A reduced front-tracking model for fingering instabilities of viscous gravity currents on planar and conical surfaces

Fingering instabilities that develop at the advancing front of viscous gravity currents originate from the competition between gravitationally driven amplification of the front and capillary regularisation. Although the large-scale evolution of gravity currents is accurately captured by similarity solutions, the physical mechanisms underlying wavelength selection are not yet fully understood. In this study, a reduced front-tracking framework is formulated by coupling lubrication theory, self-similar spreading laws and a localised description of front dynamics. The approach is first applied to the classical experiments of Huppert ( Nature , vol. 300, 1982 b, pp. 427–429), involving currents propagating down inclined planes, and is subsequently generalised to gravity currents spreading over conical surfaces. New experiments conducted in conical geometry with Newtonian fluids that span a broad range of physical properties validate the predicted self-similar spreading behaviour and yield quantitative measurements of the emergent fingering wavelength. The results demonstrate that wavelength selection arises from the interplay between the global spreading dynamics and local processes at the front. In particular, the conical geometry modifies the instability by inducing a radial expansion of the front, which diverts a portion of the incoming flux into the azimuthal direction and thereby alters the balance between destabilising and stabilising mechanisms. The proposed framework offers a physically transparent interpretation of wavelength selection and establishes a direct connection between similarity solutions, front dynamics and pattern formation in viscous gravity currents. A distinctive feature of the model is that the selected wavelength is not purely a local property of the advancing front, but depends on the cumulative amplification history of the perturbations during current propagation.