A parcel let go 300 m up comes back, and does not stop there
At its defaults it draws a parcel let go 300 m up comes back, and does not stop there. Height of a parcel released 300 m above where it started, against time, for each environment. Nothing is holding it and nothing is damping it, so a stable column does not return the parcel to where it belongs — it overshoots by as much as it was displaced, every time, which is what an oscillation is. At -5 K/km the integrated period is 4.67 min against 4.67 from 2π/N; At 0 K/km the integrated period is 5.75 min against 5.75 from 2π/N; At 6.5 K/km the integrated period is 9.95 min against 9.95 from 2π/N. The two agree to about a per cent, and the difference is the amplitude: the restoring force is computed from the full temperature difference here, which is not quite proportional to the displacement. Any rate steeper than the adiabat leaves the picture instead of oscillating.
stratified-parcel is one function in lib/figures/fluids.js —
matter that will not hold a shape, and the forces in it. Everything below came out
of it during this build, at parameters taken from the essays rather than invented for this
page. A figure here is the figure a reader meets in an essay, and if the generator changes,
this page changes with it.
At its defaults
Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.
Height of a parcel released 300 m above where it started, against time, for each environment. Nothing is holding it and nothing is damping it, so a stable column does not return the parcel to where it belongs — it overshoots by as much as it was displaced, every time, which is what an oscillation is. At -5 K/km the integrated period is 4.67 min against 4.67 from 2π/N; At 0 K/km the integrated period is 5.75 min against 5.75 from 2π/N; At 6.5 K/km the integrated period is 9.95 min against 9.95 from 2π/N. The two agree to about a per cent, and the difference is the amplitude: the restoring force is computed from the full temperature difference here, which is not quite proportional to the displacement. Any rate steeper than the adiabat leaves the picture instead of oscillating.
A parcel that leaves the air behind
The options are the ones The cloud that cools more slowly than the air passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
A parcel of air lifted from the ground at 30 °C with a dew point of 23 °C, against height, beside the air it rises through, which cools at 6.5 K per kilometre to the tropopause at 15 km. Unsaturated, the parcel cools at the dry rate and is soon colder than its surroundings; at 890 m it reaches saturation, a cloud base, and from there cools along the moist adiabat. From 2080 m it is warmer than the air round it and accelerates upward, until at 13.4 km it is cold again. The shaded area, the buoyancy integrated over that climb, is the convective available potential energy: 2543 J/kg. If all of it went into motion the updraft would reach 71 m/s, the square root of twice the CAPE. The 99 J/kg below the level of free convection is the push it needs to get started.
How fast saturated air cools as it rises
The options are the ones The cloud that cools more slowly than the air passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The rate at which a rising parcel of saturated air cools, in kelvin per kilometre, against its temperature, at pressures of 1000, 700 and 500 hectopascals, with the dry rate g/cₚ = 9.75 K/km dashed. Condensing vapour gives up its latent heat to the parcel, so saturated air cools more slowly — and how much more slowly depends on how much vapour there is to condense, which rises steeply with temperature: 3.8 K/km at 25 °C near the ground, 5.8 at 0 °C and 700 hPa, 7.7 at −20 °C and 500 hPa. Cold air holds so little water that its saturated rate approaches the dry one; warm, humid air rising through the lower troposphere cools at less than half the dry rate.
Three kinds of stability, and the one in between
The options are the ones The cloud that cools more slowly than the air passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
How a layer of air responds to a parcel displaced through it, classified by the layer's lapse rate (vertical) against its temperature at 850 hPa. Below the saturated rate (lower shading) the layer is absolutely stable: a parcel returns whether or not it is saturated. Above the dry rate it is absolutely unstable, which happens only in a thin layer next to ground heated by the sun. In between it is conditionally unstable: stable to a dry parcel, unstable to a saturated one. The standard atmosphere's 6.5 K/km (dashed) lies in that band at every temperature above −2 °C, and the band is widest in warm air, 6.2 K/km wide at 25 °C. Most of the troposphere, most of the time, is stable until something lifts air to saturation, and then is not.
Where the parcel's water goes, and what it pays
The options are the ones The cloud that cools more slowly than the air passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
For the parcel lifted from 30 °C with a dew point of 23 °C, against height: the water it still carries as vapour, in grams per kilogram of air (blue), and the warming the vapour it has already condensed has given it, in kelvin (red). It starts with 17.7 g/kg and keeps all of it to cloud base at 890 m. Above, it condenses: 8.8 g/kg are left at 5 km and 1.46 at 10 km. The latent heat released by that condensation, L/cₚ per unit of water, has warmed it by 22 K by 5 km and 40 K by 10 km relative to dry air lifted the same way. That warmth is what the parcel spends on its buoyancy, and the water that falls out as rain is the receipt.
A wind that comes down warmer than it went up
The options are the ones The cloud that cools more slowly than the air passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Air carried over a ridge 3 km high, starting at sea level at 20 °C with a dew point of 14 °C, drawn along its path with its temperature labelled. It cools at the dry rate to its condensation level at 750 m, then at the saturated rate to the crest, reaching 1.0 °C, raining out all but 5.8 of its 10.0 g/kg of water on the windward slope. Coming down the far side with no water left to evaporate, it warms at the full dry rate all the way and arrives at sea level at 30.2 °C, 10.2 K warmer than it set out, with a dew point of 5.9 °C. The heat is the latent heat of the rain left behind. That warm, dry wind on the lee of a range is the foehn of the Alps and the chinook of the Rockies.
What checks it
physicscheck asserts something about stratified-parcel that
could fail — it draws it and measures the result against a value reached some other
way.
Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
The cloud that cools more slowly than the air
Dry air carried upward cools at 9.8 degrees per kilometre, and the atmosphere, which cools at about 6.5, is stable against it: a dry parcel nudged upward ends up colder than its surroundings and sinks back. Saturated air is different. As it rises its vapour condenses, the latent heat warms it, and it cools at only four to six degrees per kilometre in the lower atmosphere. The same column of air is then unstable to it. Most of the troposphere, most of the time, is stable until something lifts air to saturation, and then it is not — and that one asymmetry runs from a fair-weather cumulus to a thunderstorm, to the warm wind that comes down the far side of a mountain.
FluidsThe lake that holds its gas until it does not
On the night of 21 August 1986, Lake Nyos in Cameroon released about a cubic kilometre of carbon dioxide in a few hours, and some 1,700 people died in their sleep in the valleys below. The lake was not boiling and it was not erupting. Its deep water had been holding the gas safely for decades, in a layering that was stable as long as nothing moved. Lift that water far enough and it cannot stop rising. The same instability, in a pipe, now drains the danger away.
FluidsThe latitude past which a tide cannot split
The ocean's internal tide carries about a terawatt, and somewhere it has to be broken into waves small enough to mix the water. One of the ways it breaks is by pumping waves at half its own frequency, the way a child on a swing pumps at twice the swing's. Those half-frequency waves cannot exist where the planet's rotation forbids oscillations that slow — poleward of 28.8° for the semidiurnal tide — so the route has an edge on the map, fixed by the Moon's period and the Earth's spin.
FluidsThe layer a parcel cannot leave
Whether a column of air overturns is not decided by its density but by a difference of two gradients — the rate the environment cools with height, and the rate a lifted parcel cools on its own. Subtract one from the other and what is left is a restoring force per unit displacement, so a stable atmosphere rings at a period of minutes and an unstable one has no period at all.
FluidsThe 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.
FluidsThe plume that stops below the surface
Hot smoke rises, oil and gas from a seabed leak rise, and in a fluid of uniform density they would go on rising for ever. Real oceans and atmospheres are stratified, denser lower down, and a rising plume carries its own downfall with it: it rises by drawing in the fluid around it, and the fluid it draws in near the bottom is denser than the fluid it is carried up into. Its buoyancy runs out, it overshoots, and it spreads sideways at a height set only by how buoyant its source is and how strongly the surroundings are layered — a quarter power of one and a minus three-quarters power of the other.
FluidsThe 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.
FluidsThe reflection that changes the wavelength
An internal wave's frequency fixes the angle its energy makes with gravity, so a sloping wall cannot send it back the way a mirror would. The reflected beam leaves at the same angle to the vertical rather than the same angle to the wall, its wavelength changes by a factor that diverges when the slope matches the ray, and in a closed basin the changes accumulate until every ray in the fluid lies on one line.
FluidsThe salt that sinks through a stable sea
Warm salty water lying on cold fresh water is heavier at the bottom at every depth, and by every test of static stability it should stay where it is. It does not. Heat leaks sideways out of a thin tongue of sinking water a hundred times faster than salt does, so the tongue arrives cooled and still salty — heavier than its new surroundings — and keeps going. A column can be stable by density and unstable by diffusion, and most of the subtropical ocean is.
FluidsThe wave that holds a ship back
In 1893 the polar ship Fram, which could make four or five knots, was held to about one in a calm Arctic sea with nothing visible in the water. The sea was layered — a metre or two of fresh meltwater over salt — and the ship was making a wave on the boundary between the layers, a wave that travels at about a knot and carries away almost all of a slow ship's power. Below that speed the drag is a hump no steady thrust can climb; above it the wave cannot keep up and the drag falls away.
FluidsThe wave that is required to stand still
A stratified fluid supports internal waves at every wavelength there is. Put a steady wind over a ridge and the requirement that the pattern stay put picks exactly one of them — 2πU/N, and nothing about the mountain appears in it. The clouds that mark the crests sit still while the air goes through them at twenty metres a second, and the momentum the wave carries away is delivered thirty kilometres up.
FluidsThe wave that picks an angle
Shake a rod slowly in a tank of salty water layered by density and the disturbance leaves along four straight beams, at an angle fixed entirely by how fast the shaking is. Change the wavelength and the angle does not move. Change the frequency and it does. Above the buoyancy frequency there is no wave at all.