Illustrated Physics
physics, drawn
Concept

Convection — where it appears

Heat carried by the bulk motion of a fluid rather than by conduction through it, driven by the buoyancy of the warmer parts. It sets a planet's temperature profile because a rising parcel cools by expanding at a rate its own heat capacity fixes, giving a lapse rate that has nothing radiative in it.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

The two holes a grain can fall through, and their exact sizes. Three equal spheres in contact, and four, drawn with the largest sphere that passes between them. The numbers are geometry and nothing else. Three mutually touching spheres put their centres on an equilateral triangle of side 2R, whose circumradius is 2R/√3, so the gap admits a sphere of radius 0.154701R — about a seventh. Four in a square admit 0.414214R, nearly half. A real packing contains both arrangements and everything between, so a grain smaller than the first threshold gets through everywhere, one larger than the second gets through nowhere, and one in between percolates slowly through the loosest routes. That is the whole size-dependence of segregation by percolation, and it is why the effect is reliable below about a seventh and erratic between a seventh and a half.

The big one comes to the top

Shake a jar of mixed grains and it sorts itself, which is the opposite of what shaking a mixture of gases does. There is no thermodynamic paradox in it because there is no temperature to speak of — and the mechanism is a piece of geometry with an exact number in it: a sphere fits through the gap between three touching spheres only below a radius ratio of 0.1547.

fluids · Granular matter The temperature a planet ought to be. Each body's measured surface temperature against the temperature at which it would radiate away exactly the sunlight it absorbs — computed from two numbers, the sunlight reaching it and the fraction it reflects, with no other property of the body used. Points on the diagonal are bodies the one-line argument gets right. the Moon: balance 270 K, surface 250 K, −20 K with no atmosphere; Mercury: balance 433 K, surface 340 K, −93 K with no atmosphere; Mars: balance 210 K, surface 210 K, +0 K; Earth: balance 255 K, surface 288 K, +33 K; Venus: balance 227 K, surface 737 K, +510 K. The airless bodies fall below the line rather than on it, and the reason is the fourth power: a surface running from noon heat to night cold radiates like its hottest parts and averages like its coldest, so a mean thermometer reading is lower than the temperature that matches the emitted flux. Mars, whose atmosphere is thin and whose surface is nearly isothermal by comparison, sits on the line. Everything with a substantial atmosphere sits above it, by tens of kelvin on Earth and hundreds on Venus, and always in the same direction. Nothing here explains why. What the figure fixes is the size and the sign of what has to be explained.

The height a planet is seen from

A body in sunlight settles where it radiates away what it absorbs, and that takes two numbers and one line of arithmetic. It gets the Moon right and Earth wrong by thirty-three kelvin. The correction is not that the atmosphere traps heat but that it moves the level space sees the planet from, and the rest is done by a lapse rate that is not a radiative quantity at all.

thermodynamics · Blackbody The deepest water reads warmer than the water above it. A column of seawater 10.9 km deep whose potential temperature — the temperature each parcel would have if brought to the surface without gaining or losing heat — falls from 20 °C at the top to 1 °C in the deep, and the temperature a thermometer lowered into it would read. The two agree near the surface and part company with depth: the thermometer reads a minimum of 1.38 °C near 3.9 km and then rises to 2.35 °C at the bottom, although the water there is the same water, with the same potential temperature, 1.00 °C, as the water two kilometres higher. The warming is compression alone; no heat reaches the deep trench from anywhere.

The water that warms on the way down

Lower a thermometer into the Mariana Trench and it reads colder and colder for four kilometres and then starts to warm, reaching about two and a half degrees at the bottom, eleven kilometres down. No heat is coming up from the sea floor to do it. The water at the bottom is the same water as the water two kilometres higher, carried down and squeezed, and squeezing warms it. In fresh water near four degrees the same squeezing does nothing at all, and below four degrees it cools — so a parcel of cold lake water pushed deep enough keeps sinking on its own.

fluids · Hydrostatics Why the mixture of two equal waters is heavier than either. A temperature–salinity diagram for seawater near the surface, with lines of equal density (curves) 0.3 kg/m³ apart. Two waters lie on the same line: one at 4 °C and 33.93 g/kg, the other at 16 °C and 36.53 g/kg, both 1026.00 kg/m³. Mixing them in any proportion moves along the straight line between them, because temperature and salt add linearly; the lines of equal density curve, because cold water's density responds less to temperature than warm water's. The straight line lies on the dense side of the curve. An even mixture, at 10 °C and 35.23 g/kg, is 0.182 kg/m³ denser than either parent, and sinks between them.

The mixture heavier than either water

Two bodies of seawater can have exactly the same density — one cold and fresh, one warm and salty — and sit side by side without either wanting to sink. Mix them, and the mixture is denser than both, and sinks. Nothing has been added and nothing removed; the extra weight comes from the fact that water's density is a curved function of temperature while mixing averages temperature in a straight line. The same curvature, at its most extreme, is why a lake overturns twice a year and freezes from the top, and why the open ocean, too salty to have a density maximum above freezing, does neither.

fluids · Stratification A pond that is hottest at the bottom and does not turn over. Temperature, salt and density against depth in a salt-gradient solar pond: a fresh upper layer 0.3 m deep at 25 °C, a gradient layer 1.2 m thick in which the salt rises from 2 to 20 per cent and the temperature from 25 to 80 °C, and a storage layer of hot brine 2 m deep (shaded bands are the two mixed layers). The density rises steadily downward, from 1011 to 1122 kg/m³, so the hottest water is also the heaviest and stays at the bottom. The dashed line is the same temperatures with no added salt: the bottom water would then be 25 kg/m³ lighter than the top, and the pond would overturn.

The pond that is hottest at the bottom

Water heated from below rises, which is why a pond in the sun is never much warmer than the air: the heat is carried to the surface and lost there. Put a metre of salt gradient in the way and the hot water at the bottom is also the heaviest, cannot rise, and keeps its heat, behind an insulating layer that sunlight passes straight through. Ponds built this way reach eighty degrees under an ordinary sky. The salt does not collect any extra light. What it buys is temperature.

fluids · Stratification Insulating a thin wire makes it lose more heat. The heat lost per metre from a wire 2 mm across held 40 K above the air, against the outer radius of the PVC insulation round it (k = 0.19 W/m K), with the surface losing 10 W/m²K to still air by convection and radiation, on a logarithmic radius axis; dashed, the bare wire. The first layers of insulation increase the loss, from 2.51 to a maximum of 12.11 W/m at an outer radius of 19 mm — the critical radius k/h — because they add surface faster than they add resistance. Beyond the critical radius the loss falls again, but so slowly — the resistance of the insulation grows only as the logarithm of its radius — that it does not get back down to the bare wire's until the PVC is 178 km thick. Any sheath a cable could carry makes it lose more heat, not less.

The insulation that makes a wire lose more heat

Wrap a hot water pipe in glass wool and it loses less heat. Wrap a thin electrical wire in PVC and it loses more — four times more, at the best thickness — and runs cooler for it. The insulation adds a resistance to heat flowing through it, but it also adds surface for the heat to leave from, and around a thin enough cylinder the surface wins. The crossover is a radius, k/h, set by the insulation and the air; below it, more insulation means more heat lost, and a small enough ball cannot be insulated by anything wrapped round it.

thermodynamics · Diffusion

Named alongside it

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

AbsorptionBuoyancyLapse rateMaximum densityStratificationThermal conductivityAdiabatic processAlbedoAtmosphereBlackbodyCabbelingCompressibility

All concepts