How much does a cloud weigh?

About 500,000 kilograms — the weight of a hundred elephants. That is the mass of the liquid water floating above you in one modest, fair-weather cumulus cloud, the fluffy cotton-wool kind you drew as a child.

This figure sounds like it cannot possibly be right. Clouds drift; elephants emphatically do not. In this paper we derive the number for ourselves with one measurement and one multiplication, and then use a little mechanics to resolve the paradox of how a hundred elephants' worth of water stays up.

How much water is in a cloud?

A cloud is not a solid lump of water — it is ordinary air containing a fine mist of liquid droplets. Meteorologists measure this as liquid water content, and for a typical cumulus cloud it is about $0.5$ grams of water per cubic metre of cloud [1]. That is remarkably little: a whole cubic metre of cloud — a box you could stand inside — contains about a tenth of a teaspoon of water.

The trick is that clouds contain a lot of cubic metres. A typical cumulus is about $1$ km across, $1$ km deep, and $1$ km tall [1], and its volume is where the calculation comes alive:

$$V = 1{,}000 \text{ m} \times 1{,}000 \text{ m} \times 1{,}000 \text{ m} = 10^{9} \text{ m}^3$$

One innocent-looking cloud is a billion cubic metres of damp air.

Now the whole calculation is a single multiplication:

$$m = 0.5 \text{ g/m}^3 \times 10^{9} \text{ m}^3 = 5\times10^{8} \text{ g} = 500{,}000 \text{ kg}$$

Half a million kilograms, or $500$ tonnes, of liquid water. An adult African elephant has a mass of around $5{,}000$ kg, so our cloud carries the water-weight of

$$\frac{500{,}000}{5{,}000} = 100 \text{ elephants.}$$

Curiously, all that water gathered into one place would be underwhelming: $500$ tonnes is $500 \text{ m}^3$, a cube less than $8$ m across — one back-garden swimming pool, stretched into a cloud a kilometre wide. (For a comparison in the other direction, we once poured all of humanity into a Scottish lake in another article.)

Why doesn't it fall?

Here is the resolution of the paradox: the cloud's water is not one $500$-tonne mass but around $10^{17}$ separate droplets, each with a radius of about $10$ micrometres ($10^{-5}$ m) and a mass of about four billionths of a gram. For objects this small, air resistance dominates completely.

A falling droplet stops accelerating when drag balances its weight — its terminal velocity. For tiny spheres this is given by Stokes' law:

$$v = \frac{2 r^2 g (\rho_w - \rho_a)}{9\mu}$$

where $r = 10^{-5}$ m is the droplet radius, $g = 9.81 \text{ m/s}^2$, $\rho_w = 1{,}000 \text{ kg/m}^3$ and $\rho_a = 1.2 \text{ kg/m}^3$ are the densities of water and air, and $\mu = 1.8\times10^{-5} \text{ Pa·s}$ is the viscosity of air. Substituting:

$$v = \frac{2 \times (10^{-5})^2 \times 9.81 \times 998.8}{9 \times 1.8\times10^{-5}} \approx 0.012 \text{ m/s}$$

The droplets are falling — at about $1.2$ centimetres per second. At that rate a droplet takes roughly $23$ hours to descend one kilometre, while the gentle updraughts inside a cumulus cloud (often around $1$ m/s, nearly a hundred times faster) carry it back up long before it gets anywhere. A cloud is less a floating object than a very slow fountain.

What the number leaves out

A typical cumulus cloud holds about 500,000 kg of liquid water — one hundred elephants — kept aloft because that mass is shattered into $10^{17}$ droplets so small that air resistance almost cancels their weight. We should be honest about what "the weight of a cloud" means, though. The air inside our cubic kilometre of cloud has a mass of about $1.2\times10^{9}$ kg — over a million tonnes, dwarfing the water by a factor of more than $2{,}000$. The water is barely $0.04\%$ of the cloud by mass. And the truce with gravity is temporary: when droplets merge and grow, terminal velocity grows with $r^2$, drag loses the argument, and the elephants arrive after all — as rain.

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References:

[1] U.S. Geological Survey, Water Science School, "How Much Does a Cloud Weigh?": usgs.gov/special-topics/water-science-school

[2] Rogers, R. R. and Yau, M. K. A Short Course in Cloud Physics, 3rd ed. (Pergamon Press, 1989).

Note: Liquid water content varies widely — from around 0.05 g/m³ in wispy stratus to several g/m³ in storm clouds — so "the" weight of a cloud spans a few orders of magnitude. A large cumulonimbus can carry millions of tonnes of water.