http://www.ck12.org Chapter 4. Technical Chapters
The power of tidal waves
This section, which can safely be skipped, provides more details behind the formula for tidal power used in the
previous section. I’m going to go into this model of tidal power in some detail because most of the official estimates
of the UK tidal resource have been based on a model that believe is incorrect.
Figure G.2 shows a model for a tidal wave travelling across relatively shallow water. This model is intended as
a cartoon, for example, of tidal crests moving up the English channel or down the North Sea. It’s important to
distinguish the speedUat which the water itself moves (which might be about 1 mile per hour) from the speedvat
which the high tide moves, which is typically 100 or 200 miles per hour.
The water has depthd. Crests and troughs of water are injected from the left hand side by the 12-hourly ocean tides.
The crests and troughs move with velocity
υ=√
gd. (G. 2 )We assume that the wavelength is much bigger than the depth, and we neglect details such as Coriolis forces and
density variations in the water. Call the vertical amplitude of the tideh. For the standard assumption of nearly-
vorticity-free flow, the horizontal velocity of the water is near-constant with depth. The horizontal velocityUis
proportional to the surface displacement and can be found by conservation of mass:
U=
vh
d. (G. 3 )
Figure G.4:Average tidal powers measured by Cartwright et al. (1980).
If the depth decreases, the wave velocityυreduces (equation (G.2)). For the present discussion we’ll assume the
depth is constant. Energy flows from left to right at some rate. How should this total tidal power be estimated? And
what’s themaximumpower that could be extracted?
One suggestion is to choose a cross-section and estimate the averageflux of kinetic energyacross that plane, then
assert that this quantity represents the power that could be extracted. This kinetic-energy-flux method was used by
consultants Black and Veatch to estimate the UK resource. In our cartoon model, we can compute the total power
by other means. We’ll see that the kinetic-energy-flux answer is too small by a significant factor.