Stochastische

Punktwirbeldynamik

-

Stochastic

Point Vortex Dynamics

als Diplomarbeit vorgelegt von

Michael Rath

Januar 2010

Westfalische-Wilhelms-Universitat Munster

Contents

Introduction 1

1 Theoretical foundations 51.1 Hydrodynamics and vortices . . . . . . . . . . . . . . . . . . . . . . . 51.2 Determinism and stochastics in hydrodynamics . . . . . . . . . . . . 10

1.2.1 The Kraichnan model . . . . . . . . . . . . . . . . . . . . . . 101.2.2 The Fokker-Planck equation . . . . . . . . . . . . . . . . . . . 111.2.3 Derivation of a Fokker-Planck equation for a finite set of Langevin

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Vortices in a Kraichnan velocity field 212.1 One vortex in a Kraichnan velocity field . . . . . . . . . . . . . . . . 212.2 Two vortices in a Kraichnan velocity field . . . . . . . . . . . . . . . 25

2.2.1 The cases Γ1,Γ2 ∈ R with Γ1 6= −Γ2 . . . . . . . . . . . . . . 272.2.2 The case Γ1 = −Γ2 ≡ Γ: Two according to amount equipol-

lent, counter-rotating vortices . . . . . . . . . . . . . . . . . . 362.3 Derivation and implementation of additional constraints . . . . . . . 41

2.3.1 Consequences of the postulation of incompressibility . . . . . 412.3.2 Consequences of the postulation of isotropic statistics . . . . 42

2.4 The relative part of the motion . . . . . . . . . . . . . . . . . . . . . 442.4.1 Change to polar coordinates . . . . . . . . . . . . . . . . . . . 472.4.2 Mode ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.4.3 Choice for θ(r) and ψ(r) . . . . . . . . . . . . . . . . . . . . . 50

2.5 Solution for vortex profiles v(r) = 12πr1−ξ . . . . . . . . . . . . . . . 52

2.5.1 Implementation of initial conditions . . . . . . . . . . . . . . 542.5.2 Graphical presentation of the results . . . . . . . . . . . . . . 59

2.6 Solution for vortex profiles v(r) = 12πr . . . . . . . . . . . . . . . . . 65

2.6.1 Asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . . 652.7 A remark on the analogy to anyon statistics . . . . . . . . . . . . . . 71

2.7.1 The two anyon system . . . . . . . . . . . . . . . . . . . . . . 71

Conclusion 75

A Appendix 77A.1 The special case ξ = 0 - direct derivation . . . . . . . . . . . . . . . . 77A.2 Fractional statistics of anyons . . . . . . . . . . . . . . . . . . . . . . 80A.3 Modified Runge-Kutta method - source code . . . . . . . . . . . . . 81

i

Contents

Bibliography 85

ii

Introduction

“Observe the motion of the surface of the water, which resembles that ofhair, which has two motions, of which one is caused by the weight of thehair, the other by the direction of the curls; thus the water has eddyingmotions, one part of which is due to the principal current, the other torandom and reverse motion.”(Leonardo da Vinci)

The subject of fluid turbulence has attracted the attention of physicists over cen-turies. Leonardo da Vinci was probably the first one to use the term “la turbolenza”to describe the corresponding flow phenomenon. Being a derivative of ‘turbare’(Latin, in English: to disorder), turbulence refers to the irregular motion of liquidsand gases. Turbulence arises in any liquid or gas provided that the flow velocityis sufficiently high. In nature as well as in engineering, it is the dominating flowregime; on the one hand, it accelerates mixing processes, on the other hand it leadsto considerable flow resistances of vehicles. Concerning the structure of a turbulentflow one observes that it is extremely complex: it is unsteady, random and chaotic.This complexity arises from the inherent non-linearities of the fluid motion, whichare essential for the creation and maintenance of turbulence. Turbulence itself isstill the most important unsolved problem of classical mechanics (cf. [28]).

Fluid turbulence as a continuum phenomenon can be described by the classical lawsof fluid motion; the flow of gases and liquids is governed by a partial differential equa-tion discovered independently by Stokes and Navier. It is the backbone of modernfluid dynamics and has to be understood as Newton’s law of motion formulated forthe flow of a fluid. The Navier-Stokes equation may describe both laminar and tur-bulent flows. Many complexities of fluid turbulence may be found in the equation inform of non-locality, viscous diffusion and dissipation. An important scientific studyof turbulence dates back to 1883, when Osborne Reynolds established pipe flow ex-periments (cf. [25]) where he showed that the appearance of eddies depends on thevalue of a particular combination of parameters - the so-called Reynolds number. Ifit exceeds a certain limit, the fluid viscosity is no longer able to stabilize the motion.

The vital constituents of all turbulent flows are whirling eddies and vortices of awide range of the size. In 1922, Richardson delivered a qualitative description of theso-called turbulent energy cascade:

“Big whorls have little whorls that feed on their velocity, and little whorls

1

INTRODUCTION

have lesser whorls and so on to viscosity.”(Lewis F. Richardson)

As this quotation shows he painted a picture in which the energy is passed downfrom the large scales of the motion to smaller scales until reaching a sufficientlysmall length scale such that the tiniest eddies - being still significantly larger thanthe intermolecular distances - eventually are viscously dissipated into heat. Addingin this context constantly energy on a large scale comparable to the size of thesystem, the so-called integral scale, we arrive in a flux balance or dynamic equilib-rium, respectively. In his theory of 1941, Kolmogorov used these qualitative ideasto establish the first quantitative theory in turbulence research (cf. [18]), withinfounding the field of “Statistical Hydrodynamics”. Postulating that for a very highReynolds number the small scale turbulent motions are statistically isotropic, hemade predictions claiming universal statistics on medium and small scales, in theso-called inertial range. These predictions hold for low order statistics.1.

The Navier-Stokes equation itself was of limited use for about more than a centuryuntil the computer processing rates became sufficiently fast such that numerical so-lutions could be obtained for some simple two-dimensional flows in the 1960s. Thespecial role of two-dimensional turbulence is that it is nowhere realized in nature orthe laboratory in a strict sense, but only in computer simulations. Nevertheless, onecan justify investigating it - it idealizes geophysical phenomena for example in theatmosphere or oceans and provides a way to model them (cf. [20]). Additionally,results may be compared to three-dimensional investigations. There were also de-veloped various experiments in the recent time to model two-dimensional turbulencein the labaratory (cf. [6]). The probably most striking difference between two- andthree-dimensional turbulence is that in two-dimensional systems one gets an inverseenergy cascade going to larger scales, detaining the system to dissipate energy atsmall scales. (cf. [28]). Nonetheless there can be found a different direct cascade,the so-called enstrophy cascade. The quantity “enstrophy” is directly related to thevorticity of the flow.

Experiments as well as numerical calculations give the indication that so-called co-herent structures characterize turbulent flow fields, whereby the concept “coherentstructure” denotes a spatiotemporal collection of vorticity, which emerges for longertimes. A still open question is why this local structures develop. Hence it is notastonishing that vortices as the basic elements of all turbulent flows are still in thefocus of investigation and interest. Unfortunately, the Navier-equation has only fewexact vortex solutions, the simplest case is a point vortex with localized vorticity inone point of the plane as a solution of the two-dimensional Navier-Stokes equationwith vanishing viscosity, the so-called Euler equation. An approach to model flow

1Compare also [10] for an extensive overview. - We may additionally note here that after beingconfronted with systematic deviations of his theory both experimental as well as numerical,embodied in the so-called phenomenon intermittency, Kolmogorov refined it in 1962.

2

INTRODUCTION

fields is to take a system consisting of a finite set of point vortices. It is possible toinvestigate its properties on the basis of a finite-dimensional Hamiltonian system.And since it is Hamiltonian, it is possible to perform a statistical treatment alongthe lines of equilibrium statistical mechanics (cf. [23]). Using this ansatz, Onsagerfound a possibility to explain theoretically the phenomenon of the tendency of vor-tices to form large structures (cf. [7]).

So coherent structures as well as fluctuating velocity fields are characteristics ofturbulent flows and should be comprised in statistical considerations of hydrody-namics. In this sense, this thesis can be seen as an attempt to get familiar withthe concept of taking in account both deterministic dynamics and methods of sta-tistical physics. Therefore we investigate a simple setup containing ingredients fromboth: We take point vortices and investigate their stochastic dynamics under theinfluence of an incompressible, stochastic Kraichnan velocity field (e. g. introducedin [8]). Kraichnan velocity fields are a useful tool to model short-time correlatedvelocity fields and are δ-correlated in time, guaranteeing essential simplifications inthe analytical calculations. It is rather a purely academic motivation, leaving theopportunity to perhaps expand and match the problem to more applied questions.So simply speaking, this thesis is about first deriving a Fokker-Planck equation for aset of point vortices in a Kraichnan velocity field, and, subsequently, solving it. Fora constant diffusion coefficient and two identical vortices with the “standard vortexprofile”- which qualitatively is proportional to the reciprocal distance to the vortexcore, this problem has already been solved by Agullo and Verga (compare [2]). Withrespect to generalization, there are some open issues to the problem of two vortices:

• One question is whether and under which circumstances it is also solvable forvortices with different circulation, also with regard to limiting cases.

• A second question would be if there exist solutions for the problem in the caseof a non constant diffusion matrix.

• Finally there is the question if it is solvable for different “non-standard” vortexprofiles, which do not necessarily have to be solutions of the basic hydrody-namical equations. This has to be seen as an attempt of generalization whichmay possibly have application in modelling vortices or eddy structures differ-ently. There might also be application in the field of screened vortices (cf.[9]).

The thesis is partitioned into two chapters.

The first chapter introduces the methods and the theoretical foundations we willwork with. After a short treatment of hydrodynamical and statistical basics, wewill derive a general formalism to carry over from the both deterministically andstochastically affected trajectories of particles or vortices in a fluctuating field to astochastic partial differential equation describing the time evolution of the system

3

INTRODUCTION

in a probabilistic manner. In mathematical nomenclature this is: We have a finiteset of n Langevin equations, each containing both deterministic interaction betweenthe ith constituent and the n− 1 other ones and a fluctuating part; these equationsare then conveyed by means of ensemble averaging to one Fokker-Planck equationgoverning the probability density function of the whole system.

The second chapter is the main part of the thesis. We first briefly consider onevortex described by one Langevin equation, lacking the deterministic interactionwith other vortices. Thus we will obtain a simple Brownian motion, and afterhaving applied the formalism derived in the previous chapter, we will see that theFokker-Planck reduces to a simple heat equation. Then we come to the treatmentof two votices. After a change to central and relative coordinates, we will apply theformalism derived before on the two Langevin equations of the vortices in order toobtain a Fokker-Planck equation describing the system. We investigate on vorticeswith arbitrary, different circulation, and decouple the relative and central motion ofthe vortex pair. It will come clear that the problem of solving the central motion isin most cases approximately and sometimes even exactly reduced to solving the heatequation. Afterwards we focus on solving the relative part for different vortex profilesand diffusion matrices. Eventually we derive solutions in the case of scaled diffusionscoefficients and a particularly matching generalized vortex profile. These solutionspass over into already known solutions in limiting cases. The direct derivation ofthe known case can be found in the appendix to compare the results. After havingsolved the equation, the solutions are graphically presented and commented. In thecase of a scaled diffusion matrix and a standard un-scaled vortex profile we give anoutline of an asymptotic solution.A last section of the second chapter describes an analogy of the two point vortexsystem to the system of two so-called anyons. The subject of anyons is a concept outof quantum field theory. Anyons are quasi particles known for example for explainingthe Fractional Quantum Hall Effect (FQHE) (cf. [24]). After a short introduction,we point out the analogy; two free anyons behave statically in an analogous way tothe one of two vortices in a fluctuating flow field.

4

1 Theoretical foundations

In this chapter the theoretical foundations of physical as well as the ones of math-ematical nature concerning the subject will be presented. We will first bring in thebasic hydrodynamic equations; especially we will derive the equations describingpoint vortex dynamics in two dimensions. Subsequently, we introduce the Kraich-nan model for velocity fields and a formalism to derive a Fokker-Planck equation fora set of point vortices in a Kraichnan velocity field.

1.1 Hydrodynamics and vortices

This section deals with the introduction of the basic hydrodynamical equations andthe derivation of equations describing vortex dynamics in two dimensions. Basichydrodynamical equations are known since the 1755, when Euler established theequations for fluids without friction. About 70 years later they were generalized byNavier and Stokes by taking into account molecular friction forces, leading to thewell-known Navier-Stokes equation

∂

∂tu(x, t) + u(x, t) · ∇u(x, t) = −∇p(x, t) + ν∆u(x, t) + f(x, t). (1.1)

Thereby we have

• the velocity field u(x, t),

• p(x, t) = P (x,t)ρ with P (x, t) : pressure and ρ : constant mass density

• ν denotes the kinematic viscosity and

• f(x, t) as the force per volume accumulates all external forces

In the following, we will only consider incompressible fluids with constant massdensity, the incompressibility condition reads

∇ · u(x, t) = 0 (1.2)

Taking the curl of the velocity, we obtain a quantity denoted by “vorticity”

ω(x, t) = ∇× u(x, t) (1.3)

In two dimensions, this quantity tends to build oval structures, like for example theso-called Red Spot on Jupiter (cf. [21]). In three dimensions, thin filaments emerge.

5

1 Theoretical foundations

Considering equation (1.2) and (1.3) as an analogon to magnetostatics, we may usethe Biot-Savart law (cf. [5]) and derive a representation for u(x, t) :

u(x, t) =∫

ω(x′, t)×K(x− x′)dx′ +∇Φ

whereby K(x− x′) = −∇G(x− x′). G(x− x′) denotes the Green’s function of theLaplacian, by definition ∆G(x − x′) = −δ(x − x′). Φ has to fulfill ∆Φ = 0 (cf.[23]). So if we knew the vorticity and corresponding boundary conditions, we couldcalculate the velocity field. Taking the curl of the Navier-Stokes equation, we arriveat the vorticity equation (cf. [15])

(∂

∂t+ u(x, t) · ∇

)ω(x, t) = ω(x, t) · ∇u(x, t) + ν∆ω(x, t) +∇× f(x, t) (1.4)

describing the time evolution of the vorticity.

Vortex dynamics in two dimensions

In two dimensions now, the vorticity has only one vorticity component perpendicularto the plane. If it is the z-component, we get

ω(x, t) · ∇u(x, t) = ωz(x, y, t)∂

∂zu(x, y, t) = 0

for the so-called vortex stretching term in (1.4). If we additionally neglect theexternal forces, our vorticity equation reduces to

∂

∂tω + u · ∇ω = ν∆ω

and for the nonviscous case, ν = 0 this entails

∂

∂tω + u · ∇ω = 0

If we now consider a vorticity field generated by n point vortices,

ω(x, t) =n∑i=1

Γiδ(x− xi(t)) (1.5)

we get

0 =n∑i=1

Γi

[− d

dtxi(t) + u(xi, t)

]∇xδ(xi(t)− x)

Assuming that the n vortices are never in contact in the time of investigation, weobtain for every time in this interval

6

1.1 Hydrodynamics and vortices

d

dtxi(t) = u(xi, t)

Under special conditions vortices may collide, the reason why we want to avoid thisscenario is, that in that particular moment the last two equations would lose theirvalidity.

If we want to obtain the velocity field from the vorticity field now, we have to applythe Biot-Savart law, yielding in our case of two dimensions (cf. [3])

u(x, t) =1

2π

∫(x− x′)⊥

|x− x′|2ω(x′, t)dx′ (1.6)

whereby (x, y)⊥ = (−y, x) by definition is describing an azimuthal trajectory. In-serting (1.5) into (1.6) entails

(a) Two vortices with different circula-tion, rotating around the center of vor-ticity

(b) Two vortices with the same circula-tion, rotating around the center of vortic-ity on the same trajectory

(c) Two vortices with counter-rotatingcirculation, their center of vorticity fol-lows a straight line

(d) Four vortices showing chaotic behav-ior

Figure 1.1: Trajectories of point vortices presented qualitatively, showing purely de-terministic as well as chaotic behavior. Derived using the Runge-Kuttaforth order method.

7

1 Theoretical foundations

d

dtxi(t) = u(xi, t) =

n∑i=1

ΓiK(x− xi(t))

whereby

K(x) =1

2πx⊥

|x|2, |x| 6= 0

is the derivative of the Laplacian Green function and describes the “standard” pointvortex profile1. Combining our results, we finally arrive at a set of differentialequations describing the evolution of the point vortex positions:

d

dtxi(t) = u(xi, t) =

n∑j=1,j 6=i

Γj2π

(xj − xi)⊥

|xj − xi|2(1.7)

With a Runge-Kutta forth order method one can exemplify the behavior of suchsystems of point vortices. Two and three vortex systems are integrable, above fourone arrives in Hamiltonian chaos.(cf. [15]) An important conserved quantity is thecenter of vorticity,

R =

∑j Γjxj∑j Γj

,∑j

Γj 6= 0,

which can be directly observed in the pictures for two point vortices, but of coursealso holds in the case of four vortices. In the case of two counter-rotating vortices,∑

j Γj 6= 0 is not valid, therefore the center of vorticity is not conserved.

Lamb-Oseen vortex

A further vortex solution can be given by the so-called Lamb-Oseen vortex. If wehave a vorticity distribution of the form ω(x, y, t) = ω(r, t)ez with r =

√x2 + y2,

the velocity field is purely azimuthal; reducing the vorticity equation to

∂ω(r, t)∂t

= ν∆ω(r, t),

in fact having the form of a heat equation for ω(r, t). In polar coordinates this is(cf. [15])

∂ω(r, t)∂t

= ν1r

∂

∂r

(r∂

∂r

)ω(r, t),

leading to the solution for the vorticity

1Later we will also allow vortex profiles of the form K(x) = 12π

x⊥

|x|2−ξ , which are of course not

necessarily a solution of the basic hydrodynamical equations.

8

1.1 Hydrodynamics and vortices

(a) Vorticity field of the Lamb-Oseen vor-tex, with viscosity ν = 0.02 after t = 3time units.

(b) Velocity field of the Lamb-Oseen vor-tex, in dimensionless units with viscosityν = 0.02, circulation Γ = 1 after t = 3time units.

ω(r, t) =Γ

4πνtexp

(− r2

4νt

)For the velocity field, one can derive

u(x, t) =Γ

2πr

(1− exp

(− r2

4νt

))eϕ

9

1 Theoretical foundations

1.2 Determinism and stochastics in hydrodynamics

The Navier-Stokes equation is a deterministic, nonlinear partial differential equa-tion. Nonlinear partial differential equation may show complex and chaotic behaviorcaused by local instabilities. If one takes in account that turbulent flows are char-acterized by many degrees of freedom which couple nonlocally, it is obvious that adescription of turbulent fluid motions by a combination of the deterministic, basichydrodynamic equations and probabilstic methods is reasonable. Like in statisticalphysics, the tool of choice in the theoretical treatment is ensemble averaging (cf. forexample [14]). In experiments and numerical simulations the ensemble average isreplaced by the time average.

As already explained in the introductory part, the main focus of this thesis is theinvestigation on two point vortices in an incompressible Kraichnan velocity field. Inthe following, we thus will introduce the concept of the Kraichnan velocity field.After bringing in the basics of the Fokker-Planck equation, we will derive a formal-ism to carry over from a finite set of Langevin equations to one multidimensionalFokker-Planck equation. The reason underlying is, that we want to use this for-malism to investigate generally on the dynamics of n vortices (or particles) in thesame stochastic field, each described by its particular Langevin equation - to even-tually derive one stochastic differential equation describing the time evolution of thelocations of the vortices in a probabilistic sense.

1.2.1 The Kraichnan model

In this section we introduce the concept of the Kraichnan velocity field. The modelof the passive scalar advection introduced by Kraichnan in 1968 (cf. [19]) attracted alot of attention in theoretical and numerical studies. The essence of the model is thatit considers a synthetic Gaussian ensemble of velocities decorrelated in time, definingthe ensemble by specifying the 1-point and the 2-point correlations of velocities:

1. The mean velocity 〈u(x, t)〉 has to vanish

2. The 2-point correlation has to fulfill the relation⟨uα(xi, t′)uβ(xj , t′′)

⟩= d

(α,β)0 − d(α,β)(xi − xj)δ(t′ − t′′)

=Q(α,β)(xi − xj)δ(t′ − t′′) = Q(α,β)ij δ(t′ − t′′)

whereby d(α,β)0 is constant and d(α,β)(xi − xj) is assumed to be ∝ rξ, but actually

only for rather short distances. The latter property models the scaling behaviorof the equal-time velocity correlation functions of realistic turbulent flows in theRe → ∞ limit[11]. It can be motivated by the loss of small-scale smoothness ofturbulent velocities when Re → ∞. The parameter ξ is taken between 0 and 2 sothat the typical velocities of the ensemble are not Lip- schitz continuous2.

2For more detailed examinations and further readings we refer to [11] and [8]

10

1.2 Determinism and stochastics in hydrodynamics

In this thesis, we will always claim isotropic turbulence. This expression means, thatall statistical properties of the flow - especially the average velocity fluctuations -are equal all over the flow field and not depending on the direction. Hence, we haveinvariance of translation and rotation.

If we make the assumption of isotropy and scaling behavior on all scales for now,the velocity 2-point function of the Kraichnan ensemble can be given by (cf. [11])

d(α,β)(r) = (A+ (d+ ξ − 1)B) δαβrξ + (A−B) ξrξrαrβr2

(1.8)

where δαβ is the Kronecker delta function, d the dimension, ξ the scaling exponentand A,B ≤ 0 are two parameters. We have A ∝

⟨(∇r · u(r, t′))2

⟩, so in the in-

compressible case, ∇r · u(r, t′) = 0, which we will consider, we get (cf. also [4] p.19)

d(α,β)(r) = rξ[B(d+ ξ − 1)δαβ −Bξ

rαrβr2

](1.9)

When deriving a Fokker-Planck equation for the Kraichnan model, we will find thatthe diffusion coefficients correspond to the spatial correlation functions, as we willsoon see in section (1.2.3).

Characteristic and of big importance is of course the time decorrelation of the ve-locity ensemble, which is not a very physical assumption. But although it will neverbe exactly fulfilled in nature, this concept is nevertheless valuable. Noting that theδ-correlation with respect to time also implies the Markovian property and thereforeis a good candidate for modelling processes with no memory, it can be taken as agood approximation for turbulent velocity fields with fast-decaying correlation func-tion. Additionally it can be seen as a long time approximation for flow fields withfinite correlation function. Eventually it makes, however, analytic studies easier.

1.2.2 The Fokker-Planck equation

The Fokker-Planck equation is a second order partial differential equation for thetime evolution of a probability density3 f(x1, ...,xn, t), which first was used byFokker and Planck to model the Brownian motion of particles fluids. For one spatialvariable x it reads (cf. [26])

∂

∂tf(x, t) = − ∂

∂xD1(x, t)f(x, t) +

12∂2

∂x2D2(x, t)f(x, t) (1.10)

3Here, we will always talk about the “distribution function” in the mathematical sense as theintegral over the “probability density function”; this denomination is unambiguous in contrastto the non-distinction common in physical literature.

11

1 Theoretical foundations

whereby D1 and 12D2 are the so-called drift and diffusion coefficients; the 1

2 is amatter of definition, often one finds 1, too. Now more generally, for N variables wehave

∂

∂tf(x1, ..., xN , t) = −

N∑i=1

∂

∂xiD

(i)1 f(x1, ..., xN , t)+

12

N∑i,j=1

∂2

∂xi∂xjD

(i,j)2 f(x1, ..., xN , t)

(1.11)with D

(i)1 = D

(i)1 (x1, ..., xN , t) and D

(i,j)2 = D

(i,j)2 (x1, ..., xN , t)

Alternatively, if we think of n variables each consisting of m components, we mayfind

∂

∂tf(x(1)

1 , ..., x(m)1 , ..., x(1)

n , ..., x(m)n , t) =

∂

∂tf(x1, ...,xn, t) =

∂

∂tf(x, t)

= −n∑i=1

m∑α=1

∂

∂x(α)i

D(i),(α)1 f(x, t) +

12

n∑i,j

m∑α,β

∂2

∂x(α)i ∂x

(β)j

D(i,j),(α,β)2 f(x, t)

= −n∑i=1

∂

∂xiD(i)

1 f(x, t) +12

n∑i,j=1

∂2

∂xi∂xjD(i,j)

2 f(x, t)

⇒

∂

∂tf(x, t) = −

n∑i=1

∂

∂xiD(i)

1 f(x, t) +12

n∑i,j=1

∂2

∂xi∂xjD(i,j)

2 f(x, t) (1.12)

as a useful notion for this case. Thereby we defined

x def= (x1, ...,xn) = (x(1)1 , ..., x

(m)1 , ..., x

(1)n , ..., x

(m)n ) and ∂

∂xi

def= ∇xi

Of course the two multidimensional descriptions are equivalent for n ·m = N vari-ables.

The Fokker-Planck equation as a continuity equation

There is a practical property of the solution of the Fokker-Planck equation: If f(x, t)is normalized for a fixed time t0, say

12

1.2 Determinism and stochastics in hydrodynamics

∫Rn·m

f(x, t0)dx = N(t0),

it stays normalized for all time, N(t) = constant ≡ N(t0). This is due to

∂

∂tN(t) =

∂

∂t

∫Rn·m

f(x, t)dx =∫

Rn·m

∂

∂tf(x, t)dx

=∫

Rn·m

dx

− n∑i=1

∂

∂xiD(i)

1 f(x, t) +12

n∑i,j=1

∂2

∂xi∂xjD(i,j)

2 f(x, t)

=∫

Rn·m

dx

− n∑i=1

∇xi ·D(i)1 f(x, t) +

12

n∑i=1

n∑j=1

∇xi ·(∇xj ·D

(i,j)2

)trf(x, t)

= −∫

Rn·m

dx ∇x ·D1f(x, t) +12

∫Rn·m

dx ∇x ·(∇x ·D2

)trf(x, t)

whereby for the Nabla operator we have

∇x =(

∂∂x1

, · · · , ∂∂xn

)=(∇x1 , · · · , ∇xn

)=(

∂

∂x(1)1

, · · · , ∂

∂x(m)1

, · · · , ∂

∂x(1)n

, · · · , ∂

∂x(m)n

)Now we can use the divergence theorem and obtain

= −∮

∂Rn·m=S

ndS ·D1f(x, t) +12

∮∂Rn·m=S

ndS ·(∇x ·D2

)trf(x, t) != 0

We have the so-called natural boundary condition letting the probability densityvanish in infinity. (cf. [26]) More exactly, we have to claim that f(|x| → ∞, t)→ 0faster than 1

|x|nm−1 → 0 (which is fulfilled in most cases) to save the normalizabilityof the probability density function f . We see that all terms have to vanish, since

13

1 Theoretical foundations

we integrate on the boundary of an n ·m-dimensional sphere in infinity. Hence, wefinally obtain

∂

∂tN(t) = 0⇒ N(t) = constant ≡ N(t0)

So in fact, the Fokker-Planck equation behaves as a continuity equation for theprobability measure,

∂

∂tf(x, t) = −∇x · j(x, t)

with probability current density function

j(x, t) = D1f(x, t)− 12(∇x ·D2

)trf(x, t).

The big advantage of this is: given a solution of the Fokker-Planck equation in de-pendence of t, we only have to show for one point in time that it is normalized.

Later, we will in fact always try to show that our solutions of the Fokker-Planckequation correspond to δ-functions for t = 0, which of course are by definition nor-malizable.

Another important feature, which can be derived via path integrals, is that thesolution of the Fokker-Planck equation stay positive if it was initially psoitive. [26]

1.2.3 Derivation of a Fokker-Planck equation for a finite set of Langevinequations

In this section we derive a multidimensional Fokker-Planck equation of the form(1.12) for a finite, n-dimensional set of m-dimensional Langevin equations.4

This derivations holds especially for Kraichnan velocity fields. Later we will applythis generally established formalism on the example of two Langevin equations forn = 2 point vortices which live of course only in m = 2 dimensions, finding them-selves in a stochastic Kraichnan velocity field.

Consider

xi(t) = Ki(x, t) + u(xi, t) , (1.13)

a quite general Langevin equation describing the time evolution of xi. Later, inour case, it will be describing the deterministically as well as stochastically affectedtrajectory of point vortex i. Thereby xi and xi(t) are the space and velocity vectors

4compare [12] for the “one-dimensional” case, say: one physical issue represented in one one-dimensional Langevin equation; by the way, this formalism could easily be generalized to nLangevin equations of not necessarily the same dimension m, but in physics this case is rarelyof interest.

14

1.2 Determinism and stochastics in hydrodynamics

of equation i - we have again x = (x1, ...,xn) = (x(1)1 , ..., x

(m)1 , ..., x

(1)n , ..., x

(m)n ) with

e.g. m = 2 in R2.

Furthermore, Ki(x, t) denotes the deterministic part which contains the determin-istic interaction between vortex i and the other vortices in the system; therefore itdepends on all spatial coordinates of the involved vortices.

u(xi, t) in contrast denotes the stochastical velocity field which is the same for everyLangevin equation evaluated at the particular corresponding coordinate xi. For thestochastic part of (1.13), we assume that it behaves like Gaussian distributed whitenoise 5 which implies for its average 〈u(xi, t)〉 = 0.Its correlation function is assumed to fulfill⟨

u(xi, t′)u(xj , t′′)⟩

= Q(xi − xj)δ(t′ − t′′),

this is a second-order tensor field; in a different notion, we have for u(xi, t) ∈ Rm

and α, β ∈ {1, · · · , m}⟨u(α)(xi, t′)u(β)(xj , t′′)

⟩= Q(α,β)(xi − xj)δ(t′ − t′′)

Note that we have the same stochastic field realized at different locations for everyLangevin equation.

We now consider P (xi, t) = δ(xi − xi(t)) which is a probabilistic representation ofthe path described by xi(t) whereas xi(t) has to satisfy the particular ith Langevinequation (1.13). Generalizing this leads to P (x, t) = δ(x − x(t)). Note that the δ-function has to be understood in terms of the product of δ-functions for every singlecomponent:

δ(x− x(t)) = δ(x1 − x1(t)) · · · δ(xn − xn(t))

= δ(x(1)1 − x

(1)1 (t)) · · · δ(x(m)

1 − x(m)1 (t))δ(x(1)

2 − x(1)2 (t)) · · · δ(x(m)

n − x(m)n (t))

Now we take the ensemble average over all possible paths and introduce the function

f(x, t) = 〈P (x, t)〉 = 〈δ(x− x(t))〉 (1.14)

To derive now a differential equation for f , let us consider a change ∆f(x, t) of f ina time interval τ :

∆f(x, t) = f(x, t+ τ)− f(x, t)= 〈δ(x− x(t+ τ))〉 − 〈δ(x− x(t))〉

5An exact definition may be found in [10] p 41, more details to Gaussian stochastic processes maybe found in [12], p. 86

15

1 Theoretical foundations

If we now put xi(t + τ) = xi(t) + ∆xi(t) and perform a Taylor expansion up topowers quadratic in ∆xi, we get6

∆f(x, t)

=

⟨(−

n∑i=1

∂

∂xiδ(x− x(t))

)∆xi(t)

⟩+

12

⟨n∑i,j

∂2

∂xi∂xjδ(x− x(t)) (∆xi(t)∆xj(t))

⟩

Considering nowt+τ∫t

xi(t′)dt′ = xi(t+ τ)− xi(t) = ∆xi(t)

we may calculate ∆xi(t). To reach this goal, we have to accept that∫ t+τt dt′Ki(x, t′) =

Ki(x, t)τ . Consider this: If Ki(x, t′) is the primitive of Ki(x, t′), then

t+τ∫t

dt′Ki(x, t′) = [Ki(x, t′)]t+τt = Ki(x, t+ τ)− Ki(x, t)?= Ki(x, t)τ =

d

dt(Ki(x, t))τ

and ? is obviously true if we consider

limτ→0

Ki(x, t+ τ)− Ki(x, t)τ

=d

dt(Ki(x, t)).

Now using (1.13), we get

∆xi(t) =

t+τ∫t

xi(t′)dt′(1.13)

=

t+τ∫t

dt′(Ki(x, t′) + u(xi, t′)

)

=

t+τ∫t

dt′Ki(x, t′) +

t+τ∫t

dt′u(xi, t′)?= Ki(x, t)τ + ∆u(xi, t) (1.15)

whereas τ is small and

∆u(xi, t) ≡t+τ∫t

dt′u(xi, t′).

6This is due to fid

dq(t)δ(q − q(t))

fl=

fi− d

dqδ(q − q(t))

fl,

compare also [12]

16

1.2 Determinism and stochastics in hydrodynamics

Now we can evaluate the first term of the Taylor expansion,or the drift, respectively:

⟨(− ∂

∂xiδ(x− x(t))

)∆xi(t)

⟩=

∂

∂xi[〈δ(x− x(t))(Ki(x, t)τ)〉+ 〈δ(x− x(t))∆u(xi, t)〉]

=∂

∂xi[〈δ(x− x(t))(Ki(x, t)τ)〉+ 〈δ(x− x(t))〉 〈∆u(xi, t)〉]

The first step is possible due to a property of the δ-function (details can be found in[26], pp. 69, 70 and pp. 81, 82). The second step may be performed, because xi(t)is determined by times t′ < t and ∆u(xi, t) by times t′ > t (compare also [12]). Nowwe take in account that the stochastic part of (1.13) has to vanish in the average,

〈u(xi, t)〉 = 0, which also implies

〈∆u(xi, t)〉 =

⟨ t+τ∫t

dt′u(xi, t′)

⟩=

t+τ∫t

dt′⟨u(xi, t′)

⟩= 0

because the integration interchanges. This reduces the drift term to

∂

∂xi(〈δ(x− x(t))(Ki(x, t)τ)〉)

The next step will be the calculation of the second term of the Taylor expansion,namely the diffusion. We have

12

⟨∑i,j

∂2

∂xi∂xjδ(x− x(t))(∆xi(t)∆xj(t))

⟩

=12

∑i,j

∂2

∂xi∂xj〈δ(x− x(t))〉 〈(∆xi(t)∆xj(t))〉

(1.15)=

12

∑i,j

∂2

∂xi∂xj〈δ(x− x(t))〉 〈[Ki(x, t)τ + ∆u(xi, t)] [Kj(x, t)τ + ∆u(xj , t)]〉

=12

∑i,j

∂2

∂xi∂xj〈δ(x− x(t))〉

⟨Ki(x, t)Kj(x, t)τ2 + ∆u(xi, t)Kj(x, t)τ

+ Ki(x, t)τ∆u(xj , t) + ∆u(xi, t)∆u(xj , t)〉

=12

∑i,j

∂2

∂xi∂xj〈δ(x− x(t))〉

(⟨Ki(x, t)Kj(x, t)τ2

⟩+ 〈∆u(xi, t)Kj(x, t)τ〉

+ 〈Ki(x, t)τ∆u(xj , t)〉+ 〈∆u(xi, t)∆u(xj , t)〉)

17

1 Theoretical foundations

Note that the second step may be performed due to the same argumentation usedabove: xi(t) is determined for times t′ < t and ∆xi for times t′ > t.

When inserting now equation (1.15) for ∆xi, one finds terms containing τ2, τ∆u(xi, t)and ∆u(xi, t)∆u(xj , t). 〈∆ui(x, t)〉 vanishes, so the only relevant terms left are thosecontaining ∆u(xi, t)∆u(xj , t) and those ∝ τ2:

⇒12

⟨∑i,j

∂2

∂xi∂xjδ(x− x(t))(∆xi(t)∆xj(t))

⟩

=12

∑i,j

∂2

∂xi∂xj〈δ(x− x)(t)〉

(〈Ki(x, t)Kj(x, t)〉 τ2 + 〈∆u(xi, t)∆u(xj , t)〉

)Note that now the derivatives act also on all other terms next to the δ-function,again this is due to the property of the δ-function already used above (details canbe found in [26], pp 69, 81, 82.)

If we take into account that the correlation function of the velocities is δ-correlatedin time, like a Kraichnan velocity field,⟨

u(xi, t′)u(xj , t′′)⟩

= Q(xi − xj)δ(t′ − t′′),we are able to evaluate

〈∆u(xi, t)∆u(xj , t)〉 =

t+τ∫t

t+τ∫t

dt′dt′′⟨u(xi, t′)u(xj , t′′)

⟩

=

t+τ∫t

t+τ∫t

dt′dt′′Qijδ(t′ − t′′) = Qijτ

If we now combine our results, we find

∆f(x, t) = −∇x · 〈δ(x− x(t))(K(x, t)τ)〉

+12

∑i,j

∂2

∂xi∂xj〈δ(x− x(t))〉

(〈Ki(x, t)Kj(x, t)〉 τ2 + Qijτ

)

⇔ ∆f(x, t)τ

=f(x, t+ τ)− f(x, t)

τ= −∇x · 〈δ(x− x(t))(K(x, t))〉

+12

∑i,j

∂2

∂xi∂xj〈δ(x− x(t))〉

(〈Ki(x, t)Kj(x, t)〉 τ + Qij

)

18

1.2 Determinism and stochastics in hydrodynamics

Finally using equation (1.14)

f(x, t) = 〈δ(x− x(t))〉

and taking the limit τ → 0 we get

∂

∂tf(x, t) = −∇x · (K(x, t)f(x, t)) +

12

∑i,j

∂2

∂xi∂xjQijf(x, t) (1.16)

So we derived a partial differential equation - the Fokker-Planck equation - for asystem of n Langevin equations. It describes the change of the probability densityof the systems state during the course of time. To illustrate this, figure 1.16 showsan example.

Figure 1.2: Example for the case f(x, t) = f(q, t) as a function of time t and onespatial variable q. The dashed line describes the most probable path.(from [12], p. 163)

Note that we only considered derivatives up to second order in our Taylor expansion.Nevertheless, the derived Fokker-Planck equation is exact in this case, because theexamined process characterized by the fluctuating Kraichnan velocity field with itsparticular properties is Gaussian, all higher terms vanish for τ → 0

In the formalism of a Kramers-Moyal expansion, one could alternatively verify thatcaused by the Pawula Theorem all moments except the first two vanish. Thus,the Kramers-Moyal expansion passes in an exact manner into the Fokker-Planckequation. We pass on here without detailed examinations and refer the reader to[12], pp. 85 ff..,164 and [26].

19

1 Theoretical foundations

20

2 Vortices in a Kraichnan velocity field

This chapter is about the analytical treatment of the dynamics of point vortices ina incompressible, statistical isotropic, Kraichnan velocity field. We will on the onehand investigate on “standard” point vortices as solution of the Euler equation withthe vortex profile

K(xj − xi) =1

2π(xj − xi)⊥

|(xj − xi)|2

whereby by convention we have (x, y)⊥ = (−y, x). It is describing the velocity fieldin point xj generated by a point vortex located at xi. On the other hand, we alsoallow vortex profiles of the form

K(xj − xi) =1

2π(xj − xi)⊥

|(xj − xi)|2−ξ

with ξ ∈ [0, 2]. For ξ = 0, we of course get the standard point vortex profile. Itis important to notice that these are not necessarily compatible with the basic hy-drodynamical equations, but may be an alternative way to model systems involvingvorticity. Note that the dimension changes when ξ 6= 0, it has to be matched forexample by a reasonable choice of the dimension of the “circulation” Γ, that thenalso does not hold the properties it does for the standard profile.

In a first step we will describe the system in terms of two Langevin equations.Visually, we modelled the system for different circulations with a modified Runge-Kutta method including white noise. After a change of coordinates, we then applythe formalism we derived in the previous chapter on the two Langevin equationsof two point vortices in order to obtain one Fokker-Planck equation describing thetime evolution of the whole system in a probabilistic manner. Subsequently, we willbreak down the derived Fokker-Planck equation analytically and finally solve it.

2.1 One vortex in a Kraichnan velocity field

The first simple case we want to treat is one point vortex in a Kraichnan velocityfield. We get the Langevin equation

x0 = u(x0, t)

whereby we treat the system of course in two spatial dimensions.

21

2 Vortices in a Kraichnan velocity field

The evolution of the system is by no means dependent on the deterministic velocityfield generated by the vortex. Hence, applying our formalism, we get

∂

∂tf(x0, t) =

12∂2

∂x20

Q00f(x0, t)

=12∇x0 ·Q(0) · ∇x0 f(x0, t)

whereby we used that Q00 = Q(x0 − x0) = Q(0)

So we obtain just the well known Brownian motion.

Assuming1 that

Q(0) =(θ0 00 θ0

)1In section (2.3.2) an equation will be derived, namely (2.23), with which can be obtained that

this is exact in the case of isotropic statistics and if u(x0, t) is incompressible.

Figure 2.1: Hurricane Katrina, on August 28th, 2005. In contrast to the point vor-tices we investigate on, hurricanes like Katrina are (fortunately) decay-ing eddies. Source: http://www.noaanews.noaa.gov/stories2005/images/katrina-08-28-2005-1545z.jpg

22

2.1 One vortex in a Kraichnan velocity field

(a) t=1 (b) t=2

(c) t=3 (d) t=4

Figure 2.2: f(x0, t) = f(x, y, t) for θ0 = 0.02 and t ∈ {1, 2, 3, 4}

we can also write

∂

∂tf(x0, t) =

12θ0∆x0f(x0, t)

whereby

∆x0 =∂2

∂x20,1

+∂2

∂x20,2

is the Laplacian. Assuming that the vortex is initially located at the origin,

f(x0, t = 0) = δ(x0)

the solution can be given by

f(x0, t) =1

2θ0πtexp

(−|x0|2

2θ0t

),

Hence we have a formal similarity to the Lamb-Ossen vortex presented in section(1.1). In [3] the point vortex solution, itself a probability density function describingthe time evolution of the location of one point vortex with strictly located vorticity is

23

2 Vortices in a Kraichnan velocity field

even identified with the Lamb-Oseen vortex, which in contrast describes a vorticityfield of the corresponding form

ωz(r, t) =Γ

4πνtexp

(− r2

4νt

)(compare for example [15]) with viscosity ν and circulation Γ. Note that this is ofcourse not the same; we in fact only have two differential equations equivalent tothe heat equation which of course the same solutions.

24

2.2 Two vortices in a Kraichnan velocity field

2.2 Two vortices in a Kraichnan velocity field

If we consider the motion of two vortices with constant in time vortex profiles un-der their mutual hydrodynamic interaction, perturbed by a fluctuating stochasticalvelocity field, the corresponding Langevin equations read

x1 = Γ2K(x1 − x2) + u(x1, t) (2.1)

x2 = Γ1K(x2 − x1) + u(x2, t)

⇐⇒

x1 = Γ2K(x1 − x2) + u(x1, t)x2 = −Γ1K(x1 − x2) + u(x2, t)

(2.2)

whereas ΓiK(xj −xi)!= −ΓiK(xi−xj) in each case is the deterministic part, which

is for now - with respect to generality - quite arbitrary.Later, it will be chosen to

Figure 2.3: Fiction: Katrina 1 + Katrina 2 attacking Florida (Edited withMS Paint). Source: http://www.noaanews.noaa.gov/stories2005/images/katrina-08-28-2005-1545z.jpg

ΓiK(xj − xi) =Γi2π

(xk − xl)⊥

|(xk − xl)|2−ξ

25

2 Vortices in a Kraichnan velocity field

whereas ξ ∈ [0, 2]. Note that we have the standard vortex profile, itself a solution ofthe Euler equation, only in the case ξ = 0. In all other cases, we have a generalizedmodel not necessarily compatible with the basic hydrodynamic equations, addition-ally following that we have to match the dimension of Γi. Although knowing thatthis stands in contrast to the ordinary dimension of the hydrodynamical circulation,we will nevertheless treat the denotions Γi and “circulation” equivalently.

The stochastical part u(xi, t) is assumed to an incompressible Kraichnan velocityfield, thus it has to fulfill the properties

〈u(xi, t)〉 = 0

and ⟨u(xi, t)u(xj , t′)

⟩= Qijδ(t− t′) = Q(xi − xj) = Q(xj − xi)

Note, that for i ∈ {1, 2}, we have

Q(x1 − x1) = Q(x2 − x2) = Q(0) = Qii

Just to emphasize once again that we have to deal with a double vectorial characterof the considered quantities, we write down the complete correlation matrix:

Q δ(t− t′) =(Q11 Q12

Q21 Q22

)δ(t− t′)

=(〈u(x1, t)u(x1, t

′)〉 〈u(x1, t)u(x2, t′)〉

〈u(x2, t)u(x1, t′)〉 〈u(x2, t)u(x2, t

′)〉

)(2.3)

=

〈u1(x1, t)u1(x1, t

′)〉 〈u1(x1, t)u2(x1, t′)〉 〈u1(x1, t)u1(x2, t

′)〉 〈u1(x1, t)u2(x2, t′)〉

〈u2(x1, t)u1(x1, t′)〉 〈u2(x1, t)u2(x1, t

′)〉 〈u2(x1, t)u1(x2, t′)〉 〈u2(x1, t)u2(x2, t

′)〉〈u1(x2, t)u1(x1, t

′)〉 〈u1(x2, t)u2(x1, t′)〉 〈u1(x2, t)u1(x2, t

′)〉 〈u1(x2, t)u2(x2, t′)〉

〈u2(x2, t)u1(x1, t′)〉 〈u2(x2, t)u2(x1, t

′)〉 〈u2(x2, t)u1(x2, t′)〉 〈u2(x2, t)u2(x2, t

′)〉

Among other things, some parts of this matrix will explicitly be calculated in thefollowing sections for different cases. After a coordinate transformation, we willprimarily focus on the upper left part of the matrix, which will contain the spatialcorrelations for the relative motion of the vortices2. As a remark, we refer to thecase x1 = x2 which describes the collision of the vortices and which is not allowedsince it would result in a break down of the equations. Two vortices initially locatedin the same place would result of course in the case of one localized vortex withaccumulated circulations, as treated in the previous section. So from now on, weassume x1 6= x2 for the whole time of investigation on the system.

2Of course it will be the upper left part only by convention, this depends directly on the sortsequence of the Langevin equations.

26

2.2 Two vortices in a Kraichnan velocity field

2.2.1 The cases Γ1,Γ2 ∈ R with Γ1 6= −Γ2

We will now set up of the corresponding Fokker-Planck equation for all cases Γ1,Γ2 ∈R with Γ1 6= −Γ2 by applying the formalism of the previous chapter. But before,we will introduce center of vorticity coordinates: if we put

r = x1 − x2 and R =Γ1x1 + Γ2x2

Γ1 + Γ2

we get

r = x1 − x2

(2.2)= Γ2K(x1 − x2) + K(x1, t)− (−Γ1K(x1 − x2) + u(x2, t))=(Γ1 + Γ2)u(r) + u(x1, t)− u(x2, t))

and

R =Γ1x1 + Γ2x2

Γ1 + Γ2

(2.2)=

1Γ1 + Γ2

(Γ1Γ2K(x1 − x2) + Γ1u(x1, t)− Γ2Γ1K(x1 − x2) + Γ2u(x2, t))

=1

Γ1 + Γ2(Γ1u(x1, t) + Γ2u(x2, t))

=Γ1

Γ1 + Γ2u(x1, t) +

Γ2

Γ1 + Γ2u(x2, t)

and obtain

r = (Γ1 + Γ2)K(r)+ um(t)R = up(t)

(2.4)

as the Langevin equations describing the interaction between the vortices and theKraichnan velocity field. Thereby we define

um(t) = u(x1, t)− u(x2, t)

and

up(t) =Γ1

Γ1 + Γ2u(x1, t) +

Γ2

Γ1 + Γ2u(x2, t).

If we now apply the formalism derived in the previous chapter, we get a Fokker-Planck equation of the form

27

2 Vortices in a Kraichnan velocity field

∂

∂tf(r,R, t) = −∇ ·

((Γ1 + Γ2)K(r)

0

)f(r,R, t) +

12

∑i;j∈{m,p}

∂2

∂xi∂xjQijf(r,R, t)

Thereby, we have

∇ =(∇r

∇R

)=(

∂∂r∂∂R

)!=

(∂

∂xm∂∂xp

),

It is important to accept that ∂∂xm

has to be identified with ∂∂r as well as ∂

∂xphas to

be identified with ∂∂R . The reason underlies already in the Langevin equation: The

first Langevin equation in (2.4) represents the component r and the second one R.The associated stochastical term for r is um(t) and for R is up(t). Index renaming toachieve an identifying context would result in an easier summation in the diffusionterm. This would be just over i, j ∈ {r,R} and the application of the introducedformalism could be accomplished without any further thoughts about connectionsbetween indices. But on the other side, for example concerning the renaming ofup(t) into uR(t), we would get a problem in an intuitive sense - the reason is that〈up(t)up(t′)〉 is dependent on r, but not on R as we will see below. Hence, theapparently missing connection is only an effect of a coordinate transformation andafter having clarified the notation, so we will not rename any variables.

We will now first concentrate on the diffusive term. To find its correct form, wehave to investigate explicitly the question what the correlation matrices of um andup do look like.

First, for 〈um(t)um(t′)〉 we get

⟨um(t)um(t′)

⟩=⟨(u(x1, t)− u(x2, t))(u(x1, t

′)− u(x2, t′))⟩

=⟨u(x1, t)u(x1, t

′)⟩

+⟨u(x2, t)u(x2, t

′)⟩−⟨u(x1, t)u(x2, t

′)⟩−⟨u(x2, t)u(x1, t

′)⟩

=2[Q(0)−Q(x1 − x2)]δ(t− t′)

def=C(r)δ(t− t′) (2.5)

Second, for 〈up(t)up(t′)〉, we have

28

2.2 Two vortices in a Kraichnan velocity field

⟨up(t)up(t′)

⟩=⟨(

Γ1

Γ1 + Γ2u(x1, t) +

Γ2

Γ1 + Γ2u(x2, t)

)(Γ1

Γ1 + Γ2u(x1, t

′) +Γ2

Γ1 + Γ2u(x2, t

′))⟩

=(

Γ1

Γ1 + Γ2

)2 ⟨u(x1, t)u(x1, t

′)⟩

+(

Γ2

Γ1 + Γ2

)2 ⟨u(x2, t)u(x2, t

′)⟩

+Γ1Γ2

(Γ1 + Γ2)2

⟨u(x1, t)u(x2, t

′)⟩

+Γ1Γ2

(Γ1 + Γ2)2

⟨u(x2, t)u(x1, t

′)⟩

=[

Γ21 + Γ2

2

(Γ1 + Γ2)2Q(0) +

2Γ1Γ2

(Γ1 + Γ2)2Q(x1 − x2)

]δ(t− t′)

def=D(r)δ(t− t′) (2.6)

and finally for the mixed terms, we have

⟨um(t)up(t′)

⟩=⟨up(t)um(t′)

⟩=⟨

(u(x1, t)− u(x2, t))(

Γ1

Γ1 + Γ2u(x1, t) +

Γ2

Γ1 + Γ2u(x2, t)

)⟩

=Γ1 − Γ2

Γ1 + Γ2[Q(0)−Q(r)] δ(t− t′)

def=G(r)δ(t− t′) (2.7)

So for the complete correlation matrix (compare (2.3)), we get

Q =(

C(r) G(r)G(r) D(r)

)(2.8)

Inserting this in our Fokker-Planck equation, we obtain

29

2 Vortices in a Kraichnan velocity field

∂

∂tf(r,R, t) +∇ ·

((Γ1 + Γ2)K(r)

0

)f(r,R, t)

=12

∑i;j∈{m,p}

∂2

∂xi∂xjQijf(r,R, t)

=12

∞∫0

dt′δ(t− t′)∑

i;j∈{m,p}

∂2

∂xi∂xjQijf(r,R, t)

=12

∞∫0

dt′∑

i,j∈{m;p}

∂2

∂xi∂xj

⟨ui(t)uj(t′)

⟩f(r,R, t′)

=12

∞∫0

dt′ [∂2

∂x2m

⟨um(t)um(t′)

⟩+

∂2

∂x2p

⟨up(t)up(t′)

⟩+2

∂2

∂xp∂xm

⟨up(t)um(t′)

⟩] f(r,R, t′)

(2.5),(2.6),(2.7)=

12

∞∫0

dt′δ(t− t′)[∂2

∂r2C(r) +

∂2

∂R2 D(r) + 2∂2

∂R∂rG(r)

]f(r,R, t′)

=

∞∫0

dt′δ(t− t′)12

[∇r · C(r) · ∇r +∇R · D(r) · ∇R +∇R · G(r) · ∇r

]f(r,R, t′)

and this finally yields3

∂

∂tf(r,R, t) +∇ ·

((Γ1 + Γ2)u(r)

0

)f(r,R, t)

=12

[∇r · C(r) · ∇r +∇R · D(r) · ∇R +∇R · G(r) · ∇r

]f(r,R, t) (2.9)

3The last step of the calculation has to be understood as

∇r · C(r) · ∇r =“

∂∂r1

∂∂r2

”·„C11(r) C12(r)

C21(r) C22(r)

«·

∂∂r1∂∂r2

!?=

2Xk,l=1

∂2

∂rk∂rlCkl(r);

? holds for incompressible flow fields, it will be shown in section (2.3.1), that ∇r ·Q = 0 dueto incompressibility, and thus

∇r · C = ∇r · D = ∇r · G = 0.

30

2.2 Two vortices in a Kraichnan velocity field

Decoupling of the central and relative motion

We will now use the divergence theorem (compare appendix, paragraph [1]) to de-couple the stochastical motion into a central and relative part. Fortunately, we willsee that some correlations in the separated parts do not give any contribution andvanish caused by a normalization argument.

First, we focus on the relative part. If we want to derive an equation spatially onlydependent on the relative distance r of the two vortices, we have to take a look atequation (2.9), and integrate over all possible values of R.4 We define∫

dRf(R, r, t) def= h(r, t),

and obtain

∫dR

∂

∂tf(R, r, t) =

∂

∂t

∫dRf(R, r, t) =

∂

∂th(r, t)

=∫dR[−∇ ·

((Γ1 + Γ2)K(r)

0

)f(r,R, t)

]+∫dR[

12

[∇r · C(r) · ∇r +∇R · D(r) · ∇R +∇R · G(r) · ∇r

]f(r,R, t)

]

=−∫dR ∇r · (Γ1 + Γ2)K(r)f(r,R, t) +

12

∫dR ∇r · C(r) · ∇rf(r,R, t)

+12

∫dR ∇R · D(r) · ∇Rf(r,R, t) +

12

∫dR ∇R · G(r) · ∇rf(r,R, t)

=−∇r · (Γ1 + Γ2)K(r)∫dRf(r,R, t) +

12∇r · C(r) · ∇r

∫dRf(r,R, t)

+12

∫dR ∇R · D(r) · ∇Rf(r,R, t) +

12

∫dR ∇R · G(r) · ∇rf(r,R, t)

=−∇r · (Γ1 + Γ2)K(r)h(r, t) +12∇r · C(r) · ∇rh(r, t)

+12

∫dR ∇R · D(r) · ∇R f(r,R, t)︸ ︷︷ ︸

I

+12

∫dR ∇R · G(r) · ∇rf(r,R, t)︸ ︷︷ ︸

II

We are now able to use the divergence theorem on I and II; exemplary for term II,the G(r) -term, this yields

4more precisely: byRdRf(R, r, t), we of course mean

R∞0dR1

R∞0dR1f(R1, R2, r1, r2, t)

31

2 Vortices in a Kraichnan velocity field

∫V (R)

∇R · G(r) · ∇rf(r,R, t)dV =∮

∂V (R)

(G(r) · ∇rf(r,R, t)

)· dS != 0

In the limit R → ∞, the integration on the boundary of V, ∂V , is the integrationover the boundary of a circle with infinite radius. But since f is a probability density,it has to vanish for R→∞, and so does II. Hence for the relative part, we finallyhave

∂

∂th(r, t) +∇r · (Γ1 + Γ2)K(r)h(r, t) =

12

[∇r ·C(r) · ∇r]h(r, t) (2.10)

This will be the equation we will later concentrate on. The study of the relative posi-tion of the two vortices gives most benefit in understanding their mutual interaction.

To get an equation solely for the motion of the center of vorticity, we integrate over

r =(r1

r2

), ∫

drf(R, r, t) def= g(R, t),

and obtain

∫dr∂

∂tf(R, r, t) =

∂

∂t

∫drf(R, r, t) =

∂

∂tg(R, t)

=∫dr [−∇r · (Γ1 + Γ2)K(r)f(r,R, t)]

+∫dr[

12

[∇r · C(r) · ∇r +∇R · D(r) · ∇R +∇r · G(r) · ∇R

]f(r,R, t)

]

=−∫dr ∇r · (Γ1 + Γ2)K(r)f(r,R, t)︸ ︷︷ ︸

I

+12

∫dr∇r · C(r) · ∇rf(r,R, t)︸ ︷︷ ︸

II

+12

∫dr ∇R · D(r) · ∇R f(r,R, t) +

12

∫dr ∇r · G(r) · ∇Rf(r,R, t)︸ ︷︷ ︸

III

I, II and III vanish due to the same argumentation accomplished above, say appli-cation of the divergence theorem and vanishing probability density on the boundaryof an infinite circle. We finally get

32

2.2 Two vortices in a Kraichnan velocity field

∂

∂tg(R, t) =

12

∫dr ∇R ·D(r) · ∇R f(r,R, t) (2.11)

For the case that D(r) = D is constant, we get

∂

∂tg(R, t) =

12

∫dr ∇R ·D · ∇R f(r,R, t) =

12∇R ·D · ∇R

∫dr f(r,R, t)

⇒

∂

∂tg(R, t) =

12∇R ·D · ∇R g(R, t) (2.12)

thus a pure diffusive equation. The solution looks exactly like in the one vortex casein section (2.1).

Numerical simulation of two point vortices with different circulation affected bywhite noise

In this section we show some stochastic affected trajectories of the whole two pointvortex system for vortices with different circulation. Their qualitative

behavior is presented in the case of a constant correlation matrix Q(0) = Q

(1 00 1

)for different values of Q. The numerical method is a modification of Runge-Kuttaforth order (compare appendix), in every calculation step we add a Gaussian dis-tributed white noise term multiplied with

√Qdt whereby dt is the time-step. The

two trajectories start at the gridpoints (0,0) and (1,1).

The special case Γ1 = Γ2 ≡ Γ: Two identical vortices

We now shortly consider the case where Γ1 = Γ2 ≡ Γ, because it has already beeninvestigated. If we put

r = x1 − x2 and R =Γx1 + Γx2

Γ + Γ=

12

(x1 + x2)

we obtain

r = 2ΓK(r)+ um(t)R = up(t)

(2.13)

whereby

um(t) = u(x1, t)− u(x2, t) and up(t) =12

(u(x1, t) + u(x2, t)).

33

2 Vortices in a Kraichnan velocity field

(a) Q=0.000001 (b) Q=0.00001

(c) Q=0.0001 (d) Q=0.001

Figure 2.4: Stochastically affected trajectories of the whole two point vortex systemfor vortices with different circulation: Qualitative behavior for differentvalues of Q

This has the consequence, that the G(r)-term does not appear at all. This becomesclear with ⟨

um(t)up(t′)⟩

=⟨up(t)um(t′)

⟩

=⟨

(u(x1, t)− u(x2, t))12

(u(x1, t′) + u(x2, t

′))⟩

=12

(⟨u(x1, t)u(x1, t

′)⟩−⟨u(x2, t)u(x2, t

′)⟩)

=12

[Q(0)−Q(0)]δ(t− t′) = 0 (2.14)

or when inserting Γ1 = Γ2 ≡ Γ in (2.7), we get G(r) = 0, respectively.

Hence for the complete correlation matrix (compare (2.3)), we get

Q =(C(r) 0

0 D(r)

)(2.15)

When setting G(r) = 0 in (2.9), we get the same two equations describing thedynamic interaction between the vortices in a probabilistic manner:

34

2.2 Two vortices in a Kraichnan velocity field

∂

∂th(r, t) +∇r · 2ΓK(r)h(r, t) =

12

[∇r ·C(r) · ∇r]h(r, t)

and

∂

∂tg(R, t) =

12

∫dr ∇R ·D(r) · ∇R f(r,R, t)

Interestingly, the structure of G(r) has no effect on the stochastic motion of the twovortices when we analyze the decoupled equations apart of each other. But it obvi-ously has an effect, if we take a look at the equation for the joint-probability (2.9).The reason underlying is the stochastic independency of the decoupled equations inthe case Γ1 = Γ2, as shown above, the correlation of the two terms vanishes and thusthey are stochastically independent. Hence, f(r,R, t) turns out to be the productof the probability density functions of the center of vorticity and the relative vortexdistance

f(r,R, t) = h(r, t) g(R, t)

So for Γ1 6= Γ2, we have thus a slight generalization of the case Γ1 = Γ2 ≡ Γ treatedin [2] and [3], emphasizing that we must not use the simple product f(r,R, t) =h(r, t) g(R, t) if we treat the general case Γ1 6= Γ2.

Written in terms of conditional probability we exactly have

f(r,R, t) = h(r, t) g(R|r, t)

or

f(r,R, t) = h(r|R, t) g(R, t),

respectively. In the sense of a Born-Oppenheim approximation, we will treat the mo-tions as stochastically independent, noting that this approximation becomes moreand more exact, if Γ1 → Γ2.

Numerical simulation of two point vortices with the same circulation affectedby white noise

This section deals with the stochastic affected trajectories of the whole two pointvortex system for vortices with the same circulation. Their qualitative behavior is

presented again in the case of a constant correlation matrix Q(0) = Q

(1 00 1

)for

different values of Q, the numerical method is again the modification of Runge-Kuttaforth order used above and the two trajectories start again at the gridpoints (0,0)and (1,1).

35

2 Vortices in a Kraichnan velocity field

(a) Q=0.00002 (b) Q=0.0002

(c) Q=0.002 (d) Q=0.04

Figure 2.5: Stochastically affected trajectories of the whole two point vortex systemfor vortices with the same circulation: Qualitative behavior for differentvalues of Q

2.2.2 The case Γ1 = −Γ2 ≡ Γ: Two according to amount equipollent,counter-rotating vortices

In this case, the Langevin equations read

x1 = −ΓK(x1 − x2) + u(x1, t) (2.16)

x2 = −ΓK(x1 − x2) + u(x2, t)

We have to use other coordinates this time5:

r = x1 − x2 and R =12

(x1 + x2)

We are used to, e. g., the center-of-mass or the center-of-vorticity transformation.But this transformation is also possible and leads to reasonable results. Now thetransformation of our system of linear differential equations entails

5If we tried to apply the equations derived in the general case |Γ1| 6= |Γ2| with center-of-vorticitycoordinates, singularities would emerge in some of the correlation matrices we calculated in theprevious section.

36

2.2 Two vortices in a Kraichnan velocity field

r = um(t)˙R = −ΓK(r)+ up(t)

(2.17)

whereas again

um(t) = u(x1, t)− u(x2, t) but this time up(t) =12

(u(x1, t) + u(x2, t)).

The Fokker-Planck equation now reads

∂

∂tf(r, R, t) = −∇ ·

(0

−ΓK(r)

)f(r, R, t) +

12

∑i;j∈{m,p}

∂2

∂xi∂xjQijf(r, R, t)

with

∇ =(∇r

∇R

)=( ∂∂r∂∂R

)=

(∂

∂xm∂∂xp

)we have

∂

∂tf(r, R, t) = +∇R · ΓK(r)f(r, R, t) +

12

∑i;j∈{m,p}

∂2

∂xi∂xjQijf(r, R, t)

Now we again have to investigate the question what Qij does look like. 〈um(t)um(t′)〉has already been calculated (compare the calculation to (2.6)), it is⟨

um(t)um(t′)⟩

= C(r)δ(t− t′)

Analogous, we find

⟨up(t)up(t′)

⟩=⟨

12

(u(x1, t) + u(x2, t))12

(u(x1, t′) + u(x2, t

′))⟩

=14

(⟨u(x1, t)u(x1, t

′)⟩

+⟨u(x2, t)u(x2, t

′)⟩

+⟨u(x1, t)u(x2, t

′)⟩

+⟨u(x2, t)u(x1, t

′)⟩)

=12

[Q(0) + Q(x1 − x2)]δ(t− t′)

≡D(r)δ(t− t′) (2.18)

37

2 Vortices in a Kraichnan velocity field

and

⟨um(t)up(t′)

⟩=⟨up(t)um(t′)

⟩=⟨

(u(x1, t)− u(x2, t))12

(u(x1, t′) + u(x2, t

′))⟩

=12

(⟨u(x1, t)u(x1, t

′)⟩−⟨u(x2, t)u(x2, t

′)⟩)

=12

[Q(0)−Q(0)]δ(t− t′) = 0 (2.19)

Finally we obtain

∂

∂tf(r, R, t) = ∇R · ΓK(r)f(r, R, t) +

12

[∇r · C(r) · ∇r +∇R ·D(r) · ∇R

]f(r, R, t′)

Note that again the stochastic terms are independent and so f(r, R, t) turns out tobe the product of the probability density functions of the center of vorticity and therelative vortex distance in an exact manner:

f(r, R, t) exact= h(r, t) g(R, t)

38

2.2 Two vortices in a Kraichnan velocity field

Decoupling of the central and relative motion

If we again perform an integration over R to decouple the equations, we obtain

∫dR

∂

∂tf(R, r, t) =

∂

∂t

∫dRf(R, r, t) =

∂

∂th(r, t)

=∫dR[∇ ·(

0ΓK(r)

)f(r, R, t) +

12[∇r ·C(r) · ∇r +∇R ·D(r) · ∇R

]f(r, R, t)

]

=∫dR

(∇R · ΓK(r)

)f(r, R, t)

+12

∫dR ∇r ·C(r) · ∇rf(r, R, t) +

12

∫dR ∇R ·D(r) · ∇Rf(r, R, t)

=∫dR ∇R · ΓK(r)

+12∇r ·C(r) · ∇r

∫dRf(r, R, t) +

12

∫dR ∇R ·D(r) · ∇Rf(r, R, t)

=∫dR ∇R · ΓK(r)︸ ︷︷ ︸

I

+12∇r ·C(r) · ∇r h(r, t) +

12

∫dR ∇R ·D(r) · ∇R f(r, R, t)︸ ︷︷ ︸

II

We again are able to use the divergence theorem on I and II to see that these termsvanish in the limit R→∞ because of normalization. So this time, we finally get apure diffusive equation for the relative part:

∂

∂th(r, t) =

12

[∇r ·C(r) · ∇r]h(r, t)

but for the center-of-mass part, we get a drift-diffusion equation:

∂

∂tg(R, t) = ∇R · ΓK(r)g(R, t) +

12[∇R ·D(r) · ∇R

]g(R, t)

39

2 Vortices in a Kraichnan velocity field

Numerical simulation of two point vortices with counter-rotating circulationaffected by white noise

Here we present the qualitative behavior of two point vortices with counter-rotatingcirculation affected by white noise. Again we only investigate the case of a constant

correlation matrix Q(0) = Q

(1 00 1

)for different values of Q with the same numer-

ical method as used before; the two trajectories also starting again at the gridpoints(0,0) and (1,1).

(a) Q=0.0001 (b) Q=0.001

(c) Q=0.01 (d) Q=0.1

Figure 2.6: Stochastic affected trajectories of the whole two point vortex system forvortices with antipodal circulation: Qualitative behavior for differentvalues of Q

40

2.3 Derivation and implementation of additional constraints

2.3 Derivation and implementation of additional constraints

The derived additional constraints in this section - namely some consequences of thepostulation of incompressibility and isotropic statistics - have general application.Hence they could have already been established in the basic theoretical foundations.Nevertheless, it may seem more reasonable to adopt them but now here to empha-size their practical application and to show their direct effect on the treated problem.

2.3.1 Consequences of the postulation of incompressibility

If we assume an imcompressible flow, we have

∇xi · u(xi, t) =∑α

∂

∂xαiu(α)(xi, t) = 0 (2.20)

Now we consider the expression

〈(∇x1 · u(x1, t))u(x2, t)〉

which is

= 〈(∇x1 · u(x1, t))u(x2, t)〉 = 〈0 u(x2, t)〉 = 0

due to (2.20), but also

= 〈(∇x1 · u(x1, t))u(x2, t)〉 =

⟨(∑α

∂

∂x(α)1

u(α)(x1, t))u(x2, t)

⟩

=∑α

⟨∂

∂x(α)1

u(α)(x1, t)u(x2, t)

⟩=∑α

∂

∂x(α)1

⟨u(α)(x1, t)u(x2, t)

⟩

= ∇x1 · 〈(u(x1, t))u(x2, t)〉 = ∇x1 ·Q(x1 − x2)δ(t− t′) = 0

Hence we get

∇x1 ·Q(x1 − x2) = 0 ∨ δ(t− t′) = 0

And finally because of

∇x1 · (Q(x1 − x2)) = ∇r ·Q(x1 − x2) = ∇r ·Q(r)

we have

41

2 Vortices in a Kraichnan velocity field

∇r ·Q(r) = 0. (2.21)

With (2.21), we of course also have the identity

∇r · C(r) = ∇r · 2[Q(0)−Q(r)] = 0 (2.22)

which will be used in the following calculations.

2.3.2 Consequences of the postulation of isotropic statistics

For isotropic turbulence, all statistical properties of the flow - especially the averagevelocity fluctuations - are equal all over the flow field and not depending on thedirection. Hence, we have invariance of translation and rotation. The translationalinvariance by the way became already visible within the fact, that the correlationmatrix does not depend on R. Claiming isotropic statistics, we may in mathemat-ical terms assume that the correlation matrix6 for the relative part C(r) is onlydependent on r (cf. [15]) and that it is symmetric:

Cαβ(r) = θ(r)δαβ + ψ(r)rαr

rβr

with θ(r), ψ(r) only dependent on |r|

With (2.22), we will now show that

∂

∂rθ(r) +

1r

∂

∂rrψ(r) = 0. (2.23)

This is an important identity connecting incompressibility of the flow and the isotropyof the statistics and has particular application in the following. To calculate it, wemake use of the identity7

∂

∂ri=rir

∂

∂r(2.24)

With ∇r · C(r) = 0 on the one hand, we get6The complete correlation matrix for the whole system was given by

Q =

„C(r) G(r)

G(r) D(r)

«7In the most general case with r = |r| =

pr21 + · · ·+ r2

i + · · ·+ r2n for r = (r1, . . . , ri . . . rn), it is

∂

∂rif(r) =

∂

∂rif(qr21 + · · ·+ r2

i + · · ·+ r2n) = 2ri·

1

2(r2

1+· · ·+r2i+· · ·+r2

n)−12∂

∂rf(r) =

r1

r

∂

∂rf(r)

or∂

∂ri=rir

∂

∂r,

respectively.

42

2.3 Derivation and implementation of additional constraints

0 =(0 0

)=(∂∂r1

∂∂r2

)·

(θ(r) + ψ(r)

r21

r2 ψ(r) r1r2r2

ψ(r) r1r2r2 θ(r) + ψ(r)

r22

r2

)i,j∈{1;2},i 6=j⇐⇒ 0 =

∂

∂riθ(r) +

∂

∂riψ(r)

r2i

r2+

∂

∂rjψ(r)

rirjr2

(2.25)

=∂

∂riθ(r) + ψ(r)

∂

∂ri

r2i

r2+r2i

r2

∂

∂riψ(r) + ψ(r)

∂

∂rj

rirjr2

+rirjr2

∂

∂rjψ(r)

=∂

∂riθ(r) + ψ(r)(

2rir2− 2r3

i

r4) +

r2i

r2

∂

∂riψ(r) + ψ(r)(

rir2−

2rir2j

r4) +

rirjr2

∂

∂rjψ(r)

=∂

∂riθ(r) + ψ(r)

3rir2− ψ(r)

2r3i

r4− ψ(r)

2rir2j

r4+r2i

r2

∂

∂riψ(r) +

rirjr2

∂

∂rjψ(r)

=∂

∂riθ(r) + ψ(r)

3rir2− ψ(r)

2rir4

(r2i + r2

j ) +r2i

r2

∂

∂riψ(r) +

rirjr2

∂

∂rjψ(r)

=∂

∂riθ(r) +

1r2

[3riψ(r)− 2riψ(r) + r2

i

∂

∂riψ(r) + rirj

∂

∂rjψ(r)

]

(2.24)=

rir

∂

∂rrθ(r) +

1r2

[riψ(r) +

r3i

r

∂

∂rψ(r) + rirj

rjr

∂

∂rψ(r)

]= 0

Since both components vanish identically, we multiply the ith component by rri

andobtain

⇔ 0 =∂

∂rθ(r) +

1r

[ψ(r) +

r2i

r

∂

∂rψ(r) +

r2j

r

∂

∂rψ(r)

]and because r2

i + r2j = r2

1 + r22 = r2 for i, j ∈ {1; 2}, i 6= j, we finally have

=∂

∂rθ(r) +

1r

[ψ(r) +

r2

r

∂

∂rψ(r)

]

=∂

∂rθ(r) +

1r

[ψ(r) + r

∂

∂rψ(r)

]

=∂

∂rθ(r) +

1r

∂

∂rrψ(r) = 0, �

43

2 Vortices in a Kraichnan velocity field

2.4 The relative part of the motion

From now on there will be only a consideration of the Fokker-Planck equation forthe relative part in the cases where Γ1 6= −Γ2 and Γ1 = Γ2 which we derived in thepenultimate section:

∂

∂th(r, t) +∇r · (Γ1 + Γ2)u(r)h(r, t) =

12

[∇r · C(r) · ∇r]h(r, t).

Again, we emphasize that to describe the whole system we need the product of theprobability density of the relative part and the conditional probability density ofthe center part, which becomes more and more unconditional for Γ1 → Γ2. Withthe result (2.23) from the previous section, we may now further reduce the diffusionterm of this Fokker-Planck-equation. We consider

[∇r · C(r) · ∇r]h(r, t)

which entails

=(∂∂r1

∂∂r2

)·

(θ(r) + ψ(r)

r21

r2 ψ(r) r1r2r2

ψ(r) r1r2r2 θ(r) + ψ(r)

r22

r2

)( ∂∂r1∂∂r2

)h(r, t)

We first take a look at the right part of it

[C(r) · ∇r]h(r, t)

=

(θ(r) + ψ(r)

r21

r2 ψ(r) r1r2r2

ψ(r) r1r2r2 θ(r) + ψ(r)

r22

r2

)( ∂∂r1∂∂r2

)h(r, t)

44

2.4 The relative part of the motion

i,j∈{1;2},i 6=j=⇒ h(r, t)

∂

∂riθ(r) + h(r, t)

∂

∂riψ(r)

r2i

r2+ h(r, t)

∂

∂rjψ(r)

rirjr2

+θ(r)∂

∂rih(r, t) + ψ(r)

r2i

r2

∂

∂rih(r, t) + ψ(r)

rirjr2

∂

∂rjh(r, t)

=h(r, t)(∂

∂riθ(r) +

∂

∂riψ(r)

r2i

r2+

∂

∂rjψ(r)

rirjr2

)︸ ︷︷ ︸

(2.25)= 0

+θ(r)∂

∂rih(r, t) + ψ(r)

r2i

r2

∂

∂rih(r, t) + ψ(r)

rirjr2

∂

∂rjh(r, t)

=θ(r)∂

∂rih(r, t) + ψ(r)

r2i

r2

∂

∂rih(r, t) + ψ(r)

rirjr2

∂

∂rjh(r, t)

⇒

[C(r) · ∇r]h(r, t) =

(θ(r) ∂

∂r1h(r, t) + ψ(r) r

21r2

∂∂r1h(r, t) + ψ(r) r1r2

r2∂∂r2h(r, t)

θ(r) ∂∂r2h(r, t) + ψ(r) r

22r2

∂∂r2h(r, t) + ψ(r) r2r1

r2∂∂r1h(r, t)

)

If we now also take the left differential operator into account, we have

[∇r · C(r) · ∇r]h(r, t)

=∑

i,j∈{1;2},i 6=j

∂

∂ri

[θ(r)

∂

∂rih(r, t) + ψ(r)

r2i

r2

∂

∂rih(r, t) + ψ(r)

rirjr2

∂

∂rjh(r, t)

]

=∂

∂r1

[θ(r)

∂

∂r1h(r, t) + ψ(r)

r21

r2

∂

∂r1h(r, t) + ψ(r)

r1r2

r2

∂

∂r2h(r, t)

]

+∂

∂r2

[θ(r)

∂

∂r2h(r, t) + ψ(r)

r22

r2

∂

∂r2h(r, t) + ψ(r)

r1r2

r2

∂

∂r1h(r, t)

]

45

2 Vortices in a Kraichnan velocity field

= (∂

∂r1θ(r))(

∂

∂r1h(r, t)) + (

∂

∂r1ψ(r)

r21

r2)(

∂

∂r1h(r, t)) + (

∂

∂r1ψ(r)

r1r2

r2)(

∂

∂r2h(r, t))

+ θ(r)∂2

∂r21

h(r, t) + ψ(r)r2

1

r2

∂2

∂r21

h(r, t) + ψ(r)r1r2

r2

∂

∂r1

∂

∂r2h(r, t)

+ (∂

∂r2θ(r))(

∂

∂r2h(r, t)) + (

∂

∂r2ψ(r)

r22

r2)(

∂

∂r2h(r, t)) + (

∂

∂r2ψ(r)

r1r2

r2)(

∂

∂r1h(r, t))

+ θ(r)∂2

∂r22

h(r, t) + ψ(r)r2

2

r2

∂2

∂r22

h(r, t) + ψ(r)r1r2

r2

∂

∂r1

∂

∂r2h(r, t)

2.25=[(∂

∂r1θ(r)) + (

∂

∂r1ψ(r)

r21

r2) + (

∂

∂r1ψ(r)

r1r2

r2)]

︸ ︷︷ ︸=0

(∂

∂r1h(r, t))

+ θ(r)∂2

∂r21

h(r, t) + ψ(r)r2

1

r2

∂2

∂r21

h(r, t) + ψ(r)r1r2

r2

∂

∂r1

∂

∂r2h(r, t)

+[(∂

∂r2θ(r)) + (

∂

∂r2ψ(r)

r22

r2) + (

∂

∂r2ψ(r)

r1r2

r2)]

︸ ︷︷ ︸=0

(∂

∂r2h(r, t))

+ θ(r)∂2

∂r22

h(r, t) + ψ(r)r2

2

r2

∂2

∂r22

h(r, t) + ψ(r)r1r2

r2

∂

∂r1

∂

∂r2h(r, t)

= θ(r)∂2

∂r21

h(r, t) + ψ(r)r2

1

r2

∂2

∂r21

h(r, t) + ψ(r)r1r2

r2

∂

∂r1

∂

∂r2h(r, t)

+ θ(r)∂2

∂r22

h(r, t) + ψ(r)r2

2

r2

∂2

∂r22

h(r, t) + ψ(r)r1r2

r2

∂

∂r1

∂

∂r2h(r, t)

46

2.4 The relative part of the motion

= θ(r)(∂2

∂r21

+∂2

∂r22

)h(r, t)+ψ(r)r2

1∂2

∂r21h(r, t) + r2

2∂2

∂r22h(r, t)

r2+2ψ(r)

r1r2

r2

∂

∂r1

∂

∂r2

h(r, t)

(2.24)= θ(r)∆rh(r, t) + ψ(r)

r41 + r4

2

r4

∂2

∂r2h(r, t) + 2ψ(r)

r21r

22

r4

∂2

∂r2h(r, t)

= θ(r)∆rh(r, t) + ψ(r)(r2

1 + r22)2

r4

∂2

∂r2h(r, t)

= θ(r)∆rh(r, t) + ψ(r)∂2

∂r2h(r, t) (2.26)

we thus have

⇒

[∇r · C(r) · ∇r]h(r, t) = θ(r)∆rh(r, t) + ψ(r)∂2

∂r2h(r, t)

and finally get

∂

∂th(r, t) +∇r · (Γ1 + Γ2)K(r)h(r, t) =

12

[∇r · C(r) · ∇r]h(r, t).

(2.26)⇐⇒

∂

∂th(r, t) +∇r · (Γ1 + Γ2)K(r)h(r, t) =

12

[θ(r)∆r + ψ(r)

∂2

∂r2

]h(r, t) (2.27)

For ψ(r) = 0 and θ(r) = 2ν = constant this equation has already been solved (cf.[2], compare also appendix, paragraph A.2.).

2.4.1 Change to polar coordinates

From now on we only want to focus on vortices with pure azimuthal velocity field,in polar coordinates this means

K(r) = K(r) = eϕ v(r).

Hence we take equation (2.27),

47

2 Vortices in a Kraichnan velocity field

∂

∂th(r, t) +∇r · (Γ1 + Γ2)K(r)h(r, t) =

12

[θ(r)∆r + ψ(r)

∂2

∂r2

]h(r, t)

and make a coordinate transformation to polar coordinates,

r =(r1

r2

)→(rϕ

)Thereby we have the relations

∇rh(r, ϕ, t) =( ∂

∂r1r∂∂ϕ

)h(r, ϕ, t),

∇r ·K(r) =1r

∂

∂r(rKr(r)) +

1r

∂

∂ϕKϕ(r),

∆r =1r

∂

∂rr∂

∂r+

1r2

∂2

∂ϕ2=

∂2

∂r2+

1r

∂

∂r+

1r2

∂2

∂ϕ2

We obtain

∂

∂th(r, ϕ, t) +∇r · (Γ1 + Γ2)K(r)h(r, ϕ, t)

=12

[θ(r)(

∂2

∂r2+

1r

∂

∂r+

1r2

∂2

∂ϕ2) + ψ(r)

∂2

∂r2

]h(r, ϕ, t)

=12

[[θ(r) + ψ(r)]

∂2

∂r2+ θ(r)

1r

∂

∂r+θ(r)r2

∂2

∂ϕ2

]h(r, ϕ, t) (2.28)

As we said before, we from now on consider the case where

K(r) = K(r) = eϕ v(r). (2.29)

Taking a look at the drift term we obtain

∇r · (h(r, ϕ, t)K(r)) = u(r) · ∇rh(r, ϕ, t) + h(r, ϕ, t)∇r ·K(r)

(2.4.1)= v(r)

1r

∂

∂ϕh(r, ϕ, t) + h(r, ϕ, t)

1r

∂

∂ϕv(r)

=v(r)1r

∂

∂ϕh(r, t)

48

2.4 The relative part of the motion

and hence

∂

∂th(r, ϕ, t)+v(r)

(Γ1 + Γ2)r

∂

∂ϕh(r, ϕ, t) =

12

[[θ(r) + ψ(r)]

∂2

∂r2+ θ(r)

1r

∂

∂r+θ(r)r2

∂2

∂ϕ2

]h(r, ϕ, t)

(2.30)

2.4.2 Mode ansatz

Until now, we have derived this quite general stochastic partial equation describ-ing the time evolution of the relative distance and position of the two vortices ina probabilistic way with quite few assumptions and simplifications. The only as-sumptions we have made so far are having an incompressible, Kraichnan velocityfield, isotropic statistics, pure azimuthal vortex profiles and that the two vorticesare of the same type. Hence, at this point, we theoretically may still insert arbi-trary functions for v(r), θ(r) and ψ(r) and try to derive a solution for the resultingequation. But as one can easily think of, this equation is not solvable for everyset of desired functions, it is only solvable for very few cases, requiring additionalinvestigation: after having eventually solved the equation, we still have a stochasticdifferential equation, which demands from us to test and verify that the obtained“solution” is a solution - meaning that it is a positive, normalized probability density.

To reduce the problem of solving a partial differential equation to the problem ofsolving an ordinary differential equation, we apply the mode ansatz

h(r, ϕ, t) = einϕe−λthn,λ(r) (2.31)

This ansatz leads to the linear eigenvalue problem

−λhn,λ(r)+v(r)inΓ1 + Γ2

rhn(r) =

12

[[θ(r) + ψ(r)]

∂2

∂r2+ θ(r)

1r

∂

∂r− n2 θ(r)

r2

]hn(r),

(2.32)superposition leads to the general solution

h(r, ϕ, t) =∫dλ∑n

a(n, λ)einϕe−λthn,λ(r),

still of course leaving the problem of determining hn,λ(r). By using this ansatz, wenow have obtained a second-order linear ordinary differential equation with formalsimilarity to the stationary Schrodinger equation. Now the left task is an appropri-ate selection of v(r), θ(r) and ψ(r) to find analytical solvable versions of (2.32).

One could now use the Sturm Lioville theory as presented in [13], if the equationwas not complex. There are solutions in the class of Fuchs’ differential equations(compare [27]), which make use of hypergeometric polynomials, but prohibit func-tions v(r) ∝ 1

r that are of essential interest for us according to the velocity field of

49

2 Vortices in a Kraichnan velocity field

point vortices. We will see that all solutions to the equation that are relevant in thecontext of point vortices are related to Bessel’s differential equation. They may befound for example in [1] or [16], and a special case we need in [22].8

2.4.3 Choice for θ(r) and ψ(r)

If we now once again consider the Kraichnan model

Q(α,β)ij = Q(α,β)(xi − xj)

=d(α,β)0 − d(α,β)(xi − xj)δ(t′ − t′′)

where in the incompressible case we had the relation (1.9),

d(α,β)(r) = d(α,β)(r) = rξ[B(d+ ξ − 1)δαβ −Bξ

rαrβr2

]and take (2.5)

C(r) = 2 [Q(0)−Q(r)] , (2.33)

we obtain for α, β ∈ {1, 2}

Cαβ(r) = 2

d(α,β)0 − d(α,β)(0)︸ ︷︷ ︸

=0

−(d(α,β)0 − d(α,β)(xi − xj))

=2d(α,β)(r) = rξ

2B(d+ ξ − 1)︸ ︷︷ ︸θ0

δαβ +−2Bξ︸ ︷︷ ︸ψ0

rαrβr2

=θ0rξδαβ + ψ0r

ξ rαrβr2

=θ(r)δαβ + ψ(r)rαrβr2

Hence, we have

8After three months of research on methods of solving differential equations, I am quite sure ofhaving found all relevant and known analytical solutions to that class of equations.

50

2.4 The relative part of the motion

θ(r) = θ0rξ , ψ(r) = ψ0r

ξ (2.34)

Thereby θ0 and ψ0 are related due to incompressibility; with (2.23),

∂

∂rθ(r) +

1r

∂

∂rrψ(r) = 0

it is

0 = ξθ0rξ−1 +

1r

(ψ0rξ + rξψ0r

ξ−1)

⇔ξθ0 + ψ0 + ξψ0 = 0

⇔ξ =−ψ0

θ0 + ψ0(2.35)

If we insert now our “definitions” of θ0 and ψ0 in (2.35), we see that

ξ =−ψ0

θ0 + ψ0=

Bξ

B(d+ ξ − 1− ξ)=

ξ

d− 1= ξ

for d = 2 dimensions. �

So we just verified that we calculated properly.

Inserting this in our Fokker-Planck equation (2.32) results in a linear eigenvalueproblem of the form

−λhn,λ(r) + v(r)inΓ1 + Γ2

rhn,λ =

12

[[θ0 + ψ0]rξ

∂2

∂r2+ θ0r

ξ 1r

∂

∂r− n2 θ0r

ξ

r2

]hn,λ

(2.36)

with hn,λ(r) =: h, ∂∂rhn,λ(r) =: h′ and ∂2

∂r2hn,λ(r) =: h′′to simplify writing, andarranging in orders of derivation, we get

51

2 Vortices in a Kraichnan velocity field

− λh+ v(r)inΓ1 + Γ2

rh =

12

[θ0 + ψ0]rξh′′ + θ0rξ−1h′ − n2

2θ0r

ξ

r2h

⇔ − λh+ in(Γ1 + Γ2) v(r)

rh =

12θ0r

ξ 1rh′ +

12

[θ0 + ψ0]rξh′′ − n2

2θ0r

ξ

r2h

⇔ −2r2λ

θ0 + ψ0r−ξh+

2in(Γ1 + Γ2) v(r) rθ0 + ψ0

r−ξh =θ0

θ0 + ψ0rh′ + r2h′′ − n2θ0

θ0 + ψ0h

⇔ r2h′′ +θ0

θ0 + ψ0rh′ − n2θ0

θ0 + ψ0h− r−ξ

[− 2r2λ

θ0 + ψ0+

2in(Γ1 + Γ2) v(r) rθ0 + ψ0

]h = 0

2.5 Solution for vortex profiles v(r) = 12πr1−ξ

We will now investigate cases in which the azimuthal velocity is

v(r) =1

2πr1−ξ (2.37)

with a parameter ξ ∈ [0, 2]

Figure 2.7: Vortex profile: r versus v(r) = 12πr1−ξ for ξ ∈ [0, 2]

Note that for ξ ∈ [0, 2] we have a spectrum of velocities reaching from the “stan-dard” point vortex profile to the one of a rotating disc, represented by the monotonic

52

2.5 Solution for vortex profiles v(r) = 12πr1−ξ

increasing straight line in the plot. A rotating disc has of course a total lack of dif-ferential rotation and is therefore not an appropriate candidate to model vortices influids. Nevertheless, it will be included in our calculations, so it may be seen as atheoretical limiting case.

For the correlations we take a corresponding behavior, namely

θ(r) = θ0rξ, ψ(r) = ψ0r

ξ,

with the same ξ. In this case it is possible to derive exact analyitical solutions.Inserting this in equation (2.36),

r2h′′ +θ0

θ0 + ψ0rh′ − n2θ0

θ0 + ψ0h− r−ξ

[− 2r2λ

θ0 + ψ0+

2in(Γ1 + Γ2) v(r) rθ0 + ψ0

]h = 0

we get

r2h′′ +θ0

θ0 + ψ0rh′ +

[r2−ξ 2λ

θ0 + ψ0− in(Γ1 + Γ2)

π(θ0 + ψ0)− n2θ0

θ0 + ψ0

]h = 0 (2.38)

This equation is solvable for 2− ξ 6= 0 (compare [22]), its particular solution is

h = hλ,n(r) = r12− θ0

2(θ0+ψ0)

[C1Jν

(2

2− ξ

√2λ

θ0 + ψ0r

2−ξ2

)+ C2Yν

(2

2− ξ

√2λ

θ0 + ψ0r

2−ξ2

)]

= hλ,n(r) = r−ξ2

[C1Jν

(2

2− ξ

√2λ

θ0 + ψ0r

2−ξ2

)+ C2Yν

(2

2− ξ

√2λ

θ0 + ψ0r

2−ξ2

)]whereby

r12− θ0

2(θ0+ψ0) = rθ0+ψ0

2(θ0+ψ0)− θ0

2(θ0+ψ0) = rψ0

2(θ0+ψ0)(2.35)

= r−ξ2

C1 and C2 are arbitrary constants to be determined by implementation of initialconditions; Jν(z) and Yν(z) are the Bessel functions of the first and second kind (cf.[5])

Jν(z) =∞∑k=0

(−1)k( z2)2k+ν

Γ(ν + k + 1)k!

and

Yν(z) =Jν(z) cos νπ − J−ν(z)

sin νπwith

ν =1ξ

√−4(−(Γ1 + Γ2)inπ(θ0 + ψ0)

− θ0n2

θ0 + ψ0

)=

2ξ

√(Γ1 + Γ2)inπ(θ0 + ψ0)

+θ0n2

θ0 + ψ0

53

2 Vortices in a Kraichnan velocity field

So our general solution finally reads

h(r, ϕ, t) =∑n∈Z

∞∫0

dλe−λteinϕ

(r−

ξ2

[C1Jν

(2r

2−ξ2

2− ξ

√2λ

θ0 + ψ0

)+ C2Yν

(2r

2−ξ2

2− ξ

√2λ

θ0 + ψ0

)])(2.39)

We remark, that the solvability condition of this equation, 2 − ξ 6= 0, excludesexactly the limiting case of the vortex profile for a rotating disc, which was realizedfor ξ = 2. But as we said before, this case would not have been a good candidatefor modeling vortices in a turbulent fluid anyway.

2.5.1 Implementation of initial conditions

This section deals with the implementation of reasonable initial conditions for theeigenvalue problem - also with regard to a graphic presentation. It is obvious thata probabilisticly sharp localization of the relative distance of the two point vorticesis such a choice, expressed by means of a Dirac delta:

h(r, ϕ, 0) = δ(r − r0, ϕ) (2.40)

So the initial distance between the two vortices is r0, and their initial inclinationis 0. This choice is possible without loss of generality, because the Fokker-Planckequation is invariant by rotation. If we manage to express the solution by a normal-ized function for a fixed point in time, say t = 0, then we have a normalized solutionfor all times. (cf. section (1.2.2))

To match the initial condition to our solution from the previous section at time t = 0,we first have to express the Dirac delta in terms of Bessel functions. Therefore wemake use of the so-called Jacobi-Anger expansion (to be found in [1] p. 361)

exp(iz cos(α)) =∞∑

ν=−∞iνJν(z) exp(iνα) (2.41)

Taking α now as the angle between k and (r− r0) and z = |k| |r− r0| = k |r− r0|,we obtain with the Fourier transform of the Dirac delta

54

2.5 Solution for vortex profiles v(r) = 12πr1−ξ

δ(r− r0) =1

4π2

∫dk exp(ik · [r− r0])

=1

4π2

∞∫0

dk k

2π∫0

dα exp(ik |r− r0| cos(α))

(2.41)=

14π2

∞∫0

dk k

2π∫0

dα

∞∑ν=−∞

iνJν(k |r− r0|) exp(iνα)

=1

4π2

∞∫0

dk k

2π∫0

dαJ0(k |r− r0|) exp(0) +∞∑

ν=−∞ν 6=0

iνJν(k |r− r0|)2π∫0

dα exp(iνα)

=

14π2

∞∫0

dk k

2πJ0(k |r− r0|) +∞∑

ν=−∞ν 6=0

iνJν(k |r− r0|)(

1iν

(1− 1))

=1

2π

∞∫0

dk kJ0(k |r− r0|)

⇒ δ(r− r0) =1

2π

∞∫0

dk kJ0(k |r− r0|) (2.42)

This expression of the Dirac delta is elementary. We now use Gegenbauer’s additiontheorem for Bessel functions (to be found in [1] p. 363) to develop J0(k |r− r0|):

J0(k |r− r0|) = J0(kr0)J0(kr) + 2∞∑ν=1

Jν(kr0)Jν(kr) cos(νφ)

With this and J−n(z) = (−1)nJn(z), we get

δ(r− r0) =1

2π

∞∫0

dk k

[J0(kr0)J0(kr) + 2

∞∑ν=1

Jν(kr0)Jν(kr) cos(νϕ)

]

=1

2π

∞∫0

dk k∑ν∈Z

Jν(kr0)Jν(kr)eiνϕ

There are now Bessel functions with argument ∝ r2−ξ

2 in our solution (2.39) of theFokker-Planck equation and Bessel functions with argument ∝ r in the expression for

55

2 Vortices in a Kraichnan velocity field

the Dirac delta which have to be matched, following the tedious task of searching formultiplication theorems to transform the more complicated terms into simpler ones.9

Fortunately, there is another possibility. First, we rename r and r0 by r and r0

δ(r− r0) =1

2π

∞∫0

dk k∑ν∈Z

Jν(kr0)Jν(kr)eiνϕ (2.43)

Considering the substitution

r = r2−ξ

2 r0 = r2−ξ

20

and inserting in (2.43), we get

δ(r2−ξ

2 − r2−ξ

20 , ϕ) =

12π

∞∫0

dk k∑ν∈Z

Jν(kr2−ξ

20 )Jν(kr

2−ξ2 )eiνϕ

The Dirac delta has a useful property now - it is (cf. [5])

δ(g(r0)) =n∑i=1

δ(r0 − ri)|g′(ri)|

whereby ri are the simple roots of g(r0). Now we consider

g(r0) = r2−ξ

2 − r2−ξ

20

with the only simple root at r = r0. Hence, with g′(r0) = 2−ξ2 r

2−ξ2−1 = 2−ξ

2 r1− ξ2−1 =

2−ξ2 r−

ξ2 , we get

δ(r2−ξ

2 − r2−ξ

20 , ϕ) = δ(r − r0, ϕ)

12−ξ

2 r−ξ2

With (2.43), one obtains

δ(r − r0, ϕ) =1

2π2− ξ

2r−

ξ2

∞∫0

dk k∑ν∈Z

Jν(kr2−ξ

20 )Jν(kr

2−ξ2 )eiνϕ (2.44)

an expression for the Dirac delta with the desired properties to match the Besselfunctions. Comparing this to equation (2.39),

h(r, ϕ, t) =∑n∈Z

∞∫0

dλe−λteinϕ

(r−

ξ2

[C1Jν

(2r

2−ξ2

2− ξ

√2λ

θ0 + ψ0

)+ C2Yν

(2r

2−ξ2

2− ξ

√2λ

θ0 + ψ0

)])9After quite a while in hopeful search of such theorems, I myself by now believe there are none.

56

2.5 Solution for vortex profiles v(r) = 12πr1−ξ

for t = 0 and φ = 0, leads to C2 = 0.

Now we substitute 22−ξ

√2λ

θ0+ψ0by k, or λ by (2−ξ)2k2(θ0+ψ0)

8 , respectively.

We havedλ

dk= k

14

(2− ξ)2(θ0 + ψ0)

which leads to

h(r, ϕ, t) = r−ξ2

∑n∈Z

einϕ∫dk k

14

(2−ξ)2(θ0+ψ0) a(n, k) exp(−(2− ξ)2(θ0 + ψ0)

8k2t

)C1Jν(kr

2−ξ2 )

and finally we identify a(n, k) with

a(n, k) =4

C1(2− ξ)2(θ0 + ψ0)1

2π2− ξ

2Jν(kr

ξ20 ),

which entails10

h(r, ϕ, t) =1

2π2− ξ

2r−

ξ2

∑n∈Z

einϕ∫dk k exp

(−(2− ξ)2(θ0 + ψ0)

8k2t

)Jν(kr

2−ξ2 )Jν(kr

2−ξ2

0 )

(2.45)With the identity

∞∫0

e−ρ2k2Jν(rk)Jν(r0k)k dk =

12ρ2

exp(− r

2 + r20

4ρ2

)Iν

(rr0

2ρ2

)

and ρ2 = (2−ξ)2(θ0+ψ0)8 t, the modified Bessel function Iν(z) = i−νJν(iz), r = r

2−ξ2

and r0 = r2−ξ

20 , we can finally write down the normalized solution of our Fokker-

Planck equation:

h(r, ϕ, t) =1

2π2− ξ

2r−

ξ2

1

2 (2−ξ)2(θ0+ψ0)t8

∑n∈Z

einϕ exp

(− r2−ξ + r2−ξ

0

4 (2−ξ)2(θ0+ψ0)8 t

)Iν

((rr0)

2−ξ2

2 (2−ξ)2(θ0+ψ0)8 t

)(2.46)

10Note again that

ν =2

ξ

s(Γ1 + Γ2)in

π(θ0 + ψ0)+

θ0n2

θ0 + ψ0∝ n

57

2 Vortices in a Kraichnan velocity field

⇐⇒ h(r, ϕ, t) = 1π(2−ξ)(θ0+ψ0)t r

− ξ2∑

n∈Z einϕ exp

(− r2−ξ+r2−ξ

012

(2−ξ)2(θ0+ψ0)t

)Iν

((rr0)

(2−ξ)2

14

(2−ξ)2(θ0+ψ0)t

)

= H(r, t)

[1 + 2<

( ∞∑n=1

einφI−10

((rr0)

2−ξ2

14(2− ξ)2(θ0 + ψ0)t

)Iν

((rr0)

2−ξ2

14(2− ξ)2(θ0 + ψ0)t

))](2.47)

whereas H(r, t) is defined by

H(r, t) =1

π(2− ξ)(θ0 + ψ0)tr−

ξ2 exp

(− r2−ξ + r2−ξ

012(2− ξ)2(θ0 + ψ0)t

)I0

((rr0)

ξ2

14(2− ξ)2(θ0 + ψ0)t

)(2.48)

Note that we have to multiply by the factor r to have a complete probability density,r is the Jacobian of the transformation to polar coordinates. If we compare this tothe result of the standard vortex profile derived in the appendix, we see that we gotequivalence for ξ = 0: It follows from ξ = 0 and ψ0 = 0 due to incompressiblity(compare previous chapter), that

h(r, ϕ, t) = H(r, t)

[1 + 2<

( ∞∑n=1

einϕI−10

(rr0

θ0t

)Iν

(rr0

θ0 + t

))]whereby

H(r, t) =1

2πθ0texp

(−r

2 + r20

2θ0t

)I0

(rr0

θ0t

)

58

2.5 Solution for vortex profiles v(r) = 12πr1−ξ

2.5.2 Graphical presentation of the results

Now we come to the graphical presentation. We have toWe plot rh(r, ϕ, t) for different values of t with mathematica, with respect to calcu-lation time we only sum up the first 20-100 terms of the infinite progression. It isquite difficult to write a code for infinite sums of bessel functions (cf. citebesselnu-merics); the numerics of mathematica have sometimes already problems handling alarge finite sum of modified Bessel functions. But one do not has to desperate - wemay take into account that the infinite progression of the modified Bessel functionsconverges: For

n→∞,

we have

|ν| ∼ n

,and hence

|Iν(cr)| ∼ In(cr)

Considering

|I0(cr) + 2<∞∑n=1

Iν(cr)einϕ| ≤ |I0(cr)|+ 2∞∑n=1

|Iν(cr)|,

the normal convergence is a consequence of the convergence of the positive series

I0(cr) + 2∞∑n=1

In(cr) = exp(cr) for cr > 0

(cf. also [3]). Assuming that the higher terms give less contribution, we only take afinite number of terms to establish plots for the probability density. Note that thelarger the value of t, the less is the contribution of the modified Bessel functions andthe less the angular part is destining the evolution of the probability density.

For our graphical presentation of r h(r, ϕ, t), we first chose ξ = 0, θ0 = 0.02, andr0 = 1.

59

2 Vortices in a Kraichnan velocity field

(a) t=5 (b) t=10

(c) t=15 (d) t=20

(e) t=40 (f) t=60

(g) t=80 (h) t=800

Figure 2.8: r h(r, ϕ, t) for different values of t. Parameters: ξ = 0, θ0 = 0.02, andr0 = 1

60

2.5 Solution for vortex profiles v(r) = 12πr1−ξ

For t = 5 we still have a quite sharp located probability density function, caused bythe initial condition δ(r−1, ϕ). Going further in time, we can observe, that a smallerangle and a bigger distance r, and a bigger angle and a smaller distance r correspondto each other in terms of the most probable events to a fixed time t. If we regardthis in a differential manner, considering changes with time, this shows a behavior∆ϕ∆t ∼ f

(1

∆r∆t

), scaled by a certain function f . So in fact we may interpreted this

observation as a description of an acceleration process if the vortices get nearer eachother and a deceleration process if the move away from each other. This seems tobe reasonable in the context of phenomenological fluid oberservations.The more we progress in time, the less is the characterizing influence of the angularpart of the probability density. What we eventually see is a change of the topologyof the probability density function - for 800 time steps we see that for large timeswe arrive in an isoptropic limes concering the angular influence of the probabilitydensity.We could have already observed this reviewing the formulas: in the asymptoticregime of t→∞, we have

h(r, ϕ, t) ∼ 12πθ0t

exp(r2 + r2

0

2θ0t

)I0(

rr0

θ0t)

which corresponds to a initially concentrated probability density on a ring of radiusr0.

If we take a closer look at the graph for t = 5 by adjusting the plot range to striclyr ≥ 0, we observe a feature of the graph which shows us the limits of our approxi-mation of taking only a finite number of the terms of the progression:

The darker blue parts denote locations where the graph gets slightly negative, al-though taking into account the first n = 500 terms of the progression and a calcu-lation time of about half an hour. For t = 10 time units already n = 150 terms areenough to have graphs without the probability density getting negative, for t = 40there even is no visible difference between n = 20 and n = 40 or higher. Now insuccession, we choose ξ = 1 for the same parameters as before θ0 = 0.02, and r0 = 1

61

2 Vortices in a Kraichnan velocity field

(a) t=10 (b) t=20

(c) t=40 (d) t=60

(e) t=80 (f) t=160

(g) t=240 (h) t=500

Figure 2.9: r h(r, ϕ, t) for different values of t. Parameters: ξ = 1, θ0 = 0.02, andr0 = 1

62

2.5 Solution for vortex profiles v(r) = 12πr1−ξ

We see that the qualitative behavior is very similar in the two cases, a hint that theansatz of the modified vortex profile is not a bad candidate for modelling, since thecase ξ = 0 corresponds to the standard vortex profile as we saw before.

The one thing one can distinguish is the time in which the vortices rotate aroundeach other and the time in which the systems finally reaches an isotropic-lookingstate. ξ = 0 is faster than ξ = 1. ξ = 1.9 does not lead to acceptable results in anappropriate amount of calculating time. One has to sum up much more terms toform a reasonable graph.

(a) ξ == 0.5 (b) ξ = 1

(c) ξ = 1.5 (d) ξ = 1.8

Figure 2.10: r h(1, ϕ, t) for different values of ξ. Parameters: r = 1, θ0 = 0.02, andr0 = 1

Here we see the angle in dependence of time - for smaller ξs corresponding to thestandard vortex profile the angle changes faster, for larger ξ it changes slower. As onecan observe, the plot for ξ is of worse quality, it although we took more calculationsteps and more to compute it. In the area of since the equation is not solvable forξ = 2, it is understandable that the terms show strange behavior if too few termsare added up.

63

2 Vortices in a Kraichnan velocity field

2.6 Solution for vortex profiles v(r) = 12πr

We will now investigate cases where the azimuthal velocity is given by

v(r) =1

2πr, (2.49)

independently of the scaling exponent in the correlation matrix. With equation(2.30), the evolution equation takes the form

r2h′′ +θ0

θ0 + ψ0rh′ − n2θ0

θ0 + ψ0h− r−ξ

[−2r2λ

θ0 + ψ0+in(Γ1 + Γ2)π(θ0 + ψ0

)]h = 0 (2.50)

This equation is not analytically solvable for ξ 6= 0, but at least we can try to treatan asymptotic case for large values of r before we solve it for the special case ofconstant spatial correlations.

2.6.1 Asymptotic behavior

To investigate on the asymptotic behavior of

r2h′′ +θ0

θ0 + ψ0rh′ − n2θ0

θ0 + ψ0h− r−ξ

[−2r2λ

θ0 + ψ0+in(Γ1 + Γ2)π(θ0 + ψ0

)]h = 0

for large values of r, we first make an ansatz u(r)w(r) = uw for h, resulting in therelations

h = uw h′ = u′w + uw′ h′′ = u′′w + 2u′w′ + uw′′ (2.51)

As a consequence, we get

r2(u′′w+2u′w′+uw′′)+θ0

θ0 + ψ0r(u′w+uw′)− n2θ0

θ0 + ψ0uw−r−ξ

[−2r2λ

θ0 + ψ0+in(Γ1 + Γ2)π(θ0 + ψ0)

]uw = 0

Now we want to eliminate the terms proportional to u′. This entails

2r2w′ + ζrw = 0

whereby we substituted

ζ =θ0

θ0 + ψ0

leading to the solution

w = r−12ζ

Hence we obtain

r2(u′′ + uw′′

w) + ζru

w′

w− n2θ0

θ0 + ψ0u− r−ξ

[−2r2λ

θ0 + ψ0+in(Γ1 + Γ2)π(θ0 + ψ0)

]u = 0,

64

2.6 Solution for vortex profiles v(r) = 12πr

For w = r−12ζ , we have w′ = −1

2ζr− 1

2ζ−1 and w′′ = (−1

2ζ)(−12ζ−1)r−

12ζ−2, following

w′′

w= r−2(−1

2ζ)(−1

2ζ − 1)

and

w′

w= r−1(−1

2ζ)

Inserting this yields

r2u′′ +ζ2

4u− n2θ0

θ0 + ψ0u− r−ξ

[−2r2λ

θ0 + ψ0+in(Γ1 + Γ2)π(θ0 + ψ0)

]u = 0

This leads to the Schrodinger-type equation

u′′ +[(

ζ2

4− n2θ0

θ0 + ψ0

)1r2− r−ξ

(−2λ

θ0 + ψ0+in(Γ1 + Γ2)π(θ0 + ψ0)

1r2

)]u = 0 (2.52)

Considering now the asymptotic behavior for large r, we get

u′′ + r−ξ2λ

θ0 + ψ0u = 0 (2.53)

Equation (2.53) is again solvable using the Bessel functions (cf. [5]):

Jν(z) =∞∑k=0

(−1)k( z2)2k+ν

Γ(ν + k + 1)k!

and

Yν(z) =Jν(z) cos νπ − J−ν(z)

sin νπ.

Then

u = C1

√rJ 1

2q

(√2λ

θ0 + ψ0

rq

q

)+ C2

√rY 1

2q

(√2λ

θ0 + ψ0

rq

q

)(2.54)

is a solution of (2.53), whereas q = 12(−ξ+2) and C1 and C2 are arbitrary constants

to be derived from given initial conditions (cf. [22]). So we have Bessel functionsindependent of n of the fixed order 1

2−ξ again.

Resubstituting, we get

h = r−12ζu

and the general asymptotic solution of the relative part is finally the superposition

65

2 Vortices in a Kraichnan velocity field

h(r, ϕ, t) =∫dλ∑n

a(n, λ)einϕe−λtr−12ζu

=∫dλ∑n

a(n, λ)einϕe−λtrζ ·

·

(C1J 1

2−ξ

(√2λ

θ0 + ψ0r

2−ξ2

22− ξ

)+ C2Y 1

2−ξ

(√2λ

θ0 + ψ0r

2−ξ2

22− ξ

))

=∫dλe−λtrζ

(C1J 1

2−ξ

(√2λ

θ0 + ψ0r

2−ξ2

22− ξ

)

+C2Y 12−ξ

(√2λ

θ0 + ψ0r

2−ξ2

22− ξ

))∑n

a(n, λ)einϕ

Attempt to implement initial conditions

The ”arbitrary“ parameters of the equation have to be chosen in a way, so thath(r, ϕ, t) is real and positive for all values of r, ϕ and t, as it must be a probabilitydensity function.

The problem now when trying to implement reasonable initial conditions expressedin terms of the in (2.44) derived version of the δ-function

δ(r − r0, ϕ) =1

2π2− ξ

2r

2−ξ2−1

0

∞∫0

dk k∑ν∈Z

Jν(kr2−ξ

20 )Jν(kr

2−ξ2 )eiνϕ (2.55)

is the constant order of the Bessel function independently of n in every summand inour solution for the differential equation, which detains us from simply comparingand matching it to the expression of the δ-function. To foreclose, we will not be ableto implement a proper initial condition

However, the one thing we can try to do here, is to go forward in an analogousway to the already accomplished calculation, hoping to find that the solution isnormalizable. Adding the multiplicative term

J 12−ξ

(√2λ

θ0 + ψ0r

2−ξ2

0

22− ξ

)by taking

a(n)J 12−ξ

(√2λ

θ0 + ψ0r

2−ξ2

0

22− ξ

)= a(n, λ)

66

2.6 Solution for vortex profiles v(r) = 12πr

leads to

h(r, ϕ, t) =∫dλe−λt

(C1J 1

2−ξ

(√2λ

θ0 + ψ0r

2−ξ2

0

22− ξ

)J 1

2−ξ

(√2λ

θ0 + ψ0r

2−ξ2

22− ξ

)

+ C2J 12−ξ

(√2λ

θ0 + ψ0r

2−ξ2

0

22− ξ

)Y 1

2−ξ

(√2λ

θ0 + ψ0r

2−ξ2

22− ξ

))∑n

a(n)einϕ

=∫dλe−λtC1J 1

2−ξ

(√2λ

θ0 + ψ0r

2−ξ2

0

22− ξ

)·

· J 12−ξ

(√2λ

θ0 + ψ0r

2−ξ2

22− ξ

)∑n

a(n)einϕ (2.56)

+∫dλe−λtC2J 1

2−ξ

(√2λ

θ0 + ψ0r

2−ξ2

0

22− ξ

)·

· Y 12−ξ

(√2λ

θ0 + ψ0r

2−ξ2

22− ξ

)∑n

a(n)einϕ (2.57)

We treat the two integrals (2.56) and (2.57) separately and substitute

22− ξ

√2λ

θ0 + ψ0by k, λ by

(2− ξ)2k2(θ0 + ψ0)8

and get

dλ

dk= k

14

(2− ξ)2(θ0 + ψ0)

First, we take a look at the second integral from above (2.57):

∫dλe−λtC2J 1

2−ξ

(√2λ

θ0 + ψ0r

2−ξ2

0

22− ξ

)Y 1

2−ξ

(√2λ

θ0 + ψ0r

2−ξ2

22− ξ

)∑n

a(n)einϕ

leading to ∫dk k

14

(2− ξ)2(θ0 + ψ0) exp(−(2− ξ)2(θ0 + ψ0)

2k2t

)·

·C1Y 1(2−ξ)

(kr(2−ξ)

2 )J 1(2−ξ)

(kr(2−ξ)

20 )

∑n∈Z

einϕ a(n) = N(r, ϕ, t)

67

2 Vortices in a Kraichnan velocity field

If we want to prove, if N(r, ϕ, t) is normalizable, we have to perform the integration

∞∫r0

rdr

2π∫0

dϕN(r, ϕ, t)

and finding a finite value for all t. Note that I 1ξ

has still constant order. Angular in-

tegration in the intervall [0, 2π] of this gives a multiplicative 2π. But even when notfinding this term in an integral table, integrating it with mathematica shows thatit diverges. So we have to choose C2 = 0 to save the possibility of normalization inthe case of keeping the ansatz we chose.

Taking a look now at (2.57),

∫dλe−λtC1J 1

2−ξ

(√2λ

θ0 + ψ0r

2−ξ2

0

22− ξ

)J 1

2−ξ

(√2λ

θ0 + ψ0r

2−ξ2

22− ξ

)∑n

a(n)einϕ

leads to

h(r, ϕ, t) =∫dk k

14

(2− ξ)2(θ0 + ψ0) exp(−(2− ξ)2(θ0 + ψ0)

2k2t

)·

·C1J 1(2−ξ)

(kr2−ξ

2 )J 12−ξ

(kr2−ξ

20 )

∑n∈Z

einϕ a(n)

With the identity

∞∫0

e−ρ2k2Jν(rk)Jν(r0k)k dk =

12ρ2

exp(− r

2 + r20

4ρ2

)Iν

(rr0

2ρ2

)

and ρ2 = (2−ξ)2(θ0+ψ0)8 t, the modified Bessel function Iν(z) = i−νJν(iz), r = r

2−ξ2

and r0 = r2−ξ

20 we obtain

h(r, ϕ, t) =1

2 (2−ξ)2(θ0+ψ0)t8

∑n∈Z

a(n)einϕ exp

(− r2−ξ + r2−ξ

0

4 (2−ξ)2(θ0+ψ0)8 t

)I 1

2−ξ

((rr0)

2−ξ2

2 (2−ξ)2(θ0+ψ0)8 t

)

⇔ h(r, ϕ, t) =4

(2− ξ)2(θ0 + ψ0)texp

(− r2−ξ + r2−ξ

012(2− ξ)2(θ0 + ψ0)t

)·

·I 12−ξ

((rr0)

2−ξ2

14(2− ξ)2(θ0 + ψ0)t

)∑n∈Z

a(n)einϕ

Angular integration in the interval [0, 2π] again gives a multiplicative 2π:

68

2.6 Solution for vortex profiles v(r) = 12πr

2π∫0

dϕh(r, ϕ, t) = N(r, t) =

8π(2− ξ)2(θ0 + ψ0)t

exp

(− r2−ξ + r2−ξ

012(2− ξ)2(θ0 + ψ0)t

)I 1

2−ξ

((rr0)

2−ξ2

14(2− ξ)2(θ0 + ψ0)t

)a(0)

The r-integration∫∞r=0 rN(r, t)dr cannot be accomplished analytically. But at least,

numerically integrating with mathematica gives different positive finite values fordifferent ξ, and since one still may choose a(0) in an appropriate way, it should benormalizable. Also the integrand looks quite integrable,

Figure 2.11: Integrand for δ ∈ {0, 12 , 1,

32 , 2}

but we are not able to give an analytical solution in this case - by the way, thecalculations last quite long. Bessel functions of fractional order seem to cause someproblems in the numerics of mathematica.

Hence, we were not able to implement a δ-function as initial condition, but at leastit seems that the solution is normalizable for all ξ and that the analogous way tothe already performed one did kind of work out here.

69

2 Vortices in a Kraichnan velocity field

2.7 A remark on the analogy to anyon statistics

Because of being a partial differential equation, the Fokker–Planck equation can besolved analytically only in special cases. The formal analogy of the Fokker–Planckequation with the Schrodinger equation allows the use of advanced operator tech-niques known from quantum mechanics for its solution in a number of cases. Aninteresting fact one may observe is that the Fokker-Planck equation we derived forthe two point vortices can be treated in analogy of the so-called two anyon system- in a consideration of our Fokker-Planck equation in imaginary time results in aSchrodinger equation, which corresponds to the two anyon problem. The timede-pendence can be separated both in the vortex and the anyon calculation, the rest ofthe calculation in cylindrical coordinates exhibits the same differential equations.

An anyon can be described as an object consisting of a spinless charge orbitingaround a flux. If the charge is proportional to the flux, this object has a fractionalspin of

s =qφ

4π

whereby φ is the flux and q is the charge. Due to the fractional spin, anyons obeyfractional statistics. In the appendix this feature is presented in the context of twoanyons.(A good introduction to the subject of anyons is found in [24]).

2.7.1 The two anyon system

The Hamiltonian of a two anyon system can be given by

H =(p1 − qa1)2

2m+

(p2 − qa2)2

2m

whereby we have the vector potentials

ai =φ

2πz × (ri − rj)|ri − rj |2

for i, j ∈ {1, 2},i 6= j. If we work in center of mass and relative coordinates similarlike we did before, the Hamilitonian reads

H =P2

4m+

(p− qarel)2

m

whereby

70

2.7 A remark on the analogy to anyon statistics

R =r1 + r2

2⇒ P = p1 + p2

r =r1 − r2

2⇒ p =

p1 − p2

2

arel =φ

2πz × r|r|2

=

(a

(r)rel

a(θ)rel

)

in cylindrical coordinates. Solving this quantum mechanical problem now for thecenter of mass motion (CM), we obtain ECM = P

4m and ψCM = eiP·R.

We now consider a bosonic charge orbiting around a bosonic flux, the wavefunctionof the system is symmetric under exchange, meaning

ψrel(r, θ + π) = ψrel(r, θ) (2.58)

For the relative motion we go over to cylindrical coordinates in the Hamiltonian,too, and for ~ = 1 we obtain

Hrelψrel(r, θ) =(p− qarel)2

mψrel(r, θ) (2.59)

=

[− 1m

(∂2

∂r2+

1r

∂

∂r

)+

1mr2

(i∂

∂θ− qφ

2π

)2]ψrel(r, θ) = Erelψrel(r, θ) (2.60)

The Hamiltonian is separable in r and θ, so we have

ψrel(r, θ) = h(r)Yl(θ)

The angular solution can be obtained by solving(i∂

∂θ− qφ

2π

)2

Yl(θ) = ηYl(θ),

leading to

Yl(θ) = eilθ

with l even to match the boundary condition for bosons (2.58). This, of course,leads to

η = (l +qφ

2π)2

with l even. Now inserting η in the remaining radial equation we obtain an differen-tial equation totally alike to the one we solved for two vortices in an incompressible,Kraichnan velocity field for δ = 0, the standard vortex profile:

71

2 Vortices in a Kraichnan velocity field

[− 1m

(∂2

∂r2+

1r

∂

∂r

)+

1mr2

(l − qφ

2π

)2]h(r) = Erelh(r)

Multiplying by −mr2 and abbreviating ∂∂rh(r) = h′ we get

r2h′′ + rh′ + (Erelmr2 − (l +

qφ

2π)2)h = 0

chapter*Conclusio Comparing this for example to equation (A.1) in the appendix,

r2h′′ + rh′ + (2λθ0r2 − (n2 +

in(Γ1 + Γ2

πθ0))h = 0

we can identify

Erelm↔2λθ0

(l +qφ

2π)2 ↔ (n2 +

in(Γ1 + Γ2

πθ0)

and find the predicted analogy between the two vortex system we investigated onand the two anyon system. The solutions we derived just would have to be modifiedby renaming, if one wanted to present the solutions for the two anyon system.

72

Conclusion

The central issue of this thesis was the investigation on the dynamics of point vorticesin an incompressible and statistical isotropic Kraichnan velocity field. We combinedboth deterministic dynamics and methods of statistical physics to model a simplesystem containing coherent structures as well as fluctuating flow fields, which arecharacteristics of turbulent fluid motion. To obtain an appropriate description of thesystem, we derived a Fokker-Planck equation describing the stochastical dynamicsand solved it subsequently.

The thesis was partitioned into two chapters. The first chapter introduced the meth-ods and the theoretical foundations we needed as a basis for our investigations. Aftera brief introduction to hydrodynamics and especially vortex dynamics, we presentedthe Kraichnan model. Despite the fact that Kraichnan velocity fields are a usefultool to model short-time correlated velocity fields, they deliver important simplifi-cations in our analytical calculations due to their decorrelation in time. Then wederived a general formalism to carry over from a finite set of Langevin equations toone Fokker-Planck equation.

The second chapter was the main part of the thesis, first, we briefly considered onevortex described by one Langevin equation. The equation was lacking the determin-istic interaction and reduced to a simple Brownian motion.In the following treatment of two vortices, we first made a change to central andrelative coordinates in the two Langevin equations and subsequently we applied theformalism derived in the theoretical foundations.We decoupled the obtained Fokker-Planck equation into a central and relative partfor vortices with arbitrary, different circulation. Solving the central motion was inmany cases approximately and sometimes even exactly reduced to solving the heatequation. Separating the motion required a consideration in terms of conditionalprobability. For vortices with the same circulation and with antipodal circulation thesolutions of the central and relative motion decouple were statistically independentand the complete joint probability density function reduced to a simple productof the relative and central part. For different circulations, one can only treat thedecoupling with conditioned probability densities, for example one can consider anexact relative motion and a conditioned central motion. In the sense of a Born-Oppenheimer approximation, we assumed now that the relative motion and thecentral motion do not affect each other that much and limited ourselves to theconsideration of a Brownian motion of the center of vorticity and an exact solutionof the relative part. Visually, we modelled the different cases with a modified Runge-

73

CONCLUSION

Kutta method including white noise.Now exactly solving the relative part was the more complicated part of the work. Wefirst derived and implemented some consecutions of incompressibility and isotropicstatistics. Then we made a change to polar coordinates and applied a mode ansatzon the obtained Fokker-Planck equation to reduce the problem of solving ordinarydifferential equations. We eventually derived solutions in the case of scaled diffu-sions coefficients and a particularly matching generalized vortex profile, solving theordinary differential equations and implementing a δ-function as a proper initialcondition. As we have shown, these solutions pass over into already known solutionsin limiting cases; the known solutions are derived in the appendix. After havingsolved the equations, the solutions were presented graphically and commented. Thecase of a scaled diffusion matrix and a standard (not scaled) vortex profile was in-vestigated asymptotically, but unsatisfyingly we were not able to implement properinitial conditions. At least, it seems that the solutions are normalizable. Finallywe described the analogy of the two point vortex system to the system of two freeanyons, which behave statistically in an analogous way.

To find out more about stochastic vortex dynamics in future, one could concernthe analogy to the anyons, for example there exist some classes of solutions in then-anyon problem [17], which perhaps may be carried over to solve similar n-vortexsystems.

74

A Appendix

A.1 The special case ξ = 0 - direct derivation

We investigate equation(2.50), but this time in the case ξ = 0. This has alreadybeen done in a similar way in ([3]), the calculation is often analogous. The benefit ofperforming it anyway, it is that it is a possibility to control, if the generalized resultsfor a not constant correlation matrix and the modified vortex profile v(r) = 1

2πr1−ξ

match to the simple case in the limit, say for δ → 0

δ00 implies also ψ0 = 0 (compare equation (2.35), leading to

r2h′′ + rh′ − n2h−[−2r2λ

θ0+in(Γ1 + Γ2)

πθ0

]h = 0

⇔ r2h′′ + rh′ +[

2r2λ

θ0− (n2 +

in(Γ1 + Γ2)πθ0

)]h = 0 (A.1)

This again can be solved in terms of the Bessel functions of the first and second kind

Jν(z) =∞∑k=0

(−1)k( z2)2k+ν

Γ(ν + k + 1)k!, Yν(z) =

Jν(z) cos νπ − J−ν(z)sin νπ

We get (cf.[22])

h(r) = C1Jν

(√2λθ0r

)+ C2Yν

(√2λθ0r

)(A.2)

as a solution of (A.1), whereas C1 and C2 are arbitrary constants to be derived from

any given initial condition and ν =√n2 + in(Γ1+Γ2)

πθ0.

So the complete and generalized solution of the relative part reads (cf. [22])

h(r, ϕ, t) =∑n∈Z

∫dλ a(n, λ)einϕe−λthn,λ(r)

=∑n∈Z

∫dλ a(n, λ)einϕe−λt

[C1Jν

(√2λθ0r

)+ C2Yν

(√2λθ0r

)](A.3)

75

A Appendix

Implementation of initial conditions

Now we again want to implement an initial condition in terms of a Dirac delta

h(r, φ, 0) = δ(r − r0, ϕ) (A.4)

and match it to our general solution (2.39) from the previous subsection. Exactly likein the previous section we need an expression for the Dirac delta in terms of Besselfunctions. We already accomplished this task using the Jacobi-Anger expansion(cf. [1] p. 361) on the Fourier transform of the Dirac delta, and then applyingGegenbauer’s additions theorem to obtain the expression

δ(r− r0) =1

2π

∞∫0

dk k

[J0(kr0)J0(kr) + 2

∞∑ν=1

Jν(kr0)Jν(kr) cos(νϕ)

]

=1

2π

∞∫0

dk k∑ν∈Z

Jν(kr0)Jν(kr)eiνϕ

Now comparing this to equation (A.3)

h(r, ϕ, t) =∑n∈Z

∫dλ a(n, λ)einϕ exp (−λt)

[C1Jν

(√2λθ0r

)+ C2Yν

(√2λθ0r

)]

for t = 0 and φ = 0, we immediately see that we have to choose C2 = 0. Further,we have to substitute

√2λθ0

by k, or λ by k2θ02 , respectively. This leads to

h(r, ϕ, t) =∑n∈Z

einϕ∫dk kθ0 a(n, k) exp

(−θ0

2k2t

)C1Jν(kr)

The last thing to do for now is to identify a(n, k) with 1C1θ0

12πJν(kr0), so we finally

obtain

h(r, ϕ, t) =1

2π

∑n∈Z

einϕ∫dk k exp

(−θ0

2k2t

)Jν(kr)Jν(kr0) (A.5)

Note that ν =√n2 + in(Γ1+Γ2)

πθ0is ∝ n - and becomes exactly equal to n for (Γ1 +

Γ2) → 0. We now additionally show that the initial condition is also verified for(Γ1 + Γ2) 6= 0: With the identity

∞∫0

e−ρ2k2Jν(rk)Jν(r0k)k dk =

12ρ2

exp(−r

2 + r20

4ρ2

)Iν

(rr0

2ρ2

)

76

A.1 The special case ξ = 0 - direct derivation

we obtain for ρ2 = θ02 t and the modified Bessel function Iν(z) = i−νJν(iz) the

solution for the relative part(compare [3], p.7)

h(r, ϕ, t) =1

2πθ0texp

(−r

2 + r20

2θ0t

)∑n∈Z

einϕIν

(rr0

θ0t

)(A.6)

= H(r, t)

[1 + 2<

( ∞∑n=1

I−10

(rr0

θ0t

)Iν

(rr0

θ0t

)einϕ

)](A.7)

(A.8)

whereas H(r, t) reads

H(r, t) =1

2πθ0texp

(−r

2 + r20

2θ0t

)I0

(rr0

θ0t

)(A.9)

which matches exactly the generalized case as shown above.

Remark In [3], the Fokker-Planck equation was given by

∂

∂tP (r, θ, t) = − 1

r2

∂

∂θP (r, θ, t) + ν

(∂2

∂r2+

1r

∂

∂r+

1r2

∂2

∂θ2

)P (r, θ, t)

and its solution by

P (r, θ, t) = G(r, t)

1 + 2<

∞∑p=1

I−10

( r

2νt

)Iµp

( r

2νt

)eipθ

with

G(r, t) =1

4πνtexp

(−r

2 + 14νt

)I0

( r

2νt

)(A.10)

whereby the choice was r0 = 1

So the results are compatible.

77

A Appendix

A.2 Fractional statistics of anyons

We consider

H =P2

4m+

(p− qarel)2

m

whereby

arel =φ

2πz × r|r|2

The center of mass motion thus follows a free translation. However the relative onedepends on whether the orbiting charge is bosonic, fermionic or corresponds to ananyon. It has reduced to a system of a single charge of mass m

2 orbiting around aflux at a distance r, proposing cylindrical coordinates r = (r, θ). Thus for the vectorpotential

arel =

(a

(r)rel

a(θ)rel

)=(

0φ

2πr

)If we take a bosonic charge orbiting around a bosonic flux, the wavefunction of thesystem is symmetric under exchange, meaning ψrel(r, θ + π) = ψrel(r, θ).If we perform a gauge transformation now

arel → a’rel = arel −∇Λ(rθ) = arel −∇(φ

2πθ

we have (a′

(r)rel

a′(θ)rel

)=

(a

(r)rel −

1r∂∂θ (φθ2π )

a(θ)rel

)=(

φ2πr −

φ2πr

0

)=(

00

)and hence the Hamiltonian can be given by

H =P2

4m+

p2

m,

the Hamiltonian of two free particles. Now we have to take into account that thewavefunction of the relative Hamiltonian has changed, we have

ψ′rel(r, θ) = e−iqΛψrel(r, θ) = exp(−i qφ2πθ)ψrel(r, θ)

Note that this wavefunction is no longer symmetric under r→ −r, because

ψ′rel(r, θ + π) = exp(−iqφ2

)ψ′rel(r, θ) = e−iαψ′rel(r, θ)

So the initially assumed two objects with bosonic charge orbiting around a bosonicflux behave like two particles developing a phase e−iα under exchange, meaning that

78

A.3 Modified Runge-Kutta method - source code

they follow fractional statistics.

A.3 Modified Runge-Kutta method - source code

PROGRAM punktwirbe l

IMPLICIT NONE

INTEGER : : n i t e r =2000 , iREAL : : dt =0.1 ,QREAL, DIMENSION(2 ) : : x1 , x2 , k1 1 , k1 2 , k1 3 , k1 4 , k2 1 , k2 2 ,

k2 3 , k2 4 , bm1, bm2

! S t a r t p u n k t ex1 =(/0 ,0/)x2 =(/1 ,1/)

Q=0.000001 ! Rauschparameter

! I t e r a t i o nOPEN(9 ,FILE=” tra . dat ” )

DO i =1, n i t e r

CALL c a l c r h s ( x1 , x2 , k1 1 , k2 1 )CALL c a l c r h s ( x1+dt /2 .0∗ k1 1 , x2+dt /2 .0∗ k2 1 , k1 2 , k2 2 )CALL c a l c r h s ( x1+dt /2 .0∗ k1 2 , x2+dt /2 .0∗ k2 2 , k1 3 , k2 3 )CALL c a l c r h s ( x1+dt∗k1 3 , x2+dt∗k2 3 , k1 4 , k2 4 )CALL gausZufallBM (bm1)CALL gausZufallBM (bm2)

x1=x1+dt /6 .0∗ ( k1 1 +2.0∗ k1 2 +2.0∗ k1 3+k1 4 ) + s q r t (2∗Q∗dt ) ∗bm1

x2=x2+dt /6 .0∗ ( k2 1 +2.0∗ k2 2 +2.0∗ k2 3+k2 4 ) + s q r t (2∗Q∗dt ) ∗bm2

PRINT∗ , ” i ” , iWRITE( 9 ,∗ ) dt∗ i , x1 ( 1 : 2 ) , x2 ( 1 : 2 )

END DOCLOSE(9 )!

79

A Appendix

<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<

ENDPROGRAM

!∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

!∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

SUBROUTINE c a l c r h s ( x1 in , x2 in , x1 out , x2 out ) ! Subrout ineRunge Kutta

IMPLICIT NONE

REAL : : gamma1=1.0 , gamma2=−1.5REAL : : r , p iREAL, DIMENSION(2 ) , INTENT(IN) : : x1 in , x2 inREAL, DIMENSION(2 ) , INTENT(OUT) : : x1 out , x2 out

p i=ACOS(−1.0)

r =(( x2 in (1 )−x1 in (1 ) ) ∗∗2+( x2 in (2 )−x1 in (2 ) ) ∗∗2) ∗∗0 .5

x1 out =((/−( x2 in (2 )−x1 in (2 ) ) , x2 in (1 )−x1 in (1 ) /) ∗gamma1)/ (2 . 0∗ pi ∗ r ∗∗2)

x2 out =((/−( x1 in (2 )−x2 in (2 ) ) , x1 in (1 )−x2 in (1 ) /) ∗gamma2)/ (2 . 0∗ pi ∗ r ∗∗2)

END SUBROUTINE

SUBROUTINE gausZufallBM ( box muel l e r ) ! Subrout ine gausscheZ u f a l l s z a h l

IMPLICIT NONE

REAL : : p iREAL, DIMENSION(2 ) , INTENT(INOUT) : : box muel l e rREAL, DIMENSION(2 ) : : randomCALL random number ( random (1) )

80

A.3 Modified Runge-Kutta method - source code

CALL random number ( random (2) )p i=ACOS(−1.0)

box muel l e r (1 ) = s q r t (−2∗ l og (1−random (1) ) ) ∗ cos (2∗ pi ∗random(2) ) ;

box muel l e r (2 ) = s q r t (−2∗ l og (1−random (1) ) ) ∗ s i n (2∗ pi ∗random(2) ) ;

END SUBROUTINE

81

A Appendix

82

Bibliography

[1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. U.S.Dept. of Commerce, National Institute of Standards and Technology, 1972.

[2] O. Agullo and A. Verga. Exact two vortices solution of navier-stokes equations.Physical Review Letters, 78:2361, 1997.

[3] O. Agullo and A. Verga. Effect of viscosity in the dynamics of two point vortices:Exact results. Phys. Rev. E, 63(5):056304, Apr 2001.

[4] D. Bernard. Turbulence for (and by) amateurs, 2000.

[5] S. Bronstein. Teubner-Taschenbuch der Mathematik. B. G. Teubner StuttgartLeipzig, 1996.

[6] H. J. H. Clercx and G. J. F. van Heijst. Two-dimensional navier–stokes turbu-lence in bounded domains. Applied Mechanics Reviews, 62:020802 (25 pages),2009.

[7] G. L. Eyink and K. R. Sreenivasan. Onsager and the theory of hydrodynamicturbulence. Rev. Mod. Phys., 78(1):87, Jan 2006.

[8] G. Falkovich, K. Gawedzki, and M. Vergassola. Particles and fields in fluidturbulence. Reviews of Modern Physics, 73:913, 2001.

[9] R. Friedrich. Vortrag am 18.1.2010. none, 2010.

[10] U. Frisch. Turbulence - The Legacy of A. N. Kolmogorov. Cambridge UniversityPress, 1995.

[11] K. Gawedzki and M. Vergassola. Phase transition in the passive scalar advec-tion. Physica D: Nonlinear Phenomena, 138(1-2):63 – 90, 2000.

[12] H. Haken. Synergetics - An Introduction. Springer-Verlag Berlin HeidelbergNew York, 1977.

[13] C. Harper. Analytical Mehtods in Physics. Wiley, 1999.

[14] W. Hoeber. Repititorium Theoretische Physik, 2. Auflage. Springerverlag BerlinHeidelberg New York, 2005.

[15] M. H. R. F. John Argyris, Gunter Faust. Die Erforschung des Chaos. Springer-Verlag Berlin Heidelberg, 2010.

83

Bibliography

[16] E. Kamke. Differentialgleichungungen. Akademische Verlagsgesellschaft Geest& Portig K.-G., 1969.

[17] A. Khare. A class of exact solutions for n anyons in a n-body potential. PhysicsLetters A, 221(6):365 – 369, 1996.

[18] A. N. Kolmogorov. The local structure of turbulence in incompressible viscousfluid for very large reynolds numbers. Proceedings: Mathematical and PhysicalSciences, 434(1890):9–13, 1991.

[19] R. H. Kraichnan. Small-scale structure of a scalar field converted by turbulence.Phys. Fluids, 11:945, 1968.

[20] R. H. Kraichnan and D. Montgomery. Two-dimensional turbulence. Reportson Progress in Physics, 43(5):547–619, 1980.

[21] J. Miller, P. B. Weichman, and M. C. Cross. Statistical mechanics, euler’sequation, and jupiter’s red spot. Phys. Rev. A, 45(4):2328–2359, Feb 1992.

[22] A. D. Polyanin and V. F. Zaitsev. Handbook of Exact Solutions for OrdinaryDifferential Equations, 2nd Edition. Chapman & Hall/CRC, Boca Raton, 2003.

[23] J. P. R. Friedrich. Fluid dynamics: Turbulence. Institute for TheoreticalPhysics, University of Muenster, Institute of Physics, Carl-von-Ossietzky Uni-versity Oldenburg.

[24] S. Rao. An anyon primer, 1992.

[25] O. Reynolds. An experimental investigation of the circumstances which deter-mine whether the motion of water shall be direct or sinuous, and of the law ofresistance in parallel channels. Philosophical Transactions of the Royal Societyof London, 174:935–982, 1883.

[26] H. Risken. The Fokker-Planck equation. Methods of Solution and Applications.Springer Verlag Berlin Heidelberg New York Tokyo, 1984.

[27] N. Svartholm. Die losung der fuchsschen differentialgleichung zweiter ordnungdurch hypergeometrische polynome. Math. Ann., 116:413–421, 1938.

[28] P. Tabeling. Two-dimensional turbulence: a physicist approach. Physics Re-ports, 362(1):1 – 62, 2002.

84

Danksagung

Abschließend mochte ich allen danken, die zum Gelingen dieser Arbeit beigetra-gen haben, insbesondere danke ich Prof. Dr. R. Friedrich fur das interessanteThema der Diplomarbeit und die freundliche Betreuung zu jedem Zeitpunkt derBearbeitungszeit. Außerdem danke ich Michael Wilczek, Oliver Kamps, RobertHahn, Christoph Honisch, Johannes Greber, Michel Voskuhle, Michael Kopf undAnton Daitche fur Hilfestellungen bei verschiedensten Problemen und fur inspiri-erende Diskussionen.Meiner Familie, und ganz besonders meinen Eltern Norbert und Monika Rath,danke ich fur deren Unterstutzung, ohne die mir mein Studium wahrscheinlich nichtmoglich gewesen ware.Bei allen Mitarbeiterinnen und Mitarbeitern des Instituts fur Theoretische Physikbedanke ich mich fur ihre Hilfsbereitschaft und die angenehme Arbeitsatmosphare.

Hiermit versichere ich, die vorliegende Arbeit selbststandig angefertigt und michkeiner anderen als der angegebenen Quellen und Hilfsmittel bedient zu haben.

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·Datum Unterschrift