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

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model regarding this property. The work initiated here points out some lines of research. We intend to use the strongly nonlinear model and the weakly nonlinear higher order model to study the interaction of large amplitude internal waves with multiscale topography profiles. The refocusing and stabilization of solitary waves for the large levels of nonlinearity allowed by these models is the goal of current research. A difficulty arises to this end, due to the nonlocal dispersive term: it is not easy to find explicit solitary wave solutions for the strongly nonlinear system, and the solitary waves suggested by the weakly one-directional lower-order theory steepen with higher nonlinearity. In [8] the strategy to overcome this difficulty is to use the weakly one-directional solitary wave from the ILW equation as an initial guess for the solitary wave profile. Then find a numerical solitary wave iteratively via the Newton-Raphson method. The profile obtained will serve as initial data for the propagating model. 87
Appendix A Approximation for the horizontal derivatives at the unperturbed interface We want to justify the use of √ β M(ξ) T[ M(˜ξ)u 1tx ] instead of ( √ β M(ξ) T[ M(˜ξ)u 1t ])x in the substitution ofη x in system (2.30), in the case of slowly varying topography. Hence we identify 1 M(ξ) T[ M(˜ξ)u 1t ] as the tangential derivative of the solution of the Laplace equation with Neumann conditions defined in the auxiliary problem ⎧ ⎪⎨ ⎪⎩ Φ xx +Φ zz = 0, on− h 2 L + h 2 L h(εx)≤z≤0, Φ z = u 1t , at z=0, Φ z −ε h 2 L h′ (εx)Φ x = 0, at z=− h 2 L + h 2 L h(εx), (A.1) 88
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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
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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: Conclusions and future work In the
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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