2次元乱流系における動的統計性について

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九州大学学術情報リポジトリ

Kyushu University Institutional Repository

2次元乱流系における動的統計性について

渡邊, 威

Graduate School of Sciences, Kyushu University

https://doi.org/10.11501/3166627

出版情報：Kyushu University, 1999, 博士（理学）, 課程博士バージョン：

権利関係：

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Dynamical Statistics of Two-Dimensional Turbulence

Takeshi Watanabe

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Thesis

Dynamical Statistics of Two-Dimensional Turbulence

Takeshi Watanabe

Department of Physics, Faculty of Science Kyushu University 33, Fukuoka 812-8581, Japan

February 18, 2000

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Mirna equation is studied in both the externally forced and freely decaying systems. Statistical natures in developing and decaying processes of turbulence are discussed from the dynamical scaling viewpoint based on the Kolmogorov-type phenomenological scaling theory in the inverse energy cascade process.

In the externally forced system, the vortical quasicrystal structure self-organizes in the potential vorticity field. It is shown that the temporal evolution of the power spectrum of the potential vorticity field, which is well characterized by the single peak position and its height, obeys the dynamical scaling law in the quasicrystalization process. The dimensional analysis with the assumption that the energy transfer rate in the inverse energy cascade range is constant temporally and spatially leads to the determination of the dynamical scaling laws, which turn out to be in good agreement with the results numerically obtained.

In the freely decaying system, the dynamical scaling laws similar to those in the externally forced case are both numerically and theoretically found in the developing process of self

organized coherent vortices under the assumption that the total energy is constant during the temporal evolution. In the physical space, the collective motion of these coherent vortice i represented by the phenomenological scaling theory on the average quantities related to them.

We numerically show that these quantities algebraically evolve with time through the mutual advection and merging among them, which is characterized by a single scaling exponent X·

Moreover we discuss the phenomenological determination of the scaling exponent from the consideration of the advection velocity scaling, and obtain x ⁼

1/2,

which turns out to be in good agreement with the numerical result.

A part of this thesis is also assigned to discuss the fundamental statistical properties of two-dimensional turbulence described by the two-dimensional Navier-Stokes equation in order to review the traditional statistical description of two-dimensional turbulence and clarify our main results.

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1 1 Introduction

Many regular patterns observed in the nature are often called coherent structures. The self

organization, which causes these structures, can be regarded as the typical nonlinear and nonequibirium phenomena. In physical systems, for example, the magnetic wall structure in the spin systems, the roll and hexagonal convection patterns in the thermal convection system, and the spiral or target patterns in the reaction diffusion and the liquid crystal systems are the typical examples of the coherent structures and have been actively investigated in the context of the ordering process and the dissipative structure

[1].

In the field of fluid mechanics, on the other hand, many kinds of flow patterns are observed in real flow fields such as in planetary atmosphere. One of the most famous examples of the flow pattern observed in fluid may be the von Karma11 vortex street which is produced in the downstream of objects in a laminar flow. Such a vortex pattern is observed over wide range of spatial scale, from atmosphere to experiment in laboratories. It is very useful to introduce dimensionless parameter to characterize the state of flow. The most important parameter may be the Reynolds number

(Re) [2],

which is determined in terms of the characteristic velocity of fluid U, the characteristic length of the system L, and the kinematic viscosity v of the working fluid as

Re=-.

UL

v

(1.1)

In the above case, U and L are estimated by the velocity of mean flow and the scale of the object respectively. It is experimentally known that the von Karman vortex street is observed in flows with the Reynolds number

Re

,...,

0(102) [3].

In this case, the flow state has the regular structure in space and produces vortices periodically in time. As the Reynolds number is increased, the vortex street breaks down and the flow pattern changes into a turbulent state.

This is the onset of turbulence. In a sufficiently large Reynolds number, the flow pattern is fully complicated, and no regular structure and patterns are observed. Not only turbulent flows are widely observed in nature, but also we use efficient transport due to turbulence even in our daily life, e.g. in mixing cream in a cup of coffee. Though turbulent flow is complex and random in space and time, what kind of the coherent structures or statistical laws exists in turbulence ?

Rapid progress in computer facility in recent years has brought about significant knowledge on turbulence. The detailed direct numerical simulation (DNS)

[4]

of the Navier-Stokes (NS) equation and the visualization of the interesting field are the powerful tools for investigations of turbulence. Moreover the additional benefit of the DNS is that we can directly and minutely observe the physical processes of the dynamics in real flow field and calculate several statistical quantities which is difficult to be analyzed with data experimentally obtained. From these studies, it has been clear that certain kinds of coherent structures exist even in high Reynolds

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2 1 INTRODUCTION

number turbulence. In homogeneous and isotropic turbulence of incompressible fluid which is described by the three dimensional (3D) NS equation, for example, we can notice that the region with large amplitude of vorticity, the tube or sheet like vortex structure, is localized in space and survives for long time [5, 6]. The existence of such coherent structures clearly indicates that in these regions strong and weak velocity fluctuations coexist in the turbulent field. This recognition have given an impact for the traditional statistical description of turbulence because the physical picture of turbulence is not consistent with the Kolmogorov 1941 theory (K41) [7]

in fully developed turbulence, which derives the universal scaling law for the moment function of the longitudinal velocity increment under the assumption of no fluctuation of the energy dissipation rate, i.e. the transfer rate E. Therefore, it is thought that the coherent structure is the origin of the fluctuation of ^Eand causes the deviation from the K41 scaling. This is called the intermittency problem, which is one of the main topics of turbulence theory, and have been discussed from the several viewpoints [3, 8, 9].

Thrbulent flow observed in our world is generally in the 3D space. Special interest on turbulenc has also been focused on how the geometry of fluid motion affects the properties of turbulent motion. For example, large scale fluid motions in the atmosphere or ocean, and th l ctrostatic fluctuation in strongly magnetized plasma can be typically dealt with the two dimen ional (2D) fluid. Therefore, one expects that the turbulence observed in these systems may b discussed by considering the characteristics of 2D turbulence. It is very important to study the several characteristics of 2D turbulence not only for the fundamental interest from fluid mechanics but also for the application to the above-mentioned systems.

Many inve tigations have been actively carried out to clarify the dynamical or statistical properties of the above-mentioned systems from the viewpoint of 2D turbulence [10, 11]. It i recognized that the fundamental properties of 2D turbulence are mostly described by the 2D-NS equation, and many aspects are mainly investigated from the theoretical point of view as well as by the DNS of 2D-NS equation. The most remarkable property of 2D turbulence is characterized by th cascade dynamics different from that of 3D turbulence represented by the K41 theory. The unique cascade dynamics of 2D turbulence originates from the difference of the conservation laws betw n the 3D- and 2D-NS equations in the invicid limit, i.e. the total energy and the total enstrophy are the quadratic conservation quantities in the 2D case, while the total energy in the 3D case. So the cascade dynamics in 2D turbulence is characterized by the dual cascade of thes quantities which has been proposed by Kraichnan [12] and Batchelor [13]. They predicted that the enstrophy i transferred to the small scale motion, while the energy to the large scale motion (inverse energy cascade). The dual cascade process is now well confirmed by the numerical simulations [14 15, 16, 17, 18, 19] or the laboratory experiments [20, 21, 22 23, 24, 25]. The en rgy transfer process toward the large scale is a peculiar interesting property of 2D turbulence because the energy of system tends to be concentrated on the largest

3

scale through the inverse energy cascade and the large cale coherent structure s If-organizes in the physical space. At present, it is one of the main topics in the investigations of 2D turbulence to clarify the formation process of coherent structure and its dynamical and statistical natures.

The first step in order to apply the knowledge in the inve tigation of the 2D- S turbulence to realistic 2D turbulent phenomena is to examine the turbulent state described by more concrete model equations for the 2D fluid motion. In this thesis, we inve tigate the 2D turbulence described by the Charney-Hasegawa-Mirna (CHM) equation which is the implest but es ntial model equation of motion to discuss several phenomena observed in the large scale fluid motion of the atmosphere or the ocean [26, 27, 28, 29] and the electrostatic fluctuation in magnetically confined plasma [30]. One of important states observed in these systems is the drift-Rossby wave turbulence [11, 31] which is well explained by the CHM equation. W all the turbulence described by the CHM equation as the CHM turbulence.

The CHM equation has a structure similar to the vorticity equation derived from the 2D-NS equation. In fact, they coincide with each other in an appropriate limit. So the conservation laws and its dual cascade dynamics in the CHM turbulence are similar to those of 2D-NS turbulence [32, 33]. However, the important difference between the CHM and 2D-NS turbulence is represented by the existence of the characteristic spatial scale present in the CHM equation, which separates the dynamical and statistical properties of these turbulent states. In the 2D-NS equation, there is no such spatial scale. The investigations concerning the CHM turbulenc in the past have mainly been discussed about the static properties of th dual ca ade proce and vortical structures in the physical space [33, 34, 35, 36, 37, 38, 39, 40, 41 ]. On of the purposes of this thesis is to clarify the dynamical properties of cascade in the CHM turbulence. The 2D turbulent state when the external force is present is quite different from that of the freely decaying turbulence because the total energy is not almost dissipated and does not reach the statistically steady state in an infinitely large system unless an effective dissipation mechanism on the large scale is included. As a result, we must consider the dynamical or statistical properties of the 2D turbulence for the externally forced and freely decaying cases separately.

In this thesis, we investigate temporal evolutions of spectra in the wavenumber space for both cases and analyze their statistical properties from the viewpoint of the dynamical scaling based on the Kolmogorov-type scaling phenomenology. Especially we investigate the inverse energy cascade process in the wavenumber space and discuss its scaling properties by considering the pattern formation process of the coherent vortex structure in the physical space [42, 43].

Furthermore the important nature of 2D turbulence is represented by the emergence of axis

symmetrical isolated vortices in the physical space. These vortices called ' coherent vortice ' was first found by McWilliams [44] in the study of the freely decaying 2D-NS turbulence. The role of coherent vortices in the 2D-NS turbulence is remarkable since the dynamics of the system is governed by the collective motion of coherent vortices. Many studies have revealed

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4 1 INTRODUCTION

the mechanisms of self-organization of coherent vortices such as their collective motion [44, 45, 46, 47, 48, 49] and the final vortical state [50] predicted by the phenomenological variational principle. In particular, it is a remarkable fact [46, 47, 49] that the existence of coherent vortices causes a large deviation of the energy spectrum in the enstrophy transfer range predicted by Kraichnan and Batchelor. In this way, it is recognized that the existence of coherent vortices has a significant influence on the traditional statistical description of 2D turbulence.

The temporal evolution of coherent vortices in the decaying 2D-NS turbulence can be di

vided into three stages. In the first stage, coherent vortices self-organize by starting with a random initial condition. In the second stage, the coherent vortices move like point vortices de

scribed by the Hamilton dynamics, and develop into larger ones through the mutual advection and the merging process among vortices with same sign of circulation. The statistical nature of this collective motion by coherent vortices is characterized by the phenomenological scaling theory [51] composed of average physical quantities related to coherent vortices. These quanti

ties algebraically develop in time, which are characterized with a single scaling exponent. The theoretical determination of this scaling exponent is an interesting open problem. At present, many numerical simulations [51, 52, 53, 54] or laboratory experiments [55, 56, 57] are discussing the validity of the scaling theory. In the last stage, one pair of vortices with opposite sign of circulation diffusely disappears with time.

On the other hand, the freely decaying CHM turbulence is also dominated by the collective motion of self-organized coherent vortices [35]. However the nature of collective motion is quite different from that of freely decaying 2D-NS turbulence because of the effect of characteristic cale present in the CHM equation. Mutual advection among coherent vortices are limited within the order of this scale range, which is remarkable characteristic in contrast to the long range interaction among them in 2D-NS turbulence. This fact greatly affects the dynamics of coherent vortices in the CHM turbulence. So another aim of this thesis is to clarify the role of coherent vortices in the CHM turbulence by investigating the conservative properties of them [43]. In addition, we construct the phenomenological scaling theory for coherent vortices from the analogy in the freely decaying 2D-NS turbulence and discuss its validity. Moreover, we phenomenologically derive the value of the scaling exponent characterizing the scaling theory by considering the advection velocity scaling of coherent vortices [54]. Furthermore, we will discuss unique properties of the decaying CHM turbulence which are not observed in the decaying 2D-NS turbulenc .

This thesis i organized as follows. In Sec.2, we survey fundamental conservative properties of the 2D-NS equation and introduce the Fourier representation for the statistical description in the wavenumber space. We briefly explain examples of the 2D fluid motion and derive the CHM equation in Sec.3. The dual cascade theory for the 2D-NS turbulence is reviewed in detail and the scaling law of the energy spectrum for the CHM turbulence is discussed in Sec.4. Sections

5

2-4 are the review part of this thesis. In Sec.5, we discuss the vortical self-organized tate in the CHM turbulence through the direct numerical simulation of the CHM equation and the visualization of the vorticity field. In connection with this vortical ordering pro ess, dynamical scaling properties in the wavenumber space are investigated both numerically and th oretically in Sec.6. Section 7 is assigned to the brief review of the phenomenology of collective motion of coherent vortices and its application for the freely decaying CHM turbulence. The theoretical derivation of the scaling exponent and its validity are discussed. We briefly summarize the main results in this thesis in Sec.8.

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6 2 FUNDAMENTAL PROPERTIES OF INCOMPRESSIBLE FLUID

2 Fundamental properties of incompressible fluid

In thi section, we show the essential difference between 3D and 2D fluid motions through the conservation properties of the Navier-Stokes (NS) equation. In addition, by introducing the Fourier representations of NS equation and the statistical quantities for the description of turbulent state, dynamical characteristics in the wavenumber space are surveyed. The details are shown in [2).

2.1 Characteristics of the Navier-Stokes equations in 3D and 2D

First we consider 3D case. The temporal evolution of an incompressible fluid is described by the Navier-Stoke (NS) equation

aui aui

1

ap a2ui

-+u·-=---+v-- at J axj paxi ax;'

^(2.1)

where

Ui (i =

1, 2, 3) indicates the

Xi

component of the velocity field ^u

(x

,

t) (x = (x1, x2, x3))

and

p(x, t)

is the pressure field. Both the kinematic viscosity

v

and the density

p

of fluid are as umed to be constant. The incompressibility of fluid is expressed as

aui =

0

axi .

_(2.2)

Fluid motion is a famous example of nonlinear dissipative phenomena. The second term of the l ft hand side of Eq.(2.1) is the inertial term which is the origin of the nonlinearity of fluid motion. On the other hand, the dissipative nature of fluid originates from the last term of Eq.(2.1).

The i-th component of vorticity field

Wi (x, t)

is defined by

(2.3)

where

a = (ajax!, a;ax2, a;ax3).

Equations (2.1) and (2.2) are reduced to the equation of vorticity

awi awi a2wi

�

+ Uj-a . = aijWj + V-a 2

¹

u�

xJ xj

where

aij

represents the rate of strain tensor defined as

(2.4)

(2.5)

In Eq.(2.4), the first term of right hand ide affects the stretching and compressing of vorticity.

The existence of thi term brings about a particular statistical or dynamical nature of 3D fully developed turbulence.

In most of this thesis, we shall take the case where the fluid is confined within the domain V with the periodic boundary condition. In this case, it is important to note that the constants

2.1 Characteristics of the Navier-Stokes equations in 3D and 2D 7

of the fluid motion are derived for the invicid case

v =

0. First the total energy of fluid, which is the integral of

ut

/2 over the whole domain V, satisfies the equation derived from Eqs. (2.1) and (2.2) as

(2.6)

This indicates that, in the case of

v =

0, the total kinetic energy is a constant of motion. On the other hand, for finite

v,

the total energy decreases with time. This originates from the last term in Eq.(2.1) or (2.4). Moreover, the total helicity of fluid defined by the integral of

Uiwi/2

over the domain Vis conserved for

v =

0 because of the equation derived from Eqs.(2.1), (2.2) and (2.4)

-

d

J

¹

-UiWidV = -v J ( aij _J + _z ^aw ^· ^aw ^· )

dV.

dt v 2 ^v

axi ax j

^(2.7)

The right hand side of Eq.(2.6), which is the integral of

wf

/2 over the domain V, is called the total enstrophy. The total enstrophy is thus closely connected to the nature of dissipation rate of total energy. The equation for temporal evolution of the total enstrophy derived from Eq.(2.4)

-^d

j

-w· dV ¹

² = j w·a· ·w

^·^d

V

^-2

v /

^{-n. dV}^{1 2}

dt V 2 ¹ V

z ZJ J

^V²

z

^(2.8)

yields the non-vanishing term for the invicid case

V =

0, where

f2i = (a

X W

)i

and the integral of

Ot

/2 over V is called the parinstrophy. Therefore, the total enstrophy is not a constant of motion in the invicid case. This result implies that the total energy will be dissipated even in the limit

v

^-+0, i.e. Re -+ oo, because the right hand side of Eq.(2.6) (the energy di sipation rate) can take the finite value under

v

^-+0 limit if the total enstrophy diverges with time by the amplification of vorticity which is caused by the first term of the right hand side in Eq.(2.8).

This is the essential feature of 3D fluid motion when one discusses statistical or dynamical properties of fully developed turbulence.

In order to introduce the 2D fluid motion, let

x1 = x, x2 = y, X3 =

^z be a right-handed coordinate system, and we define the velocity field by

ui(x, t) = Ux(x, y, t), u2(x, t) = uy(x, y, t), u3(x, t) =

0. (2.9)

The 2D fluid is confined within the domainS with periodic boundary condition. From Eqs.(2.3) and ( 2. 9), the vorticity field yields

auy aux

^_

Wx = Wy =

0,

Wz = ax - ay = W (X

¹^Y¹

t),

where one should notice that the vorticity field is a scalar function in the 2D fluid.

The equation of motion for vorticity

w(x, y, t)

is given by

aw 2

- + at

^u^·

\i'w = v\7 w

^'

(2.10)

(2.11)

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where the 2D differential operator \7 ⁼

(ojox, oj8y)

is defined. The most important difference between Eqs.(2.4) and (2.11) is that there is no compressing and stretching process of vorticity in 2D fluid motion. This is the essential feature of the 2D fluid dynamics. Consequently, in the invicid case

v =

0, the vorticity

w

for each fluid particle is conserved in time.

Furthermore, we can introduce the stream function

'1/J(x, y, t)

^via

8'1/J 8'1/J

Ux =

_-

_fJy

'

Uy

⁼

OX . The vorticity is therefore expressed by the stream function

'1/J

^as

and Eq.(2.11) is rewritten with the equation of the stream function as

where the Jacobian operator

J

(a, b)

⁼

oa 8b

^_

8a ob

OX

8y By ox

(2.12)

(2.13)

(2.14)

(2.15)

is defined. Thus, the 2D incompressible fluid motion is described by a single scalar function

1/;(x, y, t).

In the invicid case

v =

0, several constants of motion exist in the 2D fluid motion. The total energy same as Eq.(2.6) is conserved and the total helicity is a trivial constant in 2D fluid. The most important difference between 3D and 2D fluids is the conservation of the total enstrophy because of the equality Eq.(2.8)

!!_ f �w2dS = -

2

v f �n?ds dt ls

2

ls

2 z '

(2.16) where One should notice that

WiCJijWj =

0 and

nl

_{= O}

W _jfJy, n2 = _-fJwjfJx

_and

n3 =

0 from the definition. Therefor , the total enstrophy is conserved for

v

⁼0. This nature originates from the conservation of vorticity with time along the stream line, i.e. Eq.(2.11) or (2.14) with

v =

_0.

In fact, integrals of any power of vorticity

wq

^over

S

are constants of motion. Consequently, any functions of

w,

which are called Casimir, are also conserved in the 2D incompressible fluid.

The conservation of enstrophy implies that the total energy is not dissipated in the limit

v

^-+⁰

since the total enstrophy is always finite with time because of Eq. (2.16). On the other hand, the total parinstrophy atisfies the equation derived by Eq. (2.11) as

! is � n�dS =is fl;cr;jfljdS- �is ( ��:

⁺

��;) ² ^dS.

^(2.17)

The first term on the right hand side repre ents the rate of amplification of vorticity gradients by extension of iso-vorticity lines. Therefore we notice that the parinstrophy can increase with

2.2 Fourier analysis of the 2D Navier-Stokes equation 9

time in the limit ^{v -+}0. This fact implies that the total enstrophy will b dissipated even in the invicid limit because the right hand side of Eq.(2.16) (th enstrophy dissipation rate) can take the finite value by the amplification of parinstrophy. The energy con ervation and the enstrophy dissipation in the invicid limit indicates that th 2D turbulence shows quite different aspects in comparison with the 3D turbulence. In connection with this characteristics, the detailed statistical features of 2D turbulence will be discussed in a later sub-section.

2.2 Fourier analysis of the 2D Navier-Stokes equation

It is quite useful to introduce the Fourier component representation for the statistical descrip

tion of turbulence because one can recognize the various scales of fluctuation as the collection of modes with various wave lengths. Let the 2D domain

S

be a square of size

L

⁽⁰^:S

{ x, y}

^:S

L)

as

S

₌

_L2

with the periodic boundary condition. The Fourier component

'1/Jk(t)

of th stream function

'lj;(r, t), (r = _{(x, y))}

is defined as

1/J(r' t) = c; r ^� ^{1/Jk ( t)}

^exp[ik

^{. r],}

^(2.18)

1/Jk(t) = c� r ^Is 1/J(r, t)

^exp[-ik.

r] dr, '1/J�(t) = '1/J-k(t),

(2.19) (2.20) where k

= _{(kx, ky)}

and ^z*denotes the complex conjugate of ^z. From Eqs.(2.13) and (2.14), the equation of motion for

'1/Jk(t)

is given by

/:::;.

o'l/Jokt(t)

⁺^v

" k2"'

'f'

' k(t) =

_��

_M k _{( )} P, q "'· (t)"'· (t)

^'f'P ^'f'Q ^'

k=p+q

q2- p2

^/:::;. ^-

(

₂7r

) ₂

Mk(P, q) =

₂_k

₂ _(p

X

q)z, L = _L L _8k,p+q,

k=p+q p,q

(2.21)

(2.22) where lkl

= _k, _IPI

₌

_{p, lql} = _q

_and

8ij

denotes the Kronecker delta. Moreover, the Fourier component of the vorticity

w(r, t)

is represented as

wk(t) = -k2'1/Jk(t).

Therefore Eq.(2.21) is rewritten into the equation of motion for

wk(t)

^as

o wk(t) 2 �

- 0 - t -

⁺

vk wk(t)

⁼ ^�

L(p, q)wp(t)wq(t), k=p+q

L( _p, _q

⁾^__-

(p

^X₂

q)z (_!__ _{q2 p2 .}

_

_!__)

(2.23)

(2.24)

Furthermore, the Fourier representation of the total energy

E(t)

and the enstrophy

Q(t)

^are

given through

1

/,

1

(

2n

)

4 1

(

2n

)

₄

_k2 ₂

E(t) =

_{L2 £2}

2iu(r, t)i2 dr = _L � _2iuk(t)l2 = _L � 21'1/Jk(t)i ,

^(2.25)

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1

r

¹

(

^21r

) ⁴

¹

(

^21r

) ⁴ ^k4

Q(t)= £ 2 } L 2 2w(r,t)2dr= L � 21wk(t)12= L � 2��k(t)12,

^(2.26)

where

luk(t)l2 = k21�k(t)12

is used from Eq.(2.12).

Let us take the limit L _{---1 oo.} In this case, the summation of

k

is replaced by the integral of

k

^as

6.

L

^---1

f f dpdq5(k- p- q) =I 1 ^dpdq.

k=p+q

^6.

(2.27)

(2.28)

In this thesis, on the discussion of the statistical nature of 2D turbulence, we suppose that the statistical nature of velocity or vorticity field is homogeneous and isotropic in space. This assumption simplifies the statistical description of turbulence. One of the most important sta

tistical quantities in turbulence is the energy spectrum

E(k, t)

which indicates the distribution of the energy in each modes

k

defined through

(2.29)

where

(-

^·^·

)

being the ensemble average. One should notice that

(l�k(t)12)

is the function of scalar

k

from the assumption of the statistical homogeneity and isotropy. Similarly to

E(k, t),

the en trophy spectrum

Q(k, t)

is defined by

(2.30)

The equation for the temporal evolution of

E(k, t)

is derived from Eqs.(2.21) and (2.29) with L _{---t oo}as

(2.31)

(2.32)

(2.33)

where a is the angle between

p

^and

q.

The function

T(k, t)

is called the

energy transfer function,

which plays an important role for the nonlinear energy transfer dynamics in the wavenumber space. From Eq. (2.33), the detailed conservation of energy and enstrophy for triad interaction with

k = p + q

are easily obtained as

T(k,p,q) +T(p,q,k) +T(q,k,p) =

^0, ^(2.34)

(2.35)

2.2 Fourier analysis of tbe 2D Navier-Stokes equation

Consequently, the integrations of

T(k, t)

^and

k2T(k, t)

over all region of

k

^{lead to}

fooo T(k, t) dk =

^0,

fooo ^{k2T(k, t)} ^{dk =}

^0,

11

(2.36)

which just correspond respectively to the conservation of the total energy and the ens trophy. In fact, the most essential characteristics of the nonlinear interaction in the Fourier space is that the nonlinear term of the NS equation conserves not only the total energy and the en trophy as mentioned above but also the energy and the enstrophy among any three Fourier mod s of {

uk, up, uq}

^{and {}^Wk,

wp, wq}

satisfying the condition

k + p + q =

^0. ^Thisconditions are represented as

(2.37)

(2.38) which are easily proved from Eqs.(2.21) and (2.23).

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12 3

2D FLUID SYSTEMS

3 2D fluid systems

In this section, we introduce some physical systems described as the 2D fluid. Realistic flow geometry in this world is, needless to say, the 3D fluid motion. However, 3D fluid flow under certain restricted conditions can be reduced to 2D fluid flow. Here, two typical examples are introduced. One is the geophysical fluid motion and the other is the electrostatic fluctuation in magnetically confined plasma. These systems are composed of certain dynamical equations, and we will show that these equations of motion have same structures. Noting that the time scale under consideration is extremely longer than the characteristic time scale contained in these systems, we can derive the single equation of motion, i.e. the Charney-Hasegawa-Mirna (CHM) equation, resembling to the vorticity equation derived by the 2D-NS equation. The fundamental feature of the CHM equation will be discussed for investigating the geophysical or plasma turbulence.

3.1 Geophysical fluid motion

Large scale motion of the geophysical fluid such as the atmosphere and the ocean is a typical example of 2D fluid system [26]. In general, the large scale geophysical motions occur within an extraordinary thin layer of fluid. In the atmosphere, for example, the characteristic length scale L of the horizontal fluid motion is about the order of thousands of kilometers which ' is extremely large in comparison with the characteristic vertical scale D, the order of ten kilometers. In the geophysical fluid, the aspect ratio b

b = L << D 1 (3.1)

usually a very small number, which is the most characteristic condition for the geophysi

cal fluid. Therefore the geophysical fluid is approximately described by the 2D fluid motion dominated by the horizontal motion.

In addition, the geophysical fluid is within the thin layer on the surface of the rotating sphere. We define the Ross by number ^cwhich is the ratio of the horizontal time scale of fluid motion

T

^{= L}

/

U, where U is a horizontal characteristic velocity scale, and the period 1/ f of rotation of the sphere, say the earth, as

u 1

E=^-=^-

j£ JT'

^(3.2)

where f is the Coriolis parameter. The Rossby number ^cis usually very small number for the large seal geophysical motion. In other words, the definition of ' large scale' in the geophysical fluid i ^c << 1.

In order to discu s the general natures of the large scale motion of the atmosphere or the ocean, we should consider a simple but essential example, i.e. the dynamics of a shallow, rotat

ing layer of homogeneous incompressible and invicid fluid with depth

h(x, y, ^t).

The schematic

3.1 Geophysical fluid motion

z

, , , , ,

f=2Q

, , ,

, , , ,

:tY

L

h(x,y,t)

+ ^hs(x,y)

Figure 3.1: The shallow water model

figure is shown in Fig.3.1. We consider the equations of motion

\7. u

^{= 0,}

ou

¹

ot

^{+ U}^·

\7u ^+fez

^{X U =}

-p \7p- ^gez,

13

D

X

(3.3) (3.4)

where

\7

⁼

(8/ox,ojoy,ojoz), u(x,y,z,t)

⁼

(ux,uy,uz)

and

ez

represents the unit vector in the z-axis direction. The rotation axis of system coincides with the z-axis, and f is the Coriolis parameter, which is twice the rotation rate. The gravitational acceleration

g

and the density of fluid pare assumed to be constant.

From the condition Eq.(3.1), the vertical motion of fluid should be negligible in a sense that the vertical characteristic velocity W satisfies W

�

O(c5U) from Eq.(3.3). So the vertical component of the equation of motion is reduced to

op

²

oz

⁼^-^p

g ⁺

^O(c5

)

^, ^(3.5)

which is identical to the hydrostatic approximation required ^asthe definition for the shallow fluid model. The integration of Eq.(3.5) with the boundary condition

p(x, y, h, ^t)

^=Po=^const

on the surface

z

⁼

h(x, y, t)

^yields

p(x, y, z, t)

⁼

pg(h(x, y, t) - z)

^+Po,

(3.6)

so that the pressure at any point is equal to the weight of the unit column of fluid above its point plus the pressure Po on the surface

h

of the shallow fluid.

(13)

14 3 2D FLUID SYSTEMS

The horizontal pressure gradient is given from Eq.(3.6) as

(3.7)

which indicates that the horizontal accelerations are independent of z. Therefore, the horizontal velocities are independent of z if they have no z-dependence initially. Consequently, equations of the motion for the horizontal components are given by

au_l

Bt

₊(u.l · \7 .l)u.l = -g\7 .lh + Uj_ ^Xfez, ^(3.8)

where u1_(x,y,t) =

(ux,uy)

and V.l = (ajax,ajay) are defined.

Since the horizontal velocities are independent of z, Eq.(3.3) can be integrated with respect to z. The integral under the boundary condition on the rigid surface z = hB(x, y, t) of the bottom of the shallow fluid, Uz(x, y, hB, t) = (u.l · \7 .l)hB which is the condition that there is no-normal flow at z = hB, is expressed as Uz(x, y, z, t) = (hB ^- z)\7 1_ · u1_ + (u1_ · \7 1_)hB·

Combining this result with the kinematic condition Uz = (ajat + u.l · \7 1_)h at the surface z = h(x, y, t) yields

aH

at + ^\7^1_· (u.LH) = 0,

H(x, y, t) = h(x, y, t) - hB(x, y) = D + ry(x, y, t) - hB(x, y),

(3.9) (3.10)

where H(x, y, t) and D represent the total depth of shallow water and the characteristic constant one respectively, and ry(x, y,

t)

is the surface displacement in the vertical direction.

A set of shallow water equations is composed by Eqs.(3.8), (3.9) and (3.10) which are the fundamental equations to discuss the large scale geophysical dynamics. Consequently, the num

ber of dynamical equations (Eqs.(3.3) and (3.4)) with the variables

u

x(x, y, z, t),

uy

(x, y, z, t), Uz(x, y, z,

t),

p(x, y, z, t) are reduced to the 2D fluid motion for the variables

ux

(x, y, t), uy(x, y, t) and H(x, y, t) under the condition 8 << 1. The shallow water equations contain two character

istic time scales. One originates from the inertial term in Eq.(3.8) and the other is due to the Coriolis acceleration term. When the time scale of motion is extremely slower than the period

j-1,

the Rossby number satisfies ^c << 1. Under this condition, we will discuss that the shallow water equations are also reduced to a single equation of motion in a later sub-section.

3.2 Electrostatic fluctuation in magnetized plasma

Study of electrostatic fluctuations in the magnetically confined plasma is important for the fundamental research in the field of fusion plasma physics [58]. Many aspects of fluctuation in the trong magnetized plasma are theoretically discussed by utilizing the 2D fluid motion because low frequency and long wave length fluctuations in the plane perpendicular to the magnetic field are sufficiently slow in comparison with the inertial motion along the magnetic field.

3.2 Electrostatic fluctuation in magnetized plasma

Figure 3.2: The drift velocity

15

First, we start to consider a simple example of one component (electron) plasma confined in the strong, uniform magnetic field B =

B

ez, which is called the

electron guiding center plasma

[59, 60]. The equation of motion for the electron fluid is given by the electrostatic drift velocity (Fig.3.2) [61]

Vd = ExB

B2 '

^(3.11)

where vd =

(vx,Vy,O),

_B=

(O,O,B),

and E = ^{-\7 1_¢,} ^(\71_ = (ajax,ajay)). The function

¢(x, y, t) is an electrostatic potential determined by the Poisson equation

(3.12)

where -e is the electron charge and n is the density of electron fluid. The electron fluid density n satisfies the continuity equation,

an - + vd · \7 1_n = 0,

&t ^(3.13)

where we have used the incompressible condition \7 .l ·vd = 0 with Eq.(3.11). Equations (3.11), (3.12), and (3.13) lead to the equation of motion

av3._

¢

₊

J ( ""

_v2"") = o

at ^�, ^_l^� ^' ^(3.14)

where the valuables are transformed into the dimensionless form, e.g. e¢/Te ^--+ ¢ (Te is the electron temperature). This equation, which is so-called 2D

drift-Poisson equation,

coincides

(14)

16

3 2D FLUID SYSTEMS

with the 2D vorticity equation (Eq.(2.14)) with

^{v =}

0 provided that we regard the electron density n and the electrostatic potential ¢ as the vorticity

w

and the stream function 'ljJ re-

pectivcly. Therefore, we can recognize that magnetized plasmas, even in the above simple situation, the 2D fluid motion is essentially equivalent to the dynamics of electrostatic fluctua

tions. Especially, the electron guiding center plasma has been investigated in connection with the relaxation process of the 2D-NS turbulence. The most noticeable result is recently found in [62], where vortices relax to various types of the regular crystal structure through vortex merging and filamentation.

Another example is in the context of the drift wave turbulence observed in the magnetically confined plasma. We assume that the electron temperature Te is sufficiently larger than the ion temperature Ti. This implies that equations of motion for ions are dealt with the cold ion fluid in the plane perpendicular to the magnetic field. The ion fluid momentum equation accelerated by the Lorentz force is given by

av�

_�

ut + (v

^1_^•

\7 �)v �

⁼

--\7 �¢ + v � ^mi

^e ^X^Wcez,

^(3.15)

where the ion fluid velocity v�(x,y,t)

⁼

⁽

^vx,vy

⁾

^,

^and ^mi ^and

^e

are the ion mass and the ion charge re

p

ctively.

We = eB

lmi is the ion cyclotron frequency. The inertia of the ion fluid motion along the magnetic field is neglected since the neutralization of charge density along the magnetic field is sufficiently quick because of the small inertia of electron due to small mass, i.e. mi

>>

me. Consequently, the feature of electrostatic fluctuation shows approximately the 2D motion. In addition, the equation of the density conservation of ions is written as

ani at + \7 � . ( v

^l_

ⁿⁱ⁾

⁼

^0, ^(3.16)

where ni is the density of ion fluid. In order to obtain the relation between ni and ¢, we adopt the quasi-neutrality condition [61], where the ion density ni is related to the electron density ne which can be shown to obey the Boltzmann distribution, such as

n, "'n,

⁼

n0(x) exp [i]

^oo

^(na + ^n(x)) [ ¹ + i l· ^(3.17)

The density no (x) is the temporally uniform background density of electrons or ions, which is decomposed into the average density na.

⁼

(no), ( -

·

·) being the spatial average, and the fluctuation n(x) from na, i.e. no(x)

⁼

na. + ^n(x). Equations (3.15), (3.16) and (3.17) are the closed set of quation to describe 2D electrostatic fluctuation in the magnetically confined plasma for the description of drift wave turbulence.

3.3 The Charney-Hasegawa-Mirna equation in drift-Rossby wave turbulence

We can notice that the hallow water equations Eqs.(3.8), (3.9) and (3.10) have the same structure as Eqs.(3.15), (3.16) and (3.17) in magnetically confined plasma. The correspondence

3.3 Tbe Cbarney-Hasegawa-Mima equation in drift-Rossby wave turbulence

17 of variables are

u�

^H

v

^�'

¢

^H

^h,

^We^H

^j, ⁿⁱ

^+-t

H. ^(3.18)

Therefore, the dynamics of these two different systems obey the same quation of motion [11, 31]. Hereafter, we consider the case of geostrophic fluid motion. The case of magnetically confined plasma is also derived by corresponding the variable according to Eq.(3.18).

In order to proceed sy stematically, we introduce the dimensionless variables for Eqs.(3.8), (3.9) and (3.10) as

x

⁼

Lx', y

⁼

Ly',

^t⁼

Tt', u�

⁼^U

u� ^, ^'TJ

⁼

^No'T}1,

^H⁼^D

H'

^,

^(3.19)

where ()

'

indicates dimensionless variables. The characteristic param ters

^{L, T, U}⁼ ^L

I

T D

and No are the characteristic magnitudes of length, time, velocity, depth of the hallow water and the free surface elevation respectively. In this case, the shallow water equations for the dimensionless variables are obtained as

aH' ' ( 'H') - at' -+\7

^_L

, ·u�

⁼

⁰

^,

(3.20)

(3.21) 2

I

hB

( \ 2

^I ^I

⁾ _{(3 22)}

H1(x, y, t)

⁼

¹ + c:>. 'TJ - D

=

1 + ^c

^"' ^'TJ

^{- 'TJB} ^,

^·

where the parameter No is evaluated as No

⁼

f

U L

I g

=

(

U

If

L

) (!2

L2

I g) in order that th pressure gradient term must be large enough to balance the Coriolis acceleration term. The parameter c: is the Rossby number defined in Eq .(3.2), and we introduce the ratio >. of the characteristic horizontal scale

^L

and the Ross by deformation radius

^R⁼

yfgJ5 If by >.

⁼L

I

R.

We assume that hBI Dis the magnitude of order c: as hBI

D ⁼

c'TJ'e, where 'TJs

⁼

0(1). In order to clarify an outlook, we transform the above equations as follows. Substituting Eq.(3.22) into Eq.(3.21) y ields \7�. u�

⁼

-(alat' + u�. ^\7�) ^log[1 + c(>.2TJ'- 'TJs)]. Moreover, combining this equation with that of \7�

^x

{Eq.(3.20)} leads to

(�, + u� · \7�) ( \7�

^x

u ^� ^- � ^log[1 + c:(.A2'T}1- 'TJs)J )

⁼

^0. ^(3.23)

Consequently, Eqs.(3.8), (3.9) and (3.10) are rewritten as Eqs.(3.20) and (3.23).

Now we focus our attention on the motion whose time cales are extremely longer than

j-1. This slow motion is extracted by taking the small Rossby number limit c

<<

1 because of

Eq.(3.2). Under this condition, the second term in Eq.(3.23) is expanded as log(1 + c(>.2'TJ1 -

'TJs))lc:

^�

;.2'TJ1 -TJ'e· In this case, no explicit c: dependence appears in Eq.(3.23). The fast mode

with the Rossby number c has a short time scale in comparison with the slow mode with 0(1)

(15)

18 3 2D FLUID SYSTEMS

time scale in Eq.(3.23). Under the condition c << 1, we can adiabatically eliminate the variable

u�

in the above equations. Equation (3.20) is solved as

'

...,

^{"' '}

Uj_--Vj_'T] ^Xez.

Substituting this result into Eq. (3.23) with ^c << 1 leads to

(:

^t-

^(\77]

^{X ez}

⁾

^·

^\7 ) (\727]- A2'T]

+

'T]B) =

0,

(3.24)

(3.25)

where ()' and 01.. are neglected for simplicity. Equation (3.25) is called the Charney equation or the quasi-geostrophic potential vorticity equation, which describes the temporal evolution of the flow in the geostrophic equilibrium in the planetary atmosphere (26).

In the case of magnetically confined plasma, the equation corresponding to Eq.(3.25) is written as

(3.26)

which describe slow electrostatic fluctuations in comparison with the time scale

w;1.

This equation was first derived by Hasegawa and Mirna (30), which is called the Hasegawa-Mirna equation. In Eq. (3.26), ).. represents the ratio of the system size L to the ion Larmor radius

p = JT. fmdwc

which corresponds to the Rossby radius in the geophysical fluid, and

cnB(x) = n(x)/na

<< 1 is assumed.

In this the is, we call Eq.(3.25) or Eq.(3.26) the Charney-Hasegawa-Mirna equation or sim

ply th CHM equation. The CHM equation gives an important paradigm for discussing the geostrophic turbulence or the drift wave turbulence in plasma. Many investigations concern

ing with the CHM equation and its turbulent state have been made in the above fields. The spectral structure in the cascade process has been investigated theoretically and numerically in [32, 33, 37). In addition, the coherent structure of vortices and its dynamics has been revealed theoretically and numerically in [35, 36). Especially, the numerical results shown in [35, 39]

affe t our aim to investigate the turbulent properties of the CHM equation. This will be intro

duced in the S c.5 in detail. On the other hand, the studies about the dynamics of the vortex pairs, which is called the modon, has been studied by introducing the point vortex model for the CHM equation [40, 41). Moreover the several effects of the deformation radius for the turbulent state were discussed by the spectral closure approach [34) and the direct numerical simulation of the CHM equation [35). Furthermore the approach from the dynamical system viewpoint of the CHM equation by introducing the shell model like so-called GOY model [63]

has been carried out in [38]. These investigations are made from several different viewpoints. In the later sections, we will di cus the statistical properties of the formation process of coherent structure and its scaling properties.

In this thesis, we deal with the more simplified situation of the CHM equation. We assume that

'T]B

(or

nB)

is constant in spac . This corresponds to the neglect of the effect of wave,

3.3 The Charney-Ha.segawa-Mima equation in drift-Rossby wave turbulence 19

which may be realized in the strong turbulent case. In this case Eq.(3.26) (or (3.25)) yield

oq

at+

J(¢,q) =

0' (3.27)

q

(

_{x, y,}

t) = (\72- )..2)¢(x

_{y, t}

)

(3.2 )

where

q

is called the potential vorticity and

J(a, b)

is th Jacobian operator d fined in Eq.(2.15).

Here we can notice that the CHM equation resembles the vorticity equation (2.14) d rived for the 2D-NS equation with ^v

=

0, which is in fact identical to the CHM equation in the limit ).. --t 0. The CHM equation has two conserved quadratic quantities, th total energy E and

total potential enstrophy U defined by

1

!,

¹

E

=

-

- £2

£2

^-q¢dr,

2 1

!,

¹

²

U

=

2

-q\7 ¢dr,

L p2

(3.29)

(3.30)

which are identical to the energy and the enstrophy for the 2D-NS equation in the limit

A

--t 0.

In the).. =f. 0 case, the CHM equation includes a spatial scale ).. -l which separat s the statistical and dynamical features of the CHM equation from the 2D-NS equation. For the ale l << ).. ^-t, the dynamics is dominated by the feature of the 2D-NS equation, and th tatistical properties are expressed by the 2D-NS equation, while the features of the dynamics or statistics for the scale l >> ).. ^-1 are the own nature of the CHM equation. This point will b discussed in the later sections for the turbulent state described by the CHM equation (the drift-Rossby wav turbulence or the CHM turbulence) which contains the similar feature of the 2D-NS turbulenc ^· The structure of the energy spectrum for the CHM turbulence will be discus d in the next section.

In discussing the effect of the parameter ).. in the CHM turbulence, it is important to refer to concrete values of ).. in real physical systems. For the atmospheric flow on the earth, the parameter ).. is estimated as )..

=

5 "' 10. Therefore the deformation effect is small in the atmosphere, which yields the dynamics by the CHM equation is approximated by the 2D-NS equation. The parameter ).. for Jupiter's great red spot is used in [29] as

A =

10. On the other hand, the characteristic value of ).. in the oceanic flow on the earth is estimated as ).. ⁼50 [28) ^· This value is large enough. Therefore the turbulent state is sufficiently affected by the effect of the deformation radius. Furthermore, for magnetically confined plasma system, ).. ⁼0(100) is evaluated by using the typical parameter of the tokamak. This value is extremely large. So the turbulent states observed in the oceanic flow or the tokamak plasma ar th appropriate situations discussed in this thesis.

(16)

20 4 STATISTICAL PROPERTIES OF CASCADE IN THE 2D TURBULENCE

4 Statistical properties of cascade in the 2D turbulence

In this section, we survey the statistics of the high Reynolds number turbulence described by the 2D-NS equation. The most important difference between 3D and 2D turbulences is that the nonlinear term conserves the enstrophy in addition to the energy in the case of 2D-NS equation with v = 0. Statistical description of the turbulent state is discussed by the phenomenological dual cascade theory with the conserved quantities. Moreover we briefly discuss the structure of the energy spectrum in the CHM turbulence on the analogy of the dual cascade theory in the 2D-NS turbulence.

4.1 Dual cascade theory in the 2D turbulence

In Sec.2.2, the 2D-NS equation is represented by the Fourier component of the stream function '!j;as

(

) ⁶

2 2

B'!f;a� ^t

⁺

vk2'1/Jk(t)

=

L

q

2

� ²

^P

^(p

^x

q)z'l/Jp(t)'l/Jq(t).

k=p+q (

^4.1)

The right hand side of Eq.

(

4.1) indicates the nonlinear interaction among Fourier modes. One Fourier mode '1/Jk interacts with all Fourier modes satisfying k =

p

+

q.

Therefore, the nonlin

earity combines the fluctuation of a mode with those of all modes in the wavenumber space.

In addition, the nonlinear term conserves the quadratic quantities such as the energy and the enstrophy among any Fourier modes satisfying k +

p

+

q

⁼⁰as mentioned in Sec.2.2. On the other hand, the second term on the left hand side of Eq.

(

4.1) represents the dissipation term through which the energy and the enstrophy are dissipated. The ratio of the nonlinear term to the dissipation term is called the

Reynolds number Re

which measures the strength of the nonlinearity of the system. In Eq.(4.1), the Reynolds number

Re

is estimated as

Re

=

1'1/Jkl.

v (4.2)

The state whi h nonlinear term dominates the dynamics of the system is called the high R ynolds number turbulence (HReT). In HReT the dominant physical process in the wavenum

b r space is the transfer of the quadratic quantities among Fourier modes by the nonlinear term and the di ipation process in the large wavenumber region where the dissipation term is dom

inant in compari on with the nonlinear term. Thus, we recognize that HReT is a typical phenomenon associated with the nonlinear and nonequilibrium process.

The tatistical description of HReT is one of the most challenging problems in hydrody

namics. It is quite difficult to construct the statistics of HReT from the basic equation (the NS equation) because of the strong nonlin arity and the dissipation. This fact does not allow us to use the standard framework of the equilibrium statistical mechanics. The physical process of 3D HReT can be understood phenomenologically as the energy cascade process from the energy

4.1 Dual cascade theory in the 2D turbulence 21

injection scale L to the dissipation scale

ld.

For the 3D HReT under the statistical homogeneity and isotropy, the Kolmogorov 1941 theory (K41) [7] ha shown that the univ rsal statistical law would exist over the scale

ld

<< l << L, which is called inertial range. The K41 theory, in which the energy transfer per mass from the wavenumber L -l to

kd(

⁼

l;t1)

i a sumed to be constant rate t, is based on the argument of the dimensional analysis. In this case, the dimensional analysis yields t

"" k-2t-3.

Furthermore the assumption of the energy spectrum

E(k)("' t-2k-3)

in the inertial range depend only on

k

and on the rate f leads to th form (4.3)

The scaling law

(

-5/3 law) of the energy spectrum is widely observed in laboratory experim nt or atmospheric turbulence reviewed in [3]. Therefore, the assumption of uniform energy transfer seems to be correct for the inertial range statistics. However, it is well accepted that the energy transfer rate or the dissipation rate t: fluctuates both temporally and spatially, which leads to the correction from the -5/3 law. The correction from the K41 theory is called the

intermittency problem,

which is the central topic in the investigation of HReT. This intermittency problem is reviewed in [3, 8, 9] or discussed in (64, 65].

In the 2D HReT, on the other hand, the statistical nature of turbulent state is more com

plicated than that of the 3D HReT because of the conservation of the enstrophy in addition to that of the energy. This problem was discussed in Sec.2.1, where the possibility of the enstro

phy dissipation in the invicid limit is indicated in contrast with that of the energy dissipation.

Therefore the two kinds of inertial ranges, the energy and enstrophy transfer ranges, will be investigated for the statistics of 2D HReT [12, 13]. Kolmogorov type dimensional analysis yields that the scaling law of energy spectrum in the enstrophy transfer range

(4.4)

where the enstrophy transfer rate "l is assumed to be constant. The schematic figures of the energy spectrum by the above mentioned scaling theory are shown in F ig.4.1.

The inertial ranges for energy and enstrophy transfer can be analytically formulated by the energy flux function

IIE(k, t)(""

^t

)

and the enstrophy flux function

IIQ(k, t)("" "l)

defined as

IIE(k, t)

⁼

100 T(k', t) dk',

^(4.5)

IIQ(k, t)

⁼

100 k'2T(k', t) dk',

^(4.6)

T(k, t)

being the energy transfer function defined in Eq.(2.32).

IIE(O, t)

⁼

IIE(oo, t)

⁼ ⁰

and

IIQ(O, t)

⁼

IIQ(oo, t)

⁼0 are satisfied from the definition. Physically,

IIE(k, t) (IIQ(k, t))

represents the mean rate of transfer of the energy

(

enstrophy) per unit mas from wavenumbers

2次元乱流系における動的統計性について