Relaxed Variational Principle for Water Wave Modeling

RELAXED VARIATIONAL PRINCIPLE FOR WATER WAVE MODELING

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

1University College Dublin School of Mathematical Sciences

Workshop on Ocean Wave Dynamics ACKNOWLEDGEMENTS

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

TALK IS MAINLY BASED ON: Clamond, D., Dutykh, D. (2012). Practical use of variational principles for modeling water waves. Physica D: Nonlinear Phenomena, 241(1), 25-36. WATER WAVE PROBLEM -I

PHYSICAL ASSUMPTIONS: ◮ Fluid is ideal ◮ Flow is incompressible ◮ . . . and irrotational, i.e. u = ∇φ ◮ Free surface is a graph ◮ Above free surface there is void

0.5 ◮ z Atmospheric pressure is constant η (x,y,t) 0 a y O

-0.5

h(x,y,t) x h 0 -1 Surface tension can be also taken into -1.5 -2

2 account 1 0 4 -1 3 2 -2 1 0 -1 -3 -2 -3 -4 WATER WAVE PROBLEM - II

◮ Continuity equation

2 ∇x φ = 0, (x, y) Ω [ d,η], ,y ∈ × − ◮ 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 ZAKHAROV (1968) [ZAK68]; CRAIG &SULEM (1993) [CS93]

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 = ρL dx dt, L := φt + ∇x,y φ + gy dy L Z Z Z 2| | t1 Ω −d δφ: ∆φ = 0, (x, y) Ω [ d,η], ∂φ ∈ × − δφ − : + ∇φ ·∇d = 0, y = d, |y= d ∂y − δφ : ∂η + ∇φ ·∇η ∂φ = 0, y = η(x, t), |y=η ∂t − ∂y δη: ∂φ + 1 ∇φ 2 + gη = 0, y = η(x, t). ∂t 2 | |

◮ We recover the water wave problem by computing variations w.r.t. η and φ GENERALIZATION OF THE LAGRANGIAN DENSITY D. CLAMOND &D.DUTYKH,PHYS. D (2012) [CD12]

◮ Introduce notation (traces): 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

◮ Explicitly 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 µ,ν: Lagrange multipliers or pseudo-velocity ﬁeld GENERALIZATION OF THE LAGRANGIAN DENSITY D. CLAMOND &D.DUTYKH,PHYS. D (2012) [CD12]

◮ Relaxed variational principle:

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

◮ Classical formulation (for comparison): 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.

◮ Not so interesting. . . CONSTRAINING WITH FREE SURFACE IMPERMEABILITY

◮ Constraint:

ν˜ = ηt +µ ¯ ·∇η

◮ Generalized Serre (Green–Naghdi) equations [Ser53]:

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¯) − − CANNOT BE OBTAINED FROM LUKE’S LAGRANGIAN: δµ¯: u¯ = ∇φ¯ 1 v˜∇η = ∇φ¯ − 3 6 INCOMPRESSIBILITY AND PARTIAL POTENTIAL FLOW

◮ Ansatz and constraints (v = φy ): 6 µ¯ = u¯, ν˜ = v˜, u¯ = ∇φ,¯ v˜ = (η + d)∇2φ¯ −

1 2 1 2 1 3 2 2 L = φη¯ t gη (η + d)(∇φ¯) + (η + d) (∇ φ¯) − 2 − 2 6

◮ Generalized Kaup-Boussinesq equations: 1 η + ∇· (η + d)∇φ¯ + ∇2 (η + d)3(∇2φ¯) = 0, t 3 1 2 1 2 2 2 φ¯t + gη + (∇φ¯) (η + d) (∇ φ¯) = 0. 2 − 2 ◮ Hamiltonian functional: 1 1 1 H = gη2 + (η + d)(∇φ¯)2 (η + d)3(∇2φ¯)2 dx ZΩn2 2 − 6 o INCOMPRESSIBILITY AND PARTIAL POTENTIAL FLOW

◮ Ansatz and constraints (v = φy ): 6 µ¯ = u¯, ν˜ = v˜, u¯ = ∇φ,¯ v˜ = (η + d)∇2φ¯ −

1 2 1 2 1 3 2 2 L = φη¯ t gη (η + d)(∇φ¯) + (η + d) (∇ φ¯) − 2 − 2 6

◮ Generalized Kaup-Boussinesq equations: 1 η + ∇· (η + d)∇φ¯ + ∇2 (η + d)3(∇2φ¯) = 0, t 3 1 2 1 2 2 2 φ¯t + gη + (∇φ¯) (η + d) (∇ φ¯) = 0. 2 − 2

◮ Dispersion relation (c2 < 0, κd > 1/√3): 1 η = a cos κ(x ct), c2 = gd(1 (κd)2) − − 3 DEEP WATER APPROXIMATION

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

2 1 2 1 2 2κL = 2κφη˜ t gκη + u˜ + v˜ u˜ (∇φ˜ κφ˜∇η) κv˜φ˜ − 2 2 − · − − ◮ generalized Klein-Gordon equations:

1 −1 2 1 1 2 2 ηt + κ ∇ φ˜ κφ˜ = φ˜[∇ η + κ(∇η) ] 2 − 2 2 1 2 φ˜t + gη = ∇· φ˜∇φ˜ κφ˜ ∇η −2 − ◮ Hamiltonian functional: 1 1 − 1 = gη2 + κ 1[∇φ˜ κφ˜∇η]2 + κφ˜2 dx H ZΩn2 4 − 4 o DEEP WATER APPROXIMATION

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

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

◮ generalized Klein-Gordon equations:

1 −1 2 1 1 2 2 ηt + κ ∇ φ˜ κφ˜ = φ˜[∇ η + κ(∇η) ] 2 − 2 2 1 2 φ˜t + gη = ∇· φ˜∇φ˜ κφ˜ ∇η −2 − ◮ Multi-symplectic structure:

Mzt + Kzx + Lzy = ∇z (z) S 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 2 1 2 2 φ˜t + gη = (dφ˜) (∇φ˜) (ηdφ˜)∇ φ˜ (dφ˜)d(ηdφ˜). 2 − 2 − −

◮ 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) DEEP-WATER SERRE–GREEN–NAGHDI EQUATIONS CONSTRAININGWITHTHEFREE-SURFACE IMPERMEABILITY

◮ Additional constraint:

v˜ = ηt + u˜ ·∇η

◮ Lagrangian density reads:

2 1 2 1 2 2κ L = φ˜ (κηt + ∇· u˜) gκη + u˜ + (ηt + u˜ ·∇η) − 2 2

δ u˜ : 0 = u˜ +(ηt + u˜ ·∇η)∇η ∇φ,˜ − δ φ˜ : 0 = κηt + ∇· u˜,

δη : 0 = 2gκη + κφ˜t + ηtt +(u˜ · η)t + · (u˜ηt )+ · [(u˜ · η)u˜]. ∇ ∇ ∇ ∇

◮ Cannot be derived from Luke’s Lagrangian: ∇φ˜ = u˜ + v˜∇η DEEP-WATER SERRE–GREEN–NAGHDI EQUATIONS CONSTRAININGWITHTHEFREE-SURFACE IMPERMEABILITY

◮ Additional constraint:

v˜ = ηt + u˜ ·∇η

◮ Lagrangian density reads:

2 1 2 1 2 2κ L = φ˜ (κηt + ∇· u˜) gκη + u˜ + (ηt + u˜ ·∇η) − 2 2

δ u˜ : 0 = u˜ +(ηt + u˜ ·∇η)∇η ∇φ,˜ − δ φ˜ : 0 = κηt + ∇· u˜,

δη : 0 = 2gκη + κφ˜t + ηtt +(u˜ · η)t + · (u˜ηt )+ · [(u˜ · η)u˜]. ∇ ∇ ∇ ∇

◮ Incompressibility is satisﬁed identically:

0 = κηt + ∇· u˜ ∇· u + vy = 0 ⇐⇒ DEEP-WATER SERRE–GREEN–NAGHDI EQUATIONS CONSTRAININGWITHTHEFREE-SURFACE IMPERMEABILITY

◮ Additional constraint:

v˜ = ηt + u˜ ·∇η

◮ Lagrangian density reads:

2 1 2 1 2 2κ L = φ˜ (κηt + ∇· u˜) gκη + u˜ + (ηt + u˜ ·∇η) − 2 2

δ u˜ : 0 = u˜ +(ηt + u˜ ·∇η)∇η ∇φ,˜ − δ φ˜ : 0 = κηt + ∇· u˜,

δη : 0 = 2gκη + κφ˜t + ηtt +(u˜ · η)t + · (u˜ηt )+ · [(u˜ · η)u˜]. ∇ ∇ ∇ ∇

◮ Exact dispersion relation if k = κ: − η = a cos k(x ct), c2 = 2 g κ (k 2 + κ2) 1 1 − ARBITRARY DEPTH CASE NO AVAILABLE SMALL PARAMETERS. . .

◮ 0.5 Finite depth ansatz: z η (x,y,t)

0 a y cosh Y cosh Y sinh Y O κ ˜ x u κ u x -0.5 κ x φ φ( , t), ˜( , t), v h(x,y,t) v˜( , t), x h 0 ≈ cosh κh ≈ cosh κh -1 ≈ sinh κh

-1.5

cosh κY sinh κY -2

4 x x 3 µ µ˜( , t), ν ν˜( , t). 2 1

0 4 -1 3 2 -2 1 cosh h sinh h 0 ≈ κ ≈ κ -1 -3 -2 -3 -4 ◮ Finite depth Lagrangian:

1 2 1 2 sinh(2κh) 2κh L =[ ηt +µ ˜ ·∇η ] φ˜ g η +[ν ˜ v˜ v˜ ] − − 2 − 2 2κ cosh(2κh) 2κ − sinh(2κh)+ 2κh + [˜µ · u˜ 1 u˜2 + φ˜∇· µ˜ κ tanh(κh)φ˜ µ˜ ·∇η] − 2 − 2κ cosh(2κh)+ 2κ 2κh + 1 φ˜ ν˜ 1 . 2 sinh(2κh) − CONCLUSIONS & PERSPECTIVES

CONCLUSIONS: ◮ A relaxed variational principle was presented ◮ Practical usage of this principle was illustrated ◮ All models automatically possess the Lagrangian structure ◮ In most cases the Hamiltonian as well!

PERSPECTIVES: ◮ Further validation of derived models is needed ◮ Deeper study of their properties ◮ Development of variational discretizations 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 and C. Sulem. Numerical simulation of gravity waves. J. Comput. Phys., 108:73–83, 1993. J. C. Luke. A variational principle for a ﬂuid with a free surface. J. Fluid Mech., 27:375–397, 1967. F. Serre. Contribution a` l’etude´ des ecoulements´ permanents et variables dans les canaux. La Houille blanche, 8:374–388, 1953. REFERENCES II

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.