Influence of averaging coefficients on weakly nonlinear stability of shallow water flows A.A. KOLYSHKIN and S. NAZAROVS Department of Engineering Mathematics Riga Technical University 1 Meza street, Riga LATVIA Abstract: - Stability of shallow flows is analyzed in the present paper. Momentum correction coefficients are introduced in the shallow water equations in order to take into account non-uniformity of the velocity distribution in the vertical direction. Linear stability of parallel base flow is governed by the modified Rayleigh equation. Methods of weakly nonlinear theory are used in order to derive the amplitude evolution equation for the most unstable mode. It is shown that the evolution equation is the complex Ginzburg-Landau equation. Key-Words: - averaging coefficients, shallow flows, weakly nonlinear analysis, Ginzburg-Landau equation. 1 Introduction Depth-averaged equations (the Saint-Venant equations) are often used to model large-scale turbulent motions in shallow water [1]. These equations are used when the transverse length scale of the flow is much larger than water depth. Shallow water equations have been recently used for linear stability analyses of transverse turbulent motions in shallow waters [2]-[8]. Experimental and theoretical analyses in [2]-[8] show that the development of instability in shallow water is different from the case of deep water. Bottom friction in shallow flows acts as a suppression mechanism of the transverse growth of perturbations. In addition, development of three-dimensional instabilities is prevented due to limited water depth. One of the main assumptions in shallow water theory is the independence of the flow characteristics from the vertical coordinate since shallow water equations are depth-averaged equations. There are many cases, however, where this assumption is not valid. Changes in flow geometry, flow regimes or roughness of the bottom boundary can lead to large deviations from the above assumptions [9]. Averaging coefficients (momentum and pressure corrections coefficients) are introduced in [10]-[11] in order to take into account the non-uniformity of the velocity distribution in the vertical direction. The linear stability theory can only predict when a particular flow becomes unstable. In particular, the critical values of the stability parameters (critical Reynolds number for viscous flows or critical bed friction number for shallow flows) can be calculated by means of the linear stability theory. However, the evolution of the unstable disturbance above the threshold cannot be predicted by the linear theory. Weakly nonlinear theories [12]-[13] are used to take into account the effect of nonlinearities analytically in the unstable region where the parameters are very close to the critical values. As a result, an amplitude evolution equation for the most unstable mode is derived. In particular, the methods of weakly nonlinear theory are used in [8] to derive the complex Ginzburg-Landau equation which can be used to describe the dynamics of shallow flows behind obstacles (such as islands) above the threshold. Previous studies [14] indicated that the stability characteristics of shallow flows are quite sensitive to the relative magnitude of the averaging coefficients. In particular, it is shown that the averaging coefficients have significant impact on the stability domains of transverse flows in compound channels. The present paper is devoted to weakly nonlinear stability analysis of shallow flows where the averaging coefficients are taken into account. The amplitude evolution equation is derived under the assumption that the bed friction number is slightly below the critical value. It is shown that the resulting equation has complex coefficients and is of Ginzburg-Landau type. Thus, the Ginzburg-Landau model may be used to analyze the dynamics of shallow flows in a weakly nonlinear regime.