RELAXED VARIATIONAL PRINCIPLE FOR WATER WAVE MODELING DENYS DUTYKH 1 Senior Research Fellow UCD & Chargé de Recherche CNRS 1 University College Dublin School of Mathematical Sciences Workshop on Ocean Wave Dynamics
ACKNOWLEDGEMENTS COLLABORATOR: Didier Clamond: Professor, LJAD, Université 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.
0.5 0-0.5-1 -1.5-2 4 3 2 1 h(x,y,t) 0 η (x,y,t) -1-2 -3-4 -3 x -2 z -1 O 0 1 y 2 3 4 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 Atmospheric pressure is constant a h 0 Surface tension can be also taken into account
WATER WAVE PROBLEM - II Continuity equation 2 x,yφ = 0, (x, y) Ω [ d,η], Kinematic bottom condition φ y + φ d = 0, y = d, Kinematic free surface condition η t φ + φ η =, y = η(x, t), y Dynamic free surface condition φ t +1 2 x,yφ 2 +gη+σ ( η ) = 0, y = η(x, t). 1+ η 2
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: ρ η t = δh δ φ, ρ φ t = δh δη, Hamiltonian: H = η d 1 2 x,yφ 2 dy + 1 ( ) 2 gη2 +σ 1+ η 2 1
LUKE S VARIATIONAL PRINCIPLES J.C. LUKE, JFM (1967) [LUK67] First improvement of the classical Lagrangian L := K +Π: L = t 2 t 1 Ω δφ y= d : δφ y=η : ρl dx dt, L := η d δφ: φ = 0, (x, y) Ω [ d,η], δη: φ y + φ d = 0, y = d, ( φt + 1 2 x,yφ 2 + gy ) dy η t + φ η φ y = 0, y = η(x, t), φ t + 1 2 φ 2 + gη = 0, y = η(x, t). 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 ˇφ := φ(x, y = d(x, t), t): value at the bottom Equivalent form of Luke s lagrangian: L = φη t + ˇφd t 1 2 gη2 + 1 2 gd 2 η d [ 1 2 φ 2 + 1 2 y] φ2 dy Explicitly introduce the velocity field: u = φ, v = φ y L = φη t +ˇφd t 1 η [ 1 2 gη2 2 (u 2 +v 2 )+µ ( φ u)+ν(φ y v)] dy d µ,ν: Lagrange multipliers or pseudo-velocity field
GENERALIZATION OF THE LAGRANGIAN DENSITY D. CLAMOND & D. DUTYKH, PHYS. D (2012) [CD12] Relaxed variational principle: L = (η t + µ η ν) φ+(d t + ˇµ d + ˇν)ˇφ 1 2 gη2 + η d [ µ u 1 2 u 2 +νv 1 ] 2 v 2 +( µ+ν y )φ dy Classical formulation (for comparison): L = φη t + ˇφd t 1 2 gη2 η d DEGREES OF FREEDOM: η,φ; u, v ;µ,ν [ 1 2 φ 2 + 1 2 y] φ2 dy
SHALLOW WATER REGIME CHOICE OF A SIMPLE ANSATZ IN SHALLOW WATER Ansatz: u(x, y, t) ū(x, t), v(x, y, t) (y + d)(η + d) 1 ṽ(x, t) φ(x, y, t) φ(x, t),ν(x, y, t) (y + d)(η + d) 1 ν(x, t) Lagrangian density: L = φη t 1 [ 2 gη2 +(η+ d) µ ū 1 2 + 1 2ū 3 νṽ 1 ] 2 µ φ 6ṽ Nonlinear Shallow Water Equations: Not so interesting... h t + [hū] = 0, ū t +(ū )ū + g h = 0.
CONSTRAINING WITH FREE SURFACE IMPERMEABILITY Constraint: ν = η t + µ η Generalized Serre (Green Naghdi) equations [Ser53]: h t + [hū] = 0, ū t + ū ū + g h+ 1 3 h 1 [h 2 γ] = (ū h) (h ū) [ū (h ū)] h γ = ṽ t + ū ṽ = h ( ( ū) 2 ū t ū ( ū) ) CANNOT BE OBTAINED FROM LUKE S LAGRANGIAN: 1 δ µ: ū = φ 3ṽ η φ
INCOMPRESSIBILITY AND PARTIAL POTENTIAL FLOW Ansatz and constraints (v φ y ): µ = ū, ν = ṽ, ū = φ, ṽ = (η + d) 2 φ L = φη t 1 2 gη2 1 2 (η + d)( φ) 2 + 1 6 (η + d)3 ( 2 φ) 2 Generalized Kaup-Boussinesq equations: η t + [(η + d) φ ] + 1 3 2[ (η + d) 3 ( 2 φ) ] = 0, φ t + gη + 1 2 ( φ) 2 1 2 (η + d)2 ( 2 φ) 2 = 0. Hamiltonian functional: { 1 H = 2 gη2 + 1 2 (η + d)( φ) 2 1 6 (η + d)3 ( 2 φ) 2} dx Ω
INCOMPRESSIBILITY AND PARTIAL POTENTIAL FLOW Ansatz and constraints (v φ y ): µ = ū, ν = ṽ, ū = φ, ṽ = (η + d) 2 φ L = φη t 1 2 gη2 1 2 (η + d)( φ) 2 + 1 6 (η + d)3 ( 2 φ) 2 Generalized Kaup-Boussinesq equations: η t + [(η + d) φ ] + 1 3 2[ (η + d) 3 ( 2 φ) ] = 0, φ t + gη + 1 2 ( φ) 2 1 2 (η + d)2 ( 2 φ) 2 = 0. Dispersion relation (c 2 < 0, κd > 1/ 3): η = a cosκ(x ct), c 2 = gd(1 1 3 (κd)2 )
DEEP WATER APPROXIMATION Choice of the ansatz: {φ; u; v;µ;ν} { φ; ũ; ṽ ; µ; ν}e κ(y η) 2κL = 2κ φη t gκη 2 + 1 2ũ 2 + 1 2ṽ 2 ũ ( φ κ φ η) κṽ φ generalized Klein-Gordon equations: η t + 1 2 κ 1 2 φ 1 2 κ φ = 1 2 φ[ 2 η +κ( η) 2 ] φ t + gη = 1 2 [ φ φ κ φ 2 η ] Hamiltonian functional: { 1 H = 2 gη2 + 1 4 κ 1 [ φ κ φ η] 2 + 1 4 κ φ 2} dx Ω
DEEP WATER APPROXIMATION Choice of the ansatz: {φ; u; v;µ;ν} { φ; ũ; ṽ ; µ; ν}e κ(y η) 2κL = 2κ φη t gκη 2 + 1 2ũ 2 + 1 2ṽ 2 ũ ( φ κ φ η) κṽ φ generalized Klein-Gordon equations: η t + 1 2 κ 1 2 φ 1 2 κ φ = 1 2 φ[ 2 η +κ( η) 2 ] φ t + gη = 1 2 [ φ φ κ φ 2 η ] Multi-symplectic structure: Mz t +Kz x +Lz y = z S(z)
COMPARISON WITH EXACT STOKES WAVE Cubic Zakharov Equations (CZE): η t d φ = (η φ) d(ηd φ)+ 1 2 2 (η 2 d φ)+d(ηd(ηd φ))+ 1 2 d(η2 2 φ), φ t + gη = 1 2 (d φ) 2 1 2 ( φ) 2 (ηd φ) 2 φ (d φ)d(ηd φ). Phase speed : EXACT: g 1 2κ 1 2 c = 1+ 1 2 α2 + 1 2 α4 + 707 384 α6 + O(α 8 ) CZE: g 1 2κ 1 2 c = 1+ 1 2 α2 + 41 64 α4 + 913 384 α6 + O(α 8 ) GKG: g 1 2κ 1 2 c = 1+ 1 2 α2 + 1 2 α4 + 899 384 α6 + O(α 8 ) n-th Fourier coefficient to the leading order: same in gkg & Stokes but not in CZE) n n 2 α n 2 n 1 (n 1)! (the
DEEP-WATER SERRE GREEN NAGHDI EQUATIONS CONSTRAINING WITH THE FREE-SURFACE IMPERMEABILITY Additional constraint: ṽ = η t + ũ η Lagrangian density reads: 2κL = φ(κη t + ũ) gκη 2 + 1 2ũ2 + 1 2 (η t + ũ η) 2 δ ũ : δ φ : δη : 0 = ũ +(η t + ũ η) η φ, 0 = κη t + ũ, 0 = 2gκη +κ φ t +η tt +(ũ η) t + (ũη t )+ [(ũ η)ũ]. Cannot be derived from Luke s Lagrangian: φ = ũ + ṽ η
DEEP-WATER SERRE GREEN NAGHDI EQUATIONS CONSTRAINING WITH THE FREE-SURFACE IMPERMEABILITY Additional constraint: ṽ = η t + ũ η Lagrangian density reads: 2κL = φ(κη t + ũ) gκη 2 + 1 2ũ2 + 1 2 (η t + ũ η) 2 δ ũ : δ φ : δη : 0 = ũ +(η t + ũ η) η φ, 0 = κη t + ũ, 0 = 2gκη +κ φ t +η tt +(ũ η) t + (ũη t )+ [(ũ η)ũ]. Incompressibility is satisfied identically: 0 = κη t + ũ u + v y = 0
DEEP-WATER SERRE GREEN NAGHDI EQUATIONS CONSTRAINING WITH THE FREE-SURFACE IMPERMEABILITY Additional constraint: ṽ = η t + ũ η Lagrangian density reads: 2κL = φ(κη t + ũ) gκη 2 + 1 2ũ2 + 1 2 (η t + ũ η) 2 δ ũ : δ φ : δη : 0 = ũ +(η t + ũ η) η φ, 0 = κη t + ũ, 0 = 2gκη +κ φ t +η tt +(ũ η) t + (ũη t )+ [(ũ η)ũ]. Exact dispersion relation if k = κ: η = a cos k(x 1 ct), c 2 = 2 gκ(k 2 +κ 2 ) 1
0.5 0-0.5-1 -1.5-2 4 3 2 1 h(x,y,t) 0 η (x,y,t) -1-2 -3-4 -3 x -2 a z -1 O 0 1 y 2 3 4 ARBITRARY DEPTH CASE NO AVAILABLE SMALL PARAMETERS... Finite depth ansatz: φ coshκy coshκh φ(x, t), u coshκy coshκh µ coshκy sinhκy µ(x, t), ν ν(x, t). coshκh sinhκh Finite depth Lagrangian: sinhκy ũ(x, t), v ṽ(x, t), h sinhκh 0 L = [η t + µ η] φ 1 2 gη2 +[ ν ṽ 1 2 ṽ 2 sinh(2κh) 2κh ] 2κ cosh(2κh) 2κ +[ µ ũ 1 2ũ2 + φ µ κ sinh(2κh) + 2κh tanh(κh) φ µ η] 2κ cosh(2κh)+2κ [ ] + 1 φ ν 2κh 2 sinh(2κh) 1.
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
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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 à l étude des écoulements 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.