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This study evaluates the performance of several open boundary conditions applied to the Princeton Ocean Model. The focus is on passive open boundary conditions (OBCs) applied to the external mode i.e., conditions that are applied when the mean flow at the open boundary is unknown and the values of the variables must be assumed or extrapolated from the interior solution. Three types of open boundary conditions are tested: a) radiation conditions b) characteristic methods, c) relaxation schemes. The general characteristics of the OBCs conditions is summarized in Table I and those of the numerical experiments in Table II.

Table II

Value	Parameter
Density of sea water	1000 kg.m-3
Coriolis parameter	1.028 10 s
Length of basin	1000 km
Width of basin	500 km
Grid size (DX = DY)	20 km
Bottom friction coefficient	0.0025
Lateral friction	200 ms

Two sets of experiments are discussed: 1) The barotropic adjustment of an initial perturbation in the sea surface elevation; 2) Forcing by a traveling storm. The first experiment was designed to test the the reflection properties of various OBCs schemes in flows dominated by wave radiation. It consists of the free adjustment of a sea surface displacement in a fluid otherwise at rest. The initial perturbation is a symmetrical mound centered at the middle of the channel, with maximum elevation of 1 m at the southern boundary and linearly decreasing offshore (Fig. 1a). The benchmark for this test was conducted in a closed basin with the same meridional extension, and bottom topography, as the cases with OBCs but with a zonal extension of 10,000 km. The results are summarized in Fig. 1b which shows the time evolution of the sea-surface elevation at an alongshore line 20 km from the southern boundary.

The oceanic adjustment to the initial perturbation is accomplished by the propagation of surface gravity and edge waves, with the continental shelf acting as a wave guide. The initial disturbance splits into two mounds of almost equal amplitude traveling alongshore, a Kelvin wave moving to the right (facing the sea from shore), and a fundamental mode edge wave moving to the left. The phase speed of both wave packets is almost the same and equal to the non-dispersive long wave speed, rotation playing a minor role. As the depression left behind oscillates, new edge waves are generated until approximately hour 6, where most of the initial energy is lost by radiation. Subsequently, the basin adjusts to equilibrium with small oscillations of nearly 2.3 hours period. During the spin-down period, energy losses are mostly related to wave radiation, with bottom and lateral friction playing only negligible roles. The steady state is one of no motion.

The behavior of the different OBCs for this particular experiment is summarized in Fig. 2, a two dimensional version of Fig. 1b, where results of only the most representative cases are displayed. For reference, the left top panel in Fig. 2 shows the time evolution of the sea surface elevation in the benchmark experiment. The tilting of sea surface contours is associated with the phase speed of surface gravity waves. The relaxation schemes MAR and SPO, and the characteristic method HOC behave similarly to the benchmark experiment, allowing a perfect propagation of incoming waves. Among the radiation type OBCs the best performances were those of FLA and GWI, not surprisingly since they use the phase speed of the dominant signal, and SRE which outperforms all the other Orlanski type OBCs. Of the remaining OBCs the worst performance was that of MOI which starts to show reflections as soon as the first wave packets reaches the open boundaries. These reflections are noted in Fig. 2 by the change of slope of the sea surface elevation contours The remaining Orlanski type OBCs, namely SOE, ORI, and ORE (not shown), perform well until the end of the spin-down process and then start to produce spurious reflections. This behavior seems to be associated with the fact that, by the end of the spin-down, the little energy that remains makes difficult the calculation of the appropriate phase speeds.

The second experiment to be discussed investigates the oceanic response to the passage of a tropical, or extra-tropical, cyclone over a coastal region. The objective of this experiment is to test the behavior of the various OBCs schemes in flows dominated by direct wind forcing and wave radiation. The wind stress is generated by a cyclone translating at 8 m.s-1 from the northwestern to the southeastern portion of the channel. The storm has a radius of 100 km and enters the domain at approximately x= 250 km, y= 500 km, leaving the domain 24 hours later at x= 750 km, y= 0. there is no direct wind forcing on the open boundaries. Since there is no known analytical solution to this problem, the cases using OBCs will be compared with an experiment conducted in the extended domain.

Fig. 3 shows contours of sea surface elevation in the benchmark experiment at 12 hours intervals. The oceanic adjustment to the storm forcing is achieved through the generation of surface gravity waves and continental shelf waves. For a positive Coriolis parameter there are high frequency edge waves propagating alongshore in both directions (East-West), and low frequency coastal shelf modes propagating in the positive x-direction. Because of the closed offshore boundary there are also two Kelvin modes propagating with the coast to their right. After 36 hours the center of the storm is near the middle of basin, as indicated by the depression following the wake of the storm. Wind forcing over the shelf area generates continental shelf waves that propagate eastward, the first wave packet reaches the eastern boundary after approximately 50 hours. Alongshore wavelengths, calculated from the model output, vary from approximately 500 km to 600 km, with a phase speed of about 6 m.s. There is good agreement between the observed characteristics of the first wave-packet and the lowest barotropic shelf modes. For a mode-1 wave 550 km long the estimated speed is approximately 5.8 m.s. The dominance of the low-mode shelf restonse to cyclones can be related to the similarity in space and time scales of the forcing and these waves (Tang and Grimshaw, 1995). In our experiment the translational speed of the storm in the direction of propagation of the shelf waves is 5.6 m.s which is comparable with the phase speed of the mode 1 shelf wave and much less than the phase speed of Kelvin edge waves.

The results of the experiments using different OBCs are summarized in Fig. 4 which shows snapshots of sea surface elevation contours at hour 96, immediately after the first shelf wave mode has crossed the eastern boundary. The left side of Fig. 4 shows the OBCs that performed similar to the benchmark experiment, while the right side shows the OBCs with problems. MAR, SPO, and FLA produce results similar to those of the benchmark experiment, although the latter presents some tilting of the contour lines at the southeast corner. Compared to the results of the extended domain, Orlanski's type radiation conditions (only MOI and SOE are shown) show strong reflection, with waves propagating along the eastern boundary. GWI and HOC also exhibit reflections at the same boundary but their deviations from the benchmark experiment are more pronounced at the western open boundary. The difference of colors between the left and right side of Fig. 4 is associated to the spurious changes in mean sea levels produced by the OBCs.

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