Transcript RUNUP of N–WAVES OVER CONTINENTAL SHELF AND SLOPE

Nonlinear Evolution of Long Waves
Over a Sloping Beach
(Analytical solution to the benchmark problem #1)
by
Utku Kanoğlu
Department of Engineering Sciences
Middle East Technical University
Ankara 06531 Turkey
([email protected])
Nonlinear shallow water wave equations
Y
hs(t)
h(x,t)
X
b
h0(x)
u (h  h0 )x  ht  0
ut  u u x  h x  0
~
~ ~
~
~
~
x~
x / l , h0  h0 /( l tan b ), h  h~ /( l tan b ), u  u~ / g l tan b , and t  ~
t / l / g tan b
u(x,t) horizontal depth-averaged velocity
h(x,t) free-surface elevation
h(x)=x tanb
Carrier & Greenspan
(1958 Water waves of finite amplitude on a sloping beach. JFM 4, 97-109)
Nonlinear shallow water wave equations
Hodograph
transformation
(x, t)
(, )

2
 u 2

u
, x
h , h 
 , t u

16
4
2
2
  ( )  0
Second order ODE
Shoreline is at   0 in the transform space

 ( ,  )   0

 0 (1/  ) 
h (x,0)
2
J 0 ( ) sin(  ) J1 ( ) ( ) d d
?
where ( )  u ( ,0) 
4

h ( ,0)
h ( ,0)
Major difficulty
To drive equivalent initial condition over the transform space given the
initial condition in the physical space.
Carrier, Wu & Yeh
(2003 Tsunami run-up and draw-down on a plane beach. JFM 475, 79-97)
Nonlinear shallow water wave equations
Hodograph
transformation

u2
2
u   , x    h , h    , t  u  
2
2
Tuck & Hwang (1972 JFM)
(, )
(x, t)
4  ( )  0
Second order ODE
 ( ,  )    F (b)G (b,  ,  )db   P(b)G (b,  ,  )db  where P( )   ( , 0) and F ( )    ( , 0)



0
0


1
   b
0
2

 1 b  2  4(  b) 2 

1

G (b,  ,  )  b  J 0 ( ) sin(  / 2) J 0 ( b)d  
K 
 b     b

0
16b
2

  
4

 1
b
16b


K
   b
 2
   4(  b) 2  2  4(  b) 2  2
 /2
where complete ellipticintegral K (k )  
0
d / 1  k sin 2 
Proposed solution
(2004 Nonlinear evolution and runup-rundown of long waves over a
sloping beach. Accepted for publication in JFM)
u 2 
x
 
16 2
4
2
h (x,0)

 ( ,  )    0

 0 (1/  ) 
2

x
2
16
h ( ,0)
J 0 ( ) sin(  ) J1 ( ) ( ) d d where ( )  u  ( ,0) 

u2
1 
h ( ,  ) 

    2 ( )
4
2
4 0


0
4

h ( ,0)

J 0 ( ) J1 ( ) cos( ) d d
1 
 1
 
   2 ( )  
J1 ( ) J1 ( ) sin(  ) d  d 
2 0
0 
 
2
Solution for a particular time t* or a location x* Synolakis (1987 JFM)
i 1  i 
t *  ti
 1

 2
 i 1   i 

8
x*  xi


4
 u u
Comparision of the initial wave profiles
2
h(x,0) = 0.006e0.4444(x4.1209) 0.018e4.0(x1.6384) , x ≈ σ2/16
2
5
Benchmark initial condition
Proposed initial condition
h(m)
0
-5
-10
0
10000
20000
30000
x(m)
40000
50000
Comparision of the free surface elevation and
velocity profiles at given times
-20
-30
0
0
h(m)
u(m/s)
-10
200
400
600
800
-10
u(m/s)
h(m)
0
-20
-30
0
200
400
600
800
0
-200
200
400
x(m)
600
800
h(m)
200
400
600
800
t=175s
-10
-15
-20
0
u(m/s)
0
t=160s
-5
20
10
8
6
4
2
0
-2
0
0
1
0
-1
-2
-3
-4
-200
200
400
600
800
t=220s
0
200
400
x(m)
600
800
Shoreline runup-rundown motion (  0 )
h s (  )  h (0,  )  
1  2
 ( )

0
4


0

J1 ( ) cos( ) d d
1 
 1
 
   2 ( )    J1 ( ) sin(  ) d  d 
2 0
0 2
 
J0(ωσ)=1
2
lim σ ->0 J1(ωσ)/σ=ω/2
 cos(arcsin( /  )) /  2  2 ,   




J
(

)
cos(

)
d


0,
 

0 1


2
2
2
2
  /    (     ),   

1
h s (  )  h (0,  )   
4

X X 
 

X   

 ( )d   (0) 
2
0
1
1

  (0) 
2
2


 2   2
0

0
2
2

2
2

2
d( ) 

d 
d


2
d( ) 
d 
d

Comparision of the shoreline trajectory and
velocity
300
200
x(m)
100
0
-100
-200
0
50
100
150
200
250
300
350
200
250
300
350
u(m/s)
10
0
-10
-20
0
Benchmark solution
Proposed solution
Proposed simplified solution
50
100
150
t(s)
Observations
helevation( x,0)  hdepression( x,0)

 ( ,  )    0

0
elevation( )   depression( )
(1 /  )  2 J 0 ( ) sin(  ) J1 ( ) ( ) d d where ( )  u  ( ,0) 
4

h ( ,0)
elevation( ,  )  depression( ,  )

2
 u 2

u
, x
h , h 
 , t u

16
4
2
2
The extreme values of u and h for the leading-elevation and
-depression initial waveforms are the inverse of each.
Maximum runup of the elevation wave = Minimum rundown of the depression wave
However, spatial and temporal variations of them are different.
Observations
0.04
h
(a)
0.02
0
-0.02
0.04
0.02
h 0
-0.02
-0.04
0.04
0.02
h 0
-0.02
-0.04
0
10
20
30
40
(d)
0
10
20
30
(g)
0
10
20
30
40
0.1
(b)
hs 0.05
0
-0.05
-0.1
0.2
h 0.1
s 0
-0.1
-0.2
0.2
hs 0.1
0
-0.1
-0.2
0
5
10
0
5
10
0
5
10
10
R
0.4
us 0.2
0
-0.2
-0.4
0
5
0
5
0
5
10
R
H
10
~H3/4
-2
10
-2
10
-1
10
-2
10
-2
H
10
H
}
Synolakis (1987)
15
10
-1
~H
10
10
(i)
5/4
-1
15
(f)
15
(l)
-1
~H5/4
-2
10
t
R
10
0.4
us 0.2
0
-0.2
-0.4
(c)
(k)
-1
-2
15
0.1
0.05
0
-0.05
-0.1
t
(j)
10
15
(h)
x
10
15
(e)
us
Tadepalli & Synolakis (1994)
-1
Conclusions
A new solution to the IVP is developed by direct integration, without
resorting to singular elliptic integrals.
Given an initial condition in the physical space (x,t), the derivation of
the equivalent initial condition in the transform space is possible, by
linearizing the hodograph transformation for the spatial variable.
2
h (x,0)
u 2 
x
 
16 2
4

x
2
16
h ( ,0)
Simplified equations are presented for the shoreline elevation and
velocity.
Comparisons show that direct integration yields identical results with
benchmark solution.
Direct integration shows that the runup variation on the leading wave
height are same as derived from asymptotics by Synolakis and
Tadepalli & Synolakis.