Anomalous Chained Turbulence in Actively Driven Flows on Spheres

Recent experiments demonstrate the importance of substrate curvature for actively forced fluid dynamics. Yet, the covariant formulation and analysis of continuum models for nonequilibrium flows on curved surfaces still poses theoretical challenges. Here, we introduce and study a generalized covariant Navier-Stokes model for fluid flows driven by active stresses in nonplanar geometries. The analytical tractability of the theory is demonstrated through exact stationary solutions for the case of a spherical bubble geometry. Direct numerical simulations reveal a curvature-induced transition from a burst phase to an anomalous turbulent phase that differs distinctly from externally forced classical 2D Kolmogorov turbulence. This new type of active turbulence is characterized by the self-assembly of finite-size vortices into linked chains of antiferromagnetic order, which percolate through the entire fluid domain, forming an active dynamic network. The coherent motion of the vortex chain network provides an efficient mechanism for upward energy transfer from smaller to larger scales, presenting an alternative to the conventional energy cascade in classical 2D turbulence.

Figure 1

Stationary solutions of Eqs. (2) are superpositions of the form Eq, (3) with $f[-\ell (\ell +1)+4]=0$. (a) An exact stationary solution with $\ell =6$ which is also approximately realized as a transient state in the time-dependent burst solution of Fig. 2 (movie 1). (b) Complex symmetric solutions can be constructed by choosing the expansion coefficients ${\psi }_{m\ell }$ accordingly [46]. In both panels, the stream functions are normalized by their maxima; see Supplemental Material [41] for coefficients ${\psi }_{m\ell }$.

Figure 2

Phase diagrams [(a),(b)] and representative still images [(c)–(e)] from simulations showing quasistationary burst dynamics ($B$ phase), anomalous vortex-network turbulence ($A$ phase), and classical 2D turbulence ($T$ phase). (a), (b) The $A$ and $T$ phase are approximately separated by the condition $\kappa \mathrm{\Lambda }=1$ (vertical dashed line) and differ by the average number of vortices (a), the branch geometry of the tension field (b), and the energy spectra (Fig. 3). The $B$ phase arises for narrow-band energy injection $\kappa R\lesssim 1$ when only a single $\ell$ mode is active (region right below the dash-dotted line); decreasing $\kappa$ further gives a passive fluid (white region). (c)–(e) Top: Instantaneous vorticity fields normalized by their maxima. Bottom: Surface tension fields normalized by the maximum deviation from the mean. (c) Quasistationary preburst state from movie 1 resembling the exact solution in Fig. 1; see movies 2 and 3 for additional examples labeled by $*$ in panel (a). (d) For subcritical curvature and intermediate energy injection bandwidths, $R^{-1}<\kappa <{\mathrm{\Lambda }}^{-1}$, the flows develop a percolating vortex-chain network structure (movie 4), with accumulation of tension and vorticity along the edges. (e) For broadband energy injection $\kappa \mathrm{\Lambda }>1$, smaller eddies merge to create larger vortices, as typical of classical 2D turbulence (movie 5). Parameters: (a) ${\alpha }_{\omega }=0.5$; (c) $R/\mathrm{\Lambda }=2$, $\tau =4.9\mathrm{s}$, $\kappa \mathrm{\Lambda }=0.29$; (d) $R/\mathrm{\Lambda }=10$, $\tau =14.9\mathrm{s}$, $\kappa \mathrm{\Lambda }=0.5$; (e) $R/\mathrm{\Lambda }=10$, $\tau =11.7\mathrm{s}$, $\kappa \mathrm{\Lambda }=2.0$. Panels (a),(b) show steady-state time averages over $[50\tau ,100\tau ]$. Solid curves in (c)–(e) indicate stream lines of the velocity fields.

Figure 3

Time-averaged energy spectra and fluxes indicate two qualitatively different types of upward energy transport. (a) For narrow-band energy injection $\kappa \mathrm{\Lambda }<1$, the energy spectrum exhibits a peak corresponding to the dominant vortex size $\mathrm{\Lambda }$ (red curve). For broadband injection $\kappa \mathrm{\Lambda }\sim 2$, the spectra decay monotonically (blue and black curves). (b) In all four examples, the fluxes confirm inverse energy transport, albeit with different origins. For broadband energy injection (blue and black curves), the upward energy flux to larger scales is due to vortex mergers [Fig. 2; movie 5]. By contrast, for narrow-band injection (red curve), a relatively stronger upward energy flux arises from the collective motion of vortex chains [Fig. 2; movie 4]. The shaded regions indicate the energy injection ranges with colors matching those of the corresponding curves, respectively. Parameters: $R/\mathrm{\Lambda }=10$ for a unit sphere, $\tau =11.7\mathrm{s}$, time step $5\times {10}^{-4}\tau$, total simulation time $500\tau$. Spectra and fluxes were determined after relaxation by averaging over $[150\tau ,500\tau ]$. For $\kappa \mathrm{\Lambda }\gg 1$, energy steadily accumulates at larger scales and the absence of a large-scale dissipative mechanism leads to a divergent total enstrophy and kinetic energy on the sphere.