Wind farm wake losses
A Socratic walk-through of wind farm wake losses — reasoned out one step at a time, not lectured.
The question we started with
THE QUESTION #Why does the second row of a wind farm earn far less than the first from exactly the same wind?
Two turbines, same model, same hub height, standing a few hundred metres apart on flat ground. The same weather system crosses both. Over a year the downwind one earns noticeably less, and if you keep adding rows the deficit deepens for a while and then stops deepening. Nothing about the second machine is worse.
The tempting first thought is that the front row has "used up" the wind. But wind is not a tank being drained — it is a flow, continuously resupplied. So what exactly does the first row leave behind, and why does it matter so much?
Reasoning it through
REASONING #A companion piece here on three-bladed rotors works out what a single turbine can take: Betz's ceiling of 16/27 of the kinetic energy crossing its swept disc. That is silent about everything downstream. This question begins where that one stops — not with what the rotor takes, but with what it does to the air it takes it from.
Momentum is the way in. A rotor extracts energy by pushing back on the air, and by Newton's third law the air is pushed back on equally, so it leaves slower than it arrived. Classical one-dimensional momentum theory makes this precise: if the rotor slows the flow at its own plane by a fraction a of the free wind speed, the far wake settles at (1 - 2*a*) of that speed. At the Betz optimum a is one third, so the idealised far wake runs at one third of the free wind. That is the same calculation that produces the 59.3 per cent ceiling, read for its other output.
Now the amplification, which is where the money is. Kinetic energy flux through a fixed area goes as the cube of speed, because you get one power of speed from the energy per unit mass twice over and one more from the mass arriving per second. So a wake at 90 per cent of the free wind speed carries 0.9 cubed, or 72.9 per cent, of the power. A ten per cent shortfall in wind is a twenty-seven per cent shortfall in earnings. The second row does not need to be sitting in much of a hole to be sitting in an expensive one.
Then why is the deficit not the ruinous two-thirds that momentum theory predicts? Because that idealised wake is inviscid, and the atmosphere is not. The wake is a slow, turbulent column embedded in fast-moving air, and eddies at its boundary drag momentum inward from above and to the sides. The wake refills — and how fast it refills is a property of the air, not of the turbine, set by the ambient turbulence. Rough terrain and a well-mixed unstable boundary layer refill a wake quickly; a smooth sea under a stable, stratified layer refills it slowly, which is why offshore wakes are observed to persist for tens of kilometres and can reach a neighbouring farm entirely.
That gives the layout rule its shape. Wake width scales with rotor diameter, so spacing is set in diameters rather than metres, and the designer buys recovery distance with land. I will not quote a spacing rule as a constant: it is a site-specific optimisation, and published layouts vary widely.
So which constraint actually binds? Not aerodynamics — pure aerodynamics says spread the machines out until the wakes are gone. What binds is the cost of ground and of what runs under it: lease area, access roads, and the inter-array cable, all of which grow with spacing while the wake gain shrinks. Below that sits a second, less physical constraint: a farm is financed against a modelled energy yield, and the wake models in use disagree with each other, so layouts are partly chosen to be defensible to a lender rather than optimal in the air.
And the failure that gets accepted is stated plainly in the question: the back rows will underperform, permanently and by design. Nobody tries to eliminate it. The test applied to the marginal turbine is only whether its own reduced output still beats its own marginal cost, and where it does, the array grows and the average per-machine yield falls.
The analogy
THE ANALOGY #Think of a line of people wading upstream through a river, single file. The one in front meets undisturbed water; everyone behind is in the slack, churned water their leader left. Nothing was consumed — the river keeps arriving — but the water in that lane moves differently, and it takes distance for the surrounding current to smooth it out. Space the line out and the effect fades; step sideways out of the lane and it vanishes at once.
the waders take nothing from the river and only disturb it, whereas a turbine genuinely removes energy from the air it slows — and the river's smoothing depends on its own turbulence in a way the analogy makes look automatic rather than, as it is offshore, sometimes agonisingly slow.
Clarifying the model
THE MODEL #The load-bearing claim is that the loss is a momentum deficit carried downstream in a specific direction, not a depletion of the resource over the farm's footprint. That distinction is testable, and the test is cheap: it predicts that array output at a fixed wind speed should depend strongly on wind direction, collapsing when the wind runs down a row and recovering when it comes off-axis. Farms show exactly that. The refuting observation would be an array whose output at a given speed was indifferent to direction, or a measured wake deficit that failed to decay with downstream distance and failed to decay faster under high ambient turbulence. Either would kill the account.
Two honest qualifications. The (1 - 2*a*) result is one-dimensional and inviscid, so it describes the shape of the argument rather than any real wake — and the numbers I derived above are exact arithmetic on illustrative speeds, not measurements of a site. And the near wake, close behind the rotor, is dominated by discrete tip vortices rather than by the smooth deficit profile the momentum picture suggests, so the simple account is a far-wake account.
A picture of it
THE PICTURE #How to readRead left to right as the wake gets slower, and down the curve for what that costs. The line is the cube law plotted directly — each point is the speed fraction cubed — so it is arithmetic, not data. Start at the far left, where a turbine stands in clean air, then move one step to a wake at 95 per cent of the free wind, a deficit you would struggle to notice on an anemometer trace: the available power has already fallen by fourteen per cent. That steepness is the whole answer — the second row does not need to be badly sheltered to be badly paid.
What became clearer
WHAT CLEARED #The front row does not consume the wind; it slows a column of it and leaves that column lying downstream. Because power goes as the cube of speed, a deficit too small to see costs a great deal, and because the wake refills only by turbulent mixing, how long the penalty lasts is set by the atmosphere rather than by the machine. Spacing buys recovery, land and cable make spacing expensive, and the design accepts that the back of the array will earn less — checked not against the front row's yield, but against whether the marginal turbine still pays for itself.
Where to go next
ONWARD #- How wake steering trades one turbine's output for a whole row's, and why it is hard to prove in the field.
Key terms
TERMS #| Term | What it means |
|---|---|
| Wake deficit | the reduction in wind speed downstream of a rotor, relative to the undisturbed flow. |
| Axial induction factor | the fraction by which a rotor slows the flow at its own plane; the far wake runs at one minus twice this. |
| Array efficiency | a farm's actual output divided by what the same machines would produce standing alone. |
Every term the collection defines is gathered in the glossary.