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5.4 The Wind-Forced Shallow-Water Model 995.4 The Wind-Forced Shallow-Water Model5.4.1 The Governing EquationsThe shallow-water equations including wind-stress forcing and bottom friction read:∂u∂t∂v∂t∂η∂t− τ botx∂η=−g∂x + τ xwindρ o h∂η=−g∂y + τ ywindρ o h(u h) ∂ (v h)=−∂ −∂x ∂y− τ boty(5.6)where (τxwind ,τywind ) is the wind-stress vector, and (τxbot ,τybot ) is the frictional bottomstressvector. For simplicity, lateral friction, the nonlinear terms, and the Coriolisforce are not included yet in the momentum equations. This model only describesdepth-averaged effects of wind forcing and bottom friction.5.4.2 Semi-implicit Approach for Bottom FrictionUnder the exclusive action of bottom friction and using a quadratic bottom-frictionlaw, the momentum equation can be written as:∂u∂t =−ru√ (u 2 + v 2 )/h (5.7)∂v∂t =−rv√ (u 2 + v 2 )/h (5.8)where r is a non-dimensional bottom-drag coefficient Under the assumption thatthe initial f ow runs into the x-direction, an explicit approach of the bottom-frictionterm would lead to the finite-di ference equation:u n+1j,k= u n j,k (1 − ɛ) with ɛ = rΔt ∣ ∣u n ∣j,k /h uwhere h u is thickness of the water column at the u-grid point. The problem now areinstances of ɛ>1, which can happen in shallow parts of a model domain, triggeringunwanted acceleration of the fl w. Bottom friction cannot do such things. Thisproblem can be avoided when using a semi-implicit approach for bottom friction,leading to the equations:u n+1j,k= u n j,k− rΔt un+1j,k√ ( ) 2 ( )u n j,k + vn 2/huu (5.9)