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
Outline
The Serre equations
I. Derivation
II. Properties
III. Solutions
IV. Solution stability
V. Wave shoaling
John D. Carter Stability and Shoaling in the Serre Equations
Derivation of the Serre Equations
Derivation of the Serre Equations
John D. Carter Stability and Shoaling in the Serre Equations
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)
John D. Carter Stability and Shoaling in the Serre Equations
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
John D. Carter Stability and Shoaling in the Serre Equations
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
John D. Carter Stability and Shoaling in the Serre Equations
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
John D. Carter Stability and Shoaling in the Serre Equations
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)
John D. Carter Stability and Shoaling in the Serre Equations
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
John D. Carter Stability and Shoaling in the Serre Equations
Properties of the Serre Equations
Properties of the Serre Equations
John D. Carter Stability and Shoaling in the Serre Equations
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
John D. Carter Stability and Shoaling in the Serre Equations
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.
John D. Carter Stability and Shoaling in the Serre Equations
Solutions of the Serre Equations
Solutions of the Serre Equations
John D. Carter Stability and Shoaling in the Serre Equations
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.
John D. Carter Stability and Shoaling in the Serre Equations
Trivial Solution of the Serre Equations
If k = 0,
η(x , t) = a0 + a1
u(x , t) = 0
John D. Carter Stability and Shoaling in the Serre Equations
Periodic Solutions of the Serre Equations
The water surface if 0 < k < 1.
a0+a1H1-k2L
k2a1
John D. Carter Stability and Shoaling in the Serre Equations
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
John D. Carter Stability and Shoaling in the Serre Equations
Soliton Solution of the Serre Equations
The corresponding water surface
a0
a1
John D. Carter Stability and Shoaling in the Serre Equations
Stability of Solutions of the Serre Equations
Stability of Solutions of the Serre Equations
John D. Carter Stability and Shoaling in the Serre Equations
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
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)
)
John D. Carter Stability and Shoaling in the Serre Equations
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
John D. Carter Stability and Shoaling in the Serre Equations
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.
John D. Carter Stability and Shoaling in the Serre Equations
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
John D. Carter Stability and Shoaling in the Serre Equations
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
John D. Carter Stability and Shoaling in the Serre Equations
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
John D. Carter Stability and Shoaling in the Serre Equations
Wave Shoaling in the Serre Equations
Wave Shoaling in the Serre Equations
John D. Carter Stability and Shoaling in the Serre Equations
Wave Shoaling
So far we’ve assumed shallow water and a horizontal bottom.
¿What happens if the bottom varies (slowly)?
John D. Carter Stability and Shoaling in the Serre Equations
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
John D. Carter Stability and Shoaling in the Serre Equations
Wave Shoaling
We consider a slowly-varying, constant-slope bottom of the form
ξ(x) = 0 + εx +O(ε2)
Flat Bottom Slowly Sloping Bottom
John D. Carter Stability and Shoaling in the Serre Equations
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)
John D. Carter Stability and Shoaling in the Serre Equations
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.
John D. Carter Stability and Shoaling in the Serre Equations
Wave Shoaling
At the next order in ε, the equations are a big mess.
¡However, the system can be solved analytically!
John D. Carter Stability and Shoaling in the Serre Equations
Wave Shoaling
Original Surface First-Order Correction
John D. Carter Stability and Shoaling in the Serre Equations
Wave Shoaling
Combined Surface
John D. Carter Stability and Shoaling in the Serre Equations
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