Discontinuous transition from direct to inverse cascade in three-dimensional turbulence

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by  Ganapati Sahoo

A discontinuous transition from direct to inverse cascade in three-dimensional
turbulence
Ganapati Sahoo1, Alexandros Alexakis2 and Luca Biferale1
1 Department of Physics and INFN, University of Rome ’Tor Vergata’,
Via della Ricerca Scientifica 1, 00133 Rome, Italy. and
2 Laboratoire de Physique Statistique, E´cole Normale Sup´erieure, CNRS,
Universit´e Pierre et Mari´e Curie, Universit´e Paris Diderot, 24 rue Lhomond, 75005 Paris, France.
Inviscidinvariantsofflowequationsarecrucialindeterminingthedirectionoftheturbulentenergy
cascade. InthisworkweinvestigateavariantofthethreedimensionalNavier-Stokesequationsthat
shares exactly the same ideal invariants (energy and helicity) and the same symmetries (under
rotations, reflexions and scale transforms) as the original equations. It is demonstrated that the
examined system displays a change in the direction of the energy cascade when varying the value
7 of a free parameter which controls the relative weights of the triadic interactions between different
1 helical Fourier modes. The transition from a forward to inverse cascade is shown to occur at a
0 critical point in a discontinuous manner with diverging fluctuations close to criticality. Our work
2 thus supports the observation that purely isotropic and three dimensional flow configurations can
supportinverseenergytransferwheninteractionsarealteredandthatinsideallturbulentflowsthere
n
is a competition among forward and backward transfer mechanism which might lead to multiple
a
energy-containing turbulent states.
J

6
2
In turbulent systems the direction of the energy cas- cid invariants or breaking any of the symmetries of the

cade determines the macroscopic properties of the flow NSE. Our study is based on the helical decomposition
]
D leading to a finite energy dissipation rate in the case of [18–21]ofthevelocityfieldu,thatintermsofitsFourier
C a forward cascade (from large to small scales) or to the modesu˜ itiswrittenas: u˜ =u˜+h++u˜−h− whereh±
k k k k k k k
. formation of a condensate in the case of an inverse cas- aretheeigenvectorsofthecurloperatorik×h± =±h±.
n k k
cade (from small to large scales) [1]. It has been long In real space the velocity field is written as u=u++u−
i
l thought that the direction of cascade is determined by where u± is the velocity field whose Fourier transform
n
[ the dimensionality and the ideal invariants of the flow. is projected to the h± base. It is easy to realize that

Twodimensional(2D)turbulencepossessestwosigndef- in terms of the helical decomposition the nonlinear term
1
inite invariants, the energy and the enstrophy. Energy of the 3D NSE is split in 4 (8 by considering the obvi-
v
is transferred backward to larger scales while enstrophy oussymmetrythatchangesthesignofallhelicalmodes)
9
9 is transferred forward to the small scales. On the other possible classes of helical interactions, corresponding to
6 hand, in three dimensional (3D) turbulence while energy triads of helical Fourier modes, (u˜±,u˜±,u˜±) as depicted
k q p
7 is sign definite, the second invariant, the helicity, is sign by the four triadic families in Fig. 1. In the set of sim-
0
indefinite. As a result, helicity does not put any local or
.
1 global constraints on the direction of the energy transfer
0
anditisanempiricalfactthatin3Dturbulentflowsboth
7
energy and helicity are transferred to small scales [2, 3].
1
: Therearesystemswhichdevelopamorecomplexphe-
v
i nomenology, e.g. flows in thin layers, in a stratified
X
medium, in the presence of rotation or of magnetic field
r show a quasi-2D behavior [4–13] and display a bidirec-
a
tional split energy cascade: part of the energy goes to- FIG.1: (Coloronline)Sketchofthefourclassesoftriadicin-
wardssmallscales(asin3D)andparttothelargescales teractions present in the exact helical-Fourier decomposition
ofNSE.Green(red)linesdescribethebackward(forward)en-
(as in pure 2D flows). This phenomenon has been ob-
ergy transfer from the most unstable mode [21]. The thicker
served also in recent experiments [14–16] and in atmo-
line corresponds to the dominant term.
spheric measurements [17]. The reason for the appear-
ance of an inverse energy flux is ascribed to the presence
ulations performed in this paper, we change the relative
of (resonant) waves or of geometric confinement, that
weightamonghomochiraltriads(ClassI)andalltheoth-
favor the enhancement of quasi-2D Fourier interactions
ers by introducing a factor 0 ≤ λ ≤ 1 in the nonlinear
over the 3D background.
evolution. We show that by using this weighting proto-
In this work we study a model system for which the col the turbulent evolution displays a sharp transition,
interactionsintheNavier-Stokesequations(NSE)areen- for a critical value λ , from forward to backward energy
c
hanced or suppressed in a controlled way without reduc- transferbutstillkeepingthedynamicsfullythreedimen-
ing the number of degrees of freedom, altering the invis- sional, isotropic and parity invariant. By analyzing the
2
dynamics of each class separately, it was shown in [21] Run N kf ν µ Re
that the homochiral triads (Class I in Fig. 1) always N1K1 256 [10,12] 10−14 0.5 6·106
N2K1 512 [10,12] 10−16 0.5 6·108
lead to an excess of energy transfer to large scales. On
N3K1 1024 [10,12] 10−18 0.5 6·1010
theotherhand,thetransferdirectionoftriadsofClass-II
N2K2 512 [20,22] 10−16 0.5 5·106
depends on the geometry of the three interacting modes
whileClassIIIandIValwaystransferenergyforward. In TABLE I: Parameters of the numerical simulations where
[22, 23] it was shown that the above prediction is robust Re = ε1/3/(νk22/3). The large-scale friction proportional to
inj f
bydemonstratingthatiftheNSEisrestrictedonlytoho- µ was applied only for k < k = 2 in the simulations where
µ
mochiralinteractions(ClassI)itdisplaysafullyisotropic we observe a inverse energy cascade.
3D inverse energy cascade. In [24], a system that transi-
tioned from the NSE to that of homochiral triads[22, 23]
was investigated by introducing a random decimation of
direction of energy cascade from forward to inverse. The
modes with negative helicity with a varying probability,
purpose of this work is to investigate how this transition
0≤α≤1(α=0beingtheoriginalNSEandα=1being
takes place as the parameter λ is varied.
thesystemofhomochiraltriads). Inthatstudy,thetran-
We perform a systematic series of high resolution nu-
sitionfromforwardtoinverseenergycascadehappensin
merical simulations of Eq. (1) in a box of size L = 2π
a quasi-singular way such that the inverse cascade exists
with spatial resolution of N3. Energy is injected at in-
only at α ∼ 1 demonstrating that even if only a small
termediate wavenumbers k by a Gaussian forcing delta
set of interactions among helical waves of both sign are f
correlated in time with a fixed injection rate ε . We
present (Class II, III and IV), the energy transfer is al- inj
use a pseudo-spectral code, fully dealiased and with sec-
ways forward. Similar conclusions were reached by [25]
ondorderAdams-Bashforthtimeadvancingschemewith
where the amplitude of the negative helical modes was
exact integration of the viscous term. Table I lists the
controlled by a dynamical forcing.
parameters for all simulations.
In this work we investigate a variant of the original
NSE obtained by introducing different weighting of the
4 helical-Fourier classes, such as to smoothly interpo-
λ = 0.0
late from the full NSE to the reduced version where in- 101 -5/3 λ = 0.2
teractions among the u+ and the u− are forbidden of k λ = 0.25
[22,23], butwithoutremovinganymodes. Inparticular, λ = 0.275
λ = 0.3
we evolve the following system:
10-1 λ = 0.35
λ = 0.4
∂tu=P[N]−ν∆4u−µ∆−2u+F (1) k)
E(
where ν isthe coefficientofthe hyper-viscosityterm and -7/3
10-3 k
µ of the energy sink at large scale needed to arrest the
inverse cascade of energy (if any). P is a projection op-
erator to incompressible fields. The nonlinearity N is
defined as: 10-5
1 10 100
N =λ(u×w)+(1−λ)[P+(u+×w+)+P−(u−×w−)] (2) k
where w=∇×u is the vorticity, P± stands for the pro- FIG.2: (Coloronline)Log-logplotofenergyspectraatchang-
jection operator to the incompressible helical base with ingλandforfixedforcingrange(runsN2K1inTableI).The
u± =P±[u] and P=P++P−. This model, proposed in grey area denotes the forcing window. The two straight lines
correspondstothescalingpredictedinpresenceanenergycas-
[26], is graphically summarized in Fig. 1. For any value
cadeandforthehelicitycascade. Noticethatforλ>λ ∼0.3
c
of λ the inviscid system conserves the same quantities as
there is no inverse energy cascade and the spectra agree well
the3DNSE,namelytheenergyE = 1(cid:104)u2(cid:105)andthehelic- with k−5/3 in the forward range. Contrary, for λ<λ an in-
2 c
ity H = 1(cid:104)u·w(cid:105) (where the angular brackets stand for verseenergytransferdevelopandtheforwardscalingbecomes
2
spatialaverage)andhasthesamerotation,reflectionand closer to the one predicted for helicity transfer, i.e., k−7/3.
dilatation symmetries. For λ=1, N reduces to the non-
linearity of the NSE and energy cascades forward. For Figure 2 shows the energy spectra measured at the
λ = 0 the two fields u± decouple and Eq.(1) becomes steady state for different values of the parameter λ ob-
the equation examined in [22, 23]. It conserves two en- tained from simulations N2K1. Clearly for large values
ergies E± = 1(cid:104)(u±)2(cid:105) and two sign definite helicities ofλthereisnosignificantenergyinthelargescaleswhile
2
H± = 1(cid:104)u±·w±(cid:105) independently and cascades energy small scales display a spectrum compatible with k−5/3.
2
inversely. We thus expect that as λ is varied continu- Forsmallvaluesofλ,theenergyispeakedatlargescales
ously from λ = 1 to λ = 0 there will be a change in the forming a spectrum close to k−5/3 while a steeper spec-
3
2.5
N1K1
2 0
1.5 k = k /f ε2)/inj -0.2 NNN232KKK112 0.4
0.5 =k /f -0.4 ∆Π (k=k /2)
k E f
( 0.2
Π(k)E-0.5 λ = 0.0 ΠE -0.6
λ = 0.2 -0.8
0
λ = 0.25
-1.5 k f λ = 0.275 -1 0 0.2 0.4 0.6 0.8 1
2
= λ = 0.3
k λ = 0.35 0 0.2 0.4 0.6 0.8 1
-2.5 λ = 0.4 (a) λ
1 10 100
k
1 N1K1
N2K1
bFaIGnd.3s:hoPwloststhoefefnoercrgedyflraunxgfeorodfiwffearveenntuvmablueerss.ofTλh.Tehaerrgorways εinj 0.8 NN32KK12 0.1
markthewavenumbersatwhichwemeasurethefluctuations k)/f 0.6
in the flux (see insets of Fig. 4). 2
= ∆Π (k=2k)
(k 0.4 0.05 E f
E
Π
trum closer to k−7/3 is observed in the small scales. The 0.2
two behaviors suggest a change from a forward to an in-
verse cascade, which is best demonstrated by looking at 0 0 0 0.2 0.4 0.6 0.8 1
the energy fluxes depicted in Fig. 3. The energy flux is
0 0.2 0.4 0.6 0.8 1
defined as
(b) λ
Π(k)=−(cid:104)u<·N(cid:105), (3)
k FIG. 4: Normalized energy flux at a scale larger (top) and
where u< expresses the velocity field u filtered so that smaller(bottom)thentheforcingrangevsλ. Insetsshowthe
k
its Fourier transform contains only wavenumbers k sat- fluctuationsaroundthemeanvalues. Weplottedtheguiding
curvethroughdatapointsfromN1K1forwhichwehavemore
isfying |k| ≤ k, and expresses the rate energy is trans-
sample points.
ferred out of the set of wavenumbers |k| ≤ k to larger
k. Π(k) is constant in the inertial ranges k (cid:28) k (cid:28) k
µ f
and k (cid:28) k (cid:28) k (where k ∼ 1 is the hypo-viscous
f ν µ
wavenumber and k ∼N/3 the viscous-wavenumber). It the total injection rate ε . The different symbols cor-
ν inj
ispositiveifthecascadeisdirectandnegativeifthecas- respond to the different cases described in table I and
cade is inverse. correspond to an increase of Re keeping k fixed (runs
f
Asλisvariedthedirectionofcascadeischanging. For N1K1→N2K1→N3K1) or to an increase of k keeping
f
λ ≥ 0.3 the flux is almost zero for k < k , while it is Re approximately fixed (runs N1K1→N2K2). For run
f
positive and constant for k < k < k . For λ ≤ 0.2 the N1K1 the transition from forward to inverse cascade is
f ν
opposite picture holds. For k < k the flux is negative smooth, displaying a bidirectional cascade for values of
f
and constant, while k >k the flux is positive but weak. λ in the range 0 < λ < λ (cid:39) 0.3 while a pure forward
f c
For values of λ in the range 0.2 < λ < 0.3 we observe a cascade (|Π(k /2)|/ε =0 and Π(2k )/ε =1) is ob-
f inj f inj
bidirectional cascade: the co-existence of a forward and served for vales λ > λ . When Re and k are increased
c f
inverse transfer. theamplitudeoftheinversecascadeforthepointsinthe
Thebidirectionalcascadeishoweverafinitesizeeffect range 0 < λ < λ is increasing approaching the value
c
and this behavior does not survive the large Reynolds |Π(k /2)|/ε =1 while the forward cascade is decreas-
f inj
and the large box-size (k ) limit. This is demonstrated ing approaching the value Π(2k )/ε = 0. The latter
f f inj
in Figure 4 where we examine how the inverse and for- finding, suggests that at infinite Re and k the cascade
f
ward fluxes change as the Reynolds number Re and the is unidirectional and inverse for λ < λ while it is uni-
c
box size k are increased. The inverse flux (measured directional and forward for λ > λ . The transition is
f c
at the wavenumber k = k /2) as a function of λ for dif- thus discontinuous. This is at difference from what ob-
f
ferent values of the Reynolds numbers (grid-sizes) and served in quasi-2D systems where the transition occurs
different box sizes is shown in Fig. 4(a) while the for- ina continuousmanner (bya bidirectionalcascade)sim-
wardenergyflux(measuredatthewavenumberk =2k ) ilartoasecondorderphasetransitions,andatdifference
f
is shown in Fig. 4(b). Both fluxes are normalized by fromwhatobservedin[24]wherethetransitionoccurred
4
at a singular value of their model-parameter α∼1. easily measured in experiments. In particular, for the
This abrupt transition can be justified by realizing original NSE, the von Karman-Howarth equation states
that in a bidirectional cascade the two inertial ranges that the third order structure function is related to the
(k (cid:28) k (cid:28) k and k (cid:28) k (cid:28) k ) must have differ- direction of the cascade and it is negative for a forward
µ f f ν
ent physical properties to sustain different directions of transfer and positive for a backward transfer. In the
cascade. Thisispossiblewhen,anewdimensionallength form of the NSE investigated here (1) the von Karman-
scale(cid:96) isintroduced(e.g. (cid:96) isthelayerthicknessinthin Howarth equation is more complicated (see, e.g. ap-
∗ ∗
layer turbulence, or the Zeeman scale in rotating flows) pendixA.1of[23]forthecasewithλ=0). Nevertheless,
that determines the properties of the flow due to the ex- weshowinFig.6thatevenasimplemeasurementbased
ternalmechanism. Theamplitudeoftheinverse/forward on S (r) is in good agreement with the indication that
3
cascadethendependsonthe‘distance’oftheforcingscale for r >r =2π/k the sign do change by crossing λ .
f f c
(cid:96) from the critical length-scale (cid:96) . In our case, no par-
f ∗
ticularscale(cid:96) isintroducedbytheparameterλ. Onthe
∗ 2.5
contrary,theinertialrangesarescaleinvariantforallval- λ = 0.0
ues of λ. Thus, both ranges, (cid:96) > (cid:96) or (cid:96) < (cid:96) effectively 2 λ = 0.4
f f
share the same properties and have to develop either a
1.5
forwardorabackwardcascade, becausetheflowcannot
distinguish the large from the small scales. 1
0.5
0.6 r)
N1K1 S(3 0
N2K1
-0.5
0.5 N3K1
-1
)
kf 0.4 -1.5
=
k 0.01 0.1 k r 1 10
( f
E
0.3
FIG.6: Thirdorderstructurefunctionsfortwoextremecases
withonlydirectorinverseenergycascadeλ=0.4andλ=0.
0.2
Inthisworkwehavedemonstratedthatbycontrolling
the amplitude of the interactions in the NSE the energy
0 0.2 0.4 0.6 0.8 1
λ cascadecanchangedirectionfromforwardtoinverseand
vice versa. In the model used here, this change of direc-
tion is not due to previously known mechanisms, e.g., a
FIG.5: Totalenergycontentatawavenumberinsidetheforc-
ingrange(k=11)vsλ. Weplottedtheguidingcurvethrough change in the dimensionality, a change in the ideal in-
datapointsfromN1K1forwhichwehavemoresamplepoints. variants, orthebreakingofanysymmetryoftheoriginal
equations caused by the introduction of external forc-
Notsurprisinglythesystemdisplaysinterestingbehav- ing as in the presence of rotation or of a magnetic field,
iorclosetothecriticalvalueλ . InFig. 5weplot,E(k ), revealing that the fully non-linear dynamics of the 3D
c f
theintensityofthespectrumattheforcingwavenumbers NSE is more complex than what told by the accepted
vs λ and for different Reynolds numbers. The response phenomenology. In particular, we showed that the en-
ofthesystemiscritical,showingatendencyforE(k )to ergy cascade is strongly sensitive to the relative dynam-
f
divergeasλ→λ . Thisdivergenceisalsoreflectedinthe ical weight of homochiral to heterochiral helical Fourier
c
amplitudeofthefluxfluctuations∆Πshownintheinset interactions,suggestingtosearchforsimilarfootprintsof
offigures4(where∆ΠofrunN2K2ismultipliedby23/2 inverse energy transfer also in other empirical turbulent
to account for the 23 more interactions involved). The realization. Forinstance,itwouldbeinterestingtocheck
existenceofmultiplephasesforthephysicsoftheenergy ifpreferentialaccumulationofenergyinhomochiralreso-
containing eddies is an important remark that finds sup- nanttriadsisalsoobservedinrotatingturbulenceatlow
portalsoinrecentexperimentalempiricalfindingswhere Rossby numbers, in thick fluid layers, in shear flows, or
turbulent realizations with multiple states have been ob- in the presence of magnetic field. Our results indicate
served in swirling and in Taylor-Couette flows [27, 28]. that the transition becomes discontinuous in the large
The direction of the energy transfer can be also stud- Re limit. This is the first time that such a discontinu-
ied by looking at the behavior of the structure functions ous transition is reported for the cascade direction. It
S (r) = (cid:104)(δu (r))n(cid:105) where δu (r) = (u(x+r)−u(x))· is shown to be linked to the scale invariance of the sys-
n (cid:107) (cid:107)
r/r, that have the great advantage that can be also be tem that needs to be broken for a bidirectional cascade
5
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