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Turbulent flows for relativistic conformal fluids in 2+1 dimensions

Article  in  Physical review D: Particles and fields · October 2012

DOI: 10.1103/PhysRevD.86.126006 · Source: arXiv

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Federico Carrasco

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Oscar Reula

National University of Cordoba, Argentina

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Turbulent flows for relativistic conformal fluids in 2+1 dimensions

Federico Carrasco1,2, Luis Lehner2,3, Robert C. Myers2,

Oscar Reula1, Ajay Singh2,41FaMAF-UNC, IFEG-CONICET, Ciudad Universitaria, 5000, Cordoba, Argentina.2Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada.3Department of Physics, University of Guelph, Guelph, Ontario N1G 2W1, Canada.

4Department of Physics & Astronomy and Guelph-Waterloo Physics Institute,

University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

We demonstrate that relativistic conformal hydrodynamics in 2+1 dimensions displays a turbu-lent behaviour which cascades energy to longer wavelengths on both flat and spherical manifolds.Our motivation for this study is to understand the implications for gravitational solutions throughthe AdS/CFT correspondence. The observed behaviour implies gravitational perturbations of thecorresponding black brane/black hole spacetimes (for sufficiently large scales/temperatures) willdisplay a similar cascade towards longer wavelengths.

I. INTRODUCTION

The AdS/CFT correspondence [1, 2] provides a re-markable framework for studying certain strongly cou-pled gauge theories in d dimensions by mapping to weaklycoupled gravitational systems in d + 1 dimensions. Ofparticular interest is the relation between gauge the-ory plasmas and black hole geometries [3–5]. Here thecorrespondence was used to show that perturbed blackbrane geometries at the classical level have a dual de-scription in terms of fluid dynamics equations govern-ing long-wavelength perturbations about an equilibriumstate. This fluid/gravity correspondence builds on thehighly successful and ongoing program to calculate thenear-equilibrium transport coefficients for strongly cou-pled plasmas using holographic techniques, e.g., [6, 7].

This duality has been exploited in a large number ofworks which exploit known gravitational behaviour to in-fer properties of diverse systems described by stronglycoupled field theories in the large N limit. On the hydro-dynamical front, work in [8] describes how the onset ofnaked singularities can be tied to finite-time blowups inhydrodynamics. Additionally, numerical simulations areincreasingly being exploited to understand isotropizationand thermalization of systems starting far from equilib-rium, e.g., [9–15].

In all these works, the approach has been to under-stand the behaviour of relevant systems in the gravita-tional side of the duality and infer from it properties ofthe gauge theory dynamics. In the present work, we fol-low the opposite route, namely to study particular phe-nomena that might arise in the field theory to under-stand possibly unexpected phenomena on the gravita-tional side.

Our starting point is a simple and well-known ob-servation about the behaviour of turbulence of Newto-nian fluids in two spatial dimensions. Specifically in thiscase, turbulent behaviour induces an energy cascade fromshort to long wavelengths e.g., [16]. This contrasts tothe standard cascade from long to short scales charac-terizing turbulence in three and higher dimensions.[54]

The obvious question is therefore whether this effectappears in the relativistic conformal fluids relevant forthe AdS/CFT correspondence, and if so, what is thedual interpretation in the gravitational theory. A pos-itive answer would seem to distinguish AdS gravity infour dimensions from higher dimensions in a unique way.An affirmative answer is also particularly intriguing asthe two known instabilities in AdS spacetimes, super-radiance and the recently found ‘mildly-turbulent’ be-haviour in [17], both induce energy cascades from low tohigh frequencies for all d ≥ 3.To answer this question, which has also been raised

earlier[5], we examine the behaviour of specific fluid flowswhich are dual to perturbations of Schwarzschild andKerr black holes in AdS4. We study the problem forspherical horizons in global coordinates, as well as pla-nar horizons described by Poincaré coordinates. Specifi-cally, we study the relativistic Euler equations with theparticular equation of state corresponding to a confor-mal fluid on a fixed ℜ× S2 manifold for the former casewhile ℜ×T 2 for the latter. In each background, we exam-ine perturbations of the stationary configurations dual tothe corresponding stationary black holes. We then exam-ine the onset of turbulence and its cascade behaviour inthis setup, and compare it with the expected behaviourfor the case of incompressible non-relativistic flows with‘standard’ equations of state. We find that, as in theNewtonian case, turbulence leads to perturbations cas-cading from shorter to longer wavelengths for relativisticconformal hydrodynamics in 2+1 dimensions.

This work is organized as follows: in the remainder ofthis section we briefly review some well known aspectsabout turbulence in two and three spatial dimensions.Section II discusses the initial configurations considered,as well as details of our numerical implementation. Sec-tion III presents results for the cases considered for con-formal hydrodynamics on the sphere (the results on thetorus, which are qualitatively similar, are included in anappendix). We discuss the consequences of the obtainedbehaviour in section IV. (Visualizations of the fluid flowsstudied here can be found in [18]).

http://arxiv.org/abs/1210.6702v2

2

A. Turbulence

Turbulence is a ubiquitous property of fluid flows ob-served in nature [19]. Qualitatively, we might describeturbulence as a flow regime characterized by chaotic orstochastic behaviour. Most of our theoretical under-standing of turbulence comes from the study of non-relativistic incompressible fluids. Certainly, while a fullunderstanding of turbulence is not yet available, somerobust results do exist. Namely, for the inviscid case intwo spatial dimensions, a global regularity theorem hasbeen proved, together with theorems about uniquenessand existence of solutions [20] implying no singularity ofthe velocity field can develop in a finite time.For the three-dimensional case it is not yet known

whether the same holds true and resolving this issue con-stitutes a major open problem (see e.g., [21]). It is knownhowever, that qualitatively two and three dimensionalturbulent fluid flows exhibit profound differences arisingfrom the existence of a key conserved quantity which hasa radical effect in the fluid’s turbulent behaviour. Thisquantity, dubbed enstrophy , — see discussion around(18) — has been argued to imply a very different cas-cade picture in two dimensions, as compared to the three-dimensional one [22]. Small scales will support a ‘directenstrophy cascade’ towards smaller wavelengths, with allthe enstrophy dissipation taking place on the shortestscales. The energy flux towards small scales will thenbe damped and the energy will be, instead, transferredto larger scales in an ‘inverse energy cascade’. This be-haviour is observed in nature, controlled experiments andnumerical simulations.In a phenomenological theory of two-dimensional tur-

bulence [22, 23], the existence of two inertial ranges waspointed out: a direct k−3 enstrophy cascade at smallwavelengths, and an inverse energy cascade with spec-trum k−5/3, at larger scales, were predicted.Numerical simulations have shown the emergence of

strong coherent vortex structures that dominates the flowafter some time [24]. These vortices emerge as anoma-lous fluctuations at small scales, and have a lifetime muchlonger than their characteristic eddy turnover time [25].As the dynamics continues, two such vortices might col-lide and usually merge if they rotate in the same direc-tion, forming a vortex of larger scale. In the process theenergy is nearly conserved, so it acts as the mechanismof inverse energy cascade. The statistical distributionof vortices over scales leads to an energy spectrum k−3

which is much steeper than the originally expected k−5/3,and vortices were recognized as the fluctuations responsi-ble of intermittency and possible anomalous dimensions[26].The emergence and dynamics of vortices still pose a

number of difficult questions, and the relevance of theinitial data and external forcing is not yet fully under-stood. An exciting possibility, of course, would be thatholographic studies might shed new light on this problemfrom a fundamentally different point of view.

II. EQUATIONS, RATIONALE AND

NUMERICAL IMPLEMENTATION

Our goal is to study the dynamics dual to a perturbedblack hole in 3 + 1 dimensions from the global point ofview, so we consider the field theory in 2 + 1 dimensionson the sphere (or a torus, see Appendix A). To do so,and in order to deal with a well posed problem, we re-strict to the zeroth order expansion of the theory, i.e.,to the equations determined by the conservation of thestress energy tensor of a perfect fluid. One reason forthis choice is that the inclusion of viscous terms wouldyield an acausal system of equations [27–30] (for a recentdiscussion in the holographic context see e.g., [31]).[55]Hence properly incorporating these effects poses a seriouscomplication for the corresponding simulations.At first sight, working with perfect fluid equations

would seem to be a severe limitation for our study andthe conclusions that can be drawn from this work. How-ever, recall that hydrodynamics treats the conservationof a stress-energy tensor in a gradient expansion, e.g.,[32, 33]. For the conformal case, no intrinsic scales appearin defining the fluid and hence the temperature naturallycontrols the equation of state and all of the non-vanishingtransport parameters. Hence to the stress tensor for aconformal fluid, in d spacetime dimensions, takes the fol-lowing form

Tµν = αTd

(

(gµν + d uµuν)−d β

Tσµν +O(T−2)

)

(1)

where σab describes the first-order viscous contributionand α, β are dimensionless coefficients which character-ize the fluid. From this expression, we see that the vis-cous terms can be arbitrarily suppressed by increasingthe temperature. To be more precise, if the characteris-tic scales L controlling the flow are kept fixed, i.e., thesize of the sphere (or torus) in the present case, the gra-dient expansion becomes an expansion in 1/LT and thehigher order terms are suppressed by setting LT ≫ 1.Thus the perfect fluid can be seen as an arbitrarily goodapproximation in this regime. Further, as we discussedabove, turbulence is expected to generate a cascade fromshort to long scales for fluids in two spatial dimensions.In fact, our simulations will confirm this expectation andso the interesting dynamics here indeed progresses mainlytowards longer and longer scales. Hence the viscous andhigher order terms should not play a significant role inthe observed behaviour.To further support the conclusion that the viscous con-

tributions are insignificant, we will consider two ways ofincorporating effects of these terms: (i) through the ad-dition of an artificial viscosity (as defined in [34, 35])and (ii) by comparing results for higher temperatures,where the viscous and higher terms become less relevant(as described above), and showing that the observed tur-bulent behaviour remains unchanged both qualitativelyand quantitatively, e.g., see figure 5. Notice that onecould add terms up to second order in the derivative ex-

3

pansion (third order in derivatives) as suggested in [32](which provides a modern extension of the method sug-gested in [28]). This approach effectively amounts to con-trolling and reducing gradients in the solution through adynamical equation governing second order gradients ofthe flow. As we will see, the natural evolution of thesystem to longer wavelengths renders these approachesunnecessary.In what follows we describe the system of equations

considered, discuss useful monitoring quantities — in-cluding the conserved quantity associated to the enstro-phy — as well as a brief summary of stationary solutionsinferred from the dual gravitational picture.

A. Evolution Equations

The system follows from the local conservation of thestress-energy tensor of a perfect fluid,

T µν = (ρ+ p)uµ uν + p gµν , (2)

along with the condition of conformal invariance that re-quires the stress tensor to be traceless, i.e., T µµ = 0. Thislatter condition fixes the equation of state as

p = ρ/2 , (3)

for a conformal fluid in 2+1 dimensions — compare with(1) for general dimensions.The set of dynamical variables we have chosen to evolve

the system are: {ρ̃, ui}, where ρ̃ ≡ log(ρ1/3). Theremaining component of the three-velocity, u0, whichcan be identified with the Lorentz factor γ (u0 ≡ γ =(1 − v2)−1/2), is obtained at each time-step through thenormalization condition uµu

µ = −1. In equilibrium, thetemperature T of the system is related with the abovevariables by ρ = T 3, or equivalently, T = eρ̃.[56]The system of evolution equations obtained for our dy-

namical variables then reads,

∂tρ̃ =1

αγ

{

zDρ̃+ zϑ− 12Dz

}

, (4)

∂tui =1

αγ

{

[α∂i + uiD] ρ̃+ αDui − zuiϑ+ui2Dz

}

(5)

The spatial derivatives along the flow are denoted byD ≡ ukDk, while ϑ ≡ Dkuk is the spatial divergenceand Dk is the covariant derivative associated with thespatial metric hij . We also have defined, z ≡ γ2 andα ≡ 1 + γ2.

B. Equilibrium Configurations

Equilibrium configurations for the toroidal case aresimply constant temperature/constant velocity fluidflows and we adopt conditions as described in the ap-pendix. The equilibrium states for conformal fluids on

the two-sphere can be obtained straightforwardly byrequiring no entropy production within the first sub-leading (dissipative) order in the fluid expansion at equi-librium. This imposes a restrictive condition for the shearviscosity (i.e., σab = 0). A simple family of such config-urations corresponds to rigidly rotating fluids, which inour variables are given by

uθ = 0 , uφ = γ ω0 , T = γ τ (6)

where γ = (1−ω20 sin2(θ))−1/2 is the Lorentz factor. Thetwo (constant) parameters characterizing these solutionsare: the ω0, the angular rotation rate, and τ , the lo-cal temperature measured by co-moving observers. Notethat implicitly, we have set the radius of the S2 to onehere, i.e., the proper length of the equator, θ = π/2, issimply 2π.The thermodynamics and the local stress tensor of

these solutions has been found to be in precise agree-ment with thermodynamics and boundary stress tensorof the spinning black holes [36]. In this reference, theauthors compare conformal fluids on spheres of arbitrarydimensions with large rotating black holes onAdS spaces.First, global thermodynamical quantities are compared.Then, appealing to the duality, a comparison is made be-tween the local stress tensor of the fluid configuration andthe boundary stress tensor for the most general rotatingblack hole in AdSd+1, as given in [37]. The relevantblack hole solution for our case, the one correspondingto Kerr AdS4, is labeled by two parameters a and r+,related with the angular momentum (per unit mass) andthe horizon radius of the black hole, respectively. Perfectagreement was found between these two theories in thelarge r+ limit, upon the following identifications [36]:

ω0 ←→ a , (7)

τ ←→ 3 r+4π

. (8)

Thus, the static configuration with ω0 = 0 is the fluiddual to AdS4 Schwarzschild black hole, and the rotatingfluid configurations (ω0 6= 0) are dual to the AdS4 Kerrgeometry.

C. Conservation Laws

Conserved quantities are invaluable tools to analyzethe behaviour of any given system and, as discussed, theirexistence can imply a particular behaviour of the fluidflow. We here discuss these quantities which we monitorin our efforts to understand the behaviour of the system,in particular the phenomenon of turbulence. In particu-lar, given a conserved current Jµ (i.e., ∇µJµ = 0), theexistence of a conserved charge immediately follows,

∫

S2J0dΣ = const . (9)

Hence in the following, we identify various conserved cur-rents and charges for the present system.

4

Given a killing vector field Kµ and the conserved stresstensor, the current Jµ ≡ T µνKν , is automatically con-served. Thus, from the killing vectors, ξ ∼ ∂∂t andζ ∼ ∂∂φ , we construct the first two conserved quantitiesthat are identified with the total energy and angularmomentum of the fluid, respectively:

E ≡ 12

∫

S2ρ(3γ2 − 1)dΣ , (10)

Lφ ≡3

2

∫

S2ργuφdΣ . (11)

From thermodynamical considerations, one knows thatthe local entropy s ≡ ρ2/3, is conserved along the flowdirection (in a perfect fluid). Thus, the current Jµ = suµ

is conserved and total entropy conservation follows,

S ≡∫

S2ρ2/3γdΣ . (12)

Another quantity of interest is the vorticity whicharises as a purely geometric property (i.e., independentof the equations of motion). It is constructed by tak-ing the exterior derivative to the flow velocity, resultingin the vorticity two-form ωµν ≡ ∂[µuν]. In 2+1 dimen-sions, one can define a naturally conserved current justby taking its Hodge dual, i.e., we define Wα = ǫαµνωµν .The total ‘circulation’, which remains constant through-out the evolution, is then

C ≡∫

S2W 0dΣ =

∫

S2ǫ0ijωij dΣ . (13)

However, note that the conservation law is ‘topological’in this case and so this conserved charge actually vanishessince C = −

∫

S2 du = 0.Finally, we define the enstrophy, whose conservation

plays a crucial role on two-dimensional turbulence andits inverse cascade scenario. This quantity, for incom-pressible non-relativistic fluids, is just the integral of thesquare vorticity field. Carter [38] has generalized the con-cept for relativistic fluids in three spatial dimensions, andwe extend it here for two-dimensional relativistic confor-mal fluids.Let us begin with ∇µT µν = 0 with the stress tensor

given in (2) for a conformal fluid in d spacetime dimen-sions (i.e., p = ρ/(d−1). For convenience we project theseconservation equations along and orthogonal to the flowvelocity,

uµ∂µρ = −d

d− 1ρ(∇µuµ) ≡ − d

d− 1ρΘ , (14)

Pµν∂νρ = −dρuν∇νuµ ≡ −dρaµ ; (15)

where Θ is the (full covariant) divergence; aµ, the acceler-ation; and Pµν ≡ gµν+uµuν , the projector perpendicularto uµ. Next, consider the two-form,

Ωµν = ∇[µρ1/duν] , (16)

which is built to satisfy the Carter-Lichnerowicz equationof motion

Ωµνuν = 0 , (17)

which is equivalent to (15). Note that the inclusion ofρ1/d in (16) is crucial to produce this formulation of theequations of motion. Then, from the Cartan Identity, isstraightforward to show that

£λuΩ = λu · dΩ+ d(λu ·Ω) = 0 . (18)Here the first term vanishes because Ω = d(ρ1/du) isan exact form, while the second term vanishes by theCarter-Lichnerowicz equation (17). This means that thetwo-form Ω does not change along the flow direction.The latter observation motivates one to look for a new

conservation law, however, we will only be able to con-struct the desired result by using an identity which onlyholds for d = 3, i.e., two spatial dimensions. In this case,we write the following current: Jµ ≡ ρ−2/3(ΩαβΩαβ)uµ.To establish that this current is conserved, we make useof the identity,

ΩµαΩµβ =1

2Ω2Pαβ , (19)

where Ω2 ≡ ΩµνΩµν . The latter holds for two spatialdimensions since we can write, Ωµν = ǫµναl

α, for somearbitrary vector lα. The condition Ωµνu

ν = 0 then re-quires that lα be proportional to uα, i.e., lα = ξuα. Onecan then find the proportionality factor ξ in terms of Ω2

to show (19) holds. With this relation, we have

∇µJµ = ρ−23

{

Ω2[

Θ− 23ρ

uµ∂µρ

]

+ uµ∂µΩ2

}

= 2ρ−23

{

ΘΩ2 +Ωαβ uµ∇µΩαβ}

= 2ρ−23

{

ΘΩ2 − 2ΩµαΩµβ(∇αuβ)}

= 0 .

Here we have used (14) in the second line and (18) inthe third line, while the final vanishing follows from (19).Therefore, the enstrophy,

Z ≡∫

S2ρ−2/3Ω2u0dΣ =

∫

S2(ωµνω

µν +1

2aµa

µ)γdΣ

(20)is conserved, where the first term in the expression isjust the vorticity two-form squared (as expected fromthe non-relativistic version), and the second one, whichinvolves the acceleration aµ ≡ uν∇νuµ, accounts forthe fact that the fluid world-lines are not necessarilygeodesics. One important point to emphasize from (20)is that the factors of ρ cancel out in the end, so that thefinal expression for the enstrophy is independent of theenergy density.

D. Numerical approach and setup

Our goal is to study the dynamical behaviour of a per-turbed, otherwise stationary, fluid configuration dual to

5

the Kerr/Schwarzchild black hole in global AdS4 or thePoincaré patch. We next describe the different compo-nents of our numerical implementation.

1. Initial Data

Initial data for perturbations of the stationary fluidconfigurations are directly added to the fluid velocitysuch that eddys are induced. This is straightforwardlyachieved by considering perturbations which, in the co-moving frame of the fluid, describe space varying veloc-ities that change sign in a smooth manner along someparticular direction chosen. In the case of the torus, thisis straightforward as described in the appendix. For thespherical case, we adopt a perturbation in the fluid ve-locity as:

uφ → uφ = γ ω ≡ γ(ω0 + δ ωp(θ, φ)) , (21)

for some small value δ and a function ωp defined on thesphere (generally chosen to be one of the spherical har-monic basis functions Y mℓ (θ, φ)). In this expression, γis the local Lorentz factor defined for the full angularvelocity ω, including the ωp contribution. For future ref-erence, it is useful to indicate the initial vorticity densityassociated to this flow,

W 0(θ) =1√h∂θuφ =

1

sin(θ)∂θ

[

sin2(θ)γ(ω0 + δωp(θ, φ))]

= (ω0 + δωp)γ3 cos(θ)(2 − ω2 sin2(θ))

+δγ3 sin(θ)∂θωp .

From this expression, we see that the rigid rotation com-ponent (6) of the flow introduces a Y 01 component tothe vorticity at order ω0 and higher ℓ contributions alsoappear at higher orders in ω0. Thus for sufficiently fastflows, the background contribution renders analyzing tur-bulent behaviour through vorticity more delicate.

2. Grid Scheme

We now discuss details of our grid implementation andresults. Since the qualitative behaviour is the same inboth topologies considered –but the implementation ismore involved in the spherical case– from now on weconcentrate on the spherical flows to simplify the pre-sentation and defer details of the toroidal case to theappendix.Since the topology of our computational domain is S2,

we employ multiple patches to cover it in a smooth way.A convenient set of patches is defined by the cubed spherecoordinates. There are six patches with coordinates pro-jected from the sphere, and each of this patches consti-tute a uniform grid — see figure 1). These grids aredefined in a way such that there is no overlap and onlygrid points at boundaries are common to different grids

(multi-block approach). To ensure a correct transfer ofinformation among the different grids we follow the tech-nique described in [39], which relies on the addition ofsuitable penalty terms to the evolution equations to pre-serve the energy norm through the whole sphere. Thistechnique ‘penalize’ possible mismatches between valuesthe characteristic fields take at interfaces and enforce con-sistency through suitably introduced driving terms. Herewe follow the strategy introduced in [40] which is an ex-tension of the method introduced in [41, 42].

FIG. 1: Cubed Sphere Coordinates. A total of six Cartesianpatches are employed to cover the sphere. Only patch bound-aries coincide at common points.

Introducing coordinates {t, x, y} with (x, y) to labelpoints in each Cartesian patch, the metric in each onereads,

ds2 = −dt2+ 1D2

{

(1 + y2)dx2 + (1 + x2)dy2 − 2xydxdy}

where D ≡ 1 + x2 + y2. The non-vanishing Christoffel’ssymbols are given by,

Γxxx =−2xD

, Γxxy =−yD

, Γyyy =−2yD

, Γyxy =−xD

3. Numerical Scheme and Stability

In order to construct stable finite difference schemes forour initial value problem we use the method of lines [34].This means that we first discretize the spatial derivatives(constructing some suitable finite difference operators)and obtain a system of ordinary differential equationsfor the grid functions. To ensure the stability of the nu-merical scheme we use the energy method described on[35]. We employ (fourth/second-order accurate at inte-rior/boundary points) finite difference operators satisfy-ing summation by parts (the discrete analogue of integra-tion by parts) and deal with interface boundaries withappropriate penalty terms at the interfaces as describedin [39]. For the time integration, we adopt a 4th orderRunge-Kutta algorithm.All simulations were performed using 81x81 grids for

each of the six patches, giving a total number of around

6

40.000 grid points to cover the entire sphere. We haveconfirmed convergence through several tests, in partic-ular we adopted different initial data, and studied thenumerical solutions with increased resolution with eachgrid having 321x321, 161x161, 81x81 points (labeled by4,2,1 respectively). With each obtained solution, denotedby U (i)(t, θ, φ) (for i = 4, 2, 1), we calculated the conver-

gence rate p as, Q(t) ≡ ||U(4)−U(2)||L2

||U(2)−U(1)||L2≈ 2p. For the cases

considered, the obtained rate was in very good agreementwith the expected for resulting 3rd order implementationconstructed.

III. RESULTS

We analyze the dynamics of small perturbationsaround the stationary fluid configurations duals toSchwarzschild and Kerr geometries on AdS4. We sep-arately study these two cases, starting with a qualita-tive description of the system evolutions, for generic per-turbations. We show in figures 4 and 6 the sequenceon the evolution of the vorticity field for the two cases.Then, in order to gain some insight into the turbulentbehaviour and to capture the possible cascading phenom-ena, we perform a spherical harmonic decomposition (upto ℓ=12) of the relevant fields and compute their associ-ated power spectrum as a function of time.This signal processing analysis is derived simply from

a generalization of Parseval’s theorem, which states thatthe total power of a function f defined on the unit sphereis related to its spectral coefficients by,

1

4π

∫

S2|f(θ, φ)|2dΣ =

∞∑

ℓ=0

Cf (ℓ) ,

Cf (ℓ) =

ℓ∑

m=−ℓ

|Aℓm|2 ;

where Aℓm are the coefficients of the expansion of f inthe spherical harmonics. In figures 4 and 8, we plot thecoefficients C(ℓ) as a function of time, to analyze howthe different modes behave during the evolution.

A. Non-rotating case: perturbations to

Schwarzschild

One can basically classify the dynamical behaviour ofthe system into four different stages. In figure 2, we plotthe vorticity field of a representative example for each oneof them. A first stage corresponds to an initial transientperiod when the initial configuration seemingly remainsunchanged for some time interval. (This interval dependson the perturbation considered, being shorter for largerℓ’s perturbations at a fixed value of δ, but it does not

depend on the system’s temperature). Closer inspec-tion however reveals that non-trivial dynamics beginsto manifest and an exponential growth of some modessets in (see fig. 4). Notice that the initial growth rateof these modes is approximately the same though thehigher ℓ’s grow to larger values earlier. Since truncationerrors feed higher frequencies this behaviour is not sur-prising. As these modes become large enough (roughlycommensurate to the magnitude of the initial perturba-tion), the original symmetry of the system gets brokenand a number of eddies arise and move around. Such aninstant might be regarded as the beginning of turbulence(fig. 2(b)). As the dynamics continues, the eddies gradu-ally turn into individual vortices and exhibit a seeminglychaotic motion, during a stage that we will refer as fullydeveloped turbulence (fig. 2(c)). With the vortices prop-agating around the sphere, encounters of same-sign vor-tices lead to increasingly larger vortex structures. Duringthis stage, governed by non-linear effects, energy trans-fers from the higher l modes to lower ones. The processcontinues until four vortices, two of each sign, are formed(fig. 2(d)).

The end-state is found to be qualitatively the same inall the cases we have considered, regardless of the ini-tial perturbation introduced and of the system’s temper-ature. We have explored generic perturbations in thevelocity initial data, as described on (21): from differentsingle-mode perturbations (i.e., ωp ∼ Y mℓ (θ, φ) for par-ticular ℓ and m), to a random combination of all of themup to ℓ = 12. We have further considered the inclusionof a random forcing term in the evolution equations, byadding a term fν to the right hand side of the stressconservation equation as: ∇µT µν = fν . This force isgiven by a random field, both in time and space (andthus, acting on very short scales). Even in this forcedcase, the dynamical character and properties of the so-lution remain qualitatively unchanged. This (in additionto convergence studies) indicates that our grid structureis playing no significant role on the obtained behaviourand on the final configuration attained.

Main features of the long-term behaviour of the solu-tion are illustrated in figure 3. In particular, the men-tioned four dominant vortices of the resulting configura-tion. It can be noted that the temperature and energydensities attain local minimums at the vortices’ locationsand that the enstrophy is almost exclusively containedwithin them.

In figure 4 we have displayed a few representativemodes of the vorticity power spectrum for an initial per-turbation ωp(θ, φ) = Y

010(θ, φ), δ = 0.2 and τ = 100. The

coefficients C(ℓ) of the spectrum are normalized withrespect to the total power and plotted (in logarithmicscale) as a function of time. In the figure, the modesℓ = 9 and ℓ = 11 dominate the spectrum at t = 0 —as expected since the vorticity involves a derivative ofthe three-velocity and the initial data for the latter hasa single ℓ = 10 mode. The rest of the modes exhibitan exponential growth on the early stages of the evolu-

7

(a) t = 0 (b) t = 108

(c) t = 291 (d) t = 950

FIG. 2: Evolution of the vorticity field for a perturbationωp(θ, φ) = Y

0

10(θ, φ) and δ = 0.2, on Schwarzschild (ω0). (a)Initial config., (b) beginning of turbulence, (c) fully developedturbulent stage, (d) final state.

(a)Vorticity (b)Temperature

(c)Energy (d)Enstrophy

FIG. 3: Late times configuration for distinct relevant fieldsin the non-rotating case (ω0 = 0 and T ∼ 100, at t = 1200).Notice that vortices correspond to minima in the energy. Thisbehaviour is expected as stable vortex structure require alarger surrounding pressure that prevents its dispersion.

tion, as previously described. As the turbulent cascadebegin, and after a short stage in which the mode struc-ture is rather complex, the original high-ℓmodes decreasewhile the lower ones increase and gradually dominate theflow. Particularly the ℓ = 2 mode in the spectrum, whichrepresents the quadrupole contribution, is the dominantmode at late times.Note that the vortices in figure 3, i.e., the ℓ = 2 mode

above, are essentially quasi-stationary as we have ne-glected here the viscous (and higher) contributions whichwould certainly affect the very long-time behaviour of thesolution. As we noted before, one might wonder whether

0 100 200 300 400 500 600 700 800

time

0,0001

0,01

1

log(

C(l

))

l=2l=4l=6l=9l=11

FIG. 4: Relevant modes in the power spectrum of the vorticityfield for a ωp(θ, φ) = Y

0

10(θ, φ) and δ = 0.2 perturbation, inthe non-rotating case (at temperature T ∼ 100).

the observed turbulent phenomena, and the conclusionswe can draw from this study, might be significantly af-fected when such dissipative terms are taken into ac-count. However, recall that in the discussion around (1),we argued the perfect fluid equations will give a gooddescription when LT ≫ 1. We verified the accuracy ofthis statement by comparing the system’s dynamics at avariety of temperatures — recall that the radius of thesphere is fixed to be one — while keeping the initial datafor uk fixed. Figure 5 illustrates the behaviour of the L2

norm of uθ, i.e.,[∫

S2 |uθ|2dΣ]1/2

. The latter is a usefulproxy for the solution’s turbulent behaviour since in theabsence of turbulence, it would remain zero by symmetryconsiderations. As is evident in this figure, the system’sbehaviour does not change as the temperature increases.In particular, the growth rate and both onset and fulldevelopment of turbulence are essentially the same in allcases considered.

B. Rotating case: perturbations to Kerr

The qualitative behaviour of the system on the rotat-ing scenario is illustrated in figure 6, and it happens to bevery similar to the presented above for the Schwarzschildcase. However, we emphasize that these rotating solu-tions are not simply static solutions, i.e., ω0 = 0, in arotating frame. Recall that the rigid rotation introducesa background vorticity field, which one finds completelydominates the long-term behaviour. A key difference isevident immediately after the vortices are formed: thebackground rotation drags them into the main rotatingstream and gradually separates them according to the di-rection of their rotation. If the background rotation flows

8

0 100 200 300 400

time

1e-06

0,0001

0,01

log(

||uθ |

| 2)

T~100T~1000T~5000T~10000

FIG. 5: Dependence with temperature: logarithm of theL2 norm of u

θ for different temperatures, in the non-rotatingcase.

left to right as in fig. 6(c), clockwise rotating vortices ac-cumulate in the southern (lower) hemisphere while thevortices with counter-clockwise rotation migrate towardsthe opposite pole. Then, the system undergoes a merg-ing process (of co-rotating vortices) as in the non-rotatingcase, but now just one vortex of each sign remains at latetimes, as in fig. 6(d). The initial northwards/southwardspropagation of counterclockwise/clockwise vortices canbe also explained in terms of the equal-sign mergers ofvortices by regarding each of the smaller vortices gen-erated in the flow as interacting with two much largervortices induced at the north and south pole by the rigidrotation component of the flow. Note that the ‘final’two vortices are not necessarily in precise alignment withthe rotation axis, being often the case that they remainorbiting around the poles.Main features of the long-term behaviour of the so-

lutions in the rotating scenario are illustrated in figure7. In particular the mentioned two dominant vorticesas they oscillate around the poles, with the temperatureand energy attaining local minima at the vortices’ loca-tions and being concentrated near the equator where thefluid velocity is larger. The enstrophy is again almostexclusively contained within the vortices.In figure 8 we display representative modes of the vor-

ticity spectra for two different parameters of the rotat-ing solution with τ = 100 and a perturbation set byωp(θ, φ) = Y

010(θ, φ) and δ = 0.08. Here, the ℓ = 1

mode represents the background rotation contributionwhile the ℓ = 9 and ℓ = 11 modes are the ones associ-ated with the perturbation, as discussed previously. Weshould recall here that the coefficients in the spectrumare normalized to unity, and thus it becomes clear fromthe plots that the long term dynamics is being completelydominated by the rotation.For the case ω0 = 0.1, the configuration starts with

(a) t = 0 (b) t = 294

(c) t = 1010 (d) t = 3500

FIG. 6: Evolution of the vorticity field for a perturbationωp(θ, φ) = Y

0

10(θ, φ) and δ = 0.08, on a rigid rotation withω0 = 0.1. (a) Initial config., (b) beginning of turbulence, (c)fully developed turbulent stage, (d) final state.

(a)Vorticity (b)Temperature

(c)Energy (d)Enstrophy

FIG. 7: The late time configuration for distinct relevant fieldsin the rotating case (ω0 = 0.1 and T ∼ 100, at t = 3600).

a perturbation comparable in magnitude with the back-ground motion. As the vortices form and turbulence be-gins, the high-ℓ’s modes decrease displaying a cascadingphenomena into lower ℓ modes. In particular this cas-cade progresses towards the ℓ = 1 mode that eventuallygoverns the state of the system. On top of this rotationdominated flow, the two vortical structures mentionedabove persist and are represented in the spectrum by acombination of higher modes, predominantly the ℓ = 2and ℓ = 4 modes.Interestingly, for faster initial background rotations

(i.e., ω0 & 0.5) while keeping the perturbation ampli-tude δ fixed, the flow is already dominated by the ℓ = 1mode from the beginning and the turbulent stage takeslonger to appear and fully develop. It seems that a strong

9

0 100 200 300 400 500 600 700 800 900

time

0,0001

0,01

1

log[

C(l

)]

l=1l=2l=4l=9l=11

(a) ω0 = 0.1

0 100 200 300 400 500 600 700 800 900

time

0,0001

0,01

1

log[

C(l

)]

l=1l=2l=4l=9l=11

(b) ω0 = 0.5

FIG. 8: Relevant modes in the power spectrum of the vorticityfield in the rotating case (at temperature T ∼ 100), for tworotation parameters ω0. The initial perturbation is given byωp(θ, φ) = Y

0

10(θ, φ) and δ = 0.08.

rotating stream in the background hinders to a certaindegree the formation and merging of vortices.

IV. DISCUSSION AND FINAL WORDS

In this work we have analyzed relativistic conformalfluid flows on both S2 and T 2 backgrounds and foundfirst that turbulence naturally arise in these flows andsecond that it gives rise to a ‘inverse’ cascade fromshorter to longer wavelengths. These results not only ex-tend common observations in Newtonian hydrodynam-ics but also have tantalizing implications for the be-haviour of gravity in four-dimensional AdS spacetimes.Indeed, the AdS/CFT correspondence implies that forsufficiently high temperatures/length scales, the confor-mal fluid flows studied here have a dual description interms of gravitational perturbations on Schwarzschildor Kerr black holes in AdS4, either with a sphericalor planar horizon. The turbulent behaviour observed

here implies, through the duality, that gravitational per-turbations in this limit should cascade to smaller fre-quencies. This constitutes a prediction obtained withinholography which had not been previously anticipatedon firm grounds[57]. Indeed, previously identified in-stabilities in AdS, super-radiance (e.g., [43]) and ‘weak-turbulence’ [17, 45] imply a frequency shift towards higherfrequencies. Therefore the cascading phenomena ob-served here implies an altogether new behaviour on thegravitational side. Furthermore, as described in [3, 5, 36],the full metric of the corresponding spacetime can be ob-tained and the implications of this cascading behaviouranalyzed through suitable geometric quantities. We de-fer such tasks to a forthcoming work. We close notingthat this possible cascade behaviour, first raised in [5]and demonstrated here (and presented in e.g., [46]) hasnow been observed in the Poincaré patch case [47].

Of course, another field theory ‘prediction’ is that largeAdS black holes in five and higher dimensions will also ex-hibit turbulence but this chaotic behaviour will give riseto a ‘standard’ cascade to shorter wavelengths in thesecases. From a gravitational perspective, this apparentlygeneric behaviour for ‘hot’ horizons in AdS is completelyunexpected and calls for a better understanding withingravity itself.

A step towards understanding the distinction betweengravity in four and higher dimensions can be obtained byrecalling that enstrophy conservation is what drives thehydrodynamics in 2+1 dimensions to exhibit the inversecascade. One can then exploit the duality to translate theenstrophy into geometrical variables and to understandthe implications of its conservation on the gravitationalside. That is, the conservation of enstrophy in the fluiddescription implies a quasi-conserved quantity exists inthe bulk gravity theory. Further, as we have seen here,the system displays a rich dynamical vortex configura-tion that merge towards a long lived state described byrelatively few, long wavelength vortices. Isolated vorticesin conformal fluids in 2+1 dimensions have been studiedin [48] and their properties identified. Again using thefluid/gravity correspondence, one could produce a grav-itational description of these quasi-stationary vortices.That the understanding of these or other geometricalquantities might help shed new light in turbulence phe-nomena is definitively an exciting prospect (see e.g., [49]).

Certainly, the full gravitational description will nat-urally incorporate dissipative contributions in the fluidflows. While we argued and quantitatively demonstratedthat this dissipation will not modify the essential featuresof the turbulent behaviour at sufficiently high tempera-tures. In Newtonian hydrodynamics, the onset of turbu-lence is discussed in terms of the Reynolds number Re.The typical benchmark for the onset of turbulent flowsis that Re have value of a few thousand. In relativistichydrodynamics, the latter may be estimated as [50]

Re ∼ TLη/s

= 4π TL , (22)

10

where L is a characteristic length scale in the flow. In thelast expression above, we have substituted the celebratedholographic value η/s = 1/4π [6, 7, 51] — we would haveη/s = β for the general conformal fluid in (1). As thisexpression illustrates, we can produce arbitrarily largevalues of Re by increasing T (while keeping the systemsize fixed). Hence to observe turbulent behaviour, we areagain naturally pushed to the regime where the hydro-dynamic gradient expansion works well and our perfectfluid model becomes a good approximation. Of course,the viscous effects will definitely modify the very long-time behaviour observed here. For example, the conser-vation of the enstrophy is only true to first order in thehydrodynamic gradient expansion.

Of course, another interesting extension of the presentinvestigation would be studying further the details of theturbulent cascade, in particular, the Kolmogorov scal-ing exponents. Figure 13 illustrates some preliminary re-sults, which suggest that the ‘Newtonian’ kinetic energywill scale with the expected exponent of −5/3. However,various caveats must be noted. First, a clean easy-to-distinguish Kolmogorov-type reasoning applies to a sit-uation where the turbulent flow is driven by an exter-nal force at some high frequency (in the present caseof 2+1 dimensions) and viscous dissipation also dampsthe energy flow on much longer time scales. This sce-nario must be contrasted with the freely-decaying tur-bulence (i.e., without any driving force) which figure 13describes. In freely-decaying turbulence determining theintertial range is more delicate but in such range (whichshrinks as time proceeds) the −5/3 slope can be distin-guished. Additionally, Kolmogorov’s scaling argumentsare made in a Newtonian context and can at best be re-garded as approximate for the relativistic fluids studiedhere. Fortunately, the appropriate exact scaling relationsapplicable for relativistic hydrodynamic turbulence havebeen derived in [50]. The present simulations providea framework for further studying these relativistic rela-tions. More generally, however, the most exciting possi-bility would again be if a holographic perspective couldprovide new insights into the issues surrounding theseturbulent cascades.

Acknowledgments: It is a pleasure to thank P. Chesler,R. Emparan, M. Kruczenski and T. Wiseman for inter-esting discussions. This work was supported by NSERCthrough Discovery Grants and CIFAR (to LL and RCM)and by CONICET, FONCYT and SeCyT-Univ. Na-cional de Cordoba Grants (to FC and OR). F.C. thanksPerimeter Institute’s Visiting Graduate Fellows programfor hosting his stay at the Perimeter Institute, whereparts of this work were completed. Research at Perime-ter Institute is supported through Industry Canada andby the Province of Ontario through the Ministry of Re-search & Innovation. Computations were performed atSciNet.

Appendix A.

For comparison purposes, we also consider scenariosrelated to the dual case of an AdS4 black brane solu-tion where the non-radial directions are compactified ona torus of size D. In Fefferman-Graham coordinates, la-beled by (t, x, y), the asymptotic metric is simply the flatone and the equations of motion can be straightforwardlyimplemented.

A. Stationary solutions and initial data

Equilibrium configurations are given simply by con-stant flows at a given (arbitrary) temperature T . Forsimplicity we consider a torus with domain [0, D]2. Wethen restrict to initial configurations with a flow alongx, i.e., ua = δaxu0, and introduce generic perturbationsδua to this flow. For concreteness, we describe here twoparticular cases where the initial three-velocity is givenby: case (A) with a perturbation of compact support,

ua = δax(

u0 + δu sin(2πy/3) y2 (y − L)2

)

(A-1)

for 0 ≤ y ≤ L and ua = δaxu0 otherwise; and case (B)

ua = δax (u0 + δu sin(16πy/D)) . (A-2)

We note however the qualitative features observed in allcases considered remain the same.

B. Turbulence, cascading behaviour and

temperature dependence

We performed a series of numerical experiments adopt-ing u0 = 0, 0.1 and 0.5 and δu = 0.01, · · · , 0.05 (i.e.,perturbations from 2% to 50%). and considered temper-atures T ∈ [1, 103] with L = 10 and typical grid sizes of[401, 401] (though consistency of behaviour was checkedwith grids 1.5 and 2 times better resolved).As in the case of the sphere, the dynamics display a

turbulent behaviour leading to the development of largevortices. As an illustrative example, Figure 9 displays thevorticity of the system at different times for case (A) withu0 = 0.1, δu = 0.01 and T = 1. Early on, the behaviouris seemingly stationary but as turbulence develops theinitial symmetry is completely broken and vortices arisewhich grow as they merger leading to a configurationdescribed by long wavelengths.As we have done in the S2 case we also study the sys-

tem’s behaviour upon variation of temperature to ensureterms neglected in our study do not significantly affectthe dynamics obtained. As in the previous case, a use-ful proxy to monitor the evolution is the (L2) norm ofuy which, in the absence of turbulent dynamics, wouldremain zero. Figure 10 illustrates that the observed be-haviour is essentially unchanged with temperature.

11

FIG. 9: Representative snapshots of the vorticity. As timeprogresses the initial configuration is strongly disturbed bythe formation of vortices. As the dynamics continues, largervortices are formed.

0 1000 2000 3000

-8

-6

-4

-2

log 1

0(||u

y|| 2

)

T ~ 1T ~ 4.6T ~ 21T ~ 100T ~ 460T ~ 4600

FIG. 10: Logarithm of the L2 norm of uy for different tem-

peratures. Essentially the behaviour observed is unchangedwith temperature.

Last, we also considered varying the size of the torusby increasing D by factors of 1.5 and 2 in case (A) and

observed the same cascading behaviour.To study how the energy cascades as time progresses,

we compute the Fourier transform of energy densityT 00 and the vorticity density W 0. Since our computa-tional grid is discretized by points {x1i , x2j} (with i, j =1, · · · , N), we denote ~n = {i, j}, and write T 00(~x) =T 00(~n) and the vorticity W 0(~x) = W 0(~n). The Fouriertransform of these quantities is given by

T̂ 00( ~K) =∑

~n

e−2πi~K. ~n

N T 00(~n) , (A-3)

Ŵ 0( ~K) =∑

~n

e−2πi~K. ~n

N W 0(~n) . (A-4)

Here ~K = {K1,K2} and K1,K2 ∈ [0, N − 1]. Clearly,T̂ 00( ~K) and Ŵ 0( ~K) are also represented on a N×N grid.It is convenient to re-express the transformed quantities

as functions of K = | ~K|. From the functions T̂ 00( ~K) andŴ 0( ~K), we define T 00(K) and W 0(K) as follows

T 00(K) =∑

i

|T 00( ~Ki)| , (A-5)

W 0(K) =∑

i

|W 0( ~Ki)| , (A-6)

where the sums run over ~Ki’s satisfying K ≤ | ~Ki| <K+1. With this approach an effective wave-number gridof length

√2N is obtained. Figure 11 illustrates a clear

cascade to lower wave-numbers as time proceeds. In thecase of T 00(K) the dominant mode is given by K = 1while for the vorticity both K = 1 and K = 2 are ofsimilar magnitude. As in the S2 case, this is expected asthe vorticity field is obtained by taking a single derivativewhich increases the mode content by one.It is interesting to monitor whether the observed be-

haviour is consistent with the ‘standard’ Kolmogorov ex-pectation. To illustrate this, we compute the Fouriertransform of the ‘Newtonian’ kinetic energy per unit massE = 1/2v2. To make sense of this limit within our rel-ativistic description, we chose u0 = 0 and δu = 0.03,whose solution is such that |v| is bounded by ≃ 0.04 forall times, thus the fluid’s motion stays far from relativis-tic speeds. Figure 12 shows the vorticity behaviour forthis case and 13 illustrates the Fourier transformation ofthis kinetic energy for representative times. As time pro-ceeds the energy in higher frequency modes diminisheswhile the opposite behaviour is observed for the lowerones. Additionally, at intermidiate frequencies, the en-ergy exhibits a behaviour with frequency consistent witha slope of −5/3, the expected Kolmogorov exponent.

[1] Juan Martin Maldacena. The Large N limitof superconformal field theories and supergravity.Adv.Theor.Math.Phys., 2:231–252, 1998.

[2] Ofer Aharony, Steven S. Gubser, Juan Martin Malda-cena, Hirosi Ooguri, and Yaron Oz. Large N field theo-ries, string theory and gravity. Phys.Rept., 323:183–386,

12

0 100 200 300 400 500 600

time

10log[

T00

(k)]

k = 1k = 5k = 25k = 65

0 100 200 300 400 500 600

time

10

log[

W(k

)]

k = 1k = 5k = 25k = 65

FIG. 11: Fourier modes for energy density (top panel) andvorticity density (bottom panel). As time progresses, a clearcascade to lower wave numbers is obtained.

2000.[3] Sayantani Bhattacharyya, Veronika E Hubeny, Shiraz

Minwalla, and Mukund Rangamani. Nonlinear Fluid Dy-namics from Gravity. JHEP, 0802:045, 2008.

[4] Sayantani Bhattacharyya, R. Loganayagam, Ipsita Man-dal, Shiraz Minwalla, and Ankit Sharma. ConformalNonlinear Fluid Dynamics from Gravity in Arbitrary Di-mensions. Technical Report TIFR/TH/08-38, 2008.

[5] Mark Van Raamsdonk. Black Hole Dynamics From At-mospheric Science. JHEP, 0805:106, 2008.

[6] P. Kovtun, D.T. Son, and A.O. Starinets. Viscosity instrongly interacting quantum field theories from blackhole physics. Phys.Rev.Lett., 94:111601, 2005.

[7] G. Policastro, D.T. Son, and A.O. Starinets. The Shearviscosity of strongly coupled N=4 supersymmetric Yang-Mills plasma. Phys.Rev.Lett., 87:081601, 2001.

[8] Yaron Oz and Michael Rabinovich. The Penrose In-equality and the Fluid/Gravity Correspondence. JHEP,

FIG. 12: Representative snapshots of the vorticity. Again, astime progresses a turbulent behaviour is clearly displayed andan apparent cascade to lower wavelengths.

1 10 100k

10-10

10-8

10-6

10-4

10-2

FT(v

2 )

t = 928t = 1328t = 1688t = 2248t = 3048

FIG. 13: Fourier modes for the kinetic energy density for rep-resentative times. As time progresses a cascading behaviourtowards lower frequencies is manifested and in an interme-diate ‘inertial regime’ with k ∈ (8, 400), the scaling is quiteconsistent with a −5/3 slope (indicated by a dotted line inthe figure).

1102:070, 2011.[9] Paul M. Chesler and Laurence G. Yaffe. Holography

and colliding gravitational shock waves in asymptoticallyAdS5 spacetime. Phys.Rev.Lett., 106:021601, 2011.

[10] Keiju Murata, Shunichiro Kinoshita, and Norihiro Tana-hashi. Non-equilibrium Condensation Process in a Holo-graphic Superconductor. JHEP, 1007:050, 2010.

[11] Simon Caron-Huot, Paul M. Chesler, and Derek Teaney.

13

Fluctuation, dissipation, and thermalization in non-equilibrium AdS5 black hole geometries. Phys.Rev.,D84:026012, 2011.

[12] David Garfinkle, Leopoldo A. Pando Zayas, and Dori Re-ichmann. On Field Theory Thermalization from Gravi-tational Collapse. JHEP, 1202:119, 2012.

[13] Hans Bantilan, Frans Pretorius, and Steven S. Gub-ser. Simulation of Asymptotically AdS5 Spacetimes witha Generalized Harmonic Evolution Scheme. Phys.Rev.,D85:084038, 2012.

[14] Alex Buchel, Luis Lehner, and Robert C. Myers. Thermalquenches in N=2* plasmas. 2012.

[15] Mariano Chernicoff, Daniel Fernandez, David Mateos,and Diego Trancanelli. Jet quenching in a strongly cou-pled anisotropic plasma. 2012.

[16] C.-k. Chan, D. Mitra, and A. Brandenburg. Dynamicsof saturated energy condensation in two-dimensional tur-bulence. Phys. Rev. E , 85(3):036315, March 2012.

[17] Piotr Bizon and Andrzej Rostworowski. On weakly tur-bulent instability of anti-de Sitter space. Phys.Rev.Lett.,107:031102, 2011.

[18] Federico Carrasco, Luis Lehner, Robert Myers, Os-car Reula, and Ajay Singh. Conformal Fluids andTurbulent Behaviour in 2+1 Dimensions. Techni-cal Report http://spaces.perimeterinstitute.ca/2d-turbulence/content/2d-turbulence, 2012.

[19] P.A. Davidson. Turbulence: An Introduction for Scien-tists and Engineers. Oxford University Press, USA, 2004.

[20] Rose, H.A. and Sulem, P.L. Fully developed turbulenceand statistical mechanics. J. Phys. France, 39(5):441–484, 1978.

[21] Krzysztof Gawedzki. Easy turbulence. Technical ReportIHES/P/99/56, August 1999.

[22] R.H. Kraichnan. Inertial ranges in two-dimensional tur-bulence. Phys. Fluids, 10:1417–1423, 1967.

[23] G.K. Batchelor. High Speed Computing in Fluid Dynam-ics. Phys. Fluids, 12:II–233, 1969.

[24] James C. Mcwilliams. The emergence of isolated coherentvortices in turbulent flow. Journal of Fluid Mechanics,146:21–43, 1984.

[25] Vadim Borue. Inverse energy cascade in stationary two-dimensional homogeneous turbulence. Phys. Rev. Lett.,72:1475–1478, Mar 1994.

[26] R Benzi, G Paladin, S Patarnello, P Santangelo, andA Vulpiani. Intermittency and coherent structures intwo-dimensional turbulence. Journal of Physics A: Math-ematical and General, 19(18):3771, 1986.

[27] Ingo Muller. Zum Paradoxon der Warmeleitungstheorie.Z.Phys., 198:329–344, 1967.

[28] W. Israel and J.M. Stewart. Transient relativistic ther-modynamics and kinetic theory. Annals Phys., 118:341–372, 1979.

[29] Robert Geroch and Lee Lindblom. Dissipative relativisticfluid theories of divergence type. Phys. Rev. D, 41:1855–1861, Mar 1990.

[30] Robert Geroch and Lee Lindblom. Causal theories of dis-sipative relativistic fluids. Annals of Physics, 207(2):394–416, 1991.

[31] Alex Buchel and Robert C. Myers. Causality of Holo-graphic Hydrodynamics. JHEP, 0908:016, 2009.

[32] Rudolf Baier, Paul Romatschke, Dam Thanh Son, An-drei O. Starinets, and Mikhail A. Stephanov. Relativisticviscous hydrodynamics, conformal invariance, and holog-raphy. JHEP, 0804:100, 2008.

[33] Paul Romatschke. Relativistic Viscous Fluid Dynam-ics and Non-Equilibrium Entropy. Class.Quant.Grav.,27:025006, 2010.

[34] B. Gustafsson J. Oliger H. Kreiss. Time Dependent Prob-lems And Difference Methods. John Wiley and Sons, Inc.,1995.

[35] Gioel Calabrese, Luis Lehner, Oscar Reula, Olivier Sar-bach, and Manuel Tiglio. Summation by parts and dis-sipation for domains with excised regions. Technical Re-port LSU-REL-080103, 2004.

[36] Sayantani Bhattacharyya, Subhaneil Lahiri, R. Lo-ganayagam, and Shiraz Minwalla. Large rotating AdSblack holes from fluid mechanics. JHEP, 0809:054, 2008.

[37] G.W. Gibbons, H. Lü, Don N. Page, and C.N. Pope. Thegeneral kerr–de sitter metrics in all dimensions. Journalof Geometry and Physics, 53(1):49–73, 2005.

[38] Brandon Carter. Covariant theory of conductivity inideal fluid or solid media. Lecture Notes in Mathematics,1385:1–64, 1989.

[39] Luis Lehner, Oscar Reula, and Manuel Tiglio. Multi-block simulations in general relativity: high orderdiscretizations, numerical stability, and applications.Class.Quant.Grav., 22:5283–5322, 2005.

[40] Florencia Parisi and Oscar Reula. in preparation. 2012.[41] Mark H. Carpenter, Jan Nordström, and David Got-

tlieb. A Stable and Conservative Interface Treatment ofArbitrary Spatial Accuracy. Journal of ComputationalPhysics, 148(2):341–365, 1999.

[42] Jan Nordström and Mark H. Carpenter. High-Order Fi-nite Difference Methods, Multidimensional Linear Prob-lems, and Curvilinear Coordinates. Journal of Computa-tional Physics, 173(1):149–174, 2001.

[43] Elizabeth Winstanley. On classical superradiance inKerr-Newman - anti-de Sitter black holes. Phys.Rev.,D64:104010, 2001.

[44] J. Evslin, Hydrodynamic Vortices and their Gravity Du-als. Fortsch. Phys. 60:1005, 2012. [arXiv:1201.6442 [hep-th]].

[45] Oscar J.C. Dias, Gary T. Horowitz, and Jorge E. San-tos. Gravitational Turbulent Instability of Anti-de SitterSpace. Class.Quant.Grav., 29:194002, 2012.

[46] Federico Carrasco, Luis Lehner, Robert Myers, OscarReula, and Ajay Singh. Conformal Fluids and Turbu-lent Behaviour in 2+1 Dimensions. Technical Reporthttp://pirsa.org/12060025/, 2012.

[47] Paul Chesler. in progress. 2012.[48] Jarah Evslin and Chethan Krishnan. Vortices in (2+1)d

Conformal Fluids. JHEP, 1010:028, 2010.[49] Christopher Eling, Itzhak Fouxon, and Yaron Oz. Grav-

ity and a Geometrization of Turbulence: An IntriguingCorrespondence. 2010.

[50] Itzhak Fouxon and Yaron Oz. Exact Scaling RelationsIn Relativistic Hydrodynamic Turbulence. Phys.Lett.,B694:261–264, 2010.

[51] Christopher P. Herzog. The Hydrodynamics of M theory.JHEP, 0212:026, 2002.

[52] Xiao Liu and Yaron Oz. Shocks and Universal Statisticsin (1+1)-Dimensional Relativistic Turbulence. JHEP,1103:006, 2011.

[53] Irene Bredberg, Cynthia Keeler, Vyacheslav Lysov, andAndrew Strominger. From Navier-Stokes To Einstein.JHEP, 1207:146, 2012.

[54] The one dimensional case has been studied in [52].[55] This issue does not arise in the Newtonian context and

14

a holographic scenario realizing this limit has been dis-cussed in [53]. Consequently, the well known turbulentbehaviour of Newtonian fluids is obtained in this case.

[56] Here, we have dropped the constant α appearing in (1),which gives an overall normalization factor in the energy

density. Including this factor would simply produce aninconsequential constant shift of ρ̃.

[57] Note however the possibility had been raised earlier in [5,44]

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