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

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2.25 2.2 2.15 ω/k 2.1 2.05 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 k Figure 3.2: Detail from Fig. 3.1. 51
Chapter 4 Numerical results 4.1 Hierarchy of one-dimensional models For numerical implementations, we first normalize the shallow water velocity c 2 0 = ( ρ2 ρ 1 − 1 ) of system (2.27) by settingη=η ∗ , u 1 = c 0 u ∗ 1 , t= t∗ c 0 . Dropping the asterisks, the following Strongly Nonlinear Corrugated Bottom Model (SNCM) is obtained, ⎧ η t − 1 [ ] (1−η)u1 ξ ⎪⎨ M(ξ) = 0, ⎪⎩ u 1t + 1 M(ξ) u 1 u 1ξ − 1 M(ξ) η ξ= √ β ρ 2 ρ 1 1 M(ξ) T [0,2l] [ (1−η)u1 ] ξt . (4.1) We will work on the periodic domainξ ∈ Π[0, 2l], so that instead of the operatorT its periodic versionT [0,2l] appears above. See Appendix C for the definition ofT [0,2l] . The choice of a computational periodic domain was made to be able to use spectral methods. To avoid the influence of the boundaries on the evolution of the perturbationηinteracting with the bottom profile we added two 52
Page 1 and 2:
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
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: 2.6 2.4 2.2 ω/k 2 1.8 1.6 1.4 1.2
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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