Attenuation of ocean waves due to random
perturbations in the seabed profile
Hyuck Chung, Luke Bennetts, Malte Peter
School of Computer & Mathematical Sciences, Auckland U. of Tech.
School of Mathematical Sciences, Univ. of Adelaide
Institute for Mathematics, Univ. of Augsburg
KOZWAVES 2014 - Newcastle
Outline
Introduction
Method of solution
Numerical results
Summary
Schematics
x
z
φ∝
cosh k (z + h) i kx
e
cosh kh
irregular seabed
h
z = −h + b(x)
Background
1. Linear PDEs and boundary conditions
2. Multi-scale expansion: slow variables
3. Attenuation of waves by randomly irregular seabed
4. Ensemble-average and realization-dependent solutions
5. Finding the above within the linear wave theory
Mathematics of waves over irregular seabed
Linear time-harmonic waves in incompressible fluid
I
φ(x, z) : velocity potential of water
2
I ω φ
g
= ∂z φ : free surface condition
(∂x2 + ∂z2 )φ = 0
(g∂z − ω 2 )φ = 0
∂n φ = 0
for
for
for
−h + b(x) < z < 0
z=0
z = −h + b(x)
Scaling regime
Scaling regime based on wavenumber k and small ε
1. The seabed shape given by smooth random process b(x)
2. kh = O(1) and klg = O(1), lg is the correlation length of b(x)
3. lg /h = O(1), the seabed shape b(x) = O(ε), the slope of the
seabed b0 (x) = O(ε)
Realizations of seabed
Stationary process
r
b(x) = σ
M
2 X
cos (Am x + Bm )
M
m=1
Am and Bm are random variables that are determined by the
prescribed probability density and auto-correlation functions.
I
PDF : b(x) has the same normal distribution at any x
I
Auto-correlation : Gaussian function
I
Correlation/characteristic length lg is the standard deviation
(width) of the auto-correlation function
Other examples of b(x)
I
Step-functions : series of random numbers
I
Deterministic deviation from a periodic function: sin (x + εg(x))
In both cases, diffusion of a pulse over the seabed has been
observed.
Multi-scale expansion
Introduction of slow variables
x0 = x, x1 = εx, x2 = ε2 x, ....
Perturbation method
φ = φ0 + εφ1 + ε2 φ2 + · · ·
∂x = ∂x0 + ε∂x1 + ε2 ∂x2 + · · ·
Approximation of the seabed condition
∂n φ(x, −h + b(x)) = ∂z φ − b0 ∂x φ = 0
b2
b2
φz + bφzz + φzzz = b0 φx + bφxz + φxzz
2
2
Attenuation in the leading order wave
Slow-attenuating wave
φ0 satisfies the homogeneous BVP w.r.t. the fast variable x0 .
φ0 (x0 , x1 , x2 ) =
i gA(x1 , x2 ) cosh k (z + h) i kx0
e
ω2
cosh kh
where k is the real root of the dispersion equation
gk tanh kh = ω 2
It turns out A(x2 )
A(x2 ) ∼ exp (−βi + i βr )x2
Exponentially decaying w.r.t. the slow variable x2 .
Expression of φ1
Seabed condition for φ1
∂z φ1 = ∂x0 (b(x0 )∂x0 φ0 ) ,
for z = −h
Expression of φ1
Z
∞
φ1 =
∂x 0 (b(x 0 )∂x 0 φ0 )G(|x − x 0 | , −h) dx 0
−∞
G(|x − x 0 | , −h) is a Green’s function for the Laplace equation with the
seabed condition ∂z G = δ(x − x 0 ) at z = −h.
Green’s function
Green’s function for the BVP of φ1
G(ξ, −h) =
i ω 2 ei k |ξ|
2
ω 2 kh + gk sinh kh
−
∞
X
i ω 2 ei kn |ξ|
n=1
ω 2 kn h + gk sin2 kn h
where {i kn } are the imaginary roots of the dispersion equation.
Expression of φ2
Deriving the equation for A(x1 , x2 )
The ensemble average/coherent h·i part of the equation for φ2
(∂x20 + ∂z2 )hφ2 i = 2 i k ∂x2 φ0
(g∂z − ω 2 )hφ2 i = 0
∂z hφ2 i = h∂x0 (b(x0 )∂x0 φ1 )i
for
for
for
−h < z < 0
z=0
z = −h
hφ2 i is expressed using the same G(|x − x 0 | , −h). Then φ0 and φ1
are used to derive the equation for A(x2 ) w.r.t. the slow variable x2
Cg
∂A
i(βr + i βi )
=
A(x2 )
∂x2
2 cosh kh
Attenuation amplitude
Attenuation in the slow variable regime
A(x2 ) = A(0) exp (−βi + i βr )x2 /Cg
Attenuation parameters
Qatt = βi
Cg
Lloc = 2
ε βi
Attenuation rate
Localization length
Attenuation happens at ε−2 order, and is sensitive to the range of
parameters.
Numerical results
Num params: Dom length (x lg) = 200; Res (per lg) = 4; Ensemble = 1000
=1e−02; h=1 (MS)
=1e−02; h=1 (RS)
=1e−02; h=1 (RS−loc)
Bug in the codes
0.9
0.8
0.7
0.6
kQ/
2
0.5
0.4
0.3
0.2
0.1
0
−0.1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
klg
H
I
M-Scale method ∼ numerical ensemble average
Numerical results
Num params: Dom length (x lg) = 200; Res (per lg) = 4; Ensemble = 1000
=1e−02; h=1 (MS)
=1e−02; h=1 (RS)
=1e−02; h=1 (RS−loc)
Bug in the codes
0.9
0.8
0.7
0.6
kQ/
2
0.5
0.4
0.3
0.2
0.1
0
−0.1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
klg
H
I
Long lg : M-Scale method 6= realization dependent
I
Short lg : M-Scale method ∼ realization dependent
Numerical results
Num params: Dom length (x lg) = 200; Res (per lg) = 4; Ensemble = 1000
=1e−02; h=1 (MS)
=1e−02; h=1 (RS)
=1e−02; h=1 (RS:loc)
0.9
0.8
0.7
0.6
kQ/
2
0.5
0.4
0.3
0.2
0.1
0
−0.1
0
0.5
1
1.5
2
2.5
klg
I
3
3.5
4
4.5
5
M-Scale method ∼ numerical ensemble average
Numerical results
Num params: Dom length (x lg) = 200; Res (per lg) = 4; Ensemble = 1000
=1e−02; h=1 (MS)
=1e−02; h=1 (RS)
=1e−02; h=1 (RS:loc)
0.9
0.8
0.7
0.6
kQ/
2
0.5
0.4
0.3
0.2
0.1
0
−0.1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
klg
I
Long lg : M-Scale method 6= realization dependent
I
Short lg : M-Scale method ∼ realization dependent
Summary
1. Linear wave equations can lead to attenuation in the ensemble
average sense
2. The random seabed is simulated using harmonic random
process satisfying the conditions of multi-scale expansion
3. There is a big discrepancy between the ensemble average
solution and the realization dependent solution for weakly
random seabed
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