Irregular bed forms in steep, rough channels - 1. Stability analysis

A scaling analysis of the depth-integrated momentum equations tailored to the rough bed conditions of mountain streams suggests that certain velocity correlation terms that arise from depth integration, and which normally can be neglected in the case of smoother alluvial channels, can be a significant part of the momentum balance in these steep channels. By introducing the kinetic energy equation of the time-averaged motion to treat these correlation terms, which involve products of local deviations in velocity components about depth-averaged values, a flow model that suitably characterizes streamwise accelerations is obtained. A linear stability analysis using a flow model that retains the streamwise correlation terms suggests that their effect is to strengthen the initial selection of bed form wavelengths, as reflected by sharpened peaks in curves of growth rate versus bed form wavelength. Wavelengths with zero migration rate are close to wavelengths having the large;st growth rate; thus selection of fixed bars is strong. Critical width-depth ratios necessary for bed form growth are significantly less than the critical ratios that are predicted when correlation terms are neglected. Moreover, a broader band of wavenumbers can be activated at a given width-depth ratio, and bed form modes representing midchannel bars can be activated in a narrower channel than would otherwise be predicted. Thus alternate bars can initially "compete" with midchannel bars, particularly at low sediment transport rates. This competition probably contributes to the complexity of bed topography that is typical of rough, mountain channels.