Kelvin-Helmholtz instabilities in shallow water

Kelvin-Helmholtz instabilities in shallow water Propagation of large amplitude, long wavelength, internal waves Vincent Duchêne 1 Samer Israwi 2 Raafat Talhouk 2 1 IRMAR, Univ. Rennes 1 UMR 6625 2 Faculté des Sciences I, Université Libanaise, Beirut Center for Advanced Mathematical Sciences American University of Beirut Nov. 12, 2014

1. Credits : Naval Research Laboratory http://www7320.nrlssc.navy.mil/global_nlom/ Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 1 / 17 Internal gravity waves Stratification, due to variation of salinity and temperature. Figure : Temperature vs depth 1

1. Credits : St. Lawrence Estuary Internal Wave Experiment SLEIWEX) http://myweb.dal.ca/kelley/sleiwex/index.php Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 1 / 17 Internal gravity waves Stratification, due to variation of salinity and temperature. Figure : St. Lawrence Estuary 1

1. Credits : NASA s Earth Observatory Picture of the Day July 1, 2003) http://earthobservatory.nasa.gov/iotd/view.php?id=3586 Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 1 / 17 Internal gravity waves Stratification, due to variation of salinity and temperature. Figure : Sulu Sea. April 8, 2003 1

1 Motivation full Euler system Kelvin-Helmholtz instabilities 2 Asymptotic models Asymptotic models Drawbacks New systems 3 Numerical simulations 4 Well-posedness

Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 3 / 17 The full Euler system Horizontal dimension d = 1, flat bottom, rigid lid. Irrotational, incompressible, inviscid, immiscible fluids. Fluids at rest at infinity, small) surface tension.

Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 3 / 17 The full Euler system The system can be rewritten as two coupled evolution equations in using Dirichlet-Neumann operators. ζ and ψ φ 1 interface.

The full Euler system The system can be rewritten as two coupled evolution equations in using Dirichlet-Neumann operators. ζ and ψ φ 1 interface. Full Euler system Zakharov s formulation) δh t ζ = x δv, ) v = x ρ 2 φ 2 interface δh ρ 1φ 1 interface t v = x δζ, with H = 1 2 R gρ 2 ρ 1 )ζ 2 dx + ρ2 ζ 2 R d 2 φ 2 2 dzdx + ρ1 d1 2 φ R ζ 1 2 dzdx +σ R 1 + x ζ 2 1 ). This system is ill-posed without surface tension. [Ebin 88 ; Iguchi,Tanaka&Tani 97 ; Kamotski&Lebeau 05] Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 3 / 17

Modes are stable for k small iff v 0 2 < ρ 2d 1 +ρ 1 d 2 ρ 1 ρ 2 gρ 2 ρ 1 ). There are always unstable modes if σ = 0 and ρ 1, v 0 0. All modes are stable if σ 0 and ρ 1 ρ 2 v 0 2 is sufficiently small. Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 4 / 17 Kelvin-Helmholtz instabilities Linearize the system around ζ = 0, v = v 0, constant. Linearized system [Lannes&Ming] { t ζ + c v0 D) x ζ + bd) x v = 0, t v + a v0 D) x ζ + c v0 D) x v = 0, L) with b > 0 and a v0 D) = gρ 2 ρ 1 ) v 0 2 ρ 1 ρ 2 ρ 2 tanhd 1 D ) + ρ 1 tanhd 2 D ) D σ 2 x There are growing modes, e ikx ωk)t) with Iωk)) 0 iff a v0 k) < 0, i.e. v 0 2 tanhd1 k ) < + tanhd ) 2 k ) gρ2 ρ 1 ) + σ k 2). ρ 1 k ρ 2 k

Kelvin-Helmholtz instabilities Figure : Stability domains, with surface tension green) and without red) When surface tension is present, all the modes are stable provided v 0 2 < v min 2 2 ρ 1 + ρ 2 σgρ2 ρ 1 ). ρ 1 ρ 2 Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 5 / 17

Kelvin-Helmholtz instabilities When surface tension is present, all the modes are stable provided v 0 2 < v min 2 2 ρ 1 + ρ 2 ρ 1 ρ 2 σgρ2 ρ 1 ). Nonlinear generalization of this criterion : [Lannes 13] Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 5 / 17

1 Motivation full Euler system Kelvin-Helmholtz instabilities 2 Asymptotic models Asymptotic models Drawbacks New systems 3 Numerical simulations 4 Well-posedness

Asymptotic models : construction Asymptotic models may be constructed from asymptotic expansions of the Dirichlet-Neumann operators, w. r. t. given dimensionless parameters. ɛ def = a d 1, µ def = d 1 2 λ 2, γ def = ρ 1 ρ 2, δ def = d 1 d 2, def Bo = σ gρ 2 ρ 1 )λ 2. Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 7 / 17

Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 8 / 17 Asymptotic models : examples Precision Oµ) Shallow water a.k.a Saint-Venant) system + surface tension) t ζ + x w = 0, ) ) h1+γh 2 t h 1h 2 w + γ + δ) x ζ + ɛ 2 2 2 h1 γh 2 x h 1h 2) w 2 = γ+δ 2 Bo 2 x ζ x with h 1 = 1 ɛζ and h 2 = 1 δ + ɛζ and h 1 + γh 2 w def = h 1 h 2 1 h 2 t, x) ɛζt,x) 1 δ x φ 2 t, x, z) dz 1+µɛ 2 x ζ 2 ), γ 1 x φ 1 t, x, z) dz. h 1 t, x) ɛζt,x)

Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 8 / 17 Asymptotic models : examples Precision Oµ 2 ) Green-Naghdi system [Miyata 85 ; Mal tseva 89 ; Choi&Camassa 99] t ζ + x w = 0, ) ) h1+γh 2 t h 1h 2 w + µq[ζ]w + γ + δ) x ζ + ɛ 2 2 2 h1 γh 2 ) x h 1h 2) w 2 = µɛ 2 x R[ζ, w] x ζ with h 1 = 1 ɛζ and h 2 = 1 δ + ɛζ and h 1 + γh 2 w def = h 1 h 2 Q[ζ]w R[ζ, w] 1 h 2 t, x) ɛζt,x) 1 δ x φ 2 t, x, z) dz + γ+δ Bo 2 x def = 1 h 3 2 x h 3 2 x h 2 w )) + γh 1 x h 3 1 x h 1 w ))), h 2 x h 2 w )) 2 γ h 1 x h 1 w )) ) 2 def = 1 2 1+µɛ 2 x ζ 2 ), γ 1 x φ 1 t, x, z) dz. h 1 t, x) ɛζt,x) + 1 3 w h 2 2 x h 2 3 x h 2 w )) γh 2 1 x h 1 3 x h 1 w ))).

Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 9 / 17 Shallow water models and KH instabilities Qn : How well do shallow water models predict KH instabilities?

Shallow water models and KH instabilities Qn : How well do shallow water models predict KH instabilities? Figure : Stability domains for full Euler green), Saint-Venant purple) and Green-Naghdi blue) ɛ 2 wmin FE 2 1 vs ɛ 2 w GN µ Bo min 2 1 µ Bo. Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 9 / 17

Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 9 / 17 Shallow water models and KH instabilities Qn : How well do shallow water models predict KH instabilities? A : The classical GN model follows the same behavior [Choi&Camassa 99] However, whereas Saint-Venant underestimates, Green-Naghdi overestimates KH instabilities [Jo&Choi 02 ; Lannes&Ming] Qn : Can we do better? A : [Nguyen&Dias 08 ; Choi,Barros&Jo 09 ; Cotter,Holm&Percival 10 ; Boonkasame&Milewski 14 ; Lannes&Ming] Strategy : 1 Change of unknowns 2 BBM trick 3.../...

Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 10 / 17 Construction of our model The original GN system t ζ + x w = 0, ) ) h1+γh 2 t h 1h 2 w + µq[ζ]w + γ + δ) x ζ + ɛ 2 2 2 h1 γh 2 x h 1h 2) w 2 2 + γ+δ Bo 2 x ) = µɛ x R[ζ, w] x ζ 1+µɛ 2 x ζ ), 2 with h 1 = 1 ɛζ and h 2 = 1 δ + ɛζ and Q[ζ]w def = 1 h 3 2 x h 3 2 x h 2 w )) + γh 1 x h 3 1 x h 1 w ))), R[ζ, w] h 2 x h 2 w )) 2 γ h 1 x h 1 w )) ) 2 def = 1 2 + 1 3 w h 2 2 x h 2 3 x h 2 w )) γh 2 1 x h 1 3 x h 1 w ))).

Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 10 / 17 Construction of our model The GN system with improved and nonlocal!) frequency dispersion t ζ + x w = 0, ) ) h1+γh 2 t h 1h 2 w + µq F [ζ]w + γ + δ) x ζ + ɛ 2 2 2 h1 γh 2 x h 1h 2) w 2 2 = µɛ x R F [ζ, w] ) + γ+δ Bo 2 x x ζ 1+µɛ 2 x ζ ), 2 with h 1 = 1 ɛζ and h 2 = 1 δ + ɛζ and Q F [ζ]w def = 1 h 3 2 x F µ 2 h 3 2 x F µ 2 R F def [ζ, w] = 1 h 2 2 x F µ 2 h 2 w )) 2 γ h 1 x F µ 1 + 1 3 w h 2 2 x F µ 2 h 3 2 x F µ 2 where F µ i = F i µ D ). h 2 w )) + γh 1 x F µ 1 h 1 w )) 2 ) h 2 w )) γh 2 1 x F µ 1 h 3 1 x F µ 1 h 1 w ))), h 3 1 x F µ 1 h 1 w ))). Notation Fourier multiplier) : F µ i uξ) = F i µ ξ )ûξ). x = id.

Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 10 / 17 Construction of our model The GN system with improved and nonlocal!) frequency dispersion t ζ + x w = 0, ) ) h1+γh 2 t h 1h 2 w + µq F [ζ]w + γ + δ) x ζ + ɛ 2 2 2 h1 γh 2 x h 1h 2) w 2 2 Q F [ζ]w R F [ζ, w] where F µ i def = 1 h 3 2 x F µ 2 def = 1 h 2 2 x F µ 2 h 2 w )) 2 γ h 1 x F µ 1 = F i µ D ). + 1 3 w h 2 2 x F µ 2 = µɛ x R F [ζ, w] ) + γ+δ Bo 2 x h 3 2 x F µ 2 h 2 w )) + γh 1 x F µ 1 h 3 1 x F µ 1 h h 2 3 x F µ 2 Examples : F µ 1 i = or F µ 1+µθi D 2 i = h 1 w )) 2 ) h 2 w )) γh 2 1 x F µ 1 1 w ))), x ζ 1+µɛ 2 x ζ ), 2 h 3 1 x F µ 1 h 1 w ))). 3 δ i µ D tanhδ i µ D ) 3. δ 2i µ D 2

Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 11 / 17 Our systems Promotion... 1 can be tuned to reproduce formally) the formation of the Kelvin-Helmholtz instabilities, or to suppress them ; 2 preserves natural quantities : E def = I def = γ + δ)ζ 2 + ζ dx, 3 possesses symmetry groups def V i = h i w + µq F i [ζ]w dx, 2γ + δ) 1 + µɛ2 µɛ 2 x ζ Bo 2 1 ) + h 1 + γh 2 w 2 h 1 h 2 + µ γ 3 1 h3 x F µ w ) 2 1 1 + µ h 1 3 h3 2 x F µ w ) 2 2 dx h 2 x x + α, t t + α, x, u 1, u 2 ) x + αt, u 1 + α, u 2 + α) 4 enjoys a Hamiltonian structure.

1 Motivation full Euler system Kelvin-Helmholtz instabilities 2 Asymptotic models Asymptotic models Drawbacks New systems 3 Numerical simulations 4 Well-posedness

Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 13 / 17 Numerical scheme Spatial discretization : spectral method ut, x) j ut, x j) sinπx x j )/h) πx x j )/h with x j = jh. FD)u j ut, x j)fd) sinπx x j )/h) πx x j )/h = M F ut, xj ) ) j. exponential accuracy for smooth functions 2 9 = 512 points gives machine precision 10 8 ) Time evolution : High order Runge-Kutta scheme Matlab s ode45) Reasonable CFL condition t Ch High order scheme = high accuracy here, typically 10 0 ) Without surface tension With surface tension

1 Motivation full Euler system Kelvin-Helmholtz instabilities 2 Asymptotic models Asymptotic models Drawbacks New systems 3 Numerical simulations 4 Well-posedness

Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 15 / 17 Strategy ideas...) The system may be rewritten as ) ) ) ) 1 0 ζ 0 1 ζ 0 b t + w a c x w where = l.o.t a def = a 0 ɛζ) ɛ 2 ã 0 ɛζ) w 2) 1 Bo x a1 ɛζ) x ) + µ x F µ i ã1 ɛζ)w 2 x F µ i ) b def = b 0 ɛζ) µ x F µ i b1 ɛζ) x F µ i ) c def = c 0 ɛζ) µ x F µ i c1 ɛζ) x F µ i ) The symmetrizer defines a natural energy : ζ, w) 2 def = aζ, ζ ) + bw, w ). X 0 L 2 L 2 Hyperbolicity conditions If h 1 = 1 ɛζ > h 0 and h 2 = δ + ɛζ > h 0, then a 0, a 1, b 0, b 1 > 0.

Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 15 / 17 Strategy ideas...) ) ) 1 0 ζ 0 b t + w 0 1 a c ) x ζ w ) = l.o.t a def = a 0 ɛζ) ɛ 2 ã 0 ɛζ) w 2) 1 Bo x a1 ɛζ) x ) + µ x F µ i ã1 ɛζ)w 2 x F µ i ) b def = b 0 ɛζ) µ x F µ i b1 ɛζ) x F µ i ) The symmetrizer defines a natural energy : ζ, w) 2 def = aζ, ζ ) + bw, w ). X 0 L 2 L 2 Hyperbolicity conditions If h 1 = 1 ɛζ > h 0 and h 2 = δ + ɛζ > h 0, then a 0, a 1, b 0, b 1 > 0. If F i ξ) ξ σ and ɛ 2 w 2 1 + µ Bo) 1 σ ) sufficiently small, then ζ, w) 2 X ζ 2 + 1 x ζ 2 + 0 L 2 Bo L w 2 + µ x F µ 2 L 2 i w 2. L 2

) ) a 0 ζ 0 b t + w A priori estimates 0 a a c ) x ζ w ) = l.o.t Usual strategy : multiply the system with Λ s = 1 + D 2 ) s/2 and use commutator estimates to control X s norm, s > 1/2 + 1 : ζ, w) 2 ζ 2 + 1 X s H s x ζ 2 + w 2 + µ Bo H s H s x F µ i w 2. H s Here, we cannot estimate [Λ s, a] x w, Λ s ζ ) L 2 and/or [Λ s, a] t ζ, Λ s ζ ) L 2. Instead, work with ζ, w) E N α ζ 2 + 1 L 2 x α ζ 2 + α w 2 + µ x F µ Bo L 2 L 2 i α w 2. L 2 where the sum is over α N, α multi-indices in space and time. Differentiate α times, extract leading order system for α ζ, α w) = control of α ζ, α w) X 0. Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 16 / 17

Well-posedness Well-posedness of the Green-Naghdi models Assume that ζ 0, w 0 ) E N with N large enough time-derivatives given by the system) ; ɛζ 0 { L < min 1, 1 }, ɛ 2 α w 0 2 δ L + µ Bo) 1 σ µ x F µ i ) j α w 0 )< 2 L Cɛζ 0 ). α 1 j M Then there exists C, T and a unique strong solution ζ, w) to the Green-Naghdi system for t [0, T ), and t < T, ζ, w) E N t) C ζ 0, w 0 ) E N expct), and the hyperbolicity conditions remain satisfied. Remarks : This result allows to fully justify the models w.r.t. the full Euler system [Lannes 13] C, T are uniformly bounded with respect to γ [0, 1], ɛ [0, 1], µ [0, µ max], and also σ R +. The time domain is [0, T /ɛ) if a stronger hyperbolicity condition is satisfied. If σ 1 or µ = 0, the result is also valid without surface tension Bo = 0), and ζ, w) E N = ζ 2 + H w 2. N H N Vincent Duchêne Kelvin-Helmholtz instabilities in shallow water 17 / 17

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