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Page 1: John D. Carter - Seattle Universityfac-staff.seattleu.edu/carterj1/web/papers/IMACSshort.pdfJohn D. Carter Stability and Shoaling in the Serre Equations Governing Equations The 1-D

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

John D. Carter Stability and Shoaling in the Serre Equations

Page 3: John D. Carter - Seattle Universityfac-staff.seattleu.edu/carterj1/web/papers/IMACSshort.pdfJohn D. Carter Stability and Shoaling in the Serre Equations Governing Equations The 1-D

Derivation of the Serre Equations

Derivation of the Serre Equations

John D. Carter Stability and Shoaling in 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)

John D. Carter Stability and Shoaling in the Serre Equations

Page 5: John D. Carter - Seattle Universityfac-staff.seattleu.edu/carterj1/web/papers/IMACSshort.pdfJohn D. Carter Stability and Shoaling in the Serre Equations Governing Equations The 1-D

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

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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

John D. Carter Stability and Shoaling in the Serre Equations

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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

John D. Carter Stability and Shoaling in the Serre Equations

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Governing Equations

After depth averaging, the dimensionless governing equations are

ηt + ε(ηu)x = 0

ut + ηx + εu ux −δ2

(η3(uxt + εu uxx − ε(ux)2

))x

= O(δ4, εδ4)

John D. Carter Stability and Shoaling in the Serre Equations

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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

(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

Page 10: John D. Carter - Seattle Universityfac-staff.seattleu.edu/carterj1/web/papers/IMACSshort.pdfJohn D. Carter Stability and Shoaling in the Serre Equations Governing Equations The 1-D

Properties of the Serre Equations

Properties of the Serre Equations

John D. Carter Stability and Shoaling in 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

John D. Carter Stability and Shoaling in the Serre Equations

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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.

John D. Carter Stability and Shoaling in the Serre Equations

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Solutions of the Serre Equations

Solutions of the Serre Equations

John D. Carter Stability and Shoaling in 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.

John D. Carter Stability and Shoaling in the Serre Equations

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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

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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

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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

John D. Carter Stability and Shoaling in the Serre Equations

Page 18: John D. Carter - Seattle Universityfac-staff.seattleu.edu/carterj1/web/papers/IMACSshort.pdfJohn D. Carter Stability and Shoaling in the Serre Equations Governing Equations The 1-D

Soliton Solution of the Serre Equations

The corresponding water surface

a0

a1

John D. Carter Stability and Shoaling in the Serre Equations

Page 19: John D. Carter - Seattle Universityfac-staff.seattleu.edu/carterj1/web/papers/IMACSshort.pdfJohn D. Carter Stability and Shoaling in the Serre Equations Governing Equations The 1-D

Stability of Solutions of the Serre Equations

Stability of Solutions of the Serre Equations

John D. Carter Stability and Shoaling in the Serre Equations

Page 20: John D. Carter - Seattle Universityfac-staff.seattleu.edu/carterj1/web/papers/IMACSshort.pdfJohn D. Carter Stability and Shoaling in the Serre Equations Governing Equations The 1-D

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

(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)

)

John D. Carter Stability and Shoaling in the Serre Equations

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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

John D. Carter Stability and Shoaling in the Serre Equations

Page 23: John D. Carter - Seattle Universityfac-staff.seattleu.edu/carterj1/web/papers/IMACSshort.pdfJohn D. Carter Stability and Shoaling in the Serre Equations Governing Equations The 1-D

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

Page 24: John D. Carter - Seattle Universityfac-staff.seattleu.edu/carterj1/web/papers/IMACSshort.pdfJohn D. Carter Stability and Shoaling in the Serre Equations Governing Equations The 1-D

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

Page 25: John D. Carter - Seattle Universityfac-staff.seattleu.edu/carterj1/web/papers/IMACSshort.pdfJohn D. Carter Stability and Shoaling in the Serre Equations Governing Equations The 1-D

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

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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

John D. Carter Stability and Shoaling in the Serre Equations

Page 27: John D. Carter - Seattle Universityfac-staff.seattleu.edu/carterj1/web/papers/IMACSshort.pdfJohn D. Carter Stability and Shoaling in the Serre Equations Governing Equations The 1-D

Wave Shoaling in the Serre Equations

Wave Shoaling in the Serre Equations

John D. Carter Stability and Shoaling in the Serre Equations

Page 28: John D. Carter - Seattle Universityfac-staff.seattleu.edu/carterj1/web/papers/IMACSshort.pdfJohn D. Carter Stability and Shoaling in the Serre Equations Governing Equations The 1-D

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

Page 29: John D. Carter - Seattle Universityfac-staff.seattleu.edu/carterj1/web/papers/IMACSshort.pdfJohn D. Carter Stability and Shoaling in the Serre Equations Governing Equations The 1-D

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

Page 30: John D. Carter - Seattle Universityfac-staff.seattleu.edu/carterj1/web/papers/IMACSshort.pdfJohn D. Carter Stability and Shoaling in the Serre Equations Governing Equations The 1-D

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

Page 31: John D. Carter - Seattle Universityfac-staff.seattleu.edu/carterj1/web/papers/IMACSshort.pdfJohn D. Carter Stability and Shoaling in the Serre Equations Governing Equations The 1-D

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

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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.

John D. Carter Stability and Shoaling in the Serre Equations

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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!

John D. Carter Stability and Shoaling in the Serre Equations

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Wave Shoaling

Original Surface First-Order Correction

John D. Carter Stability and Shoaling in the Serre Equations

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Wave Shoaling

Combined Surface

John D. Carter Stability and Shoaling in the Serre Equations