Fluvial drainage basin area is often related to channel length and local slope through power law relationships and the relatively small range of exponents observed in these relationships is thought to result from physical mechanisms. Proposed mechanisms assume that the observed correlation between drainage area and fluid discharge is caused by precipitation. Using high resolution DEMs of channelized continental slope settings offshore Monterey, CA and Brunei Darussalam we extracted submarine channel profiles and drainage area statistics from five basins. In-situ and remote observations suggest discharge in these oceanic settings is determined by boundary conditions at the shelf-edge. In spite of substantial differences in environment and physical process, the data yield submarine scaling exponents within the range of terrestrial (fluvial) observations. The convergence in scaling relationships from two very different settings supports theoretical arguments that channel network structure results from the aggregation of random walks.