Advanced Ocean Modelling: Using Open-Source Software

98 4 2.5D Vertical Slice ModellingFig. 4.1 The 2.5d vertical ocean sliceis valid for shallow-water processes (i.e. processes of a horizontal scale exceedingtheir vertical scale by far), the latter equations can be formulated as:∂v geo∂z∂u geo∂z=+ gρ f=− gρ f∂ρ∂x∂ρ∂y(4.3)(4.4)These relations, called the thermal-wind equations, imply that geostrophic flowdisplays a vertical shear in the presence of lateral density gradients. In oceanography,the application of the thermal-wind equations for derivation of the baroclinicgeostrophic flow field from density measurements is called the geostrophic method.See Pond and Pickard (1983) for a detailed description of this method.Geostrophic surface currents, not captured by the geostrophic method, follow therelationships:− f v geo =−g ∂η∂x+ fu geo =−g ∂η∂y(4.5)(4.6)and run along lines of the same sealevel elevation. Sea-level anomalies derived fromsatellite altimetry can therefore be used to map the surface circulation of the ocean.In the case of uniform density, the latter equations describe depth-independentgeostrophic flow.The beta-plane approximation describes departures of the Coriolis parameterfrom a constant value and is given by:f = f o + βy (4.7)where β is the meridional variation of the Coriolis parameter with a value of β =2.2 × 10 −11 m −1 s −1 at mid-latitudes, and y is the distance in metres with respect