Mod elisation math ematique des vagues · Mod elisation math ematique des vagues David Lannes Institut de Math ematiques de Bordeaux et CNRS UMR 5251 Journ ee des doctorants David

Modélisation mathématique des vagues

David Lannes

Institut de Mathématiques de Bordeaux et CNRS UMR 5251

Journée des doctorants

David Lannes (IMB) Modélisation mathématique des vagues Valenciennes, 10/09/2015 1 / 30

Goal


Where do waves come from? How are they created?

Source: Les vagues en équations, Pour la Science, no 409, novembre 2011David Lannes (IMB) Modélisation mathématique des vagues Valenciennes, 10/09/2015 3 / 30

Where do waves come from? What is their speed?

Sir Isaac Newton(1642-1727)Principia Mathematica, 1687



Leonhard Euler(1707-1783)Mémoires de l’Académie royale des sciences et desbelles lettres de Berlin, 1757 Equations of fluid mechanics

ρ(∂tU + U · ∇X ,zU) =−∇X ,zP + ρgdiv U =0

This equations are very general

What do they tell us about waves?



Giuseppe Lodovico Lagrangia(Joseph Louis Lagrange)(1736-1813)Mémoire sur la théorie dumouvement des fluides, 1781



Source: Les vagues en équations, Pour la Science, no 409, novembre 2011



Comparison of Newton and Lagrange’s formulas

Lagrange: c =√gH.

All waves have same speed

Newton: c = 1√2π

√gL where L is the wave length of the wave

Waves of different wavelength propagate differently





Lagrange: c =√gH.


Newton: c = 1√2π



This is dispersion:





Lagrange: c =√gH.


Newton: c = 1√2π



Comparison for a wave a0(x) = sin(x) + 0.5 sin(2x).




Recall how waves are created


So the good formula should be

Newton: c = 1√2π

√gL


Closer to the shore Another formula!

Closer to the shore we observe:

And the relevant formula is

Lagrange: c =√gH


Closer to the shore What happens?

Siméon DenisPoisson

(1780–1840)

Augustin LouisCauchy(1789–1857)

Sir George BiddellAiry(1801–1892)

Sir George GabrielStokes(1819–1903)

A single formula with twodifferent asymptotic regimes

Lagrange’s formula in shallowwater (H/L→ 0),Newton’s formula in deep water(H/L→∞).


Modern mathematical approaches Notations


Modern mathematical approaches The free surface Euler equations

The free surface Euler equations

1 ∂tU + (U · ∇X ,z)U = −1ρ∇X ,zP − gez in Ωt

Definition

Equations (H1)-(H9) are called free surface Euler equations.

ONE unknown function ζ on a fixed domain Rd THREE unknown functions U on a moving, unknown domain Ωt




1 ∂tU + (U · ∇X ,z)U = −1ρ∇X ,zP − gez in Ωt2 div U = 0

Definition







3 curl U = 0

Definition







3 curl U = 0

4 Ωt = {(X , z) ∈ Rd+1,−H0 + b(X ) < z < ζ(t,X )}.

Definition







3 curl U = 0

4 Ωt = {(X , z) ∈ Rd+1,−H0 + b(X ) < z < ζ(t,X )}.5 U · n = 0 on {z = −H0 + b(X )}.

Definition







3 curl U = 0

4 Ωt = {(X , z) ∈ Rd+1,−H0 + b(X ) < z < ζ(t,X )}.5 U · n = 0 on {z = −H0 + b(X )}.6 ∂tζ −

√1 + |∇ζ|2U · n = 0 on {z = ζ(t,X )}.

Definition







3 curl U = 0

4 Ωt = {(X , z) ∈ Rd+1,−H0 + b(X ) < z < ζ(t,X )}.5 U · n = 0 on {z = −H0 + b(X )}.6 ∂tζ −

√1 + |∇ζ|2U · n = 0 on {z = ζ(t,X )}.

7 P = Patm on {z = ζ(t,X )}.

Definition




Modern mathematical approaches The free surface Bernoulli equations

The free surface Bernoulli equations


3 curl U = 0

4 Ωt = {(X , z) ∈ Rd+1,−H0 + b(X ) < z < ζ(t,X )}.5 U · n = 0 on {z = −H0 + b(X )}6 ∂tζ −

√1 + |∇ζ|2U · n = 0 on {z = ζ(t,X )}.

7 P = Patm on {z = ζ(t,X )}.

Definition

Equations (H1)’-(H9)’ are called free surface Bernoulli equations.

ONE unknown function ζ on a fixed domain Rd ONE unknown function Φ on a moving, unknown domain Ωt





3 U = ∇X ,zΦ4 Ωt = {(X , z) ∈ Rd+1,−H0 + b(X ) < z < ζ(t,X )}.5 U · n = 0 on {z = −H0 + b(X )}6 ∂tζ −

√1 + |∇ζ|2U · n = 0 on {z = ζ(t,X )}.

7 P = Patm on {z = ζ(t,X )}.

Definition






1 ∂tU + (U · ∇X ,z)U = −1ρ∇X ,zP − gez in Ωt2 ∆X ,zΦ = 0


√1 + |∇ζ|2U · n = 0 on {z = ζ(t,X )}.

7 P = Patm on {z = ζ(t,X )}.

Definition






1 ∂tΦ +12 |∇X ,zΦ|

2 + gz = −1ρ(P − Patm) in Ωt2 ∆X ,zΦ = 0


√1 + |∇ζ|2U · n = 0 on {z = ζ(t,X )}.

7 P = Patm on {z = ζ(t,X )}.

Definition






1 ∂tΦ +12 |∇X ,zΦ|


3 U = ∇X ,zΦ4 Ωt = {(X , z) ∈ Rd+1,−H0 + b(X ) < z < ζ(t,X )}.5 ∂nΦ = 0 on {z = −H0 + b(X )}.6 ∂tζ −

√1 + |∇ζ|2U · n = 0 on {z = ζ(t,X )}.

7 P = Patm on {z = ζ(t,X )}.

Definition






1 ∂tΦ +12 |∇X ,zΦ|


3 U = ∇X ,zΦ4 Ωt = {(X , z) ∈ Rd+1,−H0 + b(X ) < z < ζ(t,X )}.5 ∂nΦ = 0 on {z = −H0 + b(X )}.6 ∂tζ −

√1 + |∇ζ|2∂nΦ = 0 on {z = ζ(t,X )}.

7 P = Patm on {z = ζ(t,X )}.

Definition




Modern mathematical approaches Working with a fix domain

The Lagrangian approach

One parametrizes any fluid particle of Ωt by its initial position through thediffeomorphism Σ {

∂tΣ(t,X , z) = U(t,Σ(t,X , z)),

Σ(0,X , z) = (X , z).

Writing

Ũ(t,X , z) =U(t,Σ(t,X , z)),

A(t,X , z) =|∇X ,zΣ|−1

we get {∂tU + U · ∇X ,zU = −∇X ,zP + gdiv (U) = 0

in Ωt

{∂tŨ = −A∇X ,z P̃ + gTr (A∇X ,zŨ) = 0

in Ω0



The geometric approach (I)

The Lagrangian diffeomorphism{∂tΣ(t,X , z) = U(t,Σ(t,X , z)),

Σ(0,X , z) = (X , z).

is volume preserving since U is divergence free

Σ ∈ H = {Σ : Ω0 → Rd+1,Σ volume preserving.

Moreover the energy is conserved

H =1

2

∫Ωt

|U|2 + g2

∫Rdζ2

=1

2

∫Ω0

|∂tΣ|2 + gG (Σ)︸︷︷︸:=L(Σ,∂tΣ)



The geometric approach (II)

Defining TΣH

TΣH = {Σ′ Ω0 → Rd+1, div (Σ′ ◦ Σ) = 0},

the free surface Euler equations can be viewed as a critical point of theaction

L(Σ, ∂tΣ) = =1

2

∫Ω0

|∂tΣ|2 − gG (Σ).

Remark

Arnold (1966): the Euler equation for an incompressible inviscid fluid canbe viewed as the geodesic equation on the group of volume-preservingdiffeomorphisms.



Lagrangian interface formulation (I)

We consider a Lagrangian parametrization of the surface

Γt = {M(t, α), α ∈ R},

with {∂tM(t, α) = U(t,M(t, α)),

M(0, α) = (α, ζ0(α)).

One then has

∂2tM =∂tU + ∂tM · ∇X ,zU=∂tU + U · ∇X ,zU

=− gez −1

ρ∇X ,zP

And since P = Patm is constant at the surface

∂2tM + gez =1

ρ(−∂nP)n



Lagrangian interface formulation (II)

∂2tM + gez =1

ρ(−∂nP)n ∂αM1∂2tM1 + (g + ∂2tM2)∂αM2 = 0

We still need a relation between ∂tM1 and ∂tM2 !!!!

Complex analysis

(x , z) ∈ R2 x + iz ∈ CIncompressibility+Irrotationality=Cauchy Riemann for U

U is holomorphic in Ωt

∂tM = U(t,M(t, α)) is the boundary of a holomorphic function,therefore

∂tM = H(Γt)∂tM

with

H(Γt)f (t, α) =1

iπp.v.

∫f (t, α′)∂αM(t, α

′)

M(t, α)−M(t, α′)dα′.



An Eulerian approach: The Zakharov-Craig-Sulemformulation

Zakharov 68:1 Define ψ(t,X ) = Φ(t,X , ζ(t,X )) .

2 ζ and ψ fully determine Φ: indeed, the equation{∆X ,zΦ = 0 in Ωt ,Φ|z=ζ = ψ, ∂nΦ|z=−H0+b = 0.

has a unique solution Φ.3 The equations can be put under the canonical Hamiltonian form

∂t

(ζψ

)=

(0 1−1 0

)gradζ,ψH

with the Hamiltonian

H =1

2

∫Rd

gζ2 +

∫Ω|U|2






has a unique solution Φ.

3 The equations can be put under the canonical Hamiltonian form

∂t

(ζψ

)=

(0 1−1 0

)gradζ,ψH


H =1

2

∫Rd

gζ2 +

∫Ω|U|2






has a unique solution Φ.3 The equations can be put under the canonical Hamiltonian form

∂t

(ζψ

)=

(0 1−1 0

)gradζ,ψH


H =1

2

∫Rd

gζ2 +

∫Ω|U|2



Question

What are the equations on ζ and ψ???

• Equation on ζ. It is given by the kinematic equation

∂tζ −√

1 + |∇ζ|2∂nΦ|z=ζ = 0

Craig-Sulem 93:

Definition (Dirichlet-Neumann operator)

G [ζ] : ψ 7→ G [ζ]ψ =√

1 + |∇ζ|2∂nΦ|z=ζ



Question

What are the equations on ζ and ψ???

• Equation on ζ. It is given by the kinematic equation

∂tζ −√

1 + |∇ζ|2∂nΦ|z=ζ = 0

Craig-Sulem 93:

Definition (Dirichlet-Neumann operator)

G [ζ] : ψ 7→ G [ζ]ψ =√

1 + |∇ζ|2∂nΦ|z=ζ

The equation on ζ can be written

∂tζ − G [ζ]ψ = 0



• Equation on ψ. We use (H1)” and (H7)”

∂tΦ +1

2|∇X ,zΦ|2 + gz = −

1

ρ(P − Patm) AND P|z=ζ = Patm

w∂tΦ|z=ζ +

1

2|∇X ,zΦ|2|z=ζ + gζ = 0

The equation on ψ can be written

∂tψ + gζ +1

2|∇ψ|2 − (G [ζ]ψ +∇ζ · ∇ψ)

2

2(1 + |∇ζ|2)= 0.

The Zakharov-Craig-Sulem equations∂tζ − G [ζ]ψ = 0,

∂tψ + gζ +1

2|∇ψ|2 − (G [ζ]ψ +∇ζ · ∇ψ)

2

2(1 + |∇ζ|2)= 0.

Two scalar equations on the fix d-dimensional domain Rd !




∂tΦ +1

2|∇X ,zΦ|2 + gz = −

1

ρ(P − Patm) AND P|z=ζ = Patmw

∂tΦ|z=ζ +1

2|∇X ,zΦ|2|z=ζ + gζ = 0


∂tψ + gζ +1

2|∇ψ|2 − (G [ζ]ψ +∇ζ · ∇ψ)

2

2(1 + |∇ζ|2)= 0.


∂tψ + gζ +1

2|∇ψ|2 − (G [ζ]ψ +∇ζ · ∇ψ)

2

2(1 + |∇ζ|2)= 0.

Two scalar equations on the fix d-dimensional domain Rd !




∂tΦ +1

2|∇X ,zΦ|2 + gz = −

1

ρ(P − Patm) AND P|z=ζ = Patmw

∂tΦ|z=ζ +1

2|∇X ,zΦ|2|z=ζ + gζ = 0


∂tψ + gζ +1

2|∇ψ|2 − (G [ζ]ψ +∇ζ · ∇ψ)

2

2(1 + |∇ζ|2)= 0.


∂tψ + gζ +1

2|∇ψ|2 − (G [ζ]ψ +∇ζ · ∇ψ)

2

2(1 + |∇ζ|2)= 0.

Two scalar equations on the fix d-dimensional domain Rd !David Lannes (IMB) Modélisation mathématique des vagues Valenciennes, 10/09/2015 22 / 30

Local well posedness Linearized equations around the rest state

Linearized equations

{∂tζ − G [0]ψ = 0,∂tψ + gζ = 0.

andG [0] = |D| tanh(H|D|)

and therefore∂2t ζ + g |D| tanh(H|D|)ζ = 0

Newton and Lagrange’s formulas:

∂2t ζ − gH∂2x ζ = 0 in shallow water∂2t ζ + g |D|ζ = 0 in deep water.


Local well posedness Linearized equations around the rest state

Quasilinearized equations

After differentiation and change of unknowns, the structure is

(∂t + V · ∇)(ζ̃

ψ̃

)+

(0 −G [ζ]a 0

)(ζ̃

ψ̃

)= l.o.t.

Symbolic approximation

G [ζ] = |D|+ order 0

Jordan block a > 0

This is the Rayleigh-Taylor criterion (−∂zP)|z=ζ > 0.

Theorem

The (ZCS) is locally well posed.


Asymptotic expansions Nondimensionalization

Asymptotic models

Goal

Derive simpler asymptotic models describing the solutions to the waterwaves equations in shallow water.

For the sake of simplicity, we consider here a flat bottom (b = 0).

We introduce three characteristic scales1 The characteristic water depth H02 The characteristic horizontal scale L3 The order of the free surface amplitude a



Asymptotic models

Goal

Derive simpler asymptotic models describing the solutions to the waterwaves equations in shallow water.

For the sake of simplicity, we consider here a flat bottom (b = 0).We introduce three characteristic scales

1 The characteristic water depth H02 The characteristic horizontal scale L3 The order of the free surface amplitude a

Two independent dimensionless parameters can be formed from thesethree scales. We choose:

a

H0= ε (amplitude parameter ),

H20L2

= µ (shallowness parameter ).



We proceed to the simple nondimensionalizations

X ′ =X

L, z ′ =

z

H0, ζ ′ =

ζ

a, etc.


Asymptotic expansions Nondimensionalized equations

∂tζ +∇ · (hV ) = 0,

∂t∇ψ +∇ζ +ε

2∇|∇ψ|2 − εµ∇(−∇ · (hV ) +∇(εζ) · ∇ψ)

2

2(1 + ε2µ|∇ζ|2)= 0,

where in dimensionless form

h = 1 + εζ and V =1

h

∫ εζ−1

V (x , z)dz .

Shallow water asymptotics (µ� 1)We look for an asymptotic description with respect to µ of ∇ψ interms of ζ and V

This is obtained through an asymtotic description of V in the fluid.

This is obtained through an asympotic description of Φ in the fluid,Φ ∼ Φ0 + µΦ1 + µ2Φ2 + . . .At first order, we have a columnar motion and therefore∇ψ = V + O(µ).Next order approximation: Green-Naghdi



∂tζ +∇ · (hV ) = 0,

∂t∇ψ +∇ζ +ε

2∇|∇ψ|2 − εµ∇(−∇ · (hV ) +∇(εζ) · ∇ψ)

2

2(1 + ε2µ|∇ζ|2)= 0,


h = 1 + εζ and V =1

h

∫ εζ−1

V (x , z)dz .

Shallow water asymptotics (µ� 1)

We look for an asymptotic description with respect to µ of ∇ψ interms of ζ and V





∂tζ +∇ · (hV ) = 0,

∂t∇ψ +∇ζ +ε

2∇|∇ψ|2 − εµ∇(−∇ · (hV ) +∇(εζ) · ∇ψ)

2

2(1 + ε2µ|∇ζ|2)= 0,


h = 1 + εζ and V =1

h

∫ εζ−1

V (x , z)dz .






∂tζ +∇ · (hV ) = 0,

∂t∇ψ +∇ζ +ε

2∇|∇ψ|2 − εµ∇(−∇ · (hV ) +∇(εζ) · ∇ψ)

2

2(1 + ε2µ|∇ζ|2)= 0,


h = 1 + εζ and V =1

h

∫ εζ−1

V (x , z)dz .



This is obtained through an asympotic description of Φ in the fluid,Φ ∼ Φ0 + µΦ1 + µ2Φ2 + . . .

At first order, we have a columnar motion and therefore∇ψ = V + O(µ).Next order approximation: Green-Naghdi



∂tζ +∇ · (hV ) = 0,

∂t∇ψ +∇ζ +ε

2∇|∇ψ|2 − εµ∇(−∇ · (hV ) +∇(εζ) · ∇ψ)

2

2(1 + ε2µ|∇ζ|2)= 0,


h = 1 + εζ and V =1

h

∫ εζ−1

V (x , z)dz .



This is obtained through an asympotic description of Φ in the fluid,Φ ∼ Φ0 + µΦ1 + µ2Φ2 + . . .At first order, we have a columnar motion and therefore∇ψ = V + O(µ).

Next order approximation: Green-Naghdi



Saint-Venant

{∂tζ +∇ · (hV ) = 0,∂tV + εV · ∇V +∇ζ = 0.

where we dropped all O(µ) terms.



This is obtained through an asympotic description of Φ in the fluid,Φ ∼ Φ0 + µΦ1 + µ2Φ2 + . . .At first order, we have a columnar motion and therefore∇ψ = V + O(µ).

Next order approximation: Green-Naghdi



Green-Nadghi

{∂tζ +∇ · (hV ) = 0,(I + µT )

(∂tV + εV · ∇V

)+∇ζ + µQ(V ) = 0.

where we dropped all O(µ2) terms.






Justification

One needs to prove that the solution exists on a time interval [0,T/ε]with T independent of µ

One needs bounds on the solution on this time scale

The previous proof does not work!

G [ζ]ψ ∼ |D|ψ + order 0 (symbolic analysis)G [ζ]ψ ∼ µ∇((1 + εζ)∇ψ) + O(µ2) (shallow water expansion)

Symbolic analysis and shallow water expansions are not compatible

Justification OK away from wave breaking and shoreline



Justification

One needs to prove that the solution exists on a time interval [0,T/ε]with T independent of µOne needs bounds on the solution on this time scaleThe previous proof does not work!

Beware the W 1,∞\C 1(Rd) waves!!!

G [ζ]ψ ∼ |D|ψ + order 0 (symbolic analysis)G [ζ]ψ ∼ µ∇((1 + εζ)∇ψ) + O(µ2) (shallow water expansion) Symbolic analysis and shallow water expansions are not compatibleJustification OK away from wave breaking and shoreline



Justification

One needs to prove that the solution exists on a time interval [0,T/ε]with T independent of µOne needs bounds on the solution on this time scaleThe previous proof does not work!

Beware the big waves!!!

G [ζ]ψ ∼ |D|ψ + order 0 (symbolic analysis)G [ζ]ψ ∼ µ∇((1 + εζ)∇ψ) + O(µ2) (shallow water expansion) Symbolic analysis and shallow water expansions are not compatibleJustification OK away from wave breaking and shoreline



Justification

One needs to prove that the solution exists on a time interval [0,T/ε]with T independent of µ

One needs bounds on the solution on this time scale

The previous proof does not work!

G [ζ]ψ ∼ |D|ψ + order 0 (symbolic analysis)G [ζ]ψ ∼ µ∇((1 + εζ)∇ψ) + O(µ2) (shallow water expansion)

Symbolic analysis and shallow water expansions are not compatible

Justification OK away from wave breaking and shoreline


Open problems in coastal oceanography

Numerically, we can handle

Shoreline

Wavebreaking




Where do waves come from?How are they created?What is their speed?

Closer to the shoreAnother formula!What happens?

Modern mathematical approachesNotationsThe free surface Euler equationsThe free surface Bernoulli equationsWorking with a fix domain

Local well posednessLinearized equations around the rest state

Asymptotic expansionsNondimensionalizationNondimensionalized equations


0.0: 0.1: 0.2: 0.3: 0.4: 0.5: 0.6: 0.7: 0.8: 0.9: 0.10: 0.11: 0.12: 0.13: 0.14: 0.15: 0.16: 0.17: 0.18: 0.19: 0.20: 0.21: 0.22: 0.23: 0.24: anm0: 1.0: 1.1: 1.2: 1.3: 1.4: 1.5: 1.6: 1.7: 1.8: 1.9: 1.10: 1.11: 1.12: 1.13: 1.14: 1.15: 1.16: 1.17: 1.18: 1.19: 1.20: 1.21: 1.22: 1.23: 1.24: anm1: fd@rm@0:

Documents

Mod elisation math ematique des vagues · Mod elisation math ematique des vagues David Lannes Institut de Math ematiques de Bordeaux et CNRS UMR 5251 Journ ee des doctorants David