Advanced Ocean Modelling: Using Open-Source Software

3.24 Exercise 14: Positive Estuaries 913.24.3 Implementation of Variable Channel WidthThe width b of the channel varies only in the x-direction and not with depth.Wu (2007) presents dynamical equations for more complex river shapes. Horizontaland vertical velocities are width-averaged across the channel. Under theseassumptions, the momentum equations and advection-diffusion equation for scalarsremain the same except for a modification of the diffusion term. This term is nowgiven by:Diff(ψ) = 1 ( )∂ ∂ψbK h + ∂ ( )∂ψK z (3.83)b ∂x ∂x ∂z ∂zwhere ψ is a substitute for variables, b(x) is channel width, and K represents eithereddy viscosity in the momentum equations or eddy diffusivity for scalars. The continuityequation for the width-averaged flow can be written as:and vertical integration leads to:∂(b u)∂x+∂(b w)∂z= 0 (3.84)∂η∂t=− 1 b∂(b h〈u〉)∂x(3.85)where 〈u〉 is horizontal flow velocity averaged over both depth and width of thechannel. Owing to the appearance of channel width in the continuity equation, coefficientsin the S.O.R. scheme (see Sect. 3.4) are now given by:a e = b e Δz/Δx , a w = b w Δz/Δx , a t = b k Δx/Δz , a b = b k Δx/Δzwhereb e = 0.5 (b k + b k+1 ) and b w = 0.5 (b k + b k−1 )Accordingly, the source term on the right-hand side of (3.24) is given by:qi,k ∗ = ρ o [(be ui,k ∗ Δt− b w ui,k−1∗ ) (Δz + bk w∗i,k − wi+1,k) ∗ ]Δx(3.86)3.24.4 Advanced Turbulence ClosureThe vertical mixing scheme by Pacanowski and Philander (1981) is a sole functionof the Richardson number (Eq. 3.61). This scheme has been developed for tropicaloceanapplications, but we take the freedom to adopt this scheme for this exercise.Vertical eddy viscosity is calculated from