The steady-state profiles of wind speed and relative temperature are first presented and compared to the observations in Sect. 4.1, Sect. 4.2 presents the turbulent fluxes, Sect. 4.3 discusses the effect of subsidence heating and its implication for a steady-state flow simulated by the LES approach, and Sect. 4.4 finally gives a brief sensitivity analysis for the very stable simulation.
General Characteristics
The results of the WSBL and VSBL simulations are averaged over the horizontal plane and over the final hour of simulation to calculate bulk quantities and vertical profiles. The simulations reach an approximate quasi-steady equilibrium during this hour, since the relevant quantities do not change significantly. An exception is the presence of an inertial oscillation in the velocity profile with a time scale \(T_i\approx 12.6\,\mathrm {h}\). The time-averaged surface friction velocity \(u_*\), surface kinematic temperature flux \(Q_*\), surface Obukhov length L, and diagnosed boundary-layer height h are listed in Table 2. Here, we diagnose the boundary-layer height using the method used by Kosović and Curry (2000) and Beare et al. (2006). First, the height at which the total horizontal stress reaches \(5\%\) of its surface value is calculated, and is subsequently linearly extrapolated to the height at which the stress vanishes assuming a linear stress profile.
Figure 5 shows the profiles of wind speed, relative wind direction, and relative temperature for the WSBL (top row) and VSBL (bottom row) simulations compared with the observed values (red) of our selected cases. The observed wind directions are shifted to match the simulated values at \(z=1.23\) and \(z=18.11\,\mathrm {m}\) for the WSBL and VSBL cases, respectively (see below).
In general, good agreement between the simulated and observed wind speeds is found for the WSBL case. With the exception of the highest observation level, discrepancies between the simulation and observations are less than \(0.9\,\mathrm {m\,s^{-1}}\). The estimate of the geostrophic wind speed for this case appears to be realistic. As noted in Sect. 2.3, the observed wind speed in the WSBL case at \(41.2\,\mathrm {m}\) is possibly influenced by the presence of a local wind-speed maximum. Correspondingly, the simulation exhibits a jet with maximum \(13.1\,\mathrm {m\,s^{-1}}\) (about \(9\%\) of G) at \(z\approx 43\,\mathrm {m}\).
In the WSBL case, the observation tower does not capture the full extent of the boundary layer and its associated wind turning. For this reason, the observed and simulated values of the wind direction are matched at the lowest observation height. A relatively good correspondence is found for the relative wind direction for the WSBL case, where the turning of the wind with respect to height is accurately represented by the simulation with the exception of the highest observation level, which appears to deviate from the lower observations. No explanation for this observed value is found. However, a partial blocking of the aerovanes due to riming or deposition of ice cannot be excluded. The total wind turning at the surface is approximated by local linear extrapolation (over five points) of the simulated values and is approximately \(45^\circ \).
As the boundary-layer depths in the simulations and observations are similar, the relative temperature profiles are comparable, but a number of differences remain. The observed temperatures increase more rapidly with height below \(25\,\mathrm {m}\) than the simulated temperatures, but more slowly above 25 m. Except near the surface, the observed temperature profile appears to be more linearly shaped and has only a weakly pronounced inflection point. This is in contrast to the simulated temperature profile, which exhibits two pronounced inflection points at approximately \(5\hbox { m}\) and \(35\,\mathrm {m}\), resulting in a strongly ‘convex–concave–convex’ profile. At the second inflection point, the averaged temperature gradient in the simulation attains a local maximum of \(\partial _z \langle \theta \rangle =0.85\,\mathrm {K\,m^{-1}}\), which is twice as large as the observed gradient \(\partial _z T_\mathrm {obs}\approx 0.4\,\mathrm {K\,m^{-1}}\). Mirocha and Kosović (2010) show that a relatively small increase in the subsidence rate leads to an increased magnitude of the potential temperature gradient throughout the bulk of the boundary layer in addition to a lower boundary-layer height.
For the VSBL case, the agreement between the simulated and observed wind speed is remarkably good, with the difference \(< 0.5\,\mathrm {m\,s^{-1}}\) (see Fig. 5d), and the estimate of the geostrophic wind speed appears to be accurate. The hour-averaged simulation result indicates the occurrence of a weak low-level jet at \(z\approx 5.3\,\mathrm {m}\) with peak wind speed \(3.8\,\mathrm {m\,s^{-1}}\) (\(10\%\) of G). It should be noted that the full simulation of the VSBL case at steady state covered one inertial period (\(12.6\,\mathrm {h}\)) during which the strength of the jet ranges from 3.7 to \(4\,\mathrm {m\,s^{-1}}\). The inertial oscillation has a minor impact on the velocity profile below the jet (\(z\le 4\,\mathrm {m}\)) resulting in variations of \(0.1\,\mathrm {m\,s^{-1}}\) (not shown).
The wind directions are compared to the observed value at \(18\,\mathrm {m}\) as this point is situated above the bulk of the boundary layer. Close to the surface, the observed and simulated wind directions deviate by 10–\(15^\circ \), but, particularly under low wind speeds, the observed wind direction is not fully reliable. Local linear interpolation towards the surface results in a total wind turning of \(52^\circ \) in the simulation, which is slightly larger than in the WSBL case.
The simulated temperature profile in the VSBL case has the same overall shape as in the WSBL case (cf. Fig. 5c, f), but with the change in temperature distributed over a smaller total height. In the lowest \(5\,\mathrm {m}\), the horizontally-averaged temperature gradient varies between \(\partial _z \langle \theta \rangle \approx 1.4\,\mathrm {K\,m^{-1}}\) at the lowest inflection point (\(z\approx 0.4\,\mathrm {m}\)) and \(\partial _z \langle \theta \rangle \approx 7.8\,\mathrm {K\,m^{-1}}\) at the second inflection point situated at \(3.6\,\mathrm {m}\). Here, the gradient from the surface to the first grid point above the surface is excluded as MOST is applied from the surface to this level. At the same time, the observed bulk gradient between \(z = 0.7\) and \(2.9\,\mathrm {m}\) is equal to \(\varDelta T_\mathrm {obs}/\varDelta z\approx 5\,\mathrm {K\,m^{-1}}\). Although the boundary layer is under-sampled in the very stable case, since not enough measurement levels are present on the tower, the shape of the observed temperature profile is expected to be exponential (see Sect. 2.3). Both Estournel and Guedalia (1985) and Edwards (2009) show that the inclusion of radiative fluxes in a one-dimensional model indeed results in a more exponentially-shaped temperature profile for low geostrophic wind speeds, and so it is expected that the inclusion of radiative transfer in future simulations will improve the agreement with the observed temperature profile.
The simulated profiles suggest that the boundary-layer heights are approximately \(40\,\mathrm {m}\) and \(5\,\mathrm {m}\) for the weakly stable and very stable cases, respectively, where the height of the jet is used as a proxy. An accurate prediction from the observations is impossible. Whereas in the WSBL case, the tower is not high enough to capture the full boundary layer, the region \(z=4\)–\(10\,\mathrm {m}\) is under-sampled in the VSBL case. Nevertheless, the simulations and observations seem to be in agreement on the order of magnitude of the boundary-layer height. Note that, following Nieuwstadt (1984) and Banta et al. (2006), the profiles can be scaled using diagnosed values at the surface or jet maximum height. Such scaling of our simulated boundary layer results in a rather similar structure between the WSBL and VSBL cases (not shown), qualitatively resembling the non-dimensional profiles of Nieuwstadt (1984). In summary, although a number of estimates and assumptions have been made, and radiative processes have been omitted, the simulations successfully mimic the selected weakly stable and very stable regimes found during the Antarctic winter.
Turbulent Fluxes
Figures 6 and 7 show the total and resolved fluxes of momentum \(F(u_i)\) (\(i=x,y\)) and temperature \(F(\theta )\) for the WSBL and VSBL cases, respectively. Note that, for clarity, a different notation is adopted here for the vertical fluxes as compared with the tensor notation in Eq. 1 (e.g., \(F(u_i)\) instead of \(\tau _{i3}\)). In the WSBL case, the total cross-isobaric momentum flux at the surface is equal to the isobaric flux, whereas in the VSBL case, the surface cross-isobaric momentum flux is found to exceed the isobaric flux by approximately \(30\%\). Here, isobaric and cross-isobaric are defined as being parallel and perpendicular, respectively, to the direction of the geostrophic velocity aligned along the x-coordinate. The increasing ratio of the cross-isobaric to the isobaric momentum flux for increasing stratification was also reported by Sullivan et al. (2016).
Inspection of the momentum-flux profiles reveals that, at the diagnosed boundary-layer height (see Table 2), the isobaric momentum flux is reduced to \(<1\%\) of its surface value. The corresponding reduction for the cross-isobaric momentum fluxes is found to be \(\approx 5\%\). In both simulations, the relative contribution of the SFS fluxes to the total fluxes increases near the top of the boundary layer, and accounts for roughly half of the flux at the top of the SBL. In the lower half of the SBL, more than \(80\%\) of the momentum fluxes are resolved for both cases with the exception of the first gridpoint above the surface where MOST is applied.
The total temperature fluxes have a tendency towards a constant value with height close to the surface (see Figs. 6b and 7b), whereas more curvature is present in the SBL further from the surface, which contradicts the results for the typical SBL at mid-latitudes (see, e.g., Nieuwstadt 1984; Galmarini et al. 1998; Beare et al. 2006; Svensson et al. 2011), where quasi-steadiness implies a linearly decreasing temperature flux. This discrepancy with the traditional shape can be explained by the role of subsidence heating in our simulations, which is discussed further in Sect. 4.3. The kinematic temperature fluxes at the surface correspond to surface heat fluxes of \(H_0=-24.7\,\mathrm {W\,m^{-2}}\) in the WSBL case and \(H_0=-3.1\,\mathrm {W\,m^{-2}}\) in the VSBL case. Although the gradient Richardson number \({Ri}_g\) exceeds 0.25 at \(z>2.5\,\mathrm {m}\) for the VSBL case, the local shear is sufficient to maintain continuous mixing throughout the bulk of the boundary layer. As for the flux of momentum at the top of the SBL, the SFS scheme accounts for roughly half of the total flux. The explanation for this reduction in the amount of resolved fluxes may be twofold. First, somewhat unsurprisingly, the mesh size is no longer sufficient to resolve the same proportion of the flux-carrying eddies as the characteristic length of the large eddies is reduced by the increased amount of stratification with respect to the shear, viz., an increase in the gradient Richardson number. As a consequence, more flux has to be accounted for by the SFS scheme. Note also that the SFS fluxes may be partly overestimated, since the eddy diffusivities \(K_{m,h}\) may be too large at the top of the SBL, which is an artefact of the Smagorinksy–Lilly-type closure as it depends on the local strain (see Eq. 2). Therefore, there may be excessive SFS mixing in weak turbulent flow with large shear (Germano et al. 1991), but a quantification of these effects is beyond the scope of our study due to the computational requirement of higher resolutions or a change of the SFS scheme. Nevertheless, Sect. 4.4 gives a brief sensitivity analysis for the VSBL case.
Steady Versus Quasi-Steady?
Figure 8 presents the hourly- and domain-averaged vertical profile of the rate of change of the potential temperature \({\dot{\theta }}\) due to subsidence heating and divergence of the kinematic temperature flux for the VSBL simulation, illustrating that the heating by subsidence has a maximum of approximately \(1.14\times 10^{-3}\,\mathrm {K\,s^{-1}}\) around the inflection point of the temperature profile. The heating rate decreases to zero towards the surface and above the boundary layer. This decrease is caused by a decrease in the subsidence velocity towards the surface and a decrease in the temperature gradient above the SBL, respectively. Interestingly, the cooling induced by the divergence of the total heat flux almost balances the subsidence heating (cf. the black line in Fig. 8). Some residual heating and cooling \(<10^{-4}\,\mathrm {K\,s^{-1}}\) is observed in the boundary layer. Possible causes include numerical inaccuracies, i.e., discretization errors, in evaluating the divergence of the temperature flux, and the averaging procedure. Here, the results were averaged over 1 h with statistics output every two simulation seconds. It is expected that, for simulations at higher resolutions, this residual decreases and the total rate-of-changes reach zero. Note that similar results were obtained for the WSBL case (not shown).
The turbulent temperature flux and the heating by subsidence are both internally coupled to the gradient of the temperature field as they depend on and modify the temperature. However, as the heating by subsidence is a slower process, one may suppose that the temperature flux adapts to the subsidence heating. As a result, the shape of the time- and domain-averaged temperature flux profile is such that, at each height, subsidence heating is balanced (see Fig. 8), which leads to a horizontally-averaged state of thermal equilibrium in which the averaged temperature does not change in time. A general, but simple, condition for this steady state is given by integrating the evolution equation for the horizontally-averaged temperature \(\left<\theta \right>\)
$$\begin{aligned} \frac{\partial \left<\theta \right>}{\partial t}= 0 =-\frac{\partial \left<F_z(\theta )\right>}{\partial z} - w_s(z)\frac{\partial \left<\theta \right>}{\partial z} \end{aligned}$$
(4)
in which the horizontal transport terms are neglected due to horizontal homogeneity. Additionally, the change of temperature due to the divergence of the subsidence velocity \(w_s\) is also neglected (see Appendix 1). Setting the rate of change equal to zero and integrating in the vertical direction gives the condition
$$\begin{aligned} -\left<F_s(\theta ; s)\right>\Big |_{s=0}^{s=z} - \int _{0}^{z} w_s(s)\frac{\partial \left<\theta \right>}{\partial s}\,\mathrm {d}s= {constant}, \end{aligned}$$
(5)
where a dummy variable s is used to represent height. The value of this constant is zero as both contributions on the left-hand side vanish at the surface. Indeed, this is consistent with the simulated temperature flux that tends towards a constant value with respect to height near the surface for both the WSBL and VSBL simulations (see Figs. 6b and 7b) since, near to the surface, the integral contribution of subsidence heating is close to zero. Furthermore, this condition implies that the integrated amount of subsidence heating is equal to the surface flux of temperature, thereby setting an integral constraint.
The steady state at the Dome C site is different from the quasi-steady conditions sometimes encountered at mid-latitudes (Nieuwstadt 1984). In the absence of subsidence, the SBL continues to cool as a whole, whereas the shape of the vertical temperature profile remains largely unchanged in time (Derbyshire 1990; van de Wiel et al. 2012). The condition for the quasi-steady state is found by neglecting subsidence and differentiating with respect to z in Eq. 4, and changing the order of differentiation
$$\begin{aligned} \frac{\partial }{\partial t}\left( \frac{\partial \left<\theta \right>}{\partial z}\right) =0. \end{aligned}$$
(6)
As discussed in Derbyshire (1990), true quasi-steadiness is not possible in the realistic atmospheric SBL. For a quasi-steady, continuously cooling SBL with zero heat flux at the SBL top, the temperature contrast between the bulk and the top would become unlimited (and so would the local Richardson number). In the absence of any gradient-smoothing processes, e.g., radiation or molecular diffusion, this would result in a singularity at the top of the SBL. As such, even quasi-steadiness is not achievable in the mid-latitudes, which makes the present case an attractive alternative for idealized studies of the atmospheric SBL. Indeed, such quasi-steady behaviour has been approached in LES studies of the SBL without subsidence (see, e.g., Beare et al. 2006; Zhou and Chow 2011; Sullivan et al. 2016). A disadvantage, however, is that a continuous surface cooling or surface heat flux has to be prescribed to more-or-less approach this quasi-steady state.
The results indicate that the inclusion of a source term of energy by subsidence opens the possibility of attaining a true thermal steady state for LES investigations of the SBL apart from possible inertial oscillations. It is important to note that a similar conclusion was reached by Mirocha and Kosović (2010), who show that the inclusion of subsidence results in a “nearly steady behaviour” of the SBL in their LES case, which applied a cubic subsidence profile and the calculation of the heating rate per grid cell using the local thermal gradient. Interestingly, observations in the Arctic clear-sky SBL by Mirocha et al. (2005) provide compelling evidence that a significant part of the negative turbulent heat flux at the surface is balanced by warm air entrained into SBL by subsiding motions. Similarly, the importance of subsidence on the near-surface Antarctic heat budget was also found in model studies of regional climate and the general circulation (van de Berg et al. 2007; Vignon et al. 2018).
Sensitivity to Resolution
A Smagorinksy–Lilly closure with stability correction is used despite its limitations and dependence on model parameters, such as the Smagorinksy constant and grid size. However, Matheou (2016) shows that the Smagorinksy–Lilly-type closure can accurately simulate the moderately stable boundary layer and give results comparable to the reference stretched-vortex SFS model (Chung and Matheou 2014; Matheou and Chung 2014), but an a priori choice of the optimal model parameters and resolution is challenging.
A first and relevant test for consistency of the LES approach is to investigate the grid convergence by investigating whether first- and second-order statistics reach a constant value for a change in resolution. Figure 9 shows the simulation results for the wind-speed profile and kinematic temperature flux for the VSBL and VSBLc simulations at 0.08-\(\mathrm {m}\) and 0.125-\(\mathrm {m}\) resolution, respectively. While the simulated wind-speed profiles are similar with differences \(<0.03\,\mathrm {m\,s^{-1}}\), the highest resolution simulation has a slightly lower jet height than the case with a coarser resolution, with the difference in jet height approximately \(0.25\,\mathrm {m}\). Additionally, a steeper increase of the temperature is found for the VSBL simulation, than for the VSBLc simulation, with the maximum difference reaching about \(1.5\,\mathrm {K}\) at a height of \(4.7\,\mathrm {m}\) (not shown).
Some differences are also found in the second-order statistics, such as the kinematic temperature flux (see Fig. 9b). For the highest resolution of \(0.08\,\mathrm {m}\), the total and resolved fluxes are both negligibly small above the diagnosed boundary-layer height, although the SFS contribution reaches approximately \(50\%\) near the top of the boundary layer. Physically speaking, at the top of the SBL, the vertical temperature gradient (hence local gradient Richardson number \({Ri}_g\)) can become very large (cf. Fig. 5c, f), so that the integral length scale may locally become smaller than the grid size. As expected, the SFS contribution is enlarged in the case of a coarser grid. Here, the resolved temperature flux becomes zero at \(4.5\,\mathrm {m}\), whereas the SFS fluxes only become negligibly small around \(z=7\,\mathrm {m}\). First, this indicates that, in the region \(z=4.5\) to \(5.5\,\mathrm {m}\) (\(\approx h\) in the highest resolution run), the dominant turbulent length scale is reduced below the grid spacing of \(0.125\,\mathrm {m}\) due to the increased stratification. Second, it suggests that, in the coarse-grid case, the region \(z=5.5\)–\(7\,\mathrm {m}\) is influenced by excessive diffusion of the SFS scheme. The increase in boundary-layer height for lower resolutions is consistent with Beare et al. (2006) and Sullivan et al. (2016).
In contrast, the relative difference in the surface kinematic temperature flux is only \(2.5\%\), with the corresponding differences for the surface momentum fluxes of \(4.5\%\), which shows that, for a small change of resolution, the eventual surface fluxes are robust. Although the relative difference is small, it is not known how this difference changes under a further increase (or decrease) of resolution. Due to a combination of long integration times and the required number of grid points, an investigation of possible grid convergence for a doubling of the resolution to \(0.04\,\mathrm {m}\), for example, is beyond the current scope of research. However, also taking into account that both the total and the resolved flux become negligible at the same height for the 0.08-\(\mathrm {m}\) simulation, it is expected that a further increase in resolution would not significantly change the diagnosed boundary-layer height or surface fluxes, but merely increase the contribution of the resolved fluxes to the total fluxes.