a reduced model for internal waves interacting with submarine ...

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Contents Abstract 1 Resumo 2 1 Introduction 3 2 Derivation of the reduced model 8 2.1 Reducing the upper layer dynamics to the interface . . . . . . . . 11 2.2 Connecting the upper and lower layers . . . . . . . . . . . . . . . 18 2.3 Dispersion relation for the linearized model . . . . . . . . . . . . 27 2.4 Unidirectional wave regime . . . . . . . . . . . . . . . . . . . . . 29 2.5 Solitary wave solutions . . . . . . . . . . . . . . . . . . . . . . . 34 3 A higher-order reduced model 36 3.1 Higher-order upper layer equations . . . . . . . . . . . . . . . . . 36 3.2 Improved approximation for pressure at the interface . . . . . . . 41 3.3 Dispersion relation for the higher-order model. Comparison with the previous model . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 Numerical results 52 i
4.1 Hierarchy of one-dimensional models . . . . . . . . . . . . . . . 52 4.2 Method of lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 Flat bottom experiments . . . . . . . . . . . . . . . . . . . . . . 66 4.4 Periodic topography experiments . . . . . . . . . . . . . . . . . . 75 4.5 Computing solitary waves solutions . . . . . . . . . . . . . . . . 81 Conclusions and future work 86 A Approximation for the horizontal derivatives at the unperturbed interface 88 B The Dirichlet-to-Neumann operator 91 C The periodic counterpart of the operatorT 93 Bibliography 98 ii
Page 1 and 2: INSTITUTO NACIONAL DE MATEMÁTICA P
Page 3: Acknowledgements First, I would lik
Page 7 and 8: Resumo É obtido um modelo reduzido
Page 9 and 10: when the steepening of a given wave
Page 11 and 12: in particular that of solitary wave
Page 13 and 14: Chapter 2 Derivation of the reduced
Page 15 and 16: demands that η t + u i η x = w i
Page 17 and 18: function f (x, z, t), let its assoc
Page 19 and 20: Substitution of w 1 into Eq. (2.2)
Page 21 and 22: i. e. u 1z =βw 1x . Hence ( ) u (0
Page 23 and 24: fluid layer. 2.2 Connecting the upp
Page 25 and 26: Furthermore, [√ ( √ )] βφt x,
Page 27 and 28: The problem in conformal coordinate
Page 29 and 30: problem (2.22). Therefore, P x =−
Page 31 and 32: assumption was made on the wave amp
Page 33 and 34: Its linearization around the undist
Page 35 and 36: c 0 chosen, there will be a right-
Page 37 and 38: and for similar reasons η t +η x
Page 39 and 40: 3. For system (2.28) a similar unid
Page 41 and 42: Chapter 3 A higher-order reduced mo
Page 43 and 44: expression above as Let us express
Page 45 and 46: and substituting in the asymptotic
Page 47 and 48: and it is easy to take x-derivative
Page 49 and 50: which together with η tt = ( (1−
Page 51 and 52: For the weakly nonlinear model of o
Page 53 and 54: Again, ω2 k 2 large. → 0 as k→
Page 55 and 56:
2.6 2.4 2.2 ω/k 2 1.8 1.6 1.4 1.2
Page 57 and 58:
Chapter 4 Numerical results 4.1 Hie
Page 59 and 60:
Linear Weakly nonlinear Strongly no
Page 61 and 62:
is to approximate theξ-derivative
Page 63 and 64:
inary axis (where the eigenvalues o
Page 65 and 66:
2.4 2.2 RK4 AM3 AM4 2 1.8 1.6 |R(z)
Page 67 and 68:
Also, recall that K1=E(η n , u n 1
Page 69 and 70:
• T is the convolution matrix for
Page 71 and 72:
The AM4 scheme used is: η n+1 j V
Page 73 and 74:
0.6 0.4 0.2 η 0 −0.2 40 35 30 25
Page 75 and 76:
0.6 0.5 0.4 0.3 η 0.2 0.1 0 −0.1
Page 77 and 78:
where c 1 and c 2 are two functions
Page 79 and 80:
0.5 0.4 0.3 0.2 η 0.1 0 −0.1 −
Page 81 and 82:
0.5 0.4 0.3 0.2 η 0.1 0 −0.1 −
Page 83 and 84:
0.5 0.4 0.3 0.2 η 0.1 0 −0.1 −
Page 85 and 86:
0.6 0.5 0.4 0.3 η 0.2 0.1 0 −0.1
Page 87 and 88:
0.1 0.09 0.08 0.07 0.06 0.05 0.04 0
Page 89 and 90:
−η 0.1 0.08 0.06 0.04 0.02 0 60
Page 91 and 92:
Conclusions and future work In the
Page 93 and 94:
Appendix A Approximation for the ho
Page 95 and 96:
thenω 0 satisfies the Neumann prob
Page 97 and 98:
The inverseT ofT has a symbol−i t
Page 99 and 100:
Setξ=πξ/l. In the new coordinate
Page 101 and 102:
Takingξ-derivatives, φ ξ (ξ,ζ)
Page 103 and 104:
Bibliography [1] Artiles, W. & Nach
Page 105 and 106:
[16] Koop, C. G. & Butler, G., 1981
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