Stability Analysis of Ecomorphodynamic Equations
  Although riparian vegetation is present in or along many water courses of the world, its active role resulting from the interaction with flow and sediment processes has only recently become an active field of research. Especially, the role of vegetation in the process of river pattern formation has been explored and demonstrated mostly experimentally and numerically until now. In the present work, we shed light on this subject by performing a linear stability analysis on a simple model for riverbed vegetation dynamics coupled with the set of classical river morphodynamic equations. The vegetation model only accounts for logistic growth, local positive feedback through seeding and resprouting, and mortality by means of uprooting through flow shear stress. Due to the simplicity of the model, we can transform the set of equations into an eigenvalue problem and assess the stability of the linearized equations when slightly perturbated away from a spatially homogeneous solution. If we couple vegetation dynamics with a 1D morphodynamic framework, we observe that instability towards long sediment waves is possible due to competitive interaction between vegetation growth and mortality. Moreover, the domain in the parameter space where perturbations are amplified was found to be simply connected. Subsequently, we proceed to the analysis of vegetation dynamics coupled with a 2D morphodynamic framework, which can be used to evaluate instability towards alternate and multiple bars. It is found that two kinds of instabilities, which are discriminated mainly by the Froude number, occur in a connected domain in the parameter space. At lower Froude number, instability is mainly governed by sediment dynamics and leads to the formation of alternate and multiple bars while at higher Froude number instability is driven by vegetation dynamics, which only allows for alternate bars. 
Fabian B\"arenbold, Beno\^it Crouzy and Paolo Perona
physics.flu-dyn