1. Linear stability analysis of the formation of beach cusps Norihiro Izumi, Tohoku University Asako Tanikawa, Fuji Film Sofware CO. LTD Hitoshi Tanaka, Tohoku University

2. Beach cusps observed on Sendai Coast

3. Conceptual diagram of beach cusps and rip currents

4. Linear stability analysis Impose transverse perturbations on a beach Study the initial growth of perturbations ~ ~ wave setup wave setdown ~ ~ ~ ~ ~

5. Revisiting Hino’s analysis When the wave crest is parallel to the shoreline, the dominant wavenumber does not appear. Perturbations with infinitesimally small wavelengths grow fastest.

6. The boundary conditions and matching conditions in Hino’s Cross-shore velocity vanishes right at the shoreline Matching solutions at the wave breaking point The shoreline is not shifted by perturbation The wave breaking point is not shifted by perturbation Matching Cross-shore velocity vanishes when the total depth vanishes Waves break when a wave breaking condition is satisfied

7. Governing equations Momentum Eqs. Continuity Eq. of water Continuity Eq. of sediment radiation stress tensor bed shear stress vector coefficient between sediment transport rate and velocity

8. Radiation stress : energy per unit width and unit length ： wave velocity and group velocity : amplitude of waves Outside the wave breaking zone Inside the wave breaking zone

9. Bed shear stress Assuming that the incident angle of waves is zero ：bottom friction coefficient ：maximum orbital velocity near the bottom Outside the wave breaking zone Inside the wave breaking zone

10. Nondimensionalization

11. Nondimensional governing eqs. Outside the wave breaking zone Inside the wave breaking zone

12. Asymptotic expansions A：amplitude of perturbations λ：wavenumber of perturbations in the y direction ｐ ：growth rate of perturbations

13. O(1): the base state solution assumeing a linear beach profile Inside: Outsize:

14. O(A): the perturbed problem Inside the wave breaking zone Outside the wave breaking zone

15. The boundary conditions and the matching conditions Solutions inside and outside the wave breaking zone

16. Results A peak of the growth rate appears aroundl=6 Spacing of cusps The dominant wave number