This is a repository copy of Analytical theory of forced rotating sheared turbulence: The perpendicular case.

Transcription

1 This is a repository copy o Analytical theory o orced rotating sheared turbulence: The perpendicular case. White Rose Research Online URL or this paper: Version: Accepted Version Article: Leprovost, N. and Kim, E.J. (28) Analytical theory o orced rotating sheared turbulence: The perpendicular case. Physical Review E, 78 (1) ISSN Reuse Unless indicated otherwise, ulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 o the Copyright, Designs and Patents Act 1988 allows the making o a single copy solely or the purpose o non-commercial research or private study within the limits o air dealing. The publisher or other rights-holder may allow urther reproduction and re-use o this version - reer to the White Rose Research Online record or this item. Where records identiy the publisher as the copyright holder, users can veriy any speciic terms o use on the publisher s website. Takedown I you consider content in White Rose Research Online to be in breach o UK law, please notiy us by ing eprints@whiterose.ac.uk including the URL o the record and the reason or the withdrawal request. eprints@whiterose.ac.uk

2 Analytical theory o orced rotating sheared turbulence. I. Perpendicular case Nicolas Leprovost and Eun-jin Kim Department o Applied Mathematics, University o Sheield, Sheield S3 7RH, UK arxiv:physics/7116v4 [physics.lu-dyn] 12 Jun 28 Rotation and shear lows are ubiquitous eatures o many astrophysical and geophysical bodies. To understand their origin and eect on turbulent transport in these systems, we consider a orced turbulence and investigate the combined eect o rotation and shear low on the turbulence properties. Speciically, we study how rotation and low shear inluence the generation o shear low (e.g. the direction o energy cascade), turbulence level, transport o particles and momentum, and the anisotropy in these quantities. In all the cases considered, turbulence amplitude is always quenched due to strong shear (ξ = νk 2 y/a 1, where A is the shearing rate, ν is the molecular viscosity and k y is a characteristic wave-number o small-scale turbulence), with stronger reduction in the direction o the shear than those in the perpendicular directions. Speciically, in the large rotation limit (Ω A), they scale as A 1 and A 1 lnξ, respectively, while in the weak rotation limit (Ω A), they scale as A 1 and A 2/3, respectively. Thus, low shear always leads to weak turbulence with an eectively stronger turbulence in the plane perpendicular to shear than in the shear direction, regardless o rotation rate. The anisotropy in turbulence amplitude is however weaker by a actor o ξ 1/3 lnξ ( A 1/3 lnξ ) in the rapid rotation limit (Ω A) than that in weak rotation limit (Ω A) since rotation avors almost-isotropic turbulence. Compared to turbulence amplitude, particle transport is ound to crucially depend on whether rotation is stronger or weaker than low shear. When rotation is stronger than low shear (Ω A), the transport is inhibited by inertial waves, being quenched inversely proportional to the rotation rate (i.e. Ω 1 ) while in the opposite case, it is reduced by shearing as A 1. Furthermore, the anisotropy is ound to be very weak in the strong rotation limit (by a actor o 2) while signiicant in the strong shear limit. The turbulent viscosity is ound to be negative with inverse cascade o energy as long as rotation is suiciently strong compared to low shear (Ω A) while positive in the opposite limit o weak rotation (Ω A). Even i the eddy viscosity is negative or strong rotation (Ω A), low shear, which transers energy to small scales, has an interesting eect by slowing down the rate o inverse cascade with the value o negative eddy viscosity decreasing as ν T A 2 or strong shear. Furthermore, the interaction between the shear and the rotation is shown to give rise to a novel non-diusive lux o angular momentum (Λ-eect), even in the absence o external sources o anisotropy. This eect provides a mechanism or the existence o shearing structures in astrophysical and geophysical systems. PACS numbers: Jv,47.27.T-,97.1.Kc I. INTRODUCTION Rotating turbulent lows can be ound in many areas such as engineering (turbo-machinery, combustion engine), geophysics (oceans, Earth s atmosphere) or astrophysics (gaseous planets, galactic and accretion disks). Large-scale luid motions tend to appear as a robust eature in these systems, oten in the orm o shear lows (such as circulations on the surace o planets, dierential rotation in stars and galaxies or lows in a rotating machinery). There have been accumulating evidence that large-scale shear lows as well as rotation play a crucial role in determining turbulence properties and transport, such as energy transer or mixing (see below or more details). The understanding o the physical mechanism or the generation o large-scale shear lows and the complex interaction among rotation, shear lows and turbulence thus lies at the heart o the predictive theory o turbulent transport in many systems. A. Summary o previous works While both rotation and shear low apparently have a similar eect on quenching turbulent transport, the eiciency o their eects as well as the basic physical mechanisms are totally dierent. It is thus useul to contrast these in detail. 1. Sheared turbulence The main eect o shear low is to advect turbulent eddies dierentially, elongating and distorting their shapes, thereby rapidly generating small scales which are ultimately disrupted by molecular dissipation on small scales (see Fig. 1). That is, low shear acilitates the cascade o various quantities such as energy or mean square scalar density

3 2 Background shearing low X Turbulent eddy y Typical distance an eddy can transport a passvie scalar ield FIG. 1: Sketch o the eect o shear on a turbulent eddy. to small scales (i.e. direct cascade) in the system, enhancing their dissipation rate. As a result, turbulence level as well as turbulent transport o these quantities can be signiicantly reduced compared to the case without shear. Another important consequence o shearing is to induce anisotropic transport and turbulent level since low shear directly inluences the component parallel to itsel (i.e. x component in Fig. 1) via elongation while only indirectly the other two components (i.e. y and z components in Fig. 1) through enhanced dissipation. This shearing eect o shear low can be captured by time-dependent Fourier transorm where the wave number in the shearing direction (e.g. k x in Fig. 1) increases linearly in time [1, 2, 3]. It is important to emphasize that the aorementioned shearing eect (due to dierential advection) is via nonlocal interaction between large and small-scale modes, and can dominate over nonlinear local interaction between small scales or suiciently strong low shear [e.g. 4]. Thereore, the evolution o small-scale quantities can be treated as linear by neglecting local interactions compared to nonlocal interactions. This ormulation, also called the rapid distortion theory (RDT) by various previous authors [2, 5], was used to study the linear response o turbulence to a mean low with spatially uniorm gradients. The linear treatment o luctuations by incorporating strong low shear was also used in the astrophysical context by [1] by using shearing coordinates. The generation o large-scale shear lows (the so-called zonal lows) through a similar nonlocal interaction has been intensely studied in the magnetically conined plasmas, where turbulence quenching by shear low is believed to be one o the most promising mechanisms or improving plasma coninement [6, 7]. In decaying sheared turbulence, [8] have shown a surprisingly good agreement between the RDT predictions and numerical simulations. Forced sheared turbulence was proposed or the irst time by [9] in the context o twodimensional near-wall turbulence to explain the logarithmic dependence o the large scale velocity on the distance to the wall. In that case, the external orcing is provided by a continuous supply o vorticity rom intermittent coherent burst o vorticity coming rom the viscous layer. This work was later generalized to three dimensions [1, 11] with the same conclusions. Subsequently, theoretical predictions (using a quasi-linear theory) or the transport o passive scalar ields in 2D hydrodynamic turbulence by [12] and [13] have been beautiully conirmed by recent numerical simulations[14]. In particular, they have shown that turbulent transport o particles can be severely quenched inversely proportional to low shear A while turbulence level is reduced as A 5/3. Re. [3] has shown that in 3D orced HD turbulence, strong low shear can quench turbulence level and transport o particles with strong anisotropy (much weaker along the low shear which is directly aected by shearing) and has emphasized the dierence in turbulence level and transport, which is oten used interchangeably in literature. A similar weak anisotropic transport was shown or momentum transport by[15] in orced 3D HD turbulence. Further investigations have been perormed on turbulent transport in orced turbulence by incorporating the interaction o sheared turbulence with dierent types o waves that can be excited due to magnetic ields [16, 17, 18], stratiication [19] or both magnetic ields and stratiication [2].

4 3 2. Rotating turbulence Rotation has both similar and dierent eects on turbulent transport. First, rotation can reduce transport in the limit o rapid rotation (similarly to low shear), but through a physical mechanism that is dierent rom that o shear, namely by phase mixing o inertial waves [21]. It also induces only slight anisotropy in the transport (by a actor o two), much less signiicant than the strong anisotropy due to shear. Further, since phase mixing aects turbulent transport without necessarily quenching turbulence level, turbulence level may not be aected by rotation. This reduction in transport without much eect on turbulence level is a common eature o turbulence strongly aected by waves, and is also ound in MHD turbulence where magnetic ields support Alven waves [16, 17, 22] and stratiied turbulence [19] where stable stratiication excites internal gravity waves. A more striking dierence between low shear and rotation is that rotation acilitates the cascade energy to large scale, generating large-scale lows. For instance, in the extreme limit o very rapid rotation, the luid motion becomes independent o the coordinate along the rotation axis (the so-called Taylor-Proudman theorem [23, 24]). The generation o large-scale low has been shown by various numerical simulations including [25] and [26]. In particular, [26] have shown that the inverse cascade o energy is more pronounced in orced turbulence due to statistical triadic transer through nonlocal interaction. It is important to note that this nonlocal interaction leading to inverse cascade can be successully captured by inhomogeneous RDT theory which permits the eedback o the nonlinear local interaction between small scales onto the large scales via Reynolds stress (constituting the other part o quasi-linear analysis) while neglecting nonlinear local interaction between small scales or luctuations compared to nonlocal interactions. As must be obvious by comparing the Coriolis orce with nonlinear advection terms, the RDT works well or suiciently strong rotation (small Rossby number) even in the absence o shear low. For instance, the agreement o the RDT prediction with numerical results has been shown by various previous authors including [25], but mostly in decaying turbulence. However, in this case, the RDT cannot accurately capture the turbulence structure in the plane perpendicular to rotation axis where nonlinear local interactions between inertial waves seem important (see, e.g. [26]). The validity and weakness o the RDT together with comparison with various numerical simulation (without an external orcing) with/without shear lows and stratiication can be ound in excellent review by [27] and Cambon and [28], to which readers are reerred or more details. In comparison, ar much less is understood in the case o orced turbulence. In particular, the main interest in orced turbulence is a long-term time behavior where the dissipation, enhanced by shear distortion, is balanced by energy input, thereby playing a crucial role in leading to a steady equilibrium state. The computational study o this long time behavior is however not only expensive but also diicult because o the limit on numerical accuracy, as noted by [29]. Thereore, analytical theory by capturing shearing eect (such as quasi-linear theory with timedependent wavenumber) would be extremely useul in obtaining physical insights into the problem as well as guiding uture computational investigations. We note that the previous works by Kichatinov and Rudiger and collaborators [3, 31, 32, 33, 34] using quasi-linear theory are valid only in the limit o weak shear. We urther note that physically, the local nonlinear interactions in Navier-Stokes equation can be captured by an external orcing [35, 36]. B. Main objectives and methodology Our main motivation is to understand the origin o large-scale shear low and its eect on turbulent transport in rotating systems. To this end, we consider a orced turbulence and investigate the combined eect o rotation and shear low on the turbulence properties including transport o momentum and particles. Speciically, we are interested in how rotation and low shear inluence the generation o shear low (e.g. the direction o energy cascade), turbulence level, transport o particles and momentum, and the anisotropy in these quantities. Given the dierences/similarities in the eects o low shear and rotation (as discussed in Sec. IA), o particular interest is to identiy the relative strength o low shear to rotation rate or the cross-over between inverse and direct cascades and isotropic and almostisotropic turbulence/transport. Recalling that low shear o strength A acts over the time-scale A 1 while rotation induces inertial waves o requency Ω, one could naively think that low shear would dominate the eect o rotation or suiciently strong shear with A Ω while the eect o low shear may be neglected in the opposite limit A Ω. This will however be shown to be true only in the case o the transport o passive scalar ields and or the sign o eddy viscosity. That is, even in the case o weak shear compared to rotation A Ω, the shear has yet a crucial eect on determining the overall amplitude o turbulence level and momentum transport since its shearing process (generating small scales) works coherently over more than one oscillation o the waves. To complement this, we are also interested in how shear-dominated turbulence is inluenced by rotation. As will be shown later, when the system is linearly stable, weak rotation tends to make turbulence/transport more isotropic. Concerning momentum transport, another important question is the possibility o non-diusive transport. In rotating turbulence, the inverse cascade can occur not only due to a (diusive) negative viscosity, but also due to

5 non-diusive momentum transport. The latter is known as the anisotropic kinetic α-eect (AKA) [37] or as the Λ-eect in the astrophysical community. The appearance o non-diusive term in the transport o angular momentum prevents a solid body rotation rom being a solution o the Reynolds equation [38, 39], and thus act as a source or the generation o large-scale shear lows. For instance, this eect has been advocated as a robust mechanism to explain the dierential rotation in the solar convective zone. Starting rom Navier-Stokes equation, it is possible to show that these luxes arise when there is a cause o anisotropy in the system, either due to an anisotropic background turbulence (see [33] and reerences therein) or else due to inhomogeneities such as an underlying stratiication. We will show that non trivial Λ-eect can result rom an anisotropy induced by shear low on the turbulence even when the driving orce is isotropic, in contrast to the case without shear low where this eect exists only or anisotropic orcing [32]. We note that although much less attention has been paid to the eect o rotation and shear on mixing and transport o scalars(such as pollutants, heat or reacting species) compared to momentum transport, this is an important problem in understanding the distribution and mixing o a variety o physical quantities in dierent systems. For instance, observations show that the concentration o light elements at the surace o the Sun is smaller than what is expected by comparison with Earth s or meteorites abundance. As these light elements can only be destroyed below a strong shear layer (the so-called solar tachocline), their transport is subject to the eects o strong shear and rotation. The study o transport o passive scalar has been mostly limited to the purely rotating case [4, 41] or non-rotating sheared turbulence [42, 43]. For purely rotating turbulence, linear theory has shown a strong suppression o particle diusion by rotation, conirmed by numerical simulations [41]. In comparison, the study o particle diusion in sheared rotating turbulence was done only by [44], who ound that numerical simulation results agree airly well with his linear theory. The purpose o this paper is to provide theoretical prediction on these issues by considering a 3D incompressible luid, orced by a small-scale external orcing. As we are interested in the eect o low shear, we capture this eect non-perturbatively by using time-dependent wavenumber [see Eq. (3)]. By assuming either suiciently strong shear or rotation rate, we employ a quasi-linear analysis to compute turbulence level, eddy viscosity, and particle transport or temporally short-correlated, homogeneous orcing. As the computation o these quantities involve too complex integrals to be analytically tractable, they are analytically computed by assuming an ordering in time scales. In our problem, there are three important (inverse) time-scales: the shearing rate A, the rotation rate Ω and the diusion rate D = νky 2 where ν is the (molecular) viscosity o the luid and ky 1 is a characteristic small scale o the system. We irst distinguish the two cases o strong rotation (Ω A) and weak rotation (Ω A). The irst regime o strong rotation will be studied in the strong shear (A D) and weak shear (A D) regime. On the other hand, the second regime o weak rotation will be considered only in the strong shear (A D) case, as the eects o both shear and rotation disappear in the opposite limit (A D). We believe that our results would provide not only useul physical insights in understanding the complex dynamics o rotating sheared turbulence, but also serve as a guide or urther theoretical/computational works, especially considering the diiculty o numerical study o this system. The remainder o the paper is organized as ollows: in II, we ormulate our problem. Theoretical results o turbulent intensity and turbulent transport are provided in III. Some o the detailed analysis are provided only in III. We then discuss our indings in the strong shear limit in IV and provide concluding remarks in V. The eect o rotation on linear stability o shear lows and some o the detailed algebra are provided in Appendices. Since analytical analysis perormed in the paper are quite involved, some o the readers who are mainly interested in the results might wish to go to IV and V ater reading II. 4 II. MODEL We consider an incompressible luid in a rotating rame with average rotation rate Ω, which are governed by t u+u u = P +ν 2 u+f 2 Ω u, (1) u =. Following [3], we study the eect o a large-scale shear U = U (x)ĵ on the transport properties o turbulence by writing the velocity as a sum o a shear (chosen in the x-direction) and luctuations: u = U + v = U (x)ĵ + v = xaĵ+v. Without loss o generality, we assume A >. In the ollowing, we consider the coniguration o Figure (2) where the shear and rotation (in the z direction) are perpendicular and simpliy notation by using Ω = 2 Ω. Then, the Coriolis orce is simply Ω[ u y i+u x j], where i, j and k are the unit vectors associated with the Cartesian coordinates. Note that our x y coordinates are not conventional in that our x and y directions correspond to y and x in previous works (see [29] or instance). Thereore, the shearing, the stream-wise and the span-wise direction correspond to the x, y and z direction, respectively.

6 5 z Ω y A = du y dx x FIG. 2: Sketch o the coniguration in the perpendicular case To calculate turbulence amplitude (or kinetic energy) and turbulent transport, we need to solve the equation or the luctuating velocity ield. To this end, we employ the quasi-linear theory [45] where the nonlinear local interactions between small scales are neglected compared to nonlocal interactions between large and small scales and obtain: t v+u v+v U = p+ν 2 v+ Ω v, (2) v =, where p and are respectively the small-scale components o the pressure and orcing. As noted in the introduction, this approximation, also known as the RDT [2], is justiied in the case o strong shear as the latter induces a weak turbulence, leading to weak interaction between small scales which is negligible compared to the(non-local) interaction between the shear and small scales. This has in act been conirmed by direct numerical simulations, proving the validity o the predictions o quasi-linear theory with a constant-rate shear both in the non-rotating [8] and rotating unorced [29] turbulence and also or orced turbulence [14]. Further, note that the quasi-linear analysis is also valid in the limit o rapid rotation [36]. To solve Eq. (2), we introduce a Fourier transorm with a wave number in the x direction evolving in time in order to incorporate non-perturbatively the eect o the advection by the mean shear low [1, 2, 3]: v(x,t) = 1 (2π) 2 d 3 k e i[kx(t)x+kyy+kzz] ṽ(k,t), (3) where k x (t) = k x () + k y At. From Eqs. (2) and (3), we obtain the ollowing set o equations or the luctuating velocity: A τ ˆv x = ik y τˆp+ ˆ x +Ωˆv y, (4) A τˆv y Aˆv x = ik yˆp+ ˆ y Ωˆv x, A τˆv z = ik zˆp+ ˆ z, = τvˆ x + ˆv y +βˆv z. Here, the new variables ˆv = ṽexp[ν(k 2 H t+k3 x/3k y A)] and similarly or ˆ and ˆp have been used to absorb the diusive term, and the time variable has been changed to τ = k x (t)/k y. In the remainder o the paper, we solve Eq. (4) or the luctuating velocity (with a vanishing velocity as initial condition). We then use these results and the correlation o the orcing (deined in IIC) to compute the turbulence intensity and transport (deined in IIB).

7 6 A. Transport o angular momentum As the large-scale velocity is in the y direction, we are mostly interested in the transport in that direction. The large-scale equation or the y component o velocity U is given by Eq. (1) with a supplementary term R where R is the Reynolds stress given by: R = vv y. (5) To understand the eect o R on the transport o angular momentum, one can ormally Taylor expand it with respect to the gradient o the large-scale low: R i = Λ i U ν T x U δ i1 + = Λ i U +ν T Aδ i (6) Here, Λ i and ν T are the two turbulent transport coeicients rom non-diusive and diusive momentum lux, respectively. Note that the irst term in the expansion is due to the small-scale driving and the Coriolis orce in Eq. (1) which break the Galilean invariance [46]. First, ν T is the turbulent (eddy) viscosity, which simply changes the viscosity rom the molecular value ν to the eective value ν +ν T. Note that the sign o eddy viscosity represents the direction o energy cascade, with positive (negative) value or direct (inverse) cascade. Second, the irst term involving Λ i in equation (6) is proportional to the rotation rate rather than the velocity gradient. This means that it does not vanish or a constant velocity ield and thus permits the creation o gradient in the large-scale velocity ield. This term bears some similarity with the α eect in dynamo theory [47, 48] and has been known as the Λ-eect [3, 38] or anisotropic kinetic alpha (AKA)-eect [37]. Similarly to the α eect, this eect exists only i the small-scale low lacks parity invariance (going rom right-handed to let handed coordinates). However, in contrast to the α eect, the Λ eect requires anisotropy or its existence [3, 37]. B. Particle (or heat) transport To study the inluence o rotation and shear on the particle and heat transport, we have to supplement equation (1) with an advection-diusion equation or these quantities. We here ocus on the transport o particles since a similar result also holds or the heat transport. The density o particles N(x,t) is governed by the ollowing equation: t N +U N = D 2 N, (7) where D is the molecular diusivity o particle. Note that, in the case o heat equation, D should be replaced by the molecular heat conductivity χ. Writing the density as the sum o a large-scale component N and small-scale luctuations n (N = N +n), we can express the evolution o the transport o chemicals on large scales by: t N +U N = (Dδ ij +D ij T ) i j N, (8) where the turbulent diusivity is deined as v i n = D ij T jn. D ij T will analytically be computed to see the eect o rotation and low shear on turbulent transport o chemicals which can be highly anisotropic. Note that the transport o a passive scalar quantity (contrary to the angular momentum which is a vector quantity) has to be diusive due to the act that it is solely advected by the low [49]. For simplicity, we assume a unit Prandtl number D = ν and apply the transormation introduced in equation (3) to the density luctuation n to obtain the ollowing equation: τˆn = ( jn ) ˆv j. (9) A Equation (9) simply shows that the luctuating density o particles can be obtained by integrating the luctuating velocity in time. C. External orcing As mentioned in introduction, we consider a turbulence driven by an external orcing. To calculate the turbulence amplitude and transport deined in II A and II B (which involve quadratic unctions o velocity and/or density), we

8 prescribe this orcing to be short correlated in time (modeled by a δ-unction) and homogeneous in space with power spectrum ψ ij in the Fourier space. Speciically, we assume: i (k 1,t 1 ) j (k 2,t 2 ) = τ (2π) 3 δ(k 1 +k 2 )δ(t 1 t 2 )ψ ij (k 2 ), (1) or i and j = 1, 2 or 3. The angular brackets stand or an average over realizations o the orcing, and τ is the (short) correlation time o the orcing. Note that the δ correlation is valid as long as the correction time τ is the shortest time-scale in the system [i.e. τ Ω 1,A 1,1/(νk 2 )]. For most results that will be derived later, we assume an incompressible and isotropic orcing where the spectrum o the orcing is given by: ψ ij (k) = F(k)(δ ij k i k j /k 2 ). (11) It is easy to check that in the absence o rotationand shear, this orcing leads to an isotropic turbulence with intensity: v 2 = 2τ (2π) 2 F(k) ν dk, (12) where the subscript stands or a turbulence without shear and rotation. In addition to an isotropic orcing, we will also consider an anisotropic orcing in IIIA2 to examine the combined eect o rotation and anisotropy, which can lead to non-diusive luxes o angular momentum. Speciically, we consider an extremely anisotropic orcing with motion restricted to a plane perpendicular to a given direction g. The motion in this perpendicular plane is however assumed to be isotropic. Such a orcing can be modeled by the ollowing power spectrum [33]: [ ψ ij (k) = G(k) δ ij k ik j k 2 (g k)2 k 2 δ ij g i g j + g k ] k 2 (g ik j +g j k i ). (13) In that case, the turbulence without rotation or shear would have the ollowing properties: (v g) 2 =, (14) (v g) 2 = 2τ G(k) 3(2π) 2 dk. ν 7 The system (4) can be simpliied to: Here: 2 τ III. ANALYTICAL RESULTS [ (γ +τ 2 ] [ h1 (τ) ] )ˆv x + β 2 Ω( Ω 1)ˆvx = τ A Ωβ h 2(τ) A, (15) τˆv z = β γ ] Ω 1 τ[ τvx ˆ +β γ ˆv y = (τvˆ x +βˆv z ). ˆv x + h 2(τ) γa, Ω = Ω/A, β = k z /k y, γ = 1+β 2 = k 2 H/k 2 y (k 2 H = k 2 y +k 2 z), (16) h 1 (τ) = γ ˆ x τ ˆ y βτ ˆ z, h 2 (τ) = ˆ z β ˆ y. To solve the irst o equation (15) which is a non-homogeneous second order dierential equation, we need two boundary conditions. We impose a vanishing initial velocity v(τ ) = which implies ˆv x (τ ) = and τ ˆv x τ=τ = h 1 (τ )/(γ +τ 2 )A. The second boundary condition can be shown to be obtained in the intermediate steps o deriving Eq. (15). The exact solution to (15) is obtained in the appendix A, where we address the stability o the homogeneous solution o system (with = ). Computations o correlation unctions, by using this exact solution, however turns out to be too complex to be analytically tractable. To gain a physical insight into the role o inertial waves and low shear in turbulent transport, we consider the two limits (i) the strong rotation where the eect o waves dominates shearing (Ω A) and (ii) the weak rotation where shearing dominates the eects o waves (Ω A). Approximate solutions can be derived in these two regimes which can then be used or deriving analytic orm o correlation unctions or turbulence intensity and transport.

9 8 A. Rapid rotation limit: Ω A When the rotation rate is much larger than the shearing rate (Ω = Ω /A 1), the oscillation o inertial waves is roughly coherent without being damped over shearing time o A 1. Thereore, these waves can play a dominant role in determining the direction o energy cascade (sign o eddy viscosity) and transport o particles via phase mixing (i.e. by aecting the phase relation). However, as shown below, low shear can still have a non-trivial eect on turbulence by enhanced dissipation so long as it is stronger than molecular dissipation. To characterize the latter, we introduce a parameter ξ = νky/a, 2 the ratio o typical molecular dissipation rate to shearing rate. Here, k y is the characteristic wavenumber o the orcing in the stream-wise direction. We can, or instance, envision the orcing to have a spectrum peaked around this characteristic wave-number k y. In the ollowing, we examine the changes in turbulence characteristics in weak (ξ 1) and strong (ξ 1) shear limits to elucidate the eects o low shear in inertial wave-dominated turbulence. In the rapid rotation limit ( Ω A), the solution o Eq. (15) can be ound by using WKB approximation [5] as: Here, ˆv x (τ) = ˆv y (τ) = ˆv z (τ) = 1 τ A(γ +τ 2 ) 3/4 { } ĥ1 (t) (γ +t 2 ) cos[v(t,τ)]+ĥ2(t)(γ +t 2 ) 1/4 θsin[v(t,τ)] 1/4 dt, τ 1 τ { ĥ 1 (t) ( dt τ cos[v(t,τ)]+βθ ) γ +τ Aγ(γ +τ 2 ) 3/4 τ (γ +t 2 ) 2 sin[v(t,τ)] 1/4 +ĥ2(t)(γ +t 2 ) 1/4( θτ sin[v(t,τ)] β )} γ +τ 2 cos[v(t,τ)], (17) 1 τ { ĥ 1 (t) ( dt βτ cos[v(t,τ)] θ ) γ +τ Aγ(γ +τ 2 ) 3/4 τ (γ +t 2 ) 1/4 2 sin[v(t,τ)] )} +ĥ2(t)(γ +t 2 ) 1/4( θβτ sin[v(t,τ)]+ γ +τ 2 cos[v(t,τ)] Ω = Ω, ω = β Ω, θ = sign(β Ω), (18) ( s(t) = 1 1 ) ( ( ) t 1 arcsinh )+O, 2 Ω γ v(t,τ) = ω [s(t) s(τ)]. In the ollowing subsections, we compute the various correlation unctions by assuming a homogeneous and shortcorrelated orcing [see Eq. (1)]. As the system (15) involves the orcing in terms o ĥ1 and ĥ2 only [see Eq. (16)], it is convenient to use the power spectrum φ ij as: h i (k 1,t 1 ) h j (k 2,t 2 ) = τ (2π) 3 δ(k 1 +k 2 )δ(t 1 t 2 )φ ij (k 2 ), (19) or i and j = 1 or 2. In the case o an isotropic and incompressible orcing [Eq. (11)], φ ij in Eq. (19) can be written: φ 11 (k) = γ(γ +a 2 )F(k),φ 12 (k) =,φ 22 (k) = γf(k). (2) Ω Turbulence intensity We begin by examining the eects o rotation and low shear on turbulence level in wave-dominated turbulence due to strong rotation ( Ω A). The eect o shear will urther be clariied by comparing results in weak shear limit (ξ 1) with those in the strong shear limit (ξ 1). First, turbulence intensity in the shear direction can be obtained by using Eqs (17) and (19) as: v 2 x = τ (2π) 3 A d 3 k + a { dτ e 2ξ[Q(τ) Q(a)] φ11 (k) (γ +τ 2 ) 3/2 γ +a 2 cos2 [v(a,τ)] (21) }. +θφ 12 (k)sin[2v(a,τ)]+φ 22 (k) γ +a 2 sin 2 [v(a,τ)]

10 Here, a = k x /k y, β = k z /k y, γ = 1+β 2, ξ = (νky 2)/A and Q(x) = x3 /3+γx. In the case o an isotropic orcing [Eq. (2)], Eq. (21) and the turbulence intensity in the two other directions can then be derived as: vx 2 τ = (2π) 3 d 3 k γ γ +a A 2 F(k) I (k), (22) vy 2 = τ (2π) 3 d 3 k γ +a A 2 F(k) { β 2 I k)+i 2 (k) }, vz 2 = τ (2π) 3 d 3 k γ +a A 2 F(k) { I (k)+i 2 (k) }. Here: I p (k) = + a τ p e 2ξ[Q(τ) Q(a)] (γ +τ 2 ) 3/2 dτ. (23) In order to understand the eect o shearing on turbulence intensity in this wave-dominated turbulence, we irst examine (22) in the weak shear limit (ξ 1) where the shear is negligible. In this case, the integral I p in Eq. (23) takes the approximate value: I p (k) a p 2ξ(γ +a 2 ) = Aa p. (24) 5/2 2νk 2 (γ +a 2 ) 3/2 By using Eq. (24) in Eq. (22), we can then obtain the ollowing result or the turbulent intensity: vx 2 = τ (2π) 3 vy 2 = τ (2π) 3 v 2 z = τ (2π) 3 d 3 k F(k) 2νk 2 Perorming the integration over the angular variable, we obtain: vx 2 = τ (2π) 3 = 2τ 3(2π) 2 dk F(k) 2ν v 2 y = v2 z = 1 3 v2. dk F(k) ν 2π dφ γ γ +a 2, (25) d 3 k F(k) 2νk 2 β 2 +a 2 γ +a 2, d 3 k F(k) 2νk 2 1+a 2 γ +a 2. π dθsinθ ( cos 2 θ +sin 2 θsin 2 φ ) = 1 3 v2, (26) Here, v 2 is the turbulence amplitude in the absence o rotation and shear [see Eq. (12)]. These results thus show that, in the large rotation limit, the turbulence intensity is isotropic and equals to the one without rotation [see Eq. (12)] or suiciently weak shear with ξ 1. Furthermore, in this limit o a suiciently weak shear where (Ω,D) A, turbulence intensity is independent o rotation since waves do not necessarily quench turbulence level. A similar result was also obtained in MHD turbulence and stratiied turbulence where magnetic ields and gravity waves mainly aect transport without much eect on turbulence level [17, 19, 2]. We shall show below that a strong anisotropy can be induced when shearing eect is not negligible (ξ 1) even in the rapid rotation limit (Ω A). In order to understand the eect o low shear, we now consider the strong shear limit (ξ 1). In this limit, the integral (23) is simpliied as: ( ) I (k) = 1 a 1, (27) γ γ +a 2 I 2 (k) = lnξ 3. 9

11 1 By plugging Eq. (27) in Eq. (22), we obtain: vx 2 = τ (2π) 3 A vy 2 = τ (2π) 3 A vz 2 τ = (2π) 3 A d 3 k γ +a 2 F(k) ξ v 2, (28) d 3 k γ +a 2 F(k) lnξ 3 d 3 k γ +a 2 F(k) lnξ 3 ξ lnξ v 2, ξ lnξ v 2, to leading order in ξ 1. Note that in the calculation o v 2 x, we neglected the component proportional to a = k x /k y as it is odd in both k x and k y and thus vanishes ater integration over the angular variables or an isotropic orcing. The last terms in Eq. (28), expressed in terms o the turbulence amplitude in the absence o rotation and shear v 2 [see Eq. (12)], explicitly show the dependence o turbulence level on rotation and shear. That is, all the components o turbulence intensity is reduced or strong shear ξ 1. Further, the x component along shear is reduced as ξ A 1 while the other two components as ξ lnξ, with an eectively weaker turbulence in the shear direction than in the perpendicular one, by a actor o ln ξ. This shows that shear low can induce anisotropic turbulence (unlike rotation) even when the orcing is isotropic. This result is similar to that obtained in the simulation o a Couette low at high rotation rate [51] where the velocity luctuations perpendicular to the wall exceed that in the stream-wise direction. Nevertheless, Eq. (28) shows that a strong rapid rotation yet insures an isotropy in velocity luctuations in y z directions ( v 2 y = v2 z ). 2. Transport o angular momentum As noted in the Introduction, rotation tends to cascade energy to large scales while shear low to small scales. Would thus the inverse cascade be a robust eature or rapid rotation (Ω A)? I yes, what would the eect o low shear? Would there be a non-diusive momentum transport? We answer these questions by irst considering an isotropic orcing and then anisotropic orcing. The eect o shear will be elucidated by looking at the two limits o weak shear (ξ 1) and strong shear (ξ 1), as done in First, in the case o an isotropic orcing, we obtain the ollowing Reynolds stress rom equations (17) and (19): v x v y = τ (2π) 3 A d 3 k γ +a 2 F(k) I 1 (k), (29) where I 1 was deined in Eq. (23). Eq. (29) is computed in the weak and strong shear limits, below. First, in the weak shear limit (ξ 1), there is no contribution to leading order in Ω 1 as the unction I 1 is odd in a and thus vanishes ater integration over the wave vector. We thus include one higher order in Ω 1 in the expansion and obtain the ollowing result: v x v y = τ (2π) 3 A d 3 k af(k) 2ω J(k). (3) Here, we deined a unction J(k), which has the ollowing asymptotic behavior in the weak shear limit: + τe 2ξ[Q(τ) Q(a)] J(k) = sin[2ω a (γ +τ 2 ) 3/2 {s(a) s(τ)}]dτ (31) aω A 2(γ +a 2 ) 3/2 [ν 2 k 4 +ω 2 ], where ω = ω A/ γ +a 2. Plugging Eq. (31) in Eq. (3) and perorming the integration over the azimuthal angle variable φ, we obtain: v x v y = τ π A 32(3π) 2 dk k 2 F(k) dθsin 5 1 θ ν 2 k 4 2 +ω. (32) Finally, we change the integration variable rom θ to ω = Ωcosθ, obtaining the ollowing ormula: v x v y = τ A 16(2π) 2 Ω + dk k 2 F(k) Ω dω ( 1 ω 2 /Ω 2) 2 ν 2 k 4 +ω 2. (33)

12 Thereore, in the large rotation and weak shear limit, the Reynolds stress becomes purely diusive (with no Λ-eect) with the turbulent viscosity: ν T πτ 32(2π) 2 Ω + dk F(k) ν 11. (34) This result shows that the turbulent viscosity is positive and proportional to Ω 1 or large Ω. It is worth comparing Eq. (34) with Eq. (22) in [31]. To this end, we use Eq. (12), which gives the turbulence amplitude without rotation (the original turbulence o Kichatinov) in Eq. (34) to obtain the turbulent viscosity ν T π v 2 /64 Ω. Thus ν T in Eq. (34) is the same as Eq. (22) in [31] or Ω 1 and θ = π/2, but has an opposite sign. This is due to the τ-approximation used by Kichatinov which gave an unphysical result. Later, [52] showed that the viscosity is also positive at any rotation rate when derived consistently with quasi-linear approximation in the weak shear limit. In comparison, in the strong shear limit (ξ 1), the unction I 1 in Eq. (23) has the ollowing asymptotic behavior: I 1 (k) = 1 γ +a 2. (35) Plugging Eq. (35) in Eq. (29), we obtain the turbulent viscosity in the strong shear limit as: ν T = v xv y A = τ (2π) 3 A 2 d 3 k F(k). (36) Eq. (36) shows that the turbulent viscosity is negative (as F(k) > ) in the strong shear limit, in sharp contrast to the weak shear limit where ν T > [see Eq. (34)]. Furthermore, the magnitude o ν T is reduced by the shear ( A 2 ) and is independent o rotation, which should also be compared with the weak shear limit [see Eq. (34) where ν T Ω 1 ]. Thereore, the turbulent viscosity changes rom positive (or weak shear) to negative (or large shear) as the ratio o shear to dissipation increases. This result can be understood i we assume that, as in most rapidly rotating luid, the inverse cascade is associated with the conservation o a potential vorticity [53]. In the presence o strong shear (compared to dissipation), the potential vorticity is strictly conserved giving rise to an inverse cascade (negative viscosity). When the dissipation increases, the potential vorticity is less and less conserved and thus the inverse cascade is quenched. Our results show that there is a transition rom inverse to direct cascade as the dissipation is increased. A similar behavior is also ound in two-dimensional hydrodynamics (HD) where an inverse cascade can be shown to be present only or suicient weak dissipation [16]. It is important to note that the negative viscosity ν T < obtained here or strong rotation/strong shear (Ω A νky) 2 signiies the ampliication o shear low as the eect o rotation avoring inverse cascade dominates shearing (generating small scales). However, the magnitude o ν T is reduced by shear as ν T A 2 since low shear inhibits the inverse cascade. This can be viewed as sel-regulation that is, sel-ampliication o shear low is slowed down as the latter becomes stronger. The preceding results [Eqs. (34) and (36)] indicate that in the large rotation limit where rotation dominates over shear, the momentum transport is purely diusive or isotropic orcing, with opposite sign o turbulent viscosity or weak (ξ 1) and strong shear (ξ 1) or a ixed value o Ω /A ( 1). In the case o anisotropic orcing, there is however a possibility o the appearance o non-diusive momentum transport (Λ-eect). To examine this possibility, we now consider an extremely anisotropic orcing (introduced in II C) where the orcing is restricted to horizontal plane (y-z), perpendicular to the direction o the shear. Using Eq. (13) with g ij = δ i1, we obtain the ollowing Reynolds stress: v x v y = τ (2π) 3 A Here, I 1 was deined previously in Eq. (23) and: J (k) = K(k) = d 3 k + τe 2ξ[Q(τ) Q(a)] a + a γg(k) 2 γ +a 2 [{ I 1 (k) J (k) } +βθk(k) ]. (37) (γ +τ 2 ) 3/2 cos[2ω {s(a) s(τ)}]dτ, (38) e 2ξ[Q(τ) Q(a)] (γ +τ 2 ) sin[2ω {s(a) s(τ)}]dτ. We again consider the weak and strong shear limits in the ollowing. First, in the weak shear limit (ξ 1), Eq. (37) is simpliied to: v x v y = τ (2π) 3 d 3 γg(k)βθ ω k A 4(γ +a 2 ) 3/2 ν 2 k 4 2 +ω. (39)

13 Perorming the angular integration in Eq. (39) and taking the large rotation limit, we obtain the ollowing: v x v y = τ 3(2π) 3 ΩA d 3 k G(k) ν 12. (4) Equation (4) is odd in the rotation and thus represents the Λ-eect. Again, the latter avors the creation o velocity gradient rather than smoothing it out and can thus provide a mechanism or the occurrence o dierential rotation (e.g., in the sun). By using Eq. (14), one can see that the Λ-eect is proportional to the anisotropy in the turbulence without shear and rotation. This result shows that, in the large rotation limit, one needs anisotropic orcing to generate non-diusive luxes o angular momentum [as in the case without shear as shown 31]. This should be contrasted to the case o weak rotation (see IIIB) where the shear can alone give rise to an anisotropic turbulence, thereby leading to a Λ-eect even with an isotropic orcing. Finally, in the opposite, strong shear limit (ξ 1), Eq. (37) becomes: v x v y = τ (2π) 3 A d 3 k γg(k) 2(γ +a 2 ), (41) which is even in the rotation. Thus, the turbulent viscosity ν T is obviously negative. Thus, in the large shear limit (but still negligible compared to the rotation), anisotropic orcing does not induce any non-diusive luxes but just increases the magnitude o the negative turbulent viscosity. 3. Transport o particles In the large rotation limit ( Ω /A 1), inertial waves might play a crucial role in transport o particle as waves can alter the phase relation between particle density and velocity, as noted previously. How does this eect appear in orced turbulence? What is the eect o shear low on particle transport dominated by waves? These questions are answered in this subsection. In the rapid rotation limit ( Ω /A 1), turbulent particle diusivities can be obtained ater a long, straightorward analysis (see Appendix B or details about the algebra) as: D xx T D yy T = Dzz T τ 8π Ω τ 16π Ω F(k) dk, (42) ν F(k) dk 1 ν 2 Dxx T. Note that in that case, the result is not sensitive to the value o the parameter ξ and thus we do not distinguish between the weak and large shear limits. Eq. (42) shows that DT xx, Dyy T and Dzz T are all reduced as Ω 1 (with no eect o the shear) or large Ω and also that there is only a slight anisotropy in the transport o scalar: the transport in the direction o the rotation is twice larger than the one in the perpendicular direction [34]. Interestingly, this anisotropy in the transport o particles is not present in turbulence intensity [see Eq. (26)]. This is because waves can aect the phase between density luctuation and velocity, not necessarily altering their amplitude. However, it is important to note that this anisotropy is only a actor o 2, much weaker than that in sheared turbulence without rotation [3]. To summarize, in this subsection 3.1, we have examined how a shear low can aect the turbulent property when turbulence is largely dominated by inertial waves in rapid rotation limit ( Ω /A 1). In particular, the results show: 1. that shear low reduces turbulence level with a strong anisotropy [Eq. (28)], leading to an eectively weaker turbulence in the direction o the shear [which would otherwise be almost isotropic [Eq. (25)]; 2. that in comparison, transport o particles is mainly governed by waves with almost isotropic property (within a actor o 2) and quenched as Ω 1 as rotation rate Ω increases; 3. that energy cascade is inverse with negative viscosity or strong rotation/shear limit (Ω A νk 2 y) while its rate is slowed down by strong shear; 4. that momentum transport is purely diusive or isotropic orcing, with non diusive transport appearing only or anisotropic orcing.

14 13 B. Weak rotation limit: Ω A When Ω A, low shear can distort inertial waves over the period o their oscillation, dramatically weakening the eects o these waves on turbulence. Thereore, shear may take a dominant role in determining turbulence property (studied in [3]) while rotation modiies some o the properties o this shear-dominated turbulence. The investigation o this limit would thus permit us to clariy the eects o rotation as well as low shear, thereby complementing the analysis done in Sec. IIIA or strong rotation (Ω A). O particular interest is (1) to what extent the quenching and anisotropy o sheared turbulence [3] are aected by rotation, which avors isotropic turbulence; (2) how the direction o the energy cascade, which tends to be direct in 3D sheared turbulence, is aected by rotation (which preers inverse cascade); (3) whether momentum transport can occur via non-diusive luxes. To answer these questions, we expand various physical quantities in powers o Ω = Ω /A as: X(τ) = X (τ)+ω X 1 (τ)+..., (43) in the weak rotation limit (Ω A) and calculate the turbulence intensity and transport up to irst order in Ω. For the sake o brevity, we here just provide the inal results o the calculation. Note that in this limit, we are only interested in strong shear case (ξ 1) since in the opposite limit where νky 2 A Ω, the eects o both shear and rotation simply disappear to leading order. 1. Turbulence intensity By using the expansion in powers o Ω (43) and Eq. (19) and ater a long, but straightorward algebra, we can obtain the turbulence intensity in the shear direction as ollows: vx 2 = τ (2π) 3 d 3 kφ 11 (k) [ L (k)+β 2 ΩL1 (k) ]. (44) A Here: L (k) = L 1 (k) = + dτ e 2ξ[Q(τ) Q(a)] a + dτ e 2ξ[Q(τ) Q(a)] a (γ +τ 2 ) 2 T (x) = 1 γ arctan (γ +τ 2 ) 2 dτ, (45) [τ{t (τ) T (a)} 12 ( )] γ +τ 2 ln dτ, ( x γ ). γ +a 2 In the strong shear limit (ξ 1), the integrals L and L 1 in Eq. (45) can be simpliied: L (k) L 1 (k) = + a + a + a 1 (γ +τ 2 ) 2dτ = 1 2γ [ ] π 2 a T (a) γ γ +a 2, (46) [τ{t (τ) T (a)} 12 ( )] γ +τ 2 ln dτ 1 (γ +τ 2 ) 2 γ +a 2 [ τ 2γ(γ +τ 2 ) + 1 ] 2γ T (τ) {T (τ) T (a)}dτ. Note that the second ormula or L 1 in Eq. (46) was obtained by integration by part. The leading order behavior o Eq. (44) coming rom the term involving L is due to shearing eect, showing that v 2 x is quenched by low shear A 1 (see [3]). The eect o rotation appears as a correction proportional to L 1. One can see rom Eq. (46) that this correction L 1 is positive or all values o a (or a <, the negative part o the integral is always smaller than the positive one as the irst term is odd in τ and the second one is an increasing unction o a). Thereore, the turbulence intensity v 2 x in Eq. (44) increases or Ω > whereas it decreases or Ω <. This can physically be understood rom the linear instability analysis (perormed in appendix): that is, instability ( Ω > ) increases turbulence level while stability ( Ω < ) reduces it.

15 The other components o the turbulence amplitude can be obtained by ollowing similar analysis in the strong shear limit (ξ 1) as ollows: [ ( ) 2 vy 2 τ π (2π) 3 d 3 k β 2 A 2 β T (a) φ 11(k)+φ 22(k)] 2 ( ) 1/3 3 γ 3γ 2, 2ξ v 2 z [ Γ(1/3)+ Ωβ 2 Γ(4/3)( lnξ) ] (47) [ ( ) 2 ( ) 1/3 τ π (2π) 3 d 3 k β 2 A T (a) φ 11(k)+φ 22(k)] γ 3γ 2 2ξ [ Γ(1/3)+ Ωβ 2 Γ(4/3)( lnξ) ]. 14 Here, Γ is the Gamma unction. The irst terms in Eq. (47) represent the turbulence amplitude in the direction perpendicular to shear without rotation [3], which are reduced as A 2/3 or strong shear. Compared to the leading order behavior o vx 2 A 1 in shear direction, the reduction is weaker by a actor o ξ 1/3. That is, a strong anisotropy in turbulence level can be induced or strong shear. The second terms in Eq. (47) capture the eect o weak rotation on sheared turbulence, with turbulence amplitude again being increased or decreased depending on the sign o Ω. Furthermore, the correction comes with a multiplying actor lnξ > 1, which is larger compared to that or the amplitude in the shear (x) direction (which is independent o shear [Eq. (44)]). Thereore, in the stable situation ( Ω < ) o our interest, weak rotation has the eect o reducing turbulence in the y z plane more than the one in the shear direction. As a result, the anisotropy induced by low shear is weakened by rotation. Interestingly, this illustrates the tendency o rotation o leading to almost isotropic turbulence. It is also interesting to note that the leading order terms in vy 2 and vz, 2 although apparently very similar, are not exactly the same. For instance, in the case o an isotropic orcing, the angular integration gives vy 2 > v2 z. This slight anisotropy in y z (stream and span-wise) directions in sheared turbulence was also observed in numerical simulations o homogeneous turbulence subject to high shear rate: the luctuating velocity in the direction o the low is larger than the one in the direction o the shear [8]. This can be contrasted to the exact equipartition between vy 2 and vz 2 [see (28)] in the case o rapid rotation. This is another maniestation o the dierence between shear low and rotation in inducing anisotropic turbulence. In summary, in the case o a weak rotation/strong shear turbulence (A Ω and A νky 2 ), the rotation tends to reduce the anisotropy in sheared turbulence. 2. Transport o angular momentum As noted previously, a strong anisotropy in turbulence is caused by strong shear in the weak rotation limit. There is thus a possibility that this anisotropic turbulence gives rise to non-trivial non-diusive momentum transport. This will be shown to be the case below. In the strong shear limit (ξ 1), momentum lux can be derived as: v x v y τ (2π) 3 A { d 3 φ11 (k) k γ + β2 Ω 3γ ( lnξ) [ [ 1 2(γ +a 2 ) +β2 ( π β 2 2 T (a) γ ( ) ] 2 π 2 T (a) γ ] }. ) 2 φ 11(k)+φ 22(k) The momentum lux in (48) consists o a diusive part (the irst hal term in the integrand on the RHS) and a non-diusive part (the second hal term in the integrand on the RHS). First, the diusive part, independent o Ω, recovers the eddy viscosity o sheared turbulence without rotation [3], showing that its value decreases as A 2 or strong shear. This result agrees with previous studies o non-rotating sheared turbulence [1] which ound a Reynolds stress inversely proportional to the shear, leading to a log dependence on the distance to the wall or the large-scale shear low. Second, the non-diusive part, the correction due to the rotation, is proportional to Ω and is odd in the rotation. This is a non-diusive contribution to Reynolds stress the so-called Λ-eect. The origin o this non trivial Λ-eect is the strong anisotropy induced by shear low on the turbulence even when the driving orce is isotropic. It is important to contrast this to the case o rapid rotation limit where non-diusive luxes emerge only or anisotropic orcing. A similar result was also ound in IIIA2 [see Eqs. (34) and (36)]. This Λ-eect [the second term in Eq. (48)] is obviously o the same sign as Ω whereas the turbulent viscosity [the irst term in Eq. (48)] can either be positive (48)