagrangian framework. We will come back to this point in the following sections.

A possible explanation for the deviation of the measured scaling exponents from

the multifractal prediction will be given in Section .. The following section deals

with Lagrangian scaling behavior of magnetohydrodynamic flows.

..

As in the Navier-Stokes flow the tracers are advanced by the underlying velocity field.

Therefore the structure functions under consideration in magnetohydrodynamic

flows are the moments of the velocity increments (.). The computation of

the structure functions is identical to the procedure in Navier-Stokes flows (see

previous section).

The structure functions are given in Figure .(a). The inset shows the compen-

sated second order function. Again a linear scaling behavior is assumed because of

relation .. However, a plateau is absent as in the Navier-Stokes case. The peak

lies at approximately 7τη.

Figure .(b) shows the corresponding logarithmic derivatives. None of the

displayed functions has a flat region which could be associated with a scaling range.

This is not surprising as the Eulerian counterparts showed no clear scaling range,

either. In the previous section concerned with the Navier-Stokes case it has been

pointed out that Lagrangian statistics are affected by the decorrelating effect of the


1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1

0.1 1 10

Sp

τ/τη

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

0.1 1 10

S2/

(εkτ

)

(a) order p = 1–8 from top to bottom, inset:

compensated second order structure function

0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2 2.2 2.4 2.6 2.8

3 3.2 3.4 3.6 3.8

4

1 10

d lo

g(S

p)/d

log(

τ)

τ/τη

p=1p=2p=3p=4p=5p=6p=7p=8

(b) logarithmic derivative

Figure .: Lagrangian structure functions of turbulence, Rλ = 234

time dependence. Therefore one cannot expect a scaling range in the Lagrangian

structure function if there is none in the Eulerian framework.

In order to get any information about scaling exponents one has to rely on

. These functions given in Figure . are curved as in the Navier-Stokes case

as can clearly be seen from the inset of Figure .. The assumption of extended

self-similarity seems to be less fulfilled in the Lagrangian framework. It has to be

stressed that in flows no trapping appears (see Section ..) which could be

responsible for spoiling the functions. There has to be an additional effect that

spoils the assumption. However, the same procedure is applied for extracting

the scaling exponents as in the Navier-Stokes case. The computed exponents are

shown as straight lines in Figure . and listed in Table . for two different

Taylor-Reynolds numbers. They agree within the error bars so that a Reynolds

number dependence cannot be detected. Comparing them to the prediction of

the multifractal model one observes that the predicted values are smaller than the

measured ones. The multifractal prediction therefore overestimates the degree of

intermittency. In the situation is reversed compared to the Navier-Stokes

case. While for the former the multifractal model predicts a higher degree of

intermittency than can be observed the latter is more intermittent than predicted.

A possible explanation of this deviation is given in the following section.

. Structures and Lagrangian intermittency

1e-14

1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1

1e-04 0.001 0.01 0.1

Sp

S2

p=1p=3p=4p=5p=6p=7p=8

0.5 1

1.5 2

2.5 3

3.5 4

1e-04 0.001 0.01 0.1

Figure .: Relative Lagrangian velocity structure functions () in turbulence, solid lines:

computed scaling behavior, Rλ = 234, inset: logarithmic derivative


As was shown in sections .. and .. the multifractal prediction underestimates

the degree of Lagrangian intermittency in Navier-Stokes turbulence and overes-

timates the degree of intermittency in magnetohydrodynamic turbulence. The

situation is depicted in Figure .. Lagrangian Navier-Stokes turbulence is more

intermittent than turbulence. This is a striking result, because the situation is

reversed in the Eulerian framework. There the dimension of the most dissipative

structures is responsible for the deviation from the predictions. In flows

these are current and vortex sheets of dimension two while in Navier-Stokes flows

the vortex filaments are of dimension one. The She-Lévêque model differs for

neutral and charged fluids only by the co-dimension C0 of the nearly singular

structures (see (.)). The dimension determines the possibility of observing

extreme events which are responsible for anomalous scaling. Therefore is

more intermittent in Eulerian coordinates than Navier-Stokes turbulence. As can

be seen from Figure . the most dissipative structures have a different impact on

the Lagrangian intermittency than on the Eulerian one.


p ζLp /ζ2 (using ) ζL

p /ζL2 (using )

Rλ 234 107

1 0.526± 2.3 · 10−3 0.526± 4.9 · 10−3

2 1 1

3 1.407± 1.4 · 10−2 1.407± 2.2 · 10−2

4 1.730± 5.6 · 10−2 1.733± 8.2 · 10−2

5 1.960± 1.4 · 10−1 1.974± 2.0 · 10−1

6 2.109± 2.5 · 10−1 2.142± 3.5 · 10−1

7 2.207± 3.5 · 10−1 2.26± 5.5 · 10−1

8 2.280± 4.2 · 10−1 2.36± 7.5 · 10−1

Table .: Lagrangian scaling exponents ζLp for magnetohydrodynamic turbulence

That Lagrangian Navier-Stokes turbulence is more intermittent than tur-

bulence contradicts the monotonicity property of the multifractal model (see

Section .). If a hydrodynamic flow is more intermittent than a conducting flow

in Eulerian coordinates this relation should also hold in Lagrangian coordinates.

As can be seen from Figure . the prediction of the multifractal model for

agrees better with the measurement for Navier-Stokes flows and vice versa.

A possible explanation is the following: In Navier-Stokes turbulence the vortex

filaments of dimension one are responsible for trapping events (see Figure .(a)).

Tracer spend a long time in these structures and experience high accelerations and

therefore high velocity differences. Because the Lagrangian framework is based on

temporal changes the tracer feels accelerations as if it moves on a surface. From

the tracer’s point of view the filament is unfolded in time into a two-dimensional

surface. A one-dimensional filament in Eulerian coordinates corresponds to a two-

dimensional surface in Lagrangian coordinates. In flows the extreme events

correspond to tracers reflected by current and vortex sheets. These events take

mainly place where two sheets approach each other transversely (see Figure .(a)).

The structure where the strongest accelerations occure is therefore a line. The

two-dimensional sheets are responsible for one-dimensional acceleration structures.

The nearly singular structures have a completely different impact on the scaling

behavior when measured in Eulerian or Lagrangian coordinates.

The agreement of the multifractal prediction and the measurements when inter-

changing the co-dimensions of the most dissipative structures in Lagrangian Navier-


0.5

1

1.5

2

2.5

3

1 2 3 4 5 6 7 8

ζ p

p

Navier-Stokes Re 316Navier Stokes multifractal

MHD Re 270MHD multifractal

Figure .: Scaling exponents in Navier-Stokes and turbulence together with the multifractal

prediction

Stokes and turbulence is much better than for the Eulerian co-dimensions. A

perfect agreement would be surprising because the variation of the co-dimension

was the only change in the She-Lévêque model when switching from Eulerian to

Lagrangian coordinates. Other parameters such as the assumed cascade dynamics

remained unchanged.

Some important remarks have to be made concerning the presented multifractal

approach:

• Interchanging the co-dimensions is reasonable but needs more theoretical

foundation.

• The deviation from the linear scaling law of the second order structure

function (see Section ..) has to be analyzed in more detail at higher

Reynolds numbers, because it is an important assumption of the model.

• The multifractal model is based on the Eulerian singularity spectrum (see

Section .). If this spectrum is of Kolmogorov () type, also the Lagrangian

exponents would display the scaling behavior. Recent numerical simu-


lations by Kamps and Friedrich [] of two-dimensional Navier-Stokes

turbulence show anomalous Lagrangian scaling behavior measured in the

inverse cascade. This contradicts the multifractal model because the inverse

cascade is not intermittent in the Eulerian framework and obeys normal

scaling. Following the multifractal model the Lagrangian statistics should

not be intermittent, either.

. Alternative increments

In Navier-Stokes turbulence Lagrangian structure functions depend conveniently

on the velocity increment projected onto a fixed axis,

δτv = (~v(t + τ)− ~v(t)) · ~e.

In numerical isotropic simulations the normalized vector ~e is one of the coordinate

basis vectors.

There are other possibilities for the increment. In this section two different types

of increments will be considered.

.. The norm increment

Another reasonable type of a structure function variable is the norm of the velocity

increment (norm increment),

δNτ v = |(~v(t + τ)− ~v(t))|. (.)

This measures the change in the magnitude of the velocity after a time τ . It is

interesting to study the scaling behavior according to this increment because it

does not take into account pure directional changes of the velocity vector. A tracer

circulating with constant speed around a vortex filament would not contribute to

the statistics when considering the increment (.).

The left part of Figure . shows the logarithmic derivative of the structure

function of the increment (.). As for the standard increment no clear scaling

range is visible. Using the assumption of the corresponding functions are

plotted in the right part of Figure .. The computed scaling exponents are listed

in Table .. Comparing these values with the scaling exponents of the standard

increment (see Table .) it is obvious that the norm increment yields a more


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

1 10

Sp

τ/τη

p=1p=2p=3p=4p=5p=6p=7p=8

1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1e-04 0.001 0.01

Sp

S2

p=1p=3p=4p=5p=6p=7p=8

0

0.5

1

1.5

2

2.5

Figure .: Structure functions depending on the norm of the velocity increment in Navier-Stokes

turbulence at Rλ = 316, left: logarithmic derivative, right: structure functions together with

the computed scaling exponents (straight lines), inset: logarithmic derivative

intermittent statistic than the standard increment. That is a bit surprising because

as mentioned above the norm increment does not take into account the changes in

the direction of the motion. The helical trajectories arising from trapped tracers in

vortex filaments undergo rapid changes in the direction. The statistics of the norm

increment show that this is not the only important feature of the tracer motion. In

addition the tracer varies the magnitude of his velocity dramatically along its path.

A possible explanation for this might be the vortex stretching which shrinks the

diameter of the vortex and enhances the angular velocity.

p ζNp /ζN

2 (using ) . . . . . . .∆(ζN

p /ζN2 ) . . . . . . .

Table .: Scaling exponents of the norm increment in Navier-Stokes, Rλ = 316

.. The equal time increment

Another possibility for a velocity increment is the Lagrangian equal time increment

(),

δτ~v = ~u( ~X(~y, τ), τ)− ~u(~y, τ).


The difference to the standard Lagrangian velocity increment (.) is that the

second term on the right hand side is evaluated at a time τ instead of 0. The

difference is taken between the points in space ~X(~y, τ) and ~y. The subtracted

velocities at these points correspond to equal points in time.

The motivation for this increment was developed in collaboration with the group

of R. Friedrich from the Westfälische Wilhelms-Universität in Münster (Westf.),

Germany. The Lagrangian of the equal time increment takes the form

PL(~v, ~y, t) =

∫d3x′

⟨δ[~x′ − ~X(~y, t)]δ[~v − (~u(~x′, t)− ~u(~y, t))]

⟩.

If the right hand side factorizes one would end up with

PL(~v, ~y, t) =

∫d3x′

⟨δ[~x′ − ~X(~y, t)]

⟩〈δ[~v − (~u(~x′, t)− ~u(~y, t))]〉

=

∫d3x′p(~x′, ~y, t)PE(~v, ~x′ − ~y′),

PE(~v, ~x′−~y′) is the standard Eulerian (discussed in Section ..) and p(~x′, ~y, t)

denotes the probability of a tracer to end at ~x′ after a time t when it is started at ~y.

The situation is more complicated when considering the standard Lagrangian

increment. After factorizing one obtains

PL(~v, ~y, t) =

∫d3x′p(~x′, ~y, t) 〈δ[~v − (~u(~x′, t)− ~u(~y, 0))]〉 .

Now the Eulerian does not appear on the right hand side.

The equal time velocity increment is therefore an intermediate step towards

the standard increment. The logarithmic derivative of the structure functions

depending on this increment are shown in the left figure of .. They show a clear

scaling range. The corresponding absolute scaling exponents are listed in Table ..

The scaling range extends from approximately . to . dissipation times. As this

increment show scaling behavior over a broad range of scales the spoiling of the

structure functions of the standard increment has to be attributed to the different

points in times used for the evaluation.

Table . also lists the scaling exponents obtained by the assumption. They

are less intermittent than for the standard increment. The third order structure

function scales nearly linearly. This increment is related to the Eulerian spatial

increment due to the sweeping effect of small scales by large scales. However, if the

Taylor hypothesis applies one would indeed measure the She-Lévêque exponents,

but the equal time statistics are more intermittent than the Eulerian statistics.


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

1 10

d lo

g(S

p)/d

log

τ

τ/τη

p=1p=2p=3p=4p=5p=6

1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1

0.001 0.01

Sp

S2

p=1p=3p=4p=5p=6p=7p=8

0 0.5

1 1.5

2 2.5

0.001 0.01

Figure .: Structure functions depending on the equal time increment in Navier-Stokes turbulence

at Rλ = 316, left: logarithmic derivative, right: structure functions together with the computed

scaling exponents (straight lines), inset: logarithmic derivative

p ζp (absolute values) ζp/ζ3 (using ) ζp/ζ2 (using )

1 0.35± 2.7 · 10−2 0.380± 3.3 · 10−3 0.535± 2.5 · 10−3

2 0.65± 3.5 · 10−2 0.711± 3.3 · 10−3 1

3 0.91± 3.0 · 10−2 1 1.403± 7.7 · 10−3

4 1.14± 1.8 · 10−2 1.248± 6.7 · 10−3 1.749± 2.0 · 10−2

5 1.34± 1.4 · 10−2 1.457± 1.7 · 10−2 2.039± 3.8 · 10−2

6 1.50± 2.4 · 10−2 1.628± 3.2 · 10−2 2.276± 6.2 · 10−2

7 1.63± 4.7 · 10−2 1.763± 5.3 · 10−2 2.462± 9.6 · 10−2

8 1.73± 8.4 · 10−2 1.867± 8.3 · 10−2 2.600± 1.5 · 10−1

Table .: Scaling exponents of the equal time increment in Navier-Stokes turbulence, Rλ = 316

The observed scaling exponents are identical to the exponents measured in frozen

turbulence (see next section).

It would be interesting to examine the statistics of this new increment in two-

dimensional turbulence. To see whether the scaling range is also greatly extended

and to compare the results to frozen turbulence. In there might be a larger

difference as in . In two-dimensional frozen turbulence the tracers stick to the

static streamlines. Elliptical regions are topological separated from each other by

streamlines acting as separatrices. This separation is broken in the dynamical case.

This difference is absent in due to the additional dimension.


. Frozen Navier-Stokes turbulence

A step towards the understanding of Lagrangian intermittency can be taken by

looking at the intermittency of tracers advanced by a static turbulent field. The

velocity field is frozen and the seeded particles move within this time independent

field. The Eulerian statistics of the frozen field is certainly given by the She-Lévêque

formula. As was pointed out in a previous section the Eulerian scaling behavior

can be computed from a single snapshot of the velocity field. The frozen temporal

statistics are the purest translation of the Eulerian statistics into the tracer point of

view.

Figure .: Tracer trajectory and vortex filaments (grey) in frozen Navier-Stokes turbulence, color

encodes the acceleration of the tracer (red = high acceleration, blue = low acceleration)

The dynamics of the tracers differ from the dynamical case where the velocity

field changes in time according to the Navier-Stokes equations. Figure . shows

a trajectory of tracer trapped within a vortex filament. The motion is highly

symmetric and stays nearby the filament for a much longer time than in the

dynamical case. The dynamical field experienced by the tracer changes more

rapidly which decreases the time the tracer is attached to a certain structure. Finally,

the time dependence is responsible for breaking up even the coherent structures

after a certain time. Helical motions of tracers are much more probable in frozen

flows than in real turbulence.

. Frozen Navier-Stokes turbulence

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

1 10

d lo

g(S

p)/d

log(

τ)

τ/τη

p=1p=2p=3p=4p=5p=6p=7p=8

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

0 0.01 0.02 0.03 0.04 0.05 0.06

d lo

g(S

p)/d

log(

S2)

S2

p=1p=2p=3p=4p=5p=6p=7p=8

Figure .: Lagrangian structure functions in frozen Navier-Stokes turbulence, Rλ = 316, left:

logarithmic derivative, right: logarithmic derivative of functions

To measure the degree of intermittency of the flow from the tracer’s point of view

one can compute the Lagrangian structure function. The definition is the same as

for the dynamical case .. The logarithmic derivative of the structure functions

are shown in the left part of Figure .. They exhibit clear plateaus in which the

structure function obeys a scaling law. Interestingly this region is quite large. It

extends approximately from 3τη to 15τη, as indicated by the vertical bars. From

this interval the absolute values together with the errors can be extracted from the

ordinate (see Section ...). They are listed in Table .. The first impression

is that the second order exponent is far away from unity, which is the suggested

Lagrangian value. Indeed, the third order scales nearly linearly. This is similar to

the Eulerian case. However, the measured frozen exponents are more intermittent

than the Eulerian ones (see Table .). Because the Eulerian statistics enter the

frozen statistics an additional effect enhances the degree of intermittency.

To get more precise scaling exponents one can use again the assumption of .

The logarithmic derivative of the structure function of order p depending on the

second order structure function are plotted in the right figure of .. The functions

are bent as in the dynamical case. Apart from the equal time increment this an

additional example in which the structure functions exhibit a clear scaling region

whereas the assumption of extended self-similarity is spoiled. One can conclude that

the poor scaling seems to be an inherent problem of the Lagrangian coordinates.

Nevertheless one can compute the relative slopes with the same procedure applied


p ζp (absolute values) ζp/ζ3 (using ) ζp/ζ2 (using )

1 0.35± 2.5 · 10−2 0.385± 3.3 · 10−3 0.538± 2.0 · 10−3

2 0.66± 3.4 · 10−2 0.717± 3.6 · 10−3 1

3 0.92± 3.6 · 10−2 1 1.393± 6.5 · 10−3

4 1.15± 4.7 · 10−2 1.239± 5.6 · 10−3 1.723± 1.5 · 10−2

5 1.34± 6.0 · 10−2 1.436± 8.1 · 10−3 1.995± 2.0 · 10−2

6 1.49± 7.1 · 10−2 1.598± 2.0 · 10−2 2.216± 3.2 · 10−2

7 1.63± 7.6 · 10−2 1.729± 6.0 · 10−2 2.396± 8.4 · 10−2

8 1.73± 7.4 · 10−2 1.835± 1.3 · 10−1 2.540± 1.7 · 10−1

Table .: Lagrangian scaling exponents of frozen Navier-Stokes turbulence, Rλ = 316

for the dynamical case. There are at least two possibilities for the choice of the

reference structure function. From the Lagrangian point of view one would choose

the second order function. However, the third order structure function scales nearly

linearly. Similarly to the Eulerian case one could therefore use the third order

function as a reference. The results for both choices are listed in Table .. When

normalizing to the third order function the absolute and values are in fair

agreement. One has to keep in mind that the absolute third order exponent is .

and not .

Comparing the scaling exponents normalized to the second order structure

function with the dynamical scaling exponents (see Table .) one observes that the

dynamical flow is more intermittent than the frozen flow. The time dependence

increases the probability of extreme events. That is a second hint at the fact that pure

vortex trapping is not the only contribution to the intermittency of Lagrangian

turbulence, because the trapping time in frozen filaments is greatly enhanced

compared to the dynamical case. The time evolution and the vortex stretching

probably play a crucial role.

Several conclusions can be drawn for the dynamical Lagrangian statistics from

the results presented in this section:

• The scaling range observed in the logarithmic derivatives is approximately

the same as in the dynamical case. That is a further justification of the range

chosen for evaluating the Lagrangian scaling exponents.

• The large frozen scaling region compared to the small dynamical scaling

. Decorrelated Navier-Stokes

region supports the explanation of the spoiling of the dynamical statistics by

the time evolution of the flow.

• The values of the presented frozen scaling exponents are just in between the

Eulerian and Lagrangian exponents. Measuring the Eulerian intermittency

with Lagrangian coordinates in a frozen flow enhances the observed intermit-

tency. The temporal evolution has an additional influence on the Lagrangian

statistics.

• One has to be careful in applying the assumption of in Lagrangian coor-

dinates. While one could argue in the dynamical case that the functions

are bent because the scaling range is very narrow, this argumentation fails

in frozen turbulence. There one observes a clear scaling range and the

functions are bent, anyway.


In this section a decorrelated flow will be considered. The question will be addressed

if the Lagrangian statistics show anomalous scaling behavior even if the flow is

uncorrelated and shows normal scaling in the Eulerian framework.

To generate a decorrelated flow the ordinary Navier-Stokes equations are inte-

grated in time. After each time step the phases of the Fourier modes ~k are rotated

with a frequency

~ω~k = λ

√|~k|3|~k|−5/3, (.)

which corresponds to the local eddy turn-over frequency. λ is a constant parameter

and defines the amplitude of the rotation. This transformation is energy-preserving

and divergence-free. Due to the rotation of the modes the filaments are destroyed so

that no coherent structures exist in the flow. Once the structures have disappeared

normal scaling is expected for the Eulerian structure functions. Indeed, Table .

shows the Eulerian scaling exponents (using ) which are in fair agreement with

the theory (.). They do not follow exactly the prediction of a linear law

which indicates that the coherent structures are not completely destroyed.

If one now injects tracers into the flow one first observes that the turbulent

diffusion is greatly decreased. Figure . shows a trajectory of a tracer in an

ordinary Navier-Stokes and decorrelated flow. As the coherent structures are absent


-4-3-2-1 0 1

1 1.5

2 2.5

3 3.5

4 4.5

5 5.5

6 6.5

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

correlateduncorrelated

Figure .: Trajectories of tracers in a Navier-Stokes (Rλ = 234) and uncorrelated (λ = 0.5) flow

p Eulerian ζp/ζ3 Lagrangian ζp/ζ2

1 0.35± 0.007 0.51± 0.01

2 0.68± 0.008 1

3 1 1.48± 0.02

4 1.31± 0.019 1.96± 0.05

5 1.61± 0.48 2.42± 0.09

6 1.90± 0.09 2.86± 0.13

7 2.17± 0.14 3.30± 0.17

8 2.42± 0.20 3.72± 0.22

Table .: Eulerian and Lagrangian scaling exponents ζp obtained from a decorrelated (λ = 0.5)

Navier-Stokes simulation, Rλ = 122

in the latter the tracer is not swept by large scale eddies. The trajectory reflects a

standard diffusion process.

To analyze the consequences for the Lagrangian statistics Table . lists the

Lagrangian scaling exponents. They approximately follow a linear law, although

slight deviations from the prediction (.) are observable.


The conclusion is that an uncorrelated flow with scaling also shows

scaling for the Lagrangian statistics. The notion ‘uncorrelated’ has to be stressed.

In two-dimensional turbulence the scaling exponents are of type within the

inverse cascade. No intermittency occurs. However, Lagrangian structure functions

show anomalous scaling (see Kamps and Friedrich []). The flow is intermittent

when measured in the Lagrangian coordinates. The difference to the situation

considered in this section might be the coherent structures in the flow. They

are vortices with a long lifetime responsible for a large turbulent diffusion. As in

, tracers are trapped within these structures and experience strong accelerations.

This process is absent in uncorrelated flow.

Chapter

Numerical methods

Due to the increasing speed and memory of computers, numerical simulations of

the Navier-Stokes and equations have become a valuable tool for studying

turbulence. To investigate the intrinsic properties of turbulence such as the energy

cascade and intermittency one often uses periodic boundary conditions in order to

keep the influence of the geometry on the flow structure as small as possible. For

this kind of turbulence simulations pseudo-spectral codes are widely used (see e. g.

Gottlieb and Orszag [], Canuto et al. [], Vincent and Meneguzzi [],

She et al. [], Mueller and Biskamp [], Yokokawa et al. [], Chevillard

et al. [], Yeung and Borgas [], Biferale et al. [a], Bec et al. []).

The developed code is of this type and will be presented in the following sections.

This chapter is organized as follows: The basic properties of the pseudo-spectral

code used throughout this work are discussed in Section .. The implemented time

stepping schemes are presented in Section .. In order to achieve fully developed

turbulence a large number of grid points is indispensable. The dependence of the

statistical results on the grid resolution is presented in Section .. Turbulence on

huge lattices can only be simulated on super-computers with a large number of

s. The parallelization of the code is presented in Section .. The position of

tracers has to be interpolated from the Eulerian grid. For this task two schemes are

used and explained in Section ., while their influence on the numerical results

is discussed in Section .. Details of the performed simulations like the initial

conditions and forcing used to obtain a statistically stationary state are presented

in Section .. At the end of this chapter the design of the code will briefly be

reviewed.

Chapter Numerical methods

. Solving the basic equations

Pseudo-spectral codes treat the Navier-Stokes equations (.) and equations

(.), (.) in Fourier space and compute the convolutions arising from the

nonlinear terms in real space. A fast Fourier transformation () is used to switch

between the two spaces. For this task the code uses the library as explained in

Section ..

The physical fields are represented by Fourier series

~u(~x, t) =∑

~k

ei~k·~x~u~k(t),

and similar for ~B(~x, t). The Fourier coefficients are computed on a cubic grid with

N3 collocation points, with N even to achieve the highest efficiency of the .

Now lets turn to the basic equations in Fourier space. The Navier-Stokes equation

(.) reads

∂t~u + ν|~k|2~u = i~k · (~u~u), (.)

when neglecting the pressure term. The pressure assures the incompressibility

condition and is considered in Section ... In the vorticity formulation (.)

one gets

∂t~ω + ν|~k|2~ω = i~k × ( ~u× ~ω). (.)

The equations (.) and (.) become

∂t~ω + ν|~k|2~ω = i~k × ( ~u× ~ω +~k × ~B × ~B), (.)

∂t~B + η|~k|2 ~B = i~k × (

~u× ~B) (.)

The four basic equations (.), (.), (.) and (.) have a similar structure. The

dissipation terms on the left hand side can be computed exactly (see Section .),

while the convolution on the right hand are computed in real space and transferred

back to Fourier space by a .

As the Fourier modes of a real function obey the symmetry

~u−~k = ~u~k∗

only the positive z components are computed and stored by the .

A reason for using a spectral method to simulate turbulence is the high accuracy

of this method.

. Solving the basic equations

.. Accuracy of the spectral method

The spectral representation uK(x) of a function u(x),

uK(x) =K∑

k=−K

ukeikx, (.)

is very accurate. The approximation error ‖u− uK‖Lp for a periodic Cm function

u(x) is (see Canuto et al. [])

‖u− uK‖Lp(0,2π)

≤ CK−m‖u(m)‖Lp(0,2π)

. (.)

For an infinitely differentiable function, the approximation error is smaller than any

power of 1/K. Then the convergence is exponential. This behavior is commonly

called infinite accuracy.

In addition the dissipation of the numerical scheme is negligible. For example

ordinary finite difference or volume schemes show numerical dissipation because

of the error made by differentiation of a function. Using a spectral method the

derivative is exact. A very small numerical dissipation stems from the removal of

the dealiasing error as explained in the following section.

.. Dealiasing

When computing a product of two functions in real space and transforming this

product back to Fourier space an error occurs, called aliasing error. The Fourier

transformation of the real product contains modes of higher frequency than the

initial functions. These lie outside the resolved range. They are folded back into

the resolved spectrum by the fast Fourier transform. This generates an error at large

mode numbers. It is possible to remove this error completely by enlarging the grid

by % in each direction and only considering the old modes. However, using this

procedure approximately % of the computational time is spent on unneeded

modes. Therefore in this work a method called spherical truncation is applied. Using

this method all modes in a sphere with a diameter of the computational domain

are taken into account for the simulation. The residual is neglected by setting them

to zero. Vincent and Meneguzzi [] pointed out that this procedure produces a

small error in the dissipation range and is of the same order as the discretisation

error.


.. The constraint of incompressibility

In this section the issue of a solenoidal velocity and magnetic field will be discussed.

An initially zero divergence magnetic field stays solenoidal because of the

equation (.), as can be seen by taking the divergence of (.). For the velocity

field the situation is more complicated.

There are several ways to accomplish the constraint of incompressibility. The

first one is to initially omit the pressure term in ., i. e. solve the equation

∂t~u∗ + (~u∗ · ∇~u∗) = ν∆~u∗,

and afterwards to project the velocity ~u∗ onto its solenoidal fraction. This can easily

be done in Fourier space, where the projection operator takes the form

(δjl −kjkl

|~k|2).

The resulting velocity field

uj = (δjl −kjkl

|~k|2)u∗l

is solenoidal.

Another way to fulfill the condition of incompressibility is to switch from the

velocity formulation to the vorticity formulation .. The incompressibility of the

corresponding velocity field ~u has to be incorporated into the reconstruction from

the vorticity field ~ω. Assuming a solenoidal ~u, in Fourier space the reconstruction

yields

~u~k = i~k × ~ω~k

|~k|2

which certainly leads to ∇ · ~u = 0.

The implementation of the vorticity formulation is slightly faster than the velocity

formulation because of the projection needed.

. Time stepping

For the time integration of the Navier-Stokes (.) and equations (.)–

(.) a low storage Runge-Kutta third order scheme and a trapezoidal Leapfrog

second order scheme are implemented.

. Time stepping

The viscous term ν∆~u in (.) (and similarly the resistive term in (.)) can be

computed exactly. This can be seen by rewriting (.) as

∂t~u = L(~u) + ν∆~u, with

L(~u) = −(~u · ∇)~u−∇p,

or in Fourier space

∂t~u = L(~u)− ν|~k|2~u. (.)

Introducing a new variable

~u = ~u exp(ν|~k|2t), with

∂t~u = (∂t~u + ν|~k|2~u) exp(ν|~k|2t)

replaces (.) by

∂t~u = L(~u exp(−ν|~k|2t)) exp(ν|~k|2t). (.)

The dissipation term ν∆~u now enters the new equation (.) as the exponen-

tial term and therefore introduces an explicit time dependence. However, the

dissipation can now be handled exactly within every specific time scheme.

In the following two different time schemes implemented will be presented. A

Runge-Kutta third order in the following section and a trapezoidal Leapfrog second

order in Section ...

.. Runge-Kutta third order

The Runge-Kutta scheme used is a low storage integrator of third order introduced

by Shu and Osher []. Only one additional array is needed.

It takes the form

u1 = u0 + ∆tL(u0),

u2 =3

4u0 +

1

4u1 +

1

4∆tL(u1),

u3 =1

3u0 +

2

3u2 +

2

3∆tL(u2).

(.)

The operator L denotes the right hand side (rhs) of the differential equations.

Applying this scheme to the specific time-dependent equation (.), one has to be

aware about the point in time T the intermediate steps u1, u2, and u3 belong to.

Starting at time t implies T (u0) = t. Computing u1 one takes a step of ∆t which


sets the point in time of u1 to t + ∆t. In the same manner one can compute the

point in time for u2 and u3 as T (u2) = 3/4T (u0)+1/4T (u1)+1/4∆t = t+1/2∆t

and T (u3) = 1/3T (u0) + 2/3T (u2) + 2/3∆t = t + ∆t, respectively. Writing (.)

for (.) yields

~u1 = (~u0 + ∆tL(~u0)) exp(−ν|~k|2∆t),

~u2 =3

4~u0 exp(−ν|~k|2 ∆t

2) +

1

4exp(ν|~k|2 ∆t

2) +

1

4∆tL(~u1) exp(ν|~k|2 ∆t

2),

~u3 =1

3~u0 exp(−ν|~k|2∆t) +

2

3~u2 exp(−ν|~k|2 ∆t

2) +

2

3∆tL(~u2) exp(−ν|~k|2 ∆t

2).

(.)

For the computation only one backup of the field ~u0 is needed.

.. Trapezoidal Leapfrog

Beside the Runge-Kutta scheme described above, a trapezoidal Leapfrog is imple-

mented into the code. This scheme is of second order. For a time-dependent right

hand side L(~u, t) a classical Leapfrog scheme reads,

~un+1 − ~un−1

2∆t= L(~un, tn). (.)

It has two time levels, corresponding to the even and uneven n. For the purpose of

stability one has to couple these two levels. This can be done by averaging after

each time step.

The implemented scheme takes the form

~u2n = ~u2n−2 + 2∆tF (~u2n−1), (.)

~u2n+1 = ~u2n−1 + 2∆tF (~u2n), (.)

~u2n =1

2(~u2n+1 + ~u2n−1). (.)

First a preliminary field ~u2n is computed. Then the uneven level ~u2n+1 and after-

wards the final even ~u2n is calculated from the uneven levels by averaging.

To start the simulation from only one level a Euler-step is performed to obtain a

second level needed for the Leapfrog scheme.

. Need for high resolutions

The smallest structures evolving in a turbulent flow have a size of the order of the

Kolmogorov scale η. These structures have to be resolved, i. e. the grid spacing dx

. Need for high resolutions

0.01

0.1

1 10 100

E(|

k|)/

k-5/3

|k|

10243

5123

2563

0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2 2.2 2.4 2.6 2.8

3 3.2 3.4

10 100 1000

S6

S3

10243

5123

2563

Figure .: Comparison of the energy spectrum (left) and Eulerian structure functions (right) for

different numbers of grid points

has to be of the same order as the Kolmogorov length. The largest structures have a

size of the order of the simulation box L. The Reynolds number Re ∼ (L/η)4/3

contains the fraction of the size of the largest to the smallest structures. Therefore

the relation between the Reynolds number and the overall number of grid points

# is,

Re ∼ #49 . (.)

Even for moderate Reynolds numbers of a few thousand, already billions of grid

points are needed.

To investigate features of turbulence within the inertial range a clear inertial range

is needed. If the simulation is performed with an insufficient number of grid points,

no clear scaling region will be observable, because the inertial range is polluted

by the forcing and the viscous range. In Figure . the energy spectrum and the

logarithmic derivative of the Eulerian structure functions are shown for different

numbers of grid points in Navier-Stokes turbulence. In simulations using less than

10243 grid points a clear inertial range, i. e. a plateau is hardly visible. The minimal

specifications required for a simulation of this size are approximately s,

GByte memory and GByte hard disc memory. The largest simulations were

performed on a local Opteron cluster and the Regatta in Jülich.


. Parallelization

Numerical simulations of fully developed turbulence require a vast amount of

computational and memory resources, as was shown in Section .. Today’s large

computers consist of a huge number of processors with distributed memory, i. e.

every manages only a fraction of the entire memory (see ). Therefore

numerical codes for the purpose of simulating turbulence have to be designed for

massive parallel computers with distributed memory.

x

y

z

proc

ess

proc

ess

proc

ess

2π

2π

2π

Figure .: Partitioning of the parallel direction by the

In pseudo-spectral codes more than half of the computational time is spent on

the fast Fourier transforms. This work uses the (Fastest Fourier Transform in

the West) library to compute these transformations. This open source collection

of C-functions compete in efficiency with proprietary libraries. In addition it has

the advantage of being portable, i. e. one can test ones code on a small desktop

computer and recompile and run the same code on a super-computer.

The library parallelizes the computations by dividing the computational

space into slices of equal width. The resulting distribution is depicted in Figure ..

For details concerning the data layout and available routines see -. The

transforms in these slices can be performed locally. To compute the last trans-

form the entire volume has to be transposed, which requires an inter-process

communication. This shows that a fast inter-process network is indispensable.

. The interpolation

Measurements of Lagrangian turbulence need a large number of tracer particles

advanced by the underlying flow in order to obtain reliable statistical results within

a few large eddy turn-over times. Therefore the tracers are treated parallel, too. The

distribution of the entire computational space is given by the . The developed

code assigns a process to every tracer according to the computational slides it

belongs to. Figure . shows an example of partitioning the parallel direction onto

tree s.

0 1 2 3 4 5 6 7 8 9 10 11 12global indices

0physical units

process0 process1 process2

0 2 3 4 5−1−2local indices

process0 process1 process2

1

2/3π 4/3π 2π

Figure .: Partitioning of the parallel direction for the physical fields and tracer particles

If a tracer leaves the domain of his local process it is transferred to a neighboring

process via the Message Passing Interface (). The major part of computational

effort spent for the tracers is the interpolation of the fields at the tracer position.

This is done in a parallel way and presented in the following section. The par-

allelization is naturally possible, because both implemented schemes are local in

the sense, that only local values of the field are necessary for the interpolation

procedure.

. The interpolation

In order to integrate the tracer according to the surrounding flow, the velocity field

has to be interpolated at the tracer’s position. There are two constraints on the

interpolation scheme. First it has to be accurate in order to be able to follow the

nearly singular structures of a turbulent flow. Second it has to parallelize efficiently

because of the usually high number of tracers within the flow. And of course it has

to meet the specific requirements of smoothness corresponding to the intended

analysis.

There are mainly two different interpolation schemes used in literature. The first

consists of cubic splines used by Chevillard et al. [] and Yeung and Borgas

[], the second is a tri-linear scheme used by Biferale et al. [a]. Cubic

splines are accurate and smooth, but due to their non-local character it is difficult


t

w

u

(x, y, z)particle

(xi, yj, zk) (xi+1, yj, zk)

(xi, yj, zk+1)

(xi+1, yj, zk+1)

(xi, yj+1, zk+1) (xi+1, yj+1, zk+1)

(xi+1, yj+1, zk)

Figure .: Particle together with its surrounding cube of grid cells

to parallelize the computation on distributed memory computers. Because most

of the current massive parallel computers use this type of memory, a tri-cubic

interpolation scheme is implemented instead. For comparing the impact of the

interpolation scheme on the numerical results, additionally a tri-linear scheme is

implemented. The computational effort for the integration of the tracers is mostly

spent on the interpolation of the velocity at the tracer’s positions. The effort of the

tri-cubic scheme is approximately three times the effort of the tri-linear scheme.

The differences in the statistical results are presented in detail in Section ..

The overall task for both schemes is the same. They have to compute the values

of a field given on a lattice at an intermediate point. Given a point (x, y, z), one

first has to find the point surrounding grid cell. Figure . shows a tracer together

with its surrounding cube of grid points. At the vertices of this cube the values

of the underlying field f , denoted by f000, f100, f110 etc. are given. The distances

along the coordinate axis are labeled by t, u, and w. The tri-linear as well as the

tri-cubic scheme interpolates the value of f at the tracer’s position (x, y, z). Both

schemes will now be explained.

.. Tri-linear interpolation

The tri-linear interpolation is the most simple method. The sought-after function

is approximated as a linear function, which is determined solely by prescribed

values at the vertices of the tracer surrounding cube.

. The interpolation

It is useful to introduce the abbreviations,

t = x− xi, dx = xi+1 − xi, t =x− xi

dx,

u = y − yj, dy = yj+1 − yj, u =y − yj

dy,

w = z − zk, dz = zk+1 − zk, w =z − zk

dz,

(.)

and denote the function values at the vertices by f(xi, yj, zk) = f000, f(xi+1, yj, zk)

= f100, f(xi, yj+1, zk) = f010 etc. The value of f at the position (x, y, z) equals

f(x, y, z) = f(s0) + ∂xf(s0)t + ∂yf(s0)u + ∂zf(s0)w

+ ∂xyf(s0)tu + ∂xz tw + ∂yzf(s0)uw + ∂xyzf(s0)tuw

up to second order. The partial derivatives are computed numerically as

∂xf(s0) =f100 − f000

dx,

∂yf(s0) =f010 − f000

dy,

∂zf(s0) =f001 − f000

dz,

∂xyf(s0) =f110 − f010 − f110 + f000

dxdy,

∂xzf(s0) =f101 − f001 − f100 + f000

dxdz,

∂yzf(s0) =f011 − f011 − f010 + f000

dydz,

∂xyzf(s0) =f111 − f101 − f110 + f100 − f011 + f001 + f010 − f000

dxdydz.

Now the required function value at (x, y, z) can be computed solely from the

function values at the vertices,

f(x, y, z) = f000(1− t)(1− u)(1− w) + f100(1− u)(1− w)t (.)

+ f010(1− t)(1− w)u + f001(1− t)(1− u)w + f110(1− w)tu (.)

+ f101(1− u)tw + f011(1− t)uw + f111tuw (.)

This scheme is a very easy and fast interpolation method. However, one has to keep

in mind that the interpolated function is just continuous but not differentiable at

the surfaces of the cube. It is therefore not possible to compute, for example, the

acceleration of tracers from the interpolated velocity.

The results achieved with this scheme will be compared to the results of the

following scheme in ..


.. Tri-cubic interpolation

The bi-cubic interpolation scheme (for two dimensions) can be found in Press et al.

[]. For the purpose of this work this scheme is extended to three dimensions.

With the same abbreviation as in the linear case (.) the function value

f(x, y, z) at the tracer position (x, y, z) is

f(x, y, z) =4∑

i=1

4∑j=1

4∑k=1

cijkti−1uj−1wk−1 (.)

up to fourth order. These equations have unknown coefficients cijk. The particle

surrounding cube has vertices. If one specifies equations at each of these vertices

the coefficients are well defined. The first one is the function value (.) itself.

The others are the first, mixed second and mixed third derivative. One uses the

mixed derivatives for reasons of symmetry. The equations are

∂xf(x, y, z) =4∑

i=1

4∑j=1

4∑k=1

cijki− 1

dxti−2uj−1wk−1,

∂yf(x, y, z) =4∑

i=1

4∑j=1

4∑k=1

cijkj − 1

dyti−1uj−2wk−1,

∂zf(x, y, z) =4∑

i=1

4∑j=1

4∑k=1

cijkk − 1

dzti−1uj−1wk−2,

∂x∂yf(x, y, z) =4∑

i=1

4∑j=1

4∑k=1

cijk(i− 1)(j − 1)

dxdyti−2uj−2wk−1,

∂x∂zf(x, y, z) =4∑

i=1

4∑j=1

4∑k=1

cijk(i− 1)(k − 1)

dxdzti−2uj−1wk−2,

∂y∂zf(x, y, z) =4∑

i=1

4∑j=1

4∑k=1

cijk(j − 1)(k − 1)

dydzti−1uj−2wk−2,

∂x∂y∂zf(x, y, z) =4∑

i=1

4∑j=1

4∑k=1

cijk(i− 1)(j − 1)(k − 1)

dxdydzti−2uj−2wk−2.

(.)

The interpolated function is differentiable in the specified derivatives at the

boundary of a grid cell. Since only the function values are given on the grid, the

derivatives have to be computed numerically by finite differences. This is done by

centered second order differences. However, the smoothness of the interpolated

function does not depend on the accuracy of the prescribed derivatives.

. Influence of the numerical precision

The implementation is as follows. First one has to find the particle-surrounding

cube. Then the derivatives (.) are computed at the vertices. The coefficients cijk

can be efficiently computed in the following way. The equations (.) and (.)

at vertices lead to equations of the form A · c = f . This system of equations is

linear with a constant matrix A. This matrix does not depend on the underlying

function f . The inverse matrix A−1 allows for computing the coefficients cijk and

finally the sought-after function value at (x, y, z) by using (.). Because the

matrix A does not depend on the underlying field f . The inverse matrix A−1 has

only to be computed once. Furthermore because only a fourth of the matrix A−1 is

non-zero it is efficient to compute the coefficients directly from the determining

equations instead of performing a matrix multiplication.

The tri-cubic interpolation scheme is local and therefore parallelizes efficiently.

However, compared to the linear scheme it is numerically more expensive. A cubic

interpolation of a single function value needs approximately three times more

computational time than the linear does.

The results achieved with this scheme will be compared to the results of the linear

scheme in the following section. In addition the influence of the floating-point

precision will be analyzed.


The highest achievable Reynolds number depends on the number of grid points

(see (.)) and therefore on the available amount of memory. In addition the

Reynolds number has a direct impact on the extension of the inertial range. To

investigate intrinsic properties of turbulence within the inertial range, a clear

and large inertial range is indispensable. In the literature, nearly all numerical

simulations are performed with double floating-point precision. Recently, the

largest numerical simulation worldwide performed on the Earth Simulator by

Yokokawa et al. [] used single precision data for the velocity field and double

precision data for the calculation of the convolution sums in order to reduce the

amount of memory needed. For this reason it was possible to set up a simulation

of 40963 grid points.

Apart from the extension of the inertial range the accuracy of the numerical

results is of great importance in order to represent the underlying physics correctly.


fields convolutions interpolation

double precision double precision tri-cubic

single precision double precision tri-cubic

single precision single precision tri-cubic

double precision double precision tri-linear

Table .: Floating-point precision configurations and interpolation schemes for –

To investigate the impact of the floating-point precision on the numerical results,

simulations using three configurations of floating-point precision were performed

(see Table .). The first one, which is the most common approach, computes all

fields with double precision (). The second one corresponds to the configura-

tion on the Earth Simulator which uses single precision for the velocity fields and

double precision for the convolutions (). The third one uses single precision

for all fields (), which halves the needed amount of memory compared to

and therefore allows for an increased Reynolds number. The performance of

was approximately % higher than for on a Linux-Opteron cluster.

Concerning Lagrangian turbulence the trajectories of the tracer particles have

to be integrated accurately, especially to follow the motion in the vicinity of the

nearly singular structures precisely. The crucial point is the interpolation scheme

used to obtain the fluid fields at the tracer position. As described in Section .,

the interpolation process consumes most of the computational time spent on the

time integration of the tracers. There are two different schemes implemented in the

code. The first is a tri-cubic scheme, the second a tri-linear scheme. To investigate

the impact of the interpolation scheme on the Lagrangian results, a set of tracers

has been integrated in the same turbulent flow once with the tri-linear () and

with the tri-cubic interpolation scheme (–; see Table .). The initial

points of the tracers were identical both for the run using tri-linear as well as for

the run using tri-cubic interpolation.

The common parameters of the simulations are listed in the fourth column

(RunNs) of Table ..

In total 105 tracers were injected into each flow after a statistically stationary

state had been developed and integrated for approximately four large eddy turnover

times. The statistics are computed from the last three large eddy turnover times for

all runs.


As mentioned in Section .. a central point of many theories of turbulence

are the scaling laws of the energy spectrum. The left part of Figure . shows the

computed energy spectra from –. Because of the relatively low Reynolds

1e-06

1e-05

1e-04

0.001

0.01

0.1

1 10 100

E(k

)

k

RUN1RUN2RUN3

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

10 100

Sp

l/η

S2

S4

S6

RUN1RUN2RUN3

Figure .: left: energy spectra from to , right: Eulerian longitudinal structure functions

S2, S4 and S6 from top to bottom from to

number, no clear scaling range is visible. More important for the purpose of

estimating the impact of the floating-point precision on the results is the fact

that the spectra of the different runs are hardly distinguishable. Even higher order

statistics such as the Eulerian longitudinal structure functions look the same for all

types of floating-point precisions (see right part of Figure .).

The according Eulerian probability density function () of velocity increments

yields a more subtle comparison. The left part of Figure . shows the s for the

variable ux(x+2dx)−ux(x) normalized to unit variance. The s just differ within

the errors due to the finite statistical ensemble, but the overall shape is identical.

Therefore the floating-point precision has no impact on the considered Eulerian

statistical properties of turbulent flows. One can suspect that also other Eulerian

statistical quantities are unaffected by the underlying floating-point precision.

In the following the influence of the interpolation scheme on the Lagrangian

statistics will be considered. The right figure in . shows trajectories of a single

particle starting at x for to . All tracers displayed start at the same initial

position. The stays close to for the longest time as one would expect.

The sudden deviation of , and from is due to the chaotic

character of the turbulent flow. Small differences in the numerical integration add


1e-05

1e-04

0.001

0.01

0.1

1

-6 -4 -2 0 2 4 6

P(δ

lu)

δlu/σδlu

RUN1RUN2RUN3

-2-1 0 1 2

-1 0

1 2

3 4

2

3

4

5

6

7

x starting point

RUN1RUN2RUN3RUN4

Figure .: left: Eulerian probability density function for ux(x+2dx)−ux(x) from to ,

right: Trajectories of a single particle from to

up in time to significant differences in the velocity field. The trajectory of

deviates first from the others. This implies that the interpolation scheme has a

greater impact on the trajectories than the floating-point precision.

To analyze the impact of differing single particle trajectories on the statistics of an

ensemble of fluid elements the s of the Lagrangian acceleration were computed.

These s for to are shown in the left part of Figure ..

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

-30 -20 -10 0 10 20 30

P(δ

τv)

δlv/σδτv

RUN1RUN2RUN3RUN4

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

1 10 100

Sp

t/η

S2

S4

S6

RUN1RUN2RUN3RUN4

0 0.5

1 1.5

2 2.5

3 3.5

4

1 10

Figure .: left: s of Lagrangian velocity increments from to , right: Lagrangian

structure functions from to , inset: logarithmic derivative of S2 (bottom) and S4 (top)

The differences between the runs with different floating-point precisions differ only

within the statistical fluctuations. The computed with the tri-linear interpo-

. Simulations

lation in is slightly narrower than the with the tri-cubic interpolation

in . This is because the tri-cubic interpolation scheme is more capable to

follow the trajectories of the nearly singular structures (vortex tubes) which are

responsible for the stretched tails of the s. As the broadness reflects the degree

of intermittency and the Reynolds number of the turbulent flow (see Vincent and

Meneguzzi []), the tri-linear interpolation scheme might underestimate the

degree of intermittency. The right figure in . shows the corresponding Lagrangian

structure functions. These clearly display no differences for to , i. e. no

dependence on the floating-point precision.

Concerning the interpolation scheme, the Lagrangian structure functions slightly

differ for and . As for the s, the interpolation scheme has a small

impact on the shape of the measured Lagrangian structure functions. The inset of

the right figure in . shows the logarithmic derivative of the second and fourth

order structure function. Due to the limited Reynolds number it is difficult to

conclude that the differences according to the interpolation scheme are substantial

within the inertial range and would therefore yield different scaling laws.

In summary the floating-point precision used has no significant influence on the

Eulerian and on the Lagrangian statistical results. The differences regarding the

interpolation scheme are more pronounced, with the tendency that the tri-cubic

interpolation scheme is more capable to reproduce the intermittent character of

the Lagrangian statistics.

. Simulations

This section will provide detailed information about the simulations performed

and analyzes throughout this work. The simulations differ mainly in the type, i. e.

Navier-Stokes or , and the chosen Reynolds number Rλ. For hydrodynamic

and magnetohydrodynamic systems simulations with a highest achievable Reynolds

number and a more conservative resolution were performed each. The parameters

of the simulations are listed in Table ..

The Reynolds number depends on the ratio of the integral scale to the dissipation

scale (see .). The integral scale is unaffected by the specific parameters of the

simulation and is approximately half the size of the entire box. The dissipation

scale depends on the viscosity chosen. The smaller the viscosity the smaller are

the smallest structures of flow. A small viscosity implies a large Reynolds number.


However, the simulation has to be well resolved. That means that the scales on

which the dissipation mainly acts have to be resolved. However, it is possible and a

usual procedure to choose the dissipation scale smaller than the grid resolution to

examine features within the inertial range of scales. This is done for RunNs and

RunMhd. In addition two simulations with the same number of grid points but a

dissipation scale equal to the grid spacing are performed (RunNs and RunMhd).

RunNs and RunNs are Navier-Stokes simulations with a smaller number of grid

points but maximal possible Reynolds number. All simulation were performed

with a negligible magnetic helicity and cross helicity. The initial conditions of the

velocity and magnetic fields are described in the following section.

The number of tracer is chosen to obtain reliable Lagrangian statistics within a

few large eddy turn-over times. Due to fluctuations in the turbulent parameters

such as the total energy one has to average the statistics over several large eddy

turn-over times. Therefore, it is unprofitable to integrate too many tracers. The

numbers given in Table . are based on experience. On average there is a tracer in

every cube with a edge-length of grid points. The initial conditions of the tracers

are described in the following section.

All simulations make use of the Runge-Kutta third order time scheme because it

is more accurate than the trapezoidal Leapfrog scheme. The interpolation scheme

in all runs is the tri-cubic one because of its high accuracy. Only in Section . the

linear scheme is considered.

The visualization is done by using the Advanced Visual System ().

.. Initial conditions

There are some constraints on the possible initial conditions. In incompressible

Navier-Stokes flows the velocity field has to be divergence-free and in addition to

that the magnetic field in flows. The initial velocity and magnetic field are

prepared in Fourier space. Initial conditions are

~B~k = ~a~k exp(−|~k|2/k20 − iαk), ~u~k = ~b~k exp(−|~k|2/k2

0 − iβk),

with random phases αk and βk. The direction of the initial modes ~u~k and ~B~k are

chosen perpendicular to ~k which satisfies the incompressibility constraint. The

wave number k0 which is the location of the local maximum of the initial energy

spectrum is chosen as k0 = 4 for all simulations.

. Simulations

RunNs RunNs RunNs RunNs RunMhd RunMhd

Rλ 316 178 122 150 234 107

u0 0.18 0.16 0.16 0.15 0.22 0.19

εk 3.5 · 10−3 2.1 · 10−3 2.1 · 10−3 1.5 · 10−3 1.02 · 10−2 7.5 · 10−3

εm – – – – 1.46 · 10−2 11.3 · 10−3

ν 2 · 10−4 8 · 10−4 3 · 10−4 2 · 10−4 1.5 · 10−4 5 · 10−4

ηd – – – – 1.5 · 10−4 5 · 10−4

dx 6.14 · 10−3 6.14 · 10−3 1.23 · 10−2 2.45 · 10−2 1.23 · 10−2 1.23 · 10−2

η 2.45 · 10−3 6.4 · 10−3 1.1 · 10−2 8.6 · 10−3 4.3 · 10−3 1.1 · 10−2

τη 0.12 0.28 0.37 0.37 0.12 0.26

L 1.8 2 1.9 2.1 2.5 2

TL 10 11 12 14 6.3 5

N3 10243 10243 5123 2563 5123 5123

Np 5 · 106 1 · 106 1 · 106 1 · 105 1 · 106 1 · 106

Table .: Parameters of the numerical simulations. Rλ: Taylor-Reynolds number√

15u0L/ν,

u0 =√

2/3Ek, Ek: kinetic energy, Em: magnetic energy, E = Ek + Em, εk: kinetic energy

dissipation rate, εm: magnetic energy dissipation rate, ε = εk + εm, ν: viscosity, ηd: resistivity,

η: dissipation length scale (ν3/εk)1/4, τη: Kolmogorov time scale (ν/εk)1/2, L = (2/3E)3/2/ε:

integral scale, TL = L/u0: large-eddy turn-over time, N3: number of collocation points, Np:

number of particles, Navier-Stokes simulations: Run–Run, simulations: Run, Run

In turbulence the ideal invariants, the magnetic helicity (.) and cross

helicity (.) are important for the subsequent evolution of the flow dynamics.

While a dominant magnetic helicity in the initial condition ends up in a force

free configuration with ~j ‖ ~B a large initial cross helicity results in an Alfvénic

configuration with ~u ± λ~B. In addition finite magnetic helicity slows down the

turbulent energy decay (see e. g. Biskamp and Mueller []) because the magnetic

helicity is better conserved than the energy. The simplest and most dynamical con-

figuration has both vanishing magnetic helicity and cross helicity. All simulations

were done with this configuration, which can be achieved by appropriate choices

of the parameter αk and βk.

Initially the tracers are distributed randomly in the entire cube to sample the

volume homogeneously and allow for differing tracer separations. As can be seen

from Figure . the initial Poisson distribution is invariant during the simulation.

Therefore, the integration of the tracers is volume preserving.


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

5 10 15 20 25 30 35 40

P(#

)

#

T=t0T=t0+2TL

poisson PDF

Figure .: of the tracer density in Navier-Stokes turbulence, coarse grained in boxes of size

L/32, # denotes the number of particles in each box, Rλ = 316

.. Forcing

Freely decaying turbulence loses its energy due to the viscous transformation of

kinetic energy into heat. For many statistical quantities like the energy spectrum

it is appropriate to consider temporal averages in order to minimize the natural

statistical fluctuations. In order to achieve a statistical stationary state one has to put

in energy continuously. There are several possible ways of forcing. Conveniently the

forcing is applied to large scales. This work follows Biskamp and Mueller [],

who freeze the lowest order modes in Fourier space. The modes with an absolute

wave numbers |~k|,

kmin ≤ |~k| ≤ kmax,

are kept constant, i. e. are excluded from the time integration. In this work the

modes with |~k| ≤ 2 are kept constant on the one hand to minimize the impact

of the forcing on the inertial range and on the other hand to assure a reasonable

isotropic flow.

The type of forcing used in this work has the advantage that it is readily applicable

to the case of an -flow. The alignment of the magnetic and velocity field, which

switches off the interaction, is avoided by keeping constant the large flow structure.

. Code design

In addition the inverse cascade of the magnetic field would lead to condensation in

one |~k| = 1 mode. The type of forcing used also avoids this undesirable state.

In practice the simulations start from the initial state and the turbulence decays

freely for a few large eddy turn-over times. Within this period of time a generic

large scale flow has evolved. After that the forcing is switched on. Again after few

large eddy turn-over times the turbulence is statistically stationary and statistical

quantities like the total energy or total enstrophy fluctuate around a mean (see

Figure .).

0.0505

0.051

0.0515

0.052

0.0525

0.053

0.0535

0.054

0 0.5 1 1.5 2 2.5 3 3.5 4

tota

l ene

rgy

TL

31

32

33

34

35

36

37

38

0 0.5 1 1.5 2 2.5 3 3.5 4

tota

l ens

trop

hy

TL

Figure .: Fluctuating energy (left) and enstrophy (right) in statistically stationary Navier-Stokes

turbulence

. Code design

The pseudo-spectral code is developed in ++. It is based on a Fortran code used

by Biskamp and Mueller []. The Fortran code uses a trapezoidal Leapfrog

scheme to solve the equations. This code has been completely rewritten in

++ to gain more flexibility in handling different time schemes and incorporating

the advection of tracers. Apart from the the code makes no use of existing

open source libraries to assure a maximal portability.

The design applies object-oriented methods. The physical fields are matrices able

to perform s. All subtle issues concerning the parallelization and data layout are

encapsulated into the matrix class. The tracers are objects, too.


The decision which physical system (Navier-Stokes or ), time scheme

(Runge-Kutta or trapezoidal Leapfrog), interpolation scheme (tri-cubic or tri-

linear) and which floating-point precision (double or single) are used is taken at

compile time by use of templates.

As the amount of memory and data is extraordinarily high for a simulation of

10243 grid points ( GByte for the velocity field), the storage of the physical fields

has to be paid attention to. The library handling the input and output of the fields

has been developed at the Department of Theoretical Physics I in Bochum by

Holger Sebert. It is capable to rearrange the data according to the number of s

in a flexible way. Furthermore an interface to the visualization software exists.

All post processing diagnostics on the physical fields were done using .

Chapter

Summary

This work uses high resolution direct numerical simulations of the Navier-Stokes

and equations to compute the Lagrangian statistics of turbulent hydrodynamic

and plasma flows. A numerical code has been developed which is capable of

accurately integrating the fundamental equations and precisely advancing tracers

according to the velocity field.

This work focuses on the phenomenon of intermittency in turbulent flows. To

measure the degree of intermittency conveniently the scaling behavior of structure

functions is considered. These depend either on velocity differences over spatial

separations or on velocity differences along trajectories of fluid elements. While

the former approach, called Eulerian framework, has widely been used for a long

time, the latter, called Lagrangian framework, received increasing attention in the

last few years. This is due to the enormous experimental efforts required to track

individual particles in fully developed turbulent flows. Recently two experiments

by Voth et al. [] and Mordant et al. [] successfully measured Lagrangian

intermittency in Navier-Stokes flows using optical and acoustical techniques, re-

spectively. Biferale et al. [a] performed numerical simulations of Lagrangian

Navier-Stokes turbulence and proposed a model describing the anomalous scaling

behavior of the Lagrangian structure functions. Their measurements and predic-

tions deviate from the measurements of the experiments. Several major issues of

Lagrangian intermittency are still under discussion. This work aims at clarifying

and explaining the controversial points of view. Especially the prediction of the

model is reconciled with measurements of the experiments.

This work provides an important contribution to the understanding of La-

grangian intermittency. This is achieved by considering the Lagrangian statistics in

Chapter Summary

conducting turbulent flows in addition to neutral turbulent flows. The comparison

of these turbulent systems yields properties of the tracer dynamics which are dif-

ficult to observe considering only Navier-Stokes turbulence. It is worth stressing

that Lagrangian intermittency in conducting flows has not been analyzed in the

literature, yet.

In order to test the numerical methods and provide detailed information on

Eulerian intermittency of the considered flows the probability density functions

(s) and corresponding structure functions are computed. They are in good

agreement with experiments (see e. g. Anselmet et al. []).

This work investigates the Lagrangian s in Navier-Stokes and turbulence.

In both systems a transition from Gaussian distributions for large increments to

highly intermittent s for small increments is observed. In turbulence both

the velocity field as well as the magnetic field show this transition. In conclusion

the velocity field and magnetic field increments are highly intermittent variables.

The degree of intermittency is similar. The tails of the magnetic field for small

increments are as stretched as the tails of the acceleration s.

Considering Lagrangian structure functions reveals that both Navier-Stokes

turbulence and turbulence show anomalous scaling behavior. An important

result is that Navier-Stokes turbulence is more intermittent than turbulence.

The situation is reversed in the Eulerian framework. This interesting difference

can be attributed to the differing influence of the coherent small scale structures

on intermittency considered either in the Eulerian or Lagrangian framework. In

hydrodynamic flows the most dissipative structures are filaments of dimension one

while in these are current and vortex sheets of dimension two. The higher

the dimension the higher is the degree of intermittency measured in Eulerian

coordinates. In contrast to that the dimension a tracer measures while it is trapped

in a vortex filament in Navier-Stokes turbulence is two. The trapping lasts for

several dissipation times, during which the filament unfolds in a second dimension

in Lagrangian coordinates. In the tracers are reflected by the current sheets.

The strongest accelerations happen along a line of dimension one. This shows that

the dimensions of the coherent structures are exchanged when switching from the

Eulerian to the Lagrangian framework. These observations explain the deviation

of a multifractal model proposed by Biferale et al. [a] of Lagrangian Navier-

Stokes intermittency from measurements done by Mordant et al. [], Xu et al.

[].

In this work attention has also been paid to the ongoing dispute on the range for

evaluating the Lagrangian scaling exponents. A clear scaling range is not observable

even for the highest Reynolds numbers both in numerical and experimental mea-

surements (see Biferale et al. [b], Mordant et al. []), while there is a clear

range in the Eulerian framework. This work investigates the scaling behavior of

frozen turbulence. Tracers are advanced in a stationary velocity field taken from a

dynamical simulation. In this case a clear scaling range is visible. However, in the

dynamical case this range is spoiled. As an explanation the temporal decorrelation

of the flow which arises neither in the frozen nor in Eulerian statistics is suggested.

The range used in experiments and in this work, spanning from less than a dissipa-

tion time to a few dissipation times, is reasonable because it is the same as the range

displaying scaling behavior in frozen turbulence. A further argument for this range

is the clear scaling range of an additionally considered temporal velocity increment

in evolving turbulence. Again this range coincide with the range chosen for the

evaluation of the Lagrangian exponents in this work. The degree of intermittency

of frozen turbulence is just in between the Eulerian and dynamical Lagrangian

case. Therefore, the pure change to Lagrangian coordinates enhances the observed

intermittency. The temporal evolution leads to an additional contribution.

In order to investigate the influence of coherent structures in turbulent flows

on Lagrangian intermittency the case of a decorrelated flow has been analyzed.

The decorrelation is achieved by rotating the Fourier modes after each time-step.

This procedure destroys the coherent structures. The decorrelated flow shows

Kolmogorov scaling both in Eulerian and Lagrangian coordinates. Therefore no

intermittency occurs in absence of coherent structures when measured either in

the Eulerian or Lagrangian framework. In two-dimensional turbulence Lagrangian

structure functions do show anomalous scaling behavior while no intermittency

occurs in the Eulerian framework (see Kamps and Friedrich []). The analysis

of the decorrelated flow points out the importance of coherent structures for the

appearance of Lagrangian intermittency.

In addition to the standard Lagrangian temporal increments two different types

of Lagrangian increments are considered. One of these measures the increment

of the norm of the tracer velocities. The statistics are highly intermittent which

demonstrates that in addition to pure trapping events a second influence has

to be taken into account to explain the strong deviations from the scaling.

The second alternative increment considered is an equal time increment. Velocity

Chapter Summary

differences are computed at positions along the trajectories of tracers. The velocities

are taken at equal points in time. They show a clear scaling range and the same

scaling behavior as the frozen statistics.

Stationary fully developed turbulence is numerically generated on a periodic

grid up to a Taylor-Reynolds number of Rλ = 316 and integrated forward in time

using a Runge-Kutta third order scheme. The trajectories of millions of tracers

are computed by interpolating the velocity at the tracer’s position by a tri-cubic

interpolation scheme. The differences in the results arising from the use of a

tri-linear scheme are analyzed in detail. The tri-linear scheme shows a tendency

to underestimate the degree of intermittency when measured in the Lagrangian

framework. This is due to the fact that the tri-cubic scheme is more capable

of following tracer exposed to strong accelerations. However, the impact on the

statistical results is small. Furthermore the influence of the floating-point precision

used for the simulations has been analyzed. Recently the world’s largest simulation

was done by Yokokawa et al. [] on the Earth-Simulator with mixed single

and double floating-point precisions. Three different floating-point configurations,

including the Earth-Simulator setup, are considered in this work. No differences in

the statistical results can be observed when either using double or single floating-

point precision. The usage of single precision data halves the required amount of

memory compared to double precision data. Therefore higher Reynolds numbers

are achievable with single precision data.

The investigation of Lagrangian turbulence also has implications for other fields

of active research. Lagrangian statistics correspond to the statistics of small par-

ticles in turbulent flows in the limit of vanishing inertia. These arise in many

environmental and industrial application such as rain formation (see e. g. Falkovich

et al. []) or combustion devices (see e. g. Post and Abraham []). A better

understanding of the limiting case of Lagrangian turbulence will help to solve

problems concerned with real small particles.

The modeling of frozen Lagrangian statistics will be a fruitful continuation to

shed new light on the dynamical case. The proposed new time increment might

be a valuable tool which can be applied to investigate the differences between two-

and three-dimensional Lagrangian intermittency.

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