Appendix: Discussion of approximations

1. 2D geometry. Topographic reliefs such as Stellwagen Bank or the New England Shelf have finite extension and present curved fronts. Hence, 3D effects cannot be ruled out a priori. Same can be said of the shoaling region. A model of the kind discussed above that were to include 3D effects would require a domain roughly 50 Km x 50 Km x 90 m. Even assuming a coarse grid in the horizontal, apt at capturing at least the bore, no less that 512x512x100 points should be used. This represents a very large problem, even for the present generation supercomputers. Also, as we will discuss below, 3D effects are mostly likely to be important at much smaller scales, the inclusion of which would push the computational grid beyond what is currently feasible. Therefore, before attempting a 3D model, it seems reasonable to evaluate a 2D model against experimental data. Also, SAR images of the areas under consideration show that the curvature of the wave fronts is small.
2. Boussinesq and rigid lid. In a fluid governed by the the Euler equations, with density $\rho(x,z,t)$ and pressure $p(x,z,t)$, vorticity is generated by the baroclinic torque
   

   $${\nabla\rho\times\nabla p\over \rho^2}.$$ (7)

   
   
   It is straightforward to write the pressure as
   

   $$p=-\int_{\eta(x,t)}^z\,g\rho(x,s,t)\,ds+p'(x,z,t),$$ (8)

   
   
   where the first term is the hydrostatic component of the pressure, $\eta$ being the position of the free surface, while the last is due to the requirement of keeping the flow divergenceless. With this in mind, the baroclinic torque can be written as
   

   $${\nabla\rho\times\nabla p\over \rho^2}=\left[{\partial\rho\o... ...artial x}ds+{\partial p'\over\partial x}\right)\right]/\rho^2.$$ (9)

   
   
   A simple scaling argument shows that, for a flow characterized by a stratification extending over a finite length $H$ (that is $\partial\rho/\partial z\simeq \Delta\rho/H$), and such that $(\Delta\rho/\rho)(D/H)\ll 1$, where $D$ is the total depth, the dominant terms in eq. 9 are $(g\partial\rho/\partial x)/\rho$ and $(g\partial\eta/\partial x)(\partial\rho/\partial z)/\rho$. In the proposed Boussinesq approximation, only the former term is retained (albeit the latter could be in principle included). If $A$ is an estimate of the amplitude of the free surface fluctuation, the latter can be neglected if $A/H\ll 1$. Since in Massachusetts Bay $H=O(10\,\rm {m})$, $D=O(100\,\rm {m})$,and $\Delta\rho/\rho \sim 10^{-3}$ the first condition is surely met, while it is reasonable to assume that $A< 1\, \rm {m}$, while $H> 10\, \rm {m}$, so that the contribution of the free surface should be at least an order of magnitude smaller than the contribution due to the buoyancy force. The simplification provided by the Boussinesq approximation is considerable, essentially because it filters out the acoustic modes from the equations. Similarly, a surface whose position and shape is given (rather than calculated) avoids the problem of having to deal with surface gravity waves. Also, note that although the upper surface is rigid, the choice of boundary conditions used in the model implies that the the effects of advection by an unsteady tide are captured quite naturally.

Figure 1: Position (top panel) and instrumentation (lower panel) of the moorings in Massachusetts Bay during the 1998 USGS/WHOI Massachusetts Bay Internal Wave Experiment (MBIWE).

Figure 2: Environmental conditions during the MBIWE98 experiment. (a) Low-pass filtered density at mooring A; (b) Low pass filtered density at mooring C; (c) Bottom pressure at mooring C.

Figure 3: Across-slope currents (a) and temperature (b) during the passage of a mode 1 solibore at mooring C.

Figure 4: Across-slope currents (a) and temperature (b) during the passage of a modified? solibore. Note that the modal structure in the velocity current is less clear, while the temperature field shows clear evidence of overturning.

Figure 5: Across-slope currents (a) and temperature (b) during the passage of a critical solibore at mooring C. Note the reversal of the mid depth currents shortly after the passage of the initial pulse and the large warming extending to the bottom just before the sudden recover of stratification.

Figure 6: Geometry of the channel used for the experiments.

Figure 7: Density field during the first tidal cycle, non rotating case.

Figure 7: Cont'd.

Figure 8: Density field during the second tidal cycle, non rotating case.

Figure 8: Cont'd.

Figure 9: Density field during the first part of the third tidal cycle, showing the shoaling of the bore generated during the second cycle. Non rotating case.

Figure 10: Density field during the flloding phase of the first tidal cycle. Rotating case.

Figure 10: Cont'd.