Two-loop calculation of the scaling behavior of two-dimensional forced Navier-Stokes equation

5 Pages·0.12 MB·English

by  J. Honkonen

Two-loop calculation of the scaling behavior of two-dimensional forced Navier-Stokes
equation
J. Honkonena, Yu. S. Kabritsb and M. V. Kompanietsb
aTheoretical Physics Division, Department of Physical Sciences, P.O.Box 64, 00014 University of Helsinki, Finland
bDepartment of Theoretical Physics, St Petersburg University, Ulyanovskaja 1, St Petersburg, Petrodvorez, 198904 Russia.
(February 8, 2008)
sphericturbulence [11]the energyandenstrophysources
Asymptotic properties of the solution of two-dimensional
are at the outer edges of the scaling intervals, and it
randomlyforced Navier-Stokesequation with long-rangecor-
2 is not clear whether there is an energy and enstrophy
relations of the driving force are analyzed in the two-loop
0 sink between them [11] or they coexist [9]. In both cases
order of perturbation theory with the use of renormalization
0 the Kolmogorov spectrum of the inverse energy cascade
2 group. Kolmogorov constant of the energy spectrum is cal- E(k) k−5/3 for k k is observedexperimentally and
culated for both the inverse energy cascade and the direct ∝ ≪ I
n enstrophy cascade in thesecond order of theε expansion. in the majority of simulations for wave numbers smaller
a than the characteristic wave number k of the energy
I
J 47.27.Gs,05.10.Cc pumping. For the enstrophy inertial range the existing
5 data is not so clear-cut, although recent high-precision
1 numerical simulations [12] and accurate analysis of the

atmospheric data [13] seem to support the existence of
D] I. INTRODUCTION an energy spectrum k−3 in the enstrophy cascade.
∝
In this paper we have carried out a two-loop renor-
C
Asymptotic properties of the stochastic Navier-Stokes malization of the field theory generated by the two-
.
n equationhavebeenactivelystudiedwiththeaidofrenor- dimensional randomly forced Navier-Stokes equation.
li malization group (RG) during the last two decades (for We have taken into account the divergences specific of
n a comprehensive review, see [1]). A random force with two dimensions in the way proposed in Ref. [3], and cal-
[
powerlike falloff of spatial correlations has been used to culated the asymptotic expression for the energy spec-

1 maintain a stationary scale-invariant regime with a sub- trum in the second orderof the ε expansionfor both the
v sequentgenerationofanεexpansionofscalingexponents energyandenstrophyinertialranges. Forconvenienceof
5 and scaling functions. calculations, we have used the stochastic Navier-Stokes
2
Most work has been carried out in view of appli- equationwithasubsequentdimensionalregularizationof
0
cation to three-dimensional turbulence; partly because divergences in two dimensions [3] instead of the specifi-
1
0 of greater physical interest than in the two-dimensional cally two-dimensionalsetup [4] with the stream-function
2 case, partly due to additional divergences, which occur description of the fluid flow.
0 in two dimensions and prevent a direct use of the previ-
/ ously obtained general d-dimensional results. Moreover,
n
i there were some serious flaws in the early work [2] on II. RENORMALIZED FIELD THEORY FOR THE
nl the RG-analysis of the forced Navier-Stokes equation in TWO-DIMENSIONAL STOCHASTIC
NAVIER-STOKES EQUATION
: twodimensions,andonlyrecentlyhasaconsistentrenor-
v
malization procedure been put forward[3,4] and used in
i
X various problems involving randomly forced incompress- Considerthe stochasticNavier-Stokesequationforthe
r ible fluid [5–7]. flow of homogeneous incompressible fluid, which for the
a Apart from energy, in two dimensions the enstro- transverse components of the velocity field assumes the
phy (squared vorticity) is an inviscid conserved quan- form
tity quadratic in the velocity. Therefore,two self-similar
∂ v +P v ∂v =ν 2v ξ v +F , (2.1)
regimes corresponding to an inverse energy cascade to- t i ij l l j 0 i 0 i i
∇ −
wards small wave numbers and a direct enstrophy cas-
together with the incompressibility condition ∂ v = 0.
cade towards large wave numbers are expected to take i i
In Eq. (2.1) v (t,x) are the coordinates of the diver-
place [8] instead of the direct energy cascade observed i
genceless velocity field, ν is the kinematic viscosity, ξ
in three-dimensional turbulence. The energy (enstro- 0 0
isthecoefficientoffriction,andP isthetransversepro-
phy) pumping leading to a steady state with the two ij
jectionoperator(P =δ k k /k2 inthe wave-number
scaling regimes may be realized in two different ways. ij ij− i j
space), and F are the coordinates of the random force.
On one hand, in numerical simulations [9] and some ex- i
Here, and henceforth, summation over repeated indices
periments [10] the energy and enstrophy pumping takes
is implied.
place on scales in between the inverse energy cascade
In experimental realizations and simulations of a two-
andthe enstrophy cascade. Onthe other hand, in atmo-
dimensionalturbulentflow energymay be consumednot
1
only by microscale dissipation, but also by the friction theuseofGalileiinvariance,causalityandsymmetriesof
at the boundaries of the fluid layer. The friction term in the model allows to write the renormalizedaction in the
Eq. (2.1) makes it possible to maintain stationary state form
withtheanticipatedinverseenergycascadetowardssmall
1 dtdk
wave numbers and the direct enstrophy cascade towards S = v˜
2 (2π)d
large wave numbers, when the pumping is carriedout in Z
between the corresponding inertial ranges. The coeffi-
cient of friction is a mass term from the point of view of g1ν3µ2ε(k2)1−δ−εh(m/k)+g2ν3µ−2δZ2k2 v˜ (2.6)
×" #
renormalization, therefore we put ξ = 0 in the calcula-
0
tion of the renormalization constants of the solution of + dtdxv˜ ∂ v+(v )v νZ 2v ,
t 1
Eq. (2.1). As shown in Ref. [4], the friction term in Eq. · ·∇ − ∇
Z
(2.1)isnotrenormalizedandthusdoesnotaffecttheRG (cid:2) (cid:3)
where µ is the scale-setting parameter of the renormal-
equations and the subsequent asymptotic analysis.
izedmodeland2δ =d 2istheparameterofdimensional
In the applications of the stochastic Navier-Stokes −
regularization. Only two renormalization constants Z
equation(2.1)toturbulencetherandomforceisassumed 1
and Z are needed to absorb the UV divergences of the
to have a gaussian distribution with zero mean and the 2
model in two dimensions. To avoid excessive notation,
correlation function in the wave-vector space [1] of the
we have used the same symbols for both the fields and
form
their Fourier transforms in (2.6).
F (t,k)F (t′,k′) =P (2π)d/2δ(k+k′)δ(t t′)d (k). Renormalized parameters of the action (2.6) are de-
i j ij F
h i − fined by
(2.2)
ν =νZ , (2.7)
0 1
Thescalarkernelhasapowerlikeasymptoticbehaviorat D =g ν3µ2εZ−3, (2.8)
large wave numbers: 01 1 1
D =g ν3µ−2δZ Z−3. (2.9)
02 2 2 1
d (k)=D k4−d−2εh(m/k), (2.3)
F 0
We haveusedacombinationofdimensionalandanalytic
where h(x) is a well-behaved function of the dimension- regularizationwith the parameters ε and 2δ =d 2. As
−
less argument m/k ensuring the convergence of the in- a consequence, the UV divergences appear as poles in
verseFouriertransformofd (k)atsmallk andwith the linear combinationsof the regularizingparameters. Nor-
F
large k behavior fixed by the condition h(0) = 1. In a malization has been fixed by the choice of the minimal
fixed dimension above two dimensions the force correla- subtraction (MS) scheme [14].
tionfunctionisnotrenormalizedandthethekernel(2.3)
remains intact. This is, however, not the case in two di-
mensions, in which renormalization generates additional III. RENORMALIZATION-GROUP EQUATIONS
terms k2 into the force correlation function. In order AND FIXED POINTS
∝
to deal with a multiplicatively renormalizable theory –
whichisconvenienttechnically–weaddthis termtothe We set up the notation and basic equations for the
correlation kernel at the outset and use, instead of the spatialFourier transformofthe paircorrelationfunction
function (2.3), the modified function of the random velocity field
d′ (k)=D k4−d−2εh(m/k)+D k2. (2.4) ddx
F 01 02 W (t t ,k;D ,D ,ν )= 1
mn 1− 2 01 02 0 (2π)d
The force correlation function is related to two basic Z
v (t ,x )v (t ,x ) eik·(x1−x2), (3.1)
physical quantities, the energy pumping rate and the m 1 1 n 2 2
×h i
E
enstrophy pumping rate , as
B because this quantity is directly connected to the en-
E = d−2 1 (2dπk)dd′F(k), 2erg0∞y sEp(ekc)tdrukm. through the relation hvn(t,x)vn(t,x)i =
Z Independence of the unrenormalized pair correlation
= d−1 dk k2d′ (k) (2.5) fuRnction
B 2 (2π)d F
Z W (t,k;D ,D ,ν )=WR (t,k;g ,g ,ν,µ)
mn 01 02 0 mn 1 2
ind-dimensionalspace,whichallowstoconnectthe”cou-
pling constant” D with the pumping rate in the corre- of the velocity field v of the scale-setting parameter µ
01
sponding asymptotic region. gives rise to the basic RG equation
We cast the stochastic problem (2.1), (2.2), (2.4) into
afieldtheorywiththe De-Dominicis-Janssen”action”in ∂ ∂ ∂ ∂
µ +β +β γ ν WR =0 (3.2)
theusualmanner[1]. AnanalysisofUVdivergenceswith " ∂µ 1∂g1 2∂g2 − 1 ∂ν # mn
2
fortherenormalizedcorrelationfunctionWmRn. Thecoef- γ =u +u +u2 1α+ 3 r
ficientfunctions ofEq.(3.2)β ,β , andγ areexpressed 1 1 2 1 2 4 −
1 2 1 (cid:18) (cid:19)
intermsoflogarithmicderivativesoftherenormalization 3 1
constants. We use the definitions +u1u2 α+ 2 −2r +u22 −2 −r
(cid:18) (cid:19) (cid:18) (cid:19)
γi =µ∂l∂nµZi(cid:12) , βi =µ∂∂gµi(cid:12) , (3.3) γ2 = (u1+u2u2)2 + u31(1u2−r) (3.11)
(cid:12)0 (cid:12)0 5 31
where i = 1,2, and th(cid:12)(cid:12)(cid:12)e subscript ”0”(cid:12)(cid:12)(cid:12) refers to partial +u21 2α+ 4 −3r
(cid:18) (cid:19)
derivatives taken at fixed values of the bare parameters.
5 25 1
It is convenient to express the correlation function +u1u2 2α+ 4 −3r +u22 −2 −r ,
through a dimensionless scalar function R. In two- (cid:18) (cid:19) (cid:18) (cid:19)
dimensional space we define this function through the where
relation
g g
1 2
u = , u = . (3.12)
1 2
1 32π 32π
WR (t,k;g ,g ,ν)= g ν2P (k)R(τ,s;g ,g ), (3.4)
mn 1 2 2 1 mn 1 2 The constant r = 0.1685 in Eq. (3.11) comes from a
−
numerical calculation of those parts of two-loop graphs,
where s = k/µ is the dimensionless wave number, and
for which analytic results were not feasible. The con-
τ = tνk2 the dimensionless time. Solving Eqs. (3.2),
stant α = C ln(4π) = 1.9538 (C is Euler’s con-
(3.3) by the method of characteristicswe obtain the cor- − −
stant) is brought about by a 2δ = d 2 expansion of
relation function in the form −
the geometric factor S /(2π)d = 2/[Γ(d/2)(4π)d/2] =
d
1 1 [1 + αδ + O(δ2)]. The method of calculation is es-
WR (t,k;g ,g ,ν)= g ν2P (k)R tk2ν,1;g ,g , 2π
mn 1 2 2 1 mn 1 2 sentiallythesameaswasusedforthed-dimensionalcase
(cid:0) (3(cid:1).5) in Ref. [15]
Atone-loopordertheγ-functions(3.11)areexactlythe
where g are the solution of the Gell-Mann-Low equa- same as those of the d-dimensional Navier-Stokes equa-
i
tions: tion in two dimensions [3]. They also coincide with the
expressions obtained directly in two dimensions for the
dgi =β [g ,g ] , i=1,2, (3.6) correspondingstochasticvorticityequation[4]. Thus,we
dlns i 1 2 think that for calculation of the coefficient functions of
the renormalization group equation the results of the d-
and ν is the running coefficient of viscosity
dimensional model in the two-parameter expansion may
ν =νe− 1sdxγ1(g1(x),g2(x))/x. (3.7) bmeigahptplbieedsdomireectdlyisctroepthanectiweso-idnimceanlcsuiolantailoncsasein.voTlvhienrge
R composite operators due to different symmetries in two-
Writingthelatterintermsoftheunrenormalized(physi-
dimensional and general d-dimensional cases, but we do
cal)parametersandthe running couplingconstantg [1]
1 no calculate anything like that here.
as
Thefixedpointsaredeterminedbythesystemofequa-
D 1/3 tions β1 = β2 = 0. From the solution of Eqs. (3.6) near
ν = 01 k−2ε/3, (3.8) a fixed point it follows that the fixed point is infrared
g
(cid:18) 1 (cid:19) stable, when the matrix ωnm = ∂nβm is positive defi-
∗ ∗
nite. If ε < 0, then the trivial fixed point: u = u = 0
we arrive at the expression 1 2
is infrared stable. The anomalous asymptotic behavior
1 of the model at small wave numbers is governed by the
WmRn(t,k;g1,g2,ν)= 2g11/3D021/3k−4ε/3 nontrivial fixed point
×Pmn(k)R (D01/g1)1/3k2−2ε/3t,1;g1,g2 . (3.9) u∗1 = 94ε− 227(2α+5−4r),
(cid:16) (cid:17)
For the β-functions: ∗ 2 4
u = ε (α 2 3r), (3.13)
2 9 − 81 − −
β =g ( 2ε+3γ ),
1 1 − 1 at which the eigenvalues of the stability matrix are
β =g ( γ +3γ ), (3.10)
2 2 2 1
−
4 2√2
a tedious two-loop calculation, with the use of the step ω1,2 = i ε
function h(m/k) = θ(k m) in the kernel (2.4), and 3 ± 3 !
−
dimensional regularization,yields 2 4 3 2r
+ r i − √2 ε2. (3.14)
−3 − 9 ± 9
(cid:18) (cid:19)
3
Note that these eigenvalues are independent of α. The Thus,thechoiceε=2in(4.3)rendersthespectrum(4.2)
realparts of both eigenvalues(3.14) are positive and the completely scale-invariant with the Kolmogorov expo-
fixed point (3.13) infrared stable, when ε>0. nents corresponding to the energy cascade, whereas the
substitutionε=3in(4.4)leadstoscale-invariantbehav-
ior in the enstrophy cascade.
IV. TWO-LOOP CALCULATION OF The Kolmogorovconstants may be calculated in the ε
KOLMOGOROV CONSTANTS expansion from (4.2), (4.5) and (4.6). The present two-
loop calculation allows to find correction terms to the
TheconnectionbetweentheenergyspectrumE(k)and previosly found [4] expressions. The result is
the equal-time correlation function of the velocity field,
hvn(x)vn(x)i = 2 0∞E(k)dk, in the two-dimensional C(ε)=2 31/3ε1/3 1+ 2(1+r)ε , (4.7)
wave-vectorspace amounts to · 9
R (cid:20) (cid:21)
3+2r
k C′(ε)=3 21/3ε1/3 1+ ε . (4.8)
E(k)= Wnn(0,k). (4.1) · 9
4π (cid:20) (cid:21)
From (3.9) the asymptotic expression For ε = 2 we obtain from (4.7) C = 4.977. The clo-
sure model leads to the prediction C = 6.69 [8]. Re-
k1−4ε/3 sults of numerical simulations vary from C = 2.9 [16] to
E(k)=g∗1/3D012/3 8π R(0,1;g1∗,g2∗) (4.2) C ∼ 14 [17]. Experimental results [10] yield the range
3 < C < 7. The result obtained here is thus in better
∗ ∗ ∗ ∗
follows, when k 0. Here, g =32πu , g =32πu are agreement with other available data than the leading-
→ 1 1 2 2
thevaluesofthecouplingconstantsattheinfrared-stable order value of the ε expansion C = 3.634 obtained in
fixed point (3.13). Ref. [4].
Therelations(2.5)allowtoexpresstheparametersD01 For ε = 3 the value C′ = 10.29 obtained from (4.8)
and D02 in terms of the energy (or enstrophy) pumping is significantly larger than the leading-order value C′ =
rates ( ). Integratingoverthewave-numbershellm< 5.451ofRef.[4]andthediscrepancybetweenthepresent
k <ΛEwiBthh(x)=1in the kernel(2.4)we obtain,inthe result and the closure-model prediction C′ = 2.626 [8]
limit of widely separated upper and lower wave-number is larger. However, from a more detailed analysis of the
limits, model it may be concluded that calculation of the con-
′
stant C is not unambigous. The value obtained from
= D01Λ2(2−ε) + D02Λ4, (4.3) (4.8) corresponds to the case, in which the coefficient of
E 8π 2 ε 16π friction ξ = 0. If, however, this coefficient is retained,
− 0
D Λ2(3−ε) D then the asymptotic expression for the energy spectrum
= 01 + 02Λ6. (4.4)
is [4]
B 8π 3 ε 24π
−
The spectrum (4.2) should be independent of the details k1−4ε/3
oftheenergypumping,i.e. independentoftheuppercut- E(k)=g∗1/3D012/3 8π
(o4ff.3Λ),inthtihsegroaanlgiesmac≪hiekve≪d bΛy. AthcecocrhdoinicgetDothe=re0latainodn R 0,1;g∗,g∗,(g1∗)1/3ξ0k−(2−2ε/3) (4.9)
ε = 2 for the anticipated inverse energy cas0c2ade. The × 1 2 D1/3 !
01
relation (4.4), in turn, shows that the choice D = 0
02
and ε = 3 leads to scale-invariant behavior for the di- instead of (4.2). The energy spectrum (4.9) is scale-
rect enstrophy cascade. In both cases it shouldbe borne invariant for ε=3 regardless of the value ξ0. Therefore,
in mind that the bare coupling constant D is techni- the proportionality constant in the scaling law seems to
02
cally a book-keeping parameter reflecting the necessity be nonuniversal in the enstrophy inertial range, and de-
of the introduction of the short-range term in the corre- pendsonthepropertiesoflarge-scaledissipation[4]. Re-
lation function of the random force. Physically, it could cent spectral closure analysis and numerical simulations
berelatedtotheintensityofthermalfluctuations,which, have led to similar conclusions [18].
however, are irrelevant in the energy balance of station-
ary developed turbulence.
V. CONCLUSION
The Kolmogorov constants are determined from the
asymptotic relations
Inthispaperwehavecarriedoutatwo-looprenormal-
Λ 4(ε−2)/3 ization of the randomly forced Navier-Stokes equation
E(k)=C(ε) 2/3k−5/3 , (4.5)
with long-range correlated random force in two dimen-
E k
(cid:18) (cid:19) sions in view of two different patterns of scale-invariant
E(k)=C′(ε) 2/3k−3 Λ 4(ε−3)/3 . (4.6) asymptotic behavior.
B k
(cid:18) (cid:19)
4
We have calculated the Kolmogorov constant for a [14] J. Zinn-Justin,Quantum Field Theory and Critical Phe-
powerlike asymptotic energy spectrum k−5/3 of the nomena (Oxford University Press, Oxford,1989).
∝
random velocity field in the inertial range of the inverse [15] L.Ts. Adzhemyan, A.N. Vasil’ev, Yu.S. Kabrits, and
energycascadeinthesecondorderofanεexpansionwith M.V. Kompaniets, Vestnik Sankt-Peterburgskogo Uni-
the result C = 4.977, which is in reasonable agreement versiteta, Seriya4, Vypusk1, 3 (2000) [in Russian].
with other available experimental and theoretical data. [16] J.R. Herring and J.C. McWilliams, J. Fluid Mech. 153,
We have also calculated the Kolmogorov constant for 229 (1985).
thespectrum k−3 intheinertialrangeofthedirecten- [17] E.D. Siggia and H. Aref, Phys.Fluids 24, 171 (1981).
∝ ′ [18] Y.KanedaandT.Ishihara,Phys.Fluids13,1431(2001).
strophy cascade with the second-order result C =10.29
with a larger deviation than at the leading order from
results obtained by other methods. However, explicit
asymptotic expressions for the pair correlation function
of the random velocity field obtained in the present ap-
proach strongly indicate that the Kolmogorov constant
intheenstrophycascadeisnotuniversal,butdependson
the enstrophy dissipation due to large-scale friction.
ACKNOWLEDGMENTS
This work was supported in part by the Nordic Grant
for Network Cooperation with the Baltic Countries and
Northwest Russia No. FIN-18/2001.
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