1. Introduction
In this paper we report on a series of large-eddy simulations (LES) of an idealized PBL driven by radiative cooling from the top of a radiatively opaque layer of fluid—a smoke cloud. The purpose of our study is to explore and understand the sensitivity of LES to small scales, as manifest in either the resolution of the mesh or the model of subgrid-scale (SGS) motions. The smoke cloud is studied because we find that it retains sensitivities that we have also found to be evident in simulations of stratocumulus (Stevens et al. 1998) and because by only representing the essential features of radiatively driven stratocumulus, without the complications associated with the generation or consumption of latent heat through cloud microphysical processes (e.g., Lilly 1968; Moeng and Schumann 1991; Bretherton et al. 1999), it is somewhat simpler.
The smoke cloud has recently come under intense study using both LES and laboratory analogs, and results now emerging in the literature are somewhat in conflict. A key result of the laboratory work is that for such high RiB flows, the nondimensional entrainment rate is a function of the Prandtl number of the fluid (Sayler and Breidenthal 1998). The interpretation of this result is that entrainment across the strongly stratified interface, which separates the turbulent fluid from the quiescent fluid, depends on the diffusive thickening (through molecular processes) of the stratified layer, that is, Taylor layers. The conclusion that the thickness of the Taylor layers regulates the entrainment rates in high RiB flows is troubling for LES, which by nature assumes that molecular processes are inconsequential. Just how troubling this is remains uncertain, mostly because the laboratory results can be questioned on many grounds:they are low Reynolds number, the low aspect ratio of the container may bias the problem, container scale circulations may affect physical processes, and differences in how the radiative forcing is applied may affect the analogy between the laboratory flow and the LES. Thus, despite the fact that the laboratory experiments represent a real flow, it is not obvious that such experiments are any better at representing the hypothetical smoke cloud than is LES.
LESs of the smoke cloud are not only in conflict with the low Reynolds number laboratory data, they also disagree among themselves. While there is general agreement that fine vertical resolution is needed to properly represent processes at the entrainment interface (e.g., Bretherton et al. 1999; Stevens and Bretherton 1999; Lewellen and Lewellen 1998; Lock and MacVean 1999; vanZanten et al. 1999; also see section 2 and appendix C of this article), there is disagreement as to the importance of small-scale turbulence as manifest in sensitivities to horizontal resolution or SGS model assumptions.2 For instance, simulations by the Lawrence Berkeley Labs–University of Washington (LBL–UW) group (Stevens and Bretherton 1999) display an explicit sensitivity to the horizontal resolution for weakly stratified experiments (small RiB); in contrast, simulations by the West Virginia University (WVU) group (Lewellen and Lewellen 1998) are remarkable in their lack of sensitivity to horizontal resolution even for strongly stratified interfaces (large RiB). Thus while the range of resolved scales appears to be important in determining the entrainment rate in the LBL–UW LES, this is not the case for the LES by the WVU group.
Further disagreement among LES is also evident in recently published entrainment relationships (Lewellen and Lewellen 1998; Lock and MacVean 1999; vanZanten et al. 1999). Both the Lewellen and Lewellen and the vanZanten et al. studies have proposed that process partitioning provides a rational framework for describing entrainment rates deduced from LES. Roughly speaking, for the smoke cloud this closure assumption can be interpreted as stating that the extrapolated buoyancy flux in an entraining PBL (denoted by “b” in Fig. 1) is a fixed fraction of what it would be in the absence of entrainment. In other words, process partitioning closures predict that b/(a + b) or simply a/b is a universal constant. Although calculations by a number of groups suggest that their respective simulations are well constrained by such a relationship, the entrainment relations are, to a certain extent, model dependent. For instance, the IMAU (vanZanten et al. 1999) and the WVU (Lewellen and Lewellen 1998) groups predict a/b ≈ 0.4, while the UKMO group (Lock and MacVean 1999) predicts a/b closer to 0.2. Scatter in the relationships among the models tends to be larger than the scatter associated with any one model, which suggests that the precision of an individual simulation is much greater than its accuracy.
Thus recent studies raise many questions: Do the entrainment relationships found by various groups to describe the behavior of large RiB flows reflect real physical relationships? If so, what is the physical character of these relationships, and what physical processes do they underscore? Moreover, if the LES results reflect real physical processes, what explains the scatter among the various calculations? Why are calculations by some groups extraordinarily robust as a function of horizontal resolution while others are not? And how can the LES be reconciled with previous laboratory and experimental work? In this paper we begin to take a crack at some of these important and outstanding questions. We do so by attempting to summarize what ended up being scores of simulations, with two independently developed codes, on 64 × 64 × 71 grids or larger. Our focus will be on the nature of horizontal resolution sensitivities in fine vertical resolution simulations of the smoke cloud and the relationship of such sensitives to SGS models.
2. Background
a. Entrainment rates and fluxes
b. Methods
1) Setup
A description of the experimental configuration of LES of the smoke cloud is detailed in Bretherton et al. (1999). For completeness the basic thermodynamic configuration is compiled in Table 1. Unless otherwise stated it is adhered to identically here. Broadly speaking our simulations fall into three classes: SMK-S-064 simulations are smoke cloud calculations based on the Smagorinsky SGS model and with 64 points in each horizontal direction. SMK-T-064 are identically configured calculations except they are based on the Deardorff prognostic turbulent kinetic energy (TKE) SGS model. SMK-S-128 calculations are identical to SMK-S-064 calculations except twice the number of points are used to span the same domain in each horizontal direction. In the future, references to higher-resolution or finer-mesh simulations refer to simulations falling into the SMK-S-128 class. Standard simulations are denoted by the SMK-S-064 class.
All of the results actually presented in this paper are from the Colorado State University (CSU) code (e.g., appendix A) although in some select instances we test the robustness of our ideas using the substantially different nested-mesh, pseudospectral, code developed at the National Center for Atmospheric Research (NCAR;Moeng 1984; Sullivan et al. 1996). For reasons discussed below, all of the calculations with the CSU model were carried out with a fine (5 m) vertical mesh spanning at least a 100-m zone about the mean inversion.3 Away from the entrainment zone the mesh is gradually stretched to a maximum Δz = ½Δx near the surface and 100 m in the quiescent stratified layer. Tests with a uniformly fine mesh indicate that the method of stretching does not noticeably influence our results.
2) Vertical grid spacing
In addressing the sensitivity of LES to the representation of small scales it would seem natural to explore the sensitivity of the simulations to vertical resolution (e.g., Bretherton et al. 1999; Stevens and Bretherton 1999; Lock and MacVean 1999). Indeed, we initially proceeded along these lines, only to find that the resulting sensitivities in our calculations are complicated by the fact that the sharp radiative cooling profile tends to nonlinearly cool the air in the inversion as smoke is diffused into it—leading to larger entrainment rates for increasing resolution. Indeed we have constructed a simple analytical model (included for reference in appendix C) that we believe provides a plausible explanation (solely in terms of radiative effects) for the previously noted sensitivity of entrainment to vertical resolution. Because we believe that previously reported vertical sensitivities are predominantly radiative–dynamical sensitivities, rather than intrinsically dynamical sensitivities, as has been sometimes suggested (e.g., Stevens and Bretherton 1999), and because our interest is in the latter type of interaction (and in particular how it relates to SGS processes), in this paper we do not further pursue questions related to the sensitivity of our simulations to changes in vertical resolution.
Instead we fix the vertical grid spacing about the entrainment zone to be 5 m in all of our calculations. This scale was chosen because (as evident by comparing simulations with 5- and 3-m vertical grids, and as predicted by the analytic model in appendix C) the sensitivity of the calculation to further refinements in the vertical grid tended to be less than the sensitivities that interest us here. Because we used the same vertical grid in all the calculations, and because we did not alter the nature of the radiative flux parameterization [e.g., Eq. (A5)], the amount of radiative cooling in the inversion was approximately fixed in all our calculations; that is, in contrast to other studies where this was not the case, tests show that this effect does not contribute significantly to the sensitivities we explore.
3) Analysis
Statistics are compared after the simulations achieve a quasi steady state (as measured by the time evolution of the boundary layer turbulent kinetic energy TKE and the linearity of fluxes of conserved quantities). Most statistical quantities (i.e., velocity variances, the mean state, fluxes, and terms in the resolved-scale TKE budget) are computed during the integration at 30-s intervals (approximately every 30 time steps). Inevitably, unanticipated postprocessing is warranted in which case the averages are over a coarser grained time record (every 3 min for 1 h). The results are not particularly sensitive to this degree of refinement in the granularity of the time record.
To make efficient use of limited resources, many simulations are conducted by branching off a control simulation for 90 min of simulated time. Only the last 60 min of a branched integration are analyzed. By studying how flows tended to decorrelate from their initial conditions (or from each other) over this period, we found that our analysis period may begin too soon. This motivated us to selectively test our ideas using integrations started from fresh initial conditions. In these cases the variability among independent realizations, when averaged over the analysis period, was found to be much smaller than the sensitivities we are interested in. As a result we are satisfied that our analysis procedure reveals robust sensitivities of a particular code. The fact that critical conclusions were yet further tested using integrations with a completely different model suggests that our results may even have some degree of generality.
We typically plot fields versus the nondimensional height z/〈zi〉, where angle brackets denote averaging over horizontal planes and time records. For the smoke cloud simulations, zi(x, y, t) is determined to be the uppermost level at which the smoke concentration falls to one-half its value at the lowest model level; although, as noted in the text, other methods are used at times. Typically our analysis is done by averaging all time records on the computational grid and then interpolating to a 〈zi〉 normalized grid. Because weΔt ≈ 2Δz (where we is the entrainment rate, Δt is the analysis period, and Δz = 5 m is the typical grid spacing at the inversion) our results are not overly sensitive to the method of averaging.
c. What does entrainment look like in LES of the smoke cloud?
Before proceeding with a detailed study, based largely on statistical measures of the flow, it is worthwhile to familiarize ourselves with the structure of the entrainment zone as represented by LES. Figure 2 illustrates the structure of the inversion for a SMK-S-128 simulation with cs (the length scale coefficient in the Smagorinsky model) equal to 0.23. Many things are apparent in this figure: (i) the thickness of the inversion (as measured by the distance between smoke or θ contours) is variable, tending to be thicker above downdrafts; (ii) there is no evidence of contours overturning at the inversion, instead contours tend to be steepened at the edge of updrafts and peeled away at the base of downdrafts; (iii) the inversion height fluctuates only slightly across the domain and the entire jump in θ often spans no more than one or two grid points; (iv) smoke and θ contours are well correlated, but because θ has a source in radiation we do not expect them to be perfectly correlated; (v) thin layers of radiatively cooled air are most evident in the divergent layers above updrafts, and deep layers of radiatively cooled air tend to correlate with downdrafts (e.g., compare Figs. 2a and 2b).
Additional insight is gained through a perusal of conditionally sampled fields, for example, Fig. 3 derived from SMK-S-064 class simulations. The conditional sampling method we use is described by Schmidt and Schumann (1989). Because we are interested in interfacial structure, and its relation to entrainment, we define events based on downdraft velocity at zc = 0.85zi, using an exclusion distance d = 0.15zi, and a sampling threshold of w ⩽ −1.25σw(zc). The analysis is quantitatively sensitive to choices of thresholds; nonetheless, in conjunction with the snapshots it proves useful in helping one to form at least a qualitative view of entrainment in LES. The analysis shows that the most radiatively destabilized region of the downdraft is off-axis at the top (or root) of the downdraft. We attribute this to the fact that the downdraft is accelerated by the negative buoyancy of the radiatively cooled air that is converging at its base. In association with this convergence, a high-pressure maximum can be found in the inversion at the root of the downdraft. These high pressure regions are associated with a weak recirculation region that helps thicken the interface (causing the dipole-like structure in the θ′ field at zi), thereby facilitating the incorporation of inversion zone air into the PBL at the downdrafts root.
Figure 4 attempts to encapsulate our provisional view of entrainment in these relatively coarse resolution LES of radiatively driven large RiB flows. The main elements of the figure are threefold: the interface is shown to be thinned and thickened by the energy containing large eddies; the pressure maximum lies in the inversion, above the downdraft that incorporates entrained air; the radiatively cooled air initially forms above updrafts and feeds the downdrafts. Associated with the thickened interface and the high pressure region are decelerating interfacial disturbances. On our figure these disturbances are represented as a flapping and steepening of a contour. This simplified representation of interfacial disturbances can be misleading, as often contours within the inversion are observed to diverge in association with small-scale propagating disturbances.
Unfortunately, it is not straightforward to diagnose the source of entrained air. Because there are small amounts of the radiatively active tracer throughout the inversion, radiative cooling in this layer may still play a small role in setting the entrainment rate, for example, appendix C. In terms of turbulent processes, the simulations show evidence (as noted above) of filaments of inversion air being pulled away from the inversion by the pressure gradient set up by the large eddies. Although this thickening and peeling is also evident in better resolved, lower RiB flows (e.g., Sullivan et al. 1998), in our simulations, where RiB is large, this process appears more dominant. The agitation of the stable interface by the large eddies results in small-scale interfacial disturbances that propagate in and along the stratified layer. These disturbances converge on and establish the pressure maximum at the root of the downdraft, and perhaps contribute to the thickening and peeling process. While there does appear to be evidence of mixing associated with these propagating disturbances, the lack of resolution frustrates attempts to quantify their role in preconditioning inversion air for subsequent incorporation into the downdraft. Ultimately, to say that entrainment is well resolved, one would like to see a separation of scales between fluid comingling and entrainment—this is not apparent in any of our large RiB simulations.
3. The effect of horizontal resolution
Similar to the LBL–UW results, but in contrast to those by the WVU group, entrainment rates in calculations with the CSU code are sensitive to horizontal resolution. Figure 5 shows the effect of a doubling of horizontal resolution on the buoyancy and smoke fluxes. We note (in part because it is not obvious from the figure) that the better resolved flow has larger entrainment fluxes. Even less evident from the figure is that when the resolution is doubled changes in the total heat flux at zi largely reflect changes in the resolved heat flux at zi.
Changing the horizontal resolution also changes λ, the mixing length scale used by our parameterization of SGS turbulence (e.g., appendix B). Based on simulations of the small-RiB CBL, Mason (1989) argues that specifying values of cs that differ from the theoretically derived value (Cs) essentially sets a filter scale λf = csλ/Cs, and that simulations with equivalent values of λf should behave equivalently.4 Indeed, we see that doubling the resolution (i.e., halving the mesh spacing) and doubling the value of cs results in fluxes that tend toward the coarse mesh integration. A wider ranging sequence of calculations is described in Table 2. These results suggest that Mason’s argument captures the tendency of the simulations, especially in the limit as λ goes to zero for a fixed domain size. Indeed, recently Mason and Brown (1999) looked at this issue further for the case of a weakly capped CBL and demonstrated that simulations with λf fixed tend to converge with increasing cs reflecting the fact that the actual filter implied by the simulation is determined both by the SGS model and one’s choice of numerical methods. Our results tend to support their finding for the case of the smoke cloud. The approximate agreement in entrainment rates for small λ (i.e., experiments 304.2 and 302.1) also is reflected by good agreement in the velocity variances (Fig. 6); although there is some indication that higher-order moments retain a sensitivity to λ for fixed λf, particularly near the boundaries of the turbulent flow, this might merely reflect the fact that the higher-order moments are more sensitive to finite-differencing errors in these regions. In the end, because the effects of changes to λ with fixed cs are reasonably well captured by simulations in which λ is fixed, but cs is varied, we consider the sensitivity to small scales further by studying how changes in the SGS model affect our calculations.
4. The Smagorinsky (Lilly) Model
For many of the experiments we shall consider Ric and cs to be free parameters; although, given a number of assumptions as reviewed in the appendix one should set Ric equal to the turbulent Prandtl number, Pr ≈ 0.3 and cs = Cs = π−1[2/(3α)]3/4, where α ≈ 1.5 is the Kolmogorov constant. Also recall that because Ric is, from (26), a turbulent Prandtl number, it plays two roles:in unstable conditions it acts like a Prandtl number in that it determines the ratio of buoyancy and shear production of small-scale turbulence, in stable conditions it determines when the buoyancy field will be sufficiently strong to suppress the generation of small-scale turbulence by shear.
In the above form [i.e., Eq. (4)], the Smagorinsky model can be thought of as having two components: a neutral component proportional (by the constant cs) to the magnitude of the deformation and a prefactor, which depends on the local RiD. Below our discussion is loosely organized around the respective role of these two terms; although, because the deformation appears in the definition of RiD this separation is only roughly true.
a. The effect of cs
1) On entrainment rates and scalar fluxes
As discussed in the previous section, SMK-S-128 class simulations are sensitive to the value of cs in the Smagorinsky model (cf. Fig. 5). A similar result is illustrated in Fig. 7, which summarizes SMK-S-064 simulations spanning a broader range in cs.5 The tendency of calculations with larger values of cs to have smaller entrainment rates occurs despite the presence of a negative feedback. Less entrainment implies more production of TKE, which should support further entrainment. Presumably, such a feedback would impose a certain amount of rigidity on the flow, thereby lessening the sensitivity of the results to changes in parameters.
The effect of cs on entrainment rates is largely mediated by changes in the resolved scales. Consider the entrainment heat flux, which we associate with the minimum heat flux located near the inversion. As was discussed in section 2, so long as the radiative flux does not change its shape among the simulations (which to a sufficient degree of approximation is the case in our simulations) the entrainment heat flux is a good proxy for the entrainment rate. That, in an absolute sense, the change in the total entrainment heat flux is mostly accommodated by changes in the resolved scales (or a lack of change in the subgrid scales) is more clearly evident in Table 3, which lists the resolved and SGS contributions to the entrainment heat flux for calculations with varying values of cs and Δx.
The reduction of resolved entrainment associated with an increase in cs is associated with reductions in the near-interfacial values of resolved vertical velocity variance. Figure 8 illustrates how the cs = 0.35 solution has narrower tails in the velocity distribution near the inversion, and also a less agitated inversion (as evidenced by a reduced probability of the inversion being characterized by relatively weak stratification). Conditionally sampled fields and snapshots support this view, wherein larger values of cs leads to a smoother, more highly organized, flow (i.e., one in which correlations in conditionally sampled fields are stronger).
The cospectra of w and θ is given by the real part of
The view (e.g., Fig. 8) that it is the effect of cs on the velocity field that is critical in reducing the entrainment rate, and the resolved entrainment heat flux, is further supported by simulations in which cs is alternately modified in either the eddy viscosity calculation, the eddy-heat diffusivity calculation, or the eddy-smoke diffusivity calculation: changes to the viscosity lead to by far the largest impact on entrainment rates, changes to eddy-heat diffusivities affect entrainment rates only slightly, and changes to the eddy-smoke diffusivity affects entrainment not at all. Still more tests suggest that if cs is allowed to vary with height, it is the value of cs in the uppermost part of the PBL that most strongly affects the flow.
2) On SGS flux maxima
As pointed out above, in the current implementation of the model, changing cs does not significantly affect the SGS entrainment heat flux; however, the values of SGS heat fluxes outside of a small neighborhood about zi are very sensitive to cs—particularly in the destabilized region of the flow, that is, z/zi ∈ (0.8, 0.95). Figure 10 illustrates that the disproportionate increase in the SGS heat fluxes in the destabilized zone reflects a disproportionate increase in Km (and hence Kh) with increasing cs. Recall that Km ∝
3) On velocity moments
The effect of cs on our calculations is not limited to thermodynamic fluxes and entrainment rates. For instance, Fig. 11 shows that increasing cs tends to produce a more pronounced upper PBL peak in the resolved variance of the horizontal velocity. Because the SGS energy is more-or-less constant this sensitivity projects onto the total variances as well. Although not shown, a further sensitivity to cs is evident in the structure of the vertical velocity skewness near the inversion. Increasing cs tends to result in a less pronounced maxima, similar to what is evident in Fig. 6b.
b. The stability prefactor
The stability prefactor in the Smagorinsky model can be modified by either changing the value of Ric or by altering the functional form of the term multiplying (csλ)2S in Eq. (4). In our tests we consider both a modest change to the functional form of the prefactor, as well as the effects of changing Ric. In the latter (which we discuss first), we consider separate suites of experiments in which Ric is alternately modified in the stable and destabilized regions of the flow, as this helps us better delineate the role of Ric.
By reducing Ric, for RiD < 0, we can artificially increase the eddy viscosity in the destabilized part of the flow, and thus selectively increase the amount of the forcing (or buoyancy flux) that is carried by the SGS model. Changes in Ric sufficient to yield a twofold increase in the maximum value of the SGS heat flux have no noticeable effect on either the distribution of the vertical velocity at the inversion or on the resolved entrainment. This indicates, somewhat surprisingly, that there is not a strong relationship between the amount of the buoyancy flux available for driving resolved-scale motions and the value of the resolved entrainment heat flux.
Changes to Ric, for RiD > 0, also have a minor influence on the overall flow, with most of the effect being limited to the actual value of the SGS heat flux at zi. Experiments with Ric ∈ (1/3, 1/6, 1/12) result in SGS heat fluxes of (−2.0, −0.5, 0.0) W m−2, respectively, with no discernible influence on the resolved heat flux. Because the SGS flux constitutes such a small fraction of the total flux at zi, this change has little effect on the flow as a whole. Furthermore, unlike the effect of cs on entrainment, the Ric (for RiD > 0) effect saturates if Ric becomes sufficiently small. The fact that these two effects (i.e., the Ric and cs effects) are independent are further supported by tests that show that the cs sensitivity is maintained even for a SGS model with Ric = 0.
Mason (1989) has used the argument that the mixing length-scale should be reduced in stabilized portions of the flow to justify modifications to the form of the Smagorinsky model. In appendix B [i.e., Eq. (B14)] we show how a length-scale correction that accounts for possible stability effects results in a modified SGS model that approaches its cutoff Richardson number more rapidly. Using the Smagorinsky model cast in this form reduces the SGS contribution to the entrainment heat flux. Further tests designed to mimic the approach of the UKMO group were performed, in these tests instead of using Eq. (B14), we simply squared the stability term in Eq. (B13); that is, we write Km ∝ (1 − RiD/Ric).2 Such a change had an even less discernible effect on our solutions.
5. The Deardorff model
In attempting to see if the above delineated sensitivities are evident in calculations based on different algorithms (SGS and otherwise), we repeated the standard smoke cloud integration using the NCAR code. Because, in computing SGS fluxes, this code solves, following Deardorff (1980), an equation for e (the SGS TKE) it is not possible to simply change cs. Instead we change the value of cm, which relates the eddy viscosity to a length scale and
To better understand what is causing the lack of sensitivity in LES with the NCAR code, we introduced a prognostic e model into the CSU code. For the SGS length scale λ both codes use λ = (3/2)(ΔxΔyΔz)1/3. In the CSU code Δz varies with height, but for the purpose of the length scale computation it is held fixed at 7.4 m. Results from this calculation with the CSU model are shown in Fig. 12. The prognostic e model effectively eliminates the sensitivity of entrainment to changes in the eddy viscosity.
Further calculations with the CSU code indicated that the basic sensitivity of the resolved-scale entrainment fluxes is preserved in calculations with the e model, but what differs is the ability of the SGS fluxes to compensate. This is clearly evident in Fig. 12, where in the entrainment zone SGS heat fluxes contribute more substantially to the total entrainment flux, and are more sensitive than corresponding Smagorinsky model calculations to changes in the SGS model—although in the destabilized region around 0.9zi the SGS fluxes are less sensitive to changes in the SGS model. The difference between the calculations mainly reflects, as it must, the nonlocal, nonequilibrium terms in the e equation (i.e., the advective transport, the diffusive transport terms that model the third moments, and the storage term). As illustrated by the dash–dot lines in Fig. 13, the net contribution from these terms (which are neglected in the Smagorinsky model), are leading order in the entrainment zone and explain the increased value, and compensating sensitivity, of the SGS component of the entrainment heat flux (indicated by solid lines), as well as the reduction (relative to corresponding calculations with the Smagorinsky model) of the SGS heat flux around 0.9zi.
We conducted further simulations, in which we artificially modified the prognostic equation for e so as to isolate and understand the effect of the three individual processes neglected in the Smagorinsky model. We found that each of the three terms were independently capable of providing the aforementioned compensating effect; although only the advective transport terms did so robustly. The ability of either diffusion-like terms (i.e., the modeled third moment terms in the prognostic equation for e) or nonequilibrium terms to effectively (and increasingly) transport e into the inversion layer with increasing cm depended upon details of the numerical algorithm.
The effectiveness of the diffusive transport of e depends on how Ke, the eddy diffusivity of e is calculated. In both the NCAR and the CSU model e is assumed, following Mason (1989), to locate at layer interfaces (i.e., on w points on the Arakawa C-grid template). Because Ke is a function of e, diffusion calculations require averaging values of Ke from layer interfaces to layer centers (i.e., from w points to θ points). If this averaging is done arithmetically, that is, we solve for Ke at level k + ½ according to 2
For the case of the nonequilibrium terms, we found that their effectiveness in the CSU model was evident only if the flow was horizontally dealiased using an upper one-thirds wave cutoff filter, as is routinely done in the NCAR model. The justification for doing so is that the small scales are well known to be contaminated by both aliasing and finite-difference errors. Recent attempts at quantifying these errors suggest that the energy in the error power spectrum may considerably exceed the energy in the SGS power spectrum at small scales (Ghosal 1996). Introducing a spectral cutoff filter into the CSU code results in a 50% reduction in
6. Discussion
First let us emphasize that the sensitivities that we report here are, for the most part, modest. Moreover in a broad sequence of subsequent calculations such sensitivities were found to be weakened in lower RiB flows (we did experiments with Δϒ = 4 K and 2 K), when the forcing of the turbulence is moved to the surface as in the clear convective PBL, or if higher-order (but oscillatory) advection schemes are used to transport scalars. Nonetheless, our results are important because they demonstrate that the ability of the smoke cloud calculation to show convergence as a function of cs, [or if we accept the arguments of Mason and Brown (1999), which our calculations largely corroborate, as a function of grid-spacing] depends to a great extent on the behavior of the SGS model.
In Fig. 14 we attempt to synthesize our findings schematically. Here we show that increasing cs tends to damp small-scale motions (as represented in the diagram by the magnitude of the deformation). This damping has two further consequences: (i) as the inversion region becomes less energetic (e.g.,
In section 4 we suggested, that physically, radiatively driven layers such as the smoke cloud may behave more rigidly than other flows. Our reasoning was that if something acted to change the entrainment rate the resolved energetics would change in a compensating manner, that is, less entrainment implies more TKE production, which then leads to more entrainment. Given this view we see that the feedback process is partially truncated by the Smagorinsky model; because enhanced production of e just below the inversion cannot affect directly the budget of e in the inversion, the production of more e need not generate more SGS entrainment. Including advective or diffusive transport, or time-rate-of-change terms in the e model extends, in a sense, this rigidity to the SGS model, thereby leading to apparently more robust solutions. Although we must be cautious here, physicality and robustness are different concepts, and one needs not imply the other. In the end, our inability to do extensive convergence tests makes it difficult to say what the correct answer is. At best we can say that our results explain contradictory sensitivities to horizontal resolution as reported by several groups, and warrant some degree of open-mindedness when evaluating the ability of LES to properly represent entrainment scalings in large RiB flows. Two points we discuss further below.
a. Entrainment rate sensitivities and previously published work
Recall that the LBL–UW group (Stevens and Bretherton 1999) report on calculations that exhibit a sensitivity to the horizontal resolution, but the calculations by the WVU group (Lewellen and Lewellen 1998) are remarkably insensitive to horizontal resolution. The fact that the WVU group used a prognostic e model while the LBL–UW group used a Smagorinsky model is consistent with our results.7 Our results are also consistent with the extensive number of sensitivity studies done by the WVU group. For example, the lack of sensitivity of the WVU calculations to the SGS model reflects the fact that despite the number of tests conducted, the only one that led to a significant sensitivity was the one that truncated the compensating mechanism we hypothesize to be present in their model, that is, those tests they call low Reynolds number tests, which are performed by setting the eddy diffusivity and viscosity to constant values (D. Lewellen 1998, personal communication). Last, our results are also consistent with the unpublished results of the UKMO group, who find a sensitivity to horizontal resolution in their model (they have conducted unprecedented, and unpublished simulations with 5-m uniform grid spacing), that is based on the Smagorinsky closure.
Sensitivities to the eddy viscosity and vertical resolution may also explain the differences among the high vertical resolution calculations discussed in the original smoke cloud intercomparison study. That the UKMO code has the smallest entrainment (Bretherton et al. 1999, their Fig. 10) is consistent with results presented at the Smoke Cloud Workshop, which indicated that the UKMO simulations were characterized by a much larger mean eddy viscosity just below the inversion. Although we have not focused on the effect of different representations of resolved modes (i.e., different advection schemes) this could in all likelihood also have some bearing on the differences among models—particularly in so far as the advection schemes preferentially damp or amplify small scales in the momentum budget.
Because our results also indicate that differently configured LES codes are able to support different entrainment rates, it should not be surprising that entrainment heat flux ratios (upon which the process partitioning model of entrainment rests) differ among models—even if they are robust for individual models. Consider our results with the Smagorinsky model: depending on the effective viscosity near the inversion, one can arrive at different values of the ratio of the entrainment heat flux to the extrapolated heat flux (i.e., a/b in the terminology of Fig. 1). A larger mean eddy viscosity tends to favor smaller ratios, as we find that a/b varies from 0.42 to 0.25 as cs is increased from 0.15 to 0.53. These differences are on the order of the differences discussed in the introduction.
b. Can we rule out the importance of small scales?
Although the model of the heat flux described above may be justified (or perhaps does not matter) within a well-developed inertial range, its validity is open to question within a poorly represented entrainment zone—particularly when the entrainment zone is characterized by large gradients and variances in the buoyancy variable as well as diabatic processes [not included in Eq. (6)].
Another reason for open-mindedness is that our flow analysis fails to reveal clearly identifiable structures, or processes, associated with entrainment. This may well be a failure of our analysis, but there are a number of indications that the sharp gradients, and the poor resolution of interfacial processes, affects the solutions one gets. Moreover, given our finding that the sensitivity, or the tendency toward convergence of a calculation, depends on the assumptions made in the SGS model, it seems fair to speculate that robust balances in various models might in the end be artifacts of the model numerics, peculiarities in the flow configuration, and SGS model assumptions.
Yet a further reason for open-mindedness is that it remains unclear as to what component of the total entrainment heat flux is truly resolved. As discussed in section 2c, flow analysis does not support the conclusion (based on a partitioning of the flux between the SGS and resolved component) that most of the entrainment heat flux is resolved. Indeed, we cannot even rule out the possibility that numerical diffusion plays a leading role in determining the amount of resolved entrainment.
In the end, our results further argue for the need to test the entrainment relationships derived from LES. These tests should include much higher resolution simulations, a specially designed field campaign, and continually refined laboratory measurements.
7. Summary
Our major findings are twofold:
Increased values of the eddy viscosity result in smaller values of the resolved entrainment heat flux, which, depending on the behavior of the SGS model, may or may not result in changes to the net entrainment rate.
Because the reduction of the resolved entrainment through enhanced eddy viscosities in the upper part of the PBL tends to be associated with greater production of SGS TKE (e), models that allow for the e at a point in space and time to be influenced by values of the e at neighboring points (e.g., Deardorff’s SGS model and most variants) are better able to compensate for the sensitivities of the resolved-scale entrainment heat fluxes.
In addition we try to show how, for modest changes in the horizontal mesh, the SGS viscosity in the entrainment zone acts as a proxy for horizontal resolution. Our results suggest that the horizontal sensitivity reported in simulations by the LBL–UW group but absent in the calculations by the WVU group reflect different assumptions used in their SGS models. Consequently, in flows such as the smoke cloud (where commonly used resolutions poorly resolve processes in the entrainment zone) the robustness, or lack thereof, of previously reported results may have less to do with physical processes, and more to do with uncertain assumptions made in the numerical, and SGS models.
If the sensitivities we report here end up representing the degree of our uncertainty in representing entrainment in LES, then we are in good shape. Given the limitations in our understanding of a variety of other processes, an uncertainty in parameterized entrainment rates of less than 50% is certainly tolerable. However, given the uniformity of approach in most LES models, our inability to rule out numerical diffusion as contributing significantly to the resolved entrainment, and the lack of a good theory for the behavior of small-scale turbulence in the stratified entrainment zone, only time, and further work, will tell if the sensitivities discussed here are accurate measures of our current uncertainty.
Acknowledgments
BS’s contributions were supported by NCAR’s Advanced Study Program. Al Cooper is thanked for helping to make it a very nice program. Revisions were made while the first author was visiting the Max-Planck-Institut für Meteorologie in Hamburg Germany, as a fellow of the Alexander Humboldt Foundation. Peter Duynkerke is thanked for pointing out the difference among entrainment flux ratios by various groups. Chris Bretherton, Andrea Brose, Peter Duynkerke, Vanda Grubisic, Piotr Smolarkiewicz, Zbignew Sorbjan, Dave Stevens, Shouping Wang, and three anonymous reviewers are also all thanked for their comments on earlier versions of this manuscript. GEWEX and members of working group one of GCSS are thanked for helping to initiate and further supporting this work. Mike Shibao is thanked for drafting Fig. 4. Last, the Tex project and the Free Software Foundation (GNU) are thanked for donating the word processor and text editor used in this analysis and subsequent writeup.
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APPENDIX A
Description of the CSU Model and its Algorithms
The equations are solved using finite differences. The momentum equations are marched forward by a leapfrog scheme with an Asselin filter while scalar terms are integrated using a forward-in-time method. Advection terms in the momentum equation are solved using fourth-order centered differences except in regions of grid stretching where second-order centered differences are used. In the scalar equations, advection terms are calculated in one of two ways. One method uses the fourth-order generalization of the Lax–Wendroff differences (Tremback et al. 1987), the other method interpolates between the fourth-order fluxes and a first-order scheme in order to ensure monotonicity in the solution (Zalesak 1979). The former scheme is variance friendly, the latter is variance diminishing. Pressure is solved in Fourier space in the horizontal and by inverting a tridiagonal system in the vertical. We periodically check to ensure that the solver satisfies the continuity equation to machine accuracy throughout the course of a simulation. The equations are solved on a staggered (Arakawa C) grid, which is compressed and or stretched in the vertical. Integrations with a fine vertical mesh throughout were compared to integrations with a compressed mesh near the top of the PBL and a stretched mesh above the PBL. Sensitivities to the moderate stretching ratios (1.1) used were looked for, but not found. The eddy diffusivity is calculated following the procedure outlined by Mason (1989), similarly, to avoid vertical averaging of SGS viscosities and diffusivities, the SGS TKE is collocated with w on the staggered mesh. Boundary conditions are periodic in x and y. Boundary conditions at the top of the domain are constant gradient for scalar variables and free slip for momentum.
APPENDIX B
SGS Models
Prognostic SGS–TKE models
The above models for the SGS energy equation yields a closed set of equations given values for cϵ, cm, ch, and a length scale l. The length scale is often related to a generalized measure of the grid spacing, which we denote by λ, and/or the local stability.
These stability corrections are necessary when using the prognostic SGS–TKE model. In the absence of such corrections entrainment rates and SGS fluxes are dramatically altered. This differs from some previous studies (e.g., Schumann 1991) and what we found using the Smagorinsky model (cf. section 5b). These issues have been investigated further and are discussed in more detail by Stevens et al. (1999).
In presenting the above closure assumptions we have not endeavored to defend their veracity. Indeed it has long been known that fundamental aspects of these equations are fallacious (e.g., Clark et al. 1979). But in many respects, the fallacy of some assumptions does not appear to have a dramatic effect on the fidelity of many aspects of the flow (Schmidt and Schumann 1989;Mason 1989; Nieuwstadt et al. 1991). Thus we are less interested in whether the SGS model is correct, and more interested in whether it is incorrect in a way that matters.
Equilibrium models
APPENDIX C
Spurious Radiative–Dynamical Interactions
We consider an idealized situation in which an interface capping a turbulent underlying smoke fluid moves steadily through our vertical domain. We suppose that the entrainment process is layerwise and diffusive, that is it is characterized by a successive replacement of the lowest layer of smokefree, warmer, “free tropospheric” fluid by smoke-full, cooler, turbulent boundary layer fluid. We further suppose that the rate of growth of the smoke layer (we = dzi/dt) is, in the absence of other affects, set externally, perhaps by the large-scale energetics of the underlying fluid, as for instance is advocated by Lewellen and Lewellen (1998). As shown from our flow snapshot in Fig. 2 and by the PDF of maximum temperature gradients in Fig. 8b, even for Δx = 25 m and Δz = 5 m, the interface is to a first approximation confined to one or two grid levels. This view of the interface is corroborated by the snapshots of a similar flow at higher resolution (e.g., Stevens and Bretherton 1999), the almost stepwise growth of the height of the minimum buoyancy flux (e.g., Fig. 4 in Lock and MacVean 1998), as well as results from the original smoke cloud intercomparison (Bretherton et al. 1999). Simulations performed with coarser vertical or horizontal resolution, are presumably even better characterized by this model.
More typical situations live between these two limits. For instance, in the case studied in this paper κ = 0.02 m2 kg−1, Δϒ ≈ 4 K (as taken from the mode of the distribution in Fig. 8b with Δz = 5 m), F0/cp ≈ 0.06 K kg m−2 s−1, and we ≈ 0.0025 m s−1 (e.g., corresponding to experiment 304.2 in Table 2) yields ξ = 8.33. From (C10) it follows that R(25) ≈ 0.54 and R(5) ≈ 0.81, thus implying an increase in entrainment rate between a Δz = 5 m calculation and a Δz = 25 m calculation of 45%. Entrainment rate ratios between the 5- and 25-m calculations by the UKMO, WVU, and ARAP models [i.e., R(25)/R(5) − 1] are 46% as reported by Bretherton et al. (1999).8 The extent of agreement is probably fortuitous; there is a certain amount of arbitrariness in specifying Δϒ, for instance, choosing Δϒ = 7 K leads to a somewhat smaller prediction of R(25)/R(5) − 1 = 33%. Moreover our use of a linearized exponential function for the radiative flux is actually an upper bound on the effect. Because our model indicates that setting a threshold on the minimum smoke path necessary for radiation to become active can effectively eliminate the cooling in the radiatively ambiguous layer, it provides an explanation for the reduced sensitivity to vertical resolution in calculations that set a critical smoke path threshold of about s ≈ 0.5Δz before radiative fluxes become active.9 This effect is not explained by competing ideas (e.g., Stevens and Bretherton 1999; Bretherton et al. 1999). Overall the degree of explanatory power of the model, and its physical and quantitative basis lead us to believe that it is the most plausible explanation for the sensitivity of smoke cloud simulations to vertical resolution.
Last, we emphasize that while the purely spurious effect discussed here may make entrainment rates artificially sensitive to vertical resolution in poorly resolved smoke cloud simulations, this should not be confused with in some senses a similar (albeit physical) effect associated with radiative cooling in the time-mean interfacial layer of finite thickness evident, for instance, in simulations with well-resolved cloud-top undulations (Moeng et al. 1999).
Heat flux from LES showing the relationship between the entrainment heat flux and the extrapolated heat flux used in process partitioning parameterizations of entrainment.
Citation: Journal of the Atmospheric Sciences 56, 23; 10.1175/1520-0469(1999)056<3963:LESORD>2.0.CO;2
Snapshots from an SMK-S-128 simulation at t=9000 s of θ (upper panel, contours at 287.5, 288.0, 289.0, 290.0, 291.0, 292.0, 293.0, 294.0, 294.5 K), w (middle panel, contours at −1.00, −0.50, −0.25, −0.10, 0.00, 0.10, 0.25, 0.50, 1.00, with negative contours dashed and zero contour thickened), and s (lower panel, contours at 0.05, 0.20, 0.35, 0.50, 0.65, 0.80, 0.95, 0.98, 1.00). Shading is applied to help accentuate features.
Citation: Journal of the Atmospheric Sciences 56, 23; 10.1175/1520-0469(1999)056<3963:LESORD>2.0.CO;2
Conditionally sampled fields from an SMK-S-064 class simulation. The sampling is about downdrafts at zc = 0.85zi. In defining events we use d = 0.15zi and a threshold of 1.25σw (z = zc) (see Schmidt and Schumann for more details about the method). Here θ perturbations (upper panel, contours at −0.25, −0.15, −0.10, −0.05, 0.00, 0.05, 0.10, 0.15, 0.25 K), with superposed velocity vectors; smoke concentration perturbations (middle panel, unevenly spaced contours at −0.0100, −0.0050, −0.0025, 0.0000, 0.0025, 0.0050, 0.0100, 0.0200, 0.0400), and p′ (lower panel, contours every 0.01). Negative contours are dashed and zero contour is thickened. Shading is applied to help accentuate features.
Citation: Journal of the Atmospheric Sciences 56, 23; 10.1175/1520-0469(1999)056<3963:LESORD>2.0.CO;2
Schematic figure of the nature of entrainment in a radiatively forced layer as represented by our LES. In part this figure represents the dynamics as we see them superimposed on a conditionally averaged view of the interface.
Citation: Journal of the Atmospheric Sciences 56, 23; 10.1175/1520-0469(1999)056<3963:LESORD>2.0.CO;2
Total and SGS smoke/heat fluxes for an SMK-S-064 class simulation with cs = 0.23 (solid), and SMK-S-128 class simulations with cs = 0.23 (dashed) and cs = 0.46 (dotted). (c) A replot of (a) in a manner that better illustrates features in the entrainment heat flux.
Citation: Journal of the Atmospheric Sciences 56, 23; 10.1175/1520-0469(1999)056<3963:LESORD>2.0.CO;2
Sensitivity to Δx for fixed λf for smoke cloud simulations: (a) velocity variances (w2 curves peak near 0.5z/zi) and (b) skewness of w. Solid line, Δx = 25 m; dashed line Δx = 50 m; dotted line Δx = 100 m.
Citation: Journal of the Atmospheric Sciences 56, 23; 10.1175/1520-0469(1999)056<3963:LESORD>2.0.CO;2
Smoke cloud sensitivity of (a) theta and (b) smoke fluxes to changes in cs for a suite of SMK-S-064 class simulations with cs = 0.15 (solid), 0.23 (dashed), 0.35 (dotted), 0.53 (dash–dot).
Citation: Journal of the Atmospheric Sciences 56, 23; 10.1175/1520-0469(1999)056<3963:LESORD>2.0.CO;2
PDFs of (a) w at zi (thick) and zi − Δz (thin lines) and (b) (Δθ/Δz)max for SMK-S-064 class calculations with cs = 0.15 (solid line) and cs = 0.35 (dotted line).
Citation: Journal of the Atmospheric Sciences 56, 23; 10.1175/1520-0469(1999)056<3963:LESORD>2.0.CO;2
Two-dimensional cospectra of w and θ∗, where θ∗ is the value of θ interpolated, on the basis of our advection algorithms, to a w point, and κ = (
Citation: Journal of the Atmospheric Sciences 56, 23; 10.1175/1520-0469(1999)056<3963:LESORD>2.0.CO;2
Sensitivity of (a) Km and (b) RiD to changes in cs for a suite of SMK-S-064 class calculations. Lines as in Fig. 7. Included for reference are results from some SMK-S-128 class simulations with cs = 0.23 (dash–dot–dot), cs = 0.46 (fine dots).
Citation: Journal of the Atmospheric Sciences 56, 23; 10.1175/1520-0469(1999)056<3963:LESORD>2.0.CO;2
Resolved variances in horizontal winds. Lines as in Fig. 7.
Citation: Journal of the Atmospheric Sciences 56, 23; 10.1175/1520-0469(1999)056<3963:LESORD>2.0.CO;2
Heat and smoke fluxes for SMK-T class calculations with cm = 0.1 (solid) and cm = 0.25 (dashed).
Citation: Journal of the Atmospheric Sciences 56, 23; 10.1175/1520-0469(1999)056<3963:LESORD>2.0.CO;2
Budgets of SGS TKE from some SMK-T class simulations. Buoyancy production (solid line), shear production (dotted line), dissipation (dashed line), other terms (i.e., advective plus diffusive transport and storage terms) (dash–dot). Here cm = 0.1, (upper panel), and cm = 0.25 (lower panel).
Citation: Journal of the Atmospheric Sciences 56, 23; 10.1175/1520-0469(1999)056<3963:LESORD>2.0.CO;2
A schematic relationship between cs (or any parameter setting an effective filter scale), resolved entrainment and SGS entrainment in SGS–TKE-based SGS models. The tendency of a field to increase or decrease with a change in cs is indicated by the direction of the triangles in the box, as well as the shading of the boxes.
Citation: Journal of the Atmospheric Sciences 56, 23; 10.1175/1520-0469(1999)056<3963:LESORD>2.0.CO;2
(a) Entrainment heat fluxes and (b) vertical velocity variances from SMK-S-064 class simulations with cs = 0.15 (solid), 0.23 (dashed), 0.35 (dotted), 0.53 (dash–dot), and an SMK-S-128 class simulation with cs = 0.23.
Citation: Journal of the Atmospheric Sciences 56, 23; 10.1175/1520-0469(1999)056<3963:LESORD>2.0.CO;2
Initial sounding. In all cases the mean wind and all surface fluxes are set to zero.
Value of we for simulations with differing values of λ but with λf held constant at 50 m.
Resolved and SGS heat flux minima (in W m−2) for simulations with differing values of cs.
RiB is similar to the inverse of the square of the bulk Froude Number as earlier defined by Rouse and Dodu (1955). If the generic scale velocity used in Eq. (1) is replaced by Deardorff’s convective scale velocity, then our RiB is identical to Deardorff’s Ri∗. For most definitions of a ϒ∗ (the convective temperature scale), RiB = Δϒ/ϒ∗.
In discussions of LES results we often use the words small scales to refer to scales on the order of the grid scale, which is orders of magnitude larger than the viscous or diffusive scales postulated to be important in low Reynolds number high RiB laboratory flows.
Calculations with the NCAR model utilized a fine mesh with 8.33-m vertical spacing throughout the entrainment zone.
Here we distinguish between Cs the theoretical value of the Smagorinsky constant, and cs the value we use in our SGS parameterization.
Strictly speaking values cs < Cs imply a filter scale smaller than ones grid scale. However, only for such small values of cs do we generate a −5/3 energy spectrum down to the grid scale—a traditional goal in SGS modeling, although as pointed out by Mason and Brown (1999) not necessarily a well-founded one. Larger values of cs tend to produce spectra that fall off more rapidly. For this reason, and because associating λ with Δx is only a rough statement of the filter scale, we believe the cs = 0.15 experiments are worth considering.
For archival purposes we also note here two further effects of spectral filtering: (i) the maximum of the horizontal variances below zi are reduced; (ii) the subgrid entrainment heat flux minimum tends to become more peaked in the inversion, as a result just below zi resolved and subgrid fluxes are of opposite sign, an effect in better accord with the resolved cospectra just below zi (cf. Fig. 9b).
Calculations that use no SGS model (or for which the SGS model is not relied upon to provide the dissipation), but rely on limiters in the advection algorithms to dissipate energy and bound the flow should (because of their inherently local nature), according to our arguments, behave more like calculations based on the Smagorinsky model.
We picked these calculations because they were the only ones that used a 5-m vertical grid in their high vertical resolution calculations. The CSU and NCAR high-resolution simulations were performed with Δz = 12.5 m.
These calculations were first presented by Hans Cuijpers, and subsequently reproduced by us. Hans Cuijpers deserves credit for originally proposing what essentially amounts to this explanation. Peter Duynkerke is thanked for his tireless (and in the end contagious) advocacy of essentially these ideas.
