Numerical outputs of the three-layer model. Top: real part of the latitudinal velocity v against y. The quantities v_num, v_I, v_R, v_T, v_F, and v_W are the numerical, incident, reflected, transmitted, first-order Frobenius, and Whittaker velocities, respectively. Bottom: wave action flux against y. The quantities , , , and are the numerical, transmitted, incident and reflected, and Frobenius wave action fluxes, respectively. For all panels, θ₀ = 0, the mean flow is linear in the grey-shaded shear regions, and the critical level is marked by dashed lines. The horizontal wave numbers are set to k_x = k_z = 0.1. From left to right: (i) ω = 0.02 and R_o = 0.3 (R > 1∕4), (ii) ω = 0.002 and R_o = 0.8 (R > 1∕4), and (iii) ω = 0.09 and R_o = 1.8 (R < 1∕4).