A remarkable diversity exists in the morphology and dynamics of submarine fans, which influence the transport of microplastics, burial of organic carbon, subsea geo-hazards, and their potential to house geofluids and high-resolution paleo-environmental records. Like river deltas, submarine fan morphology is a product of evolving fluid and sediment transport fields, but unlike their terrestrial counterparts, we lack a unifying framework to predict their morphology. Here, we simplify critical environmental forcings, like regional slopes and sediment properties, through a dimensionless framework defined by the densimetric Froude number (ratio of inertial to gravitational forces) and Rouse number (ratio of settling velocity of sediments to shear velocity) of turbidity currents. We explore this framework by leveraging a depth-averaged numerical model and measure fan rugosity as a proxy for their morphological complexity. We show a systematic increase in rugosity by either increasing the densimetric Froude number or decreasing the Rouse number of the simulated flows. These changes reflect observed gradients in the dynamics of channel migration and help discriminate submarine fans that have the potential to impact global climate through sequestration of organic carbon.