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

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0.1 0.09 0.08 0.07 0.06 −η 0.05 0.04 0.03 0.02 0.01 0 0 10 20 30 40 50 60 ξ Figure 4.19: Dashed line: initial condition, dotted line: initial wave recovered at t=51.0509, error=9.8510×10 −4 . 0.1 0.05 −η 0 −0.05 120 100 80 60 40 t 20 0 0 5 10 15 20 25 30 35 40 45 50 ξ Figure 4.20: Fission of a single solitary wave until t=102.1018. 85
Conclusions and future work In the present work, a one-dimensional strongly nonlinear variable coefficient Boussinesq-type model for the evolution of internal waves in a two-layer system is derived. The regime considered is a shallow water configuration for the upper layer and an intermediate depth for the lower layer. The bottom has an arbitrary, not necessarily smooth nor single-valued profile generalizing the flat bottom model derived in [8]. This arbitrary topography is dealt with by performing a conformal mapping as in [24]. In the unidirectional propagation regime the model reduces to an ILW equation when a slowly varying topography is assumed. The adjustment for the periodic wave case and its computational implementation are also performed. We study the interaction of internal waves with periodic bottom profiles and the evolution of approximate solitary wave solutions. The expected qualitative behaviour is captured. A higher-order reduced one-dimensional model is also obtained, though it has not being implemented computationally yet. Both reduced models have dispersion relations that reproduce correctly the limit from the Euler equations for the shallow water (long waves) regime in the upper layer. The higher-order model, by taking into account the nonhydrostatic correction term for the pressure, approximates better the full dispersion relation. It will be interesting to study numerically the behaviour of the weakly nonlinear higher-order 86
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INSTITUTO NACIONAL DE MATEMÁTICA P
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Acknowledgements First, I would lik
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4.1 Hierarchy of one-dimensional mo
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Resumo É obtido um modelo reduzido
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when the steepening of a given wave
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in particular that of solitary wave
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Chapter 2 Derivation of the reduced
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demands that η t + u i η x = w i
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function f (x, z, t), let its assoc
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Substitution of w 1 into Eq. (2.2)
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i. e. u 1z =βw 1x . Hence ( ) u (0
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fluid layer. 2.2 Connecting the upp
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Furthermore, [√ ( √ )] βφt x,
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The problem in conformal coordinate
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problem (2.22). Therefore, P x =−
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assumption was made on the wave amp
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Its linearization around the undist
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c 0 chosen, there will be a right-
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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: −η 0.1 0.08 0.06 0.04 0.02 0 60
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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