A hierarchy of non-hydrostatic models for free-surface flows

A hierarchy of non-hydrostatic models for free-surface flows

Yohan Penel ¹

1

Team ANGE (CEREMA, Inria, UPMC, CNRS)

Joint work with

E. Fern´ andez-Nieto (Univ. Sevilla, Spain) M. Parisot & J. Sainte-Marie (ANGE)

Collaborators: E. Audusse (Univ. Paris 13), T. Morales de Luna (Univ. C´ ordoba, Spain)

XVI International Conference on Hyperbolic Problems

Theory, Numerics, Applications

Aachen – August, 4th. 2016

Outline

1 Introduction

2 Derivation of the hierarchy of models

3 Energy

4 Linear wave analysis

5 Conclusion

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

Literature about free-surface flows

Free-surface incompressible Euler equations

Shallow water equations Hydrostatic pressure

Homogeneous velocity

Shallow water assumption

Literature about free-surface flows

Free-surface incompressible Euler equations

Shallow water equations Hydrostatic pressure

Homogeneous velocity

Shallow water assumption

A. Barr´e de Saint-Venant,Th´eorie du mouvement non permanent des eaux, avec application aux crues des rivi`eres et `a l’introduction des mar´ees dans leurs lits(C. R. Acad. Sci.73, 1871)

J.-F. Gerbeau, B. Perthame,Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation (Discrete Contin. Dyn. Syst. Ser. B1(1), 2001)

S. Ferrari, F. Saleri,A new two-dimensional Shallow Water model including pressure effects and slow varying bottom topography (Math. Model. Numer. Anal.38(2), 2004)

F. Marche,Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects (Eur. J. Mech. B Fluids26(1), 2007)

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

Literature about free-surface flows

Free-surface incompressible Euler equations

Shallow water equations hhh hhh hh h

Hydrostatic pressure

Homogeneous velocity

Shallow water assumption

F. Serre,Contribution `a l’´etude des ´ecoulements permanents et variables dans les canaux(La Houille Blanche6, 1953) A.E. Green, P.M. Naghdi,A derivation of equations for wave propagation in water of variable depth(J. Fluid Mech.78(2), 1976) M.-O. Bristeau, J. Sainte-Marie,Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems (Discrete Contin. Dyn. Syst. Ser. B10(4), 2008)

D. Lannes, P. Bonneton,Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation(Phys. Fluids 21(1), 2009)

Peregrine ’67, Madsenet al.’91 ’96 ’03 ’06, Nwogu ’93, Casulliet al.’95 ’99, Yamazakiet al.’09, . . .

Literature about free-surface flows

Free-surface incompressible Euler equations

Shallow water equations Hydrostatic pressure

hhh hhh Homogeneous velocity hhh hhh hhh

hhh h

Shallow water assumption

E. Audusse, M.-O. Bristeau, B. Perthame, J. Sainte-Marie,A multilayer Saint-Venant system with mass exchanges for Shallow Water flows.

Derivation and numerical validation(Math. Model. Numer. Anal.45(1), 2011)

F. Bouchut, V. Zeitlin,A robust well-balanced scheme for multi-layer shallow water equations(Discrete Contin. Dyn. Syst. Ser. B13(4), 2010) E.D. Fern´andez-Nieto, E.H. Kon´e, T. Morales de Luna, R. B¨urger,A multilayer shallow water system for polydisperse sedimentation(J. Comput.

Phys.238, 2013)

Castroet al.’01 ’04 ’10, Narbonaet al.’09 ’13, Rambaud ’11, . . .

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

Literature about free-surface flows

Free-surface incompressible Euler equations

Shallow water equations hhh hhh hh h

Hydrostatic pressure

hhh hhh Homogeneous velocity hhh hhh hhh

hhh h

Shallow water assumption

Derivation of multilayer non-hydrostatic models

M. Zijlema, G.S. Stelling,Further experiences with computing non-hydrostatic free-surface flows involving water waves(Int. J. Numer. Methods Fluids48(2), 2005)

Y. Bai, K.F. Cheung,Dispersion and nonlinearity of multi-layer non-hydrostatic free-surface flow(J. Fluid Mech.726, 2013)

Fluid domain

z = z

(x) z = η(t, x)

H(t,x) u = (u,w)

x z

Water height:

H(t , x ) = η(t , x) − z _b (x)

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

Euler equations

Model



 

 

∂ x u + ∂ z w = 0

∂ _t u + ∂ _x (u ² + p) + ∂ _z (uw) = 0

∂ t w + ∂ x (uw) + ∂ z (w ² + p) = −g set in the domain Ω(t ) =

(x, z ) ∈ R ²

z b (x ) ≤ z ≤ η(t, x) Boundary conditions

∂ _t η(t, x) + u t, x , η(t, x)

∂ _x η(t, x) − w t, x, η(t , x)

= 0 p t, x, η(t , x)

= p ^atm (t, x) u t, x, z _b (x)

z _b ⁰ (x) − w t , x, z _b (x )

= 0 together with well-prepared initial conditions

Pressure fields p(t, x, z ) = p ^atm (t , x) + g η(t, x) − z

+ q(t , x , z )

Euler equations

Model



 

 

∂ x u + ∂ z w = 0

∂ _t u + ∂ _x (u ² + q) + ∂ _z (uw) = −∂ x (g η + p ^atm )

∂ t w + ∂ x (uw) + ∂ z (w ² + q) = 0 set in the domain Ω(t ) =

(x, z ) ∈ R ²

z b (x ) ≤ z ≤ η(t, x) Boundary conditions

∂ _t η(t, x) + u t, x , η(t, x)

∂ _x η(t, x) − w t, x, η(t , x)

= 0 q t, x, η(t, x)

= 0 u t, x, z _b (x)

z _b ⁰ (x) − w t , x, z _b (x )

= 0

together with well-prepared initial conditions

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

Multilayer framework

z = z

(x) = z

(x) z = η(t, x) = z

(t, x)

z = z

(t, x) z = z

(t, x) H(t, x) h

(t, x)

x z

Height decomposition: h _α (t, x) = ` <sub>α</sub> H(t , x) with ` _α ∈ (0, 1) and P L

α=1 ` _α = 1

Multilayer framework

z = z

(x) = z

(x) z = η(t, x) = z

(t, x)

z = z

(t, x) z = z

(t, x) H(t, x) h

(t, x)

x z

Height decomposition: h _α (t, x) = ` <sub>α</sub> H(t , x) with ` _α ∈ (0, 1) and P L

α=1 ` <sub>α</sub> = 1 Homogeneous mesh: ` _α = 1

L

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

Multilayer framework

z = z

(x) = z

(x) z = η(t, x) = z

(t, x)

z = z

(t, x) z = z

(t, x) H(t, x) h

(t, x)

x z

Notations

[[f ]] _α+1/2 = f _α+1/2 ⁺ − f _α+1/2 ⁻ , e f _α+1/2 = γ _α+1/2 f _α+1/2 ⁻ + (1 − γ _α+1/2 )f _α+1/2 ⁺

Multilayer framework

z = z

(x) = z

(x) z = η(t, x) = z

(t, x)

z = z

(t, x) z = z

(t, x) H(t, x) h

(t, x)

x z

Notations

hf i _α (t, x) = 1 h α (t, x)

Z z

(t,x) z

(t,x)

f (t , x, z ) dz

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

Multilayer framework

z = z

(x) = z

(x) z = η(t, x) = z

(t, x)

z = z

(t, x) z = z

(t, x) H(t, x) h

(t, x)

x z

Notations

n _α+1/2 = (−∂ _x z _α+1/2 , 1) ^T

Discontinuous Galerkin framework

Let us be given a velocity field satisfying

[[u]] _α+1/2 · n _α+1/2 = 0 ⇐⇒ [[w ]] _α+1/2 = [[u]] _α+1/2 ∂ x z _α+1/2 Toy model

∂ t R + ∂ x (u R + P ) + ∂ z (w R + Q ) = S (1) where R , P , Q and S take values in R ^p

Semi-discrete formulation over each layer L α = (z _α+1/2 , z _α−1/2 )

∂ t (h α R α ) + ∂ x (h α [u R α + P α ]) + F α+1/2 ^R − F α−1/2 ^R = h α S α

where

F α+1/2 ^R = Υ _α+1/2 R e α+1/2 − P f α+1/2 ∂ _x z _α+1/2 + Q e α+1/2

Υ _α+1/2 = w e _α+1/2 − ∂ t z _α+1/2 − e u _α+1/2 ∂ x z _α+1/2

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

Discontinuous Galerkin framework

Let us be given a velocity field satisfying

[[u]] _α+1/2 · n _α+1/2 = 0 ⇐⇒ [[w ]] _α+1/2 = [[u]] _α+1/2 ∂ x z _α+1/2 Toy model

∂ t R + ∂ x (u R + P ) + ∂ z (w R + Q ) = S (1) where R , P , Q and S take values in R ^p

Semi-discrete formulation over each layer L α = (z _α+1/2 , z _α−1/2 )

∂ t (h α R α ) + ∂ x (h α [u R α + P α ]) + F α+1/2 ^R − F α−1/2 ^R = h α S α

where

F α+1/2 ^R = Υ _α+1/2 R e α+1/2 − P f α+1/2 ∂ _x z _α+1/2 + Q e α+1/2

Υ _α+1/2 = w _α+1/2 ⁻ − ∂ t z _α+1/2 − u ⁻ _α+1/2 ∂ x z _α+1/2

Discontinuous Galerkin framework

Let us be given a velocity field satisfying

[[u]] _α+1/2 · n _α+1/2 = 0 ⇐⇒ [[w ]] _α+1/2 = [[u]] _α+1/2 ∂ x z _α+1/2 Toy model

∂ t R + ∂ x (u R + P ) + ∂ z (w R + Q ) = S (1) where R , P , Q and S take values in R ^p

Semi-discrete formulation over each layer L α = (z _α+1/2 , z _α−1/2 )

∂ t (h α R α ) + ∂ x (h α [u R α + P α ]) + F α+1/2 ^R − F α−1/2 ^R = h α S α

where

F α+1/2 ^R = Υ _α+1/2 R e α+1/2 − P f α+1/2 ∂ _x z _α+1/2 + Q e α+1/2

Υ _α+1/2 = w _α+1/2 ⁺ − ∂ t z _α+1/2 − u ⁺ _α+1/2 ∂ x z _α+1/2

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

Main idea

Discontinuities across layer interfaces allowed provided jump conditions

∂ _t Z[[ R ]] _z=Z + ∂ _x Z[[u R + P ]] _z=Z − [[w R + Q ]] _z=Z = 0 or equivalently

Υ[[ R ]] − ∂ x Z[[ P ]] + [[ Q ]] = 0 Integrating Eq. (1) over a layer L α yields

h α h S i _α = ∂ t (h α h R i _α ) − R _α+1/2 ⁻ ∂ t z _α+1/2 + R _α−1/2 ⁺ ∂ t z _α−1/2

+ ∂ _x (h _α hu R + P i _α ) − (u ⁻ _α+1/2 R _α+1/2 ⁻ + P _α+1/2 ⁻ )∂ _x z _α+1/2 + (u _α−1/2 ⁺ R _α−1/2 ⁺ + P _α−1/2 ⁺ )∂ x z _α−1/2

+ w _α+1/2 ⁻ R _α+1/2 ⁻ + Q ⁻ _α+1/2 − w _α−1/2 ⁺ R _α−1/2 ⁺ + Q ⁺ _α−1/2

Main idea

Discontinuities across layer interfaces allowed provided jump conditions

∂ _t Z[[ R ]] _z=Z + ∂ _x Z[[u R + P ]] _z=Z − [[w R + Q ]] _z=Z = 0 or equivalently

Υ[[ R ]] − ∂ x Z[[ P ]] + [[ Q ]] = 0 Integrating Eq. (1) over a layer L α yields

h α h S i _α = ∂ t (h α h R i _α ) + ∂ x (h α hu R + P i _α ) + h

R _α+1/2 ⁻ Υ _α+1/2 − P _α+1/2 ⁻ ∂ _x z _α+1/2 + Q _α+1/2 ⁻ i

− h

R _α−1/2 ⁺ Υ _α−1/2 − P _α−1/2 ⁺ ∂ _x z _α−1/2 + Q ⁺ _α−1/2 i

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

Main idea

Discontinuities across layer interfaces allowed provided jump conditions

∂ _t Z[[ R ]] _z=Z + ∂ _x Z[[u R + P ]] _z=Z − [[w R + Q ]] _z=Z = 0 or equivalently

Υ[[ R ]] − ∂ x Z[[ P ]] + [[ Q ]] = 0 Integrating Eq. (1) over a layer L α yields

h α h S i _α = ∂ t (h α h R i _α ) + ∂ x (h α hu R + P i _α ) + h

R e α+1/2 Υ _α+1/2 − P f α+1/2 ∂ _x z _α+1/2 + Q e α+1/2

i

− h

R e α−1/2 Υ _α−1/2 − P f α−1/2 ∂ _x z _α−1/2 + Q e α−1/2

i

Spaces of approximation

q

, e u

, w e

q

, e u

, w e

z = z

z = z

z = z

w

w

w

u

,q

u(t , x) =

L

X

α=1

u _α (t, x) 1 {L

(t,x)} (z ) + E L

w (t, x) =

L

X

α=1

w _α (t, x) − z − z _α (t , x )

∂ _x u _α (t, x)

1 {L

(t,x)} (z ) + E _L ⁰

q continuous over the water column

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

Core of the models

Applying the previous semi-discretisation to the Euler equations leads to



 



 



∂ t h α + ∂ x (h α u α ) + Υ _α+1/2 − Υ _α−1/2 = 0

∂ t (h α u α ) + ∂ x h α u ² _α + h α q _α

+ U α+1/2 − U _α−1/2 = −h α ∂ x (g η + p ^atm )

∂ _t (h _α w _α ) + ∂ _x (h _α u _α w _α ) + W _α+1/2 − W _α−1/2 = 0

with 





U _α+1/2 = e u _α+1/2 Υ _α+1/2 − ∂ _x z _α+1/2 q _α+1/2 W _α+1/2 = w e _α+1/2 Υ _α+1/2 + q _α+1/2 and

Υ α+1/2 = w e α+1/2 − ∂ t z α+1/2 − e u α+1/2 ∂ x z α+1/2

Core of the models

Applying the previous semi-discretisation to the Euler equations leads to



 



 



∂ t h α + ∂ x (h α u α ) + Υ _α+1/2 − Υ _α−1/2 = 0

∂ t (h α u α ) + ∂ x h α u ² _α + h α q _α

+ U α+1/2 − U _α−1/2 = −h α ∂ x (g η + p ^atm )

∂ _t (h _α w _α ) + ∂ _x (h _α u _α w _α ) + W _α+1/2 − W _α−1/2 = 0

with 





U _α+1/2 = e u _α+1/2 Υ _α+1/2 − ∂ _x z _α+1/2 q _α+1/2 W _α+1/2 = w e _α+1/2 Υ _α+1/2 + q _α+1/2 and

Υ α+1/2 =

L

X

β=α+1

∂ x h β

u β − u

where u =

L

X

α=1

` α u α

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

Core of the models

Applying the previous semi-discretisation to the Euler equations leads to



 



 



∂ t H + ∂ x Hu

= 0

∂ t (h α u α ) + ∂ x h α u ² _α + h α q _α

+ U α+1/2 − U _α−1/2 = −h α ∂ x (g η + p ^atm )

∂ _t (h _α w _α ) + ∂ _x (h _α u _α w _α ) + W _α+1/2 − W _α−1/2 = 0

with 





U _α+1/2 = e u _α+1/2 Υ _α+1/2 − ∂ _x z _α+1/2 q _α+1/2 W _α+1/2 = w e _α+1/2 Υ _α+1/2 + q _α+1/2 and

Υ α+1/2 =

L

X

β=α+1

∂ x h β

u β − u

where u =

L

X

α=1

` α u α

Requirements for the P ¹ choice

Additional equation

∂ _t (zw) + ∂ _x (zuw) + ∂ _z z (w ² + q)

= w ² + q.

which is discretised as

∂ t (h α hzw i _α ) + ∂ x (h α u α hzw i _α ) + F α+1/2 ^zw − F _α−1/2 ^zw = h α

w ² + q

α

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

Requirements for the P ¹ choice (S ₁ model)

Additional equation

∂ _t (zw) + ∂ _x (zuw) + ∂ _z z (w ² + q)

= w ² + q.

which is discretised as

∂ t (h α hzw i _α ) + ∂ x (h α u α hzw i _α ) + F α+1/2 ^zw − F _α−1/2 ^zw = h α

w ² + q

α

Let us introduce the signed standard deviation σ _α = − ^h

^∂

^u

2 √

3 such that w ²

α = w ² _α + σ _α ² , hzwi _α = z α w α + h _α σ _α 2 √

3

Requirements for the P ¹ choice (S ₁ model)

Additional equation

∂ _t (zw) + ∂ _x (zuw) + ∂ _z z (w ² + q)

= w ² + q.

which is discretised as

∂ t (h α hzw i _α ) + ∂ x (h α u α hzw i _α ) + F α+1/2 ^zw − F _α−1/2 ^zw = h α

w ² + q

α

Then

∂ _t (h _α z _α w _α ) + ∂ _x (h _α z _α u _α w _α ) + ∂ _t h ² _α σ α

2 √ 3

+ ∂ _x

h ² _α σ α u α

2 √ 3

+ z _α+1/2 ( w e _α+1/2 Υ _α+1/2 + q _α+1/2 ) − z _α−1/2 ( w e _α−1/2 Υ _α−1/2 + q _α−1/2 )

= h _α w ² _α + σ _α ² + q _α

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

Requirements for the P ¹ choice (S ₁ model)

Additional equation

∂ _t (zw) + ∂ _x (zuw) + ∂ _z z (w ² + q)

= w ² + q.

which is discretised as

∂ t (h α hzw i _α ) + ∂ x (h α u α hzw i _α ) + F α+1/2 ^zw − F _α−1/2 ^zw = h α

w ² + q

α

Then

∂ _t (h _α σ _α ) + ∂ _x (h _α σ _α u _α ) = 2 √ 3

q _α − q _α+1/2 + q _α−1/2 2

− Υ _α+1/2

h α ∂ x u α

12 + w e α+1/2 − w α

2

+Υ _α−1/2

h α ∂ x u α

12 + w α − w e _α−1/2 2

Requirements for the P ¹ choice (S _1/2 model)

Additional equation

∂ _t (zw) + ∂ _x (zuw) + ∂ _z z (w ² + q)

= w ² + q.

which is discretised as

∂ t (h α hzw i _α ) + ∂ x (h α u α hzw i _α ) + F α+1/2 ^zw − F _α−1/2 ^zw = h α

w ² + q

α

Rather using a Hermitte interpolation leads to

zw _|L

≈ z α w α + (z − z α )(w α − z α ∂ x u α ), w _|L ²

≈ w ² _α − 2(z − z α )w α ∂ x u α

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

Requirements for the P ¹ choice (S _1/2 model)

Additional equation

∂ _t (zw) + ∂ _x (zuw) + ∂ _z z (w ² + q)

= w ² + q.

which is discretised as

∂ t (h α hzw i _α ) + ∂ x (h α u α hzw i _α ) + F α+1/2 ^zw − F _α−1/2 ^zw = h α

w ² + q

α

Then

∂ _t (h _α z _α w _α ) + ∂ _x (h _α z _α u _α w _α ) + z _α+1/2 q _α+1/2 − z _α−1/2 q _α−1/2 + Υ _α+1/2

z _α+1/2 w e _α+1/2 + H ²

4L ² (∂ ] x u) _α+1/2

− Υ _α−1/2

z _α−1/2 w e _α−1/2 + H ²

4L ² (∂ ] x u) _α−1/2

= h α w ² _α + q _α

Requirements for the P ¹ choice (S _1/2 model)

Additional equation

∂ _t (zw) + ∂ _x (zuw) + ∂ _z z (w ² + q)

= w ² + q.

which is discretised as

∂ t (h α hzw i _α ) + ∂ x (h α u α hzw i _α ) + F α+1/2 ^zw − F _α−1/2 ^zw = h α

w ² + q

α

Then

q _α = q _α+1/2 + q _α−1/2

2 + Υ _α+1/2 H

4L (∂ ] x u) _α+1/2 + w e _α+1/2 − w α

2

− Υ _α−1/2 H

4L (∂ ] _x u) _α−1/2 + w α − w e _α−1/2 2

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

S ₁ model

∂ _t H + ∂ _x Hu

= 0

∂ _t (h _α u _α ) + ∂ _x h _α u ² _α + h _α q _α

+ U _α+1/2 − U _α−1/2 = −h _α ∂ _x (g η + p ^atm )

∂ t (h α w α ) + ∂ x (h α u α w α ) + W _α+1/2 − W _α−1/2 = 0

∂ t (h α σ α ) + ∂ x (h α σ α u α ) = 2 √ 3

q _α − q _α+1/2 + q _α−1/2 2

−Υ α+1/2

h _α ∂ _x u _α

12 + w e _α+1/2 − w _α 2

+ Υ _α−1/2

h _α ∂ _x u _α

12 + w _α − w e _α−1/2 2

∂ _x u _α + w _α+1/2 ⁻ − w α

h _α /2 = 0 σ _α = − h α ∂ x u α

2 √ 3 w _α−1/2 ⁺ − u α ∂ x z _α−1/2 +

α−1

X

β=1

∂ x (h β u β ) = 0

S _1/2 model

∂ _t H + ∂ _x Hu

= 0

∂ _t (h _α u _α ) + ∂ _x h _α u ² _α + h _α q _α

+ U _α+1/2 − U _α−1/2 = −h _α ∂ _x (g η + p ^atm )

∂ t (h α w α ) + ∂ x (h α u α w α ) + W _α+1/2 − W _α−1/2 = 0 q _α = q _α+1/2 + q _α−1/2

2 + Υ α+1/2

H

4L (∂ ] x u) _α+1/2 + w e _α+1/2 − w _α 2

− Υ _α−1/2 H

4L (∂ ] x u) _α−1/2 + w α − w e _α−1/2 2

∂ x u α +

w _α+1/2 ⁻ − w α

h α /2 = 0 σ α = − h α ∂ x u α

2 √ 3 w _α−1/2 ⁺ − u _α ∂ _x z _α−1/2 +

α−1

X

β=1

∂ _x (h _β u _β ) = 0

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

S ₀ model

∂ _t H + ∂ _x Hu

= 0

∂ _t (h _α u _α ) + ∂ _x h _α u ² _α + h _α q _α

+ U _α+1/2 − U _α−1/2 = −h _α ∂ _x (g η + p ^atm )

∂ t (h α w α ) + ∂ x (h α u α w α ) + W _α+1/2 − W _α−1/2 = 0 q _α = q _α+1/2 + q _α−1/2

2

w α − u α ∂ x z α +

α−1

X

β=1

∂ x (h β u β ) + 1

2 ∂ x (h α u α ) = 0

Energy inequality

Denoting K = ^u

^+w ₂

, smooth solutions to the Euler equations satisfy

∂ t

Z η z

K + g η + z b

2 + p ^atm

dz

+ ∂ x

Z η z

u(K + q + g η + p ^atm ) dz

= H∂ t p ^atm . Proposition

Let us assume that γ _α+1/2 − ¹ ₂

Υ _α+1/2 ≥ 0. If (H, u α , w α , q _α ) are smooth solutions to the S 1 -model, then with K α = ^u

^+w ₂

^+σ

∂ _t

" _L X

α=1

h _α K _α + gz _α + p ^atm

#

+ ∂ x

" _L X

α=1

h α u α K α + q _α + g η + p ^atm

#

≤ H∂ t p ^atm .

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

Comment

Constraint γ _α+1/2 − ¹ ₂

Υ _α+1/2 ≥ 0 is equivalent to taking

γ _α+1/2 = 1

2 1 + λ sign(Υ _α+1/2 ) for any λ ≥ 0, which gives

R e α+1/2 Υ _α+1/2 = R _α+1/2 ⁺ + R _α+1/2 ⁻

2 Υ _α+1/2 − λ

2 |Υ _α+1/2 |

R ⁺ _α+1/2 − R _α+1/2 ⁻

.

The energy inequality is satisfied in particular for γ _α+1/2 = ¹ ₂ (λ = 0) and for

γ _α+1/2 = 1 {Υ

≥0} (λ = 1).

Hydrodynamic balances

Proposition

Let (H, u α , w α , q _α+1/2 ) be smooth solutions to the S 1 model. Then

§ The conservation of global volume: ∂ t

Z

R

H(t, x) dx

= 0

§ The balance of horizontal momentum:

∂ t

Z

R

H(t, x)u(t , x ) dx

= − Z

R

H(t , x )∂ _x p ^atm (t , x ) + gH(t, x) + q _1/2 (t, x)

∂ _x z _b (x) dx

§ The balance of vertical momentum:

∂ t

Z

R

H(t, x)w (t , x) dx

= − Z

R

q 1/2 (t , x ) dx

1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

Dispersion relations

Let us linearise around the so-called lake-at-rest steady state (H 0 , 0, 0, 0).

Proposition

There exists a plane wave solution

H, ˆ ˆ u _α , w ˆ _α , q ˆ _α

e ^i(kx−ωt) to the linearised S 1

system provided the following dispersion relation holds

ω ² = k ² c _sw ² A ⁻¹ _kH

0

e, ` where c sw = √

gH 0 , ` = (` 1 , . . . , ` L ) ∈ R ^L , e = (1, . . . , 1) ∈ R ^L and

A x = I L + x ² B, with B αβ = − ` ² _α

6 δ αβ + ` β





` _max{α,β}

2 +

L

X

γ=max{α,β}+1

` γ





Dispersion relations

Let us linearise around the so-called lake-at-rest steady state (H 0 , 0, 0, 0).

Proposition

There exists a plane wave solution

H, ˆ ˆ u _α , w ˆ _α , q ˆ _α

e ^i(kx−ωt) to the linearised S 0

system provided the following dispersion relation holds

ω ² = k ² c _sw ² A ⁻¹ _kH

0

e, ` where c sw = √

gH 0 , ` = (` 1 , . . . , ` L ) ∈ R ^L , e = (1, . . . , 1) ∈ R ^L and

A x = I L + x ² B, with B αβ = − ` ² _α

4 δ αβ + ` β





` _max{α,β}

2 +

L

X

γ=max{α,β}+1

` γ





1. Introduction 2. Derivation of the hierarchy of models 3. Energy 4. Linear wave analysis 5. Conclusion

Phase velocity

Conclusion

§ Summary

à Derivation of a class of multilayer non-hydrostatic models as semi-discretisations of the Euler equations

à Proof of physical properties (energy, hydrodynamic balances, dispersive effects)

§ On-going works

à Convergence of the dispersion relation à Numerical simulations

à Incorporation of viscous effects à Enriching the physics

E. Fern´ andez-Nieto, M. Parisot, Y. Penel & J. Sainte-Marie, Layer-averaged

approximations for inviscid flow models (preprint).

Thank you for your attention