Asymptotic Theory for the Probability Density Functions in Burgers Turbulence

4 Pages·0.12 MB·English

by  Weinan E

Asymptotic Theory for the Probability Density Functions in Burgers Turbulence
Weinan Ea) and Eric Vanden Eijndenb)
Courant Institute of Mathematical Sciences
New York University
New York, New York 10012
A rigorous study is carried out for therandomly forced Burgers equation in theinviscid limit. No
closure approximationsare made. Instead theprobability density functionsofvelocity and velocity
gradientarerelated tothestatisticsofquantitiesdefinedalongtheshocks. Thismethodallows one
to compute the anomalies, as well as asymptotics for the structure functions and the probability
density functions. It is shown that the left tail for the probability density function of the velocity
gradient has to decay faster than |ξ|−3. A further argument confirms the prediction of E et al.
[Phys. Rev. Lett. 78, 1904 (1997)] that it should decay as |ξ|−7/2.
PACS numbers: 47.27.Gs, 05.40.-a, 02.50.Ey
9
9
9
In this Letter, we focus on statistical properties of so- are generated near shock creation. Their analysis gives
1
lutions of the randomly forced Burgers equation a rigorous upper-bound for α: α ≤ 7/2. In [5], it was
n
claimedthatthisboundisactuallyreached,i.e.,α=7/2.
a
J ut+uux =νuxx+f, (1) FinallyGotohandKraichnan[6]arguedthattheviscous
termisnegligibletoleadingorderforlarge|ξ|,i.e. F ≈0
8
wheref isazero-mean,statisticallyhomogeneous,white- for |ξ| ≫ B1/3. This approximation leads to α = 3 and
in-time Gaussian process with covariance 1
1 β =1. For other approaches, see e.g. [7,8]. In this letter
v
we proceed at an exact evaluation of (4) and we prove
6 hf(x,t)f(y,s)i=2B(x−y)δ(t−s), (2)
that α has to be strictly larger than 3 (a result which
0
0 where B(x) is smooth. We are particularly interested doesnotrequirethatsteadystatebereached). Atsteady
1 in the probability density function (pdf) of the velocity state, we prove that β = 1 and we give an argument
0 gradient ξ(x,t) = u (x,t), since it depends heavily on which supports strongly the prediction of [5], namely,
9 x
the intermittentevents createdby the shocks. Assuming α=7/2.
9
/ statistical homogeneity, and letting Q(ξ;t) be the pdf of To begin with, let us remark that it is established in
n ξ(x,t), it can be shown that Q satisfies the mathematics literature that the inviscid limit
y
d Q =ξQ+ ξ2Q +B Q −ν hξ |ξiQ , (3) u0(x,t)= limu(x,t), (6)
- t ξ 1 ξξ xx ξ ν→0
o
a where B =−B(cid:0) (0)(cid:1). hξ |ξi is the (cid:0)ensemble-(cid:1)averageof existsforalmostall(x,t). Sinceu0willingeneraldevelop
h 1 xx xx
ξ conditional on ξ. The explicit form of this term is shocks, say, at x = y, we may have u0 ∝ δ(x−y), and
c xx x
: unknown, leaving (3) unclosed. There have been several one cannot simply drop the viscous term in the Burgers
v
proposalsonhowtoapproximatelyevaluatethequantity equationwithoutgivingsomemeaningtou0u0 atshocks.
i x
X ThiscanbedoneusingBV-calculus[9],whichallowsone
ar F(ξ;t)=−νli→m0ν hξxx|ξiQ ξ. (4) towriteanequationforu0 andgivesrulesformanipulat-
ing the terms entering this equation and computing the
(cid:0) (cid:1)
Atsteadystate,theyallleadtoanasymptoticexpression effect of the viscous term in the inviscid limit. An alter-
of the form native, more intuitive, way of accessing the effect of the
C−|ξ|−α as ξ →−∞, viscous shock on the velocity profile outside the shock
is to carry out an asymptotic analysis near and inside
Q∼ (5)
( C ξβe−ξ3/(3B1) as ξ →+∞, the shock. Here we will take the second approach and
+
refer the interested reader to [10] for the first approach
forQ,butwithavarietyofvaluesfortheexponentsαand withBV-calculus. Itisimportanttoremarkthatthetwo
β (here the C±’s are numerical constants). By invoking approaches lead to the same results.
the operator product expansion, Polyakov [1] suggested Before considering velocity gradient, it is helpful to
that F = aQ+bξQ, with a = 0 and b = −1/2. This study the statistics of velocity itself. Let R(u;t) be the
leads to α = 5/2 and β = 1/2. Boldyrev [2] considered pdf of u(x,t). Assuming statistical homogeneity, R sat-
thesameclosurewith−1≤b≤0,whichgives2≤α≤3 isfies
and β = 1 + b. The instanton analysis [3,4] predicts
the right tail of Q without giving a precise value for β, Rt =B0Ruu−ν huxx|uiR u, (7)
but has notgivenany specific prediction for the left tail.
(cid:0) (cid:1)
E et al. [5]made a geometricalevaluationofthe effectof where B0 = B(0). To compute −ν huxx|uiR u, let us
F,basedontheobservationthatlargenegativegradients notethatforν ≪1,thesolutionsof(1)consistofsmooth
(cid:0) (cid:1)
1
pieceswhere the viscouseffectisnegligible,separatedby u−s/2
limνhu |uiR=−ρ ds du¯ (u−u¯)T(u¯,s;t).
thin shock layers inside which the viscous effect is im- ν→0 xx Z Zu+s/2
portant. Let u (x,t) be the solution of the Burgers
out
(13)
equationoutside the viscousshock layer;u canbe ob-
out
tained as a series expansion in ν. To leading order in ν,
This equation gives an exact expression for the viscous
u satisfiesRiemann’sequation,u +uu =f. Inorder
out t x contribution in the limit ν → 0 in terms of certain sta-
to deal with the shock layer, say at x=y, define
tistical quantities associated with the shocks. Of course,
using (13)in (7) does notlead to a closedequationsince
x−y
uin(x,t)=v ,t , (8) T remains to be specified. However, information can al-
ν
(cid:18) (cid:19) ready be obtained at this point without resorting to any
and write v = v + νv + O(ν2). To leading order, closure assumption. For instance, using (13) in (7) and
0 1
v (z,t) satisfies (v −u¯)v = v , yielding v (z,t) = takingthesecondmomentoftheresultingequationyields
0 0 0z 0zz 0
u¯−(s/2)tanh(sz/4) where u¯=dy/dt and s is the jump hu2it =2B0−2ǫ with
across the shock. Consequently we have the following
1
generic velocity profile inside the shock layer: ǫ= limνhu2i= ρh|s|3i. (14)
ν→0 x 12
s s(x−y)
uin(x,t)=u¯− tanh +O(ν). (9) In particular, at steady state ρh|s|3i=12B0.
2 4ν
(cid:18) (cid:19) Similar calculations can be carried out for multi-point
pdf’s and, in particular, for W(w;x,t), the pdf of the
Theactualvaluesofu¯andsareobtainedfromthematch-
velocity difference w(x,z,t) = u(x + z,t)− u(z,t). It
ing conditions between u and u . In terms of v and
in out
leads to an equation of the form
the stretched variable z, they are
w
s ′ ′
z→lim±∞v0 =x−lyim→0±uout =u¯± 2. (10) Wt=−wWx−2 −∞dw Wx(w ;x,t) (15)
Z
+2[B −B(x)]W +H(w;x,t),
0 ww
It is well-known that s≤0.
We willuse(9)toevaluatetheviscoustermin(7). By where, to O(x), H is given by
definition [11],
H=ρ wS(w;t)+hsiδ(w)
1 L w (16)
νhuxx|uiR=ν lim dx uxxδ[u−u(x,t)]. (11) +2ρ(cid:2) dw′ S(w′;t)−(cid:3)2ρθ(w)+O(x).
L→∞2LZ−L Z−∞
Inthelimitν →0onlysmallintervalsaroundtheshocks Here θ(w) is the Heaviside function and S(s;t) =
will contribute to the integral. So, we can split the inte- du¯T(u¯,s;t) is the conditional pdf of s(y,t). By direct
gralinto smallpiecesinvolvingonlythe shocklayersand
substitutionitmaybe shownthatthesolutionof(15)is,
usethe genericformofuin inthe layersto evaluatethese Rto O(x2), [12]
integrals. To O(ν), this gives
1 w
W ∼(1−ρx) Q ;t +ρxS(w;t)+O(x2). (17)
νhuxx|uiR x x
N 1 (cid:16) (cid:17)
=ν lim dx u δ[u−u (x,t)] ThefirstterminthisexpressioncontainsQ(ξ;t),thepdf
L→∞2LN j Zj−th layer inxx in (12) of the non-singular part of the velocity gradient, to be
X +∞
consideredbelow(see(20)). Thistermaccountsforthose
=ρ dsdu¯ T(u¯,s;t) dz v δ[u−v (z,t)],
0zz 0 realizationsoftheflowwherethereisnoshockinbetween
−∞
Z Z
z and x+z (an event of probability 1−ρx+O(x2)).
where in the second integral we picked any particu- This term also leads to the consistency constraint that
lar shock layer and we went to the stretched variable limx→0W = δ(w) (using limx→0Q(w/x;t)/x = δ(w)).
z = (x−y)/ν. Here N denotes the number of shocks The next term in (17), ρxS(w;t), accounts for the real-
in [−L,L], ρ = ρ(t) = limL→∞N/2L is the shock den- izations of the flow where there is a shock in between z
sity,andT(u¯,s;t)istheprobabilitydensityofu¯(y,t)and and x+z (an event of probability ρx+O(x2)). Equa-
s(y,t) conditional on the property that there is a shock tion(17)canbeusedtocomputethestructurefunctions,
atpositiony (T isindependentofy becauseofstatistical h|w|ai= dw|w|aW. To leading order this gives
homogeneity). The last integral in (12) can of course be
evaluated using the explicit form of v0. Another, more R xah|ξ|ai+O(x) if 0≤a<1,
elegant, way to proceed is to use the equation for v , h|w|ai∼ (18)
0 (xρh|s|ai+O(x1+a) if 1<a,
(v −u¯)v = v , and change the integration variable
0 0z 0zz
from z to v using dzv = dv v /v = dv (v −u¯).
0 0zz 0 0zz 0z 0 0 where h|ξ|ai = dξ|ξ|aQ. Using ρh|s|3i = 12B , we get
The result is 0
Kolmogorov’srelation for a=3
R
2
h|w|3i∼12xB . (19) Notice that the exponential terms are irrelevantin these
0
expression since s≤0. Equation (26) implies
Wenowgobacktothevelocitygradient. Observefirst
that, in the limit ν → 0, the velocity gradient can be 2u¯ s 2f
t t x
written as z→lim±∞v1z =∓ s − s ± s =ξ±, (27)
ux(x,t)=ξ(x,t)+ s(yj)δ(x−yj), (20) where the last equality is just the definition of ξ±. Note
j that (27) can be rewritten as
X
where the y ’s are the locations of the shocks, ξ is the s s
non-singularjpartofu . Assuminghomogeneity,adirect st =−2 ξ−+ξ+ , u¯t = 4 ξ−−ξ+ +fx. (28)
x
consequence of (20) is (cid:0) (cid:1) (cid:0) (cid:1)
In the limit ν → 0 these are the equations of motion
hu i=hξi+ρhsi=0. (21) along the shock.
x
We can now evaluate the viscous contribution using
Unlike the viscous case where ξ =u , hence hξi=0, we
x
haveintheinviscidlimithξi=−ρhsi=6 0.Notealsothat 1 L
νhξ |ξiQ=ν lim dx ξ δ[ξ−ξ(x,t)]. (29)
the inviscid limit of the solutions of (3) converge to the xx L→∞2LZ−L xx
pdf of ξ only, which is still going to be denoted by Q.
ToevaluateF,therearetwowaystoproceed. Oneisto The calculation is similar to the one for the velocity and
rewrite(15)intermsofthepdfof(u(x+z,t)−u(z,t))/x eventually leads to
andtakethelimitasxgoestozero. Thisistheapproach
ρ
taken in [10]. The other is to evaluate (4) directly. The F(ξ;t)= ds s V−(ξ,s;t)+V+(ξ,s;t) , (30)
2
two approaches amount to different orders of taking the Z
(cid:2) (cid:3)
limit x→0,ν →0, and give the same result. Hence the
whereV±(ξ,s;t)arethe conditionalpdf’s ofξ±(y,t)and
twolimitingprocessescommute. Wewilltakethesecond
s(y,t). Theappearanceofξ± in(30)isofcourseadirect
approach and evaluate (4) using the same basic idea as
result of the O(1) term in (22).
above. Here,however,wehavetoproceedmorecarefully
Wenowuse(30)in(3)andanalyzesomeconsequences
withthe shocklayeranalysis. Differentiationof(9)gives
of
s2 s(x−y)
ξin(x,t)=−8νsech2 4ν +O(1). (22) Qt =ξQ+ ξ2Q ξ+B1Qξξ+F(ξ;t). (31)
(cid:18) (cid:19)
(cid:0) (cid:1)
Taking the first moment of (31) leads to
While the next order term in (9) was negligible in the
dliommitinνat→es t0h,ethOe(νO−(11))ccoonnttrriibbuuttiioonnattotξhien(bxo,rtd)earcotufatlhlye hξit = ξ3Q +−∞∞+ ρ2[hsξ−i+hsξ+i], (32)
shock layer because the latter falls exponentially fast as
(cid:2) (cid:3)
theouterregionisapproached,whereastheformertends where we used dξ ξF = ρ[hsξ−i+hsξ+i]/2. On the
to constants, say, ξ±. In particular, the matching be- other hand, averaging the first equation in (28) gives
R
tween ξ (x,t) and ξ (x,t) involves the O(1) terms. To
out in ρ
see how matching takes place, differentiating the expres- ρhsi t =−2[hsξ−i+hsξ+i]. (33)
sion for u , we have ξ = ν−1v +v +O(ν). The
in in 0z 1z (cid:0) (cid:1)
matching condition between ξ and ξ reads This equation uses the fact that shocks are created at
in out
zero amplitude, and shock strengths add up at collision.
z→lim±∞v1z =x−lyim→0±ξout ≡ξ±. (23) These are consequences of the fact that the forcing is
smooth in space [13]. Since hξi = −(ρhsi) from (21),
t t
The equation for v1 is the comparison between (32) and (33) tells us that the
boundary term in (32) must be zero. Since Q ≥ 0, ξ3Q
v +(v −u¯)v +v v =v +f , (24)
0t 0 1z 1 0z 1zz x has different sign for large positive and large negative
and, from the above argument, the only information we values of ξ. Therefore we must have limξ→+∞ξ3Q = 0
really need about v is its values at the boundaries z → and limξ→−∞ξ3Q = 0. This proves that Q goes to zero
1 faster than |ξ|−3 as ξ →−∞ and ξ →+∞.
±∞. Since v falls exponentially fast for large |z|, (24)
0z
The analysis can be carried out one step further for
reduces to
the stationary case (Q = 0). In this case, treating (31)
s s t
u¯ ± t ± v =v +f , z →±∞, (25) as an inhomogeneous second order ordinary differential
t 2 2 1z 1zz x
equation,wecanwriteitsgeneralsolutionasQ=C Q +
1 1
where we used the asymptotic values of v . Thus, as C Q +Q ,whereC andC areconstants,Q andQ are
0 2 2 3 1 2 1 2
z →±∞, two linearly independent solutions of the homogeneous
equationassociatedwith (31), andQ is some particular
3
2u¯ s 2f
v ∼∓ tz− tz± xz+c±+c±e±sz/2. (26) solutionofthis equation. One suchparticularsolutionis
1 s s s 1 2
3
Q = ξ dξ′ ξ′F(ξ′) − ξe−Λ ξ dξ′ eΛ′G(ξ′), (34) Even though this argument gives only a lower bound for
3 Z−∞ B1 B1 Z−∞ F at large negative values of ξ, further arguments pre-
sented in [10] indicate that this lower bound is actually
where Λ=ξ3/(3B1) and sharp.
We thank Bob Kraichnanand Stas Boldyrev for stim-
ξ ′ ξ′F(ξ′) ulating discussions. The work of W. E is supported by
G(ξ)=F(ξ)+ξ dξ . (35)
Z−∞ B1 a Presidential Faculty Fellowship from the National Sci-
ence Foundation. The work of E. V. E. is supported by
Withthisparticularsolution,itcanbeshown(see[10]for
U.S. Department of Energy Grant No. DE-FG02-86ER-
details)thattherealizabilityconstraintsimplythatC =
1 53223.
C =0, i.e. the only non-negative, integrable solution is
2
Q = Q . Furthermore, in order that Q actually be non-
3
negative, F must satisfy
0≥F ≥Cξ2e−ξ3/(3B1) as ξ →+∞, (36)
for some constant C <0. Substituting into (34), we get a) Electronic address: [email protected]
b) Electronic address: [email protected]
C−|ξ|−3 −ξ∞dξ′ ξ′F(ξ′) as ξ →−∞, [1] A. M. Polyakov, Phys. Rev. E 52, 6183 (1995). Here
Q∼ (37)
( C ξe−ξ3R/(3B1) as ξ →+∞, we followed the presentation of [6]. It is worth noting
+ thatPolyakovneverdiscussesvelocitygradients.Instead
he studies the pdf of velocity increment over the iner-
which confirms the result Q ∼ C−|ξ|−α with α > 3 as tial range of scales. However, his predictions imply the
ξ →−∞, and gives β =1.
results stated here, as a consequence of the discussions
The actual value of the exponent α depends on the before (22).
asymptotic behavior of F. The latter can be obtained [2] S. A. Boldyrev, Phys.Rev.E 55, 6907 (1997).
fromfurtherconsiderationsonthedynamicsoftheshock [3] V.GurarieandA.Migdal,Phys.Rev.E54,4908(1996).
(28). This is ratherinvolvedandwillbe left to [10]. The [4] E. Balkovsky, G. Falkovich, I. Kolokolov, and V. Lebe-
result gives α = 7/2 which confirms the prediction of dev, Phys.Rev.Lett. 78, 1452 (1997).
[5]. Herewewillrestrictourselvestoaninterpretationof [5] W.E,K.Khanin,A.Mazel,andYa.G.Sinai,Phys.Rev.
Lett. 78, 1904 (1997).
the current approach in terms of the geometric picture.
[6] T. Gotoh and R. H. Kraichnan, Phys. Fluids 10, 2859
Observe that the largest values of ξ± are achieved just
(1998).
aftertheshockformation. Assumethatashockiscreated
[7] J.-P.Bouchaud,M.M´ezard,andG.Parisi, Phys.Rev.E
at time t = 0, position x = 0, and with velocity u = 0. 52, 3656 (1995).
Then, locally [8] J.-P. Bouchaud and M. M´ezard, Phys. Rev. E 54, 5116
(1996).
x=ut−au3+···. (38) [9] A. I. Vol’pert, Math. USSR-Sbornik2, 225 (1967).
[10] W. E and E. VandenEijnden, in preparation.
It follows that for t ≪ 1 the solutions of 0 = ut−au3,
[11] Here we use ergodicity with respect to spatial average.
u±, behave as This restricts us to working on the entire line in which
case the existence of a stationary state remains unclear.
t t However,spatialaverageisusedonlyforsimplicityofthe
u± =∓ ⇒ s=−2 , (39)
a a argumentandcanbeavoidedasisdonein[10].Thenone
r r
can work on a finite system for which case the existence
and ξ±, solutions of 1=ξt−3au2ξ, behave as of a stationary state is proved in: W.E, K. Khanin, A.
MazelandYa.Sinai,”Invariantmeasuresfortherandom-
1
ξ± =− . (40) forced Burgers equation”, submitted to Ann.Math..
2t [12] Equation (17) uses relation (21).
[13] Incases(notconsideredhere)whenshocksarecreatedat
Assuming that these give the dominant contribution to
finiteamplitude(e.g.intheunforcedcasewithpiecewise
F(ξ) for large negative values of ξ, the asymptotic form
linear initial data), an additional term appears at the
of F is
righthandsideof(33).Consequentlyfrom(32),Qdecays
∞ like |ξ|−3 when ξ→−∞.
F ∼C dt s(t) δ[ξ−ξ−(t)]+δ[ξ−ξ+(t)] , (41)
Z0
(cid:8) (cid:9)
where C is some constant related to the statistics of the
shock life-time and a, and s(t), ξ±(t) are given by (39),
(40). The evaluation of (41) gives F ∼ C|ξ|−5/2, and,
hence,
Q∼C−|ξ|−7/2 as ξ →−∞. (42)
4