Communications of Computational Fluid Dynamics, No.3, pp.167-177, 2007 www.cfdchina.com

Numerically simulation of nonlinear internal wave

propagation in the strait of Gibraltar

Liqun Xu zhaoting(Institute of Physical Oceanography, OUC, qingdao, 266071)

（中国海洋大学物理海洋研究所）Abstract:

A rotation modified KP equation derived in Grimshaw (1985) is applied for the simulation of internal wave

propagation in the strait of Gibraltar. Special attention is put on the effects of rotation and lateral boundary.

Numerical results show that rotation and lateral boundary effects can partly explain the asymmetry characters of

internal wave along the central axis of the strait in available satellite images.

Key word: internal solitary wave, KP model, rotation, lateral boundary

1. Introduction

Internal waves are found everywhere in the stratified oceans of the world. They arise due to disturbances induced in the essentially stable ocean stratification by wind fluctuations, variation in atmospheric pressure fields, heavy rain, turbulence, wave action or flow over topography. In particular, large amplitude internal waves (which may evolve into isolated waves of fixed form called solitons) are commonly produced at continental shelf breaks (and other local topographic features) by tidal forcing. These waves have amplitudes of order tens of meters and orbital velocities of order tens of cm/s and therefore significantly affect the local stratification and current shear.

There is considerable mass of data on internal bores and high frequency internal solitary waves in the Gibraltar strait (Lacombe et al., 1964; Ziegenbein, 1969, 1970; Kinder, 1984; La Violette &Arnone, 1988; Armi & Farmer, 1988; Watson & Robinson 1990; Wesson & Gregg, 1994). These observation indicate that eastward traveling, large amplitude internal bores/internal solitary wave trains with ranges from crest to trough of more than 100m are formed downstream of the Camarinal Sill during tidal outflow. And Brandt (1997) used a two layer, nonhydrostatic Boussinesq model for the simulation of the generation and propagation process. Also a nonhydrostatic primitive equation model was used for internal waves in the strait Morrove (2000). Both model result similar features to observation data.

With advent of remote sensing techniques, satellite/ship-board radar images of internal waves in this strait show a complex two dimensional character. Watson &Robinson (1990) developed a linear ray model for the simulation of the internal waves propagation in Gibraltar Strait, and comparison was made with the available ship/board radar images. They conclude that current shear play the most important role in determine the propagation character of these internal waves.

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Pierini (1989) applied the KP model to the simulation of internal waves propagation into the Alboran Sea, which founded a firm underpinnings for the applicability of KP type equations in the simulation of two dimensional characters of internal waves propagation. Recently, Boussinesq type equation has also developed and used for the internal wave evolution in two dimensional L. &L (2000).

However, the models mentioned above ignore the influence of rotation, especially with lateral boundary presented. In laboratory experiments conducted in a rotating channel with stratified flow, Maxworthy (1983) first pointed out that rotation plays an important role in the dynamics of internal solitary wave evolution. It causes the amplitude varies exponentially across the channel and, the wave crest curved backward due to the nonlinear character. After that, Renouard et al. conducted similar experiment. Apart from confirming the observation result of Maxworthy, some quantitative information was obtained. And Katsis & Akylas numerically produced the experiment result of Renouard et al with a rotation modified KP model. Similarly, Tommoson (1990) developed a two layer Boussinesq type equation, which was used to study the evolution of internal waves in a channel.

Although the KP type equation has been derived and extensively studied theoretically, its practical use, unlike KdV equation, is very limited. In the present paper, the rotation modified KP equation for internal wave derived in Gimshaw (1985) is applied for the simulation of internal wave evolution in Gibraltar Strait, and qualitative comparison with available remote sensing data is made. Part 2 presents the model equation and numerical scheme used. The model results and analysis are in part 3. The final part is the conclusion.

2. Rotation modified KP equation 2.1 Model equation and boundary conditionTo consider the effects of rotation and lateral boundary on internal wave propagation, a

rotation modified KP equation derived in Grimshaw (1985) has been used. The model equation is:

(1)

Where is the amplitude function, , , and are separately linear phase velocity,

nonlinear coefficient, and dispersion coefficient, is the Coriolis parameter. For simplicity, in a

two layer fluid the coefficients can be calculated as follows:

, , (2)

In which, is the reduced gravity, , is the upper and lower layer thickness,

is the average density of two layers, and is the density difference of the upper and lower layer.

For internal wave propagation in a channel, the lateral boundary conditions are:

, at (3)

In solving numerically the rotation modified KP equation subjected to the boundary

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conditions (3), it is convenient to make a transformation of variable (Katsis & Akylas, 1987),

(4)

So that, in terms of , the boundary conditions of (3) can be transformed into a simple form:

, at (5)

With this transformation, the evolution equation (1) becomes:

(6)

Equation (6) together with the boundary conditions (5) will be numerical integrated. 2.2 Numerical scheme

The method used to numerically solve the KP model equation is an extension of the method

used by Liu A.K. to solve the variable coefficient KDV equation. Here, due to two-direction

dependence of KP equation, a splitting operator method will be used. So the model equation will

be splitted in to the following coupled equation.

(7)

(8)

Equation (7) is solved using the method of pseudo-spectral method. The x-derivatives are

evaluated using FFT and the equation is then integrated forward in time using leap-frog method.

The second equation is integrated using a Crank-Nicolson type scheme, and this equation is then

approximated by

(9)

where

In these equations, , refer to the x and y space steps respectively, and is the time

step. In the following numerical experiments, , , .3. Numerical experiments on internal waves evolution

3.1 Initial condition and simulation regionFor simplicity, a constant depth with two layers is used in the calculation, and some

parameters are showed in table 1. To model the evolution of nonlinear internal wave, an initial

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condition similar to Tommoson (1990) is used, i.e. a bulge like depression is given by,

(10)

And some parameters used in the numerical simulation are presented in table 1. In figure 1, a SAR image of internal wave train together with the topography contour is presented, the wave exhibits a significant asymmetry across the strait.

Table 1. Parameters used in the numerical experiments

Upper layer depth 200m

Lower layer depth 1000m

Amplitude of initial disturbance 50m

Parameters in initial conditions 2.0km

Parameters in initial conditions 12.0km

Parameters in initial conditions 0

Internal wave distribution map has been showed in figure 2. From this map, a clear phenomenon is that most waves toward right hand side when propagation out the Gibraltar strait. Several explanations have been considered for this. For example, Apel et al (2000) concluded that internal waves are refracted by a combination of pycnocline and total water depths, and when their wavelengths (which are near 1000–2000 m) become of the order of the depth, refraction acts strongly. In addition, wave phase speeds are 1–2 m/s and thus solitons can be advected and distorted by tidal and coastal currents. All these processes govern the wavefront shapes.

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Figure 1. ERS-1 SAR image acquired on 24 March 1996 at 2239 UTC (orbit 24583, frame 0711). The image

shows a newly formed soliton packet propagating through the Strait of Gibraltar. Five oscillations are visible. The

lead crest is approximately 30 km from the generation point at the Camarinal Sill. Original image ©ESA 1996.

(www.internalwaveatlas.com) The color broken lines are the topography contour in the Gibraltar strait and

Alboran Sea. (Smith and Sandwell, 1997).

Figure 2. Internal wave distribution map in the Gibraltar Strait and Alboran Sea.

(www.internalwaveatlas.com)

3.2 numerical results and analysis

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(a)

(b)Figure 3. Two simulation results of internal wave propagation and disintegration in the strait, (a) result at

2.9h, (b) results at 6.5h

Figure 3 shows two pictures of the simulation results at different time. The presented images are the gradient of upper layer horizontal velocity, which is calculated approximately form the

amplitude by . It is clear that a disintegration process has been developed during the

wave propagation from 2.9h to 6.5h, which could be predicted by KdV theory (C. Y. Lee and R. C. Bearsley). However, compared with previous simulation results in Pierini (1989), the incorporation of rotation effect has cause an asymmetry feature of the wave across the strait. The amplitude varies exponentially across the strait, and due to nonlinear effect, the smaller the amplitude, the slower the phase velocity. So the waves curve back toward the left side of the channel. Meanwhile, at 6.5h the leading wave has propagated about 40km, which is just about at the mouth of the strait at the eastern part.

Compared with the internal wave train showed in figure 1, some qualitative familiar features can be found, for example, the wave train in figure 1 has also a asymmetry structure across the central axis of the strait, and a little bending back of the wave crest at the left side. However, some difference should be mentioned here, i.e. the wave train in figure 1 also bend back at the right hand side, even though it is not so significant compared with the other side. This difference may be attributed to the friction effect of the lateral boundary and a shallow topography towards the two side of the strait which could not be considered with the model used here.

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Figure 4, (a) Horizontal velocity gradient at 8.4h, and (b) Internal bore just about to exit the Strait of

Gibraltar. From an (ERS-2) image, 1102 UTC 18 Mar 1998. Photo courtesy of the (ESA).

The velocity gradient at 8.4h, when the waves have propagated out the strait, is presented in figure 4. In figure 4 (b), a SAR image on internal wave is presented, in which, an internal bore has been captured. Without the lateral boundary influence, the waves continue evolution forward, and with two sides bending back. Even though it is not significant like the wave in the SAR image, the asymmetry feature is kept during this process, which causes a tendency for the waves to propagate toward the rightward direction. Figure 5 is an astronaut photograph taken from the Space Shuttle on 11 October 1984 which captured an internal wave train is showed. In this picture, the tendency for the wave train to run toward the right hand side direction is clear. Although several other processes may cause such feature, such as circulation in the Alboran Sea, the influence of the rotation effect with lateral boundary presented in the strait before it run out can be thought to be a possible candidate.

Figure 5. A satellite image of internal wave trains just run out from the Gibraltar Strait

4. ConclusionIn this paper, a rotation modified KP type equation has been used for the simulation of

internal wave propagation in the strait of Gibraltar. Particular attention is on the rotation effect

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with lateral boundary presented. From the calculation results and preliminary comparison with available satellite images, a possible explanation can be found if rotation is considered.

Internal waves are ubiquitous features in the ocean; especially large amplitude internal solitary waves (or ISW trains) have been captured by SAR or other remote sensing techniques. These waves belong to mesoscale phenomenon in the ocean, and can propagate hundreds of kilometers with little energy dissipation. For such a character, rotation effect definitely plays an important role in its evolution process.

Even though a lot of work has been down on ISW evolution, most of them have been focus on the one dimensional character, such as KdV theory, which has been widely used for ISW simulation. With the available data of satellite image, a horizontal two dimensional feature has been presented. However, corresponding theoretical model is still very limited. And duo to its nonlinear and nonhydrostatic character, numerical simulation with primitive equation is time consuming and impractical.

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