数理解析研究所講究録
第 769 巻 1991 年 156-171

156

乱流のカスケードと非ガウス分布 (要約)東大 理 神部 勉$X<\overline{R}L\ell\rangle^{\backslash }$看 -z p る

仁$\overline{\eta}_{k}\sim\ovalbox{\tt\small REJECT}\nearrow^{\backslash }\prime r_{\acute{f}}^{7}4^{\backslash }$$,$$f_{L}$$\wedge$う 一つり$Z_{/’\overline{p}_{\tau^{\backslash /L_{\Psi}’(}}^{i,\prime}}^{\prime_{y}\nearrow\acute{a}_{\vee}^{k}}tb_{\iota^{\backslash 1}\triangleright t\lambda^{\backslash }’\tau}^{4t:}(t_{\backslash }\backslash$

竜

力 X )

$A’l’b\sim\#_{//b}\eta$と7$\tau 1\ell z\nearrow i^{b_{\backslash }^{\backslash }}-\mathscr{J}_{T}’$と$-\vdash$”$\ovalbox{\tt\small REJECT}_{K}^{>}|\sim\backslash 7^{\overline{\theta}b\eta}\backslash 5\not\in p’$$\prime P^{\sim\backslash }/|7^{-}\backslash ^{\backslash }\backslash /\eta/\triangleright t_{(\overline{\star}<\tilde{\gamma}’}’-1\iota/\ovalbox{\tt\small REJECT}_{\backslash }\Phi$(ブヒ。 こら毛$\overline{\gamma}^{1}/V/\gamma$$\ovalbox{\tt\small REJECT}^{\underline{\backslash }}|\nwarrow$$\%^{p}\nearrow g\ovalbox{\tt\small REJECT}’-\uparrow^{7}+p\wedge 7\succeq_{J}\wedge?\not\in\not\in \mathfrak{x}_{\overline{A}^{\ell\eta\lambda \mathcal{T}}}^{\vee/}- l\triangleright,f_{\theta 4^{\sqrt{}}}^{\epsilon}<^{-}\epsilon^{4}\circ h^{\ovalbox{\tt\small REJECT} d2}\lambda^{\backslash }$$/^{\sim}7\succeq 7- k- i_{7}$a($\ovalbox{\tt\small REJECT} 5\backslash$し,$’-4|6^{\frac{\backslash }{\nearrow 7}r^{J_{\overline{\Downarrow}}}}\triangleleft\cross$%$p^{\backslash }\backslash 4\triangleleft$‘/Jh‘毒-) 7 遇*\leftarrow !

よう$f+\backslash \neq p4\varphi’.d\sim’\nearrow$$\not\geq\ovalbox{\tt\small REJECT} lT^{5\zeta}$の$Z^{\backslash \backslash }$みろ.

$\circ-\backslash$$\triangleleft\pi^{B}$$rt’/a^{\wedge},-\theta^{t}\prime 7$$\emptyset\sqrt{}*,r_{-}\overline{\beta}^{l^{\sim}}p*r^{-}-$

ぜる ことが$7_{c}^{t}\delta b$。

$\underline{|/}\eta X7\backslash \ovalbox{\tt\small REJECT}^{t^{p}}7\cdot\cdot\triangleleft\overline{/}-/\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}’\cdot\tau A\not\in p\triangleleft$

デ)\mbox{\boldmath $\psi$}亥3}工或が かわた$\#\backslash$$\overline{f}_{f}^{z}$$f^{\wedge}g^{r}\emptyset oeg_{b}A_{3}J$$\circ$本$\overline{<}$カスブ$-\vdash$$k^{\lrcorner_{\check{\prime}}}{}_{\prime_{\Psi}^{-}}C\underline{)\backslash 7_{\vee^{\backslash }}}7$デ)\check /*種綜々分$\backslash ’\dagger\}^{\backslash }|_{\sim}^{/d}-//\pi’(\vee/7\wedge\circ p_{\epsilon*\overline{\wedge}}^{\alpha.h}$と$|(|i$\S

獲嬬 vLZ 分$?\not\in|^{\prime\{<^{t^{\sim}}\beta}4k$$t\eta_{\iota^{\backslash \backslash }}^{\{\text{え詳鎗}}$$\mathscr{J}_{\vee}l^{-}\sim\neq/d, 4\sigma\sigma$($p^{\backslash }$$\ovalbox{\tt\small REJECT} p_{\beta}-r$

Aff$3/\}x^{f}\#^{\sim}E\overline{\prime T}\backslash -7$$\Gamma_{d^{\sim}}\mathcal{Z}/tfg_{[X}$a$\ovalbox{\tt\small REJECT}_{A}^{\sim\succ!\underline{\hslash}^{\backslash }r}\backslash ^{\backslash }\sim$。

$1|Z\backslash - rarrow h7_{-}$$|$r_{\tilde{t}}$

ユレ

い。

–

$\backslash /\succeq yC’\ovalbox{\tt\small REJECT}_{\overline{A}}^{\backslash }$$\mathscr{J}_{2}\vee-$七$\mu(f_{0\Gamma}$

[

$Z$

)

157

A Dynamical Theory of Cascade in Turbulenceand Non-Gaussian Statistics
T. KAMBE
Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113

1. Introduction

A dynamical mechanism is considered which connects the cascade with nongaussian statistics of velocity gradients. Turbulence is characterized by the continuous excitation of all scales, but in the Fourier space of the velocity field, the excitedamplitude decreases rapidly with increasing wave numbers so that contribution to
the total kinetic energy from the small scale components is negligibly small. Roughly
speaking by the central limit theorem, the sum of a large number of Fourier modes
is distributed normally when the Fourier amplitudes of different wave numbers are
independent in the energy-containing eddies (Batchelor 1953).
However it is well-known that non-gaussian statistics are observed at small scales.
Two simplest measures of non-gaussianity are the skewness and flatness.
$(a)$ Experimental observations concerning the small-scale motion show that the flatness of the probability distribution of the various velocity derivatives increases steeply
with the order of the derivative. For example, the flatness factor of the n-th order“
for a coordinate and the corresponding veloclongitudinal derivative, shows the value of about 3.9, 4.9 and 5.9 (for $n=1$ ,ity , defined as
2 and 3 respectively: Batchelor 1953; Monin &Yaglom 1975) which are greater than
the gaussian value 3, where the over-bar denotes the ensemble average. A large flatness factor of a distribution implies that the probability density function has a higher
central peak and broader skirts than the gaussian function of the same standard deviation. These properties are considered to be intrinsic to the micro-structures ofturbulence in general.
$(b)$ The measurements of the skewness factor
suggest that the value isaround $-0.3\sim-0.5$ . The probability density of is characterized by the properties
are more probable than srriall negative values, butthat small positive values ofthis positive contribution tois less than balanced by the larger contribution ofnegative large values of as compared with positive cmes.
In the study of numerical simulation of incompressible Navier-Stokes turbulence
at moderately large Reynolds number, She et al. (1988) isolated the contributions$g_{n}=\partial^{n}u/\partial x$

$u$

$x$

$F_{n}=g_{n}^{4}-/(g_{n}^{2})^{2}-$$g_{1}^{3}-/(g_{1}^{2})^{3/2}-$$g_{1}$$g_{1}^{3}$$g_{1}^{3}-$$g_{1}^{3}$

158

of different flow scales by considering Fourier-band filtering of the velocity field, andinvestigated the non-gaussian behaviors. In particular the probability distributionfunctions are found to have near-exponential forms for the velocity and its derivatives.
Exponential behaviors of the distribution functions are observed in various context
of experiments and numerical simulations ( $e.g$ . Van Atta&Chen 1970; Sheih et al.
1971; Yamamoto &Hosokawa 1988; Kida &Murakami 1989; Gagne 1990). These
properties are examined also by the statistical analysis (Yamamoto&Kambe 1991)
about the contributions from different Fourier subspaces of a decaying turbulence
obtained numerically.
Kraichnan (1990) considered a heuristic model for evolution of the probability
distribution of velocity gradient and an exponential distribution. The idea is thatan initial gaussian field is mapped dynamically into a non-gaussian field. The initialstatistics of the fluctuations are preserved during the dynamical cascade. The dynamics is governed by a nonlinear ordinary differential equation that models competition
between the inertial straining and viscous damping and produces small scales. Similar
nonlinear evolution equation has been considered by She (1991). His model equation
is distilled from the vorticity equation by taking account of a random backgroundfield and a dynamical exponent for a local self-stretching mechanism with a scalingargument.
Study of the vorticity dynamics based on the vorticity equation for a viscous incompressible fluid was performed previously for arbitrary flat shear layers (Kambe
1983, 1986) or an arbitrary axisymmetric shear layer (Kambe 1984). In these analyses, motions of rotational (shear) layers superimposed on irrotational straining field
are investigated, in which the dynamical evolution of the vorticity proceeds under
the influence of three elemental processes: convection, stretching and viscous diffusion. With using local expressions of the background straining velocity field which
are linear with respect to the space coordinates, it is found to be able to represent
the evolution of the rotational layers exactly in terms of the initial vorticity distribution. This formulation can describe exponential growth of the vorticity when theviscous term is neglected. However this is contrasted with the algebraically explosive behaviors which are known in the solution of the quadratically nonlinear model
equation of the vorticity (Rose&Sulem 1978) or the equation for the enstrophy inthe statistical theory of turbulence with the quasi-normal approximation (Proudman&Reid 1954). The present study is an endeavor to improve the previous study byincorporating the property of scale invariance of turbulence field described below.’$\alpha$1The idea has been inspired by the presentation of She $(1991$a $, b)$ at the international workshop
held in Japan, October 1990. His model equation includes the effect of scale invariance.

159

We investigate a dynamical mechanism that connects the cascade in turbulence with
the near-exponential tail in the distribution of velocity and their derivative fields.
It will be shown below that the difference in behavior between the lateral and longitudinal derivatives may lead to different probability distributions: slower-than or
steeper-than exponential decay.
One of the remarkable symmetries of the Euler equation is the scale invariancewhich is considered by Frisch&Parisi (1985) in relation to a multifractal model of
turbulence that they introduced. This property is as follows. The Euler equationand theis invariant if we simultaneously scale the distance by , the velocity by), where is an arbitrary local scaling exponent.
(the pressure bytime byRecent wavelet analysis of an experimental data suggests-0.5 $<h<1.0$ (Bacry et al.
1990). The Kolmogorov’s self-similar cascade corresponds to the value
which
describes scale invariance of the rate of energy transfer between eddies of different.
scales.
$\lambda^{h}$$\lambda$$\lambda^{1-h}$$\lambda^{2h}$$h$$h= \frac{1}{3}$2. Straining of a test field by large-scale fields

2.1 Local representation
Taking account of the features of turbulence described in the introduction, we try
to construct a model of cascade in turbulence. We consider the vorticity equation,$\omega_{t}+(v\cdot\nabla)\omega-(\omega .

\nabla)v=\nu\nabla^{2}\omega$,(1)

where is the velocity and $\omega=\nabla\cross v$ is the vorticity. Suppose that the velocity
field $(u, v, w)$ is localy represented in the cartesian coordinate system $(x, y, z)$ as
$v$

$u$

$=$

$Ax-\Psi_{z}(x, z)-\psi_{y}(x,y, t)$

,

$v$

$=$

By

,

$w$

$=$

$Cz+\Psi_{x}(x,z)$$+\psi_{x}(x,y,t)$(2)

near the origin (0,0,0), where $A,$ $B,C$ are constants with the constraint relation$A+B+C=0$ . The function $\Psi(x,z)$ satisfies the equation, $\Psi_{xx}+\Psi_{zz}=0$ , hence$\Psi(x,z)$ being a streamfunction describing a steady incompressible irrotational flow inplane, while $\psi(x,y,t)$ is a time-dependent streamfunction describing a rotational, and considered to be a rotational perturbation
flow in $x,y$ plane, hence$\Psi(x,z)$
to the steady irrotational field. Obviously the velocity field (2) satisfies thesolenoidal condition, $divv=0$ . The velocity consists of three components: a regular
irrotational field $v^{(r)}=$ ( $Ax$ , By, $Cz$ ), a singular irrotational field$x,$ $z$$\psi_{xx}+\psi_{yy}\neq 0$$v^{(\cdot)}=(-\Psi_{z},0, \Psi_{x})$160whose functional form is given in the next subsection, and an unsteady test field$v^{(t)}=(-\psi_{y}, \psi_{x}, 0)$ . It is readily shown that the vorticity has only thecomponent
associated with :$z$$\psi$$\omega=(0,0, \omega)$,$\omega(x,y,t)=\psi_{xx}+\psi_{yy}$We consider the evolution of the test field $\psi(x,y,t)strainedbythebackgroundsingu-$, which are supposed to be of the scale of inertial
lar field
and the regular fieldisrange and larger one, respectively. It is important that the singular component
introduced here to represent the turbulent field having the scale invariance mentioned
would be quasi-steadyin the end of the previous section. In turbulence, the fieldin comparison with the rapid cascading process of fluctuations.
The and components of the vorticity equation (1) vanish identicaUy. Using. the explicit expression of the velocity (2), we have$v^{(i)}$$v^{(\ell)}$$v^{(s)}$$v^{(s)}$

$x$

$y$

$\frac{\partial\omega}{\partial t}+Ax\frac{\partial\omega}{\partial x}+By\frac{\partial\omega}{\partial y}-C\omega-\frac{\partial}{\partial x}(\Psi_{z}\omega)+\frac{\partial(\psi,\omega)}{\partial(x,y)}=\nu\nabla^{2}\omega$for the

$z$

,

(3)

component $\omega=\psi_{xx}+\psi_{yy}$ .

2.2 Singular irrotational field
Based on the view that straining of the test field by only the regular component
is insufficient to describe the cascade in turbulence, we introduce a singularcomponent
having the scale-invariant property. Assume that$v^{(r)}$$v^{(\ell)}$$\Psi(x,z)=\sum_{n=0}^{\infty}a_{n}x^{h-2n_{Z}2n+1}=ax^{h_{Z}}(1+\zeta_{1}\frac{z^{2}}{x^{2}}+(2\frac{z^{4}}{x^{4}}+\cdots)$

where is a fractional parameter in the range-O.5 $<h<1.0,$ $a=a_{0}$ and
The irrotationality condition $\Psi_{xx}+\Psi_{zz}=0$ leads to$h$(4)$\zeta_{n}=a_{n}/a_{0}$.

,$\zeta_{n}\equiv\frac{a_{n}}{a_{0}}=(-1)^{n}\frac{\{2n-1-h\}}{(2n+1)!}$

where $\{n-h\}\equiv(n-h)(n-1-h)\cdots(1-h)(-h)$ for an integer . This series solution
converges for $|z/x|<1$ , but has a singular behavior at $x=0$ due to the terms of theform $x^{h-2n}$ with a fractional power $h-2n$ . Here we restrict our consideration to thepart of positive $x:0<|z|<x$ . Clearly the solution (4) satisfies the viscous vorticity
equation identically in the region of definition because of the vanishing vorticity.
The velocity fields, $i.e$ .
and components, represented by the streamfunction$\Psi(x,z)$ are
$n$

$x$

$-\Psi_{z}$

$=$

$z$

$-ax^{h}(1+3 \zeta_{1}\frac{z^{2}}{x^{2}}+5\zeta_{2}\frac{z^{4}}{x^{4}}+\cdots)$

,

161

$\Psi_{x}$$ax^{h-1}z(h+(h-2) \zeta_{1}\frac{z^{2}}{x^{2}}+(harrow 4)\zeta_{2}\frac{z^{4}}{x^{4}}+\cdots)$$=$

respectively. It is evident that the above velocities are scaled byneous scaling of and by .
$x$

$z$

$\lambda^{h}$for the simulta-$\lambda$

2.3 Test field $\psi(x,y, t)$Consider a test field whose vorticity is localized and characterized by a lengthis supposed to decrease by the straining of the backgroundscale. The scale
fields. First we assume that the fluctuation $\psi(x,y,t)$ is represented in the form$\omega$$\lambda(t)$$\lambda(t)$(5)$\psi(x,y,t)=F(x,t)$

for simplicity. (Later we will consider the case, $\psi(x, y, t)=yF(x,t)$ which shows anadditional property of the test field). Further we assume that the vorticity $\omega(x, t)=$is given by the form,$F_{xx}$$F_{xx}(x, t)=s_{0}( \frac{\lambda_{0}}{\lambda(t)})^{\gamma}f(\xi)$

$w$

,

(6)

here
$\xi=\frac{x-\lambda\xi_{0}}{\lambda}$

,

or

(7)

$x=\lambda(\xi+\xi_{0})$being a constant. The coefficient denotes a typical initial value of the test fieldvorticity. Note that the assumption (6) is given a special form to describe the cascade
process in which the scale parameteris supposed to decrease. This is, in a sense,a phenomenological setup on which the subsequent consideration is based.
$\xi_{0}$$s_{0}$$\lambda(t)$2.4 Velocity fields
Now the velocity components are given by$u=Ax-\Psi_{z}$,$v=By+F_{x}$ ,(8)$w=Cz+\Psi_{x}$where is approximated by $ax^{h}z$ with a positive parameter . From this we obtainthe longitudinal derivatives,$\Psi$$a$$\frac{\partial u}{\partial x}=A-\Psi_{xz}$,$\frac{\partial v}{\partial y}=B$,(9)$\frac{\partial w}{\partial z}=C\dashv-\Psi_{xz}$where $\Psi_{xz}=ahx^{h-1}$ . The lateral derivatives are$\frac{\partial u}{\partial z}=-\Psi_{zz}\sim-x^{h-2}z$,$\frac{\partial v}{\partial x}=F_{xx}$,$\frac{\partial w}{\partial x}=\Psi_{xx}\sim x^{h-2_{Z}}$

.

(10)

162

The other lateral derivatives vanish. Since the vorticity has only the component,
the rate of vortex stretching is given by tlie longitudinal derivative $\partial w/\partial z=C+\Psi_{xz}$ .
Consider a favorable case for the vortex stretching, that is $C>0$ and $A,$ $B<0$ .
Cubic sum of the longitudinal derivatives is$z$$S_{3}=(A-\Psi_{xz})^{3}+B^{3}+(C+\Psi_{xz})^{3}=3ABC+3(C^{2}-A^{2})\Psi_{xz}-3B\Psi_{xz}^{2}$(11)

where $A+B+C=0$ is used. Although each derivative of (9) takes both negative orof the three cubic terms is found to be positive since eachpositive value, the sum
term on the right hand side is positive due to the assumed properties. Considerationof the relation to the turbulence statistics is given in the section 3.2.
Let us consider a fluid particle advected passively by the flow, whose motion isdenoted as $(X(t), Y(t),$ $Z(t))$ . The $x,$ positions must satisfy the equation,$S_{3}$$y$$\dot{X}=AX-\Psi_{z}(X, Z)$,$\dot{Y}=BY+F_{x}(X,t)$,where the over dot means the time derivative with respect to the fixed particle in. A small x-separation $\triangle x(t)=X_{1}(t)-X_{2}(t)$motion andis approximated by
between two nearby particles marked as 1 and 2 will be governed by$ax^{h}$$\Psi_{z}$$\dot{\Delta}_{X}=(A-\Psi_{xz}(\overline{X}, Z))\Delta_{X}$

,

(12)

where $\overline{X}=(X_{1}+X_{2})/2$ .

2.5 A model of cascade
Intermittency in turbulence is considered to be an outcome of the dynamicalcascade of the turbulent fluctuations. So far, we have described analytically a local
representation of the turbulence field. We now try to derive a model equation ofcascade of the fluctuation given above. Substituting (6) to in (3) and using (4) and(5), we obtain$\omega$(13)$-g( \xi)\dot{\lambda}=p(\xi)\lambda+G(\xi)\lambda^{h}+\nu f^{n}(\xi)\frac{1}{\lambda}$

by dropping the common factor$s_{0}\lambda_{0}^{\gamma}\lambda^{-\gamma-1}$$g(\xi)=\gamma f(\xi)+(\xi+\xi_{0})f’(\xi)$,, where$G( \xi)=\frac{d}{d\xi}[\Psi_{z}(\frac{x}{\lambda}, \frac{z}{\lambda})f(\xi)]$$p(\xi)=Cf(\xi)-A(\xi+\xi_{0})f’(\xi)$

where$\Psi_{z}\approx ax^{h}$,,. Let us now introduce the normalized variable$\kappa(t)=\lambda(0)/\lambda(t)$

(14)

163

signifying a typical wave number of the test field with unit initial value, $\kappa(0)=1$ .
asis readily obtained from (13) by multiplying
The equation for$\kappa(t)$$\lambda_{0}/\lambda^{2}$(15)$\frac{d}{dt}\kappa=\frac{p(\xi)}{g(\xi)}\kappa+\lambda_{0}^{h-1}\frac{G(\xi)}{g(\xi)}\kappa^{2-h}+\nu\frac{1}{\lambda_{0}^{2}}\frac{f^{n}(\xi)}{g(\xi)}\kappa^{3}$

From now we examine only the general feature of the test field, $i.e$ . the scaling. So that we approximate the function $f(\xi)$ by
andparameter
restrict our consideration to the behavior at $\xi=0$ (and $z=0$ ). Since $f”(O)=$$-b^{-2}f(O)$ , the parameter. In addition, we havesignifies the magnitude of$f’(0)=0,$ $g(0)=\gamma f(0),$ $G(0)\approx ah\xi_{0}^{h-1}f(0),$ $p(0)=Cf(0)$ . Thus we obtain from(15)$f( O)\exp(-\frac{1}{2}\xi^{2}/b^{2})$$\kappa(t)$$b^{-2}$$f^{u}(\xi)$$\frac{d}{d\tau}\kappa(\tau)=C\kappa+L_{0}\kappa^{2-h}-\nu k_{0}^{2}\kappa^{3}$

,

(16)

where $\tau=t/\gamma,$ $L_{0}=\Psi_{xz}(x_{0},0)\approx ahx_{0}^{h}‘ i=ah(\lambda_{0}\xi_{0})^{h-1}$ and $k_{0}=1/b\lambda_{0}$ . Since$-0.5<h<1.0$ , the exponent of the second term is $1<2-h<2.5$ , faUing in between
the first and third terms.
It is interesting to find that the form of the equation (16) for is equivalent
to that of She (1991) except for the coefficients. Especially noted that the same
exponent $2-h$ appears at the second term. Kraichnan’s model equation takes theform, $j=|s_{0}|J^{2}-\nu k_{d}^{2}J^{3}$ , where $J(s_{0},t)=s(t)/s_{0}$ for a transverse shear , andis a characteristic dissipation wave number for . One of the major differences liesin that the fractional exponent is not included in this model, instead a quadraticterm is taken into account to express the nonlinear straining. In the present analysis(and She’s model, too), the fractional exponent is adopted to represent the fact thatthe turbulent straining has a singular nature not expressible by an integer power.
The idea of Kraichnan (1990) is as follows. The field variables like the velocity
gradients develop under the nonlinear dynamical equation representing the cascadeand viscous damping, during which the probabihty measure of the distribution is kept
unchanged, probably due to the rapid process of the cascade. So that the distribution.
is distorted when expressed in terms of the dynamically evolving variable, due toa nonlinear dependence on the initial value. This is a dynamical mapping of the
distribution function (the mapping closure, due to Kraichnan). Let denote a field
variable, and suppose that has an initial variablewith the initial distribution$P_{0}(s_{0})\sim\exp[-s_{0}^{2}]$given by the gaussian function,. The probability distributionfunction will be given by$\kappa$$s$$k_{d}$$s_{0}$

$h$

$s$

$s$

$s_{0}$$P(s)=P_{0}(s_{0}) \frac{\partial s_{0}}{\partial s}$.

Kraichnan represented as $s=s_{0}J(s_{0}, t)$ and sought the solution $J(s_{0}, t)$ by his nonlinear equation.

164

3. An improvement of the model dynamics

3.1 Improved representation
In the previous form of the test field (5) which has only the component of velocity$v^{(t)}=F_{x}(x, t)$ , the longitudinal velocity derivative
vanishes identically. In
order to examine more general properties of the cascade including the longitudinal
derivative as well as self-interaction, we consider the test field in the form,$y$$\partial v^{(t)}/\partial y$.

$\psi(x,y,t)=yF(x,t)$(17)

Then the vorticity is given by$\omega=yF_{xx}(x,t)$,,$=s_{0} \frac{y}{\lambda_{0}}(\frac{\lambda_{0}}{\lambda(t)})^{\gamma}f(\xi)$(18)

where the function $F_{xx}(x, t)$ is assumed to be of the same form as (6) before. The
velocity fields in this case take the form,$u=Ax-\Psi_{z}-F(x,t)$ ,$v=By+yF_{x}(x,t)$ ,$w=Cz+\Psi_{x}$

.

(19)

Therefore the longitudinal derivatives are$\frac{\partial u}{\partial x}=A$

一$\Psi_{xz}-F_{x}$,$\frac{\partial v}{\partial y}=B+F_{x}$

,

(20)

$\frac{\partial w}{\partial z}=C+\Psi_{xz}$and the lateral derivatives are$\frac{\partial u}{\partial z}=-\Psi_{zz}\sim-x^{h-2}z$,$\frac{\partial v}{\partial x}=yF_{xx}=\omega$,$\frac{\partial w}{\partial x}=^{--}\Psi_{xx}\sim x^{h-2}z$

.

(21)

Substituting (18) in (3) and using (17), we obtain$-g( \xi)\dot{\lambda}=p(\xi)\lambda+G(\xi)\lambda^{h}+q(\xi)\lambda^{-\gamma+2}+\nu f^{n}(\xi)\frac{1}{\lambda}$

where the third termwith the definition
The equation of

newly appears to represent the self-interaction, and$q\lambda^{-\gamma+2}$$p(\xi)$$=$$(C-B)f(\xi)-A(\xi+\xi_{0})f’(\xi)$$q(\xi)$$=$$s_{0}[f’(\xi)\phi(\xi)-f(\xi)\phi’(\xi)]$$\phi’’\equiv f(\xi)$(22). The coefficients$\kappa(\tau)=\lambda(0)/\lambda(\tau)$

,

(23)

,

and $G(\xi)$ are the same as before.
corresponding to (16) is now obtained as$g(\xi)$$\frac{d}{d\tau}\kappa(\tau)=(C-B)\kappa+L_{0}\kappa^{2-h}+q_{0}\kappa^{\gamma}-\nu k_{0}^{2}\kappa^{3}$

,

(24)

165

where $q_{0}=s_{0}f(O)=s_{0}$ , by taking as $f(O)=1$ , and is a free parameter unspecified
so far. The coefficient $q_{0}=s_{0}$ denotes the initial magnitude of the test field vorticity,that is the transverse shear.
$\gamma$

3.2 Remarks$(a)$Scaling exponent
The equations for the vorticity$\gamma$$\dot{\omega}=(C+\Psi_{xz})\omega$,$\omega$and its derivativeare$\omega_{x}$$\dot{\omega}_{x}=2(C+\Psi_{xz})\omega_{x}+B\omega_{x}+\Psi_{xxz}\omega$

,

(25)

from (3) for the inviscid motion $(\nu=0)$ , where the over-dot denotes the convective
material derivative. Hence we have$-2 \frac{\dot{\omega}}{\omega}+\frac{\dot{\omega}_{x}}{\omega_{x}}=B+\Psi_{xxz}\frac{\omega}{\omega_{x}}$Assuming that
ing , we obtain$\omega\sim\lambda^{-\gamma}(t),$$\omega_{x}\sim\lambda^{-\gamma-1}(t)$

.

(26)-

and $\Psi_{xxz}\sim ah(h-1)(\lambda\xi_{0})^{h-2}$ and neglect-$B$$( \gamma-1)\frac{\dot{\lambda}}{\lambda}\approx ah(h-1)(\lambda\xi_{0})^{h-2}\lambda$.

(neglecting ). Hence we have
The equation (12) implies A$1+c(1-h)$ where $c=1/\xi_{0}>0$ . Since $h<1,$ $\gamma>1$ . It is likely that $2-h\geq\gamma(>1)$ ,since the vortex stretching by the singular component will be more effective than the
self-stretching for the cascade. Here we take $c=1$ , therefore$A$$/\lambda\sim-ah(\lambda\xi_{0})^{h-1}$$\gamma=2-h$$\gamma\approx$

.

(27)

This means that the vorticity (18) has the same behavior as the lateral derivative ofas decreases (see (21)). For the value of (27), the second andvarying likethird terms on the right hand side of (24) are put together with the common factor
and the coefficient given by $(L_{0}+s_{0})$ which is taken to be positive. The twoterms are considered to be of the same order:.
$(b)$
Skewness
The cubic sum of the longitudinal derivatives is$x^{h-2}$$v^{(\ell)}$$x$$\kappa^{2-h}$$L_{0}\sim s_{0}$$\Sigma_{3}$$=$$(A-\Psi_{xz}-F_{x})^{3}+(B+F_{x})^{3}+(C+\Psi_{xz})^{3}$$=$$S_{3}+3(-(A-\Psi_{xz})^{2}+B^{2})F_{x}-3(C\dashv-\Psi_{xz})F_{x}^{2}$,is given by (11). In the case $C>0$ and $F_{x}>0$ (given above), the thirdwhere
term on the right hand side is negative, while the second term becomes negative for$A\leq B<0$ . In particular it is expected that at the peak position of the test field,$S_{3}$

166

the new negative contributions exceed in magnitude the positive , although eachof the three terms of (20) takes positive‘or negative values. Thus we find that ourfield is simulating qualitatively the property $(b)$ described in the introduction for the
probability density function. Over most points (in the neighborhood of the origin
and $|z|<x$ ) the cubic sum will be given approximately bywhich is positive,
but not very large. However at points in a region near the peak of the test field
becomes negative. This
which could be intensified by the cascade, the cubic sum
infers qualitatively the relation of the skewness factor and the distribution function
mentioned in \S 1, by supposing that the time average of the longitudinal derivativeat a fixed point in the experiment corresponds to the spatial average in the present
model.
$S_{3}$$S_{3}$$\Sigma_{3}$4. Distribution functions of velocity derivatives

4.1 Near-exponential distributions of lateral and longitudinal derivatives
Growth of the scale parameter$(a)$$\kappa$

As far as the condition for the initial growthis satisfied,tends to a
non-zero stationary value
at which the right hand side of (24) vanishes.
For large values of $L_{0}+s_{0}$ , the scale parameter
will grow sufficiently so that thestationary value
is given by the balance of the last three terms of (24), that is wehave. Thus we obtain$\dot{\kappa}(0)>0$$\kappa(t)$$\kappa_{*}=\kappa(\infty)$$\kappa(t)$$\kappa_{*}$$[(L_{0}+s_{0})\kappa_{*}^{2-h}-\nu k_{0}^{2}\kappa_{*}^{3}]\approx 0$

十 $h$$\kappa_{*}\approx(\frac{L_{0}+s_{0}}{\nu k_{0}^{2}})^{1/(1}$

).

(28)

Lateral derivative
As shown in (21), the lateral shear $s(t)$ is given by $\omega=yF_{xx}$ . Therefore from theexpression (18) we find the scaling relation (for $\xi=0$ ):
$(b)$

(29)

$\frac{s(t)}{s_{0}}\approx\kappa^{2-h}(t)$

for

$y$

of the order of$\lambda_{0}$, using (27). For the stationary state, we obtain$\frac{s}{s_{0}}*\approx\kappa_{*}^{2-h}\sim s_{0}^{(2-h)/(1+h)}$where $s_{*}=s(\infty)$ , assumi-ng
Hence$L_{0,-}\sim s_{0}$$s_{0}\sim s_{*}^{\alpha(h)/2}$,,in (36) (except for the proportionality constant).

with$\alpha(h)=\frac{2}{3}(1+h)$

.

(30)

167

Thus the distribution of$s_{*}$

will be(31)$P(s_{*})=P_{0}(s_{0}) \frac{\partial s_{0}}{\partial s_{*}}\sim s_{*}^{\mu_{\alpha}}\exp[-s_{*}^{\alpha}]$with $\mu_{\alpha}=-(2-h)/3$ . $For-\frac{1}{2}<h<1$ , we obtain
The Kolmogorov
scaling $h=1/3$ gives $\alpha=8/9$ . This leads to a slower-than exponential $(i.e$ .
and predicts an upward flare of the skirt of the1) distribution with respect todistribution observed in the linear-log plot from the simulations of turbulence (Sheet al. 1988; Hosokawa&Yamamoto 1989; Kraichnan 1990; Vincent &Meneguzzi1991).
$(c)$ Longitudinal derivative
The longitudinal derivative concerned with the cascade is given byorof (20), which include . The component of intermittencyis$\frac{1}{3}<\alpha(h)<\frac{4}{3}$$\alpha<$$s_{*}$‘$\partial u/\partial x$$g$$F_{x}$$\theta v/\partial y$$\sigma(t)$$\sigma(t)=F_{x}=s_{0}(\frac{\lambda_{0}}{\lambda(t)})^{1-h}\phi’(\xi)$,$\phi’(\xi)=\int_{-\infty}^{\zeta}f(\xi’)d\xi’$(32), while $\partial v/\partial yincludes+F_{x}$ . An argumentfrom (27). Note that
includessimilar to that for the lateral derivative suggests the relation$-F_{x}$$\partial u/\partial x$(33)$\sim(\frac{L_{0}+s_{0}}{\nu})^{(1-h)/(1+h)}$$\frac{\sigma_{*}}{\sigma_{0}}\approx\kappa_{*}^{1-h}$

where $\sigma_{0}=s_{0}\phi’(0)>0$ andpositive, if
is negative.
For the derivative$\sigma_{*}=\sigma(\infty)$. Note that, for a large$\sigma_{0},$$\partial v/\partial y$will be$\partial u/\partial x$, the initial value $wiU$ be negative with the value$g_{0}=A-\Psi_{xz}(x_{0},0)-\sigma_{0}\approx-\sigma_{0}(1+c_{-})$ where $c_{-}=|A-\Psi_{xz}|/\sigma_{0}$ . The final value$g=\partial u/\partial x$

will be(34)$g_{*}\approx-\sigma_{*}\approx-\sigma_{0}\overline{\kappa}_{*}^{1-h}$from (33). Since $L_{0}=\Psi_{xz}(x_{0},0)$ , we have $g_{0}\approx-L_{0}-s_{0}\phi’(0)+A\approx-L_{0}-s_{0}$ ,assuming $\phi’(0)=O(1)$ and neglecting . The equation (28) suggests.
Thus we have
(35)
$A$

$\kappa_{*}\sim|g_{0}|^{1/(1+h)}$$|g_{*}|\sim\sigma_{0}|g_{0}|^{e(h)}\approx(1+c_{-})^{e(h)}\sigma_{0}^{2/\beta(h)}$from (42), where$\beta(h)=1+h$ ,$\epsilon(h)=\frac{1-h}{1+h}$

.

(36)

This corresponds to the negative side of the distribution. On the other hand, theinitial value ofwill be positive with $g_{0}=B+\sigma_{0}\approx\sigma_{0}(1-c_{+})>0$ for a$g_{*}$$g=\partial v/\partial y$168large fluctuation , where $c+=|B|/\sigma_{0}$ . It is assumed that $1>c+>0$ and $c+<c_{-}$ .
The final value will be$\sigma_{0}$$g_{*}\approx+\sigma_{*}\sim\sigma_{0}g_{0}^{e(h)}\approx(1-c_{+})^{e(h)}\sigma_{0}^{2/\beta(h)}$

,

(37)

corresponding to the positive side of .
is a gaussian random variable with the probability
Suppose that
. Then the distribution of will be$g_{*}$$\sigma_{0}$$\exp[-\sigma_{0}^{2}]$$Q_{0}(\sigma_{0})\sim$$g_{*}$(38)$Q(g_{*})=Q_{0}( \sigma_{0})\frac{\partial\sigma_{0}}{\partial g_{*}}\sim\{\begin{array}{l}g_{*}^{\mu_{\beta}}exp[-k_{-}g_{*}^{\beta}](g_{x}<0)g_{*}^{\mu_{\beta}}exp[-k_{+}g_{*}^{\beta}](g.>0)\end{array}$where $\mu_{\beta}=-(1-h)/2$ ,$k_{-}=( \frac{1}{1+c_{-}}I^{1-h},$$k_{+}=( \frac{1}{1-c+}I^{1-h}$

,

(39)

being smaUer than . This kind of asymmetry is observed in both experiments( $e.g$ . Gagne 1990) and numerical simulations ( $e.g$ . Hosokawa&Yamamoto 1989;
Yamamoto&Kambe 1991). $For-\frac{1}{2}<h<1$ , we obtain $-<\beta(h)<2$ . In the Kolmogorov scaling $h=1/3$ , we have $\beta=4/3$ . This leads to a steeper-than exponential$(i.e. \beta>1)$ distribution with respect to the longitudinal derivative. This behavior
too is observed in the numerical simulations as well as in the experiment (Makita
1991).
$k_{-}$$k_{+}$$\sigma_{*}$

4.2 Flatness factors
Once the probability distribution functions (pdfs, in short) are found by themethod described above, it is possible to estimate the flatness factors of the velocity derivatives. Now we consider the n-th order longitudinal derivative. The (normalized) pdf is$g_{n}(t)\equiv$$g(n, t)=\partial^{n}u/\partial x^{n}$$Q_{0}( \sigma_{0})d\sigma_{0}=\frac{1}{\sqrt{\pi}}e^{-\sigma^{\beta}}\cdot\frac{\beta}{2}\sigma^{\frac{1}{*2}\beta-1}d\sigma_{*}arrow P(x)dx$

where$P(x)= \frac{1}{\sqrt{\pi}}x^{-\frac{1}{2}}e^{-x}$,$\beta=\beta(n, h)=\frac{2(1+h)}{\delta(n)+1+h}$(40)$x=|\sigma_{*}|^{\beta}$,$\delta(n)=n-h$

,

(41)

169

( takes only positive values, while
Setting $g_{*}(n)=k_{n}\sigma_{*}(n)$ , we obtain$x$$g_{*}$

takes either of positive or negative values).

$<g_{*}(n)^{2}>=k_{n}^{2} \int_{0}^{\infty}x^{\frac{2}{\beta}}P(x)dx=\frac{k_{n}^{2}}{\sqrt{\pi}}\Gamma(\frac{3}{2}+\frac{\delta(n)}{1+h})$,$<g_{*}(n)^{4}>=k_{n}^{4} \int_{0}^{\infty}x^{4}\not\supset P(x)dx=\frac{k_{n}^{4}}{\sqrt{\pi}}\Gamma(\frac{5}{2}+\frac{2S(n)}{1+h})$,where (40) and (41) are used, andfactor is given by$\Gamma(x)$is the gamma function. Thus the flatness$F_{n}(h)= \frac{<g_{n}^{4}>}{<g_{n}^{22}>}=\sqrt{\pi}\frac{\Gamma(\frac{5}{2}+2D(n,h))}{[\Gamma(\frac{3}{2}+D(n,h))]^{2}}$,$D(n, h)= \frac{\delta(n)}{1+h}$

.

(42)

takes the gaussian value 3. Hence the
It is readily shown that, if $D(n, h)=0,$
function $D(n, h)=S(n)/(1+h)$ characterizes the degree of non-gaussian statistics.
In the above case, we have$F_{n}$$\delta(n)=n-h$,$D(n, h)= \frac{n-h}{1+h}$

.

(43)

For $h=1$ , we obtain $F_{1}(1)=3$ . The value $h=1/3$ of the Kolmogorov scaling leads
to $D(n, \frac{1}{3})=(3n-1)/4$ . Therefore$F_{1}( \frac{1}{3})=\sqrt{\pi}\frac{\Gamma(\frac{7}{2})}{[\Gamma(2)]^{2}}\approx 5.9$,

$F_{2}( \frac{1}{3})=\sqrt{\pi}\frac{\Gamma(5)}{[\Gamma(\frac{11}{4})]^{2}}\approx 16.4$,$F_{3}( \frac{1}{3})=\sqrt{\pi}\frac{\Gamma(\frac{13}{2})}{[\Gamma(\frac{7}{2})]^{2}}\approx 46.2$.

These values appear to be high, compared with the measurements quoted in $(a)$ of
the introduction. However a numerical simulation yielded $F_{1}=7.55,$ $F_{2}=14.4$ and$F_{3}=16.1$ (Yamamoto&Kambe 1991), the first two being not far from the present
results.
5. Summary and discussion
A dynamical model of intermittency in turbulence is presented in which the velocity field is represented locaUy in physical space and consists of three components: aregular field, a singular field with a fractional scaling exponent and a test field. The
first two are supposed to represent large-scale inertial range velocity field and the last
one an intermittency which is a cascading component. Nonlinear ordinary differential
equations are derived to describe growth of the inverse scale of the test field. Finalequilibrium value of rc depends on the initial valueof a velocity derivative, thefractional exponent and the viscosity. In a particular case, the equation reduces$h$$\kappa$$s_{0}$$h$170

to that of She (1991). The present analysis is based on an analytical representationof the velocity field. This suggests that the lateral derivatives of the velocity of the, while the scaling (dynamical) exponent ofintermittency component scales like).
She is $1-h$ (according to his notation, scaling likeof the cascading test
the present formulation the longitudinal derivative
field is characterized by a scaling exponent smaller by one than that of the lateraland the corresponding pdf isderivatives. Therefore the variable scales likegiven by $\exp(-|\sigma|^{\beta})$ with $\beta=1+h$ . In the Kolmogorov scaling we have $\beta=4/3$
which corresponds to a steeper-than exponential pdf. The present model can predict
the asymmetry of the distribution function of the longitudinal derivative, which isconsistent with the observed statistics.
From the analytical expression of the intermittency component, we have an estimate of the n-th order derivative. The flatness factor obtained from the pdf for then-th order derivative increases with the order . The estimated values (for $n=1$ and2) coincide with the corresponding ones of numerical simulation, but the observed
values are smaller than the present one.
One of the points of the present study is that the velocity field includes a component which has a fractional scaling exponent, giving a singular behavior. This isbased on the views that the fractional scaling behavior is predicted by the argument
given at the end of \S 1, further that straining of the test field only by the regularcomponent (Kambe 1983, 1984) is insufficient to describe the cascade in turbulence.
The singular component
gives rise to alocally intensified rate of vortex stretchingthat could result in the intermittency. This formulation may be considered to be arealization in an analytical form of the idea of She (1991 a).
At first sight, it may appear that the present representation of the velocity isgiven a particular form. However the following arguments indicate that the velocity
has a fairly general character more than its appearance. The regular fieldhas a
is taken in thegeneral local expression. The direction of stretching of the field
direction that does not hurt its generality, but it is assumed to have a two-dimensional
form that is not always the case. Another restriction is that the vorticity of thetest field coincides with the axis. In turbulent field there is of course substantial
probability of such an arrangement. Once the assumed field is realized, then the
cascade mechanism becomes active and a large amplitude fluctuation appears witha fair probability. In unfavorable arrangements the fluctuations will stay at small
amplitudes with near-gaussian probability.
$\kappa^{2-h}$$J^{1-\alpha}$$\ln$$\sigma$$\kappa^{1-h}$$\sigma$$n$$v^{(\ell)}$$v^{(r)}$$v^{(\iota)}$

$z$

$z$

171

References
Bacry, E., Arneodo A., Frisch U., Gagne, Y. &Hopfinger, E. (1990) Wavelet analysis
of fully turbulent data and measurement of scaling exponents, in Turbulence andcoherent structures (eds. M\’etais&Lesieur, Kluwer acad. publ.).
Batchelor, G.K. (1953) The Theory of Homogeneous Turbulence, chap. viii,(Cambridge University Press, London).
Batchelor, G. K. &Townsend, A. A. (1947) Proc. R. Soc. A 190, 534-550.
Frisch, U. &Parisi, G. (1985) in Turbulence and Predictability in Geophysical Fluid
Dynamics and Climate Dynamics (eds. Ghil, Benzi&Parisi, North-Holland)

84-88.
Gagne, Y. (1990) Properties of fine scales in high Reynolds number turbulence,in Advances in Turbulence 3 (European Turbulence Symposium at Stockholm).
Hosokawa, I. &Yamamoto, K. (1989) J. Phys. Soc. $Jpn$ . $58,20- 23$ .
Kambe, T. (1983) J. Phys. Soc. $Jpn$ . $52,834- 841$ .
Kambe, T. (1984) J. Phys. Soc. $Jpn$ . $53,13- 15$ .
Kambe, T. (1986) Fluid $Dyn$ . Res. 1, 21-31.
Kida, S. &Murakami, Y. (1989) Fluid $Dyn$ . Res. 4, 347-370.
Kraichnan, R. H. (1990) Phys. Rev. Lett. 65, 575-578.
Makita, H. (1991) Fluid $Dyn$ . Res. 8.
Monin, A.S. &Yaglom, A.M. (1975) Statistical Fluid Mechanics vol. 2 (MIT Press).
Proudman, I. &Reid, W.H. (1954) Phil. Trans. $Roy$ . Soc. A 247, 163-189.
Rose, H.A. &Sulem, P.L. (1978) Journal de Physique 39, 441-484.
She, Z.-S. (1991 a) Phys. Rev. Lett. 66, 600-603.
She, Z.-S. (1991 b) Fluid $Dyn$ . Res. 8.
She, Z.-S., Jackson, E. &Orszag, S.A. (1988) J. Sci. Comput. 3, 407-434.
She, Z.-S. &Orszag, S.A. (1991) Phys. Rev. Lett. 66, 1701-1704.
Sheih, C.M., Tennekes, H. &Lumley, J.L. (1971) Phys. Fluids 14, 201-2215.
Van Atta, C.W. &Chen, W.Y. (1970) J. Fluid Mech. 44, 145-159.
Vincent, A. &Meneguzzi, M. (1991) J. Fluid Mech. 225, 1-20.
Yamamoto, K. &Hosokawa, I. (1988) J. Phys. Soc. $Jpn$ . $57,1532- 1535$ .
Yamamoto, K. &Kambe, T. (1991) Fluid $Dyn$ . Res. 8.