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Paul Billant - Jean-Marc Chomaz - Patrick Huerre - Sabine Ortiz - Peter Schmid
Observations in nature as well as in laboratory flows reveal that turbulence in stably stratified media exhibits very peculiar features. The turbulent flows that prevail in such situations are organised in thin horizontal and decoupled layers in which vortices have a typical pancake shape. Until now, no theory has been able to satisfactorily explain the physical mechanism responsible for these layered structures and to predict their characteristic thickness. In order to do so, the dynamics of a prototype flow has been investigated in our laboratory and simulated numerically.
Direct numerical simulations of the zigzag instability of a pair of contra-rotating vortices were performed for high Reynolds numbers (Deloncle, Chomaz & Billant 2007). These DNS show that the amplitude of the zigzag instability is growing dramatically, producing very intense vertical shear, but two different scenarios have been observed following the parameter R=Re Fr2, where Fr is the Froude number and Re. the Reynolds number.
When R is small, the exponential growth of the zigzag instability is stopped when vertical viscous effects become dominant as already observed in the laboratory. There is therefore a direct transfer of energy from large scales to small dissipative scales. By contrast, when the parameter R is large, the vertical shear reaches values such that the local Richardson number falls below 1/4. A secondary Kelvin-Helmholtz instability appears and leads a transition to small scale tri - dimensional turbulence (Figure 10).
Figure 10 - Direct Numerical Simulation of the zigzag instability of a pair of counter-rotating vortices in a stratified fluid. 3D visualization of the vertical vorticity of the two vortices (a) and vertical cross section (b) showing the appearance of small transverse vortices due to shear instability (Deloncle, Bill & Chomaz 2007).
Large-scale geophysical flows are in general subjected to both a stable stratification and planetary rotation. The extension of the previous study to the dynamics of a pair of counter-rotating vortices under the combined effects of rotation and stratification has therefore been investigated experimentally on the rotating table facility of the National Center for Meteorological Research. These two effects are in general antagonistic: stratification tends to structure flows in uncoupled horizontal layers whereas rotation makes the flow two-dimensional. Furthermore, rotation is known to destabilize anticyclonic vortices (rotating in a direction opposite to the planet rotation). Different three-dimensional instabilities have been identified on the cyclonic and anticyclonic vortices and they are observed in well-defined ranges of Rossby number and Froude number. For instance, as shown in Figure 11, two kinds of disturbances are present on the anticyclonic vortex for Rossby numbers of order unity : an axisymmetric centrifugal instability (a) and an oscillatory instability (b) of azimuthal wavenumber m=1.
Figure 11 - Axisymmetric centrifugal instability (a) and oscillatory asymmetric instability (b) affecting the anticyclonic vortex of vortex pair. The cyclonic vortex on the right is stable.
These experimental observations have been corroborated by the results of a numerical investigation performed in the case of the so-called Lamb-Oseen vortex.
Figure 12 - Experimental set-up for the generation of a pancake dipole in a tank filled with stably stratified fluid. The pair of vertical flaps to the right is suddenly closed and the ejected fluid forms a columnar dipole. The diaphragm then chops the columnar vortex into a pancake dipole of the desired thickness.
Experiments have recently been conducted to study the dynamics of a single layer of pancake dipole in a strongly stratified fluid. The pair of counter-rotating vertical vortices generated by suddenly closing two hinged vertical flaps is chopped by a diaphragm which only allows a single slice to go through, thereby giving rise to a pancake dipole (Figure 12). Particle Image Velocimetry (PIV) measurements are displayed in a vertical plane in Figure 13. The magnitude of the horizontal velocity is represented at two different times following the generation of the pancake dipole. The vertical thickness of the dipole is seen to decrease with time. Two boundary layers are clearly observable near the dipole upper and lower boundary (Figure 13b). As the dipole moves, these layers form a wake which is responsible for the thinning of the pancake. Viscosity is therefore responsible for decorrelating the flow in the vertical direction, but the thinning out process stops once the two layers have merged, thereby setting a limiting viscous vertical scale. This mechanism might explain the long-time behavior of turbulent stratified flows in specific parameter ranges.
Figure 13 - Particle Image Velocimetry (PIV) measurements in the vertical midplane of a pancake dipole. The colors represent the magnitude of the horizontal velocity at two different times after the generation of the pancake dipole. The screen is located on the right of each image and is out of view. The dipole moves from right to left.
In a recent study, a new instability arising in a pair of co-rotating (in contrast to counter-rotating) vortices has been observed and analysed both from an experimental and theoretical point of view. The co-rotating vertical vortices are generated by quickly rotating two flaps immersed in a tank filled with stably stratified fluid. As shown in Figure 14, a zig-zag instability develops, which symmetrically distorts the vortices. The horizontal distance between the vortices varies periodically along the vertical direction. At some heights, merging is accelerated, whereas at others, it is delayed. This spectacular phenomenon results in a complex twisting of the vortex lines (right snapshot in Figure 14).
Figure 14 - Side views of the zig-zag instability affecting a pair of co-rotating vortices in a strongly stratified fluid. Time increases from left to right.
Direct numerical simulations have also been performed (Figure 15). The three-dimensional development of the instability is found to be in good agreement with both experimental observations and linear stability theory.
Figure 15 - Direct numerical simulation of the development of the zig-zag instability experienced by a pair of co-rotating vortices in a strongly stratified fluid. Time increases from left to right. Compare with Figure 14.
Dynamics of stratified turbulence;. To better understand the structure and dynamics of stratified turbulence, direct numerical simulations (Figure 16) of strongly forced stratified turbulence have been performed during the 3 month stay at LadHyX of G. Brethouwer and in collaboration with E. Lindborg (KTH). A forcing at horizontal two-dimensional scales was used (Brethouwer, Billant, Lindborg & Chomaz 2007).
The results were interpreted on the basis of dimensional analysis of strongly stratified flows (small horizontal Froude number). This reasoning shows that the number controlling the dynamics of strongly stratified flows is the buoyancy Reynolds number R. Depending on the value of R, the physical phenomena which determine the vertical scale are totally different, even if the Reynolds number is large and the Froude number small. When R is large, the viscous effects are negligible and the vertical scale is U/N , where U is the scale of horizontal velocity and N the Brunt-Väisälä. frequency. In this case, the dynamics is inherently three-dimensional but anisotropic. When R is small, the viscous effects due to vertical shear are dominant. In this case, the vertical scale is . The dynamics is two-dimensional horizontal but with viscous dissipation due to vertical shear of the same order as the inertial effects.
Alternatively, the parameter R can be written , where is the Ozmidov scale and the Kolmogorov dissipation scale. This shows that there are 2 ranges of scales in a turbulent stratified flow ( ). ). For large scale , the local Froude number is small and the dynamics is highly stratified. We then have an anisotropic energy cascade from large horizontal scales to small scales with a horizontal spectrum of horizontal kinetic energy of the form , where is the horizontal wavenumber (Lindborg, Journal of Fluid Mechanics, vol. 550, pp. 207-242, 2006). For scales between the Ozmidov scale and the Kolmogorov scale, , the Froude number is greater than 1 and becomes increasingly large as the scale becomes smaller. The dynamics tends to that of the classical homogeneous isotropic turbulence.
Figure 16 - Direct numerical simulation of stratified turbulence for the same horizontal Froude number Fr = 0.015 and various Reynolds numbers Re increasing from bottom to top; vertical sections of the field of density fluctuations (Brethouwer, Billa, Lindborg & Chomaz 2007 ). These simulations show the importance of the parameter R comparing the intensity of vertical transport to dissipation. Even if Re is large, R must be large too to be in the regime of stratified turbulence. The R number is increasing from bottom to top, R = 0.3, 0.7, 1.8, 3, 9.6.