Jets or vortices—What flows are generated by an inverse turbulent cascade?

Abstract

An inverse cascade -energy transfer to progressively larger scales -is a salient feature of twodimensional turbulence. If the cascade reaches the system scale, it creates a coherent flow expected to have the largest available scale and conform with the symmetries of the domain. In a doubly periodic rectangle, the mean flow with zero total momentum was therefore believed to be unidirectional, with two jets along the short side; while for an aspect ratio close to unity, a vortex dipole was expected. Using direct numerical simulations, we show that in fact neither the box symmetry is respected nor the largest scale is realized: the flow is never purely unidirectional since the inverse cascade produces coherent vortices, whose number and relative motion are determined by the aspect ratio. This spontaneous symmetry breaking is closely related to the hierarchy of averaging times. Longtime averaging restores translational invariance due to vortex wandering along one direction, and gives jets whose profile, however, can be deduced neither from the largest-available-scale argument, nor from the often employed maximum-entropy principle or quasi-linear approximation.

Condensate in quasi-two-dimensional turbulence

“…The vorticity profile of these vortices has been shown to displays universal features, independent of the forcing mechanism which produces the inverse cascade [21,23]. Changing the shape of the domain from square to rectangular, the emergence of jets in the condensed state with a complex phenomenology has also been observed [25,26].…”

“…A fraction of the energy injected at scale L f goes to the large scales where it is not dissipated by viscosity and, for finite horizontal extensions, accumulates producing a large-scale vortex system called the condensate [16,17]. The statistics of the condensate has been investigated in details by experiments [8,[18][19][20] and numerical simulations [21][22][23][24][25] in the 2D limit S = 0. In the case of a square box with periodic boundary conditions, the condensate is a pair of system-size vortices of opposite sign.…”

How vortices and shocks provide for a flux loop in two-dimensional compressible turbulence

“…On the contrary, large-scale motions are usually less dissipative, so that an inverse cascade can proceed unimpeded, either producing larger and larger scales or reaching the box size and creating a coherent mode of growing amplitude. That process is now actively studied in 2D incompressible turbulence [3,4,[9][10][11][12][13], including in a curved space, where vortex rings rather than vortices are created [14]. The energy of an incompressible flow in an unbounded domain grows unlimited when the friction factors go to zero at a finite energy input rate.…”

Interaction between mean flow and turbulence in two dimensions

“…In other words, apart from the zonal flow, there must be some other strong jet-size coherent patterns pumped by an inverse cascade, which contribute energy and enstrophy fluxes but not the vorticity flux vω into a zonal average. Those coherent patterns were found to be vortices in DNS of an inverse cascade in a double-periodic domain [42]. Figure 1 shows that zonal flow appears when the aspect ratio differs from unity yet vortices remain.…”

“…The resulting mean flow may thus depend on the choice of forcing. On the contrary, the focus here and in [42] is on the universal limit of small-scale forcing and developed inverse cascade. Even though mean flows with straight lines cannot appear out of inverse cascade, they can be driven by external forces, for instance, in pipes and channels.…”