Earth

Tides in an estuary

Watch the evening tide run up a funnel-shaped estuary past the lighthouse: the channel narrows towards the river, so the wave rises as it goes and the range at the head is far larger than at the mouth, the flood is shorter than the ebb, and the salt sea water pushes tens of kilometres inland against the river. Raise the river’s discharge and the salt is pushed back towards the sea; make the funnel sharper or straighten it; close the head and the channel resonates as a quarter-wave, the head’s tide a / cos(kL) of the sea’s. Calm closed water stays exactly calm, a salt estuary with no river stays exactly salt, and the water and salt balance against what crossed the mouth and the head.

By Mykhailo Makarov

Actual application image

What happens

A 60 km channel 8 m deep whose width narrows from 3 km as exp(−x/25 km), in 120 cells on a staggered grid: the surface at cell centres, the velocity at faces. Forward-backward steps of 20 s update the surface from the face discharges, then the velocity from the new surface slope, with upwind advection and implicit Manning friction. The sea’s M2 tide of 1.5 m comes in at the mouth over two periods; the river brings 100 m³/s at the head. Salt rides the same discharges upwind with a tidal dispersion of 150 m²/s, so a salt estuary with no river stays salt and calm closed water stays calm. A closed prismatic channel checks the quarter-wave a/cos(kL).

Try this

Watch the range grow up the funnel and the flood run shorter than the ebb; raise the river and push the salt out; straighten or sharpen the funnel; close the head and compare with the quarter-wave.

The model and its limits

A one-dimensional, well-mixed channel of uniform depth, no drying of the flats, salinity without density effects, a single tidal constituent. Ranges belong to this idealized estuary.

Equations used in the module

Equations
b·∂η/∂t + ∂(b·H·u)/∂x = 0,  b = b0·e^(−x/L_b)
∂u/∂t + u·∂u/∂x = −g·∂η/∂x − g·n²·|u|·u/H^(4/3)
∂(b·H·S)/∂t + ∂(b·H·u·S − b·H·K·∂S/∂x)/∂x = 0
closed prismatic channel: a_head = a/cos(kL),  k = ω/√(g·h)

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