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Earth

Water cycle

Run a year of weather over a catchment of forest, town and lake: moisture blows in, rains out in proportion to each land’s share, the canopy catches a quarter and breathes it back, forest soil drinks fifty millimetres a day while pavement takes five, soils evaporate and percolate, the groundwater seeps to the river over three hundred days and the river drains to the sea in two. Drop a 50 mm storm on day 100 and compare the flood peak of the forested catchment with the paved one; pump the aquifer hard and watch the baseflow die; cut off the advected moisture and the stores run down together. Nine stores plus exports minus imports keep the initial total to rounding, and the groundwater recession is the closed-form exponential.

By Mykhailo Makarov

Water cycle — Actual application image
Actual application image

What happens

The water cycle of a catchment as nine lumped stores in millimetres over the whole area: the air above, the forest canopy, forest surface and soil, town surface and soil, groundwater, river and lake. Advected moisture and a storm feed the air, rain falls in proportion to the land shares, the canopy intercepts a quarter and evaporates it, surfaces infiltrate at the forest’s or the town’s rate and run off at their own pace, soils evaporate and percolate, groundwater feeds the river and is pumped, the lake takes half the forest runoff and evaporates, the river drains to the sea, the air exports vapour. Explicit stepping at a hundredth of a day with a store’s outflows scaled together so none goes negative; one second of the experiment is one day.

Try this

Watch the gauges settle, then the storm on day 100 raise the river; pave the catchment and compare the peak; pump the aquifer and watch the baseflow die; check the recession against the exponential.

The model and its limits

Lumped linear and saturating stores, one storm, no snow, no seasonal evaporation, no routing delays beyond the store time constants. A teaching catchment, not a hydrological model.

Equations used in the module
Equations
dS_i/dt = Σ F_in − Σ F_out,  R = S_atm/τ_rain
I = K_inf·f·S_surf/(S_surf + S_½)·(1 − S_soil/S_max)
Q_base = S_gw/τ_gw,  S_gw(t) = S_gw(0)·e^(−t/τ_gw) (no recharge)
Σ S_i + E_out − I_in = const

Related experiments

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Rainfall and runoff — Actual application image

Earth

Rainfall and runoff

Make it rain over a catchment. Compare permeable soil with rapid surface runoff; then stop the rain and watch drainage.

River meanders — Actual application image

Earth

River meanders

Let a 20 m wide river with a two-metre wiggle run for two thousand years: the near-bank flow feels the curvature a little upstream of where it is, so the outer bank of every bend erodes while the inner bank builds a point bar with the same volume, bends 400 m long grow fastest, the closed-form wavelength of the bank law, and the channel’s sinuosity climbs past two until necks are cut and oxbows left. A dead-straight channel never moves; double the erodibility and the first centuries move twice as fast; the circumcircle curvature of a test arc is exactly 1/R.

River delta — Actual application image

Earth

River delta

Send a river of 200 m³/s carrying a hundredth of a cubic metre of sand a second onto a shelf and watch a century of delta grow: the water spreads to every lower neighbour in proportion to slope, each cell of sea drops a share 1 - exp(-depth / 4 m) of what passes and rises until it stands above the sea with a gentle seaward slope, the shoreline advances and the river finds several mouths. Triple the sediment and the lobe sweeps across the whole shelf; deepen the shelf and the same sand builds less land; raise the sea a centimetre a year and the gain is smaller still. Sediment supplied equals deposited plus exported to rounding.