Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Rossby number — where it appears

The ratio of a flow's inertia to the Coriolis force acting on it. It is enormous for a bathtub and small for an ocean gyre, which is why the earth's rotation decides one of them and not the other.

Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.

The bath is ten thousand times too small. The Rossby number — the ratio of the inertial term to the Coriolis term in the momentum equation — for eight flows at 45 degrees, on a logarithmic axis. Above one, rotation is a correction; below one, it is the physics. A draining bath sits at 3.2e+3 and a mid-latitude depression at 1.9e-1. The Coriolis term is not absent from the bath: it is present, computable, and four orders of magnitude smaller than the terms that decide what happens. Saying so is not the same as saying it is zero.

The bath does not know the hemisphere

The Coriolis term in a draining bath is not absent. It is present, computable, and about four hundred times smaller than what a hand left in the water an hour ago is still doing — which is a ratio rather than an opinion, and it is the same kind of argument as a Reynolds number.

misconceptions · Coriolis scale The current at the surface is 45° from the wind, and nothing sets that angle. The Ekman spiral drawn as a hodograph: each point is the velocity at one depth, and depth runs along the curve. At the surface the flow is at exactly 45 degrees to the wind that drives it — not approximately, exactly, and independently of the wind, the viscosity and the latitude. By one Ekman depth the flow has turned another radian and lost 1/e of its speed; by three it is a hundredth of the surface value and pointing back the way it came. The angle is a property of the equation having two terms in it, and nothing else.

The layer that stops at a depth

Every other boundary layer grows. This one does not — rotation supplies a frequency, the balance against diffusion supplies a length, and the transport that comes out contains the stress on the surface and not the viscosity underneath it.

viscous · Rotating 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 The number that settles the argument. How many turns the earth's contribution gets through in the time the vessel takes to drain: the draining time divided by the rotation period at the drain. A bathtub manages a tenth of one and there is nothing to see. The apparatus that settled it manages nearly nineteen hundred, and that is the whole of what its enormous area-to-drain ratio was for.

The bath that was only ever a wait

The Coriolis force is far too weak to steer a draining bath, and it does steer a large enough tank that has been left alone long enough. Both are true, and the number that separates them is not a force ratio — it is how many turns the earth's contribution gets through before the vessel is empty.

misconceptions · Coriolis scale The barrier is pinched to nothing in one direction. Where the inverse cascade is stopped, drawn in the plane of wavenumbers. The curve is the scale at which a Rossby wave oscillates as fast as an eddy turns over, and inside it the cascade cannot proceed. It is not a circle: it closes to a point on the axis of modes with no variation in longitude, so the cascade runs on unimpeded towards the largest scales in exactly one direction — and what it makes there is a band.

A cascade that arrives as stripes

Two-dimensional turbulence sends its energy upward in scale and, on a plane, piles it into a pair of vortices filling the box. On a rotating planet one extra term in the vorticity equation blocks that in every direction but one, and what arrives at the large scales is not a vortex at all but a set of bands.

turbulence · Two-dimensional The Earth turns a straight river into a very gentle bend. The transverse circulation through the depth of a straight river running at 1.2 m/s at latitude 50°N, in thousandths of the main flow, against height above the bed — and the circulation of the same river without rotation but bent to the left with a radius of 2V/f, 21 kilometres. The two nearly coincide: the Earth's rotation pushes the fast surface water right and the slow bed water left, exactly as a bend does, and for a river it is a bend of about twenty kilometres' radius everywhere at once.

The Earth bends every river a little

In 1860 Karl Ernst von Baer claimed that the Earth's rotation makes rivers in the northern hemisphere cut into their right banks. The Coriolis force on a river does drive a helix that throws surface water towards the right bank, the same helix a bend drives. For a mountain stream it is a ten-thousandth of what the stream's own bends do. For the great slow rivers of the far north it is a fifth to two-fifths, and there Baer's claim is not absurd.

misconceptions · Coriolis scale A slow current divides round the fluid over a hill (δ = 4). A uniform stream, left to right, over a Gaussian hill on the floor of a rotating layer, at δ = h₀/(H·Ro) = 4, where h₀/H is the hill's height against the depth and Ro the Rossby number U/fa; the onset for this shape is δ = 3.134. The dashed circle is the hill's e-folding radius. Water crossing the hill is squashed and, keeping its potential vorticity, spins clockwise; that anticyclone adds to the stream on one side and opposes it on the other. Here it stops the stream: the outlined region, 1.54 square radii, holds fluid that circles for ever and never leaves, and it sits beside the summit rather than on it, on the side where the swirl runs against the current.

The hill a slow current will not climb

The Taylor–Proudman theorem says a rotating fluid goes round an obstacle rather than over it, as if a solid column stood above it. Conserving potential vorticity turns that limit into a threshold. A current is stopped over a hill once the hill's height against the depth exceeds a fixed multiple of the Rossby number — 2 for a flat-topped hill, 3.13 for a Gaussian one, 16/3 for a cone — and the fluid it holds sits beside the summit rather than on it.

regimes · Geostrophy The Earth moves the whole train to the right. The centreline of a river like the lower Ob, in its own widths, flowing left to right with its right bank below: now, and 100 years later with and without the Earth's rotation. In that time the bends grow and move 0.77 widths downstream, identically in both. The Earth adds only a steady shift towards the right bank, 0.39 widths a century — the bend migration rate times the Earth's 39% share of the helix.

A river drifts right, and only a reach can show it

Add the Earth's rotation to the law by which a meandering river migrates and it changes nothing about the meanders: they grow and travel downstream exactly as before. It adds one thing, a steady sideways drift of the whole river towards its right bank, at the bend migration rate times the Earth's share of the helix — four-tenths of a width a century on the lower Ob. No single bend can show it, because each bend moves several times further on its own. A survey of about ten to twenty bends can.

misconceptions · Coriolis scale Stratification makes a hill easier to block. The strength at which a current over a Gaussian hill first stops and traps fluid, δc = (h₀/H)/Ro at onset, against the stratification B = NH/fa — the depth over the height fa/N to which a rotating stratified flow feels the hill. Homogeneous, it is the 3.134 of the unstratified layer. As B grows the hill's anticyclone gathers at the bottom, where it is stronger, and the threshold falls; in deep water it falls as 2.243/B, which is a Froude number: the current is blocked when N h₀/U exceeds 2.243, and the rotation has dropped out.

A stratified sea blocks a current sooner and traps less

A slow current in a rotating layer stops over a hill once the hill's height over the depth, divided by the Rossby number, passes 3.134, and a column of water over the hill is trapped from floor to surface. In a stratified sea the hill's anticyclone gathers near the floor and fades upward over a height of fa/N. It is stronger there, so the current stops sooner; and it is confined there, so what it traps is a cap, not a column. In deep water the threshold loses the rotation altogether and becomes a Froude number: a current is blocked when N h₀/U exceeds 2.24.

regimes · Geostrophy

Named alongside it

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

CoriolisModel limitRotating frameScalingGeostrophic balanceMisconceptionRegimeThresholdAngular momentumDimensionlessEddy viscosityMeasurement

All concepts