Advanced Ocean Modelling: Using Open-Source Software

112 4 2.5D Vertical Slice ModellingOceanographers frequently use this relation to estimate slopes of the sea level,driving the surface geostrophic flow, from the observed slope of the pycnocline. Onthe other hand, the thickness of the surface mixed layer is given by:h = h o + η 1 − η 2where h o is the undisturbed thickness. Given that the magnitude η 2 typically exceedsthat of η 1 by far, the barotropic pressure gradient can be formulated according to thereduced-gravity concept as:−g ∂η 1∂x=−g′∂h∂xwhere reduced gravity is defined by g ′ = (ρ 2 − ρ 1 )/ρ 2 g. Using the reduced-gravityconcept for a two-layer ocean, the offshore distance a of the outcrop of the densityinterface for full upwelling can be estimated from (see Cushman-Roisin, 1994):a =I| f | − R (4.18)where R = √ g ′ h 1 / | f | is the internal Rossby radius of deformation, and theso-called “wind impulse” is defined by:I = 1 τ dtρ o h 1∫eventwith τ being the alongshore component of upwelling favorable wind stress. Whenaveraging this wind-stress component over a certain time span t ∗ , the latter equationcan be expressed as:I = 〈τ〉ρ o h 1t ∗According to Eq. (4.18), the transition between partial and full upwelling occurswhen:I = | f | RThe latter two equations can be combined to yield an estimate of the time span ittakes for full upwelling to develop; that is,t ∗ = ρ o h 1√g′ h 1〈τ〉(4.19)The width of a fully developed upwelling front is of the order of the internalradius of deformation R (Cushman-Roisin, 1994).