Illustrated Physics
physics, drawn
Series

Stratification — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Everything is decided against one line at 9.76 K per kilometre. Temperature against height for five environments, with the dry adiabat drawn heavy. A parcel lifted from the ground cools along the adiabat, at g/c_p = 9.76 K/km — a number with no meteorology in it, only gravity and the heat capacity of air. If the environment cools faster than that, a lifted parcel finds itself warmer than its surroundings and keeps going; if it cools more slowly, the parcel finds itself colder and sinks back. -5 K/km gives N² = 5.02e-4 s⁻², a period of 4.7 min; 0 K/km gives N² = 3.32e-4 s⁻², a period of 5.7 min; 6.5 K/km gives N² = 1.11e-4 s⁻², a period of 9.9 min; 9.8 K/km gives N² = -1.43e-6 s⁻², an e-folding time of 835 s; 12 K/km gives N² = -7.63e-5 s⁻², an e-folding time of 114 s. The classification is a comparison of two slopes and nothing else: no density appears in it, and the same cold air is stable under one profile and unstable under another.

    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.

    part 1 · fluids
  2. The cross a shaken cylinder leaves in a stratified fluid. The beams radiated by a small body oscillating in a fluid of buoyancy frequency 1.053e-2 per second — a period of 9.9 minutes — at 0.3, 0.6, 0.9 times that frequency. The disturbance does not spread in circles. It leaves along four rays, and the angle of those rays to the horizontal is fixed entirely by the ratio of the driving frequency to the buoyancy frequency: 17.5° at 0.3N, 36.9° at 0.6N, 64.2° at 0.9N. Nothing about the size of the body, the amplitude of the shaking or the wavelength enters. Drive it faster and the beams stand up; drive it slower and they lie down; drive it above N and there are no beams at all, because the dispersion relation ω = N cos φ has no solution. The short arrows across each beam are the wavevector, which is perpendicular to the beam — the dot products drawn here are 6e-17 — so the crests travel sideways across the ray while the energy travels along it, and a fluid doing this looks, in a photograph, as though its waves are moving at right angles to where they are going.

    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.

    part 2 · fluids
  3. The pattern that stands still while the air goes through it. Streamlines of a steady 20 metre-per-second wind over a bell-shaped ridge 800 metres high and 6.0 kilometres wide, in air whose buoyancy frequency is 1.05e-2 per second. Nothing in the picture is moving: the air crosses it from left to right at 20 metres a second and the waves stay where they are, because standing still is what selects them. The vertical wavelength is 2πU/N = 11.9 kilometres, and the crests lean upstream as they rise — checked here by finding the maximum displacement a quarter of a wavelength up, which sits well upstream of the ridge. That tilt is the signature of energy travelling upward, and it is the feature every hand-drawn version of this picture gets backwards. Cloud forms at the crest of each wave where the air is highest and coldest, which is why the lens-shaped clouds sit stationary in a moving airstream.

    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.

    part 3 · fluids
  4. A reflection that keeps the angle to gravity and not to the wall. An internal wave beam of frequency 0.5N reflecting from a slope of 12°, with the wavelength ratio for slopes of 12°, 20°, 28° computed beside it. The frequency fixes the angle the energy makes with the horizontal — 30.0° here — because the restoring force is gravity and gravity is vertical, so the reflected beam must leave at that same angle whatever the wall is doing. Incident and reflected rays are therefore not mirror images, and the wavelength changes on reflection by sin(θ+α)/sin(θ−α). A flat floor gives one, checked exactly. A slope approaching the ray's own angle gives infinity, checked as the limit, and that is where a basin's internal tide is compressed until it breaks.

    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.

    part 4 · fluids
  5. The latitude past which the tide cannot shed its energy by halves. 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°.

    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.

    part 5 · fluids
  6. A thrust that cannot push past the hump. The resistance a hull meets against its speed in knots, in two layers whose fastest interfacial wave travels at 1.02 knots: ordinary friction rising as the square of the speed, plus the drag of the interfacial waves, whose hump sits just below the wave speed. The horizontal lines are 2 steady engine thrusts. A ship settles where its thrust meets the resistance curve. Thrust 0.6 meets it at 0.81 knots; Thrust 1.3 meets it at 0.94, 1.00, 1.82 knots. A ship accelerating from rest reaches the first crossing and stops gaining speed there, below the hump, even when a faster crossing exists beyond it — which is the dead water sailors reported, a ship held to a fraction of its usual speed by a wave it cannot see.

    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.

    part 6 · fluids

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