Generator

A parcel let go 300 m up comes back, and does not stop there

One function in the fluids library, called 39 times across 6 essays. Below: what it draws at its defaults, what it draws at every branch an essay asks for, whether the site's own gate puts a claim to it, and everywhere it is called.

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.

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.

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.

The latitude past which the tide cannot shed its energy by halves

The options are the ones The latitude past which a tide cannot split passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Frequency in cycles per day against latitude. The curve is the inertial frequency, 2Ω sin(latitude), below which no internal wave can oscillate; the shaded region above it is where internal waves exist. The horizontal lines are the semidiurnal and diurnal tides and the frequencies half of each, where a parametric instability would put the waves the tide decays into. Each line ends where it meets the curve, which is its critical latitude: M2, semidiurnal at 1.932 per day, 74.5°; M2 ÷ 2 at 0.966 per day, 28.8°; K1, diurnal at 1.003 per day, 30.0°; K1 ÷ 2 at 0.501 per day, 14.5°. Equatorward of 28.8° the semidiurnal tide can feed waves at half its frequency; poleward of it those waves cannot exist and that route is closed. The diurnal tide's subharmonic is confined within 14.5° of the equator, and the diurnal tide itself cannot propagate as a free internal wave poleward of 30°.

Pumping at one frequency grows waves at half of it

The options are the ones The latitude past which a tide cannot split passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Pumping at one frequency grows waves at half of it. Where a small oscillation whose restoring force is modulated at the tidal frequency grows rather than stays bounded, against its own natural frequency as a fraction of the tide's and the strength of the modulation. The shaded tongues are the unstable regions, found from the Floquet multipliers of an oscillator whose squared frequency is modulated by a fraction ε at the tidal frequency, over one tidal period. The widest tongue is centred on half the tidal frequency, and at small strength it is a quarter of ε wide, as perturbation theory predicts; at ε = 0.19 it spans 0.478 to 0.525 and grows by 0.15 e-folds per tidal period at its centre. The second tongue, at the tidal frequency itself, opens only as the square of the strength and is far narrower. A tidal wave straining the ocean is this pump, and the small waves riding on it are the oscillators: the ones that grow fastest are those at half its frequency.

Where a small oscillation whose restoring force is modulated at the tidal frequency grows rather than stays bounded, against its own natural frequency as a fraction of the tide's and the strength of the modulation. The shaded tongues are the unstable regions, found from the Floquet multipliers of an oscillator whose squared frequency is modulated by a fraction ε at the tidal frequency, over one tidal period. The widest tongue is centred on half the tidal frequency, and at small strength it is a quarter of ε wide, as perturbation theory predicts; at ε = 0.19 it spans 0.478 to 0.525 and grows by 0.15 e-folds per tidal period at its centre. The second tongue, at the tidal frequency itself, opens only as the square of the strength and is far narrower. A tidal wave straining the ocean is this pump, and the small waves riding on it are the oscillators: the ones that grow fastest are those at half its frequency.

Growth that stops at a latitude, and how sharply

The options are the ones The latitude past which a tide cannot split passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Growth that stops at a latitude, and how sharply. The fastest growth, in e-folds per tidal period, available to any internal wave the semidiurnal tide can pump parametrically, against latitude, for modulation strengths of 0.1 and 0.3. At each latitude the natural frequency is searched over everything above the local inertial frequency, and the growth is the Floquet rate at the best of them. Equatorward of 28.8° a wave at exactly half the tidal frequency is allowed and the growth is the resonant value; poleward of it the nearest allowed wave is detuned by the inertial frequency, and growth continues only while the detuning fits inside the tongue. With ε = 0.1 it stops at 30.0°; with ε = 0.3 it stops at 31.5°. A weak pump switches off almost exactly at the critical latitude; a strong one reaches a degree or two past it.

The fastest growth, in e-folds per tidal period, available to any internal wave the semidiurnal tide can pump parametrically, against latitude, for modulation strengths of 0.1 and 0.3. At each latitude the natural frequency is searched over everything above the local inertial frequency, and the growth is the Floquet rate at the best of them. Equatorward of 28.8° a wave at exactly half the tidal frequency is allowed and the growth is the resonant value; poleward of it the nearest allowed wave is detuned by the inertial frequency, and growth continues only while the detuning fits inside the tongue. With ε = 0.1 it stops at 30.0°; with ε = 0.3 it stops at 31.5°. A weak pump switches off almost exactly at the critical latitude; a strong one reaches a degree or two past it.

Near its critical latitude a wave stops carrying its energy away

The options are the ones The latitude past which a tide cannot split passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Near its critical latitude a wave stops carrying its energy away. The horizontal group speed of an internal wave at four tidal frequencies, as a fraction of its value at the equator, against latitude, from the hydrostatic dispersion relation — the squared frequency is f squared plus N squared times the squared ratio of horizontal to vertical wavenumber — at fixed vertical wavelength, where the fraction is the square root of 1 − (f/ω) squared. Each falls to zero at its critical latitude, because a wave whose frequency approaches the inertial frequency becomes an inertial oscillation, turning in place rather than travelling. For a vertical wavelength of 500 m in water with N = 0.002 s⁻¹, a wave at half the semidiurnal frequency moves 14 km a day at the equator, 10 at 20° and 3 at 28°. Energy handed to such waves near the critical latitude therefore stays where it was handed over, as vertical shear, which is what a place that mixes the ocean needs.

The horizontal group speed of an internal wave at four tidal frequencies, as a fraction of its value at the equator, against latitude, from the hydrostatic dispersion relation — the squared frequency is f squared plus N squared times the squared ratio of horizontal to vertical wavenumber — at fixed vertical wavelength, where the fraction is the square root of 1 − (f/ω) squared. Each falls to zero at its critical latitude, because a wave whose frequency approaches the inertial frequency becomes an inertial oscillation, turning in place rather than travelling. For a vertical wavelength of 500 m in water with N = 0.002 s⁻¹, a wave at half the semidiurnal frequency moves 14 km a day at the equator, 10 at 20° and 3 at 28°. Energy handed to such waves near the critical latitude therefore stays where it was handed over, as vertical shear, which is what a place that mixes the ocean needs.

The same pump, three latitudes

The options are the ones The latitude past which a tide cannot split 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 same pump, three latitudes. A small internal wave pumped by the semidiurnal tide at a strength of 0.2, integrated for 30 tidal periods, at 20°, 29.5°, 33°, its amplitude on a logarithmic axis. At each latitude the wave takes the frequency nearest half the tide's that is still above the local inertial frequency. At 20° that is 0.500 of the tidal frequency; it grows by 0.157 e-folds a period, matching its Floquet rate, and ends 1.9 decades up; at 29.5° that is 0.511 of the tidal frequency; it grows by 0.145 e-folds a period, matching its Floquet rate, and ends 1.9 decades up; at 33° that is 0.565 of the tidal frequency; it lies outside the unstable tongue and only beats. The first grows because it is resonant; the second, just past 28.8°, grows more slowly because it is detuned but still inside the tongue; the third cannot reach resonance at all.

A small internal wave pumped by the semidiurnal tide at a strength of 0.2, integrated for 30 tidal periods, at 20°, 29.5°, 33°, its amplitude on a logarithmic axis. At each latitude the wave takes the frequency nearest half the tide's that is still above the local inertial frequency. At 20° that is 0.500 of the tidal frequency; it grows by 0.157 e-folds a period, matching its Floquet rate, and ends 1.9 decades up; at 29.5° that is 0.511 of the tidal frequency; it grows by 0.145 e-folds a period, matching its Floquet rate, and ends 1.9 decades up; at 33° that is 0.565 of the tidal frequency; it lies outside the unstable tongue and only beats. The first grows because it is resonant; the second, just past 28.8°, grows more slowly because it is detuned but still inside the tongue; the third cannot reach resonance at all.

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 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.

The 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.

The 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.

The 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.

The 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.

The 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.

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