A NUMERICAL STUDY OF DAILY TRANSITIONS IN THE CONVECTIVE BOUNDARY LAYER. Zbigniew Sorbjan

Transcription

1 A NUMERICAL STUDY OF DAILY TRANSITIONS IN THE CONVECTIVE BOUNDARY LAYER Zbigniew Sorbjan Department of Physics, Marquette University, Milwaukee, WI 53201, U.S.A. address: Marquette University, Department of Physics Wehr Phys. Blg., 540 North 15th Street, Milwaukee, WI tel. (414) , fax: (414) , Submitted to Boundary-Layer Meteorology on August 15, 2006, revised on Oct 28, 2006 Abstract. The paper examines daily (morning-afternoon) transitions in the atmospheric boundary layer based on large-eddy simulations. Under consideration are the effects of the stratification at the top of the mixed layer and of the wind shear. The obtained results describe the transitory behaviour of temperature and wind velocity, their second moments, the boundary layer height Z m (defined by the peak of the potential temperature gradient) and its standard deviation σ m, the mixed layer height z i (defined by the minimum of the potential temperature flux), entrainment velocity W e, and the entrainment flux H i. The entrainment flux and the entrainment velocity are found to lag slightly in time with respect to the surface temperature flux. The simulations imply that the atmospheric values of velocity variances, measured at various instants during the daytime, and normalized in terms of the actual convective scale w *, are not

2 expected to collapse to a single curve, but to produce a significant scatter of observational points. The measured values of the temperature variance, normalized in terms of the actual convective scale Θ *, are expected to form a single curve in the mixed layer, and to exhibit a considerable scatter in the interfacial layer. Keywords: Convective boundary layer, Daily transitions, Interfacial layer, Large-eddy simulations, Mixed layer. 1. Introduction The atmospheric boundary layer (ABL) is an inherently complex and heterogeneous system, which is under permanent transition, enforced by a variety of internal and external factors. The external forcing related to the solar cycle is most essential, as it defines two distinct regimes of the ABL, convective during the day, and stable at night. The first large-eddy simulations (LES) of transitory processes in the ABL were performed by Deardorff (1974 a, b), who considered daily transformation from 1000 to 1600 LT on day 33 of the Wangara experiment. During subsequent years, however, the subject of daily transitions was put aside, and most of the LES studies focused on the convective boundary layer (CBL) under idealized "quasi-steady" conditions, characterized by a constant surface heat flux. The trend was eventually interrupted by the LES analysis of the decaying mixed layer, performed by Nieuwstadt and Brost (1986). The authors investigated a windless and dry case, with turbulence decaying as a result of a sudden shut-off of the upward surface heat flux. 2

3 A decade later, Sorbjan (1997) considered evening transition with a gradual change of the heat flux with time, in response to the decreasing sun s elevation. His study was followed by the work of Cederwall and Street (1999), Shaw and Barnard (2002), and Edwards et al. (2004). Acevedo and Fitzjarrald (1999, 2001) examined additional effects of moistening near the Earth's surface during the early evening transition. Pino et al. (2004) included the effects of shear and the inversion strength during the evening decay. The numerical investigations of the decaying mixed layer were subsequently followed by observational work. Grant (1997), LeMone and Grossman (1999) examined evening evolutions of the ABL. Mahrt et al. (1999) observed that the latent heat flux during evening events decreased at a slower rate than the strength of turbulence and the boundary layer depth, which led to the significant moistening of the surface layer. Acevedo and Fitzjarrald (1999, 2001) reported the occurrences of specific humidity jumps during evening transitions, and the drops in surface temperature, accompanied by an abrupt decay in wind velocity. Grimsdell and Angevine (2000, 2002) used the wind profiler data to characterize the late afternoon transformations. Processes resulting from daily transformations during morning and early afternoon conditions obtained relatively lesser observational and theoretical attention, and have not yet been adequately examined. Existing work on the subject includes the simulations of Sorbjan (1997), who considered "free-encroachment", which takes place when the morning mixed layer rapidly grows into a residual of the well-mixed layer from the previous day, and also the observational studies of Angevine et al. (2001), Silva et al. (2002), and LeMone, et al. (2002). The purpose of this paper is to further examine daily (morning-afternoon) transitions in the convective boundary layer, especially their self-similar structure, controlled by the wind shear and thermal stratification at the top of the mixed layer. The work focuses on three daily 3

4 regimes, which will be referred to as "morning expansion", "mid-day convection", and "afternoon weakening. Free encroachment and evening decay are not considered in the present study. During "morning expansion", the surface heat flux H s monotonically increases with time. Convective thermals erode the nocturnal temperature inversion and form a quickly growing (expanding) mixed layer. "Mid-day convection" is developed around the solar noon, when the surface flux H s is approximately constant within a period of a few hours. The universal properties of this quasi-steady regime were discussed by Sorbjan (2005, 2006) in cases of both free and forced convection. The mid-day convection eventually evolves into the "afternoon weakening" regime, when the surface heat flux H s decreases in response to the declining sun s elevation. The structure of the paper is as follows. The performed large-eddy simulations are outlined in Section 2. The results, describing the daily behaviour of temperature, wind, and their second moments, are introduced in Section 3. The final remarks are presented in Section Simulations The LES model of Sorbjan (1996a) was employed to simulate four cases of the atmospheric mixed layer. The considered cases were defined by two values of the geostrophic wind, 0 m s -1 and 10 m s -1, and by two types of temperature stratification at the top of the mixed layer. The first type involved the presence of an interfacial layer with a sharp temperature jump. Such a jump is usually formed above mixed layers within long-lived anticyclonic systems. In the second type, the interfacial layer was absent, which is often observed during an early-morning erosion of the nocturnal temperature inversion. Hereafter, the four considered cases are referred to as R0g, 4

5 R0s, R10g, and R10s. The numbers 0 and 10 specify the assumed values of the geostrophic wind, while the letters "g" and "s" indicate a "graduate" or "sharp" temperature trend at the top of the initial mixed layer. The simulations employed a mesh with 128 x 128 x 100 grid points. The grid increments were Δx = Δy = 25 m, and Δz = 20 m. The roughness parameter was z o = 0.01 m, while the Coriolis parameter was f = 10-4 s -1. In order to reduce the simulation time and avoid unnecessary efforts of generating nocturnal conditions (which would require large jobs with about 10 5 time steps, and with a much finer resolution than is needed for convective runs), each case included a relatively short, convective "start-up" run. It assumed the presence (some time after a sunrise) of a 300m-deep mixed layer (as the initial condition), with a uniform potential temperature of T o = 300 K. The temperature gradient in the free-atmosphere was set to be Γ = 3 K km -1. The interfacial layer in runs R0s and R10s was initially 100 m thick, with the temperature gradient γ i equal to 30 K km -1. The "start-up" runs were executed for 4000 time steps in cases R0g and R0s, and for 9000 time steps in cases R10g and R10s, assuming a constant surface temperature flux H o = K m s -1. At the end of such initialization, a quasi-steady "morning" ABL was developed. After the initialization, simulations were restarted and continued for the next 6200 time steps in run R0g, for 6000 time steps in run R0s, for and time steps in runs R10g and R10s respectively. The surface temperature flux H s was varied with time t as: H s /H o = sin [π (t - t o )/6000], where t o indicates time (in seconds) at the end of each initialization stage. Because the time increments during the simulations were automatically adjusted based on the Courant-Friedrichs-Levy stability condition, the total simulation length was different for each run. The surface temperature flux H s in all simulations grew from H o to the maximum value of 5

6 2H o, and then decreased to the initial value H o, or slightly below it in runs R0s and R10s, as shown in Figure 1. Statistics of turbulence were obtained by a standard procedure in LES modelling, which involves horizontal averaging. No time averaging has been applied, which resonates well with the fact that simulated turbulence is non-stationary, i.e. its statistical characteristics (probability distributions, means and variances) change over time. Most statistical methods dealing with nonstationary processes are based on the assumption that the time evolution can be rendered stationary through the use of mathematical transformations. For example, subtracting the periodic mean function from the data creates a series of anomalies, which might occur to be stationary. Anomalies could also be normalized and as a result show stationary properties. Equivalent procedures were tested in the presented paper. Statistical moments were calculated at instants indicated in Figure 1 by letters ("O, A, B, C, D" in the "g" runs, and "o, a, b, c, d, e" in the "s" runs), and normalized in terms of current values of the convective scales of Deardorff (1970). Specifically, they were obtained at the end of the initialization period, at instants "O" in runs R0g and R10g, and as "o" in runs R0s and R10s. The instants "A" in runs R0g and R10g, and "a" in runs R0s and R10s describe "morning expansion". The instants, referred to as "B" or "b", were chosen near the moment, at which the surface temperature flux reached the maximum value ("mid-day convection"). The instants "C", "D", "c", "d", or "e", were obtained for decreasing values of the heat flux H s, and therefore mark the "afternoon weakening regime. In run R0g, the calculations consisted of partial runs performed in the following sequence: s-o, O-A, O-B, O-C, C-D, where "s" represents the beginning of the initialization run. After each partial run, the model was restarted from the stored values. In run R0s, the sequence 6

7 of calculations was: s-o, o-a, o-b, b-c, b-d. The run R10g was executed as a similar sequence: s- O, O-A, O-B, B-C, B-D, while run R10s was performed as a sequence s-o, o-a, o-b, o-c, o-d, o-e. The restarting caused the butterfly effects (due to deterministic chaos) in runs R0g, R10g, and R10s, which can be noticed in the history plots presented in Section 3. All history data in the paper (including those in Figure 1) were recorded every 20 time steps. The resulting characteristics of the ABL, obtained in all partial runs, are summarized in Table 1. In the table, z i is the mixed layer height, defined by the minimum value of the temperature flux, w * = (β H s z i ) 1/3 and Θ * = H s /w * are the convective scales, β = g/t o is the buoyancy parameter, g is the gravity acceleration, τ * = z i /w * is the eddy overturning time scale, and u * is the friction velocity. 3. Results 3.1. MEAN PROFILES The initial and resulting profiles of the potential temperature are depicted in Figures 2a and 2b. The figures show that the potential temperature in the mixed layer increased during all simulations, and the mixed layer extended vertically. In runs R0g and R10g (with a graduate temperature stratification on the top), the mixed layer growth was larger than in runs R0s and R10s. However, the increase of temperature in the mixed layer was by K lower. At the end of all simulations, the cooling effects can be noticed in the figures, as the temperature profiles near the surface coincide or cross at instants C, D, c, d, and d, e. The vertical cross-sections (not displayed) showed that the depth of the mixed layer, and the temperature gradients at the top of the mixed layer, increased in areas dominated by updrafts, and decreased in areas of sinking air. Consequently, after horizontal averaging, the interfacial layer, initially absent in runs R0g and R10g, appeared in these runs at the top of the mixed layer. 7

8 The layer can be identified in the figure by the increased values of the temperature gradient (hereafter, the maximum value of the temperature gradient in the interfacial layer will be denoted as γ i ). Its depth is in the range of m. Note in Figure 2 that the horizontal averaging is also responsible for smoothing out the initially sharp edges of the individual temperature profiles in "s" runs. In runs R0s and R10s, the initial 3K-temperature jump above the mixed layers was reduced to about K, while the temperature gradients γ i decreased (e.g., to 10 K km -1 in the interval c-d of R0s). In runs R0g and R10g, the resulting temperature gradients γ i exceeded the initial gradient of 3 K km -1 (e.g., γ i = 4.1 K km -1 in the interval C-D of R0g). Such changes in run R0s can be explained based on the change rate equation for the temperature jump ΔΘ at the top of the mixed layer. The equation can be written as d"# dt $ % dz i dt & d# m, where Θ m is the mixed- dt layer potential temperature. It can be verified based on Figure 2a that in the interval o-d the first term is about 0.15 K, while the second term of the equation is about 0.7 K, indicating that ΔΘ decreases with time. The gradients γ i in all runs did not change significantly with time. Thus, it can be concluded that convection does not generate strong temperature jumps at the top of mixed layers. There must be, therefore, external factors responsible for the formation of very abrupt changes of temperature at the top of the mixed layer. One of the factors can be identified as subsidence. Note that the convective term (w Θ/ z) in the potential temperature equation is zero within the mixed/residual layer. The term is negative above the mixed layer, which eventually can lead to the formation of large temperature jumps in sufficiently long-lived high-pressure systems. The profiles of wind velocity components in run R10s are shown in Figure 3 (velocity profiles in other runs look similar, and therefore are not presented). The figure indicates that 8

9 during the daily transition the components of the wind velocity remain unchanged in the lower half of the mixed layer. In the upper half of the mixed layer, and in the interfacial layer, the wind changes in response to the growth of the mixed layer. As a result, the vertical gradients of wind components gradually decrease with time. The daily transformation of the temperature and wind can be further analyzed based on their dimensionless vertical gradients. Dimensionless gradients are expected to have universal properties in stationary conditions (e.g., Sorbjan, 2005, 2006). In Figures 4a, b, the dimensionless gradients z i d" " * dz and z i du u * dz are presented for the potential temperature and the longitudinal component of the wind velocity in run R10s. Both dimensionless gradients reach large values (positive or negative) near the surface, and peak values in the interfacial layer. The dimensionless temperature gradients in Figure 4a hardly change with time in the lower half of the mixed layer. This indicates that convective scaling is effective during daily transitions in this region of the CBL. Variations of the dimensionless temperature gradient in the interfacial layer are substantial. The interfacial profiles are shifted toward smaller values at instants "a", "b", and "c", and then toward larger values at instants "d" and "e". Similar tendency can be observed in run R10g, which is not shown This behaviour is caused by the variation of the ratio z i /Θ * in time, which can be verified in Table 1 (e.g., in run R10g, z i /Θ * = 21,771 m K -1 at instant O, while z i /Θ * = 10,930 m K -1 at instant A, z i /Θ * = 9,244 m K -1 at instant B, z i /Θ * = 14,965 m K -1 at instant C, and z i /Θ * =18,420 m K -1 at instant D). The dimensionless velocity gradients in Figs 4b are nearly constant with time in the lower part of the mixed layer. They diverge in the upper portion of the mixed layer and in the interfacial layer THE ABL GROWTH 9

10 The time history of a parameter Z m and its standard deviation σ m are depicted in Figures 5a, b. Z m is defined as an altitude, at which the vertical gradient of the potential temperature reaches a maximum at the top of the mixed layer. It was obtained for each individual temperature profile in the domain, and then horizontally averaged. The parameter Z m indicates approximately the middle of the interfacial layer, and the height at which the temperature variance reaches its peak. Therefore, it can be regarded as an estimated height of the boundary layer, which consists of both the mixed layer and the interfacial layer. Sullivan et al. (1998) found Z m to be representative of the entrainment interface dynamics. The mixed layer height z i (defined by the peak of the horizontally-averaged temperature flux) is also depicted in Figure 5a. The standard deviation σ m can be interpreted as being proportional to the depth of the interfacial layer. Note that the early portion of the time history, in which the model adjusts to the initial condition, is not presented in Figures 5. Lorenz's "butterfly effects" can be noticed in the time evolution of parameters, Z m and σ m in Figures 5 b. They are caused by the sensitivity of the nonlinear LES system to small variations of the boundary conditions. Figure 5a shows that the boundary layer height Z m exceeds the mixed layer height z i. For example, in run R0g at the instant A, z i = 709 m, while Z m is about 775 m (i.e., z i /Z m 0.9). The values of Z m in the figure monotonically increase with time in all four runs (note that the effects of subsidence were not included here). The mixed layer height z i follows this trend, but exhibits larger fluctuations. The growth of Z m in time in run R0g is faster than in run R0s, which can be associated with the strength of the topping inversion. Similarly, the growth of Z m in run R10g is faster than in run R10s (Figure 5b). The standard deviation σ m increases very slowly with time in runs R0s and R10s (effects of the topping temperature inversion). In runs R0g and R10g, σ m follows the temporal trend of 10

11 the surface temperature flux H s. The values of σ m are lagged in time by about 500 s with respect to the surface heat flux. The lagging interval is comparable to the eddy overturning time scale τ * (see Table 1). The standard deviation σ m can be related to the height scale of the interfacial layer S h = w * /N i, where N i = [β γ i ] 1/2 is the interfacial Brunt-Väisäla frequency (Sorbjan, 2004, 2005). The evolution of the scale S h in runs R0g and R0s is depicted in Figure 5a. The figure indicates that σ m S h, or equivalently: " m = c m w * N i (1) where c m is an empirical constant. Note that the maxima of σ m and S h in the figure are slightly shifted in time. It can be expected that σ m is additionally impacted by shear effects, but such influence is relatively minor and can be neglected. Following Sorbjan (2005), the entrainment velocity w e = dz i dt in shearless conditions can be expressed as: w e /w * (z i /S h ) -2 2 w * z 2 2. As the temporal variations of the boundary layer i N i height Z m are much smoother than those of z i, and also more consistent with flow visualizations (e.g., Sullivan et al., 1998), Z m should be used in practical applications to characterize the entrainment velocity. Switching from w e and z i to W e = dz m dt and Z m, and taking the effects of shear into consideration, yields: W e = c e W * 3 Z m 2 N i 2 f e(ri) (2) 11

12 where W * = (β H s Z m ) 1/3 is another vertical velocity scale (analogous to w * ), Ri = N i 2 /S i 2 is the interfacial Richardson number, S i 2 = (du/dz) i 2 +(dv/dz) i 2 is the shear based on the maximum velocity gradients in the interfacial layer, c e is a constant, and f e (Ri) is an empirical function of the interfacial Richardson number Ri. In shearless cases, Ri, and f e (Ri) 1. Expression (2) is tested in Figures 6a, b in shear-less runs R0g and R0s, assuming f e (Ri) = 1 and c e = 0.6. The entrainment velocity W e was obtained from the LES values of Z m displayed in Figure 5a, by calculating finite differences in adjacent points, and by subsequently smoothing the results using a 4-point filter (i.e., the arithmetic average of two adjacent points on both sides of a given point). The fluctuations of W e in Figure 6 are caused by convective activities at the top of the mixed layer (fluctuations of the entrainment velocity calculated based on z i are much larger). The figure indicates that the curve obtained from Equation (2) reproduces the trend of the LES-based entrainment velocity W e, after its high-frequency fluctuations are removed. In the start-up run S-O, when the surface flux H s is constant, the values of W e based on Equation (2) change slightly, adjusting to the variation of the interfacial temperature gradient γ i. During the interval O-A-B-C-D, W e responds to the variation of the surface heat flux. It can be noted, that the maximum values of W e, obtained from Equation (2), occur at the moment, when the surface heat flux H s reaches its maximum. The LES values of the entrainment velocity lag by about 500 s (i.e., about τ * ) in both runs R0g and R0s. Figure 6c depicts the entrainment velocity in run R10s. The curve obtained from Equation (2) (for f e (Ri) = 1, i.e., no shear effects included) is marked in the figure by a dotted line. Note that the Richardson number Ri is relatively large in this case, in the range , which allows to ignore the effects of shear. As in Figures 6a, b, the LES values of the 12

13 entrainment velocity in Figure 6c occur about 400 s later than the maximum of the surface heat flux (i.e., about τ * ) SECOND MOMENTS Profiles of the temperature flux H (total, i.e., resolved plus subgrid) scaled by the current values of the surface flux H s in runs R10g are presented in Figure 7. The flux profiles in other runs look alike, and therefore are not displayed. The profile at instant O (in the quasi-steady run) is linear. Profiles at other instants are non-linear due to non-stationarity, especially near the surface. Linearity of the flux profile in stationarity conditions follows from the temperature equation Θ/ t = - H/ z. Differentiating this equation with respect to z, and applying the assumption that the temperature gradient does not change with time, indicates that the second derivative of H is zero, and that the heat flux is linear (has a zero curvature). In non-stationary conditions, the morning "heating" near the surface will cause the temperature gradient (negative) to decrease with time. As a result H will have a positive curvature (its second derivative will be positive). The afternoon "cooling" near the surface will cause that the temperature gradient (negative) to increase with time, and H to have a negative curvature. Consequently, gradually in time, the flux profiles become shaped like the letter S. At the top of the mixed layer (z/z i = 1), the dimensionless fluxes are nearly constant with time, except at instant D, when the entrainment flux ratio H i /H s is larger. This effect is due to the decreased values of the surface heat flux H s. The quadrant analysis of the resolvable temperature flux <w Θ > (not shown) indicates that the maximum of the flux at z/z i = 0.1 (at instant D in Figure 7) is mainly due to the contribution of quadrant I with warmer air rising (w > 0, Θ > 0). The partial flux in this quadrant 13

14 increases with height up to z/z i = 0.4, where it reaches a maximum of 0.7H s. Above, it decreases to about 0.05H s at z/z i = 0.9. The contribution of quadrant III (colder air sinking) is also positive, but smaller, and decreases with height in the lower half of the mixed layer, from 0.5H s to 0.15H s at z/z i = 0.9. The effects of quadrant II (colder air rising) and IV (warmer air sinking) are negative, almost uniform with height, and significantly smaller in the lower half of the mixed layer (~ -0.05H s ). According to Sorbjan (2006), the entrainment flux can be expressed as H i S w S θ f Η (Ri), or: 2 N H i = c H w i * " f H (Ri) (3) where S w = w * and S θ = w * N i /β are the interfacial scales for vertical velocity and temperature, f Η (Ri) is an empirical function of the interfacial Richardson number Ri, and c H is a constant. In shear-less case, Ri, and f Η (Ri) 1. The above expression was tested by Conzemius and Fedorovich (2006) using quasi-steady LES runs. In order to verify it in transitory cases, history plots of the entrainment flux obtained in runs R0g and R0s are presented in Figures 8a, b. The flux H i, obtained from Equation (3) for c H = and f H (Ri) = 1, is represented in the Figure by the smoother curve. The entrainment flux in the windless run R0g changes in Figure 8a from about K m s -1 to about K m s -1 within the start-up run S-O, when the surface flux is constant. The resulting flux ratio -H i /H s is in the range of The steady decrease of the entrainment flux during the start-up run can be associated with the increase of the interfacial temperature gradient γ i. Within the interval O-B-C-D, the entrainment flux seems to follow the negative values of the surface temperature flux H s. This result could be expected, since a common 14

15 assumption in the bulk modelling of the mixed layer is that -H i /H s = const. The resulting flux ratio -H i /H s within the interval O-B-C-D is in the range of The entrainment flux in run R10g is shown in Figure 8b. Its values are slightly larger than in run R0g, and change from about K m s -1 to K m s -1 within the start-up run S-O. In the interval O-B-C-D, the flux has a similar pattern as in run R0g, and the resulting flux ratio -H i /H s is in the range 0.1 to 0.2. The flux minimum is reached between instants B and C. The flux H i obtained from Equation (3) is also depicted in the figure. It was obtained for the same value of c H = , and for the empirical function f H (Ri) = ( Ri ) /(1+ 1 Ri )0.5, proposed by Sorbjan (2006). Since Ri in run R10g varies in the range of , the value of the shear correction function f H (Ri) The curve based on Equation (3) reproduces the trend of the LES based entrainment flux H i, after the high-frequency oscillations are disregarded. Its minimum, however, appears at instant B (when the surface temperature flux has its maximum), which is about 500 s earlier than in the LES curve. In other words, the LES values of the entrainment flux lag by about 500 s (i.e., about τ * ) with respect to the values of the surface flux. Profiles of the dimensionless variances of temperature and wind velocity components in run R0g are presented in Figure 9. Profiles of the dimensionless variance σ θ2 /Θ * 2 in Figure 9a coincide in the mixed layer, which indicates that convective scaling is effective in this region. At the top of the mixed layer, the dimensionless variances diverge. Since the temperature gradient γ i increases in time (Figure 2), it is expected that the peak temperature variance will also increase (Sorbjan, 2004, 2005, 2006). This trend is disturbed in the figure by the variation of the scaling factor Θ -2 *, which decreases in the interval O-B (see Table 1), and increases in the interval B-D. The dimensional values of the velocity variances σ 2 u /w 2 * and σ 2 w /w 2 * in run R0g are depicted in Figure 9b. The figure shows that both horizontal and vertical dimensionless variances 15

16 evolve in time. The profiles approximately coincide at instants O and B, both representing quasisteady conditions. The remaining profiles are shifted toward smaller values at instant A, and toward larger values at instants C and D. This fact implies that convective scaling is valid only in quasi-steady conditions, and is ineffective during daily transitions. Figures 9 indicate that values of the velocity variances, measured in the atmosphere at various instants during a day, and normalized in terms of the actual convective scale w *, will not collapse to a single curve, but will produce a significant scatter of observational points. On the other hand, the measured values of the temperature variance, normalized in terms of the actual convective scale Θ *, will form a single curve in the mixed layer, but will exhibit a considerable scatter in the interfacial layer. 3. Conclusions Daily transitions in the atmospheric boundary layer were examined based on large-eddy simulations. The examined transition regimes included "morning expansion", "mid-day convection", and "afternoon weakening. The simulations indicated that the values of the wind velocity components remain unchanged during the daily transition in the lower half of the mixed layer. In the upper half of the mixed layer, and in the interfacial layer, the wind gradients decrease in response to the growth of the mixed layer. The dimensionless temperature gradients z i d" " * dz are constant with time in the lower half of the mixed layer and evolve in time above. Profiles of the dimensionless temperature flux H/H s are non-linear during daily transitions, especially near the surface. The flux profiles gradually become shaped like the letter 16

17 S during afternoon weakening. As a result, the temperature flux increases with height in the surface layer, which indicates cooling generally by warm updrafts. The minimum entrainment flux H i is lagged (by about the value of the eddy overturning scale τ * ) with respect to the surface temperature flux maximum. The entrainment flux can be predicted in terms of the interfacial scales (Equation 3). It reproduces the trend of H i quite closely, even though its mid-day peak occurs earlier in time. Profiles of the dimensionless temperature variances σ 2 2 θ /Θ * coincide in the lower part of the mixed layers, which indicates that convective scaling is effective in this region. In the interfacial layer, the dimensionless variances evolve in time, and the convective scaling is ineffective in this region. The dimensional velocity variances σ 2 u /w 2 * and σ 2 w /w 2 * also evolve in time. Consequently, convective scaling is valid only in quasi-steady conditions (mid-day regimes), and is ineffective during daily transitions. The boundary layer height Z m monotonically increases with time in all four runs. The mixed layer height z i generally follows this trend, but exhibits much larger fluctuations. The entrainment velocity W e = dz m dt can be predicted by Equation (2). The maximum values of W e, obtained from this equation occur at the moment, when the surface heat flux reaches maximum. The LES values of W e lag behind by about the value of the eddy overturning scale τ *. The standard deviation σ m of the ABL height Z m is found to be proportional to the interfacial height scale S h. Acknowledgments: The paper is based on research which has been supported by the National Science Foundation grant No. ATM

18 References Acevedo, O. and Fitzjarrald, D.R.: 1999, 'Observational and numerical study of turbulence during early evening transition. Proc. 13th Symposium on Boundary Layers and Turbulence', January, 1999, Dallas, Texas. American Meteorological Society, 45 Beacon St., Boston, MA, pp Acevedo, O. and Fitzjarrald, D.R.: 2001, 'The early evening surface-layer transition: Temporal and spatial variability. J. Atmos. Sci., 58, Angevine, W.M., Klein Baltink, H., and Bosveld, F.C.: 2001, Observations of the morning transition of the convective boundary layer. Boundary-Layer Meteorol., 101, Cederwall, R.T, and Street, R.L.: 1999, 'A study of turbulence in an evolving stable atmospheric boundary layer using large-eddy simulation'. First International Symp. on Turb. and Shear Phenom., Santa Barbara, CA, Sept , Conzemius, R.J., and Fedorovich, E.: 2006, Dynamics of sheared convective boundary layer entrainment. Part II: Evaluation of bulk model prediction of entrainment flux. J. Atmos. Sci., 63, Deardorff, J. W.: 1970, 'Preliminary results from numerical integration of the unstable planetary boundary layers'. J. Atmos. Sci., 27, Deardorff, J.W.: 1974a, Three-dimensional numerical study of the height and mean structure of a heated planetary boundary layer. Boundary-Layer Meteorol., 7, Deardorff, J.W.:1974b, Three-dimensional numerical study of turbulence in an entraining mixed layer. Boundary-Layer Meteorol., 7,

19 Edwards, J.M, Beare, B., and Lapworth, A.: 2004, 'Modelling transition boundary layers'. Proc. 16th Symposium on Boundary Layers and Turbulence, 9 13 August 2004, Portland, Maine. American Meteorological Society, 45 Beacon St., Boston, MA. Grimsdell, A.W. and Angevine, W.M.: 2000, 'Afternoon transition of the continental convective boundary layer'. J.Appl.Meteorol., 41, Grimsdel, A.W., and Angevine, W.M.: 2002, 'Observations of the afternoon transition of the convective boundary layer'. J. Atmos. Sci., 41, Grant, A.L.M. : 1997, An observational study of the evening transition boundary-layer. Quart. J. Roy. Meteorol. Soc., 123, Hignett, P.: 1991, 'Observations of diurnal variation in a cloud-capped marine boundary layer'. J. Atmos. Sci., 48, LeMone, M.A. and Grossman, R.L.: 1999, 'Evolution of potential temperature and moisture during the morning: CASES-97'. Proc.13th Symp. on Boundary Layers and Turbulence, Dallas Texas, January, 1999, AMS. LeMone, M.A., Grossman, R., McMillen, T., Liou, K., Ou, S.C., McKeen,S., Angevine, W., Ikeda, K., and Chen, F.: Cases-97, 2002: 'Late-morning warming and moistening of the convective boundary layer over Walnut River watershed'. Boundary-Layer Meteorol, 104, Mahrt, L., Sun, J., MacPherson, J.I., Dobosy, R., Kustas, W., and Prueger, J. :1999: 'Diurnal Boundary layer transitions'. Proc. 13th Symposium on Boundary Layers and Turbulence, January, 1999, Dallas, Texas, American Meteorological Society, 45 Beacon St., Boston, MA, pp

20 Nieuwstadt, F.T.M. and Brost, R.A. : 1986, 'The decay of convective turbulence'. J. Atmos. Sci., 43, Pino, D., Jonker, H.J.J., de Arellano, J.V.-G., and Dosio, A.: 2004, 'Role of the shear and inversion strength during sunset turbulence over land: characteristic length scales'. Proc. 16th AMS Symposium on Boundary Layers and Turbulence, 9 13 August 2004, Portland, Maine. American Meteorological Society, 45 Beacon St., Boston, MA. Shaw, W.J., and Barnard, J.C.: 2002, 'Scales of turbulence decay from observations and direct numerical simulations. Proc. 15th Symp. on Boundary Layers and Turbulence'. July, Wageningen, The Netherlands, American Meteorological Society, 45 Beacon St., Boston, MA. Silva, R., Acevedo, O., Morales, O., Fitzjerrald, D., Sakai, R., Czaikowsky, M., Staebler, R., 2002: 'Turbulent fluxes, temperature and humidity convergence after sunrise'. Proc. 15th Symp. on Boundary Layers and Turbulence, July, Wageningen, The Netherlands, American Meteorological Society, 45 Beacon St., Boston, MA. Sorbjan, Z.: 1996 a, 'Numerical study of penetrative and "solid-lid" non-penetrative convective boundary layers'. J. Atmos. Sci., 53, Sorbjan, Z., 1996 b: 'Effects caused by varying the strength of the capping inversion based on a large eddy simulation model of the shear-free convective boundary layer'. J. Atmos. Sci., 53, Sorbjan, Z.: 1997, 'Decay of convective turbulence revisited'. Boundary-Layer Meteor. 82, Sorbjan Z.: 2004, Large-eddy simulations of the baroclinic boundary layer. Boundary-Layer Meteorol., 112,

21 Sorbjan Z.: 2005, Statistics of scalar fields in the atmospheric boundary layer based on largeeddy simulations. Part I: Free convection. Boundary-Layer Meteorol., 116, 3, Sorbjan Z.: 2006, Statistics of scalar fields in the atmospheric boundary layer based on largeeddy simulations. Part II: Forced convection. Boundary-Layer Meteorol., 119, 1, Sullivan, P.P, Moeng, C.-H., Stevens, B., Lenschow, D.H., and Mayor, S.D.:, 1998: Structure of the entrainment zone capping the convective atmospheric boundary layer. J. Atmos. Sci, 55,

22 Figure 1. Time history of the surface temperature flux H s assumed during the performed simulations. The letters indicate instants, at which statistics of turbulence were calculated. 22

23 23

24 Figure 2. Profiles of the potential temperat ure obtained at instants marked by the letters in runs: a) runs R0g and R0s, and b) runs R10g and R10s. 24

25 Figure 3. Profiles of wind velocity components obtained in run R10g, at instants marked by the letters. 25

26 26

27 Figure 4. Profiles of dimensionless gradients at instants marked by the letters in run R10s: a) for the potential temperature z i " * d" dz, and b) for the component of wind velocity z i u * du dz. 27

28 28

29 Figure 5. Time history of the boundary layer height Z m and its horizontal standard deviation σ m : a) for runs R0g and R0s; note that the mixed layer height z i, and the interfacial height scale S h are also shown in the figure and marked by dotted lines, and b) runs R10g and R10s. 29

30 30

31 31

32 Figure 6. Time history of the entrainment velocity W e = dz m dt estimate based on Equation (2): a) for run R0g, b) run R0s, and f) run R10s., obtained from LES, and its 32

33 Figure 7. Profiles of the dimensionless temperature flux H/H s in run R10g at instants marked by the letters. 33

34 34

35 Figure 8. Time history of the entrainment flux H i (solid lines): a) for run R0g, b) for run R10g. The estimates obtained based on Equation (3) are marked by dotted lines. 35

36 36

37 Figure 9. Profiles of dimensionless variances in run R0g at instants marked by the letters: a) for the potential temperature σ θ2 /Θ *2, and b) for veloc ity variances σ u2 /w *2 and σ w2 /w *2. 37

38 TABLE 1. Characteristics of the resulting mixed layers, calculated at the indicated instants of time. In the table Θ * is the temperature scale in [K], w * is the vertical velocity scale in [m s -1 ], z i is the height scale in [m], τ * is the eddy overturning time scale [s], u * is the friction velocity scale in [m s -1 ]. Run Parameter Instant O Instant A Instant B Instant C Instant D R0g Θ * = w * = z i = τ * = R10g Θ * = w * = z i = τ * = u * = Instant o Instant a Instant b Instant c Instant d Instant e R0s Θ * = w * = z i = τ * = R10s Θ * = w * = z i = τ * = u * =