Page 72 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

64 Fundamentals of Ocean Renewable Energy

friction, advective inertia force, and diffusion due to turbulence. As a result of
these nonlinear forces, the tidal signal in shallow water regions is more complex
than its deep water counterpart, and sometimes asymmetrical. Therefore, the
tidal signal can no longer be reconstructed by combining/superimposing astro-
nomical (e.g. diurnal and semidiurnal) components.
To understand the generation of overtides, consider a simple case: an estuary
in which the open (seaward) boundary is forced by an M2 signal (Fig. 3.17).
Considering the 1D equations of motion for the estuary, the water elevation and
averaged velocity (over estuarine cross-sections) can be evaluated by solving
the continuity equation

∂η ∂[(h + η)V]
+ = 0 (3.22)
∂t ∂x
and the momentum equation
2
∂V ∂V ∂η n V|V|
+ V + g + g = 0 (3.23)
∂t ∂x ∂x (h + η) 4/3
where h is the depth relative to MSL, η is water elevation relative to this
datum, and n is the Manning coefficient (which depends on the bed roughness).
The momentum equation shows the balance of linear and nonlinear forces:
2
∂η n V|V|
the pressure force (linear; g ), friction force (nonlinear; g ), and the
∂x (h+η) 4/3
inertia force, which has linear ( ∂V ) and nonlinear (V ∂V ) terms. In deep
∂t ∂x
water, the frictional force approaches zero, because water depth is in the
denominator of the friction term. Also, the spatial variation of velocity ∂V is
∂x
negligible for tides in deep waters, so we can neglect convective acceleration.
Therefore, tidal propagation in deep water is governed by the linear momentum
equation
∂V ∂η
+ g = 0 (3.24)
∂t ∂x

FIG. 3.17 Tidal asymmetry in an idealized estuary.