1. 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)

2. Transition of dune formation Weakly nonlinear stability analysis (Yamaguchi & Izumi, 2003) - The subcritical bifurcation appears in the transition Water depth variation during flood (Simons and Richardson, 1961) Hysteresis in the transition

3. Transition of dune formation Weakly nonlinear analysis (Yamaguchi & Izumi, 2003) - the subcritical bifurcation in the transition - effect of bed inclination in sediment transport formulas - Kovacs-Paker ’ s formula The subcritical bifurcation does not appear. - Fredsoe ’ s formula The subcritical bifurcation appears when the effect of bed inclination is small. | The gravity effect on inclined bed is estimated to be large. [ the gravity effect ]

4. Effect of 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 bed inclination Pressure distribution on dune form (Raudkivi, 1963) [the gravity] [resistance due to the pressure gradient]

5. 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)

6. 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 Sediment particle velocity v p the force balance in the tangential direction to the bed

7. Force balance on the bedload layer Volume of particle in the bedload layer  Effect of pressure gradient on bedload transport the force balance in the tangential direction to the bed bed shear stress gravity force the sum of the pressure p 1 - p 4 grain stress fluid shear stress at the bottom ( )

8. Effect of pressure gradient on bedload transport Bedload sediment transport rate particle velocity the coordinate system volume of particle in the bedload layer critical Shields shear stress - Effect of pressure gradient on bedload sediment transport : decrease : increase - Balance between the gravity and pressure gradient tangential to the bed

9. Application to analysis of dunes the coordinate system - Formulations flow (2D Reynolds equations ) time variation of bed elevation - Weakly nonlinear stability analysis - Linear stability analysis

10. Application to analysis of dunes - Linear stability analysis  : growth rate, k : wave number - Weakly nonlinear stability analysis the growth rate expansion method Landau equation Landau constant  1 < 0 : supercritical bifurcation  1 > 0 : subcritical bifurcation | hysteresis in the transition FcFc a b Subcritical bifurcation

11. Results of the linear stability analysis neutral curve Kovacs-Parker ’ s formula the present formula incorporating pressure gradient Instability diagram stable unstable S = 0.002 ( average bed slope )  c = 0.84 ( dynamic Coulomb friction coefficient )  * co = 0.047 ( the critical Shields shear stress for a horizontal bed )

12. Results of the weakly nonlinear stability analysis Kovacs-Parker ’ s formula The present formula incorporating pressure gradient Landau constant  1 0 subcritical bifurcation | hysteresis in the transition supercritical bifurcation Balance between the gravity and pressure gradient out of phase | The effect of the gravity is reduced by the pressure gradient gravity : 0.500  pressure gradient : -0.546  phase difference (from the phase of bed form) the phase of linear solutions

13. 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.