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

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forηto propagate in only one direction. Moreover, c 2 (k)=̂η 0 (k)= ˆη(k, 0). The exact solution is ⎧ ⎪⎨ ̂η 0 (0), k=0, ˆη(k, t)= ⎪⎩ ̂η 0 (k) exp ( −iω(k)t ) , k0. Although the amplitude of each mode is preserved as time advances, each component of the wavetrain travels with its own phase velocityω(k)/k. As a result, an initial Gaussian shape will disperse into an oscillatory train as shown in the numerical experiments. The exact solution can be employed to test the numerical scheme precision and stability properties. We consider the following Gaussian function as the initial perturbation of the interface, η 0 (ξ)=0.5e −a(ξ−4π)2 /128 with a=50. The LFM parameters arel=16π,ρ 1 = 1,ρ 2 = 2,β=0.05,α=0. The numerical parameters are N= 256,∆ξ=2l/N= 0.3927,∆t=∆ξ=0.3927 . See Fig. 4.7 were the numerical solution for t=78.5398 is depicted together with the initial condition. The numerical solution shows very good agreement with the exact solution. The same experiment with a finer grid, N= 512,∆ξ=2l/N= 0.19635,∆t=∆ξ=0.19635 is depicted in Fig. 4.8. The result is the same as the one obtained in the first experiment, which indicates convergence of the numerical method. 73
0.5 0.4 0.3 0.2 η 0.1 0 −0.1 −0.2 0 20 40 60 80 100 120 ξ Figure 4.7: Pulse propagating over a flat bottom in the linear dispersive regime. Dotted line: numerical solution for the LFM using RK4 for t = 78.5398 and N= 256, dashed line: initial condition, solid line: exact solution. 0.5 0.4 0.3 0.2 η 0.1 0 −0.1 −0.2 0 20 40 60 80 100 120 ξ Figure 4.8: Pulse propagating over a flat bottom in the linear dispersive regime. Dotted line: numerical solution for the LFM using RK4 for t = 78.5398 and N= 512, dashed line: initial condition, solid line: exact solution. 74
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INSTITUTO NACIONAL DE MATEMÁTICA P
Page 3 and 4:
Acknowledgements First, I would lik
Page 5 and 6:
4.1 Hierarchy of one-dimensional mo
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Resumo É obtido um modelo reduzido
Page 9 and 10:
when the steepening of a given wave
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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: where c 1 and c 2 are two functions
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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