Dispersion of passive tracers in velocity field with non delta-correlated noise

14 Pages·0.55 MB·English

by  P. Castiglione

Dispersion of passive tracers in velocity field with non delta-correlated noise
P. Castiglione and A. Crisanti
Dipartimento di Fisica, Universit`a di Roma “La Sapienza”, and INFMsezione Roma I
P.le A.Moro 2, 00185 Roma, Italy.
(Physical Review E, in press)
Thediffusivepropertiesinvelocityfieldswhosesmallscalesareparameterizedbynonδ-correlated
noise is investigated using multiscale technique. The analytical expression of the eddy diffusivity
tensorisfoundfora2Dsteadyshearflowanditisanincreasingfunctionofthecharacteristicnoise
decorrelation time τ. In order to study a generic flow v, a small-τ expansion is performed and
the first correction O(τ) to the effective diffusion coefficients is evaluated. This is done using two
9 differentapproachesanditresultsthatattheorderτ theproblemwithacolorednoiseisequivalent
9 to the white-in-time case provided by a renormalization of the velocity field v 7→ v˜ depending on
9 τ. Two examples of 2D closed-streamlines velocity field are considered and in both the cases an
1 enhancement of thediffusion is found.

n PACS number(s): ;
a
J

0
2 I. INTRODUCTION

1
v The problem of diffusion in a given velocity field has both theoretical and practical relevance in many different
7 fields of science and engineering as, e.g., transport processes in chemical engineering and combustion studies [1].
9 The tracers transport, in particular the evolution of their concentration, plays an important role in many aspects of
1 geophysics. For the oceanic flows, satellite data indicate that the mesoscale features, like eddies and cold filaments,
1
advect temperature and nutrients over spatial and temporal scales longer than those of the geostrophic turbulence.
0
The diffusion enhancement by a given velocity field has attracted a lot of works in the last years. In particular the
9
role of the velocity field properties has been largelyinvestigatedwhile the effects of small scales parameterizationare
9
/ not understood.
t
a In this paper we will focus on the effects of a finite noise correlationtime. This problemis relevant in studying the
m transportin the ocean since in this system the noise term comes from unresolvedvelocity scales which are correlated
in time.
-
d In Section 2, by using the multiscale technique, we study the diffusion properties of the model proposed in Ref.
n [3] for transport in the upper mesoscale ocean. The transport is described by a Langevin equation with a Gaussian
o
colored noise in time.
c
The aim is to understand whether a finite noise correlation time τ enhances or depresses the dispersion process in
:
v a given velocity field v(x,t) with respect to the delta-correlated case (τ =0).
i
X Exploiting the scale separation in the dynamics we derive, using the multiscale technique [4], an effective diffusive
equation for the macrodynamics, the calculation of the effective diffusivity second-order tensor is reduced to the
r
a solution of one auxiliary partial differential equation [5], [6], [8].
In Section 3 we consider a shear flow, in this case the diffusion coefficient increases with τ. The solution of the
auxiliaryequationis,ingeneral,quitedifficult,therefore,toinvestigatetheroleofthefiniteτ inSection4weperform
a small–τ expansion. An alternative method is presented in the Appendix A.
In Section 5 we study the case of two closed-streamlines fields that mimics the transport in the Rayleigh-B´enard
system: the quasi-two-dimensionalflow studied by Shraiman in [17] and the AB flow. In both the cases the presence
of a small correlation time enhances the diffusion process.
Conclusions are reserved for the final Section 6.
II. EFFECTIVE DIFFUSION EQUATION FOR GAUSSIAN COLORED NOISE
We consider large scale and long time dynamics of the model proposed in [2] and already studied in [3] for the
transport of a fluid particle in the upper mesoscale ocean:
d
x=v(x,t)+s(t) (1)
dt
1
where v is a d-dimensionalincompressible velocity field (∇ v =0), for simplicity, periodic both in space and in time
·
and s is a Gaussian random variable of zero mean and correlationfunction
si(t)sj(t′) = σ2 δije−|t−τt′|. (2)
h i τ
The term v(x,t) represents the part of the velocity field that one is able to resolve, i.e., the larger scale mean flow,
whereas s(t) represents the part of the velocity field containing the subgridscale flow processes, e.g. the small-scale
turbulence. The plausibility of such a description is discussed in [9], [10], [11].
In the limit τ 0, resulting e−|t−t′|/τ/τ δ(t t′), the (2) reproduces the widely studied delta-correlated case
→ → −
s (t)s (t′) =σ2 δ δ(t t′) (3)
i j ij
h i −
the diffusive properties of which we would like to compare with the τ-correlated noise case.
To study the dispersion of tracers evolving according to eqs. (1) and (2) on large scales and long times we use the
multiscaletechnique. Thisisapowerfulmathematicalmethod, alsoknownashomogenization,forstudyingtransport
processes on time and spatialscales much largerthan those of the velocity field v. It has been alreadyapplied to the
delta-correlated case [4] and it has been shown that the motion on large time and spatial scales is diffusive and it is
described by an effective diffusion tensor which takes into account the effects of the advecting velocity on the bare
diffusion coefficient σ2.
Toapplythis methodtothecaseofGaussiancolorednoise,wefirstwriteeqs. (1)and(2)intoaMarkovianprocess
by enlarging the state space considering s(t) as a variable evolving according to the Langevin equation:
d 1
s= s+w(t) (4)
dt −τ
where now the noise w(t) is a white noise with correlation functions
σ 2
w (t)w (t′) = 2 δ δ(t t′). (5)
i j ij
h i τ −
(cid:16) (cid:17)
Wehavenowatwo-variable(x,s)MarkovianprocesswhoseassociatedFokker-Planckequationcanbeeasilyobtained.
Indeed introducing
x w v+s 0 0
Y = ; W = ; V = ; A= (6)
s w 1s 0 1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) −τ (cid:19) (cid:18) (cid:19)
b
the equations (1) and (4) become
d
Y =V(Y,t)+A W(t). (7)
dt ·
The associated Fokker-Planckequation is b
σ 2 1
∂ Θ= ∂ 2Θ (v+s) ∂ Θ+ ∂ (sΘ) (8)
t ss x s
τ − · τ ·
(cid:16) (cid:17)
where Θ=Θ(x,s,t) denotes the probability density.
The doubling of the space dimension is the price to pay for having a Fokker-Planck equation. In the Appendix A
wediscussadifferentapproachto the problemwhichdoesnotdoublethe dimensionofthespace,butleadsingeneral
to a non-Markovianmaster equation.
We cannowapply the multiscale technique. Following[8]in additiontothe fast variablesx andt weintroduce the
slow variables defined as X = ǫx and T = ǫ2t where ǫ 1 is the parameter controlling the separation between the
≪
smallscales relatedto the velocityfield v and the largescale relatedto the Θ variation. The two sets of variablesare
considered independent and so we have to make the substitution
∂ ∂ +ǫ∂ ; ∂ ∂ +ǫ2∂ . (9)
x x X t t T
7→ 7→
The solution of the Fokker-Planck equation (8) is sought as a perturbative series
Θ(x,t,X,T,s)=Θ(0)+ǫΘ(1)+ǫ2Θ(2)+... (10)
2
where the functions Θ(n) depend on both fast and slow variables. By inserting (9) and (10) into the Fokker-Planck
equation(8), equating terms of equalpowers in ǫ andchoosing the solutions which have the same periodicities as the
velocity field, we obtain a hierarchy of equations the first three of which are:
Θ(0) =0 (11)
D
Θ(1) = (v+s) ∂ Θ(0) (12)
X
D − ·
Θ(2) = (v+s) ∂ Θ(1) ∂ Θ(0) (13)
X T
D − · −
where the operator is defined as
D
1 σ 2
=∂ +(v+s) ∂ ∂ (s ) ∂ 2. (14)
t x s ss
D · − τ − τ
(cid:16) (cid:17)
In order to solve eq. (11) we make use of scale separation and we write the solution Θ(0) as the sum of two terms:
Θ(0) (X,T,s) depending on the slow variables and Θ(0)(x,t,s) depending on the fast variables. Here and in the
h i
following the indicates the average over the fast variables.
h·i
Equation (11) then splits into the two equations e
Θ(0) =0 (15)
D
e
σ 2 1
∂ 2 Θ(0) + ∂ (s Θ(0) )=0. (16)
ss s
τ h i τ h i
(cid:16) (cid:17)
One can show [8] that the solution Θ(0) will relax to a constant with respect to fast variables, so we can simply take
e Θ(0)(x,t,X,T,s)= Θ(0) (X,T,s) (17)
h i
and write the solution of eq. (16) as
Θ(0) =Q(X,T) (s) (18)
h i P
where (s) is defined as
P
e−τs2/2σ2
(s)= (19)
P (2πσ2/τ)d2
with d the dimension of the x-space.
By using (18) we see that eq. (12) can be written as
Θ(1) =f(x,t,s) G(X,T) (20)
D ·
with
f(x,t,s)= (v+s) (s), G(X,T)=∂ Q(X,T) (21)
X
− P
with solution
Θ(1)(x,t,X,T,s)=χ(x,t,s) G(X,T). (22)
·
The vector field χ is called the auxiliary field and it solves the auxiliary equation
χ=f. (23)
D
Finally by averagingeq. (13) over the fast variables , , and integrating over s , ( ), we obtain
h·i ·
∂ (v+s)Θ(1) = ∂ Θ(0) = ∂ Q(X,T) (24)
X T T
h i − h i −
which, using eq. (22), becomes
3
∂ Q= (v +s )χ ∂2 Q=DE ∂2 Q. (25)
T −h j j ii XiXj ij XiXj
This is the diffusion equation describing the large–scale dynamics, i.e., the dynamics in the slow variables. The
effective eddy diffusivity tensor DE is given by
ij
1
DE = (v +s )χ + (v +s )χ . (26)
ij −2 h i i j i h j j i i
(cid:16) (cid:17)
From the auxiliary equation (23) one can show that the DE is positive definite. Indeed, if we consider the i-th and
ij
the j-th component of (23), multiply by χ and χ respectively, sum the two terms and average the result over the
j i
periodicities, , and integrate over the random variable s we end up with
h·i
σ 2
DE = dds (∂ χ ) (∂ χ ) (s) 0. (27)
ij τ sh ii sh ji P ≥
(cid:16) (cid:17) Z
This result can be extended to non-periodic velocity field following the prescriptions in [7].
III. A SOLVABLE CASE: THE STATIONARY SHEAR FLOW
The resolution of the auxiliary field equation for a generic v is not an easy task. Therefore not trivial solvable
cases are useful to understand the properties of the solution. In particular the auxiliary equation can be resolved for
parallel flows, which in two dimensions have the form
v(x,y)=(v(y),0) (28)
where v(y) is an arbitrary function of y. Note that these flows automatically satisfy ∇ v = 0. To evaluate the
·
effective diffusion coefficients we first write the solution of th auxiliary equation as
χ (x,t,s,x′,t′,s′)= dt′ dx′ ds′ (x,t,s,x′,t′,s′) (v (x′)+s′) (s′) (29)
i − G i i P
Z
where is the Green function of the operator :
G D
= δ(x x′)δ(t t′)δ(s s′). (30)
DG − − −
Inserting (29) into eq. (26) we have:
DE = dtdxdsdt′ dx′ ds′ (v (x)+s ) (x,t,s,x′,t′,s′) v (x′)+s′ (s′). (31)
ij − i i G j j P
Z
(cid:0) (cid:1)
Now we note that the Green function can be written as
G
= δ(x x(t;x′,t′))δ(s s(t;s′,t′)) w (32)
G h − − i
where the averageis over the realizations of the white noise w, x(t;x′,t′) and s(t;s′,t′) are the solutions of eqs. (1),
(4) and(28)with initial condition x′ =x(t′;x′,t′) ands′ =s(t′;s′,t′). For the velocityfield (28) the solutions of the
equations (1) and (4) can be written as
x(t)=x′+ tdt v(y(t ))+ tdt s (t )
t′ 1 1 t′ 1 1 1
y(t)=y′+ Rtt′dt1 s2(t1) R (33)
s(t)=s′e−(Rt−t′)/τ + tt′dt1 w(t1)e−(t−t1)/τ.
Inserting (32) and (33) into (31),after some straightforwRardstraightforwardalgebra one obtains:
D1E1 =σ2+ 21π dk |v(k)|2 tl→im∞ tdt′ exp −σ2 k2 (t−t′)−τ 1−e−t−τt′ (34)
Z Z0 (cid:16) h (cid:16) (cid:17)i(cid:17)
and b
DE =0 , DE =σ2 (35)
12 22
4
where v(k) is the Fourier transform of v(y). The same result can be obtained directly from the definition
1
b DE = lim (x (t) x ) (x (t) x ) (36)
ij t→∞2th i −h ii j −h ji i
using (33). Now, because of the inequality
exp σ2 k2τ 1 e−t−τt′ 1 (37)
− ≥
h (cid:16) (cid:17)i
one has
DE(τ) σ2+ 1 dk v(k) 2 lim tdt′ e−σ2k2(t−t′)
11 ≥ 2π | | t→∞
Z Z0
1 v(k) 2
=σ2+ dk| b | =DE(0) (38)
2π σ2 k2 11
Z
b
Therefore for a stationaryparallelflow a colorednoise produces an enhancement of the dispersion. Similar equations
canbeobtainedforatimedependentshearflow. Howeverinthiscaseitisnotsimpletoseethesignofthecorrection.
The results will be reported elsewhere. It is trivial to show that the results in this section hold also for a 3-d shear
flow :
v(x,y,z)=(v(y,z),0,0) (39)
IV. EDDY DIFFUSIVITY FOR SHORT NOISE CORRELATION TIME
By using multiscale technique, the calculation of the eddy diffusivities has been reduced to the solution of the
auxiliary equation (23). Numerical methods are generally needed to solve it but to do so we have to work in a
2d-dimensional space. In general, this is not feasible so, to get more insight the generic v case we study the small
τ case and expand the auxiliary field χ in a power series of τ. A typical time of the physical system which can be
compared to τ is τ =λ/ v2 1/2 i.e. the average time it takes a particle to travel a characteristic length λ.
s
h i
Taking
α =√τ s
i i
∞
τ
χi = 2πσ2 χ(ik)(x,α,t)τk2
k=0
X
we obtain the following expression for the eddy diffusivity tensor
DiEj =−21 2π1σ2 " τk2 hviχj(k)i+hvjχi(k)i +τk−21 hαiχj(k)i+hαjχi(k)i # (40)
Xk (cid:16) (cid:17) (cid:16) (cid:17)
and for the auxiliary equation:
τk2 (∂t+v·∂x)+τk−21 α·∂x−τk−22 ∂α(α )−σ2∂α2α χi(k) =− vi+ √ατi e−2ασ22. (41)
Xk h (cid:0) (cid:1)i (cid:18) (cid:19)
From the expression (40) we see that in order to determine the correction of order τ to DE, we need the quantities
ij
χ (0) , χ (0), χ (1) , χ (1), χ (2) , χ (2), χ (3) . The fields χ (k) obey to the equations
i i i i i i i i
h i h i h i h i
χ (0) =0 (42)
α i
O
αχi(1) =α ∂xχi(0)+αie−2ασ22 (43)
O ·
αχi(2) =(∂t+v ∂x)χi(0)+α ∂xχi(1)+vie−2ασ22 (44)
O · ·
χ (h) =(∂ +v ∂ )χ (h−2)+α ∂ χ (h−1) h 3 (45)
α i t x i x i
O · · ≥
5
where the operator is defined as
α
O
=∂ (α ) σ2∂2 . (46)
Oα α· − αα
The solutions of eqs. (42), (43) and (44) can be written in the form
χi(0) =χi(0)(x,t)e−2ασ22 (47)
χi(1) =(a1iα1+a2iα2+a3i) e−2ασ22 (48)
e
χi(2) = b1iα21+b2iα22+b3iα1α2+b4iα1+b5iα2+b6i e−2ασ22 (49)
the coefficients a and b are fu(cid:0)nctions of x and t while for the field χ (3) it r(cid:1)esults that
ij ij i
χi(3) =c(x,t)e−2ασ22. (50)
h i
By inserting expressions (47), (48) and (49) into eqs. (42), (43), (44) and (45) we can determine the coefficients a
ij
for i =1,2 and b for i =1,..,5 and then by integrating over α the equations for χ (2), χ (3) and χ (4) respectively
ij j j j
we finally have the equations for the remaining functions χ(0), a and b
i 3i 6i
χ (0) = v (51)
xt i i e
O −
a =0 (52)
xt 3i
O
e 3
b = σ2 ∂2v +∂2v ∂ χ(0) +2σ2∂ v ∂2 χ(0) (53)
Oxt 6i 2 i j j i j m jm i
h i
where e e
=∂ +v ∂ σ2∂2. (54)
xt t
O · −
The DE coefficients at the first order in τ read
ij
1
DE =σ2δ v χ(0) + v χ(0)
ij ij − 2 h i j i h j i i
τ σ2 (cid:16) (cid:17)
− 2 2 hvi ∂2χe(j0)i+hvj ∂e2χ(i0)i +hvi b6ji+hvj b6ii . (55)
(cid:20) (cid:16) (cid:17) (cid:21)
We note that instead of the 2d-dimensional eqeuation (26) wee have now a system of two d-dimensional equations (51)
and (53) without the random variable s. Of course this is numerically much more convenient.
We note that by defining new velocity and auxiliary fields as
σ2τ
v =v ∂2v (56)
− 2
χ=χ(0)+τ b +σ2∂2χ(0) (57)
e 6
(cid:16) (cid:17)
and negletting terms of (τ2) equationse(51),e(53) and (55) can bee written as
O
∂ +v∂ σ2∂2 χ =v (58)
t i i
−
and (cid:0) (cid:1)
e e e
1
DE =σ2δ ( v χ + v χ ) (59)
ij ij − 2 h i ji h j ii
formally equivalent to the Gaussian white noise result. e e e e
In the Appendix A we use a different method to obtain the expression of the eddy diffusivity tensor for small τ
up to (τ2). By starting from the Master equation associated with the Langevin equation (1) and using a small τ
O
expansion we derive the Fokker-Planck equation
∂ Θ(x,t)= ∂ [v (x,t)Θ(x,t)]+ ∂2 [ (x,t)Θ(x,t)] (60)
t − i i ij Dij
6
with
τ
(x,t)=σ2 δ + [∂ v (x,t) + ∂ v (x,t)] . (61)
ij ij i j j i
D 2
h i
Now, by applying multiscale technique to (60) we end up with the following equations
w(0) = v (62)
Oxt i − i
w(1) =σ2 ∂2v + ∂2v ∂ w(0)+ ∂ v ∂2 w(0) (63)
Oxt i i j j i k j kj i
(cid:16) 1 (cid:17)
DE =σ2δ v w(0) + v w(0)
ij ij − 2 h i j i h j i i
τ (cid:16) (cid:17)
v ∂2w(0) + v ∂2w(0) + v w(1) + v w(1) (64)
− 2 h i j i h j i i h i j i h j i i
h(cid:16) (cid:17) i
which differ from the previous ones.
This resultposes the questionabout the validity ofthe two sets ofequations andhence of the expansions. In order
to answer to this question we firstly note that the two sets can be considered as a particular case of the generalized
equations
w(0) = v (65)
Oxt i − i
w(1) =σ2 (p q)∂2v +(p q)∂2v ∂ w(0)+p∂ v ∂2 w(0) (66)
Oxt i − i − j j i k j kj i
(cid:16) 1 (cid:17)
DE =σ2δ v w(0) + v w(0)
ij ij − 2 h i j i h j i i
τ (cid:16) (cid:17)
qσ2 v ∂2w(0) + v ∂2w(0) + v w(1) + v w(1) (67)
− 2 h i j i h j i i h i j i h j i i
h (cid:16) (cid:17) i
obtained starting from the Master equation
∂ Θ(x,t)= v (x,t)∂ Θ(x,t)+∂2 [ (x,t)Θ(x,t)] (68)
t − i i ij Dij
where
e
v =v +q σ2τ∂2 v (69)
i i i
pτ
=σ2 δ + (∂ v + ∂ v ) (70)
ij ij i j j i
De 2
(cid:16) (cid:17)
and by applying the multiscale technique.
The role of the two free parameters p and q is of generalizing the Master equation (60) being our feeling that
there exists a family of different microscopic process specified by v˜ and that correspondto the same macroscopic
ij
diffusive process specified by DE. D
ij
The first set consisting of the equations (51), (53) and (55) corresponds to q = 1/2 and p = 2, whereas the
second one consisting of the equations (62), (63) and (64) corresponds to q = 0 and p = 1. Now a closer analysis of
the diffusion coefficients (70) reveals that it may take negative values, introducing unphysical singularities into the
problem. Moreover the multiscale technique requires a DE definite positive. This means that (68) do not constitute
ij
genuine Markovian process with a well defined corresponding Langevin equation driven by a Gaussian white noise.
In general this expansion do not converge uniformly in x and the range of validity is restricted to τ 1, τ/σ2 1
≪ ≪
(in dimensionless unit), thus they are asymptotic estimate for τ 0. In this range of equations validity we do not
→
have any a priori arguments for choosing one of the two sets of equations and it is for this reason that we expect to
obtain the same value of the eddy diffusivity tensor using the two sets. This is indeed the case for the parallel shear
flows (see Appendix B). This result can be thought as a first indication of the equivalence (with respect to the value
of the diffusivity) of the generalized equations.
In general we do not expect that all possible choices of q and p give the same DE but we can think to select the
ij
class of equivalent process by applying the generalized equations to the parallel shear flow and imposing DE equal to
ij
the known expression (38) independent on both q and p. This calculation is reported in Appendix B and it ends up
in the following condition
p=2q+1 (71)
7
All these considerations suggest that for short τ there exists a class of equivalent equations depending on the
parameter p that lead to the same eddy diffusivity tensor.
Ifthisisthecase,amongtheallpossiblechoicesofqandpconsistentwith(71),wecanchoose: p=0andq = 1/2.
In this case =σ2δ and (60) reduces to a truly Markovianprocess with the associated Langevin equation −
ij ij
D
d
x=v(x,t)+ξ(t) (72)
dt
where ξ is a Gaussian white noise. This is the micreoscopic Markovian process which approximates the long time
and large space transport properties of a colored noise process with short noise correlation time. In other words to
study the diffusion properties of (1)–(2) for smallτ we canreplace the originalcolorednoise process with the process
described by (72). This will give the correct diffusion coefficients up to (τ2).
O
We havecheckednumerically that for the AB flow the eddy diffusivity tensor assumes the same value for the three
different choices of the parameters q and p consistent with (71): p = 2 and q = 1/2, p = 1 and q = 0, p = 0 and
q = 1/2. This make us confident with our conclusions.
−
V. FLOWS WITH CLOSED STREAMLINES
We apply now our analysis to two models for the Rayleigh-B´enard steady convection: the first one consists of
an horizontal extent of convection cells much larger than its height so that the flow can be considered quasi-two-
dimensional; the second one is the two-dimensional AB flow made of a structure periodically repeated in the space.
A. A quasi-two-dimensional flow
We consider the flow discussed by Shraiman [17]. This is described by the stream function
vL πk π
ψ(x,y)= sin x sin y (73)
πk L L
(cid:18) (cid:19) (cid:16) (cid:17)
with v being the characteristic velocity, L the height of the cell (y [0,L]) and L/k the x-periodicity of the roll
∈
pattern. The top and the bottom plates of the cell are assumed impermeable for the passive scalar so that the
appropriate boundary conditions for the tracers density function Θ are ∂ Θ = 0. The streamlines of the flow
y y=0,L
|
are illustrated in Fig. (2). Using Fokker-Planckequation
∂ Θ= v ∂Θ+σ2∂2Θ, (74)
t
− ·
for large P´ecletnumber 1 Pe=vL/σ2, k=2 and τ =0 the eddy diffusivity coefficient DE(0) has been calculated in
11
[17] and it is
σ2 vLσ2
DE(0)= √Pe= . (75)
11 √π π
r
According to the results of the previous section, the colored noise case with small τ is described up to order (τ2)
O
by the same Fokker-Planck equation provided the velocity field is renormalized as eq. (69) with q = 1/2. Taking
−
into account that v = ∂ ψ, v =∂ ψ, and using (73), we have
x y y x
−
∂2ψ = (π/L)2 (1+k2)ψ.
−
Therefore up to order (τ2) the diffusion is described by
O
1The case studied byShraiman corresponds to thedelta-correlated case ( τ =0 )
′ 2 ′
hsi(t)sj(t)i=σ δijδ(t−t)
where σ2 is now themolecular diffusivity of theconsidered system.
8
σ2τ π 2
ψ(x,y)= 1+ 1+k2 ψ(x,y)=c(σ2,τ,k)ψ(x,y). (76)
2 L
(cid:18) (cid:16) (cid:17) (cid:0) (cid:1)(cid:19)
Since c does notdependeonx andt, we canrepeatthe calculationofShraimanandobtainunder the same conditions
of (75) the expression for the eddy diffusivity coefficient
σ2 5σ2τ π 2
DE(τ)= √cPe=DE(0) 1+
11 √π 11 s 2 L
(cid:18) (cid:16) (cid:17) (cid:19)
5σ2τ π 2
=DE(0) 1+ + (τ2). (77)
11 4 L O
(cid:18) (cid:16) (cid:17) (cid:19)
We then conclude that, in this case, a small τ enhances the diffusion coefficient.
The same result can be deduced from the multiscale equations (58) and (59): in fact because of the structure of
these two equations it is not difficult to show that if we changeonly the module of the velocity field (v =cv) and we
know the explicit form of DE =f(v) as a function of v we have DE =f(v). For large P´eclet number and τ =0 the
ij ij
function f(v) is given by eq. (75) from which (77) follows. e
e e
B. The AB flow
The AB flow is given by the velocity field
v(x,y)=(Bcos(y), Acos(x)). (78)
For A=B =1 the streamlines form a closed periodically repeated structure made of four cells as shown in Fig. (3).
We expect that the diffusive behavior of such a system is similar to the previous case, in fact we know [15] that for
small P´eclet number Pe the eddy diffusivity tensor is proportional to √Pe like in the quasi-two-dimensionalcase.
In the Figure (1) the behavior of the quantity ∆=[DE(τ) DE(0)]/[DE(0)τ] versus σ2 is shown. The correction
−
∆ has been calculated by integrating numerically the equations (51) and (53) and evaluating the quantity
1 σ2 DE(τ) DE(0)
v ∂2χ(0) + v b = − =∆ (79)
− DE(0) 2 h 1 1 i h 1 61i DE(0)τ
(cid:20) (cid:21)
for different values of σ2. The equations are seolved by using a pseudo-spectral method [13] in the basic periodicity
cell with a grid mesh of 64 64 points. De-aliasing has been obtained by a proper circular truncation which ensures
×
better isotropy of numerical treatment.
It is evident that also in this case the numerical results follow a linear behavior with a positive angular coefficient
and so we can conclude that the introduction of the colored noise leads to an enhancement of the diffusion. In
particularwe cansee that the numericalresults follow very wellthe line ∆=σ2/4. This is not surprisingbecause we
know that for large Pe
DE(0)=DE(0)=DE(0)=C √Pe=C √v (80)
11 22 1 2
therefore using the same arguments of the previous section we can deduce that
1 1
DE(τ) = DE(0) 1+ σ2 τ = DE(0) 1+ σ2τ + (τ2) . (81)
2 4 O
r (cid:18) (cid:19)
in a very good agreement with the numerical results.
VI. SUMMARY AND CONCLUSIONS
In this paper we have studied the transport properties in velocity fields whose small scales are parameterized by
Gaussian colored noise. We analyzed in particular the effects of a finite noise correlation time τ on the diffusive
properties for large time and spatial scales. In this limit, using the multiscale technique, we derive the diffusion
equation(25)andtheassociatedeffectivediffusiontensor. Thelatterisobtained,oncethevelocityfieldv isgiven,by
the solution of an auxiliary equation, see (23) and (26), of the same structure of the original Fokker-Planckequation
9
[19]. The former is, however,an exact result for the diffusive regime valid for very long times, thus avoiding all finite
time effects of the Fokker-Planck equation or the associated Langevin equation (1),(2).
Theauxiliaryequationcannotbe solvedforagenericvelocityfield,neverthelesstherearenontrivialflowsforwhich
the solution can be found. This is the case for the steady parallel flow for which the effective diffusion coefficient is
an increasing function of τ. To study in more details the effects of a small noise correlation time for a generic v we
have performed a small-τ expansion and evaluatedthe first correction (τ) to the effective diffusion coefficient. This
O
is done by using two different approach. We find that to order (τ) there exist a one-parameter family of flows with
O
the same diffusion properties. This invariance can be used to pick up the most convenient microscopic dynamics,
frombothanalyticalandnumericalporpoises. We applythe small-τ resultstotwotwo–dimensionalmodelflowswith
closed streamlines. In both the cases we find an enhancement of the diffusion.
The enhancement of the diffusion for small τ has been interpreted in [18] in terms of interference mechanism
betweenturbulentandmoleculardiffusion. The colorednoisemakesthe diffusion particlesforgottenoftheir previous
positions less rapidly than in the white noise case, thus the Lagrangian correlation time increases and so does the
eddy-diffusivity.
The study of the problem for τ not small is object of current work.
ACKNOWLEDGMENTS
We warmly thank Angelo Vulpiani, Massimo Vergassola and Andrea Mazzino for extensive, stimulating and very
useful discussions and suggestions. PC is grateful to the European Science Foundation for a TAO exchange grant.
This work has been partially supported by INFM (Progetto Ricerca Avanzata- TURBO) and by MURST (program
no. 9702265437).
APPENDIX A: THE MASTER EQUATION FOR COLORED NOISE AND SMALL τ
In this appendix we derive the master equation for the probability density Θ(x,t) in the limit of small τ for the
process described by the Langevin equation (1) and (2). If the random variable s is a Gaussian white noise of zero
mean, Θ(x,t) satisfiesthe Fokker-Planckequation. We addresshere the case ofa colorednoise. Now the processx is
non-Markovianand no exact simple equation for Θ is known. Let us consider an Ornstein-Uhlenbeck process s, that
is a zero mean Gaussian process with correlations [c.f.r eq. (2)]
Cij(t,t′)= si(t)sj(t′) = σ2 δije−|t−τt′| (A1)
h i τ
where τ is the correlationtime. The probability density is given by
Θ(x,t)= δ(x(t) x) (A2)
h − i
where x(t) is a solution of (1) for a given realization of s and for a given initial condition. The average is taken over
the noise realizations. Taking the time derivative of (A2) and using eq. (1) one gets
∂ Θ(x,t)= ∂ [v (x,t)Θ(x,t)]+∂ s (t)δ(x(t) x) . (A3)
t i i i i
− h − i
Taking advantage of the Gaussian nature of s the average in (A3) can be rewritten as
∂tΘ(x,t)=−∂i[vi(x,t)Θ(x,t)]+ ∂i2j tdt′ στ2 e−t−τt′ δδsxj((tt′)) δ(x(t)−x) . (A4)
Zt0 (cid:18) (cid:19) (cid:28) i (cid:29)
Because of the δ-function a closed equation is possible only if the functional derivative either does not involve the
process x or depends on it solely at the “Markovian” end-point t = t′. At this stage thus we cannot simplify the
master equation any further. In the limit of small correlationtime τ a closed equation can be derived by performing
the largely used small τ expansion. If the noise is close to the white noise limit (τ = 0) it is reasonable to expand
the functional derivative about its Markovian value, i.e., the one obtained for the δ-correlated noise. The Taylor
expansion of δx (t)/δs (t′) around the Markovianend point t′ =t is
j i
δx (t) δx (t) d δx (t)
j = j + j (t′ t) + ...
δs (t′) δs (t′) dt′δs (t′) −
i i (cid:12)t′=t i (cid:12)t′=t
=δij ∂i(cid:12)(cid:12)vj(x,t)(t′ t) +(cid:12)(cid:12) (A5)
− (cid:12) − ·(cid:12)··
10