What does the quantification of causes and effect look like, for clouds in offshore wind turbine wakes?

Question

At Horns Rev windfarm off the coast of Denmark, sometimes in winter, clouds appears in the wake of the turbines. I've only seen photos of the phenomenon when the wind direction is exactly aligned with the grid layout - that is, it's blowing directly from a turbine to its closest neighbour. That may be because it's most picturesque then (and thus most likely to be photographed); or it may be that there's something going in the fluid dynamics that requires that alignment for the phenomenon to occur.

I guess there are several things at work here: that wake losses are highest when wind is exactly aligned with one axis of the turbine grid; that air temperatures vary with height above water; that the temperature is low enough to be close enough to form fog anyway (and in the photo, it looks like they're a layer of mist just above the sea's surface); that the turbine's wake is mixing air from different altitudes

A study started in early 2011 at Lawrence Livermore National Laboratory on turbine wakes, following on from a study by DONG energy on wakes at Horns Rev (787 kB pdf here)

I'm wondering if it's possible to predict when the phenomenon in the photo here might occur. So my question is - what's the specific formulation of what's going on, here: what does the quantification of causes and effect look like?

Answer 1

Mister S.Emeis from Institute for Meteorology and Climate Research, Karlsruhe Institute of Technology, gives an answer: Link here

Here is the abstract of his explanation:

The occurrence of wake clouds at Horns Rev wind farm is explained as mixing fog. Mixing fog forms when two nearly saturated air masses with different temperature are mixed. Due to the non-linearity of the dependence of the saturation water vapour pressure on temperature, the mixed air mass is over-saturated and condensation sets in. On the day in February 2008 (the figure in the question), when the wake clouds were observed at Horns Rev, cold and very humid air was advected from the nearby land over the warmer North Sea and led to the formation of a shallow layer with sea smoke or fog close above the sea surface. The turbines mixed a much deeper layer and thus provoked the formation of cloud trails in the wakes of the turbines.

Answer 2

Thanks to @Martin Gales for the lead to the answer. Here's the quantification, from http://www.dewi.de/dewi/fileadmin/pdf/publications/Magazin_37/07.pdf

The temperature dependence of the saturation water vapour pressure E is described theoretically by the Clausius-Clapeyron equation ... Approximately, the dependence of water vapour saturation pressure in hPa on temperature can be described more simply by Magnus’ formula ... $$ E(t) = 6.107 \times 10^{\frac{a t}{(b+t)}} $$

Here t denotes air temperature in °C. a and b are two constants (over water: a=7.5, b=235) ... Mixing of two saturated air masses of different temperature leads to a mixed air mass which is always oversaturated, because each straight line which connects two separate points on the saturation water vapour curve runs through the space above the curve between the two points

Now, that's the gist of it. There's more to it than that, that's not covered in the paper, as to why the condensation appears to be emerging from a point that's about 70m downwind from the hub, and at hub height (about 70m ASL)

Answer 3

Here is a paper published in Energies, freely accessible:

http://www.mdpi.com/1996-1073/6/2/696