  We consider a numerical approach for a covariant generalised Navier-Stokes
equation on general surfaces and study the influence of varying Gaussian
curvature on anomalous vortex-network active turbulence. This regime is
characterised by self-assembly of finite-size vortices into linked chains of
anti-ferromagnet order, which percolate through the entire surface. The
simulation results reveal an alignment of these chains with minimal curvature
lines of the surface and indicate a dependency of this turbulence regime on the
sign and the gradient in local Gaussian curvature. While these results remain
qualitative and their explanations are still incomplete, several of the
observed phenomena are in qualitative agreement with experiments on active
nematic liquid crystals on toroidal surfaces and contribute to an understanding
of the delicate interplay between geometrical properties of the surface and
characteristics of the flow field, which has the potential to control active
flows on surfaces via gradients in the spatial curvature of the surface.
