---
abstract: 'Simulations of vortex tube dynamics reveal that the non-Gaussian nature of turbulent fluctuation originates in the effect of random advection. A similar non-Gaussian distribution is found numerically in a simplified statistical model of random advection. An analytical solution is obtained in the mean-field case.'
author:
- 'Y-h. Taguchi'
- Hideki Takayasu
---

v

Non-Gaussian distribution in Random advection dynamics

Institut für Festkörperforschung, Forschungszentrum Jülich,\
D-5170 Jülich, Germany\
and\
Department of Physics, Tokyo Institute of Technology,\
Oh-okayama, Meguro-ku, Tokyo 152, Japan[@present]

Department of Earth Science, Kobe University, Kobe 657, Japan.

In the study of statistical physics for non-equilibrium systems the deviation from the Gaussian distribution has been the central issue. We are expecting the existence of universal mechanisms of producing non-Gaussian distributions as in the case of thermal equilibrium systems, however, our knowledge has been still in the elementary level.

Non-Gaussian probability distribution function (NGPDF) is especially important in fluid turbulence. Turbulence is a typical far-from-equilibrium phenomena as it is characterized by the energy cascade which violates both the detailed balance and equi-partition. The appearance of NGPDF is so common and we can not construct any theory without taking the non-Gaussian nature into account[@Batchelor; @Monin]. Latest technique of direct observation of turbulence[@Goldburg] and also direct numerical integration of Navier-Stokes equation[@Kida] are clarifying the details NGPDF from experimental viewpoints.

Recently, theoretical investigation to NGPDF in turbulence is attracting much attention. An approach is to assume the multifractality in the geometry of turbulent velocity field as an ideal limit[@Benzi], and another one called the mapping closure conjectures the existence of a kind of smooth map which describes the time evolution of statistical quantities[@She91; @Kraichnan].

Yakhot et al.[@Yakhot] also present a phenomenological excellent approach in order to derive a NGPDF for vorticities in the fluid turbulence. Although it is fairly reasonable, physical meanings in their assumptions are not clear.

In this paper we first introduce a numerical model of vortex dynamics on lattice and show that the vorticity distribution in a turbulent steady state is actually far from Gaussian but closer to a symmetric exponential distribution. In order to clarify the origin of this NGPDF we modify the dynamics so that vortex tubes move randomly by passive advection. It is shown that this modification does not seem to affect the vorticity distribution, which implies that the random advection is playing an important role. Then, we focus our attention to the effect of random advection and introduce a random scalar advection model on lattice. Surprisingly the scalar also follows a distribution which is almost identical to the former NGPDFs. A mean-field version of the scalar model is then solved analytically showing a clear evidence of NGPDF.

Let us first introduce our lattice vortex model[@Tag] which is equivalent to the following set of vorticity equations of incompressible fluid in the continuum limit

$$\begin{aligned}
\frac{\partial \w}{\partial t} + (\v \cdot \nabla ) \w &=& (\w \cdot \nabla)\v
+ \nu \Delta \w, \label{vor}\\
\rot \v &=& \w, \label{rot}\\
\div \v &=& 0,  \label{div}\\ \nonumber\end{aligned}$$

where $\v, \w$ and $\nu$ are the velocity, vorticity and viscosity, respectively. We assign vorticities on a simple cubic lattice. Every bond has only one vorticity component along its direction. This means that an $x$-bond has only $x$ component of vorticity, i.e., a bond can be viewed as a vortex tube. For a given configuration of vorticities the vorticity field is calculated by using the Biot-Savier law (equivalent to eqs.(\[rot\]) and (\[div\]). The fluid velocity of a bond at $\r,\v(\r)$, is estimated at the mid-point of the bond. Dynamics is defined so as to satisfy the vorticity equation(\[vor\]).

Due to the advection term $(\v \cdot \nabla) \w$ in eq. (\[vor\]) the vorticity flows with the fluid as shown by the name of Kelvin’s theorem[@Landau], the vorticity $\omega_z$ at a $z$-bond is transported to its six neighboring $z$-bonds. By this effect the value of $\omega_z$ at $\r \pm \hi$ is increased by $\pm J_{iz} \Delta t / 2$, where $\hi$ is a unit vector directing either $x$, $y$ or $z$ direction, and $J_{ij}$ is the flux of vorticity, $\tvi \wj$. The coefficient $1/2$ of the vorticity flux and the signs $\pm$ are introduced to keep the spatial symmetry. The first term on the right hand side of eq.(\[vor\]) shows the vortex stretching term, so we have to modify the vorticities around the advectively transported bonds so that modified vorticities make loops as shown in Fig.\[dyna\]. In this procedure we add $\mp  J_{iz} \Delta t/2$ to the bonds at $\r \pm \hi/2 \pm \hz/2$, namely, $x$ and $y$ component appear due to the elongation of vortex tube. The diffusion term in eq. (\[vor\]) is discretized by the usual finite difference method. It has been shown that this dynamics is equivalent to eq.(\[vor\]) in the continuum limit and numerical simulation of this vortex tube dynamics on a periodic $24 \times 24 \times 24$ lattice has reproduced many basic properties of fluid turbulence[@Tag].

As for the probability distributions we obtained the following results:

1.The velocity components (for example $v_x$) follows nearly Gaussian.

2\. The distribution of differentiated quantities, such as $\partial v_x /\partial x, \partial v_x / \partial y $ and $\omega_x$, are non-Gaussian and close to exponential (see Fig.\[wx\]).

3\. By removing lower wave number components, the distribution of $v_x$ also becomes closer to an exponential as firstly found by She et al[@She88].

We also observe a kind of conditional distributions for velocities on bonds whose vorticities are in a fixed range. The distribution are nearly identical both for large $\omega$ and for small $\omega$, indicating that the velocity on a bond is nearly independent of its vorticity strength.

We now modify our model to seek the origin of the NGPDF. The modification is to randomize the velocity field at each time step keeping its distribution. Namely, we do not use Biot-Savier law, but the vorticities are transported and elongated by random advection. The probability distribution of $\omega$ after several hundreds time steps is quite similar to that of original lattice vortex model (see Fig.\[wx\]). This result clearly shows that the appearance of NGPDF is independent of the details of velocity field. As suggested by Sinai and Yakhot[@Sinai; @Yakhot] NGPDF may be caused by random advection, and if so, we can expect a large universality class including another exponential-like distribution in thermal convection flow[@Siggia; @Yanagita].

In order to see the effect of random advection more clearly we introduce a scalar model on lattice. The model is governed by the following stochastic equation $$\omega (r_0,t + \Delta t) = \omega (r_0,t) - v(r_0,t) \omega (r_0,t) \Delta t
+ \sum_{r} P(r_0,r,t) v(r,t) \omega (r,t) \Delta t,
\label{eq:adv}$$

where $v(r,t)$ and $P(r_0,r,t)$ are independent random variables. $v(r,t)$ takes non-negative values, and $P(r_0,r,t)=1$ when a flow from site $r$ to $r_0$ exists at time $t$, otherwise $P(r_0,r,t)=0$ and it is normalized as $\sum_r P(r_0,r,t)=1$. The one-dimensional version of eq.(\[eq:adv\]) is defined by the special case that either $P(r_0,r_0-1,t)$ or $P(r_0,r_0+1,t)$ is equal to 1 with probability $1/2$. We perform the simulation on a 1-dimensional lattice of size $10000$ with the periodic boundary condition. The random number $v(r,t)$ is in the range of $[0,0.5]$ and $\Delta t =0.5$. For a given random initial condition of ${\omega (r,0)}$ we observe the distribution of $\omega(r,t)$ at sufficiently large $t$. The result is also plotted in Fig.\[wx\]. The distribution is far from Gaussian and is very close to those of former cases. Now it seems obvious that the origin of NGPDF is in the simple random advection transport.

A mean-field version of eq.(\[eq:adv\]) can be solved analytically. We consider the situation that a site is interacting with a mean-field site. To avoid unimportant complexity we assume the case that the random number $v$ takes either 0 or $j_0/\Delta t$ with probability $1/2$ ($j_0 \in [0,1]$). In this situation the stochastic eq.(\[eq:adv\]) becomes $$\omega (t+t_0) = \left \{ \begin{array}{ll}
 \omega (t), & {\rm Prob.} 1/4 \\
(1-j_0) \omega (t), & {\rm Prob.} 1/4 \\
\omega (t) + j_0 \omega_M, & {\rm Prob.} 1/4 \\
(1-j_0) \omega (t) + j_0 \omega_M, & {\rm Prob.} 1/4
\end{array}, \right.
\label{eq:mean}$$ where $\omega_M$ is an independent random number having the same distribution as $\omega (t)$. By introducing the characteristic function $$Y(\rho,t)=\int^{\infty}_{-\infty} e^{i\rho \omega} P(\omega,t) d \omega,$$ eq.(\[eq:mean\]) becomes $$Y(\rho, t+\Delta t) = \frac{1}{4} \{ Y(\rho,t) + Y(\rho-\rho j_0,t)\}
\{ 1 + Y ( \rho j_0,t) \}.
\label{eq:mean2}$$

Taylor expansion of eq.(\[eq:mean2\]) in terms of $\rho$ gives a set of equations for the moment functions $\{ < \omega^n > \}$, and it can be easily shown that a steady state exists when $< \omega > \neq 0$.

The steady state solution is obtained with the aid of algebraic calculation by computer. In Fig.\[fig:j0\] we plot the cumulants for orders up to $7$ together with those for one-sided exponential distribution for comparison. Higher order cumulants are not zero in any case and showing a tendency to diverge as the order goes to infinity, which clearly demonstrates that the steady state distributions are not Gaussian.

In the special case of $j_0 = 1/2$ we can estimate the functional form of $Y(\rho)$ as follows. By denoting $Y_n = Y(\rho 2^n)$ eq.(\[eq:mean2\]) becomes $$Y_{n+1}-Y_n = 2 \cdot \frac{Y_n^2 -Y_n}{3 - Y_n}.
\label{eq:j2}$$ Supposing that $n$ can be a continuous number and we approximate the left hand side by a derivative $dY/dn$. Then eq.(\[eq:j2\]) can easily be integrates and we have the following cubic equation for $Y(\rho)$ with $< \omega > = 1$, $$\rho^2 Y(\rho)^3 + (Y(\rho)-1)^2 =0,$$ which can be solved explicitly by Cardano’s formula. The asymptotic behavior for $\rho \rightarrow \infty$ is given as $$Y(\rho) \propto \rho^{-2/3}.$$

This power law decay is very different from the case of a Gaussian distribution where the characteristic function decays faster than any power of $\rho$. In the case of exponential distribution the characteristic function is given by $1/(1 + i \rho)$ (asymmetric case) or $1/(1+\rho^2)$ (symmetric case), which also shows a power law decay as $\rho \rightarrow \infty$. In this sense our distribution is far from Gaussian, but has a similarity to the exponential distribution.

For smaller $j_0$ the cumulants are smaller, and in the limit of $j_0 \rightarrow 0$ it can be shown that the normalized cumulant of order $n$, $< \omega^n >_c / < \omega^2 >_c^{n/2}$, vanishes for $n \geq 3$. Namely, the distribution converges to a Gaussian in this limit, which agrees with the well-known fact that the distribution function satisfying the usual diffusion equation in the continuum limit is a Gaussian since eq.(\[eq:adv\]) becomes a usual diffusion equation in the limit of $\Delta t \rightarrow 0$. This indicates that the finiteness of $\Delta t $ is very important for the appearance of NGPDF in our random advection dynamics.

One may think that these results for $\Delta t \neq 0$ are meaningless because we must take the limit of $\Delta t \rightarrow 0$ in order to recover eq.(\[vor\]). This is a wrong argument neglecting the finiteness of correlation time in the real dynamics with $\Delta t =0$. It is known that the fastest mode in real turbulent flows has a characteristic time approximately given by $\sqrt{\nu / \varepsilon}$, where $\varepsilon$ is the mean energy dissipation rate [@visco]. For time scales less than this value advections can not be treated as random. In our random advection models $\Delta t$ should be viewed as the correlation time in real dynamics, therefore, it should take a finite value.

Our theoretical approach is successful only in the case of $< \omega > \neq 0$, while observed distributions are symmetric with $< \omega > = 0$. At present we have no rigorous way of connecting these two cases, but we are now considering the effect of spatial fluctuation as the key of solving this difficulty. Even in the case of $< \omega > =0$ averaged over the whole space, $<\omega >$ can be nonzero if the average is taken over a finite area due to the spatial fluctuation. So, if the distribution of $\omega$ is determined rather locally not using the information from the whole space, then the distribution for $< \omega > \neq 0$ may have direct consequence to the real distributions.

In summary, we showed that the random advection dynamics creates far-from Gaussian fluctuations whose distribution functions are closer to the exponential. As shown in the discussions of the statistical scalar model on lattice and its mean-field analysis, the mechanism of NGPDF is so simple that a wide application may be expected.

H.T. thanks M. Takayasu for useful discussions. NEC software (c) is also acknowledged for allowing Y.T. to use EWS-4800/220, with which all calculations were performed. This work is supported by Grant-Aid for Scientific Research on Priority Areas, “Computational Physics as a New Frontier in Condensed Matter Research”, from the Ministry of Education, Science and Culture, Japan.

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