We live in a world of three spatial dimensions, and most of the physics we observe reflects that dimensionality. In some circumstances, however, the degrees of freedom in one (or more) dimensions are suppressed, leading to an effectively lower-dimensional system. This change of dimensionality can have profound impact on the physics of such systems. In fluid dynamics, a reduction in dimensionality is important because there are many mechanisms that generate large spatial anisotropy with two spatial dimensions dominant compared with the third: thin fluid layers, stratification, rotation, magnetic field, etc. Here, we consider the dramatic changes that strong dimensional anisotropy imposes on the character of fluid turbulence.
Figure 1. Examples of ideal and quasi-2D fluid turbulence that emphasize the role of coherent vortex structures in thin layers: (a) numerical simulation of ideal 2D turbulence (vorticity) (Boffetta & Ecke 2012), (b) laboratory quasi-2D turbulence (vorticity) (Boffetta & Ecke 2012) and (c) Atlantic Gulf Stream eddies (streak image) (Sirah 2012).
So what is the difference between two-dimensional (2D) turbulence (Boffetta & Ecke 2012) and the more familiar three-dimensional (3D) version (Frisch 1995)? In 3D, when energy is injected at large scales, it is transferred on average to smaller scales in a forward cascade with energy dissipated at the smallest scales by viscous forces. If the spatial scales of forcing and dissipation are widely separated, energy is conserved at intermediate scales, leading to a
power-law relationship between energy and wavenumber. In 2D, this transfer of energy to small scales cannot occur due to an additional conservation law related to the condition in 2D that the vorticity contained within a spatial contour is conserved; the vorticity can only be rearranged in space but not increased or reduced as is possible in 3D. Thus, a characteristic of 2D turbulence is the persistence of vortical structures; see figure 1. The constraint on vorticity implies that the mean-square vorticity is transferred to small scales whereas the energy flow is inverse, i.e. it is transferred to large scales. This leads to a double cascade in 2D, with energy cascading towards scales larger than the forcing scale whereas the mean-square vorticity cascades towards smaller scales. In many circumstances, the ideal limits of 3D or 2D turbulence are not met and there is a bidirectional cascade where different transfer mechanisms dominate at different length scales and no inertial range may exist. The relationship between the ideal limits in 2D and 3D with a bidirectional cascade for thin but 3D layers is what we consider here.
Much research on 3D and 2D turbulence has focused on ideal isotropic homogeneous turbulence in the respective spatial dimensions. There is, however, a large amount of interesting physics in the anisotropic regime of 3D turbulence where one direction is suppressed compared with the other two degrees of freedom. This is the regime recently addressed by Benavides & Alexakis (2017), which significantly extends earlier work exploring the crossover from 3D turbulence to 2D turbulence (Celani, Musacchio & Vincenzi 2010). Benavides and Alexakis numerically compute a model with highly resolved 2D components in the lateral direction coupled to a truncated single-mode vertical component. The system is forced on a lateral scale
that is large compared with the vertical height
. The control parameter is
, where larger values correspond to thinner, more 2D systems. The energy flux is computed and its behaviour as a function of
is considered. The resulting vorticity field reveals the quantitative feature of dominant vertical vorticity, as seen in figure 2. The insets show the 3D energy density and reflect the trend that as
is increased, the overall 3D energy density decreases, becoming highly singular in space for higher
is varied, the forward energy flux characteristic of 3D turbulence and the 2D inverse energy flux behave in unexpected ways, namely they exhibit sharp transitions at specific values of
. This behaviour is common in bifurcations in low-dimensional dynamical systems and in transitions from laminar to turbulent flow, but sharp transitions in turbulent flows are unusual. Benavides and Alexakis show that there is net inverse energy transfer that increases from zero at a well-defined value
; see figure 3. At higher
, the 3D energy flux goes to zero sharply at a critical value of
that depends on the Reynolds number
of the flow. The empirically observed
is consistent with scaling arguments by Benavides & Alexakis (2017) and with a linear stability analysis of 2D flows with respect to 3D perturbations (Gallet & Doering 2015). Finally, Benavides & Alexakis (2017) argue that the critical points should persist, as additional vertical modes are included in a fully resolved 3D numerical computation.