Linear stability analysis of the formation of beach cusps

Norihiro Izumi, Tohoku University

Asako Tanikawa, Fuji Film Sofware CO. LTD

Hitoshi Tanaka, Tohoku University

Beach cusps observed on Sendai Coast

Conceptual diagram of beach cusps and rip currents

Linear stability analysis

wave se

tdownwave setup

Impose transverse perturbations on a beach Study the initial growth of

perturbations~

~

~

~

~

~~

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.

The boundary conditions and matching conditions in Hino’s

　0~0~ xu at

0~~

0~ ζhu when

~~~2

h

a whenMatching

Cross-shore velocity vanishes right at the shorelineMatching solutions at the wave breaking point

The shoreline is not shifted by perturbation

BLx~~ atMatching

Cross-shore velocity vanishes when the total depth vanishes

Waves break when a wave breaking condition is satisfied

The wave breaking point is not shifted by perturbation

Governing equations

Momentum Eqs.

Continuity Eq. of water

Continuity Eq. of sediment

:~

,~

,~

yyxyxx SSS

:sC

radiation stress tensor

bed shear stress vector

coefficient between sediment transport rate and velocity

y

S

x

S

hhxg

y

uv

x

uu

xyxxx~

~

~

~

)~~

(

1

)~~

(

~

~

~

~

~~

~

~~

y

S

x

S

hhyg

y

vv

x

vu

yyxyy~

~

~

~

)~~

(

1

)~~

(

~

~

~

~

~~

~

~~

0~

~~~

~

~~~

y

hv

x

hu

y

vC

x

uC

t

h ss~

~

~

~~

~

:~,~yx

Radiation stress

　 : energy per unit width and unit length

： wave velocity and group velocity

: amplitude of waves

Inside the wave breaking zone

1

Outside the wave breaking zone

~~~2 ha

gchgc ~)~~

(~ 2/1 0~

~~(

16

1~2

1~

~~(

16

3~2

3~

22

22

xy

yy

xx

S

hgES

hgES

0~

2

1~

~~~

,2

1~

~2~~

xy

gyy

gxx S

c

cES

c

cES ,

E~

gcc ~,~

a~

2~2

1ag

0~

0~

0~

xy

yy

xx

S

S

S

Bed shear stress

： maximum orbital velocity near the bottom

fC ： bottom friction coefficient

Inside the wave breaking zone Outside the wave breaking zone

Assuming that the incident angle of waves is zero

vhgC fy~)

~~(~ 21

vuwC fx~cossin~cos1~2~ 2

vuwC fy~sin1~cossin~2~ 2

vwCy~~2~

uwCx~~4~

w~

uhgC fx~)

~~(2~ 21

uhk

aC fx

~~~sinh

~~4~

vhk

aC fy

~~~sinh

~~2~

Nondimensionalization

tHgC

LHt

H

gHkk

ahHahyxLyx

wccvuHgwccvu

Bs

BB

BB

BB

gBg

2/1

2/1

21

)~

(

~~~

~,~

/~

),,(~

)~,~

,~

(),,(~

)~,~(

),,,,()~

()~,~,~,~,~(

hxy

uv

x

uu bx

hyy

vv

x

vu by

x

S

hhxy

uv

x

uu xxbx

1

y

S

hhyy

vv

x

vu yyby

1

y

v

x

u

t

h

Nondimensional governing eqs.

Inside the wave breaking zone Outside the wave breaking zone

0

y

hv

x

hu

),2()(2

),( 21 vuhC

yx

)1,3()(16

),( 2 hSS yyxx

),2(),( vuCwyx 2/12/1

12sinh

2

sinh

kh

kh

kh

kaw

,0

y

vh

x

uh

y

v

x

u

t

h

B

Bf

H

LCC ~

~2

Asymptotic expansions

yexAhxhh

yexAx

yexAvv

yexAuu

pt

pt

pt

pt

cos)()(

cos)()(

sin)(

cos)(

10

10

1

1

A ： amplitude of perturbationsλ ： wavenumber of perturbations

in the y directionｐ ： growth rate of perturbations

Outsize: Inside: 　　　　　　　　　　　　　　　

O(1): the base state solution x

h

x d

)(d

8

3

d

d 002

0

)1)(1(0 xZK M 00

MM ZxZh )1(0assumeing a linear beach profile

Outside the wave breaking zone

0d

d

0d

d'

0

02

11

1

101

010

00

01

10

01

vx

uph

vhx

uhuh

vh

Cw

uh

Cw

dx

d

O(A): the perturbed problem

Inside the wave breaking zone

0d

d

0)(d

d)()''(

08

18)(

0d

d

8

31

d

d

8

3

)(

11

1

1001

00100

1112/100

1112/1

00

vx

uph

vhx

uhuh

hvh

C

xx

hu

h

C

0

as 0

0

1

1

1

xh

v

The boundary conditions and the matching conditions

0at 01 xu

101101

11

11

11

)1(')1()1(')1(

)1()1(

)1()1(

)1()1(

ooii

oi

oi

oi

hh

vv

uu

)]1(')1('[)1('2

)]1()1([)1(2

000

1111 ooo

ooo

ha

ha

:,oi Solutions inside and outside the wave breaking zone

Results

A peak of the growth rate appears around =6

The dominant wave numberSpacing of cusps

B

Bf

H

LCC

2

6cc

BC

LL

~

2~ BC LL

~~