The theory of Kolmogorov is the foundation of turbulence research . It states that at sufficiently large Reynolds numbers and under the assumption of local isotropy, there exist a range of scales where statistics of velocity take a self-similar form. Further, within this range, a sub-range of scales of size r, where η≪r≪L, can be distinguished. This will be further referred to as the “inertial range”. In this range of scales, statistics of turbulence depend not on viscosity but only on the dissipation rate ϵ. It follows that the wavenumber spectrum of the vertical wind velocity component can be expressed as 3E⟂(κ,t)=C⟂κ-5/3ϵ2/3, where C⟂≈0.65 is a constant and κ is the wavenumber. Equivalently, the same can be presented in terms of the second-order structure function, which for the vertical velocity component w reads 〈δw2〉=〈(w(x+r,t)-w(x,t))2〉. Within the inertial range and under the assumption of local isotropy, this function takes the form 4〈δw2〉=Cϵr2/3, where C≈2.86. Equations () and () form the basis of various schemes for the estimation of the energy dissipation rate, including from lidar measurements .

Equations () and () work well when turbulence is close to isotropic, at least locally, and is stationary. Recently, extensions of Kolmogorov's theory towards unsteady turbulence were put forward by . These extensions predict that the rate of energy transfer across scales in the inertial range is affected by turbulence decay and is not constant. expressed the turbulence kinetic energy spectrum as a sum of the equilibrium, Kolmogorov spectrum and a non-equilibrium correction. They derived a formula similar to Eq. () and argued that deviations of the spectra from Kolmogorov's scaling are related to the deviations from Cϵ=const in Eq. (). focused on the large-scale part of the turbulence kinetic energy spectrum during the non-equilibrium decay and found that it has the self-preserving form 5E(κ,t)∝ϵL3f(κL). introduced the notion of “balanced nonstationary turbulence”, where the transfer across the scales is proportional to dissipation, albeit with a proportionality constant smaller than 1. This assumption led to a modified form of the energy spectrum in the inertial range: 6E(κ,t)=Ckϵ(t)2/3κ-5/31-c(κ+L(t))-2/32, where c is a dimensionless constant. The spectra followed the above formula even during the equilibrium decay with Cϵ=const. The function in square brackets in the above equation reaches the value 1 asymptotically, at large wavenumbers (small scales), where the spectra remain close to the Kolmogorov -5/3 form.

investigated how the inertial range of the structure functions is affected during non-equilibrium decay of turbulence. They concluded that in the case of decaying turbulence, the second- and the third-order structure functions are closest to Kolmogorov's predictions at the small-scale end of the inertial range. For larger scales, the structure functions increasingly deviate from equilibrium, even at very large Reynolds numbers. The authors considered Lundgren's formula for the structure functions, derived with the use of matched asymptotic expansions. For very high Reynolds numbers, it reads 7〈δu2〉=Cϵr2/31-A2(r/L)2/3, where A2 is a dimensionless constant of the order 1. Under the assumption of local isotropy, the above formula is mathematically equivalent to Eq. (), provided that the bracketed term in Eq. () can be expanded in the Taylor series.

We now discuss turbulence statistics, in particular, the integral length-scale change in time according to the theory of equilibrium and non-equilibrium decay. Based on this, we can identify which type of parametrization more adequately describes the collapse of the convective boundary layer before sunset. Here we mostly refer to the recent papers by and , who investigated decaying turbulence using numerical experiments and derived non-equilibrium decay laws.

Under the assumption of horizontal homogeneity, the transport equation for the turbulence kinetic energy k=〈ui′ui′〉/2=3/2U2 in the ABL reads 8∂k∂t=-T-ϵ+P+B, where T=1ρ∂〈p′w′〉∂z+12∂〈ui′ui′w′〉∂z,P=-〈ui′w′〉∂〈ui〉∂z,B=〈w′b′〉. T, P and B stand for the turbulent transport, shear production and buoyancy forcing, respectively. Above, p′ denotes the pressure fluctuations and b′ the fluctuations in buoyancy. The buoyancy flux will be defined as 9〈w′b′〉=g〈θv〉〈w′θv′〉, where θv is the virtual potential temperature and g stands for the gravity acceleration.

Turbulence production due to shear P is an important part of the budget close to the Earth's surface. Buoyancy B plays a dominant role during daytime convection; finally, the role of turbulent transport T is to transfer the kinetic energy produced close to the surface to higher altitudes.

At the beginning of the evening transition, the buoyancy flux B=0 and the convective boundary layer collapse rapidly. Turbulence is still produced by the shear P in the surface layer; however we can assume that at higher altitudes, turbulence starts to decay freely such that the time derivative of k on the left-hand side (LHS) of Eq. () is balanced mainly by the dissipation ϵ. Under such assumptions, Eq. () rewritten in terms of the velocity scale U2=2/3k is 10dU2dt=-23ϵ(t).

During the equilibrium decay, the dissipation rate is described by Taylor's law (Eq. ) and Kolmogorov's type of turbulence kinetic energy spectrum (Eq. ). For further comparisons with experimental data, it is convenient to express the rate of change of velocity statistics as a function of the turbulence Reynolds number, Re=UL/ν. It is of note that the product of velocity and length scales UL is proportional to the eddy viscosity νT. Hence, the Reynolds number Re in fact expresses the ratio of the eddy viscosity and the molecular viscosity Re∼νT/ν.

After substituting Eq. () into the right-hand side (RHS) of Eq. () and further rearrangements, we obtain 11dU-2dt=23Cϵ1ννUL=23Cϵ1ν1Re. To derive the corresponding equation for the rate of change in L, the equilibrium law (Eq. ) is differentiated over time: 121ϵdϵdt=31UdUdt-1LdLdt. To express the derivative dϵ/dt, the predictions of the classical k-ϵ turbulence model can be used : 131ϵdϵdt=C01U2dU2dt, where C0=1.9 is a model constant. Substituting Eq. () into Eq. (), we obtain 141LdLdt=32-C01U2dU2dt. After multiplying both sides by 2L2, using Eqs. () and (), the following relation is derived: 151νdL2dt=AeULν=AeRe, where Ae=4/3(C0-3/2)Cϵ. Hence, the classical theory predicts that the integral length scale increases with time during the decay of turbulence and that the decay of L2 slows down when the Reynolds number Re decreases.

Even though L is expected to increase in time, the product UL and also the turbulence Reynolds number Re are expected to decrease in time. Indeed, after rearranging and combining Eqs. () and (), we obtain the following formula: 16dULdt=UdLdt+LdUdt=-13CϵU2+Ae2U2=-115CϵU2<0.

The scenario of non-equilibrium decay predicts that the dissipation rate scales according to the relation of Eq. (). After introducing this relation into Eq. () and further rearrangements, we obtain 17dU-2dt=23CneU0L01ν2νUL2=23CneRe01ν1Re2, where Re0=U0L0/ν. Equation () is different than the corresponding Eq. () derived for the equilibrium decay. It scales with Re-2 instead of Re-1, and, additionally, it depends on the initial conditions through Re0. During the turbulence decay, the Reynolds number Re decreases with time, hence the ratio Re0/Re≥1. This implies that decay rates predicted by Eq. () increase in time more sharply than those predicted by its equilibrium counterpart (Eq. ). Fast anomalous changes in turbulence kinetic energy during non-equilibrium decay, which gradually decrease at later times, were observed experimentally by .

As argued by , in the initial stages of decay, after the forcing is stopped and long-range correlations suddenly disappear, the integral length scale starts to decrease with time. Using the self-similar form of the spectra (Eq. ) and relation (Eq. ), derived the following formula for the time derivative of the length scale: 181νdL2dt=Ane-BneRe, where Ane and Bne are coefficients related to the initial conditions and integrated spectral function. We note that the formula presented in the original paper by was written in terms of the Reynolds number based on the Taylor length scale. However, in the non-equilibrium decay, this length scale becomes proportional to the integral length scale L, which follows directly from Eq. (); see also the discussion by . Equation () is qualitatively different than its equilibrium counterpart (Eq. ). For large turbulence Reynolds numbers Re, the time derivative of L2 can be negative, causing L to decay in time. As the Reynolds number decreases during the decay of turbulence, the RHS of Eq. () will eventually become positive and L will start to increase with time until dL2/dt reaches its maximum. At this point, the system arrives at its equilibrium state and the statistics further follow the equilibrium Taylor relation (Eq. ), albeit with a larger value of Cϵ≈1. This also implies that L further grows in time according to Eq. ().

Results of the numerical experiment presented in confirmed that during the non-equilibrium decay the time derivative of L2 was a decreasing function of the turbulence Reynolds number. Moreover, in the initial stages of decay, dL2/dt<0.

Our purpose is to show, based on the experimental evidence, that the non-equilibrium form of decay is present in the atmospheric turbulence before sunset. During non-equilibrium decay, estimating the dissipation rate from the low-wavenumber part of the wind velocity spectra (Eq. ) or with the use of Eq. () becomes questionable and leads to underpredictions of the dissipation rate. On the other hand, the resolution of the Doppler lidar is not sufficient to measure small turbulent motions, which are less affected by the non-equilibrium correction. Hence, unlike in laboratory experiments, direct verification of Eqs. () and () is not possible. For this reason, the non-equilibrium will be detected indirectly, by recording changes in the scaling of the frequency spectra and the structure functions. According to Eqs. () and (), increasing deviations from the Kolmogorov scaling can be explained by the decrease in the integral length scale. This is in contrast to the theory of equilibrium decay where the integral length scale should increase with time according to Eq. () and the scaling of spectra and structure functions should become closer to the Kolmogorov one. We calculated both frequency spectra and structure functions from time series of the vertical velocity component measured by the Doppler lidar and estimated the scaling exponents (slopes) using least-squares fitting.

Apart from non-equilibrium correction, the slopes can be affected by insufficient resolution in time and space and high noise-to-signal ratios . investigated modifications of spectra due to instrumental noise, aliasing and space averaging. In particular, in the Fourier space, the latter modification affects the whole range of scales and not only the highest wavenumbers. Moreover, the modification depends on the horizontal wind speed U‾. In order to convert time coordinates to space coordinates, Taylor's frozen-eddy hypothesis is used, with x=U‾t. This relation is justified only if the turbulence intensity defined as U/U‾, where U‾ is the mean horizontal wind speed, is small enough. As U‾ decreases, frequency spectra become more affected.

Because of these possible modifications of the spectra, in this work we focus on detecting changes in the slopes rather than on their exact values. We filter out data with high instrumental noise and data where U/U‾>0.15. Although the frequency spectra and structure functions are mathematically equivalent, they may respond differently to different sources of errors. In parallel to the slopes, we present the calculated integral length scales and mean wind velocity to verify whether the changes in the scaling are due to changes in the integral length scale, as predicted by Eqs. () and () or, rather, are affected by changes in the mean wind speed.

As we focus on the evening hours in this work, we denote the sunset time as t=0. Turbulence statistics are calculated as 0.5 h averages or medians relative to the sunset time for both Rzecin and Warsaw for the same months during the summer. We present the statistics as functions of time to analyze how they change during rapid decay of turbulence.

We first present the evolution of the heights of the ABL and convective boundary layer (CBL) in Fig. . The uncertainties were estimated as a standard error of the mean (SEM) of the half-hour time intervals; see Appendix . As is observed, the ABL height at the rural site is lower and decreases more rapidly compared to the urban site. This may indicate the influence of the urban heat island effect in large cities. The behavior of the CBL height is similar; i.e., much higher values are observed in Warsaw. In both the Rzecin and the Warsaw sites, the CBL collapses rapidly ca. 2.5 h before sunset. The results are in agreement with the long-term study by .

In order to estimate the beginning of the evening transition, we present the buoyancy fluxes, their median values and standard errors in Fig. . At t=-2 h, the median values of the fluxes are still positive. Afterwards they cross the zero level at the Rzecin site and become close to zero at the Warsaw site.

The buoyancy fluxes and the CBL height are further used to calculate the characteristic scales which govern the flow during daytime convection. The convective Deardorff scale w* and the corresponding timescale τ are defined as 26w*=D〈w′b′〉01/3,τ=Dw*, where the CBL height is denoted by D and 〈w′b′〉0 is the surface value of the buoyancy flux. We present both scales in Fig.  and additionally compare w* with the friction velocity u*. A description of the error estimates is given in Appendix . As is seen in Fig. , both in Rzecin and in Warsaw, w* is still larger than u* at t=-2 h and the timescale τ increases sharply around t=-2 h. This timescale is larger in Warsaw, which suggests that the turbulence decay is slower in the urban surface layer. Shortly after t=-2 h, u* becomes larger than or comparable to w* (within the confidence intervals), which implies that turbulence production due to shear P becomes dominant in the surface layer. We will assume that at higher altitudes the contribution of the shear production is negligible and that, to the leading order, the evolution of the turbulence kinetic energy is described by Eq. (). Under such assumptions, the changes in U and L should be described by formulas derived in Sect.  and . For this reason we will further treat statistics at t=-2 h as initial conditions and focus our attention on the time interval of ±2 h relative to the sunset.

At 18:30 UTC, with convection still present, the wind energy spectrum was measured at a height of 195 ma.g.l., i.e., within the mixed layer (see Fig. a). In this case the spectrum is visibly steeper than -5/3. The steep slopes of frequency spectra in the convective regime were also reported by other authors . speculated that these steep slopes could be linked to the presence of inverse cascades at large scales, which leads to the -11/5 Bolgiano–Obukhov scaling . However, as argued by , in the case of poorly resolved data, the steepening of slopes may also be caused by the artificial instrumental dissipation due to effective low-pass filtering.

Figure b shows that before sunset, when the convective layer rapidly decays, the frequency spectrum becomes less steep than the Kolmogorov's prediction. This observation is not related to instrumental artificial dissipation, which rather causes the opposite effect (steepening of the spectra).

In Fig. , the time–height evolution of the slopes of the frequency spectra and velocity structure functions and the estimate of the CBL height are shown. The regions marked by red and orange colors in Fig. a and yellow and light green in Fig. b exhibit scaling steeper than that of Kolmogorov. Before sunset, when the CBL collapses, the turbulence kinetic energy decays rapidly and a sharp decrease in the slopes at all heights is observed. Decrease in the slopes is also seen in the upper part of the convective ABL, where the rising updrafts become weaker. Therein, stable stratification possibly alters the spectra and structure functions. The stratification effects on spectral slopes were included in a recent model by . This issue is, however, beyond the scope of the present work, as we focus instead on the modification of the spectra due to nonstationarity.

We next investigated how different values of the detrending window influence the results in order to choose the most appropriate value. The size of the detrending window can potentially affect the turbulence velocity scale and the integral length scale, calculated from Eqs. () and (), respectively. To estimate the errors, we took into account the standard error of the mean and errors in velocity estimates from the HALO lidar toolbox (see Appendix  for details). We calculated and compared both mean and median values, and their comparison is shown in Fig. . Median values are advantageous if the probability distribution of the data is non-Gaussian or in the presence of rare but very large or very small values (outliers), which affect the mean. As can be seen in Fig. , all the mean values are larger then the corresponding medians. Both the mean and the median values of U are not much affected by the change in the detrending window in the algorithm. They decrease with time during the evening transition. Mean and median values of the integral length scales increase with increasing size of the detrending window. The data become considerably scattered for the largest window. This is to be expected, as L converges slowly and a large sample is needed to estimate it with sufficient accuracy . The time span needed to calculate statistics is proportional to L; hence larger length scales require longer time spans. As a result, 0.5 h averages may be insufficient to reduce random errors to acceptable levels needed to calculate the time derivatives from Eqs. () and (). Independent of the size of the detrending window, we find that the median values of integral length scales first decrease with time and next increase or become constant. This conclusion seems to be universal and allows further analysis to be performed using only one set value for the detrending window, which has been chosen to be 600 s. The median values of the product UL (and also the turbulence Reynolds number Re) decrease with time, as expected, for all detrending windows. The corresponding mean values decrease in time before sunset for the two smaller detrending windows.

To investigate how the turbulence properties change not only in time but also with altitude, we divided results into three altitude ranges: 75–345 ma.g.l., 345–585 ma.g.l. and 585–855 ma.g.l. According to Fig. , the measurements are within the residual layer, and in Rzecin the level 75–345 ma.g.l. corresponds to the lower and middle part of the ABL, 345–585 m.a.g.l. reaches the top of the layer, and the third level of 585–855 m.a.g.l. is placed partly above the mean top of the ABL. The ABL is higher in Warsaw such that the three levels corresponds to its lower, middle and upper part, respectively. In spite of the scatter of results for individual 0.5 h averages, median values of the slopes of frequency spectra determined for the range f∈[0.15s-1, 0.3s-1] (Fig. ) clearly increase from values close to Kolmogorov's -5/3 to around -1 after sunset. Analogously, slopes of the structure functions (Fig. ) determined for the range r∈[40m, 150m] decrease with time for both sites, Rzecin and Warsaw. The inertial ranges of structure functions tend to be smaller than those for wind frequency spectra. Hence, due to the finite temporal resolution of the measurements, the calculated structure functions may be affected by large eddies from beyond the inertial range. As a result, the slopes of structure functions are gentler than the Kolmogorov 2/3 predictions even at t=-2h, especially for the Rzecin site. The slopes also show the altitude-dependent relation. What can be observed from both figures (Figs.  and ) is that for Warsaw the median values of slopes at higher altitudes are much closer to Kolmogorov's predictions than they are for the rural case. This can be partly explained by the urban heat island effect and differences between the urban and rural morphologies. The turbulent flows are generated due to an intense shear at the top of the canopy layer H . In the urban environment, this is at the height of the rooftops of the buildings. In Warsaw, in a 1 km × 1 km area centered on the RS-Lab, H≈12.2 m. In Rzecin, the top canopy layer height reaches H=2–2.5 m. As far as the thermally driven turbulent flows are concerned, the city centers are usually warmer than the suburbs and areas of urban agglomeration because of the urban heat island phenomenon and, related to this, have relatively high heat capacity compared to the rural environment. This implies that the urban surface can still emit heat even after sunset. Heat emitted from a warm urban surface generates convection and mixes the air in the urban canopy layer. It also generates a dome of warm air in higher parts of the boundary layer. The temperature profile of this dome is quasi-adiabatic and similar to the temperature profiles around midday . Results presented in Figs.  and  suggest, however, that the heat island effects are most significant 2 h before sunset at the highest altitude range of 585–855 ma.g.l. Afterwards, a rapid change in the scaling exponent is observed.

In order to examine if the changes in the scaling exponent are induced by changes in the mean velocity, we analyzed how U‾ changes with time and height. If turbulence were Kolmogorov-like and the spectra were affected only by spurious modifications due to insufficient resolutions in time and space, then a decrease in mean velocity would increase the absolute value of the slopes by introducing artificial dissipation. An increase in the mean velocity, on the other hand, would bring the scaling closer to the Kolmogorov -5/3 (or 2/3 for the structure function) but not below this value. Figure  presents the mean horizontal velocity U‾. As is seen in the figure, U‾ increases with time mostly close to the surface and only after sunset. At higher altitudes, it has only a slight tendency to increase. At the same time, the absolute values of slopes in Figs.  and decrease considerably below -5/3 and 2/3, respectively. Hence, we conclude that changes in the slopes are not primarily affected by the changes in the mean wind speed. Instead, the collapse of the largest convective motions possibly leads to the non-equilibrium states of turbulence as predicted by Eqs. () and (). Later on, the size of the inertial range decreases and is shifted towards small scales (large wavenumber) which are not detected by the lidars.

A difference between rural and urban sites observed in Fig.  is that in the latter, mean velocity starts to slightly increase before sunset at low altitudes. As described by , when turbulence decreases rapidly, the airflow becomes more influenced by the surface heterogeneity and horizontal temperature variations. The temperature variations lead to local horizontal pressure gradients. These, in turn, could induce stronger horizontal winds to increase the horizontal temperature transport.

discussed the formation of early-evening calm periods during which the mean velocity decreased in the surface layer and had a tendency to increase at higher altitudes. , on the other hand, observed a decrease in the mean speed at altitudes of up to 500 m. The early-evening calm periods in the surface layer were also recorded in other studies . No systematic decrease in the mean wind speed is observed in Fig.  before sunset. However, we recall that, to calculate turbulence properties, we removed data for which the Taylor frozen-eddy hypothesis was not satisfied, that is data with a high U/U‾ ratio. This procedure could, to some extent, affect the tendencies observed in Fig. .

The dependence of the integral length scales on time and altitude is presented in Fig. . The vertical integral length scales increase with the altitude. Moreover, at the urban site they are larger at higher altitudes, which follows from stronger convection and more shear-driven turbulent flows in this part of the ABL. As argued by , the roughness surface layer in cities may be higher than previously expected and can reach up to z/H=30, where H is the mean height of buildings (at the Warsaw site, H≈12.2 m).

According to Fig. , the turbulent velocity scale decreases with time for both rural and urban cases. In connection with the stable horizontal wind velocity, before and after sunset the figure shows that the observed turbulent flows are mostly driven by the convection, which agrees with our observational statement about the decay of the ABL and the presence of the decaying convection-driven turbulent flows before sunset.

To study decay of turbulence with reference to the values measured 2 h before sunset, normalized 15 min median values of L/L0 and U/U0, where L0=L(t=-2h) and U0=U(t=-2 h), are plotted in Fig. . In this figure we include only data measured before sunset. It is seen that both turbulence kinetic energy and the integral length scale have a tendency to decrease with time before sunset. As follows from Eqs. () and (), a decrease in the integral length scale during the decay of turbulence will cause increasing deviations from the Kolmogorov scaling. This appears to be the case in the observed decay of the convective boundary layer and explains the changes in the slopes, presented in Figs.  and .

We finally calculated the time derivatives dU-2/dt and dL2/dt from the 0.5 h median values (for both sites and at all heights) for -2 h ≤t≤0 (evening hours) and present them in Fig.  as functions of Re. We compare experimental data with the classical predictions of Eqs. () and () with Cϵ=0.5 and the non-equilibrium predictions () and (). The coefficients Ane, Bne and Cne in Eqs. () and () were estimated from linear regression as the best fit.

The errors in the estimates in Fig.  are considerable such that it is difficult to identify the scaling of dU-2/dt. However, dU-2/dt is clearly a decreasing function of the turbulence Reynolds number Re, as predicted by both equilibrium and non-equilibrium relations.

As far as dL2/dt is concerned, points follow the non-equilibrium scaling. As the Reynolds number decreases, both equilibrium and non-equilibrium predictions for dL2/dt become close, so we expect that at even lower Re values, the data will follow the equilibrium predictions. Hence, by analysis of both panels in Fig. , it can be concluded that the non-equilibrium decay is likely to be present in the initial stages.

Due to large deviations from the equilibrium scaling, the energy dissipation estimates with the use of Eq. () or Eq. () are not reliable at t>-2 h. Therefore we used Eq. () to calculate the profile of ϵ only at t=-2 h and for z/D<0.6. An example of the second-order structure function is presented in Fig. a, and the dissipation rate, non-dimensionalized with the use of w* and the CBL height D, is presented in Fig. b and c, as a function of z/D. We use these estimates for comparison with the predictions of Eqs. () and () in Fig.  and mark them as red dots at t=-2 h. Figure  presents median dissipation values before sunset obtained using the classical Taylor law (Eq. ), with Cϵ=0.5 and assuming the non-equilibrium scenario. In the latter, the dissipation rate is expected to follow the non-equilibrium relation (Eq. ), until its values become equal to the predictions of the classical Taylor law (Eq. ) but with a higher value of the constant Cϵ=1, which is the upper bound of the dissipation coefficient . The dissipation rate further follows the equilibrium Taylor law (Eq. ) with Cϵ=1. During this latter period, the frequency spectra will still deviate from Kolmogorov's scaling at low wavenumbers . The same is true for the structure functions; see Eq. (). The equilibrium -5/3 and 2/3 slopes of spectra and structure functions will be reached asymptotically at high wavenumbers which are not measured by the lidar system. By comparing both scaling laws, it can be concluded that the equilibrium Taylor law underpredicts the dissipation rate of turbulence kinetic energy by up to a factor of 2.

The dissipation values decrease with altitude and in time. In Rzecin, the decrease is most rapid closer to the surface. Under the conditions of radiative cooling over a rural area, surface-based inversion is developed, which stops the air mixing above the ground faster than at the urban sites. As a consequence of very similar meteorological conditions for both stations (dry and warm conditions), we assume that the differences in turbulence properties result mainly from differences in surface morphology between both environments. Due to the difference in surface morphology, at the urban site the effect of wind shear and the presence of shear-driven turbulent flows are still significant at altitudes of 75–345 ma.g.l. On the other hand, Fig.  suggests that at higher altitudes, the dissipation rates decrease in time much faster over the urban site. Shortly before sunset, the heat island effects are possibly much more limited in height.