Three-Dimensional Vortices Generated by Self-Replication in Stably Stratified Rotating Shear Flows

A previously unknown instability creates space-filling lattices of 3D vortices in linearly stable, rotating, stratified shear flows. The instability starts from an easily excited critical layer. The layer intensifies by drawing energy from the background shear and rolls up into vortices that excite new critical layers and vortices. The vortices self-similarly replicate to create lattices of turbulent vortices. The vortices persist for all time. This self-replication occurs in stratified Couette flows and in the dead zones of protoplanetary disks where it can destabilize Keplerian flows.

Figure 1

${\omega }_z/f\equiv \mathrm{Ro}$ of the anticyclonic (blue) vortices and cyclonic (red) vortex layers in the $x\mathrm{\text{−}}y$ plane. The initial perturbing vortex at the origin is above the plane shown here ($z=-0.4$). The first generation zombie vortices form at $|x|\le 1$, and sweep outward in $x$. The Rossby number Ro of these vortices is $\sim -0.2$. (The color is reddest at ${\omega }_z/f=0.2$, e.g., near $x=1/3$ at the bottom of panel d; bluest at ${\omega }_z/f=-0.2$, e.g., near $x=0.6$ in panel d; and green at ${\omega }_z/f=0$). $f/\overline{N}=1$ and $\sigma /\overline{N}=-3/4$. The $x\mathrm{\text{−}}y$ domain is $|x|\le 4.7124$; $|y|\le 2.3562$, and is larger than shown. Movies of Figs. 1, 2 are in the Supplemental Material [25]. (a) $t=64/\overline{N}$. (b) $t=256/\overline{N}$. (c) $t=576/\overline{N}$. (d) $t=2240/\overline{N}$. See text for details.

Figure 2

Zombie vortices sweep outward from the perturbing vortex at the origin in the $x\mathrm{\text{−}}z$ plane (at $y=0$). Anticyclonic ${\omega }_z$ is black (darkest is ${\omega }_z/f=-0.2$) and cyclonic is white (lightest is ${\omega }_z/f=0.2$). This is the same flow as in Fig. 1. The domain has $|z|\le 4.7124$ and is larger than shown. (a) $t=128/\overline{N}$. Critical layers with $s=0$ and $|m|=1$, 2, and 3 are visible. Diagonal lines are internal inertio-gravity waves with shear, not critical layers. (b) $t=480/\overline{N}$. 1st generation vortices near $|x|=1$ and $1/2$ have rolled up from critical layers with $s=0$ and $|m|=1$ and 2, respectively. (c) $t=1632/\overline{N}$. 2nd generation $|m|=1$ vortices near $|x|=0$ and 2 were spawned from the 1st generation vortices near $|x|=1$. Another 2nd generation of $|m|=1$ vortices is near $|x|\simeq 1/2$ and $3/2$, which were spawned by the 1st generation near $|x|=1/2$. (d) $t=3072/\overline{N}$. 1st, 2nd, and 3rd generation vortices.