v (m/s) v (knots) Friction power density (W/m2) tide farm power density R1 = 0.01 R1 = 0.003 (W/m2) 0.5 1 1.25 0.4 1 1 2 10 3 8 2 4 80 24 60 3 6 270 80 200 4 8 640 190 500 5 10 1250 375 1000

## Estimating the tidal resource via bottom friction

Another way to estimate the power available from tide is to compute how
much power is already dissipated by friction on the sea floor. A coating of
turbines placed just above the sea floor could act as a substitute bottom,
exerting roughly the same drag on the passing water as the sea floor used
to exert, and extracting roughly the same amount of power as friction used
to dissipate, without significantly altering the tidal flows.

So, what’s the power dissipated by “bottom friction”? Unfortunately,
there isn’t a straightforward model of bottom friction. It depends on the
roughness of the sea bed and the material that the bed is made from –
and even given this information, the correct formula to use is not settled.
One widely used model says that the magnitude of the stress (force per
unit area) is R1ρU2, where U is the average flow velocity and R1 is a di-
mensionless quantity called the shear friction coefficient. We can estimate
the power dissipated per unit area by multiplying the stress by the velocity.
Table G.9 shows the power dissipated in friction, R1ρU3, assuming
R1 = 0.01 or R1 = 0.003. For values of the shear friction coefficient in this
range, the friction power is very similar to the estimated power that a tide
farm would deliver. This is good news, because it suggests that planting a
forest of underwater windmills on the sea-bottom, spaced five diameters
apart, won’t radically alter the flow. The natural friction already has an
effect that is in the same ballpark.

## Tidal pools with pumping

“The pumping trick” artificially increases the amplitude of the tides in a
tidal pool so as to amplify the power obtained. The energy cost of pumping
in extra water at high tide is repaid with interest when the same water is
let out at low tide; similarly, extra water can be pumped out at low tide,
then let back in at high tide. The pumping trick is sometimes used at La
Rance, boosting its net power generation by about 10% (Wilson and Balls,
1990). Let’s work out the theoretical limit for this technology. I’ll assume

Table G.9. Friction power density R1ρU3 (in watts per square metre of sea-floor) as a function of flow speed, assuming R1 = 0.01 or 0.003. Flather (1976) uses R1 = 0.0025–0.003; Taylor (1920) uses 0.002. (1 knot = 1 nautical mile per hour = 0.514 m/s.) The final column shows the tide farm power estimated in table G.6. For further reading see Kowalik (2004), Sleath (1984). 