Kelvin-Helmholtz instabilities in shallow water
|
|
- Marjory McDaniel
- 5 years ago
- Views:
Transcription
1 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 Faculté des Sciences I, Université Libanaise, Beirut Center for Advanced Mathematical Sciences American University of Beirut Nov. 12, 2014
2 1. Credits : Naval Research Laboratory 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
3 1. Credits : St. Lawrence Estuary Internal Wave Experiment SLEIWEX) 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
4 1. Credits : NASA s Earth Observatory Picture of the Day July 1, 2003) 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,
5 1 Motivation full Euler system Kelvin-Helmholtz instabilities 2 Asymptotic models Asymptotic models Drawbacks New systems 3 Numerical simulations 4 Well-posedness
6 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.
7 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.
8 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.
9 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
10 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
11 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
12 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
13 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
14 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
15 1 Motivation full Euler system Kelvin-Helmholtz instabilities 2 Asymptotic models Asymptotic models Drawbacks New systems 3 Numerical simulations 4 Well-posedness
16 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
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 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)
18 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 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 = µɛ 2 x ζ 2 ), γ 1 x φ 1 t, x, z) dz. h 1 t, x) ɛζt,x) w h 2 2 x h 2 3 x h 2 w )) γh 2 1 x h 1 3 x h 1 w ))).
19 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?
20 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
21 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.../...
22 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.../...
23 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 h1 γh 2 x h 1h 2) w 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 = w h 2 2 x h 2 3 x h 2 w )) γh 2 1 x h 1 3 x h 1 w ))).
24 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 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 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.
25 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 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 ) 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
26 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 ) 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.
27 1 Motivation full Euler system Kelvin-Helmholtz instabilities 2 Asymptotic models Asymptotic models Drawbacks New systems 3 Numerical simulations 4 Well-posedness
28 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
29 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
30 1 Motivation full Euler system Kelvin-Helmholtz instabilities 2 Asymptotic models Asymptotic models Drawbacks New systems 3 Numerical simulations 4 Well-posedness
31 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.
32 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.
33 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 µ Bo) 1 σ ) sufficiently small, then ζ, w) 2 X ζ x ζ L 2 Bo L w 2 + µ x F µ 2 L 2 i w 2. L 2
34 ) ) 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 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 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
35 ) ) 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 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 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
36 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
37 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
38 Thank you for your attention!
A new class of two-layer Green-Naghdi systems with improved frequency dispersion
A new class of two-layer Green-Naghdi systems with improved frequency dispersion Vincent Duchêne Samer Israwi aafat Talhouk January, 6 Abstract We introduce a new class of Green-Naghdi type models for
More informationarxiv: v2 [math.ap] 18 Dec 2012
Decoupled and unidirectional asymptotic models for the propagation of internal waves Vincent Duchêne arxiv:08.6394v [math.ap] 8 Dec 0 October 3, 08 Abstract We study the relevance of various scalar equations,
More informationAsymptotic models for internal waves in Oceanography
Asymptotic models for internal waves in Oceanography Vincent Duchêne École Normale Supérieure de Paris Applied Mathematics Colloquium APAM February 01, 2010 Internal waves in ocean The large picture Figure:
More informationASYMPTOTIC SHALLOW WATER MODELS FOR INTERNAL WAVES IN A TWO-FLUID SYSTEM WITH A FREE SURFACE
SIAM J MATH ANAL Vol 4 No 5 pp 9 60 c 00 Society for Industrial and Applied Mathematics ASYMPTOTIC SHALLOW WATER MODELS FOR INTERNAL WAVES IN A TWO-FLUID SYSTEM WITH A FREE SURFACE VINCENT DUCHÊNE Abstract
More informationModified Serre Green Naghdi equations with improved or without dispersion
Modified Serre Green Naghdi equations with improved or without dispersion DIDIER CLAMOND Université Côte d Azur Laboratoire J. A. Dieudonné Parc Valrose, 06108 Nice cedex 2, France didier.clamond@gmail.com
More informationOn the Whitham Equation
On the Whitham Equation Henrik Kalisch Department of Mathematics University of Bergen, Norway Joint work with: Handan Borluk, Denys Dutykh, Mats Ehrnström, Daulet Moldabayev, David Nicholls Research partially
More informationKelvin Helmholtz Instability
Kelvin Helmholtz Instability A. Salih Department of Aerospace Engineering Indian Institute of Space Science and Technology, Thiruvananthapuram November 00 One of the most well known instabilities in fluid
More informationDynamics of Strongly Nonlinear Internal Solitary Waves in Shallow Water
Dynamics of Strongly Nonlinear Internal Solitary Waves in Shallow Water By Tae-Chang Jo and Wooyoung Choi We study the dynamics of large amplitude internal solitary waves in shallow water by using a strongly
More informationThe effect of a background shear current on large amplitude internal solitary waves
The effect of a background shear current on large amplitude internal solitary waves Wooyoung Choi Dept. of Mathematical Sciences New Jersey Institute of Technology CAMS Report 0506-4, Fall 005/Spring 006
More informationFinite Differences for Differential Equations 28 PART II. Finite Difference Methods for Differential Equations
Finite Differences for Differential Equations 28 PART II Finite Difference Methods for Differential Equations Finite Differences for Differential Equations 29 BOUNDARY VALUE PROBLEMS (I) Solving a TWO
More informationContents (I): The model: The physical scenario and the derivation of the model: Muskat and the confined Hele-Shaw cell problems.
Contents (I): The model: The physical scenario and the derivation of the model: Muskat and the confined Hele-Shaw cell problems. First ideas and results: Existence and uniqueness But, are the boundaries
More informationThe Whitham Equation. John D. Carter April 2, Based upon work supported by the NSF under grant DMS
April 2, 2015 Based upon work supported by the NSF under grant DMS-1107476. Collaborators Harvey Segur, University of Colorado at Boulder Diane Henderson, Penn State University David George, USGS Vancouver
More informationKELVIN-HELMHOLTZ INSTABILITIES AND INTERFACIAL WAVES
KELVIN-HELMHOLTZ INSTABILITIES AND INTERFACIAL WAVES DAVID LANNES 1. Introduction 1.1. General setting. We are interested here in the motion of the interface between two incompressible fluids of different
More informationarxiv: v1 [math.ap] 13 Feb 2019
WAVES INTERACTING WITH A PARTIALLY IMMERSED OBSTACLE IN THE BOUSSINESQ REGIME arxiv:192.4837v1 [math.ap] 13 Feb 219 D. BRESCH, D. LANNES, G. MÉTIVIER Abstract This paper is devoted to the derivation and
More informationNumerical Modeling of Stratified Shallow Flows: Applications to Aquatic Ecosystems
Numerical Modeling of Stratified Shallow Flows: Applications to Aquatic Ecosystems Marica Pelanti 1,2, Marie-Odile Bristeau 1 and Jacques Sainte-Marie 1,2 1 INRIA Paris-Rocquencourt, 2 EDF R&D Joint work
More informationThe Kelvin-Helmholtz Instabilities in Two-Fluids Shallow Water Models
The Kelvin-Helmholtz Instabilities in Two-Fluids Shallow Water Models David Lannes and Mei Ming To Walter Craig, for his 60th birthday, with friendship and admiration. Abstract The goal of this paper is
More informationModel Equation, Stability and Dynamics for Wavepacket Solitary Waves
p. 1/1 Model Equation, Stability and Dynamics for Wavepacket Solitary Waves Paul Milewski Mathematics, UW-Madison Collaborator: Ben Akers, PhD student p. 2/1 Summary Localized solitary waves exist in the
More informationThe inverse water wave problem of bathymetry detection
Under consideration for publication in J. Fluid Mech. 1 The inverse water wave problem of bathymetry detection V I S H A L V A S A N A N D B E N A D D E C O N I N C K Department of Applied Mathematics,
More informationNumerical Study of Oscillatory Regimes in the KP equation
Numerical Study of Oscillatory Regimes in the KP equation C. Klein, MPI for Mathematics in the Sciences, Leipzig, with C. Sparber, P. Markowich, Vienna, math-ph/"#"$"%& C. Sparber (generalized KP), personal-homepages.mis.mpg.de/klein/
More informationarxiv: v1 [math.ap] 18 Dec 2017
Wave-structure interaction for long wave models with a freely moving bottom Krisztian Benyo arxiv:1712.6329v1 [math.ap] 18 Dec 217 Abstract In this paper we address a particular fluid-solid interaction
More informationShallow water approximations for water waves over a moving bottom. Tatsuo Iguchi Department of Mathematics, Keio University
Shallow water approximations for water waves over a moving bottom Tatsuo Iguchi Department of Mathematics, Keio University 1 Tsunami generation and simulation Submarine earthquake occures with a seabed
More informationRecapitulation: Questions on Chaps. 1 and 2 #A
Recapitulation: Questions on Chaps. 1 and 2 #A Chapter 1. Introduction What is the importance of plasma physics? How are plasmas confined in the laboratory and in nature? Why are plasmas important in astrophysics?
More informationPeriodic Solutions of the Serre Equations. John D. Carter. October 24, Joint work with Rodrigo Cienfuegos.
October 24, 2009 Joint work with Rodrigo Cienfuegos. Outline I. Physical system and governing equations II. The Serre equations A. Derivation B. Justification C. Properties D. Solutions E. Stability Physical
More informationMath 575-Lecture 26. KdV equation. Derivation of KdV
Math 575-Lecture 26 KdV equation We look at the KdV equations and the so-called integrable systems. The KdV equation can be written as u t + 3 2 uu x + 1 6 u xxx = 0. The constants 3/2 and 1/6 are not
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Modified Equation of a Difference Scheme What is a Modified Equation of a Difference
More informationPartial Differential Equations
Partial Differential Equations Analytical Solution Techniques J. Kevorkian University of Washington Wadsworth & Brooks/Cole Advanced Books & Software Pacific Grove, California C H A P T E R 1 The Diffusion
More informationARTICLE IN PRESS. Available online at Mathematics and Computers in Simulation xxx (2011) xxx xxx
Available online at www.sciencedirect.com Mathematics and Computers in Simulation xxx (0) xxx xxx Suppression of Rayleigh Taylor instability using electric fields Lyudmyla L. Barannyk a,, Demetrios T.
More informationPHYS 432 Physics of Fluids: Instabilities
PHYS 432 Physics of Fluids: Instabilities 1. Internal gravity waves Background state being perturbed: A stratified fluid in hydrostatic balance. It can be constant density like the ocean or compressible
More information2.29 Numerical Fluid Mechanics Spring 2015 Lecture 13
REVIEW Lecture 12: Spring 2015 Lecture 13 Grid-Refinement and Error estimation Estimation of the order of convergence and of the discretization error Richardson s extrapolation and Iterative improvements
More informationSAMPLE CHAPTERS UNESCO EOLSS WAVES IN THE OCEANS. Wolfgang Fennel Institut für Ostseeforschung Warnemünde (IOW) an der Universität Rostock,Germany
WAVES IN THE OCEANS Wolfgang Fennel Institut für Ostseeforschung Warnemünde (IOW) an der Universität Rostock,Germany Keywords: Wind waves, dispersion, internal waves, inertial oscillations, inertial waves,
More information1 POTENTIAL FLOW THEORY Formulation of the seakeeping problem
1 POTENTIAL FLOW THEORY Formulation of the seakeeping problem Objective of the Chapter: Formulation of the potential flow around the hull of a ship advancing and oscillationg in waves Results of the Chapter:
More informationMy doctoral research concentrated on elliptic quasilinear partial differential equations of the form
1 Introduction I am interested in applied analysis and computation of non-linear partial differential equations, primarily in systems that are motivated by the physics of fluid dynamics. My Ph.D. thesis
More informationDerivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations
Derivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations H. A. Erbay Department of Natural and Mathematical Sciences, Faculty of Engineering, Ozyegin University, Cekmekoy 34794,
More informationTheory of linear gravity waves April 1987
April 1987 By Tim Palmer European Centre for Medium-Range Weather Forecasts Table of contents 1. Simple properties of internal gravity waves 2. Gravity-wave drag REFERENCES 1. SIMPLE PROPERTIES OF INTERNAL
More informationStability of traveling waves with a point vortex
Stability of traveling waves with a point vortex Samuel Walsh (University of Missouri) joint work with Kristoffer Varholm (NTNU) and Erik Wahlén (Lund University) Water Waves Workshop ICERM, April 24,
More informationPartial differential equations
Partial differential equations Many problems in science involve the evolution of quantities not only in time but also in space (this is the most common situation)! We will call partial differential equation
More informationFinite-time singularity formation for Euler vortex sheet
Finite-time singularity formation for Euler vortex sheet Daniel Coutand Maxwell Institute Heriot-Watt University Oxbridge PDE conference, 20-21 March 2017 Free surface Euler equations Picture n N x Ω Γ=
More informationNumerical Studies of Backscattering from Time Evolving Sea Surfaces: Comparison of Hydrodynamic Models
Numerical Studies of Backscattering from Time Evolving Sea Surfaces: Comparison of Hydrodynamic Models J. T. Johnson and G. R. Baker Dept. of Electrical Engineering/ Mathematics The Ohio State University
More informationapproach to Laplacian Growth Problems in 2-D
Conformal Mapping and Boundary integral approach to Laplacian Growth Problems in 2-D Saleh Tanveer (Ohio State University) Collaborators: X. Xie, J. Ye, M. Siegel Idealized Hele-Shaw flow model Gap averaged
More informationOn the well-posedness of the Prandtl boundary layer equation
On the well-posedness of the Prandtl boundary layer equation Vlad Vicol Department of Mathematics, The University of Chicago Incompressible Fluids, Turbulence and Mixing In honor of Peter Constantin s
More information7 EQUATIONS OF MOTION FOR AN INVISCID FLUID
7 EQUATIONS OF MOTION FOR AN INISCID FLUID iscosity is a measure of the thickness of a fluid, and its resistance to shearing motions. Honey is difficult to stir because of its high viscosity, whereas water
More informationNonlinear shape evolution of immiscible two-phase interface
Nonlinear shape evolution of immiscible two-phase interface Francesco Capuano 1,2,*, Gennaro Coppola 1, Luigi de Luca 1 1 Dipartimento di Ingegneria Industriale (DII), Università di Napoli Federico II,
More informationGreen-Naghdi type solutions to the Pressure Poisson equation with Boussinesq Scaling ADCIRC Workshop 2013
Green-Naghdi type solutions to the Pressure Poisson equation with Boussinesq Scaling ADCIRC Workshop 2013 Aaron S. Donahue*, Joannes J. Westerink, Andrew B. Kennedy Environmental Fluid Dynamics Group Department
More informationOn the limiting behaviour of regularizations of the Euler Equations with vortex sheet initial data
On the limiting behaviour of regularizations of the Euler Equations with vortex sheet initial data Monika Nitsche Department of Mathematics and Statistics University of New Mexico Collaborators: Darryl
More informationA Numerical Study of the Focusing Davey-Stewartson II Equation
A Numerical Study of the Focusing Davey-Stewartson II Equation Christian Klein Benson Muite Kristelle Roidot University of Michigan muite@umich.edu www.math.lsa.umich.edu/ muite 20 May 2012 Outline The
More informationLecture 1: Water waves and Hamiltonian partial differential equations
Lecture : Water waves and Hamiltonian partial differential equations Walter Craig Department of Mathematics & Statistics Waves in Flows Prague Summer School 208 Czech Academy of Science August 30 208 Outline...
More informationSome problems involving fractional order operators
Some problems involving fractional order operators Universitat Politècnica de Catalunya December 9th, 2009 Definition ( ) γ Infinitesimal generator of a Levy process Pseudo-differential operator, principal
More informationarxiv: v1 [math.ap] 27 Jun 2017
Solitary wave solutions to a class of modified Green-Naghdi systems Vincent Duchêne Dag Nilsson Erik Wahlén arxiv:706.0885v [math.a] 7 Jun 07 November 8, 08 Abstract We provide the existence and asymptotic
More informationNumerical dispersion and Linearized Saint-Venant Equations
Numerical dispersion and Linearized Saint-Venant Equations M. Ersoy Basque Center for Applied Mathematics 11 November 2010 Outline of the talk Outline of the talk 1 Introduction 2 The Saint-Venant equations
More informationOn the weakly nonlinear Kelvin-Helmholtz instability of tangential discontinuities in MHD
On the weakly nonlinear Kelvin-Helmholtz instability of tangential discontinuities in MHD John K. Hunter Department of Mathematics University of California at Davis Davis, California 9566, U.S.A. and J.
More informationDispersion in Shallow Water
Seattle University in collaboration with Harvey Segur University of Colorado at Boulder David George U.S. Geological Survey Diane Henderson Penn State University Outline I. Experiments II. St. Venant equations
More informationRecovering point sources in unknown environment with differential data
Recovering point sources in unknown environment with differential data Yimin Zhong University of Texas at Austin April 29, 2013 Introduction Introduction The field u satisfies u(x) + k 2 (1 + n(x))u(x)
More informationUSC Summer School on Mathematical Fluids. An Introduction to Instabilities in Interfacial Fluid Dynamics. Ian Tice
USC Summer School on Mathematical Fluids An Introduction to Instabilities in Interfacial Fluid Dynamics Ian Tice Scribe: Jingyang Shu July 17, 2017 CONTENTS Ian Tice (Scribe: Jingyang Shu) Contents Abstract
More informationNonlinear Modulational Instability of Dispersive PDE Models
Nonlinear Modulational Instability of Dispersive PDE Models Jiayin Jin, Shasha Liao, and Zhiwu Lin Georgia Tech ICERM workshop on water waves, 4/28/2017 Jiayin Jin, Shasha Liao, and Zhiwu Lin Georgia Tech
More informationA non-stiff boundary integral method for 3D porous media flow with surface tension
A non-stiff boundary integral method for 3D porous media flow with surface tension D. M. Ambrose 1 and M. Siegel 1 Department of Mathematics, Drexel University, Philadelphia, PA 191 Department of Mathematical
More informationThe Shallow Water Equations
If you have not already done so, you are strongly encouraged to read the companion file on the non-divergent barotropic vorticity equation, before proceeding to this shallow water case. We do not repeat
More information1. Comparison of stability analysis to previous work
. Comparison of stability analysis to previous work The stability problem (6.4) can be understood in the context of previous work. Benjamin (957) and Yih (963) have studied the stability of fluid flowing
More informationAST242 LECTURE NOTES PART 5
AST242 LECTURE NOTES PART 5 Contents 1. Waves and instabilities 1 1.1. Sound waves compressive waves in 1D 1 2. Jeans Instability 5 3. Stratified Fluid Flows Waves or Instabilities on a Fluid Boundary
More informationOn quasi-richards equation and finite volume approximation of two-phase flow with unlimited air mobility
On quasi-richards equation and finite volume approximation of two-phase flow with unlimited air mobility B. Andreianov 1, R. Eymard 2, M. Ghilani 3,4 and N. Marhraoui 4 1 Université de Franche-Comte Besançon,
More informationMartín Lopez de Bertodano 1 and William D. Fullmer 2. School of Nuclear Engineering, Purdue University, West Lafayette, IN 47907, USA 2
Effect of viscosity on a well-posed one-dimensional two-fluid model for wavy two-phase flow beyond the Kelvin-Helmholtz instability: limit cycles and chaos Martín Lopez de Bertodano 1 and William D. Fullmer
More informationAn integrable shallow water equation with peaked solitons
An integrable shallow water equation with peaked solitons arxiv:patt-sol/9305002v1 13 May 1993 Roberto Camassa and Darryl D. Holm Theoretical Division and Center for Nonlinear Studies Los Alamos National
More informationLecture 4: Birkhoff normal forms
Lecture 4: Birkhoff normal forms Walter Craig Department of Mathematics & Statistics Waves in Flows - Lecture 4 Prague Summer School 2018 Czech Academy of Science August 31 2018 Outline Two ODEs Water
More informationFlows Induced by 1D, 2D and 3D Internal Gravity Wavepackets
Abstract Flows Induced by 1D, 2D and 3D Internal Gravity Wavepackets Bruce R. Sutherland 1 and Ton S. van den Bremer 2 1 Departments of Physics and of Earth & Atmospheric Sciences, University of Alberta
More information2 The incompressible Kelvin-Helmholtz instability
Hydrodynamic Instabilities References Chandrasekhar: Hydrodynamic and Hydromagnetic Instabilities Landau & Lifshitz: Fluid Mechanics Shu: Gas Dynamics 1 Introduction Instabilities are an important aspect
More informationSurface Tension Effect on a Two Dimensional. Channel Flow against an Inclined Wall
Applied Mathematical Sciences, Vol. 1, 007, no. 7, 313-36 Surface Tension Effect on a Two Dimensional Channel Flow against an Inclined Wall A. Merzougui *, H. Mekias ** and F. Guechi ** * Département de
More informationLecture 4: Numerical solution of ordinary differential equations
Lecture 4: Numerical solution of ordinary differential equations Department of Mathematics, ETH Zürich General explicit one-step method: Consistency; Stability; Convergence. High-order methods: Taylor
More informationGlobal well-posedness and decay for the viscous surface wave problem without surface tension
Global well-posedness and decay for the viscous surface wave problem without surface tension Ian Tice (joint work with Yan Guo) Université Paris-Est Créteil Laboratoire d Analyse et de Mathématiques Appliquées
More informationMath 7824 Spring 2010 Numerical solution of partial differential equations Classroom notes and homework
Math 7824 Spring 2010 Numerical solution of partial differential equations Classroom notes and homework Jan Mandel University of Colorado Denver May 12, 2010 1/20/09: Sec. 1.1, 1.2. Hw 1 due 1/27: problems
More informationA second-order asymptotic-preserving and positive-preserving discontinuous. Galerkin scheme for the Kerr-Debye model. Abstract
A second-order asymptotic-preserving and positive-preserving discontinuous Galerkin scheme for the Kerr-Debye model Juntao Huang 1 and Chi-Wang Shu Abstract In this paper, we develop a second-order asymptotic-preserving
More informationRELAXED VARIATIONAL PRINCIPLE FOR WATER WAVE MODELING
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
More informationCHAPTER 19. Fluid Instabilities. In this Chapter we discuss the following instabilities:
CHAPTER 19 Fluid Instabilities In this Chapter we discuss the following instabilities: convective instability (Schwarzschild criterion) interface instabilities (Rayleight Taylor & Kelvin-Helmholtz) gravitational
More informationSurprising Computations
.... Surprising Computations Uri Ascher Department of Computer Science University of British Columbia ascher@cs.ubc.ca www.cs.ubc.ca/ ascher/ Uri Ascher (UBC) Surprising Computations Fall 2012 1 / 67 Motivation.
More informationGap size effects for the Kelvin-Helmholtz instability in a Hele-Shaw cell
PHYSICAL REVIEW E, VOLUME 64, 638 Gap size effects for the Kelvin-Helmholtz instability in a Hele-Shaw cell L. Meignin, P. Ern, P. Gondret, and M. Rabaud Laboratoire Fluides, Automatique et Systèmes Thermiques,
More information5 Linear Algebra and Inverse Problem
5 Linear Algebra and Inverse Problem 5.1 Introduction Direct problem ( Forward problem) is to find field quantities satisfying Governing equations, Boundary conditions, Initial conditions. The direct problem
More informationarxiv: v1 [physics.flu-dyn] 30 Oct 2014
The Equation as a Model for Surface Water Waves arxiv:4.899v physics.flu-dyn] 3 Oct 4 Daulet Moldabayev, Henrik Kalisch Department of Mathematics, University of Bergen Postbox 78, 5 Bergen, Norway Denys
More informationNumerical Methods for Conservation Laws WPI, January 2006 C. Ringhofer C2 b 2
Numerical Methods for Conservation Laws WPI, January 2006 C. Ringhofer ringhofer@asu.edu, C2 b 2 2 h2 x u http://math.la.asu.edu/ chris Last update: Jan 24, 2006 1 LITERATURE 1. Numerical Methods for Conservation
More informationCOMPUTATIONALLY EFFICIENT NUMERICAL MODEL FOR THE EVOLUTION OF DIRECTIONAL OCEAN SURFACE WAVES
XIX International Conference on Water Resources CMWR 01 University of Illinois at Urbana-Champaign June 17-,01 COMPUTATIONALLY EFFICIENT NUMERICAL MODEL FOR THE EVOLUTION OF DIRECTIONAL OCEAN SURFACE WAVES
More information1/27/2010. With this method, all filed variables are separated into. from the basic state: Assumptions 1: : the basic state variables must
Lecture 5: Waves in Atmosphere Perturbation Method With this method, all filed variables are separated into two parts: (a) a basic state part and (b) a deviation from the basic state: Perturbation Method
More informationThe Nonlinear Schrodinger Equation
Catherine Sulem Pierre-Louis Sulem The Nonlinear Schrodinger Equation Self-Focusing and Wave Collapse Springer Preface v I Basic Framework 1 1 The Physical Context 3 1.1 Weakly Nonlinear Dispersive Waves
More informationMagnetohydrodynamics Stability of a Compressible Fluid Layer Below a Vacuum Medium
Mechanics and Mechanical Engineering Vol. 12, No. 3 (2008) 267 274 c Technical University of Lodz Magnetohydrodynamics Stability of a Compressible Fluid Layer Below a Vacuum Medium Emad E. Elmahdy Mathematics
More informationTHE CONVECTION DIFFUSION EQUATION
3 THE CONVECTION DIFFUSION EQUATION We next consider the convection diffusion equation ɛ 2 u + w u = f, (3.) where ɛ>. This equation arises in numerous models of flows and other physical phenomena. The
More informationStabilité des systèmes hyperboliques avec des commutations
Stabilité des systèmes hyperboliques avec des commutations Christophe PRIEUR CNRS, Gipsa-lab, Grenoble, France Séminaire EDP, Université de Versailles 9 octobre 2014 1/34 C. Prieur UVSQ, Octobre 2014 2/34
More informationExistence and stability of solitary-wave solutions to nonlocal equations
Existence and stability of solitary-wave solutions to nonlocal equations Mathias Nikolai Arnesen Norwegian University of Science and Technology September 22nd, Trondheim The equations u t + f (u) x (Lu)
More informationWaves on deep water, II Lecture 14
Waves on deep water, II Lecture 14 Main question: Are there stable wave patterns that propagate with permanent form (or nearly so) on deep water? Main approximate model: i" # A + $" % 2 A + &" ' 2 A +
More informationBoundary layers for Navier-Stokes equations with slip boundary conditions
Boundary layers for Navier-Stokes equations with slip boundary conditions Matthew Paddick Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie (Paris 6) Séminaire EDP, Université Paris-Est
More informationModel adaptation in hierarchies of hyperbolic systems
Model adaptation in hierarchies of hyperbolic systems Nicolas Seguin Laboratoire J.-L. Lions, UPMC Paris 6, France February 15th, 2012 DFG-CNRS Workshop Nicolas Seguin (LJLL, UPMC) 1 / 29 Outline of the
More informationarxiv: v1 [math.na] 27 Jun 2017
Behaviour of the Serre Equations in the Presence of Steep Gradients Revisited J.P.A. Pitt a,, C. Zoppou a, S.G. Roberts a arxiv:706.08637v [math.na] 27 Jun 207 a Mathematical Sciences Institute, Australian
More informationReview of Fundamental Equations Supplementary notes on Section 1.2 and 1.3
Review of Fundamental Equations Supplementary notes on Section. and.3 Introduction of the velocity potential: irrotational motion: ω = u = identity in the vector analysis: ϕ u = ϕ Basic conservation principles:
More informationConcepts from linear theory Extra Lecture
Concepts from linear theory Extra Lecture Ship waves from WW II battleships and a toy boat. Kelvin s (1887) method of stationary phase predicts both. Concepts from linear theory A. Linearize the nonlinear
More informationBlow-up or No Blow-up? the Role of Convection in 3D Incompressible Navier-Stokes Equations
Blow-up or No Blow-up? the Role of Convection in 3D Incompressible Navier-Stokes Equations Thomas Y. Hou Applied and Comput. Mathematics, Caltech Joint work with Zhen Lei; Congming Li, Ruo Li, and Guo
More informationEvolution equations with spectral methods: the case of the wave equation
Evolution equations with spectral methods: the case of the wave equation Jerome.Novak@obspm.fr Laboratoire de l Univers et de ses Théories (LUTH) CNRS / Observatoire de Paris, France in collaboration with
More informationy = h + η(x,t) Δϕ = 0
HYDRODYNAMIC PROBLEM y = h + η(x,t) Δϕ = 0 y ϕ y = 0 y = 0 Kinematic boundary condition: η t = ϕ y η x ϕ x Dynamical boundary condition: z x ϕ t + 1 2 ϕ 2 + gη + D 1 1 η xx ρ (1 + η 2 x ) 1/2 (1 + η 2
More informationMath 46, Applied Math (Spring 2009): Final
Math 46, Applied Math (Spring 2009): Final 3 hours, 80 points total, 9 questions worth varying numbers of points 1. [8 points] Find an approximate solution to the following initial-value problem which
More informationViscous capillary fluids in fast rotation
Viscous capillary fluids in fast rotation Centro di Ricerca Matematica Ennio De Giorgi SCUOLA NORMALE SUPERIORE BCAM BASQUE CENTER FOR APPLIED MATHEMATICS BCAM Scientific Seminar Bilbao May 19, 2015 Contents
More informationInverse Scattering with Partial data on Asymptotically Hyperbolic Manifolds
Inverse Scattering with Partial data on Asymptotically Hyperbolic Manifolds Raphael Hora UFSC rhora@mtm.ufsc.br 29/04/2014 Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/2014
More informationBounds on Surface Stress Driven Flows
Bounds on Surface Stress Driven Flows George Hagstrom Charlie Doering January 1, 9 1 Introduction In the past fifteen years the background method of Constantin, Doering, and Hopf has been used to derive
More informationBoundary Layers. Lecture 2 by Basile Gallet. 2i(1+i) The solution to this equation with the boundary conditions A(0) = U and B(0) = 0 is
Boundary Layers Lecture 2 by Basile Gallet continued from lecture 1 This leads to a differential equation in Z Z (A ib) + (A ib)[ C + Uy Uy 2i(1+i) ] = 0 with C = 2. The solution to this equation with
More informationHydrodynamic Instability. Agenda. Problem definition. Heuristic approach: Intuitive solution. Mathematical approach: Linear stability theory
Environmental Fluid Mechanics II Stratified Flow and Buoyant Mixing Hydrodynamic Instability Dong-Guan Seol INSTITUTE FOR HYDROMECHANICS National Research Center of the Helmholtz Association www.kit.edu
More informationNumerical Methods for ODEs. Lectures for PSU Summer Programs Xiantao Li
Numerical Methods for ODEs Lectures for PSU Summer Programs Xiantao Li Outline Introduction Some Challenges Numerical methods for ODEs Stiff ODEs Accuracy Constrained dynamics Stability Coarse-graining
More informationThe purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and engineering.
Lecture 16 Applications of Conformal Mapping MATH-GA 451.001 Complex Variables The purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and
More information