LETTERS PUBLISHED ONLINE: XX MONTH XXXX | DOI: 10.1038/NPHYS1910 Upscale energy transfer in thick turbulent fluid layers H. Xia 1 , D. Byrne 1 , G. Falkovich 2 and M. Shats 1 * Flows in natural fluid layers are often forced simultaneo usly at 1 scales smaller and much larger than the depth. For example, 2 the Eart h’s atmosp heri c flows are pow ered by gra dien ts of3 solar heating: vertical gradients cause three-dimensional (3D) 4 convect ion whereas horizontal gradients drive planetary scale 5 flows. Nonlinear interactions spread energy over scales 1,2 . The 6 question is whether intermediate scales obtain their energy 7 from a lar ge- sca le 2D flo w or fro ma small -sc ale 3D tur bul ence. 8 The paradox is that 2D flows do not transfer energy downscale 9 whereas 3D turbulence does not support an upscale transfer. 10 Here we demonstr ate experimentally how a large-scal e vortex 11 and small-scale turbulence conspire to provide for an upscale 12 energy cascade in thick layers. We show that a strong planar 13 vortex suppresses vert ical moti ons, thus faci lita ting an upscale 14 energy cascade. In a bounded system, spectral condensation 15 into a box -size vortex provides for a self-organi zed planar flow 16 that secur es an upsc ale ener gy tran sfer . 17 Turbulencein thin layers is quasi-two-dimensio nal and supports 18 an inv ers e ene rg y cas cad e 3 , as ha s be en conf ir me d in ma ny19 exper iments in elect rolyt es 4–6 andsoap fil ms 7,8 . In bou nded sys tems 20 the inv ers e cascade may lea d to a spectral conde nsa tion, tha t 21 is, the formation of a flow coherent over the entire domain 4–6 . 22 One expects that in thick layers the flow is 3D and there is no 23 inverse energy cascade. Indeed, as has been demonstrated in 3D 24 nume rical modellin g, when the layer thic knes s, h, exceeds half25 the forc ing scale, lf, h/lf> 0.5, the ons et of vertical mot ion s 26 destroys the quasi-two-dimensionality of the turbulence and stops 27 the upscale energy transfer 9,10 . 28 In this Letter we report new laboratory studies of turbulence 29 in layers which show that a large-scale horizontal vortex, either 30 imp osed ext ern all y or gen erated by spe ctral con de nsa tio n in 31 tur bulen ce, sup pre sse s ve rti cal motio ns in thi ck lay ers. Thi s lea ds to 32 a rob ustinve rseenerg y cas cad e eve n inthi ck lay erswith h/lf> 0.5. 33 In our exp eri ments, tur bul enc e is ge ner ate d by the int era cti on of34 a large number of electromagnetically driven vortices 11–13 . The d.c. 35 electric current flowing through a conducting fluid layer interacts 36 with the spat ially variab le vert ical magneti c field . The field is 37 produced by an array of 900 magnets placed beneath the fluid 38 cell, the size of which is 0 .3 × 0.3 m 2 . The flow is visualized using 39 seeding particles, which are suspended in the fluid, illuminated 40 using a horizontal laser slab, and filmed from above. Particle image 41 velocimetry (PIV) is used to derive the turbulent velocity fields. 42 To visualize the vertical flows, a vertical laser slab is used and the 43 particle motion is filmed from the side. To quantify the velocity44 fluctuations in 3D, a defocusing PIV technique has been developed 45 tha t allowsall three vel oci ty compon ent s to be me asured 14 . 46 We use two dif fer ent confi gur ati ons: (1) a sin gle lay er of47 ele ctr oly te on a sol id botto m, and (2) a lay er of ele ctr oly te 48 on top of ano the r lay er of a non-co ndu cti ng hea vie r liq uid , 49 1 Research School of Physics and Engineering, The Australian National University, Canberra ACT 0200, Australia, 2 Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel. *e-mail: [email protected]. whic h subs tanti ally decr eases friction . For a thin doub le-la yer 50 configuration in the presence of a boundary, a 2D inverse cascade 51 was shown to form a spec tral conden sate, or the box-siz e vort ex, 52 which dominated the flow in a steady state 11–13 . The statistics of53 the increment δVL across a distance rof the velocity component 54 para llel to rwere meas ured using hund reds of instantaneous 55 velocity fields 11–13 . The second moment ofδVL determines the 56 spectrum, whereas the third moment, S 3L = (δVL ) 3 , is related in 57 isotropic turbulence to the value and the direction of the energy58 fluxε across the scale r: 59 ε = −(2/3) S 3L /r(1) 60 The sign ofS 3 thus determines the flux direction: ε is positive in 61 3D (direct cascade) and negative in 2D (inverse cascade). Indeed, 62 positive linear S 3 (r) has been observed in quasi-2D turbulence 63 in soap films 8 and in electrolyte layers 11–13 . For turbulence with 64 the conde nsa te, it is crucial to subtra ct a coherent flo w from 65 the instantaneous velocity fields to recover the correct statistics 66 of the turbulent velocity fluctuations 13,15 . The point is that the 67 velocity differences δVcontain contributions from the spatially68 inhomogeneous vortex flowδ ¯ V, in addition to the contributions 69 from the turbulent velocity fluctuations δ ˜V: δV= δ ˜V+ δ ¯ V. The 70 vort ex cont ribu tionthen ente rs the high er-or der moments: δV2 = 71 δ ˜V2 + δ ¯ V2 , δV3 ≈ δ ˜V3 + δ ¯ V3 + 3δ ¯ Vδ ˜V2 , and so on. It 72 has been demonstrated that the presence of the large-scale flows 73 substantially modifies S 3 andeven aff ect s its sig n 12,13 . 74 We now present the new results, starting with a thick two-layer 75 conf igur ation when the top laye r thick ness h t exce eds half the 76 forcing scale lf. The forcing scale in this experiment is lf= 9mm, 77 and the thicknesses of the bottom and top layers are h b = 4 mm 78 and h t = 7 mm respe cti vel y. The size of the boun dar y box is 79 L = 120mm. Visualization in both horizo ntal ( x– y) and vertical 80 (z– y) pla nesshows tha t theflow is sub sta nti all y 3D at the ear ly sta ge 81 of the evolution, as seen in Fig. 1a,b, in agreement with numerics. 82 However, if one waits long enough, one sees the appearance of a 83 large planar vortex, the diameter of which is limited by the box84 size (numerical simulations 10 were done on a shorter timescale). 85 Apparently, a residual inverse energy flux, existing even in the 86 presence of 3D motions, leads to the spectral condensation and to 87 the generation of the large coherent vortex which dominates the 88 flow in a steady state ( Fig. 1c). In the vertical cross-section, the 89 flow is now close to planar ( Fig. 1d). The vortex gives a strong 90 spectral peak at k ≈ k c , as seen in Fig. 1e, where k fis the forcing 91 wavenumber. The subtraction of the temporal mean reveals the 92 spectrum of the underlying turbulence, see Fig. 1g, which shows a 93 reaso nabl y good agre emen t with the Kraic hnan k −5/3 inve rse energy94 cascade. Simi larly , befo re the mean subt racti on, S 3 is nega tive 95 NATURE PHYSICS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturephysics 1
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LETTERSPUBLISHED ONLINE: XX MONTH XXXX | DOI: 10.1038/NPHYS1910
Upscale energy transfer in thick turbulent
fluid layersH. Xia1, D. Byrne1, G. Falkovich2 and M. Shats1*
Flows in natural fluid layers are often forced simultaneously at1
scales smaller and much larger than the depth. For example,2
the Earth’s atmospheric flows are powered by gradients of 3
solar heating: vertical gradients cause three-dimensional (3D)4
Figure 1 | Structure of turbulence in the double layer configuration.
Turbulence is forced electromagnetically by the array of magnetic dipoles,
lf =9 mm, in the top layer of electrolyte (ht =7 mm) resting on the 4 mm
layer of non-conducting heavier liquid (Fluorinert). Particle streaks are
filmed in the horizontal x–y plane with an exposure time of 2 s (a,c) and in
the vertical z–y plane with an exposure time of 1 s (b,d) during the
development of the large scale vortex. Images (a,b) correspond to the time
interval centred around t=5 s after the flow forcing is switched on; (c,d)
correspond to t=20 s. The size of the horizontal plane box is 120 mm. The
spectrum of the steady flow (e) and the third-order structure function S3L
(f ) are dominated by the contribution from the large coherent vortex (c),
which develops as a result of turbulence spectral condensation. Subtraction
of the mean velocity field reveals the spectrum (g) and S3L (h), which are
similar to those in quasi-2D turbulence.
and is not a linear function of r (Fig. 1f). The subtraction of the1
mean vortex flow reveals positive S3, which is a linear function2
of r , in agreement with equation (1) (Fig. 1h). The slope of that3
linear dependence gives the value of the upscale energy flux,4
ε ≈ 1.8 × 10−6 m2 s−3. This value was independently verified by 5
the energy balance analysis, as described in ref 13: ε is compared6
with the large-scale flow energy dissipation rate εd. For the present7
experiment this agreement is within 10%.8
The results of Fig. 1 indicate that, contrary to expectations,9
turbulence in a thick layer (ht/l f ≈0.78) supports the inverse energy 10
cascade, as evidenced by the S3 result and also by the generation of 11
the strong spectral condensate fed by the cascade. Similar resultsare obtained in even thicker layers, up to ht = 12 mm, or h/l f ≈ 1.3.Strong spectral condensates, the energy of which constitutes above90%of thetotalflow energy, are observed in those cases.
After the development of the condensate, particle streaks arealmost planar, at least in the statistical sense (Fig. 1d). Thus,there must be a mechanism through which the vortex securesits energy supply by suppressing vertical motions and enforcingplanarity. To investigate the effect of a large flow on 3D turbulent
motion, we perform experiments in a single layer of electrolyte,h = 10 mm, with a much larger boundary, L = 300 mm, to avoidspectral condensation. In this case the large-scale vortex is imposedexternally. The timeline of the experiment is shown schematically in Fig. 2a. First, turbulence is excited. Then a large vortex (150 mmdiameter) is imposed on it by placing a magnetic dipole 2 mmabove the free surface. After the large magnet is removed, thevortex decays, while turbulence continues to be forced. As themagnet blocks the view, the measurements are performed duringtheturbulence stage andduring thedecayof thevortex.
Figure 2b shows the profiles V z rms(z ) of the vertical velocity fluctuations, measured using defocusing PIV. Before the vortex is imposed, vertical fluctuations are high, V z rms ≈ 1.6m m s−1.When the vortex is present, they are reduced by a factor of about
four, down to V z rms ≈ 0.4m m s−1. This is also confirmed by thedirect visualization of the flow in the vertical z – y plane. Particlestreaks are shown in Fig. 2c–e. Strong vertical eddies are seen inthe turbulence stage, before the vortex is imposed (Fig. 2c). Shortly after the large magnet is removed, at t –t 0 = 1 s, the streaks show novertical excursions. As the large vortex decays, the 3D eddies start toreappear near the bottom (Fig. 2e).
We also study the statistics of the horizontal velocities. Thekinetic energy spectra and the third-order moments are shownin Fig. 2f –h. For turbulence without the vortex, the spectrum issubstantially flatter than k−5/3 (Fig. 2f). After the large vortex isimposed, the spectrum shows a strong peak at low wavenumbers.However, the mean subtraction recovers the k−5/3 2D spectrum, asshown in Fig. 2g. The third-order moment undergoes an even more
pronounced change after the imposition of the large vortex: S3Lcomputedafterthe mean subtraction is much largerthan during theturbulence stage, andit is a positive linear function of r , Fig. 2h, justas in thecase of thedouble-layer experiment. Thus theimposedflow enforcesplanarityand strongly enhancesthe inverse energy flux.
The strong suppression of vertical eddies in the presence of animposed flow must be due to the vertical shear LS = dV h/dz ,which destroys vertical eddies for which the inverse turnover timeis less than LS. This shear is obtained from the z -profiles of thehorizontal velocity measured using defocusing PIV. In a single-layerexperiment, the averaged shear of the horizontal velocity due to thepresence of the strong imposed vortex is LS ≈1.6 s−1. Such a shearis sufficient to suppress vertical eddies with inverse turnover timesτ −1 ∼V z rms/h≈ 0.16 s−1. In the spectrally condensed turbulence,
an inverse energy cascade is sustained in layers that are substantially thicker than are possible in unbounded turbulence. A possiblereason for this is also the shear suppression of the 3D vortices. Herethe vertical shear is lower, LS ≈ 0.5 s−1, yet it substantially exceedsthe inverse turnover time for the force-scale vortices in the double-layer experiment, τ −1 ∼V z rms/h≈ 0.06 s−1. One might think thatthe pronounced change in the flow field in the presence of thelarge vortex is due to the global fluid rotation. However, the Rossby numbers in the reported experiments are larger (especially in thedouble layer experiment, Ro>3) than those at which quasi-2D flow properties were observed in rotating tanks(Ro<0.4; ref. 16).
The suppression of vertical motions by the shear flow and theonset of the inverse cascade observed in this experiment may berelevant for many natural and engineering applications. In the solartachocline17, a thin layer between the radiative interior and the
Figure 2 | Effects of externally imposed large-scale flow on turbulence in a single fluid layer (thickness h= 10 mm, forcing scale l f =8 mm). The size of
the horizontal plane box is 300 mm. a, The time line of the experiment: turbulence forcing is switched on, then, 100 s later, a large scale vortex (150 mm
diameter) is imposed on top of the turbulence. The vortex forcing is then removed at t0. The large vortex decays, while the turbulence is still forced.
Measurements using the defocusing PIV technique show that vertical velocity fluctuations are substantially reduced by the large-scale vortex ( b). Error
bars correspond to the 10% instrumental accuracy of the technique. Streaks of seeding particles filmed in the vertical ( z–y ) cross-section (c) show strong
3D motion before the large-scale vortex is imposed (exposure time is 2 s). Particle streaks filmed during the decay of the large-scale vortex (exposure time
is 1 s) show that vertical motions are suppressed (d). As the vortex decays further, 3D eddies re-emerge from the bottom of the cell ( e). The spectrum of
horizontal velocities is flat in turbulence (f ). During the decay of the large vortex the spectrum is dominated by the low-k spectral feature (open diamonds
in g). The subtraction of the mean flow recovers the k −5/3 cascade range (open circles in g). The third-order structure function S3L computed after the
mean subtraction is a positive linear function of the separation distance (open circles in h). S3L in the turbulence stage is much smaller (open squares in h).
outer convective zone, turbulence is expected to be 2D, despite1
being excited by radial convection17. Turbulence suppression by 2
the shear in the tachocline has been considered theoretically 18.3
Present results indicate that only one velocity component may be4
strongly suppressed, making the turbulence 2D. Another interesting5
example is the wavenumber spectrum of winds in the Earth6
atmosphere measured near the tropopause2, which shows E (k) ∼7
k−5/3 in the mesoscale range (10–500 km) and a strong peak at8
the planetary scale of 104 km. Numerous hypotheses have been9
proposed to explain the mesoscale spectrum, with most arguments10
centred on the direct versus the inverse energy cascade2,19–23. The11
shape of the spectrum alone cannot resolve this issue because both12
the 3D Kolmogorov direct cascade and the 2D Kraichnan inverse13
cascade predict E (k) ∼ k−5/3. Direct processing of atmospheric14
data gave S3L(r ) < 0 for some range of r in the mesoscales, thus 15
favouring the direct cascade hypothesis24. However, the subtraction 16
of the mean flows, necessary for the correct flux evaluation, has 17
not been done for the wind data. This leaves the question about 18
the source of the mesoscale energy unresolved. Estimates of the 19
vertical shear due to the planetary scale flow 2, represented by the 20
spectral peak at 104 km, show that the shear suppression criterion 2
can be satisfied for small-scale eddies with sizes less than 10 km. 22
Thus, it is possible that thesuppression of 3D verticaleddies induces 23
an inverse energy cascade through the mesoscales in the Earth 24
8. Belmonte, A. et al . Velocity fluctuations in a turbulent soap film: The third21
moment in two dimensions. Phys. Fluids 11, 1196–1200 (1999).22
9. Smith, L. M., Chasnov, J. R. & Fabian, W. Crossover from two- to23
three-dimensional turbulence. Phys. Rev. Lett. 77, 2467–2470 (1996).24
10. Celani, A., Musacchio, S. & Vincenzi, D. Turbulence in more than two and less25
than three dimensions. Phys. Rev. Lett. 104, 184506 (2010).26
11. Shats, M. G., Xia, H., Punzmann, H. & Falkovich, G. Suppression of turbulence27
by self-generated and imposed mean flows. Phys. Rev. Lett. 99, 164502 (2007).28
12. Xia, H., Punzmann, H., Falkovich, G. & Shats, M. Turbulence-condensate29
interaction in two dimensions. Phys. Rev. Lett. 101, 194504 (2008).30
13. Xia, H., Shats, M. & Falkovich, G. Spectrally condensed turbulence in thin31
layers. Phys. Fluids 21, 125101 (2009).32
14. Willert, C. E. & Gharib, M. Three-dimensional particle imaging with a single33
camera. Exp. Fluids 12, 353–358 (1992).34
15. Chertkov, M., Connaughton, C., Kolokolov, I. & Lebedev, V. Dynamics of energy condensation in two-dimensional turbulence. Phys. Rev. Lett. 99,
084501 (2007).16. Hopfinger, E. J., Browand, F. K. & Gagne, Y. Turbulence and waves in a
rotating tank. J. Fluid Mech. 125, 505–534 (1982).17. Spiegel, E. A. & Zahn, J-P. The solar tachocline. Astron. Astrophys. 265,
106–114 (1992).18. Kim, E-J. Self-consistent theory of turbulent transport in the solar tachocline.
Astron. Astrophys. 441, 763–772 (2005).19. Frisch, U. Turbulence: The Legacy of A.N. Kolmogorov
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AcknowledgementsThis work was supported by the Australian Research Council’s Discovery Projects
funding scheme (DP0881544) and by the Minerva Foundation and the Israeli Science
Foundation.
Author contributionsH.X., D.B. and M.S. designed and performed experiments; H.X. and D.B. analysed the
data. M.S. and G.F. wrote the paper. All authors discussed and edited the manuscript.
Additional informationThe authors declare no competing financial interests. Reprints and permissions
information is available online at http://npg.nature.com/reprintsandpermissions.
Correspondence and requests for materials should be addressed to M.S.
Query 1: Line no. 1Please note that the first paragraph has been editedaccording to style.
Query 2: Line no. 59 Please check S3L =(δV L)3 and ε=−(2/3)S3L/r is
ok as given as it is not clear in pdf.Query 3: Line no. 62S3L and S3 used inconsistently please check. If thereis no difference in meaning then changing to S3L
throughout would avoid the neeed to change thefigures.