COHERENT PHENOMENA IN STOCHASTIC DYNAMICAL SYSTEMS

COHERENT PHENOMENA IN STOCHASTIC DYNAMICAL SYSTEMS V. I. KLYATSKIN AND D. GURARIE Abstract. The paper develops the basic ideas of statistical topography of random processes and ¯elds to analyze coherent phenomena in simple dynamical systems. Such phenomena take place with probability one, and provide links between individual realizations and statistical characteristics of system at large. We con¯ne ourselves to several examples: the transfer phenomena in singular dynamic systems under the action of random forces; dynamic localization of plane waves in randomly strati¯ed media; clustering of randomly advected passive tracers; formation of caustic structures for wave ¯elds in randomly inhomogeneous media. All these phenomena are studied based on the analysis of one-point (space-time) probability distribution functions.

Contents 1. 2.

Introduction Examples of dynamical systems, problem formulation, and special features of solutions 2.1. Particles in random velocities and forces. 2.2. Plane waves in randomly layered media 2.3. Passive tracer advection by random velocities 2.4. Waves in random multi-D media 2.5. Equations of geophysical °uid dynamics 2.6. Solution dependence on initial parameters and coe±cients of equations 3. Indicator function and Liouville equation 3.1. Ordinary di®erential equations 3.2. First order partial di®erential equations 3.3. Statistical topography of random ¯elds 4. Statistical analysis of dynamical systems 4.1. Forward and backward Fokker-Planck equations 4.2. Typical realizations of random processes 4.3. Log-normal processes 5. Transfer phenomena in dynamical systems with blow-up singularities 5.1. Simple ODE case 5.2. Caustics in random media 5.3. Re°ection phase for plane waves in layered media 6. Localization of plane waves in randomly strati¯ed media 6.1. Single layer medium.

2 4 4 6 12 13 16 17 19 19 19 21 23 23 26 27 28 28 30 33 34 34

Date: (To appear in \Uspekhi Fizicheskikh Nauk", 1999). The research was sponsored by the Russian Foundation for Basic Research (projects 98-0564479, 96-15-98527, 96-05-65347) and the CRDF grant #RMI-272. The second author enjoyed the hospitality of NCAR, Boulder, CO, while completing the work. 1

2

V. I. K LYATSKIN AND D. GURARIE

6.2. Strati¯ed two-layer model 6.3. Localization in parabolic wave-guides 7. Passive tracers in random velocity ¯elds 7.1. Lagrangian description 7.2. Eulerian description 7.3. Additional factors 8. Caustical structure of wave-¯elds in random inhomogeneous media. 8.1. Region of weak °uctuations 8.2. Strong intensity °uctuations 9. Conclusion References

37 39 40 42 45 49 56 58 60 61 62

1. Introduction Many physical processes take place in the complex media, whose parameters could be viewed as space-time realizations of chaotic (or stochastic) ¯elds. Such dynamic problems are too complex to allow an explicit mathematical solution for speci¯c realizations of the media. However, one is often interested in generic features of random solutions, rather than particular details. So one is naturally inclined to adopt the well developed machinery of random ¯elds and processes, that is to replace individual realizations with statistical (ensemble) averages. Nowadays, such approach is commonly used in many problems of the atmospheric and oceanic physics. Randomness of the medium gives rise to stochastic physical (solution) ¯elds. Thus a typical realization of, say 2D scalar (density) ¯elds ½ (R; t) with R = (x; y) would resemble a complex mountain terrain with randomly distributed peaks, valleys, passes, - etc. But ¡ the¢® standard statistical tools, like means h½ (R; t)i, and moments ½ (R; t) ½ R0 ; t0 , where h:::i - indicates ensemble averaging over random parameters, would often smooth out some important qualitative features of individual realizations. So the resulting \mean ¯elds" would bear little likeness to a typical (individual ) realization, and sometimes give con°icting predictions. For instance, ensemble average of randomly advected passive tracer would often yield a di®usive process, that smooths out the mean-(solution) ¯eld, whereas individual realizations would tend to evolve very rugged and fragmented shape. Thus, standard statistical means could reasonably predict some \global" spatialtemporal scales and parameters of solutions, but tell little about the ¯ne (small scale) structure and details of evolution. Such details (for advected tracer) could strongly depend on some special properties of random velocities, for instance, compressibility vs. incompressibility. In the former case the turbulent transport would typically bring about the formation of clusters - regions of high tracer concentration surrounded by low density \voids" [1, 2]. However, all statistical moments of the particle's position (or separation-function) show exponential increase in time, which implies \statistical dispersion" of particles [2 - 4]. Similarly, optical rays in random media diverge exponentially in the \mean" [3 - 5], yet the caustics are formed at ¯nite distances within the layer with probability one [6 - 9].

COHERENT PHENOMENA

3

Another example of the kind is the dynamic localization of plane waves by randomly layered media. When a plane wave is incident upon randomly layered halfspace, its intensity decreases exponentially in the distance from the boundary, for almost all realizations. Yet all statistical moments show an exponential increase [10 - 13]. We shall call physical phenomena that occur with probability one and characterize \typical realizations" coherent . Such statistical coherence could be viewed as a way to \organize" dynamic complexity, and identify its \statistically stable" properties, by analogy with the usual notion of coherence, as self-organization of complex, multi-component systems, arising from their chaotic interaction (cf. [14]). Although our notion of coherence di®ers from the standard usage, we ¯nd it natural. In general there is no simple way to assertain that a given phenomenon occurs with probability 1. However, it becomes possible to do it theoretically for certain problems with simple models of °uctuating parameters. In other cases one could do it by numeric modeling, or analyzing physical experiments. Of course, our notion of coherence makes it more a mathematical problem, rather than the physical one. Let us also remark that in many cases we don't have full understanding of the physical causes that lead to coherence. For instance, the above clustering of Lagrangian particles advected by random potential velocities represents by itself a purely kinematic phenomenon in the absence of any real particle interaction. The complete statistics (e.g. all n-point moments) would allow in principle a complete description of the dynamical system. But in practice one could handle only a few simple statistics, typically expressed through the one-point probability distributions (PDF) in space-time variables. The natural problem then is to deduce some important qualitative and quantitative characteristics of individual realizations from such limited data. It takes on a particular signi¯cance for the atmospheric and oceanic problems, where \statistical ensembles" do no exist in the strict sense, or require long temporal exposure (time series) and experimentalists deal most often with individual realizations. A possible answer to the problem is suggested by the methods of statistical topography. The name statistical topography was ¯rst coined in book [15], though the main ideas go back to papers [16 - 19] (see also survey [20], for a complete bibliography). The early works applied statistical topography of random ¯elds to the statistical analysis of the rough sea surface, and the radar and TV images. Applications of statistical topography to the turbulent transport problem came later [2, 21 - 23], while [24] adopted these ideas for the wave propagation in random media. The methods of statistical topography call into question the basic \philosophy" of statistical analysis of stochastic dynamical systems. We believe such approach could be useful for experimentalists who apply statistical tools to large experimental data. In the present paper we shall exploit some basic ideas of statistical topography of random ¯elds and processes to analyze coherent phenomena in a few model examples, chosen from a great variety of such systems, namely

² transfer phenomena in (nonlinear) dynamical systems with \singular solutions" and random forcing. Similar transfer phenomena for a general class of

4


randomly forced dynamical systems with ¯nite/ discrete set of stable/ unstable equilibria, is well known and discussed in many textbooks ² dynamic localization of plane waves in randomly strati¯ed (1-D) media ² clustering of passive tracers by random (turbulent) velocity ¯elds ² formation of caustic for wave propagation in multi-D random media All these phenomena could be approached through a uni¯ed method based on the analysis of the one-point probability distribution functions (PDF), that result from their dynamic evolution. We shall start with the dynamic description of model systems, and discuss speci¯cs of their solutions in the presence of random parameters. The statistical analysis will follow. Though our examples are drawn mostly from the statistical hydrodynamics, radio-physics and acoustics, - the areas of interest to the authors, we believe the methods developed in the paper could ¯nd applications in other areas of physics. 2. Examples of dynamical systems, problem formulation, and special features of solutions 2.1. Particles in random velocities and forces. A particle moving in a (random) velocity ¯eld is described by an ordinary di®erential system (2.1)

d dt r(t)

= U (r; t); r(t0 ) = r 0 :

Here U (r; t) = u0 (r; t) + u(r; t) is made of the deterministic (mean) component u0 (r; t), and the random perturbation u(r; t). In the absence of randomness (u (r; t) = 0), and constant u0 we get a simple rectilinear motion r(t) = r0 + u0 (t ¡ t0 ): The same system (2.1) could also describe the particle motion under the action of random forces. Indeed, a simple linear friction law yields d dt r(t) d v(t) dt

(2.2)

=

v(t);

=

¡¸v(t) + f (r; t);

r(0) =

r0 ; v(0) = v 0 :

Once again in the absence of friction and forcing we get a simple rectilinear motion v(t) = v 0 ; r(t) = r0 + v 0 t: Let us discuss some qualitative features of stochastic system (2.1) in the absence of the mean °ow. Formally equation (2.1) describes the motion of independent particles, as no interaction takes place. If however, ¯eld u (r; t) has ¯nite correlation radius lcor , then particles within lcor - proximity of each other lie in the common domain of in°uence of velocity u (r; t). Hence, they could exhibit a collective behavior. In general, velocity ¯eld u (r; t) is made of the solenoidal (div u = 0), plus potential (u = rÁ (r; t)) components. Numeric simulations [22, 25] of multiparticle systems, driven by (2.1) show marked di®erence between the two cases. Fig. 1a shows a divergent-free random ¯eld u (r; t) advecting a uniformly distributed set of particles over the disk. Here the total area enclosed by the deformed contours is conserved, and the particles ¯ll the area in a (approximately) \uniform" manner.

COHERENT PHENOMENA

5

Observe, however, that contours become increasingly more rugged and \fractallike". In the presence of potential component (ru (r; t) 6= 0), the initial uniform distribution of particles (over the square) evolves into clusters - compact regions of high concentration amidst low-density voids. The results of numeric simulations are shown in ¯g. 1b. Let us stress here, the kinematic nature of this e®ect. Indeed, the ensemble averaging over velocity realizations could completely obliterate it. Let us also note, that numeric simulations of ¯g. 1 were conducted for stationary (time-independent) ¯elds u(r). Such clustering of particle systems was ¯rst observed in papers [26, 27], via computer simulation of a simple model of atmospheric dynamic, based on the so called EOLE experiment. This global experiment was conducted in Argentina in 1970-71, and involved launching 500 air balloons of constant density, that spread over the entire Southern hemisphere at the altitude roughly 12 km.. Fig. 2 shows numeric simulation of the distribution of balloons 105 days after the beginning [27], and clearly exhibits their convergence into clusterized groups. We shall came back to \turbulent dispersion" later, but now let us turn attention to another stochastic feature of randomly stirred dynamic systems (2.1), the so called transfer phenomena. The well known examples of transfer phenomena involve systems with ¯nitely many stable equilibria (see, for instance [3, 28]). Here we shall con¯ne ourselves to a simple case, that arises in the statistical theory of wave propagation (below), and exhibits singular in time solutions (2.3)

dx dt

= ¡¸x 2 (t) + f (t); x(0) = x0 ; ¸ > 0:

In the absence of forcing (f (t) = 0) it has an exact solution of the form x(t) =

1 1 ; t0 = ¡ : ¸ (t ¡ t0 ) ¸x0

If initial point x 0 > 0, hence t0 < 0, solution x(t) converges monotonically to 0, as t ! 1. If x 0 < 0, then solution x(t) approaches ¡1, i.e. \blows up" in ¯nite time t0 = ¡1=x0 . In the latter case random forcing would play no signi¯cant role. It becomes signi¯cant only for positive x 0, near \semistable equilibrium" x = 0. Indeed, as x (t) grows su±ciently small, random force f (t) could \transfer" it (\kick over") into the negative half-line (an attraction basin of f¡1g), where it would be dragged to ¡1 in ¯nite time, by dominating nonlinearities. Thus stochastic forcing could turn any initial state x 0 (not only [¡1; 0]) into an unstable (\blow up") solution, that reaches ¡1 in ¯nite time t0 . Fig. 3 gives a schematic view of a particular solution realization x (t), for various times: t < t0 , t > t0 , that exhibits a \quasi-periodic" pattern. The principal coherent phenomenon here is the blow up of realizations of the process, and the pertinent statistics. The above example involves additive random noise f . The simplest case of a multiplicative noise is given by the stochastic parametric resonance model - a second order di®erential equation d2x dt2

(2.4)

+ ! 20 [1 + z(t)]x(t) = 0;

x(0) = x 0 ; dx (0) = v 0 dt

with random potential (process) z (t). It appears in many areas of physics. From the physical standpoint dynamic system (2.4) is subjected to a parametric excitation, since process z (t) could contain harmonics of all frequencies, including

6


f2! 0 =n : n = 1; 2; :::g. Those modes could produce a parametric resonance, as for the well known periodic (Mathieu) case z (t) (see, for instance [3, 5, 10]). 2.2. Plane waves in randomly layered media. In the previous section we considered two examples of the initial value problems described by ordinary di®erential equations. Next we shall review a simple boundary value problem, namely the 1D stationary wave problem. Let us consider a inhomogeneous layered medium occupying strip L0 < x < L. A plane wave of unit amplitude u 0 (x) = e¡ ik¢(x ¡L) is incident upon it from the right half-space x > L (¯g. 4a). The wave ¯eld in the strip obeys the Helmholtz equation (2.5)

d2 u(x) dx2

+ k2 [1 + "(x)]u(x) = 0

with function " (x) representing inhomogeneities of the media. We assume " = 0 outside the strip, and " (x) = " 1 (x) + i° within, the real part "1 (x) responsible for the wave scattering, while imaginary one ° describing wave attenuation by the media. In the right half space x > L the wave ¯eld is made of the incident and re°ected components u(x) = e¡ik(x¡ L) + RL eik(x¡L) where RL is the (complex) re°ection coe±cient. In the left half x < L0 we have u(x) = T L eik(L0¡ x) with the (complex) transmission coe±cient T L . The boundary d conditions for (2.5) are continuity relations for u and its derivative dx at x = L; L0 ¯ i du + u¯ x=L = 2; k dx ¯ i du (2.6) ¡ u¯ = 0: k dx

x=L 0

So the wave ¯eld in the inhomogeneous medium is determined by the boundary value problem (2.5) -(2.6), as opposed to the initial value problem (2.4) with the same di®erential equation. If parameter "1 is random, one is interested in the statistics of the re°ection and transmission coe±cients: RL = u(L) ¡ 1, and T L = u(L0 ), that depend on the boundary values of the wave-¯eld (2.5) -(2.6), as well as the ¯eld intensity I(x) = ju(x)j2 within the layer (statistical radiative transport ). Equation (2.5) implies the energy conservation (dissipation) law at x < L kI(x) =

d dx S(x);

where S (x) denotes the energy-density °ux £ ¤ 1 d ¤ d S(x) = 2ik u(x) dx u (x) ¡ u¤ (x) dx u(x) : Furthermore, one has

S(L) = 1 ¡ jRL j2 ; S(L0 ) = jT L j2 : If the medium does not dissipate waves (° = 0), then the energy-conservation yields (2.7)

jRL j2 + jT L j2 = 1:

COHERENT PHENOMENA

7

Let us turn to some special features of the stochastic boundary value (2.5) (2.6). In the absence of medium °uctuations, "1 (x) = 0, and su±ciently small attenuation, the ¯eld intensity decays exponentially inside the layer as I(x) = ju(x)j2 = e¡ k°(L¡x) :

(2.8)

Fig. 5 shows numeric simulations of two wave intensities in a su±ciently thick layer, that come from two di®erent realizations of the medium [29]. Skipping further details and parameters of the problem, let us only note a clearly perceived exponential fall-o® trend accompanied by large intensity °uctuations, directed both ways (to zero and to in¯nity). They result from the multiple scattering processes in randomly inhomogeneous media, and demonstrate the so called dynamic localization. Similarly a point-source located inside the strip is described by the boundary value Green's function of the Helmholtz equation

(2.9)

d2 G(x; x 0 ) + k 2 [1 + "(x)]G(x; x 0 ) = 2ik(x dx2 ¯ i d ¯ k dx G(x; x 0 ) + G(x; x0 ) x=L = 0; ¯ i d G(x; x 0 ) ¡ G(x; x 0 )¯x=L = 0: k dx 0

¡ x 0 );

Here the exterior wave ¯eld (outside the layer) consists of outgoing waves (¯g. 4b) with the transmission coe±cients T 1 = G(L; x 0 ); T 2 = G(L0 ; x 0 ): Moreover, wave ¯eld in the left half-space x < x0 is proportional to the plane wave incident from the half-space x > x0 upon the layer (L 0; x 0 ) [10], i.e. G(x; x0 ) » u(x; x 0 ); x · x 0 : Let us also remark that the scattering problem (2.5)-(2.6) corresponds to putting the source (2.9) on the boundary x 0 = L, i.e. taking u(x) = G(x; L). Boundary value problems (2.5), (2.6) and (2.9) could be solved by the imbedding method of [30 - 32], that reformulates them as initial value problems in parameter L- the right boundary end of the strip [11]. Thus the re°ection coe±cient RL of (2.5), (2.6) obeys the Riccati equation in L, (2.10)

d R dL L

= 2ikRL +

ik "(L) (1 2

2

+ RL ) ; RL0 = 0;

whereas ¯eld u(x) = u(x; L) inside the layer obeys the linear equation d u(x; L) dL

(2.11)

= iku(x; L) +

u(x; x)

ik "(L) (1 + 2

R L) u(x; L);

= 1 + Rx :

Indeed, writing R in the phase-amplitude form R = AeiÁ , we can recast Riccati equation (2.10) as a coupled system 8 d £ ¡ ¢ ¤ < dL Á = k (2 + ") A + "2 1 + A2 cos Á :

d dL A

¡ ¢ = k "2 1 ¡ A2 sin Á

Hence follows the equation for the squared modulus of the re°ection coe±cient WL = jRL j2 (2.12)

d WL dL

=

¡2k°WL ¡

W L0

=

0:

ik " 1 (L) (RL ¡ R¤L ) (1 ¡ W L ) ; 2

8


If boundary L 0 is completely re°ective, the initial condition becomes W L0 = 1. So in the absence of damping (° = 0) the incident wave is fully re°ected, W L = 1. Hence the re°ection coe±cient R L = expfiÁL g, and equation (2.10) would imply the following evolution of the \re°ection phase" d Á dL L

(2.13)

= 2k + k"1 (L) (1 + cos ÁL )

valid through the entire range (¡1; 1) of variable ÁL . On the other hand equation (2.11) for the wave-¯eld u involves only trigonometric functions of ÁL . In order to make a transition from (¡1; 1) to the natural range [¡¼; ¼ ] of ÁL , we introduce another function [13, 33] µ ¶ ÁL (2.14) zL = tan 2 that obeys a nonlinear dynamic evolution of type (2.3) ¡ ¢ d (2.15) z = k 1 + z2L + k" 1(L) dL L

with singular-type (exploding) solutions. When the left boundary L0 is moved to ¡1, and the media is non dissipative (° = 0), there exists a \stationary" solution WL = 1, independent of L, which corresponds to the complete re°ection of incident waves. Such solution will be shown to appear in the stochastic problem with probability one [10]. Researchers often deal with multidimensional situations, when certain wavetypes generate other types due to spatial inhomogeneity of the medium. In some cases such media could be approximately divided into a discrete set of distinct strata, with continuously changing parameters within each stratum. As an example we mention large scale/ low frequency oscillations in the atmosphere and ocean, known as Rossby waves. These modes are derived in the context of the quasigeostrophic model, where the atmosphere (or ocean) is viewed as an aggregation of thin multi-layer ¯lms, strati¯ed by density variations and thicknesses [34]. The role of \localizing media" for the Rossby waves is played by the (inhomogeneous) bottom topography. The simplest one-layer model (for the so called barotropic modes ) could be reduced to the Helmholtz equation, while the two-layer system could account for the baroclinic e®ects [35 - 37]. We shall consider a simple two-layer wave-model given by system [38] (2.16)

d2 dx2 Ã 1 d2 Ã dx2 2

+ k2 Ã 1 ¡ ® 1 F (Ã 1 ¡ Ã 2 ) = 0 ; + k2 [1 + "(x)] Ã 2 + ® 2 F (Ã 1 ¡ Ã 2 ) = 0

In the context of Rossby waves it appears after the reduction of the full 2D wavesystem to a single latitudinal (Northward) variable x, assuming an idealized topographic pro¯le z = H + h (x), i.e. mountain ridges extending East-to-West, and considering Fourier-modes of a ¯xed longitudinal wave-number: ei· y Ã (x) : Here parameters ® 1 = 1=H1 ; ® 2 = 1=H2 designate recipro cals of the layers' 2 depth, while coupling constant F = fg0 ±½ ½ involves local Coriolis value f 0 = 2- sin µ 0 at latitude µ ( - = Earth rotation frequency) and the density variation between ½ +½ two layers ±½ = ½2 ¡ ½1 over the mean density ½0 = 1 2 2 . Function " (x) depends on (random) topographic bottom pro¯le h (x), Coriolis f 0 . and latitudinal and longitudinal wave numbers (see [34] for details). As before we assume random function " (x) to vanish outside the ¯nite interval [L0 ; L]. The schematic geometry of problem (2.16) is shown in ¯g. 6. The boundary

COHERENT PHENOMENA

9

conditions include radiation condition at 1, and the continuity of wave ¯elds along with their derivatives at boundary points L0 and L. Parameter F encodes the vertical strati¯cation, and gives the horizontal length scale for the \cross-mode"generation. The speci¯c form of equations (2.16), like coe±cients ® i measuring layers' thicknesses etc., arise in the geophysical °uid setup [35 - 37]. The basic equations and parameters could change, depending on the physical system in question. But one essential feature should remain, namely the linear form of wave interactions. System (2.16) could be formally reduced to a single-layer case by setting F = 0; Ã 1 = 0, and the corresponding wave equation takes on the \Helmholtz form" (2.5). Another way to reduce the system to a \single-layer" is via limit H1 ! 0, hence Ã 1 = Ã 2 . Let us remark however, that the order of two limiting pro cedures, L0 ! ¡1 (half-space) and Hi ! 0 (single layer), cannot be interchanged in the statistical problem [35]. The layers' relative thickness could be made arbitrarily small, but should remain nonzero. Let us consider the equations for the Green's function of (2.16) (2.17)

d2 Ã dx22 1 d dx2 Ã 2

+ k2 Ã 1 ¡ ® 1 F (Ã 1 ¡ Ã 2 ) = ¡v 1 ±(x ¡ x 0 ); + k2 [1 + "(x)] Ã 2 + ® 2 F (Ã 1 ¡ Ã 2 ) = ¡v2 ±(x ¡ x0 )

with sources in either the upper or the lower layer respectively. Introducing vector ¯eld Ã(x; x 0 ) = fÃ 1(x; x 0 ); Ã 2 (x; x0 )g and vector v = fv1 ; v 2 g, we can recast (2.17) in the vector form h 2 i d 2 2 (2.18) + A + k "(x)¡ Ã(x; x0 ) = ¡v±(x ¡ x0 ); dx2

that resembles the scalar Helmholtz equation (2.5). Here matrix µ 2 ¶ k ¡ ®1F ®1F 2 A = ®2F k 2 ¡ ®2 F plays the role of a \uniform background" (constant refraction index), while "¡ µ ¶ 0 0 ¡= 0 1 represents medium inhomogeneities. The square-root matrix µ ¶ ® ~2 + ¸® ~ 1 (1 ¡ ¸)~ ®1 (2.19) A=k ; (1 ¡ ¸)~ ®2 ® ~ 1 + ¸® ~2 has entries depending on relative thicknesses of the two layers ® ~ 1 = ®1 = (® 1 + ®2 ) = H2 =H0 ; ® ~ 2 = ®2 = (® 1 + ®2 ) = H1 =H0 ; ® ~1 + ® ~ 2 = 1: © 2 ª and the eigenvalues ¸ ; 1 of the \refractive index matrix" · ¸ F 2 ¸ = 1 ¡ (® 1 + ®2 ) 2 k is assumed to be positive, ¸2 > 0.

1 A2 , k2

where

10


Next we consider the fundamental matrix-solution of (2.18) h 2 i d (2.20) + A2 + k2 "(x)¡ ª(x; x 0 ) = ¡E ±(x ¡ x 0); dx2 that gives all other \vector-solutions" Ã as (2.21)

Ã(x; x0 ) = ª(x; x 0 )v:

The columns fÃ 11 ; Ã 21 g ; fÃ 12 ; Ã 22 g of matrix ª describe waves generated by the point sources fv1 ; 0g; f0; v 2 g, in the upper and lower layers respectively. Fundamental matrix ª satis¯es boundary conditions ¡d ¢ ¯ ¡d ¢ ¯ (2.22) ¡ iA ª(x; x0 )¯ x=L = 0; dx + iA ª(x; x 0 )¯x= L = 0: dx 0

Following [38] we shall place the wave source on the boundary x0 = L. The corresponding boundary value problem, with the \jump-condition" at the source gives h 2 i d 2 2 + A + k "(x)¡ ª(x; L) = 0; 2 (2.23) ¯ ¯ ¡ dx ¢ ¡d ¢ d ¯ ¡ iA ª(x; L) = E; + iA ª(x; L)¯ = 0: dx

dx

x=L

x=L0

The latter could be further simpli¯ed by diagonalizing constant matrix-coe±cient A (2.19) via transformation · ¸ · ¸ 1 ¡1 ® ~1 1 ¡1 K= ;K = : ® ~2 ® ~1 ¡~ ®2 1 The transformed coe±cients A and ¡ become · ¸ · ¸ 0 ® ~2 ¡1 ~ ~ B=A=k ; ¡ = K¡K = 0 1 ¡~ ®1® ~2

¡1 ® ~1

¸

and the transformed ª (2.24) obeys the equation (2.25)

ª ! U(x; L) = ¡2iK ª(x; L) K ¡ 1 B; h

i

d2 2 2 ~ 2 + B + k "(x)¡ U (x; L) = ¡ dx ¢ d ¡ iB ¢ U (x; L) jx= L = ¡2iB; ¡ dx d dx + iB U (x; L) jx= L0 = 0:

0;

Boundary-value problem (2.25) describes interaction and generation of k- and ¸waves of unit amplitude (labeled according to the diagonal entries of B) . Here the incident ¸-wave U11 generates k-wave U21 , whereas incident k-wave U22 generates ¸-wave U12 . It follows from (2.25) that the amplitude of the generated k-wave U21 is proportional to ± = ¸~ ®1® ~ 2 = ¸H1 H2 =H02 : Parameter ± is always less than ¸=4. In the real (geophysical) media ± becomes small parameter, as ® ~ 1® ~ 2 0 and 0 for y < 0 F (¿ ; y) = P (y(¿ ) < y) = h£(y ¡ y(¿ ))i

We call a typical realization of the process a deterministic median curve, computed from an algebraic equation F (¿ ; y ¤ (¿ )) = 12 : The motivation for this de¯nition comes from the properties of the median. Namely, for any time-interval [¿ 1 ; ¿ 2 ] process y (¿ ) \winds around" the median in such a way that it spends on average half of the time above it, y (¿ ) > y ¤ (¿ ), and half of time below, y (¿ ) < y ¤ (¿ ) (see ¯g. 9 and ref. [12, 13]) ® ® 1 T y(¿ )>y¤(¿ ) = T y(¿ ) 0 solution goes monotonically to zero, but x 0 < 0 blow up to ¡1 in ¯nite time t0 . Solution of the stochastic problem (5.1) is determined by the forward and backward Fokker-Planck equation in terms of the di®erence t ¡ t0 , that we shill be relabeled as t

(5.2)

@ p(x; tjx 0) @t @ p(x; tjx 0) @t

=

p(x; 0jx 0)

=

=

2 @ x 2 p(x; tjx 0 ) + D @@x2 p(x; tjx 0 ); @x @2 ¡x20 @x@ 0 p(x; tjx 0) + D @x 2 p(x; tjx0 ); 0

±(x ¡ x 0 ):

COHERENT PHENOMENA

29

Here variables x; p(x; tjx0 ) and D have dimensions [x] = t¡ 1 ; [p] = t;[D] = t¡3 , that allows to bring (5.2) to the dimensionless form

(5.3)

=

@ p(x; tjx0 ) @t

@ = ¡x20 @x p(x; tjx0 ) + 0

p(x; 0jx0 )

@ 2 @x x p(x; tjx 0 )

@2 @x2 p(x; tjx0 );

@ @t p(x; tjx0 )

+

@2 p(x; tjx 0 ); @x2 0

= ±(x ¡ x0 ):

Next we set up boundary conditions for (5.3) at singular points §1. Two kinds of problems are of special interest to us. First type appears, if curve x (t) is assumed to terminate at t0 by turning to ¡1. It corresponds to vanishing of the PDFs °ux density (5.4)

J (t; x) = x 2 p(x; tjx0 ) +

@ @x p(x; tjx 0)

at x = 1. Thus we get the boundary conditions J(t; x) !

p(x; tjx 0 ) !

In this case the total space-integral

G(tjx0 ) =

Z1

0, as x ! 1;

0, as x ! ¡1:

dxp(x; tjx 0 ) 6= 1

¡1

de¯nes the probability that function x (t) would stays ¯nite through the entire range (¡1; 1), i.e. the probability of a \non-singular value" of x (t), P (t < t0 ), t0 being less that the (random) blow-up time. So the probability to reach the singularity in ¯nite time is given by Z1 P (t > t0 ) = 1 ¡ dxp(x; tjx0 ) ¡1

(see, for instance [28]). Its density function (5.5)

p(tjx 0 ) =

@ P (t @t

> t0 ) =

@ ¡ @t

Z1

dxp(x; tjx 0 )

¡1

obeys the backward FP-equation that follows from (5.3), (5.6)

@ p(tjx 0 ) @t

= ¡x20 @x@ 0 p(tjx 0 ) +

@2 p(tjx 0 ): @x2 0

Let us estimate the mean \blow-up" time for the system to transfer from state 1 R fx0 g to f¡1g, hT (x0 )i = tdtp(tjx0 ). This function solves the time-independent 0

DE, a consequence of (5.6) (5.7)

¡1 = ¡x20 dxd 0 hT (x 0 )i +

d2 dx2 0

hT (x 0 )i ;

with the boundary conditions: hT (x 0 )i = 0 at x 0 = ¡1; and hT (x 0 )i -bounded at x0 = +1 . The latter is easily integrated to get Zx0 Z1 © ¡ ¢ª (5.8) hT (x0 )i = d» d´ exp 13 »3 ¡ ´ 3 : ¡1

»

30


Thus we get from (5.8) the mean time between two subsequent singular points (\blow-up" events of ¯g. 3) p 1=6 hT (1)i = ¼ 123 ¡(1=6) ¼ 4:976: Let us also remark that hT (0)i = 23 hT (1)i corresponds to a transition from x0 = 0 to x 0 = ¡1. A di®erent set of boundary conditions arises, if one lets x (t) to be discontinuous, and considers it over the entire range of t, so as x (t) reaches f¡1g at time t = t0 ¡0, it immediately reappears at f1g at t = t0 + 0. The corresponding boundary conditions for (5.3) impose continuity of the PDF °ux at §1 J(t; x)jx=¡1 = J (t; x)jx=+1 : In this case the limiting (t ! 1) stationary probability distribution is independent of the initial state x0 Zx © ¡ ¢ª (5.9) P (x) = J d» exp 13 » 3 ¡ x3 ; ¡1

and has normalizing coe±cient J = 1= hT (1)i - the stationary probability °ux . From (5.9) we get asymptotic formula for P at large x P (x) ¼

(5.10)

1 : hT (1)i x 2

This asymptotics is produced by the distribution of jumps of x (t) x(t) =

1 t ¡ tk

and randomness plays little role here. Indeed, given su±ciently large t À hT (1)i and x, one ¯nds PDF ¶À 1 ¿ µ 1 X 1 1 X p(x; tjx0 ) = ± x¡ = 2 h± (t ¡ tk )i t ¡ tk x k=0

=

1 2¼x 2

k=0

Z1

¡1

d·e

¡i·t

1 X

k=0

-

i·tk

e

®

1 = 2¼ x2

Z1

d·e¡i·t

¡1

©0 (·) 1 ¡ ©(·)

- i·t ® expressed in terms of two characteristic functions: © (·) = e 0 - ¯rst \blow-up" 0 - i·T ® point, and ©(·) = e - time window between two subsequent blow-ups. As t ! 1 we get an asymptotic formula 1 P (x) = ¡ 2¼ix 2 hT (1)i

Z1

¡1

d·e¡ i· t

1 J = 2; · + i0 x

that coincides with (5.10). 5.2. Caustics in random media. The problem of caustics in random media is similar to the previous one. Indeed, the wave-front curvature in the 2D-planar case, (x; y), is described by the equation (5.11)

d u(x) dx

= ¡u 2 (x) + f (x); u(0) = u0 ;

COHERENT PHENOMENA

31

2

@ with source f (x) = 12 @y 2 "(x; y(x)), while vertical displacement y (x) obeys di®erential system (2.49), both determined by variations of the refraction index, " along the ray. If Gaussian, homogeneous, isotropic ¯eld " is delta-correlated in x (the axis of propagation)

h"(x; y)"(x 0; y 0 )i = ±(x ¡ x0 )A(y ¡ y 0 )) its one-point PDF of the curvature-function is independent of the vertical displacement, and obeys the FP-equation (5.12)

@ @x P (x; u)

=

@ @u J (x; u); P (0; u)

Here the probability °ux density J(x; u) = u 2 P (x; u) + D2

= ±(u ¡ u 0 );

@ @u P (x; u);

and the di®usion coe±cient D=

1 @4 4 @y 4 A(0)

=¼

Z1

· 4 d·© " (0; ·):

0

where © " (0; ·) denotes the 2D spectral density of random ¯eld "(x; y), along its vertical (y -wave-number) direction. Equation (5.12) was studied in the previous section. We have shown random process u (x) to be discontinuous and drop to ¡1 at ¯nite distances x 0 = x (u0 ), determined by the initial condition u0 (here x plays the role of time t in the previous section). This \blow-up" means the focusing of optical rays by randomly inhomogeneous media. The means focusing distance hx (u 0 )i is given by the integral Zu0 Z1 © 2 ¡ 3 ¢ª 2 hx(u 0 )i = D d» d´ exp 3D » ¡ ´3 ¡1

»

hence [6 - 8] (5.13)

D1=3 hx(1)i ¼ D

1=3

hx(0)i =

6:27; 2 1=3 3D

hx(1)i ¼ 4:18:

Here quantity hx (0)i gives the mean \focusing distance" of the incident plane wave (zero curvature), while hx (1)i - measures the mean distance between two follow-up focusing events. Remark 3. In the method of smooth perturbations [5, 40, 41] one utilizes the level function of intensity Â (x; y) = ln I(x; y) to study its °uctuations. Al l statistical - ® 2 characteristics of Â depend on its variance ¾ 2 = Â2 ¡hÂi , which is approximately cubic in x; ¾ 2 ¼ Dx3 , to ¯rst order . One distinguishes two regions of intensity °uctuations: ¾ 2 ¿ 1 (weak °uctuations), and ¾ 2 À 1 (strong ones). As formula (5.13) shows random focusing occurs in the region of strong °uctuations. Equation (5.12) has limiting stationary PDF, as x ! 1, that corresponds to the constant probability °ux density, and looks similar to (5.9) Zu © 2 ¡ 3 ¢ª (5.14) P (u) = J d» exp 3D » ¡ u3 ; ¡1

32


where J = 1= hx(1)i . From (5.12) we get large-u asymptotics of P P (u) ¼

1 hx(1)i u 2

which indicates that the stationary statistics is determined by the behavior of u (x) in the vicinity of jumps u(x) =

1 : x ¡ xk

The wave-¯eld intensity has a similar structure due to (2.49), xk I(x) = : jx ¡ x k j

The asymptotics of its PDF at large I and x takes the form 2x P (x; I) = ; hx(1)i I 2

depending on the distance x, travelled by the wave. The probability of focusing at a distance x, found in the previous section, was Z1 P (x > x 0 ) = 1 ¡ duP (x; u): ¡1

Hence its PDF is related to the °ux density by the following equation [6-8] Z1 @ @ p(x) = @x P (x > x0 ) = ¡ @x duP (x; u) = lim J (x; u): ¡1

u !¡1

To ¯nd asymptotic dependence of p (x) on small parameter D ¿ 1, we adopt the standard analytic techniques for parabolic problems with small leading coe±cient (see, for instance [28]). Solution of (5.12) is represented as exponential © ª (5.15) P (x; u) = C (D) exp ¡ D1 A(x; u) ¡ B(x; u) : After substitution of (5.15) into (5.12) we select terms of order D 0 ; D ¡1 ; ::: and get pde's for unknown functions A and B. Constant factor C (D) is determined by the known probability distribution (of the plane wave) at x ! 0 ½ ¾ u2 1 P (x; u) = (2¼Dx) exp ¡ : 1=2 2Dx That gives an estimate C (D) ¼ D¡ 1=2 . Substitution of (5.15) in the expression of \focusing" probability we get £ ¤ (5.16) p(x) = lim P (x; u) u2 ¡ 12 @@u A(x; u) : u!¡1

Exponential form (5.15) along with the probability distribution of \focusing", immediately yield the structural dependence of p (x) on x from the dimensional analysis [9]. Indeed, u; D and P (x; u) have dimensions [u] = x ¡1 ; [D] = x ¡3 ; [P (x; u)] = x: Hence from (5.15) - (5.16) we get the scaling law © ª p(x) = C 1 D¡1=2 x ¡5=2 exp ¡C 2 =Dx3 ;

COHERENT PHENOMENA

33

with unknown coe±cients C 1 ; C2 . The latter were computed in [6] and the ¯nal result takes the form © ª ¡ 1=2 ¡ 5=2 (5.17) p(x) = 3® 2 (2¼D) x exp ¡® 4 =6Dx 3 ;

with ® = 1:85. The applicability condition for (5.17) is Dx 3 ¿ 1. However, numeric simulations [6] show that (5.17) gives accurate prediction of PDF p, also for Dx 3 » 1. In 3D case one could apply the dimensional analysis to compute the PDF of caustic formation [9] as © ª p(x) = ®D ¡1 x ¡4 exp ¡¯=Dx 3 ;

with certain constants ®; ¯. This scaling law with ® = 1:74, ¯ = 0:66 was obtained in [7]. 5.3. Re°ection phase for plane waves in layered media. In some cases the above discontinuous random processes prove to be useful in stochastic systems with no apparent discontinuity of solutions. One such example are °uctuations of phasefunctions in wave-propagation. For instance, we have shown in section 2.2, that the incident plane wave in randomly layered medium, has the re°ection coe±cient with phase function ÁL that obeys equation (2.13). we want to ¯nd the distribution law of random variable ÁL. The problem with solution (2.13) is that it gives ÁL as real-valued function on the entire line (¡1; 1). However, in applications we need distribution of ÁL on its natural domain [¡¼; ¼]. Furthermore, the half-space limit of ÁL should be independent of L (uniform distribution). To get the ¯nite-interval distribution we replace phase ÁL with a trigonometric function zL = tan(ÁL =2), that possesses singular poles. The dynamic equation for zL has form (2.15). We assume random coe±cient "1 (L) (the real part of the complex refraction index) to be Gaussian, delta-correlated with variance ¾ 2" and correlation radius l0 , so h"1 (L)i = 0; h"1 (L)"1 (L0 )i = 2¾ 2" l0 ±(L ¡ L0 ); Then the stationary (L-independent) half-space limiting PDF P (z) =

lim

L 0!¡ 1

P (L; z)

solves the equation (5.18)

¡ ¢ d ¡· dz 1 + z2 P (z) +

d2 dz2 P (z)

=0

whose coe±cients, · = k=2D, and D = k2 ¾ 2" l0 =2 depend on the incident wave number k and the media parameters. Solution of (5.18) with constant probability °ux at the boundary (1) has the form [11, 33] P (z) =

J(·)

Z1 z

(5.19)

J ¡ 1 (·) =

n h d» exp ¡·» 1 +

³ ¼ ´1=2 Z1 ·

0

»2 3

io + z(z + » ) ;

n ³ » ¡1=2 d» exp ¡· » +

»3 12

ó

:

34


The PDF of such phase-function on ¯nite interval [¡¼; ¼] ¯ 2 ¯ P (Á) = 1+z : 2 P (z) ¯ z= tg(Á=2)

is shown in ¯g. 11 for di®erent values of parameter ·: · = 0:1 (A), · = 1 (B), · = 10 (C). As · grows large, · À 1, we get asymptotic expansion P (z) ¼ 1=¼(1 + z 2 ) on ¡1 < z < 1, that corresponds to a uniformly distributed phase P (Á) =

1 ; 2¼

¡¼ < Á < ¼:

6. Localization of plane waves in randomly stratified media Our main goal here is the wave-propagation in the single-layer and multi-layer media. The issues of wave localization by randomly strati¯ed media with or without dissipation were actively pursued and debated over the last few years (see review articles [11 - 13]). However, di®erent physical system and conditions produced inconclusive and often inconsistent results. The reason is largely due to complex spatial structure of ¯eld realizations inside the media. Namely, the general decay of intensity away from the source is accompanied by increasingly strong and rare bursts due to the coherent superposition of multiple scattering evens (¯g.5). As the result one could observe dynamic localization for almost all individual realizations, whereas statistical averages (like the mean intensity and higher moments) show no statistical (energy) localization on the level of ensembles of realizations. 6.1. Single layer medium. The one-layer, 1D model of wave propagation and scattering was set up in section 2.2. Here the statistical problem has two parts: statistical analysis of the re°ection and transmission coe±cients on the boundary of random layer, and statistical analysis of the ¯eld intensity inside the layer. 6.1.1. Re°ection and transmission coe±cients. The imbedding method gives closed form equations (2.10) - (2.12) for the squared modulus of the re°ection coe±cient W as function of the (right) moving boundary parameter L. Once again we assume the random component of the medium refraction coe±cient "1 (x) to be Gaussian, and use delta-correlation approximation with variance ¾ 2" and correlation radius l0 h"1 (L)i = 0; h"1 (L)"1 (L)i = B" (L ¡ L) = 2¾ 2" l0 ±(L ¡ L): The resulting Fokker-Planck equation for PDF P (L; W ) = h± (WL ¡ W )i has the form [10] @ P (L; W ) @L

(6.1)

P (L0 ; W )

@ @ = 2k° @W (W P ) ¡ 2D @W [W (1 ¡ W )P ] + @ D @ @W (1 ¡ W )2 W @W P;

= ± (W ) ;

with the di®usion coe±cient D = k2 ¾ 2" l0 =2. Derivation of (6.1) exploits averaging over fast oscillations of the \re°ection phase", and is valid under an additional constraint k=D À 1 (see sect. 5.3). Taking into account ¯nite correlation radius l0 of " 1 (L) and using the di®usion approximation, the FP-equation (6.1) doesn't change its form, but the di®usion

COHERENT PHENOMENA

35

coe±cient becomes (6.2)

k2 D= 4

Z1

d» B"(») cos(2k» ) =

¡1

k2 © "(2k); 4

where © " denotes the spectral function of random process " (L). Di®usion approximation assumes that "1 (x) has negligible e®ect on the wave-¯eld on the scale of correlation radius l0 , that is Dl0 ¿ 1 . In the absence of dissipation (° = 0) the FP-equation (6.1) could be integrated [10]. Assuming random layer [L0 ; L] to be su±ciently wide, ¿ = D(L ¡ L0 ) À 1, we get the following asymptotics for the moments of the transmission coe±cient jTL j2 = 1 ¡ WL ³ ¿´ ® [(2n ¡ 3)!!]2 ¼ 5=2 ¡3=2 jT L j2n ¼ ¿ exp ¡ : 2 2n¡1 (n ¡ 1)! 4

Thus all statistical moments of the transmission/re°ection coe±cients have a universal asymptotic dependence on the (dimensionless) layer width ¿ , but their coe±cients vary with n. As all statistical moments of jT j converge to zero with increasing ¿ , we get the re°ection modulus jRj ! 1 with probability one. Hence randomly strati¯ed half-space is fully re°ective. Nonzero dissipation ° does not allow a closed form solution of (6.1) for a ¯nite width interval. However, the half-space limit (L0 ! ¡1; ¿ ! 1) has stationary 2 (L/ ¿ -independent) distribution for W L = jRL j µ ¶ 2¯ 2¯W (6.3) P (W ) = exp ¡ ; (1 ¡ W ) 2 1¡W where ¯ = k°=D represents the dimensionless absorption coe±cient. Pdf (6.3) has an obvious physical meaning. Coe±cient ¯ = 0 gives ±-source at W = 1, corresponding to the full re°exivity by su±ciently wide random layer. All other (non zero) ¯ give continuously distributed PDFs over [0; 1] (¯g. 14), so only a fraction of the incident wave-¯eld would \rer°ect back" with positive probability P (W ) > 0, while P (1) is always 0, due to a partial absorption of the wave by the medium. Pdf (6.3) yields all moments of WL = jRL j2 , in particular it gives asymptotic formula for the mean ½ 1 ¡ 2¯ ln(1=¯); ¯ ¿ 1; (6.4) hW i ¼ 1=2¯; ¯ À 1; as well as the recurrence for higher moments ® ® (6.5) n W n+1 ¡ 2(¯ + n) hW n i + n W n¡1 = 0 (n = 1; 2; :::):

If we place the source inside the strati¯ed media, the wave-¯eld solves the boundary value problem (2.9). In the in¯nite space limit (L0 ! ¡1; L ! 1), the mean intensity at the source becomes hI(x 0 ; x0 )i = 1 + ¯ ¡1 [5,10]. Its unlimited growth, as ¯ ! 0, indicates the accumulation of energy by randomly strati¯ed media.

36


6.1.2. Wave intensity within random layer. Let us study the wave-¯eld structure inside the chaotically disordered layer. If layer [L0 ; L] is su±ciently wide (¿ = D(L ¡ L 0) À 1), and dissipation-free (° = 0), one has the so called stochastic parametric resonance, manifested by the initial exponential growth of all moments fhIn (x; L)i : n > 1g, of the intensity (I(x; L) = ju(x; L)j2 ), in variable L [10]. They their reach maximum somewhere in the middle of the layer (¯g. 12). In the half-space limit (L0 = ¡1), the range of exponential (explosive) growth extends through the entire half-line, but the mean intensity remains ¯xed hI(x; L)i = 2. In this case the realizations of wave-intensity have the form (6.6)

I(x; L) = 2W (x; L) (1 + cos ÁL ) ;

where phase-function ÁL solves a stochastic equation (2.13) (see section 5.3). Amplitudefunction W (x; L) = exp f¡ [q(L) ¡ q(x)]g ; is expressed through q, that solves related stochastic equation d dL q(L)

(6.7)

= k"1 (L) sin ÁL :

The inial condition for (6.7) is imposed at a far-¯eld point B, q (B) = 0, and the random variable ÁB is uniformly distributed [10]. Random functions q(L) and ÁL are statistically independent here. Average PDF of q, P (L; q) = h± (q(L) ¡ q)i over the fast oscillations we get the Fokker-Planck equation (6.8)

@ P (L; q) @L

2

@ = ¡D @@q P (L; q) + D @q 2 P (L; q); P (B; q) = ±(q):

Hence random function q(L) is Gaussian, while distributionW (x; L) is log-normal, having all moments starting with the second one to grow exponentially inside the random layer (6.9)

hW (x; L)i = 1; hWn (x; L)i = expfn(n ¡ 1)D(L ¡ x)g:

Its typical realization, due to log-normality, becomes (6.10)

W ¤ (x; L) = expf¡D(L ¡ x)g;

So its realizations obey the inequality W (x; L) < 4 expf¡D(L ¡ x)=2g: with probability 12 , and the latter is valid in the half-space limit. If the physics of disordered systems such exponential fall-o® in variable » = D(L¡x) for a typical realization (6.10) is associated with the dynamical localization [11, 53, 54], the localization length being llo c = 1=D However, the energy is not localized in the statistical (mean) sense here. To conclude: one-point PDFs yield the detailed evolution of the ¯eld intensity on the level of individual realizations, and allow one to estimate the parameters of evolution through the statistics of the media. Dissipation arrests exponential growth of the moments (6.8), and at large » À 4(n ¡ 1=2) ln(n=¯), those reverse to the decaying asymptotic law of [13] ³ ´ ¡ (n¡ 1=2) hW (» )i » ln(1=¯)»¡3=2 exp ¡ »4 : = An ¯

COHERENT PHENOMENA

37

As for the mean ¯eld intensity of a point source in the layer, one ¯nds [13] hI(x; x 0 )i =

³ ´ ¼ 5=2 ¡3=2 » exp ¡ »4 ; 8¯

at large distance from the source ( » = D(x ¡ x 0 ) À 1). So one has both statistical and dynamical localization in this case. 6.2. Strati¯ed two-layer model. The simplest model of wave propagation in the strati¯ed (two-layer) media was outlined in section 2.2. It satis¯es the di®erential system (2.17). The di®usion approximation technique was used in papers [37, 38] to derive the Fokker-Planck equations for PDFs of the intensities of re°ection coe±cients Wij = jRij j2 . Unlike the one-layer case, they involve now four di®usion coe±cients, expressed through the spectral function of random process " (x) as follows D1 (6.11)

D3

µ

¶2 µ ¶2 k H1 k H2 = ©" (2¸k); D2 = ©" (2k); 2¸ H0 2 H0 µ ¶2 µ ¶2 k k = ©" (k(1 + ¸)); D4 = ©" (k(1 ¡ ¸)): 2¸ 2¸

For small scale inhomogeneities (relative to the wave length), kl0 ¿ 1, all di®usion coe±cients are proportional to a single D, determined by (6.2). Namely, (6.12)

D1 =

µ

1 H1 ¸ H0

¶2

D; D2 =

µ

H2 H0

¶2

D; D3 = D4 =

1 ¸2

D:

Let us also notice that the di®usion approximation requires Di l0 ¿ 1: As we mentioned in part 2 the problem has a small parameter ±, that could be used to expand and simplify it. We write perturbation expansion in ± and maintain only ¯rst order terms , i.e. neglect the secondary radiation e®ects. In such approximation the squared moduli of the diagonal re°ection coe±cients W 11 and W22 are statistically independent and their PDFs P (L; W 11 ); P (L; W22 ) obey © @ £ ¤ @ 2 @L P (L; W 11 ) = @W ¡D1 (1 ¡ W11 ) + 2±(D3o+ D4 )W 11 @2 2 + D1 @W P (L; W11 ); 2 (1 ¡ W 11 ) W 11 11 n £ ¤ @ @ 2 P (L; W 22 ) = @ W ¡D (1 ¡ W ) + 2±(D + D )W 2 22 3 4 22 @L 22 o (6.13) @2 2 + D2 @W P (L; W22 ): 2 (1 ¡ W22 ) W22 22

Those resemble equation (6.1) of the one-layer model, but have an extra term @ 2±(D3 + D4 ) @W [W P (L; W )]

This means that the generation of cross-modes (\k-incident to ¸-scattered", or \¸-incident to k-scattered") is statistically equivalent to the e®ective damping (dissipation) in the original stochastic problem for incident wave-¯elds U11 ; U22 . In other words the dissipation-free refraction index "(x) should be replaced by the complex one, "(x) + i±(D3 + D4 ).

38


Furthermore, the half-space limit (L0 ! ¡1) has stationary (L-independent) solutions of (6.13) µ ¶ 2° 1 2°1 W 11 P (W 11 ) = exp ¡ ; (1 ¡ W 11 )2 1 ¡ W11 µ ¶ 2° 2 2° W 22 (6.14) P (W 22 ) = exp ¡ 2 2 (1 ¡ W 22 ) 1 ¡ W22 with parameters

D3 + D4 D3 + D4 ; °2 = ± : D1 D2 Those parameters measure the \e®ective dissipation" due to cross-generation of the wave-modes, as opposed to the \e®ective wave di®usion" resulting from multiple scattering by random obstacles. For small scale inhomogeneities the damping parameters H2 2 H1 (6.16) ° 1 = 2¸ ;° = H1 2 ¸ H2 (6.15)

°1 = ±

are determined by the relative thicknesses of two strata (for a ¯xed wave length ¸), and bear no other dependence on the statistics of the media. Furthermore, one has ° 1° 2 = 4, so two °'s are reciprocals of each other. Probability distributions (6.14) yield statistics of the re°ection coe±cients of incident waves. It follows from our discussion that a su±ciently wide random band [L0 ; L] (or the limiting half-space) has the transmission rates jT 11 j2 and jT 22 j2 be almost surely 0. So the incident ¸- and k-waves are localized. Their localization length are determined either by the \di®usion" or \dissipation"coe±cients, whichever is stronger . So case °1 ¿ 1 (°2 À 1) yields the localization length for two modes proportional to the localization length llo c of the one-layer model µ ¶2 1 ¸H0 1 ¸H0 (1) (2) lloc = = lloc ; llo c = = lloc : D1 H1 2±(D3 + D4 ) 4H1 H2 In the opposite case: ° 1 À 1 (° 2 ¿ 1), we get (1)

lloc =

1 ¸H0 1 (2) = lloc ; lloc = = 2±(D3 + D4 ) 4H1 H2 D2

µ

H0 H2

¶2

lloc :

Statistics of the \o®-diagonal" modes W 12 poses more challenging problem, since it involves coupling of W12 with W11 and W 22 . The analysis of the corresponding Fokker-Planck equations shows that special combinations T1 = 1 ¡ W 11 ¡ ±W 12 and T 2 = 1 ¡ W22 ¡ ±W 12 that arise in (2.26) and determine the transmission rates of the generated cross-modes, have no stationary (half-space) solutions, of the form P (Ti ) = ± (T i ). This mean the absence of localization for the generated cross-modes [38]. Coming back to the source-problem in the upper or lower strata, at the boundary x0 = L, of the random band, one could show that the transmission coe±cients di®er from zero in both strata (the upper and the lower one), so there is no localization whatsoever. The basic model could be further compounded by introducing inhomogeneities in both strata (the upper and the lower one), or by taking di®erent types of waved coupling , or by changing °uctuating parameters, e.g. " (x) to dx "(x), etc., all of

COHERENT PHENOMENA

39

them relevant to certain geophysical problems [35-37]. Such changes could further complicate and modify the Fokker-Planck equations, as well as their dependence on the basic geometry and statistics of the problem. But the main conclusion - the absence of the dynamic localization should still hold. 6.3. Localization in parabolic wave-guides. Wave beam propagating in the parabolic wave-guide with randomly varying curvature is described by system (2.44). Function A (x) could be represented in the form A(x) = k®a2

1 + Ã(x) exp (¡2i®x) : 1 ¡ Ã(x) exp (¡2i®x)

where Ã(x) obeys an equation derived from (2.44) d Ã(x) = dx Ã(0) =

¢2 i ¡ i®x e ¡ Ã(x)e¡ i®x z(x); 2®k 1 ¡ k®a2 : 1 + k®a2

¡

Let us introduce the phase and amplitude of Ã(x) via s w(x) ¡ 1 ifÁ( x)¡2®xg Ã(x) = e ; w ¸ 1: w(x) + 1 Then we get a system of equations for functions w (x) and Á (x) p d 1 w(x) = ¡ ®k z(x) w 2 (x) ¡ 1 sin (Á(x) ¡ 2®x) ; dx ¡ ¢ 1 w(0) = 2k®a 1 + k 2 ®2 a4 ; 2 Ã ! w d 1 p cos (Á(x) ¡ 2®x) ; dx Á(x) = ®k z(x) 1 ¡ w 2 (x) ¡ 1 (6.17)

Á(0) =

0:

Hence the wave-¯eld intensity on the wave-guide axis (2.46) assumes the form (6.18)

I(x; 0) =

w(x) +

®ka2

p

w 2 (x) ¡ 1 cos (Á(x) ¡ 2®x)

:

As before we assume z(x) to be Gaussian, delta-correlated with parameters hz(x)i = 0; hz(x)z(x0 )i = 2¾ 2 l ±(x ¡ x 0 ): Furthermore, we assume the variance of z-°uctuations to be su±ciently small, ¾ 2 ¿ 1. Then statistical properties of w(x) and Á(x) change slowly on scales of order 1=®. Hence to determine statistics of the wave intensity (6.18) we could average it over rapid oscillations, regarding functions w(x) and Á(x) statistically independent, and over the uniformly distributed phase Á(x). The resulting PDF of w(x), P (x; w) = h± (w(x) ¡ w)i obeys the Fokker-Planck equation ¡ 2 ¢ @ @ @ (6.19) P (x; w) = D @w w ¡ 1 @w P (x; w); P (0; w) = ± (w(0) ¡ w) @x with the di®usion coe±cient D = ¾ 2 l=2® 2 k2 . Under the same assumptions we could ¯nd the moments of intensity hI n (x; 0)i on the wave-guide axis. Here the averaging works in two stages. On the ¯rst step

40


we average over the fast phase-oscillations, and get ¿µ ¶n À I (6.20) = P n¡ 1(w): ®ka2 Á Pn (w) being the n-th Legendre polynomial. On the second step we average (6.20) over the distribution of w (6.19). The ¯nal result [5, 10] becomes ¿µ ¶n À I (6.21) = Pn¡1 (w 0 ) exp fDn(n ¡ 1)xg : ®ka2 If the wave-beam parameters are adjusted to the wave-guide (see (2.43)), then w0 = 1 and formula (6.21) becomes (6.22)

hI n (x; 0)i = exp fDn(n ¡ 1)xg ;

The latter means that I (x; 0) is distributed according to the log-normal law. However, the typical realization of process I (x; 0) decays exponentially inside the medium I ¤ (x; 0) = exp f¡Dxg ; So the radiation should spread over in the cross-sectional directions (away from the axis) for speci¯c realizations, which means dynamic localization in x. The typical realization gives the standard Gaussian cross-sectional intensity (2.54) modulated along the axis n 2 o I ¤ (x; ½) = I ¤ (x; 0) exp ¡ a½2 I ¤ (x; 0) : 7. Passive tracers in random velocity fields Now we shall consider the statistical problem of the passive tracer di®usion in random velocity ¯eld, introduced in section 2.3. We shall study general compressible velocity ¯elds (ru (r; t) 6= 0), assumed to be Gaussian homogeneous, isotropic in space, stationary in time with correlationfunction and spectral tensor

(7.1)

hu i(r; t)uj (r0 ; t0 )i = Bij (r ¡ r0 ; t ¡ t0 ) Z 1 E ij (k; t) = ( 2¼) N drBij (r; t)e¡ikr ;

N = 2; 3 being the space dimension. Spectral tensor could be decomposed in the usual solenoidal, E s (k; t) plus potential, E p (k; t) components ³ ´ k k k k (7.2) E ij (k; t) = E s (k; t) ± ij ¡ ki 2j + E p (k; t) ik2j : The practically important cases include

² Incompressible °uid °ow : ru(r; t) = 0; E p (k; t) = 0; ² Potential velocities: (E s (k; t) = 0); ² Mixed case.

An example of the latter is a °oating tracer [2, 22]. Indeed, if such tracer moves over the surface z = 0 of an incompressible 3D °uid, with horizontal and vertical velocity components: u(r; t) = (U ; w)., then surface elevation z = z (R; t) creates an e®ective compressible horizontal °ow with the divergence r R U = ¡ @z wjz=0 .

COHERENT PHENOMENA

41

We assume the entire 3D spectral tensor of u (R; t) to be ³ ´ k k (7.3) Eij (k; t) = E(k; t) ±ij ¡ ki 2j : and write tracer density as

½(r; t) = ½(R; t)±(z); r = (R; z); R = (x; y): After substitution in (2.29), and integration overt variable z, the reduced 2D system would evolve according to ³ ´ @ @ (7.4) + U (R; t) ½(R; t) = 0; ½(R; 0:) = ½0 (R): @t @R

The resulting horizontal (compressible) ¯eld U (R; t) is clearly Gaussian, homogeneous, isotropic with spectral tensor µ ¶ Z1 k? ®k?¯ 2 2 E ®¯ (k? ; t) = dkz E(k? + kz ; t) ±®¯ ¡ 2 k? + kz2 (7.5)

®; ¯

=

¡1

1; 2:

expressed in terms of the horizontal and vertical wave-numbers k = (k? ; kz ). Comparing (7.5) with (7.2) we get the solenoidal and potential velocity components on the z = 0 - plane [2], Z1

s

E (k? ; t) =

(7.6)

¡1 Z1

E p (k? ; t) =

dkz E (k2? + k2z ; t); dkz E (k2? + k2z ; t)

¡1

k2?

k2z : + k2z

Let us now go back to the general case. In what follows we shall assume velocity u (r; t) to be delta-correlated in time, and approximate its space-time correlations ef f Bij (r; t) = 2Bij (r)±(t);

(7.7)

with the e®ective correlation-tensor Z1 Z1 ef f 1 Bij (r) = 2 dtBij (r; t) = dtBij (r; t): ¡1

0

The homogeneity and isotropy of u, yield the following standard relations for the matrix-function B ef f , along with its ¯rst, second and fourth derivatives f Bef kl (0) =

D0 ± ; N kl

ef f @ @ri Bkl (0) @ 2B ef f (0) ¡ @rkl i @r j

(7.8)

=

Ds 2

= 0;

[(N + 1)± kl ±ij ¡ ±ki± lj ¡ ± kj ±li] + ¤ ±kl ±ij + ±ki± lj + ±kj ±li ;

N( N+2)

D2p N(N +2)

£

eff

@ 4 Bkl (0) @ri @rk @rj @rl

p

=

D4 N

±ijkl :

Here we used the standard convention of summing over the repeated indices.

42


We shall also de¯ne the e®ective spectral densities Z1

~ s(k) = E

dtE (k; t); E~ p (k) = s

0

Z1

dtE p (k; t)

0

of the solenoidal and potential components, and introduce the following \di®usion coe±cients" Z h i ~ s(k) + E ~ p (k) ; D0 = dk (N ¡ 1)E

(7.9)

Ds2

=

Dp2

=

Dp4

=

Z Z Z

dkk 2E~ s (k);

dkk 2E~ p (k); dkk 4E~ p (k):

Co e±cient D0 combines time-integrated (solenoidal and potential ) kinetic energies , while Dp2 ; Ds2 give time-integrated total enstrophies of two components. For the sake of presentation we shall con¯ne ourselves to the 2D motion , i.e. consider equation (7.4). As we mentioned in section 2.3. ¯rst order PDEs of type (7.4) are solved by the method of characteristics. Introducing characteristic rays R (t) (2.30) (\particle tra jectories"), we can rewrite (7.4) in the form (2.31), which in our case turns into a di®erential system d dt R(tj»)

=

U(R; t); R(0j» ) = » ;

@U (R; t) ½(tj» ); ½(0j» ) = ½0 (» ): @R The above equations describe the Lagrangian transport of tracer's particles. When supplemented by the evolution of the Jacobian divergence (7.10)

d dt ½(tj»)

¡

=

= @ U (R ;t) j(tj»); @R

d j(tj») dt

(7.11)

j(0j») = 1

they yield solution ½(tj») ´ ½0 (»)/ j(tj»):

(7.12)

7.1. Lagrangian description. The Lagrangian indicator function © Lag (t; R; ½; j j») = ±(R(tj») ¡ R)±(½(tj») ¡ ½)±(j(tj») ¡ j) satis¯es the Liouville equation @ © (t; R; ½; jj») @t Lag

=

h ¡ +

(7.13)

© Lag (0; R; ½; jj»)

@ U (R; t) @R

@ U (R ;t) @R

³

@ @½ ½

¡

@ @j

j

í

© Lag (t; R; ½; jj»);

= ±(» ¡ R)±(½0 (») ¡ ½)±(j ¡ 1):

We shall start our analysis with such important statistics, as particles' positions and their density. Averaging (7.13) over the ensemble of velocity realizations fU g, gives the Fokker-Planck equation for the one point Lagrangian PDF

COHERENT PHENOMENA

43

P (t; R; ½; j j» ) = h©Lag (t; R; ½; jj»)i, [2, 23],

(7.14)

n

2

D0 @ 2 @R 2

+

p D2

³

@ @t P (t; R; ½; jj»)

@ 2 @ @ ½ ½ @½

2

= 2

@ ¡ 2 @½@j ½j + @@j 2 j 2

ó

P (t; R; ½; jj»);

P (0; R; ½; j j» ) = ±(R ¡ »)±(½0 (») ¡ ½)±(j ¡ 1):

Here the initial state could be visualized as highly localized tracer (near source ») of the mean speci¯c concentration ½. Accordingly, solution of (7.14) could be factored into the product of three terms µ ¶ ½ (») (7.15) P (t; R; ½; jj») = P (t; Rj»)P (t; jj»)± ½ ¡ 0 ; j The ¯rst one is the standard Gaussian propagator in variable R, for a single randomly di®using particle initiated at point R0 , with the e®ective di®usivity D0 n³ ´ o h i 0 0 (R ¡R )2 1 @2 1 (7.16) P (t; RjR 0) = exp D t ±(R ¡ R ) = exp ¡ 0 2 2 2¼D 0t 2D 0t @R The second factor n³ ´ o n o ln 2(je¿ ) @2 2 1 p (7.17) P (t; j j» ) = exp Dp2 @j j t ±(j ¡ 1) = exp ¡ 2 2j ¼ ¿ 4¿ describes a similar evolution of the Jacobian density (its PDF), generated by the @2 2 di®erential operator @j on the half-line 0 · j < 1: 2j Here and henceforth we shall often use the dimensionless time ¿ = Dp2 t The product form of solution (7.15) implies statistical independence of the particle's position R(tj») and the Jacobian (°ow) divergence j(tj»). Furthermore, Gaussian distribution (7.16) corresponds to the standard Brownian motion with parameters ® (7.18) hR(tj»)i = »; ¾ 2®¯ (t) = [R® (t) ¡ » ®][R¯ (t) ¡ » ¯ ] = D0 ± ®¯ t whereas Jacobian (divergence) factor j has log-normal distribution,with constant mean hj (tj» )i = 1, and exponentially increasing higher moments (7.19)

hj n (tj»)i = exp [n(n ¡ 1)¿ ] ; n = §1; §2; :::

The moments of the Lagrangian density ½ have similar exponential growth due to (7.12) h½n (tj»)i = ½n0 (») exp [n(n + 1)¿ ] ;

(7.20)

So both the (Lagrangian) mean density h½i and its higher moments increase in time. Furthermore, the joint PDF of ½ and j has the form (7.21)

P (t; ½; jj») = P (t; jj») ± (½ ¡ ½0 (»)/ j) ;

with log-normal P (t; jj») of (7.17). Integrating out variable j (7.21) we get the PDF of the Lagrangian density alone ½ ¾ ln 2 (½e¡¿ /½0 (») ) 1 p (7.22) P (t; ½j») = 2½ ¼¿ exp ¡ : 4¿ The same solution could also be obtained by the direct reduction of the general Fokker-Planck equation (7.14) (7.23)

@ P (t; ½j») @t

@ 2 @ = D2p @½ ½ @½ P (t; ½j»); P (0; ½j») = ±(½0 (») ¡ ½):

44


Some unexpected statistical features of the tracer and Jacobian densities, like exponential increase of their moments, etc., ¯nd simple explanation on terms of the log-normality. Thus the \typical realization" of random Jacobian density is given by an exponential curve j ¤ (t) = exp(¡¿ ):

(7.24)

Furthermore, random process j(tj») admits majorant (envelop) estimates, like j(tj») < 4 exp(¡¿ =2) holding for all ¿ 2 [0; 1] with probability 12 . As for the tracer density it has increasing \typical median", and admits minorant estimates from below ½¤ (t) = ½0 exp(¿ );

½(tj») > ½0 exp(¿ =2)=4:

Let us notice marked di®erences of Lagrangian statistics in the compressible and incompressible cases. The latter have j(tj») ´ 1, so the density remains constant along (characteristic) particle trajectories ½(tj») = ½0 (») = const . The above results con¯rm our main thesis on the dominant role of large °uctuations from the \typical behavior", that shape the essential statistics of random processes j(tj») and ½(tj»). As for the random \particle position" process both cases (compressible and incompressible) don't show much di®erence. Next we shall study the relative displacement of two particles. The separationfunction l (t) = R1 (t) ¡ R 2 (t) obeys the dynamic evolution d l (t) dt

= U (R 1 ; t) ¡ U (R2 ; t) ; l(0) = l 0 ;

and the corresponding Fokker-Planck equation (7.25)

@ P @t

(t; l) =

@2 D (l)P (t; l); @l® @l¯ ®¯

P (0; l) = ±(l ¡ l 0 ):

h i f ef f The di®usion tensor D®¯ (l) = 2 Bef ®¯ (0) ¡ B®¯ (l) involves the structure ma-

trix of vector ¯eld U (R; t) at two di®erent points. In general equation (7.25) has no closed form solution. However, for a short initial displacement, relative to the correlation radius of U , l0 ¿ lco r , the coe±cients D®¯ (l) could be expanded in the Taylor series in l, to get the ¯rst order approximation ¯ f ¯ @ 2 Bef ®¯ (l) ¯ D®¯ (l) = ¡ l° l± : ¯ @ l° @ l± ¯ l=0

The corresponding di®usion tensor D®¯ could be further simpli¯ed using symmetries of the problem (homogeneity and isotropy), that express through the e®ective di®usion rates (7.8), (7.26)

D®¯ (l) =

1 8

p

[3Ds2 + D2 ] l2 ±®¯ ¡

1 4

p

[Ds2 ¡ D2 ] l®l¯ :

To get a closed form equation for the moments of l, we substitute (7.26) in (7.25), multiply by l n and integrate over l. This yields a simple linear evolution [2], d dt

hln (t)i = 18 n [n (Ds2 + 3Dp2 ) + 2 (Ds2 ¡ Dp2 )] hln (t)i ;

COHERENT PHENOMENA

45

whose solutions grow exponentially in time for all n = 1; 2; :::. Furthermore, random process l(t) has log-normal PDF with parameters l0 ¿ · ¸À l (t) 2 ln = 14 (Ds2 ¡ Dp2 ) t; ¾ ln[l(t)=l = 14 (Ds2 + 3Dp2 ) t: 0] l0 Hence, a typical realization of the \particle-separation" © p ª l¤ (t) = l0 exp 14 (D2s ¡ D2 ) t ;

grows or decreases exponentially in time, depending on sign of D2s ¡ Dp2 . In particular, incompressible °ows (Dp2 = 0) have exponentially increasing typical realizations, which means exponential divergence of particles at short distances and times. This result holds as long as µ ¶ lcor 1 s D t ¿ ln ; 4 2 l0 i.e. the Taylor expansion (7.26) remains valid. At the opposite end stand pure potential velocities (D2s = 0). Here a typical realization of particle-separation would exponentially decrease, so the particles tend to coalesce. Such tendency of the °ow to \bring particles together" could lead to formation of clusters - regions of high tracer concentration interspersed within large voids. Indeed, our conclusion is consistent with some numeric studies (illustrated in ¯g. 1b), although our model of random velocities is di®erent from those used in computations. This suggests that clustering is largely insensitive to speci¯cs of the model, which quali¯es it as a coherent phenomenon. Let us remark that the \Eulerian clustering" (to be discussed in the next section) is possible even in the case (Ds2 ¡ D2p ) > 0 , as long as velocities have a potential component. 7.2. Eulerian description. In the Eulerian form of turbulent transport we take the indicator function ©E ul (t; R; ½) = ±(½(t; R) ¡ ½) , that obeys the Liouville equation (3.11) ³ ´ @ U (R ;t) @ @ @ + U (R; t) © Eu l (t; R; ½) = @t @ ½ ½©Eu l (t; R; ½); @R @R (7.27)

©E ul (0; R; ½) =

±(½0(R) ¡ ½):

Then we get the PDF for the Eulerian tracer-¯eld, P (t; R; ½) = h©E ul (t; R; ½)i by averaging (7.27) over ensemble, or alternatively using (3.10), as in [2, 22]. The resulting FP equation takes the form ³ ´ 2 @ @2 2 ¡ 12 D0 @ 2 P (t; R; ½) = Dp2 @½ 2 ½ P (t; R; ½); @t @R (7.28) P (0; R; ½) = ±(½0 (R) ¡ ½): The equation for moments follows directly from (7.28) ³ ´ p @ 1 @2 n D2 n(n ¡ 1) h½n (R; t)i ; @t ¡ 2 D0 @ R 2 h½ (R; t)i = (7.29) h½n (R; 0)i = ½n0 (R)

46


In particular, the mean tracer density obeys the evolution @ @t

(7.30)

h½(R; t)i = 12 D0

@2 2 @R

h½(R; t)i

consistent with that of a single-particle PDF (7.14). Notice however, that the \mean" di®usion coe±cient D0 (7.30) gives only global (large scale) characteristics and (spatial) scales of the tracer distribution, and carries little information about the ¯ne (local) structure and details of realizations. Solutions of the moment equations (7.29) are given by the convolution with the standard Gaussian propagator, Z n (7.31) h½ (R; t)i = exp [n(n ¡ 1)¿ ] dR 0 P (t; RjR 0 )½n0 (R 0) whereas the FP equation (7.28) combines the Gaussian di®usion in variable R with \log-normal di®usion" in ½ 2 (0; 1). In particular, initial uniform density, ½0 (R) = const, yields log-normal PDF P (t; ½), independent of R, along with its integrated probability distribution (7.32)

P (t; ½) F (t; ½)

½

¾ ln2 (½e¿ =½0 ) = 2½ ¼¿ exp ¡ ; 4¿ µ ¶ ln (½e¿ =½0 ) = © p - error-function. 2 ¿ 1 p

The corresponding mean-¯eld and moments grow in ¿ as (7.33)

h½(R; t)i = ½0 ;

h½n (R; t)i = ½n0 exp [n(n ¡ 1)¿ ] ;

whereas a typical realization falls o® exponentially ½¤ (t) = ½0 exp(¡¿ ):

(7.34)

This shows that the Eulerian statistics are also caused by large °uctuations about a typical realization (7.34), and suggests the clustering of tracer density for compressible °ows. So far we have studied one-point PDF of the tracer concentration ½, and deduced some its spatial-temporal properties. There are few other ¯ne-scale structures of realizations, that could be gleaned from the FP equation (7.28) and its PDF-solutions. In section 3.3 we mentioned some important geometric characteristics of ½, related to iso-contours ½ (R; t) = ½-const, and the corresponding statistical means expressed through its PDF (7.28). Examples include the mean area enclosed by contours: ½ (R; t) ¸ ½, and the mean \enclosed mass " (7.35)

hS(t; ½)i

=

Z1 ½

hM (t; ½)i

=

Z1 ½

d~½

Z

~½d~½

Z

dR P (t; R; ~½);

dR P (t; R; ~½):

COHERENT PHENOMENA

47

One could easily derive the closed form equations for both quantities, based on (7.28), and ¯nd exact solutions [2], Z µ ¶ ln (½0 (R)e¡ ¿ =½) (7.36) hS(t; ½)i = dR © p 2 ¿ µ ¶ Z ln (½0 (R)e¿ =½) p hM (t; ½)i = dR ½0 (R) © 2 ¿ At large time ¿ À 1, the total area of high density concentration (above ½) decreases in time according to Z p 1 (7.37) hS(t; ½)i ¼ p exp (¡¿ =4) dR ½0 (R); ¼¿ ½ whereas the enclosed mass within the ½-area r Z p ½ (7.38) hM (t; ½)i ¼ M ¡ exp (¡¿ =4) dR ½0 (R) ¼¿ converges monotonically to the total mass of the system Z M = dR½0 (R):

The last result con¯rms our earlier conclusion regarding clustering of tracer in the tightly bounded regions of high density. Let us illustrate a few dynamic features of the clustering process with an example of a uniformly distributed density ½0 (R) = ½0 -const. In this case we use speci¯c area (per unit \2D-volume") of high concentration ½(R; t) ¸ ½, equal to µ ¶ Z1 ln (½0 e¡ ¿ =½) (7.39) s(t; ½) = P (t; ~½)d~½ = © p ; 2 ¿ ½

in terms of R-independent solution P (t; ½) of (7.28) computed in (7.32). At the same time speci¯c mass (per unit \2D-volume") enclosed in such regions µ ¶ Z1 1 ln (½0 e¿ =½) (7.40) m(t; ½) = ½~P (t; ~½)d~½ = © p : ½0 2 ¿ ½

In follows from (7.39)-(7.40) that the speci¯c area drops o® exponentially ³ ¿´ p 1 (7.41) s(t; ½0 ) = ©(¡ ¿ =2) ¼ p exp ¡ ¼¿ 4 as the tracer's mass aggregates (almost entirely) when ¿ ! 1 ³ ¿´ p 1 (7.42) m(t; ½0 ) = ©( ¿ =2) ¼ 1 ¡ p exp ¡ : ¼¿ 4

Furthermore, their large-time asymptotics are independent of the ratio ½=½0 . However, the evolution leading to the asymptotic clustered state (7.41)-(7.42) crucially depends on this ratio ½=½0 . If we take a low \test-level" ½=½0 < 1, which corresponds to initial values of speci¯c quantities: s (0; ½) = 1, m (0; ½) = 1, and let system evolve, we ¯rst see small regions formed, where concentration drops below ½, ½(R; t) < ½. Initially they would cover a small area, but as time goes on, those regions would rapidly grow in size, while the enclosed mass would \leak away" to the outer (cluster) regions, eventually reaching the asymptotic state (7.41),

48


(7.42). At some particular moment ¿ ¤ = ln (½0=½), the speci¯c area would drop to s(¿ ¤ ; ½) = 1=2 - half the initial size. In the opposite case ½=½0 > 1 (high test-level ½), the initial values: s (0; ½) = 0 and m (0; ½) = 0. Here the evolution would ¯rst create a few cluster regions of high concentration, ½(R; t) > ½. Those would continue \sucking in" a sizable portion of the tracer mass, as their area contracts, but the enclosed mass grows, and gradually pass to the same asymptotic state (7.41), (7.42) (see ¯g. 13). As was pointed out in section 2.3 to get further (¯ne scale) geometric structure of the tracer concentration we need to consider its gradient p(R; t) = r½(R; t), and higher derivatives. The tracer gradient evolves according to (2.36), hence its indicator function ©(t; r; ½; p) = ± (½(R; t) ¡ ½) ± (p(R; t) ¡ p) satis¯es equation (3.18), where r was replaced by R. Averaging (3.18) over the velocity ensemble we get the FP-equation for the joint PDF of the tracer concentration and its gradient P (t; R; ½; p) = h©(t; R; ½; p)i

Namely, @ P (t; R; ½; p) @t

(7.43)

=

P (0; R; ½; p) =

h

2 1 D @ 2 0 @R 2

+ Dp2

@2 ½ @R @p

2

@ 2 + Dp2 @½ 2½

^ s (p) + 1 Dp L ^ p (p) + 3D p @ p @ ½ + 18 Ds2 L 2 @ p @½ 8 2 i 2 p + 12 D4 @@p2 ½2 P (t; R; ½; p); ± (½0 (R) ¡ ½) ± (p(R) ¡ p) ;

Here we introduced di®erential operators in the \gradient variable" p ³ ´2 ^ s(p) = 3 @ 22 p2 ¡ 2 @ p ¡ 2 @ p = 3 @ 22 p2 ¡ 2 @ 2 p k p l ; L @p @p @p @p @pk p l ³ ´ 2 ^ p (p) = @ 22 p2 + 18 @ p + 10 @ p: (7.44) L @p @p @p

which represent the contributions of the solenoidal and potential components (superscripts s and p). In general equation (7.43) has no closed-form solution. In case of divergence-free velocities it could be reduced to [21 - 23], h i @ 1 @2 1 s ^s (7.45) P (t; R; ½; p) = D + D L (p) P (t; R; ½; p): 0 2 @t 2 8 2 @R One could show that evolution (7.45) conserves the mean value of the gradient, Z hp(R; t)i = p P (t; R; ½; p)dRdp d½ while the gradient's norm has log-normal PDF. So its typical realization as well as higher moments grow exponentially in time. Furthermore, - 2 ® p (R; t) » p20 exp fDs2 tg : as

Besides, the average contour length ½(R; t) = ½ ¡ const also grows exponentially hL(t; ½)i = l0 exp fDs2 tg ;

starting from initial length l0 , as a consequence of (7.45) and (3.16).

COHERENT PHENOMENA

49

Let us remark that the divergent-free velocities conserve the number of contours, as those cannot appear or disappear, but only evolve in time from their initial value and distribution in space. In the process of evolution initially smooth tracer distribution acquires increasingly complicated spatial structure, its gradients grow and sharpen, and the contours undergo distortion and fractalization. We have shown the schematic view of the process in ¯g. 1a, based on numeric simulations for a somewhat di®erent velocity ¯eld. Once again we see, that the qualitative features of the process are insensitive to speci¯cs of the model. Remark 4. We have studied statistics of equation (7.4) in the Lagrangian and Eulerian formulation, and showed the clustering of the tracer in the presence of the potential velocity component, both on the level of particles and the (Eulerian) ¯elds. Alongside dynamic equation (7.4) one is sometimes interested in the nonconservative tracer transport (see for instance, [47]) ³ ´ @ @ + U (R; t) ½(R; t) = 0; ½(R; 0) = ½0 (R): @t @R The same type is also equation (2.59). Since the Lagrangian equations for particles coincide with (7.10), the clustering" occurs on the level of particles. However, continuous Eulerian ¯elds do not cluster here. This case is similar to a divergentfree one in that it conserves the mean contour number, the mean area, and the R tracer mass M (t; ½) = ½ (R; t) dS, enclosed inside contours ½(R; t) > ½.

7.3. Additional factors. So far we have studied the transport problem by random velocities in the absence of the mean °ow and the molecular di®usion. Furthermore, we used delta-correlated approximations in time for random velocities. While such simpli¯cations are justi¯able under certain conditions, those missing factors should be taken into account at some stage, and spatial-temporal scales. Their incorporation brings about some new features and physical e®ects. In this section we shall brie°y outline a few examples that include additional factor for divergent-free velocities. 7.3.1. Linear shear °ow. We shall consider velocity pro¯les that combines linear (horizontal) shear v x = ®y; vy = 0: with the random component [23]. In this case equation (7.45) is transformed into ³ ´ 2 @ @ + ®y @x ¡ 12 D0 @ 2 P (t; R; ½; p) = @t @R ½ · ¸¾ ³ ´2 2 @ 1 s @ 2 @ @ (7.46) ®px @py + 8 D2 3 @ p2 p ¡ 2 @ p p ¡ 2 @ p p P (t; R; ½; p):

To derive the ¯rst and second moments of p we observe that ensemble means hp ii; hp i pj i, etc. are computed from the one-point PDF (7.46) by integrating out variables p, Z hpi (R; t)i = p iP (t; R; ½; p) dp; ::: Furthermore, evolution equation (7.46) naturally splits into the spatial-temporal part, given by operator @t + L (with di®usive-transport L = ®y@x ¡ D20 ¢) in the

50


l.h.s., and the \p-operator" M p = ®px @p@ y + ::: in the r.h.s. The moments of p evolve according to ¿ À ¿ µ ¶À p1 p1 (@t + :::) = M p¤ p2 p2 0 1+ * + * 2 p1 p 21 (@t + :::) p1 p 2 = M p¤ @ p 1 p2 A p 22 p 22 where

¤

designates the adjoint di®erential operator i D2 h 2 2 Mp¤ = ¡®p1 @p@ 2 + 3p ¢p ¡ 2 (p ¢ r) + 2 (p ¢ r) 8 Applying Mp¤ to the linear and quadratic functions in p we get µ ¶ µ ¶ p1 0 Mp¤ = p2 ¡®p 1 0 1 0 ¡ ¢ 1 2 p1 D2 14 p21 + 34 p 22 A Mp¤ @ p1 p 2 A = @ ¡®p21 ¡ D¡22 p 1 p 2 ¢ 2 3 2 1 2 p2 ¡2®p1 p 2 + D2 4 p 1 + 4 p 2

Hence follows the evolution of moments µ ¶ µ ¶ hp 1 i 0 (7.47) (@t + L) = hp 2 i ¡® hp1 i 0 - 2® 1 0 - 2 ® 3D2 - 2 ® 1 D2 p + 4 p2 p1 4 - 1® 2 D2 A = @ A (@t + L) @ hp 2i - ® -1 p2 ®2 i - ¡® ® p 1 ¡ 2 hp1 pD 3D 2 2 2 2 p2 p 1 ¡ 2® hp 1 p2 i + 4 p 2 4

Equations (7.47) are of the type: (@t + L) X = M X, with 2 £ 2 or 3£3 matrix M .(for vector X of the \¯rst" and \second" moments, respectively). Since operator L and matrix M act on independent variables, hence commute, solutions of (7.47) combine di®usive transport in the R-space with matrix-exponential etM . If we drop di®usive term L = 0, i.e. assume statistically homogeneous (R- independent) state of the tracer concentration: h½i ; hr½i ::: = const, equations (7.47) turn into ordinary di®erential systems. In the 2D-case (¯rst moments) solutions hp x (t)i = hpy (t)i =

p x(0) p y (0) ¡ ®p x(0)t:

increase linearly in t. So the mean gradient is \tipped over" by the shear into the negative vertical position of increasing strength (horizontal stretching of isocontours). The corresponding ODE system for the second moments has matrix 2 1 3 3 0 4 4 M = D2 4 ¡¯ ¡ 12 0 5 3 ¡2¯ 14 4 expressed through the dimensionless parameter ¯ = D®2 - mean \shear strength" over \time-integrated enstrophy" of the random component (7.9). Its eigenmode solutions e ¸t X have eigenvalues ¸, determined by ¡ ¢2 (7.48) ¸ + 12 D2 (¸ ¡ D2 ) = 32 ®2 D2 ;

COHERENT PHENOMENA

51

For small ratio ¯ = ®=D2 ¿ 1 (\weak mean shear" compared to the \integrated enstrophy" of °uctuations) the roots of (7.48) are approximated by ¡ ¢ ¸1 = D2 1 + 23 ¯ 2 ¡1 ¢ ¸ 2;3 = ¡D2 2 § ij¯j

Hence at large t the gradient's evolution is dominated by the °uctuating velocities. At the other extreme, ¯ = ®=D2 À 1, equation (7.48) has approximate eigenvalues ¡ ¢1=3 ¡ ¢1=3 i(2=3)¼ ¡ ¢1=3 ¡i(2=3)¼ ¸1 ¼ 32 ®2 Ds2 ; ¸ 2 ¼ 32 ® 2 Ds2 e ; ¸ 2 ¼ 32 ® 2 Ds2 e : Since ¸ 2;3 have negative real part, we get an asymptotic solution of (7.47) of the form n¡ - 2 ® ¢1=3 o p (t) » exp 32 ®2 D2s t

¡ ¢1=3 valid on time scales 32 ® 2 Ds2 t À 1. Three components of the principal eigenmode (corresponding to the largest eigenvalue ¸ 1 ) are plotted below as functions of shear rate ¯ (¯g. 15). - dimensionless ® - ® Starting with an isotropic state at ¯ = 0 : p 2x = p 2y = 1, hp x py i = 0, the system becomes increasingly anisotropic: - 2® - ® p y ¼ 1; p 2x ! +0; hpx p y i ! ¡0 as ¯ increases. So strong shear alines vertical gradients and decorrelates horizontal ones and the cross-terms, consistent with the general pattern of stretching and °attening of iso-contours by shearing.

7.3.2. Molecular di®usivity. As we mentioned earlier random velocity °uctuations cause initially smooth tracer density to develop small scale structures with steepening gradients. In the real physical situation the molecular di®usion °ux would tend to smooth out small scales, so the above pure \transport-dynamics" could last only for a limited time. From the mathematical standpoint molecular di®usion would turn the problem into a second order stochastic partial di®erential equation (2.37), whose one-point PDFs have no closed-form FP equation. Here we shall con¯ne ourselves with estimating the time scale of the initial (\explosive") phase, and follow the exposition of [23]. To this end we consider powers ½n (R; t); n = 1; 2; :::; that obey a nonclosed system of equations that follow from (2.37) ³ ´ @ @ + U (R; t) ½n (R; t) = ·¢½n (R; t) ¡ ·n(n ¡ 1)½n¡2 (R; t)p 2 (R; t): @t @R The ensemble average over random velocities yields (non-closed) equations for statistical moments ¡ ¢ ® (7.49) @@t h½n (R; t)i = 12 D0 + · ¢ h½n (R; t)i ¡ ·n(n ¡ 1) ½n¡ 2 (R; t)p 2 (R; t) Assuming · ¿ D0 we could write (7.49) in the integral form h i 2 h½n (R; t)i = exp 12 D0 t @ 2 ½n0 (R) ¡ @R Zt h i® 2 (7.50) ¡·n(n ¡ 1) d¿ exp 12 D0 (t ¡ ¿ ) @ 2 ½n¡ 2 (R; ¿ )p 2 (R; ¿ ) @R 0

52


To estimate the last term of (7.50) we exploit the FP equation (7.45) for the joint PDF of ½ and p = r½, derived for zero molecular di®usivity. ® As the result we get an (approximate) closed-form equation for moments ½n¡2 p2 , whose solution h i - n¡2 ® 2 (7.51) ½ (R; t)p2 (R; t) = exp Ds2 t + 12 D0 t @ 2 ½n¡2 (R)p 20 (R) 0 @R

Substitution of (7.51) in (7.50) yields the conditions when the last term in the r.h.s. of (7.50) gives negligible contribution to evolution of h½n i, and could be dropped. They impose certain constraints on spatial scale R 0 of the initial tracer concentration ½0 , and the time range. Namely, [23] Ds2 R20 : ·n2 More detailed analysis could be given in special cases, e.g. constant mean gradient1 of ½, [55] This case corresponds to solving equations (2.29), (2.37) with the initial data Ds2 R20 À ·n(n ¡ 1); D2s t ¿ ln

½0 (R) = GR; p0 (R) = G: As above we shall con¯ne ourselves to the 2D case. Breaking ½ into its mean and °uctuating components ½(R; t) = GR + e½(R; t)

we get the equation for ~½ ³ ´ @ @ (7.52) + U (R; t) e½(R; t) = ¡GU (R; t) + ·¢e½(R; t); e½(R; 0) = 0: @t @R

Unlike the previous problems, this one has a stationary (limiting) probability distribution as t ! 1, both for the tracer and its gradient. Lately this problem has drawn a considerable attention both from the theoretical and experimental standpoint, see papers [56 - 62]. Their authors have used computer simulations and phenomenology to derive among other things \long exponential tails" of the \gradient's PDF", while paper [62] has shown a \slowly decaying tail" for ½ itself in its limiting (stationary) state. Coming back to (7.52) let us notice that the variance of the gradient °uctuations p(R; t) = re e ½(R; t) satis¯es ® D G2 lim pe2 (R; t) = 0 ; t!1 2· whereas the variance of the tracer concentration is given by (7.53)

(7.54)

D

½ Zt E 2 e (R; t) = D0 G ½ d¿ 1 ¡ 2

0

2· D0 G 2

¾

-

® e2 (R; ¿ ) p

Formulae (7.53), (7.54) allow one to estimate the relaxation time (to a stationary regime) · ¸ D0 + 2· s D2 T0 » ln 2· 1This model could represent tracer distribution between a constant-strength (pollution) source and the dissipation-region, subjected to random wind-shear

COHERENT PHENOMENA

53

as well as the variance of the stationary (limiting) state · ¸ D E D D0 + 2· lim e ½2 (R; t) » 0s G2 ln : t!1 D2 2·

Here coe±cients: D0 » ¾ 2ut0 and D0=Ds2 » l20 , are expressed through statistics of random velocities, namely its variance - ¾ 2u , and spatial, temporal correlation radii t0 , l0 . So we get the relaxation time µ ¶ l2 ¾ 2 t0 T 0 ¼ 20 ln 1 + u ¾ u t0 2· logarithmically dependent on the molecular di®usivity ·. As for the stationary variance of e ½ we get an estimate · 2 ¸ D E ¾ t0 2 2 2 ½ » G l0 ln u e · provided · ¿ ¾ 2u t0 .

7.3.3. Velocity °uctuations with ¯nite temporal correlation. One way to account for the ¯nite temporal correlation radius was mentioned in section 4.1, based on the diffusion approximation method. This method would still require some constraints on the correlation-radius, though less restrictive than the \delta-correlated" ones. One could also discover some new physical phenomena, due to the ¯nitude of temporal correlations. Those would be illustrated by two problems: sedimentation [23, 63], and the °oating tracer on random surface z (R; t), with statistically independent \driving velocities". The ¯rst problem has manifestly anisotropic di®usion tensor relative to the sedimentation direction. We shall denote the sedimentation velocity by v, and assume its °uctuating component to have the energy spectrum E (k; t) = E (k) expf¡jtj=¿ 0 g; to fall-o® exponentially in time with correlation radius. ¿ 0 The limiting di®usion tensor, as t ! 1, was computed in [23, 63] Z 1 p 1 Dij (v) = dkE(k)¢ij (k) ; 2 v k 1 + p cos2 µ µ ¶ ki kj ¢ij (k) = ±ij ¡ 2 ; k Here cos µ = kv=kv measures the angle between, while function p(k; v) =

(kv)¿ 0 : 1 + ·k2 ¿ 0

After we project tensor Dij (v) on the vertical (sedimentation) direction marked by k, and the transverse (perpendicular) plane ?, the corresponding di®usion rates Dk (v)

=

4¼ v

Z1

kdkE (k)fk (k; v);

4¼ v

Z1

kdkE (k)f? (k; v );

0

D? (v)

=

0

54


could be expressed through the pair of functions · µ ¶¸ 1 1 fk (k; v) = arctan(p) + arctan(p) ¡ 1 ; p p · µ ¶¸ 1 1 f ? (k; v) = arctan(p) ¡ arctan(p) ¡ 1 p p depending on p = p(k; v). For small values p; i.e. sedimentation velocity v ¿ l0 =¿ 0 - spatial over temporal velocity correlations, both functions fk (k; v); f? (k; v) approach values close to 2p , 3 corresponding to isotropic di®usion una®ected by the sedimentation drift v. For large values of p (v À l0 =¿ 0 ) the strong anisotropy shows up fk (k; v) = 2 f ? (k; v) » = ¼=2:

Let us stress that the resulting anisotropy is due to the ¯nitude of temporal velocity correlations, and would disappear in the delta-correlated case. Let us turn to the second problem. Here the basic transport equation (7.4) has to be modi¯ed to include an additional random parameter z @ @t ½(R; t)

+ @ @R fU (R; z(R; t); t)½(R; t)g = 0; R = fx; yg: We assume the total mass Z dR½(R; t) = 1 (7.55)

The analysis of (7.55) would involve two \averaging steps", over the fU g- and fzg-ensembles. First we shall average over random velocities fU (R; z; t)g, and use the di®usion approximation. This yields the FP equation for the mean concentration R(R; t) = h½(R; t)iU @R @t @2 @ R®@ R¯

Zt

dt0

0

@ i@R ®

Zt 0

(7.56)

Z

dk? 0

dt

Z

Z1

¡1

dk?

(R; t) =

dkz E ®¯ (k2? + k2z ; t ¡ t0 ) expfikz Z(R; t; t0 )gR(R; t) ¡ Z1

¡1

2 kz dkz E ®¯ (k? + kz2 ; t ¡ t0 ) expfikz Z(R; t; t0 )g @z( R;t) @R¯ R(R; t)

where E ®¯ designates the velocity spectral (energy) tensor, as function of the horizontal and vertical wave-vectors µ ¶ k? ®k?¯ 2 2 2 2 E ®¯ (k? + kz ; t) = E(k? + kz ; t) ±®¯ ¡ 2 k? + kz2 while Z(R; t; t0 ) = z(R; t) ¡ z(R; t0 ) measures surface variation at di®erent times t; t0 . We view (7.56) as stochastic equation in random parameter Z(R; t; t0 ).

COHERENT PHENOMENA

55

Surface elevation z is assumed to be Gaussian ¯eld with the mean and correlation function ® hz(R; t)i = 0; z(R; t)z(R 0 ; t0 ) = Bz (R ¡ R 0 ; t ¡ t0): Introducing new function

F (R; t; t0 ; kz ) = expfk2z Dz (t ¡ t0 ) + ikz Z(R; t; t0 )gR(R; t); linked to R through the \random" phase Z, and deterministic Dz (t) = Bz (0; 0) ¡ Bz (0; t), so that R(R; t) = expf¡k2z Dz (t ¡ t0) ¡ ikz Z(R; t; t0)gF (R; t; t0; kz ) FP equation (7.56) takes the form @ R(R; t) @t

=

@2 @R® @R¯

Zt 0

dt

0

Z

dk ?

Z1

dkz E ®¯ (k2? + k2z ; t ¡ t0 )

¡1

expf¡kz2 Dz (t ¡ t0)gF (R; t; t0 ; kz ) ¡ Zt Z Z1 @ 0 i @R dt dk? kz dkz E ®¯ (k2? + kz2 ; t ¡ t0 ) ®

¡1

0

(R; t) expfikz Z(R; t; t )g @z@R R(R; t): 0

(7.57)

¯

Field R(R; t) is a functional of the random surface elevation, e e R(R; t) = R[R; t; z(R; ¿ )]

e e and the mean value of functional F (R; t; t0; kz ) over random ¯eld Z( R; ¿ ) is computed via ® hF (R; t; t0 ; kz )iz = expfk2z Dz (t ¡ t0 ) + ikz Z(R; t; t0 )gR(R; t) z = D n oE e e¿ ) + ikz Bz (R e ¡ R; e e ¡ R; e R[(R; t; z(R; ¿ ¡ t) ¡ B z (R ¿ ¡ t 0) ] : z

So it depends on the mean density subjected to a \functional shift". We are interested in the dispersion-tensor ¿Z À § ®¯ (t) = R ®R¯ R(R; t)dR = 12 ¾ 2 (t)±®¯ z

reduced to scalar ¾ 2 due to isotropy. The later is derived by averaging R over the ensemble of z-realizations, hRiz . To this end we need to average equation (7.57), in fzg. Dropping higher order terms, ¾ 4z , one could get the following expression for the di®usion coe±cient D(t) = 2

Zt 0

d¿

Z

dk?

Z1

¡1

dkz E ®® (k2?

+

@ 2 ¾ (t) @t

=

2 k2z ; ¿ )e¡k zD z( ¿)

Z

dR hF (R; t; t ¡ ¿ ; kz )iz :

Thus surface elevation has double e®ect on the tracer di®usion. On the one hand it changes the e®ective (random) velocity spectrum, on the other hand it transforms the tracer density. Both e®ects are due to ¯nitude of the temporal correlation radius. As for the delta-correlated case the tracer statistics become

56


independent of the surface elevation and have the same form as for the tracer in an ideal °uid °ow. 8. Caustical structure of wave-fields in random inhomogeneous media. We have studied wave propagation in random media in section 2.4. using parabolic approximation (2.38). From the mathematical stand-point this problem is similar to the passive tracer di®usion by potential velocity ¯elds. We have discovered a fundamental property of potential °ows - the clustering of tracer. For wave problems clustering (of wave intensity) shows up as caustic structures brought about by random focusing and defocusing of wave ¯elds by the media. The basic techniques of delta-correlated approximation, used in the previous section for the statistical analysis of \cluster structures", is not applicable anymore. Indeed, the longitudinal correlation-radius (in the beam-direction) for the beam phase is now comparable to the beam track [5, 40, 41]. So the dynamic equation (2.39) for the wave intensity, has little practical value for the analysis of random media. However, the one-point PDF of the wave intensity, derived from (2.38) allows one to adopt some ideas of the statistical topography [24]. In this regard let us mention pioneer papers [64, 65] (see also [66]), that applied the theory of large deviations to the analysis of wave propagation in turbulent media. We consider an iso-contour of constant intensity I (x; R) = I, in the ¯xed crosssectional plane2 x = const, and its singular indicator (8.1)

©(x; R; I) = ± (I(x; R) ¡ I) ;

as a \functional of the media". The ensemble-mean of (8.1) de¯nes the one-point PDF (8.2)

P (x; R; I) = h©(x; R; I)i = h± (I(x; R) ¡ I)i :

Other quantities could be expressed through (8.2), like mean area enclosed by contour: I(x; R) > I Z1 Z e e (8.3) hS(x; I)i = dI dRP (x; R; I) I

and the mean ¯eld power enclosed within iso-contours Z1 Z e e e (8.4) hE (x; I)i = IdI dRP (x; R; I): I

Further information on the ¯eld intensity and its ¯ne structure could be derived from the cross-sectional gradient p(x; R) = rR I(x; R), and the joint PDF of I and p, Indeed, (8.5) 2The

P (x; R; I; p) = h± (I(x; R) ¡ I) ± (p(x; R) ¡ p)i : hL(x; I)i =

Z

dR

Z

dpjp(x; R)jP (x; R; I; p)

role of time in optical problems is often played by the longitudinal variable x.

COHERENT PHENOMENA

57

would give the mean contour length at level I(x; R) = I-const. The higher (second) derivatives of I would yield an estimate of the mean number of closed contours at level I, given approximately (modulo non-closed contours) by the integral hN (x; I)i = (8.6)

hNin (x; I)i ¡ hNo ut (x; I)i = Z 1 dR h·(x; R; I)jp(x; R)j± (I(x; R) ¡ I)i ; 2¼

Here N in (x; I), N out (x; I) count contours with the inward- and outward-looking gradient p, (i.e. \mountain peaks" and closed \valleys" of topography I (x; R), truncated at the level I), and · designates the curvature of the contour

(8.7)

@ 2I (x;R ) ¡p 2y (x; R) @z2

·(x; R; I) = @ 2I (x;R) @ 2I( x;R) ¡ p2z (x; R) @y 2 + 2p y (x; R)pz (x; R) @y@z p3 (x; R)

:

Here and henceforth we shall consider a plane incident wave, whose one-point PDFs will be independent of R due to the (transverse) spatial homogeneity. Thus statistical averages (8.3)-(8.7) could be viewed as speci¯c (per unit area) values of the relevant quantities. The natural length scale in the cross-sectional plane x p= const independent of the media is the size of the ¯rst Fresnel zone Lf (x) = x=k, that measures the light-shadow di®raction region of a non-transparent screen (see for instance, [40, 41]). Then the mean speci¯c values of the contour lengths and the number of contours are given by the following dimensionless expressions hl(x; I)i = Lf (x) hjp(x; R)j± (I(x; R) ¡ I)i ;

(8.8)

(8.9)

hn(x; I)i =

1 L2 (x) h·(x; R; I)jp(x; R)j± 2¼ f

(I(x; R) ¡ I)i :

In fact, formula (8.9) gives the mean excess of contours with the opposite orientation of gradient (normal) vector, within the ¯rst Fresnel zone. As above we assume random ¯eld " (x; R) to be Gaussian, homogeneous, isotropic with the correlation and spectral functions B" (x 1 ¡ x2 ; R 1 ¡ R 2 ) = h"(x 1 ; R1 )"(x2 ; R 2 )i = Z1 Z = dqx dq© "(qx; q) expfiqx(x 1 ¡ x 2 ) + iq(R1 ¡ R2 )g; ¡1

(8.10)

© "(qx ; q) =

1 (2¼)3

Z1

¡1

dx

Z

dRB" (x; R) expf¡iqx x ¡ iqRg:

The x-axis can be roughly divided into three regions, depending on the properties of the ¯eld intensity ² region of weak (intensity) °uctuations; ² region of strong focusing; ² region of strong °uctuations.

58


8.1. Region of weak °uctuations. In the region of weak °uctuations the statistics of intensity are well described by the method of smooth perturbations. Here one writes a perturbation expansion of the amplitude level function (pre-exponent of amplitude) {(x; R) =

1 2

ln I(x; R)

that could be viewed as Gaussian random ¯eld. In this case the mean and the variance of the amplitude level are related by h{(x; R)i = ¡¾ 2{ (x):

Let us introduce an important parameter, called scintillation index ¯ 0 (x) = 4¾ 2{ (x) (see for instance, [40, 41]). For small ¯ 0 (x) ¿ 1, the variance of the wave intensity ® ¾ 2I (x) = I 2 (x; R) ¡ 1 ' ¯ 0 (x) is approximated by ¯ 0 , so the intensity is a log-normal, random process with the one-point PDF ½ ³ 1 ´¾ 1 1 (8.11) P (x; I) = p exp ¡ ln 2 Ie 2 ¯0(x) 2¯ 0 (x) I 2¼¯ 0 (x)

The region of weak °uctuations refers to ¯ 0 (x) · 1. Here the typical realization of log-normal process (8.11) falls o® exponentially © ª I ¤ (x) = exp ¡ 12 ¯ 0 (x) and the essential statistics (like moments hI n (x; R)i) are formed by large deviations about I ¤ . Furthermore, random process I (x) admits the majorant estimates, for instance, half of all realizations (probability 1=2) obey the inequality © ª I(x) < 4 exp ¡ 14 ¯ 0 (x)

at any distance x along the beam. This clearly suggest an onset of the \clustering" of the wave intensity. The knowledge of PDF (8.11) yields some quantitative characteristics of the \cluster structure", such as speci¯c mean area hs(x; I)i and speci¯c mean power he(x; I)i, enclosed by iso-contours I(x; R) > I, see [24]. The evolving cluster-structure follows changing scintillation index ¯ 0(x), and depends strongly on the level I. In the most interesting (high intensity) case I > 1 , we have hs(0; I)i = 0; he(0; I)i = 0 at the initial plane. As ¯ 0 (x) grows small cluster regions of high intensity I(x; R) > 1, are formed,, that persist over certain distances and rapidly absorb a signi¯cant fraction of the wave power. Further down the path, their area contracts, as ¯ 0 (x) increases, but the entrailed power keeps growing, creating bright spots within dark regions. So one observes the e®ect of random focusing of radiation by inhomogeneous patches of the media. In the region of weak °uctuations the gradient of the amplitude level rR Â(x; R) is statistically independent of Â(x; R) itself. This allows one to compute the speci¯c mean contour length I(x; R) = I , and to estimate the speci¯c mean number of contours. Indeed, in the region of weak °uctuations the gradient q(x; R) = rR Â(x; R) has Gaussian PDF, and one has [24] q (8.12) hl(x)i = 2Lf (x) hjq(x; R)ji IP (x; I) = Lf (x) ¼¾ 2q (x)IP (x; I);

COHERENT PHENOMENA

hn(x; I)i = (8.13)

59

® @ ¡ 1¼ L2f (x) q 2 (x; R) I @I IP (x; I) = 2 2 ³ ´ L f (x)¾ q (x) 1 ln Ie 2 ¯ 0(x) IP (x; I): ¼¯ 0 (x)

Let © us observe ª formula (8.13) turns to zero for the typical realization I = I0 (x) = exp ¡ 12 ¯ 0 (x) . This means that the \typical intensity" has statistically equal numbers of contours I(x; R) = I0 with di®erent orientations (bright \peaks" and dark \valleys"). The above discussion was based on the fundamental parameter ¯ 0 (x) determined by the media. If one assumes delta-correlated refraction "(x; R) in x, the correlation function (8.10) is approximately given by Z1 B" (x; R) = ±(x)A(R); A(R) = dxB" (x; R) ¡1

(see for instance, [5, 40, 41]). In the turbulent media [5, 40, 41] one could compute ¯ 0 (x) in terms of the 3D spectral function of coe±cient ", ©" (0; q) = © "(q), - a function of the planar wave-vector q, Z1 h ³ 2 í 2 2 2 (8.14) ¯ 0 (x) = 4¾ { (x) = 2k ¼ x dqq©" (q) 1 ¡ qk2x sin qkx 0

provided the turbulence \¯lls in" the entire space. If random inhomogeneous "(x; R) is concentrated in a narrow band ¢x ¿ x (random phase screen), then Z1 h ³ 2 í 2 2 2 (8.15) ¯ 0 (x) = 4¾ { (x) = 2k ¼ ¢x dqq©" (q) 1 ¡ cos qkx : 0

The typical dielectric permeability " in (8.14), (8.15) has spectral function © "(q) = AC"2 q¡ 11=3 expf¡(q 2 =· 2m )g with constant A = 0:033, coe±cient C"2 , depending on the ambient °ow parameters, and the turbulence microscale · m : Assuming large value of the so called wave parameter D(x) = · 2m x=k À 1 we derive the following scaling law for ¯ 0 (x) in two cases ½ 0:307C"2 k7=6 x 11=6 ; ¢x = x - continuous media (8.16) ¯ 0 (x) = : 0:563C"2 k 7=6 x 5=6 ¢x; ¢x ¿ x - phase screen Hence follows the variance of the amplitude level-gradient Z1 h ³ 2 í 2 1=6 k 2¼2 x ¾ q (x) = 2 dqq 3 ©" (q) 1 ¡ qk2x sin qkx = L1:476 (x)¯ 0 (x): 2 (x) D f

0

Finally, one could ¯nd the dependence of the mean (speci¯c) length and contournumber hl(x; I)i ; hn(x; I)i, (8.12) and (8.13), on parameters ¯ 0 (x) and D(x) . Their dependence on the turbulent microscale indicates the presence of small ripples superimposed on the large-scale random landscape of intensity. The ripples do not a®ect the distribution of high concentration areas and the wave-power within them, but they could bring about the roughening and fragmentation of iso-contours.

60


As we mentioned earlier the above description holds for small scintillation values ¯ 0 (x) · 1. As ¯ 0 (x) grows larger (with x), the method of smooth perturbations ceases to work, and one has to deal with the fully nonlinear \complex phase" equation. This region of strong focusing poses di±cult analytic problem, and we won't discuss it here. The further growth of ¯ 0 (x), e.g. ¯ 0 (x) ¸ 10 , would take one into the region of strong intensity °uctuations, where the statistics of intensity saturate. 8.2. Strong intensity °uctuations. In the region of strong °uctuations the intensity PDF could be approximated by ( ) Z1 1 (ln z ¡ (¯(x) ¡ 1)=4)2 (8.17) P (x; I) = p dz exp ¡zI ¡ ¯(x) ¡ 1 ¼(¯(x) ¡ 1 0

® (see for instance, [24, 67, 68]), with a di®erent scintillation index ¯(x) = I 2 (x) ¡1. In the turbulent media, either continuous, or localized within the phase screen, one ¯nds ( ¡2=5 1 + 0:861¯ 0 (x); ¢x = x; (8.18) ¯(x) = ¡2=5 1 + 0:429¯ 0 (x); ¢x ¿ x;

(see for instance, [5]), with ¯ 0 (x) given by (8.16). Let us remark a gradual \slide down" of ¯(x) along with the increase of ¯ 0 (x) . Indeed, the long-distance limit ¯ 0 (x) ! 1, yields ¯ (x) = 1 , while moderate values ¯ 0 (x) = 1 corresponds to ¯(x) = 1:861. Let us also remark that PDF (8.17) is not applicable in the narrow vicinity of I = 0, the larger ¯ 0 (x) the smaller is the window about 0. It has to do with in¯nitely large values of the \negative" moments 1=I n (x; R), that follow from (8.17). However, the ¯nite values of ¯ 0 (x), no mater how large, give ¯nite moments h1=I n (x; R)i, hence vanishing probability of I = 0, P (x; 0) = 0. The existence of the narrow band near I = 0, has no e®ect on the behavior of \positive" moments hI n (x; R)i at large ¯ 0 (x). Asymptotic formulae (8.17) (8.18) describe a transition into the region of saturated intensity °uctuations, ¯(x) ! 1. In this region the mean speci¯c area of high intensity I(x; R) > 1, and speci¯c power enclosed inside, are given by (8.19)

P (I) = e¡I ; hs(I)i = e ¡I ; he(I)i = (I + 1)e¡ I :

Hence the fraction of the mean area and the mean power depend only on level I. For large I, these fractions are insigni¯cant. Exponential distribution of PDF (8.19) implies that complex random ¯eld u(x; R) is Gaussian. Furthermore, the gradient of the cross-sectional amplitude is statistically independent of the wave ¯eld intensity, and is also Gaussian [24]. Besides it has no statistical dependence on the second derivatives of intensity in cross-variables. In this case, one could compute the mean speci¯c contour length and the contour number explicitly q hl(x)i = Lf (x) 2¼¾ 2q (x)Ie¡ I ; (8.20)

hn(x)i

=

2L2f (x)¾ 2q (x) ¡ ¼

I¡

1 2

¢

e ¡I ;

COHERENT PHENOMENA

61

where (8.21)

¾ 2q (x) =

1:476 D 1=6 (x)¯ 0 (x): L2 f ( x)

Let us remark that formula (8.20) does not apply in the narrow vicinity of zero intensity I = 0. For the latter we expect hn(x; 0)i = 0. Formula (8.20) shows the mean contour length and contour number continue to grow with ¯ 0 (x), though the total contour-area and the power enclosed therein, remain constant. Such process leads to increasing roughness and fragmentation of iso-contours. The explanation lies in the dominant role played here by the interference of partial waves coming from various directions. The dynamics of iso-contours depends strongly on the balance between focusing and defocusing of radiation by di®erent patches of turbulent media [69]. The focusing by large scale inhomogeneities results in high intensity peaks on the random landscape of I. In the regime of maximal focusing (¯ 0 (x) » 1) about half of the total power is concentrated in such high narrow peaks. As ¯ 0 (x) increases further the defocusing prevails, that tends to smooth out high peaks and create highly fragmented (interferential) landscape with large number of small peaks near I » 1. Besides scintillation parameter ¯ 0 (x) the mean contour length and the number of contours depend on the wave parameter D(x), hence both would grow as the microscale of inhomogeneities goes down. This process re°ects a superposition of the large scale intensity landscape with small ripples, due to the wave scattering on small scale inhomogeneities. In this section we attempted to give a qualitative explanation of the cluster (caustic) structure of the wave ¯eld in the cross-sectional plane, for incident plane wave in the turbulent media, and to quantify and estimate some its statistical topography. In general, such problems have many parameters. However, when con¯ned to a ¯xed cross-section, the solution is described by a single parameter ¯ 0 - the variance of the intensity-level in the region of weak °uctuations. We analyzed two extreme asymptotic cases, that correspond to the weak and strong (saturated) °uctuations of intensity. We should remark that the above asymptotic formulae could apply only certain limits depending on the intensity level I. As I goes down the applicability range of these formulae should extend. The most interesting case from the standpoint of applications involves the region of the developed caustical structure. Its detailed analysis would require the knowledge of joint PDFs of I and cross-sectional gradients at arbitrary distances along the beam. Such analysis could be done either using approximate methods for all possible values of essential parameters [61], or by the numeric simulation as in papers [44, 45]. 9. Conclusion We conclude with some general \philosophical" comments, that follow from our work ² For many dynamical systems statistical (mean) characteristics show little resemblance to individual realizations, and sometimes gives contradictory results. For such systems the traditional \moment-based" description carries little information. Instead they require a statistical description in terms of PDFs (at least \single-point" or \single-level")

62


² However, such stochastic dynamics could often exhibit some statistically coherent physical phenomena, that occur with probability one, i.e. for almost all realizations of the process. One could suspect coherent phenomena to be abundant in the nature, as they appear in the most simple systems, described by ordinary di®erential equations, like (2.1). No other physical model would beat it in simplicity! The basic statistical model for positive-valued parameters of coherent processes is furnished by the log-normal process. ² The coherent phenomena are largely independent of the speci¯c model of °uctuating parameters. In some cases they allow a complete characterization in terms of single-point (in space-time) PDFs of the process, which could be deduced by the method of statistical topography. Of course, certain parameters of a particular phenomenon (like characteristic time and space scales for clustering) could depend on the speci¯c random model. ² Coherent phenomena could be particularly relevant to physical problems, where the \ensemble means" (and \ensembles" themselves) are not available, and experimentalist deals most often with speci¯c realizations. This applies, in particular to the physics of the atmosphere and ocean. References [1] Klyatskin V I Uspekhi Fiz. Nauk 37 531 (1994) [Physics - Uspekhi 37, 501 (1994)]. [2] Klyatskin V I, Saichev A I Zh. Eksp. Teor. Fiz. 111 1297 (1997) [JETP 84 716 (1997)]. [3] Klyatskin V I, Statisicheskoe Opisanie Dinamicheskich Sistem s Flukyuiruyuschimi Parametrami (Statistical Description of Dynamical Systems with Fluctuating Parameters) (Moscow: Nauka, 1975). [4] Gurbatov S, Malakhov A, Saichev A Nelineinie Sluchaini Volni v Sredach bes Dispersii (Moscow: Nauka, 1990) [Translation into English Nonlinear Random waves and Turbulence in Nondispersive Media: Waves, Rays and Particles (Manchester: Manchester University Press, 1991)]. [5] Klyatskin V I Stochasticheskie Uravneniya i Volni v Sluchaino - Neodnorodnoi Srede (Stochastic Equations and Waves in Randomly Inhomogeneous Medium) (Moscow: Nauka, 1980); Ondes et equations stochastiques dans les milieux aliatoirement non homogenes (Besancon Cedex France: Les editions de physique, 1985). [6] Kulkarny V A, White B S Phys. Fluids 25 1770 (1982). [7] White B S SIAM J. Appl. Math. 44 127 (1983). [8] Zwillinger D, White B S Wave Motion 7 207 (1985). [9] Klyatskin V I Waves in random media 3 93 (1993). [10] Klyatskin V I Metod Pogrugeniya v Teorii Rasprostraneniya Voln (The Imbedding Method in Wave Propagation Theory) (Moscow: Nauka,1986). [11] Scattering and Localization of Classical Waves in Random Media (ed. P Sheng) (Singapore: World Science, 1990). [12] Klyatskin V I, Saichev A I Uspekhi Fiz. Nauk 162 161 (1992) [Soviet Physics Usp. 35 231 (1992)]. [13] Klyatskin V I , in Progress in Optics XXXIII (ed. E. Wolf) (Amsterdam: North-Holland, 1994) p.1. [14] Nicolis G, Prigogin I Exploring Complexity, an Introduction (New York: W H Freeman and Company, 1989). [15] Ziman J M Models of Disorder, The theoretical physics of homogeneously disordered systems (Cambridge: Cambr. Univ. Press, 1979). [16] Kac M Bull. Amer. Math. Soc. 49 314 (1943). [17] Rice S O Bell Syst. Tech. J. 23 282 (1944); 24 46 (1945), Selected Papers on Noise and Stochastic Processes (ed. N Wax) (New York: Dover, 1954). [18] Longuet-Higgins M S Proc. Cambridge Philos. Soc. 52 234 (1956), 53 230 (1957), Philos. Trans. R. Soc. London Ser. A249 321 (1957), A250 157 (1957). [19] Swerling P I.R.E. Tranc. Inf. Theory IT-8 315 (1962).

COHERENT PHENOMENA

63

[20] Isichenko M B Rev. Modern Phys. 64 961 (1992). [21] Klyatskin V I, Woyczynski W A, in Stochastic Models in Geosystems (eds. S A Molchanov, W A Woyczynski) (New York: Springer-Verlag, 1996) p. 171. [22] Saichev A I, Woyczynski W A, in Stochastic Models in Geosystems (eds. S A Molchanov, W A Woyczynski) (New York: Springer-Verlag, 1996) p. 359. [23] Klyatskin V I, Woyczynski W A, Gurarie D, in Stochastic Modelling in Oceanography (eds. R J Adler, P Muller, B L Razovsky (Boston: Birkhauser, 1996) p. 221, J. Stat. Phys. 84 797 (1996). [24] Klyatskin V I, Yakushkin I G Zh. Eksp. Teor. Fiz. 111 2044 (1997) [JETP 84 1114 (1997)]. [25] Zirbel C L, Cinlar E, in Stochastic Models in Geosystems (eds. S A Molchanov, W A Woyczynski) (New York: Springer-Verlag, 1996) p.459. [26] Mesinger F J. Atmos. Sci. 22 479 (1965). [27] Mesinger F, Mintz Y Tech. Rep. 4, 5 (Los Angeles: Dep. Meteorology, Univ. of California, 1970). [28] Gardiner C W Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Berlin: Springer - Verlag, 1985). [29] Yaroshchuk I O Chislennoe Modelirovanie Pasprostraniniya Ploskich Voln v Sluchaino Sloistich Lineinich i Nelineinich Sredach (Numerical Modeling of the Propagation of Plane Waves in Randomly Layered Linear and Nonlinear Media), Ph.D dissertation (Vladivostok: Paci¯c Oceanological Institute, 1986). [30] Casti J, Calaba R Imbedding Methods in Applied Mathematics (Reading, MA: AddisonWesley, 1973). [31] Bellman R, Wing G M An Introduction to Invariant Imbedding (Philadelphia: SIAM, 1992). [32] Kagiwada H H, Kalaba R Integral Equations Via Imbedding Methods (Reading, MA: AddisonWesley, 1974). [33] Guzev M A, Klyatskin V I, Popov G I Waves in Random Media 2 117 (1992). [34] Pedlosky J Geophysical Fluid Dynamics (New York: Springer-Verlag, 1982). [35] Klyatskin V I Izvestiya RAN, Fiz. Atm. i okeana 32 824 (1996) [Atmospheric and Oceanic Physics 32 757 (1996)]. [36] Klyatskin V I, Gurarie D, in Stochastic Modelling in Oceanography (eds. R J Adler, P Muller, B L Razovsky) (Boston: Birkhauser, 1996) p. 149. [37] Graynik N V, Klyatskin V I Izvestiya RAN, Fiz. Atm. i okeana, 33 723 (1997) [Izvestiya, Atmospheric and Oceanic Physics 33 (1997)]; Graynik N V, Klyatskin V I , Gurarie D., Wave Motion, 28 333, ( 1998). [38] Graynik N V, Klyatskin V I Zh. Eksp. Teor. Fiz. 111 2030 (1997) [JETP 84 1106 (1997)]. [39] Monin A S, Yaglom A M Statisticheskaya Hydrodinamika, (Moscow: Nauka, Vol. 1, 1965, Vol. 2, 1967) [Translation into English Statistical Fluid Mechanics (Cambridge, Ma.: MIT Press, 1975)]. [40] Tatarskii V I Rasproctranenie Voln v Turbulentnoi Atmocfere (Moscow: Nauka, 1967) [Translation into English The E®ects of the Turbulent Atmosphere on Wave Propagation (Spring¯eld, Va.: National Technical Information Service, 1977)]. [41] Rytov S M, Kravtsov Yu A, Tatarskii V I Vvedenie v Statisticheskuyu Radio¯ziku (Moscow: Nauka, Vol.1,1977, Vol. 2, 1978) [Translation into English Principles of Statistical Radiophysics v.1-4 (Berlin: Springer-Verlag, 1987-1989)]. [42] Guzev M A, Klyatskin V I Waves in Random Media 1 275 (1991). [43] Gurvich A S, Kallistratova M A, Martvel' F E Izv. Vyssh. Uchebn. Zav. Radio¯z. 20 1020 (1977) [Radiophys. & Quantum Electron. 20 705 (1977)]. [44] Flatte S M, Wang G Y, Martin J JOSA A10 2363 (1993). [45] Flatte S M, Bracher C, Wang G Y JOSA A11 2080 (1994). [46] Whitham G B Linear and Nonlinear Waves (New York: John Wiley & Sons Inc., 1974). [47] Landau L D, Lifshits E M Gidrodinamika (Moscow: Nauka, 1986) [Translation into English Fluid Mechanics, 2nd edition (London: Pergamon Press (1987)]. [48] Hopf E J. Ration. Mech. Anal. 1 87 (1952). [49] Furutsu K J J. Res. NBS D 667 303 (1963). [50] Novikov E A Zh. Eksper. Teor. Phys. 47 1919 (1964) [Sov. Phys. JETP 20 1290 (1965)]. [51] Donsker M D Proc. Conference on Theory and Applications of Analysis in Functional Space (M.I.T. Press, 1964), p. 17.

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[52] Klyatskin V I, in Mathematics of Random Media (eds. W Kohler, B S White) Lectures in Appl. Math. 27 447 (AMS, Providence RI. 1991). [53] Lifshits I M, Gredeskul S A, Pastur L A Vvedenie v Teoriyu Neuporyadochennich Sistem (Moscow: Nauka, 1982) [Translation into English Introduction to the Theory of Disordered Solids (New York: John Wiley, 1988)]. [54] Anzygina T N, Pastur L A, Slusarev V A Fizika nizkich temper. 7 5 (1981) [Sov. J. Low Temp. Phys. 7 1 (1981)]. [55] Klyatskin V I, Woyczynski W A Zh. Eksper. Teor. Phys. 108 1403 (1995) [JETF 81 770 (1995)]. [56] Pumir A, Shraiman B, Siggia E Phys. Rev. Lett. 66 2984 (1991). [57] Gollub J, Clarke J, Gharib M, Lane B, Mesquita O Phys. Rev. Lett. 67 3507 (1991). [58] Holzer M, Pumir A Phys. Rev. E47 202 (1993). [59] Pumir A Phys. Fluids A6 2118 (1994). [60] Holzer M, Siggia E Phys. Fluids A6 1820 (1994). [61] Kerstein A, McMurtry P A Phys. Rev. E49 474 (1994). [62] Shraiman B I, Siggia E D Phys. Rev. E49 2912 (1994). [63] Klyatskin V I, Nalbandyan O G Izvestiya RAN, Fiz. Atm. i okeana 32 291 (1997) [Izvestiya, Atmospheric and Oceanic Physics 32 (1997)]. [64] Bunkin F V, Gochelashvili K S Izv. Vyssh. Uchebn. Zav. Radio¯z. 11 1864 (1968)[Radiophys. & Quantum Electron. 11 1059 (1977)]. [65] Bunkin F V, Gochelashvili K S Izv. Vyssh. Uchebn. Zav. Radio¯z. 12 875 (1969)[Radiophys. & Quantum Electron. 12 697 (1977)]. [66] Gochelashvili K S, Shishov V I Volni v sluchaino neodnorodnoi srede (Waves in Randomly Inhomogeneous Media) (Moscow: VINITI, Soviet Academy of Sciences, 1981). [67] Dashen R Opt. Letters 9 110 (1984). [68] Churnside J H, Cli®ord S F JOSA A4 1923 (1987). [69] Yakushkin I G, Zavorotny V U Waves in Random Media 2 165 (1992).

COHERENT PHENOMENA

65

Figure captions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Particle dynamics in solenoidal (a) and potential (b) velocity ¯elds Distribution of air balloons in the atmosphere 105 days after the launch Typical realization of solutions of the equation (2.3) Incident plane wave on a layer of random medium (a), and the wave-source inside the medium (b) Numeric modeling of the dynamic localization for two realizations of random media Two-layer medium Cross-section of laser beam in turbulent media Caustic structure in the swimming pool Typical realization of a random process Probability distribution function of a log-normal process Dynamics of cluster formation for ½=½0 = 1:5 (a), and ½=½0 = 10 (b) Wave-guide with optically dense core: evolution of concave and convex wavefronts Compressible °ow advecting iso-contours of passive tracer (solid) and the Jacobian factor (dashed) PDFs (6.3) of the re°ection coe±cient in 1D strati¯ed media, for di®erent values of the absorption coe±cient Three principal components of the gradient correlation-tensor (sec.7.3.1) in the linearly sheared °ow as functions of the normalized shear rate ¯

66


Figures

Fig.1

Fig.2

COHERENT PHENOMENA

Fig. 4

Fig. 5

67

68


Fig.6

Fig.7

Fig.8

COHERENT PHENOMENA

Fig.9

Fig.10

69

70


Fig.11

COHERENT PHENOMENA

Fig.12

Fig. 13

Fig. 14

71

72


Fig.15 (1)Institute

of Atmospheric physics, Russian Academy of Sciences, Moscow, Russia, Oceanological Institute, Russian Academy of Sciences (Far East Division),, Vladivostok, Russia E-mail address: [email protected] (2)Pacific

Math. Department, CWRU, Cleveland, OH 44106 Current address : NCAR, P.O. Box 300, Boulder, CO 80303 E-mail address: [email protected] URL: http://www.cwru.edu/artsci/math/gurarie/home.html

COHERENT PHENOMENA IN STOCHASTIC DYNAMICAL SYSTEMS

COHERENT PHENOMENA IN STOCHASTIC DYNAMICAL SYSTEMS

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