Hysteretic transitions between quasi-two-dimensional flow and three-dimensional flow in forced rotating turbulence

Abstract

Conflict between formation of a cyclonic vortex and isotropization in forced homogeneous rotating turbulence is numerically investigated. It is well known that a large rotation rate of the system induces columnar vortices to result in quasi-two-dimensional (Q2D) flow, while a small rotation rate allows turbulence to be three-dimensional (3D). It is found that the transition from the Q2D turbulent flow to the 3D turbulent flow and the reverse transition occur at different values of the rotation rates. At the intermediate rotation rates, bistability of these two statistically steady states is observed. Such hysteretic behavior is also observed for the variation of the amplitude of an external force.Formation of columnar structures parallel to a rotation axis is one of the most fundamental and distinctive phenomena in flows subject to rotation. The emergence of columnar vortices in the rotating turbulence makes a threedimensional (3D) flow into a quasi-two-dimensional (Q2D) flow. The Taylor-Proudman theorem has succeeded in explaining the cylindrical flow in laboratory experiments and field observations in terms of the Taylor column. However, the theorem cannot describe transitions between the Q2D and 3D flows, because energy is exchanged between the Q2D mode and the 3D mode by nonlinear mechanisms [1]. The energy transfers to the Q2D modes were demonstrated by an instability analysis [2] and weak-turbulence theory in the large-rotation limit [3]. The Coriolis term breaks the parity invariance of the governing equation of the flow, and introduces a scale-independent time scale which induces two-dimensionalization at larger scales more effectively. Therefore, the Coriolis effect originates cyclone-anticyclone asymmetry with enhanced stretching of cyclonic vorticity and destabilization of anticyclonic one due to the centrifugal instability and the vortex tilting [4].To classify the flow properties in rotating systems, the Rossby number Ro, which is the ratio between the linear and nonlinear time scales, has been used [5]. Note that though various definitions of Ro are used in literature, the following facts are independent of its detailed definition. When the Coriolis force is weak relative to turbulence, i.e., Ro ≫ 1, the 3D Kolmogorov turbulence is obtained. When Ro ∼ 1, only cyclonic vortices appear at large scales, and the flow becomes Q2D. When Ro ≪ 1, both cyclonic and anticyclonic vortices appear, and the flow fields are almost completely two-dimensionalized. The transitions between the Q2D turbulence and the 3D turbulence by changing the system's rotation rate Ω were numerically studied [6]. It was reported that Ro-dependence of turbulent statistics is not monotonic in the range Ro ∼ 1, where the coherent vortices and inertial waves at small wave numbers and the turbulence at large wave numbers coexist [7,8]. The two-dimensionalization and the cyclone-anticyclone asymmetry depend on the external forces and the boundary conditions (e.g., Ref.[9]).Recently, Ref.[10] reported a phase diagram for statistically st...

Condensates in thin-layer turbulence

“…Further simulations at larger Re and possibly smaller K (larger boxes) are required to resolve this issue. Similar hysteretic behaviour has recently been reported in rotating turbulence, see (Yokoyama & Takaoka 2017). More generally, multistability is observed in many turbulent flows, see (Weeks et al 1997;Ravelet et al 2004) as examples.…”

Condensates in rotating turbulent flows

“…This was indeed found to be the case in Alexakis (2015). Further more, recently Yokoyama & Takaoka (2017) were able to follow the hysteresis diagram of the subcritical bifurcation. Finally we also note that in this regime a strong asymmetry between co-rotating and counter rotating vortexes is expected, (see for example Hopfinger et al 2014)).…”

“…It is also important to note that our forcing is invariant along the axis of rotation and thus the forcing acts only on the slow manifold (that consists of all the Fourier velocity modes for which k z = 0). This in contrast with the case examined in (Alexakis 2015;Yokoyama & Takaoka 2017) where a Taylor-Green forcing was used that has zero average along the vertical direction. Thus, while the Taylor-Green forcing does not inject energy directly to the slow manifold, the Roberts forcing used here injects energy only to the slow manifold.…”

“…In this work, we try to determine the behaviour of a forced rotating flow at late times when the flow has reached a steady state, in the absence of any large scale dissipative mechanism. Due to the long computational time required to reach a steady state, very few investigations have focused on this regime like the early low resolution studies in Bartello et al (1994) and more recently the studies in (Alexakis 2015;Dallas & Tobias 2016;Yokoyama & Takaoka 2017), where turbulent rotating flows at steady state were investigated. Experiments on the other hand for which long times are realizable have investigated this steady state limit Campagne et al (2014); Yarom & Sharon (2014); Campagne et al (2016); Machicoane et al (2016).…”

Experimental study of the nonlinear saturation of the elliptical instability: inertial wave turbulence versus geostrophic turbulence

“…Beyond this plateau, a slight decrease of both the rms and the central vorticity of the mean flow is observed at higher Ro i , at least for Ekman numbers E = 5.0 × 10 −6 and E = 7.4 × 10 −6 . It is possibly linked to a transition from rotating to isotropic turbulence as the Rossby number draws closer to 1 (Yokoyama & Takaoka 2017). As observed for instance by Barker & Lithwick (2013), increasing the amplitude of the forcing leads to the destabilisation of the strong vortices produced by the saturation of the elliptical instability.…”