Now is the time on Do the Math when we scan the energy landscape for viable alternatives to fossil fuels. In this post, we’ll look at tidal power, which is virtually inexhaustible on relevant timescales, is less intermittent than solar/wind (although still variable), and uses old-hat technology to make electricity. For this exercise, we mainly care about the scale at which the alternatives can contribute, leaving practical and economic considerations sitting in the cold for a bit (spoiler alert: most are hard and expensive). Last week, we looked at solar and wind, finding that solar can satisfy our current demand without batting an eyelash, and that wind can be a serious contributor, although apparently incapable of carrying the load on its own. Thus we put solar in the “abundant” box and wind in the “useful” box. There’s an empty box labeled “waste of time.” Any guesses where I’m going to put tidal power? Don’t get upset yet.

## The Sphere Makes a Good Point

Tides are simply a consequence of putting an extended body in the gravitational field from another body. We exert tides on each other, in fact—though don’t try to use this as an excuse for the bulge that forms around your waist this holiday season!

Some gravity background: since Newton’s time, we have understood gravity to vary as the inverse square of the distance between masses. So gravity scales like 1/*r*², where *r* is the distance between sources. Even Einstein’s General Relativity (replaces Newtonian Gravity) respects this relationship, and we have tested that it is accurate to better than a part in ten billion using the lunar orbit. One gnarly consequence of the inverse-square law is that the gravitational force from a spherical body (planet, moon, star, etc.) is **exactly the same** as if *all the mass* were located **in a point** *at the center of the body*. In other words, the dirt under your feet plays some role in tugging you down. That dirt is very close, so 1/*r*² is large, but there is not much dirt right under you. Meanwhile, dirt on the other side of the Earth also exerts a pull. There’s more of it (within a given cone angle, for instance), but its pull is much weaker by the same factor. It all evens out to produce an effective gravitational pull toward the center of the Earth, as if all the mass were located there.

As an aside, if the Sun turned into a black hole—keeping its present mass in the process—Earth’s orbit would not change. The Sun is already *acting* like a gravitational point as far as the Earth is concerned. All that matters is mass and distance to the center, as far as gravity is concerned. Of course in this scenario, I would have to drastically revise my statements about the abundance of solar energy in last week’s post.

## Tidal Origin

What does this have to do with tides? Well, the Moon—siting about 60 Earth-radii away— pulls on the Earth as if from a point. And the extended size of the Earth means that if we say the gravitational pull from the Moon at Earth’s center has strength 1/60², the pull on the near side is 1/59², while the pull at the far side is 1/61². In other words, the Moon’s pull varies by ±3.4% as we cross the Earth. Compared to the average response of the Earth (its center), the side facing the Moon *really* wants to get closer to the Moon, while the side opposite doesn’t understand what all the fuss is about, and is more sluggish in its attraction to the Moon. The result is an eager bulge on one side and a lethargic bulge on the other side. This is why a location sees two high tides per day, as the Earth rotates under the Moon-pointing bulge (but interaction with continental shelves/coastlines can delay it significantly, so that seeing the Moon high in the sky only means you’re at high tide in the middle of the open ocean).

Meanwhile, the Sun is 23,500 Earth-radii away, so its gravity varies by only ±0.0083% across Earth. But the Sun’s gravity on Earth is about 180 times stronger than that from the Moon, so the absolute force variation from the Sun across the Earth is about 45% as much as it is for the Moon (180×0.000083/0.034). During new and full moon, the Earth, Moon, and Sun are in a line and the bulges add (spring tides). At quarter moon, the high/low from the Sun partly fills in the low/high from the Moon, diminishing the amplitude (neap tides).

For the mathy among you, because tides deal with a *difference* of force across a small change in distance, tidal force is just the *derivative* of the underlying gravitational force times the displacement distance. Differentiating 1/*r*² gives 2/*r*³, so that the force difference is proportional to 2*ΔR*/*r*³, where *ΔR* is the displacement from the nominal (center) point. For numerical simplicity, I expressed everything above in units of Earth’s radius, so *ΔR* = 1.

## Slosh Factor

For a perfect fluid body (oceanic Earth), the lunar tides would result in a peak-to-trough tidal amplitude of approximately one meter. To get the scale, we realize that the Moon’s mass is one eightieth that of Earth’s and 60 times farther away from the oceans as the Earth center. So lunar gravity is 1/(80×60²) times that of Earth gravity. Since we saw earlier that lunar tides constitute a 3.4% variation of lunar gravity, we end up with the Moon’s gravity varying by ±1.2×10^{−7} times Earth’s gravity. A small number, yes, but we multiply by the radius of Earth (6378 km) to get the deformation height that establishes potential energy balance. Now we have 0.75 m. Add in solar tides, and we’re close to a meter.

Most of us have seen tides well in excess of a meter. Some special places exceed 10 m, but even run-of-the-mill places like Puget Sound have 4 m tides, and San Diego gets 2–3 m. These are all exhibiting a sloshing that happens in shallow water or geographic restrictions. Take a look at tides in Hilo, Hawaii for contrast. Popping out of deep water in the middle of the Pacific Ocean, Hawaii gets open ocean tides with an amplitude of about—wait for it—a meter.

In the middle of the ocean, water does not have to move very far to raise the tide. Everyone just scrunches a little closer together laterally, forcing some water up. But introduce a continental shelf or an inlet, and we have a pinch point so that lots of water must actually *flow* past the bottleneck to raise a tide on the other side of the restriction. The flow can get carried away and rise up on the shore more than the tides actually demanded. The sloshing sometimes gets really out of hand when the geography produces a resonant cavity: when the natural fill/drain period is close to the six-hour tidal span. It’s just like how pushing on a playground swing at the resonant frequency produces big motion with little force (kids hate when I push because I like to experiment with out-of-phase pushes).

## Putting Tides to Work

These special resonant-sloshy places are natural candidates for harnessing tidal power. Open the gates for the flow, close it off and exact a toll on the ebb. If you’re really clever, you can charge both ways. As we saw in the post on pumped storage, the power available to a hydro dam, in Watts, is *P* = *ρFgh*, where *ρ* = 1000 kg/m³ is the density of water, *g* = 10 m/s² is surface gravity, *F* is the flow rate of water in cubic meters per second, and *h* is the height of the water behind the dam.

Compared to most hydroelectric facilities, whose heights can be over 200 m, the tidal *h* is puny. The only other knob we have is the flow, *F*. And this is determined by the area impounded, the height of the tide, and the timescale over which the height difference is exploited. We have little choice over the latter, as time and tide wait for no dam.

In fact, this brings up the interesting point of choreography. We need a height difference on either side of the barrier in order to extract energy, so the inside tide and outside tide cannot be perfectly in phase with each other: this is what happens when no barrier is present. And they can’t be perfectly out of phase, since the only way the inner tide can rise is if the outer tide is bigger at that moment. It turns out, the only cases that make physical sense have the inner tide delayed by zero to one-quarter cycle—where a cycle is about 12 hours, or one high tide to the next. The maximum power extracted happens at a one-eighth cycle (45°) delay. The following (busy) chart shows the optimal sequence.

The black curve is the outside tide, going from high to low and back to high. The solid blue curve is the inner tide delayed by 1/8 cycle. Note that it reaches its max/min right when it crosses the outer tide. This must be true if the difference between outer and inner determines the direction of flow: the blue curve descends (drains) when it is higher than the black curve, and ascends (fills) when the black curve is higher. Note that the blue curve must sacrifice some amplitude (it’s not as high as black curve, or external tide). The dashed blue curve is the height difference between outer and inner tides. The red curve indicates power extraction: highest when the height difference *and* flow (rate of change of the solid blue curve) are at their maximum. Power goes slack when the height difference and flow momentarily stop. Note that the average power is half the peak power. For real stations peak power is experienced during the largest spring tides, so the average power across all tides will be less than half the peak capability.

The average power extracted in the ideal scenario is *P*_{avg} = (1/8)*πρgAH*²/*T*, where *A* is the area impounded, *H* is the peak-to-peak tidal amplitude, and *T* is the period (about 12 hours). This works if there is no bottleneck in getting water into and out of the barrier, and I have not factored in conversion efficiency (around 90% for hydro).

Compared to the naïve picture of emptying a volume, *V* = *AH*, of water with an average height of ½*H* in time *T*, the 45° phase scheme gets about three-quarters the average power of the dumb guess (*P*_{avg} = ½*ρgAH*²/*T*).

## Real and Imagined Installations

The Rance tidal power plant in France, built in 1966, has a peak capacity of 240 MW, impounding 22.5 km² on an average tidal amplitude of 8 m. Because it cannot always be generating at full capacity, it effectively gets 40% of this on average, or 96 MW. The dummy model above computes 125 MW from these parameters, so not bad. The Rance station was recently surpassed as the largest tidal power plant in the world by the Sihwa facility in South Korea, edging our Rance with 254 MW of peak capacity and 30 km² of impound.

That these modest plants are the largest yet built is some indication of the limitations of tidal power. If it were easy to snatch this freebee from nature, we would see the straightforward technology implemented all over. South Korea is currently building another, larger plant expected to peak at 1.3 GW and 21% capacity factor for 2.4 TWh of annual electricity.

But some people speak of bigger plans. The large Penzhina Bay at the armpit of the Kamchatka Peninsula in Siberia (not a comment on the region or people: merely a geographic analogy) has an area of 20,000 km² and 10 m tides. *Cha-ching*! It is thought this could provide as much as 87 GW if fully developed. A list of proposed future concepts total about 115 GW.

## Global Potential

Clearly there are special locations where tidal power can be easily trapped and put to use. In this sense, tidal is a lot like geothermal power: mostly diffuse, but with a few hotspots. By all means, grab it where it’s easy. To put things in context, the world operates today on an energy diet of 13 TW. The 115 GW of dreams mentioned above—when combined with a typical 30% capacity factor— represent 0.26% of our global demand. Hopefully, we’ve overlooked 400 times more potential than what we see in the best spots. Hmmm.

Let’s get bold with math and break free of the 115 GW shackles! How much **could** conceivably be harnessed? We will first escape the confines of shorelines and go for the bulk of tides: *the open ocean, baby*! Don’t ask me how, but let’s trap the one-meter-high bulge on each side of the globe twice daily. The surface area of Earth is *πR*² ≈ 5×10^{14} m². Trapping half the ocean means about 2×10^{14} m². We then have 2×10^{17} kg of mass raised, on average, 0.5 m for our 1 m tidal height. The potential energy, *mgh* is then 10^{18} J. Releasing this twice daily (every 45,000 seconds) constitutes 22 TW for a whole-Earth-scale project.

Sorry, am I being ridiculous? It’s not a real proposal, just a mechanism to get a handle on scale. Only *slightly* less ridiculous is to use the increased tidal amplitude over continental shelves and ring our continents with walls to impound and tax the ebb and flow. So let’s imagine walls 100 km out to sea where shelves exist. Not all shorelines have shelves, but imagine that such shorelines are enough to circle the globe once, or 40,000 km. For a tidal height of 2 m (average 1 m) and 4 million square kilometers, we get 8×10^{16} J of energy. Dumped twice a day and we get about 2 TW total. And you thought the Great Wall in China was impressive.

If you’re thinking that I should have ringed the world *twice* instead of once to get all applicable shorelines, then just like when a student comes to me with a complaint about the partial credit grading on an exam problem, I’ll point out that I did not exact realistic efficiencies, capacity factors, and phase choreography effects. Want me to make it worse? Still, this scheme is a completely impractical proposition.

A final measure of scale comes from my own line of research. The Moon is slipping away from us at a rate of 3.8 cm per year due to tidal dissipation on Earth. Friction as the continents rotate through the water-bulge like a super-slow egg-beater drags the bulge slightly off the Earth-Moon line, providing a gravitational “carrot” urging the Moon to accelerate forward in its orbit. The extra angular momentum imparted to the Moon (robbed from Earth as its rotation slows) causes the orbit to swell. I’ll spare you the potential/kinetic energy calculation here, but the result is a change of 4×10^{18} J every year, working out to 0.1 TW. So natural barriers interacting with all the tides around the globe siphon 0.1 TW from the lunar orbit. The wild schemes above would exceed this amount, which is certainly possible—i.e., the current 0.1 TW is not a top-down constraint. Extracting tidal power at the rate of 115 GW—the total of proposed projects—would double the egress rate of the Moon and further slow Earth’s rotation. Not to worry though: I’m guessing you don’t notice or care about the current rates of egress/slowing, so why would doubling either be cause for concern?

## Put it in a Box

I have tidal power dangling from my fingers. I hover over the “abundant” box momentarily simply for amusement. I move over to “useful,” where we put wind power due to its practical capability of producing perhaps several TW of power and offsetting more than a sliver of our present demand. Tidal doesn’t fit here, so I’ll drop it in the “waste of time” box.

Who labeled these boxes? I want to register a complaint! Tidal *is* useful, in that it is practical to extract energy from nature in this way. In some select places, tidal power lights *real* lights, toasts *real* croissants, and displaces *real* fossil fuels. But my overall goal is to assess which forms of power can take on a *substantial fraction* (possibly up to a quarter) of our power needs. Only those sources *capable* of expansion at this scale stand any chance of achieving even half of that. Tidal is not one of those players.

To be clear, I’m not trying to discourage pursuit of *any* viable alternative to fossil fuels. What I *do* want to discourage is the sense of comfort we get because we’ve heard of lots of solutions to our energy problems (tidal, wave, geothermal, energy from trash, etc.). When we imagine a smorgasbord of options in front of us, we think we’ll never go hungry. But when the plate arrives and it’s a raisin here, a crumb of bread there, and a speck of cheese there, the variety alone is no longer a source of satisfaction. It’s happened to me in shi-shi restaurants.

As I said in the post on the meaning of sustainability, it is as unlikely that a hundred 1% solutions will satisfy us as it is that we could strap enough gerbils together to make a serviceable pony. We need a few solid, scalable, reliable solutions to fall back on. And tidal is not one of those. It’s more like a decoration than a foundation. Let’s use it where we can, but I don’t want anybody sleeping better.