COMPARISON OF ANALYTICAL AND
NUMERICAL APPROACHES FOR
LONG WAVE RUNUP
by
ERTAN DEMİRBAŞ
MAY, 2002
The runup phenomena is one of the
important
subject
for
coastal
development in coastal engineering. The
hazard of long waves generated by
earthquakes have in many cases causes
deaths and extensive destructions near
the coastal regions.
On this basis many studies on long wave
runup phenomena have been presented
numerically and analytically.
In this thesis, the runup of long waves
is investigated using a numerical
model. The results are presented and
discussed
with
analytical
and
experimental studies.
For different wave profiles;
Solitary wave and
N-wave
the runup characteristics have been
investigated.
INTRODUCTION
Different from wind generated waves, the
length of long waves are longer
comparing to water depth.
Wind waves show orbital motion, on the
other hand long waves show translatory
motion.
It losses very little energy while it is
propagating in deep water. The velocity
is directly proportional to the square root
of the depth.
C = √(g x d)
As the water depth decreases, the
speed of the long wave starts to
decrease. However the change of the
total energy remains constant.
Therefore while the speed is
decreasing, the wave height grows
enormously.
The Study of Long Wave
Runup Phenomena
The study of long wave runup has direct
consequence to tsunami hazard assessment and
mitigation in coastal region.
Generally the long waves have been modeled as
Solitary Waves. Some examples are Carrier &
Greenspan (1958), Shuto (1967), Pedersen &
Gjevik (1983), Synolakis (1987).
Recently N-waves have been modeled to
describe the long wave characteristics (Tadepalli
and Synolakis, 1994).
The Necessity of Numerical Studies
The earlier studies on long wave runup
relied largely on analytical approaches.
Although the analytical studies provide
simple
analytical
solutions,
their
applications are limited due to
Complex beach geometry,
Different generation parameters, and
Different wave parameters
Therefore the numerical studies are
necessary to simulate propagation and
coastal amplification of long waves in
irregular topographies.
This would enable us to evaluate the
risks near coastal regions and mitigate
the
possible
regions.
hazards
on
coastal
H
O
W
E
V
E
R
The problem is to
develop an adequate
numerical model to
describe the physical
phenomena accurately.
LITERATURE SURVEY
When
studying
long
wave
countermeasures, much attention is
paid to runup and inundation of the
beach.
The runup of long waves have been
studied
using
analytical
and
numerical approaches as well as
experimental studies.
Analytical Approaches
The runup of long waves have been
studied analytically by,
 Synolakis (1987)
 Pelinovsky and Mazova (1991)
 Tadepalli and Synolakis (1994)
 Pelinovsky, Kozyrev and Troshina (1995)
 Kanoğlu and Synolakis (1998)
Numerical Approaches
 Lin, Chang and Liu studied a combined
experimental and numerical effort on
solitary wave runup and rundown on
sloping beaches (1999).
 Titov and Synolakis (1995) developed a
finite difference model using Godunov
method to simulate the long wave runup
of breaking and non-breaking solitary
waves.
 Also Zelt (1991), Kobayashi (1987) and
Liu (1995) studied the same problem.
Our Numerical Model
In this study the numerical model
TUNAMI-N2 is used to simulate different
cases.
TUNAMI-N2 is one of the key tools
incorporating the shallow water theory
consisting of non-linear wave equations
for developing studies with different
initial conditions.
Governing Equations
The basic equations used in the model
are the nonlinear form of long wave
equations as follows.
η  uh + η  v h + η


0
t
 x
 y
u
u
u
  x
u
v
g

0
t
x
y
x 
v
v
v
  y
u
v
g

0
t
x
y
y 
Those equations above sometimes do not
satisfy the conservation of mass principle.
Therefore in the model the equations below
satisfying both the conservation of mass and
momentum principles are used.
  M  N


0
t x y
 M   M 2    MN 
 gn 2

 

 7/3 M M 2  N 2  0

  gD
 t  x D   y D 
x D
 N   MN    N 2 
 gn 2

  gD

 7/3 N M 2  N 2  0


 t  x D   y D 
 y D
ANALYTICAL APPROACHES FOR
SOLITARY WAVE RUNUP
The key goal in analytical approaches is
to introduce a relation between Runup (R)
and Wave Height (H).
Analytical
studies
provide
simple
solutions however their applications are
generally limited to idealized cases.
Runup of Solitary Waves
Synolakis
(1987)
presented an
empirical
relationship
between the
normalized
runup and the
normalized
wave height.
y
z
x
Hinput
Gauges
d
Toe

Runup Law
1
2
R
H
 2.831(cot β) ( )
d
d
5
4
The normalized maximum runup of Solitary Waves
up a 1:19.85 beach versus the normalized wave
height
Obviously,
the runup
variation is
different for
breaking and
non-breaking
solitary
waves as
shown in
figure
(Synolakis,
1987).
The breaking criterion of
Solitary Waves derived by
Gjevik & Pedersen (1981)
H
 0.479(cot β)
d
10
9
Runup of N-waves
N-Wave
Tadepalli
and
Synolakis
(1994)
introduced N-waves firstly in the
modeling of long waves.
They discussed relatively general Nwave profile and two special cases
of N-waves as the isosceles N-wave
and the double N-wave
Generalized N-Wave
H elevation
Generalized N-wave
H depression
The maximum runup of the
generalized N-wave is
R  2.831ε cot β0 H
5
4
X
1
H elevation >H depression
 X 2  0.366 / γ s   0.618/γ s
Isosceles N-Wave
Isosceles N-wave
H elevation
The maximum runup of the
isosceles N-wave is
1
2
R  3.86(cot β 0 ) H
5
4
H depression
H elevation =H depression
Numerical Applications
For linear basins, more than 300 different
simulations were carried out.
The aim is to discuss the non-linear
numerical results with the linear and also
a few non-linear analytical approaches
and experimental studies.
Selected Basins
Three different basins are used to simulate different
initial conditions. The slopes are selected as 1:10,
1:20 and 1:30.
The grid size and time step is selected as 20 m and
0.25 seconds respectively in order to satisfy
stabilities.
10 000
cot 
d=30
5 000
5 000
1
Initial Wave
0.5
0.6
0.4
0.4
0.3
0.2
0.2
0
0.1
-0.2
0
-0.4
4
5
6
7
Solitary Wave
8
4
5
6
N-Wave
7
8
Climbing of Solitary Wave
The climb of a solitary wave with H/d=0.019 up a 1:19.85 beach
(at the toe of the slope)
Climbing of Solitary Wave
The climb of a solitary wave with H/d=0.019 up a 1:19.85 beach
Climbing of Solitary Wave
The climb of a solitary wave with H/d=0.019 up a 1:19.85 beach
Runup of Solitary Waves
R/d
1,0E+00
1,0E-01
1,0E-02
1,0E-03
1,0E-02
1,0E-01
H/d
Run-up Law (1/10)
Sol Ws (1/10) Non-Breaking
Sol Ws (1/10) Breaking
1,0E+00
Runup of Solitary Waves
R/d
1,0E+00
1,0E-01
1,0E-02
1,0E-03
1,0E-02
1,0E-01
H/d
Run-up Law (1/10)
Sol Ws (1/10) Non-Breaking
Sol Ws (1/10) Breaking
1,0E+00
Runup of Solitary Waves
R/d
1,0E+00
1,0E-01
1,0E-02
1,0E-03
1,0E-02
1,0E-01
H/d
Run-up Law (1/20)
Sol Ws (1/20) Non-Breaking
1,0E+00
Runup of Solitary Waves
R/d
1,0E+00
1,0E-01
1,0E-02
1,0E-03
1,0E-02
1,0E-01
H/d
Run-up Law (1/20)
Sol Ws (1/20) Non-Breaking
Sol Ws (1/20) Breaking
1,0E+00
Runup of Solitary Waves
R/d
1,0E+00
1,0E-01
1,0E-02
1,0E-03
1,0E-02
1,0E-01
1,0E+00
H/d
Run-up Law (1/20)
Sol Ws (1/20) Non-Breaking
Sol Ws (1/20) Breaking
Lab Data Non-Breaking
Lab Data Breaking
Linear Data
Runup of Solitary Waves
R/d
1,0E+00
1,0E-01
1,0E-02
1,0E-03
1,0E-02
1,0E-01
1,0E+00
H/d
Run-up Law (1/20)
Sol Ws (1/20) Non-Breaking
Sol Ws (1/20) Breaking
Lab Data Non-Breaking
Lab Data Breaking
Linear Data
Runup of Solitary Waves
R/d
1,0E+00
1,0E-01
1,0E-02
1,0E-03
1,0E-02
1,0E-01
H/d
Run-up Law (1/20)
Lab Data Non-Breaking
Sol Ws (1/20) Non-Breaking
Lab Data Breaking
Sol Ws (1/20) Breaking
Linear Data
1,0E+00
Runup of Solitary Waves
R/d
1,0E+00
1,0E-01
1,0E-02
1,0E-03
1,0E-02
1,0E-01
H/d
Run-up Law (1/20)
Lab Data Non-Breaking
Sol Ws (1/20) Non-Breaking
Lab Data Breaking
Sol Ws (1/20) Breaking
Linear Data
1,0E+00
Runup of Solitary Waves
R/d
1,0E+00
1,0E-01
1,0E-02
1,0E-03
1,0E-02
1,0E-01
H/d
Run-up Law (1/30)
Sol Ws (1/30) Non-Breaking
Sol Ws (1/30) Breaking
1,0E+00
Runup of Solitary Waves
R/d
1,0E+00
1,0E-01
1,0E-02
1,0E-03
1,0E-02
1,0E-01
H/d
Run-up Law (1/30)
Sol Ws (1/30) Non-Breaking
Sol Ws (1/30) Breaking
1,0E+00
Runup of Generalized N-waves
1,0E+00
R/d
1,0E-01
1,0E-02
1,0E-02
1,0E-01
2.831e(cot o )1/2H5/4[(-a-0.366/g)+0,618/g]
Leading Depression N-Wave (1:10)
1,0E+00
Runup of Generalized N-waves
1,0E+00
R/d
1,0E-01
1,0E-02
1,0E-02
1,0E-01
1,0E+00
2.831e(cot o )1/2H5/4[(-a-0.366/g)+0,618/g]
Leading Depression N-Wave (1:10)
Leading Elevation N-Wave (1:10)
Runup of Generalized N-waves
1,0E+00
R/d
1,0E-01
1,0E-02
1,0E-02
1,0E-01
1,0E+00
2.831e(cot o )1/2H5/4[(-a-0.366/g)+0,618/g]
Asymptotic Expression
Leading Depression N-Wave (1:10)
Leading Elevation N-Wave (1:10)
Runup of Generalized N-waves
1,0E+00
R/d
1,0E-01
1,0E-02
1,0E-02
1,0E-01
2.831e(cot o )1/2H5/4[(-a-0.366/g)+0,618/g]
Leading Depression N-Wave (1:20)
1,0E+00
Runup of Generalized N-waves
1,0E+00
R/d
1,0E-01
1,0E-02
1,0E-02
1,0E-01
1,0E+00
2.831e(cot o )1/2H5/4[(-a-0.366/g)+0,618/g]
Leading Depression N-Wave (1:20)
Leading Elevation N-Wave (1:20)
Runup of Generalized N-waves
1,0E+00
R/d
1,0E-01
1,0E-02
1,0E-02
1,0E-01
1,0E+00
2.831e(cot o )1/2H5/4[(-a-0.366/g)+0,618/g]
Asymptotic Expression
Leading Depression N-Wave (1:20)
Leading Elevation N-Wave (1:20)
Runup of Generalized N-waves
1,0E+00
R/d
1,0E-01
1,0E-02
1,0E-02
1,0E-01
2.831e(cot o )1/2H5/4[(-a-0.366/g)+0,618/g]
Leading Depression N-Wave (1:30)
1,0E+00
Runup of Generalized N-waves
1,0E+00
R/d
1,0E-01
1,0E-02
1,0E-02
1,0E-01
1,0E+00
2.831e(cot o )1/2H5/4[(-a-0.366/g)+0,618/g]
Leading Depression N-Wave (1:30)
Leading Elevatsion N-Wave (1:30)
Runup of Generalized N-waves
1,0E+00
R/d
1,0E-01
1,0E-02
1,0E-02
1,0E-01
1,0E+00
2.831e(cot o )1/2H5/4[(-a-0.366/g)+0,618/g]
Asymptotic Expression
Leading Depression N-Wave (1:30)
Leading Elevatsion N-Wave (1:30)
Runup of Isosceles N-wave
1,0E-01
R/d
1,0E-02
1,0E-03
1,0E-03
1,0E-02
1,0E-01
1/2
5/4
3.86(cot o ) H
Asymptotic Expression
Isos. Leading Depression (10)
Isos. Leading Elevation (10)
Runup of Isosceles N-wave
1,0E-01
R/d
1,0E-02
1,0E-03
1,0E-03
1,0E-02
1,0E-01
1/2
5/4
3.86(cot o ) H
Asymptotic Expression
Isos. Leading Depression (10)
Isos. Leading Elevation (10)
Runup of Isosceles N-wave
1,0E-01
R/d
1,0E-02
1,0E-03
1,0E-03
1,0E-02
1,0E-01
1/2
5/4
3.86(cot o ) H
Asymptotic Expression
Isos. Leading Depression (10)
Isos. Leading Elevation (10)
Runup of Isosceles N-wave
1,0E-01
R/d
1,0E-02
1,0E-03
1,0E-03
1,0E-02
1,0E-01
1/2
5/4
3.86(cot o ) H
Asymptotic Expression
Isos. Leading Depression (20)
Isos. Leading Elevation (20)
Runup of Isosceles N-wave
1,0E-01
R/d
1,0E-02
1,0E-03
1,0E-03
1,0E-02
1,0E-01
1/2
5/4
3.86(cot o ) H
Asymptotic Expression
Isos. Leading Depression (20)
Isos. Leading Elevation (20)
Runup of Isosceles N-wave
1,0E-01
R/d
1,0E-02
1,0E-03
1,0E-03
1,0E-02
1,0E-01
1/2
5/4
3.86(cot o ) H
Asymptotic Expression
Isos. Leading Depression (20)
Isos. Leading Elevation (20)
Runup of Isosceles N-wave
1,0E-01
R/d
1,0E-02
1,0E-03
1,0E-03
1,0E-02
1,0E-01
1/2
5/4
3.86(cot o ) H
Asymptotic Expression
Isos. Leading Depression (30)
Isos. Leading Elevation (30)
Runup of Isosceles N-wave
1,0E-01
R/d
1,0E-02
1,0E-03
1,0E-03
1,0E-02
1,0E-01
1/2
5/4
3.86(cot o ) H
Asymptotic Expression
Isos. Leading Depression (30)
Isos. Leading Elevation (30)
Runup of Isosceles N-wave
1,0E-01
R/d
1,0E-02
1,0E-03
1,0E-03
1,0E-02
1,0E-01
1/2
5/4
3.86(cot o ) H
Asymptotic Expression
Isos. Leading Depression (30)
Isos. Leading Elevation (30)
Discussion
In overall approach the numerical results
show the same trend with analytical and
experimental approaches.
Especially the climb of the solitary wave
up a 1:19.85 slope beach shows that the
numerical model results almost similar
values according to the available
experimental study.
For the runup calculations, the numerical
model results lower runup values
compared with analytical studies in both
Solitary Waves and N-waves.
The trend of the relation between the
normalized runup and initial wave
amplitude at the toe of the slope is
consistent for slopes steeper than 1:30 for
non-breaking solitary wave.
The underestimation observed in
numerical results is
believed to be the effect of the
difference between the actual
runup and calculated numerical
runup
thought to be the result of higher
reflection
For
N-waves,
the
analytical
approach gives an approximate
upper
limit.
Furthermore
the
analytical results have not been
supported by experimental studies
yet.
The numerical results are found
below the upper limit as expected.
One important discussion for N-waves
is also the effect of wave form on wave
runup.
Surprisingly, the maximum computed
runup of leading depression N-wave is
calculated as higher than the runup of
leading elevation N-wave.
The numerical model can be improved
for the calculation of wave runup.
This application can be extended to
milder slopes.
This
study
can
be
extended
to
investigate the runup of the long waves
for piecewise linear topographies.
THANK YOU…
Ertan Demirbas - 2002