Abstract
Flows in natural fluid layers are often forced simultaneously at scales smaller and much larger than the depth. For example, the Earth’s atmospheric flows are powered by gradients of solar heating: vertical gradients cause threedimensional (3D) convection whereas horizontal gradients drive planetary scale flows. Nonlinear interactions spread energy over scales^{1,2}. The question is whether intermediate scales obtain their energy from a largescale 2D flow or from a smallscale 3D turbulence. The paradox is that 2D flows do not transfer energy downscale whereas 3D turbulence does not support an upscale transfer. Here we demonstrate experimentally how a largescale vortex and smallscale turbulence conspire to provide for an upscale energy cascade in thick layers. We show that a strong planar vortex suppresses vertical motions, thus facilitating an upscale energy cascade. In a bounded system, spectral condensation into a boxsize vortex provides for a selforganized planar flow that secures an upscale energy transfer.
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Main
Turbulence in thin layers is quasitwodimensional and supports an inverse energy cascade^{3}, as has been confirmed in many experiments in electrolytes^{4,5,6} and soap films^{7,8}. In bounded systems the inverse cascade may lead to a spectral condensation, that is, the formation of a flow coherent over the entire domain^{4,5,6}. One expects that in thick layers the flow is 3D and there is no inverse energy cascade. Indeed, as has been demonstrated in 3D numerical modelling, when the layer thickness, h, exceeds half the forcing scale, l_{f}, h/l_{f}>0.5, the onset of vertical motions destroys the quasitwodimensionality of the turbulence and stops the upscale energy transfer^{9,10}.
In this Letter we report new laboratory studies of turbulence in layers that show that a largescale horizontal vortex, either imposed externally or generated by spectral condensation in turbulence, suppresses vertical motions in thick layers. This leads to a robust inverse energy cascade even in thick layers with h/l_{f}>0.5.
In our experiments, turbulence is generated by the interaction of a large number of electromagnetically driven vortices^{11,12,13}. The d.c. electric current flowing through a conducting fluid layer interacts with the spatially variable vertical magnetic field. The field is produced by an array of 900 magnets placed beneath the fluid cell, the size of which is 0.3×0.3 m^{2}. The flow is visualized using seeding particles, which are suspended in the fluid, illuminated using a horizontal laser slab, and filmed from above. Particle image velocimetry (PIV) is used to derive the turbulent velocity fields. To visualize the vertical flows, a vertical laser slab is used and the particle motion is filmed from the side. To quantify the velocity fluctuations in 3D, a defocusing PIV technique has been developed that allows all three velocity components to be measured^{14}.
We use two different configurations: (1) a single layer of electrolyte on a solid bottom, and (2) a layer of electrolyte on top of another layer of a nonconducting heavier liquid, which substantially decreases friction. For a thin doublelayer configuration in the presence of a boundary, a 2D inverse cascade was shown to form a spectral condensate, or the boxsize vortex, which dominated the flow in a steady state^{11,12,13}. The statistics of the increment δ V_{L} across a distance r of the velocity component parallel to r were measured using hundreds of instantaneous velocity fields^{11,12,13}. The second moment of δ V_{L} determines the spectrum, whereas the third moment, S_{3L}=〈(δ V_{L})^{3}〉, is related in isotropic turbulence to the value and the direction of the energy flux ɛ across the scale r:
The sign of S_{3L} thus determines the flux direction: ɛ is positive in 3D (direct cascade) and negative in 2D (inverse cascade). Indeed, positive linear S_{3L}(r) has been observed in quasi2D turbulence in soap films^{8} and in electrolyte layers^{11,12,13}. For turbulence with the condensate, it is crucial to subtract a coherent flow from the instantaneous velocity fields to recover the correct statistics of the turbulent velocity fluctuations^{13,15}. The point is that the velocity differences δ V contain contributions from the spatially inhomogeneous vortex flow , in addition to the contributions from the turbulent velocity fluctuations . The vortex contribution then enters the higherorder moments: , , and so on. It has been demonstrated that the presence of the largescale flows substantially modifies S_{3L} and even affects its sign^{12,13}.
We now present the new results, starting with a thick twolayer configuration when the top layer thickness h_{t} exceeds half the forcing scale l_{f}. The forcing scale in this experiment is l_{f}=9 mm, and the thicknesses of the bottom and top layers are h_{b}=4 mm and h_{t}=7 mm respectively. The size of the boundary box is L=120 mm. Visualization in both horizontal (x–y) and vertical (z–y) planes shows that the flow is substantially 3D at the early stage of the evolution, as seen in Fig. 1a,b, in agreement with numerics. However, if one waits long enough, one sees the appearance of a large planar vortex, the diameter of which is limited by the box size (numerical simulations^{10} were done on a shorter timescale). Apparently, a residual inverse energy flux, existing even in the presence of 3D motions, leads to the spectral condensation and to the generation of the large coherent vortex that dominates the flow in a steady state (Fig. 1c). In the vertical crosssection, the flow is now close to planar (Fig. 1d). The vortex gives a strong spectral peak at k≈k_{c}, as seen in Fig. 1e, where k_{f} is the forcing wavenumber. The subtraction of the temporal mean reveals the spectrum of the underlying turbulence, see Fig. 1g, which shows a reasonably good agreement with the Kraichnan k^{−5/3} inverse energy cascade. Similarly, before the mean subtraction, S_{3L} is negative and is not a linear function of r (Fig. 1f). The subtraction of the mean vortex flow reveals positive S_{3L}, which is a linear function of r, in agreement with equation (1) (Fig. 1h). The slope of that linear dependence gives the value of the upscale energy flux, ɛ≈1.8×10^{−6} m^{2} s^{−3}. This value was independently verified by the energy balance analysis, as described in ref 13: ɛ is compared with the largescale flow energy dissipation rate ɛ_{d}. For the present experiment this agreement is within 10%.
The results of Fig. 1 indicate that, contrary to expectations, turbulence in a thick layer (h_{t}/l_{f}≈0.78) supports the inverse energy cascade, as evidenced by the S_{3L} result and also by the generation of the strong spectral condensate fed by the cascade. Similar results are obtained in even thicker layers, up to h_{t}=12 mm, or h/l_{f}≈1.3. Strong spectral condensates, the energy of which constitutes above 90% of the total flow energy, are observed in those cases.
After the development of the condensate, particle streaks are almost planar, at least in the statistical sense (Fig. 1d). Thus, there must be a mechanism through which the vortex secures its energy supply by suppressing vertical motions and enforcing planarity. 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 avoid spectral condensation. In this case the largescale vortex is imposed externally. The timeline of the experiment is shown schematically in Fig. 2a. First, turbulence is excited. Then a large vortex (150 mm diameter) is imposed on it by placing a magnetic dipole 2 mm above the free surface. After the large magnet is removed, the vortex decays, while turbulence continues to be forced. As the magnet blocks the view, the measurements are performed during the turbulence stage and during the decay of the vortex.
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}≈1.6 mm s^{−1}. When the vortex is present, they are reduced by a factor of about four, down to 〈V_{z}〉_{rms}≈0.4 mm s^{−1}. This is also confirmed by the direct visualization of the flow in the vertical z–y plane. Particle streaks are shown in Fig. 2c–e. Strong vertical eddies are seen in the 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 no vertical excursions. As the large vortex decays, the 3D eddies start to reappear near the bottom (Fig. 2e).
We also study the statistics of the horizontal velocities. The kinetic energy spectra and the thirdorder moments are shown in Fig. 2f–h. For turbulence without the vortex, the spectrum is substantially flatter than k^{−5/3} (Fig. 2f). After the large vortex is imposed, the spectrum shows a strong peak at low wavenumbers. However, the mean subtraction recovers the k^{−5/3} 2D spectrum, as shown in Fig. 2g. The thirdorder moment undergoes an even more pronounced change after the imposition of the large vortex: S_{3L} computed after the mean subtraction is much larger than during the turbulence stage, and it is a positive linear function of r, Fig. 2h, just as in the case of the doublelayer experiment. Thus the imposed flow enforces planarity and strongly enhances the inverse energy flux.
The strong suppression of vertical eddies in the presence of an imposed flow must be due to the vertical shear Ω_{LS}=d〈V_{h}〉/dz, which destroys vertical eddies for which the inverse turnover time is less than Ω_{LS}. This shear is obtained from the zprofiles of the horizontal velocity measured using defocusing PIV. In a singlelayer experiment, the averaged shear of the horizontal velocity due to the presence of the strong imposed vortex is Ω_{LS}≈1.6 s^{−1}. Such a shear is sufficient to suppress vertical eddies with inverse turnover times τ^{−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 possible reason for this is also the shear suppression of the 3D vortices. Here the vertical shear is lower, Ω_{LS}≈0.5 s^{−1}, yet it substantially exceeds the inverse turnover time for the forcescale vortices in the doublelayer experiment, τ^{−1}∼〈V_{z}〉_{rms}/h≈0.06 s^{−1}. One might think that the pronounced change in the flow field in the presence of the large vortex is due to the global fluid rotation. However, the Rossby numbers in the reported experiments are larger (especially in the double layer experiment, R o>3) than those at which quasi2D flow properties were observed in rotating tanks (R o<0.4; ref. 16).
The suppression of vertical motions by the shear flow and the onset of the inverse cascade observed in this experiment may be relevant for many natural and engineering applications. In the solar tachocline^{17}, a thin layer between the radiative interior and the outer convective zone, turbulence is expected to be 2D, despite being excited by radial convection^{17}. Turbulence suppression by the shear in the tachocline has been considered theoretically^{18}. Present results indicate that only one velocity component may be strongly suppressed, making the turbulence 2D. Another interesting example is the wavenumber spectrum of winds in the Earth atmosphere measured near the tropopause^{2}, which shows E(k)∼k^{−5/3} in the mesoscale range (10–500 km) and a strong peak at the planetary scale of 10^{4} km. Numerous hypotheses have been proposed to explain the mesoscale spectrum, with most arguments centred on the direct versus the inverse energy cascade^{2,19,20,21,22,23}. The shape of the spectrum alone cannot resolve this issue because both the 3D Kolmogorov direct cascade and the 2D Kraichnan inverse cascade predict E(k)∼k^{−5/3}. Direct processing of atmospheric data gave S_{3L}(r)<0 for some range of r in the mesoscales, thus favouring the direct cascade hypothesis^{24}. However, the subtraction of the mean flows, necessary for the correct flux evaluation, has not been done for the wind data. This leaves the question about the source of the mesoscale energy unresolved. Estimates of the vertical shear due to the planetary scale flow^{2}, represented by the spectral peak at 10^{4} km, show that the shear suppression criterion can be satisfied for smallscale eddies with sizes less than 10 km. Thus, it is possible that the suppression of 3D vertical eddies induces an inverse energy cascade through the mesoscales in the Earth atmosphere. Moreover, our results may be relevant not only for thin layers but also for boundary layer flows with turbulence generated by surface roughness, convection or other sources. It may also shed light on the nature of velocity correlations at horizontal distances far exceeding the distance from the ground, which is important for wind farms and in solving many other problems.
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Acknowledgements
This work was supported by the Australian Research Council’s Discovery Projects funding scheme (DP0881544) and by the Minerva Foundation and the Israeli Science Foundation.
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H.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.
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Xia, H., Byrne, D., Falkovich, G. et al. Upscale energy transfer in thick turbulent fluid layers. Nature Phys 7, 321–324 (2011). https://doi.org/10.1038/nphys1910
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DOI: https://doi.org/10.1038/nphys1910
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