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Weakly nonlinear analysis of dunes by the use of a sediment transport formula incorporating the pressure gradient Satomi Yamaguchi (Port and airport Institute, Japan) Norihiro Izumi (Tohoku University, Japan) Transition of dune formation Water depth variation during flood (Simons and Richardson, 1961) Hysteresis in the transition Weakly nonlinear stability analysis (Yamaguchi & Izumi, 2003) - The subcritical bifurcation appears in the transition Transition of dune formation Weakly nonlinear analysis (Yamaguchi & Izumi, 2003) - the subcritical bifurcation in the transition - effect of bed inclination in sediment transport formulas [ the gravity effect ] - Fredsoe’s formula The subcritical bifurcation appears when the effect of bed inclination is small. - Kovacs-Paker’s formula The subcritical bifurcation does not appear. | The gravity effect on inclined bed is estimated to be large. Effect of pressure gradient Pressure distribution on dune form (Raudkivi, 1963) Effect of bed inclination [the gravity] [resistance due to the pressure gradient] In this study, a bedload transport formula incorporating the pressure gradient on the basis of Kovacs-Paker’s formula for the weakly nonlinear analysis of dunes. Effect of pressure gradient on bedload transport The coordinate system - on the basis of Kovacs-paker’s formula - 2D (bed inclination in the streamwise direction) Effect of pressure gradient on bedload transport Force balance on a sediment particle drag force gravity force resistance due to pressure gradient dynamic Coulomb friction force the force balance in the tangential direction to the bed Sediment particle velocity vp Effect of pressure gradient on bedload transport Force balance on the bedload layer bed shear stress gravity force the sum of the pressure p 1 - p 4 grain stress fluid shear stress at the bottom ( ) the force balance in the tangential direction to the bed Volume of particle in the bedload layer x Effect of pressure gradient on bedload transport Bedload sediment transport rate volume of particle in the bedload layer the coordinate system critical Shields shear stress particle velocity - Effect of pressure gradient on bedload sediment transport : decrease : increase - Balance between the gravity and pressure gradient tangential to the bed Application to analysis of dunes - Formulations flow (2D Reynolds equations ) time variation of bed elevation - Linear stability analysis - Weakly nonlinear stability analysis the coordinate system Application to analysis of dunes - Linear stability analysis W : growth rate, k : wave number - Weakly nonlinear stability analysis the growth rate expansion method Landau equation Landau constant a 1 < 0 : supercritical bifurcation a a 1 > 0 : subcritical bifurcation | hysteresis in the transition Fc b Subcritical bifurcation Results of the linear stability analysis Instability diagram S = 0.002 (average bed slope) stable mc = 0.84 (dynamic Coulomb friction coefficient) t*co= 0.047 (the critical Shields shear stress for a horizontal bed) unstable neutral curve Kovacs-Parker’s formula the present formula incorporating pressure gradient Results of the weakly nonlinear stability analysis The present formula incorporating pressure gradient Kovacs-Parker’s formula a1 < 0 Landau constant supercritical bifurcation a1 > 0 subcritical bifurcation | hysteresis in the transition Balance between the gravity and pressure gradient phase difference (from the phase of bed form) out of phase : 0.500p | The effect of the gravity is reduced by the pressure gradient gravity pressure gradient : -0.546p the phase of linear solutions Conclusions - We propose a bedload formula incorporating pressure gradient for the analysis of dunes. - The weakly nonlinear analysis with the use of the proposed formula shows that the subcritical bifurcation occurs in the transition between dune-covered and flat beds even if the effect of bed inclination is reasonably large. - It is found that the pressure gradient reduces the gravity effect in the bedload formula.