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I wonder at what density of wind farms they begin to interfere with each other's generating capacity. Presumably within one farm the turbines are sufficiently spaced that the reduction in power is minimal, but how large can an individual one get? Or is the amount of energy they take out of the wind negligable?

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You pretty much answered your own question: compared to the size of the coast, the amount of energy these wind farms capture is negligible. You might see a reduction of power if you built a dense farm up and down an entire coast, but even then, the ocean is big compared to these farms.

Edit to add: the Hornsea wind farm featured in this article is 2.5 GW and about 400 sq miles. [0]. The total energy capacity of existing generation assets is on the order of 7,500 GW [1]. Let's double that, so 15,000 GW, which would be about 2,400,000 sq miles. There's 1,015,756 linear miles of coast [2]. We know Hornsea is roughly square, so a 20 mile deep set of turbines doesn't interfere with each other, so that gives us ~20,000,000 sq miles of usable coast for wind, and again, if we double the existing electrical generation for the earth, we'd cover 2,400,000 sq miles. (obviously not all of that is usable, but we're talking orders of magnitude here) There's really no conceivable situation where we'd build enough wind farms to interfere with each other.

[0] https://en.wikipedia.org/wiki/Hornsea_Wind_Farm

[1] https://www.statista.com/statistics/267358/world-installed-p...

[2] https://en.wikipedia.org/wiki/Ocean

A maximum of 60% of the winds power can be extracted from wind (Bet'z law [1]) and modern turbines are only capable of catching 80% of that. There's lots of research in optimal spacing.

I read that a rule of thumb was 4-5 diameter widths between turbines at right angles to prevailing wind and then 7 diameter widths between rows facing the wind

[1] https://en.wikipedia.org/wiki/Betz%27s_law

Wikipedia has a short paragraph on the topic with some references for more details: https://en.wikipedia.org/wiki/Wind_farm#Turbine_spacing

I don't have the answer handy, but I do know there is a ton of research that goes into wind farm optimization. Location of the towers, cost to build, etc.

I am curious how much small elevation differences impact optimal positioning of turbines, as that is a non-issue with offshore wind. The turbines in land-based wind farms are not in a nice grid like the offshore wind farm. https://openinframap.org/#9.68/41.6112/-92.4971

The excerpt from an abstract below mentions the wake turbulence can cause other turbines to shut down.

"...The distance between the turbines is among other things dependent on the recovery of wind energy behind the neighboring turbines and the increased wind load. Models for the mean wind speed and turbulence intensity in wind turbine parks are considered with emphasis on modeling the spatial correlation. Representative limit state equations for structural failure of wind turbine towers are formulated. The probability of failure is determined taking into account that wind turbines are parked for wind speeds larger than 25 m/s resulting in reduced wind loads. An illustrative example is presented where illustrative models for the spatial correlation is taken into account"

https://www.tandfonline.com/doi/full/10.1080/102866006011566...

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