Fig. 1. Sketch of the geometry for the two-dimensional flow of undisturbed depth D. The uniform background flow is denoted by U. The functions ρ(x,z,t), ψ(x,z,t) represent surfaces of constant density and streamfunction, respectively. The undisturbed level is indicated by the dashed horizontal line. The elevation of the topography is given by z=h(x) and its maximum height by a. The characteristic length scale of the topography is L. Note that for the case of the solitary waves with vortex cores the uniform flow is reversed and the topography vanishes, i.e. a=0.

Fig. 2. The last term in Eq. (11) can be thought of as the vector product of the density gradient ρ=(ρ_x,0,ρ_z) with the acceleration of fluid particles a=(a₁,0,a₃), given by ρ×a. Notice that since the flow is two dimensional the vorticity vector ω=ωj has only one component in the y-direction.

Fig. 3. Schematic picture of the three zones: (I) outer zone, (II) inner zone, and (III) recirculation zone. The recirculation region can be assumed to be stagnant to leading order.

Fig. 4. Schematic picture of (A) the coordinate system for the stratified flow in a channel, (B) cross-section of the density field at the centre of the recirculation zone, with nonzero vorticity and static instability of the density field and (C) the same as (B) but with constant density. The fluid inside the recirculation region is stagnant to leading order, thus streamfunction and density are constant inside the recirculation region to leading order.

Fig. 5. Plot of the phase speed versus the amplitude A_*<A<A_*+μ for the solitary wave with a vortex core and a KdV outer solution (dashed), and the phase speed of the traditional KdV solitary wave (solid). c₀ is the linear phase speed.

Fig. 6. Contour plot of the streamfunction for K=1.2 and H^*=0.1 at the times Ut/D=0.0,7.66,15.32,22.98,30.64. Breaking occurs at Ut/D=23.74. The width of the domain is denoted along the horizontal and the depth along the vertical.

Fig. 7. The K^*−H^* parameter space diagram based on the hydrostatic Long's model solution for flow over two-dimensional obstacles. Crosses denote the cases K^*=0.95,1.0,1.1,1.2, plotted in Fig. 8, Fig. 9 and Fig. 10. The hatched region denotes the region of instability for values of K^* and H^* which do not satisfy Eq. (67).

Fig. 8. Plot of the amplitude function A(x,t) for the resonant mode of vertical displacement as computed by the spectral model for the case with H^*=0.1, L/D=2.0 and (a) K=1.2, corresponding to the case shown in Fig. 6, (b) K=1.1 and (c) K=1.0. The obstacle is centred at x/D=20. The corresponding breaking times are t_br=23.7, 28.8, 88.2.

Fig. 9. Plot of the amplitude function A(x,t) for the resonant mode of vertical displacement as computed by the FALW model for the case with, H^*=0.1, L/D=2.0 and (a) K=1.2, (b) K=1.1 and (c) K=1.0. The obstacle is centred at x/D=20. The corresponding breaking times are t_br=21.1, 25.9, 77.9.

Fig. 10. Plot of the amplitude function A(x,t) for the resonant mode of vertical displacement as computed by (a) the spectral model and (b) the FALW model for the case with, H^*=0.1, L/D=2.0 and K=0.95 up to the time Ut/D=115 and 160, respectively. The obstacle is centred at x/D=20.

Fig. 11. Plot of the maximum absolute amplitude normalized by the maximum possible amplitude |A/A_*|_max as a function of time, corresponding to the calculations for K=0.95, 1.0, 1.1 and 1.2: The solid lines indicate the spectral model and the dashed lines the FALW model.

Fig. 12. Vertical position of wave breaking for several cases, 1.0K1.2, note that the breaking location is either at the top or at the bottom and sets in downstream or upstream of the hill.

Fig. 13. Plot of the drag as a function of time on the hill of height H^*=0.1 corresponding to the calculations for K=0.95, 1.0, 1.1 and 1.2: The solid lines indicate the spectral model and the dashed lines the FALW model.

Fig. 14. Plot of the streamfunction for a KdV outer solution at the depth z=2/3D for μ=0.99μ_max, σ=0.01 and τ₂=0.

Fig. 15. Density (left) and streamfunction (right) contour plots of the KdV outer solution with μ=0.99μ_max, corresponding to Fig. 14, for the nondimensional times t_n=0, 2.8, 5.7, 8.6 and 11.5. Note that width is denoted along the horizontal and depth along the vertical.

Fig. 16. Plot of the streamfunction for a mKdV outer solution at the depth z=2/3D for μ=0.99μ_max, σ=0.01 and τ₁=0.

Fig. 17. Density (left) and streamfunction (right) contour plots of the mKdV outer solution with μ=0.99μ_max, corresponding to Fig. 16, for the nondimensional times t_n=0, 2.5, 5.0, 7.5 and 10.1. Note that width is denoted along the horizontal and depth along the vertical.

Fig. 18. Plot of maximum adverse velocity u_adv versus nondimensional time t_n for the KdV outer solution given in Fig. 15.

Fig. 19. Plot of the maximum adverse velocity u_adv versus nondimensional time t_n for the mKdV outer solution given in Fig. 17.

Fig. 20. Plot of the absolute error in phase speed Δc=c_exp−c of the phase speed in the numerical model c_exp to the theoretical phase speed c, versus the amplitude A_*<A<A_*+μ of the solitary wave with a KdV outer solution. Notice that the error is constant over the time of integration, see, for example the streamfunction plot in Fig. 14, where the upstream propagation of the wave is noticeable.

Table 1. Table of nondimensional breaking times t_br for the non-Boussinesq model (nB) and the finite amplitude long-wave model (FALW)