ON SOLITARY WAVES FOR BOUSSINESQ’S TYPE MODELS S. Bellec and M. Colin April, the 3rd 2015
S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
1 / 27
Introduction
Context Aim: Modeling wave transformation in near-shore zones. dispersive effects; nonlinear effects;
enhanced Boussinesq-type models, Green-Nadghi, ...
Sumatra 2004 tsunami reaching the coast of Thailand (from Madsen et al.2008) S. Bellec(IMB, INRIA)
Undular tidal bore − Garonne 2010 (from Bonneton et al.2011) WATER WAVES MODELING
April, the 3rd 2015
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Introduction
Modeling Several BT models : approximation of Euler equations Design properties of these models : linear dispersion relation shoalling coefficients Two kinds of BT models : amplitude-velocity system amplitude-volume flux system degenerate to same linearized system but differ from high order derivatives terms ! cf : M.W. Dingemans S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
3 / 27
BT Models
Euler system of equations Euler system: ∂x p =0 ∂t u + u∂x u + w ∂z u + ρ ∂z p ∂t w + u∂x w + w ∂z w + =0 ρ ∂x u + ∂z w = 0 ∂z u − ∂x w = 0
(1)
B.C. : - in z = η: - in z = −d:
w = ∂t η + u∂x η, P = 0 w = −u∂x d
S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
4 / 27
BT Models
Parameters
nonlinearity parameter ε = dispersion parameter σ =
a d0 d0 L
Weakly nonlinear models ε=O(σ 2 ) Obtention of Boussinesq equations : x z x˜ = , z˜ = , t˜ = L d0
√
gd0 η d t, η˜ = , d˜ = , L a d0
d0 L 1 p , u˜ = √ u, w ˜= √ v , p˜ = a gd0 gd0 ρ a gd0
S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
5 / 27
BT Models
Euler system of equations Euler system: εu˜t˜ + ε2 u˜u˜x˜ + ε2 w ˜ u˜z˜ + p˜x˜ = 0 εσ 2 w ˜t˜ + ε2 σ 2 u˜w ˜x˜ + ε2 σ 2 w ˜w ˜z˜ + p˜z˜ + 1 = 0
(2)
u˜x˜ + w ˜z˜ = 0 ˜x˜ = 0. u˜z˜ − σ 2 w B.C. : - in z˜ = ε˜ η: w ˜ = η˜t˜ + εu˜η˜x˜ , p˜ = 0 ˜ - in z˜ = −d: w ˜ = −d˜x˜ u˜
cf: O. Nwogu (1993), Alternative form of Boussinesq equations for nearshore wave propagation.
S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
6 / 27
BT Models
Nwogu Integrate incompressibility equation and plugg in the irrotational condition ! Z z˜ 2 ∂ u˜ 2 ∂ = −σ ud ˜ z˜ . ∂ z˜ ∂ x˜2 −d˜ Taylor expansion of z˜ 7→ u( ˜ t˜, x˜, z˜) around z˜ = z˜α (U˜ = u( ˜ t˜, x˜, z˜α )) (˜ z − z˜α )2 ∂ 2 u˜ ∂ u˜ u˜ = U˜ + (˜ z − z˜α ) |z˜=˜zα + |z˜=˜zα + · · · ∂ z˜ 2 ∂ z˜2 integrate and plugg ∂2 u˜z˜ = −σ ∂ x˜2 2
S. Bellec(IMB, INRIA)
"
z − z˜α )2 (d˜ + z˜α )2 ˜ U˜ + (˜ (˜ z + d) − 2 2
WATER WAVES MODELING
#
∂ u˜ |z˜=˜zα + · · · ∂ z˜
!
April, the 3rd 2015
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BT Models
Nwogu Apply ∂z˜ and ∂z˜2 : u˜z˜z˜ = −σ 2
∂ 2 U˜ + O(σ 4 ), u˜z˜z˜z˜ = O(σ 4 ). ∂ x˜2
Coming back to Taylor expansion u˜ = U˜ − σ 2
z − z˜α )2 ∂ 2 U˜ ∂2 ˜ + (˜ (˜ z − z˜α ) 2 [(d˜ + z˜α )U] ∂ x˜ 2 ∂ x˜2
! + O(σ 4 ).
plugg in the second equation of Euler and integrate between ε˜ η and z˜ + B.C. ∂2 p˜ = ε˜ η − z˜ + εσ ∂ t˜∂ x˜ 2
S. Bellec(IMB, INRIA)
z˜2 ˜ ˜ (d z˜ + )U + O(εσ 4 , ε2 σ 2 ). 2
WATER WAVES MODELING
April, the 3rd 2015
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BT Models
Nwogu equations Plugg in the first equation of Euler Nwogu 1: U˜t˜ + εU˜U˜x˜ + η˜x˜ + σ 2
z˜α2 ˜ U˜ + z˜α [d˜U˜t˜]x˜x˜ 2 t x˜x˜
= O(εσ 2 , σ 4 ).
(4)
Integrate incompressibility equation using the expression of u: ˜ Nwogu 2: h i ˜ U˜ + σ 2 ∂ η˜t˜ + (ε˜ η + d) ∂ x˜ x˜
S. Bellec(IMB, INRIA)
˜ 2 (d) ˜ x˜x˜ d˜z˜α + [d˜U] 2 ! d˜z˜2 ˜ 3 (d) α + − U˜x˜x˜ = O(εσ 2 , σ 4 ). 2 6
WATER WAVES MODELING
April, the 3rd 2015
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BT Models
Nwogu-Abbott To go further : h˜ = d˜ + ε˜ η , Q˜ := h˜U˜ = d˜U˜ + O(ε),
U˜t˜ =
Q˜ d˜ + ε˜ η
! = t˜
(5)
Q˜t˜ + O(ε). d˜
!
U˜t˜x˜x˜ = U˜x˜x˜x˜ =
Q˜t˜ + O(ε), d˜ x˜x˜ ! Q˜ + O(ε). d˜ x˜x˜x˜
Multiply Nwogu 1 by h˜ and Nwogu 2 by εU˜
S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
10 / 27
BT Models
Nwogu-Abbott ! d˜z˜2 ˜ 3 Q˜ ˜ 2 ( d) ( d) α = O(εσ 2 , σ 4 ). − d˜z˜α + Q˜x˜x˜ + 2 2 6 d˜ x˜x˜ ! ! z˜α2 Q˜t˜ Q˜ 2 2˜ ˜ ˜ ηx˜ + σ d z˜α Qt˜x˜x˜ + + h˜ = O(εσ 2 , σ 4 ), 2 d˜ x˜x˜ h˜
∂ η˜t˜ + Q˜x˜ + σ ∂ x˜ 2
Q˜t˜ + ε
x˜
Back to physical variables : Nwogu-Abbott: (Filippini, Bellec, Colin, Ricchiuto) Q 2 3 =0 ηt + Qx + β1 d Qxx + β2 d d xx x 2 Q Q + ghηx + α1 d 2 Qtxx + α2 d 3 = 0, Qt + h x d txx β1 = θ + 1/2, β2 = θ2 /2 − 1/6, α1 = θ, α2 = θ2 /2. S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
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BT Models
Other asymptotic model Peregrine : ηt + [(η + d)u] ¯ x = 0. u¯t + u¯u¯x + g ηx +
d2 d u¯txx − [d u¯t ]xx 6 2
(6) = 0.
(7)
Abbott ((η, q) version of Peregrine) : ηt + qx = 0. qt +
q2 d +η
S. Bellec(IMB, INRIA)
+ g (d + η)ηx + x
(8)
d3 q d2 − qtxx = 0. 6 d txx 2
WATER WAVES MODELING
April, the 3rd 2015
(9)
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BT Models
Beji-Nadaoka : ηt + [(η + d)u] ¯ x = 0. d2 d u¯t + u¯u¯x + g ηx + (1 + αB ) u¯txx − [d u¯t ]xx 6 2 2 d d ηxxx − [dηx ]xx = 0. + αB g 6 2
(10)
(11)
Beji-Nadaoka-Abbott (close to Madsen and Sorensen) : ηt + qx = 0.
(12)
3 Q2 d Q d2 Qt + + g (d + η)ηx + (1 + αB ) − Qtxx d +η x 6 d txx 2 3 d d + αB ηxxx − [dηx ]xx = 0. 6 2
S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
(13)
13 / 27
BT Models
Cauchy Problem Nwogu-Abbott with constant bathymetry 2 2 ∂t η + Qx + βσ d Qxxx= 0, Q2 + g (d + εη)ηx = 0 (1 − ασ 2 d 2 ∂xx )Qt + ε d + εη x Nowgu-Abbott: Theorem : Well posedness for T = O( ε1 ). η ∈ C ([0, T ]; L2 (R)) ∩ L∞ (0, T ; H 2 (R)), Q ∈ C ([0, T ]; L2 (R)) ∩ L∞ (0, T ; H 4 (R)) Conditions : d + εη0 (x) > 0 on R Q2
0 g (d + εη0 ) − ε (d+εη 2 > 0 on R 0)
Two difficulties : loss of derivatives : Bona, Chen, Saut h do not vanish ! S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
14 / 27
Solitary waves
Motivations
Question : Why ? create a hierarchy of models verification of numerical schemes practical applications : shoaling, ... provide existence and uniqueness result + free software to compute!
Main Contribution : exhibit the relation between c and A ! Condition : constant bathymety
S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
15 / 27
Solitary waves
Beji-Nadaoka Equations: ¯x =0 ηt + [hu]
u¯t + g ηx + u¯u¯x − γd 2 u¯txx − αB gd 2 ηxxx = 0 γ = αB +
1 3
Solitary wave η(t, x) = ηc (x − ct) u(t, ¯ x) = u¯c (x − ct), (
ηc =
d u¯c c−u¯c
,
−c u¯c + g ηc + Remark : c0 =
√
u¯c2 2
00
00
+ cγd 2 u¯c − αB gd 2 ηc = 0
gd.
S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
16 / 27
Solitary waves
Beji-Nadaoka Case 1 : γ = 0. −ηc00 = g (ηc ). 3 g (s) = gd 2
c 2d 2 s c2 + c02 − 2(d + s)2 2 d
Z , G (s) :=
s
g (t) dt 0
Th : Beretycki-Lions (1983). Let f beR a locally Lipschitz continuous real function with f (0) = 0 and z F (z) = 0 f (s)ds. −u” = f (u), u ∈ C 2 (R), lim u(x) = 0, u(x0 ) > 0 for some x0 ∈ R. x→±∞
has a unique solution u ∈ H 1 (R) ∩ C 2 (R) if and only if ξ0 = inf{ξ > 0, F (ξ) = 0} exists, and satisfies ξ0 > 0, f (ξ0 ) > 0. Remark : ξ0 = max u. R S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
17 / 27
Solitary waves
Beji-Nadaoka
g (s) = 0 ⇐⇒ s = 0 or s = s1 or s = s2 , where 2
s1 := d
( cc 2 − 4) − 0
q 2 8 cc 2 + 0
4
c4 c04
2
, s2 := d
( cc 2 − 4) +
q
0
4
2
8 cc 2 + 0
c4 c04
.
c ≤ c0 , we have s2 ≤ 0 : NO SOLUTION ! c > c0 : UNIQUE SOLUTION! G (A) = 0 ⇔ c 2 = c02
S. Bellec(IMB, INRIA)
d +A d
WATER WAVES MODELING
April, the 3rd 2015
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Solitary waves
Beji-Nadaoka Case 2 : γ 6= 0. 2cc02 αB d 2 02 u¯ − c u¯c (c − u¯c )2 (c − u¯c ) c u¯2 (c − u¯c )2 +c02 u¯c (c − u¯c ) + c = 0. 2 00
(cγd 2 (c − u¯c )2 − αB d 2 cc02 )u¯c −
QUASILINEAR ! Idea : u¯c = f (vc ) s = f (s) − c 2 > c02 S. Bellec(IMB, INRIA)
c02 αB c 2 αB + 0 . γ(c − f (s)) cγ
αB ⇒ f is a diffeomorphism γ WATER WAVES MODELING
April, the 3rd 2015
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Solitary waves
Beji-Nadaoka 00
−vc =
1 cc02 f (vc )2 − cf (vc ) + − c02 + . 2 cγd c − f (vc ) 2
Conclusion : Assume that one of the following alternative is satisfied : i) γ = 0, c > c0 , ii) γ > 0, αB ≤ 0, c > c0 , 2−
αB
iii) γ > 0, αB > 0, c ∈ (c0 , c0 q αγB ). γ
Then BN admits a unique solitary wave of the form (ηc (x − ct), u¯c (x − ct)). A2 A+d A αB A3 A2 4 2 2 c γ − +c c0 γ log( )−γ + 6(d + A)3 2(d + A)2 d d +A 2 d(d + A) 2 α BA − c04 = 0. 2d 2 Conversely, if γ < 0, αB < 0 and c ≥ 0, then BN has no positive solutions of the previous form. S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
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Solitary waves
Beji-Nadaoka
Figure: Solitary waves for Beji-Nadaoka equations (ηc is on the left and u¯c on the right), with d = 1, ηc0 = 0.2 and αB = 1/15.
S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
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Solitary waves
Madsen-Sorensen
Equations: (
ηt + qx = 0, 2 q¯t + ghηx + q¯h − Bd 2 q¯txx − βgd 3 ηxxx = 0 x
Conclusion : For all c > c0 , MS admit a unique solitary wave of the form (ηc (x − ct), q¯c (x − ct)). In addition, the relation between parameter c and the amplitude A of ηc is given by A2 A3 2 + 6d c 2 = c02 . dA − d 2 log( d+A d )
S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
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Solitary waves
Nwogu Equations:
ηt + [(η + d)U]x + βd 3 Uxxx = 0, Ut + UUx + g ηx + αd 2 Utxx = 0. r
Conclusion : For all α ∈ (− 21 , 0) and c > max c0 , c0
β2 α2 β 2− α
! , Nwogu admit a
unique solitary wave of the form (ηc (x − ct), Uc (x − ct)). In addition, the amplitude A of Uc and parameter c satisfy 2 c βc 2 − αc 2 c − c2 βc 2 − αc 2 (βc02 − αc 2 )2 1 3 A + + 0 A2 + 0 − 0 2 − A − 18α 4α 12cα 3α 2α 6α2 c 2 (βc02 − αc 2 )4 (βc02 − αc 2 )3 (c02 − c 2 )(βc02 − αc 2 )2 cα + + − log(1+ 2 A) 6c 3 α3 2cα3 3cα2 βc0 − αc 2 = 0. Conversely, if c ≤ c0 , then Nwogu have no positive solutions of the previous form. remark : no result for Nwogu-Abbott..... S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
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Solitary waves
Shoaling A. Filippini, S. Bellec, M .Ricchuito, MC.
Figure: Shoaling of a solitary wave; computational configuration and gauges position (Grilli and al).
d0 = 0.44m,
S. Bellec(IMB, INRIA)
a0 A = 0.2m slope = 1 : 35, ε = ∈ [0.2; 2.2] d0 h
WATER WAVES MODELING
April, the 3rd 2015
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Solitary waves
Shoaling 0.5 0.45 0.4
Experimental data A BNA MS NA g9 g7
0.35
η/h0
g5
g3
g1
0.3 0.25 0.2 0.15 0.1 0.05 0 37
38
39
40
41 t/(h0/g)1/2
42
43
44
45
Figure: Nonlinear shoaling. Comparison between computed wave heights. Models in amplitude-flux form.
0.5 0.45 0.4
Experimental data P BN MSP N
g9 g7 g5
0.35
g3 g1
η/h0
0.3 0.25 0.2 0.15 0.1 0.05 0 37
38
39
40
41
42
43
44
45
t/(h /g)1/2 0
Figure: Nonlinear shoaling. Comparison between computed wave heights. Models in amplitude-velocity form. S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
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Solitary waves
Shoaling 1.6
1.4
max
/h
0
1.2
Exp. data P BN MSP N A BNA MS NA
η
1
0.8
0.6
0.4 20
21
22
23
24
25
26
27
x/h0
Figure: Nonlinear shoaling. Comparison between computed wave peak evolution.
S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
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Solitary waves
THANK YOU
S. Bellec(IMB, INRIA)
WATER WAVES MODELING
April, the 3rd 2015
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