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4.2 Long Surface Gravity Waves 734.2.10 First-Order Shapiro FilterAs will be shown below, the finite-di ference equation presented above are subjectto oscillations developing on wavelengths of 2Δx. Some of these oscillations mightrepresent true physics, others might be artificia numerical waves. To remove thesesmall-scale oscillations, the following first-orde Shapiro filte (Shapiro, 1970) canbe used:η n+1k= (1 − ɛ)η ∗ k + 0.5ɛ(η∗ k−1 + η∗ k+1 ) (4.21)where ηk ∗ are predicted from (4.19) and ɛ is a smoothing parameter. This methodremoves curvatures in distributions to a certain degree. The smoothing parameter inthis scheme should be chosen as small as possible.4.2.11 Land and CoastlinesLand grid points are realised by requesting absence of fl w on land. In additionto this, no f ow is allowed across coastlines unless a special floodin algorithm isimplemented (see Sect. 4.4). The layer thickness h can be used as a control as towhether grid cells are “dry” or “wet” . Then, we can set u k to zero in grid cellswhere h k ≤ 0. Owing to the staggered nature of the grid (see Fig. 4.5), coastlinesrequire the additional condition that u k has to be zero if h k+1 ≤ 0.4.2.12 Lateral Boundary ConditionsThe model domain is define such that the prediction ranges from k =1tok = nx.Values have to be allocated to the f rst and last grid cells of the model domain; thatis, to k = 0 and k = nx+1 (Fig. 4.7). One option is to treat these boundaries as closed.Advective lateral flu es of any property are eliminated via the statements:u n 0 = 0u n nx = 0Fig. 4.7 The boundary grid cells of the model domain used for implementation of lateral boundaryconditions