A hybrid model for the wind profile for the whole boundary layer Sven-Erik Gryning1 Ekaterina Batchvarova1,2 1 DTU Wind Energy, Risø Campus, Technical University of Denmark, Denmark 2 National Institute of Meteorology and Hydrology, Sofia, Bulgaria

The turning of the wind has recieved little attention in the wind energy community although it can result in reduced energy production and reduced lifetime We present work on a parameterization that accounts for the effect of the: Boundary-layer height Baroclinity (horizontal temperatute gradient) Atmospheric stability. Preliminary results suggest that applying non-dimensional scaling of the turning of the wind, (z/h) versus non-dimensional Ekman scaling are in relatively good agreement.

(well known to all I presume ) Basic equations (well known to all I presume )   𝑓 𝑣− 𝑣 𝐺 = 𝜕 𝑢 ′ 𝑤 ′ 𝜕𝑧 −𝑓 𝑢− 𝑢 𝐺 = 𝜕 𝑣 ′ 𝑤 ′ 𝜕𝑧 Momentum flux divergence Coriolis force Pressure force − 𝑢 ′ 𝑤 ′ =𝐾 𝜕𝑢 𝜕𝑧 − 𝑣 ′ 𝑤 ′ =𝐾 𝜕𝑣 𝜕𝑧 first order closure

Applying the lower boundary conditions 𝑢=𝑣=0 at 𝑧=0   𝑢=𝑣=0 at 𝑧=0 and approximating the K profile with a power law, 𝐾= 𝐾 0 𝑧 𝑚 the angle α between the surface and the geostrophic wind is shown below (Köhler 1933)

Taylor – Ekman analytical solution 𝑓 𝑣− 𝑣 𝐺 = 𝜕 𝑢 ′ 𝑤 ′ 𝜕𝑧 −𝑓 𝑢− 𝑢 𝐺 = 𝜕 𝑣 ′ 𝑤 ′ 𝜕𝑧 − 𝑢 ′ 𝑤 ′ = 𝐾 0 𝜕𝑢 𝜕𝑧 − 𝑣 ′ 𝑤 ′ = 𝐾 0 𝜕𝑣 𝜕𝑧 first order closure with K= 𝐾 0 Applying the angle α as lower boundary conditions arctan 𝑣 𝑢 =𝛼 at 𝑧=0 (and not 𝑢=𝑣=0 )

Taylor-Ekman 𝑢= 𝑢 𝐺 1− 2 𝑠𝑖𝑛 𝛼 𝑒 −𝛾𝑧 𝑐𝑜𝑠 𝛾𝑧−𝛼+ 𝜋 4   𝑢= 𝑢 𝐺 1− 2 𝑠𝑖𝑛 𝛼 𝑒 −𝛾𝑧 𝑐𝑜𝑠 𝛾𝑧−𝛼+ 𝜋 4  𝑣= 𝑢 𝐺 2 𝑠𝑖𝑛 𝛼 𝑒 −𝛾𝑧 𝑠𝑖𝑛 𝛾𝑧−𝛼+ 𝜋 4 𝛼≦ 𝜋 4 Note the wind direction is counterclock wise.

For 𝛼≧ 𝜋 4 the following equations are suggested (mirror 𝛼= 𝜋 4) 𝑣= 𝑢 𝐺 2 𝑐𝑜𝑠 𝛼 𝑒 −𝛾𝑧 𝑠𝑖𝑛 𝛾𝑧+𝛼− 𝜋 4 𝑡𝑎𝑛 2 𝛼 𝑢= 𝑢 𝐺 1− 2 𝑐𝑜𝑠 𝛼 𝑒 −𝛾𝑧 𝑐𝑜𝑠 𝛾𝑧+𝛼− 𝜋 4  

How to deal with baroclinic effects? Høvsøre site) In a baroclinic atmosphere the geostrophic wind generally has vertical shear, which is related to the horizontal temperature difference. The effect on the wind is perpendicular to the temperature gradient. The baroclinic effects on the turning of the wind may be very pronounced and can be stronger than the Coriolis forces. At Høvsøre (Floors 2013) during the winter (land cold and sea relatively warmer) the thermal wind was directed towards the south; during the summer (land warm and sea relatively cold) it was directed towards northeast.

Introducing baroclinicity in the Taylor-Ekman solution. 𝑓 𝑣− 𝑣 𝐺 = 𝜕 𝑢 ′ 𝑤 ′ 𝜕𝑧 −𝑓 𝑢− 𝑢 𝐺 = 𝜕 𝑣 ′ 𝑤 ′ 𝜕𝑧   𝑣 𝐺 = 𝑣 𝐺0 + 𝜆 𝑦 𝑧 𝑢 𝐺 = 𝑢 𝐺0 + 𝜆 𝑥 𝑧 𝜆 𝑦 = 𝑔 𝑇 𝑓 𝜕𝑇 𝜕𝑥 𝜆 𝑦 = 1 𝑓 𝜕 Φ 𝑧 − Φ 0 𝜕𝑥 𝜆 𝑥 =− 𝑔 𝑇 𝑓 𝜕𝑇 𝜕𝑦 𝜆 𝑥 =− 1 𝑓 𝜕 Φ 𝑧 − Φ 0 𝜕𝑦 or or   where 𝑢 𝐺0 and 𝑣 𝐺0 are surface geostrophic wind and 𝜆 𝑥,𝑦 represent the thermal wind components. 𝑢= 𝑢 𝐺 1− 2 𝑠𝑖𝑛 𝛼 𝑒 −𝛾𝑧 𝑐𝑜𝑠 𝛾𝑧−𝛼+ 𝜋 4 + 𝜆 𝑥 𝑧  𝑣= 𝑢 𝐺 2 𝑠𝑖𝑛 𝛼 𝑒 −𝛾𝑧 𝑠𝑖𝑛 𝛾𝑧−𝛼+ 𝜋 4 + 𝜆 𝑦 𝑧

Boundary-layer height   Boundary-layer height Derive K at 10% of the boundary layer height (K= 𝑢 ∗ 0.1 ℎ 𝑟𝑒𝑎𝑙 ) ℎ 𝐸𝑘 =𝜋 2𝐾 𝑓 ℎ 𝑟𝑒𝑎𝑙 =0.1 𝑢 ∗ 𝑓 Neutral conditions (neutral) ℎ 𝐸𝑘 =𝜋 2 𝜅 𝑢 ∗ 0.1 ℎ 𝑟𝑒𝑎𝑙 𝑓 ℎ 𝑟𝑒𝑎𝑙 = 𝐶 𝑁 2 𝜅 𝑢 ∗ 0.1 ℎ 𝑟𝑒𝑎𝑙 𝑓 ℎ 𝑟𝑒𝑎𝑙 = 𝐶 𝑁 2 2 𝜅 𝑢 ∗ 0.1 𝑓 =0.1 𝑢 ∗ 𝑓 𝐶 𝑁 =1.12 ℎ 𝑟𝑒𝑎𝑙 ℎ 𝐸𝑘 = 𝐶 𝑁 𝜋 ≈0.4

Boundary-layer height   Boundary-layer height We derive K at 10% of the boundary layer height ℎ 𝑟𝑒𝑎𝑙 =0.46 𝑢 ∗ 𝐿 𝑓 ℎ 𝐸𝑘 =𝜋 2𝐾 𝑓 Stable conditions: (stable) ℎ 𝐸𝐾 =π 2 𝜅 𝑢 ∗ 0.1 ℎ 𝑟𝑒𝑎𝑙 4.7 0.1 ℎ 𝑟𝑒𝑎𝑙 𝐿 𝑓 ℎ 𝑟𝑒𝑎𝑙 = 𝐶 𝑆𝑇 2 𝜅 𝑢 ∗ 0.1 ℎ 𝑟𝑒𝑎𝑙 4.7 0.1 ℎ 𝑟𝑒𝑎𝑙 𝐿 𝑓 ℎ 𝑟𝑒𝑎𝑙 = 𝐶 𝑆𝑇 2 𝜅 4.7 𝑢 ∗ 𝐿 𝑓 ℎ 𝑟𝑒𝑎𝑙 =0.46 𝑢 ∗ 𝐿 𝑓 𝐶 𝑆𝑇 =1.12 Then when 𝐶 𝑆𝑇 ≈ 𝐶 𝑁 (0.46 is the mean value from COST710 report)

Apply the measurements from ”The Høvsøre tall wind-profile experiment” Validation Apply the measurements from ”The Høvsøre tall wind-profile experiment” Peña, Floors and Gryning 2015 (BLM) Case ℎ (m) 𝑢 ∗ (m s-1) 𝑢 𝑡 (s-1) 𝑣 𝑡 G L 𝛼 𝑚𝑒𝑎 (°) 1 100 0.19 0.0017 -0.00088 6.9 286 61 Stable  2 250 0.37 -0.0014 -0.00029 10.8 -3333 26 Neutral 3 350 0.38 -0.0061 -0.0021 14.6 130 42 4 850 0.45 -0.00051 -0.00025 21.3 222 50 5 1200 0.70 0.00063 0.000021 19.3 -2000 25  Neutral 6 1300 0.62 -0.00014 0.00031 20.4 -476 14 Unstable  7 750 0.56 -0.00087 -0.0019 14.9 ∞ 27 8 700 -0.0047 13.9 370 -12 Backing  9 0.26 -0.0012 -0.000085 5.4 -25 15 10 950 0.0046 -0.0037 11.8 71 12

Conclusions The real boundary-layer height (in physical units) typically is 40%-50% of the Ekman boundary-layer height. Use of standard boundary-layer scaling (z/h) versus non-dimensional Ekman scaling are in relatively good agreement. The Taylor-Ekman solution with baroclinic effects added, in combination with the geostrophic drag law, is a good candidate for further work on a parametrization of the turning of the wind. Needs to be validated on a larger data-set. The stability corrections on A and B in the geostrophic drag law need improvements.

Thanks for your attention