1. Introduction
Cloud models are essential tools for the study of cloud processes. Cloud models can be split into two major components: the dynamical core and the physics. The dynamical core includes the numerical representation of the governing equations for momentum and scalars, including time-stepping and advection. Model physics represents parameterized physical processes such as microphysics, radiation, and diffusion.
Since diffusion is a physical process, a subgrid-scale physical parameterization should be used to represent diffusion in a model. In reality, however, numerical diffusion exists in virtually all cloud models, as a result of the numerical schemes used for advection. Numerical diffusion can cause numerical entrainment, which dilutes and evaporates clouds.
One way to reduce numerical diffusion is to use a small grid spacing. This method works but is expensive. For instance, Khairoutdinov et al. (2009) simulated deep convective clouds with a 100-m horizontal grid spacing for a 204.8-km domain width with 256 layers. This simulation was 256 times as expensive as a run with a 1.6-km horizontal grid spacing, if the other parameters are unchanged.
For cloud models, both momentum and scalars must be advected. The momentum advection scheme is often formulated so as to satisfy momentum and kinetic energy conservation. The scalar advection scheme is often designed to satisfy scalar conservation, shape preservation, and monotonicity.
The second- and third-order advection schemes have been widely used for cloud models [e.g., the multidimensional positive definite advection transport algorithm (MPDATA; Smolarkiewicz and Grabowski 1990), the piecewise parabolic method (PPM; Carpenter et al. 1990), the Uniformly Third-Order Polynomial Interpolation Algorithm (UTOPIA; Leonard et al. 1993), and the Monotone Upstream-centered Schemes for Conservation Laws (MUSCL; van Leer 1997). Recently higher-than-third-order schemes have been developed and tested (e.g., Skamarock 2006; Blossey and Durran 2008; Wang et al. 2009)].
In this study, we focus on the effects of numerical accuracy on scalar advection in the context of cloud simulations. We have developed a monotonic multidimensional higher-order advection scheme by utilizing the methods created by Leonard (1991) and Leonard et al. (1996). For this new scheme, only odd-order schemes were used; odd-order schemes are space-uncentered schemes, which minimize numerical dispersion. Leonard (1991) showed that even-order schemes gives highly oscillatory solution, and they are much more sensitive to Courant number than odd-order schemes. We compared new third-, fifth-, and seventh-order schemes with the second-order scheme originally used in a cloud model by simulating idealized scalar fields, stratocumulus clouds [large-eddy simulation (LES)], and deep convective clouds [in a cloud-system resolving model (CSRM)]. The results are also compared with the previously mentioned deep cumulus LES of Khairoutdinov et al. (2009).
In the next section, the cloud model is briefly described. A new higher-order scalar advection scheme is presented, and tested with idealized scalar fields in section 3. The results of a stratocumulus LES are studied in section 4, and those of a deep cumulus CSRM are studied in section 5. This paper closes with a summary and conclusions in section 6.
2. Cloud model
We use the System for Atmospheric Modeling (SAM; Khairoutdinov and Randall 2003), which has been widely used (e.g., Caldwell and Bretherton 2009; Cheng and Xu 2009; Khairoutdinov et al. 2009; Yamaguchi and Randall 2008), and has participated in several Global Energy and Water Cycle Experiment (GEWEX) Cloud System Study (GCSS) LES intercomparison studies (Moeng et al. 1996; Siebesma et al. 2003; Stevens et al. 2005; Ackerman et al. 2009).
SAM is a nonhydrostatic model based on the anelastic equations. It predicts the three velocity components, liquid water static energy, total nonprecipitating water mixing ratio (vapor, cloud water, and cloud ice), total precipitating water mixing ratio (rain, snow, and graupel), and subgrid-scale turbulence kinetic energy (TKE). It uses finite differences on a staggered (Arakawa C) grid. It employs a three-phase bulk microphysics parameterization. SAM utilizes the Message Passing Interface (MPI) with horizontal domain decomposition.
The standard scalar advection scheme of SAM is based on MPDATA (Smolarkiewicz and Grabowski 1990) with IORD = 2. IORD is the number of iteration to gain complementary effect of the antidiffusive flux. SAM–MPDATA is a second-order accurate monotonic scheme. Smolarkiewicz and Grabowski (1990) recommend IORD = 4 to derive the maximum benefit from the antidiffusive flux.
3. Monotonic ULTIMATE–MACHO scheme
Our higher-order scalar advection scheme is a combination of the Universal Limiter for Transient Interpolation Modeling of the Advective Transport Equations (ULTIMATE; Leonard 1991) and Multidimensional Advective-Conservative Hybrid Operator (MACHO; Leonard et al. 1996). ULTIMATE was designed as a one-dimensional scheme with an arbitrary order of accuracy based on the Lagrange polynomial function. It can be made monotonic by using the universal limiter developed by Leonard (1991). The multidimensional version of the third-order ULTIMATE is called UTOPIA (Leonard et al. 1993). MACHO is a multidimensional advection scheme based on direct use of a one-dimensional scheme. It is attractive for its simple algorithm, conservation, and shape preservation. The combined scheme, ULTIMATE–MACHO, has an advantage of the expandability for any order. MACHO is not strictly monotonic with a one-dimensional universal limiter; Leonard et al. (1996) called it “essentially monotonic.” The result is monotonic for a constant velocity field, but weakly nonmonotonic for a varying velocity as in turbulence. We added a monotonic property to the ULTIMATE–MACHO scheme, as described below.
a. ULTIMATE
An example of the fifth-order ULTIMATE is given in Fig. 1. A resolution sensitivity study for ULTIMATE is presented in appendix A. The ULTIMATE formulas for nonuniform grids are derived and tested in appendix B.
Advection test of the fifth-order ULTIMATE. The gray line is the initial profile based on Leonard et al. (1995), which is a rectangular box, sine-squared, semi-ellipse, triangle, and Gaussian shapes from left to right. The black line shows the result after one rotation with a constant Courant number of 0.6, which requires 250 steps.
Citation: Monthly Weather Review 139, 10; 10.1175/MWR-D-10-05044.1
b. MACHO
c. Monotonicity
In an attempt to achieve monotonicity preservation, we introduced a one-dimensional universal limiter applied at steps 1, 3, and 5 of (4), a three-dimensional universal limiter developed by Thuburn (1996), and a three-dimensional flux-corrected transport (FCT; Zalesak 1979).
Our test consists of two cases: a rotating split cylinder, distortion of a sphere in turbulence. The rotating split cylinder is based on Zalesak (1979). For the distortion of a sphere in turbulence, the initially sphere-shaped tracer is advected with a background turbulence developed with a stratocumulus LES of the Second Dynamics and Chemistry of the Marine Stratocumulus field study (DYCOMS-II; Stevens et al. 2005).
In agreement with the study of Leonard et al. (1996), we obtained essentially monotonic results with a one-dimensional universal limiter; monotonic for the rotating split cylinder and weakly nonmonotonic for the distortion of a sphere in turbulence.
We are not satisfied with the results obtained with the three-dimensional universal limiter for the rotating split cylinder. The scheme was monotonic, but exhibited significant numerical diffusion.
The application of three-dimensional FCT resulted in a monotonic scheme for the two tests. The numerical diffusion was minimal. For this reason, we decided to use the three-dimensional FCT as a standard limiter for the ULTIMATE–MACHO scheme.
d. Two-dimensional rotating split cylinder
The SAM–MPDATA, third-, fifth-, and seventh-order ULTIMATE–MACHO schemes were tested. We call them SM-2, UM-3, UM-5, and UM-7, respectively. The details of each test and its results with the three-dimensional FCT are discussed below.
The initial shape and the shapes after one rotation for each of the four advection schemes are presented in Fig. 2. Only the 402 grid-box region near the cylinder is shown in the figure. Although all four schemes are monotonic, SM-2 loses the symmetry of the slot, while the UM schemes preserve it. As expected, the higher-order schemes maintain the sharp gradients better than the lower-order schemes.
Result of the rotating split cylinder test after one rotation for four advection schemes, and (bottom) the initial condition. Exact 1 and 2 values are colored black.
Citation: Monthly Weather Review 139, 10; 10.1175/MWR-D-10-05044.1
The time series of the normalized total variance, which is the total variance divided by the initial total variance, is shown in Fig. 3. Switching from SM-2 to UM-3 does not improve the results much. There is, however, a significant improvement between UM-3 and UM-5. The large dissipation for earlier time steps for all schemes is due to deformation of the edge of the split cylinder. We define the numerical dissipation rate as the difference of the normalized total variance per second. The mean dissipation rate for the last 100 steps is 10.4 × 10−5 s−1 for SM-2, 8.9 × 10−5 s−1 for UM-3, 5.2 × 10−5 s−1 for UM-5, and 4.6 × 10−5 s−1 for UM-7. UM-5 is about 2 times less dissipative than SM-2, and UM-7 is about 2.3 times less dissipative than SM-2. The difference between UM-5 and UM-7 is small.
Time series of the total variance normalized with the initial total variance.
Citation: Monthly Weather Review 139, 10; 10.1175/MWR-D-10-05044.1
e. Distortion of a sphere in turbulence
To generate a background turbulence, we simulated DYCOMS-II (Stevens et al. 2005). SAM was configured following Stevens et al. (2005), and ran for 2 h, which is long enough to generate turbulence. The horizontal domain width was 3.36 km wide, and the domain depth was 1.6 km. The horizontal grid spacing was 35 m, and the vertical grid spacing was 5 m. The time step was 0.5 s.
When SAM was restarted, a sphere-shaped tracer was introduced with its center at (1697.5, 1697.5, and 747.5 m), and with a radius of 400 m. Tracer values inside the sphere were set to 1, and those outside were set to 0. An inversion layer existed at approximately the 850-m level, so that approximately the upper one-third of the sphere was in the free atmosphere, which was not turbulent. The only process acting on the tracer was advection, and the new advection schemes were applied only to the tracer advection. That is, the PBL turbulence was the same for all four cases (SM-2, UM-3, UM-5, and UM-7). For this test, we ran an additional 600 time steps with a 0.5-s time step.
Figure 4 presents the initial shape and the shape after 600 steps (i.e., 5 min) is presented for four advection schemes. The 0.1, 0.5, and 0.9 isosurfaces are shown. The gaps between isosurfaces are tighter with the higher-order schemes. The vertical cross section of the tracer shows that the difference between UM-5 and UM-7 is very small while SM-2 and UM-3 significantly dissipate the tracer value of 0.9 in the boundary layer, compared with UM-5 and UM-7.
Distortion of a sphere in a turbulence test after 600 steps, with four advection schemes. (bottom) The initial condition and a vertical cross section of tracer with 0.5 and 0.9 contours are also shown. The isosurface with the lightest gray represents 0.1, mid-dark gray is 0.5, and darkest gray is 0.9. Total variance is normalized with the initial value.
Citation: Monthly Weather Review 139, 10; 10.1175/MWR-D-10-05044.1
Based on the normalized total variance shown in Fig. 5, the mean numerical dissipation rates for the last 100 steps are 5.7 × 10−5 s−1 for SM-2, 5.0 × 10−5 s−1 for UM-3, 4.3 × 10−4 s−1 for UM-5, and 4.1 × 10−4 s−1 for UM-7. By this measure, SM-2 is approximately 1.3 times as dissipative as UM-5 and UM-7.
As in Fig. 3, but for the distortion of a sphere in turbulence.
Citation: Monthly Weather Review 139, 10; 10.1175/MWR-D-10-05044.1
f. Numerical cost
With the same grid spacing and time step, ULTIMATE–MACHO is relatively more expensive than SAM–MPDATA. Our timing tests show that UM-3 is 1.21 times, UM-5 is 1.26 times, and UM-7 is 1.31 times as expensive as SM-2.
4. Marine stratocumulus clouds
Stratocumulus clouds observed during DYCOMS-II (Stevens et al. 2005) were simulated with the four scalar advection schemes (i.e., SM-2, UM-3, UM-5, and UM-7). SAM was configured following Stevens et al. (2005) as described in section 3e, and ran for 4 h. Liquid water static energy, total nonprecipitating water mixing ratio, and subgrid-scale TKE were advected with these schemes. This was a nonprecipitating stratocumulus case, so total precipitating water mixing ratio was not computed. Each simulation covered 4 h of observed time.
The horizontal mean vertical profiles of the liquid water potential temperature, total water mixing ratio, cloud water mixing ratio, and cloud fraction are shown in Fig. 6. These profiles were averaged over the last hour. There is a distinct difference between SM-2 and the UMs. The UMs maintain a well-mixed profile, while SM-2 produces slightly less mixed profile in the cloud layer. It is possible that the difference comes from the larger numerical diffusion of SM-2. This results in a larger maximum cloud water and cloud amount with the UMs. The difference between UM-5 and UM-7 is very small. Compared with the observations shown in Figs. 3 and 4 of Stevens et al. (2005), UM-5 and UM-7 are closest: 289 K for the liquid water potential temperature, 9 g kg−1 for total water mixing ratio in the mixed layer, and a maximum cloud water mixing ratio greater than 0.3 g kg−1 of and around 850 and 625 m for the cloud top and base heights, respectively.
The 1-h-averaged horizontal mean vertical profile of (top left) liquid-water potential temperature, (top right) total water mixing ratio, (bottom left) cloud water mixing ratio, and (bottom right) cloud fraction. The average was taken for the last hour.
Citation: Monthly Weather Review 139, 10; 10.1175/MWR-D-10-05044.1
The time series for the cloud water path, vertically integrated TKE, and diagnosed entrainment rate are presented in Fig. 7a. The entrainment rate was diagnosed using the method described in appendix C, which is an improved version of the method developed by Yamaguchi and Randall (2008). Thirty-minute running mean profiles were used as input to obtain a smooth time series for the entrainment rate. The TKE and cloud water path are considerably larger with the higher-order schemes. The entrainment rate tends to be smaller with the higher-order schemes, even though the differences in the entrainment rate among the advection schemes are small. The vertical profiles of the buoyant production of TKE, variance, and third moment of vertical velocity in Fig. 7b also show a more turbulent mixed layer with the higher-order schemes.
(a) Time series of the (left to right) liquid water path, vertically integrated TKE, and entrainment rate. (b) Hourly averaged vertical profiles of (left to right) resolved-scale buoyant production of TKE, the variance of the vertical velocity, and its third moment.
Citation: Monthly Weather Review 139, 10; 10.1175/MWR-D-10-05044.1
The simulated clouds are different because the advection schemes are different, but the entrainment rates are similar. The model may adjust to produce similar entrainment rates, as discussed by Bretherton and Wyant (1997), Zhu et al. (2005), and Uchida et al. (2010). Bretherton and Wyant (1997) and Zhu et al. (2005) find that entrainment rates for mixed layer models are similar while their liquid water paths are strongly different. Zhu et al. (2005) discuss that as long as the cloud is solid, changed entrainment efficiency is reflected in changed liquid water path rather than changed entrainment velocity. Uchida et al. (2010) obtain similar entrainment rates for their LESs with different droplet concentration (30, 50, and 150 cm−3), which produces different turbulence and liquid water path.
The above results are similar to those obtained in LESs with greater vertical resolution. Increasing the vertical resolution results in a cooler, moister mixed layer with more cloud water and a thicker cloud (Stevens et al. 2005; M. Khairoutdinov 2010, personal communication). The entrainment rate becomes smaller. The turbulence becomes stronger and more downdraft dominated; the variance of vertical velocity increases and the skewness of the vertical velocity becomes negative throughout the mixed layer.
5. Deep convective clouds
Deep convective clouds were simulated based on the idealized Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment (GATE) case of Khairoutdinov et al. (2009). Simulations were performed with SM-2, UM-3, and UM-5, and compared with the Giga-LES of Khairoutdinov et al. (2009). The Giga-LES used a 100-m horizontal grid spacing. SAM was configured as the Giga-LES, but for the horizontal grid spacing, which was 1.6 km, 16 times larger. The nonuniform-grid version of ULTIMATE was used for the vertical direction, since the case used a stretched vertical grid spacing. The horizontal domain width is 204.8 km, and the domain depth is 30 km. The vertical grid spacing is 50 m in the lowest 1.2 km, stretched to 100 m between 1.2 and 5 km, 100 m between 5 and 18 km, stretched to 300 m between 18 and 27 km, and 300 m above 27 km. The number of levels is 256. The duration is one simulated day, with a 2-s time step due to vertical Courant–Friedrichs–Lewy (CFL) condition. The new advection scheme was not applied to precipitation fallout, which can be thought of as vertical advection by the terminal velocity. That issue is left for a future study. The precipitation fallout was computed as SAM normally computes it, using SAM–MPDATA with time splitting for stability.
The evolutions of the horizontal-mean cloud water and ice for the three advection schemes and the Giga-LES are shown in Fig. 8. The results generally show the trimodal vertical profile observed by Johnson et al. (1999), which consists of shallow cumulus rising to the trade inversion, congestus rising to the freezing level, and cumulonimbus rising to the tropopause. The cloud water–ice amount at the time of formation of the first deep convective cloud, around 7 h for the UMs, is smaller than other two cases. The UMs do not have cloud water–ice greater than 0.032 g kg−1 except during the third hour in the PBL. UM-5 has, however, more cloud water–ice than UM-3 between 4 and 7 h. In this way, it resembles the result obtained with refined horizontal resolution. This can be seen in Fig. 10 of Khairoutdinov et al. (2009), which shows that smaller horizontal grid spacing results in larger cloud water–ice amounts between 5 and 7 h, for the horizontal grid spacings of 100, 200, 400, and 800 m. For the last 12 h, SM-2 is the wettest, especially in the first peak of the trimodal profile between 8 and 10 km, and the third peak in the PBL. Compared with the Giga-LES, the UMs have more cloud water–ice amount in the second peak of the trimodal profile around the 5-km level. UM-3 is drier around the 6-km level than the Giga-LES. Around the first peak of the trimodal profile between 8 and 10 km, UM-5 has cloud water–ice amount most similar to those of the Giga-LES.
Evolution of horizontally averaged cloud water and ice for the tested advection schemes and (bottom right) Giga-LES [cf. Fig. 10 of Khairoutdinov et al. (2009)].
Citation: Monthly Weather Review 139, 10; 10.1175/MWR-D-10-05044.1
The horizontal mean profiles averaged over the last 12 h for all cases are presented in Fig. 9. The variables shown in Figs. 11 and 12 of Khairoutdinov et al. (2009) were plotted for comparison. The cloud fractions below 10 km are very much alike for all of the UMs and Giga-LES. The UMs produce less cloud between 10 and 12 km than the Giga-LES, and more cloud between 12 and 16 km. The trimodal structure of cloud water–ice is in much better agreement between the UMs and Giga-LES than between SM-2 and the Giga-LES. As noted for the evolution of cloud water–ice, the second peak of the trimodal profile is overestimated by the UMs. There is no large difference for the relative humidity with respect to liquid water saturation mixing ratio among the cases, but UM-5 best matches the Giga-LES profile between 8 and 10 km. The precipitation rate and nonprecipitating water flux (vapor and cloud water–ice) are generally similar among the cases. However, UM-3 is a little better than UM-5 below 4 km, and UM-5 is a little better than UM-3 above 4 km, when both are compared with the Giga-LES. The zonal momentum flux for SM-2 and Giga-LES exhibits very good agreement, while UMs overestimate it compared with Giga-LES. This agreement between SM-2 and Giga-LES is misleading because the cloud core variables for these cases in this figure are clearly different from one another. On the other hand, UM-5 has similar magnitude to the Giga-LES at all levels. UM-3 overestimates the updraft core velocity between 5 and 10 km. The excess of total water in updraft cores is computed as the horizontal mean total water mixing ratio minus the core-averaged total water mixing ratio. Compared with Fig. 12 of Khairoutdinov et al. (2009), the improvement due to UM-5 is roughly equivalent to the simulation with a 200–400-m horizontal grid spacing with SM-2. UM-5 produces results similar to those obtained with at least 4 times smaller horizontal grid spacing with SM-2. The computational cost is reduced by more than a factor of 10.
Horizontal mean vertical profiles of parameters shown in Figs. 11 and 12 of Khairoutdinov et al. (2009) for the tested advection schemes and Giga-LES. Each profile is time averaged over the last 12 h. The updraft cores were defined as cloudy points with vertical velocity exceeding 1 m s−1.
Citation: Monthly Weather Review 139, 10; 10.1175/MWR-D-10-05044.1
6. Summary and conclusions
A multidimensional higher-order scalar advection scheme was developed and tested in cloud simulations. The ULTIMATE–MACHO scheme was made monotonic using three-dimensional FCT. With idealized scalar advection, the fifth- and seventh-order versions of ULTIMATE–MACHO produce significantly less numerical diffusion compared to the original second-order scheme and the third-order ULTIMATE–MACHO scheme.
The impact of the higher-order schemes in a simulation of stratocumulus clouds was studied using LESs. The fifth- and seventh-order schemes better maintained mixed layer profiles and produced a thicker cloud layer than the original second-order scheme. With the higher-order schemes, the entrainment rate becomes smaller and the mixed layer is more turbulent and energetic. The results are similar to those obtained with greater vertical resolution. The results for fifth- and seventh-order ULTIMATE–MACHO were comparable.
A sensitivity study was also carried to see the effects of the schemes in simulations of deep convective clouds. The results with higher-order schemes were similar to those obtained with the low-order scheme and increased horizontal resolution. The fifth-order ULTIMATE–MACHO with a 1.6-km horizontal grid spacing produced results comparable to those of a simulation with 200–400-m horizontal grid spacing using the second-order scheme; the latter is at least 10 times more expensive.
The improvement obtained with the seventh-order ULTIMATE–MACHO was minor. We recommend the fifth-order scalar ULTIMATE–MACHO scheme as optimal among those tested here.
Acknowledgments
The authors thank Peter Blossey, Cris Bretherton, and Hiroaki Miura for helpful discussions and comments. This study has been supported by the Department of Energy (DOE) Scientific Discovery through Advanced Computing (SciDAC) Grant DE-FC02-06ER64302, the National Science Foundation (NSF) Science and Technology Center for Multi-Scale Modeling of Atmospheric Processes (CMMAP), managed by Colorado State University under Cooperative Agreement ATM-0425247, and the Physics of Stratocumulus Top (POST) project funded by NSF under Grants ATM-0735118 and ATM-0735121. Computing time was provided by the National Center for Atmospheric Research (NCAR), which is sponsored by the NSF.
APPENDIX A
Resolution Sensitivity Test for ULTIMATE
Generally, as the resolution increases, the accuracy of advection also increases. This should be expected for ULTIMATE. A steady state tracer advection test was designed and performed for one-dimensional advection.
We tested five grid spacings of 32, 16, 8, 4, and 2 m. The Courant number was set to 0.25 with a constant velocity of 2 m s−1. The fifth-order ULTIMATE was used for this test. All grid spacings were tested with and without FCT. Each case ran for five time steps.
The result for the 32-m grid spacing is presented in Fig. A1. We define the advection error as the difference of the grid value from analytic value. The grid value after five steps looks accurate (Fig. A1a); it is, however, not accurate enough in terms of the advection error (Figs. A1b,c). The advection error per second around the extrema is approximately 0.0012 s−1, which is only one order of magnitude smaller than the source value. The advection error for the FCT run is distorted but the magnitude is the same order as the no-FCT run.
The result for the 32-m grid spacing. (a) Grid value of tracer for the initial (plus signs) and that after 5 steps (gray dots) for no FCT run, and analytic solution (solid line) between 80 and 432 m. (b) Advection error for no FCT case. The advection error is defined as the analytical value minus the grid value. Black is for the first step, and lightest gray is for the fifth steps. (c) Advection error for the FCT case.
Citation: Monthly Weather Review 139, 10; 10.1175/MWR-D-10-05044.1
The results for the other grid spacings are presented in Fig. A2a. As expected, the advection error is smaller for the smaller grid spacing. If the error threshold is set at 0.001, then from Fig. A2, the grid spacing should be smaller than 4 m. Very high resolution is required, even in this simple case. The FCT forces the extreme value to be monotonic, and it ends up causing a slightly larger error. To minimize the effects of the limiting procedure, we plan to implement the selective monotonicity preservation method of Blossey and Durran (2008). The domain-mean absolute advection error per second for the first step for all runs is presented in Fig. A2b. The error decreases linearly in a log–log relationship, as the grid spacing becomes smaller; in other words, the order of accuracy increases.
(a) Advection error for different grid spacings. (b) Domain-mean absolute advection error after one step for different grid spacings. (bottom right) Black (gray) plus sign represents the run without (with) FCT.
Citation: Monthly Weather Review 139, 10; 10.1175/MWR-D-10-05044.1
APPENDIX B
ULTIMATE on a Nonuniform Grid
In this appendix, we derive the ULTIMATE formulas for nonuniform grids up to fifth-order, and compared it to its uniform-grid formula with a simple one-dimensional advection test.
The result for the fifth-order formula for uniform grid with 0.6 constant Courant number is shown in Fig. 1.
The result after one rotation for the fifth-order formula for nonuniform and uniform grids is shown in Fig. B1. The nonuniform grid formula dissipates the Gaussian shape and sharp peaks stronger than the uniform formula. The result of the nonuniform grid formula resembles the result of the third-order uniform grid formula (not shown).
The result of the one-dimensional advection test for the fifth-order formula for nonuniform (black solid line) and uniform (gray solid line) grid. The initial profile is shown as a dotted line.
Citation: Monthly Weather Review 139, 10; 10.1175/MWR-D-10-05044.1
APPENDIX C
Diagnostic Method for Determining the Entrainment Rate
The new method is based on the profile of the third moment of the liquid water static energy. The third moment for DYCOMS-II at 4 h is shown in Fig. C1a with the diagnosed zB+ and zB. The vertical profile of the third moment is negative in the upper inversion layer and positive in the lower inversion layer. The upper inversion layer is negatively skewed because only small fractions of the air are as cool as the air in the mixed layer. The lower inversion layer is weakly positively skewed because most of the air is cool. This skewed profile was also confirmed with observational data (S. Krueger 2009, personal communication). Although the minimum and maximum third moments could be used in (C4), a very small positive maximum could prevent us from locating zB. The variance has only one positive peak in the inversion layer, and the third moment is near zero where the variance is near zero. The positive peak also tends to locate around the maximum vertical gradient of sl and r. Thus, it is a good reference variable to identify zB+ and zB with a reasonable threshold. The diagnosed zB+ and zB are reasonably located in the profile of the liquid water static energy and total water mixing ratio. The diagnosed entrainment rate and the storage of the liquid water static energy shown in Fig. C1b are comparable for both methods. The new method is, however, more physically based, less arbitrary, and simpler. This method can be used with other variables such as moist static energy or virtual dry static energy, which have the same property in their second and third moments.
(a) Diagnosed zB+ and zB (dashed lines) with the vertical profile of (left to right) the first to third moment of the liquid water static energy, and total water mixing ratio. (b) Time series of entrainment rate and storage of the liquid water static energy for (dashed line) Yamaguchi and Randall (2008) and (solid line) the new method. The 30-min running mean profiles were used to obtain smooth time series.
Citation: Monthly Weather Review 139, 10; 10.1175/MWR-D-10-05044.1
REFERENCES
Ackerman, A. S., and Coauthors, 2009: Large-eddy simulations of a drizzling, stratocumulus-topped marine boundary layer. Mon. Wea. Rev., 137, 1083–1110.
Blossey, P. N., and D. R. Durran, 2008: Selective monotonicity preservation in scalar advection. J. Comput. Phys., 227, 5160–5183.
Bretherton, C. S., and M. C. Wyant, 1997: Moisture transport, lower-tropospheric stability, and decoupling of cloud-topped boundary layers. J. Atmos. Sci., 54, 148–167.
Caldwell, P., and C. S. Bretherton, 2009: Large eddy simulation of the diurnal cycle in southeast Pacific stratocumulus. J. Atmos. Sci., 66, 432–449.
Carpenter, R. L., K. K. Droegemeier, P. R. Woodward, and C. E. Hane, 1990: Application of the Piecewise Parabolic Method (PPM) to meteorological modeling. Mon. Wea. Rev., 118, 586–612.
Cheng, A., and K.-M. Xu, 2009: A PDF-based microphysics parameterization for simulation of drizzling boundary layer clouds. J. Atmos. Sci., 66, 2317–2334.
Johnson, R. H., T. M. Rickenbach, S. A. Rutledge, P. E. Ciesielski, and W. H. Schubert, 1999: Trimodal characteristics of tropical convection. J. Climate, 12, 2397–2418.
Khairoutdinov, M. F., and D. A. Randall, 2003: Cloud resolving modeling of the ARM summer 1997 IOP: Model formulation, results, uncertainties, and sensitivities. J. Atmos. Sci., 60, 607–625.
Khairoutdinov, M. F., S. K. Krueger, C.-H. Moeng, P. A. Bogenschutz, and D. A. Randall, 2009: Large-eddy simulation of maritime deep tropical convection. J. Adv. Model. Earth Syst., 1, doi:10.3894/JAMES.2009.1.15.
Leonard, B. P., 1991: The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection. Comput. Methods Appl. Mech. Eng., 88, 17–74.
Leonard, B. P., M. K. MacVean, and A. P. Lock, 1993: Positivity-preserving numerical schemes for multidimensional advection. NASA TM 106055, ICOMP-93-05, Lewis Research Center, Cleveland, OH, 62 pp.
Leonard, B. P., A. P. Lock, and M. K. MacVean, 1995: The NIRVANA scheme applied to one-dimensional advection. Int. J. Numer. Methods Heat Fluid Flow, 5, 341–377.
Leonard, B. P., A. P. Lock, and M. K. MacVean, 1996: Conservative explicit unrestricted-time-step multidimensional constancy-preserving advection schemes. Mon. Wea. Rev., 124, 2588–2606.
Lilly, D. K., 1968: Models of cloud-topped mixed layers under a strong inversion. Quart. J. Roy. Meteor. Soc., 94, 292–309.
Moeng, C. H., and Coauthors, 1996: Simulation of a stratocumulus-topped planetary boundary layer: Intercomparison among different numerical codes. Bull. Amer. Meteor. Soc., 77, 261–278.
Siebesma, A. P., and Coauthors, 2003: A large eddy simulation intercomparison study of shallow cumulus convection. J. Atmos. Sci., 60, 1201–1219.
Skamarock, W. C., 2006: Positive-definite and monotonic limiters for unrestricted time-step transport schemes. Mon. Wea. Rev., 134, 2241–2250.
Smolarkiewicz, P. K., and W. W. Grabowski, 1990: The multidimensional positive definite advection transport algorithm: Nonoscillatory option. J. Comput. Phys., 86, 355–375.
Stevens, B., and Coauthors, 2005: Evaluation of large-eddy simulations via observations of nocturnal marine stratocumulus. Mon. Wea. Rev., 133, 1443–1462.
Thuburn, J., 1996: Multidimensional flux-limited advection schemes. J. Comput. Phys., 123, 74–83.
Uchida, J., C. S. Bretherton, and P. N. Blossey, 2010: The sensitivity of stratocumulus-capped mixed layers to cloud droplet concentration: Do LES and mixed-layer models agree? Atmos. Chem. Phys., 10, 4097–4109.
van Leer, B., 1997: Towards the ultimate conservative difference scheme. J. Comput. Phys., 135, 229–248.
Wang, H., W. C. Skamarock, and G. Feingold, 2009: Evaluation of scalar advection schemes in the advanced research WRF model using large-eddy simulations of aerosol–cloud interactions. Mon. Wea. Rev., 137, 2547–2558.
Yamaguchi, T., and D. A. Randall, 2008: Large-eddy simulation of evaporatively driven entrainment in cloud-topped mixed layers. J. Atmos. Sci., 65, 1481–1504.
Zalesak, S. T., 1979: Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys., 31, 335–362.
Zhu, P., and Coauthors, 2005: Intercomparison and interpretation of single-column model simulations of a nocturnal stratocumulus-topped marine boundary layer. Mon. Wea. Rev., 133, 2741–2758.
