Far away is near at hand in images of elsewhere
Interestingly the magnitude of this effect will depend on the vertical position of the sluice. The pressure acting on each side of the gate varies from 0 at the water surface, increasing linearly to a maximum at the base. The rate of increase depends on the water density, which as we know is greater for salt water than fresh. This pressure gradient can be thought of as a right angle triangle, with its apex at the water surface, and the slope of the hypotenuse depending on the density. The total force on each side of the gate is the area of the triangle for that side.
The reason that sluice position is relevant is that the pressure must be equal at that position, otherwise water would flow through the sluice.
If the sluice is very close to the water surface (unlikely in a real lock) then the level would be equal on both sides of the gate. Because salt water is denser its pressure gets greater faster as we go down the gate, so the triangle on that side is larger. This results in a force acting to push the gate towards the fresh water side (which would open the gates at Heybridge).
If on the other hand the sluice is at the bottom of the gate (a more likely scenario) then the effect will be reversed. The pressure at the base must be equal, so the water level on the salty side will be lower. The triangles will have equal base length but the salt one, being shorter, will have a smaller area. Thus there is a force acting to push the gate towards the salt water, holding it closed. When the gate is opened slightly the levels will attempt to equalise, resulting in rapid flow into the lock.
Of course at some sluice depth between the two extremes the forces will balance, though there will still be flow as the gates are opened. Fresh water will flow into the lock at the top, and salt water will flow out at the bottom.
The resultant force depends on difference in density, height of water (squared) and width of the gate. For a 4m deep lock, gate width 3.75m this could give a force of 10 kN ~ 1 Tonne which I guess most of us would notice.
That's a great explanation. It's good to know that there is at least one intelligent person in East Anglia.
Far away is near at hand in images of elsewhere
The p = - g ρ z of alandalus11’s cryptic set of formulae at #18 did give the gist of it: the calculation of the pressure at depth z. The σt bit just refers to the way oceanographers subtract 1000 from the density ρ in kg/m3 to give easy-to-handle two-digit density numbers - but the Total Dissolved Solids (TDS) bit was confusingly irrelevant.
But I suspect you may perhaps have surmised that - and of course he may not be in East Anglia.
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