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

Outline

The Serre equations

I. Derivation

II. Properties

III. Solutions

IV. Solution stability

V. Wave shoaling


Derivation of the Serre Equations

Derivation of 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)


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


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


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


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)


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


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



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.


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.


Trivial Solution of the Serre Equations

If k = 0,

η(x , t) = a0 + a1

u(x , t) = 0


Periodic Solutions of the Serre Equations

The water surface if 0 < k < 1.

a0+a1H1-k2L

k2a1


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


Soliton Solution of the Serre Equations

The corresponding water surface

a0

a1


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


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)

)



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



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.



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



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



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


Wave Shoaling in the Serre Equations

Wave 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)?


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


Wave Shoaling

We consider a slowly-varying, constant-slope bottom of the form

ξ(x) = 0 + εx +O(ε2)

Flat Bottom Slowly Sloping Bottom


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)


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.


Wave Shoaling

At the next order in ε, the equations are a big mess.

¡However, the system can be solved analytically!


Wave Shoaling

Original Surface First-Order Correction


Wave Shoaling

Combined Surface


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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