Klyatskin V.I. Lectures on Dynamics of Stochastic Systems

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266 Lectures on Dynamics of Stochastic Systems

We note that the inﬁnitely thin rings move with an inﬁnite velocity. However, actual

vortex rings move with a ﬁnite velocity signiﬁcantly smaller than the sound velo-

city, but the dynamics of relative motion of rings only slightly differs from that we

just obtained. The fact that angular velocity (10.36) coincides with the corresponding

angular velocity in the case of vortex lines shows that the points lying on rings at equal

polar angles revolve relative the center point of the line connecting them at the rota-

tional speed coinciding with the rotational speed of vortex lines located at the same

distance and having the same intensity as the rings.

To study the structure of sound radiated by the system of rings, we need to know the

velocity ﬁeld for large distances from the system. We associate the coordinate system

with the point located at the center of the segment connecting the centers of the rings.

The velocity of ﬂow outside the rings is given by the formula

v(r, t) =

(t)

2π

dφ

× r

(t)

2π

dφ

× r

. (10.37)

Here, s

and s

are the unit vectors tangent to the vortex rings and r

and r

are the

vectors specifying observation point position relative to points lying on the rings. For

large distances from the rings, the oscillating parts of the velocity have the form (in

the cylindrical coordinates)

(1)

(t) =

3κ



− R



+ z

)

3/2



−a



, v

(1)

(t) = 0,

(1)

(t) =

3κ



− 3R



)

3/2



−a



(10.38)

Introducing potential by the formula v

(1)

= −∇ϕ

(1)

, we obtain

(1)

(t) = −

− 2z

+ z

)

5/2



− a



. (10.39)

Substituting Eq. (10.35) in Eq. (10.39) and changing to spherical coordinates, we can

rewrite Eq. (10.39) in the complex form

(1)

(t) = i

1 − 3 cos

2iωt

. (10.40)

Now, we will proceed similar to the case of two vortex lines. We encircle the origin

by a sphere of radius L such that a

 L  λ, where λ is the wavelength of radiated

sound waves. We will use the fact that the motion of ﬂow inside the sphere approx-

imately coincides with the motion of noncompressible ﬂow. Outside the sphere, the

equations of motion will have the form of Eq. (10.21) with the only difference that

now 1 is the spatial Laplace operator.

Some Other Approximate Approaches to the Problems of Statistical Hydrodynamics 267

Represent potential ϕ in the form

ϕ(r, θ) = f (r, θ)e

2iωt

Substituting this expression in Eq. (10.21) and taking into account that we must obtain

divergent spherical waves for r  λ , we obtain f (r, θ) in the form

f (r, θ) =

∞

n = 0

√

(2)

n+1/2



2ωr



(

cos θ

)

where H

(2)

n+1/2

(z) is the Hankel function of the second kind and P

(z) is the Legendre

polynomial. Comparing potential ϕ(r, θ) for r  λ with ϕ

(1)

(r, θ ), we obtain that

n = 2 and

√

κγ

5/2

Consequently, potential and pressure in the wave zone have the forms

ϕ(r, θ) =

2κ







1 − 3 cos



exp





ωt−

ωr

−

3π



(r, θ ) =

4iκ







1 − 3 cos



exp





ωt−

ωr

−

3π



so that the angular distribution of energy radiated per unit time is

I(θ ) =

8π







1 − 3 cos



Vortex rings radiate energy as a quadrupole; the main portion of energy is radiated in

a cone about z-axis with corner angle 106

◦

in both positive and negative z-directions.

Integrating over θ, we obtain that total energy radiated per unit time is given by the

expression

I =

64π





Thus, intensity of sound radiated by a pair of vortex rings is proportional to M

. In

the case of statistically distributed system of pairs of vortex rings, this proportionality

will remain valid for certain portion of space, which agrees with estimates obtained

in [58–60].

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Part III

Examples of Coherent Phenomena in

Stochastic Dynamic Systems

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Lecture 11

Passive Tracer Clustering and Diffusion

in Random Hydrodynamic and

Magnetohydrodynamic Flows

11.1 General Remarks

Diffusion of such passive ﬁelds as scalar density (particle concentration) ﬁeld and

magnetic ﬁeld is an important problem of the theory of turbulence in magnetohydro-

dynamics. The basic stochastic equations for the scalar density ﬁeld ρ(r, t) and the

magnetic ﬁeld H(r, t) are the continuity equation



∂

∂t

∂

∂r

U(r, t)



ρ(r, t) = µ1ρ(r, t), ρ(r, 0) = ρ

(r), (11.1)

and the induction equation (1.59), page 30, subject to the dissipative effects



∂

∂t

∂

∂r

U(r, t)



H(r, t) =



H(r, t) ·

∂

∂r



U(r, t) + µ

1H(r, t), (11.2)

where µ

is the dynamic molecular diffusion coefﬁcient, and µ

= c

/4πσ is the

dynamic diffusion coefﬁcient associated with the conductivity of the ﬂow σ .

In these equations, U(r, t) = u

(r, t) + u(r, t); u

(r, t) is the deterministic com-

ponent of the velocity ﬁeld (mean ﬂow), and u(r, t) is the random component. In

the general case, random ﬁeld u(r, t) can be composed of both solenoidal (for which

div u(r, t) = 0) and potential (for which div u(r, t) 6= 0) components. Field H(r, t) is

here nondivergent, i.e., div H(r, t) = 0.

Dynamic systems (11.1) and (11.2) are conservative and both the total scalar mass

M = M(t) =

drρ(r, t) =

drρ

(r) = const

Lectures on Dynamics of Stochastic Systems. DOI: 10.1016/B978-0-12-384966-3.00011-8

272 Lectures on Dynamics of Stochastic Systems

and the magnetic ﬂux

dr H(r, t) =

dr H

(r) = const

remain constant during the evolution. For homogeneous initial conditions, the fol-

lowing equalities are a corollary of the conservatism of dynamic systems (11.1) and

(11.2):

ρ(r, t)

= ρ

H(r, t)

= H

where

···

denotes averaging over an ensemble of realizations of random ﬁeld

{u(r, t)}. If initial conditions are homogeneous, a statistical average of a quantity

depending, for example, on magnetic ﬁeld

f (H(r, t))

changes over in the gen-

eral case to

f (H(r, t))

⇔

f (H(r, t))

which means that quantity

f (H(r, t))

is a speciﬁc quantity per unit volume and is,

hence, an integral quantity. For example, quantity



(r, t)



is the average speciﬁc

energy per unit volume in the case of homogeneous initial conditions, but it is the

average density of energy in the case of inhomogeneous initial conditions. Integral

quantities characterize dynamic systems at large, they allow separating the processes

of ﬁeld generation, which offers a possibility of discarding dynamic details related to

advection of these quantities by random velocity ﬁeld [64].

Our interest consists in the temporal evolution of different one-point statistical

characteristics of density ﬁeld ρ(r, t), its gradient p(r, t) = ∇ρ(r, t), magnetic ﬁeld

H(r, t), and its different spatial derivatives from given initial distributions ρ

(r) and

(r) (and, in particular, homogeneous ones, ρ

(r) = ρ

and H

(r) = H

). Such

characteristics are probability densities of these ﬁelds by themselves, isotropization

(decrease of anisotropy) of vector correlations, helicity of these vector ﬁelds, and their

dissipation.

The effect of dynamic diffusion can be neglected during the initial stages of the

development of diffusion. In this case, Eqs. (11.1) and (11.2) become simpler and

assume the form



∂

∂t

+ U(r, t)

∂

∂r



ρ(r, t) +

∂U (r, t)

∂r

ρ(r, t) = 0, (11.3)



∂

∂t

+ U(r, t)

∂

∂r



H(r, t) +

∂U (r, t)

∂r

H(r, t) =



H(r, t) ·

∂

∂r



U(r, t). (11.4)

In these conditions, such nonstationary and stochastic phenomena occur as mixing,

generation and rapid increase (in time) of smaller-scale disturbances (e.g., stochas-

tic (turbulent) dynamo), and, in certain situations, clustering in the phase or physical

space. Clustering of a ﬁeld (such as tracer density, magnetic ﬁeld energy, etc.) is the

Passive Tracer Clustering and Diffusion 273

appearance of compact regions, where the content of this ﬁeld is increased, which are

surrounded by regions with decreased content of this ﬁeld.

For example, in the case of homogeneous initial conditions, the density ﬁeld

remains constant in time if liquid is noncompressible (nondivergent velocity ﬁeld)

and, consequently, the gradient of density is identically equal to zero at any spatial

point. On the contrary, in compressible ﬂows (divergent velocity ﬁeld), density ﬁeld

is always clustered, which naturally results in great gradients of the density ﬁeld. We

have seen a similar pattern in the example of the simplest problem whose solution is

given in Fig. 1.12, page 27. The characteristic time of the formation of cluster struc-

ture of the density ﬁeld is determined by the relationship D

t ∼ 1, where quantity D

given by Eq. (4.59), page 108, depends on the potential component of spectral density

of the velocity ﬁeld.

As for the magnetic ﬁeld, clustering of its energy can occur under certain condi-

tions, and we have seen an example of such clustering in Fig. 1.14, page 33, in the case

of the simplest potential velocity ﬁeld. However, general generation of magnetic ﬁeld

occurs even in noncompressible ﬂows of liquid; this generation causes intense vari-

ability of the ﬁeld structure, and different moment functions of both magnetic ﬁeld by

itself and its spatial derivatives increase exponentially in time. As a result, ﬁeld dissi-

pation related to higher-order derivatives is rapidly increased at a certain time instant,

and the effects of dynamic diffusion become governing.

The above equations correspond to the Eulerian description of the evolution of the

density ﬁeld and magnetic ﬁeld.

Equations (11.3), (11.4) are the ﬁrst-order partial differential equations and can be

solved by the method of characteristics. Introducing characteristic curves r(t) satisfy-

ing the equations of particle motion

r(t) = U(r, t), r(0) = r

, (11.5)

we can change over from Eq. (11.3) to the ordinary differential equation

ρ(t) = −

∂U (r, t)

∂r

ρ(t), ρ(0) = ρ

). (11.6)

Solutions to Eqs. (11.5) and (11.6) have an obvious geometric interpretation. They

describe the concentration behavior around a ﬁxed tracer particle moving along tra-

jectory r = r(t). As may be seen from Eq. (11.6), the concentration in divergent ﬂows

varies: it increases in regions where medium is dense and decreases in regions where

medium is rareﬁed.

Solutions to system (11.5), (11.6) depend on characteristic parameter r

(the initial

coordinate of the particle)

r(t) = r(t|r

), ρ(t) = ρ(t|r

), (11.7)

which we will separate by the bar. Components of vector r

are called the Lagrangian

coordinates of a particle; they unambiguously specify the position of arbitrary parti-

cle. Equations (11.5), (11.6) correspond in this case to the Lagrangian description of

274 Lectures on Dynamics of Stochastic Systems

concentration evolution. The ﬁrst of equalities (11.7) specify the relationship between

the Eulerian and Lagrangian descriptions. Solving it in r

, we obtain the relationship

that expresses the Lagrangian coordinates in terms of the Eulerian ones

= r

(r, t). (11.8)

Then, using Eq. (11.8), to eliminate r

in the last equality in (11.7), we turn back to

the concentration in the Eulerian description

ρ(r, t) = ρ(t|r

(r, t)) =

ρ(t|r

)j(t|r

)δ

(

r(t|r

) − r

)

, (11.9)

where we introduced a new function called divergence

j(t|r

) = det ||j

(t|r

)|| = det



∂r

(t|r

)

∂r



which is the quantitative measure of the degree of compression (extension) of physi-

cally inﬁnitely small liquid particles. One can easily obtain that it satisﬁes the equation

j(t|r

) =

∂U (r, t)

∂r

j(t|r

), j(0|r

) = 1. (11.10)

Comparing Eq. (11.6) with Eq. (11.10), we see that

ρ(t|r

) =

)

j(t|r

)

. (11.11)

Thus, we can rewrite Eq. (11.9) as the equality

ρ(r, t) =

)δ

(

r(t|r

) − r

)

(11.12)

specifying the relationship between the Lagrangian and Eulerian characteristics. Delta

function in the right-hand side of Eq. (11.12) is the indicator function for the posi-

tion of the Lagrangian particle; as a consequence, after averaging Eq. (11.12) over

an ensemble of realizations of random velocity ﬁeld, we arrive at the well-known

relationship between the average concentration in the Eulerian description and the

one-time probability density

P(r, t|r

) =

(

r(t|r

) − r

)

of the Lagrangian particle

ρ(r, t)

(

r, t|r

)

Passive Tracer Clustering and Diffusion 275

The relationship between the spatial correlation function of the concentration ﬁeld

in the Eulerian description

R(r

, r

, t)

ρ(r

, t)ρ(r

, t)

where R(r

, r

, t) = ρ(r

, t)ρ(r

, t), and the joint probability density function of posi-

tions of two particles

P(r

, r

, t|r

, r

) =

(

(t|r

) − r

)

(

(t|r

) − r

)

can be obtained similarly

R(r

, r

, t)

)ρ

)P(r

, r

, t|r

, r

For the nondivergent velocity ﬁeld (div U(r, t) = 0), both particle divergence and

particle concentration are invariant, i.e.,

j(t|r

) = 1, ρ(t|r

) = ρ

We assume in the general case that the random component of the velocity ﬁeld

is the divergent (div u(r, t) 6= 0) statistically homogeneous Gaussian random ﬁeld,

which is spherically symmetric (but generally not possessing reﬂection symmetry) in

space and stationary in time with zero-valued average (

u(r, t)

= 0) and correlation

and spectral tensors given by the formulas

(r − r

, τ ) =



(r, t)u

, t

)



dk E

(k, τ )e

ik(r−r

)

(k, τ ) =

(

2π

)

dr B

(r, τ )e

−ikr

where d is the dimension of space. In view of the suggested symmetry condition,

correlation tensor B

(r, τ ) has vector structure (4.49), page 105, (r − r

→ r)

(r, τ ) = B

iso

(r, τ ) + C(r, τ )ε

ijk

where

iso

(r, τ ) = A(r, τ )r

+ B(r, τ )δ

is the isotropic portion of the correlation tensor and ε

ijk

is the pseudotensor antisym-

metric with respect to an arbitrary pair of indices. The isotropic portion of the corre-

lation tensor corresponds to the spatial spectral tensor of the form

(k, τ ) = E

(k, τ ) + E

(k, τ ),