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The Sensitivity of Numerical Simulations of Cloud‐Topped Boundary Layers to Cross‐Grid Flow

The Sensitivity of Numerical Simulations of Cloud‐Topped Boundary Layers to Cross‐Grid Flow IntroductionLarge Eddy Simulation (LES) is an effective tool for simulating cloud‐topped boundary layers. Simulations with state of the art LES run at high‐resolution compare reasonably well with many observational cases (e.g., McGibbon & Bretherton ; Siebesma et al., ; Stevens et al., ). Any finite‐difference scheme will produce dispersive and/or diffusive errors, with the largest errors associated with the shortest wavelengths that can be represented by the grid (e.g., Lele, ). In LES, a common strategy to minimize horizontal advective errors and to increase the maximum time step for numerical stability is the translation of the simulation grid with a velocity characteristic of the mean horizontal wind. However, many situations arise where this is not practical, e. g., for large domain simulations with substantial variability of the near‐surface wind, simulations with large vertical mean wind shear, or superparameterized simulations.Matheou et al. () note a substantial reduction of cloud amount and LWP in trade‐cumulus simulations when the domain is not translated and cross‐grid flow is substantial. Their simulations use an isotropic grid. Here we examine the effects of cross‐grid flow using nonisotropic grids for three different marine‐boundary‐layer cloud cases: nocturnal stratocumulus, nonprecipitating trade cumulus, and a stratocumulus‐to‐cumulus transition. Simulations with anisotropic grids generally maintain more stratocumulus clouds even at coarse resolutions and better match observations (Pedersen et al., ). We are especially interested in LES sensitivity to cross‐grid flow in “gray‐zone” resolutions (Arakawa & Wu, ; Wyngaard, ) where energy‐containing eddies that span the depth of the boundary layer are barely resolved. These coarse resolutions are used for computational expediency in current global modeling approaches including superparameterization (Khairoutdinov et al., ), ultraparameterization (Parishani et al., ), and global cloud resolving models (CRM's) (e.g., Miyamoto et al., ). Here we study the impact of minimizing horizontal cross‐grid flow by translating the grid in our horizontally periodic domains to approximately match the mean boundary layer flow.Experiment SetupModel PhysicsWe use the System for Atmospheric Modeling (SAM, Khairoutdinov & Randall, ) version 6.10.6. SAM uses a second‐order centered differencing scheme for momentum advection. For scalar advection, we primarily use the scheme of Blossey and Durran (, hereafter BD). Experiments were also performed using the default SAM scalar advection scheme, MPDATA, and the Ultimate Macho flux corrected fifth‐order advection scheme (UM5, Yamaguchi et al., ). All of the simulations use a prognostic 1.5‐order subgrid TKE closure to calculate subgrid eddy diffusivities and viscosities (Khairoutdinov & Randall, ). The model formulations for subgrid‐scale fluxes and surface fluxes used here are invariant with respect to a spatially uniform translation velocity.Case DescriptionsWe examine three cases, summarized in Table . We first focus on the 6 h GCSS DYCOMS RF01 nocturnal nonprecipitating marine stratocumulus case (Stevens et al., ), which uses an idealized longwave radiation scheme. We also perform experiments for a 6 h BOMEX nonprecipitating shallow‐cumulus case (available as a standard case with SAM), specified following Siebesma et al. (). Last, we study the GEWEX GCSS stratocumulus‐to‐cumulus transition case (de Roode et al., ; Sandu & Stevens ) forced by a SST rise of 1.67 K d−1. Radiative fluxes in this case are computed with the Rapid Radiative Transfer Model for GCMs (RRTMG) (Iacono et al., ; Mlawer et al., ) with a full diurnal cycle of solar radiation. The translation velocities in each case, also given in Table , were selected to minimize the domain relative horizontal velocity components within the boundary layer.Summary of Case Physics and Setup With Coarse Grid of 250 m × 250 m × 20 m and a Fine Grid of 35 m × 35 m × 5 mDYCOMSBOMEXSc‐CuSurface fluxesFixedBulkBulkMicrophysicsNoneNoneKhairoutdinov and Kogan ()RadiationSimplified longwave onlyNoneRRTMGGrid translation velocity (U,V) in m s−1(6.5, −4.8)(−8.0, 0.0)(−2.0, −4.0)CoarseFineCoarseFineCoarseFineHorizontal domain size (km)8.06.724.09.08.06.7Refined zone base (m)725775660775695635Refined zone top (m)1,0009251,8252,0002,0252,775Coarsest Δ z below cloud (m)401550255015Time step (s)2.00.31.50.3VariesVariesFor all of the cases, we generally will treat the runs with (Δx, Δy, Δz) = (35 m, 35 m, 5 m) (hereafter denoted as “35 × 5”) grids as reference runs. This grid resolution has provides reasonable simulations in many GCSS test cases (e.g., de Roode et al ; Stevens et al., ;). For prototype coarse‐resolution simulations, we will use a (Δx, Δy, Δz) = (250 m, 250 m, 20 m) (“250 × 20”) grid. 250 m horizontal resolution is on the small side of the range of gray‐zone resolutions for simulating the cloud‐topped boundary layers. For each case, the 5 and 20 m vertical grid resolution refers to the region from near cloud‐base to above the highest cloud‐top. Above and below this zone, the fixed grid is smoothly transitioned to coarser grids for computational efficiency. The vertical grid specifications for each case are outlined in Table .ResultsDYCOMS Simulation OverviewWe begin by examining the effects of cross‐grid flow on DYCOMS nocturnal stratocumulus case for both the high‐resolution 35 × 5 and gray‐zone 250 × 20 domains. For each domain, we compare runs with stationary grid “SG,” and moving grid “MG.” The horizontal‐mean velocity profiles for these runs are plotted in Figure . For moving‐grid cases, the simulated velocities plus the translation velocities are plotted. For all cases, the boundary layer winds drift slowly away from the initial constant velocities. For the moving‐grid cases, the horizontal mean simulated u and v remain within 1.0 m s−1 of the translation U and V, minimizing horizontal cross‐grid flow. There are only slight differences between the SG and MG 35 × 5 profiles, whereas the SG and MG 250 × 20 profiles differ by as much as 0.3 m s−1.Mean total (left) u and (right) v velocity profiles for stationary grid (thick) and moving grid (thin) DYCOMS cases. (a) and (b) are 35 × 5 cases and (c) and (d) are 250 × 20 cases. The constant translation velocity for the latter case is shown as a dashed black line.More substantial effects of cross‐grid flow are evident in LWP, cloud fraction, and inversion height (Figure ). The latter is determined as the level where vertically interpolated horizontal‐mean relative humidity drops to 50%. The results at each resolution start to diverge after 30 min during the model spin‐up.(a) DYCOMS LWP, (b) cloud fraction, and (c) inversion height for stationary and moving‐grid cases.For the moving‐grid 35 × 5 case (MG 35 × 5), LWP increases fairly steadily from 45 min onward, while LWP gently declines in the stationary grid case (SG 35 × 5), suggesting the large cross‐domain flow in the SG 35 × 5 case is increasing cloud‐top entrainment, drying, and thinning the stratocumulus cloud. The cloud fraction steadily decreases in both cases, but about three times as quickly for SG 35 × 5 than for MG 35 × 5.With 250 × 20 resolution, the impact of the moving grid is larger and opposite in sign suggesting that the moving grid supports more efficient entrainment and consequently a thinner stratocumulus cloud. In the SG case, LWP decreases only slightly after spin‐up, while in the MG case LWP plummets to less than 60% of the stationary‐grid value. After 6 h, the MG cloud‐fraction drops to about 0.57 while the SG cloud fraction only drops to about 0.72. The boundary layer deepens about 50% more in MG 250 × 20 than in SG 250 × 20.The sensitivity of the cloud state to cross‐grid flow is not particularly tied to the advection scheme, as shown in Figure . For each scalar advection scheme, BD, UM5, and MPDATA, the 35 × 5 simulations have lower mean LWP and cloud fraction using a stationary grid while the 250 × 20 simulations have higher LWP and cloud fraction using a stationary grid.Mean liquid water path and mean cloud fraction for various scalar advection schemes in DYCOMS simulation hours 2.0–6.0. Squares and triangles indicate stationary grid (SG) and moving grid (MG) simulations, respectively. Filled and open symbols represent 35 × 5 and 250 × 20 simulations, respectively. Small squares represent SG simulations with halved time step.Because the numerical impacts of strong cross‐grid flow are substantial, we also tested the sensitivity of the SG runs to time step. The impact of changing the time step is small, especially for the 35 × 5 cases, and in all cases much smaller than the grid‐translation sensitivity. This suggests that time discretization errors are relatively unimportant in these simulations compared to spatial discretization errors. In particular, the relative importance of the time discretization error associated with the cross‐grid flow in the stationary grid runs should scale with the associated horizontal Courant number (u Δt/Δx), which is typically very small, on the order of 0.05, due to the large aspect ratio of the grid.Effects of Cross‐Grid Flow in Inversion StructureTo better quantitatively understand the sensitivity of cloud amount and thickness to the amplitude of the cross‐grid flow, we performed a series of DYCOMS experiments for both 35 × 5 and 250 × 20 configurations all with the BD advection scheme (Figure ). The grid translation velocities (U,V) varied from (6.5, −4.8) to (–6.5, 4.8), with the latter effectively doubling the cross‐grid flow of the SG experiment. For both coarse and fine grids, the strongest sensitivity of LWP and cloud fraction occurs at the smallest grid‐relative wind speeds. For 35 × 5 simulations, LWP and cloud cover decrease with increasing grid‐relative wind speed up to 6 m s−1. The 250 × 20 simulations have increasing LWP and cloud fraction with increasing wind speed up to about 7 m s−1 wind speed, above which LWP and cloud decrease with increasing wind‐speed.Sensitivity of DYCOMS mean (a) LWP and (b) cloud fraction to mean cross‐grid flow at cloud level. Statistics are averages over hours 2.0–6.0. MG and SG simulations are labeled.We now discuss the mechanisms driving the sensitivity of MBL‐top entrainment to cross‐grid flow, as the cloud‐top entrainment is the ultimate control of the stratocumulus cloud thickness and amount. The first mechanism is the influence of cross‐domain flow on the inversion structure. Cross‐grid flow appears to spread the mean temperature and moisture inversion over a larger number of vertical grid points. An example of this tendency is shown in snapshots of total water mixing ratio (qt) at 2 h of the cloud‐top inversion regions of SG and MG runs at both 35 × 5 and 250 × 20 grid resolutions (Figure ). At both resolutions, the SG simulations have a broader inversion. In SG 35 × 5, the main part of the inversion extends over 150%–200% as many grid points as in MG 35 × 5. At 2 h, the SG 250 × 20 also has a slightly thicker inversion than MG 250 × 20. We can estimate the evolution of inversion thickness calculated as the mean distance between the heights of qt‐isosurfaces of 2.5 and 8.0 g kg−1 using three‐dimensional simulation snapshots (Figure a). For all cases, the inversion thickens with time—especially during the first two simulation hours. With 35 × 5 resolution, the inversion is consistently 30–40% thicker in the SG case with cross‐grid flow but only a few percent thicker for 250 × 20 resolution.DYCOMS 2 h x‐z snapshots of total water (qt) in g kg−1 in the upper MBL, with, qt contours at 2.5 and 8.0 g kg−1 (black), u and w wind anomalies (black arrows, relative to domain mean), and cloud water contours (white) at 0.01, 0.1, 0.3, and 0.5 g kg−1 for (a) SG 35 × 5, (b) MG 35 × 5, (c) SG 250 × 20, and (d) MG 250 × 20. Note (a) and (b) show only a small x portion of their respective 6.7 km domains.(a) Inversion thickness and (b) standard deviation of inversion height in m derived from simulation snapshots.The continued presence of a broader, more diffuse inversion should enhance MBL‐top mixing and entrainment through several mechanisms—by allowing more dry warm air to be mixed down into the lower MBL by large eddies, by allowing updrafts to push higher into the inversion, and by enhancing near‐inversion horizontal gradients of heat and moisture. Another important difference between 35 × 5 and 250 × 20 resolutions is the degree to which horizontal undulations in the inversion height are resolved. After 2 h, typical standard deviation of the inversion height (Figure b) is 4–5 and 5–7 m for 35 × 5 and 250 × 20 m, respectively. For 35 × 5, this is one vertical grid level, but for 250 × 20 this is only 25–30% of a vertical grid level.Given this distinction, we hypothesize that the enhanced inversion thickening is due to horizontal advection acting on horizontal undulations of the inversion height, especially on the 35 × 5 grid. The undulations create strong horizontal gradients of heat and moisture which can then be mixed by numerical diffusion. By this reasoning, the cross‐grid flow in the 35 × 5 case has a much stronger inversion smearing effect because it resolves the inversion undulations much more effectively than the 250 × 20 case. We believe this effect explains most of the sensitivity of the 35 × 5 runs to cross‐grid flow.Entrainment EfficiencyAnother key difference between these resolutions is the extent to which entrainment responds to resolved turbulence. One approach to quantifying the relative strength of stratocumulus cloud‐top entrainment to turbulence is by computing cloud‐top entrainment efficiency Ae (e.g., Stevens, ). Here we will utilize the following definition of entrainment efficiency,1Ae=weΔbzi/w′2¯32where we is cloud‐top entrainment rate, Δb is the jump in buoyancy across the inversion, calculated using the method of Yamaguchi and Randall (), zi is the inversion depth, and w′2¯ is the mean boundary layer velocity variance averaged between the surface and zi. A commonly used alternative version of Ae uses MBL integrated buoyancy flux in the denominator in place of w′2¯; this leads to similar conclusions as the formula used here.In Figure , we show w′2¯ and Ae for the DYCOMS cases. To reduce noise, we compute Ae using 30 min means. The coarser 250 × 20 runs have smaller w′2¯ and larger Ae, showing an undesirably strong sensitivity of turbulence and entrainment efficiency to grid resolution. They also show the largest impact of cross‐grid flow. In DYCOMS, Ae is fairly stable after startup for all 35 × 5 configurations and is not significantly affected by cross‐grid flow. In contrast SG 250×20 has about double the entrainment efficiency of MG 250 × 20. Despite stronger w′2¯ and a slightly broader inversion, the SG 250 × 20 simulation entrains substantially less (see Figure c) than the MG 250 × 20 simulation. We explore the causes of this below.Time series of (a) mean boundary layer w′2¯ in m2 s−2 and (b) entrainment efficiency.To better explore impacts of changing horizontal and vertical resolution independently, we also simulate the DYCOMS case using several additional grid sizes: 35 m × 20 m, 100 m × 20 m, 500 m × 20 m, and 100 m × 5 m. The cloud fraction and entrainment efficiency Ae averaged over DYCOMS hour 2.0 to hour 6.0 are plotted against LWP for all the DYCOMS cases (Figure ). As in Figure , moving‐grid cases are marked with a triangle and stationary cases with a square. As noted in many studies, both vertical and horizontal resolution (e.g., Cheng et al., , Parishani et al., ) and grid aspect ratio (Pedersen et al., ) have a large impact on cloud water and cloud fraction. Coarsening the horizontal resolution strongly increases LWP and cloud fraction, while coarsening the vertical resolution causes a large reduction in LWP and cloud fraction. For all of the 20 m vertical resolution experiments, cross‐grid flow induces a large increase in cloud fraction and LWP which is associated with a rough halving of entrainment efficiency (Figure b). In contrast, for the 5 m vertical resolution cases, cross‐grid flow induces small decreases in cloud fraction and LWP and has a minimal effect on entrainment efficiency.Mean hour 2.0–6.0 of (a) cloud fraction versus LWP (g m−2) and (b) entrainment efficiency vs LWP. Squares represent stationary grids, triangles represent moving grids. Solid and hollow symbols represent 5 or 20 m vertical grid resolution, respectively.Eddy Filtering and EntrainmentWe now focus on the filtering effect that cross‐grid flow has on turbulence and the substantial impact this has on cloud‐top entrainment, entrainment efficiency, and MBL evolution. Figure shows snapshots of the horizontal distribution of LWP at hour 2 of the DYCOMS experiments, with SG runs on the left and MG runs on the right. For both 35 × 5 and 250 × 20 resolutions, the large cross‐grid flow in the SG simulations appears to eliminate features at the scale of the horizontal grid resolution. This filtering emerges very rapidly in the simulations and is not only confined to cloud features. Figures a–d show horizontal power spectra in the x‐direction of vertical velocity vs height for the same simulations also at hour 2. The cross‐grid flow in the SG simulations has the effect of reducing power at the highest wave numbers for both 35 × 5 and 250 × 20 resolutions throughout the boundary layer. The horizontal power spectra of the horizontal velocity components (not shown) have similar behavior. For the 250 × 20 case, it appears that the smallest eddies induce much of the MBL‐top entrainment. The numerical damping of these eddies by cross‐grid flow greatly reduces entrainment and entrainment efficiency.LWP in kg m−2 at hour 2.0.X‐axis power spectra of vertical velocity versus height at hour 2.0 for (a) SG 35 × 5, (b) MG 35 × 5, (c) SG 250 × 20, and (d) MG 250 × 20 DYCOMS simulations. Spectra for translating‐grid runs with hyperdiffusion applied are also plotted: (e) 35 × 5 and (f) 250 × 20.To test this idea, we can artificially filter out the finest‐scale eddies by applying a hyperdiffusive term to the momentum equations of the form2du→dt=−k∇4u→where k is a constant and the ∇4 operator only applies in the x and y directions. For the 35 × 5 and 250 × 20 cases we choose k to be 1.5 × 104 and 1.0 × 107m4 s−1, respectively. These values of k would lead to 2Δx oscillations in u to be damped on a timescale of 1 and 4 s, respectively. This filter is applied to all four of the DYCOMS cases and has the intended effect of filtering out some of the smallest eddies as seen in hour 2 w power spectra of the MG runs (Figures e and f). Since SAM uses a centered difference scheme for momentum advection, the inclusion of hyperdiffusion for velocity will have broadly similar consequences as the use of dissipative numerics for momentum advection would. The latter choice is advocated by Pressel et al. (), whose highest fidelity simulations of this DYCOMS case use dissipative numerics for momentum.The effects of hyperdiffusion on mean LWP and entrainment efficiency are shown in Figure . For all runs hyperdiffusion increases LWP by reducing cloud‐top entrainment. However the LWP impact is largest for the MG 250 × 20 case (empty triangles), and comparatively small for the SG 250 × 20 case (empty squares) where cross‐grid flow has already eliminated the smallest eddies. Similarly, hyperdiffusion reduces entrainment efficiency in all cases, but has a larger fractional impact in the MG cases, where small eddies are present. This experiment supports the notion that numerical damping of the small eddies in the 250 × 20 case is the main cause of the cross‐grid flow impacts.Hour 2.0–6.0 Average of entrainment efficiency Ae and LWP (g m−2) for 250 × 20 grid (hollow symbols) and a 35 × 5 grid (filled symbols). Control runs in blue and runs with hyperdiffusion in gold. SG and MG simulations are represented by squares and triangles, respectively.SAM's subgrid‐scale mixing scheme will also somewhat preferentially reduce small‐scale resolved turbulence. However, for DYCOMS the typical subgrid TKE is 30–100 times smaller than the resolved TKE, so we do not expect this has much impact on the cross‐grid flow sensitivity. SAM uses Δz for the subgrid length scale, not (Δx Δy Δz)1/3 as some LES models do. Given the grid aspect ratios (7 and 12.5) of these simulations, other models using the latter formulation would have subgrid length scales 4 or 5 times larger. In some cases, these models could have reduced sensitivity to cross‐grid flow.BOMEXIn the BOMEX trade‐cumulus case, we expect to see different sensitivities of cloud to cross‐grid flow because of the differences between trade‐cumulus boundary layers and stratocumulus‐topped boundary layers. In trade‐cumulus MBL's, cloud fraction is much smaller compared to stratocumulus MBLs. Figure shows the 6 h evolution of the four BOMEX experiments with fine and coarse grids and with stationary and translating grids. After a 90 min spin‐up period the cloud fraction reaches relatively steady values, while domain mean LWP still varies substantially with time.(a) Mean liquid water path (g m−2) and (b) total cloud fraction for four BOMEX simulations.For both 250 × 20 and 35 × 5 runs, the time‐mean domain‐mean LWP is increased significantly by cross‐grid flow. Cloud fraction due to cross‐grid flow is increased in the after‐spin‐up mean by about 0.01 for the 250 × 20 case, while it is decreased slightly in the 35 × 5 case. The overall increase of cloud liquid water with cross‐grid flow may be related to a substantial change in updraft and cloud structure. For the BOMEX SG cases, the cross‐grid flow is 8 m s−1 aligned with the x direction. In these cases, the cumulus cloud shapes are also elongated in the x direction. This is seen in Figure in horizontal cross‐sections at hour 2.0, at a height of 700 m, about 100 m above the base of the cumulus cloud layer in all simulations. In the MG 250 × 20 case (Figure d), the cloud population is dominated with clouds of one or a few grid points, while the SG 250 × 20 clouds (Figure c) are typically several grid points wide in the x direction, but mostly one grid point wide, in the y direction. For the 35 × 5 cases, where the cloud updrafts are represented by many more grid columns, some elongation in the x direction is seen in the SG run. The cloud shapes closely mimic the shapes of the updrafts at all heights in the cloud layer which are similarly elongated (not shown).Hour 2.0 x‐y snapshots of BOMEX cloud water (g kg−1) at 700 m for (a) SG 35 × 5, (b) MG 35 × 5, (c) SG 250 × 20, and (d) MG 250 × 20 cases. The mean cross‐grid horizontal velocity at 700 m is shown as an arrow for each case.Associated with these effects are changes in overall cloud population and cloud size. Figure shows vertical mean profiles averaged over hours 2–6 of cloud water, cloud fraction, and cloud number and diameter of clouds. Cloud number and size statistics are computed by using a cloud water concentration threshold of 0.01 g kg−1 and aggregating all directly adjacent cloudy grid points into cloud objects at each vertical level. The equivalent diameter (D) of the cloud objects is computed using D = 2(A/π)1/2 where A is the horizontal area of each cloud element. The shape of mean cloud fraction profiles is generally similar between all of the runs, but the shape of the cloud water shows substantial sensitivity to cross‐grid flow. The MG runs have a gentle decline of cloud water with height in the cloud layer while the SG runs have a secondary peak in cloud water at 1,350–1,400 m. The population and diameter statistics also show strong cross‐grid flow effects at all levels in the cloud layer for both 250 × 20 and 35 × 5 resolutions. The cloud number counts have a similar shape to the cloud fraction, but at both resolutions the cloud‐number in the SG runs is half that in the MG runs. The mean equivalent diameter of cloud elements (shown only where cloud fraction > 0.0005) is quite sensitive to grid resolution, about twice as large for a 35 × 5 grid as for a 250 × 20 grid, and the 35 × 5 grid has more of an increase of mean cloud‐diameter with height in the upper cloud layer. However, the cross‐grid flow sensitivity of D is consistent between resolutions; simulations with cross‐grid flow have 40–50% larger diameter of cloud elements.Horizontal mean, time mean hour 2.0–6.0 profiles of (a) cloud water (g kg−1), (b) cloud fraction, (c) number of cloud elements, and (d) mean equivalent diameter of cloud elements for four BOMEX simulations.At both resolutions and at all levels in the cloud, the MG simulations are able to maintain a larger population of small clouds. The SG simulations have fewer but larger clouds. The larger size of these clouds may insulate them from lateral entrainment and allow them to maintain larger liquid water as they ascend. The differences in the cloud structure in these simulations can be traced again to the turbulence impacts of cross‐grid flow. Examination of the turbulence spectra (not shown) indicates that the small eddies are damped by cross‐grid flow as was the case in DYCOMS, but in this case the filtering happens only along the x axis along the cross‐grid flow direction. This causes the elongated updrafts in the x direction and larger cloud shapes. The reduction of small‐scale turbulence also inhibits the formation of small clouds at the base of the cumulus cloud layer.Sc‐to‐Cu TransitionWe finally turn to simulations of the GCSS Sc‐to‐Cu transition case, which include microphysics, full radiation physics, and a steadily increasing SST over a multiday timescale. For this case, a smaller translation velocity is needed (U = −2.0 m s−1, V = −4.0 m s−1) to approximately match the PBL‐mean wind velocity. For these experiments, the MPDATA advection scheme was used, though preliminary experiments with other schemes show similar results. An adaptive time step was used for computational expediency.The LWP, cloud fraction, inversion height, and cloud base height for the SG and MG runs are shown in Figure . The Sc‐Cu simulations include the diurnal cycle which induces a large variation in boundary layer turbulence and LWP. Solar heating in the stratocumulus top of the MBL reduces LWP during the daytime and, for 250 × 20 cases, reduces cloud fraction. Superimposed on this diurnal forcing is steady boundary layer deepening due to the increasing SST. As the boundary layer deepens, it decouples and a cumulus layer develops below the stratocumulus cloud‐top (e.g., Bretherton & Wyant, ).Domain mean (a) liquid water path (g m−2), (b) cloud fraction, and (c) inversion height and domain minimum cloud‐base height (m) for four Sc‐Cu transition simulations. Nighttime intervals are shaded.Cross‐grid flow has a big effect on LWP at 250 × 20 resolution. The stationary SG 250 × 20 case has larger LWP than MG 250 × 20 through most of the simulation, about 60% higher averaged over the simulation. The daytime gaps in cloud coverage are also reduced in the SG 250 × 20 case. This behavior is consistent with both DYCOMS and BOMEX 250 × 20 results presented above, where the scale‐filtering effects of cross‐grid flow tend to increase LWP and cloud coverage.For the 35 × 5 cases, cloud fraction is nearly 100% for the entire simulation period despite the transition from a single cloud layer to a cumulus‐under‐stratocumulus regime and a decline in LWP. For the first 40 h, cross‐grid flow appears to only slightly reduce LWP. After that time, the cross‐grid flow appears to reduce the nighttime LWP significantly. This is consistent with the inversion‐spreading impacts of cross‐grid flow described in the 35 × 5 DYCOMS experiments, resulting in relatively more cloud‐top entrainment and reduced cloud water.Discussion and ConclusionsUsing LES experiments, we have identified substantial impacts of horizontal cross‐grid flow in three different MBL cases. These effects include changes in LWP and cloud fraction and depend on grid size. A driver of these changes is that cross‐grid flow filters out small scale turbulence, reducing horizontal variability in the cross‐grid flow direction. In the DYCOMS stratocumulus case, this change to the turbulent structure reduces cloud‐top entrainment, increasing LWP and cloud cover. A second effect of cross‐grid flow in stratocumulus is to spread the inversion over more grid levels, enhancing cloud‐top entrainment and decreasing LWP and cloud cover. At coarse resolution (250 × 20) the former effect is dominant, while at finer resolution (35 × 5), where the inversion undulations are better resolved, the latter effect is more important. The vertical grid spacing is more important than the horizontal spacing in determining sensitivity to cross‐grid flow. For DYCOMS, we also used alternative scalar advection schemes, UM5 and MPDATA. These schemes demonstrate qualitatively similar sensitivity to cross‐grid flow as the BD advection scheme. Cross‐grid flow effects were not sensitive to the choice of time step. The sensitivity of cloud properties to the amplitude of cross‐grid flow seems to diminish as the amplitude is increased above 7 m s−1 in the DYCOMS case and other tests we have tried, but this tendency should be explored more thoroughly.Similar LWP and cloud fraction sensitivities to cross‐grid flow are present in longer stratocumulus‐to‐trade‐cumulus simulations that include microphysics, more realistic longwave radiation, and a diurnal cycle of shortwave radiation.In trade‐cumulus simulations based on BOMEX, the filtering associated with cross‐grid flow affects the structure of convection at both resolutions studied. Cloud updrafts in the cumulus layer are elongated and enlarged with the cross‐grid flow, and the population of small clouds is greatly reduced. The increase in cloud size decreases the efficiency of lateral entrainment, and increases overall liquid water, primarily in the upper parts of the cloud layer.Because SAM uses a centered‐difference momentum scheme, the leading error terms in the momentum advection equations will tend to be dispersive rather than dissipative. Models with an odd‐order momentum scheme may instead promote dissipation of turbulent energy at grid scales. This smoothing by an odd‐order scheme, which is proportional to the local velocity across the grid, could result in a qualitatively different sensitivity to cross‐grid flow.For scalar advection, it is illuminating to theoretically estimate the numerical diffusion due to cross‐grid flow. In SAM, the scalar advection is treated using forward‐in‐time schemes that combine the time and space discretization. Each of these uses a flux‐corrected‐transport scheme to preserve monotonicity (either everywhere for UM5 or MPDATA, or selectively for Blossey‐Durran). When flux correction is applied, the numerical flux is relaxed back towards a first‐order upwind approximation, so that it seems appropriate here to analyze the effect of cross‐grid flow on a first‐order upwind scheme because it will be the effective scheme operating on extrema as they are advected horizontally across the grid when flux correction is applied. In this case, the numerical scheme can be represented by the modified equation (e.g., Smolarkeiwicz & Szmelter, ): 3∂φ∂t+∂u′φ∂x≈∂∂x(νeff∂φ∂x)where u′ = u – U, and U is the grid‐translation velocity. The numerical diffusivity, νeff, can be written as4νeff=12u′Δx−u′2Δt=12u′Δx(1−CFL)where Δt is the time step and CFL is the horizontal courant number |u′|Δt/Δx. For our simulations, this CFL number is almost always smaller than 0.1. Therefore, the numerical diffusivity is approximately proportional to the size of the cross‐grid flow, |u′|. As monotonic flux correction will operate at extrema of the scalar field when new extrema might be created, this numerical diffusivity will be most prominent in simulations with strong cross‐grid flow.It is not always practical to prevent substantial cross‐grid flow in simulations, for example, due to large mean vertical wind‐shear, due to mesoscale circulations within a model domain, when simulating with fixed topographical features such as in a regional model, or when performing global CRM experiments. In these cases, care must be taken to understand the cross‐grid flow impacts on the mean cloud state. These impacts are important for stratocumulus clouds at typical LES resolutions, and are especially important for stratocumulus and small cumulus clouds at gray‐zone resolutions, where filtering effects of cross‐grid flow are more acute.AcknowledgementsFunding for this study was provided by the U. S. Department of Energy under DE‐SC0012451. Thanks to Mike Pritchard and Hossein Parishani whose UPCAM experiments motivated this work and for many helpful discussions. We also thank Marat Khairoutdinov for the use of SAM and for his helpful comments. We would also like to thank the two anonymous reviewers, whose comments helped improve this manuscript. The simulation data used to produce the figures for this study are archived at the University of Washington Research Works, http://hdl.handle.net/1773/40990.ReferencesArakawa, A., & Wu, C.‐M. (2013). A unified representation of deep moist convection in numerical modeling of the atmosphere. Part I. Journal of the Atmospheric Sciences, 70, 1977–1991. https://doi.org/10.1175/JAS-D-12-0330.1Blossey, P. N., & Durran, D. R. (2008). Selective monotonicity preservation in scalar advection. Journal of Computational Physics, 127, 5160–5183. https://doi.org/10.1016/j.jcp.2008.01.043Bretherton, C. S., & Wyant, M. C. (1997). Moisture transport, lower‐tropospheric stability, and decoupling of cloud‐topped boundary layers. Journal of the Atmospheric Sciences, 54, 148–167. https://doi.org/10.1175/1520-0469(1997)054<0148:MTLTSA>2.0.CO;2Cheng, A., Xu, K.‐M., & Stevens, B. (2010). Effects of resolution on the simulation of boundary ‐layer clouds and the partition of kinetic energy to sub‐grid scales. Journal of Advances in Modeling Earth Systems, 2, 3. https://doi.org/10.3894/JAMES.2010.2.3de Roode, S. R., Sandu, I., van der Dussen, J. J., Ackerman, A. S., Blossey, P., Jarecka, D., et al. (2016). Large‐eddy simulations of EUCLIPSE‐GASS Lagrangian stratocumulus‐to‐cumulus transitions: Mean state, turbulence, and decoupling. Journal of the Atmospheric Sciences, 73, 2485–2508. https://doi.org/10.1175/JAS-D-15-0215.1Iacono, M., Delamere, J., Mlawer, E., Shephard, M., & Clough, S. (2008). Radiative forcing by long‐lived greenhouse gases: Calculations with the AER radiative transfer models. Journal of Geophysical Research, 113, D13103. https://doi.org/10.1029/2008JD009944Khairoutdinov, M., & Kogan, Y. (2000). A new cloud physics parameterization in a large‐eddy simulation of marine stratocumulus. Monthly Weather Review, 128, 229–243.Khairoutdinov, M. F., & Randall, D. A. (2003). Cloud resolving modeling of the ARM summer 1997 IOP: Model formulation, results, uncertainties, and sensitivities. Journal of the Atmospheric Sciences, 60(4), 607–625.Khairoutdinov, M., Randall, D., & DeMott, C. (2005). Simulations of the atmospheric general circulation using a cloud‐resolving model as a superparameterization of physical processes. Journal of the Atmospheric Sciences, 62, 2136–2154.Lele, S. K. (1992). Compact finite difference schemes with spectral‐like resolution. Journal of Computational Physics, 103, 16–42.Matheou, G., Chung, D., Nuijens, L., & Stevens, B. (2011). On the fidelity of large‐eddy simulation of shallow precipitation cumulus convection. Monthly Weather Review, 139, 2918–2939.McGibbon, J., & Bretherton, C. S. (2017). Skill of ship‐following large‐eddy simulations in reproducing MAGIC observations across the Northeast Pacific stratocumulus to cumulus transition region. Journal of Advances in Modeling Earth Systems, 9, 810–831. https://doi.org/10.1002/2017MS000924Miyamoto, Y., Kajikawa, Y., Yoshida, R., Yamaura, T., Yashiro, H., & Tomita, H. (2013). Deep moist atmospheric convection in a subkilometer global simulation. Geophysical Research Letters, 40, 4922–4926. https://doi.org/10.1002/grl.50944Mlawer, E., Taubman, S., Brown, P., Iacono, M., & Clough, S. (1997). RRTM, a validated correlated‐k model for the longwave. Journal of Geophysical Research, 102, 16663–16682.Parishani, H., Pritchard, M. S., Bretherton, C. S., Wyant, M. C., & Khairoutdinov, M. (2017). Towards low cloud‐permitting cloud superparameterization with explicit boundary layer turbulence. Journal of Advances in Modeling Earth Systems, 9, 1542–1571. https://doi.org/10.1002/2017MS000968Pedersen, J. G., Malinowski, S. P., & Grabowski, W. W. (2016). Resolution and domain‐size sensitivity in implicit large‐eddy simulation of the stratocumulus‐topped boundary layer. Journal of Advances in Modeling Earth Systems, 8, 885–903. https://doi.org/10.1002/2015MS000572Pressel, K. G., Mishra, S., Schneider, T., Kaul, C. M., & Tan, Z. (2017). Numerics and subgrid‐scale modeling in large eddy simulations of stratocumulus clouds. Journal of Advances in Modeling Earth Systems, 9, 1342–1365. https://doi.org/10.1002/2016MS000778Sandu, I., & Stevens, B. (2011). On the factors modulating the stratocumulus to cumulus transitions. Journal of the Atmospheric Sciences, 68, 1865–1881. https://doi.org/10.1175/2011JAS3614.1Siebesma, A., Bretherton, C. S., Brown, A., Chlond, A., Cuxart, J., Duynkerke, P. G., et al.. (2003). A large eddy simulation intercomparision study of shallow cumulus convection. Journal of the Atmospheric Sciences, 60(10), 1201–1219.Smolarkeiwicz, P., & Szmelter, J. (2009). Iterated upwind schemes for gas dynamics. Journal of Computational Physics, 228, 33–54. https://doi.org/10.1016/j.jcp.2008.08.008Stevens, B. (2002). Entrainment in stratocumulus‐topped mixed layers. Quarterly Journal of the Royal Meteorological Society, 128(586), 2663–2690. pStevens, B., Moeng, C.‐H., Ackerman, A. S., Bretherton, C. S., Chlond, A., de Roode, S., et al. (2005). Evaluation of large‐eddy simulations via observations of nocturnal marine stratocumulus. Monthly Weather Review, 133, 1443–1462.Wyngaard, J. C. (2004). Toward numerical modeling in the Terra Incognita. Journal of the Atmospheric Sciences, 61, 1816–1826.Yamaguchi, T., & Randall, D. A. (2012). Cooling of entrained parcels in a large‐eddy simulation. Journal of the Atmospheric Sciences, 69, 1118–1136.Yamaguchi, T., Randall, D. A., & Khairoutdinov, M. F. (2011). Cloud modeling tests of the ULTIMATE‐MACHO scalar advection scheme. Monthly Weather Review, 139, 3248–3264. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Advances in Modeling Earth Systems Wiley

The Sensitivity of Numerical Simulations of Cloud‐Topped Boundary Layers to Cross‐Grid Flow

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Abstract

IntroductionLarge Eddy Simulation (LES) is an effective tool for simulating cloud‐topped boundary layers. Simulations with state of the art LES run at high‐resolution compare reasonably well with many observational cases (e.g., McGibbon & Bretherton ; Siebesma et al., ; Stevens et al., ). Any finite‐difference scheme will produce dispersive and/or diffusive errors, with the largest errors associated with the shortest wavelengths that can be represented by the grid (e.g., Lele, ). In LES, a common strategy to minimize horizontal advective errors and to increase the maximum time step for numerical stability is the translation of the simulation grid with a velocity characteristic of the mean horizontal wind. However, many situations arise where this is not practical, e. g., for large domain simulations with substantial variability of the near‐surface wind, simulations with large vertical mean wind shear, or superparameterized simulations.Matheou et al. () note a substantial reduction of cloud amount and LWP in trade‐cumulus simulations when the domain is not translated and cross‐grid flow is substantial. Their simulations use an isotropic grid. Here we examine the effects of cross‐grid flow using nonisotropic grids for three different marine‐boundary‐layer cloud cases: nocturnal stratocumulus, nonprecipitating trade cumulus, and a stratocumulus‐to‐cumulus transition. Simulations with anisotropic grids generally maintain more stratocumulus clouds even at coarse resolutions and better match observations (Pedersen et al., ). We are especially interested in LES sensitivity to cross‐grid flow in “gray‐zone” resolutions (Arakawa & Wu, ; Wyngaard, ) where energy‐containing eddies that span the depth of the boundary layer are barely resolved. These coarse resolutions are used for computational expediency in current global modeling approaches including superparameterization (Khairoutdinov et al., ), ultraparameterization (Parishani et al., ), and global cloud resolving models (CRM's) (e.g., Miyamoto et al., ). Here we study the impact of minimizing horizontal cross‐grid flow by translating the grid in our horizontally periodic domains to approximately match the mean boundary layer flow.Experiment SetupModel PhysicsWe use the System for Atmospheric Modeling (SAM, Khairoutdinov & Randall, ) version 6.10.6. SAM uses a second‐order centered differencing scheme for momentum advection. For scalar advection, we primarily use the scheme of Blossey and Durran (, hereafter BD). Experiments were also performed using the default SAM scalar advection scheme, MPDATA, and the Ultimate Macho flux corrected fifth‐order advection scheme (UM5, Yamaguchi et al., ). All of the simulations use a prognostic 1.5‐order subgrid TKE closure to calculate subgrid eddy diffusivities and viscosities (Khairoutdinov & Randall, ). The model formulations for subgrid‐scale fluxes and surface fluxes used here are invariant with respect to a spatially uniform translation velocity.Case DescriptionsWe examine three cases, summarized in Table . We first focus on the 6 h GCSS DYCOMS RF01 nocturnal nonprecipitating marine stratocumulus case (Stevens et al., ), which uses an idealized longwave radiation scheme. We also perform experiments for a 6 h BOMEX nonprecipitating shallow‐cumulus case (available as a standard case with SAM), specified following Siebesma et al. (). Last, we study the GEWEX GCSS stratocumulus‐to‐cumulus transition case (de Roode et al., ; Sandu & Stevens ) forced by a SST rise of 1.67 K d−1. Radiative fluxes in this case are computed with the Rapid Radiative Transfer Model for GCMs (RRTMG) (Iacono et al., ; Mlawer et al., ) with a full diurnal cycle of solar radiation. The translation velocities in each case, also given in Table , were selected to minimize the domain relative horizontal velocity components within the boundary layer.Summary of Case Physics and Setup With Coarse Grid of 250 m × 250 m × 20 m and a Fine Grid of 35 m × 35 m × 5 mDYCOMSBOMEXSc‐CuSurface fluxesFixedBulkBulkMicrophysicsNoneNoneKhairoutdinov and Kogan ()RadiationSimplified longwave onlyNoneRRTMGGrid translation velocity (U,V) in m s−1(6.5, −4.8)(−8.0, 0.0)(−2.0, −4.0)CoarseFineCoarseFineCoarseFineHorizontal domain size (km)8.06.724.09.08.06.7Refined zone base (m)725775660775695635Refined zone top (m)1,0009251,8252,0002,0252,775Coarsest Δ z below cloud (m)401550255015Time step (s)2.00.31.50.3VariesVariesFor all of the cases, we generally will treat the runs with (Δx, Δy, Δz) = (35 m, 35 m, 5 m) (hereafter denoted as “35 × 5”) grids as reference runs. This grid resolution has provides reasonable simulations in many GCSS test cases (e.g., de Roode et al ; Stevens et al., ;). For prototype coarse‐resolution simulations, we will use a (Δx, Δy, Δz) = (250 m, 250 m, 20 m) (“250 × 20”) grid. 250 m horizontal resolution is on the small side of the range of gray‐zone resolutions for simulating the cloud‐topped boundary layers. For each case, the 5 and 20 m vertical grid resolution refers to the region from near cloud‐base to above the highest cloud‐top. Above and below this zone, the fixed grid is smoothly transitioned to coarser grids for computational efficiency. The vertical grid specifications for each case are outlined in Table .ResultsDYCOMS Simulation OverviewWe begin by examining the effects of cross‐grid flow on DYCOMS nocturnal stratocumulus case for both the high‐resolution 35 × 5 and gray‐zone 250 × 20 domains. For each domain, we compare runs with stationary grid “SG,” and moving grid “MG.” The horizontal‐mean velocity profiles for these runs are plotted in Figure . For moving‐grid cases, the simulated velocities plus the translation velocities are plotted. For all cases, the boundary layer winds drift slowly away from the initial constant velocities. For the moving‐grid cases, the horizontal mean simulated u and v remain within 1.0 m s−1 of the translation U and V, minimizing horizontal cross‐grid flow. There are only slight differences between the SG and MG 35 × 5 profiles, whereas the SG and MG 250 × 20 profiles differ by as much as 0.3 m s−1.Mean total (left) u and (right) v velocity profiles for stationary grid (thick) and moving grid (thin) DYCOMS cases. (a) and (b) are 35 × 5 cases and (c) and (d) are 250 × 20 cases. The constant translation velocity for the latter case is shown as a dashed black line.More substantial effects of cross‐grid flow are evident in LWP, cloud fraction, and inversion height (Figure ). The latter is determined as the level where vertically interpolated horizontal‐mean relative humidity drops to 50%. The results at each resolution start to diverge after 30 min during the model spin‐up.(a) DYCOMS LWP, (b) cloud fraction, and (c) inversion height for stationary and moving‐grid cases.For the moving‐grid 35 × 5 case (MG 35 × 5), LWP increases fairly steadily from 45 min onward, while LWP gently declines in the stationary grid case (SG 35 × 5), suggesting the large cross‐domain flow in the SG 35 × 5 case is increasing cloud‐top entrainment, drying, and thinning the stratocumulus cloud. The cloud fraction steadily decreases in both cases, but about three times as quickly for SG 35 × 5 than for MG 35 × 5.With 250 × 20 resolution, the impact of the moving grid is larger and opposite in sign suggesting that the moving grid supports more efficient entrainment and consequently a thinner stratocumulus cloud. In the SG case, LWP decreases only slightly after spin‐up, while in the MG case LWP plummets to less than 60% of the stationary‐grid value. After 6 h, the MG cloud‐fraction drops to about 0.57 while the SG cloud fraction only drops to about 0.72. The boundary layer deepens about 50% more in MG 250 × 20 than in SG 250 × 20.The sensitivity of the cloud state to cross‐grid flow is not particularly tied to the advection scheme, as shown in Figure . For each scalar advection scheme, BD, UM5, and MPDATA, the 35 × 5 simulations have lower mean LWP and cloud fraction using a stationary grid while the 250 × 20 simulations have higher LWP and cloud fraction using a stationary grid.Mean liquid water path and mean cloud fraction for various scalar advection schemes in DYCOMS simulation hours 2.0–6.0. Squares and triangles indicate stationary grid (SG) and moving grid (MG) simulations, respectively. Filled and open symbols represent 35 × 5 and 250 × 20 simulations, respectively. Small squares represent SG simulations with halved time step.Because the numerical impacts of strong cross‐grid flow are substantial, we also tested the sensitivity of the SG runs to time step. The impact of changing the time step is small, especially for the 35 × 5 cases, and in all cases much smaller than the grid‐translation sensitivity. This suggests that time discretization errors are relatively unimportant in these simulations compared to spatial discretization errors. In particular, the relative importance of the time discretization error associated with the cross‐grid flow in the stationary grid runs should scale with the associated horizontal Courant number (u Δt/Δx), which is typically very small, on the order of 0.05, due to the large aspect ratio of the grid.Effects of Cross‐Grid Flow in Inversion StructureTo better quantitatively understand the sensitivity of cloud amount and thickness to the amplitude of the cross‐grid flow, we performed a series of DYCOMS experiments for both 35 × 5 and 250 × 20 configurations all with the BD advection scheme (Figure ). The grid translation velocities (U,V) varied from (6.5, −4.8) to (–6.5, 4.8), with the latter effectively doubling the cross‐grid flow of the SG experiment. For both coarse and fine grids, the strongest sensitivity of LWP and cloud fraction occurs at the smallest grid‐relative wind speeds. For 35 × 5 simulations, LWP and cloud cover decrease with increasing grid‐relative wind speed up to 6 m s−1. The 250 × 20 simulations have increasing LWP and cloud fraction with increasing wind speed up to about 7 m s−1 wind speed, above which LWP and cloud decrease with increasing wind‐speed.Sensitivity of DYCOMS mean (a) LWP and (b) cloud fraction to mean cross‐grid flow at cloud level. Statistics are averages over hours 2.0–6.0. MG and SG simulations are labeled.We now discuss the mechanisms driving the sensitivity of MBL‐top entrainment to cross‐grid flow, as the cloud‐top entrainment is the ultimate control of the stratocumulus cloud thickness and amount. The first mechanism is the influence of cross‐domain flow on the inversion structure. Cross‐grid flow appears to spread the mean temperature and moisture inversion over a larger number of vertical grid points. An example of this tendency is shown in snapshots of total water mixing ratio (qt) at 2 h of the cloud‐top inversion regions of SG and MG runs at both 35 × 5 and 250 × 20 grid resolutions (Figure ). At both resolutions, the SG simulations have a broader inversion. In SG 35 × 5, the main part of the inversion extends over 150%–200% as many grid points as in MG 35 × 5. At 2 h, the SG 250 × 20 also has a slightly thicker inversion than MG 250 × 20. We can estimate the evolution of inversion thickness calculated as the mean distance between the heights of qt‐isosurfaces of 2.5 and 8.0 g kg−1 using three‐dimensional simulation snapshots (Figure a). For all cases, the inversion thickens with time—especially during the first two simulation hours. With 35 × 5 resolution, the inversion is consistently 30–40% thicker in the SG case with cross‐grid flow but only a few percent thicker for 250 × 20 resolution.DYCOMS 2 h x‐z snapshots of total water (qt) in g kg−1 in the upper MBL, with, qt contours at 2.5 and 8.0 g kg−1 (black), u and w wind anomalies (black arrows, relative to domain mean), and cloud water contours (white) at 0.01, 0.1, 0.3, and 0.5 g kg−1 for (a) SG 35 × 5, (b) MG 35 × 5, (c) SG 250 × 20, and (d) MG 250 × 20. Note (a) and (b) show only a small x portion of their respective 6.7 km domains.(a) Inversion thickness and (b) standard deviation of inversion height in m derived from simulation snapshots.The continued presence of a broader, more diffuse inversion should enhance MBL‐top mixing and entrainment through several mechanisms—by allowing more dry warm air to be mixed down into the lower MBL by large eddies, by allowing updrafts to push higher into the inversion, and by enhancing near‐inversion horizontal gradients of heat and moisture. Another important difference between 35 × 5 and 250 × 20 resolutions is the degree to which horizontal undulations in the inversion height are resolved. After 2 h, typical standard deviation of the inversion height (Figure b) is 4–5 and 5–7 m for 35 × 5 and 250 × 20 m, respectively. For 35 × 5, this is one vertical grid level, but for 250 × 20 this is only 25–30% of a vertical grid level.Given this distinction, we hypothesize that the enhanced inversion thickening is due to horizontal advection acting on horizontal undulations of the inversion height, especially on the 35 × 5 grid. The undulations create strong horizontal gradients of heat and moisture which can then be mixed by numerical diffusion. By this reasoning, the cross‐grid flow in the 35 × 5 case has a much stronger inversion smearing effect because it resolves the inversion undulations much more effectively than the 250 × 20 case. We believe this effect explains most of the sensitivity of the 35 × 5 runs to cross‐grid flow.Entrainment EfficiencyAnother key difference between these resolutions is the extent to which entrainment responds to resolved turbulence. One approach to quantifying the relative strength of stratocumulus cloud‐top entrainment to turbulence is by computing cloud‐top entrainment efficiency Ae (e.g., Stevens, ). Here we will utilize the following definition of entrainment efficiency,1Ae=weΔbzi/w′2¯32where we is cloud‐top entrainment rate, Δb is the jump in buoyancy across the inversion, calculated using the method of Yamaguchi and Randall (), zi is the inversion depth, and w′2¯ is the mean boundary layer velocity variance averaged between the surface and zi. A commonly used alternative version of Ae uses MBL integrated buoyancy flux in the denominator in place of w′2¯; this leads to similar conclusions as the formula used here.In Figure , we show w′2¯ and Ae for the DYCOMS cases. To reduce noise, we compute Ae using 30 min means. The coarser 250 × 20 runs have smaller w′2¯ and larger Ae, showing an undesirably strong sensitivity of turbulence and entrainment efficiency to grid resolution. They also show the largest impact of cross‐grid flow. In DYCOMS, Ae is fairly stable after startup for all 35 × 5 configurations and is not significantly affected by cross‐grid flow. In contrast SG 250×20 has about double the entrainment efficiency of MG 250 × 20. Despite stronger w′2¯ and a slightly broader inversion, the SG 250 × 20 simulation entrains substantially less (see Figure c) than the MG 250 × 20 simulation. We explore the causes of this below.Time series of (a) mean boundary layer w′2¯ in m2 s−2 and (b) entrainment efficiency.To better explore impacts of changing horizontal and vertical resolution independently, we also simulate the DYCOMS case using several additional grid sizes: 35 m × 20 m, 100 m × 20 m, 500 m × 20 m, and 100 m × 5 m. The cloud fraction and entrainment efficiency Ae averaged over DYCOMS hour 2.0 to hour 6.0 are plotted against LWP for all the DYCOMS cases (Figure ). As in Figure , moving‐grid cases are marked with a triangle and stationary cases with a square. As noted in many studies, both vertical and horizontal resolution (e.g., Cheng et al., , Parishani et al., ) and grid aspect ratio (Pedersen et al., ) have a large impact on cloud water and cloud fraction. Coarsening the horizontal resolution strongly increases LWP and cloud fraction, while coarsening the vertical resolution causes a large reduction in LWP and cloud fraction. For all of the 20 m vertical resolution experiments, cross‐grid flow induces a large increase in cloud fraction and LWP which is associated with a rough halving of entrainment efficiency (Figure b). In contrast, for the 5 m vertical resolution cases, cross‐grid flow induces small decreases in cloud fraction and LWP and has a minimal effect on entrainment efficiency.Mean hour 2.0–6.0 of (a) cloud fraction versus LWP (g m−2) and (b) entrainment efficiency vs LWP. Squares represent stationary grids, triangles represent moving grids. Solid and hollow symbols represent 5 or 20 m vertical grid resolution, respectively.Eddy Filtering and EntrainmentWe now focus on the filtering effect that cross‐grid flow has on turbulence and the substantial impact this has on cloud‐top entrainment, entrainment efficiency, and MBL evolution. Figure shows snapshots of the horizontal distribution of LWP at hour 2 of the DYCOMS experiments, with SG runs on the left and MG runs on the right. For both 35 × 5 and 250 × 20 resolutions, the large cross‐grid flow in the SG simulations appears to eliminate features at the scale of the horizontal grid resolution. This filtering emerges very rapidly in the simulations and is not only confined to cloud features. Figures a–d show horizontal power spectra in the x‐direction of vertical velocity vs height for the same simulations also at hour 2. The cross‐grid flow in the SG simulations has the effect of reducing power at the highest wave numbers for both 35 × 5 and 250 × 20 resolutions throughout the boundary layer. The horizontal power spectra of the horizontal velocity components (not shown) have similar behavior. For the 250 × 20 case, it appears that the smallest eddies induce much of the MBL‐top entrainment. The numerical damping of these eddies by cross‐grid flow greatly reduces entrainment and entrainment efficiency.LWP in kg m−2 at hour 2.0.X‐axis power spectra of vertical velocity versus height at hour 2.0 for (a) SG 35 × 5, (b) MG 35 × 5, (c) SG 250 × 20, and (d) MG 250 × 20 DYCOMS simulations. Spectra for translating‐grid runs with hyperdiffusion applied are also plotted: (e) 35 × 5 and (f) 250 × 20.To test this idea, we can artificially filter out the finest‐scale eddies by applying a hyperdiffusive term to the momentum equations of the form2du→dt=−k∇4u→where k is a constant and the ∇4 operator only applies in the x and y directions. For the 35 × 5 and 250 × 20 cases we choose k to be 1.5 × 104 and 1.0 × 107m4 s−1, respectively. These values of k would lead to 2Δx oscillations in u to be damped on a timescale of 1 and 4 s, respectively. This filter is applied to all four of the DYCOMS cases and has the intended effect of filtering out some of the smallest eddies as seen in hour 2 w power spectra of the MG runs (Figures e and f). Since SAM uses a centered difference scheme for momentum advection, the inclusion of hyperdiffusion for velocity will have broadly similar consequences as the use of dissipative numerics for momentum advection would. The latter choice is advocated by Pressel et al. (), whose highest fidelity simulations of this DYCOMS case use dissipative numerics for momentum.The effects of hyperdiffusion on mean LWP and entrainment efficiency are shown in Figure . For all runs hyperdiffusion increases LWP by reducing cloud‐top entrainment. However the LWP impact is largest for the MG 250 × 20 case (empty triangles), and comparatively small for the SG 250 × 20 case (empty squares) where cross‐grid flow has already eliminated the smallest eddies. Similarly, hyperdiffusion reduces entrainment efficiency in all cases, but has a larger fractional impact in the MG cases, where small eddies are present. This experiment supports the notion that numerical damping of the small eddies in the 250 × 20 case is the main cause of the cross‐grid flow impacts.Hour 2.0–6.0 Average of entrainment efficiency Ae and LWP (g m−2) for 250 × 20 grid (hollow symbols) and a 35 × 5 grid (filled symbols). Control runs in blue and runs with hyperdiffusion in gold. SG and MG simulations are represented by squares and triangles, respectively.SAM's subgrid‐scale mixing scheme will also somewhat preferentially reduce small‐scale resolved turbulence. However, for DYCOMS the typical subgrid TKE is 30–100 times smaller than the resolved TKE, so we do not expect this has much impact on the cross‐grid flow sensitivity. SAM uses Δz for the subgrid length scale, not (Δx Δy Δz)1/3 as some LES models do. Given the grid aspect ratios (7 and 12.5) of these simulations, other models using the latter formulation would have subgrid length scales 4 or 5 times larger. In some cases, these models could have reduced sensitivity to cross‐grid flow.BOMEXIn the BOMEX trade‐cumulus case, we expect to see different sensitivities of cloud to cross‐grid flow because of the differences between trade‐cumulus boundary layers and stratocumulus‐topped boundary layers. In trade‐cumulus MBL's, cloud fraction is much smaller compared to stratocumulus MBLs. Figure shows the 6 h evolution of the four BOMEX experiments with fine and coarse grids and with stationary and translating grids. After a 90 min spin‐up period the cloud fraction reaches relatively steady values, while domain mean LWP still varies substantially with time.(a) Mean liquid water path (g m−2) and (b) total cloud fraction for four BOMEX simulations.For both 250 × 20 and 35 × 5 runs, the time‐mean domain‐mean LWP is increased significantly by cross‐grid flow. Cloud fraction due to cross‐grid flow is increased in the after‐spin‐up mean by about 0.01 for the 250 × 20 case, while it is decreased slightly in the 35 × 5 case. The overall increase of cloud liquid water with cross‐grid flow may be related to a substantial change in updraft and cloud structure. For the BOMEX SG cases, the cross‐grid flow is 8 m s−1 aligned with the x direction. In these cases, the cumulus cloud shapes are also elongated in the x direction. This is seen in Figure in horizontal cross‐sections at hour 2.0, at a height of 700 m, about 100 m above the base of the cumulus cloud layer in all simulations. In the MG 250 × 20 case (Figure d), the cloud population is dominated with clouds of one or a few grid points, while the SG 250 × 20 clouds (Figure c) are typically several grid points wide in the x direction, but mostly one grid point wide, in the y direction. For the 35 × 5 cases, where the cloud updrafts are represented by many more grid columns, some elongation in the x direction is seen in the SG run. The cloud shapes closely mimic the shapes of the updrafts at all heights in the cloud layer which are similarly elongated (not shown).Hour 2.0 x‐y snapshots of BOMEX cloud water (g kg−1) at 700 m for (a) SG 35 × 5, (b) MG 35 × 5, (c) SG 250 × 20, and (d) MG 250 × 20 cases. The mean cross‐grid horizontal velocity at 700 m is shown as an arrow for each case.Associated with these effects are changes in overall cloud population and cloud size. Figure shows vertical mean profiles averaged over hours 2–6 of cloud water, cloud fraction, and cloud number and diameter of clouds. Cloud number and size statistics are computed by using a cloud water concentration threshold of 0.01 g kg−1 and aggregating all directly adjacent cloudy grid points into cloud objects at each vertical level. The equivalent diameter (D) of the cloud objects is computed using D = 2(A/π)1/2 where A is the horizontal area of each cloud element. The shape of mean cloud fraction profiles is generally similar between all of the runs, but the shape of the cloud water shows substantial sensitivity to cross‐grid flow. The MG runs have a gentle decline of cloud water with height in the cloud layer while the SG runs have a secondary peak in cloud water at 1,350–1,400 m. The population and diameter statistics also show strong cross‐grid flow effects at all levels in the cloud layer for both 250 × 20 and 35 × 5 resolutions. The cloud number counts have a similar shape to the cloud fraction, but at both resolutions the cloud‐number in the SG runs is half that in the MG runs. The mean equivalent diameter of cloud elements (shown only where cloud fraction > 0.0005) is quite sensitive to grid resolution, about twice as large for a 35 × 5 grid as for a 250 × 20 grid, and the 35 × 5 grid has more of an increase of mean cloud‐diameter with height in the upper cloud layer. However, the cross‐grid flow sensitivity of D is consistent between resolutions; simulations with cross‐grid flow have 40–50% larger diameter of cloud elements.Horizontal mean, time mean hour 2.0–6.0 profiles of (a) cloud water (g kg−1), (b) cloud fraction, (c) number of cloud elements, and (d) mean equivalent diameter of cloud elements for four BOMEX simulations.At both resolutions and at all levels in the cloud, the MG simulations are able to maintain a larger population of small clouds. The SG simulations have fewer but larger clouds. The larger size of these clouds may insulate them from lateral entrainment and allow them to maintain larger liquid water as they ascend. The differences in the cloud structure in these simulations can be traced again to the turbulence impacts of cross‐grid flow. Examination of the turbulence spectra (not shown) indicates that the small eddies are damped by cross‐grid flow as was the case in DYCOMS, but in this case the filtering happens only along the x axis along the cross‐grid flow direction. This causes the elongated updrafts in the x direction and larger cloud shapes. The reduction of small‐scale turbulence also inhibits the formation of small clouds at the base of the cumulus cloud layer.Sc‐to‐Cu TransitionWe finally turn to simulations of the GCSS Sc‐to‐Cu transition case, which include microphysics, full radiation physics, and a steadily increasing SST over a multiday timescale. For this case, a smaller translation velocity is needed (U = −2.0 m s−1, V = −4.0 m s−1) to approximately match the PBL‐mean wind velocity. For these experiments, the MPDATA advection scheme was used, though preliminary experiments with other schemes show similar results. An adaptive time step was used for computational expediency.The LWP, cloud fraction, inversion height, and cloud base height for the SG and MG runs are shown in Figure . The Sc‐Cu simulations include the diurnal cycle which induces a large variation in boundary layer turbulence and LWP. Solar heating in the stratocumulus top of the MBL reduces LWP during the daytime and, for 250 × 20 cases, reduces cloud fraction. Superimposed on this diurnal forcing is steady boundary layer deepening due to the increasing SST. As the boundary layer deepens, it decouples and a cumulus layer develops below the stratocumulus cloud‐top (e.g., Bretherton & Wyant, ).Domain mean (a) liquid water path (g m−2), (b) cloud fraction, and (c) inversion height and domain minimum cloud‐base height (m) for four Sc‐Cu transition simulations. Nighttime intervals are shaded.Cross‐grid flow has a big effect on LWP at 250 × 20 resolution. The stationary SG 250 × 20 case has larger LWP than MG 250 × 20 through most of the simulation, about 60% higher averaged over the simulation. The daytime gaps in cloud coverage are also reduced in the SG 250 × 20 case. This behavior is consistent with both DYCOMS and BOMEX 250 × 20 results presented above, where the scale‐filtering effects of cross‐grid flow tend to increase LWP and cloud coverage.For the 35 × 5 cases, cloud fraction is nearly 100% for the entire simulation period despite the transition from a single cloud layer to a cumulus‐under‐stratocumulus regime and a decline in LWP. For the first 40 h, cross‐grid flow appears to only slightly reduce LWP. After that time, the cross‐grid flow appears to reduce the nighttime LWP significantly. This is consistent with the inversion‐spreading impacts of cross‐grid flow described in the 35 × 5 DYCOMS experiments, resulting in relatively more cloud‐top entrainment and reduced cloud water.Discussion and ConclusionsUsing LES experiments, we have identified substantial impacts of horizontal cross‐grid flow in three different MBL cases. These effects include changes in LWP and cloud fraction and depend on grid size. A driver of these changes is that cross‐grid flow filters out small scale turbulence, reducing horizontal variability in the cross‐grid flow direction. In the DYCOMS stratocumulus case, this change to the turbulent structure reduces cloud‐top entrainment, increasing LWP and cloud cover. A second effect of cross‐grid flow in stratocumulus is to spread the inversion over more grid levels, enhancing cloud‐top entrainment and decreasing LWP and cloud cover. At coarse resolution (250 × 20) the former effect is dominant, while at finer resolution (35 × 5), where the inversion undulations are better resolved, the latter effect is more important. The vertical grid spacing is more important than the horizontal spacing in determining sensitivity to cross‐grid flow. For DYCOMS, we also used alternative scalar advection schemes, UM5 and MPDATA. These schemes demonstrate qualitatively similar sensitivity to cross‐grid flow as the BD advection scheme. Cross‐grid flow effects were not sensitive to the choice of time step. The sensitivity of cloud properties to the amplitude of cross‐grid flow seems to diminish as the amplitude is increased above 7 m s−1 in the DYCOMS case and other tests we have tried, but this tendency should be explored more thoroughly.Similar LWP and cloud fraction sensitivities to cross‐grid flow are present in longer stratocumulus‐to‐trade‐cumulus simulations that include microphysics, more realistic longwave radiation, and a diurnal cycle of shortwave radiation.In trade‐cumulus simulations based on BOMEX, the filtering associated with cross‐grid flow affects the structure of convection at both resolutions studied. Cloud updrafts in the cumulus layer are elongated and enlarged with the cross‐grid flow, and the population of small clouds is greatly reduced. The increase in cloud size decreases the efficiency of lateral entrainment, and increases overall liquid water, primarily in the upper parts of the cloud layer.Because SAM uses a centered‐difference momentum scheme, the leading error terms in the momentum advection equations will tend to be dispersive rather than dissipative. Models with an odd‐order momentum scheme may instead promote dissipation of turbulent energy at grid scales. This smoothing by an odd‐order scheme, which is proportional to the local velocity across the grid, could result in a qualitatively different sensitivity to cross‐grid flow.For scalar advection, it is illuminating to theoretically estimate the numerical diffusion due to cross‐grid flow. In SAM, the scalar advection is treated using forward‐in‐time schemes that combine the time and space discretization. Each of these uses a flux‐corrected‐transport scheme to preserve monotonicity (either everywhere for UM5 or MPDATA, or selectively for Blossey‐Durran). When flux correction is applied, the numerical flux is relaxed back towards a first‐order upwind approximation, so that it seems appropriate here to analyze the effect of cross‐grid flow on a first‐order upwind scheme because it will be the effective scheme operating on extrema as they are advected horizontally across the grid when flux correction is applied. In this case, the numerical scheme can be represented by the modified equation (e.g., Smolarkeiwicz & Szmelter, ): 3∂φ∂t+∂u′φ∂x≈∂∂x(νeff∂φ∂x)where u′ = u – U, and U is the grid‐translation velocity. The numerical diffusivity, νeff, can be written as4νeff=12u′Δx−u′2Δt=12u′Δx(1−CFL)where Δt is the time step and CFL is the horizontal courant number |u′|Δt/Δx. For our simulations, this CFL number is almost always smaller than 0.1. Therefore, the numerical diffusivity is approximately proportional to the size of the cross‐grid flow, |u′|. As monotonic flux correction will operate at extrema of the scalar field when new extrema might be created, this numerical diffusivity will be most prominent in simulations with strong cross‐grid flow.It is not always practical to prevent substantial cross‐grid flow in simulations, for example, due to large mean vertical wind‐shear, due to mesoscale circulations within a model domain, when simulating with fixed topographical features such as in a regional model, or when performing global CRM experiments. In these cases, care must be taken to understand the cross‐grid flow impacts on the mean cloud state. These impacts are important for stratocumulus clouds at typical LES resolutions, and are especially important for stratocumulus and small cumulus clouds at gray‐zone resolutions, where filtering effects of cross‐grid flow are more acute.AcknowledgementsFunding for this study was provided by the U. S. Department of Energy under DE‐SC0012451. Thanks to Mike Pritchard and Hossein Parishani whose UPCAM experiments motivated this work and for many helpful discussions. We also thank Marat Khairoutdinov for the use of SAM and for his helpful comments. We would also like to thank the two anonymous reviewers, whose comments helped improve this manuscript. The simulation data used to produce the figures for this study are archived at the University of Washington Research Works, http://hdl.handle.net/1773/40990.ReferencesArakawa, A., & Wu, C.‐M. (2013). A unified representation of deep moist convection in numerical modeling of the atmosphere. Part I. Journal of the Atmospheric Sciences, 70, 1977–1991. https://doi.org/10.1175/JAS-D-12-0330.1Blossey, P. N., & Durran, D. R. (2008). Selective monotonicity preservation in scalar advection. Journal of Computational Physics, 127, 5160–5183. https://doi.org/10.1016/j.jcp.2008.01.043Bretherton, C. S., & Wyant, M. C. (1997). 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