Advanced Ocean Modelling: Using Open-Source Software

3.3 Surface Gravity Waves 25where η o is wave amplitude, λ is wavelength and T is wave period, the solution ofEqs. (3.3, 3.4, and 3.5) are surface gravity waves that for an ocean of uniform depthh obey the dispersion relation (e.g. Pond and Pickard, 1983):√c = λ T = gλ(2π2π tanh h )λ(3.7)where c is the phase speed of the wave. Figure 3.4 displays the phase speed ofsurface gravity waves as a function of total water depth for selected wavelengths.The dispersion relation includes two different breeds of surface gravity waves thatexist in the ocean. The ratio between wavelength and total water depth determineswhich breed dominates. The first breed are shallow-water waves (or long waves)which can be characterised by λ>20h. These waves are almost barotropic; that is,horizontal flow under a wave is uniform with depth, and attain a phase speed of:c long = √ gh (3.8)Shallow-water waves are almost hydrostatic, which implies ∂ P/∂z = 0inEq. 3.3. Accordingly, horizontal pressure gradients imposed by a tilted sea surfacedo not vary with depth for such waves. This hydrostatic assumption is the basis ofthe shallow-water layer models employed in Kämpf (2009).Nonhydrostatic effects lead to a second breed of gravity waves, called deep-waterwaves or short waves. Short waves can be classified by λ