Density of Inertial Particles: Exactly Solvable 2D Models

TL;DR

This paper derives exact bounds for the mean number of caustics in 2D inertial particles under Gaussian noise, providing insights into particle clustering behavior with verified numerical accuracy.

Contribution

It introduces exactly solvable 2D models for inertial particles and establishes bounds on caustic formation, advancing understanding of particle dynamics in stochastic flows.

Findings

Bounds for caustic numbers are established for different forcing types.

Numerical methods confirm the efficiency of the derived bounds.

The models enhance understanding of inertial particle behavior in stochastic environments.

Abstract

Inertial particles in 2D driven by a Gaussian white noise forcing are considered. For two examples of the forcing (compressible and incompressible) upper and lower bounds are found for the mean number of caustics as a function of Stokes number. Efficiency of the bounds is verified by numerical methods.

Equations

$$d\mathbf{r}=\mathbf{v}dt,d\mathbf{v}=-(\mathbf{v}/\tau )dt+d\mathbf{w}(t,\mathbf{r}),\mathbf{r}(0,\mathbf{a})=\mathbf{a},\mathbf{v}(0,\mathbf{a})={\mathbf{u}}_0(\mathbf{a})$$

$$E\mathbf{w}(t_1,{\mathbf{r}}_1)d\mathbf{w}(t_2,{\mathbf{r}}_2{)}^T=\min (t_1,t_2)\mathbf{B}({\mathbf{r}}_1-{\mathbf{r}}_2)$$

$$d\mathbf{v}=-(\mathbf{v}/\tau )dt+\mathbf{B}(0{)}^{1/2}d\mathbf{w}(t)$$

$$J(t,\mathbf{a})=\det \left(\frac{\partial \mathbf{r}}{\partial \mathbf{a}}\right)$$

$$\lambda =\underset{t\to \infty }{\lim }\frac{1}{t}\underset{|\mathbf{a}|\to 0}{\lim }\ln \left(\frac{|\mathbf{r}(t,\mathbf{a})-\mathbf{r}(t,0)|}{|\mathbf{a}|}\right)$$

$$\pi (t,\mathbf{r}(t,\mathbf{a}))={\pi }_0(\mathbf{r}(t,\mathbf{a}))/J(t,\mathbf{a})$$

$$\frac{\partial \mathbf{u}}{\partial t}+\mathbf{u}\cdot \nabla \mathbf{u}+\mathbf{u}/\tau =\frac{\partial \mathbf{w}}{\partial t},\mathbf{u}{|}_{t=0}={\mathbf{u}}_0(\mathbf{x})$$

$$\frac{\partial \mathbf{u}}{\partial t}+s\mathbf{u}\cdot \nabla \mathbf{u}+\mathbf{u}=\frac{\partial \mathbf{w}}{\partial t}$$

$$s=\frac{U\tau }{L}$$

$$dx=udt,du=-(u/\tau )dt+dw(t,x),x(0)=a,u(0)=u_0(a)$$

$$Ew(t_1,x_1)dw(t_2,x_2)=min(t_1,t_2))b(x_1-x_2)$$

$$J(t,a)=\frac{\partial x}{\partial a}$$

$$dJ=J_{1}dt,d{J}_1=-(J_1/\tau )dt+Jd\tilde{w}(t)$$

$$J_1(t,a)=\frac{\partial u}{\partial a},E\tilde{w}(t)=0,E\tilde{w}(t{)}^2=-b^{\prime \prime }(0)t$$

$$dp=(-p-p^2)dt+\sqrt{2}sdw$$

$$s=\sqrt{-b^{\prime \prime }(0){\tau }^3}$$

$$\underset{t\to S-0}{\lim }p(t)=-\infty ,$$

$$P(S<\infty )=1$$

$$E(S)=2{\pi }^2\sqrt{z}M(z{)}^2,z=s^{-4/3}/4$$

$$M(z)=\sqrt{Ai(z{)}^2+Bi(z{)}^2},$$

$$L\mu (x)=-1,\mu (-\infty )=0$$

$$E(S)=\mu (\infty ).$$

$$L=(-x-x^2)\frac{\partial }{\partial x}+s^2\frac{{\partial }^2}{\partial x^2}$$

$$\bar{F}(t)=P(S>t)$$

$$\bar{F}(t)=u(t,\infty )$$

$$\frac{\partial u(t,x)}{\partial t}=L_{x}u(t,x),u(0,x)=1$$

$$u(t,-\infty )=0,u_x(t,\infty )=0$$

$$\underset{t\to S+0}{\lim }p(t)=\infty$$

$$\lambda =\bar{p}/\tau ,\bar{p}=\underset{t\to \infty }{\lim }\frac{1}{t}{\int }_0^{t}p(s)ds$$

$$\bar{p}=-1+\frac{M^{\prime }(z)}{2\sqrt{z}M(z)}$$

Full text

Density of Inertial Particles: Exactly Solvable 2D Models

Leonid I. Piterbarg

Department of Mathematics, University of Southern California

Kaprielian Hall, Room 108, 3620 Vermont Avenue, Los Angeles, CA 90089-2532

**Abstract **

Inertial particles in 2D driven by a Gaussian white noise forcing are considered. For two examples of the forcing (compressible and incompressible) upper and lower bounds are found for the mean number of caustics as a function of Stokes number. Efficiency of the bounds is verified by numerical methods.

Keywords: Inertial particles, Caustics, Lagrangian stochastic models

Introduction

In compressible fluid flows the density of Lagrangian particles finite at an initial moment can go to infinity in finite time as a flow is evolving. Such density explosions are related to occurence of caustics in the underlying velocity field. One of the most important such phenomena is motion of inertial particles in turbulence ( a comprehensive review can be found in [1]).

In this work we address the mean frequency (intensity) of caustic occurence $\nu$ in the framework of a stochastic flow modeling Lagrangian motion in a turbulent velocity field. The stochastic flow concept allows one for describing joint statistics of several particles or even a particle continuum at any moment.

The main goal is to introduce two particular models in two dimensions, very simple but not trivial, for which the dependence of $\nu$ on Stokes number $s$ (the ratio of Lagrangian and Eulerian time scales, [2]) can be analytically investigated to a certain extent. Namely, first, lower and upper bounds for $\nu(s)$ are provided yielding an exact asymptotic as $s\to 0$ and, second, $\nu(s)$ is accurately evaluated for a wide range of $s$ by solving a $1D$ parabolic equation numerically.

A general version of a stochastic flow addressed here has been introduced in [3] and assumes finite particle velocities and a white noise type of accelerations. Thus, it is not a Kraicnan model of turbulence widely used in physics of fluids [2]. More exactly, assume that the position $\mathbf{r}(t,\mathbf{a})$ of a particle at time $t$ with the initial position $\mathbf{a}$ and its velocity $\mathbf{v}(t,\mathbf{a})$ are described by the following Ito equations

[TABLE]

where $\mathbf{u}_{0}(\mathbf{x})$ is an initial (Eulerian) velocity field, $\mathbf{w}(t,\mathbf{r}$) is a random forcing with zero mean which is a Brownian motion in time and a statistically homogeneous random field in space , i.e.

[TABLE]

where $\mathbf{B}(\mathbf{r})$ is its space covariance matrix, all the vectors are two-dimensional, and finally $\tau$ is the Lagrangian correlation time scale.

In particular, the motion of a single particle in the framework (1) is covered by a Langevin equation for the velocity

[TABLE]

where $\mathbf{w}(t)$ is a standard Brownian motion in 2D, thereby the Lagrangian velocity of the particle is simply a well known (two-dimensional) Ornstein-Uhlenback process and its position is an integrated Ornstein-Uhlenback process.

In general the motion of any number $n$ of the particles is described by a diffusion process in $R^{2n}$ for the vector of positions/velocities $(\mathbf{r}_{1},\mathbf{v}_{1},...,\mathbf{r}_{n},\mathbf{v}_{n})$ with a drift and diffusivity matrix expressed in terms of $\tau$ and $\mathbf{B}$, [3].

In this work we mostly address the Jacobian

[TABLE]

and partially the top Lyapunov exponent (LE)

[TABLE]

which depends on the direction of $\mathbf{a}$ for anisotropic flows.

A consideration of two-particle motion suffices for studying both characteristics .

An important physical meaning of the Jacobian can be seen from the following. Let the initial position $\mathbf{a}$ be a random variable independent of the flow with pdf $\pi_{0}(\mathbf{a})$, then for the conditional pdf $\pi(t,\mathbf{r})$ conditioned on the flow is given by

[TABLE]

Hence, if for some $t_{0}$ the Jacobian takes zero value, the density becomes infinite at the point $\mathbf{r}(t_{0},\mathbf{a})$ what we will call an occurrence of a caustic at this point.

We show that for a certain class of forcings caustics arise with probability one, they form a stationary process in time and sensible bounds for the mean number of caustics are given.

In Eulerian terms equations (1) cover Lagrangian motion in the Eulerian velocity field $\mathbf{u}(t,\mathbf{x})$ satisfying the following equation

[TABLE]

By the method of characteristics one can find that for some short interval $(0,t)$ the solution of this equation exists and unique since at the initial moment $J(0,\mathbf{a})=1$. However after some time the uniqueness is lost (the Jacobian takes a zero value) due to a very weak dissipation modeled by the last term on the left hand side (LHS). Thus, estimates of the mean of the first moment when $J(t,\mathbf{a})$ hits zero would provide us with an idea when (4) looses uniqueness. In terms of the non-linear wave theory, one can treat that as the first moment of wave breaking.

Some of our results can be predicted by simply procceding to dimensionless variables in (4). Let $U$ and $L$ be some typical velocity and length scales respectively. Changing $\mathbf{u},\mathbf{w},\mathbf{x},t$ to $\mathbf{u}/U,\mathbf{w}/U,\mathbf{x}/L,t/\tau$ one gets from (4)

[TABLE]

where the dimensionless parameter

[TABLE]

is called Stokes number.

Thus, if $s\to 0$ then the underlying Eulerian velocity field satisfies a linear equation and hence the intensity of caustics should tend to zero. In the opposite case $s\to\infty$ the non-linear term dominates and one may expect that the number of caustics per time grows indefinitely.

The paper is organized as follows. A similar one-dimensional problem is exactly solved in Section 1. The solution is essentially used in investigating 2D phenomena. A deterministic case ($\mathbf{w}=\mathbf{0}$) is briefly discussed in Section 2. In Section 3 we derive a system of equations containing $J$ in the general case of an arbitrary homogeneous forcing. For particular forms of the forcing rigorous estimates for the mean number of caustics $\nu(s)$ are found in Section 4. Also $\nu(s)$ is computed by solving a simple $1D$ parabolic equation. The case of an isotropic forcing (unsolved yet) is mentioned in Section 5. Conclusions are gathered in Section 6 and some details are brought to Appendix.

1. One-dimensional case

A part of results from this section is well known from physical literature, [1]. We formulate them in rigorous form and give more details.

Consider

[TABLE]

with $Ew(t,x)=0$ and

[TABLE]

For

[TABLE]

one can get by direct differentiation (6) in $a$

[TABLE]

where

[TABLE]

By setting $p=\tau J_{1}/J$, introducing dimensionless time $t=t/\tau$, and using Ito formula obtain

[TABLE]

where $w$ is a standard Wiener process and

[TABLE]

. Worth noting that if the velocity and length scales of the flow are chosen as $U=\sqrt{b(0)\tau}$ and $L=\sqrt{-b(0)/b^{\prime\prime}(0)}$ respectively, then (8) coincides with the earlier introduced Stokes number (5).

Assume that zeros of $J(t)$ are prime that is typical for stationary processes, then they can be identified with moments of explosion of $p(t)$.

Proposition 1.

Process $p$ is explosive(e.g. [4]), more exactly

[TABLE]

where the explosion time $S$ is finite

[TABLE]

and its expectation can be explicitly computed

[TABLE]

where

[TABLE]

$Ai(z),Bi(z)$ are the Airy functions and $M(z)$ their modulus.

Explosiveness of $p(t)$ follows from the Feller’s criteria, [4].

To prove (9) introduce $\mu(x)$ as the mean time to explosion under condition $p(0)=x$, then solve

[TABLE]

and set

[TABLE]

where

[TABLE]

is the generator of (7). Details of computations can be found in [5].

For the purpose of studying $2D$ models the knowledge of the mean explosion time is not enough. Introduce

[TABLE]

and let $u(t,x)$ be the same probability under condition $p(0)=x$, then

[TABLE]

In [6] it is shown that $u(t,x)$ satisfies the initial value problem

[TABLE]

To ensure uniqueness of solution of (10) we add the natural boundary conditions assuming that the limit $u(t,\infty)$ exists similarly to $m(\infty)$.

[TABLE]

A simplest Euler scheme was used for solving (10) numerically. Some details and validation of the choice of the scheme parameters are given in Appendix. Notice that our goal is not to evaluate and minimize the error of the numerical computations, but rather to illustrate that (10) can be solved quite accurately and efficiently with very simple tools.

Graphs of $\bar{F}(t)$ for few values of $s$ are shown in Fig.1

After explosion process $p(t)$ can be continued by starting over, i.e. by solving same eq. (7) with the initial condition

[TABLE]

The positive sign follows from relation $J_{1}=dJ/dt$ and the fact that zeros of $J(t)$ are prime. Thus $p(t)$ takes different signs on different sides of a zero of $J(t)$.

LE for flow (7) is expressible in terms of the ergodic mean of $p$

[TABLE]

as follows from (3) and definition of $p(t)$, see [3] for details. It also can be exactly found

Proposition 2.

The ergodic mean of $p$ is given by

[TABLE]

The statement is proven in Appendix and here we just make some comments. Notice that the density of the stationary distribution $\pi(p)$ of $p(t)$ does exist and is given by

[TABLE]

where $C$ is a normalized constant.

One can find the following asymptotic

[TABLE]

Thus, the integral for the invariant mean

[TABLE]

is formally divergent, however if the integral is meant as a Cauchy principal value then it coincides with the RHS of (11).

For bounded functions of $p$ the ergodicity holds true with a conventional interpretation of the integral over the invariant density.

Proposition 3

Process $p(t)$ is ergodic , i.e. for any bounded function $f(p)$

[TABLE]

**2. Deterministic 2D Case **

Let us assume $\mathbf{w}=0$ in (1) and proceed to dimensionless variables as described in Introduction before eq. (5). As a result the expression for the Jacobian (2) is found in the explicit form

[TABLE]

where $s$ is Stokes number defined in (5)

[TABLE]

and $(a,b)$, / / ($u_{0},v_{0})$ are the initial position and velocity respectively related by $u_{0}=u_{0}(a,b),\ \ v_{0}=v_{0}(a,b)$.

Let

[TABLE]

and $S$ be the first time when $J(S)=0$. Easy to find that

[TABLE]

where

[TABLE]

In particular if $Q=0$ and $P<0$ do not depend on $(a,b)$ at all, then $G=R^{2}$ and $S$ , defined for all $s>0$, does not depend on the initial point as well. Thus, the equation

[TABLE]

looses uniqueness exactly at moment $t=S$. If one defines $u(t,x)$ after $t>S$ as a solution of (13) with the initial condition

[TABLE]

then $\nu=1/S$ could be interpreted as the intensity of caustics in time. In Figure 2 we show few curves $\nu(s)$ for different $s_{0}$ in order to compare them visually with a stochastic case addressed in the next sections.

If $S$ depends on the initial point, then its interpretation in terms of Eulerian set up (13) is not that simple and will not be discussed here.

3. System of Equations for Jacobian in general 2D Case

Now we return to the stochastic model (1). In few works it was pointed out that there is a closed equation for the matrix

[TABLE]

Namely, e.g. [3]

[TABLE]

An analysis of that equation led to some general important conclusions, but it is of little help for our purposes because in general it cannot be reduced to efficiently handled scalar equations.

Let us first rewrite (1) in the coordinate wise form with $\mathbf{r}=(x,y),\mathbf{a}=(a,b)$

[TABLE]

where

[TABLE]

and $b^{ij}$ are entries of $\mathbf{B}$ The goal is to investigate time behavior of the Jacobian

[TABLE]

It is not possible to obtain a closed equation for $J(t)$, but it can be included in a system of four equation as it is shown in Appendix. Namely, for dimensionless time $t=t/\tau$ denoted by the same letter, Jacobian can be represented as

[TABLE]

where dimensionless random function $m_{1}(t)$ is included into the following system

[TABLE]

$s$ is Stokes number defined similarly to (8) and an exact expression for it is given in Appendix.

To define processes $\eta_{i}(t)$ we nondimensionalize $w^{(i)}(t)$ and set

[TABLE]

where the subs mean derivatives. Thus $\eta$’s are dependent Wiener (non-standard) processes with a covariance matrix $Ed\eta^{(i)}(t,x,y)d\eta^{(j)}(t,x,y)=\sigma^{ij}_{\eta}dt$ given by

[TABLE]

where $b^{ij}$ is a dimensionless version of $b^{ij}$ and partial derivatives are taken at $x=0,y=0$.

4. Two special cases

Under certain conditions imposed on the forcing in (14) the first moment $S$ of $J(t)$ to hit zero turns out to be the minimum from the first explosion moments for two independent 1D processes described by (7).

In this section we assume a zero initial velocity field

[TABLE]

that implies zero initial conditions $m_{k}(0)=0,\ \ k>1$ in (15)

Model 1

Assume

[TABLE]

where $w^{(1)}(t,x),w^{(2)}(t,y)$ are independent and

[TABLE]

From (16,18) it follows that

[TABLE]

Then from (16,17,19) $E\eta_{1}^{2}=E\eta_{2}^{2},\ \ \ E\eta_{1}\eta_{2}=0$ and hence

[TABLE]

are independent identically distributed Wiener processes.

Next due to the zero initial conditions $m_{3}=m_{4}\equiv 0$. Introduce

[TABLE]

By adding and subtracting first two equations in (15) we get two separated equations

[TABLE]

for independent identically distributed processes $p(t)$ and $q(t)$

Model 2

In this model we assume

[TABLE]

with independent $U$ and $V$ such that

[TABLE]

From (16, 17, 21, 22) it follows that

[TABLE]

and

[TABLE]

are independent identically distributed Wiener processes since $\eta_{1}=-2(U^{\prime}+V^{\prime})$ and $\eta_{3}=-2(U^{\prime}-V^{\prime})$

Thus $m_{2}=m_{4}\equiv 0$ and introducing

[TABLE]

arrive at the same equations (20).

Below we show simulated velocity fields at a particular time moment for both models (Fig.3) where periodic in space forcings were used.

Let $S_{1}$ and $S_{2}$ be the first explosion moments for $p(t)$ and $q(t)$ respectively, then

[TABLE]

Certainly the knowledge of $E{S_{i}}$ is not enough to find $E(S)$, but the latter can be expressed in terms of $\bar{F}(t)=P\left(S_{1}>t\right)$ as

[TABLE]

Manipulating with this formula it is not difficult to present an example where the expected value of the minimum of two independent identically distributed random variables $X,Y$ is finite while the expectation of each variable is infinite due to heavy tails of $X,Y$ distribution. So, theoretically speaking $E{S_{i}}$ and $E(S)$ may differ in an order. Fortunately it is not the case here since the tail of distribution of $S_{1}$ is exponential and can be evaluated [5].

Namely from Theorem 1.1 in [7] it follows that the exponential moment

[TABLE]

is finite for all $\gamma=\gamma(s)$ satisfying

[TABLE]

and

[TABLE]

Hence it is reasonable to assume that

[TABLE]

with $\gamma$ satisfying (24). In view of this assumption

[TABLE]

On the other side obviously that

[TABLE]

Thus for the mean number of caustics $\nu(s)=1/E(S)$ one gets from (9)

[TABLE]

For small $s$ asymptotics of the lower bound and upper bound coincide and we get

[TABLE]

while for large $s$ the corresponding asymptotics differ just by a constant

[TABLE]

Approximation (25) is quite speculative and indeed it greatly overestimates $\nu(s)$. It can be seen by comparing the bounds with exact curve $\nu(s)$ obtained from solving the corresponding PDE for $\bar{F}(t)$ (Section 1). The upper and lower bounds for $\nu(s)$ are shown in Figure 4 as well as its numerical version.

Finally, notice that LE in Model 1 is given by the same expression (11) as in $1D$ case. Indeed, in this case each component of the separation process $\Delta\mathbf{r}=\mathbf{r}(t,\mathbf{a})-\mathbf{r}(t,\mathbf{0})$ is the separation process for one dimensional flow (6). Hence $|\Delta\mathbf{r}|\approx|\mathbf{a}\exp{(\lambda t)}$ for small $|\mathbf{a}|$ and large $t$, where $\lambda$ is LE for (6). Then our claim follows from definition (3).

In Model 2 LE cannot be found by such simple tools because the equations for $x$ and $y$ components are not split.

5. Isotropic forcing

Assume that in the original equations the forcing is isotropic, then its covariances are given by [2]

[TABLE]

where $r=\sqrt{x^{2}+y^{2}}$. Assume smooth longitudinal and normal correlation functions

[TABLE]

with $b_{0},\beta_{L},\beta_{N}>0$. Introduce $s_{L}=\sqrt{2\beta_{L}\tau^{3}},\ s_{N}=\sqrt{2\beta_{N}\tau^{3}}$

Then from (17) it can be derived that (15) takes form

[TABLE]

here $w_{j},\ \ j=1,2,3,4$ are independent standard Wiener processes.

No essential progress is made in analyzing this system yet because none of variables can be eliminated. The only reason for presenting it here is a great importance of the isotropic case for applications.

6. Conclusion and Discussion

While one dimensional stochastic models for inertial particles have been comprehensively studied, [1,8] there was a lack of $2D$ examples with a significant analytical advances. Here we suggested two such examples, where the mean time to explosion in the particle density , $\mu(s)$, can be analytically estimated for the full range of Stokes number $s$ and can be accurately evaluated by solving an initial/boundary problem for a one dimensional parabolic equation. The reported advance in the proposed models is due to reduction in the number of unknowns in the system (15) from four to two. Alas, for the most interesting isotropic case such a reduction is not possible, but a hope for finding asymptotics of $\nu(s)$ still remain. For both models the reciprocal $\nu(s)=1/\mu(s)$, interpreted as the intensity of caustics, increases monotonically from zero to infinity that is not a surprise at all.

Regarding to a Riemann type equation for the underlying velocity field (4), $\mu(s)$ can be viewed as a time scale on which a solution remains unique. It should be recognized that the interpretation of $\mu$ and $\nu$ in terms of the Eulerian velocity field is not clear enough except the deterministic case with the initial velocity field $\mathbf{u}_{0}(\mathbf{x})$ for which $|\partial\mathbf{u}_{0}\partial\mathbf{x}/|$ and $\nabla\cdot\mathbf{u}_{0}$ do not depend on $\mathbf{x}$ at all.

Appendix

A1. Proof of Proposition 2

The proof is based on the following statement

**Lemma **. If there exist real $m$ and smooth bounded function $V(x)$ such that

[TABLE]

and

[TABLE]

then with probability one

[TABLE]

**Proof **. From (A2) and Ito formula it follows that $V(p(t))$ satisfies

[TABLE]

By integrating both sides we get

[TABLE]

The second term on the right hand side goes to zero as $T\to\infty$ due to the large numbers law. The difference on the left hand side (LHS) of (A4) also converges to zero because of the boundness of $V$ and condition (A1). Lemma is proven.

Notice importance of (A1). If $\quad V(\infty)\neq V(-\infty)$ then the jumps of $V$ at explosion moments can accumulate leading to a non-zero limit of LHS.

Now we directly construct such a function $V(x)$ and determine $m$.

First, we take a bounded solution of (A2)

[TABLE]

for which $V(-\infty)=0$. To ensure (A1) one should set

[TABLE]

where

[TABLE]

The integral for $M$ is meant as Cauchy principal value. That allows for changing the order of integration after substitution

[TABLE]

Integration in $y^{\prime}$ leads to

[TABLE]

where

[TABLE]

To complete the proof one should account for, [5],

[TABLE]

This relation also can be derived from the fact that the both sides satisfy the same differential equation and same initial conditions, [9].

Finally, notice that the expression for $m$ after changing the order of integration in (A4) turns to the mean of $p(t)$ with respect to the invariant measure $\pi(p)$ given in (12)

Proposition 2 is proven. Proposition 3 can be proven by the same arguments.

A2. Details of the computational algorithm for solving (10)

Let $h$ and $\Delta t$ be space and time respectively, $X>0$ a big enough number, $x_{i}=-X+(i-1)h,\ \ n=(n-1)\Delta t$ the corresponding space/time grid, $i=1,...,M,\ \ n=1,...,N$ where $h=2X/M,\ \ \Delta t=t/N$. Set $b_{i}=\Delta t(x_{i}+x_{i}^{2}+s^{2}/h),\ \ a=\Delta ts^{2}/h$. Then a standard Eulerian scheme for (23) is written as

[TABLE]

Notice that (A5) is well posed because $u_{i}(1)=1,\ i=1,...,M,\ \ \ u_{1}(n)=0,\ \ y_{M}(n)=0,\ \ n=1,...,N$

The choice of parameters $X,h,\Delta t$ was dictated by standard conditions on ratio $\Delta t/h$ and constraints

[TABLE]

where $\mu(s)=E(S)$ is the exact value of the expectation of $S$ found from (9). In particular $\mu(1/\sqrt{2}$ =8.7735 while under our choice of $X=7.5,\Delta t=0.001,M=61$ it was obtained that

[TABLE]

A similar comparison for other $s$ can be seen from Figure 4

A3. Derivation of (15)

For the purpose of nondimensionalizing let us redenote the forcing on RHS of (14) by capital letters $W^{(i)}(t,x,y),\ \ i=1,2$. Then by differentiating (14) in $a$ and $b$ we get

[TABLE]

where

[TABLE]

and the covariance matrix of $\xi$’s is given by

[TABLE]

where $B^{ij}$ is the space covariance matrix of $\left(W^{(1)},W^{(2)}\right)$

Introduce

[TABLE]

Applying Ito formula we obtain

[TABLE]

To proceed to dimensionless variables assume

[TABLE]

where $w^{(i)}$ are dimensionless Wiener processes, thereby in dimensionless variables

[TABLE]

with

[TABLE]

Introduce a Stokes number similarly to (8)

[TABLE]

The result is

[TABLE]

[TABLE]

where

[TABLE]

We do not need a stochastic equation for $J_{5}$ since it can be found from the following equation

[TABLE]

which follows from

[TABLE]

We can reduce the number of freedom degrees by introducing $m_{k}=I_{k}/J,\quad k=1,...,5$ where $m_{5}$ is eliminated by means of (A6). The result is system (15)

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