Advanced Ocean Modelling: Using Open-Source Software

160 5 3D Level ModellingCombinations of the latter equations yield a single equation governing the meridionalstructure of V (y):∂ 2 (V ω 2∂y + − β 2 y 2− βk )2 g ′ h o ω − k2 V = 0 (5.29)It can be shown that this equation has the solutions:( ) ( )yV (y) = H n exp − y2R eq 2Req2(5.30)where H n is a so-called Hermite polynomial of the order of n with the first modesbeing given by:H 0 (ψ) = 1H 1 (ψ) = 2ψH 2 (ψ) = 4ψ 2 − 2Notice that even polynomials are symmetric about the equator, whereas those ofodd order are antisymmetric. All waves are trapped in vicinity of the equator on atrapping distance given by the equatorial radius of deformation and their dispersionrelation is given by:ω 2− k 2 − βkg ′ h o ω=(2n + 1)β√g′ h o(5.31)where n is a positive integer including zero. Accordingly, waves are composed of adiscrete set of modes. For n ≥ 1, the waves subdivide into two classes. One branchof waves are relatively fast propagating equatorially trapped inertia-gravity wavesthat follow a dispersion relation according to:√2n + 1ω ≈T 2 eq+ k 2 g ′ h o (5.32)where the so-called equatorial inertial period is defined by:√1T eq =β √ (5.33)g ′ h oTypical values for T eq are 2–3 days. This branch of equatorially trapped wavesincludes a modified form of equatorial Kelvin Waves that, unlike the wave solutiondescribed in the previous section, involves nonzero meridional flow. Figure 5.27illustrates the surface pressure field and currents of such waves that, again, can onlypropagate eastward along the equator.