Variances and co. (summary)

After having read again the previous post on turbulence intensity, I realised that it is, in fact, more of a collection of notes that an article. It leaves several questions unanswered and does not provide a clear picture of what I have arrived to. In the present post, I will try to clarify things a bit, and discuss three main items:

The importance of the averaging period (10, 20, 30 or 40 minutes) for the computation of the momentum flux and the variance;
The change of variance with the elevation, and how it relates to the sea state;
How to relate the friction velocity from the air-sea experiments and the Charnock relationship to the surface friction velocity from the log law.

The averaging period

As highlighted in the previous post, increasing the averaging period leads to increasing the variance, because of the presence of low-frequency eddies lying in the mesoscale gap and containing much energy. As illustrated below, the friction velocity is less sensitive to such effect for neutral and convective conditions (the 10 min value is about 3% smaller than the others), but for stable conditions, the differences are noticeable. More importantly maybe, the ratio between the friction velocity and the standard deviation is not constant, it increases significantly for small values of friction velocity. This means that, a possible relationship between the two quantities is:

$$ \sigma_U^2=\sigma_{U,wave}^2+\sigma_{U,atm}^2 $$

Where

$$\sigma_{U,wave} \propto u_* $$

It follows that

$$ \sigma_{U,atm} =\sqrt{\sigma_U^2-A\cdot u_{*}^2}$$

Using $A = 2.25$ , this is what I have plotted below in the (2,2,4) subplot. With this, I find:

$$ \sigma_{U,atm} \simeq 0.1 \text{ m/s}$$

Change of variance with elevation

We recall now that the vertical profiles of standard deviation look like this: they follow a power-law, starting with positive values and then decreasing with wind speed. All the plots below are for neutral conditions.

This is likely related to the seastate. See below how the value of the power-law exponent nicely correlates with the ratio between the peak wave phase velocity $c_{p}$ and the friction velocity (plot to the left). The latter has been computed using the Charnock relationship with a coefficient of 0.018 (and solved iteratively using the wind speed time series as input).

The plot to the right shows the power-law exponent as a function of the ration between $c_{p}$ and the 3 mASL wind (extrapolated from the mast measurements). What this plot shows is that, as reported elsewhere (Semedo, 2010), swell waves seem to “damp” the wind fluctuations. But the topic is still being discussed, as I (maybe) understand from (Ortiz-Suslow et al., 2021).

The two red lines above are the fits I have used for computing $\beta(z)$ as function of the friction velocity $u_{*,0}$ :

$$\beta(u_{*,0}) = u_{*,0}(\frac{z}{z_{ref}})^{\beta(u_{*,0})}$$

Matching friction velocity, and surface friction velocity

As listed in the “Air-Sea experiments” spreadsheet, measurements of the wind speed, and of the friction velocity, have been carried out and reported a various elevations, ranging from 3 to 30+ mASL. We saw in the earlier post that the friction velocity from the Vindeby RASEX experiment also decreases with the elevation. I am reposting the plot below: the values derived from the Charnock relationship do not look like they match the 10 mASL measured values. Actually, they match the 3 mASL values more closely, and it turns out that the 3 mASL values are reported in the RASEX paper. So, in the equation above it seems fair to state that:

$$z_{ref} = 3 \text{ mASL}$$

When I use all of the above, I am getting to the following “algorithm”:

Solve the log law expression (neutral) with the Charnock relationship and $\alpha_{Ch} = 0.018$ , using a wind speed time series as input, to derive $u_{*,0}$ ;
Compute $u_{z}$ from $u_{*,0}$ and the power-law expression above, with the wave-age criteria (I have used wave buoy data btw, for $c_{p}$ and $z_{ref} = 3$ mASL;
Compute $\sigma(z)$ from the expression stated earlier.