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 ~ ~ ~ ~ ~
Linear stability analysis

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 Outside the wave breaking zone
　 : 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.
Inside the wave breaking zone
Outside 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 around l=6 Spacing of cusps
The dominant wave number