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Part 8: A moving sea

Wind, salt and ice: the heat the sea holds and carries, and how its water moves.

Part 7's wet planet lay on a sea without currents, a slab of water 10 m deep that warmed and cooled where it lay. The climate model's sea is deeper, it holds salt, and it moves. It stores the summer's heat and gives it back in winter, its water floats or sinks by its temperature and its salt, the wind drives it and the turning planet steers it, and in the polar seas it freezes.

The model builds its ocean as it builds its atmosphere, as a stack of layers on the grid of Part 3 moved by the rules of Part 4. The air hands it heat, fresh water and the push of the wind, and it hands back the temperature of its surface and the depth of the stirred water that shares it, which is all the air ever sees of the sea.

This part follows the heat that one column of the model's sea stores through a year, the density of salty water, the drift that the wind drives, the slope of the sea's surface, the gyres that walls make and the ice that caps the polar sea. The last two sections set the whole sea moving over Part 7's planet and follow the heat it carries toward the poles. The first six figures run the model's own ocean or ice code live in your browser, and the few pieces written only for a figure are labeled where they appear; the moving sea of the last two sections was computed in advance.

The sea holds heat

The sea's first effect on the climate is the heat it holds. A cubic meter of seawater takes 4.08 MJ to warm by one kelvin, its volumetric heat capacity, about 3,400 times as much as a cubic meter of air near the ground. The whole column of air above a square meter, about 10,000 kg of it, holds as much heat for each kelvin as the top 2.5 m of the sea. Part 7's slab, 10 m deep, held the heat of four such columns.

The model's sea is far deeper, but only its top is stirred together: a mixed layer, the sea's counterpart of the air's in Part 1, kept at one temperature and one salinity by the wind and by the cooling of its surface. Beneath it lie 44 classes of water of fixed density, which the next section introduces, each named here by the temperature of its water. The mixed layer deepens by stirring in water of the class beneath it and grows shallower by leaving its water behind, but it is never shallower than 50 m nor deeper than 600 m. At its 50 m floor it holds the heat of 20 air columns.

The figure runs the model's own ocean, the mixed layer and all 44 classes, in one column through a year from the March equinox, begun as the model's own starting sea lies at 45°N: a mixed layer 60 m deep at 13.0 °C over the 12.6 °C class. Its heating is written for this figure, not computed by the climate model: ±133 W/m² that follows the sunlight a sea at 45°N absorbs through the year, strongest at the June solstice, less 10 W/m² for each kelvin the surface is warmer than at the start. A wind stress of 0.1 N/m² stirs the mixed layer, its currents are held at rest, and Part 7's slab takes the same heating.

In spring warmed water is lighter than the water beneath it, and this wind can stir the warmth down only about 17 m. So from day 21 to day 151 the mixed layer stays within 2 m of its 50 m floor while its surface temperature rises from 13.0 to 16.8 °C, warmest on day 150, 59 days after the strongest heating: a seasonal lag. The slab, with a fifth of the heat capacity, peaks sooner and higher, at 23.7 °C on day 129. Yet the mixed layer stores far more heat. By day 165 the top 100 m of the column holds 1.03 GJ/m² more than in March, enough to warm the whole air column above it by 100 K, while the slab stores at most 0.45 GJ/m², because its warm surface gives so much of the heating back.

From day 166 the column loses more heat than it gains. Cooled water at the surface is denser than the water just beneath it, so it sinks and keeps the mixed layer stirred, the overturning of Part 1 driven from the top, which in the sea is called convective mixing. But the mixed layer can sink into the class beneath it only once it is as dense as that class, and until then only the wind deepens it, from 55 m on day 166 to 99 m on day 293, about a third of a meter a day. By then it has cooled to the density of the 12.6 °C class, and the cooling eats into that class, nearly 10 m a day at first and more slowly as the mixed layer deepens: it reaches 153 m on day 300 and 324 m two days before the March equinox. Its surface hardly cools meanwhile, from 12.7 to 12.3 °C, because each loss of heat is shared with all the water it has taken in: 324 m of sea holds the heat of 129 air columns for each kelvin.

In the last days of the year the heating wins again, and the mixed layer retreats far faster than it grew, from 324 m to 173 m in two days, leaving the winter's water behind in the class, below the reach of the next summer. Over the year its surface swings by 4.5 K and the slab's by 20.8 K, within the 20 to 26 K that Part 7's slab swung near 45°N. Part 7's slab was warmest three months after the June solstice, though, and this one only 38 days after it: the figure takes back 10 W/m² for every kelvin the surface warms, and so strong a pull keeps a shallow sea in step with its heating. Land keeps its seasons' heat in its top few meters and swings further still, which is why, as Part 1 put it, land warms faster than the sea beside it.

Set the wind stress to 0, and nothing stirs cooler water up into the mixed layer: it lies on its floor from about day 9 to day 265, and its surface warms to 17.7 °C and swings by 5.5 K over the year. Slide the latitude to 15°N instead, where the heating swings by only ±49 W/m², and the surface swings by 2.0 K and the slab's by 7.7 K.

Two of the model's simplifications show here. All of the sunlight heats the mixed layer, and the sea has no thin skin that warms by day, so a day of sunlight averaging 250 W/m² warms and cools a 50 m mixed layer by only 0.06 K. And the 50 m floor is deeper than much of the real tropical mixed layer, 20 to 60 m by a climatology of 2004; in the Southern Ocean's winter the real one reaches 100 to 400 m.

What the model computes

The ocean takes a step of its own every few of the atmosphere's, the “longer step of its own” of Part 4: every four steps of the air on grids coarser than N = 64, which is every 1,800 seconds at Part 7's N = 16, and every 22.5 minutes at N = 128. Between its steps the atmosphere's surface update warms or cools the open sea's surface temperature T_s as it would a slab's, at the heat capacity of the mixed layer, h_0 deep, that the ocean last handed it:

At its own step the ocean takes T_s as its mixed layer's temperature T_0, and after the step it hands its new T_0 back. It never sees a flux of heat, only what the flux did to the temperature. The figure does the same every 3 hours, with its prescribed heating in place of the atmosphere. From the change in T_0 since its last step, and the salt that evaporation, rain and ice added, the ocean works out how fast its surface water is losing buoyancy, and it remembers that rate over about a day:

Here α and β are how much seawater's density changes with its temperature and with its salinity, from the equation of state of the next section, and \delta S is the salt the step added to each square meter of the column, in psu times meters. A positive \overline{B} means the surface water is growing denser. The mixed layer then changes its depth by three rules, each taking water from the class beneath it or giving water back:

The first is the wind's stirring, after the model of Kraus and Turner of 1967: the wind's stress τ sets the friction velocity u_*, and the stirring lifts water of the class beneath across the step in buoyancy between them, \Delta b = g\,(\rho_k - \rho_{\text{ml}})/\rho_0, less readily the deeper the mixed layer. Over a 50 m mixed layer at 45°N a stress of 0.1 N/m² draws in 0.70 m a day. The second acts while the surface cools. It erodes the class beneath at a rate set by how fast the surface loses buoyancy and how stable the class is, N^2 = g\,(\rho_k^{+} - \rho_{\text{ml}})/(\rho_0\,h_k), with \rho_k^{+} the densest water the class holds, halfway to the next class's label; it can begin only once the mixed layer is at least as dense as the lightest water the class holds, halfway to the label above, and until then only the wind deepens it. Where the mixed layer is already denser than a class beneath it, it swallows the class outright, at up to 100 m a day. The third acts while the surface warms: the mixed layer leaves its water behind, over about a day, down to the depth L to which the stirring could still carry the warmth, the sea's Monin–Obukhov length, but never above its 50 m floor. The water left behind goes to the class nearest its density, and a mixed layer thinner than 50 m is filled back up from the classes beneath.

Seawater

Seawater holds about 35 grams of salt in every kilogram. That is its salinity, given here in practical salinity units, psu, which for seawater are close to grams per kilogram. Salt makes water heavier and warmth makes it lighter, and heavier water sinks beneath lighter, so the sea is stacked by density. The model computes the density of seawater from the simplified equation of state that Fabien Roquet and his colleagues published in 2015, with the coefficients of the NEMO ocean model. The figure draws it over the ocean's temperatures and salinities, with lines along which the density is the same, the sea's isopycnals.

The probe opens on water at 25 °C and 35 psu, like the surface of a subtropical sea, at 1022.41 kg/m³. Real seawater of that temperature and salinity weighs 1023.3 kg/m³: the model's formula makes every water about 0.9 kg/m³ lighter, which does no harm, because only differences in density push the water. A kelvin of warming lightens it by 0.313 kg/m³, its thermal expansion, and a psu of salt makes it heavier by 0.729 kg/m³, so there a psu is worth 2.3 K. Drag the probe down to −1.8 °C at 34 psu, like a polar sea in winter, at 1026.47 kg/m³. There a kelvin changes the density by only 0.047 kg/m³, so in the model's formula water near 25 °C expands 6.7 times as much for each kelvin as at its freezing point, while a psu still adds 0.795 kg/m³: near freezing a psu is worth 17 K.

So heat layers the warm sea and salt layers the cold one. In the tropics warm water floats on cooler across the thermocline, the depths where the temperature falls fastest; in the polar seas fresher water floats on saltier across a halocline. Fresh water is densest at 4 °C, but in the model's formula water of 35 psu would be densest at −6.8 °C, so seawater grows denser all the way down to its freezing point, and a cooling sea keeps overturning until its surface freezes.

Because the lines curve, mixing makes water denser. Choose “mix two waters”: water at 0 °C and 34.54 psu and water at 15 °C and 37.33 psu have the same density, 1026.800 kg/m³, but half and half they make water at 7.50 °C and 35.93 psu that is 0.303 kg/m³ denser than either, heavy enough to sink beneath both. This is cabbeling. It comes mostly from the way the thermal expansion grows with temperature: the warm water gains more density by cooling 7.5 K than the cold water loses by warming 7.5 K.

The dots are the model's own layers, 44 classes of fixed density from 1020.5 to 1028.0 kg/m³, each at the temperature and salinity of its label. They are packed where the ocean holds its water: 0.5 kg/m³ apart for the warm pool of the tropics, 0.25 down the thermocline, 0.1 for the intermediate water below it and only 0.03 from 1026.65 to 1026.98, where the deep waters differ by little, so that North Atlantic Deep Water, near 1026.89, and Antarctic Bottom Water, near 1026.93, fall in separate classes.

The classes are the ocean's counterpart of the σ layers of Part 3: a class holds the water of one density wherever that water goes. Below the reach of the sunlight and the winds little makes water denser or lighter, so the sea's water moves along its isopycnals, as air that is neither heated nor cooled rides the isentropes of Part 4.

Switch to the classes by latitude, the ocean the model builds when it has no atlas to start from: here a sea 4,000 m deep with no land, a mixed layer 60 m deep and the classes stacked beneath it, thicker toward the equator and colder and fresher poleward of 45°. At the equator 34 classes are thicker than 5 m, and the rest are kept as tokens 1 cm thick. The probe opens on the equator in the 1026.2 class, 954 to 1,041 m down at 8.13 °C. Follow it poleward. It rises, and at 60.1°N it lies right under the mixed layer, from 60 to 246 m, at −1.00 °C: the same density, 9.1 K colder and 1.2 psu fresher. Like the isentropes, the classes climb toward the poles, and where they reach the mixed layer the air and the ice can change their water.

The next three figures use only four classes, 1023.0, 1025.0, 1026.5 and 1026.95 kg/m³, at 23.05, 15.23, 5.80 and 0.78 °C. Choose “the four of the next figures” to see them on the plane.

What the model computes

The density depends on temperature and salinity alone. It is the density each water would have at the pressure of the surface, its potential density. Real water is squeezed by its own weight, nearly 2 percent denser 4 km down, but the squeezing changes which of two waters is the heavier only for waters of nearly the same density, so the model compares all of them at the surface. With T_a the temperature less 10 °C and S_a the salinity less 35 psu, the formula is

with a_0 = 0.1655 kg/m³ per K, b_0 = 0.76554 kg/m³ per psu, \lambda_1 = 0.05952 per K, \lambda_2 = 7.4914 × 10⁻⁴ per psu and \nu = 2.4341 × 10⁻³ kg/m³ per K per psu. The 1026 kg/m³ it starts from is its density at 10 °C and 35 psu, 0.95 kg/m³ less than real seawater's, and across the ocean's temperatures and salinities it stays about 0.85 to 1.1 kg/m³ below the real density. Its thermal expansion and haline contraction, which the probe reads, are

The term in \lambda_1 makes \alpha grow with temperature, by about 6 percent of its value at 10 °C for every kelvin. When two waters mix half and half, the terms that are straight lines average away, and what is left is the cabbeling:

where \Delta T and \Delta S are the differences between the two waters. The formula has no term for pressure, so it also misses the way cold water is squeezed more than warm, which matters most for the deep waters. At the surface, water like North Atlantic Deep Water, at 2.5 °C and 34.92 psu, and water like Antarctic Bottom Water, at −0.5 °C and 34.66 psu, are almost equally dense, the warmer heavier by 0.01 kg/m³; 4 km down the colder is heavier by 0.3 kg/m³, which is why it lies beneath. The model orders its deep classes, 0.03 kg/m³ apart, by their density at the surface alone. Near freezing the formula makes water expand nearly twice as much for each kelvin as real seawater, 0.46 × 10⁻⁴ against 0.25 × 10⁻⁴ per kelvin at −1.8 °C and 34 psu, so real polar water is layered by its salt even more than the model's. And it is poor for fresh water: at 0 psu it puts the density maximum at 1.85 °C instead of 3.98, which is why the plane starts at 30 psu.

Each of the 44 classes has a fixed label density, a label salinity and the temperature that gives that density at that salinity. A class pushes on the water around it through its label density alone, so its temperature and salinity only ride along with its water. A class with no water in a column keeps a token 1 cm thick at its label, and a class thinner than 5 m counts as thin: the model passes over it when it reads its fields at a depth, and the figure does not draw it. Water changes class in only two ways. The mixed layer swallows the class beneath it as it deepens, and gives its water back to the class whose density is nearest its own. And after every step, any class more than 0.01 kg/m³ from its label mixes in water from the nearest class on the other side of its label, closing the gap over about two days. Because of cabbeling such a mixture comes out denser than a straight-line estimate predicts, so the model corrects slowly and stops 0.01 kg/m³ short, which keeps it from overshooting.

Pushed by the wind

Wind blowing over the sea drags on it with its stress, a force on each square meter of sea. The model works it out in every cell from the wind of its lowest layer of air, about 20 m up, by the similarity theory of Part 7's evaporation: a wind of 8.9 m/s in neutral air pushes with 0.1 N/m², and one of 13.5 m/s three times as hard. The push goes into the mixed layer, shared evenly through its depth, and the mixed layer moves as one slab.

Pushed steadily, the water does not keep moving along the push. Part 2's parcel, pushed steadily by the pressure, ended up drifting at right angles to it, and the mixed layer does the same under the wind, to the right of the wind in the northern hemisphere and to the left in the southern. Vagn Walfrid Ekman worked this out in 1902, to explain why Fridtjof Nansen had seen the Arctic's sea ice drift 20 to 40° to the right of the wind, and the drift is named after him, the Ekman transport.

The figure runs the model's own ocean on the N = 8 grid, a mixed layer 60 m deep over the four classes of the last section, at rest until a band of wind about 20° wide starts to blow east along one latitude, and averages everything around that circle. One panel shows the paths a float would take in the mixed layer and in the water beneath it. Another shows the mixed layer's velocity against the water beneath, its tip tracing a hodograph, with the two forces on the layer, the wind's push and the Coriolis force.

At 45°N under 0.1 N/m², the wind first pushes the mixed layer east, and the Coriolis force turns it to the right. The tip of its velocity swings around a circle once every 16.9 hours, Part 2's inertial circle seen as a velocity, and the swing shrinks by about half with each of the first turns, so that within three days only small loops are left around a steady drift, 1.49 cm/s, 83° to the right of the wind. The two forces then nearly cancel: the Coriolis force holds the layer against the wind, as it held Part 2's parcel against the pressure.

That balance sets how much water the drift carries. The Coriolis force on the layer is f times its speed times its depth, and it must match the wind's push, τ/ρ₀, so the speed times the depth, the water crossing each meter of a line each second, is τ/(ρ₀f). The readout's 0.907 m²/s is 1.00 times the 0.903 m²/s that τ/(ρ₀f) gives under the band of wind. The depth drops out: a layer twice as deep would drift half as fast and carry the same water. The drift falls 7° short of a right angle, mostly because the mixed layer drags on the water beneath it: with that drag taken away, the model's drift turns 88°. Toward the equator f shrinks, so the drag and the smoothing of the currents count for more, and on the equator f vanishes and there is no Ekman balance at all. On the N = 8 grid the cells averaged for 5°N already reach from 1° to 8°N, and there the drift turns only 47° from the wind and carries 0.63 of τ/(ρ₀f).

Ekman's own theory does not move the water as one slab. It passes the wind's momentum down from depth to depth by turbulence, an eddy viscosity, and the Coriolis force turns each depth a little further, so the current turns and weakens downward in Ekman's spiral, drawn in the figure's third panel. At an eddy viscosity of 0.02 m²/s the surface current runs at 6.8 cm/s, 45° to the right of the wind, and 62 m down it runs the opposite way at 4 percent of that speed, yet its water adds up to the slab's transport. The model has only the slab: it gets the transport right and knows nothing of how the current turns within the mixed layer.

Switch to the whole planet, with the trade winds blowing west within 30° of the equator and the westerlies east from 30° to 65°. Under the trades the drift runs away from the equator on both sides, and water rises from below to replace it, 0.67 m a day on the equator: upwelling. Away from the equator f grows and the trades weaken, so their drift slows as it goes and the water it brings piles up and sinks: set the latitude to 15°N, and there it sinks 0.28 m a day. Under the westerlies the drift runs toward the equator, and near 30°, where it meets the trades' drift, the water sinks again, 0.11 m a day. On the N = 8 grid the sinking is patchy, and between 20° and 25° the water even rises a little, but it is this sinking that drives the gyres of the section after next.

What the model computes

The wind's stress acts on the mixed layer alone, as a force spread through its depth. On each edge, the mixed layer's velocity \mathbf{u}_0 changes by

where h_e is the layer's thickness at the edge, but never less than 50 m, \mathbf{u}_1 is the velocity of the first class beneath it that holds more than 5 m of water, and r = 2 × 10⁻⁴ m/s is the drag between the two. The dots stand for the pressure, the turning by the flow's own spin and the smoothing, which the next section takes up. The class beneath feels the same drag with the opposite sign, so the drag hands momentum down without making or losing any. Over still water, with a level sea and no drag, the layer settles where the Coriolis force matches the wind's push, and its transport \mathbf{M} is

\boldsymbol{\tau}\times\hat{\mathbf{z}} points at a right angle to the right of the wind, and dividing by f, which is negative in the south, turns it to the left there. The coupled model hands the ocean the stress of its lowest layer of air, \rho\,C_D\,|\mathbf{U}|\,\mathbf{U}, once at each of the ocean's steps. Sea ice in the model does not move, and where it covers a share c of the sea, 80 percent of the stress passes through it to the water. The same stress stirs the mixed layer through its friction velocity, 0.99 cm/s under 0.1 N/m², by the first rule of the first section; in the figure the mixed layer deepens from 60 to 61.1 m in four days.

Ekman's theory, which the model does not use, balances the Coriolis force at each depth against the eddy viscosity's drag between depths, with the wind's stress at the surface:

For a wind of stress τ blowing east in the northern hemisphere, with z negative downward, its solution is the spiral

Added up over all depths, u comes to zero and v to −τ/(ρ₀f), the slab's transport to the right of the wind.

A sloping sea

In Part 2 the wind ended up blowing along the lines of equal pressure, held there by the Coriolis force. The sea does the same, and its pressure is simpler to find: at any depth it is the weight of the water above, set by the height of the sea surface and the density of every layer between. Where water piles up, the pressure beneath it is higher than around it, and the current that balances the outward push runs around the pile instead, a geostrophic current.

The figure runs the model's own ocean with the four classes on the N = 16 grid, over a flat floor 4,000 m deep, with no wind and no heating. Under a mixed layer 60 m deep at 27 °C the light 1023 class is 41 m thick, so the thermocline, the base of the light water, lies 101 m down. At the start light water has taken the place of the dense 1026.5 class over a patch about 3,000 km across near 45°N, so at its center the thermocline lies 60 m deeper, but the sea surface is flat.

The column under the light water is now lighter than the columns around it, so deep down the pressure there is lower, and the deep water flows in from every side. Within five hours the sea surface at the center has risen 20.4 cm, and after an overshoot and a few days of ringing it stands 19.4 cm high on day 10, over a thermocline 59.4 m deeper: a faint mirror of the thermocline. Its rise is the thermocline's depression times the difference in density between the light water and the 1026.5 water it pushed aside, 3.5 kg/m³ out of 1025, which comes to 20.3 cm.

The surface current circles the pile clockwise, as Part 2's air circled a high, at 0.89 cm/s on day 10, and the Coriolis force on it matches the slope of the sea surface to within half a percent, the readout's 1.00 times the geostrophic current. Below, the currents fade: the 1025 class moves at 0.35 cm/s and the two densest at only 0.07. At their depth the two pushes cancel, the extra height of the surface adding weight to every column near the center and the pile's lighter water taking the same weight away. That is Part 5's thermal wind in water.

Now make a bump in the sea surface instead, as Part 5 dropped its stone, by making the deepest class 1 m thicker over the same patch. Within three hours it has collapsed into a ring of gravity waves, whose leading edge the readout times at 181 to 193 m/s, close to the 198 m/s of a wave in water 4,000 m deep. An eddy 18.7 cm high is left, circled by every layer alike, and after a day only 31 percent of the raised water is still within 3,000 km: the radius Part 5 found, the wave's speed over f, is here about 1,900 km, close to the bump's own size.

The lens stays because its radius is far smaller. A wave along the thermocline, where water a few kilograms per cubic meter lighter rests on denser water, travels at only 2.93 m/s, and divided by f that gives a deformation radius of 28 km at 45°, a hundredth of the pile's width. That radius is also the size of the eddies that the real ocean's currents shed, its mesoscale eddies. Even the climate model's N = 128 grid, with cells 60 km apart, cannot draw them, so the model's ocean makes none of its own, and an eddy transport flattens the slopes of its classes as real eddies would; here it lifts the lens's thermocline by only 0.4 m in 30 days.

What the model computes

Each class is a layer of water like Part 5's single layer, with its own thickness h_k at the cell centers and its own velocity across each edge, and the model steps both with Part 4's rules. A class's thickness changes only by what flows in and out through the walls:

Across an edge an interior class carries the smaller of its two cells' thicknesses, \min(h_a, h_b), so it never flows into a cell where it has no water, and the mixed layer carries the thickness of the cell its water leaves. A whole column carries no more water across an edge than both sides hold, which matters at a coast or over a ridge. Nothing in these rules moves water from one class to another, so there is no third term like Part 4's. The velocity follows Part 4's momentum equation, with the same colors:

The turning is Part 4's, computed the same way from the layer's potential vorticity (\zeta_k + f)/h_k and its flux h_k\mathbf{u}_k, and K_k = \tfrac{1}{2}|\mathbf{u}_k|^2. The forcing \mathbf{F}_k gathers the wind's stress on the mixed layer and the drag between layers of the last section, a drag C_D|\mathbf{u}|\mathbf{u}/h on the deepest layer with water, with C_D = 3\times 10^{-3}, and Part 4's smoothing -\nu_4\nabla^4\mathbf{u}, which damps the grid scale in 12 hours, with \nu_4 = 1.27\times 10^{16} m⁴/s at N = 16. The pressure is in \Phi_k, the pressure the layer feels, which takes the place of Part 4's geopotential. Inside class k the pressure is the weight of the mixed layer's h_0 of water at its density \rho_{\text{ml}}, of the classes above with their thicknesses h_j and densities \rho_j, and of the class's own water down to that depth. Divided by \rho_0 = 1025 kg/m³, the part of it that changes from place to place at a fixed depth is

where \eta is the height of the sea surface. It is called the Montgomery potential. In its first term the class's own density has been rounded to \rho_0, by at most 0.3 percent here, an approximation the model makes throughout, the Boussinesq approximation. The mixed layer alone gets a second pressure term,

because its density follows its own temperature and salinity and can change along it, while every interior class has one density everywhere. When the currents change slowly, the turning balances the push, and the current runs along the lines of equal \Phi_k:

In the mixed layer \Phi_0 = g\eta, so the surface current follows the contours of the sea surface at g|\nabla\eta|/f: a slope of 1 cm in 100 km drives 0.95 cm/s at 45°. Under the lens the 1026.5 class, class 3, stays still only if its \Phi_3 is the same everywhere. Of the layers above it only the light class, class 1, changed, so

which is the readout's mirror.

The fastest wave in the sea runs along its surface, at 198 m/s over 4,000 m of water. Rather than take every step that short, the model splits it off. The layers take one RK4 step with the sea surface held still. Then the sea surface and the transport summed through the depth take short steps of their own, each no longer than

where d_{\min} is the shortest distance between two cell centers and D_{\max} the deepest water. At N = 16 the closest centers are 417 km apart, so the short step is at most 736 seconds, and the ocean's step of 1,800 seconds there takes 3 of them. The layers are then stretched to the averaged sea surface and shifted to the averaged transport, and as a last resort any current faster than 5 m/s is cut back to 5 m/s, a cap that does not conserve momentum. With all 45 layers at N = 128 the ocean holds 44.2 million numbers, the thickness, heat and salt of every layer in every cell and its velocity across every edge, against the 24 million of the dry atmosphere in Part 4.

Walls make gyres

The trade winds of Part 5's dry sphere blew from the east at 15 to 20° of latitude and its westerlies around 45°, as the Earth's do, and two sections back their drifts met between them and pressed the water down. Under a wind stress that turns from 0.1 N/m² westward at 15° to as much eastward at 45°, the water near 30°N is pressed down by 11 cm a day.

On a spinning sphere that water drifts toward the equator. A column of water squashed from above grows wider and spins more slowly, as a skater does with her arms out, and nearly all of its spin is the planet's own, f, so it can spin more slowly only by moving to where f is smaller. Harald Sverdrup worked this out in 1947, and the relation between the wind and the drift is the Sverdrup balance. It depends on f growing toward the pole, the β that made Part 2's loops creep west.

The water comes back toward the pole in a narrow, fast current along the western side of the basin, like the Gulf Stream off North America and the Kuroshio off Japan. Such currents are measured in sverdrups, Sv, a million cubic meters of water a second, and the Gulf Stream carries about 30 Sv through the Straits of Florida. The figure runs the model's own ocean in a basin like the North Atlantic's subtropics, 80° of longitude wide, from 12° to 48°N and 4,000 m deep, with a mixed layer at 27 °C over the four classes, all at rest, under a wind that blows west over 15 to 30°N and east over 30 to 45°N. The bars under the map follow the water across 30°N.

Within days a current starts north along the western wall, and by day 20 it has settled. Averaged from the surface to the floor over 25 to 35°N, it runs north at 1.21 cm/s in a band 920 km wide that carries 38.1 Sv across 30°N, while the interior drifts south at 0.079 cm/s, 15 times slower. Sverdrup's balance asks for 0.116 cm/s and 35.7 Sv. The western current carries more than that because some of its water turns back south right beside it, under the two bars just east of it, 16 Sv in all. The broad interior, still catching up with Sverdrup's balance, carries the other 25 Sv south, about two-thirds of what the balance asks. Switch the arrows to the current averaged over the depth, and the whole basin turns clockwise, an ocean gyre like the Earth's in the northern subtropics. By day 100 the western current carries 39.5 Sv and the interior drifts south at 0.096 cm/s, 13.0 times slower, and doubling the wind doubles every speed and transport.

Now make f the same everywhere, its value at 30°N, and Sverdrup's balance is gone: the water pressed down has no reason to drift south, and the western wall is no longer special. The gyre turns symmetric, north along the western wall at 1.71 cm/s on day 100 and south along the eastern at 1.52, and it never settles: only weak friction resists the spin the wind gives it, and it spins up from 75 Sv each way across 30°N on day 20 to 261 Sv on day 100. Henry Stommel showed in 1948 that the crowding against the western shore comes from the change of f with latitude.

Take the walls away instead, and there is no gyre at all: on day 100 only 1.5 Sv crosses 30°N northward. The water runs around the planet with the wind, west under the trades at 6.73 cm/s near 17.5°N and east under the westerlies at 5.77 cm/s near 37.5°N, still speeding up when the figure ends. The Earth's Southern Ocean is open like this, and there the westerlies drive the Antarctic Circumpolar Current all the way around Antarctica.

The model's western current is far broader and slower than the Gulf Stream, which is about 100 km wide and runs at up to 2.5 m/s: at N = 16 the model's cells are 481 km apart, and its current spreads over two of them and all 4,000 m of depth. Yet it carries about as much water, because Sverdrup's balance sets how much must come back from the wind alone. The 100 days show only the fast part of the gyre; the slow Rossby waves of Part 5 that would tilt the thermocline beneath it into a bowl need about eight years to cross the basin at 30°N.

The figure has four classes under the mixed layer instead of 44, its wind is prescribed rather than the model's atmosphere's, no heat or fresh water crosses its surface, and it starts at rest at one temperature.

What the model computes

Sverdrup's balance comes from the layer equations, added up from the floor to the surface. Far from the walls the flow is slow and broad, so its own spin, its momentum and friction all matter little, and the turning balances the pressure gradient and the wind. The curl of that balance removes the pressure, and since f changes with latitude, the turning leaves behind

where V is the northward transport per meter of width, the speed added up from the floor to the surface, \beta = \partial f/\partial y = 2\Omega\cos\varphi/a is how fast f grows northward, and \boldsymbol{\tau} is the wind stress. The figure's wind, \tau_x = -\tau_0\cos\left(\pi\,\frac{\varphi - 15^\circ}{30^\circ}\right) between 15° and 45°N, has a curl of -6\tau_0/a at 30°N, −9.42 × 10⁻⁸ N/m³ for \tau_0 = 0.1 N/m². There β is 1.98 × 10⁻¹¹ per meter per second, and V is −4.63 m²/s: 35.7 Sv south across the 7,700 km of the basin at 30°N, or 0.116 cm/s spread over 4,000 m. The water there sinks through the base of the mixed layer at w = -\frac{1}{\rho_0 f}\,\frac{\partial \tau_x}{\partial y}, where the wind itself is zero: −1.26 × 10⁻⁶ m/s, the 11 cm a day of the first paragraph.

Stommel added a friction that grows with the current's speed and found the return squeezed against the western wall, in a layer whose width the friction sets. In the model that friction is mostly the smoothing of the currents, with \nu_4 = 1.27 × 10¹⁶ m⁴/s at N = 16. A current along a wall then has a natural width of about (\nu_4/\beta)^{1/5}, 230 km at 30°N, half a cell, so the grid sets the width instead: 920 km on day 20, from the coast to where the flow turns south. In a separate run with the smoothing four times weaker, the current narrows only to 680 km. At N = 128, with cells 60 km apart, the natural width is 66 km, close to the Gulf Stream's 100.

The frozen sea

At the model's freezing point, −1.8 °C, seawater freezes into ice that floats, and the ice changes the sea on both sides of it. It reflects sunlight that open water would absorb, Part 7's ice–albedo feedback. It slows the sea's loss of heat, which must now pass up through the ice. And it leaves most of its salt behind: the model's ice keeps 5 psu, about a seventh of the salt in water of 34 psu, and the rest goes into the water beneath, which is called brine rejection.

The figure runs the model's own ice code on one column at 75°N, over a mixed layer at its 50 m floor. The air above it is a toy written for this figure, not the model's: it takes more heat from a warmer surface, 200 W/m² at 0 °C and 2.1 W/m² more for each kelvin warmer, lets through 70 percent of the model's sunlight, and adds a fixed 85 W/m² for the heat the winds bring from warmer latitudes. The column starts as open water at 0 °C on the September equinox and runs for 20 years.

The water starts with 368 MJ/m² above freezing, and it takes 45 days to lose it. After that the water under the ice stays at −1.8 °C, and the sea loses heat only as fast as the ice conducts it up to the cold air. The ice's thermal conductivity is 2.0 W/m/K, so a meter of ice whose top is 20 K below freezing conducts 40 W/m², which the water pays for by freezing 1.1 cm of new ice a day onto its base. On the first December solstice the ice is 1.21 m thick over 90 percent of the sea, the rest open water between the floes, called leads. Thicker ice conducts less, so it grows ever more slowly, by the square-root law of the Stefan problem.

When the sun returns, the model's own ice reflects 62 percent of it while the ice is cold and 48 percent as it melts, and after its first summer it never melts away again, because each summer melts less than the winter before grew: in its twentieth year the sea holds 3.6 to 5.3 m of ice for each square meter, and more every year. Now choose Part 7's dark ice, which reflects 7 percent. It grows through the polar night exactly as the bright ice does, but it is gone 11 days before the June solstice, and the open water absorbs the summer's sunlight instead. By its tenth year the dark ice lasts only three months a year and is at most 0.94 m thick, over 85 percent of the sea, and the water under it warms to 5.3 °C in summer.

That is Part 7's feedback in its starkest form, since the toy air brings the same heat whatever lies below it, and no current brings any. Bring 60 W/m² instead, and the bright ice thickens without end, to 16 m after 20 years; bring 92, and even the bright ice melts away each summer, while the dark ice stops forming after four winters. Back at 85 W/m², turn the snow on: the toy's summer cannot melt snow that reflects 70 percent of the sunlight, so the ice beneath it thickens every year, bright or dark, to 12 m by the twentieth. Part 7's run darkened only its bare ice; its snow stayed bright until the summer melted it, which is why its feedback was weaker but not gone.

As the ice grows, the water beneath it grows saltier: every meter of ice frozen out of a 50 m mixed layer at 34 psu adds 0.53 psu to it and 0.42 kg/m³ to its density. The gauge sets that density against the 1026.95 class, the densest water the model's ocean starts with on this planet without land. Water at −1.8 °C reaches it at 34.60 psu, after 1.13 m of ice, and by the end of the first winter the column's water is 0.59 kg/m³ denser than the class. In the model such water sinks, deepening the mixed layer through the classes beneath it, and Antarctic Bottom Water forms this way, from the brine of the sea ice around Antarctica. The figure's column never mixes down, so under the bright ice its water grows saltier year after year, to 36.97 psu by the twentieth.

The deeper the mixed layer, the more summer heat it holds and the later it freezes. Over Part 7's 10 m slab the dark ice's open water warms to 22.0 °C yet holds only 1.00 GJ/m² above freezing, while the 50 m layer warms only to 5.3 °C but holds 1.46 GJ/m², enough to melt 4.8 m of ice, and under the dark ice a 200 m layer makes no ice at all after its third winter.

The figure leaves out the model's ocean around the ice: no current brings heat under it, and its mixed layer never mixes down. The model's ice itself never moves, and its leads open and close only by melting and freezing.

What the model computes

The model's sea ice follows the zero-layer model that Albert Semtner published in 1976, with a cover A: in each cell the ice is h thick over a share A of the sea, and the rest, the leads, is open water held at the freezing point T_f = 271.35 K. A skin at the top of the ice, with a small heat capacity C_s = 2 × 10⁵ J/m²/K, takes the net heat \class{term-e}{F} that the surface gains from the air: the sunlight it absorbs, its infrared, and its sensible and latent heat together. Heat conducts up through the ice and any snow on it, and the base freezes or melts by whatever of that heat the water beneath does not supply:

Here T_s is the skin's temperature, k_i = 2.0 W/m/K and k_s = 0.31 W/m/K are the conductivities of ice and snow, m_s is the snow's mass on each square meter of ice and \rho_s = 300 kg/m³ its density, and F_o is the heat the ocean hands to the ice's base. \rho_i L_f = 917 kg/m³ × 3.34 × 10⁵ J/kg = 3.06 × 10⁸ J/m³ is the latent heat of freezing a cubic meter of ice. A skin that would warm past 0 °C stays at 0 °C, and the extra heat melts first the snow and then the ice, from the top. The leads take their own share of the cell's heat. What they gain melts ice from the side, and the cover shrinks by half the volume's relative loss, as in the model that William Hibler published in 1979; what they lose freezes new ice, which closes them as if it were 0.3 m thick. The air's radiation sees the cell's albedo, its parts blended by area:

\alpha_w is the open water's albedo, 0.02 under a high sun, 0.3 near the horizon and 0.06 for diffuse light. \alpha_i is the bare ice's, 0.62 while the ice is cold, falling over the last kelvin below melting to 0.48, which stands for melting ice with its ponds of meltwater; Part 7's dark ice set it to 0.07, cold or melting. \alpha_n is the snow's: fresh snow reflects 0.85, and near melting it closes about a fifth of its gap to 0.70 each day, until a fresh fall brings it back.

Under the ice lies the sea's mixed layer, not Part 7's slab: the ice's update takes the mixed layer's heat capacity, \rho_0 c_p\,h_0, in the slab's place. Wherever a cell has any ice, the ocean holds its mixed layer at the freezing point at each of its steps, hands the heat the mixed layer held above freezing to the ice's base, and salts the mixed layer by the change in the ice's volume, its cover times its thickness:

h_0, S_0 and T_0 are the mixed layer's depth, salinity and temperature, and \Delta t is the ocean's step. The ratio \rho_i/1000 kg/m³ turns a volume of ice into the depth of fresh water it holds, and 5 psu is the ice's own salt. Melting runs the formula backward and freshens the mixed layer. The water's volume never changes, because the salt it gains or loses stands in for the fresh water the ice takes and gives back. The freezing point stays at −1.8 °C however salty the water is.

A moving sea

The run below lets Part 7's sea move. It is Part 7's wet planet again, with the same air, grid and dark ice, but its slab of water 10 m deep is replaced by the model's own ocean, a mixed layer over 44 classes on a flat floor 4,000 m down. There is still no land, so the sea has no coasts. The sea started at rest, its classes stacked as in the section on seawater, and the run, on the model as it stood on October 11, 2026, went on for ten years before it recorded the next, which begins at a March equinox.

The globe opens on the temperature of the sea's surface at the March equinox. Over the year it is warmest on either side of the equator, 29.8 and 29.9 °C in the two bands that border it, and coolest at the poles, 3.3 °C, and it hardly changes through the seasons: within 5° of the equator by about half a kelvin, and near 45°N by 3.8 K, from 11.9 °C on March 20 to 15.7 °C on August 27, where Part 7's surface, partly frozen by March, swung by 20 to 26 K.

Switch to the current: the mixed layer runs west under the trades, 31 to 32 cm/s on the equator over the year, and east under the westerlies, 10 cm/s near 35°. Switch to the upwelling: cell by cell it is a speckle at the scale of the grid, but averaged around the equator the water 50 m down rises 0.13 to 0.15 m a day, and between 12.5° and 27.5° on either side it sinks at 0.05 to 0.09 m a day.

Switch to the sea level. Over the year it stands highest under the subtropics, 86 and 83 cm above the mean at 17.5°S and 17.5°N, dips to 50 cm over the equator, and falls to 111 and 109 cm below the mean at 52.5°S and 52.5°N: about 2 m from hump to trough, the planet's ocean surface topography. The currents run along its slopes, as in the sloping sea: the fall of 51 cm between 32.5° and 37.5° would balance the Coriolis force on a current of 10.8 cm/s east, within about a tenth of what the mixed layer runs there.

Switch to the mixed-layer depth. On the equator, and in each hemisphere's summer, the mixed layer sits on its 50 m floor. In winter the cooling surface mixes it down, as in the first section's column, to 142 m at 37.5°N in January and 156 m at 77.5°N in March, averaged around the latitude, and no single cell goes deeper than 376 m. Switch to the surface salinity: it is highest, 35.5 psu, at 22.5° in both hemispheres, near where evaporation most exceeds the rain. Switch to the rain: through the year its band stays between 14°S and 13°N, where Part 7's ranged from 18°S to 23°N.

Show the first 60 days and switch to the current: on the equator the sea runs east at first, 4.6 cm/s on day 10, turns west by day 20 and runs west at 18.5 cm/s on day 60. The run's starting ice did not last: the north had ice for the last time on May 17 of the second year and the south on November 10 of the third, and it never formed again. The moving sea brings the polar seas heat that no current brought the frozen-sea column, 40 to 46 W/m² poleward of 55° over the recorded year, most of it stored below the mixed layer, and their water never cools to freezing: the year's coldest cell is at 0.9 °C, 2.7 K above freezing. Without its ice the planet is 7.1 K warmer than Part 7's, 292.5 K over the recorded year, and 22.0 K warmer at the South Pole.

The second figure averages the sea around each circle of latitude, class by class from the surface to the floor, with the overturning drawn as lines along which the water moves. It opens on the temperature over the year.

The warm water is a lens over the tropics and subtropics, less than a kilometer thick. At 2.5°N, where the probe opens, the mixed layer is 51 m deep at 29.9 °C, the top of the 1024.0 class lies 237 m down at 19.7 °C, and the water near the floor is at 1.2 °C. Switch to the salinity: under the subtropics the water is saltier than 35 psu down to 600 m, and the polar seas are freshest at the top, 33.6 psu over deep water of 34.7 psu.

Switch to the eastward current, and the currents of the globe reach down to the floor. On the equator the water runs west at 31 cm/s in the mixed layer, 15 cm/s at the top of the 1024.0 class and still 3 to 5 cm/s in every class below 1,000 m. As in the sea without walls of the section on gyres, there are no gyres, only belts that run around the planet, 831 Sv west between 25°S and 25°N and 196 and 175 Sv east under the westerlies. Nothing like a wall holds them back, and the run's records cannot show what balances the push of the winds instead.

The white lines follow the water. Under the trades the mixed layer carries 58.4 Sv south across 10°S and 52.4 Sv north across 10°N, and nearly all of it comes back toward the equator just beneath: across 10°S the flow above each depth falls from 58 Sv at the mixed layer's base to 12 Sv at 170 m and 4 Sv at 470 m. These are two shallow cells, which rise on the equator, the upwelling of the globe, and sink in the subtropics. Under the westerlies two more cells turn the other way, 35.1 Sv north across 35°S and 34.6 Sv south across 35°N, and under the polar easterlies two weaker ones turn as the tropical cells do, 11 Sv south across 55°S and 12 Sv north across 55°N.

Choose September, and the winter hemisphere's tropical cell grows and reaches across the equator, as Part 7's Hadley cell did: the mixed layer carries 90.1 Sv south across 10°S and only 24.1 Sv north across 10°N. Across the equator 71 Sv runs south in the top 100 m, and 9 Sv of it comes back north in the next 300 m, 29 Sv between 400 and 1,000 m and 33 Sv below 1,000 m, 18 Sv of it below 3,000 m, so the lines run down to the floor. Only over the year does the return stay shallow.

Poleward of 45° the lines close around two more cells near 3,000 m, 33 Sv at 55°N and 26 Sv at 55°S, in which the densest water, 1026.95 kg/m³, creeps toward the equator over the floor and lighter water moves poleward above it. On the Earth such cells are fed by water that the winter makes dense enough at the surface to sink to the floor, part of the thermohaline circulation. In the recorded year nothing makes water that dense: the densest water at the surface is 1026.0 kg/m³. The cells are the densest water of the run's start draining from the polar seas, which lose about 2 percent of it a year, as fast as it flows toward the equator, while lighter water takes its place: the deep sea still settling, not a circulation that would last.

The sea carries heat

On Part 7's planet the air carried all the heat that crossed each latitude, 4.2 PW north across 40°N and 5.0 PW south across 40°S. Now the sea carries some of it. A band of sea that takes in more heat through its surface than it stores hands the rest to its currents, so the sea's share can be added up across each latitude as Part 6 added up the planet's and Part 7 the air's. The figure shows the heat going into the sea and the heat it stored over the recorded year, and below them the heat carried across each latitude.

The sea takes in most of its heat under the rain band, 36 to 38 W/m² on either side of the equator over the year, 2.2 PW in all between 10°S and 10°N. Show the surface's budget to see why. The sunlight it absorbs hardly changes across the tropics, but under the band the sea evaporates 3.6 mm of water a day and pays 103 to 105 W/m² for it, while under the trades, 12.5° to 17.5° from the equator, it evaporates 4.7 to 4.9 mm, pays 137 to 143 W/m² and loses heat.

The dashed line is the heat the sea stored, most of it poleward of 55°, 41.5 W/m² in the north and 37.5 W/m² in the south. That sea loses heat through its surface, so its heat comes by current, much of it given up by the sea between 40° and 50°, which is cooling: the deep sea is still settling from its start. Not all of the warming has a source. Over the year the sea and the air stored 1.95 W/m², averaged over the planet, while only 0.42 W/m² came in at the top of the air. The run's books cannot place the other 1.54 W/m²: 0.91 W/m² on the air's side, about the size of Part 7's whole gap, and 0.63 W/m² that the sea gained through no flux the books can see, as it has in every year since the third. Why is not yet known. With the heat the books cannot place and the 0.42 W/m² still coming in at the top, the sea's surface still warms by 0.12 K a year. The figure spreads the unplaced heat evenly over the globe; where it really belongs could move the crossing at the equator by 0.4 PW either way.

Below, the planet carries 6.1 PW north across 50°N and 5.7 PW south across 50°S, but 1.9 and 1.6 PW of that is heat the polar sea is storing, which a sea that had finished warming would not need. From the top of the air alone the total would peak at 5.1 PW at 35° in each hemisphere, against 4.6 and 4.7 PW on Part 7's planet: across 35° the planet with a moving sea carries 0.45 PW more in the north and 0.41 PW more in the south, about a tenth more in each. In the north the extra heat comes from the sunlight: the planet absorbs 12.9 W/m² more of it than Part 7's, most of it in the tropics, under clouds that reflect 12 to 26 W/m² less, but only 4.9 W/m² more poleward of 35°N. In the south it comes from the infrared: poleward of 35°S, where the surface is 12.8 K warmer without its ice, the planet sends out 17.3 W/m² more, against 11.3 W/m² over the whole planet.

The air carries 4.4 PW across 40° in each hemisphere. It holds 45 percent more vapor than Part 7's air, and its vapor carries about half again as much heat, 3.0 and 3.1 PW across 35°. The sea carries the rest across 40°, 1.0 PW north and 1.1 PW south, a fifth of the total, but 0.6 and 0.7 PW of that is heat on its way into the polar sea's storage: leaving its storage out, the sea would carry only 0.4 PW across 40°, under a tenth of the total, while the air's 4.4 PW would hardly change. Toward the equator the sea's share grows: across 10°S it carries 1.45 PW south, 55 percent of the total, and from 11°S to 2°N more than the air. Across 10°N it carries only 0.46 PW, because over this year the southern tropical sea stored heat and the northern gave some up; leaving its storage out, it would carry 1.15 PW south and 0.97 PW north, nearly a mirror image.

Show the sea's overturning to see how. In the tropics the shallow cells of the last section do the work. Across 10°S the mixed layer takes 58 Sv south, and the classes lighter than 1026 kg/m³ bring nearly all of it back. The water going south carries its heat as water of 30.0 °C would, and the water coming back as water of 24.7 °C would: the same water goes out warm and comes back cool, and carries 1.3 PW south. Across 30° to 40° the cells carry none of the sea's heat poleward: in the south they carry next to nothing, and in the north the cells under the westerlies carry 0.2 PW back toward the equator. The sea's poleward heat there moves within single classes: the water moving poleward is a little warmer than the water of the same density coming back. Across 50° most of it still moves that way, 1.5 of 2.1 PW in the north and 1.3 of 1.8 PW in the south.

On the Earth the atmosphere carries 5.0 PW at its peak, near 43°N, and about as much near 40°S. The total peaks near 35°, where the atmosphere carries 78 percent of it in the north and 92 percent in the south (Trenberth and Caron, 2001). Leaving out the heat its polar sea is storing, this planet's sea would carry 13 percent of the total across 35°N and 11 percent across 35°S, against the Earth's ocean's 22 and 8 percent. Much of the Earth's ocean heat travels in the narrow, fast boundary currents that walls make, the Gulf Stream among them. This sea has none, and carries its heat in its broad tropical cells and, farther out, within its classes.

From here to the land

The model's sea stores the seasons' heat, moves its water along classes of density, drifts across the wind, gathers into gyres against walls and freezes in the polar seas. Let go on Part 7's planet, it carried about half the heat that crosses 10°S, and its dark starting ice melted and never formed again. Parts of it are not finished. On the equator the real Pacific carries an eastward undercurrent of 0.5 to 1 m/s; the climate model's runs of late September ran east there at 0.23 m/s at most, over a mixed layer often stuck on its 50 m floor, and that floor, the 600 m cap and the stirring's constants are first guesses. At N = 128 the sea's depth-averaged flow ran away over the Southeast Indian Ridge, and the sea surface's short steps are still undamped. This run gains heat its books cannot place, its sea has no coasts, and its deep sea, still draining the densest water of its start, takes decades to centuries to settle. The next part adds the land, whose coasts shape the real ocean's gyres.