List of Figures

• Effect of the rotation on the discretization of a square domain.
• Effect of a poor resolution on the geometry of a strait.
• Elevation field for the Kelvin retardation problem in presence of steps along the walls at two different resolutions.
• The three major horizontal staggerings for the primitive equations.
• Triangulation of the domain.
• $\phi _i$, the basis function related to the node M_i.
• $\phi _2$ is the basis function related to the node x₂=0.
• The discontinuous linear non conforming basis function for the P₁^NC-P₁ element of Hua and Thomasset hua84.
• The discontinuous constant basis function for pressure over the macro-element of LLS.
• Local non-orthogonal coordinates in a given triangle
• Example of Legendre polynomials $\Phi _i=L_2(\xi _1) L_3(\xi _2)$
• Remeshing strategies. The triangle to be refined is in grey.
• Transformation of one triangle intro a curved triangle
• Transformation of one triangle intro a curved triangle with the coordinate system used in the computation of the integrals
• The wave test experiment
• Convergence with resolution of the normalized error in the u-component for second order C-grid formulation (FDM), O-FDM4 and R-FDM4 models.
• CPU cost with the normalized error in u-component for the second order C-grid (FDM) and R-FDM4 models.
• The four FE models (LW, HT, PZM, LLS) are tested against the analytical solution with increasing resolution.
• Convergence of the normalized error in v with respect to the resolution for the LW-FE and SE models
• Convergence of the normalized error in u with respect to the resolution for the C-grid FD, LW-FE and SE models.
• Variation in the u-component normalized error as a function of CPU cost for five models.
• Grids for the circular domain for the FD models.
• Convergence with resolution of the normalized elevation error for the second order C-grid FD, O-FDM4 and R-FDM4 models in a circular domain.
• Elevation error in a circular domain for four FE models.
• Normalized elevation error for the C-grid FD, LW-FE and SE models for a circular domain.
• Total energy after 18 days of simulation for the C-grid FD and the lumped LW, delumped LW, PZM and LLS FE models and the SE model for the geostrophically balanced eddy.
• Elevation field after a six year simulation in a non-rotated basin using O-FDM4 and R-FDM4.
• As for Figure 3.13 but for rotated basin.
• Kinetic energy during a 6 year spin-up for the C-grid FD, the lumped LW, HT, PZM and LLS FE models.
• Elevation field after a 6 year spin-up for the C-grid FD, the lumped LW, delumped LW, HT, PZM and LLS FE models for the single gyre wind forcing problem.
• Single gyre wind forcing experiments for the delumped LW FE model compared to the C-grid FD model.
• Kinetic energy during spin-up for the single gyre Munk problem with $\nu =2000$ m²s^-1 for the C-grid FD, the delumped LW FE and SE models.
• Elevation field for the SE model after 6 years from spin-up for the single gyre Munk problem with $\nu =2000$ m²s^-1, n_c=5.
• As for Fig. 3.18 but with $\nu =700$ m²s^-1.
• As for Fig. 3.19 but with $\nu =700$ m²s^-1.
• Convergence with resolution for the nonlinear Munk problem of the normalized kinetic energy error for the solution from the C-grid FD and the SE models.
• As for Fig. 3.22 but for the convergence of the normalized error with CPU cost.
• Solutions after a 6 year spin-up for the Munk problem using the adaptive SE model with n_c=5.
• Locations of variables near a step for the SW C-grid model and the QG model.
• Local advective flux along the boundary at 20 km resolution in a square basin for the enstrophy conserving formulation.
• Northward flow past a forward step. The shaded area is the model domain.
• Elevation fields in meters after a 6 year spin-up for 20 km and 10 km resolution. Shown are results from the A and B combination with or without a 3.44^o rotation angle of the basin.
• (a) Kinetic energy after spin-up and (b) ratio of $\mathcal{F}_{adv}$ to $\mathcal{F}_{i}$for the four combinations combinations.
• Kinetic energy after spin-up for the B combination in 10¹⁰ m⁵/s².
• Ratio of $\mathcal{F}_{adv}$ to $\mathcal{F}_{i}$ for the B combination.
• Convergence of $\mathcal{F}_{adv}$ with resolution for 0^o, 20^o, 45^o rotation angle for the B combination.
• Kinetic energy during spin-up for six runs using the J₁ Jacobian at 6 different rotation angles.
• Kinetic energy after spin-up for (a) J₃ at 0^o rotation, (b) J₇ at 0^o, (c) J₃ at 30^o, (d) J₇ at 30^o, (e) J₃ at -30^o, (f) J₇ at -30^o.
• Ratio of $\mathcal{F}'_{adv}$ to the wind input for (a) J₃ at 0^o rotation, (b) J₇ at 0^o, (c) J₃ at 30^o, (d) J₇ at 30^o.
• Ratio of $\mathcal{F}_{c}$ to the wind input. (a-d) as described in Fig. 4.10.
• Ratio of $\mathcal{F}_{adv}= \mathcal{F}'_{adv}+ \mathcal{F}_{c}$ to the wind input. (a-d) as described in Fig. 4.10.
• Semi-advective flux, < $\mathcal{F}'_{adv}$, and beta contribution, $\mathcal{F}_{c}$, to the vorticity budget for the J₃ Jacobian at -30^o rotation angle.
• Notation corresponding to the curvilinear coordinates.
• The five geometries used for our application of the SE method.
• Elevation fields in the Geometry V for the C-grid model after 3 years of spin-up.
• Total energy during spin-up for the A and B combinations of the FD C-grid model and for the SE model at n_c=5 (SPOC 5) in Geometry V. $\nu =100$ m²s^-1.
• Mean elevation fields for the five geometries using the SE model.
• Mean vorticity field for the Geometries IV and V using the SE model.
• Power input by the wind using the mean fields with respect to the boundary Reynolds number.
• Mean standard deviation of the elevation.
• Mean standard deviation of the elevation for frequencies with period above 200 days.
• Mean standard deviation of the elevation for frequencies with period between 17 and 200 days.
• Mean standard deviation of the elevation for frequencies with period between .6 and 17 days.
• Hovmöller diagram of the filtered elevation with respect to time and location along the boundary.
• Amplitude of the Kelvin wave in meters along the boundary averaged over 6 years.
• Time series of the amplitude of the oscillations at (x=500 km, y=0 km).
• Total energy for the last 6 years of simulation.
• Instantaneous vorticity field in the vicinity of the recirculation.
• (a) Local wind input to the vorticity in Geometry V. (b) Local divergence of the eddy transport of vorticity in Geometry V. (c) Vector plot of the eddy transport of vorticity normal to the streamlines in Geometry V.
• The region in grey represents where the absolute vorticity is approximately conserved.
• Maximum of the mean elevation for the three geometries. The maximum elevation is a good proxy for the strength of the recirculation.
• Kinetic energy of the mean fields with respect to the boundary Reynolds number.
• Mesh for the original and the refined runs.
• Total energy for the last 6 years of simulation for the original and refined meshes.
• Amplitude of the fast oscillations at (x=500 km, y=0 km) along the boundary for the original and refined meshes.
• Mean elevation fields for the original mesh and the refined mesh.
• Mean vorticity fields for the original mesh and the refined mesh.