Advanced Ocean Modelling: Using Open-Source Software

158 5 3D Level Modelling5.10 Equatorial Waves5.10.1 BackgroundThe existence of the equator gives rise to special kinds of oceanic waves that otherwisewould not exist. The dynamical reason for such waves is that the Coriolisforce changes sign across the equator, giving rise to equatorial inertial oscillationsdescribed in the previous section. However, there are other wave types existing invicinity of the equator. The most basic equations describing these waves rely onthe reduced-gravity concept for a two-layer ocean (e.g. Cushman-Roisin, 1994)in which the bottom layer always adjusts such that the lateral flow in this layervanishes. If we describe the thickness of the upper layer as the sum of a constantpart h o and a fluctuating part η ∗ , the reduced-gravity concept leads to the followingequations for the lateral currents in the upper layer:∂η ∗∂t∂u∂η∗− β y v =−g′∂t ∂x∂v∂η∗+ β yu =−g′∂t ∂y( ∂u+ h o∂x + ∂v )= 0∂y(5.23)where g ′ is reduced gravity. The equatorial beta-plane approximation has been usedhere together with the assumption that variations of the top-layer thickness remainsmall, so that a constant thickness h o can be used in the volume-conservation equation.Note that η ∗ is the negative of the interface displacement, simply becauseinterface displacements and layerthickness changes are opposite to each other.5.10.2 Equatorial Kelvin WavesThe first breed of equatorial waves to be discussed are equatorial Kelvin wavesthat can be extracted from the above equations with the assumption of vanishingmeridional flow; that is, v = 0, everywhere in the domain. In this case, the aboveequations can be written as:∂η ∗∂t∂u∂t=−g ′ ∂η∗∂xβ yu=−g ′ ∂η∗∂y∂u+ h o∂x = 0(5.24)