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

A scaling analysis of the depth-integrated momentum equations tailored to t he 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 significa nt part of the momentum balance in these steep channels. By introducing the kinetic energy equation of the time-averaged motion to treat these correla tion terms, which involve products of local deviations in velocity componen ts about depth-averaged values, a flow model that suitably characterizes st reamwise accelerations is obtained. A linear stability analysis using a flo w model that retains the streamwise correlation terms suggests that their e ffect is to strengthen the initial selection of bed form wavelengths, as re flected by sharpened peaks in curves of growth rate versus bed form wavelen gth. Wavelengths with zero migration rate are close to wavelengths having t he large;st growth rate; thus selection of fixed bars is strong. Critical w idth-depth ratios necessary for bed form growth are significantly less than the critical ratios that are predicted when correlation terms are neglecte d. Moreover, a broader band of wavenumbers can be activated at a given widt h-depth ratio, and bed form modes representing midchannel bars can be activ ated in a narrower channel than would otherwise be predicted. Thus alternat e bars can initially "compete" with midchannel bars, particularly at low se diment transport rates. This competition probably contributes to the comple xity of bed topography that is typical of rough, mountain channels.