Advanced Ocean Modelling: Using Open-Source Software

26 3 Basics of Nonhydrostatic ModellingThe folder “Miscellaneous/Dispersion Relation Calculator” on the book’s ftp sitecontains a SciLab script with which the reader can compute either phase speed orperiod of surface gravity waves from user-specified values of total water depth andwavelength.3.3.3 Orbital Motions of Water Particles and Wave PressureOrbital motions of water parcels under a pure sine wave are described by the equations(e.g. Pond and Pickard, 1983):u = 2π η (oT exp −2π z∗λw = 2π η (oT exp −2π z∗λ)(sin)cos)2π x λ − 2π t T(2π x λ − 2π t )T(3.10)(3.11)where z ∗ is (positive) distance from the sea surface. The pressure field that drivesthese motions is given by:) (P = ρ o g η o exp(−2π z∗sin 2π x λ λ − 2π t )(3.12)Twhich is consistent with Eq. (3.5).For deep-water waves, water particles move in circular orbits with a radius oforbits decreasing rapidly (exponentially) with depth. At a depth z ∗ = λ, for instance,the orbit’s radius is only 0.2% of that at the surface. This implies that such wavesattain vanishingly small orbital speeds near the seafloor.Orbital motion in shallow-water waves are elliptical near the sea surface and horizontal(simply back and forth) at the sea bottom. Shallow-water waves, if energeticenough, are capable of eroding sediment from the sea floor.3.4 Nonhydrostatic Solver3.4.1 Splitting Pressure into PartsFor convenience, dynamic pressure P is split into two parts:P = p + q (3.13)where (lower-case) p refers to the hydrostatic pressure field with reference toan undisturbed (horizontal) sea level, and q includes pressure components both