**G. I. Shapiro ^{1}, M. E. Inall^{2} and T. J. Sherwin^{2}**

^{1}P. P. Shirshov Institute of Oceanology, Moscow
^{2}University of Wales, Bangor, UK

It is well known that progressive non-linear surface waves generate a residual mass transport in the direction of their propagation (the Stokes' drift). This paper considers the similar problem for tidally generated internal waves in the case of approximately two-layer stratification. Such waves are an ubiquitous feature of the world's continental margins and are thought to play an important role in shelf sea/open ocean exchange. Theoretical expressions are consistent with the Kortweg de Vries wave equation and derived specifically to allow for comparison to be made with fixed point measurements. The theory expresses mass transports in terms of a time-varying interface displacement and can be applied to other forms of data, such as those from thermistor chains. Provided the barotropic flow is small compared to the phase speed of the waves, velocities and transports can be estimated from the interface displacement without a direct knowledge of the non-linear phase speed. Careful analysis of moored acoustic doppler current profiler data from a site 5 km shoreward of the edge of the Malin Shelf, west of Scotland, gives a value of 1.6 m2 s-1 for the mass transport in each layer. Presented theory account for approximately 70% of the observed transport.

It is generally accepted that in the ocean shelf, break regions are particularly dynamic places (see e.g. Huthnance, 1995). They attract large-scale phenomena such as ocean currents, their attendant eddies and frictional boundary layers and upwelling filaments are driven from them. Their sloping topography makes them prone to wave activity, and internal waves of all frequencies (from the order of minutes to the order of days) can be found there. Despite the degree of dynamic activity, shelf breaks also provide a barrier to exchange between shelf seas and the ocean due to the rotational constraint on geostrophic flows to follow contours of f/H. However, a contribution to ocean-shelf exchange is made by all ageostrophic processes including non-linear internal waves (NIWs), the subject of this investigation. Indeed non-linearly generated mean currents (e.g. averaged over a tidal cycle) may be important for long-term water fluxes between shelves and deep ocean, and the movement of fish larvae and sea-floor sediments (e.g. Lamb, 1997; Friedrichs and Wright, 1995; Shanks and Wright, 1987; Huthnance, 1981).

The NIWs of interest are large amplitude, short wavelength internal waves which are often found on continental shelves, within about 50 km of the break, propagating toward the coast in packets. The maximum number of waves in a packet is governed by the balance between dispersion, nonlinearity and dissipation (Barenblatt et al., 1985). Individual waves may have a length scale of order 500 m, a phase speed of order 1 m s-1 and period of order 10 minutes; within a packet the waves tend to be rank ordered with the largest (and fastest) wave leading.

A comprehensive discussion of observations reported worldwide is inappropriate here but see, for example, Ostrovsky and Stepanyants (1989). However, along the Atlantic edge of the European Shelf they have been observed in several locations; on La Chapelle Bank in the Celtic Sea waves with an amplitude of 50 m and propagation speed of 70 cm s-1 have been observed (see e.g. Pingree and Mardell, 1985, Pingree and New, 1995, and many other papers by these authors). In the Strait of Gibraltar waves up to 60 m deep have been recorded, (Watson and Robinson, 1990), whilst on the Portuguese Shelf they have an amplitude approaching 50 m and phase speed of 50 cm s-1 (Jeans, 1997).

Internal waves may significantly contribute to cross-shelf exchange. Kinder (1984) estimated the average mass transport from observations of NIWs in the strait to be 3.7 m2 s-1. However, there appear to be no direct estimates of the contribution of NIWs to shelf-edge exchange through mass transport. Huthnance (1995) suggests a value for a surface layer of order 1 m2 s-1 which makes NIWs comparable to many of the other processes mentioned in the first paragraph, in particular Ekman drainage of the slope current and upwelling filaments. It is possible that NIWs make an important contribution to mixing and exchange at the shelf edge but that, whilst they are relatively easy to measure, they are very difficult to model in general circulation models, requiring further work on the theoretical understanding of the waves.

Present work focuses on mass transport onto the shelf. Estimation of
mass transport by NIWs using nonlinear KdV type theory is developed and
compared with observations. Given that, in general, most observations are
made by in situ Eulerian instruments, the theory avoids terms involving
the horizontal spatial derivative, which is not easy to measure in the
sea.

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