Stability of vortex lattices in rotating flows

TL;DR

The paper investigates the dynamical stability of vortex lattices in three-dimensional rotating turbulence by embedding controlled triangular lattices into turbulent backgrounds. Using direct numerical simulations, it shows that lattices are not truly stationary but exist in a finite region of parameter space near the transition between forward and inverse energy cascades; their lifetimes follow memoryless statistics and are maximized by an optimal Ekman-drag value. By dynamically tuning Ekman drag to maintain a target 2D energy, the authors demonstrate metastable, long-lived lattice states and reveal how lattice longevity depends on system size and forcing. The findings provide insights into long-lived vortex patterns in planetary atmospheres and motivate experimental tests of lattice stability in rotating flows.

Abstract

Vortex lattices -- highly ordered arrays of vortices -- are known to arise in quantum systems such as type II superconductors and Bose-Einstein condensates. More recently, similar arrangements have been reported in classical rotating fluids. However, the mechanisms governing their formation, stability, and eventual breakdown remain poorly understood. We explore the dynamical stability of vortex lattices in three-dimensional rotating flows. To that end we construct controlled initial conditions consisting of vortex lattices superimposed on turbulent backgrounds. We then characterize their evolution across different Rossby numbers and domain geometries. By introducing an Ekman drag we are able to reach a steady state where vortex lattices persist with near constant amplitude up until spontaneous breakup of the lattice, or an equivalent of ``melting,'' occurs. We examine an ensemble of runs in order to determine the mean lifetime of the lattice as a function of the system parameters. Our results reveal that the stability of the lattices is a memory-less random process whose mean life-time depends sensitively on the system parameters that if finely tuned can lead to very long lived lattice states. These metastable states exhibit statistical properties reminiscent of critical systems and can offer insight into long-lived vortex patterns observed in planetary atmospheres.

Figures (11)

• Figure 1: The simplest triangular lattice in a domain with aspect ratio $\lambda_\perp = 1/\sqrt{3}$. The colors show $\left< \omega_z \right>_z \tau_f$ (i.e., the vertically averaged vertical vorticity in units of $\tau_f^{-1}$) in a simulation with $\textrm{Ro} = 0.70$ and $(L_x,L_y,L_z)=(1/2,\sqrt{3}/2,1)2\pi L_0$. The Bravais lattice vectors are indicated by the black arrows.
• Figure 2: Vertically averaged vertical vorticity, $\left< \omega_z \right>_z$, in units of $\tau_f^{-1}$, as a function of $x$ and for a value of $y$ that goes across the center of a vortex in a simulation with $\textrm{Ro} = 0.70$, $\lambda_\perp=\sqrt{3}$ and $\lambda_\parallel = 1.52$. Light blue shows the initial condition, and dark blue corresponds to a later time.
• Figure 3: Peak vertical vorticity as a function of time of the two vortices in simulations with $N_x=256$, $N_y=432$, $N_z=512$, and with different values of $\textrm{Ro}$ from $0.43$ to $1.15$. For each color, corresponding to a value of $\textrm{Ro}$, the light solid and dashed lines indicate the instantaneous value of $\max \{\omega_z\}$ in each vortex, while the dark lines give the smoothed evolution.
• Figure 4: Total (solid lines) and 2D (dashed lines) energies in the simulations with size $(L_x,L_y,L_z)=(1/2,\sqrt{3}/2,1)2\pi L_0$, with values of $\textrm{Ro}$ from $0.43$ to $1.15$. The energies are normalized by the initial value. A value $E/E(0)=2.5$ is marked by a dashed black line as a reference.
• Figure 5: Phase space in terms of $\lambda_\parallel$ and $\mathrm{Ro}^{-1}$, where the different colored markers represent the end state of the simulation after a time $T/\tau_f = 273$. Points marked as "direct" correspond to cases in which $E$ decays, points marked as "inverse" correspond to cases in which $E$ grows and the lattice breaks up, while "stable" indicates cases in which at time $T$ the lattice was still visible in the simulations.