Abstract

A nonhydrostatic stretched-grid (SG) model is used to analyze the large-scale errors generated by stretching horizontal grids and their influence on a region of interest. Simulations by a fully compressible, nonhydrostatic global atmospheric model, the Nonhydrostatic Icosahedral Atmospheric Model (NICAM), and its SG regional model, stretched-NICAM, were performed for the months of March, April, and May of 2011 using various resolutions and stretching factors. A comparison of week-long accumulative precipitation amounts between the Tropical Rainfall Measuring Mission (TRMM) satellite data and the quasi-uniform and SG simulations showed that a stretched run better represents storms and associated precipitation because the errors generated in the outer regions with coarser grid spacing do not significantly affect the inner domain centered at the focal point. For season-long simulations, in one particular set of stretched runs with the focal point located in the eastern United States, the artificial suppression of baroclinic development of midlatitude eddies in the Southern Hemisphere weakened the eddy-driven polar-front jet (PFJ), which yielded a cold bias at mid- to high latitudes. However, in the Northern Hemisphere, in contrast, the aforementioned changes are less apparent. Therefore, for the SG runs, the mean temperature was maintained at the region of interest, and an increased amount of moderate to heavy precipitation, which is also frequently found in the TRMM data, was observed; thus, the benefits of increased resolution were realized. However, careful attention must be given when applying the SG model because a regional climate response to the change in the large-scale circulations may not be fully accounted for.

1. Introduction

A rapid increase in computational power has allowed fully compressible, nonhydrostatic atmospheric global circulation models (Satoh et al. 2008, 2014; Putman and Suarez 2011; Skamarock et al. 2012) to run with horizontal grid spacing of a few kilometers (Miura et al. 2007) or less (Miyamoto et al. 2013). However, these simulations are still considered expensive, and even with the best computational capabilities available, such simulations cannot be performed for longer time periods (e.g., decadal climate variability). Therefore, further technological advancements are required to utilize the full potential of these models in climate research. Currently, commonly used methods of applying nonhydrostatic, fully elastic equations in climate modeling include employing a nested regional climate model to represent mesoscale features (Lo et al. 2008; Skamarock and Klemp 2008). This process usually involves running limited domain regional climate models, which are referred to as limited area models (LAMs), with finer grid sizes, better dynamical frameworks, and cloud microphysics (Giorgi 1990; Giorgi and Mearns 1999). At the outer boundary, the values are replaced with global model outputs, such as the Coupled Model Intercomparison Project (CMIP) climate projections or reanalysis data. This method is called dynamical downscaling because large synoptic-scale flows represented in the global model are transferred through the boundaries into the finer-resolution regional model that can resolve mesoscale features. However, several difficulties still remain when using dynamical downscaling, which arise primarily from errors generated from the lateral boundary conditions (LBCs) (Warner et al. 1997).

A practical way to circumvent the problems of LBCs is to use variable-resolution GCMs (VRGCMs). VRGCMs are global models that can concentrate horizontal grid points within a particular region of interest. Thus, the intended region of interest can have a finer resolution, whereas regions far from the region of interest are represented with coarser grid intervals to conserve computational resources. Fox-Rabinovitz et al. (2000) noted that this scheme is also appropriate for dynamical downscaling of large-scale motions produced in regions far from the focus. These motions are gradually transferred into the region of finer resolution, where smaller motions dominate as if repeated nesting has been applied. Therefore, this method may be regarded as a global LAM without boundaries. Unlike LAMs, which may require multiple nesting processes to obtain the desired resolution in the target domain, VRGCMs contain smoother transitions of grid sizes in just a single simulation with self-consistent interactions between planetary and regional-scale motions because of the sufficient representation of long waves (Fox-Rabinovitz et al. 2001, 2005, 2006). Therefore, no LBC problem arises. However, inaccuracies originating from the coarser grids, such as underrepresented orography, will be transferred into the targeted domain without the use of proper nudging techniques. There are several types of grid formulations used in VRGCMs. For example, in stretched-grid (SG) GCMs, which will be used in this study, the grid intervals continuously increase with distance from a point of focus. An alternative, which has increasingly being used in recent years, is nonstretched-grid variable-resolution models (Harris and Lin 2013; Rauscher et al. 2013; Rauscher and Ringler 2014; Zarzycki et al. 2014) that introduce a localized mesh refinement in one or multiple locations to create uniform, high-resolution grid spacing over a region of interest. Different from SG-GCMs, increased resolution in one region is not constrained by the decreased resolution of another region (Rauscher et al. 2013); however, deformed or unstructured grids at the narrow transition zone may create more localized errors compared with that in VRGCMs with more gradual refinements (Harris and Lin 2013). Another technique is adaptive mesh refinement (AMR), which can locally translate, add, or remove grid points and adapt the time stepping according to a refinement criterion at the runtime, where the grid sizes dynamically change in time to track moving flow features—for example, tropical cyclones and topographic flows (Skamarock and Klemp 1993; Jablonowski et al. 2006).

SG-GCMs have shown promising results in analyzing anomalous regional climate events (Berbery and Fox-Rabinovitz 2003; Deque and Piedelievre 1995; Fox-Rabinovitz 2000; Fox-Rabinovitz et al. 2000, 2001, 2002) and in multiyear ensemble studies of regional climates (Fox-Rabinovitz et al. 2005). Other studies that have used SG-GCMs include a climate change study (Gibelin and Deque 2003) and an intercomparison test (Fox-Rabinovitz et al. 2006). Nevertheless, SG models have not been widely used yet (Flato et al. 2013) in regional climate studies because a lack of appropriately scale-aware parameterizations may result in different responses at varying resolutions (Zarzycki et al. 2014; Rauscher et al. 2013; Sakaguchi et al. 2015), restrictions on the maximum stretching factors and grid intervals at the antifocal point (Deque and Piedelievre 1995; Caian and Geleyn 1997; Fox-Rabinovitz et al. 2006), and the marginally greater computational resources required for SG models compared with that required for LAMs. As a consequence, SG models are still considered as a “working compromise” rather than a “perfect solution” (Fox-Rabinovitz et al. 2000). On the other hand, given the constraints on the use of a high-resolution uniform-grid nonhydrostatic GCM in a regional variability study and the fact that LAMs require lateral boundary conditions at each time step, there is a need to apply nonhydrostatic SG-GCMs to study regional climates (e.g., to predict the frequency of local extreme events). Therefore, it is important to understand how an error grows with respect to the amount of stretching, which is proportional to the computational savings achieved by instead running uniform-grid models with identical resolutions at the focus.

In this study, we evaluate an SG model of a global cloud-system-resolving model called the Nonhydrostatic Icosahedral Atmospheric Model (NICAM) (Tomita and Satoh 2004; Satoh et al. 2008, 2014) using various resolutions. As a global nonhydrostatic model, NICAM can be used for a relatively wide range of horizontal scales. Using quasi-uniformly distributed grids enables the translation of grid points. Grid points near the focal point are pulled together systematically to create a gradual change in horizontal grid intervals from fine intervals at the focal point to coarser intervals for points farther away; this is called a stretched-NICAM (S-NICAM) (Tomita 2008b; Goto et al. 2015), which is the SG mode of NICAM. Therefore, we can effectively use less computational resources while concentrating the grids to allow a fine resolution at the area of interest. The objectives of this study are to assess the benefits of using SG nonhydrostatic GCMs for a short-term (week long) prediction and season-long situations to analyze how a global circulation error generated by varying degrees of grid stretching in SG-GCMs increases or decays in a season-long simulation and to classify the types of errors that are generated by stretching the grids.

2. Model descriptions

The models used in this study are the global cloud-system-resolving NICAM and its derivation, S-NICAM. The model settings are similar to those in previous studies that utilized these models (Satoh and Kitao 2013; Satoh et al. 2010, 2013, 2014; Roh and Satoh 2014). The models share identical dynamical and physical processes: a dynamic core of fully compressible, nonhydrostatic equations; a single moment microphysical parameterization scheme (Tomita 2008a); a two-stream k-distribution radiative transfer code; mstrnX, which was retuned for the Fifth Assessment Report (AR5) (Sekiguchi and Nakajima 2008); and an improved Mellor–Yamada level 2 turbulent closure model proposed by Nakanishi and Niino (2004) (Nakanishi and Niino 2006, 2009; Noda et al. 2010). The initial condition is set using the 1° × 1° resolution National Centers for Environmental Prediction (NCEP) final (FNL) operational global analysis data. Minimal Advanced Treatments of Surface Interaction and RunOff (MATSIRO; Takata et al. 2003) is used for the land surface model, which was initialized with the International Satellite Land Surface Climatology Project, Initiative II (ISLSCP II) for the albedo and with the USGS Global Land Cover Characterization (GLCC) Simple Biosphere 2 Model (GSiB22) for the vegetation data. The boundary conditions for the bulk surface flux over the ocean were calculated based on the work of Louis (1979) (Satoh et al. 2012). NICAM uses a mixed-layer ocean model with a mixed-layer depth of 15 m and a nudging time scale of 7 days. The SST in the model was nudged to the NCEP FNL, which updates every 6 h, and the sea ice fraction and mass were obtained from the CMIP3 (Phase 3 of Coupled Model Intercomparison Project) model ensemble mean of the monthly sea ice averaged over 1979–99. We used 40 vertical levels with the lowest four layers at 161-, 335-, 523-, and 726-m levels up to the highest layer at 39 620 m. NICAM solves for the zonal velocity u, meridional velocity υ, and vertical velocity w and for thermodynamic variables, including temperature T, pressure P, density , water vapor qυ, cloud water qc, cloud ice qi, rainwater qr, snow qs, and graupel specific mass qg. Here, water vapor and cloud water represent the quantities of water suspended in the atmosphere, whereas rain, snow, and graupel represent the amount of water that falls as a result of gravity.

A global NICAM distributes the horizontal grids nearly equally over the globe based on an icosahedron-shaped object composed of twenty triangular faces and twelve vertices. These vertices can be regarded as horizontal grids on the globe. We define the icosahedron-shaped object as a global level (g level) of zero. The mesh refinement processes for the horizontal grids can be performed through a recursive step by introducing new grids at the middle of the two existing grids (Tomita et al. 2001) such that the length of the horizontal interval is halved at each increment of the g level. The horizontal grid mesh resulting from the Nth division is called g-level N. For this paper, we prepared g-level six (g6), g-level seven (g7), and g-level eight (g8) simulations, in which the average horizontal grid size is approximately 112, 56, and 28 km, respectively. The resulting grid points are readjusted using spring dynamics to increase both the homogeneity of the grid system and numerical accuracy (Tomita et al. 2002). Hereafter, we refer to this standard grid system of quasi-homogeneous grids as a uniform grid.

S-NICAM, a variable-resolution model, uses a formulation of NICAM grids as a starting point, and then a Schmidt transformation is applied to concentrate the grid points toward a target point (Tomita 2008b; Goto et al. 2015). As the grid points approach the target point, their horizontal intervals gradually decrease. This process is equivalent to dynamical downscaling but realized within a single model without dealing with LBC problems. The strength of such stretching is controlled by a parameter called the total global stretching factor (Tomita 2008b), which is defined as the ratio of the smallest to the largest horizontal grid distance. In this paper, we call this factor the stretching ratio (s ratio), and moderate and relatively large s ratios of 16 (g7-str16), 36 (g7-str36), and 64 (g7-str64) were applied to the g7 simulations. The horizontal grid intervals vary from 14 to 223 km for the g7-str16 run, from 9.3 to 336 km for the g7-str36 run, and from approximately 7 to 448 km for the g7-str64 run. Thus, the majority of the horizontal resolutions are less than 4°. In addition, these simulations adopted a local stretching ratio for two adjacent grids of less than 8%, as suggested by Cote et al. (1993) and Fox-Rabinovitz et al. (2001). Table 1 provides a quick reference for the simulations performed in this paper, including information on the g levels, horizontal resolutions, focus locations, and time steps.

3. Comparison between the uniform and stretched-grid models

For the S-NICAM simulations of g7-str16, g7-str36, and g7-str64, the center of interest was set over the North American continent at 41.90°N, 87.64°W (near Chicago, Illinois). Although these are not the exact grids used in this study (only every fourth grid point is represented in contrast to the g7 grids), Fig. 1 shows illustrations of the uniform g5 grids (Fig. 1a), the stretched g5-str16 grids at the same focal point mentioned above (Fig. 1b), the stretched g5-str64 grids at the focus (Fig. 1c), and the stretched g5-str64 grids at the antifocal point (Fig. 1d). The simulations were performed for 90 days from 1 March to 30 May 2011. The time steps, which are constrained to the smallest grids in the domain, were set at 15 s for all the uniform and stretched g7 grid simulations for consistency; therefore, the computational cost is identical for those simulations and at 30 s for the g6 and g8 uniform-grid runs. These time steps are well within the CFL condition to minimize time step sensitivities on the dynamical and physical processes. All of the NICAM runs were performed without any internal nudging unless otherwise stated. The NICAM results were output every hour for two-dimensional variables and every 3 h for three-dimensional variables. Additionally, when the results are compared with 0.25° × 0.25° resolution merged Tropical Rainfall Measuring Mission (TRMM) satellite and other satellite estimation data (Huffman et al. 2007) or 1° × 1° resolution NCEP FNL data, the NICAM outputs, which are in Cartesian coordinates, are interpolated linearly onto a 0.25° × 0.25° or a 1° × 1° latitude–longitude coordinate, respectively. This conversion enables direct comparisons even when different uniform or stretched-grid sizes are used.

In previous studies, SG models showed improvements in reproducing extreme events. For example, in simulations of summer droughts in the United States (Fox-Rabinovitz 2000; Fox-Rabinovitz et al. 2001, 2002), SG models were shown to reproduce the precipitation rate closer to the observation at the area of interest than the corresponding uniform-grid global climate models with the same number of global grid points. These improvements resulted from a better representation of the orography and atmosphere–land interactions within the region of interest. To verify the supposed short-term improvements due to refining the grids around the region of interest, we simulated week-long heavy storms on 4 March 2011 over the eastern United States that caused snowstorms in the north and large-scale floods in the south (Pierce 2011; Soltow 2011). Figure 2 shows the TRMM and NICAM plots of the accumulated precipitation (in millimeters) from 0000 UTC 4 March to 0000 UTC 12 March. The TRMM figure shown in Fig. 2a was obtained by summing a given 3-hourly precipitation rate from 0.25° × 0.25° data, and all of the NICAM simulations were calculated by summing hourly average 0.25° × 0.25° data. Figure 2f of the g7-str64 plot most closely resembles the TRMM reanalysis data; the plot matches the location and magnitude of the intense precipitation core (measured at approximately 300 mm) better than any other NICAM run. Additionally, the g7-str36 plot (Fig. 2e) more or less realistically reproduces the tilt of the precipitation band observed from the south to the northeast United States. Furthermore, all of the stretched runs simulated the precipitation more realistically in intensity and distribution than the uniform g7 simulation. Also, an examination of the liquid water path indicated that a small-scale structure developed near the focus within a few hours after the simulation was initialized (not shown), which demonstrates that initializing with coarser NCEP FNL data is adequate for the purpose of this study. As long as the influence of the initial conditions is strong, S-NICAM can take full advantage of its higher resolutions at the domain, which can better represent localized, vigorous convective storms. Because important aspects of the cloud-resolving model include the reproducibility of realistic flows on a short-term basis, it is extremely useful to have the ability to apply relatively strong s ratios to the SG model to maximize the computational resources.

It was also necessary to validate the consistency of the NICAM simulation against the NCEP FNL data on a longer time period. Figure 3 shows scatterplots of the 70-day mean 500-hPa temperature from 20 March through 30 May 2011 between the NICAM simulations and the NCEP FNL. The black dots and green crosses represent the data at grid points in the Northern Hemisphere (NH) and Southern Hemisphere (SH), respectively. The linear correlations and regressions indicated in the panels are based only on the NH data. The dots and crosses overlap for all the uniform runs (Figs. 3a–c), which show no systematic differences between the NH and SH. The increasing regression coefficient from g6 (0.847) to g8 (0.983) indicates the tendency for the weak cold bias at higher latitudes in the NICAM to decrease with its horizontal resolution. Notably, the coefficients of both the linear regression (0.983) and correlation (0.993) are close to 1 in the g8 run, which demonstrates that the NICAM can reproduce the midtropospheric thermal field globally as the seasonal mean.

However, as evident in Figs. 3d–f, the stretched runs give distinctively different regression coefficients between the NH and SH; the seasonal-mean temperature distributions in the SH exhibit a strong cold bias at the mid–high latitudes (colder temperatures). In the SH, where the cold bias is obvious, the regression coefficients for the g7-str16, g7-str36, and g7-str64 simulations are 0.711, 0.695, and 0.741, respectively, and the corresponding correlations are 0.988, 0.983, and 0.980, respectively (not shown). The cold bias for the seasonal mean in the SH does not necessarily increase with decreasing resolution because the regression coefficients are similar among the stretched runs and are much smaller compared with the relatively coarse grids of the g6 run.

In the NH, the regression lines of the g7-str16, g7-str36, and g7-str64 runs are 0.924, 0.918, and 0.851, respectively, and the corresponding correlations are 0.987, 0.984, and 0.977. These values are comparable to those in the uniform runs, which implies that, unlike in the SH, the influence of a strong cold bias is kept minimal in the NH with respect to the seasonal average. Nevertheless, the weak decreasing tendency in the regression coefficient with increasing s ratio indicates the emergence of a weak cold bias at high latitudes even within the NH. In fact, the g6 run also has a lower regression coefficient (0.847) than that of the g7 and g8 runs, which suggests that this cold bias, which exists in the NH, likely originates from the same hemisphere. The global distribution of the seasonal (70 day) mean 500-hPa temperature based on the NCEP FNL is plotted with dashed contours in Fig. 4, where the seasonal bias of the g7 (Fig. 4a), g6 (Fig. 4b), and g7-str36 (Fig. 4c) simulations relative to the NCEP FNL is contoured with filled colors. In a manner consistent with Fig. 3e, the g7-str36 run displays a particularly large bias, which exceeded 6 K poleward of 40°S (Fig. 4c). This suggests that poleward heat transport by atmospheric eddies may be severely underestimated. However, biases of similar magnitudes were observed in both the g7 and g7-str36 runs in the NH, including the area of interest in North America; note that the coarse grids in the g7-str36 run could contaminate the entire NH. Also, an occurrence of a cold bias at high latitudes shown in Fig. 3c is clearly visible for the g6 run (Fig. 4b). Additionally, the stretched runs appear to be slightly colder in the deep tropics than in the uniform runs.

To understand how errors grow over time, a time series of the quasi-global root-mean-square error (RMSE) of the 500-hPa temperature between the NICAM simulations and NCEP FNL is shown in Fig. 5a. Additionally, the corresponding time series of the contributions to the RMSE from the zonal-mean component (Fig. 5b) and individual zonal wavenumber components (Figs. 5c–i) are plotted. In each hemisphere, the zonal-mean temperature distribution is related through the thermal wind balance to the subtropical jets (STJ) and (eddy driven) polar-front jet (PFJ) in the upper troposphere. The STJ is maintained by the Hadley cell, whereas the PFJ is driven by eddies developing along the midlatitude storm track (Palmén and Newton 1969; Nakamura et al. 2004). Zonal wavenumbers 1, 2, and 3 correspond to planetary waves, and zonal wavenumbers 4 through 9 correspond to midlatitude synoptic-scale disturbances. Figure 5a shows that the quasi-global RMSE in a statistically steady state is less than 5 K in all of the uniform runs, whereas it is approximately 2–3 K greater in the stretched runs. Among the stretched runs, the steady-state RMSE is smallest in the run with the smallest s ratio (g7-str16 run). Figure 5a indicates that the uniform and stretched runs require approximately 15 and more than 30 days, respectively, to reach a statistical steady state, in which the quasi-global RMSE is largely caused by the “model bias.” Figure 5b reveals that errors in the steady-state RMSE arise primarily from the zonal-mean temperature bias, which is also shown by the relatively similar error evolutions in the eddy components in Figs. 5c–i. As evident in Fig. 6, which displays the same RMSE separately for NH (Fig. 6a) and SH (Fig. 6b), this RMSE enhancement arises from the SH, where the grid spacing is artificially enlarged in the stretched runs, and the error generated in the SH exerts only a slight influence on the NH. Additionally, Fig. 6c displays the same RMSE of a longer 10-month period for the g7 and g7-str16 runs, in which the state of consistent asymptotic equilibrium continues for the entire simulation.

To determine the best possible mechanism for the bias in the SH and to analyze how the model bias increases as a result of the stretching effects, the time series of the zonal-mean bias at the 500-hPa temperature is plotted in Fig. 7. The bias in the uniform runs is particularly large at high latitudes and more or less symmetric between the NH and SH (Figs. 7a–c). The high-latitude cold bias becomes less obvious in both hemispheres as the resolution increases from g6 (Fig. 7c) to g8 (Fig. 7a), which indicates that the bias is reduced in the high-resolution NICAM. For the stretched runs (Figs. 7d–f), by contrast, the cold bias tends to be stronger in the SH than in the NH. The cold bias in the SH is initiated at approximately 50°S, where the eddy-driven PFJ tends to form (Nakamura and Shimpo 2004). The seasonal zonal-mean temperature profile (not shown) displays a strong cold bias poleward of 40°S in the stretched runs with its peak value of −12 to −15 K at 70°S between the 700- and 500-hPa levels, which indicates that the bias is not directly caused at the surface boundary but encompasses more of the dynamical problems.

Figure 8 shows the time–latitude sections of the zonal-mean zonal wind velocity u at the 200-hPa level, around which jet cores are normally observed. The NCEP FNL (Fig. 8a) depicts a well-defined PFJ throughout the period at approximately 50°S, whose core velocity reaches approximately 30 m s−1. In addition, an STJ emerged at 25°–30°S after mid-April. The observed evolution of the PFJ and STJ in the SH is reproduced in the g8 (Fig. 8b) and g7 (Fig. 8c) simulations. The evolution of the two jets is consistent with their climatological seasonal march; the SPJ is most intense from January to March, whereas the STJ begins to form in March and then intensifies into the austral winter (Nakamura and Shimpo 2004). However, the stretched runs cannot reproduce this double-jet structure observed after mid-April and do not capture the SPJ at approximately 50°S. Rather, the stretched runs simulate the seasonal intensification of a single broad jet that appears to correspond to the STJ. The diminished signature of the eddy-driven PFJ in the stretched runs is consistent with that of the aforementioned cold bias located poleward of 40°S, which suggests underestimated eddy activity in the midlatitudes.

In the NH, the NCEP FNL shows the STJ was strong at 30°N in March and then weakened until it diminished in late May (Fig. 8a). This seasonal weakening of the STJ in the NH is reproduced, at least qualitatively, in all of the NICAM runs; however, in the heavily stretched runs of g7-str36 (Fig. 8f) and g7-str64 (Fig. 8g), the intensity of the STJ is underestimated, and the jet is too broad meridionally. These stretched runs also simulate the upper-tropospheric westerlies throughout the period at the equator, which is also unrealistic.

The typical vertical structure of the westerly jets is depicted in meridional sections (Fig. 9) of the seasonally averaged zonal-mean zonal wind velocity u. Figure 9 indicates that all of the uniform runs reproduce the STJ-PFJ double-jet structure in the SH (Figs. 9a–d); however, the reproducibility is somewhat lower in the g6 run because of its relatively coarse resolution (Fig. 9d). In the stretched runs (Figs. 9e–g), by contrast, the PFJ is merged into the STJ to form a single upper-tropospheric westerly jet in the SH. While the g7-str36 run (Fig. 9e) still simulates the deep westerly jet with the well-defined surface westerlies as in the uniform runs, the upper-level jet in the g7-str36 (Fig. 9f) and g7-str64 (Fig. 9g) runs accompanies no surface westerlies. This unrealistic feature reflects the missing signature of the eddy-driven PFJ under the unrealistically weak eddy activity, which is discussed below. In the NH, the g7-str36 and g7-str64 runs (Figs. 9f,g) simulate unrealistically weak STJ in the upper troposphere.

Figure 10 shows meridional cross sections of the zonal-average meridional streamfunction based on the NCEP FNL and NICAM runs. In a manner dynamically consistent with the high reproducibility of the STJs in the two hemispheres (Fig. 9), all of the uniform runs (Figs. 10b–d) reproduce the meridional extent and intensity of the Hadley cells in the two hemispheres (Fig. 10a), including their symmetry about the equator. Consistent with the PFJ in the SH (Fig. 9), the SH Ferrel cell tends to weaken as the model resolution in the SH decreased from the g6 run to the g8 run. For the stretched runs (Figs. 10e–g), by contrast, the Hadley cells between the two hemispheres became less symmetric about the equator with the stronger (weaker) and broader (narrower) SH (NH) cell and weaker as the s ratio increased.

The aforementioned unrealistic features of the Hadley cells in the stretched runs are consistent with those of tropical convective activity as depicted in the 90-day mean precipitation (Fig. 11). In the tropics, precipitation is associated primarily with cumulus convection, and the distribution of convective heating can influence large-scale circulation. The uniform runs of g8 (Fig. 11b) and g7 (Fig. 11c) reproduce, at least qualitatively, the intertropical convergence zone observed by the TRMM satellite; however, the g8 run tends to depict smaller-scale features. The reproducibility of convective precipitation along the intertropical convergence zones (ITCZs) in the uniform runs leads to the reproduction of the Hadley cells (Fig. 10). The 90-day precipitation in the NH was simulated in the stretched run of g7-str16 (Fig. 11d), and although the convection activity was greater because of the increased grid intervals, the precipitation rate in the tropics and the NH were well reproduced, which demonstrates its overall consistency with its observational counterpart. However, the g7-str36 and g7-str64 runs (Figs. 11e,f) failed to reproduce precipitation along the ITCZs in the Atlantic and Pacific and simulated too much precipitation over Southeast Asia and the tropical western North Pacific. As a likely result of the smaller grid spacing in the NH than in the SH, this unrealistic northward displacement of the core region of convective rainfall, as can be confirmed in the zonally averaged precipitation rate as the overestimation between 10° and 30°N (Fig. 11g), should cause the misrepresentation of the Hadley cells in several of the stretched runs. Specifically, the unrealistically weak NH cell and strong SH cell should be caused by the misrepresented convective precipitation as observed in the boreal summer.

Over the midlatitude ocean basins, zonally extending bands of modest precipitation that form along “storm tracks” are reproduced in the uniform runs (Figs. 11a–c). A “storm track” is a region where synoptic-scale baroclinic eddies develop recurrently and, thus, is identified as a region where there are subweekly fluctuations in pressure, winds, and temperature (Nakamura 1992). In Fig. 12, the 90-day mean field of the 850-hPa northward heat flux is plotted, where a prime denotes deviations from the 90-day mean; an 8-day high-pass filtering was applied beforehand to extract the subweekly fluctuations. In the NCEP FNL (Fig. 12a) and uniform NICAM runs (Figs. 12b–d), well-defined storm tracks at approximately 50°S associated with the PFJ are marked as large negative values of the flux over the South Pacific, Atlantic, and Indian Oceans (Nakamura and Shimpo 2004; Nakamura et al. 2004). In contrast, the stretched NICAM runs severely underestimated synoptic-eddy activity and the associated heat flux (Figs. 12e–g) along the SH storm tracks and thereby also underestimated the intensity of the eddy-driven PFJ (Figs. 9e–g) and midlatitude precipitation (Figs. 11d–f). The poleward heat by synoptic eddies acts to reduce the thermal gradient and subsequently the thermal wind to maintain the surface westerlies as a surface manifestation of the PFJ (Nakamura et al. 2004). In fact, the zonal-mean surface wind speed at 50°S is approximately 6–10 m s−1 for the g8 and g7 runs, which is realistic, approximately 4–6 m s−1 for the g7-str16 run, and 0–2 m s−1 for the g7-str36 and g7-str64 runs.

In the NH, the storm tracks are normally observed over the Pacific and Atlantic along the Kuroshio, Gulf Stream, and their extensions, respectively, where recurrent cyclone development occurs (Nakamura et al. 2004; Fig. 12a). The two storm tracks are well reproduced in the uniform runs of g7 and g8 (Figs. 12b,c); however, the eddy activity is underestimated in the g6 run (Fig. 12d). The stretched runs can reproduce the storm track over the Atlantic where the grids are concentrated (Figs. 12e–h); however, the reproducibility of its counterpart over the North Pacific is sensitive to the s ratio and tends to lower as the ratio increases.

Recent studies noted that efficient restoration of the gradient of the surface air temperature (SAT) under the frontal SST gradient along the oceanic fronts through the differential surface heat flux across the gradient is essential for baroclinic instability for the formation of storm tracks (Nakamura et al. 2004; Sampe et al. 2010; Hotta and Nakamura 2011). Figure 13 shows the meridional gradient of the 90-day-averaged SAT simulated in the (Fig. 13a) g7 and (Fig. 13b) g7-str16, (Fig. 13c) g7-str36, and (Fig. 13d) g7-str64 runs. The influence of the SH subarctic frontal zone at 45°S along the Antarctic Circumpolar Current is reproduced in the g7 run (Fig. 13a), whereas the SAT gradient becomes increasingly weaker in the stretched run (Figs. 13b–d) because of its coarse resolution. Additionally, the 15-day-average zonal-mean latent heat flux (Fig. 13e) at the surface of the g7 run is approximately 100 W m−2 at 40°S, which ranges from 60 to 140 W m−2 around the latitude circle; however, the flux is much less in the stretched runs. Thus, the stretched run severely underestimates the surface evaporation in the midlatitude southern ocean, which is important for the amplification of extratropical cyclones. For those runs, coarser grid intervals can lead to severe underestimation of the SST/SAT gradient and moisture supply from the ocean, which are essential for baroclinic development of synoptic-scale eddies, and thereby underestimate the midlatitude PFJ and associated surface westerlies. The unrealistically weak surface winds can in turn further reduce the surface flux and thereby weaken the baroclinic eddy development.

To analyze the amount of energy transported poleward by the mean meridional circulation and eddies separately, we calculated the seasonally averaged zonal-mean northward transport of moist static energy {, where h = cpT + gz + Lqυ is the moist static energy (MSE), the square brackets indicate the zonal average, and the overbars represent the 90-day time average} and the following decomposed contributions: the heat transport by the mean meridional circulations , the transient eddy transport , and the stationary eddy transport , where primes indicate deviations from the seasonal mean and asterisks indicate the eddy component defined as deviations from the zonal mean. Then, the terms are vertically integrated. Each of these decomposed terms has a different physical association: the transport by a Hadley cell is a primary factor in the mean meridional (MM) term, whereas the transient eddy (TE) term primarily represents the transport by midlatitude synoptic-scale eddies, and the stationary eddy (SE) term represents the transport by planetary waves forced primarily by land–seas thermal contrasts and large-scale orography (Magnusdottir and Saravanan 1999; Mori et al. 2013; Numaguti 1995). Figure 14 shows the 90-day mean northward transport of MSE in green and its decomposed terms: the MM term is the dashed red line, the TE term is the dotted–dashed magenta line, and the SE term is the dotted blue line. The SE term for all runs is overall smaller than that of the other two terms, which is possibly because it is for boreal spring. In the stretched runs, the unrealistically weak eddy development in the extratropical SH results in the TE term being reduced from −1.4 PW m−2 for g8 and −1 PW m−2 for g7-str16 to approximately −0.55 PW m−2 for g7-str64, whereas the peak transport is displaced toward the equator from 45° to 15°S. In the NH, the TE transport is comparable to its SH counterpart because the peak magnitude is approximately 0.82 PW m−2 in the g7-str64 run, and the largest is 1.18 PW m−2 in the g8 run, where, in each, the TE energy transport peaks at approximately 45°N. In the g7-str64 run (Fig. 14f), the artificial suppression of midlatitude synoptic-scale eddy activity over Eurasia and the North Pacific was found to cause rather modest reduction in the zonal-mean TE energy transport under the substantial contribution by the Atlantic storm track. Consistent with the zonal-mean streamfunction (Fig. 10), the magnitude of the peak MM transport in the SH Hadley cell increased from 2.3 PW m−2 in the g7-str16 run to 3.2 PW m−2 in the g7-str64 run and decreased in the NH from 2 PW m−2 in the g7-str16 to 1.5 PW m−2 in the g7-str64 run.

4. Nudging scheme applied to the stretched grids

In this study, we have shown that the artificially suppressed activity of midlatitude eddies and/or tropical convection within the domain of particularly coarse grid spacing in the stretched NICAM runs can severely affect the reproducibility of large-scale westerly jets and the associated temperature gradient. This degradation may potentially be suppressed by making use of a nudging scheme and/or a resolution-dependent convection scheme. It has been shown that we can further improve the g7-str64 results by better simulating synoptic-scale flows (Fig. 2g). We can achieve this improvement by nudging the variables u, υ, T, P, , and qυ to the 6-hourly NCEP FNL at all grid points located more than 2300 km away from the focal point to accommodate large-scale flows. The standard Newtonian relaxation scheme is used with a 1-h relaxation time and applied at every time step in the field far from the focus. We refer to this experiment shown in Fig. 2g as g7-str64N. The total precipitation amount and the tilt of the frontal systems appear to be even more realistic, particularly around upstate New York, where over two feet of snow was recorded (Soltow 2011). The time series of the RMSE of the 500-hPa temperature, which is shown in Fig. 5a with yellow lines, is nearly constant in time at approximately 1 K owing to being strongly constrained by the NCEP FNL data. Also, the RMSE of the nonnudged region (30°–50°N, 100°–80°W) displayed a similar pattern (not shown) of the nearly constant value of 1 K in time except there were a few spikes of 2–2.5 K in magnitude that were observed. Likewise, the corresponding RMSE in the 200-hPa zonal wind velocity is also nearly constant at approximately 5 m s−1 (not shown). Additionally, the seasonal-mean biases in temperature and velocities (not shown) were also greatly reduced. For the g7-str64N run (Fig. 12h), the locations and intensities of the NH storm tracks and westerly jet streams closely resemble those in the NCEP FNL data, and thus, the peak MM and TE transports of the moist static energy in the NH are also realistic (2.14 PW m−2 and 0.86 PW m−2, respectively; Fig. 14c).

5. Dependency of the focal point location

Thus far, the focal point of the stretched-grid system in our NICAM runs has been in the NH, which tends to yield some degradation in the reproduction of midlatitude jets and storm tracks in the SH and convective activity in the tropics. In our next attempt, as shown below, the focal point is moved to the equator (at 150°W) to yield a horizontal resolution that is symmetric between the NH and SH. The s ratios of 16 (g7-str16E), 36 (g7-str36E), and 64 (g7-str64E) were assigned. Figures 3d–f display scatterplots of the seasonal (70 day) mean 500-hPa temperature in the NCEP FNL against that simulated in the g7-str16E (Fig. 3g), g7-str36E (Fig. 3h), and g7-str64E (Fig. 3i) runs. Unlike in the stretched runs with the focal point in the NH, the simulations with the focal point at the equator simulate the 500-hPa temperature, whose reproducibility is comparable between the two hemispheres. However, the regression coefficient for the NH is approximately 0.8 for those runs, which is smaller than its counterpart (~0.9 or greater) for either the uniform or the corresponding stretched runs of g7-str16, g7-str36, and g7-str64. The lower regression coefficients are due to the cold bias at a higher latitude in both hemispheres, as indicated in Figs. 7g–i.

The mean meridional streamfunction plotted in Fig. 10 for g7-str16E (Fig. 10h), g7-str36E (Fig. 10i), and g7-str64E (Fig. 10j) displays Hadley cells, which are nearly symmetric about the equator in strength and meridional width. This symmetry results from the symmetrical grid size at the equator and the comparable activity of tropical convection between the NH and SH. Unlike the stretched runs of g7-str16, g7-str36, and g7-str64, the intensity of the Hadley cells in both hemispheres tended to decrease with increasing s ratio (Figs. 10h–j). Additionally, Fig. 11g shows that the g7-str36E and g7-str64E runs display double peaks in the zonal-mean seasonal precipitation rate in the tropics because coarser grid intervals are unfavorable for simulating active convection over the Congo and Amazon basins and the Maritime Continent.

In Fig. 12, a northward eddy heat flux is shown for g7-str16E (Fig. 12g), g7-str36E (Fig. 12h), and g7-str64E (Fig. 12i) runs, all of which simulate the strongest midlatitude storm-track activity in the North and South Pacific, which are domains that are relatively close to the focal point. Even in these regions, the storm-track activity tended to decrease with increasing s ratio. Furthermore, the reproducibility of the storm-track activity is even lower over extratropical regions that are a distance from the focal point, including Eurasia and the South Atlantic and Indian Oceans. This decreasing intensity in the MM and eddy transports of MSE for larger s ratios is also confirmed in Fig. 14 for the g7-str16E (Fig. 14g), g7-str36E (Fig. 14h), and g7-str64E (Fig. 14i) runs. The maximum intensity of the MM and TE transport terms weakens equally in both hemispheres as the s ratio increases. Therefore, the location of the focal point can affect the large-scale circulation in stretched runs through the reproducibility of midlatitude eddies and tropical convections.

Hemispheric aspects of the MSE transport for all the NICAM runs are summarized in Fig. 15. The top two panels display the average poleward MSE flux in black, and the average MM and TE fluxes are in blue and red, respectively, for the NH (Fig. 15a) and SH (Fig. 15b). The SE flux term is ignored for this case because its influence is much smaller compared with the other two. The bottom two panels display the fractions of the MM and TE fluxes to the total MSE flux, which are evaluated from the statistics shown in Figs. 15a and 15b in blue and red, respectively, for the NH (Fig. 15c) and SH (Fig. 15d). Select trends are clearly visible when the horizontal grids are stretched. For example, a comparison among the g7-str16, g7-str36, and g7-str64 run reveals increasing and decreasing tendencies for the MM and TE fluxes (and their fractions), respectively, with an increasing s ratio in the SH (Figs. 15b and 15d), which is in agreement with the stronger Hadley cell (Fig. 10) and weaker storm-track activity (Fig. 12) for larger s ratios. It shows an apparent compensation in the MSE transport between the MM and TEs. In the NH (Figs. 15a and 15c), by contrast, the aforementioned tendencies are less apparent; however, the g7-str64 run has a weaker transient eddy production. Interestingly, among the g7-str16E, g7-str36E, and g7-str64E runs, the MM transport tends to decrease with an increasing s ratio nearly in proportion to the corresponding decrease of the TE flux, thus rendering their fractions to the total MSE transport rather insensitive to the s ratio in both hemispheres.

6. Simulated features around the focal point

Figure 16 shows the histograms of the probability density of 3-hourly gridded data (1° intervals in both latitude and longitude) of the 500-hPa temperature sampled over the 90-day period within the domain (30°–50°N, 100°–70°W) for the NCEP FNL (Fig. 16a) and the NICAM simulations of g8 (Fig. 16b), g7 (Fig. 16c), g7-str16 (Fig. 16d), g7-str36 (Fig. 16e), g7-str64 (Fig. 16f), and g7-str64N (Fig. 16g). Centered at the focal point, the domain covers the eastern United States. Because its meridional gradient gradually decreases from March to May, the temperature histogram is skewed negatively. Unlike the g7-str64 run (Fig. 16f), which simulates a cold bias as much as 2.3 K, the stretched runs of g7-str16 (Fig. 16d) and g7-str36 (Fig. 16e) simulate temperature histograms that are similar to those based on the NCEP FNL (Fig. 16a) with negligible biases. In the g7-str64 run, the intensity of the eddy-driven PDF in the North Atlantic is underestimated, which affects the mean bias and distribution of the temperature within the domain, and the situation is similar in the uniform g7 run (Fig. 16c). Another uniform g8 run simulated a smaller bias but still overestimated the variance (Fig. 16b). Additionally, the histograms for the NCEP FNL (Fig. 16a) and g7-str64N run (Fig. 16g) are extremely similar, which indicates that the stretching scheme works well when used in combination with a nudging scheme.

Figure 17 displays the same histograms as in Fig. 16 but for precipitation of 100 mm day−1 or greater within the domain (30°–50°N, 105°–70°W) over the continental United States east of the Rocky Mountains. A precipitation rate of 100 mm day−1 or greater is counted from the 90-day (0000 UTC 30 April–0000 UTC 15 May 2011) period of the 3-hourly 0.25° × 0.25° resolution TRMM data (Fig. 17a) and the 3-h averaged 0.25° × 0.25° resolution NICAM results in the domain (Figs. 17b–g). Because of the smaller grid spacing around the focal point, which is more favorable for mesoscale convection, the frequency of heavy precipitation greater than 150 mm day−1 is more realistic in the stretched runs (Figs. 17d–f) than in the uniform run of g7 (Fig. 17c). Furthermore, the seasonal-mean precipitation and number of grids found to be precipitating at a rate of 100 mm day−1 or greater during the 90-day period increased as the s ratio increased. For this seasonal simulation, we find that the stretched runs produce results comparable to or better than those from g7, particularly for reproducing the moderate to heavy rain on regional scales.

7. Summary and conclusions

In this paper, we used fully compressible, nonhydrostatic global circulation models with quasi-uniform grids (NICAM) and stretched grids (S-NICAM) for week-long and season-long simulations, which were compared with the NCEP FNL and TRMM data. Comparison of week-long accumulative precipitation amounts between the TRMM data and the stretched simulations showed that a larger s ratio for a stretched run leads to better representation of storms and associated precipitation because the errors generated in the outer regions with coarser grid spacing do not seriously affect the inner domain centered at the focal point. The comparison suggests that one can take full advantage of a higher spatial resolution of the inner domain for a week-long simulation to reduce the computational time.

For the seasonal simulations, however, we found that stretched runs can be seriously affected by large-scale errors that arise from unrealistic representations of midlatitude synoptic-scale eddies and/or tropical convections in the regions of coarse grid spacing that are located away from the focal point. In our particular set of stretched runs with the focal point located in the eastern United States, the artificial suppression of baroclinic development of midlatitude eddies in the SH weakens the eddy-driven PFJ and associated surface westerlies, yielding a cold bias at mid- to high latitudes. The coarser grid spacing in the SH also acts to suppress tropical convection, resulting in an unrealistic shift of the rising branch of Hadley cells into the NH tropics during boreal spring as if it were boreal summer. In another set of our stretched runs for boreal spring with the focal point located in the equatorial Pacific, the Hadley cells are more symmetric about the equator where their intensity decreased with increasing s ratio, which is a measure of the strength of the stretching. In addition, stretching also acts to suppress the development of midlatitude eddies more equally over the two hemispheres. For seasonal simulations, the grid stretching can thus affect midlatitude storm-track activity and tropical convective activity, and the former further affects the PFJs and the surface westerlies, whereas the latter affects the Hadley cells, STJs, and the surface trades. The influence of grid stretching also extends into the poleward MSE transport by meridional overturning cells, including the Hadley cells, and by midlatitude transient eddies.

Finally, we assessed the reproducibility of the 500-hPa temperature and that of moderate and heavy precipitation within the focal region of the stretched runs compared with those of the corresponding uniform runs. The reproducibility is overall higher in the stretched runs, particularly with moderate grid stretching (i.e., g7-str16 and g7-str36 runs). Furthermore, the reproducibility can become even higher in the stretched runs around the focal point if combined with a nudging scheme.

Although our assessment of the performance of the stretched NICAM simulations was rather detailed, it was based on the limited choices of the focal point and simulation period; therefore, it is still difficult to identify the relationship between the error growth and s ratio within a particular domain for a given simulation. Furthermore, caution must be exercised when using SG models. For example, for a sensitivity study to analyze how an increase in the prescribed SST can influence the local climate, SG models may not be the best choice because the possible influence on large-scale circulation as a model response (such as a change in location of the jets) may not be well represented.

In summary, SG grid models with a moderate or strong s ratio are most effective when used with a nudging scheme. If global data are nudged, for example, one can output data of a uniform-grid model that shows a reasonably high reproducibility of tropical convections and midlatitude eddies. Then, an SG model can provide the domain around the focal point with a horizontal resolution that is sufficient to resolve mesoscale features while circumventing the boundary issues the nesting scheme must face. When nudging is not an option, we recommend setting the s ratio to moderate values and selecting a focal point to allow extratropical transient eddies and tropical convections to develop realistically within the hemisphere of interest to reduce the mean bias.

Acknowledgments

The authors thank Dr. D. Goto for useful discussions. Parts of the research were supported by funds from MOE/GOSAT2, JST/CREST/EMS/TEEDDA, JAXA/EarthCARE&GCOM-C, MEXT/RECCA/SALSA, MEXT/KAKENHI/Innovative Areas 2409, KAKENHI 25287120, MOEJ/ERTDF/S-12 and the 5-1501, and K-computer HPCI System Research project (hp140046, 150156).

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