In Obukhov we trust (variances and co., suite et fin)

This post deals, once more, with the characterisation of offshore wind speed profiles, both their mean- and turbulent components. I make use of simple, known analytical models such as the log-law and the Charnock relationship. Results from these models are compared with measurements from met masts and floating LiDARs in the North Sea and the Baltic Sea.

In this post, I would like to continue discuss some of the items I concluded with earlier, and in particular:

Better understand how the surface friction velocities derived from the Charnock relationship correlate with the Wind-Sea component of the wave spectra;
Open-up to non-neutral atmospheric conditions, and better understand our use and validations of the Monin-Obukhov Similarity Theory (MOST) and the bulk formulations associated to it for the Obukhov length.

Overall, this post aims at framing, for engineering purposes, the validity of the MOST and help derive some indicative values for the surface friction velocity, and the longitudinal wind speed variance, two parameters which many are using for various analyses (see this introduction to my first post on this topic).

How the surface friction velocity correlates with Wind-Sea waves

As we saw earlier, the friction velocity $u_{*}$ relates to the wind stress $\tau$ (transport of vertical wind fluctuations) as follows:

$$ \tau = -\rho\overline{u’w’} = \rho u_{*}^{2}$$

The friction velocity $u_{*0}$ at the surface is the key parameter for wave-growth source terms parameterisations in spectral wave models, see for instance Section 5.1 of “Ocean waves in geosciences” (Ardhuin, 2021). We expect it to correlate well with the Wind-Sea part of the directional wave spectra.

To illustrate this, I have chosen the HKZA floating LiDAR dataset in the Southern North Sea (the Netherlands). It is a Fugro Wavescan buoy with a LiDAR on top, basically. The data are available via RVO’s website. Here is what I have done:

Take the lowest LiDAR wind measurement time series, at 30 mASL;
Compute the Monin-Obukhov length from ERA5;
Derive the friction velocity by solving the wind profile equations (discussed earlier) with a constant Charnock parameter equal to 0.018;
Extracted the Wind-Sea component from the directional wave spectra derived with the Maximum Entropy Method from (Lygre and Krogstad, 1986), using the wave-age criteria described in DHI’s Hollandse Kust Nord study (see its Section 3.3.1.4).

Here are the results for two wind directional bins, one for winds coming from the coast, and one with a very long fetch (Strait of Dover). For illustration, the Swell component (which is the non-Wind-Sea part of the spectra) is shown in light grey.

From this plots we see that the model predicts friction velocities that seem very realistic. The correlation between the friction velocities and the significant wave heigths seem to be relatively unsensitive to the stability; i.e. regardless of the stability the wave heights seem to grow in a similar fashion, which is something we expected.

This shows, in my opinion, that:

the MOST and its associated parametrisations aren’t totally off;
the predicted friction velocities seem indeed to be valid very close to the surface.

A closer look at the MOST in stable conditions

We just confirmed that if the MOST is broken, at least it isn’t broken much. After all, as far as I am aware all models use it in the surface layer. And, above the surface layer, some simple correction exists, see this earlier post.

Yet, the literature tells us that, in stable conditions:

The friction velocities at a given elevation in the surface layer $u_{*}(z)$ are much smaller than the surface value $u_{*0}$ , in comparison with stable conditions. See for instance Figure 5 of “On drag coefficient parameterization with post processed direct fluxes measurements over the ocean” (Oh et al. 2010);
The non-dimensional shear predicted by the MOST seem:
- to be underestimated when using sonic anemometer measurements (i.e. no bulk formulation) at FINO1, above 30 mMSL, see Figure 1 of “Atmospheric stability assessment for the characterization of offshore wind conditions” (Sanz et al. 2015);
- not to be matching the expected values, see Figure 6 of “Turbulence in a coastal environment: the case of Vindeby”, preprint from (Putri et al., 2021)

We recall that the non-dimensional shear $\phi_{M}$ derives from the wind shear. We have seen earlier that the log-law can be derived from the following expression, in neutral conditions:

$$\frac{dU} {dz} =\frac{u_{*0}}{l} \text{ where } l = \kappa z \text{ is a length scale in m}$$

The MOST (see Obukhov’s paper from 1954) introduces a scaling function called $\phi_{M}$ , which changes $l$ the characteristic length scale:

it increases when the surface layer becomes more unstable (convective);
it decreases when the surface layer becomes more stable.

This is written:

$$\frac{dU} {dz} =\frac{u_{*0}}{l} \text{ where } l = \frac{\kappa z}{\phi_M} $$

$\phi_{M}$ is a function of the ratio between the Obukhov length $L$ and the elevation $z$ , and from the above it is expressed as:

$${\phi_M}(z/L) =\frac{\kappa z}{u_{*0}}\frac{dU}{dz} $$

So, now back to the literature. Due to these apparent mismatch between I have noticed a tendency to quickly allude to the MOST “failing”, and I would like to point two issues which cannot be overlooked.

The first issue is the fact that we are measuring at fixed elevations, and therefore we are not evaluating $\frac{dU}{dz}$ but instead $\frac{\Delta U}{\Delta z}$ . This poses the question of which reference $z$ value should be used when computing the non-dimensional shear from two measurement elevations. See below an example with some typical floating LiDAR data (sonics at 4 mASL, and then LiDAR from 30 mASL): close to the surface, it is better to choose a reference $z$ computed as the mean of the log of the elevations.

The second issue is: which $u_{*}$ to use for computing $\phi_{M}$ ? Should it be $u_{*0}$ or $u_{*}(z)$ ? As we saw earlier, $u_{*}$ decreases with the elevation, following a power-law, which seems to be stability-dependent: see below the data from the RASEX campaign (Vindeby):

The first plot shows the friction velocities and standard deviations for different wind speed bins and in neutral conditions (small difference between air- and water temperature);
The second plot shows the same, but for the timestamps where the air is warmer than the water (a proxy for stable conditions).

The diamonds show the values derived from the Charnock relationship with a constant Charnock parameter equal to 0.018, and plotted at the surface (nominal 0.1 m – I could have chosen 1 m).

Of course, there are only very few data points for the “stable” conditions, but at least three wind speeds have some data (8 to 14 m/s) showing a nicely-behaved power law for both friction velocity and variance, with a clearly smaller exponent for the former than for the later.

If I choose two values of power law exponents, $\beta = -0.14$ and $\beta = -0.06$ , we see that whether the nominal Charnock $u_{*0}$ reference level is set to 0.1 or 1 m, there is still a large difference between the surface friction velocity and the ones at higher elevations, in particular in stable conditions. See the plots below (where the black dashed line shows 20% from the value of $u_{*}(z = 10)$ :

This relatively large difference between $u_{*0}$ and $u_{*}(z)$ may explain the apparent mismatch between the values of $\phi_M$ in stable conditions, in my opinion. At least this needs to be discussed and adressed I think.

A last fun plot to finish this Section: an illustration of the stability dependence of the standard deviation of the wind speed (including the sea-state dependence, via the wind speed, analysed in the previous post):

In Obukhov I trust

In the previous Sections I have discussed about:

the validity of the MOST+Charnock-derived friction velocity values, which seem to correlate very well with Wind-Sea measurements;
two caveats about validating the MOST with surface measurements.

I remain to show that the MOST works well for the purpose of most offshore-related analysis. For this, I have taken two floating LiDAR datasets from the Dutch North Sea: HKZA and TNWA, available from https://offshorewind.rvo.nl/. For determining the atmospheric stability I have used two methods:

Computing the Obukhov length direclty from the ERA5 data (fluxes);
Computing the Obukhov length via the Bulk Richardson number and the parametrisation from (Grachev and Fairall, 1997).

The results are shown below (1-hour averages), where blue line is taken from the parameterisations reports in (Stull, 1988). At least for these two sites, it shows that the methods works well. Of course, there are locations where the ERA5 fluxes are incorrect, due to for instance errors in air- or sea surface temperatures, so measurements of the surface conditions is always desirable in connection with a floating LiDAR measurement campaign.