Linear long wave propagation over submerged topography
Ravi Shankar, Yan Sheng, Megan Golbek Tucker Hartland, Mark Brandao
Keywords: linear shallow-water equations, discontinuous submerged topography
Introduction
The study of water waves over variable underwater topography has numerous practical
applications. Underwater reefs are built for several purposes, including protection from harbor
damage, beach erosion, and as first lines of defense against destructive tsunami waves.
Analytical solutions have been obtained for piecewise linear topographies [1][4][8] and
internal waves [10]. In this study, a numerical method is developed that can be easily
implemented while giving quantitative agreement with experimental data. The one-dimensional
linear shallow water equations, over discontinuous submerged topography, are solved with finite
difference methods. The problem of the discontinuous boundary conditions is approached by
introducing extra boundary conditions or applying a characteristic decomposition. For the
underwater shelf and obstacle, the numerical solution is compared with analytical solutions and
experimental data available in the literature.
Governing Equations
We introduce the relevant variables and physical constants in Table 1. Unless otherwise
specified, we will use the subscript0 to denote a characteristic dimensional quantity and the
superscript’ to denote a dimensional variable. A characteristic feature of a tsunami wave [3] is
the smallness of the parameter h𝜆 ≪ 1, where h0 is the characteristic water depth of the ocean and
0
λ is the wavelength. The presence of this small parameter allows us to use the shallow water
equations [11][13], for modeling tsunami wave dynamics.
Using the quantities introduced in table 2, non-dimensional variables can be written as:
1
𝑡=
𝑙𝑗′
𝑡′
𝑥′
𝜂 ′ (𝑥 ′ , 𝑡 ′ )
𝑢′ (𝑥 ′ , 𝑡 ′ )
𝜑 ′ (𝑥 ′ ) 2
ℎ𝑖
𝑑0
,𝑥 =
, 𝜂(𝑥, 𝑡) =
, 𝑢(𝑥, 𝑡) =
, 𝜑(𝑥) =
, 𝑘𝑖 =
, 𝑑=
, 𝑙𝑗 =
𝑡0
𝜆0
𝜂0
𝑢0
ℎ0
ℎ0
𝜆0
𝜆0
After substitutions with non-dimensional variables, the linear shallow water equation become the
equation (1).
ℎ
We define 𝑘𝑖 = √ℎ 𝑖 , where ℎ𝑖 is the depth of the water above the sea-floor in the i-th
0
region (see Figure 1); 𝑘0 = 1 by our definition.
To account for the discontinuous changes in the sea-floor depth at each interface between
two boundaries (see Figure 1), the spatial domain is partitioned into n + 1 regions; the space to
the right of the x = 0 boundary, which we will call the right region, and n subdivisions of the
space to the left of x = 0. For continuity of pressure and mass from equations (1) [5], the
following conditions are imposed at the interface between the i-th and i+1-th region can be seen
in equation (2).
An isolated wave over a shelf (Figure 1)
In variables 𝑣 = 𝑥 + 𝑡, 𝜉 = 𝑥 − 𝑡, 𝑧 = 𝑥 − 𝑘𝑡, 𝑎𝑛𝑑 𝜁 = 𝑥 + 𝑘𝑡, analytical solutions can be
seen in equations (3) and (4). The obtained numerical solutions should completely agree with
analytical solutions (3) and (4).
Numerical solution using Riemann variables
As an alternative method to verify the first numerical solution obtained with extra
boundary conditions, the numerical solution was also obtained in Riemann variables. To simplify
⃗ = (𝜂 ). Using unknown functions, 𝑝(𝑥, 𝑡), 𝑞(𝑥, 𝑡), and the
notation, we introduce the vector, 𝑉
𝑢
vector ⃗⃗𝑟 = (𝑝𝑞), we obtain a system a decoupled set of PDEs. Solutions to this decoupled system
of equations can be found using the method of characteristics, and are of the form 𝑝(𝑥, 𝑡) =
𝛾(𝑥 − 𝑘1 𝑡), 𝑞(𝑥, 𝑡) = 𝛾(𝑥 + 𝑘1 t) for some arbitrary function 𝛾, where 𝑞(𝑥, 𝑡) describes waves
2
that propagate right to left. We can solve for, the initial conditions in Riemann variables, strictly
in terms of 𝑓(𝑥)and 𝑔(𝑥) can be seen in (5). By solving equation (2) for 𝑝0 and 𝑞1 algebraically
we have our boundary values, seen in (6).
Description of numerical methods
The numerical solutions were constructed using Mathematica 9 and utilized several
second-order accurate finite differencing schemes[14]. For all of our numerical solutions, we
2
used the Gaussian function, 𝑓 (𝑥) = 𝑎𝑒 −(𝑥−𝑑0 ) , centered at 𝑥 = 𝑑0 in creating our initial profile.
Since computers have finite memory, we must restrict ourselves to artificial boundaries 𝑥𝐿
and 𝑥𝑅 . Thus we have three steps in our computational process; we must be able to calculate the
solutions at our artificial boundaries, satisfy the given boundary conditions and calculate the
intermediate values.
⃗
⃗
⃗ near 𝑥 = 0 we use (2), 𝑘12 𝜕𝑉1 = 𝜕𝑉0 . We first approximate the
To compute the values of 𝑉
𝜕𝑥
𝜕𝑥
derivatives in the previous equation, with a finite backward differencing scheme for the left
region and a forward differencing scheme for the right region [6]. We use bold numerical
subscripts to indicate the regions and use the common alphanumerical superscripts and subscripts
𝑛
𝑛
⃗𝟏 = 𝑉
⃗𝟎 ,
to denote the time and spatial steps in our approximation, respectively. Since 𝑀𝑉
0
0
where the matrix 𝑀 = (
𝑛
⃗𝟏 ,
solve for 𝑉
0
1 0
), we can make a substitution into the previous expression and
0 𝑘1 2
𝑛
2
3
𝟏
2∆𝑥𝟎
⃗ 𝟏 = ( 3𝑘𝟏 𝐼 +
𝑉
0
2∆𝑥
𝑛
𝑛
2
𝑛
𝑛
2
2𝑘
⃗𝟎 − 1 𝑉
⃗
⃗𝟏 − 𝑉
⃗ 𝟏 ),
𝑀 )−1 (∆𝑥 𝑉
+ ∆𝑥1 𝑉
−1
𝑗−2
1
2∆𝑥 𝟏 −1
𝟎
𝟏
1
where ∆𝑥1 and ∆𝑋0 are the spatial steps for the left and right regions, respectively and 𝐼 =
(
𝑛
1 0
⃗𝟏 =
). Finally, we can solve for the boundary values at 𝑥 = 0 on the right side using 𝑀𝑉
0
0 1
𝑛
⃗ 𝟎 . We apply a characteristic decomposition to determine the values at the artificial endpoints.
𝑉
0
3
Depending on the directions of the characteristic curves of the Riemann invariants, we either set
the endpoint invariants to zero or determine their values by an upwind scheme.
For the solution with extra boundary conditions, we first convert the 𝜂 and 𝑢 values at the
endpoints to Riemann variables, determine the endpoints, and then convert back to physical
variables. We use the Beam-Warming method [6] with one-sided differencing to upwind. Below
is the scheme for the left region; the right- region formula can be obtained with a simple change
of indices, can be seen in (7), where 𝐴 = (0 𝜑(𝑥)).
1
0
Lastly, we used the Lax-Wendroff method [6] with center differencing to calculate the
intermediate values. The scheme for both regions can be seen in (8).
Single obstacle (Figure 2)
We may think of this configuration as the shelf-configuration with one extra region. We
must impose similar boundary conditions to obtain a physical model, while keeping in mind that
there is an additional boundary to consider. The vector notation can be seen in (9) and (10).
Analogous to the shelf-model, we must satisfy boundary conditions (9) and (10), as well as
calculate both the intermediate and artificial boundary values. We can find formulas to calculate
the boundary values at 𝑥 = 0 and 𝑥 = 𝑥𝑀 using the methods discussed in the Riemann variable
section. Equation (10) can be rewritten using derivative approximations, a substitution from (9),
𝑛
⃗𝟐 :
and after solving for 𝑉
𝑏
𝑛
⃗𝟐 = (
𝑉
𝑏
3𝑘𝟐2
3𝑘𝟏2 −1
2𝑘𝟏2 𝑛
𝑘2
2𝑘 2 𝑛
𝑘2 𝑛
𝑛
⃗𝟏 − 𝟏 𝑉
⃗𝟏 + 𝟐 𝑉
⃗𝟐 − 𝟐 𝑉
⃗
𝐼+
𝑀𝟏 𝑀𝟐 )−1 (
𝑉
)
2∆𝑥𝟐
2∆𝑥𝟏
∆𝑥𝟏 2 2∆𝑥𝟏 2 ∆𝑥𝟐 −1 ∆𝑥𝟐 𝟐 −2
The intermediate and artificial boundary values were calculated in the same way as for the
shelf-model, with taking into consideration the additional boundary at, 𝑥 = 𝑥𝑀 .
Two-obstacles configuration (Figure 4)
4
As is the case for the one obstacle model, the configuration with two obstacles is an
extension of the shelf-model. The derivative conditions at the boundaries can be seen in (11) and
(12)
Discussion
Numerical solutions were compared with analytical solutions if available. It is clear that
the constructed numerical solutions accurately approximate the analytic solution (Figure 4). Thus
it appears that our solutions accurately capture the physical behavior of this system. We can see
from Figures 6 and 7 that our numerical models eventually diverge from experimental results as
the amplitude of the incident wave becomes sufficiently large. This is because our models were
constructed from the linear shallow water equations and neglect non-linear effects [14]. To
determine a benchmark for when our models are appropriate, we examined the intervals over
which our models began to diverge from the experimental data. Benchmarks were calculated for
the shelf, values ranged from .8% (for H = 18.1cm) to 13.8% (H = 30.0cm,), and obstacle
configuration, the obstacle ranged from 9.7-18.8% of H, of water depth. These results show that
our linear shallow water theory is valid to 10% difference for relative wave amplitudes ℎ𝜂00 ≼ 0.1.
Conclusions
An easily implementable numerical method was developed to model the dynamics of
isolated linear long waves propagating over arbitrary underwater geometries. Despite its
simplicity, the model was able to provide quantitative agreement with experimental data. This
model, valid for long incident waves with small amplitude relative to the water depth, can
accurately simulate discontinuously varying water depths. The one-dimensional linear shallow
water equations over discontinuous submerged topography were solved with finite difference
methods to construct this model. Several underwater configurations were simulated to test the
5
model: a piecewise constant underwater shelf, a flat sea- floor with a symmetric rectangular
obstacle, a series of obstacles and a sloping beach approximated by stairs. Numerical solutions
for the shelf configuration were verified against analytical solutions and reflection and
transmission coefficients obtained from the shelf and obstacle simulations were found to be
comparable to those in literature [7][9]. Coastal engineers can use this code to find the optimal
amount of barriers used along the coast to prevent future damage caused by tsunami waves.
7 Acknowledgments
The authors are supported in part by NSF award DMS-0648764, and the Undergraduate
Research Opportunities Center of California State University, Monterey Bay.
Tables and Figures
Table 1: Relevant variables and physical constants.
𝜂
vertical elevation of water from quiescent position
𝑢
depth-averaged horizontal flow velocity
ℎ
distance from water surface to sea-floor
ℎ0
dimensional water depth of sea-floor away from obstacle
ℎ𝑖
dimensional water depth over i-th region
𝑘𝑖
dimensionless water depth over i-th region; 𝑘𝑖2 = ℎℎ0𝑖
𝑥
horizontal position coordinate
𝑡
time variable
𝑓
initial waveform distribution
𝜑
𝑙𝑗 𝑑
𝜂0
𝑢0
𝑐0
𝜆0
𝜆
𝑡0 = 0
𝑐0
ocean floor topography
length of j-th obstacle
distance of solitary wave peak from the x = 0 boundary
characteristic wave amplitude
characteristic fluid velocity
characteristic velocity of wave propagation
characteristic length of initial wave
characteristic time of propagation
Table 2:
η0
u0
c0
λ0
t0 =
characteristic wave amplitude
characteristic fluid velocity
characteristic velocity of wave propagation
characteristic length of initial wave
λ0
c0
characteristic time of propagation
6
Equations:
1
xL
0
x 0
Η0
xL
xR
2
Λ0
h1
1
xM
h1
h2
d0
Η0
Λ0
d0
h0
Figure 1: Schematic of a solitary wave, with
passing over a shelf (from right to left).
xB2
xM
xB1
xR
l1
h0
xL
0
x 0
Figure 2: Schematic of a solitary wave
passing over an obstacle.
xR
x 0
Η0
h3
h4
Λ0
h1
h2
l1
d0
h0
Figure 3: Schematic of a
solitary wave passing over two
obstacles.
l2
4
3 2
1
0
Plots of Numerical solutions:
7
Figure 4: Plots of numerical solutions with analytic solutions (3) and (4) at various moments in time (t = 0 and t = 5,
respectively). The strong overlapping suggests an accurate approximation by the numerical solutions.
Figure 5: Plots of numerical solution in Riemann variables for two-obstacle configuration (t = 0, t = 5, t = 11,
respectively). Note the rippling effect caused by multiple transmissions and reflections.
Shelf Configuration:
Figure 6: Plots of transmitted wave amplitudes against incident wave amplitudes. Physical parameters were
converted to dimensionless forms.
Single Obstacle Configuration:
8
The graph on the left (Figure 7) is a comparison reflected wave amplitudes from our numerical method and
experimental data found in literature. On the right (Figure 8) is a plot of transmission coefficients against obstacle
height; single obstacle with rectangular shape [7]. Physical parameters were converted to dimensionless forms.
Citations:
.
.
.
.
.
.
.
.
.
.
.
.
.
.
[1] Chang, H. K., & Liou, J. C. (2007). Long wave reflection from submerged trapezoidal breakwaters. Ocean
Engineering, 34(1), 185-191.
[2] Courant, R., Friedrichs, K., & Lewy, H. (1967). On the partial difference equations of mathematical physics.
IBM journal of Research and Development, 11(2), 215-234.
[3] Goring, D. G. (1978). Tsunamis–the propagation of long waves onto a shelf (Doctoral dissertation, California
Institute of Technology).
[4] Kˆanoglu, U., & SYNOLAKIS, C. E. (1998). Long wave runup on piecewise linear topographies. Journal of
Fluid Mechanics, 374, 1-28.
[5] Lamb, H. (1947). Hydrodynamics, Gostekhizdat. Moscow and Leningrad.
[6] LeVeque, R. J., & Le Veque, R. J. (1992). Numerical methods for conservation laws (Vol. 132). Basel:
Birkha ̈user.
[7] Lin, P. (2004). A numerical study of solitary wave interaction with rectangular obstacles. Coastal Engineering,
51(1), 35-51.
[8] Lin, P., & Liu, H. W. (2005). Analytical study of linear long-wave reflection by a two-dimensional obstacle of
general trapezoidal shape. Journal of engineering mechanics, 131(8), 822-830
[9] Seabra-Santos, F. J., Renouard, D. P., & Temperville, A. M. (1987). Numerical and experimental study of the
transformation of a solitary wave over a shelf or isolated obstacle. Journal of Fluid Mechanics, 176, 117-134.
[10] Simanjuntak, M. A., Imberger, J., & Nakayama, K. (2009). Effect of stair step and piecewise linear topography
on internal wave propagation in a geophysical flow model. Journal of Geophysical Research: Oceans (19782012),
114(C12).
[11] Stoker, J. J. (2011). Water waves: The mathematical theory with applications (Vol. 36). John Wiley & Sons.
[12] Vincent, G. (1989). Breakwater Choices. Civil EngineeringASCE, 59(7), 64-66.
[13] Voltsynger N.E., Klevannyy K.A., Pelinovskiy E.N., Dlinnovolnovaya dinamika pribrezhnoy zony. L.:
Gidrometeoizdat, 1989. 272 p.
[14] Whitham, G. B. (2011). Linear and nonlinear waves (Vol. 42). John Wiley & Sons.
9