Asymptotic models for internal waves in Oceanography

Transcription

1 Asymptotic models for internal waves in Oceanography Vincent Duchêne École Normale Supérieure de Paris Applied Mathematics Colloquium APAM February 01, 2010

2 Internal waves in ocean The large picture Figure: Sulu Sea. April 8, Credits: NASA s Earth Observatory (Picture of the Day July 1, 2003)

3 Internal waves in ocean Figure: St. Lawrence Estuary 1 1 Credits: St. Lawrence Estuary Internal Wave Experiment (SLEIWEX)

4 Two layers of immiscible, homogeneous, ideal, incompressible fluids d 1 z ζ 1 (t, x) λ a 1 ρ 1 g 0 ζ 2 (t, x) a 2 ρ 2 d 2 x ɛ 1 a 1 d 1, ɛ 2 a 2 d 1, µ d2 1 λ 2, γ ρ 1 ρ 2, δ d 1 d 2 ɛ 1 = ɛ 2 = µ ε 1, γ (0,1), 0 < δ min δ δ max < +

5 Outline 1 The full Euler system The governing equations Reduction of the equations 2 Some asymptotic models The Boussinesq/Boussinesq models Symmetric Boussinesq models The KdV approximation 3 The dead water phenomenon Presentation of the problem Asymptotic models

6 1 The full Euler system The governing equations Reduction of the equations 2 Some asymptotic models The Boussinesq/Boussinesq models Symmetric Boussinesq models The KdV approximation 3 The dead water phenomenon Presentation of the problem Asymptotic models

7 The governing equations Hypotheses on the fluids The assumptions The fluid is irrotational The fluid is homogeneous, incompressible The fluid is inviscid The fluid particles do not cross the bottom The fluid particles do not cross the surface The particles of the two fluids do not cross the interface. The equations v i = x,zφ i (i = 1,2) x,zφ i = 0 tφ i x,zφi 2 = P ρ i zφ 2 = 0 on Γ b gz tζ 1 = 1 + xζ 1 2 nφ 1 on Γ 1 tζ 2 = 1 + xζ 2 2 nφ 1 = 1 + xζ 2 2 nφ 2 on Γ 2.

8 The governing equations Hypotheses on the fluids The assumptions The fluid is irrotational The fluid is homogeneous, incompressible The fluid is inviscid The fluid particles do not cross the bottom The fluid particles do not cross the surface The particles of the two fluids do not cross the interface. The equations v i = x,zφ i (i = 1,2) x,zφ i = 0 tφ i x,zφi 2 = P ρ i zφ 2 = 0 on Γ b gz tζ 1 = 1 + xζ 1 2 nφ 1 on Γ 1 tζ 2 = 1 + xζ 2 2 nφ 1 = 1 + xζ 2 2 nφ 2 on Γ 2.

9 The governing equations Hypotheses on the fluids The assumptions The fluid is irrotational The fluid is homogeneous, incompressible The fluid is inviscid The fluid particles do not cross the bottom The fluid particles do not cross the surface The particles of the two fluids do not cross the interface. The equations v i = x,zφ i (i = 1,2) x,zφ i = 0 tφ i x,zφi 2 = P ρ i zφ 2 = 0 on Γ b gz tζ 1 = 1 + xζ 1 2 nφ 1 on Γ 1 tζ 2 = 1 + xζ 2 2 nφ 1 = 1 + xζ 2 2 nφ 2 on Γ 2. Additional assumptions The fluid is at rest at infinity The pressure P is constant at the surface, and continuous at the interface There is no surface tension

10 Reduction of the equations Dirichlet-Neumann operators The equations can be reduced to evolution equations located on the surface and on the interface thanks to the following operators φ 1 = ψ 1 φ 2 = ψ 2 ( 2 x + 2 z ) φ 1 = 0 n2 φ 1 = n2 φ 2 ( 2 x + 2 z ) φ 2 = 0 zφ 2 = 0 x,zφ 2 = 0 in Ω 2, φ 2 = ψ 2 on Γ 2 {z = ζ 2 }, zφ 2 = 0 on Γ b {z = d 2 }, φ 2 x,zφ 1 = 0 in Ω 1, φ 1 = ψ 1 on Γ 1 {z = d 1 + ζ 1 }, n2 φ 1 = n2 φ 2 on Γ 2. φ 1

11 Reduction of the equations Dirichlet-Neumann operators The equations can be reduced to evolution equations located on the surface and on the interface thanks to the following operators φ 1 = ψ 1 φ 2 = ψ 2 ( 2 x + 2 z ) φ 1 = 0 n2 φ 1 = n2 φ 2 ( 2 x + 2 z ) φ 2 = 0 zφ 2 = 0 x,zφ 2 = 0 in Ω 2, φ 2 = ψ 2 on Γ 2 {z = ζ 2 }, zφ 2 = 0 on Γ b {z = d 2 }, φ 2 x,zφ 1 = 0 in Ω 1, φ 1 = ψ 1 on Γ 1 {z = d 1 + ζ 1 }, n2 φ 1 = n2 φ 2 on Γ 2. φ 1

12 Reduction of the equations Dirichlet-Neumann operators The equations can be reduced to evolution equations located on the surface and on the interface thanks to the following operators Definition (Dirichlet-Neumann operators) The following operators are well-defined: G 2 ψ x ζ 2 2 n φ 2 interface, G 1 (ψ 1,ψ 2 ) 1 + x ζ 1 2 n φ 1 surface (, H(ψ 1,ψ 2 ) x φ 1 z=ζ2 ). Therefore, the system is entirely defined by ζ 1 ; ζ 2 ; ψ 1 φ 1 surface ; ψ 2 φ 2 interface.

13 Reduction of the equations The full Euler system Thanks to the the previous definitions, and after an adapted change of variables (obtained through the study of the linearized system), one obtains The dimensionless full Euler system (Σ) t ζ 1 1 ε G 1(ψ 1, ψ 2 ) = 0, t ζ 2 1 ε G 2ψ 2 = 0, t x ψ 1 + x ζ 1 + ε 2 x( x ψ 1 2 ) ε 2 x N 1 = 0, t ( x ψ 2 γh(ψ 1, ψ 2 )) + (1 γ) x ζ 2 + ε 2 x( x ψ 2 2 γ H(ψ 1, ψ 2 ) 2 ) ε 2 x N 2 = 0, Solutions of this system are exact solutions of our problem. We construct then asymptotic models, and therefore look for approximate solutions.

14 1 The full Euler system The governing equations Reduction of the equations 2 Some asymptotic models The Boussinesq/Boussinesq models Symmetric Boussinesq models The KdV approximation 3 The dead water phenomenon Presentation of the problem Asymptotic models

15 State of the art The one-layer problem in the long wave regime Justification of asymptotic Boussinesq model and KdV approximation: [Craig 1985], [Schneider, Wayne 2000], [BenYoussef, Colin 2000], [Bona, Colin, Lannes 2005], [Alvarez-Samaniego, Lannes 2008] The two-layer problem, with a rigid lid KdV equations for internal waves [Keulegan, 1953], [Long, 1956] Boussinesq-type models [Miyata 1985], [Choi, Camassa 1996] Hamiltonian formulation [Lvov, Tabak 2004], [Craig, Guyenne, Kalisch 2005] Consistency of Boussinesq-type models [Bona, Lannes, Saut 2008] (d = 1 or 2, with topography) Stability of the flow [Chumakova, Menzaque, Milewski, Rosales, Tabak, Turner 2004] The two-layer problem, with a free surface The KdV equations [Peters, Stoker 1960] Boussinesq-type models [Matsuno 1993], [Choi, Camassa 1996]

16 State of the art The one-layer problem in the long wave regime Justification of asymptotic Boussinesq model and KdV approximation: [Craig 1985], [Schneider, Wayne 2000], [BenYoussef, Colin 2000], [Bona, Colin, Lannes 2005], [Alvarez-Samaniego, Lannes 2008] The two-layer problem, with a rigid lid KdV equations for internal waves [Keulegan, 1953], [Long, 1956] Boussinesq-type models [Miyata 1985], [Choi, Camassa 1996] Hamiltonian formulation [Lvov, Tabak 2004], [Craig, Guyenne, Kalisch 2005] Consistency of Boussinesq-type models [Bona, Lannes, Saut 2008] (d = 1 or 2, with topography) Stability of the flow [Chumakova, Menzaque, Milewski, Rosales, Tabak, Turner 2004] The two-layer problem, with a free surface The KdV equations [Peters, Stoker 1960] Boussinesq-type models [Matsuno 1993], [Choi, Camassa 1996]

17 State of the art The one-layer problem in the long wave regime Justification of asymptotic Boussinesq model and KdV approximation: [Craig 1985], [Schneider, Wayne 2000], [BenYoussef, Colin 2000], [Bona, Colin, Lannes 2005], [Alvarez-Samaniego, Lannes 2008] The two-layer problem, with a rigid lid KdV equations for internal waves [Keulegan, 1953], [Long, 1956] Boussinesq-type models [Miyata 1985], [Choi, Camassa 1996] Hamiltonian formulation [Lvov, Tabak 2004], [Craig, Guyenne, Kalisch 2005] Consistency of Boussinesq-type models [Bona, Lannes, Saut 2008] (d = 1 or 2, with topography) Stability of the flow [Chumakova, Menzaque, Milewski, Rosales, Tabak, Turner 2004] The two-layer problem, with a free surface The KdV equations [Peters, Stoker 1960] Boussinesq-type models [Matsuno 1993], [Choi, Camassa 1996]

18 The Boussinesq/Boussinesq models 1 The full Euler system The governing equations Reduction of the equations 2 Some asymptotic models The Boussinesq/Boussinesq models Symmetric Boussinesq models The KdV approximation 3 The dead water phenomenon Presentation of the problem Asymptotic models

19 The Boussinesq/Boussinesq models Asymptotic expansion of the operators Proposition Let s > 1, ζ 1, ζ 2, ψ 1, ψ 2 H s+t (R). Then one has G 2 ψ 2 + ε x (h 2 x ψ 2 ) + ε 2 1 3δ 3 3 x x ψ 2 ) H s G 1 (ψ 1, ψ 2 ) + ε x (h 1 x ψ 1 + h 2 x ψ 2 ) ( 1 + ε 2 x 3 3 xψ 1 + ( 1 3δ 3 + 2δ) 1 ) H x ψ 2 s ( 1 H(ψ 1, ψ 2 ) x ψ 1 ε x 2 2 xψ ) δ Hs xψ 2 ε 3 C ε 3 C, ε 2 C, Notations: h 1 = 1 + εζ 1 εζ 2 and h 2 = 1 δ + εζ 2.

20 The Boussinesq/Boussinesq models The Boussinesq/Boussinesq models A Boussinesq/Boussinesq model t ζ 1 + x (h 1 x ψ 1 ) + x (h 2 x ψ 2 ) = ε ( xψ 1 + ( ) 3δ 3 2δ ) 4 xψ 2, t ζ 2 + x (h 2 x ψ 2 ) = ε 1 3δ 3 x 4ψ 2, t x ψ 1 + x ζ 1 + ε 2 ( x x ψ 1 2) = 0, t x ψ 2 + (1 γ) x ζ 2 + γ x ζ 1 + ε 2 ( x x ψ 2 2) = ε t x 2 ( γ δ xψ 2 + γ 2 ) xψ 1, t U + A 0 x U + ε ( A 1 (U) x U + B 2 x tu + C 3 x U ) = 0 with U = (ζ 1,ζ 2, x ψ 1, x ψ 2 ). Proposition (consistency) The full Euler system is consistent with the Boussinesq/Boussinesq model, with precision O(ε 2 ).

21 The Boussinesq/Boussinesq models The Boussinesq/Boussinesq models A Boussinesq/Boussinesq model (M B ) t U + A 0 x U + ε ( A 1 (U) x U + B 2 x t U + C 3 xu ) = 0. Proposition (consistency) The full Euler system is consistent with the Boussinesq/Boussinesq model, with precision O(ε 2 ). Let U be a strong solution of the full Euler system (Σ), uniformly bounded in a sufficiently high Sobolev norm. Then U satisfies the Boussinesq/Boussinesq model, up to some residuals, bounded (in H s norm) by ɛ 2 C 0. Open questions: Well-posedness of any Boussinesq/Boussinesq system? Convergence of its solutions towards solutions of the full Euler system?

22 Symmetric Boussinesq models 1 The full Euler system The governing equations Reduction of the equations 2 Some asymptotic models The Boussinesq/Boussinesq models Symmetric Boussinesq models The KdV approximation 3 The dead water phenomenon Presentation of the problem Asymptotic models

23 Symmetric Boussinesq models Symmetrization The system can be written under the compact form t U + A 0 x U + ε ( A 1 (U) x U + B 2 x t U + C 3 xu ) = 0. Multiply by adapted S S 0 + εs 1 (U) εs 2 2 x, and withdraw O(ε2 ) terms. One obtains a perfectly symmetric model of the form: The symmetric Boussinesq/Boussinesq model (S B ) ( S 0 + ε ( S 1 (U) S 2 x) ) ( 2 t U + Σ 0 + ε ( Σ 1 (U) Σ 2 x) ) 2 x U = 0, with the following properties: Matrices S 0, S 2, Σ 0, Σ 2 M 4 (R) are symmetric. S 1 ( ) and Σ 1 ( ) are linear mappings, with values in M 4 (R), and for all U R 4, S 1 (U) and Σ 1 (U) are symmetric. S 0 et S 2 are definite positive.

24 Symmetric Boussinesq models Consistency The full Euler system is consistent with the symmetric Boussinesq/Boussinesq model (S B), with precision O(ε 2 ). Well posedness The symmetric system is well-posed in H s+1 (s > 3/2) over times of order O(1/ε). Moreover, one has the estimate ( U(t) 2 + ε ) H U(t) 2 1/2 U 0 s H = U(t) H H s+1 s+1 C 0 ε s+1 ε U 0. 1 C 0 tε Hε s+1

25 Symmetric Boussinesq models Consistency The full Euler system is consistent with the symmetric Boussinesq/Boussinesq model (S B), with precision O(ε 2 ). Well posedness The symmetric system is well-posed in H s+1 (s > 3/2) over times of order O(1/ε). Moreover, one has the estimate ( U(t) 2 + ε U(t) ) 2 1/2 H s H = U(t) U 0 H s+1 H s+1 C 0 ε s+1 ε U 0. Convergence 1 C 0 tε Hε s+1 The difference between any solution U of the full Euler system (Σ), and the solution U B of the symmetric Boussinesq/Boussinesq model (S B) with same initial data, satisfies t [0, T/ε], U UB L ε 2 tc ([0,t];Hε s+1 ) 1. Numerical simulation

26 The KdV approximation 1 The full Euler system The governing equations Reduction of the equations 2 Some asymptotic models The Boussinesq/Boussinesq models Symmetric Boussinesq models The KdV approximation 3 The dead water phenomenon Presentation of the problem Asymptotic models

27 The KdV approximation The WKB expansion We seek an approximate solution of system (S B ): ( S 0 + ε ( S 1 (U) S 2 x) ) ( 2 t U + Σ 0 + ε ( Σ 1 (U) Σ 2 x) ) 2 x U = 0 of the form U app (t,x) U 0 (εt,t,x) + εu 1 (εt,t,x). At order O(1) : (S 0 t + Σ 0 x )U 0 = 0. There exists a basis e i R 4 (i = 1..4), diagonalizing S 0 and Σ 0 : = U 0 = 4 u i e i, with u i (τ,t,x) = u i (τ,x c i t). i=1 At order O(ε) : We split the equation in S 0 τ U 0 + Σ 1 (U 0 ) x U 0 + S 1 (U 0 ) t U 0 Σ 2 3 x U 0 S 2 2 x tu 0 = (S 0 t + Σ 0 x )U 1. τ u i + λ i u i xi u i + µ i x 3 i u i = 0, ( t + c i x )e i U 1 + (j,k) (i,i)α ijk u k x u j + β ij xu 3 j = 0. j i

28 The KdV approximation The KdV Approximation Definition Let U be a solution of the full Euler system (Σ). We define then the KdV approximation as U KdV = 4 i=1 u ie i, with u i solution of (KdV) { t u i + c i x u i + ελ i u i x u i + εµ i 3 x u i = 0, u i t=0 = u 0 i,

29 The KdV approximation Well-posedness There exists a unique strong solution U 0 (τ, t, x), uniformly bounded in L ([0, T] R; Hε s+2 ). Then, there exists an explicit residual U 1 C 1 ([0, T) R; H s ). Secular growth of the residual (τ, t) [0, T] R, U1 (τ, t, ) H s C 0 t. Moreover, if (1 + x 2 )U 0 H s+1, then one has the uniform estimate U 1 (τ, t, ) H s C 0, Consistency U 0 (εt, t, x) + εu 1 (εt, t, x) satisfies the symmetric Boussinesq/Boussinesq model (S B ), with precision O(ε 3/2 ) (and O(ε 2 ) if (1 + x 2 )U 0 H s+1 ). = convergence towards the solution of (S B ).

30 The KdV approximation Convergence towards solutions of the full Euler system The difference between any solution U of the full Euler system (Σ), and U KdV 4 i=1 u ie i, with u i solutions of (KdV), satisfies U UKdV L ([0,t];Hε s+1 ) ε tc 0, Moreover, if (1 + x 2 )U t=0 H s+4, then one has the uniform estimate U UKdV L ([0,T/ε];H s+1 ε ) εc 0.

31 The KdV approximation Why localization in space is relevant? Propagation of a soliton Splitting of a bell curve

32 The KdV approximation Why localization in space is relevant? Propagation of a soliton Splitting of a bell curve 3 x relative difference relative difference t T/ε t T/ε Comparison between free surface and rigid lid configurations. Big density difference: γ = 1/4 Small density difference: γ = 0.9

33 1 The full Euler system The governing equations Reduction of the equations 2 Some asymptotic models The Boussinesq/Boussinesq models Symmetric Boussinesq models The KdV approximation 3 The dead water phenomenon Presentation of the problem Asymptotic models

34 Presentation of the problem Fridtjof Nansen, 1898 This peculiar phenomenon [...] manifests itself in the form of larger or smaller ripples or waves stretching across the wake, the one behind the other, arising sometimes as far forward as almost midships. When caught in dead water, Fram appeared to be held back, as if by some mysterious force, and she did not always answer the helm. In calm weather, with a light cargo, Fram was capable of 6 to 7 knots. When in dead water she was unable to make 1.5 knots. We made loops in our course, turned sometimes right around, tried all sorts of antics to get clear of it, but to very little purpose. Vilhelm Bjerknes, 1900 In my reply to Prof. Nansen I remarked that in the case of a layer of fresh water resting on the top of salt water, a ship will not only produce the ordinary visible waves at the boundary between the water and the air, but will also generate invisible waves in the salt-water freshwater boundary below; I suggested that the great resistance experienced by the ship was due to the work done in generating these [internal] waves. (...) In December 1899 I consequently suggested a pupil of mine, Dr. V. Walfrid Ekman (...) that he should do some simple preliminary experiments. video of the experiments of Vasseur-Mercier-Dauxois, 2008

35 Presentation of the problem z Rigid lid, but not flat : ζ 1 (t,x) ζ 1 (x c s t) d 1 ζ 1 a 1 0 ζ 2 a 2 λ g d 2 x ɛ 1 a 1 d 1, ɛ 2 a 2 d 1, µ d2 1 λ 2, γ ρ 1 ρ 2, δ d 1 d 2, Fr = c s c 0. ɛ 2 = µ = ɛ 1 /ɛ 2 ε 1, γ (0,1), δ [δ min,δ max ], Fr (0,+ ).

36 Presentation of the problem Wave (making) resistance suffered by the body Definition (Wave resistance) R W P ( e x n) ds = Γ ship R P d1 +ζ 1 x ζ 1 dx. where Γ ship is the exterior domain of the ship, P is the pressure, e x is the horizontal unit vector and n the normal unit vector exterior to the ship. As a solution of the Bernoulli equation, the pressure P satisfies P(x, z) ρ 1 = t φ 1 (x,z) 1 2 x,zφ 1 (x,z) 2 gz. Using the previous change of variables, we define the dimensionless wave resistance coefficient C W. In our regime (ɛ 2 = µ = ɛ 1 /ɛ 2 ε 1), the first order approximation is C W = ζ 1 x ζ 2 dx + O(ε). R

37 Presentation of the problem Wave (making) resistance suffered by the body Definition (Wave resistance) R W P ( e x n) ds = Γ ship R P d1 +ζ 1 x ζ 1 dx. where Γ ship is the exterior domain of the ship, P is the pressure, e x is the horizontal unit vector and n the normal unit vector exterior to the ship. As a solution of the Bernoulli equation, the pressure P satisfies P(x, z) ρ 1 = t φ 1 (x,z) 1 2 x,zφ 1 (x,z) 2 gz. Using the previous change of variables, we define the dimensionless wave resistance coefficient C W. In our regime (ɛ 2 = µ = ɛ 1 /ɛ 2 ε 1), the first order approximation is C W = ζ 1 x ζ 2 dx + O(ε). R

38 Presentation of the problem Wave (making) resistance suffered by the body Definition (Wave resistance) R W P ( e x n) ds = Γ ship R P d1 +ζ 1 x ζ 1 dx. where Γ ship is the exterior domain of the ship, P is the pressure, e x is the horizontal unit vector and n the normal unit vector exterior to the ship. As a solution of the Bernoulli equation, the pressure P satisfies P(x, z) ρ 1 = t φ 1 (x,z) 1 2 x,zφ 1 (x,z) 2 gz. Using the previous change of variables, we define the dimensionless wave resistance coefficient C W. In our regime (ɛ 2 = µ = ɛ 1 /ɛ 2 ε 1), the first order approximation is C W = ζ 1 x ζ 2 dx + O(ε). R

39 Presentation of the problem Numerical simulation

40 Asymptotic models The dimensionless full Euler system The governing equations can be deduced from the full Euler system in the free surface case: The full Euler system when ζ 1 (t,x) ζ 1 (x Fr t) ε t ζ 1 1 ε G 1(ψ 1, ψ 2 ) = 0, t ζ 2 1 ε G 2ψ 2 = 0, t x ψ 1 + x ζ 1 + ε 2 x( x ψ 1 2 ) ε 2 x N 1 = 0, t ( x ψ 2 γh(ψ 1, ψ 2 )) + (1 γ) x ζ 2 + ε 2 x( x ψ 2 2 γ H(ψ 1, ψ 2 ) 2 ) ε 2 x N 2 = 0,

41 Asymptotic models The dimensionless full Euler system The governing equations can be deduced from the full Euler system in the free surface case: The dimensionless full Euler system with rigid lid εfr x ζ 1 1 ε G 1(ψ 1, ψ 2 ) = 0, ( Σ) t ζ 2 1 ε G 2ψ 2 = 0, ) t ( x ψ 2 γh(ψ 1, ψ 2 ) + (γ + δ) x ζ 2 + ε 2 x ( x ψ 2 2 γ H(ψ 1, ψ 2 ) 2) = ε 2 x N 2, Using that ζ 1 is a fixed parameter, the system reduces to two evolution equations for (ζ 2,v), with v the shear velocity defined by ( (φ2 ) ) v x γφ 1 = z=εζ 2 x ψ 2 γh(ψ 1,ψ 2 ).

42 Asymptotic models The Boussinesq-type models Plug the asymptotic expansion of the operators G 1, G 2, H into the full Euler system ( Σ), and withdraw O(ε 2 ) terms. Boussinesq/Boussinesq model { t ζ δ+γ xv + ε δ2 γ (γ+δ) 2 x (ζ 2 v) + ε 1+γδ 3δ(δ+γ) xv 3 = ε Fr γ 2 δ+γ xζ 1, t v + (1 γ) x ζ 2 + ε δ 2 γ ( 2 (γ+δ) 2 x v 2 ) = 0. The Boussinesq/Boussinesq model system can be written as tu + A 0 xu + εa 1(U) xu εa 2 3 xu = εb 0(x Fr t),

43 Asymptotic models The Boussinesq-type models Boussinesq/Boussinesq model { t ζ δ+γ xv + ε δ2 γ (γ+δ) 2 x (ζ 2 v) + ε 1+γδ 3δ(δ+γ) xv 3 = ε Fr γ 2 δ+γ xζ 1, t v + (1 γ) x ζ 2 + ε δ 2 γ ( 2 (γ+δ) 2 x v 2 ) = 0. The Boussinesq/Boussinesq model system can be written as tu + A 0 xu + εa 1(U) xu εa 2 3 xu = εb 0(x Fr t), When we multiply by adapted symmetrizer S(U) S 0 + εs 1(U) εs 2 x, 2 one gets ( ) S 0 + εs 1(U) εs 2 x 2 tu + ( Σ 0 + εσ 1(U) εσ 2 ) xu = εb(x Fr t), The symmetric Boussinesq model is consistent at order O(ε 2 ), The symmetric Boussinesq model is well-posed, + energy estimate, = The solutions of the Boussinesq model converge towards solutions of the full Euler system ( Σ), at order O(ε 2 t), for t [0, T/ε].

44 Asymptotic models The Boussinesq-type models Boussinesq/Boussinesq model { t ζ δ+γ xv + ε δ2 γ (γ+δ) 2 x (ζ 2 v) + ε 1+γδ t v + (1 γ) x ζ 2 + ε δ 2 γ ( 2 (γ+δ) 2 x v 2 ) = 0. 3δ(δ+γ) 2 3 x The Boussinesq/Boussinesq model system can be written as v = ε Fr γ δ+γ xζ 1, tu + A 0 xu + εa 1(U) xu εa 2 3 xu = εb 0(x Fr t), When we multiply by adapted symmetrizer S(U) S 0 + εs 1(U) εs 2 2 x, one gets ( S 0 + εs 1(U) εs 2 x 2 ) tu + ( Σ 0 + εσ 1(U) εσ 2 ) xu = εb(x Fr t), The symmetric Boussinesq model is consistent at order O(ε 2 ), The symmetric Boussinesq model is well-posed, + energy estimate, = The solutions of the Boussinesq model converge towards solutions of the full Euler system ( Σ), at order O(ε 2 t), for t [0, T/ε].

45 Asymptotic models The KdV approximation Definition Let U = (ζ 2,v) be a solution of the full Euler system ( Σ). We define then the KdV approximation as U KdV = ( η + + η, (γ + δ)(η + η ) ), with η ± solution of { t η ± ± x η ± ± ε 3 δ 2 γ 2 γ+δ η ± x η ± ± ε 1 1+γδ 6 δ(γ+δ) 3 xη ± = εfrγ x ζ 1, η ± t=0 = η0 ±, Convergence theorem The difference between any solution U of the full Euler system ( Σ), and U KdV is bounded by U U KdV L ([0,t];H s+1 ε ) ε tc 0. Moreover, if (1 + x 2 )U t=0 H s+4, then one has the uniform estimate U UKdV L ([0,T/ε];H s+1 ε ) εc 0.

46 Asymptotic models A simple application Lemma Let u be the solution of t u + c x u + ελu x u + εν 3 x u = ε xf (x c 0 t), with u t=0 = εu 0 H s+3, s > 3/ There exists T( c c 0 ) and C( c c 0 ) > 0 such that u L Cε. ([0,T/ε];H s ) The transport equation tv + c xv = ε xf (x c 0t) leads to v = εu 0 (x ct) + ε ( f (x c0t) f (x ct) ). c 0 c The result is obtained by comparison with this function. As a consequence, the dead-water phenomenon will always be small if the velocity of the body is away from the critical velocity ( Fr = 1).

47 Asymptotic models A simple application As a consequence, the dead-water phenomenon will always be small if the velocity of the body is away from the critical velocity ( Fr = 1) C W F r C W F r Wave resistance coefficient C W at time T = 10, depending on the Froude number Fr, with δ = 1 and 2 (γ = 0.9, ε = 0.1).

48 For more details: Coupled models (Boussinesq, shallow water, Green-Naghdi): Asymptotic shallow water models for internal waves in a two-fluid system with a free surface, SIAM J. Math. Anal., 42 (2010) KdV approximation: Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation, to appear in Math. Model. Numer. Anal. (M2AN) Dead-water phenomenon: Asymptotic models for the generation of internal waves by a moving ship, and the dead-water phenomenon, preprint Arxiv: Thank you for your attention!