Concept

Discriminant — where it appears

The quantity whose sign decides whether the roots of an equation are real, and therefore whether a solution of the stated kind exists at all. Thresholds set by one are exact and sharp rather than approximate, and they land on algebraic values like an eighth or a quarter.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Where the number says, and where it happens. Fourteen dimensionless groups on one logarithmic axis. The open circle on each row is the value at which the two terms the group compares are equal, which is one by the way the group is formed; the filled mark is the value at which the thing a reader cares about first changes by 1%. The bar between them is the distance the folklore phrase "of order one" hides, and it runs from nothing at all to a factor of 594.

What "of order one" is worth

A dimensionless group is built by comparing two terms, so it is one when the terms are equal — and that is the only thing it says. Where the behaviour actually changes is a separate question with a separate answer, and across fourteen groups on this site the two numbers differ by factors from one to five hundred and ninety-four.

regimes · Crossover $The error in the balance is the number itself. The fractional error in the geostrophic wind, against the Rossby number, on logarithmic axes. It is a straight line of slope one through the origin, and that is not an approximation: keeping the centripetal term gives V_g/V = 1 ± Ro exactly, so the error and the number are the same quantity. The geostrophic wind is one per cent right at Ro = 0.01 and a hundred per cent wrong at Ro = 1, which is the value the number is named for and is quoted as the boundary of the approximation.$

The balance that is its own error

Geostrophic balance is licensed by the Rossby number being small, and the fractional error in the geostrophic wind is the Rossby number — exactly, not approximately. So the balance everybody uses at Ro of order one is a hundred per cent wrong, and the same quadratic has a hard limit at a quarter that no anticyclone can pass.

regimes · Geostrophy

Every gradient in one plane, and where each one goes. The two invariants of a trace-free velocity gradient, R across and Q up. Under the restricted Euler equation the combination 27R²/4 + Q³ never changes, so every trajectory is one of these curves, and R never decreases, so every curve is travelled from left to right. Vieillefosse's line, where the combination is zero, has two branches: the left one runs into the origin and is the only way to reach it, and the right one is where every other trajectory ends, at infinity, in finite time.

A gradient left to itself

Follow a parcel's velocity gradient with nothing acting on it but its own square and the part of the pressure a single parcel can know about. Every starting gradient but a set of measure zero reaches infinity in a finite time, and it gets there as a sheet with its spin lying in the sheet.

kinematics · Deformation

Moving air needs a quarter of the inertia still air does. The first crossing against Stokes number for a cloud thrown into still air — time in units of the cloud's fastest convergence rate — and for a cloud carried by air converging on a line, in units of the air's strain rate. Still air only brakes: its threshold is one. Converging air keeps pushing: its threshold is a quarter, and heavy particles in it cross in about one strain time. Dots: particles integrated in the converging sine flow; the curve is the linearised oscillator.

Drag decides whether a cloud folds, and the air decides at what

A cloud of free particles keeps its velocities, and wherever it converges its paths cross in finite time: a caustic, the density infinite along a sheet. Give each particle a drag on the air and the answer depends on what the air is doing. In still air the drag only brakes, and a cloud folds only if its Stokes number is above one. In air that is itself converging the drag keeps pushing, and a quarter is enough — the same quarter that decides whether a droplet hits a wing.

kinematics · Acceleration

Named alongside it

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

Threshold Dimensionless Model limit Tolerance Asymptotics Characteristics Closure problem Coriolis Crossover Cyclone Eigenvalue Flow map

All concepts