1. Introduction
The model gray zone refers to the range of grid spacing that is comparable to the characteristic length scale of the flow (Wyngaard 2004; Arakawa et al. 2011). As the characteristic flow becomes partially resolved and partially subgrid scale (SGS) in the gray zone, ensemble-based schemes are inappropriate to parameterize the unresolved flow because of the lack of homogeneity within the grid cell. On the other hand, turbulence closures for large-eddy simulations (LESs) are inapplicable because the most energetic flow is too poorly resolved. As such, no conventional choices are available to parameterize the SGS motions on the gray-zone grid. Therefore, the gray zone is also named terra incognita by Wyngaard (2004).
In the daytime convective boundary layer (CBL), the boundary layer depth
Grid nesting has often been used to improve the representation of critical layers or zoom in to small-scale flow features of interest. It represents a computationally efficient and economical way of allocating grid cells where they are needed (Zhong and Chow 2012). In particular, grid nesting has been adopted to increase the resolution of the surface layer for LESs, where the size of turbulent eddies is strongly suppressed by the presence of the boundary (Sullivan et al. 1996; Huq et al. 2014). For example, Sullivan et al. (1996) applied a two-way nested grid in an LES to enhance the resolution of near-wall turbulence in a strongly sheared and weakly buoyant CBL. They observed improvements of turbulence statistics and spectra within the surface layer but no appreciable difference above, to which they suggested that small-scale features (relative to the LES grid) in the surface layer do not propagate far upward into the mixed layer.
This work is different from the previous studies that utilize grid refinement as a wall model for LES. On the LES grid, the most energetic eddies are well resolved. Higher resolution in the surface layer improves the representation of small-scale turbulence but hardly affects the overall boundary layer simulation as remarked by Sullivan et al. (1996). However, on the gray-zone grid, the dominant convective eddies are only partially resolved, let alone smaller-scale motions. Therefore, a resolved surface layer qualitatively changes the dynamics of the surface layer, which in turn influences the mixed layer above, because the large boundary layer convective eddies do span the entire depth of the CBL (Hunt et al. 1988).
2. Numerical methods
A two-way vertical nest algorithm is implemented in the Advanced Regional Prediction System (ARPS; Xue et al. 2000, 2001). ARPS is developed at the Center for Analysis and Prediction of Storms at the University of Oklahoma and is a nonhydrostatic finite-difference model suitable for simulations/predictions ranging from LES through regional numerical weather prediction. More details about ARPS are documented in Xue et al. (2000, 2001). To our knowledge, most community atmospheric models do not have a two-way vertical nesting capability. The Regional Atmospheric Modeling System (RAMS; Pielke et al. 1992) is one of a few models with two-way interactive nesting that can be limited in the vertical extent (Walko et al. 1995), although the purpose of its grid nesting had not been for improving CBL turbulent mixing. ARPS allows for one-way nesting, where the parent and the nested simulations are performed sequentially. The nest domain can be shallower than the parent domain. The grid configuration of the nest domain is also independent from that of the parent domain. The Weather Research and Forecasting (WRF) Model allows for both one-way and two-way nesting (Skamarock et al. 2008). In the latter, the extent of the vertical domain and the vertical grid levels have to be identical in the parent and the nested grids. Recently, Lundquist et al. (2016) developed algorithms that allow the nest domain to have denser vertical levels, which alleviates numerical errors due to large aspect ratios on coarse grids and improves the accuracy on the nest domain.
A two-way vertical nesting capability implemented for the purpose of improving gray-zone turbulence parameterization is potentially useful for other modeling applications also. For example, for the nighttime stable boundary layer (SBL) flow over the Great Plains, the boundary layer depth usually ranges from 100 to 500 m (Stull 1988, chapter 12.1.2) and, in some extremely stable cases, is of order 1 m (Mahrt 2014). When simulating such an SBL, it is not necessary to have a nest domain that extends to the full depth of the model domain. A shallow vertically nested domain is a much more computationally efficient way of allocating grid points or, in other words, limited computational resources.
a. Grid setup
A 2D schematic of the grid setup is presented here in Fig. 1. The nested grid extends from vertical level 1 to
A 2D schematic of the vertically nested gray-zone grid. Thick and thin lines represent the coarse and fine grids, respectively. The nested grid occupies the bottom 10 levels, the top 5 of which belong to the relaxation zone. The horizontal spacings are 1000 and 200 m for the parent and nest grid, respectively, with a refinement ratio of 5. The vertical spacing is 50 m. The parent grid above 600 m is not shown for brevity.
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0346.1
b. Solution algorithm
The step-by-step procedures of the nested grid algorithm are outlined in Fig. 2. Details are given below.
Flowchart of the solution algorithm. The numbers indicate the sequence of the flowchart.
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0346.1
1) Step 1
Prognostic variables on the nested grid are initialized with bilinear interpolation in the horizontal directions from the coarse parent grid. The initialization can be performed at any user-specified time. Here, the nested grid is initiated at time zero from a horizontally uniform base state to study its effects on the onset of convection. In this case, higher-order schemes yield the same results as linear interpolation and therefore are not tested in this work. Random potential temperature perturbations of ±0.1-K magnitude are applied at the lowest model level to initiate turbulent flows.
2) Steps 2–3


When surface fluxes are computed locally on the nest domain, additional heterogeneity is introduced through locally varying surface fluxes. This is favorable for the development of finescale turbulence. However, biases in surface winds and temperature can also arise because of deficiencies of turbulence closures near the wall (Moeng et al. 2007; Mirocha et al. 2013). Ideally, as the flow transitions between the parent and nested grids, the partition of fluxes between the resolved and subgrid scale changes based on the grid resolution, while the total flux is conserved. However, since SGS turbulence closures are often deficient near the bottom boundary where the flow is poorly resolved (Chow et al. 2005), the total flux might change, leading to different turbulent mixing, hence creating biases in prognostic variables. To investigate the effects of locally formulated fluxes, a separate test is performed, and the results are discussed in section 4.
3) Steps 4–5
ARPS uses a mode-splitting time integration scheme (Klemp and Wilhelmson 1978). The large time steps













With the choice of the interpolation formulas in Eqs. (3) and (7), the conservation condition of Eq. (4) is satisfied by all second-order vertical fluxes except the vertical fluxes of vertical velocity
If horizontal nesting is applied, step 5 also interpolates the lateral boundary conditions from the coarse to the fine grid. In this work, the lateral boundaries of the parent and nest domains overlap. Identical lateral boundary conditions are applied to both domains.
4) Steps 6–8







The small time steps for acoustic waves are also decreased proportionally by R times
5) Step 9










3. SGS turbulence parameterization














Besides consistency considerations, an additional reason for favoring higher-order parameterizations is because of the design of the PBL schemes. Traditional PBL schemes are designed to operate on a vertical model column and are unaware of the horizontal grid spacing
In the grid nested setup, convection is explicitly resolved in the nested grid and anterpolated back to the parent grid. As the parent grid falls inside the gray zone, the anterpolated fluxes are partially resolved and nonnegligible. Since the SGS fluxes from the PBL scheme are grid insensitive, adding a nonzero amount of resolved fluxes (because of anterpolation) can lead to an overprediction of the total fluxes. The same problem, known as double counting of fluxes (e.g., Xue et al. 1996), occurs when applying PBL schemes to fine, convection-resolving grids. The erroneous heat fluxes can lead to changes in the mean state of
Traditional higher-order PBL schemes are also grid unaware by design. Higher-order moments, such as TKE in Eq. (15), are also solved along a vertical model column. However, their reliance on higher-order moments provides a means of adaptation to scale- or grid-aware formulations. For example, when a 1.5-order TKE closure is used in the LES grid, the predicted TKE is certainly grid aware. When anterpolated back to the parent grid, additional fluxes of resolved scale to the nest but of subgrid scale to the parent grid are also added [see Eq. (12)]. This builds in grid awareness to the parent grid TKE, and in turn, the eddy diffusivity


4. Simulation setup
The test case is based on the well-studied Australian Wangara Experiment (Clarke et al. 1971). It describes a time-evolving CBL from 0900 to 1800 LST for day 33 (16 August 1967) of the experiment. The case setup follows Yamada and Mellor (1975) and is not repeated here for brevity. All simulations are performed on a 36 km × 36 km × 2.5 km domain. A list of key model parameters is presented in Table 1. A 50-m stand-alone high-resolution LES (SA50) is performed to serve as a benchmark “truth”1 for the Wangara CBL. The 900-m stand-alone simulation (SA900) is used as the control simulation. All simulations have 50 vertical levels with uniform
List of model parameters. Stand-alone and nested simulations are denoted by “SA” and “N” in the run names; “S” and “LF” at the end of the run names stand for stretched vertical grid and locally computed fluxes on the nest domain, respectively. Parent and nest variables are distinguished by capital and small letters. Nested simulations are named by their respective grid spacing on the nested grid followed by the vertical levels of the nest. The five-point sponge zone is not counted in the vertical nest level
Nested simulations are performed on varying horizontal grid spacing
5. Results and discussion
a. Mean profiles and onset of resolved convection
Figure 3 presents the vertical profiles of horizontally averaged potential temperature
Comparison of horizontally averaged potential temperature at 1200, 1500, and 1800 LST.
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0346.1
Time series of horizontally averaged boundary layer depth. Data shown every 900 s.
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0346.1
Despite the overall agreement,
To show the onset timing of explicit convection, time–height contours of
Time–height contours of
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0346.1
In Fig. 5a, between 1200 and 1300 LST, a strong patch of
To better compare
Horizontally averaged resolved (a)
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0346.1
In Fig. 6, SA900 produces the smallest variance profile of
Figures 5 and 6a together suggest that organized convection initiates in the unstable surface layer. As thermals ascend into the mixed layer, they continue to gain strength by merging smaller plumes. If the nest domain is limited to the surface layer only, as in N180-L6, the onset and the initial structure of organized convection is improved. But to optimally nurture the growth of thermals above the surface layer, the nest domain should extend to the lower half of the mixed layer so that plume-merging processes can be captured. But even so, the nested simulations still do not solely determine the characteristics of convection in the CBL. As evidenced in Fig. 6a, the maximum
The vertical profiles of
Profiles of
The total and the SGS components of TKE are presented in Fig. 6f alongside the resolved components in Figs. 6a and 6b. The total and the SGS TKE profiles from SA50 rather than filtered SA50 are plotted for direct comparison. Compared to SA50, all other runs overpredict the total TKE below ~0.3
Once resolved convection emerges, overshooting thermals contribute to the entrainment of the free tropospheric air. To investigate the effects of nesting on the entrainment flux, the entrainment flux ratio
Time series of the horizontally averaged entrainment flux ratio
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0346.1
b. Flow visualization and spectra
Figure 8 shows the convective structures of the stand-alone gray zone and the two nested simulations in terms of w in the vertical and horizontal cross sections at 1200 LST, the early stage of CBL development. The filtered SA50 contours are also presented as a benchmark in the rightmost column. The contours of the nested simulations are from the parent 900-m grid. As previously shown at 1200 LST, the SA900 run has just reached its onset of resolved convection (see Fig. 5). This is evidenced in the lack of convective structures in the leftmost column of Fig. 8 for the SA900 run. In Figs. 8e, 8i, and 8m, the horizontal convective cells are barely forming. The magnitude of resolved w is about twice as small as in the nested runs at the same time. In comparison, contours of N180-L6 in the second column show much more organized convective structures within (
Contours of w from (a),(e),(i),(m) SA900, (b),(f),(j),(n) N180-L6, (c),(g),(k),(o) N180-L16, and (d),(h),(l),(p) the filtered SA50 at 1200 LST. Here, (a)–(d) are along the x–z vertical cross section at y = 0; (e)–(h), (i)–(l), and (m)–(p) are along the horizontal cross sections at
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0346.1
Compared to a 6-level nest, a 16-level nest in the third column reveals even finer-resolved convective structures, which strongly resembles the idealized filtered-LES field in the last column. The size of the organized vertical convective motions is smaller for N180-L16 than N180-L6, as seen most clearly from the increased number of up- and downdrafts in the vertical cross section in Fig. 8c compared to Fig. 8b. A close comparison of the third and fourth columns, especially Figs. 8k and 8l, reveals that the convective structures on the parent grid of N180-L16 are “too” organized, while the filtered SA90 contours appear more random.
To quantitatively assess the simulated convective structures in Fig. 8, the energy spectra of
Horizontal energy spectra of
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0346.1
At
The convective structures are examined again at 1500 LST, when all simulations produce vigorous resolved convection. For SA900 in the first column of Fig. 10, the model domain is filled with large convective cells that resemble real convective structures. In Figs. 10e and 10i, the convective cells are from about 5 to 6
As in Fig. 8, but at 1500 LST.
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0346.1
The length scales of convection are again assessed through the energy spectra of
c. Additional tests
Numerical weather prediction (NWP) applications almost always adopt vertical grid stretching to allow for more resolution in the boundary layer. To test the nesting configuration in the presence of vertical grid stretching, an additional run (N180-L6-S; see Table 1) is performed. In N180-L6-S, the vertical grid is stretched using a hyperbolic tangent function, with minimum surface spacing of 50 m and average spacing of 100 m (Xue et al. 1995). The convective structures at 1200 LST are presented in Fig. 11 and are similar to their uniformly spaced counterparts N180-L6 in Fig. 8. In the vertical cross section in Fig. 11d, the vertical convection cells are continuous across the nest interface. First- and higher-order mean profiles are also close to those of N180-L6 (not shown). Overall, the vertical nesting procedure works well under a vertically stretched grid.
As in Fig. 8, but for N180-L6-S at 1200 LST.
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0346.1
Another test (N180-L6-LF; see Table 1) is performed whereby the surface momentum fluxes on the nest grid are computed locally according to Eq. (1) rather than interpolated from the parent grid as is done in section 2b(2). Other aspects of the N180-L6-LF setup are identical to that of N180-L6. Note that the heat flux is prescribed as boundary conditions according to section 4 and is left unchanged. The impact of locally computed momentum fluxes on wind speed is examined first. Time series of wind speed at the first grid point above the surface, where the largest impacts are expected (Moeng et al. 2007), are presented in Fig. 12a. Compared to the control run N180-L6, surface wind speeds are slightly larger when momentum fluxes are computed locally. Although the positive bias largely persists throughout the daytime, its magnitude is quite moderate. More significant changes are found in the variances of horizontal velocity. The vertical profiles of
(a) Time series of horizontally averaged surface wind speed
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0346.1
To investigate the source of the increased variance, horizontal contours of the perturbation horizontal velocity
d. Computational costs
The use of increased resolution within the nested near-surface layer increases computational cost. To give a quantitative assessment, the normalized CPU times of nested runs with horizontal nesting ratios of 5:1 and 9:1 are given in Fig. 13 relative to the CPU time of SA900. The six-level nested simulations are about 50 and 300 times more expensive than SA900 for 5:1 and 9:1 nesting ratios. The 16-level nest is about twice as expensive as the 6-level nest. The added computational expenses are certainly significant. However, compared to running the model at full resolutions, the nested grid approach is still more economical. For the 5:1 nesting ratio, the cost of a nested run is about 70% and 20% cheaper compared to a stand-alone run for 6 and 16 levels of nest, respectively. The savings further increase to about 90% and 85% for a 9:1 nesting ratio.
Normalized CPU time of the stand-alone and nested runs.
Citation: Journal of the Atmospheric Sciences 75, 4; 10.1175/JAS-D-17-0346.1
Note that the code has not been optimized for computational efficiency. For example, we made extensive use of existing ARPS subroutines for the ease of coding. Many variables, such as those related to the land surface module, were defined and carried around without any utility to avoid changes to the existing subroutines. Furthermore, when parallel capability was implemented, we used the same processor for the parent subdomain to host the same nest subdomain. These aspects of the code can all be redesigned to improve performance.
Compared to the empirically tuned gray-zone PBL schemes (e.g., Shin and Hong 2015; Ito et al. 2015; Efstathiou et al. 2016), the proposed method is probably too expensive to be of practical use in a high-resolution NWP model in its current form. However, it is an alternative approach to gray-zone modeling that does not require any empirical tuning. It is a general method based on flow dynamics by resolving instability in the critical layers. The method could be extended to modeling the shear instability gray zone, for example, by laying a fine nest layer over the critical shear layer. In this regard, our method is better than the modified PBL schemes based on empirical flux partition functions.
6. Summary and overall assessment
This paper implements a grid-refinement-based approach for improving CBL simulations in the gray zone of PBL parameterization. The core component of the algorithm is a two-way interactive vertical nest of increased horizontal resolution near the surface. Such a fine-resolution nest serves to significantly improve the modeling of near-surface turbulent convective eddies in the CBL and associated vertical fluxes. Effectively, the strategy employs an LES-type simulation within the surface and near-surface layer while relying on PBL schemes for vertical mixing in the CBL. Special care is taken in the interpolation and anterpolation procedures between the parent and the nested grids to ensure the conservation of mass and vertical fluxes. As a result, smoothness and continuity are obtained in the simulated vertical flux profiles across the nesting interface. CBL parameterization aside, the algorithm is also applicable to general grid nesting practices. Future work is planned to expand the two-way interactive nesting capability in ARPS to all three dimensions.
A posteriori tests are performed to simulate an idealized CBL. Compared to the stand-alone simulations with 900-m horizontal grid spacing, the nested simulations explicitly resolve convection in the near-surface nest domain early on and produce more realistic convective structures in the entire CBL. Most notably, the onset of convection and early CBL growth (Figs. 5 and 6) are significantly improved, and the unphysical pulsing of SA900 is ameliorated. This is an important improvement, especially considering potential feedbacks with clouds and the land surface models. Between the two nest domain depths tested, the 16-level nest outperforms the 6-level nest in terms of the agreement in higher-order statistics with those of the benchmark LES. Further increasing the nesting ratio from 5:1 to 9:1 produces modest improvements, indicating a certain degree of convergence of model results for the given parent grid.
One notable deficiency of the proposed approach is the contamination of convective structures due to the coarse-resolution parent grid, especially when the nest domain is shallow. The N180-L6 run with a 5:1 nesting ratio and a six-level nest produces a spectrum that initially agrees with the high-resolution LES benchmark for wavelength resolvable by the nested grid, but the spectrum can be contaminated by the distorted convection on the 900-m parent grid. The spectrum exhibits double-peak behavior with a physical one at a shorter wavelength and a spurious longer-wavelength peak that is influenced by the coarse-resolution grid. Later in the afternoon, the spurious peak can overtake the physical peak even within the nested high-resolution domain. In this case, the simulated flow fields show convective cell structures that are similar to those in SA900 but with more details at the small scales. In comparison, using 16 levels of nesting that covers about half of the CBL depth in N180-L16, the simulation results are much improved. In this case, no spurious peak is found even at half of the CBL depth. The simulated spectra agree well with those of the LES benchmark for wavelengths that can be resolved by the grid, although there tends to be some excessive energy near the longer-wavelength end of the spectra.
Overall, the grid-refinement-based approach improves the CBL simulation in the gray zone of PBL parameterization. The method does not rely on empirically determined partition functions as used in some recently proposed grid-aware PBL schemes, and in theory, it can work with all existing PBL schemes. It is suggested that a combination of a TKE-based PBL scheme on the coarse-resolution parent grid and a consistent TKE-based LES closure on the nested grid is preferred over first-order parameterizations because of the additional interaction of the two grids through prognostic TKE. An assessment of the added computational cost of the proposed approach is presented. Further testing of the approach in realistic simulations is required to fully demonstrate its potential.
Acknowledgments
We are grateful to Prof. Tina Chow for her help with reviewing the manuscript. We gratefully acknowledge the High Performance Computing Center (HPCC) of Nanjing University for doing the numerical calculations in this paper on its IBM Blade cluster system. Bowen Zhou is supported by the National Natural Science Foundation of China (Grant 41505047), the Dengfeng Program of Nanjing University, and a collaborative project on PBL parameterizations with the Beijing Urban Meteorology Institute. Kefeng Zhu is also supported by the Foundation of China Meteorological Administration (Grant GYHY201506006) and the National Science Foundation of China (Grant 41405100). All authors are also supported by the Research 973 Program of China (2013CB430103).
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