The bath does not know the hemisphere

The claim that bath water spirals one way in the northern hemisphere and the other way in the southern is one of the most widely repeated pieces of physics in circulation, and it is wrong. That much is well known.

What is less well handled is the correction. The usual reply — “the Coriolis force is far too weak at that scale” — is right, and it is an assertion rather than an argument unless somebody says how weak, compared with what, and what would have to be true for it to win. All three are computable.

The number that says which term matters #

The Coriolis parameter is f = 2Ω sin φ, which at 45° is 1.03 × 10⁻⁴ per second — an angular rate of about one turn per seventeen hours.

The Rossby number is U/fL: the size of the inertial term U²/L against the size of the Coriolis term fU. It is exactly the same construction as the Reynolds number, which is the inertial term against the viscous one, and it is read the same way. Above one, the rotation is a small correction to something else. Below one, the rotation is what is happening.

For a bath — a tenth of a metre per second over a third of a metre — it is 3,200. For a mid-latitude weather system it is 0.19, and that is why weather maps show the wind blowing along the isobars rather than across them.

The bath is on the wrong side of the line by three and a half orders of magnitude. That is the argument, and everything below is a way of making it concrete.

What the planet actually supplies #

Saying the term is small is not saying it is zero, so it is worth computing what the rotation offers a bathtub.

Water sitting perfectly still in a bath is not still. Seen from the rotating frame it carries the planet’s own circulation: the local vertical component of Ω is Ω sin φ, so the frame is a solid-body rotation at rate f/2, and the circulation round a ring of radius R is f·πR².

Now let that ring drain inwards, conserving its circulation. At radius r the swirl velocity is f(R² − r²)/2r, which for a half-metre bath draining to a one-centimetre plughole is 1.3 millimetres per second.

That is not zero. It is measurable with a float and a stopwatch, and it is the whole of what the planet has to offer.

What a hand left in the water offers instead #

Fill a bath and the water is moving. Not visibly, after a few minutes — but a residual circulation of a centimetre per second at the rim is entirely ordinary, and it is concentrated by exactly the same contraction.

Twenty-six micrometres per second. That is the standard the water has to meet before the hemisphere decides anything — a quarter of a millimetre a minute, which is far below what filling, temperature differences, or a hand withdrawn ten minutes earlier will leave behind.

How long the water has to sit still #

The residual does decay, and how fast is a computable time scale rather than a guess — though it is a borrowed one, and the site says so.

Viscous diffusion through the depth would take weeks. What actually spins a container down is the thin layer at the bottom, which sweeps fluid inwards and overturns the interior in about H/√(νf).

Four hours. Which is precisely why the experiments that do see the effect are constructed the way they are: Shapiro’s 1962 experiment in Boston and Trefethen’s 1965 replication in Sydney used a symmetric circular tank, filled it, covered it, and left it for twenty-four hours before pulling the plug — with a remote release, so that nobody was near the water. Under those conditions the rotation is reproducible: anticlockwise in Boston, clockwise in Sydney.

The effect is real. It is simply not what is being observed when a bath empties.

The inertial circle, and the scale rotation actually works at #

There is a second way to see the mismatch of scales, and it does not need a drain at all.

Give a parcel of water a push and let nothing act on it but the rotating frame. It goes round a circle — an inertial circle — of radius U/f, at the period 2π/f.

Five kilometres. That is the natural length scale of Coriolis motion at ordinary bath speeds, and a bath is a third of a metre across. The rotation would need ten thousand bath-widths to bend the flow appreciably, and the water reaches the drain in about a second.

That is the same argument as the Rossby number, seen as a distance rather than as a ratio, and it is perhaps the more vivid form: the planet’s rotation is not a weak effect at bath scale, it is an effect that operates at a completely different scale.

The same numbers nearer the equator #

Latitude enters everything through sin φ, so the comparison can be run again where the planet offers less.

The equatorial limit is worth stating because it is the cleanest refutation available. At the equator f is exactly zero, so a bath there has no planetary circulation whatever — and baths at the equator drain in a perfectly ordinary swirl, in whichever direction the last disturbance left them. If the hemisphere story were true, the water there would have to hesitate.

The shape of this error #

The misconception is worth classifying, because this field’s other entries share the shape.

It is not a false statement about physics. Everything in it is true — the Coriolis force exists, it acts in opposite senses in the two hemispheres, and it does influence the sense of rotation of large flows. What is wrong is the assignment of a mechanism to an outcome it does not control, with no comparison of magnitudes anywhere in the argument.

Equal transit time makes the same move with a true observation about a photograph. The Coandă effect applied to a wing takes a real identity and hands it a job it does not do. In every case the missing step is a comparison: this term against that one, in a single ratio, computed.

A third instance sits in the applied field rather than this one: the suction a jet pump appears to work by is real, measurable, and not what raises the pressure. The pattern is a true small term promoted to the explanation, and the fix in every case is the same division.

The habit that prevents it is the whole of the regimes field. Before asking whether an effect matters, form the dimensionless number that says so. It is the same discipline whether the competing term is viscosity, compressibility, gravity or the rotation of a planet, and it turns an argument about plausibility into an arithmetic.

The half of the rotation that was quietly dropped #

Every number above uses $f = 2\Omega\sin\varphi$, and that expression is not the Coriolis term. It is the vertical component of it, and keeping only that component is an approximation nobody named while it was being made — which, in an essay about arguments that skip a comparison of magnitudes, deserves the same treatment as everything else here.

The planet’s rotation vector at latitude $\varphi$ has a vertical part $\Omega\sin\varphi$ and a horizontal part $\Omega\cos\varphi$ pointing north. The first turns horizontal motion within the horizontal plane, which is the effect this whole essay has been sizing. The second couples horizontal motion to vertical motion: it pushes eastward-moving fluid upwards and westward-moving fluid downwards, and it pushes rising fluid westwards.

Geophysics drops it, and the justification is a ratio of exactly the kind used above. The horizontal component acts on the vertical velocity, and in the atmosphere a typical vertical velocity is a centimetre a second against ten metres a second horizontally — so the neglected term is smaller than the retained one by about a thousand. That is the traditional approximation, it is made in every weather model, and it is defensible for the same reason and by the same arithmetic as everything else here.

It is not zero, and it is largest exactly where the retained term vanishes. At the equator $\sin\varphi = 0$ and $\cos\varphi = 1$, so the component this essay has been computing disappears and the discarded one reaches its maximum — which qualifies the equatorial claim above. What is true there is that the planet supplies a bath no circulation about a vertical axis, not that the rotation has stopped acting.

The discarded component is measurable and is measured. An object moving east is pushed slightly upwards and therefore weighs slightly less; moving west, slightly more. The size is $2\Omega\cos\varphi \,u$, which at the equator for something travelling ten metres a second east comes to $1.5\times10^{-3}$ m/s² — about fifteen parts in a hundred thousand of gravity.

That sounds ignorable and is not. Marine and airborne gravity surveys work to a fraction of a milligal, which is $10^{-8}$ of $g$, and the correction above is a hundred and fifty milligals — four orders of magnitude larger than the signal being looked for. The Eötvös correction is applied to every gravity measurement taken from a moving platform, and it depends on the vessel’s eastward speed, so a survey ship’s navigation record is part of its gravimetry.

Which is a pleasing symmetry with the bath. The same term is a thousand times too small to matter in one instrument and ten thousand times too large to ignore in another, and only the division says which.

What is not computed here #

No drain is modelled. The contraction argument assumes a ring of fluid moving inwards conserving its circulation, which is an idealisation: a real drain has a boundary layer on the floor that transports fluid radially inwards and carries angular momentum with it, and the vortex above it is maintained by that layer rather than by inviscid contraction alone.

No free surface. The dimple, the air core and the sudden increase in swirl as a bathtub vortex forms are all free-surface effects and none of them is here. What the site does own about the vortex itself is what a concentrated line of vorticity does once it exists and why a free vortex is irrotational everywhere except at its centre, neither of which needs a planet.

The spin-down time is borrowed. It is a scaling from rotating-flow theory, quoted to produce a time scale, and no boundary layer on a rotating floor is solved anywhere on this site.

The centrifugal term is absorbed rather than dropped. In a rotating frame there are two apparent forces; the centrifugal one is folded into an effective gravity, as it is in every geophysical treatment, and this site follows that convention rather than pretending the term is absent.

Where rotation does decide things #

Everything above is about why a bath is the wrong place to look. It is worth ending on where the same term is not merely present but dominant, because the contrast is the point.

At Rossby numbers well below one — a weather system, an ocean gyre, the flow in the Earth’s core — the inertial term is negligible and the balance is between the pressure gradient and the Coriolis force alone. That is geostrophic flow, and its signature is that the wind blows along the isobars rather than down the pressure gradient, which is why a weather map can be read as a streamline plot.

The stretching term is the mechanism, and it is the same one a tornado runs on: a column of fluid that is squeezed radially spins faster, whether the spin it started with came from a weather system or from the planet.

It also produces a conservation law worth naming beside this site’s own: the absolute vorticity of a column of fluid, planetary plus relative, is conserved as the column stretches. That is why air crossing a mountain range acquires a spin it did not have, and it is the same circulation argument this essay used for a drain, applied at the scale where it is the largest term rather than the smallest.

What the misconception gets right, and why that keeps it alive #

An error this durable usually has something true holding it up, and this one has two things.

The direction is right. Where the Coriolis term does decide a rotation, the sense is exactly the one the story claims: anticlockwise in the northern hemisphere, clockwise in the southern. Cyclones obey it, ocean gyres obey it, and the carefully-run tank experiments obey it. So the story is not merely plausible; it is correct about the sign, which makes it very hard to dislodge.

The mechanism exists. There is no need to explain away a fictitious force or a misreading of a photograph. The Coriolis term is in the momentum equation, it acts on the water in the bath, and the circulation it supplies is computable and non-zero. Somebody defending the story is defending real physics applied at the wrong scale, which is a much stronger position than defending equal transit time.

What is missing is a single division. The error is not in any of the physics; it is in the absence of a comparison. Every quantity in the correct argument is elementary — a rotation rate, a length, a speed — and the whole refutation is that one number is three thousand times another.

That is worth stating as a general lesson about this field. Some misconceptions are false statements and can be corrected by supplying the true one. This kind cannot: every statement in it is true, and the repair is a ratio that nobody thought to form. There is no way to spot such an error by reading more carefully. It has to be computed.

Who found it, and when #

Gaspard-Gustave de Coriolis published the acceleration in 1835, in a paper about waterwheels and other rotating machines rather than about the planet. William Ferrel applied it to atmospheric motion in 1856. Carl-Gustaf Rossby gave the number its name in the 1930s, at the founding of dynamic meteorology.

The bathtub question was settled experimentally in the 1960s and has been repeated by students ever since, and it goes on being repeated in popular science because the correct answer is a comparison of two magnitudes and the incorrect one is a story. That asymmetry is why this field exists.

The version of the question that is worth asking #

The bathtub question is a bad question with a good one hiding behind it, and the good one is where this rung earns its place in a collection about fluids rather than about folklore.

At what size does the planet’s rotation take over? Set the Rossby number to one and solve for the length: L = U/f. At a metre per second in mid-latitudes that is about ten kilometres. Below that scale a flow is inertial and the rotation is a perturbation; above it, the rotation dominates and the flow becomes geostrophic.

Ten kilometres is a useful number to carry. It says that a river is not affected in any interesting way, that a large lake is marginally, that a sea breeze is, and that any weather system is entirely. It also explains why the effect is invisible in ordinary experience and inescapable in meteorology, without any appeal to the strength of the force at all.

The same construction answers the related question about a tornado, which is often supposed to be Coriolis-driven for the same reason a bath is. A tornado at sixty metres a second across three hundred is at a Rossby number of two thousand: its rotation comes from the parent storm, and the storm’s comes from the shear of the wind field it grew in. A tornado does not know which hemisphere it is in either — although the storms that spawn them do, which is why the sense is consistent in practice and not in principle.

Where this ladder goes next #

One more explanation of the same shape, and the last: the account of a siphon in which the atmosphere pushes the liquid over the hump. The atmosphere is real, the pressure is real, and the flow rate does not contain the hump’s height at all.