Here we describe the main data sets used in this study and provide details about screening and treatment of the data. Turbulence data (20 Hz) of wind components (u, v and w) and sonic temperature Ts measured with Campbell Scientific anemometer–thermometers (CSAT) at the divergence site tower as well as ultrahigh-frequency (UHF) wind profiler data are downloadable from http://bllast.sedoo.fr/database/ .

The governing equation for TKE in a sheared convective boundary layer under the assumption of horizontally homogeneous turbulence and no advection is given by : ∂E∂t︸TKE tendency=-u′w′‾∂U∂z︸Shear production+gθ‾w′θv′‾︸Buoyancy production--∂w′E′‾∂z︸Turbulent transport -∂w′p′/ρ0‾∂z︸Pressure transport -ϵ︸Dissipation.

Here TKE (=E) denotes 1/2u′2‾+v′2‾+w′2‾, where u′, v′ and w′ are instantaneous deviations of, respectively, the along-wind, cross-wind and vertical wind components from their respective mean values. U is the magnitude of the mean wind, which varies with height z; g is acceleration of gravity; θ‾ is mean absolute temperature; θv′ is the instantaneous deviation of virtual temperature from its mean value; ρ0 is air density; p′ is the instantaneous deviation of air pressure; and ϵ is the mean dissipation rate of TKE.

We have given the buoyancy term the subscript buoyancy production of TKE since we limit our study to the afternoon period before stable stratification starts. Hence, it is always a positive term in our case. The physical interpretation of the six terms in Eq. (), from left to right, is hence the local time rate of change of TKE, shear production of TKE, buoyancy production of TKE, vertical divergence of the turbulent transport of TKE, vertical divergence of the pressure transport of TKE, and dissipation rate of TKE.

In Fig. , we present the observed hourly TKE budget for each afternoon transition period from 12:00 UTC (normalized time 0) to zero-buoyancy flux (normalized time 1) for all four levels of the small divergence site tower. The measurement levels (2.23, 3.23, 5.27 and 8.22 m) are shown as dashed, dash-dotted, full and dotted lines, respectively.

For buoyancy production (in blue), only very small height variations are observed near the surface and a general decrease with time during the afternoon is observed for all days. On 30 June, this general picture is partly interrupted by the presence of clouds changing the energy balance.

Also, the dissipation rate (in black) is observed to have a general decrease during the afternoon transition for 8 out of 10 IOP days. Most significant deviations are found on days with an increase in shear production during the afternoon, leading to a clear increase in dissipation. Hence, shear production plays an important role near the surface in the TKE budget for most of these 10 IOP days. It has the most pronounced height dependence out of all budget terms, with higher values near the surface. The strongest dissipation rate is also found closest to the surface, but the height variation in dissipation is smaller.

Given that the TKE tendency (in green) is much smaller (2 orders of magnitude) than the other budget terms this implies that the sum of turbulent and pressure transport (in magenta) compensates for remaining height variation in the budget. Because the tendency term of TKE is much smaller than the other budget terms, we will refer to the hourly TKE as evolving in a quasi-stationary way. Here, we use the term quasi-stationarity to mean that the tendency of TKE is small in comparison to the other budget terms. This result of quasi-stationarity is consistent with the observed slowly evolving mean TKE levels in LES for a large part of the afternoon of 20 June as described in . Although the TKE tendency then increased somewhat in the late afternoon in the LES, a threshold of about -1.1×10-5 m2s-3 was used in to indicate the faster decay, and this is still quite a small TKE tendency.

The height variation in transport is found to mainly be linked with local shear production. The transport term is consistently a negative term in the TKE budget. This implies transport of near-surface-produced turbulence to the surrounding environment and upper parts of the boundary layer. Only a few occasions with positive transport term were observed in connection to changing cloud cover and more variable dissipation estimates.

To investigate general differences between the different days, we calculated statistics for each budget term during the afternoon period. These statistics are provided in Appendix B and some of the most important findings are discussed in Appendix B and only briefly restated here.

Variations in shear production between afternoons in Tables  and were found to be significantly larger than buoyancy production. Variations in dissipation and transport between different afternoons were thereby found to be mostly related to varying shear production this close to the surface. Larger variations were observed in both the transport and dissipation term compared to the buoyancy term, meaning that buoyancy alone cannot explain differences in mean values between different afternoons for these terms. The three lowest TKE mean values in Table  occurring on 30 June and 2 and 5 July had the lowest wind speed and 25 June, which had the highest wind speed, also had the highest mean afternoon TKE value.

There are, of course, exceptions to the rule that a higher wind speed leads to a higher TKE level; this topic needs to be further discussed. In Fig. , we show the mean wind profiles for the 10 afternoons and have placed the same color on the two most similar profiles to facilitate further discussions to come. It is directly clear that 24 June and 5 July (in red) have essentially equal mean wind for the afternoon as a whole, yet from Table  we note that average TKE values are higher for 24 June. This is likely related to a higher mean buoyancy production of about 3.4×10-3 m2s-3 (the highest in the data set) in comparison to about 1.9×10-3 m2s-3 for 5 July, which is the lowest in the data set. Hence, several terms need to be considered to understand the observed variations in TKE.

Nevertheless, it is interesting to note that a relatively high negative correlation (-0.69) between the mean afternoon TKE tendency and mean afternoon buoyancy production exists, as shown in Fig. a. This is interpreted to imply that, in the case of a strong buoyancy production (both before and during the afternoon), TKE levels at midday are higher and therefore TKE decay rate during the afternoon can become higher. However, it is always small in comparison to other budget terms. A weaker positive correlation (0.33) is found between TKE tendency and shear production, implying that turbulence will decay more slowly during a more shear-driven afternoon as seen in Fig. b. This is in general agreement with reduced TKE decay rates for the afternoon found in LES when including wind shear , and it is also discussed using a theoretical spectral model and LES data by . Best linear fit expressions have been included in both panel a and b. Attempts were made to non-dimensionalize the surface layer TKE tendency itself with measurement height and friction velocity and correlate it with various non-dimensional parameters such as z/L, zi/L, but it gave decreased correlation in comparison to relating tendency directly to buoyancy production as in Fig. a.

We do a broad summarizing classification of the 10 different afternoons in Table  based on the TKE budget mean values of Tables  and . In Part 2, when attempting to model TKE and TKE decay, we discuss more details and variations.

For this broad classification we take as a starting point the terms of largest variation at the 2 m level as a reference level for this classification. The days were placed into three categories (higher, moderate and weaker) in terms of mean wind speed, with 20, 25, 26 and 27 June having the higher mean wind speeds and 30 June and 2 July the weakest winds of the data set. An “X” marker denotes placement in a category. When the variation within the afternoons justifies only one part of the afternoon belonging to a given category, we denote this with parentheses, e.g., “(p)”. For the moderate category, we also indicate with “l” or “h” whether the variable mainly departs toward the lower or higher category. In a similar way shear production, transport and dissipation are classified into three categories (higher, moderate and weaker). For buoyancy production, the variations were smaller and only two categories (higher and moderate) are used. For dissipation, we also mark the special cases of 27 June and 5 July with increasing dissipation during the afternoon with “inc” within parentheses.

If the mean value of shear production at the 2 m level is above 3.5×10-3 m2s-3, it is considered higher (marked with bold font), and if it is lower than 2.0×10-3 m2s-3, it is considered weaker (underlined). The moderate category is marked in italics. These arbitrary limits illustrate an expected correspondence between the mean afternoon wind speed and classification based on mean shear production for these afternoons, but it is clearly a relative classification since mean afternoon wind speed was always below 3 ms-1.

For transport, a mean value below -2.5×10-3 m2s-3 at the 2 m level was considered stronger transport out of the near-surface layers and a mean value above -1.5×10-3 m2s-3 is marked as weaker. Bold font and italics are added on the days with higher shear production to illustrate that on these afternoons the transport is also higher or moderate. Underlining is instead added for days with weaker or moderate shear production with partly lower shear production during the afternoon, and it can be seen that these have weaker or moderate transport values.

For dissipation, a mean value equal to or lower than -4.5×10-3 m2s-3 at the 2 m level is classified as having higher dissipation, and above -3.5×10-3 m2s-3 it is considered to have lower dissipation. Bold font and underlining are added for days with higher shear production and these are found to have higher or moderate dissipation, whereas the two days with weakest shear production had the weakest dissipation (underlined and in italics). However, also 5 July, which had variable wind during the afternoon, had weaker dissipation and 19 June had higher dissipation, despite its moderate to partly lower shear production. For 19 June, it is hence not possible to draw the conclusion that higher dissipation rate is caused by high shear production; rather, it may be the higher buoyancy production that is the cause.

Finally, for buoyancy production, we have classified higher buoyancy production to imply a mean value for the afternoon of above 2.5×10-3 m2s-3 and moderate to mean below this limit.

To compare these new measurements and estimated TKE budget terms in the context of earlier studies, we first investigate the behavior of each term in the budget after normalization by friction velocity u∗ and measurement height z, as suggested in Monin–Obukhov similarity theory. Here friction velocity was defined from longitudinal shear stress, u∗2=-u′w′‾.

After normalization of Eq. () with friction velocity and measurement height and including a von Kármán constant value k (set equal to 0.4 in the analysis), the governing equation for TKE reads kzu*3∂E∂t︸Tendency=-kzu*∂U∂z︸Shear production+kzu*3gθw′θv′‾︸Buoyancy production--kzu*3∂w′E′‾∂z︸Turbulent transport -kzu*3∂w′p′/ρ0‾∂z︸Pressure transport -kzu*3ϵ,︸Dissipation which can be rewritten in Monin–Obukhov similarity notation: kzu*3∂E∂t︸Tendency=ϕm︸Shear production+ϕb︸Buoyancy production++ϕT︸Transport +ϕϵ.︸Dissipation

Here, we have lumped together pressure and turbulent transport terms into one total transport term ϕT. In Fig. , we show the normalized TKE budget terms as a function of the stability parameter z/L. Included in the plot are fitted expressions for the budget terms (neglecting the small TKE tendency term).

For buoyancy production, the expression by definition simply reads -z/L. ϕb=-z/L

For shear production, we note that a commonly used form of (1-Az/L)b with A equal to 15 and b equal to 1/4 fits the data sufficiently well. However, in neutral conditions our data approach a mean value of about 0.7 rather than 1.0. Our fitted expression thus reads ϕm=0.7(1-15z/L)-1/4.

Normalized shear production was thus found to be low in the present data set in comparison to previously reported results. The scatter in our data was, however, found to be large enough that a von Kármán constant value of 0.4 was found to be within a 95 % confidence interval for neutral stratification. The reason for low normalized shear production is unclear, but it could be a reflection of measurement uncertainty, non-stationarity and heterogeneity.

In Fig. , we have replotted the buoyancy production term (in blue circles) and shear production term (in red circles) as a function of gradient Richardson number. Here, data outside the afternoon transition period are also included to show the behavior also in slightly stable conditions. Two larger horizontal ellipses encircle data for which the buoyancy production term is very small. An average shear production for this group is about 0.7 as observed for the near-neutral data during the afternoon transition just before stable stratification has started. As discussed in , at this site, there is a delay period between when the buoyancy flux becomes zero and when the vertical virtual potential temperature gradient becomes zero. Therefore, this group of data has a range of Richardson numbers between about -0.4 and -0.2. Here, Richardson number is the gradient Richardson number, Rig=g∂θ‾∂z∂U∂z. This result may, however, not be a general feature of the afternoon and evening transition as discussed by , who obtained different results with other data sets. It is interesting to note, however, that for this data set, when the Ri number is close to zero and the buoyancy flux is close to zero, such as for the data encircled with the smaller vertical ellipses in Fig. , a mean value of shear production of about 1.0 is observed. These observations may be interpreted to imply that, in more stationary neutral conditions (when both flux and gradient are small), we observe the consensus value of 1.0, but in the case of still transitional behavior from convective eddies in the afternoon transition until and around the time of zero-buoyancy flux, we observe lower values of normalized shear production.

For dissipation, we note a variety of different results in the literature . Here, we choose to fit a linear expression to z/L. Our fitted expression becomes ϕϵ=0.45(1-1.2z/L), which suggests a weaker normalized dissipation rate in near-neutral conditions (of about 0.5). and find a value of 1.0, which would imply no total transport in neutral conditions (assuming the normalized shear production in neutral conditions is 1). Our value is closer to the value 0.61 suggested by and , and considering our observed low shear production and measurement uncertainty, these numbers may be considered comparable.

Both our shear production relationship and dissipation relationship was determined by first producing least-squares fitted expressions, but these were slightly adjusted to ensure that the transport data in the TKE budget could also still be reasonably well fitted by a residual expression. For the sum of turbulent and pressure transport term (to be consistent with observed small TKE tendency), our expressions in Eqs. ()–() then suggest ϕT=0.46z/L-0.7(1-15z/L)-1/4+0.45.

For z/L below -1, this is approximately a linear equation, 0.5z/L, and implies somewhat lower transport than a study focused on this imbalance term by , who found a best-fit linear relationship of 0.69z/L using an extensive oceanic data set. In the neutral limit, our fitted value of -0.25 implies a larger transport than suggested by (0.0) and (-0.17) but lower than the value suggested from of -0.39. In a near-neutral range our expression is nonlinear as a consequence of the nonlinearity of the shear production term. A similar nonlinearity is also suggested by the expression given by to come both from shear production and their expression of dissipation rate. In their case, the transport term also becomes positive for a range of near-neutral z/L values. also observed positive transport values in an extensive data set of near-neutral conditions under steady conditions (not transitions). This was found to be related to a large pressure transport of turbulence into the surface layer which also led to an unusually large normalized dissipation of 1.24 . As previously discussed, we only observed a few occasions of positive transport values related to clouds and/or larger uncertainty in the dissipation estimates, and this effect is not included in our mean expression.

An alternative way to express dissipation in models is to relate it to the TKE (E) or subgrid-scale energy (e) and a dissipation length scale lϵ. For instance, use a relationship of -2E3/2/zi for dissipation corresponding to a length scale of zi/2; see also the more generalized case in and their Eq. (2.3). Near the surface, the expectation is that dissipation becomes dependent on the distance above the ground z, and we will explore these aspects based on our field measurements.

In Fig. , dissipation is shown as a function of E3/2/z averaged for the afternoon. Here we first carry out an investigation of the dissipation dependence on measurement height and boundary layer depth using data averaged for full afternoons. Then later we also test our findings using data with a shorter averaging time of 1 h to be consistent with our hourly TKE budget analysis. The height dependence of the data is displayed in Fig. a by assigning different colored circles (black, blue, magenta and red) to the four measurement heights 2.23, 3.23, 5.27 and 8.22 m. A higher dissipation rate is found closer to the ground, and at any given measurement level there is a variation in dissipation related to the characteristics of each afternoon. Two best-fit linear relationships are included. One of them (full line) is forced through origin because it may be natural to assume that dissipation is zero when TKE is zero. In Fig. b, however, a colored symbol is assigned to each afternoon and it becomes clear that the dissipation dependence on the variable E3/2/z is weaker for each afternoon than implied by the full line forced through origin. It is in fact closer to the dependence implied by the dashed line y=-0.0060x-0.0019, which is a best fit on all measurement points. The slope value -0.0060 lies within the 1 standard deviation range of the mean -0.0044±0.0017 that was found when fitting each afternoon independently to the expression y=kx+A and then taking an average of all the fitted slope values k. For the intersect values A with the y axis, a mean value of -0.0023 m2s-3 with standard deviation 9.3×10-4 was found by this procedure. Thus, we can conclude with some certainty that non-zero intersection values with the y axis exist in this representation. We interpret this to imply that a variation in dissipation exists which should not be related to height above the surface.

In Fig. , we further explore this non-local variation in dissipation by plotting the intersection values A as a function of -E3/2/zi. Here, mean afternoon TKE values and mean boundary layer depth zi determined from lidar and UHF profiler were used. For 26 June no boundary layer height data were available from the lidar. Larger symbols are used to denote when lidar data have been used, and each afternoon is color-coded and uses the same symbols as in previous figures. It can be seen that a positive correlation between the parameters exists, and two best-fit lines are included. The full line based on zi determined from UHF profiler data suggests a slope value of about 2.1 and the dashed line corresponding to lidar data suggests a slope value of 2.2. Both expressions have a small negative intersection value for the y axis of -1.6×10-4 and -1.1×10-4 m2s-3, respectively, which cannot be concluded to differ much from a value of 0 given the uncertainty in the variables. We note that the slope value of 2.2 corresponds to less deviation from zero of its intersection value with the y axis, and therefore we use this as a slope value representative of the data set.

Our final alternative form for expressing dissipation as a function of TKE and a dissipation length scale then becomes D=-E3/2lϵ=-E3/22.2zi+0.006z when combining the fitted slope values in Figs. and . Here, the suggestion is that the distance from the ground z and boundary layer depth zi act in parallel to decide the governing dissipation length scale lϵ. It is worth noting that our coefficient value of 2.2 does not depart very much from the proposed value of 2.0 by or 1.92 by based on other data sets, suggesting it may have some general validity. Equation () also implies that, for heights higher than about 2.73 % of the boundary layer depth, the contribution from the z-dependent term is less than 10 % of the zi-dependent term. The expression then differs only by about 10 % of what used when modeling dissipation in very convective situations.

Figure  shows dissipation estimated from Eq. () (in b) and from Eq. () (in a): D=-u∗3kz0.45(1-1.2z/L). Equation () is implied by the fitted linear relationship of normalized dissipation to the stability parameter z/L in Eq. (). In this final evaluation we have used all 53 h of data during the afternoon transition period for which all required parameters for both models were available. Boundary layer depth estimates from the UHF wind profiler were used to also be able to include data from 26 June.

Both models behave relatively similar for cases with low observed dissipation (> -0.0025), whereas the z/L model has a tendency to overestimate dissipation for larger observed values of dissipation and a bias of -9.3×10-4 m2s-3 was found. The bias for the TKE–length scale parametrization was -4.9×10-4 m2s-3, also suggesting a slight overestimation of dissipation rate. The centered root-mean-square difference was 1.8×10-3 m2s-3 for the z/L model and about half (0.93×10-3 m2s-3) for the TKE–length scale model. The linear correlation coefficient between measurement and model was lower for the z/L model (0.70) compared to the TKE–length scale model, which had 0.80. Finally, the standard deviation of the z/L model was found to be 2.5×10-3 and 1.4×10-3 m2s-3 for the TKE–length scale model, which should be compared to the observed standard deviation of 1.5×10-3 m2s-3. In four out of four skill scores the TKE length scale model, which takes into account boundary layer depth and height above the surface, was hence found to better represent the observed dissipation than the stability-dependent z/L model. It should be noted that both models include two fitting parameters and that no explicit stability dependence has been included for the TKE length scale model. However, it may be argued that an implicit stability dependence should be included since the magnitude of TKE depends on stability. It should also be recognized that only afternoon data are considered here and other parts of the diurnal cycle such as morning transitions could be studied in future work.