Inertial particles driven by a telegraph noise
G. Falkovich,1 S. Musacchio,1L. Piterbarg,2and M. Vucelja11Weizmann Institute of Science, Rehovot, 76100 Israel 2
University of Southern California, Los Angeles, California 90089, USA 共Received 29 March 2007; published 22 August 2007兲
We present a model for the Lagrangian dynamics of inertial particles in a compressible flow, where fluid velocity gradients are modeled by a telegraph noise. The model allows for an analytic investigation of the role of time correlation of the flow in the aggregation-disorder transition of inertial particles. The dependence on the Stokes number St and the Kubo number Ku of the Lyapunov exponent of particle trajectories reveals the presence of a region in parameter space共St, Ku兲, where the leading Lyapunov exponent changes sign, thus signaling the transition. The asymptotics of short- and long-correlated flows are discussed, as well as the fluid-tracer limit.
DOI:10.1103/PhysRevE.76.026313 PACS number共s兲: 47.52.⫹j, 47.55.Kf, 47.27.eb
I. INTRODUCTION
The spontaneous formation of clusters of particles sus-pended in chaotic flows may originate from two different physical processes: compressibility of the fluid flow and par-ticle inertia. In the first case parpar-ticles are trapped in regions of ongoing compression, while in the second case inertia causes their ejection from vortical regions. The underlying link between these two phenomena is manifested in the limit of weak inertia, in which particle dynamics in incompress-ible flows can be approximated by that of tracers in a weakly compressible velocity field关1,2兴.
Clustering processes have been extensively studied, both in the case of compressible flows关3–6兴 and inertial particles 关2,7–9兴. Fractal dimension of the clusters, obtained from the ratios of Lyapunov exponents of particle trajectories, has re-vealed a powerful tool to quantify the intensity of the clus-tering关3,7兴, and has motivated further studies of Lyapunov exponents of inertial particles关9,10兴.
Aggregation is an extreme form of clustering, which oc-curs when the trajectories of different particles tend to point-like clusters. In this situation the senior Lyapunov exponent of particle trajectories become negative, signaling that nearby particles do not separate exponentially as expected in chaotic flows, but their paths eventually coalesce. This phe-nomenon has been described for fluid tracers in compressible flows关11,12兴, as well as for light particles in incompressible flows关7兴.
Surprisingly, it has been shown that in compressible flows, where fluid trajectories coalesce, large enough inertia can induce a transition from the strong clustering regime into a weak clustering one, where particle trajectories remain cha-otic and the senior Lyapunov exponent is positive 关13–16兴. Analytic results on this aggregation-disorder transition have been obtained under the assumption that the carrier flow is short correlated in time.
It is therefore natural to ask how time correlations of real turbulent flows can influence this phenomena. As pointed out by recent numerical studies of fluid tracers in compressible flows关6兴, time correlations are responsible for an increase in the level of compressibility required for the transition to the strong-clustering regime. In the case of inertial particles it
has been found关10兴 that time correlations cause an increase of the共positive兲 Lyapunov exponent 共i.e., make particle flow more chaotic兲 already at weak inertia, which cannot be pre-dicted in the framework of short-correlated flows关9,17–19兴. Here we discuss a simple one-dimensional, time-correlated model flow, recently introduced in关20兴, which al-lows one to obtain a deeper insight on the interplay between inertia and compressibility, and to investigate analytically the dependence on Stokes and Kubo numbers of the Lyapunov exponent of particle trajectories.
II. MODEL
The dynamics of two small inertial particles, whose den-sity is much larger than the fluid denden-sity, is dominated by the viscous drag. Hence, the equation for their separation R共t兲 = X1共t兲−X2共t兲 and relative velocity V reads 关21兴
R˙ = V, V˙ = −1
关V − ⌬U共X1,X2,t兲兴, 共1兲 where is the Stokes time of the particles, and ⌬U is the difference of fluid velocities at particle positions.
To investigate the behavior of the Lyapunov exponent one has to consider separations smaller than the viscous length scale of the flow. At these scales the fluid velocity difference can be written in terms of the velocity gradients Sij=jUias ⌬Ui= SijRj. The Lyapunov exponent is determined by the contributions of the local gradients Sij共t兲 experienced by the two nearby trajectories. Note that the statistics of velocity gradients in the reference frame of an inertial particle differs from the Eulerian statistics due to preferential concentration. Let us now consider an idealized one-dimensional flow, in which the fluid velocity gradient in the reference frame of a particle is modeled by a telegraph noise s共t兲, i.e., a noise that switches randomly between two fixed values ±s. Stokes number can be defined in term of the fluid gradient intensity as St= s. Equations共1兲 reduce to
R˙ = V, V˙ = − V
+
s共t兲
R. 共2兲
When s共t兲= +s the system is expanding, and evolves to-wards the asymptote V =␣R, where
␣= 1
2共− 1 +
冑
1 + 4s兲. 共3兲 The particles separate exponentially with smaller rate than fluid trajectories␣⬍s. When s共t兲=−s the system is contract-ing, and two different scenarios are observed, according to the relative intensity of fluid gradients and inertial drag. For small Stokes numbers St⬍1/4 the system evolves toward the asymptote V =R, where= 1
2共− 1 +
冑
1 − 4s兲 for s⬍ 1/4. 共4兲 Particles slow down less efficiently than the fluid, and hence their separation goes to zero with a faster exponential rate 兩兩⬎s. For large Stokes number St⬎1/4 the system has two complex conjugate eigenvalues= 1
2共− 1 ± i
冑
4s− 1兲 for s⬎ 1/4, 共5兲 and the solution decays exponentially with a clockwise spiral motion in phase space共R,V兲. This means that particles can cross the R = 0 axis, i.e., collide, with nonzero relative veloc-ity, giving origin to a shock关22兴. Notice that the solution is properly defined also for R⬍0. The change of the sign of R can be interpreted as the fast particle overcoming the slow one.Shocks start to appear for the critical value St*= 1 / 4. The presence of such critical Stokes number is characteristic of flows where the fluid velocity gradients are bounded, and its value is determined by the intensity of the strongest negative gradient 共here −s兲. Conversely, if the statistics of the fluid velocity gradient is unbounded, shocks can appear for arbi-trarily small Stokes number, but they are exponentially sup-pressed in the limit St→0 关18,22,23兴.
The transition rates1 from −s to s and2 from s to −s determine the fraction of time spent by the particle pair in regions of ongoing compression and expansion共respectively,
2/共1+2兲 and 1/共1+2兲兲. The mean value of the noise s共t兲 is
s0⬅ 具s共t兲典 = − s⌬/, 共6兲 where =1+2 and ⌬=2−1. Noise fluctuations s˜共t兲 ⬅s共t兲−s0 are exponentially correlated:
具s˜共t兲s˜共t
⬘
兲典 =4s2122 exp共−兩t − t
⬘
兩兲. 共7兲 The ratio between the correlation time of the flow 1 / and the Lagrangian separation time 1 / s defines the Kubo number Ku= s−1.Notice that in potential flows both fluid tracers and iner-tial particles spend more time in regions of a local compres-sion than in expanding regions, and hence⌬⬎0. The actual value of⌬can be determined by assuming statistical homo-geneity and isotropy of the flow. Thanks to these symmetries, the mean position of a particle is solely determined by its initial position具X共t兲典=X0and hence the distance R is statis-tically conserved.
The peculiar nature of the telegraph noise, namely the fact that its square is deterministic, allows one to obtain closed equations for具R共t兲典. Averaging over different realizations of the noise, and using the derivative formula关24兴
d
dt具tFt共兲典 =
冓
t ddtFt共兲
冔
−具tFt共兲典, 共8兲 which hold for a generic noise with zero mean and correla-tion具tt⬘典⬃exp共−兩t−t⬘
兩兲, one obtains the linear systemd dty = 1 My, 共9兲 where y =
冤
具R典 具V典 具s˜R典 2具s˜V典冥
, M =冤
0 1 0 0 s0 − 1 1 0 0 0 − 1 共s2− s 0 2兲2 0 − s 0 −共1 +兲冥
. 共10兲This linear system allows for a constant solution 具R典=R0 when s0共1+兲+s2= 0, which together with Eq.共6兲 gives
⌬= s
1 +. 共11兲
Notice that inertia reduces the trapping of the particle in compressing regions. This effect is encoded in the behavior of the parameter ⌬, which is a decreasing function of Stokes time, and vanishes in the limit→⬁.
The positivity of 1 requires ⌬⬍, and gives an upper bound for the Kubo number achievable in the system:
Ku艋 Ku*=
冑
2 St1 + 4 St − 1. 共12兲
For longer correlation time, particles are irreversibly cap-tured by contracting regions. In the limit of fluid tracers →0 the constraint becomes Ku⬍1. The accessible region in 共St, Ku兲 parameter space is shown in Fig.1.
III. STATISTICS OF PARTICLE-VELOCITY GRADIENTS From the system共2兲 one can obtain a closed equation for the particle-velocity gradient= V / R,
˙ = −2−−1关− s共t兲兴. 共13兲 The Lyapunov exponent of particle trajectories can be ob-tained from Eq.共13兲 as an ensemble average 具典 over differ-ent realizations of the noise. Equations for the stationary dis-tribution of x =+ 1 /共2兲 are easily obtained with the same procedure adopted to derive Eqs.共9兲 and 共10兲,
冋
冉
1 42− x2+ s0
冊
p册
x冋
冉
1 42− x 2− s0 冊
q册
x +冉
s 2− s 0 2 2冊
px+q = 0, 共15兲 where q共x兲=兰s˜p共x,s˜兲ds˜/ and p共x,s˜兲 is the joint stationary probability density function共PDF兲 of x and s˜. From the first equation one obtains q = −关C+共1/42− x2+ s0/兲p兴, where C is the mean flux of x, and finally
冋
s2 2−冉
1 42− x 2冊
2册
px+冋
共4x −兲冉
1 42− x 2冊
−s0 册
p + C共2x −兲 = 0. 共16兲The asymptotic behavior p⬃C/x2 共C⬎0兲 for large 兩x兩 gives the probability of strong particle velocity gradients, and is therefore related to the probability of shocks. Different solutions are found in the small-Stokes-number and large-Stokes-number regime.
A. Small Stokes number
When St⬍St*, the unique positive integrable solution of Eq.共16兲 is obtained under condition of zero flux 共C=0兲,
p = C1
共w − x兲m−1共x − w˜兲m˜ −1
共w + x兲m+1共x + w˜兲m˜ +1, x苸 共w˜,w兲, 共17兲 and zero otherwise, where
w =
冑
1 + 4s/2, m =共+⌬兲/4w,w˜ =
冑
1 − 4s/2, m˜ =共−⌬兲/4w˜. 共18兲 The solution is localized in the compact interval共w˜,w兲. Its shape is determined by the values of m and m˜ . For m˜ ⬍1 共low frequency兲 the PDF is peaked around the two bor-der values. For m⬍1⬍m˜ 共intermediate frequency兲 it van-ishes at w˜ , and finally when m⬎1 共high frequency兲 itvan-ishes both at w˜ and w共see Fig.2兲. Indeed, when St⬍St*, the solution of the linear system for 共R,V兲 oscillates between two asymptotes V =共w−1/2兲R and V=共w˜−1/2兲R accord-ing to the sign of the noise. If the noise frequency is low the system has enough time to become close to the two asymp-totes and the PDF is peaked around them. Conversely, when the sign of the noise switches frequently, the system does not have enough time to reach the asymptotes and oscillates rap-idly around the mean value.
B. Large Stokes number
When St⬎St* the solution of Eq. 共16兲 consists of two different parts,
p共x兲 = Cp1共x兲 + C2p2共x兲. 共19兲 The first one is the solution of Eq.共16兲 with C=1,
p1= 兩w − x兩m−1 兩w + x兩m+1共x2+ w˜2兲e n关arctan共x/w˜兲兴 ⫻
冕
−w x dy兩w + y兩 m+1共− 2y兲 兩y − w兩m−1共w2− y2兲e−n关arctan共y/w˜兲兴, 共20兲 while the second one is the right tail of the solution of the homogeneous共fluxless兲 equationp2=
兩w − x兩m−1 兩w + x兩m+1共x2+ w˜2兲e
n关arctan共x/w˜兲兴
Iw,⬁共x兲. 共21兲 Here Iw,⬁共x兲 is the characteristic function 共indicator兲 of the interval共w,⬁兲, and
w =
冑
4s+ 1/2, m =共+⌬兲/4w, w˜ =
冑
4s− 1/2, n =共−⌬兲/2w˜. 共22兲 Conditions for determining C and C2 are冕
p共x兲dx = 1,冕
q共x兲dx = 0. 共23兲 These conditions guarantee the continuity of the PDF in the limit St→St* 共see the Appendix兲. The PDF obtained in the FIG. 1. Parameter space共St, Ku兲. Shape transitions in theprob-ability density function共PDF兲 of x occur for St=St*= 1 / 4共vertical solid line兲, m˜ = 1共dashed line兲, and m=1 共dash-dotted line兲. Labels refers to the PDFs shown in Figs.2and 3. The gray region is not accessible by the model because Ku does not fulfill condition共12兲.
Crosses represent the boundary of the chaotic region.
0 0.2 0.4 0.6 0.8 1 3 3.5 4 4.5 s P(x) xs-1 (a) (b) (c)
FIG. 2. PDF of x in the small Stokes regime共St=1/8兲 for dif-ferent value of the Kubo number: Ku= 1 / 8共a兲, Ku=1/16共b兲, and Ku= 1 / 24共c兲.
regime St⬎St* is extended over all real x, with power-law tails p⬃C/x2 for large 兩x兩 that are due to shocks, which occur for large negative values of s共t兲. For short-correlated noise 共m⬎1兲, the PDF is characterized by an asymmetric core localized between −w and w. When m⬍1 a singular peak arises at x = w共see Fig.3兲. This behavior is easily un-derstood in term of the solution of the linear system for 共R,V兲. When s共t兲= +s the solution converges toward the as-ymptote V =␣R =共w−1/2兲R. This produces the peak at x = w, provided that the correlation time of the noise is long enough共m⬍1兲 to become close to the asymptote. The large Stokes number regime is hence characterized by an infinite series of shocks alternated to “quiet” phases in which the particle-velocity gradient relaxes toward␣.
The schematic in Fig.1summarizes the different regions in the parameter space共St, Ku兲 where shape transitions oc-curs in the PDF of x.
IV. LYAPUNOV EXPONENTS
The Lyapunov exponent of inertial-particle trajectories can be written in terms of the mean value x¯ as=x¯−1/2. For St⬍St*the mean value x¯ can be written as
x ¯ = w˜共w − w˜兲m˜ 共m + m˜兲 ⫻ F1共m˜ + 1,m + 1,m˜ + 1,m˜ + m + 1,u,v兲 F1共m˜ + ,m + 1,m˜ + 1,m˜ + m,u,v兲 , 共24兲 where u =w˜ − w w˜ + w, v = w˜ − w 2w˜ , 共25兲
and F1is the hypergeometric function of two variables关25兴. The behavior of the Lyapunov exponent as a function of the Stokes and Kubo numbers is shown in Fig.4. The Lyapunov exponent is negative for small Stokes, and decreases ap-proximatively as s−1⬃−Ku at increasing Ku numbers. In the limit St→0 it recovers the actual value for fluid tracers 关20兴,
0⬅ lim →0 = − s
2/. 共26兲
Notice that fluid tracers are always in the aggregation regime within this model, as signaled by the negative value of0. A sharp negative minimum =−2s is found for St=1/4 at Ku=共
冑
2 + 1兲/2. It corresponds to the maximum aggregation of the particle. As the correlation time of the flow decreases, the minimum becomes less pronounced, and it moves to larger Stokes numbers. A region of positive Lyapunov expo-nents is present in the parameter space for Stⲏ4.3. The iso-line of vanishing Lyapunov exponent which border this re-gion represent the transition from the strong clustering regime共⬍0兲 to the chaotic regime 共⬎0兲. Indeed, Fig.5 shows that, as Stokes number increases, the interval of Kubo numbers appears where the Lyapunov exponent grows, and eventually becomes positive. This can be understood as fol-lows: to achieve an effective mixing the correlation time of the fluid gradients must be long enough to provide substan-tial stretching of particle trajectories, but not too long, to avoid particle segregation in compressing regions. Therefore the chaotic region is confined in a window of Ku numbers10-3 10-2 10-1 1 101 -4 -3 -2 -1 0 1 2 3 4 s P(x) xs-1 (d) (e)
FIG. 3. PDF of x in the large Stokes regime共St=1兲 for different values of Kubo number, Ku= 0.1共d兲, Ku=1共e兲.
0.01 0.1 1 10 0.01 0.1 1 10 100 Ku St λ>0 λ<0
FIG. 4. The isolines of the Lyapunov exponent on the plane of Stokes and Kubo numbers. The isolines are spaced every 0.02 s 共dotted and solid lines兲. The boundary of the chaotic region 共 ⬎0兲 is represented by the dot-dashed line.
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.01 0.1 1 10 λ s -1 Ku St=1 St=2 St=10
FIG. 5. The Lyapunov exponent as a function of the Kubo number. An interval of positive Lyapunov exponents appears for Stⲏ4.3. The dashed line represents the asymptotic behavior for Ku= Ku*.
between a lower and an upper bound determined by stretch-ing efficiency and particle trappstretch-ing respectively.
Let us now consider the behavior of the Lyapunov mo-ments␥n, defined as具Rn典⬃exp共␥nt兲. The evolution of 具Rn典, for n positive integer, is determined by a closed system of the 2共n+1兲 equation,
d dt具R
n−iVi典 = 共n − i兲具Rn−i−1Vi+1典 −i
具Rn−iVi典
+ i
共s˜ + s0兲具Rn−i+1Vi−1典, 共27兲 d
dt具s˜R
n−iVi典 = 共n − i兲具s˜Rn−i−1Vi+1典 −
冉
i+
冊
具s˜Rn−iVi典− i
s0具s˜Rn−i+1Vi−1典, 共28兲 where i = 0 , 1 , . . . , n. The nth Lyapunov moment ␥n is the largest solution of the 2共n+1兲th equation
det
冋
A CD B
册
= 0; 共29兲A and B are tridiagonal matrices with the elements
Ai,i=␥n+ i/, i = 1,n, Ai,i−1= − is0/, i = 2,n, Ai,i+1= i − n, i = 1,n − 1, Bi,i=␥n++ i/, i = 1,n, Bi,i−1= is0/, i = 2,n, Bi,i+1= i − n, i = 1,n − 1, 共30兲 and C and D are subdiagonal matrices with the elements Ci,i−1= −i / for i = 2 , n and Di,i−1= i共s0
2
− s2兲/ for i = 2 , n. In the limit St→0 one recovers the Lyapunov moments of fluid tracers关20兴, ␥n=
冑
冉
2冊
2 + s2共n2− n兲 − 2. 共31兲Notice that for fluid tracers the sign of the separation R is preserved, and hence the exponents␥ncoincide with the ex-ponents␥˜n defined by具兩R兩n典⬃exp共␥˜nt兲. The asymptotic lin-ear behavior of␥nfor large n is the hallmark of the presence of an upper bound for velocity gradients.
A. Short-correlated flows
Let us now examine the limit of short-correlated flow. The limit→⬁ must be taken keeping constant s2/= D, in order to recover ␦-correlated noise fluctuations 具s共t兲s共t
⬘
兲典 = 2D␦共t−t⬘
兲. In other words, while Kubo number Ku=s/ tends to zero, Stokes number St= s must grow, so that the productKu St = s2/= D, 共32兲 remains constant. In this sense the short-correlated limit for inertial particles correspond always to the large-inertia case. In the␦-correlated limit the relevant time scale associated to fluid velocity gradients is given by the Lyapunov exponent of fluid tracers 0= −s2/= −D. This is confirmed by the col-lapse of particle Lyapunov exponents in the short-correlated limit once their intensity and Stokes times are rescaled with 兩0兩 共see Fig. 6兲. A noticeable minimum is observed for 兩0兩= Ku St⯝0.05 and a transition to chaos, i.e., from nega-tive to posinega-tive, occurs for Ku Stⲏ1.6. These features are in qualitative agreement with previous analytic and theoreti-cal results obtained in the framework of ␦-correlated flows 关14,18兴. Notice that in those studies Gaussian statistics is assumed for velocity gradients, at variance with our model in which only the two values ±s are allowed. Therefore quan-titative details such as the exact position of the minimum can be different.
We remark that in the short-correlated asymptotics, the fluid-tracers limit becomes singular. Indeed one has
lim →⬁s0= lim→⬁− s2 共1 +兲=
再
− D = 0, 0 ⬎ 0,冎
共33兲 which signals that fluid tracers are still preferentially at-tracted by regions of ongoing compression, while inertial particles with arbitrary finite are not. Notice that in the limit →⬁ fluid velocity gradients becomes unbound be-cause s→⬁, and for fluid tracers we recover the quadratic behavior of Lyapunov moments␥n= D共n2− n兲.B. Long-correlated flows
Time correlation of fluid gradient is bounded by the con-dition共12兲. When Ku=Ku*the transition rate
1from −s to s vanishes, while the transition rate2 from s to −s reaches its maximum *=共
冑
1 + 4s− 1兲/共2兲. Particles initially seeded in expanding regions are gradually captured by con-tracting ones, where they remain trapped forever. Therefore the population of expanding regions decreases exponentially as -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.001 0.01 0.1 1 10 λ |λ0 | -1 St Ku Ku=0.02 Ku=0.04 Ku=0.08P共t兲 ⬃ exp共−*t兲. 共34兲 Moments of particle separations will hence evolve asymp-totically according to
具兩R兩n典 ⬃ en␣t
P共t兲 + „1 − P共t兲…en Re共兲tf„Im共兲t…, 共35兲 whereis given by Eq.共5兲 and f共t兲 is 2periodic function. Notice that*=␣and therefore for n = 1 the decreasing frac-tion of particle in expanding regions is exactly balanced by the exponential growth of their separation. From Eqs. 共34兲 and共35兲 one obtains the Lyapunov moments
␥
˜n=␣共n − 1兲 for n 艌 „␣− Re共兲…−1,
␥
˜n= Re共兲 for n 艋 共␣− Re共兲兲−1. 共36兲 In the limit of fluid tracers one obtains
␥
˜n= s共n − 1兲 for n 艌 1/2,
␥
˜n= − sn for n艋 1/2, 共37兲 in agreement with Eq.共31兲.
Finally, the Lyapunov exponent along the critical line Ku= Ku*is=兩共␥˜
n/兲兩n=0= Re共兲. In Fig.7we compare its behavior with that of the Lyapunov exponent along the line Ku= 1. For small Stokes number both of them recover the fluid-tracers limit→0= −s, but with different power-law behavior. On the line Ku= Ku* we have 兩−
0兩s−1⬃St, while for Ku= 1 we have兩−0兩s−1⬃St2 共see lower inset of Fig. 7兲. For large Stokes number Lyapunov vanishes as s−1⬃−St−1 on the line Ku= Ku* and as s−1⬃St−2/3 for Ku= 1共see upper inset of Fig. 7兲. In between these two as-ymptotics a sharp minimum appears for St= St*.
Notice that the asymptotic decay St−2/3, here shown for long-correlated flows, have been already predicted and ob-served also for ␦-correlated flows 关17,19兴. The agreement between these results confirms that in the large Stokes
number asymptotics the role of time correlation becomes negligible and particles behave as if suspended in
␦-correlated-in-time flows.
V. CONCLUSIONS
We discussed the Lagrangian dynamics of inertial par-ticles in a simple time-correlated compressible flow, in which fluid velocity gradients in the reference frame of the particle are modeled by a one-dimensional telegraph noise. In spite of its simplicity, the model allows one to take into account consistently the different Lagrangian weights of regions of ongoing compression and expansion, and it reproduces the phenomenon of trapping of particles in compressing regions for long-correlated flows.
The peculiar nature of the telegraph noise allows one to investigate analytically the effects of time correlation of ve-locity gradients on the chaoticity of particle trajectories, and to study the dependence on Stokes and Kubo numbers of Lyapunov exponents. We discussed both the asymptotics of long- and short-correlated flows, as well as the fluid-tracers limit.
For large Stokes number, a regime characterized by the formation of shocks, we found a chaotic region in parameter space 共St, Ku兲, where the leading Lyapunov exponent be-comes positive. Inertia is therefore responsible for a transi-tion from a strong clustering regime, originated by the com-pressible nature of the flow, to a chaotic regime. The latter is observed in a range of Kubo numbers such that the time correlation of fluid gradients is long enough to provide sub-stantial stretching, but not too long to cause particles trapped in compressing regions.
This simple model clearly does not allow one to repro-duce all the complex phenomena which occur in a realistic flow. In particular it does not include the effects of preferen-tial concentration in hyperbolic regions, which are respon-sible for the increase of chaoticity of inertial particle trajec-tories in turbulent flows 关10兴. A two-dimensional extension of the model would be required for investigating these ef-fects. Further, we remind the reader that the Lyapunov sta-tistics is able to describe the clustering of inertial particles at the dissipative scales of turbulent flows, but does not allow one to characterize the structures which are observed in par-ticle distribution at scales within the turbulent inertial range 关26兴.
Note that the substitution R =⌿ exp共−t/2兲 turns Eq. 共2兲 into the Schrödinger equation,⌿¨−s⌿/=⌿/42, with space replacing time. The telegraph noise model for the Schrödinger equation was used by Bendesrkii and Pastur 关27兴, who applied the methods of 关28兴 to evaluate the density of states.
ACKNOWLEDGMENTS
We are grateful to K. Gawe¸dzki who called our attention to关27兴. This work has been supported by the ONR Grant No. N00014-99-0042 and by the grant of the Israeli Science Foundation. -2 -1.5 -1 -0.5 0 0.01 0.1 1 10 100 λ s -1 St 10-2 10-1 100 10-2 10-1 100 103 102 101 100 10-3 10-2 10-1
FIG. 7. Lyapunov exponent for Ku=1 共solid line兲 and on the line Ku= Ku*共dashed line兲, corresponding to the longest correlation
of velocity gradients achievable within the model. Upper inset: asymptotic behavior for large Stokes numbers−1⬃St−2/3共dotted
line兲, 共Ku=1兲. Lower inset: asymptotic behavior for small Stokes number兩−0兩s−1⬃St共dotted line兲.
APPENDIX
To prove the continuity of the PDF for St= St* we first notice that the limit St→St*is equivalent to w˜→0.
The limit w˜→0 of the solution 共17兲 in the small-Stokes number regime is p共x兲=C1f共x兲I0,w共x兲, where
f共x兲 = 共w − x兲 m−1
共w + x兲m+1x2e−共−⌬兲/2x, 共A1兲 and I0,w共x兲 is the characteristic function of the interval 共0,w兲. To study the limit of the solution共19兲, let us consider the limits of p1共x兲 and p2共x兲 when w˜→0. The latter is easily obtained as
p2共x兲 ⬃ e共−⌬兲/4w˜f共x兲. 共A2兲 Now let us rewrite p1共x兲 as
p1= 兩w − x兩m−1 兩w + x兩m+1共x2+ w˜2兲e −n关arctan共w˜/x兲兴 ⫻
冕
−w x 兩w + y兩m+1共− 2y兲dy 兩y − w兩m−1共w2− y2兲 e n关arctan共w˜/y兲兴. 共A3兲 As w˜→0, then p1共x兲=O共1兲 if x⬍0 andp1共x兲 ⬃ 2e共−⌬兲/4w˜w˜2f共x兲, 共A4兲 for x⬎0. To obtain the last asymptotic one takes into ac-count that the main contribution in the integral in Eq.共A3兲 is brought by the right small vicinity of y = 0, say 共0,s兲, and apply
冕
0s
exp关− arctan共y/␦兲/␦兴dy ⬃
冕
0s
exp共− y/␦2兲dy ⬃␦2 共A5兲 for all s⬎0 and small␦.
The continuity of the solution of Eq.共16兲 in the limit w˜ →0 is therefore guaranteed by the conditions
C = C1
2w˜2e共−⌬兲/4w˜, C2= 2Cw˜
2. 共A6兲
This conditions are indeed equivalent to the normalizations conditions
冕
p共x兲dx = 1,冕
q共x兲dx = 0, 共A7兲 where q = −关C+共1/42+ s0/− x2兲p兴.
To show this we notice that Eq.共16兲 can be written as
再
冋
s2 2−冉
1 42− x 2冊
2册
p + Cx2冎
x +q = 0. 共A8兲 It follows that冕
−x x q共y兲dy = −1 冋
s2 2−冉
1 42− x 2冊
2册
关p共x兲 − p共− x兲兴. 共A9兲 Thus, the second condition in Eq.共A7兲 is equivalent tolim x→⬁
x4关p共x兲 − p共− x兲兴 = 0. 共A10兲 According to Eq.共19兲 p共x兲 is written as the sum of p1and p2, whose asymptotic behavior is
p1共x兲 ⬃ 1 x2+ P1+ x4, x→ ⬁, p1共x兲 ⬃ 1 x2+ P1− x4, x→ − ⬁, 共A11兲 and p2共x兲 ⬃ P2+ x4, x→ ⬁, p2共x兲 ⬃ P2− x4, x→ − ⬁. 共A12兲 Notice that P2−= 0, P2+= e共−⌬兲/4w˜. Thus Eq.共A10兲 becomes
C2= C共P1+− P1−兲 P2+ . 共A13兲 In the limit w˜→0, P1−= o共P1+兲, P1+⬃ 2w˜2P2+, 共A14兲 and we obtain C2⬃ 2w˜2C, 共A15兲
which implies the continuity of the PDF. Normalization of p共x兲 in the limit w˜→0 is equivalent to the first condition in Eq.共A6兲.
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