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