The sea-breeze front runs till evening only near the equator

At thirty degrees the Earth keeps time with the sea breeze priced the push that drives a sea breeze. Where warm air over land meets cool air over the sea, surfaces of constant pressure and of constant density cross, and Kelvin’s theorem fails in exactly the way Bjerknes showed it must: a circulation grows out of still air. That essay’s breeze was a column of air across the coast, damped and turned by the Earth’s rotation, swelling and fading everywhere at once. It found that the evening, not friction, stops the breeze growing, and that at thirty degrees, where the inertial period is a day, the turning keeps time with the push.

It ended on what that picture leaves out. A real sea breeze arrives. Someone standing twenty kilometres inland feels nothing all morning and then, within a few minutes, a cool wind off the sea, a drop in temperature and a rise in humidity, often under a line of cumulus that marks where the warm air is being lifted over the cool. The cool sea air runs inland underneath the warm air as a gravity current, the same kind of flow as cold air spilling out of an opened refrigerator or a muddy river running along the floor of a lake, and it has a head — a leading edge that moves at a speed of its own.

The question that essay left was whether the front’s speed, which depends on the cool layer’s depth and density rather than on the push, or the push’s daily cycle decides how far inland a sea breeze reaches. The answer depends on the latitude. Near the equator the daily cycle does: the head runs for as long as the land is warmer, at the speed its depth allows, and stops only in the evening. In mid-latitudes neither does. The Earth’s rotation stops the head hours before evening, at a distance fixed by the cool layer’s Rossby radius, and nothing about the length of the afternoon can move it.

A head that sets its own speed #

The model is the simplest that has a front. The cool marine air is a single layer of shallow water, 600 metres deep offshore, under a deep layer of warmer air that stays still. At the start of the day the air over the land and over the sea are the same temperature, so the cool layer has no reason to move. As the land warms, the air over it becomes lighter than the marine air, and the marine layer feels a reduced gravity in proportion to the temperature contrast: the contrast rises to 3 kelvin in the middle of a twelve-hour afternoon and falls back to nothing at its end.

The marine layer is free to slump inland, and where it meets the warm air there is a head. A gravity current’s head is not governed by the equations in the interior of the current; it is a moving boundary with a condition of its own, that its speed is a fixed multiple of the square root of the reduced gravity times its own depth. That multiple is the head’s Froude number. Benjamin showed in 1968 that for a current much shallower than the fluid above it, with no energy lost at the head, the number is the square root of two. Real heads lose energy to the turbulent billows that curl back over them, and Huppert and Simpson’s measurements put it at 1.19, which this calculation uses; Benjamin’s value is kept as an upper bound. It is the same Benjamin whose reading of swirling flows as subcritical or supercritical the swirl that holds a wave still follows, and the head is the same kind of object: a place where the flow’s own wave speed, not anything imposed from outside, decides what can happen.

Behind the head, the interior is shallow-water flow, and information travels along its characteristics. The forward one carries a combination of the velocity and the wave speed — a Riemann invariant, the same object the jump the equations allow is built from — out of the deep reservoir offshore and into the head. Combined with the head’s own condition, it fixes the head’s depth and speed. With a constant contrast and no rotation, the head runs at a steady speed, 1.19 times the wave speed at its own depth, at a depth a little under four tenths of the reservoir’s, and the calculation reproduces both to three parts in a thousand.

At the equator the front runs until evening #

The first figure is the front’s distance inland against the hours since the land first became warmer, at four latitudes. The equatorial front starts slowly, since the contrast starts at nothing, and speeds up through the afternoon: 7 kilometres in the first hour, 33 by the third and 87 by the sixth, when its head is moving at 5.4 metres a second. It slows as the contrast fades and stops, 185 kilometres inland, only when the land is no longer warmer.

That reach is almost exactly what a much simpler estimate gives. Take the steady head speed of a current fed from a deep reservoir, at each hour’s contrast, and integrate it over the afternoon: 189 kilometres. The full calculation reaches 98 per cent of it. The head is quasi-steady, adjusting to each hour’s contrast far faster than the contrast changes, so at the equator the reach is simply the head’s speed carried through the afternoon. Both of the candidate answers to the question decide it, as a product.

The second figure shows the cool layer’s depth across the coast as the day goes on. Behind the head the layer thins back towards the sea in a long, smooth slope, the rarefaction that feeds the head with the cool air it carries inland. The dial moves the latitude.

At the equator the head is a step about two hundred metres high for most of the afternoon, deepening late in the day as the contrast falls and the same head speed needs more depth to sustain it. At 45° the picture is different from the fifth hour on: the head thins rather than deepens, to thirty metres by the eighth hour and to nothing by the eleventh, and the layer behind it no longer slopes smoothly back to the sea.

In mid-latitudes the Earth stops it #

The first figure’s other curves are the same front at 30, 45 and 60 degrees. All four start alike, since rotation needs time to act, but by the fourth hour the high-latitude fronts are falling behind, and they stop well before the contrast ends: at 45 degrees the front has covered 99 per cent of its final 99 kilometres by hour nine, and at 60 degrees, 80 kilometres by hour eight. At 30 degrees it is 133 kilometres, reached just before the end.

The mechanism is geostrophic adjustment. The cool air running inland is turned to its right in the northern hemisphere by the Coriolis force, so a wind along the coast grows behind the head, and the Coriolis force on that along-shore wind pushes back on the current’s inland motion. Given long enough, a current released across a coast settles into a balance in which its slope is held up by the along-shore wind rather than spreading further: the flow becomes a coastal jet, and the edge of the cool air stops. The distance at which it stops is set by the Rossby radius, the distance a wave of the layer travels in one over the Coriolis parameter of time.

The adjustment is the same turning that the drift a rotating planet takes back found acting on a wave’s drift: any flow left to itself on a rotating planet is swung round at the inertial frequency until a pressure gradient holds it. The difference here is that the current builds its own pressure gradient as it goes, in the slope of the cool layer, so the swing does not have to be completed; the head stops as soon as the along-shore wind behind it is strong enough to hold the slope in balance. That takes about half an inertial period — 16.9 hours at 45 degrees, so a little over eight — which is why the high-latitude fronts stop within eight or nine hours of the contrast beginning, while it still has hours to run.

The third figure plots the reach against latitude. Near the equator the reach is the quasi-steady integral, since the Coriolis parameter vanishes there. Poleward of about thirty degrees it follows a single curve, 1.33 times the Rossby radius of the cool layer at the afternoon’s peak contrast: 1.33 at 45 degrees, 1.32 at 60 and 1.31 at 70. Between the two, where the Rossby radius and the afternoon’s run are comparable, the reach moves smoothly from one regime to the other.

An observer on the ground cannot tell the two regimes apart for most of the day. Twenty kilometres inland the front arrives two and a quarter hours after the contrast begins at the equator and at 45 degrees alike; fifty kilometres inland, at 4.2 hours and 4.4. Ninety kilometres inland the difference shows, 6.3 hours against 7.7, and a hundred kilometres inland the mid-latitude breeze never arrives at all while the equatorial one passes with half its afternoon still to run. The regimes differ in where the breeze ends, not in how it begins.

The ratio of 1.33 is a property of how a rising and falling contrast releases a current, not of the numbers chosen. It is the same for a cool layer 300 metres deep and 1,000 metres deep, and the same for a contrast of 1.5 kelvin or 6, because the Rossby radius carries both, as the square root of the reduced gravity times the depth over the Coriolis parameter.

The afternoon’s length matters only where the Earth does not #

The fourth figure tests the two answers directly. On the left it lengthens the afternoon from eight hours to sixteen; on the right it deepens the cool layer from 200 metres to a kilometre.

At the equator the reach grows in proportion to the afternoon: from 154 kilometres for ten hours to 216 for fourteen. At 45 degrees it does not move: 97.5 against 98.9. The extra hours come after the front has stopped, and a stopped front has nothing to carry forward. That is the refuted claim’s error in its plainest form. A longer, hotter afternoon carries a sea breeze further inland only where the Coriolis parameter is small enough that the front is still running in the evening; elsewhere a hotter afternoon helps only through the larger contrast it brings, which raises the Rossby radius as its square root.

Both regimes share the right-hand panel. The reach grows as the square root of the cool layer’s depth at both latitudes — at the equator because the head speed does, at 45 degrees because the Rossby radius does. Depth decides the reach everywhere; the length of the push decides it only near the equator.

The linear breeze’s resonance has no counterpart here. That essay found the breeze’s reach unbounded at thirty degrees, where the Earth’s inertial period equals the day and the push and the turning keep step for day after day. A front released each morning and stopped each evening never gets the chance to resonate: at thirty degrees its reach is 133 kilometres, between its two neighbours, and nothing in the third figure marks the latitude out.

Friction works against rotation #

The fifth figure adds the friction the linear essay used, a linear drag that spins the layer down in three hours.

At the equator friction does the obvious thing: it slows the current and shortens the reach, from 185 kilometres to 133. At 45 degrees it does the opposite, from 99 kilometres to 107, and at 60 degrees from 80 to 97. The reason is that friction damps the along-shore wind as well as the inland one, and it is the along-shore wind’s Coriolis force that stops a rotating current. A friction that removes it undoes part of the arrest, and in high latitudes that matters more than the slowing. The two effects cancel between 35 and 40 degrees, and a frictional breeze’s reach varies far less with latitude than an inviscid one’s — from 133 to 97 kilometres between the equator and 60 degrees, against 185 to 80.

The same competition between rotation and something that dissipates appears in the hill a slow current will not climb, where the Earth’s rotation holds a column of fluid over a hill and the current goes round it. Rotation is a constraint on motion across the lines it organises, and anything that breaks the balance, friction included, lets the fluid cross them.

What was checked #

A constant contrast with no rotation was checked against the reservoir solution: head speed and head depth to 0.2 and 0.3 per cent. The grid was halved and doubled at 45 degrees, from 150 points to 600, and the reach moved from 98.7 to 99.5 kilometres — the kind of check the drift was the instrument argued every scheme owes, since a solver can lose what it is meant to carry without any single number looking wrong. The equatorial reach was checked against the quasi-steady integral, 0.98 of it. And the head’s Froude number was changed from 1.19 to Benjamin’s square root of two: at the equator the reach rises by a tenth, as the head speed does, while at 45 degrees it rises by three per cent, since the Rossby radius does not depend on the head at all. That last check is the cleanest test of which regime is which.

What the model leaves out #

The warm air does not move. The upper layer is deep and still, so the return flow aloft, the compensating offshore wind above the breeze, is not computed. It is weak when the warm layer is much deeper than the cool one and would slow the head a little where it is not.

The contrast is imposed. In a real day the land’s heating builds a convective boundary layer whose temperature and depth both change, and the breeze itself cools the land it reaches. The model’s contrast is a sine over the afternoon.

The coast is straight and the land flat. Headlands, bays and hills bend and block fronts, and a valley can channel one far inland.

No other wind blows. An offshore wind of a few metres a second holds the head near the shore, and an onshore one carries it further than any number here, a point a speed nobody imposed makes about buoyant flows in general: their own speed is a property of the buoyancy, and any other flow simply adds to it.

Who worked it out #

Gravity currents were studied in the laboratory from the 1940s, and Benjamin’s analysis of the head of 1968 is the classic theoretical account, with Huppert and Simpson’s 1980 measurements of the head’s Froude number. Sea-breeze fronts were mapped by observers in southern England and in Australia through the twentieth century, and John Simpson’s book on sea breezes collects the measurements. Geostrophic adjustment and the Rossby radius are Rossby’s, from the 1930s; the stopping of a rotating gravity current near a Rossby radius was studied in the laboratory and in shallow-water models from the 1980s onward.

Still open: the night the front leaves behind #

Every number here ends in the evening. When the land cools below the sea’s temperature the contrast reverses, and the cool marine air now lying inland becomes warmer than the cooling land air, so the front stops and the layer behind it begins to drain back. In mid-latitudes the stopped current carries an along-shore jet that the Earth’s rotation keeps turning, and it is still turning when the contrast reverses. The next calculation would carry the model through the night: whether the stalled front retreats as a land-breeze current of its own, how much of the marine layer is left inland by dawn, and whether the inertial oscillation the evening leaves behind makes the next morning’s breeze start sooner or later than the first.