5.1 Neutral ABL—Leipzig test case
In this case, the Leipzig wind profile is modeled with the ASL and ABL models run steady-state. The necessary simulation parameters are shown in Table 2, and the computational grid is described in Section 3.2.1. Results are shown in Figure 1 together with the measurements. A comparison of the ASL and ABL models results shows the influence of the Coriolis effect: the additional body force in the ABL model induces a velocity component v perpendicular to the direction of the geostrophic wind G and causes the wind to veer with height. In the ABL model, also the height of the ABL is now limited to about 1300 m, because of the applied length-scale limiter (seen in the middle of Figure 1, where the velocity component v approaches zero). With the chosen value of l0 = 41.8 m, the ABL height is however slightly overpredicted. This length scale is slightly larger than the one suggested by Apsley and Castro, who used l0 = 36 m for their simulation. It is generally accepted that the Leipzig experiment was actually conducted in slightly stable conditions, and when using a lower length scale of l0 = 28 m, the measured and simulated profiles agree perfectly (not shown here). However, the goal was not to match the simulation to a single observation, and the ABL model predicts the flow reasonably well and simulated results are significantly improved compared with the ASL model.
Figure 1. Results of the Leipzig test case using the ASL model (dashed blue) and the ABL model (solid red), shown together with the Leipzig wind profile (gray symbols). Left: wind component u parallel to geostrophic wind G plotted over height; middle: wind component v perpendicular to geostrophic wind G plotted over height; right: turbulent mixing length scale l (equation (5)) plotted over height.
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5.2 Neutral ABL—Cabauw test case
The neutral ABL over flat terrain at the Cabauw site in the Netherlands is simulated with the ASL and ABL models run steady-state. The necessary simulation parameters are shown in Table 2, and the computational grid is described in Section 3.2.1. The results are shown in Figure 2. The non-dimensional geostrophic wind components (u − ug) ∕ u * and (v − vg) ∕ u * are shown as functions of the non-dimensional height zfc ∕ u * , plotted with a logarithmic scale. Annual averages from the Cabauw site are shown for three classes of the geostrophic wind (G = 5,10,15 m s − 1) at heights 10, 80 and 200 m, together with simulation results using G = 10 m s − 1. When plotted using the aforementioned non-dimensional form, the simulation results for other geostrophic winds collapse on the same line. As for the Leipzig test case, the turning of the wind with height induced by the Coriolis force and the limitation of the ABL height by the length-scale limiter can be seen when comparing the simulated profiles of the ABL model with the ASL model. Results from the ABL model agree well with the measurements, and the chosen modifications prove applicable.
Figure 2. Results of the Cabauw test case using the ASL model (dashed blue) and the ABL model (solid red), shown together with annual averages of the Cabauw site (gray symbols) for three classes of geostrophic wind speeds (5, 10 and 15 m s − 1). Bars denote standard deviations. Left: dimensionless wind components (u − ug) ∕ u * and (v − vg) ∕ u * plotted over non-dimensional height zfc ∕ u * ; right: turbulent mixing length scale l (equation (5)) plotted over height.
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5.3 Non-neutral ABL—GABLS2 test case
This section focuses on assessing how well the ABL model performs in representing non-neutral ABL flows. Observations from the GABLS2 test case held in Kansas, USA with a strong diurnal cycle are chosen to validate the ABL model. In the study of Svensson et al., the observational dataset is compared against simulation results from 30 different models, and simulating the described diurnal cycle has shown to represent a challenging test case for ABL models.
The simulation uses the computational grid from Section 3.2, and the ABL model is run transient, where non-neutral conditions are induced by a prescribed time-varying ground temperature. A time step of 1 s is used. The initial conditions for the simulation of the diurnal cycle are given in Svensson et al.
Results are shown in Figure 3. Figure 3(a) is showing the time varying ground temperature that is used as a model input together with the resulting diurnal evolution of the computed potential temperature field that adopts to the changing surface conditions. The surface stability conditions are influencing the generation of turbulence and the turbulent mixing. Figure 3(b) shows the friction velocity u * at the surface (on the left axis) and the velocity variation at the 10 m level (on the right axis) over one diurnal cycle together with measurements (symbols), and the spread of the different model results from the model intercomparison study of Svensson et al. (shaded region).
Figure 3. Diurnal evolution of flow properties. a) Surface temperature θ0 (black line) that is given as an input to the ABL model and the resulting potential temperature field within the first 300 m ; b) wind speed at 10 m (green) and friction velocity u * at the surface (blue). Symbols and lines denote measurements and simulation results respectively; c) wind profiles of the ASL model (dashed blue) and ABL model (solid red) compared against analytical profiles from MOST (equation (25)) for different times of the day.
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In both plots, Figures 3(a) and 3(b), a clear transition between daytime and nighttime is visible after sunrise around 8:00 and after sunset around 18:00. Stable nighttime conditions before 8:00 are characterized by small turbulence levels and a low ABL depth of around 100 m. The stable stratification suppresses the generation of turbulence and results in small values for the friction velocity as shown in Figure 3(b). The air close to the ground is colder then the air above and, because of the small amount of mixing, only penetrates up to heights of about 100 m, where a steep temperature gradient is visible in Figure 3(a).
During daytime between 12:00 and 18:00, unstable conditions are induced by the heating of the ground. Large amounts of convective turbulence lead to a well-mixed ABL with a greater depth. Because of convection, warm air is rising upwards and penetrates the strong stable temperature gradient that is capping the ABL during night. After 12:00, the stable gradient is not existent anymore, and the ABL continues to grow in height. It is during this period that the turbulent length scales and the friction velocity at the surface reach their maximum values. At around 14:00, the maximum temperature is reached, and before returning to the stable nighttime regime, the ABL flow is close to neutral at around 18:00, where the potential temperature is nearly constant with height.
Also shown in Figure 3(b) is the evolution of the wind speed at the 10 m level. Higher wind speeds are observed during daytime, where the increased turbulence is effective at mixing momentum downward close to the ground and vice versa. The shaded areas indicate the model spread of the 30 models that were intercompared within the study of Svensson et al. Computed results generally are within the observed range, and a clear diurnal pattern is visible. Svensson et al. report that all models underestimate the 10 m wind speed after the morning transition and tend to overestimate the wind speed toward the end of the day. One obvious reason for this is that the geostrophic wind during the simulation was kept constant in space and time, whereas both,observations and mesoscale simulations shown in Svensson et al. show a decrease of the geostrophic wind during the observational period. Also note that the measured turbulent kinetic energy shows a sudden increase at about 3:00, which was reported to be a local disturbance not included in the model forcing and therefore not present in the computed results.
Figure 3(c) shows the computed wind profiles in the first 100 m at different times of the day compared against the observations and the standard logarithmic profiles from the ASL model. Also shown are the theoretical profiles from MOST, where the computed surface heat flux H0 together with equations (22) and (26) are used to determine the Obukhov length L and the modified logarithmic wind profile. Stable conditions at night result in smaller wind speeds close to the ground and higher wind speeds above, when compared against the logarithmic solution of the neutral ASL model. During unstable conditions, the wind speed increases rapidly over the first few meters, whereas it is almost constant with height above. The agreement is good, and the developed model captures the observed and theoretical non-neutral behavior.
In Figure 4, MOST (Section 4) is used to assess the performance of the ABL model. Theoretical φm functions from equation (24) are shown together with simulation results and experimental data from several field campaigns.[12-14] To decrease the spread of the experimental φm values, the data needs to be selected carefully. Especially during transitional periods in the morning and evening, the assumptions underlying MOST (stationary and horizontally homogeneous flow with constant ζ over height) are violated in real ABL flows. The shown simulation results are therefore selected accordingly: only cases for fully developed flow away from the transitional periods are shown. MOST was derived for the range | ζ | < 2, and for higher values, it can be seen that experimental and simulation results start to deviate from MOST.
Figure 4. φm from observations (gray symbols: Businger et al., Li et al., and Klipp and Mahrt), analytical expression from equation (24) with δm = 5 and γm = 15 (solid black) and for the range of analytical solutions where δm varies from 4 to 10 and γm varies from 15 to 28 (shaded area), and results from the ABL model. Note that results during transitional regimes around 8:00 (sunrise) and 18:00 (sunset) are omitted, since during these conditions, z ∕ L is not constant with height, which is a necessary assumption for MOST to be valid.
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Note that in this case, the ABL model is run for several days, cyclically repeating the surface temperature from the GABLS2 test case, until a cyclical solution is reached. This ensures that the whole flow field is in equilibrium with the model equations and that the solution is independent of the initial conditions. For the GABLS2 model intercomparison, the spin-up time of the models given in Svensson et al. (time before numerical and observational data is compared) is only 8 h. We find that the solution in this case is still significantly dependant on the initial temperature field (also given in Svensson et al.), and when compared against MOST, the agreement is not as good as for the fully converged results. This indicates that the flow field in the ABL after 8 h of spin-up time is not yet in equilibrium with the surface forcing at the ground. In a recent study, Sogachev also found that a spin-up time of several days is needed, depending on the initial conditions. In summary, we find model results to agree best with MOST when allowing the solution to fully converge to a cyclical solution. Results for the GABlS2 test case are best when following the instructions from Svensson et al. This indicates that if information on a large scale (such as the geostrophic wind or the vertical temperature profile) is available from measurements, those conditions should be used in the model to compare numerical and observational data. However, conditions in the real ABL are often non-stationary and horizontally non-homogeneous and are therefore not necessarily in agreement with empirical theories like MOST or with the model equations of numerical models.
5.4 Non-neutral flow over a steep hill
In this test case, neutral and stably stratified boundary-layer flow over a steep hill is simulated and compared against simulations and wind-tunnel measurements.[16, 37] The wind-tunnel experiment was designed to represent realistic ABL flow over a two-dimensional steep hill. Wind-tunnel flows cannot fully resemble real ABL flow at full scale, and the Coriolis effect is negligible. However, this test case allows to study stability effects under controlled conditions and is chosen to test the applicability and performance of the ABL model for flows over well-defined terrain.
The hill is steep enough to induce flow separation and represents a challenging test case for the developed ABL model. The simulation is run in transient mode with a time step of 0.1 s.
Because of the small scale of the wind-tunnel, the Coriolis effect is neglected in the model, hence equation (14) to determine a maximum length scale le cannot be applied, and no length-scale limitation is used.
The necessary input parameters to simulate the wind-tunnel flow are summarized in Table 2. Two cases are simulated: neutral and stably stratified flow with a relatively weak stratification of about 10 K m − 1 in the lowest 0.5 m and a much stronger stratification of about 40 K m − 1 above. The neutral simulations are solved steady-state and the stable simulations are run transient for 150 s of model time (equivalent to 15 tunnel flow-through times) after which the computed flow has reached a quasi-steady state.
The initial conditions for the neutral simulation are specified by the logarithmic wind profile with a surface roughness of z0 = 0.23 mm (Table 2). For the stably stratified flow, the inlet profiles at the upstream boundary are generated by running a precursor simulation: the experimental velocity and temperature profiles from the wind-tunnel given in Ross et al. are run through the ABL model using the wind tunnel specified in the succeeding text in the absence of the hill. This ensures that the inlet profiles are in equilibrium with the model equations.
Computed wind speed and turbulence properties of the neutral and stably stratified flow are compared against experimental results and simulation results[16, 37]. Figure 5 shows contour plots of the streamwise velocity u in a vertical plane perpendicular to the hill, and Figure 6 shows contour plots of the momentum flux . Results for neutral flow are shown in the left column, and results for stable flow are shown on the right. In both cases, flow separation occurs at the lee side of the hill. For the stable case, the depth of the wake region is slightly increased because the stable stratification acts to suppress vertical motion. Above 0.5 m, the strong temperature gradient of about 40 K m − 1 is effectively capping the flow, and turbulence is limited to the lower part of the domain and is significantly reduced when compared with the neutral case where the momentum fluxes are an order of magnitude larger. The ABL model captures the general effects induced by stratification, although the size of the wake is different. For both neutral and stably stratified flows, the recirculation region in the lee side of the hill is significantly overpredicted compared with the wind-tunnel experiment, whereas it agrees well with the LES results. The velocity above of the hill is generally predicted well. The values in Figure 6 are generally predicted well and agree with both wind-tunnel and LES results from Wan and Port-Agel. However, for the neutral case, the values upstream of the hill are found to be too high when compared with the wind-tunnel values. Similar findings were reported by Wan et al.: the model is found to be too dissipative in this region, which leads to an increased upwards deflection of the flow induced by the hill and leads to a slower velocity recovery downstream of the hill and hence an overestimated wake region.
Figure 5. Contour plots of non-dimensional streamwise velocity u ∕ uf in a vertical plane across the hill for neutral (left column) and stable flow (right column). Simulation results are shown along with measurements and RANS results from Ross et al. and LES results from Wan et al.
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Figure 6. Contour plots of momentum flux (in m2s − 2) in a vertical plane across the hill for neutral (left column) and stable flows (right column). Simulation results are shown along with measurements and RANS results from Ross et al. and LES results from Wan et al.
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Since the ABL model was developed for ABL flows at full scale, we cannot expect the model to reproduce the wind-tunnel measurements. Because of the small scale of the wind-tunnel, the Coriolis effect is neglected, and the length-scale limiter is not applied. The implemented turbulence closure has been developed for steady ABL flows, and it cannot be expected that the unsteady wake region in the lee of the hill is predicted correctly, and it is not the aim of the ABL model. The wake region has shown to be sensitive to changes in the model constants. No coefficients were adjusted, and all test cases are run with the same set of constants from Table 2. Although this test case is of limited value to verify the developed ABL model, it is shown that the model can be applied on curvilinear grids without any modification and that general effects of stratification on the flow are captured correctly.