Momentum ﬂuxes from airborne wind measurements in three cumulus cases over land

. This study combines airborne Doppler Wind Lidar (DWL) observations with high-frequency in situ wind measurements from a gust probe, a combination that to our knowledge has not been used before. The two measurement techniques show a similar mean in the wind components throughout the ﬂights and are then used to study momentum transport in relation to shallow cumulus over land. We present three case studies ranging from forced cumulus humilis to thicker clouds associated with stronger popcorn-like convection after a cold front passage. The wind proﬁles obtained with the DWL are helpful in 5 explaining the momentum ﬂuxes that are calculated from the 100 Hz in situ data using the eddy covariance method. Most of the momentum ﬂux proﬁles revealed down-gradient momentum transport that was generally strongest within the mixed-layer and decreasing towards cloud tops. Comparing clear-sky and cloud-topped transects, the cloudy skies revealed a substantial enhancement in the mixed-layer momentum ﬂux (more than twice as much). On one track during the third ﬂight, after a post-cold-front passage and displaying thicker clouds, shows a momentum ﬂux proﬁle that did not decrease linearly with height as 10 expected from shear-driven small-scale turbulence. The momentum in the mixed layer was very small, but a very strong ﬂux has been observed in the cloud layer. Moreover, the updraft contribution to the ﬂux was much larger in this case than in all other tracks that have been ﬂown during the campaign. Last, we look into how much ﬂux the different scales contribute to the overall transport. There we ﬁnd that the largest scales (up to 7 km) usually carry most ﬂux. However, sometimes the larger scales have opposite contribution to the ﬂux than the scales smaller than 7 km, which can then result in a smaller or almost no 15 net ﬂux. processes that inﬂuence wind variability, including large-scale dynamics and small-scale processes. In this paper we com-bine state-of-the-art airborne wind lidars combined with traditional in situ turbulence measurements to measure the proﬁle of 20 wind and turbulent wind ﬂuctuations within cloud-topped boundary layers, in which thermally-driven (convective) plumes are thought to play an important role in transporting wind. By measuring wind proﬁles at levels beyond meteorological towers and ground-based operational Doppler wind lidars, we aim to investigate the role of convection and clouds in setting the proﬁle of momentum ﬂux.

the cloud layer and near cloud top. Employing the downward staring Doppler wind LiDARs, the DLR Falcon remained around 11 km altitude throughout the flight. The instruments are described next.

In situ turbulence probe
The DLR Cessna Grand Caravan was equipped with (i) a meteorological sensor package (METPOD) that measures temperature, humidity, pressure, and wind, and (ii) the IGI systems' AEROcontrol system, which combines measurements of a 95 Differential Global Positioning System (DGPS) with a high-accuracy inertial reference system (IRS). Calibration of the devices before the flight and applying corrections afterwards result in a horizontal wind measurement uncertainty of 0.3 m s −1 and 0.2 m s −1 for the vertical wind component. Further details on the instrument specifics, calibration, correction procedure, and uncertainties can be found in Mallaun et al. (2015).
The high-frequency 100 Hz wind measurements, taken with a boom-mounted Rosemount model 858 AJ air velocity probe, 100 are used for flux calculations. The aircraft movements are corrected using IGI. A linear fit is subtracted from the data before flux calculations. All scales from 10 −2 Hz are included in this calculation, unless stated otherwise.

Energy spectra
To check the quality of the measurements, we calculated the power spectral density (based on the Fast Fourier Transform), after subtracting a linear trend from the data. Welch method was used with a Hann window with 10000 samples and 50% overlap to 105 reduce noise in the spectrum. The spectrum of the u-wind component for one of the flight days is shown in Figure 2. Each line denotes a flight leg, whereby the legs flown in the sub-cloud layer and in cloud layer (light blue and medium blue, the latter mostly hidden behind the former) generally contain more energy than the legs flown near cloud base and cloud top (yellow and dark blue). Comparing the three wind components (not shown), turbulence appears to behave anisotropic: from 0.01-1 Hz, w contains more energy than u and v. Between 1-10 Hz, u has most energy, and w least. The characteristic 5/3 slope of the 110 inertial sub-range (dashed line) is seen from ∼ 0.2 -15 Hz (equivalent to a spatial resolution of 350 m down to 5 m, assuming a typical cruising speed of 65-75 m s −1 ). From 15 Hz onward, the dampening of the fluctuations in the tube becomes visible and the signal falls of faster, except for one peak at 30 Hz, which is attributed to propeller effects (Mallaun et al., 2015).

Eddy-covariance fluxes
The time series are partitioned in leg-averaged values φ and fluctuating parts φ conform the Reynolds averaging technique.
Fluxes and variances are then calculated by multiplying and averaging the fluctuations of w and φ over a specific time window, known as the eddy-covariance method. For instance, the leg average flux of φ is given by: The smallest resolved frequency depends on the length of the leg, i.e. on the number of samples N : it is reasonable to assume that a static turbulent field is sampled. However, the statistical representation of the low frequencies is poor and therefore needs cautious interpretation.

Airborne Doppler wind LiDAR
Doppler wind LiDARs (DWLs) are the international standard for wind measurements and have been used for among other things 1) data assimilation experiments (Horányi et al., 2015;Pu et al., 2017;George et al., 2021, e.g.,), 2) to study for instance 125 turbulence, gravity waves, orographic effects (Yuan et al., 2020;Gisinger et al., 2020;Baidar et al., 2020, e.g.,), and 3) to monitor the flow in wind farms (Käsler et al., 2010;Wagner et al., 2017;Zhan et al., 2020;Schneemann et al., 2021, e.g.,). The coherent detection DWL employed in this study has a wavelength of 2022.54 nm (approximately 2 µm), being eye-safe and operating in the Rayleigh scattering regime. The (vertical) resolution of the wind measurements depends on both the duration of the pulse, also called pulse width, and the distance that the signal can travel during the sampling time. The shorter the pulse, 130 the better the spatial resolution, although a reasonable sampling duration is needed to ensure sufficient accuracy of the velocity estimation (Liu et al., 2019). With a pulse width of ∼ 400 ns and an averaging time of 1 s, we have a vertical resolution of 100 m (Witschas et al., 2017). Furthermore, the aircraft speed influences the horizontal resolution. Flying with approximately 200 m s −1 and having a sampling frequency of ∼ 40 s, the horizontal resolution is about 8 km. Pulsed LiDARs have a blind spot of tens to hundreds of meters near the beam source, depending on the pulse duration and range gate width (Liu et al., 2019).

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Therefore, although flying at 11 km, the first wind velocities are obtained from approximately 7 km altitude down to about 500 m. The DWL employed in this study has previously been compared to dropsonde measurements, in which the systematic error has been found to remain below 0.1 m s −1 and the random error to vary between 0.92 and 1.5 m s −1 (Weissmann et al., 2005;Chouza et al., 2016;Schaefler et al., 2018;Witschas et al., 2020).
The Velocity-Azimuth Display technique (Browning and Wexler, 1968) with an off-nadir angle of 20 degrees, is used to 140 retrieve all three wind components. The processing algorithm that is applied to retrieve the wind vectors from one revolution of line-of-sight measurements is described in Witschas et al. (2017). The top of the boundary layer that is around 2 km altitude is clearly visible in the w fluctuations, with larger fluctuations below, and smaller above. The top of the boundary layer is marked by predominantly blue colours, indicating negative velocities produced by overshooting thermals that become negatively buoyant. Within the boundary layer updrafts generate the largest 150 fluctuations, while a few downdrafts extending to the surface are also evident. It appears that the DWL can at least to some extent observe the coherent convective features that are responsible for mass transport of scalars and momentum.
For one of the legs in Figure 3, the histograms of the sub-cloud layer u, v, and w wind are compared in Figure 4. Mean horizontal winds over this leg are comparable, although slightly overestimated, but despite the much coarser resolution and missing v winds < 2.5 m s −1 , the wind variance observed by the DWL is only slightly overestimated. This gives us confidence  are absent. It also tells us that horizontal wind fluctuations are dominated by scales larger than 1-2 km (the effective horizontal resolution of the DWL is ∼ 8.4 km). On the other hand, the vertical wind shows much less variation than the in situ measurements. This is explained by the much larger area that is measured by the DWL: it can only see the average vertical velocity in this area, which on average is much lower than the vertical velocity of vertical transient small eddies than can be better 160 captured by the in situ measurements.

Updraft detection algorithm
Using conditional sampling we identify updrafts, following the method described and tested by Lenschow and Stephens (1980).
We conditionally sample on updrafts (w > 0 & w > 0) that are wider than 100 m, and that have an excess in absolute humidity This method is more robust than using virtual temperature or buoyancy, and can be applied both in the sub-cloud 165 and cloud layer. Table 1 shows the updraft statistics of the legs flown on 4 June 2019. It lists the number of updrafts, the relative length of the leg that they occupy, the average horizontal size and the average updraft velocity. We find that the fraction of the leg    base), and wind speed increased up to 2500 m in a layer extending through cloud base ( Figure 5). In contrast, temperature and humidity were very well-mixed vertically. The atmosphere was relatively dry, with a pronounced inversion in temperature and moisture starting near 2.2 km ( Figure 6).

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Considerably stronger wind speeds, but far less wind shear were present during the second flight (May 27) when we sampled air masses ahead of a cold front located SW-NE across eastern Germany (Figures 1(b), 5 (b,e)). The air masses were somewhat warmer and moister, but with a thermodynamic structure and a cloud base very similar to that of the first flight ( Figure 6.
Besides shallow convection, there was plenty of mid-and upper levels cloud, which we encountered at the end of the first flight leg towards the north. Later, the front seemed to break up and skies were clearer, especially towards the southeast. Eventually, 195 also in the southeastern area of our operations, shallow cumulus made way for stratocumulus layers, with only rare sights of clear sky and sunshine.
During the third flight (June 4th), we measured an extended field of shallow cumulus clouds that developed behind a cold front over northwestern Germany in air masses that were considerably colder and moister (Figure 6 (c,f)), with much lower with maximum tops near 2 km. The cloud field was organised in patches of alternating cloudy and cloud-free air masses. As the clouds were getting deeper towards the northern parts of the leg, the relative sizes of the patches increased. Near-surface winds were weak and from the south, with strong shear and a turning from southeasterly to southwesterly winds right around cloud base ( Figure 5(c,f)).
Based on the wind profiles, the three flights could be classified as having weak wind and strong shear either in the sub-cloud 205 layer (Flight 1) or in the cloud layer (Flight 3), and having strong wind but little shear (Flight 2). In the next section, we will explore the associated turbulent statistics of these flights and evaluate whether the derived momentum flux profiles are in line with our expectations e.g., that momentum fluxes throughout the mixed layer and cloud layer increase with wind shear as predicted by K-theory.

Sub-cloud and cloud layer profiles
In Figures 7 and 8 we consider the profiles of wind and momentum flux for the vector wind components u and v separately.
As in Figure 5, the wind speed is shown for both the DWL (in blue) and the in situ turbulence probe at the flight levels (circles, squares). A guideline for the flux profiles in cloudy conditions are indicated with solid black lines, which are linearly interpolated between leg averaged values at the different flight levels (and are sometimes averaged over two legs the same 215 level).
If shear-driven turbulent stress dominate the momentum flux, we expect the flux to behave as in K-theory or eddy diffusivity theory, which is mathematically expressed as: and similarly for v, in which K, the diffusivity coefficient, is strictly positive. K-theory is often used as a closure technique in 220 models to denote so-called down-gradient momentum transport by small-scale turbulence, which refers to momentum being transport from regions with high to low momentum "down" the gradient, thereby acting to reduce the gradient.
On May 24th, ignoring the strong gradients in u below ∼ 700 m, ∂ z u > 0. This implies that air parcels that are displaced upward (w > 0) generally have a negative u perturbation compared to their environment (u < 0). According to Equation 2 this leads to u w < 0. This holds generally for all flight days. Negative u perturbations are in particular evident from the 225 actual wind in air masses sampled within updrafts, which tend to be several m s −1 slower (pink triangles in Figure 7 and 8).
Similarly, the meridional momentum fluxes are also down-gradient. For example, the gradient ∂ z v < 0 above 1 km on May 24th, corresponding to a positive meridional momentum flux (v w > 0), and ∂ z v > 0 on June 4th, corresponding to v w < 0. 2 km) and thus were thicker than the clouds that were encountered on the eastern track. The thicker clouds do not only have larger momentum transport in the cloud layer, but also a much larger (percentage) contribution of the updraft to the total flux than any of the other measurements ( Figure 6). The fraction of the leg that was occupied by updrafts was also significantly larger than in all other cases. The deeper clouds may have been accompanied by wider updrafts with better protected cores that 245 may be responsible for carrying larger fluxes. Looking carefully, one might see that in the mixed-layer u and w peak at different times and that u has a different sign in various updrafts ( Figure 10). This could explain a much lower momentum flux. We discuss this further in the next sections, where we explore the fluxes sampled on (cloudy) updrafts, as well as how eddies of different scales contribute to the fluxes.

Scale contributions to flux
Large Eddy Simulations of various cases indicate that the momentum flux carried by small-scale shear-driven turbulent eddies (with a size smaller than ∼ 200 m) can contribute more than 50% of momentum fluxes. Small scale turbulence may also transport momentum in an opposite direction than larger more coherent eddy structures (Zhu, 2015). This is particularly true for the lower mixed-layer and near cloud tops. However, in shallow cumulus cases, especially from the middle of the mixed-260 layer (sub-cloud layer) to the middle of the cloud layer, the net momentum fluxes are almost entirely carried by eddies with scales greater than 400 m.
In Figure 12, the total (net) momentum flux is shown for the legs in the sub-cloud layer, near cloud base, within the cloud layer and near cloud top for June 4 th . The momentum flux at different scales is calculated using a high-pass filter filters that with increasing cut-off frequency removes larger scales. The flux all the way to the right is for instance carried by eddies up to The same is true for u w near cloud top in the eastern leg with thinner clouds, and to a lesser extent, in the flux of v w in that leg within the mixed layer and near cloud tops. In the leg with thick clouds, the change in sign of the v w flux takes place already between 0.7 -2.8 km. In other words: the profiles deviate from a profile where fluxes linearly decrease with height 275 when scales beyond 1-2 km play an important role.

Conclusions
In this paper we aimed to investigate the role of convection and clouds in setting the profile of momentum flux, guided by the We address these questions using three case studies. The first case considers clear-sky and convective cumulus humilis conditions that developed over the Swabian Alps after a number of overcast and rainy days. Clouds were approximately 500 m thick and formed near an altitude of 2 km. Winds were quite calm, although a strong turning was present near 1.4 km and a temperature/moisture inversion was clearly present near 2.2 km. A second day provided less shear, both in speed and direction, 285 but with much stronger winds. Thermodynamic structure and cloud base height were similar to the first flight, although many more clouds were present -also at mid-and higher altitudes. The approaching cold front needed us to move to keep targeting cumulus clouds. The last flight received most attention in our paper. There, clouds were randomly distributed and having many diverse cloud tops, strong increasing wind speed in the cloud layer, but much steady turning throughout the mixed-layer up to cloud top. Two tracks were flown with very similar thermodynamics as well as wind profiles, but with thicker clouds and lower 290 cloud base on one of the tracks -an ideal situation to compare somewhat different clouds in similar conditions. Below we will summarize our findings for the questions that we posed at the start of the study: 1. Comparing the Doppler Wind Lidar measurements to the in-situ "truth", we find that the leg means correspond well.
Having a much larger resolution, 7 km opposed to 70 m, and much faster traveling speed, it is hard to quantitatively compare the DWL with the in-situ measurements. We find that the DWL is able to capture the mean and standard 295 deviation of the horizontal wind components quite well, despite its much coarser footprint. This shows that the total variance in wind across a ∼ 100 km transect is dominated by scales of several kilometres and larger.
Furthermore, we benefit from the DWL profiles, as this allows us to better interpret the momentum fluxes using eddy diffusion and looking at the DWL anomaly values (measurement with subtracted mean), it shows the structure of the wind in the cloud layer and large parts of the mixed layer, as well as the location of up and downdrafts which revealed that 2. Most of the momentum flux profiles revealed down-gradient momentum transport that was generally strongest within the mixed-layer and decreasing towards cloud tops. On the same transect, flying from clear-skies to below cloudy skies 305