AVARIATIONAL APPROACH FOR WATER WAVE MODELLING

DENYS DUTYKH1 Senior Research Fellow UCD & Charge´ de Recherche CNRS

1University College Dublin School of Mathematical Sciences

NUMERIWAVES Group Meeting ACKNOWLEDGEMENTS

COLLABORATOR: Didier Clamond: Professor, LJAD, Universite´ de Nice Sophia Antipolis

COLLABORATORS: ◮ Dimitrios Mitsotakis: University of California, Merced, USA ◮ Paul Milewski: University of Bath, UK WATER WAVE PROBLEM J. STOKER: WATER WAVES: THE MATHEMATICAL THEORY WITH APPLICATIONS [STO57]

◮ Continuity equation

∇2 x x,y φ = 0, ( , y) ∈ Ω × [−d,η],

◮ Kinematic bottom condition ∂φ + ∇φ ·∇d = 0, y = −d, ∂y

◮ Kinematic free surface condition ∂η ∂φ +∇φ·∇η = , y = η(x, t), ∂t ∂y ◮ Dynamic free surface condition ∇ ∂φ 1 2 η + |∇x,y φ| +gη+σ∇· = 0, y = η(x, t). ∂t 2  1 + |∇η|2  p HAMILTONIAN STRUCTURE A. PETROV (1964) [PET64]; V. ZAKHAROV (1968) [ZAK68]

CANONICAL VARIABLES: η(x, t): free surface elevation φ˜(x, t): velocity potential at the free surface φ˜(x, t) := φ(x, y = η(x, t), t)

EVOLUTION EQUATIONS:

∂η δH ∂φ˜ δH ρ = , ρ = − , ∂t δφ˜ ∂t δη

HAMILTONIAN:

η 1 2 1 2 2 H = |∇x,y φ| dy + gη + σ 1 + |∇η| − 1 Z 2 2 q  −d LUKE’S VARIATIONAL PRINCIPLES J.C. LUKE, JFM (1967) [LUK67]

First improvement of the classical Lagrangian (L := K − Π):

t2 η 1 2 L = ρL dx dt, L := φt + |∇x,y φ| + gy dy Z Z Z 2 t1 Ω −d

δφ: ∆φ = 0, (x, y) ∈ Ω × [−d,η], ∂φ ∇ ·∇ δφ|y=−d : ∂y + φ d = 0, y = −d, ∂η ∇ ·∇ ∂φ x δφ|y=η : ∂t + φ η − ∂y = 0, y = η( , t), ∂φ 1 ∇ 2 x δη: ∂t + 2 | φ| + gη = 0, y = η( , t).

◮ We obtain the water wave problem by varying η and φ GENERALIZATION OF THE LAGRANGIAN DENSITY D. CLAMOND &D.DUTYKH,PHYSICA D, (2012), [CD12]

φ˜ := φ(x, y = η(x, t), t): quantity at the free surface φˇ := φ(x, y = −d(x, t), t): value at the bottom EQUIVALENT FORM OF LUKE’S LAGRANGIAN:

η L ˜ ˇ 1 2 1 2 1 ∇ 2 1 2 = φηt + φdt − gη + gd − | φ| + φy dy 2 2 Z h2 2 i −d

INTRODUCE THE VELOCITY FIELD: u = ∇φ, v = φy

η 1 2 1 2 2 L = φη˜ t +φˇdt − gη − (u +v )+µ·(∇φ−u)+ν(φy −v) dy 2 Z−d h2 i GENERALIZATION OF THE LAGRANGIAN DENSITY D. CLAMOND &D.DUTYKH,PHYSICA D, (2012), [CD12]

RELAXED VARIATIONAL PRINCIPLE:

1 L =(η +µ ˜ ·∇η − ν˜)φ˜ +(d +µ ˇ ·∇d +ν ˇ)φˇ − gη2 t t 2 η 1 2 1 2 + µ · u − u + νv − v +(∇· µ + νy )φ dy Z h 2 2 i −d

CLASSICAL FORMULATION:

η L ˜ ˇ 1 2 1 ∇ 2 1 2 = φηt + φdt − gη − | φ| + φy dy 2 Z h2 2 i −d

DEGREES OF FREEDOM: η,φ; u, v ; µ,ν SHALLOW WATER REGIME CHOICE OF A SIMPLE ANSATZ IN SHALLOW WATER

◮ Ansatz:

− u(x, y, t) ≈ u¯(x, t), v(x, y, t) ≈ (y + d)(η + d) 1 v˜(x, t) − φ(x, y, t) ≈ φ¯(x, t),ν(x, y, t) ≈ (y + d)(η + d) 1 ν˜(x, t)

◮ Lagrangian density:

1 2 1 2 1 1 2 L = φη¯ t − gη +(η + d) µ¯ · u¯ − u¯ + ν˜v˜ − v˜ − µ¯ · ∇φ¯ 2 h 2 3 6 i ◮ Nonlinear Shallow Water Equations:

ht + ∇· [hu¯] = 0,

u¯t +(u¯ ·∇)u¯ + g∇h = 0. CONSTRAINING WITH FREE SURFACE IMPERMEABILITY

CONSTRAINT:

ν˜ = ηt +µ ¯ ·∇η

◮ Generalized Serre equations [Ser53, GN76]:

ht + ∇· [hu¯] = 0, 1 − u¯ + u¯ ·∇u¯ + g∇h + h 1∇[h 2γ˜] = (u¯ ·∇h)∇(h∇· u¯) t 3 −[u¯ ·∇(h∇· u¯)]∇h

2 γ˜ = v˜t + u¯ ·∇v˜ = h (∇· u¯) − ∇· u¯t − u¯ ·∇(∇· u¯)
IT CANNOT BE OBTAINED FROM LUKE’S LAGRANGIAN: ∇ ¯ 1 ∇ ∇ ¯ δµ¯: u¯ = φ − 3 v˜ η 6= φ CONSTRAINING WITH FREE SURFACE IMPERMEABILITY

CONSTRAINT:

ν˜ = ηt +µ ¯ ·∇η

◮ Generalized Serre equations [Ser53, GN76]:

ht + ∇· [hu¯] = 0, 1 − u¯ + u¯ ·∇u¯ + g∇h + h 1∇[h 2γ˜] = (u¯ ·∇h)∇(h∇· u¯) t 3 −[u¯ ·∇(h∇· u¯)]∇h 2 γ˜ = v˜t + u¯ ·∇v˜ = h (∇· u¯) − ∇· u¯t − u¯ ·∇(∇· u¯)
SOLITARY WAVE SOLUTION (+ CNOIDAL WAVE):

κ − η = a sech2 (x − ct), c 2 = g(d + a), (κd)2 = 3a(d + a) 1 2 SERRE EQUATIONS -I TWO-DIMENSIONALCASE (1DH)

◮ Noncanonical Hamiltonian structure [Li02]: δH ht  δq˜  ∂x h 0 1 3 = J , J = − , q˜ = hu¯− 3 (h u¯x )x q˜t  δH ∂x q˜ + q˜∂x h∂x     δh  H = 1 hu¯2 + 1 h3u¯2 + gη2 dx 2 Z 3 x R  

◮ Space translations, time translation and Galilean boost lead invariants: ηq˜ Q = dx, H , tq˜ − xη dx Z d + η Z R R
SERRE EQUATIONS - II NUMERICALDISCRETIZATIONOFTHE SERRE EQUATIONS

◮ The fully discrete 2nd order in space FV scheme:

dh¯ 1 (1) (1) = − F (v¯) − F− (v¯) , dt ∆x +
dv¯ ∆t (2) (2) (I − M) · = − F (v¯) − F− (v¯) + D(v¯). dt ∆x +
◮ Pseudo-spectral solver:

ikd ikdF N(η, u¯) ηˆ + qˆ = −ikF{ηu¯}− , t 1 2 1 2 1 + 3 (kd) 1 + 3 (kd) 1 2 1 2 2 qˆt + ikg ηˆ = ikF 2 u¯ + 2 (d + η) u¯x − qu .  REFERENCE [DCMM12]: D. Dutykh, D. Clamond, P. Milewski, D. Mitsotakis. Finite volume and pseudo-spectral schemes for the fully nonlinear ’irrotational’ Serre equations, 2012 SERRE EQUATIONS - III VALIDATIONOFTHENUMERICALSCHEME

◮ Exact solutions:

0.06 0.06

0.05 0.05

0.04 0.04

0.03 0.03

0.02 0.02 (x, 0) (x, 0) η 0.01 η 0.01

0 0

-0.01 -0.01

-0.02 -0.02

-0.03 -0.03 -40 -30 -20 -10 0 10 20 30 40 -40 -30 -20 -10 0 10 20 30 40 x x

(a) Solitary wave (b) Cnoidal wave SERRE EQUATIONS - III VALIDATIONOFTHENUMERICALSCHEME

◮ Convergence to the SW solution under the mesh refinement:

0 10

-1 10

-2 10

-3 10 ∞ ε

-4 10

-5 10

-6 10 FV UNO2 N-2 Spectral method -7 10 1 2 10 10 N SERRE EQUATIONS - III VALIDATIONOFTHENUMERICALSCHEME

◮ Accuracy of invariants preservation:

-2 -2 10 10

|H-H | |Q-Q | ex ex

N-2 N-2 -3 -3 10 10

-4 -4 10 10 H(T) Q(T)

-5 -5 10 10

-6 -6 10 10 1 2 3 1 2 3 10 10 10 10 10 10 N N

(c) Hamiltonian H (d) Momentum Q SERRE EQUATIONS - III VALIDATIONOFTHENUMERICALSCHEME

0.4

0.2 (x,t) η 0

-0.2 40

20 0 10 0 20 -20 30 -40 40 t x HEAD-ON COLLISION EXPERIMENTAL VALIDATION: D.HENDERSON et al. (2006) [CGH+06]

−3 −3 x 10 x 10

Simulation Simulation 18 Experimental data 18 Experimental data

16 16

14 14

12 12 ) )

t 10 t 10 , , x x

( 8 ( 8 η η 6 6

4 4

2 2

0 0

−2 −2

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x x (e) t = 18.5 s (f) t = 18.6 s HEAD-ON COLLISION EXPERIMENTAL VALIDATION: D.HENDERSON et al. (2006) [CGH+06]

−3 −3 x 10 x 10

Simulation Simulation 18 Experimental data 18 Experimental data

16 16

14 14

12 12 ) )

t 10 t 10 , , x x

( 8 ( 8 η η 6 6

4 4

2 2

0 0

−2 −2

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x x (g) t = 18.7 s (h) t = 18.8 s HEAD-ON COLLISION EXPERIMENTAL VALIDATION: D.HENDERSON et al. (2006) [CGH+06]

−3 x 10

Simulation 18 Experimental data Simulation 0.025 Experimental data 16

14 0.02 12 ) ) t 10 0.015 t , , x x ( 8 ( η η 6 0.01

4 0.005 2

0 0 −2

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x x (i) t = 18.92 s (j) t = 19.0 s HEAD-ON COLLISION EXPERIMENTAL VALIDATION: D.HENDERSON et al. (2006) [CGH+06]

Simulation Simulation 0.025 Experimental data 0.025 Experimental data

0.02 0.02

) 0.015 ) 0.015 t t , , x x ( ( η 0.01 η 0.01

0.005 0.005

0 0

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x x (k) t = 19.05 s (l) t = 19.1 s HEAD-ON COLLISION EXPERIMENTAL VALIDATION: D.HENDERSON et al. (2006) [CGH+06]

−3 −3 x 10 x 10

Simulation Simulation 18 Experimental data 18 Experimental data

16 16

14 14

12 12 ) )

t 10 t 10 , , x x

( 8 ( 8 η η 6 6

4 4

2 2

0 0

−2 −2

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x x (m) t = 19.15 s (n) t = 19.19 s HEAD-ON COLLISION EXPERIMENTAL VALIDATION: D.HENDERSON et al. (2006) [CGH+06]

−3 −3 x 10 x 10

Simulation Simulation 18 Experimental data 18 Experimental data

16 16

14 14

12 12 ) )

t 10 t 10 , , x x

( 8 ( 8 η η 6 6

4 4

2 2

0 0

−2 −2

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x x (o) t = 19.33 s (p) t = 19.5 s HEAD-ON COLLISION EXPERIMENTAL VALIDATION: D.HENDERSON et al. (2006) [CGH+06]

−3 −3 x 10 x 10

Simulation Simulation 18 Experimental data 18 Experimental data

16 16

14 14

12 12 ) )

t 10 t 10 , , x x

( 8 ( 8 η η 6 6

4 4

2 2

0 0

−2 −2

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x x (q) t = 19.85 s (r) t = 20.0 s HEAD-ON COLLISION EXPERIMENTAL VALIDATION: D.HENDERSON et al. (2006) [CGH+06]

x 10−3 10

8

6 )

t 4 , x ( η 2

0

−2

0 10 20 30 40 50 60 70 80 90 100 110 x

FIGURE: Dispersive tail after overtaking collision of two solitary waves (strong interaction) at T = 120.0. DEEP WATER APPROXIMATION

◮ Choice of the ansatz: − {φ; u; v; µ; ν} ≈ {φ˜; u˜; v˜ ;µ ˜;ν ˜}eκ(y η)

1 1 2κL = 2κφη˜ − gκη2 + u˜ 2 + v˜ 2 − u˜ · (∇φ˜ − κφ˜∇η) − κv˜φ˜ t 2 2

◮ generalized Klein-Gordon equations:

1 − 1 1 η + κ 1∇2φ˜ − κφ˜ = φ˜[∇2η + κ(∇η)2] t 2 2 2 1 φ˜ + gη = − ∇· φ˜∇φ˜ − κφ˜2∇η t 2  

1 1 − 1 H = gη2 + κ 1[∇φ˜ − κφ˜∇η]2 + κφ˜2 dx ZΩn2 4 4 o DEEP WATER APPROXIMATION

◮ Choice of the ansatz: − {φ; u; v; µ; ν} ≈ {φ˜; u˜; v˜ ;µ ˜;ν ˜}eκ(y η)

1 1 2κL = 2κφη˜ − gκη2 + u˜ 2 + v˜ 2 − u˜ · (∇φ˜ − κφ˜∇η) − κv˜φ˜ t 2 2

◮ generalized Klein-Gordon equations:

1 − 1 1 η + κ 1∇2φ˜ − κφ˜ = φ˜[∇2η + κ(∇η)2] t 2 2 2 1 φ˜ + gη = − ∇· φ˜∇φ˜ − κφ˜2∇η t 2  

Mzt + Kzx + Lzy = ∇z S(z) COMPARISON WITH EXACT STOKES WAVE

◮ Cubic Zakharov Equations (CZE):

ηt − dφ˜ = −∇· (η∇φ˜) − d(ηdφ˜)+ 1 ∇2 2d ˜ d d d ˜ 1 d 2∇2 ˜ 2 (η φ)+ (η (η φ)) + 2 (η φ), ˜ 1 d ˜ 2 1 ∇ ˜ 2 d ˜ ∇2 ˜ d ˜ d d ˜ φt + gη = 2 ( φ) − 2 ( φ) − (η φ) φ − ( φ) (η φ).

◮ Phase speed : − 1 1 2 2 1 2 1 4 707 6 8 EXACT: g κ c = 1 + 2 α + 2 α + 384 α + O(α ) − 1 1 41 913 2 2 1 2 4 6 8 CZE: g κ c = 1 + 2 α + 64 α + 384 α + O(α ) − 1 1 899 2 2 1 2 1 4 6 8 GKG: g κ c = 1 + 2 α + 2 α + 384 α + O(α )

− ◮ nn 2αn n-th Fourier coefficient to the leading order: − (the 2n 1(n−1)! same in gKG & Stokes but not in CZE) CONCLUSIONS

WATER WAVES: ◮ We presented a generalized Lagrangian variational principle ◮ Some derived models are known ◮ Some proposed models are new

◮ The proposed method is alternative to asymptotic expansions ◮ In some cases the equivalence can be established ◮ The choice of ansatz allows for much freedom THANK YOU FOR YOUR ATTENTION!

http://www.denys-dutykh.com/ REFERENCES I

D Clamond and D Dutykh. Practical use of variational principles for modeling water waves. Physica D: Nonlinear Phenomena, 241(1):25–36, 2012. W Craig, P Guyenne, J Hammack, D Henderson, and C Sulem. Solitary water wave interactions. Phys. Fluids, 18(5):57106, 2006. D. Dutykh, D. Clamond, P. Milewski, and D. Mitsotakis. Finite volume and pseudo-spectral schemes for the fully nonlinear ’irrotational’ Serre equations. Submitted, 2012. REFERENCES II

A E Green and P M Naghdi. A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech., 78:237–246, 1976. Y A Li. Hamiltonian structure and linear stability of solitary waves of the Green-Naghdi equations. J. Nonlin. Math. Phys., 9(1):99–105, 2002. J C Luke. A variational principle for a fluid with a free surface. J. Fluid Mech., 27:375–397, 1967. A A Petrov. Variational statement of the problem of liquid motion in a container of finite dimensions. Prikl. Math. Mekh., 28(4):917–922, 1964. REFERENCES III

F Serre. Contribution a` l’etude´ des ecoulements´ permanents et variables dans les canaux. La Houille blanche, 8:374–872, 1953. J. J. Stoker. Water Waves: The mathematical theory with applications. Interscience, New York, 1957. V E Zakharov. Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys., 9:190–194, 1968.