Rape and Gang Rape

but also for marine operational reasons. Although this phenomenon has been intensively stud-ied, some issues still remain incomplete, for example, how are the waves generated and radiated in the LS, especially with a complex 3D bathymetry? What factors have the most significant influence on the generation and propagation of internal waves in this region? How do the waves evolve above the wide shoaling continental slope-shelf after taking shape in the deep water?

And what role does the 3D topography and rotation play in such a process?

This dissertation aims to, basically by means of numerical modelling, explore characteristics of internal wave dynamics in the northern SCS which, specifically, consists of a complete under-standing of:

• 3D generation processes and mechanisms of internal waves at LS, with a highlight of 3D modeling and the generated wave structures of multiple modes.

• the role played by various factors that determine the resultant wave fields; such factors include western ridge interference, asymmetric barotropic tides, rotational effects, etc.

• the wave shoaling process over the wide northern SCS slope and shelf that involves the evolution of both a first mode and a second mode ISW and addresses the effects of 3D topography and rotation.

To achieve this seemingly ambitious goal, the 3D Massachusetts Institute of Technology gen-eral circulation model (MITgcm), which features fully nonlinear, nonhydrostatic capacities, serves as the main tool to accomplish the numerical experiments. Model results are supported and corroborated by the classic weakly nonlinear theories, SAR imagery, and limited in-situ measurements.

1.4 Thesis outline

The structure of this dissertation is arranged as follows:

• Chapter 2 first looks back at the linear internal wave and internal tide generation problem.

Nondimensional parameters that modulate the generation regimes and characteristics are listed and interpreted. Second, classic KdV theory and fully nonlinear theory of ISWs are briefly summarized. Third, the progress of internal wave dynamics in the northern SCS is reviewed.

• Chapter 3 begins with a briefing of MITgcm, followed by an introduction of SAR.

• Chapter 4 is the core section of the dissertation, viz., 3D modelling of ISW generation mechanism and process near the LS. Model setup, results, and some sensitivity experi-ments are presented, and this part serves as a foundation and starting point for the next few chapters.

• Chapter 5 focuses on a particular phenomenon that is delineated in Chapter 4: a specific structure of ’first mode ISW followed by a second mode ISW, on which some short first mode IWs ride’. A series of SAR images are analyzed and compared to the model results, and a robust consistency is reached.

1.4. THESIS OUTLINE

• Chapter 6 emphasizes the role that played by the irregular and multi-harmonic barotropic tidal constituents in the LS and the resultant altering of ISW field. An alternation of A and B ISWs (as will be defined later) and a mutual transition are simulated, with the verification from both in-situ measurements and analytical linear formulations.

• Chapters 7 and 8 form the other key point of the dissertation: ISW shoaling process over the northern SCS slope and shelf. Chapter 7 tentatively studies the evolution of a second mode ISW over a two-dimensional (2D) shoaling slope that is typical of the northern SCS bathymetry, whereas in Chapter 8 experiments of a first mode ISW over realistic 3D bottom are performed, with emphases on the rotational and topographic effects on the shoaling wave profiles.

• Chapter 9 presents the summary and conclusions.

1.4. THESIS OUTLINE

Chapter 2

Theoretical background

2.1 Linear internal wave theory

Internal waves can be treated as a perturbation of a background static state that only varies vertically,

p= p₀(z) + p⁰(x, y, z,t) (2.1)

ρ = ρ₀(z) + ρ⁰(x, y, z,t) (2.2)

where p and ρ are pressure and density, respectively. The fields with a prime denote the motion of internal wave. The background fields p0(z), ρ₀(z) satisfy the hydrostatic balance:

d p₀

dz = −ρ₀g (2.3)

where g is the acceleration due to gravity.

The linearized Euler equations with the Boussinesq approximation in a continuously stratified fluid read

where u = (u, v, w) is the (perturbed) velocity field; ρ∗is a constant reference quantity; b is the buoyancy defined as

b= −gρ⁰/ρ∗ (2.5)

f is the Coriolis frequency, and N is the buoyancy frequency defined as

2.1. LINEAR INTERNAL WAVE THEORY

N²(z) = −g ρ∗

dρ0

dz (2.6)

The five equations of (2.4) can be reduced to a single equation for w which reads

∂²

∂ t²∇²w+ N²∇²_Hw+ f²∂²w

∂ z² = 0 (2.7)

where the Laplace operator ∇²and its horizontal component ∇²_H read

∇²= ∂²

An energy equation can be derived by manipulating equations (2.4):

1 2ρ∗

∂

∂ t[u²+ v²+ w²+ b²/N²] + u · ∇p⁰= 0 (2.9) The energy density E, which is the sum of kinetic and potential energy, is

E= 1

2ρ∗[u²+ v²+ w²+ b²/N²] (2.10) The term u · ∇p⁰= ∇ · (up⁰) in equation (2.9) denotes the divergence of the energy flux. Upon integrating equation (2.9) over a certain volume, it can be shown that the temporal variation of the total energy (kinetic and potential) is balanced by the fluxes through coming in/out of the faces of the volume.

Supposing the wave is temporally sinusoidal with a frequency ω, i.e.,

w(x, y, z,t) = ¯w(x, y, z,t)exp(−iωt) (2.11) Substituting it into equation (2.7), the following equation can be derived

∇²_hw¯−ω²− f² N²− ω²

∂²w¯

∂ z² = 0 (2.12)

Without loss of generality, taking ∂ /∂ y = 0, equation (2.12) can be written as

¯

w_xx−ω²− f²

N²− ω²w¯_zz= 0 (2.13)

The dispersion relation can be obtained by substituting ¯w∼ exp i(kx + mz) into (2.13),

2.1. LINEAR INTERNAL WAVE THEORY

ω²=N²

K²k²+ f²

K²m² (2.14)

where K =√

k²+ m²is the length of the wavevector. The above expression can be formulated in polar coordinates, assuming k = (k, m) = K(cosθ , sinθ ), where θ is the angle between the wavevector and its horizontal component. Then 2.14 becomes

ω²= N²cos²θ + f²sin²θ (2.15)

An important conclusion follows from the linear dispersion relation of internal waves, that is, the phase velocity cp=^ω_k is perpendicular to the group velocity cg=^{∂ ω}_{∂ k}, i.e.,

cp· cg= 0 (2.16)

Equation (2.13) can be solved in two approaches: the method of characteristics and the method of vertical modes. The latter method requires a surface and a bottom boundary. For the latter, assume ¯w= W (z)exp ikx, and substitute it into (2.13), it gives

Wzz+ k²N²(z) − ω²

ω²− f² W= 0 (2.17a)

W= 0 z= 0, −H (2.17b)

This is a Sturm-Liouville problem. Its solution consists of a series of eigenvalues kn and the corresponding eigenfunctions Wn. For some special profiles of N(z), Wn can be obtained ana-lytically, otherwise the equation has to be solved numerically.

After obtaining W , the other variables, u, v, p, and b can be expressed similarly, following equations (2.4),

A simple and special case lies in the assumption of uniform stratification, i.e., N = const. In this case the solution of (2.17) is obvious and it takes the simple form of

W_n= sin(nπz

H ) n= 1, 2, 3, · · · (2.19)

which is independent of ω with N = const. The dispersion relation is

k_n= ±nπ

H (ω²− f²

N²− ω²)^1/2 n= 1, 2, 3, · · · (2.20) which can be rewritten as