Illustrated Physics
physics, drawn
Concept

Dimensionless number — where it appears

A ratio of two terms in one equation, saying which is being neglected and by how much. It settles a question about regime that no argument about importance can: the Rossby number decides whether rotation shapes a flow, the Reynolds number whether viscosity does, and each is read off rather than debated.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

Where the sky's blue goes. How many times more strongly a sphere scatters 450 nm light than 650 nm light, against its radius, with the radius on a logarithmic axis running from a couple of nanometres to twenty micrometres. On the left the ratio sits at 4.35, which is the fourth power of the wavelength ratio and the whole reason the daytime sky is blue. It does not stay there. By a radius of 245 nm the preference has halved, and by a micrometre it has essentially gone: a particle comparable with the wavelength scatters every visible colour within a few per cent of equally, which is why a cloud is white, why fog is white, why milk is white and why the exhaust of a cold diesel is white while the smoke of a cigarette — whose particles are ten times smaller — is blue. Nothing about the material changed between one end of this axis and the other; only the size did.

When the particle is the size of the wave

The sky is blue because small things scatter short wavelengths far more strongly. A cloud is made of the same water and scatters every colour alike. Nothing about the material changed — only the size, and one dimensionless number crossing one.

optics · Scattering Pushed one way, travelling another. A parcel released from rest under a steady pressure-gradient acceleration of 2.0e-4 metres per second squared pointing east, integrated for 36 hours at 60°, 30°, 10°. Without rotation it would accelerate east indefinitely. With it, the motion is an inertial circle about a mean velocity at right angles to the push, and the mean over a whole number of inertial periods is measured off each path and matches the geostrophic value a/f to two per cent. At high latitude the loops are tight and the drift is almost purely along the isobars; near the equator the parcel travels a long way down the gradient before the rotation has had time to turn it, which is why geostrophic balance is a high-latitude statement and the tropics need a different set of approximations.

The ratio that decides whether the planet is turning

Whether the rotating terms matter is not a question about size. It is one dimensionless ratio, U over fL, and it runs from ten thousand in a teacup to a millionth in the Earth's core. Where it is small the pressure gradient stops accelerating the fluid and starts balancing a force on fluid already moving across it — so the flow runs along the pressure contours instead of down them, and a weather map is a streamline plot.

mechanics · Circular motion How a layer of grains moves down a slope. The velocity through a steady layer of 0.5 mm glass beads 20 grains deep flowing down slopes of 22°, 25° and 28°, against height above the rough base in grain diameters; the dashed curve is a Newtonian liquid's parabola, scaled to the 25° surface speed. Because every depth must have friction tan θ, every depth has the same inertial number, and the shear rate grows with the square root of the pressure above it: Bagnold's profile, u ∝ h^(3/2) − (h − z)^(3/2). Its surface speeds are 8.0, 41.0, 113.7 cm/s; compared with a liquid's parabola the shear is spread more evenly through the depth, because the shear rate falls off only as the square root of the depth below the surface rather than in proportion to it.

The sand whose friction depends on how fast it flows

A heap of sand stands at its angle of repose and starts to slide a few degrees steeper; between those angles there are steady avalanches, and their speed is decided by one number. The inertial number compares how long a grain takes to fall into a gap opened under it with how long the flow takes to open the gap. Friction rises with it, from the static value to a ceiling, and that single dependence explains why grains flow steadily on only a twelve-degree window of slopes, why a flowing layer has the velocity profile Bagnold measured, and why a deeper avalanche is faster — but not as a liquid would be.

fluids · Granular matter

Named alongside it

The objects these essays reach for when they reach for this one.

ApproximationAsymmetry parameterAvalancheCoriolis forceDipole radiationEquilibriumExtinctionFictitious forceFrictionGeostrophic balanceGranular matterGranular temperature

All concepts