  Recent experiments demonstrate the importance of substrate curvature for
actively forced fluid dynamics. Yet, the covariant formulation and analysis of
continuum models for non-equilibrium 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 non-planar
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 anti-ferromagnetic 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.
