Introduction

  

Figure: The figure depicts a typical internal wave with oceanic characteristics. The streamline interface DEF marks the boundary between the denser lower fluid region and the lighter upper fluid domain whilst the deformation of the free-surface, ABC, is very small as $\rho _2 \sim \rho _1$. The outskirts flow velocity, c, in the co-ordinate system moving with the wave is also (minus) the wave velocity in the system where the outskirts fluid is at rest.

Oceanic internal waves arise principally because the density of the ocean water is not constant. To a fair approximation we may think of an upper, warmer layer of density, $\rho_2$, and undisturbed depth h₂, above a deeper, colder and more saline layer of density $\rho_1 \gtrsim \rho_2)$ and depth, h₁, as depicted in Fig. 1. These layers meet at the ``thermocline'' or ``pycnocline'' which is normally marked by a sharp change in water temperature with depth. Internal waves have amplitudes that distort the thermocline and the surface profile, though, in oceanic situations the latter deformation is much smaller and often ignored in theories that assume a ``rigid-lid'' (i.e. flat top surface) boundary condition. In the deeper marginal seas where internal waves are observed, h₁ is normally much larger than h₂. This implies negative amplitude at the thermocline and, consequently, negative total mass (Evans and Ford 1996a).

Though not as destructive as ``tsunamis'', the larger oceanic internal waves carry a considerable amount of energy and the associated current flows can be strong enough to be an important factor in the design of coastal oil platforms and similar sea structures. Hence it is important to estimate the ``worst scenario'' i.e. the question of ``how large$\ldots$?'' can various properties be is of paramount importance. This puts the emphasis on estimating the size and associated currents, vorticity etc. of the maximal internal wave in any given oceanic locality. Of course, this presupposes the maximal internal wave form can be generated by the prevailing, natural formation forces. This is by no means clear and, hence, there is also the allied need to understand the mechanism(s) of formation of oceanic internal waves. Some time ago, Osborne and Burch 1980; Osborne 1990 observed very large internal waves in the tropical Andaman sea. These were prevailingly easterly moving and tended to recur after the semi-diurnal period of 12 hrs 26 mins which indicates convincingly that tidal forces play a key part in their formation mechanism. Unlike normal tides, there is no evidence of any correlation between internal wave size and the phases of the Moon. If, indeed, the formative forces are primarily tidal in origin, then the waves should be highly predictable with virtually no randomness. This would suggest that the extremal wave is limited by the magnitude of the prevailing tidal forces rather than by the maximal stable internal wave solution corresponding to the parameters at that location.

However, the latter is certainly an upper bound on the extremal form at any given location and is also, mathematically, a well-defined problem. Accordingly much attention has been devoted to this task. The nature of the solutions sought invalidate the small amplitude/long wavelength conditions required for an accurate description via the KdV theory. Consequently, integral equation techniques whose numerical solutions should be exact within potential flow have

  

Figure: The profiles of the range of internal wave solutions for h₁/h₂=3, $\rho _2/\rho _1=0.997$ (note the surface is at +1 and the bottom at -3 on the vertical scale). The waves shown correspond to the amplitudes, -n₁(0)/h₂ = 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.925, 0.95, 0.975 and 0.99. The horizontal line (- - -) marks the conjugate flow ``maximum'' amplitude of -0.998498h₂.

been utilised in order to gain insight on these extremal forms. Assuming rigid-lid boundary conditions, Turner and Vandenbroeck (1988) computed extremal forms of internal waves for parameters, h₂/h₁= 1, $\rho_2/\rho_1= 0.1$ (which are decidedly not typically oceanic) and concluded that the limiting form was one of finite amplitude and very large width - the middle part being essentially predictable as a uniform ``2-layer-flow'' domain that is ``conjugate'' to the outskirts flow where the fluid flow velocities in both layers is c. Evans and Ford (1996b) utilised a different integral equation form to investigate the limiting internal wave with oceanic parameters, h₂/h₁=1/3 and $\rho _2/\rho _1=0.997$. Their results are shown in Figure 2. Again it was seen that, at the largest amplitudes, the `width' appears to diverge and, moreover, the largest solution amplitude, $\eta_1\sim -0.99h_2$, agreed precisely with the predicted value from ``conjugate flow'' considerations viz. -0.9985h₂ (shown as a horizontal dotted line in the figure).