The pond that is hottest at the bottom

In 1901 a Hungarian geologist, Alexander von Kalecsinszky, put a thermometer down into the Medve-tó, the Bear Lake, at Sovata in Transylvania, and found the water at the surface at the temperature of a summer day and the water about one and a third metres down at close to seventy degrees. Nothing in the lake was volcanic. The lake was salt — it lies on a salt dome and had flooded within living memory — with fresh water from streams lying over saturated brine, and the sun, shining through the fresh layer, had heated the brine beneath it. The hottest water in the lake was at the bottom, a metre of cooler water lay on top of it, and it stayed there.

That arrangement looks impossible for a reason everybody knows: hot water rises. A pan on a stove is heated from below and the whole pan stirs itself, carrying the heat to the top. A pond in the sun absorbs most of the light in its upper metre or two and is mixed by wind and by its own cooling at night, so whatever heat it gathers reaches the surface and leaves by evaporation and radiation. An ordinary pond is never much warmer than the air above it. The Bear Lake breaks that rule with one ingredient, and engineers have copied it since the 1950s, building ponds that run at eighty or ninety degrees under an ordinary sky.

The density that decides which water is on top #

What rises is not hot water but light water, and temperature is only one of the things that sets how light it is. Fresh water at 25 °C has a density of 997 kg/m³ and at 80 °C of 972 — heating it by 55 degrees makes it 2.5 per cent lighter. Dissolving salt makes water heavier, and much more strongly: brine at 20 per cent sodium chloride is about 15 per cent denser than fresh water at the same temperature. A pond whose salt rises downward fast enough can have its hottest water at the bottom and its densest water there too, and then there is nothing to make the hot water move.

The solar pond is built in exactly these three layers. At the top is a thin layer of nearly fresh water, stirred by the wind and at the air’s temperature. At the bottom is a deep layer of strong brine, stirred by its own heating and at a uniform high temperature: this is where the heat is stored. Between them is the part that matters, a layer a metre or so thick in which the salt concentration rises steadily with depth. The figure computes the density from the temperature and salt at each depth — fresh water’s density as a function of temperature, plus a salt term fitted to tabulated brines — and the density rises without a break from the surface to the floor. The hot layer is the heaviest layer, and the pond is upside down only in temperature.

It is the same stratification that the layer a parcel cannot leave found in the atmosphere, where a parcel of air pushed upward finds itself denser than its new surroundings and is pushed back. Here a parcel of water nudged upward from the gradient carries its salt with it into fresher water and is too heavy to stay, and a parcel pushed down is too light. The gradient layer cannot overturn, so it cannot carry heat by moving. In the pond with no added salt — the dashed line in the figure — the bottom water would be 25 kg/m³ lighter than the top, and the same metre would be in motion within minutes.

A metre of water that cannot stir #

A layer of water that cannot move carries heat only by conduction, and water is a poor conductor: about 0.6 watts per metre per kelvin, a fortieth of steel and a seventieth of copper. Through a gradient layer 1.2 m thick with 55 degrees across it, the heat flux is

$$q = \frac{k\,\Delta T}{L} = \frac{0.6 \times 55}{1.2} \approx 27\ \text{W/m}^2.$$

That is the whole of the pond’s loss upward. It is the same arithmetic as the ice that grows more slowly the thicker it gets, with the conducting layer held still by salt rather than by being solid. In insulating terms, a metre and a fifth of still water is about as good as eight centimetres of glass wool — not remarkable. What is remarkable is that it is transparent. Glass wool would keep the heat in and the sunlight out. Still water keeps the heat in and lets a third of the sunlight through.

That is the design principle of every solar collector, from a greenhouse to a flat-plate water heater on a roof: a cover that passes sunlight and blocks the heat flowing back out. A glazed collector does it with a sheet of glass and an air gap, and the glass has to be bought, sealed and kept clean, and it breaks. The solar pond does it with a layer of the collector’s own working fluid, held still by salt, and the storage tank is the bottom of the same pond. Nothing in it is manufactured except the hole in the ground and its plastic liner.

The instability that density does not forbid #

A gradient with density increasing downward should be stable, and against ordinary convection it is. But the salt that sinks through a stable sea found that a column holding both heat and salt can convect even when its density increases downward, because the two diffuse at very different rates — heat about a hundred times faster than salt — and a moving parcel of water exchanges one with its surroundings and keeps the other. In the ocean case, warm salty water lies over cold fresh water and thin fingers of it sink. The solar pond is the other arrangement: hot salty water below, cool fresher water above, with the temperature unstable and the salt stable.

Here the failure is not a finger but an oscillation. A parcel displaced upward into cooler, fresher water is too heavy because of its salt and starts back down — but on the way up it has lost some of its excess heat to the cooler water around it, faster than it lost any salt, so it returns a little heavier than it left and overshoots its starting level. Below it gains heat from the warmer water, returns a little lighter, and overshoots upward. Each swing is larger than the last. The density gradient is positive throughout and the layer still convects, slowly, with a motion that grows over many oscillations.

The condition that stops it was worked out for solar ponds by H. Weinberger in 1964. Written for the gradients of salt fraction $S$ and temperature $T$ with depth, it asks that

$$\frac{\partial\rho}{\partial S}\,\frac{dS}{dz} \;\ge\; \frac{\mathrm{Pr} + 1}{\mathrm{Pr} + \tau}\left(-\frac{\partial\rho}{\partial T}\right)\frac{dT}{dz},$$

where $\tau$ is the ratio of salt’s diffusivity to heat’s, about one hundredth, and Pr is the Prandtl number, the ratio of the water’s viscosity to its thermal diffusivity. Without the factor in front it is the plain statement that salt must outweigh heat; the factor is the extra salt needed to damp the oscillation as well. Because hot water is much less viscous than cool water, its Prandtl number falls from about 7 at 20 °C to about 2 at 80 °C, and the factor rises: 1.18 at 30 °C, 1.45 at 80 °C. The hotter the pond is meant to run, the more salt its gradient has to carry.

A working pond is designed several times above the line, and the figure shows why the margin is spent rather than wasted. The working pond’s 15 per cent of salt per metre sits at four times the requirement for its 46 K per metre of temperature gradient. The margin is what is left after the gradient erodes from both ends — the wind stirring down into the top of it, the hot brine stirring up into the bottom of it — and after the salt has been diffusing upward for a few years. A pond that starts at the line does not stay there.

How thick the gradient should be #

The thickness of the gradient layer is the pond’s one real design variable, and the two things it does pull in opposite directions. A thicker layer conducts less heat back up, in inverse proportion. But the light that heats the brine has to pass through it first, and water absorbs sunlight strongly — not uniformly, but band by band. The infrared quarter of sunlight is absorbed in the first millimetre, the near infrared in the first few centimetres, red light within a few metres; only blue and green get through deep water. The share reaching a depth $x$ is commonly fitted, after Rabl and Nielsen, as a sum of four exponentials,

$$\tau(x) = \sum_{i=1}^{4} \eta_i\,e^{-\mu_i x},$$

with absorption lengths from about three centimetres to thirty metres. It is the depth an ultrasound image pays for its sharpness in a different medium: the light that penetrates is the light that is absorbed least, so the transmitted share falls steeply at first and then slowly. Through the fresh top layer and one metre of gradient, 34 per cent of the sunlight survives; through two metres, about 29 per cent.

The kept share is the light that reaches the bottom minus the conduction back up, both as fractions of the incoming sunlight:

$$\eta(L) = \tau(d + L) - \frac{k\,\Delta T}{L\,I},$$

with $d$ the fresh top layer and $I$ the sunlight averaged over day and night. Thin gradients lose by conduction, thick ones by absorption, and the best thickness lies where the two slopes cancel. At 60 K it is 1.9 metres and keeps 22 per cent of the sunlight. The optimum is remarkably flat: from one metre to three, the share changes by only a few per cent, because the transmitted light falls slowly at those depths. Hotter brine pushes the optimum deeper and lowers the share, since conduction rises with the temperature difference.

Twenty-two per cent sounds poor beside a photovoltaic panel’s twenty, until it is remembered that the panel’s twenty per cent is electricity in daylight and the pond’s is heat delivered round the clock, from storage, at any time of year. Measured ponds have reached fifteen to twenty per cent averaged over a year, close to this figure, the difference being mostly the walls and the ground.

The heat that takes a season to arrive #

The hot layer is two metres of brine, with a heat capacity of nearly four megajoules per cubic metre per kelvin. That is a great deal of thermal mass for 80 watts per square metre to heat, and it gives the pond its other useful property: it cannot follow the weather. A cloudy week moves the brine’s temperature by a fraction of a degree. Its time constant is the stored heat per kelvin divided by the loss per kelvin, about two months, and the season is what it follows.

A two-month time constant against a twelve-month driving cycle means the brine lags the sun by about two months and swings by much less than the sunlight would suggest, the same filtering that the summer that reaches the cellar in December found in the ground, with the pond’s insulating gradient playing the part of the soil above the cellar. In the second year of this model, drawing 40 watts per square metre continuously, the brine runs from 33 °C in February to 78 °C in late August. Its first year is spent warming up, and a pond filled in spring delivers little until autumn.

The figure also shows the opposite danger. Draw too little heat and the brine boils in its second summer, at which point vapour bubbles rise through the gradient and wreck it. A pond in a sunny climate has to be used, or deliberately cooled, and the model puts the minimum draw near 16 W/m². Real ponds lose more to their walls and to groundwater than this lumped model allows, and the margin is wider in practice — but several experimental ponds have had to be shaded or drained in their first summer for exactly this reason.

What the salt buys #

None of this collects more light. A fresh pond of the same depth absorbs more sunlight than the salt pond keeps — most of the light that enters it is absorbed somewhere in the water or on the bottom, about 85 per cent of it. The salt pond keeps a third. What the salt changes is where the absorbed heat can go.

In the fresh pond the absorbed heat is carried straight to the surface by convection, and the surface loses it in three ways: by radiation to a sky that is colder than the air, by convection to the air, and above all by evaporation, which grows steeply with the water’s temperature. The puddle that could dry in a second found how fast a warm surface can lose water when nothing limits it. At 25 °C a pond surface in a light breeze and half-saturated air evaporates enough water to carry away about 150 W/m² of latent heat; at 40 °C, more than 400. A mixed pond therefore settles where its surface losses match its absorption, in this model 2.8 degrees above the air. It holds its heat at almost no temperature at all.

In the salt pond the surface layer is just as leaky, and it is at the air’s temperature, losing exactly as a fresh pond does; but it only ever has to lose the heat absorbed in the top few centimetres. The heat absorbed below the gradient cannot reach it except by conduction through a metre of still water. Undrawn, the brine would sit 74 K above the air. And even when it is drawn down to the air’s temperature, the salt pond hands over more heat than the fresh one — 82 against 58 watts per square metre — because none of its stored heat is being evaporated.

The salt has turned a large quantity of heat at the temperature of the surroundings, which is useless, into a smaller quantity at seventy or eighty degrees, which is not. What matters to anyone wanting to use heat is its temperature, as the ceiling on every engine found, and in those terms the salt pond does a thing the fresh pond cannot do at all.

A gradient is a current of salt #

The gradient looks like a state, but it is a flow. Salt diffuses down its own concentration gradient, upward from the brine towards the fresh layer, at a rate set by its diffusivity — about $1.5\times10^{-9}\ \text{m}^2/\text{s}$ at room temperature and roughly twice that at 70 °C. Across a 1.2 m gradient holding about 200 kilograms of salt per cubic metre of difference, the flux is

$$J = D\,\frac{\Delta C}{L} \approx 1.5\times10^{-9} \times \frac{200}{1.2} \approx 2.5\times10^{-7}\ \text{kg m}^{-2}\,\text{s}^{-1},$$

about eight kilograms per square metre per year, and more in a hot pond. Left alone, the gradient would relax towards uniform salt over the diffusion time $L^2/D$, which for 1.2 metres is about thirty years — slow, as the equation that only runs forwards would predict for a molecular process over a metre, but not slow enough for a working plant.

So a solar pond is maintained the way a river delta is: by a steady supply at one end and a steady removal at the other. Salt is added to the bottom, usually by dissolving it in a “salt charger” that tops up the brine, and fresh water is fed gently to the surface to flush away the salt that has diffused up. The wind is the other enemy. It stirs the top layer and eats down into the gradient, and real ponds float nets or rings of plastic on the surface to break up the waves. Kept this way, the gradient is a steady state rather than an equilibrium, like the profile of temperature through a wall with heat flowing through it — held in place by the very flux that would destroy it if the supply stopped.

The flux is also why the material matters. Sodium chloride is cheap and works, but magnesium chloride brines, left over from salt and potash works, are denser and stay put better, and several of the large ponds were built beside potash evaporation pans for exactly that reason. Where the salt has to be bought, it is the largest running cost of the plant.

Lakes that did it without anyone #

The Bear Lake was not unique. Around the world, wherever a salt lake receives fresh inflow on top, the same layering occurs, and several lakes are hot at the bottom for no other reason. The most extreme is Lake Vanda in the Dry Valleys of Antarctica, permanently covered by three or four metres of ice, whose bottom water, sixty-odd metres down and saturated with calcium chloride, sits at about 25 °C — far warmer than the summer air above the ice. The sunlight that heats it passes through the ice and the fresh upper water, and the salt gradient below has kept the heat in for centuries. Some geothermal contribution has been argued for, but the profile is that of a solar pond.

The same arrangement occurs in the ocean on a vast scale, with less heat and less salt. Beneath the Arctic sea ice, cold fresh water lies over warmer, saltier water of Atlantic origin, and the boundary between them is arranged as a staircase of mixed layers separated by thin sheets — the oscillating instability of the earlier figure, grown to finite amplitude and organised into layers. How fast heat leaks upward through that staircase towards the ice is one of the open quantities in the Arctic’s heat budget, and it is the same Weinberger factor, in reverse: the ocean sits below the line, where the solar pond is designed to sit above it.

Turning warm brine into work #

The obvious use for eighty-degree water is as heat — for washing, for drying crops, for greenhouses, for process heat in a food plant, for desalination. Most working ponds have done that. A pond of about 3,000 square metres at El Paso, Texas, ran from the mid-1980s for more than fifteen years supplying process heat to a food cannery nearby, and drove a small desalination unit and a heat engine. In India, a 6,000-square-metre pond at Bhuj supplied hot water to a dairy.

Making electricity from it is possible and unrewarding. Between brine at 80 °C and cooling water at 25 °C, the Carnot limit is

$$1 - \frac{298}{353} \approx 15.6\ \text{per cent},$$

and a real low-temperature engine — an organic Rankine cycle, boiling a refrigerant rather than water — achieves a third to a half of that. The largest attempt was in Israel, at Beit HaArava beside the Dead Sea, where a pond of about a quarter of a square kilometre fed a turbine rated at five megawatts through the 1980s. The Dead Sea supplied the brine for nothing, which is the circumstance in which the economics come closest to working, and even there the plant was closed when oil became cheap again. The sun’s heat is easy to store in a pond and hard to sell as electricity.

A pond with no walls, no groundwater and clear water #

The figures treat the pond as infinitely wide, with no walls, so that heat moves only vertically. Real ponds are tens of metres across and two or three deep, and their walls lose heat sideways; the smaller the pond, the more that matters, and below a few hundred square metres it dominates. The ground beneath is modelled as a single loss coefficient, where in reality it warms up over the first year or two and then loses heat more slowly — unless groundwater flows beneath the pond, which can carry heat away far faster than any coefficient allows and has ruined at least one site.

The sunlight is averaged over day and night and taken to fall vertically; slanting sun has a longer path through the water and is absorbed more before reaching the brine, which favours ponds near the tropics. The light model is for clear water, and turbidity — algae, dust, dead insects — can halve the transmission through the gradient in a few weeks if the water is not treated. The density model is fitted to sodium chloride at room temperature and extended to 80 °C by assuming the salt’s contribution does not change with temperature, which is good to about a per cent. And the stability criterion is for a uniform, infinite gradient; real gradients fail at their edges, where the convecting layers above and below scour into them, and the rate at which that erosion advances is measured rather than computed.

Still open: whether a gradient can be held without tending #

The salt gradient is the cheapest transparent insulation known, and it needs looking after: salt added, fresh water flushed, nets maintained, algae controlled. Designs that keep the principle and lose the maintenance have been tried — ponds whose upper layer is a transparent polymer gel rather than graded brine, ponds divided into horizontal layers by thin transparent membranes, and saturated ponds using salts whose solubility rises with temperature, so that a saturated brine is automatically denser where it is hotter. Each solves the erosion problem and introduces another: gels yellow in sunlight, membranes collect dirt and sag, saturated salts deposit crystals that block the light. Whether any of them can run for decades as reliably as a well-tended salt gradient has not yet been shown at a scale that would settle it.

What the Bear Lake shows is the general idea. A fluid heated from below rises because its heat makes it light; put something heavier where the heat is, and the fluid keeps still, and a still fluid is an insulator. The salt in the bottom of a solar pond does no work and collects no light. It holds the hot water down, and that is enough to turn a pond that can never be warmer than the air into one that can boil.