- Calculate water stores, fluxes and residence times
- Find river discharge with the velocity–area method and the Manning equation
- Explain what drives ocean currents and tides
A water molecule you drink today may have evaporated from the ocean a few days ago and may have been frozen in a glacier thousands of years ago. Hydrology studies how water moves on land, and oceanography studies the World Ocean. Both rely on conservation of mass: water does not disappear, it only moves from one store to another.
The water cycle budget
| Store | Volume, km³ (approx.) | Share | Residence time |
|---|---|---|---|
| Oceans | 1,338,000,000 | 96.5% | thousands of years |
| Ice sheets and glaciers | 24,000,000 | 1.7% | centuries to hundreds of millennia |
| Groundwater | 23,400,000 | 1.7% | weeks to tens of millennia |
| Atmosphere | 12,900 | 0.001% | ≈ 9 days |
| Rivers | 2,100 | 0.0002% | weeks to months |
- τmean residence time, years (or days)
- Vvolume of the store, km³
- Fflux into (= out of) the store, km³/year
Residence time of a store in steady state.
The atmosphere holds ≈ 13,000 km³ of water, and ≈ 500,000 km³ of precipitation falls on Earth each year. The ocean holds ≈ 1.338 billion km³, and ≈ 413,000 km³ evaporate from it each year. Find τ for both stores.
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Ocean: τ = 1,338,000,000 / 413,000 ≈ 3200 years.
So atmospheric water is renewed several times a month, while ocean water takes millennia.
The global budget (approximately, thousand km³/year): 413 evaporate from the ocean and 373 fall on it as precipitation; 113 fall on land, 73 evaporate and 40 return to the ocean through rivers and groundwater. For any river basin the same idea is written as the water balance equation.
- Pprecipitation, mm/year (or km³/year)
- ETevaporation plus transpiration (evapotranspiration)
- Rrunoff (river and groundwater flow)
- ΔSchange in storage in the basin (soil, groundwater, snow); ≈ 0 over many years
- Crunoff coefficient (dimensionless); for the global land 40/113 ≈ 0.35
River discharge
Discharge Q is the volume of water passing through a river's cross-section each second. It is measured with the velocity–area method: the cross-section is divided into vertical strips, the depth and mean velocity (usually at 0.6 of the depth) are measured in each strip, and the strip discharges are added up.
- Qdischarge, m³/s
- Across-sectional area of flow, m²
- v̄mean velocity in the section, m/s
- bᵢ, hᵢ, vᵢwidth (m), mean depth (m) and mean velocity (m/s) of strip i
A 15 m wide river section is divided into three strips of 5 m each. Mean depths: 0.8, 1.6 and 1.0 m. Mean velocities: 0.4, 0.9 and 0.5 m/s. Find Q and the mean velocity.
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Discharges: 4 × 0.4 = 1.6; 8 × 0.9 = 7.2; 5 × 0.5 = 2.5 → Q = 11.3 m³/s.
v̄ = Q / A = 11.3 / 17 ≈ 0.66 m/s.
- nManning roughness coefficient (≈ 0.025–0.05 for natural rivers)
- Rhydraulic radius, m
- Pwetted perimeter (length of bed and banks in contact with water), m
- Sslope (dimensionless)
The Manning equation (SI units): estimates velocity from channel shape when there are no measurements.
A rectangular channel is 20 m wide with water 2 m deep, a slope of 0.0005 and n = 0.03. Find the mean velocity and the discharge.
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R^(2/3) ≈ 1.406; S^(1/2) ≈ 0.0224.
v̄ = (1 / 0.03) × 1.406 × 0.0224 ≈ 1.05 m/s.
Q = 40 × 1.05 ≈ 42 m³/s.
The way discharge changes through the year is a river's regime. In Caucasus rivers such as the Kura and the Araz, high water comes in spring and early summer from snowmelt and rain; reservoirs such as Mingachevir regulate the flow. A hydrograph shows how discharge rises and falls after a storm: the smaller, steeper and more urbanised the basin, the sooner and higher the peak.
Ocean currents
Surface currents are driven mainly by winds. Because of the Coriolis effect, the net transport of water (Ekman transport) is 90° to the right of the wind in the Northern Hemisphere. As a result, huge circular currents — gyres — form: clockwise in the Northern Hemisphere and anticlockwise in the Southern. On the western sides of oceans the currents are narrow, fast and warm (Gulf Stream, Kuroshio); on the eastern sides they are cold (Canary, California, Peru, Benguela), and water moving away from the coast causes nutrient-rich deep water to rise (upwelling), creating rich fishing grounds.
Deep circulation is controlled by differences in water density — temperature and salinity (thermohaline circulation). In the North Atlantic and around Antarctica, cold, salty, heavy water sinks and spreads through the oceans; a full loop of this “conveyor belt” takes on the order of a thousand years. Ocean currents are measured in sverdrups: 1 Sv = 10⁶ m³/s.
All the world's rivers bring ≈ 40,000 km³ of water to the ocean each year. Express this in sverdrups and compare it with the Gulf Stream in the Florida Strait (≈ 30 Sv).
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Q ≈ 4 × 10¹³ / 3.16 × 10⁷ ≈ 1.27 × 10⁶ m³/s ≈ 1.3 Sv.
30 / 1.3 ≈ 23 — in the Florida Strait the Gulf Stream carries over 20 times more water than all rivers combined.
Tides
Tides are caused by the Moon's and the Sun's gravity pulling differently on different parts of the Earth. The tide-generating force falls off with the cube of distance, so although the Sun is 27 million times more massive than the Moon, the Moon's effect is about 2.2 times stronger. Two “bulges” of water form on opposite sides of the Earth; since the lunar day lasts 24 h 50 min, most coasts have a high tide every 12 h 25 min. At new and full moon, when the Sun and Moon are in line, tides are strongest (spring tides); at the quarters they are weakest (neap tides). In the Bay of Fundy (Canada) the range reaches 16 m, while in enclosed seas such as the Caspian it is only a few centimetres.
- Fₜtidal force on a mass m, N
- Ggravitational constant = 6.67 × 10⁻¹¹ N·m²/kg²
- Mmass of the Moon or the Sun, kg
- rEarth's radius, m
- ddistance from the Earth to the Moon or the Sun, m
(7.35 × 10²² / 1.99 × 10³⁰) × (1.496 × 10¹¹ / 3.84 × 10⁸)³ ≈ 2.2 — the Moon dominates the tides.
Key points
- 96.5% of water is in the oceans; residence time τ = V / F: ≈ 9 days in the atmosphere, ≈ 3000 years in the ocean.
- Basin water balance: P = ET + R + ΔS; runoff coefficient C = R / P.
- Discharge Q = A · v̄ (with mean velocity); without measurements use Manning: v̄ = (1/n)·R^(2/3)·S^(1/2).
- Surface currents are driven by winds and the Coriolis effect, deep currents by density differences; 1 Sv = 10⁶ m³/s.
- Tidal force ~M/d³; the Moon's effect is ≈ 2.2 times the Sun's; high tides come every 12 h 25 min.
Check yourself
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