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

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The pressure p 2 (x,η, t) should be non-dimensionalized in the same fashion as p 1 , that is, p 2 =ρ 1 U 2 0 ˜p 2. Define P(x, t)= p 2 ( x,η(x, t), t ) . Then p 1 = P(x, t)−(z−η)+O(β), which immediately yields p 1 x=P x (x, t)+η x + O(β). By averaging we get p 1 x= 1 ∫1 η 1 η P x (x, t) dz+η x + O(β), = P x (x, t)+η x + O(β), ( ( ) ) = p 2 x,η(x, t), t x + O(β). (2.11) x+η An approximation for P x will be obtained later from the Euler equations for the lower fluid layer. We now approximate the mean squared horizontal velocity in terms of u 1 andη. In order to express u 1· u 1 as a function of u 1 andη, it should be pointed out that the irrotational condition in non-dimensional variables is U 0 h 1 u 1z = √ β U 0 L w 1x, 15
i. e. u 1z =βw 1x . Hence ( ) u (0) 1 = 0 and as expected for shallow water flows u(0) z 1 is independent from z: u (0) 1 = u (0) (x, t). (2.12) 1 We now correct this first order approximation. By using u 1 = u (0) 1 +βu (1) 1 + O(β 2 ) (2.13) and Eq. (2.12) it is straightforward that ∫ 1 η ∫1 u 2 1 dz= u 2 1 = u(0) 2 1 + 2βu (1) 1 u(0) 1 + O(β 2 ), η u (0) 1 ∫1 2 dz+2β η u (1) 1 u(0) 1 dz+O(β 2 ), and η 1 u 1· u 1 = u (0) 2 1 (1−η)+2η1 βu (0) 1 u(1) 1 + O(β 2 ), so that Also from Eq. (2.13), u 1· u 1 = u (0) 1 · u (0) 1 + 2βu (0) 1 u(1) 1 + O(β 2 ). (2.14) ∫1 1 u 1 dz=u (0) 1 +βu (1) 1 + O(β 2 ), η 1 η u 1 = u (0) 1 +βu (1) 1 + O(β 2 ), u 1· u 1 = u (0) 1 · u (0) 1 + 2βu (0) 1 u(1) 1 + O(β 2 ). (2.15) 16
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: Substitution of w 1 into Eq. (2.2)
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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