At thirty degrees the Earth keeps time with the sea breeze

What survives being wound up proved Kelvin’s circulation theorem — in an ideal, barotropic fluid under conservative forces, the circulation round a loop of marked fluid never changes — and then broke it three ways; the essay after it, the drift was the instrument, showed that where it holds it holds exactly. The second was a fluid whose surfaces of constant pressure and constant density cross, so that the integral of $\mathrm{d}p/\rho$ round a loop no longer vanishes and circulation grows out of a fluid at rest. It named the sea breeze as the first thing that term makes, and said, without computing it, that the rate it found “scales to something that spins up a coastal breeze in an hour”.

This essay does the scaling, and finds the claim right in its mechanism and wrong in what limits it. The push Kelvin’s broken theorem provides is large — large enough, if nothing else intervened, to drive a wind of twenty-six metres a second within an hour. Nothing like that blows. What intervenes first is not friction, which is the usual answer, but the day: the push reverses every evening, and the breeze can only grow for part of a day before it is pushed back. Friction — the brake whose time scale a fluid’s spin-down sets — then decides the hour of the breeze, and the Earth’s rotation, which turns it, decides something stranger — at one latitude the rotation keeps time with the push.

The push, from a loop #

Take a loop in the vertical plane across a coast: from the sea surface inland along the ground, up over the land to some pressure level aloft, back out to sea at that level, and down to the surface again. In hydrostatic air, $\mathrm{d}p/\rho = RT\,\mathrm{d}p/p$ on the vertical legs, and nothing on the horizontal ones if they are drawn along isobars. Round the whole loop,

$$\frac{\mathrm{d}\Gamma}{\mathrm{d}t} = -\oint \frac{\mathrm{d}p}{\rho} = R \ln\frac{p_0}{p_1}\,(T_\text{land} - T_\text{sea}),$$

which is Bjerknes’ circulation theorem of 1898 for an ideal gas: the warmer column is taller, and the tilted pressure surfaces it makes cross the density surfaces and drive the air round from the cool side at the bottom towards the warm side. The integral by direct quadrature matches the closed form to six parts in $10^{12}$.

The textbook numbers are a contrast of 10 K between land and sea, a loop reaching from the surface at 1000 hPa to 900 hPa — about 900 metres — and twenty kilometres inland and out. The circulation then grows at 302 m²/s per second, and spread along a loop about forty-two kilometres long that is an acceleration of 7.2 millimetres per second per second. In an hour, twenty-six metres per second; in four hours, over a hundred. A real sea breeze blows at a few metres per second. The estimate is right about the mechanism and at least five times wrong about the result, and the question is what the arithmetic has left out.

Part of the answer is the loop. A real land–sea contrast of 10 K is found only in the lowest few hundred metres, on a hot day, and the circulation spreads over fifty kilometres or more; three kelvins through the lowest 435 metres over fifty kilometres gives a push of 0.44 millimetres per second per second, sixteen times smaller. But even that push, left alone for a day, would build a wind of forty metres a second. Something limits the breeze in time, not only in size.

What stops it first is the evening #

The first figure is the breeze of Haurwitz’s model at the equator, where there is no Coriolis force. The push follows the land–sea contrast: strongest in the early afternoon, when the land is hottest, zero in the evening and reversed at night, when the land is colder than the sea and the land breeze blows. Measure the wind in units of the push divided by the daily frequency, $A/\omega$ — the speed the push would give in $1/\omega$ of a day, about 3.8 hours — and every figure here holds for any strength of push.

With no friction at all, the breeze does not grow without limit. It swings with the day, reaching exactly one unit at its strongest, and it peaks six hours after the push does — at the moment the push has fallen to zero and is about to reverse. The push has built the wind for six hours, then spends six hours stopping it, then drives it the other way. The daily reversal limits the breeze to about four hours’ worth of push. For the realistic contrast above, that is 6.0 metres per second, which is the size sea breezes are.

So the evening, not friction, is what first stops a sea breeze. Friction lowers it further — with a spin-down time of twelve hours the peak is 0.95 of the frictionless one, with three hours 0.62 — but the day has already done most of the work, and the right estimate of a breeze’s strength is the push times a quarter of a day, not the push divided by the friction.

Friction sets the hour #

What friction does decide is the timing, and the second figure shows it. A frictionless breeze lags its push by a quarter of a day: strongest at dusk if the heating is strongest after midday, which is not when sea breezes blow hardest. A breeze with a strong brake follows its push closely, in phase with it. Between the two, the lag is the angle whose tangent is the daily frequency over the friction, and for the spin-down times of a few hours that a turbulent, sunlit boundary layer over land gives, it is two to three hours: the strongest sea breeze comes in mid-afternoon, a few hours after the land is hottest.

That is what is observed, and it is a better use of the friction than setting the breeze’s size. A spin-down time of three hours puts the peak 2.5 hours after the strongest heating; one of an hour, just under one; twelve hours, nearly five. At higher latitudes the rotation shifts the onshore peak later again, because it turns part of the wind along the coast.

The Coriolis force turns it #

Away from the equator the breeze is not just onshore and offshore. A rotating planet turns every moving parcel of air to its right in the northern hemisphere, at a rate set by the Coriolis parameter $f = 2\Omega\sin\varphi$, and a wind that blows onshore for hours is turned to blow along the coast. The third figure draws the tip of the wind vector through a day. At 15° it traces an ellipse elongated across the coast; at 45°, an ellipse again; and in both the wind turns clockwise through the day, as sea-breeze hodographs recorded at coastal stations do, reaching every compass direction once.

At 30° the ellipse is nearly a circle, and larger: its radius is 1.82 units where the ellipses at 15° and 45° reach 1.19 and 1.16. The turning there does not fight the push; it keeps pace with it.

Thirty degrees #

The fourth figure puts the breeze’s size against latitude, and for weak friction it has a sharp peak at 29.9°. That is the latitude where the Coriolis parameter equals the daily frequency — where a parcel of air set moving and left alone would swing round in an inertial circle that takes exactly a day. A wind turning at the rate the Earth’s rotation turns it, pushed by a force that turns at the rate the day turns it, is a pushed oscillator at resonance. The push keeps adding in phase with the turning, and only friction limits the result: with a spin-down of a day, the breeze at 30° is 3.4 times as strong as at the equator.

With the strong friction of a sunny afternoon over land, the resonance all but disappears: a three-hour spin-down gives nearly the same breeze at every latitude. The resonance is therefore strongest where friction is weakest, which is over the sea, at night, and aloft. Inertial oscillations in the ocean’s surface layer, driven by winds that turn through the day, show the same peak near 30°, where the diurnal cycle and the inertial period coincide; the sea breeze’s own form of it is the land breeze and sea breeze reaching far offshore.

Why only one sense of turning resonates #

The push is a straight line: onshore by day, offshore by night, never along the coast. A straight to-and-fro push is the sum of two rotating ones, each of half the strength — one turning clockwise once a day, the other anticlockwise once a day — in the same way that a pendulum’s swing is the sum of two circular motions in opposite senses. In the northern hemisphere the Coriolis force turns a moving parcel clockwise, and a wind left to itself swings round clockwise once per inertial period. The clockwise half of the push therefore finds a wind already turning its way, and at the latitude where the inertial period is a day it stays in step with it indefinitely, adding to it every hour; the anticlockwise half fights the turning at every latitude and stays small.

That is why the hodographs at 30° are nearly circles traced clockwise: the resonant, clockwise half has grown until the other half hardly shows. It is also why the resonance is at 30° in both hemispheres but traced anticlockwise in the southern one, where the Coriolis force turns parcels to the left and the other half of the push is the one that keeps step. The Coriolis force on a breeze is as real as it is on a river’s bend, and unlike the force a bathtub is supposed to feel, it has hours to act and a push in time with it.

How far out to sea #

The fifth figure is the same resonance seen in space. The model so far is a single column of air at the coast. Rotunno showed in 1983 that for a stably stratified atmosphere, with buoyancy frequency $N$ and depth $H$, the circulation’s horizontal extent is $NH/\sqrt{|f^2 - \omega^2|}$. Poleward of 30°, where $f$ exceeds $\omega$, the circulation is trapped near the coast, within about a hundred and forty kilometres at 45° for a layer a kilometre deep. Equatorward of 30°, where $\omega$ exceeds $f$, the square root’s argument changes sign and the response is not trapped but radiated: the breeze sends waves of that wavelength away from the coast, and its influence is felt far out to sea and far inland. At 30° the scale has no bound at all, and in the linear theory the breeze reaches as far as there is water.

Both halves of the figure are the same fact as the fourth. At 30° a parcel’s own rotation is in step with the day, and a disturbance at the coast is felt as far as friction and nonlinearity allow; away from 30° the mismatch between the two frequencies confines it.

Why “hot air rises” misses it #

The sea breeze is usually explained as warm air rising over the land and cool air flowing in underneath to replace it, and that is a fair description of what a parcel of air does. It is not what drives it. Warm air over land is not rising because it is warm; a warm column in hydrostatic balance is simply taller than a cool one, and on its own it goes nowhere. What starts the motion is that the taller column’s pressure surfaces are lifted relative to the cool column’s, so that aloft there is higher pressure over land than over sea at the same height and at the surface the reverse, and the surfaces of constant pressure are no longer parallel to the surfaces of constant density. That crossing is the whole of the torque, and it acts on the loop as a whole — an inviscid flow that is not irrotational because its pressure and density have come apart. The rising air over land is one leg of a circulation; the sinking air over the sea is another, and neither is the cause of the other.

The land breeze is the same equations at night #

Nothing in the model distinguishes day from night except the sign of the push, and the land breeze — the gentler offshore wind of a clear night — is the other half of the same periodic solution. It is gentler because the push is: the land cools less at night than it warms by day, and the air over it becomes stable, which strengthens the friction. Its hodograph is the same ellipse traced through the other half of the day. Sailors along tropical coasts have always known the pair as one thing, a wind that comes in by day and goes out by night, and on coasts near 30° the pair turns right round the compass in a day.

What was checked #

The sixth figure is the ledger. The loop integral $-\oint\mathrm{d}p/\rho$, computed by quadrature in hydrostatic ideal-gas air, matches Bjerknes’ closed form to six parts in $10^{12}$ for three loops. The periodic solution of Haurwitz’s equations, written in closed form with a complex wind, matches a fourth-order Runge–Kutta integration started from rest and run for thirty days, at three latitudes, to one part in $10^{11}$. And with neither friction nor rotation the breeze is exactly $(A/\omega)\sin\omega t$, the frictionless swing of the first figure.

What the linear breeze leaves out #

The front. A real sea breeze arrives as a front: the cool sea air pushes inland as a gravity current with a sharp leading edge, a line of rising air and often a line of cloud, moving at a speed set by the density difference and the depth of the cool layer. The linear model has no front; it has a wind that rises and falls everywhere at once.

A push that changes shape. The heated layer deepens through the day, the loop’s size changes with it, and the land–sea contrast depends on the breeze itself, which carries cool air inland. The model’s push is a fixed cosine.

Linear friction. A spin-down time is the simplest stand-in for a turbulent boundary layer whose friction depends on the wind and on the heating.

Everything else that blows. Any large-scale wind across the coast adds to the breeze, and an offshore wind of a few metres a second can hold the front at the shore all day.

The convention the numbers depend on #

The push $A$ is the loop-mean acceleration, and wind speeds are in units of $A/\omega$, with $\omega$ the daily frequency, $2\pi$ per 24 hours. Friction is a linear drag with a spin-down time $1/\kappa$. The Coriolis parameter is $2\Omega\sin\varphi$ with the Earth’s sidereal rotation rate. Hours are counted from the time of the strongest land–sea contrast. The reach uses $N = 0.01$ per second and $H = 1$ km.

Who found it, and when #

Bjerknes’ circulation theorem is from 1898, and the sea-breeze loop has been its standard illustration since his textbooks. Jeffreys in 1922 described winds, like the sea breeze, in which the pressure force is balanced by friction rather than by rotation, and called them antitriptic. Haurwitz in 1947 wrote the damped, rotating, daily-forced model computed here and explained the breeze’s turning through the day. Rotunno’s linear theory of 1983 found the change of character at 30° latitude, between a trapped and a radiating response.

Still open: the front the linear breeze does not have #

The model’s breeze swells and fades everywhere at once. A real one arrives: the cool, dense sea air runs inland under the warm air as a gravity current, and its head moves at a speed set by the square root of the reduced gravity times the depth of the cool layer. The next calculation replaces the column with two layers of shallow water across a coast, heats the land side through the day, and asks when the front forms, how fast and how far it travels inland before evening stops it, and whether its speed — which depends on the depth, not on the push — or the push’s daily cycle decides how far inland a sea breeze reaches.