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Stability and Shoaling in the Serre Equations
John D. Carter
March 23, 2009
Joint work with Rodrigo Cienfuegos.
John D. Carter Stability and Shoaling in the Serre Equations
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Outline
The Serre equations
I. Derivation
II. Properties
III. Solutions
IV. Solution stability
V. Wave shoaling
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Derivation of the Serre Equations
Derivation of the Serre Equations
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Governing Equations
Consider the 1-D flow of an inviscid, irrotational, incompressible fluid.
Let
I η(x , t) represent the location of the free surface
I u(x , z , t) represent the horizontal velocity of the fluid
I w(x , z , t) represent the vertical velocity of the fluid
I p(x , z , t) represent the pressure in the fluid
I ε = a0/h0 (a measure of nonlinearity)
I δ = h0/l0 (a measure of shallowness)
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Governing Equations
The 1-D flow of an inviscid, irrotational, incompressible fluid
z0 at the bottom
zh0 undisturbed level
h0
a0
l0
x
z
Η
zΗ, free surface
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Governing Equations
The dimensionless governing equations are
ux + wz = 0, for 0 < z < 1 + εη
uz − δwx = 0, for 0 < z < 1 + εη
εut + ε2(u2)x + ε2(uw)z + px = 0, for 0 < z < 1 + εη
δ2εwt + δ2ε2uwx + δ2ε2wwz + pz = −1, for 0 < z < 1 + εη
w = ηt + εuηx at z = 1 + εη
p = 0 at z = 1 + εη
w = 0 at z = 0
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Depth Averaging
The Serre equations are obtained from the governing equations by:
1. Depth averaging
The depth-averaged value of a quantity f (z) is defined by
f =1
h
∫ h
0f (z)dz
where h = 1 + εη is the location of the free surface.
2. Assuming that δ << 1
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Governing Equations
After depth averaging, the dimensionless governing equations are
ηt + ε(ηu)x = 0
ut + ηx + εu ux −δ2
3η
(η3(uxt + εu uxx − ε(ux)2
))x
= O(δ4, εδ4)
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The Serre Equations
Truncating this system at O(δ4, εδ4) and transforming back tophysical variables gives the Serre Equations
ηt + (ηu)x = 0
ut + gηx + u ux −1
3η
(η3(uxt + u uxx − (ux)2
))x
= 0
where
I η(x , t) is the dimensional free surface elevation
I u(x , t) is the dimensional depth-averaged horizontal velocity
I g is the acceleration due to gravity
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Properties of the Serre Equations
Properties of the Serre Equations
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Properties of the Serre Equations
The Serre equations admit the following conservation laws:
I. Mass
∂t(η) + ∂x(ηu) = 0
II. Momentum
∂t(ηu) + ∂x
(1
2gη2 − 1
3η3uxt + ηu2 +
1
3η3u2
x −1
3η3u uxx
)= 0
III. Momentum 2
∂t
(u−ηηxux−
1
3η2uxx
)+∂x
(ηηtux+gη−1
3η2u uxx+
1
2η2u2
x
)= 0
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Properties of the Serre Equations
The Serre equations are invariant under the transformation
η(x , t) = η(x − st, t)
u(x , t) = u(x − st, t) + s
x = x − st
where s is any real parameter.
Physically, this corresponds to adding a constant horizontal flow tothe entire system.
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Solutions of the Serre Equations
Solutions of the Serre Equations
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Solutions of the Serre Equations
η(x , t) = a0 + a1dn2(κ(x − ct), k
)u(x , t) = c
(1− h0
η(x , t)
)κ =
√3a1
2√
a0(a0 + a1)(a0 + (1− k2)a1)
c =
√ga0(a0 + a1)(a0 + (1− k2)a1)
h0
h0 = a0 + a1E (k)
K (k)
where k ∈ [0, 1], a0 > 0, and a1 > 0 are real parameters.
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Trivial Solution of the Serre Equations
If k = 0,
η(x , t) = a0 + a1
u(x , t) = 0
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Periodic Solutions of the Serre Equations
The water surface if 0 < k < 1.
a0+a1H1-k2L
k2a1
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Soliton Solution of the Serre Equations
If k = 1,
η(x , t) = a0 + a1 sech2(κ(x − ct))
u(x , t) = c(
1− a0
η(x , t)
)κ =
√3a1
2a0√
a0 + a1
c =√
g(a0 + a1)
h0 = a0
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Soliton Solution of the Serre Equations
The corresponding water surface
a0
a1
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Stability of Solutions of the Serre Equations
Stability of Solutions of the Serre Equations
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Stability of Solutions of the Serre Equations
Transform to a moving coordinate frame
χ = x − ct
τ = t
The Serre equations become
ητ − cηχ +(ηu)χ
= 0
uτ − cuχ + u uχ + ηχ−1
3η
(η3(uχτ − cuχχ + u uχχ− (uχ)2
))χ
= 0
and the solutions become
η = η0(χ) = a0 + a1dn2(κχ, k
)u = u0(χ) = c
(1− h0
η0(χ)
)John D. Carter Stability and Shoaling in the Serre Equations
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Stability of Solutions of the Serre Equations
Consider perturbed solutions of the form
ηpert(χ, τ) = η0(χ) + εη1(χ, τ) +O(ε2)
upert(χ, τ) = u0(χ) + εu1(χ, τ) +O(ε2)
where
I ε is a small real parameter
I η1(χ, τ) and u1(χ, τ) are real-valued functions
I η0(χ) = a0 + a1dn2(κχ, k)
I u0(χ) = c(
1− h0η0(χ)
)
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Stability of Solutions of the Serre Equations
Without loss of generality, assume
η1(χ, τ) = H(χ)eΩτ + c .c.
u1(χ, τ) = U(χ)eΩτ + c.c .
where
I H(χ) and U(χ) are complex-valued functions
I Ω is a complex constant
I c .c . denotes complex conjugate
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Stability of Solutions of the Serre Equations
This leads the following linear system
L(
HU
)= ΩM
(HU
)where
L =
(−u′
0 + (c − u0)∂χ −η′0 − η0∂χ
L21 L22
)
M =
(1 00 1− η0η
′0∂χ − 1
3η20∂χχ
)
and prime represents derivative with respect to χ.
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Stability of Solutions of the Serre Equations
where
L21 = −η′0(u′
0)2 − cη′0u
′′0 −
2
3cη0u
′′′0 + η′
0u0u′′0 −
2
3η0u
′0u
′′0
+2
3η0u0u
′′′0 +
(η0u0u
′′0 − g − η0(u′
0)2 − cη0u′′0
)∂χ
L22 = −u′0 + η0η
′0u
′′0 +
1
3η2
0u′′′0 +
(c − u0 − 2η0η
′0u
′0 −
1
3η2
0u′′0
)∂χ
+(η0η
′0u0 − cη0η
′0 −
1
3η2
0u′0
)∂χχ +
(1
3η2
0u0 −1
3cη2
0
)∂χχχ
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Stability of Solutions of the Serre Equations
L(
HU
)= ΩM
(HU
)Solved numerically using the Fourier-Floquet-Hill Method.
0.000 0.002 0.004 0.006 0.008 0.010ÂHWL0
2
4
6
8
ÁHWL
0.000 0.002 0.004 0.006 0.008 0.010ÂHWL0.60
0.65
0.70
0.75
0.80ÁHWL
For a0 = 0.3, a1 = 0.1, k = 0.99
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Stability of Solutions of the Serre Equations
Qualitative observations:
I Not all solutions are stable
I If k and/or a1 is large enough, then there is instability
I Most/all instabilities have complex growth rates
I As k increases, so does the maximum growth rate
I As a1 increases, so does the maximum growth rate
I As a0 decreases, the maximum growth rate increases
I As a1 and/or k increase, the number of bands increases
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Wave Shoaling in the Serre Equations
Wave Shoaling in the Serre Equations
John D. Carter Stability and Shoaling in the Serre Equations
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Wave Shoaling
So far we’ve assumed shallow water and a horizontal bottom.
¿What happens if the bottom varies (slowly)?
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Wave Shoaling
The Serre equations for a non-horizontal bottom
ηt + (hu)x = 0
hut + huux + ghηx +(h2(1
3P +
1
2Q))
x+ ξxh
(1
2P +Q
)= 0
P = −h(uxt + uuxx − (ux)2
)Q = ξx(ut + uux) + ξxxu
2
where
I z = ξ(x) is the bottom location (ξ ≤ 0 for all x)
I z = η(x , t) is the location of the free surface
I h(x , t) = η(x , t)− ξ(x) is the local water depth
I u = u(x , t) is the depth-averaged horizontal velocity
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Wave Shoaling
We consider a slowly-varying, constant-slope bottom of the form
ξ(x) = 0 + εx +O(ε2)
Flat Bottom Slowly Sloping Bottom
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Wave Shoaling
In order to deal with the slowly-varying bottom,
ξ(x) = 0 + εx +O(ε2)
we assume
h(x , t) = h0(x , t) + εh1(x , t) +O(ε2)
u(x , t) = u0(x , t) + εu1(x , t) +O(ε2)
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Wave Shoaling
The solution to the leading-order problem is
h0(x , t) = a0 + a1 sech2(κ(x − ct)
)u0(x , t) = c
(1− a0
h0(x , t)
)κ =
√3a1
2a0√
a0 + a1
c =√
g(a0 + a1)
Note: we only consider solitary waves here.
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Wave Shoaling
At the next order in ε, the equations are a big mess.
¡However, the system can be solved analytically!
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Wave Shoaling
Original Surface First-Order Correction
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Wave Shoaling
Combined Surface
John D. Carter Stability and Shoaling in the Serre Equations