The Advanced Research WRF is used to examine the sensitivity of simulations of Typhoon Shanshan (2006) to changes in horizontal grid spacing at gray-zone resolutions (7.5–1 km) and to choices of convective parameterization (CP) schemes. It is illustrated that fine resolution (3 to 1 km) runs feature a weak model convergence in tropical cyclone (TC) intensity than coarse resolution (7.5 to 5 km) runs, and the simulations using the Grell 3D ensemble CP (GR3D) exhibit a relatively strong model convergence, compared to the other three CP schemes. Further analyses reveal that model convergence relies on a number of factors that limit or promote TC intensification as resolution varies. Smaller radius of maximum wind (RMW) and thus smaller eyewall slope, along with smaller surface energy flux outside the eyewall and smaller eyewall asymmetries, are associated with finer resolution runs. These factors, especially the smaller RMW, promote the radial gradients of kinematical and microphysical attributes such as wind, pressure, latent heating, and mixing ratio of hydrometeors near the eyewall, causing growth of storm intensity, resulting in a weak model convergence. However, the impacts of the factors promoting storm intensification are notably different under different CP conditions, which are appreciably weaker in GR3D runs than in other three groups of CP experiments and responsible for a relatively strong model convergence in GR3D simulations. Overall, improvement of current CP schemes in simulation of TCs at gray-zone resolutions is needed and targeted by a further study in future.
 Current numerical weather predictions (NWPs) use high-resolution models with horizontal grid spacing of 1–10 km [Lean et al., 2008; Roberts and Lean, 2008]. For such high-resolution models, further refinements of both physical parameterizations and numerical techniques are required [Steppeler et al., 2003]. One of the major problems in NWP from a physics perspective is the treatment of deep moist convection. Due to the assumptions and the closure hypotheses upon which convective parameterization (CP) is based, many CP schemes may not be suitable for the NWP models with a grid spacing of a few kilometers [Molinari and Dudek, 1992; Hammarstrand, 1998]. It has been debated whether CP is necessary for numerical predictions of convective systems at those grid spacings. Gerard  used the term “gray-zone resolution” to refer to this type of resolution. Yu and Lee  suggested that this type of resolution can range from 1 km to 5 km. Generally, the grid spacings below the gray-zone resolution are considered sufficient for explicit simulation of convective weathers, and thus CP is not necessary [Liu et al., 1997]. In contrast, many studies showed that there will be some problems in high-resolution simulations without CP such as irresolvable convection and underprediction or overprediction of precipitation, and thus use of CP is also recommended for higher-resolution simulations [Kotroni and Lagouvardos, 2004; Deng and Stauffer, 2006]. Without using CP in high-resolution runs (e.g., 2.8 or 1 km grid runs), the model cannot properly reproduce some convective processes in evolving cumulus clouds [Niemelä and Fortelius, 2005; Craig and Dörnbrack, 2008]. Similar problems could also be encountered in tropical cyclone (TC) simulations. Rogers et al.  cited inadequate computational resources to run operational models at sufficiently high spatial resolution, along with incomplete representation of important physical processes, as two reasons for the slow improvement in TC intensity prediction. Without improvements in representation of model physics, the decrease of grid spacing alone could not significantly and continuously improve TC intensity forecast [Fierro et al., 2009]. Mass et al.  stated that “decreasing grid spacing in mesoscale models to less than 10–15 km generally improves the realism of the results but does not necessarily significantly improve the objectively scored accuracy of the forecasts.” Gentry and Lackmann  (GL10) suggested that, as grid spacing is decreased, the structure and evolution of Hurricane Ivan (2004) undergo significant changes, with the finest resolution, the 1 km resolution, exhibiting a markedly different structure from those of the other simulations. It is assumed that perfect models should produce convergent TC intensity and structure as the grid spacing decreases although the converged solution does not necessarily approach the truth; otherwise, the model is considered not convergent. The GL10 results indicate that the model solution has not yet converged at the grid spacings ranging from 8 km to 1 km. Bryan et al.  also found that the model did not converge with grid spacings well below 1 km in their convective system simulation. In fact, it is impossible for the simulated storm to intensify indefinitely as the grid spacing decreases, i.e., the model will for sure converge at a certain very high resolution, but this resolution may not be practical to run with under the present-day computer resources. Moreover, it is still a central question, as addressed by Kain et al. , whether the extra computer resources required to run high-resolution models produce a worthwhile increase in forecast accuracy.
 Why is there an absence of model convergence in high-resolution simulations? It might be expected that the magnitude of vertical motions would increase as smaller grid spacing is used, as updrafts are more adequately resolved. However, the conclusions of Bryan et al.  suggested that higher resolution could eventually weaken the magnitude of vertical motions as detrimental processes such as entrainment with grid-scale turbulence begin to be resolved. GL10 focused on what changes occur in the representation of physical processes important to TC intensity as grid spacing decreases and implied that these changes may contribute to the additional and unnecessary strengthening of the simulated TC at high resolution.
 Furthermore, CP schemes are not generally designed for the finer grid lengths utilized in this study, nor in the inner-core regions of TCs [Molinari and Dudek, 1992; GL10]. The use of CP would implicitly account for a portion of the eyewall updraft, thereby weakening the upward branch of the secondary circulation and the compensating grid-scale subsidence within the eye [Gentry, 2007]. This problem is important because many studies on convection organized on the mesoscale (including TCs) systematically make use of horizontal grid spacing of less than 10 km within the innermost mesh. Also, the future real-time hurricane forecasts using mesoscale models will increasingly make use of cloud-resolving grid spacing (i.e., grid spacing < 5 km) as computer power increases. Thus, it is urgent to find a reasonable CP scheme suitable for the high-resolution model to obtain more realistic results with a stronger model convergence and thus a better forecast value.
 Despite some studies concerning the weak model convergence in simulation of TC intensity as grid spacing is decreased, they did not provide detailed and systematic analysis on the reasons for the weak convergence, much less the impact of CP schemes on this model convergence. Even it is not clear yet whether CP could have a remarkable impact on this model convergence. In this study, through a series of sensitivity experiments, the following three issues will be addressed: (1) possible reasons for the weak model convergence in simulating TC intensity at various gray-zone model resolutions, (2) impacts of different CP schemes on this convergence, and (3) whether it is possible to find a reasonable CP scheme that works better at high resolution to make this convergence stronger. The results obtained herein could have potential applications to widely used numerical research and forecast models.
 The rest of the paper is organized as follows. Section 2 describes the numerical model and the design of numerical experiments. Section 3 discusses the sensitivity of TC track, intensity, and structure to horizontal resolution and CP schemes. The possible reasons for the weak model convergence in simulations of TC intensity are analyzed in section 4. A summary and conclusions are given in the final section.
2 Model Configuration and Experimental Design
 Typhoon Shanshan was generated over the western Pacific (16.7°N, 134.9°E) at 1200 UTC 10 September 2006. After its occurrence, it moved westward with its intensity increased unceasingly. At 1800 UTC 14 September, it moved to the east of Taiwan (20.7°N, 124.6°E) and developed into an extremely severe typhoon. Then, it turned northeastward and made a landfall in Japan at about 1000 UTC 17 September. Shanshan has the typical turning track over the sea with the characteristics of high intensity, long duration, and fast development, which are illustrated by the best track data obtained from the Joint Typhoon Warning Center (JTWC) (http://www.usno.navy.mil/NOOC/nmfc~ph/RSS/jtwc/best_tracks/wpindex.html).
 The model used in this study is the Advanced Research Weather Research and Forecasting (WRF-ARW) model. The model domain is triply nested (DM1, DM2, and DM3) for most cases, with an added nest (DM4) for 1 km resolution runs. The inner meshes (DM2, DM3, and DM4) automatically move to follow the model storm [Skamarock et al., 2008]. The initial and lateral boundary conditions were obtained from the 1° × 1° National Centers for Environmental Prediction final analysis data at 6 h intervals (http://dss.ucar.edu/dsszone/ds083.2). The TC Bogus scheme in the WRF model was adopted to build the initial field [Skamarock et al., 2008]. In order to consider the effect of the cool upwelling on TC intensity [Chan et al., 2001; Zhong and Zhang, 2006], daily sea surface temperature (SST) was updated using the Tropical Rainfall Measuring Mission (TRMM) Microwave Imager (TMI) level-1 standard product at a 0.25° × 0.25° resolution, provided by the Earth Observation Research Center (EORC)/National Space Development Agency (NASDA) (http://www.eorc.nasda. go.jp/TRMM).
 The model integration started at 0000 UTC 14 September 2006 and ended at 1200 UTC 16 September 2006, with a total of 60 h for both the intensification period and the steady-state period of Shanshan. The Yonsei University nonlocal-K planetary boundary layer (BL) scheme [Hong et al., 2006] and Monin-Obukhov surface layer scheme [Paulson, 1970; Dyer and Hicks, 1970; Webb, 1970; Beljaars, 1994] are used in this study. Readers are referred to Sun et al.  for the model configuration in detail. It should be mentioned that the present study uses the hybrid (double moment for ice crystals and single moment for all other species) Thompson et al.  microphysical scheme, which is a new scheme with ice, snow, and graupel processes suitable for high-resolution simulations. In the Thompson scheme, riming growth of snow is required to exceed depositional growth of snow by a factor of 3 before rimed snow transfers into the graupel category, resulting in more realistic values for graupel mixing ratio in the eyewall [Fierro et al., 2009].
 Our study consists of four groups of simulations: simulation with (1) no CP (NOCP), (2) Betts-Miller-Janjić CP (BMJ) [Betts, 1986; Betts and Miller, 1986; Janjić, 1994; Janjić, 2000], (3) Kain-Fritsch CP (KFEX) [Kain and Fritsch, 1990; Kain, 2004], and (4) Grell 3D ensemble CP (GR3D) [Grell and Devenyi, 2002; Skamarock et al., 2008] in the finest mesh. Each group contains four simulations at 1, 3, 5, and 7.5 km horizontal resolutions in its finest mesh. For example, GR3D1, GR3D3, GR3D5, and GR3D7.5 represent the GR3D simulations with grid sizes of 1, 3, 5, and 7.5 km, respectively. In addition, the GR3D scheme was first introduced in WRFV3.0, and so is new, and not yet well tested in many situations. Based on an ensemble mean approach, it shares a lot in common with the Grell and Devenyi  scheme, but the quasi-equilibrium approach is no longer included among the ensemble members. The scheme is distinguished from other cumulus schemes by allowing subsidence effects to spread to neighboring grid columns, making the method more suitable to grid sizes < 10 km than other options [Skamarock et al., 2008]. Thus, in the following sections, it is worthy of more attention to its performance in the high-resolution runs.
 In this study, we adopted a reasonable approach to maintain the consistency of model design and thus effectively study the impact of horizontal resolution and CP in a nested domain configuration: The DM1, DM2, and DM3 in all cases cover the same areas with horizontal dimensions of 2475 × 3375, 1500× 1500, and 785× 785 km2, respectively (Table 1), and the choice made in the physics and dynamical setup except for the CP in the innermost domain is also consistent with each other cases. In addition, the DM3 and DM4 in 45-15-5-1 cases used the same CP. In this study, we wish to put emphasis on the need to keep consistency between all cases. Also, the analysis presented herein entirely focuses on the innermost-domain data, whose dimensions were chosen to cover the great majority of the TC's convection in all cases.
Table 1. Dimensions of the Domains for the Four Groups of Simulations With Different Horizontal Resolutions in Their Innermost Mesh
55 × 75
100 × 100
105 × 105
55 × 75
100 × 100
157 × 157
55 × 75
100 × 100
261 × 261
55 × 75
100 × 100
157 × 157
501 × 501
3 Sensitivity of TC Activity to Horizontal Resolution and CP Schemes
3.1 Storm Track
 Figure 1 compares the storm tracks from these sensitivity experiments overlaid with the JTWC best track. It is important that WRF-ARW was able to capture the track of Shanshan well, and in all cases the simulated storms followed a similar track to that of the JTWC, except for the 7.5 km runs which translated a little more rapidly than the observed in the last 24 h simulation. Including those in 7.5 km runs, the simulated time-averaged track errors are no more than 100 km. This is not a surprising result and can be attributed to the consistency of model design for DM1 and DM2 in all cases: As Marks and Shay  suggested that, unlike storm intensity, track prediction depends more on larger-scale processes that can be resolved with coarse grid prediction models. Namely, in all cases, the similarity of simulated tracks results from the similar larger-scale environmental fields produced by the DM1 and DM2 simulation. On the other hand, the similar simulated tracks, in turn, further ensure the similar larger-scale environmental fields. Therefore, due to the consistency in model design (including domain sizes and the choice made in the physics and dynamical setup) and thus TC tracks and larger-scale processes, it is more reliable to study the impact of horizontal resolution and CP on storm intensity and structure without interference from other factors.
 A further comparison of the simulated tracks among different resolution runs reveals that the track error decreases notably as the horizontal resolution of the innermost domain decreases from 7.5 km to 5 km. This is because the storm track depends on not only larger-scale steering flow but also storm structure [Fiorino and Elsberry, 1989], and the location of storm center will be more accurate due to the more realistic storm structure produced in higher-resolution runs. In terms of the storm center position, the higher-resolution runs (i.e., 3 km and 1 km runs) in the four CP groups all presented well. However, the simulated track error changes little when the resolution further decreases from 5 km to 1 km. In addition, in the fine-mesh runs (resolution < 5 km), the choice of CP had little effect on storm track; the turning of the track from northwestward to northeastward was captured decently by all the four CP groups of runs. Thus, in the fine-resolution cases (such as resolution < 5 km), the decrease of resolution and the choice of CP in the WRF-ARW cannot contribute to the notable improvement in simulation of TC track.
3.2 Storm Intensity
 Though some recent studies have noticed the impact of grid spacing on TC intensity at grid spacings below 5 km, no final conclusion has yet been reached on this matter. Fierro et al.  proposed that the features in finer-resolution simulations that tend to weaken TCs (i.e., smaller area of high surface fluxes and weaker total updraft mass flux) compensate for the features that favor TC intensity (i.e., weaker eyewall asymmetries and larger radial gradients); thus, this will result in a similar intensity between finer- and coarser-resolution runs. However, GL10 suggested that, due to changes in the representation of physical processes important to TC intensity, the simulated TC at higher resolution could be additionally strengthened. In this study, our results are basically consistent with GL10 in that the finer-resolution cases do exhibit stronger TC intensity than those at coarser resolution, and the model solution has not yet converged at these grid spacings.
 To demonstrate the impact of grid spacing and CP on TC intensity, we compare in Figures 2 and 3 the time series of minimum sea level pressure (MSLP) and maximum surface wind speed (MWS) for the sensitivity experiments and the JTWC best track. Davis et al.  found that, TC intensity did show notable changes as grid spacing was decreased from 4 km to 1.3 km (about 20 hPa in MSLP and 13 m s−1 in MWS). The similar result was also obtained in GL10. In our simulations, except for the BMJ cases, the other three groups of cases basically follow this principle: TC intensity underwent a relatively small change as the grid spacing decreases from 5 km to 3 km, while a significant increase was observed from 3 km to 1 km. However, in BMJ cases, TC intensity increases slightly as the grid spacing decreases from 7.5 km to 5 km but an extreme increase is observed from 5 km to 1 km. In addition, except for BMJ cases, due to the enhanced convection at grid scale, some deepening of the TC does occur as the grid length decreases from 7.5 km to 5 km, which is consistent with the results of GL10 for grid lengths from 8 km to 6 km. Thus, in all cases, as grid spacing is decreased to below 3 km, more deepening per decrease in grid spacing is realized. This indicates that based on the present CP, the model solution has not yet converged, especially with grid spacing below 3 km. Moreover, though the model convergence in GR3Ds is relatively strong and better than in NOCPs, BMJs, and KFEXs, the simulated peak intensity in its 1 km run is still about 10 hPa and 5 m s−1 stronger than that in its 3 km run.
 Overall, the increase in storm intensity is not uniform across similar reductions in grid spacing, and there is a significant increase in intensity as the grid spacing is decreased from 3 km to 1 km. On the other hand, due to the significant impact of CP on TC intensity and thus the model convergence, we expect to find a suitable and reasonable CP scheme to tackle the problem on the weak model convergence.
 Comparing the storm intensity in the sensitivity experiments with that in the best analysis finds that in most cases, the MSLP values were much stronger than the observed while the MWS speeds were notably weaker than the observed. Thus, it is difficult to make the simulated results consistent with the observed in terms of both MSLP and MWS. For example, in some higher-resolution cases (such as NOCP3, BMJ1, KFEX3, and GR3D1), though the simulated MWSs were relatively closer to the observed, the simulated MSLPs were quite different from the observed. Zhu and Zhang  found that the maximum wind can be affected by localized convective activities while the minimum sea level pressure (SLP) tends to be a system-integrated quantity that tends to be a more reliable measure of the vortex intensity. In general, comparing GR3Ds with the other three CP groups reveals that the evolutions of MSLP and MWS in GR3Ds were more similar to those observed. However, as a new CP scheme designed for high-resolution simulation, GR3D still has some problems to be solved. The simulated storms in GR3Ds presented a much earlier and faster intensification than that observed during the development period (about 40 hPa and 10 m s−1 from 1200 UTC 14 September to 0600 UTC 15 September).
3.3 Storm Structure
 In performing the comparisons of storm structure as functions of grid spacing and CP scheme, we focus our analysis on a single time in the mature stage here (e.g., 0000 UTC 16 September 2006). Selection of a single time will allow for a much detailed examination of spatial structure of the storm.
 Before the comparisons of storm structure, we first analyze the observed storm structure at the mature stage. The lack of intensity variation during this stage allows us to highlight the notable difference in storm structure between different cases. Figure 4 shows the rainfall rate from TMI and Precipitation Radar (PR) (at an approximately 4 km × 4 km resolution) at 0431 UTC 16 September 2006. The radius of the eyewall is about 40 km. The observed inner-core structure of TC presents strong asymmetry. The inner spiral rainbands are mainly distributed in the north of TC center while part of outer spiral rainbands appears west, southwest, and east of the TC.
 To verify the realism of simulated TC structure in those sensitivity experiments, we plotted Figure 5. Figure 5 shows the cross sections of simulated instantaneous rainfall rate at the mature stage of TC Shanshan. Compared to the observation in Figure 4, the simulated rainfall over the TC eyewall is notably larger than the observed. It needs to be stated here that relative to the magnitude of the simulated rainfall, we are more concerned with the spatial distribution of the TC rainfall. Moreover, following the decrease of grid spacing, the simulated rainfall rate did not get closer to the observed, but even became worse in NOCPs, BMJs, and KFEXs because the radii of eyewall and the asymmetric feature in their 1 km runs were quite different from the observed. Compared with the other three groups of CP cases, GR3Ds reproduced a relatively realistic spatial distribution of spiral rainbands, especially in the higher-resolution runs (e.g., GR3D1) in terms of the radius and location of spiral rainbands. Furthermore, the radii of the eyewall decrease notably as the grid lengths decrease from 7.5 km to 1 km (especially from 3 km to 1 km) in those four groups of CP experiments. This may be responsible for the difference in TC intensity between the coarser-resolution run and higher-resolution run, which will be further discussed in the next section.
4 Possible Reasons for the Weak Model Convergence in Simulation of TC Intensity
4.1 Energy and Mass Exchange
4.1.1 Energy Exchange at Air-Sea Interface
 Previous studies suggested that TCs intensify and maintain themselves against surface frictional dissipation by extracting energy from the underlying oceans. Thus, energy exchange at the air-sea interface is the key to the intensity change of a TC [Malkus and Riehl, 1960; Black and Holland, 1995]. The wind-induced surface heat exchange, which describes a positive feedback between the increase in surface energy flux (SEF) and the surface wind speed in the near-core region of a TC, is viewed as the dominant process that controls the rapid intensification of a TC [Emanuel, 1986; Rotunno and Emanuel, 1987]. SEF is the sum of sensible-heat (SH) and latent-heat fluxes (LH). As suggested by Sun et al. , SH is dependent on the surface wind speed and air-sea temperature difference (ASTD), which can be defined as SST minus air temperature at 2 m, while LH is dependent on the surface wind speed and air-sea moisture difference (ASMD), which can be defined as the saturation specific humidity minus air specific humidity at 2 m.
 In this study, in order to find the possible reasons for the weak convergence of the WRF-ARW in simulation of storm intensity mentioned above, area-integrated kinetic energy and SEF were calculated. Figure 6 shows the temporal evolutions of area-integrated SEF within a 200 km radius. Owing to the effect of abnormal warm SST and other favorable conditions, the area-integrated SEF increase continuously from 1200 UTC 14 September to 0300 UTC 16 September. Note that a significant reduction process of SEF occurred at about 0300 UTC 16 September 2006 in these experiments, for the simulated TC passed over a cold pool near Miyako Island during that period. In our experiments, the area-integrated SEFs presented an increasing trend as the horizontal spacing decreased from 7.5 km to 5 km except for KFEXs. The increased SEF in BMJ5 and GR3D5 could be attributed to its stronger area-averaged surface wind speeds (figure omitted), while the increased SEF in NOCP5 could be attributed to not only its higher surface wind speed but also its higher ASTD and ASMD (figure omitted). However, there was no significant difference between the integrated SEF in 5 km runs and 3 km runs in all the four groups of CP experiments. As the grid spacing further decreased from 3 km to 1 km, the integrated SEF in 1 km runs was not notably higher and even weaker because of the weaker area-averaged surface wind speeds (figure omitted), which is a result from the smaller RMW and thus the smaller extent of strong wind region in 1 km runs (shown in section 4.2). Thereby, the view from integrated SEF might explain the difference in storm intensity between 7.5 km runs and 5 km runs, but could not explain the difference between 3 km runs and 1 km runs, and thus other factors must be promoting storm intensity in 1 km runs.
 The radial distribution of the azimuthal mean SEF and SLP averaged in the TC's mature stage is shown in Figure 7. Similar to the difference in storm intensity between different resolution runs, the simulated SEF (SLP) in 5 km runs are notably greater (lower) than those in 7.5 km runs, especially in NOCPs and GR3Ds. As the grid spacing further decreased from 5 km to 3 km, except for BMJs, there were no substantial differences between 5 km run and 3 km run in terms of the SEF and SLP. Moreover, following the grid spacing decrease from 3 km to 1 km, except for GR3Ds, the maximum values of azimuthal mean SEF in the other three CP groups of experiments never underwent a significant change, but their radial positions shifted inward notably and the values of SEF outside the eyewall decreased substantially. Most importantly, this will lead to sharper radial gradients of SEF at the inner edge of the eyewall and thus sharper SLP gradients, which will bring about a more intense storm [Wang, 2009]. However, in GR3D1, although the maximum value of SEF also shifted inward, the SEF outside the eyewall did not decrease remarkably. This could not lead to a notable increase in the radial SEF and SLP gradients near the eyewall, neither to the increase of the simulated storm intensity in GR3D1.
 Above all, from the viewpoint of energy exchange at the air-sea interface, it is inferred that the sharp radial gradient of SEF near eyewall may be an important reason for the weak model convergence in storm intensity between 3 km runs and 1 km runs in NOCPs, BMJs, and KFEXs. As a new CP which is more suitable to high-resolution models, GR3D presents a relatively better model convergence in the simulation of storm intensity due to smaller radial gradients of SEF and SLP.
4.1.2 Inflow and Vertical Mass Fluxes in the BL
 Drawing on the works by Hendricks et al.  and Montgomery et al. , a secondary vortex enhancement mechanism was identified and demonstrated in Tory et al. . System-scale intensification was found related to the large-scale response to the net vertical mass flux driven by diabatic heating in the convective cores. This led to the enhancement of the secondary circulation in a manner akin to the classical Eliassen model of a balanced vortex driven by heat sources.
 The averaged inflow and vertical mass fluxes, and integrated BL kinetic energy are listed in Table 2. As the grid spacing decreased from 7.5 km to 5 km, the strength of the secondary circulation, measured by the inflow and vertical mass fluxes, showed a notable intensifying in nearly all the four groups of CP experiments (except for the inflow mass flux in KEFXs), consistent with the larger area-integrated SEF (Figure 6), larger hydrometeor mass aloft, and stronger updraft speeds presented in section 4.2 (Figures 12 and 13). A similar trend was simulated for the BL kinetic energy except for that in KFEXs, which also indicted the stronger near-surface wind fields. Although the BL kinetic energy in the 3 km runs is slightly higher than that in 5 km runs, the strength of the secondary circulation showed a weakening trend as the grid spacing decreased from 5 km to 3 km. However, compared with those in 3 km runs, the simulated storms in 1 km runs produced much smaller low-level inward mass flux, which by virtue of mass conservation must be consistent with smaller vertical mass flux in the eyewall and contributed to a smaller BL kinetic energy. This shows that, as the grid spacing is less than 5 km, the secondary circulation is weaker at finer resolution, which is consistent with smaller area-integrated SEF (Figure 6), and unfavorable for storm intensity. As that in area-integrated SEF, the smaller integrated mass flux and BL kinetic energy in higher-resolution runs may be a result from the smaller RMW and thus the smaller extent of strong wind region in 1 km runs (shown in section 4.2). However, it appears that at finer resolution, these unfavorable conditions for storm intensity (e.g., the smaller area-integrated SEF and mass flux), could be counteracted by other favorable conditions (e.g., larger radial gradient and other factors), and result in a stronger storm, which will be discussed later in the text.
Table 2. Time-Averaged Inflow (Second Column) and Vertical Mass Flux (kg m s−1) (Third Column) Out of a Cylinder of Radius of 100 km and Height of 1 km MSL From the Storm's Center in the TC Mature Stagea
Inflow Mass Flux (× 109 kg m s−1)
Vertical Mass Flux (× 109 kg m s−1)
BL Kinetic Energy (× 1013 m2 s−2)
The rightmost column shows the time-averaged BL integrated kinetic energy (m2 s−2) within a cylinder of height of 1 km above MSL and a radius of 200 km from the storm's center in the TC mature stage from 1800 UTC 15 September to 0600 UTC 16 September.
4.2 Differences in the Storm Structures
4.2.1 Asymmetric Structure
Fierro et al.  showed that during the steady-state period of a TC's development, the coarser-resolution cases exhibited larger-amplitude eyewall asymmetries, which has been shown to act as a potential brake on storm intensity increase [Peng et al., 1999; Yang et al., 2007]. To quantify the amplitude of the eyewall asymmetries in our sensitivity experiments, the simulated 10 m wind speed (in plane Cartesian coordinate) is first interpolated onto polar coordinate grids. The projected pole is taken as the TC center (the point with minimum wind speed). Make a fast Fourier transform (FFT) to the polar coordinate grid of the wind speed during the mature stage of the TC and then separate the wind into symmetric flow and perturbation flow that correspond to each azimuthal wave number, similar to what was done in Sun et al. . In addition, consistent with Fierro et al. , there is almost no difference between the results of FFT analysis on wind speed and that on radar reflectivity; thereby, the eyewall asymmetries could be represented by the asymmetric structure of wind speed.
 The amplitudes of wave number 0–4 of wind speed at 10 m for 3 km and 1 km runs averaged in the mature stage were provided in Table 3. In NOCPs, BMJs, and KFEXs, as the grid spacing is decreased, the amplitudes of asymmetric waves were reduced somewhat, which is consistent with the findings of Fierro et al. . Moreover, as suggested by Peng et al.  and Yang et al. , the presence of asymmetric structure is a dynamical limiting factor to TC intensity. Thereby, conversely, the weakening of asymmetric waves in the three 1 km runs could increase the storm intensity and thus the difference in TC intensity between those 3 km runs and 1 km runs. However, in GR3Ds, in contrast to the previous three CP group cases, the amplitude of wave number 1 increased drastically as the grid spacing decreased from 3 km to 1 km. This may be a limiting factor to storm intensity in GR3D1 and thus a possible reason for the relatively smaller difference in TC intensity between GR3D3 and GR3D1. In addition, the larger-amplitude eyewall asymmetries in GR3D1 also explain why axisymmetric means (i.e., symmetric flow) are generally smaller in GR3D1 than those in GR3D3 (Table 3).
Table 3. Amplitude of Fourier Spectral Decomposition of Wind Speed at 10 m for 3 km and 1 km Runs Averaged in the TC Mature Stagea
NOCP3 / NOCP1
BMJ3 / BMJ1
KFEX3 / KFEX1
GR3D3 / GR3D1
All values are in m s−1.
52.00 / 51.85
45.36 / 50.76
54.93 / 54.81
50.11 / 47.90
Wave number 1
19.67 / 11.36
8.88 / 7.17
14.14 / 9.73
5.21 / 11.84
Wave number 2
5.81 / 3.60
2.91 / 4.36
3.91 / 2.34
4.14 / 4.47
Wave number 3
1.98 / 1.57
1.64 / 1.11
1.59 / 1.05
1.90 / 1.23
Wave number 4
0.66 / 0.77
1.02 / 0.61
0.92 / 0.74
0.47 / 0.69
 Due to the specificity of asymmetric structure in GR3Ds, it is necessary to further investigate the spatial distribution of the asymmetric structure of wind speed at 10 m in GR3Ds, which is shown in Figure 8. Compared with GR3D3, although the distribution of wind speed in GR3D1 is somewhat similar to that in GR3D3, the maximum wind zone in GR3D1 is notably closer to the TC eye (Figures 8a and 8e). The calm zone of the symmetric flow in GR3D1 is notably smaller, and the amplitude of symmetric flow in GR3D1 is significantly smaller though its maximum wind speed is larger than that in GR3D3 (Figures 8b and 8f). The wind speed perturbations of wave number 1 in the two runs all have two wave peaks, namely, inner wave peak and outer wave peak (Figures 8c and 8g). Nevertheless, the amplitude, phase, and radial position of the two wave peaks in GR3D1, especially the inner wave peak, are notably different from those in GR3D3, which are the most important reason for the difference in asymmetric structures between GR3D3 and GR3D1. The amplitude and phase of wave number 2 in GR3D1 are basically consistent with those in GR3D3, but similar to the situation in wave number 1, the radial position of wave peak in GR3D1 is notably closer to the storm center (Figures 8d and 8h). Furthermore, the inward shifts of wave peaks in wave number 0–2 could also be observed in the other CP groups of experiments (figures omitted), which could result in the contraction of the eyewall and favor the development of the storm.
4.2.2 Thermodynamic Structure
 Latent heating is the direct force that drives the eyewall convection and the secondary circulation of a hurricane, which causes the storm to maintain itself and develop further. In this study, the latent heating was found to be sensitive to the choice of CP scheme and horizontal resolution. Figure 9 shows theazimuthal- and time-averaged cross sections of the model-simulated latent heating in the TC mature stage. As expected, due to the more resolved convection, the higher-resolution runs produced larger magnitude of latent heating near the eyewall. However, compared with those in coarser-resolution runs, the simulated latent heating in higher-resolution runs were concentrated in much smaller areas closer to the eye. Hack and Schubert  showed that the smaller radius at which latent heating occurs, the more significant contribution to the central pressure fall is. Thereby, the simulated latent heating at smaller radius in higher-resolution runs contributes to more central pressure falls, leading to a larger increase in the TC intensity (Figures 2 and 3).
 On the other hand, the simulated latent heating is also quite different under different CP conditions. The four CP groups with the most simulated latent heating, in order, are KFEXs, NOCPs, BMJs, and GR3Ds, which is consistent with their rank in order of the simulated TC intensity (Figures 2 and 3). As the grid spacing decreased from 7.5 km to 5 km, except for the BMJ cases, the simulated latent heating increased significantly, resulting in the increase of TC intensity. However, in BMJs, compared with BMJ7.5, the magnitude of latent heating in BMJ5 was similar to that in BMJ7.5, and the factor promoting storm intensity in BMJ5 (i.e., wider extent of latent heating) was offset by the factor limiting storm intensity (i.e., larger radius at which latent heating occurs). These may be responsible for the similarity of TC intensity between BMJ7.5 and BMJ5. Moreover, except for BMJs, there are no significant differences in latent heating and thus TC intensity between 5 km runs and 3 km runs in the other three CP groups. The radius at which the latent heating occurs in BMJ3 is much smaller than that in BMJ5, which may contribute to the significant difference in TC intensity between BMJ5 and BMJ3. Furthermore, as the grid spacing further decreased from 3 km to 1 km, the larger magnitude and smaller radius of latent heating in 1 km runs may be responsible for stronger TC intensity in 1 km runs than in 3 km runs. Moreover, Figure 9 clearly shows that GR3Ds produced the least change in the vertical distribution of latent heating especially as resolution shifted from 7.5 km to 3 km, demonstrating the least resolution dependence of the GR3D CP scheme. The same result can also be found in the vertical distributions of vertical velocity and hydrometeor mixing ratio (see Figures 12 and 13 in section 4.2.3). It should be noted that, the latent heating could also affect the pressure gradient force near the eyewall and thus RMW and TC intensity, which will be discussed next.
4.2.3 Dynamic Structure
 In the previous section, we showed that the radii of eyewall decrease notably as the grid spacings decrease from 7.5 km to 1 km (especially from 3 km to 1 km). Stern and Nolan  found a linear relationship between the size of the radius of maximum wind (RMW) (which basically corresponds to the radius of eyewall) and outward slope of eyewall based on both observational data analysis and theoretical deduction [Emanuel, 1986]. Moreover, many studies have further investigated the relationship between the outward slope of eyewall and TC intensity, but the results were various. Yang et al.  showed that storms with larger eyewall tilt (or slope) were more intense because this tilt allowed more low-θe downdraft air to reach the subcloud inflow layer which in turn increased the air-sea entropy difference there and therefore the energy input from the sea. They suggested that storms exhibiting less vertical tilt (smaller eyewall slope) were less intense because of enhanced inward potential vorticity mixing from the eye to the eyewall. However, Fierro et al.  found that, despite larger eyewall slopes, the coarser-resolution runs still did exhibit similar TC intensity to those at finer resolution. Furthermore, in the work of GL10, it is implied that the TCs with a smaller eyewall slope in finer-resolution cases were notably stronger than that with a larger eyewall slope in coarser-resolution cases. Our results are basically consistent with GL10: due to the larger radii of the eyewall and thus the larger eyewall slope, the coarser-resolution cases do exhibit weaker intensity than those at finer resolution. This may be an important reason for the larger differences in TC intensity between 3 km runs and 1 km runs (Figures 2 and 3).
 To provide a more general description of the radii of eyewall, the evolution of the overall inner-core size of the simulated storms were shown in Figure 10 in terms of azimuthal mean RMW at 10 m. Since the eyewall of the simulated storm can be asymmetric from time to time [Wang, 2007], we only use the azimuthal mean as a proxy of the location of the overall eyewall. Basically, it is found that the RMW increased with time in all the coarser-resolution runs (i.e., 7.5 km and 5 km runs) and some higher-resolution runs, while decreased throughout the simulation in the other higher-resolution runs (i.e., NOCP1, BMJ1, BMJ3, and KFEX1). Most of all, except for BMJ5 and GR3D3, the RMWs for the remained cases decreased notably in general as the grid spacings decreased, which is consistent with variations of model-simulated rainfall shown in Figure 5 and has been noted by many studies [Yau et al., 2004; Davis et al., 2008; GL10].
 We can see from Figure 7 that, as a result of more grid-resolved convection in finer-resolution runs, the low-level pressure gradient near and outside the RMW was larger than that in coarser-resolution runs. To understand how the RMW responds to changes in the radial pressure gradient associated with grid-resolved convection, we performed a radial momentum budget analysis below. Similar to the work of Gopalakrishnan et al. , the budget equation for the azimuthal mean radial winds can be approximated by
where ur and vλ are azimuthal mean radial and tangential winds; r is radius; ρ and p are air density and pressure; f is the Coriolis parameter; and Dur is the parameterized subgrid-scale diffusion, including friction, of radial winds. In the absence of friction and the forces that constitute balance, equation ((1)) reduces to the gradient wind equation (term A).
 Figure 11 provides a Hovmöller diagram of the azimuthal-averaged net radial forcing term without diffusion [i.e., term A in equation ((1))] at the 100 m level. Starting with the earliest work on the evolution of a balanced vortex by Eliassen , several theoretical models have assumed that the acceleration and diffusion terms in equation ((1)) that describe the secondary circulation may be neglected so that the vortex is in a state of gradient wind balance [e.g., Emanuel, 1986; Willoughby, 2009]. Within the inflow layer in Figure 11, winds were subgradient in the outer radii but became supergradient in the eyewall region where the inflow diminished in magnitude and the convective updraft erupted, which is consistent with the results of Gopalakrishnan et al. . As the grid spacing decreased, updrafts were more adequately resolved, especially from 3 km to 1 km cases (Figure 11). At a grid spacing of 1 km, as suggested by GL10 and Yu and Lee , features within the eyewall (i.e., an ensemble of updraft cores within the eyewall) began to be somewhat resolved. Together with the increase of the latent heating in the eyewall (Figure 9), the central pressure was decreased (Figure 7), leading to increased pressure gradient near the eyewall. Subsequently, under the stronger pressure gradient force and thus stronger net radial force (Figure 11), the eyewall in 1 km runs would contract inward, resulting in a much smaller RMW (Figures 5 and 10). This mechanism could also explain the reduction of RMW as grid spacing decreases from 7.5 km to 5 km. However, due to the similar grid-resolved convections for the 5 km runs and 3 km runs except for BMJs (Figure 11), there is no essential difference in net radial force between 5 km runs and 3 km runs, resulting in similar RMWs. It should be emphasized that the smaller RMW in finer-resolution runs also in turn contributes to the sharper radial pressure gradient. Thereby, it appears that as the grid spacing decreases, there is a feedback between the reduction of RMW and the increase of radial pressure gradient, which contributes to the further reduction of RMW. In fact, at the finer-resolution runs, the reduced RMW has also strengthened the centripetal force in term A, thus offsetting the pressure gradient force and limiting the further reduction of RMW.
 To illustrate the relationship between the outward slope of the eyewall and the size of the RMW in these simulations, Figure 12 shows the azimuthal- and time-averaged cross sections of positive vertical motion and tangential velocity in the mature stage of the TC. As expected, due to the decreased RMW, the higher-resolution runs are characterized by more upright (i.e., smaller slope) eyewalls in terms of vertical and tangential velocity. Stern and Nolan  also showed that the outward slope of the eyewall with height is directly proportional to the size of the RMW. In addition, as the grid spacings decreased, the vertical velocities in those runs were not all strengthened and even weakened (e.g., from 5 km runs to 3 km runs), but the storm intensities were almost all increased. This is consistent with the findings of Bryan et al.  in that higher resolution could eventually weaken the magnitude of vertical motions as detrimental processes, such as entrainment with grid-scale turbulence, begin to be resolved. Thereby, as the grid spacing decreased, the intensification of TC could not be purely attributed to the strength of convection.
 Comparison of the slope of eyewall (Figure 12) with the TC intensity (Figures 2 and 3) reveals a close relationship between the difference in the slope of eyewall and the difference in TC intensity. In general, the simulated storm with less (more) eyewall slope in the finer- (coarser-) resolution runs are notably stronger (weaker), which is also confirmed in GL10. Moreover, in this study, we found that, as the reduction of eyewall slope is not uniform across similar decrease in grid spacing, the increase in TC intensity is also not uniform. Specially, as the grid spacing decreased, if the change of eyewall slope was relatively smaller, the TC intensity would change relatively little (such as in cases of NOCP5-3, BMJ7.5-5, KFEX5-3, and GR3D5-3), while in the other cases (such as in cases of NOCP7.5-5, NOCP3-1, BMJ5-3, BMJ3-1, KFEX3-1, GR3D7.5-5, and GR3D3-1), the eyewall slope changed notably, and thus a notable TC intensity change occurred. Based on the above analysis, this study suggested that there exists an evident and robust relationship between eyewall slope and TC intensity.
 Furthermore, the relationship between eyewall slope and TC intensity may be attributed to the impact of the radial gradient. The ascending air parcels in the eyewall normally experience a reduction of the inward-directed pressure gradient force (in terms of tangential velocity), leading to an outward centrifugal displacement with increasing height, namely, the eyewall slope. In other word, the larger (smaller) eyewall slope corresponds to a less (more) upright tangential velocity contours near the eyewall (Figure 12). Moreover, a less (more) upright tangential velocity often comes out with a weaker (stronger) radial gradient of tangential velocity, as suggested by Fierro et al. , which would limit (promote) storm intensity.
4.2.4 Microphysical Structure
 After analyses on the asymmetric, thermodynamic, and dynamic structure of the simulated storm, this section will be devoted to its microphysical structure, which will help to further identify factors promoting/limiting storm intensity and thus the reasons for the weak model convergence in simulation of TC intensity. Figure 13 shows the azimuthal- and time-averaged cross section of total mixing ratio of model-simulated hydrometeors and wind vector in the mature stage of the TC. The hydrometeors are distributed mainly near the 8 km height level, which is consistent with the distribution of vertical velocity in Figure 12 and the wind vectors in Figure 13. Previous studies suggested that, this is because the upper-level maximum of convection could be a consequence of water unloading and buoyancy [Samsury and Zipser, 1995; Trier et al., 1996].
 On the other hand, as grid spacing is decreased, the horizontal gradient of mixing ratio of hydrometeors and radar reflectivity became stronger and more concentrated near the eyewall, and the area occupied by the primary circulation becomes more compact in the horizontal and elongated in the vertical. Moreover, these changes were not uniform across similar decrease in grid spacing (Figure 13), which is consistent with the results in Figure 12. The band in NOCP7.5 showed a weak convective system with anvil clouds at a maximum height below 7 km, which is due to less grid-resolved convection in the 7.5 km runs without any CP. However, in other three 7.5 km runs (i.e., BMJ7.5, KFEX7.5, and GR3D7.5), the convective systems were fully developed with anvil clouds reaching more than 15 km in height. As the grid spacing decreased from 7.5 km to 5 km, except for KFEX5, due to enhanced grid-resolved convection, the extent and strength of the convective band were all increased markedly, which corresponded to the significant increase of TC intensity, especially in NOCP5. This result is expected, because the coarser grid spacing (≥5 km) cannot resolve most individual cells without CP. As seen in the 3 km runs, with an exception of BMJ3, the distribution and intensity of the convective bands were basically similar to those in its 5 km runs, and thus associated with similar TC intensity.
 As the grid spacing was further decreased to 1 km, the strong bands were concentrated in a narrow area near the eyewall, showing strong radial gradients and thus a stronger storm. This may be related to the weak model convergence in simulation of storm intensity at 3 km and 1 km resolutions. Furthermore, compared with the other three CP groups, GR3D1 showed a relatively small difference in microphysical structure form GR3D3, which contributes to the similar intensity between GR3D3 and GR3D1 and thus a relatively strong model convergence.
4.3 Possible Mechanisms for the Dependence of Model Convergence on Resolution and CP
 The possible physical mechanisms responsible for the dependence of model convergence on resolution in simulation of TC intensity are identified through comprehensive diagnostics and are schematically summarized in Figure 14.
 As mentioned above, it is not the extra area-integrated mass or SEF input into TC that is responsible for the weak model convergence in TC intensity as the grid spacing decreased, but the change of TC structure or distribution of variables in TC system. As the grid spacing is decreased, the weaker asymmetric structure, the smaller SEF outside the eyewall, the smaller RMW, and more latent heating near the eyewall are all responsible for larger TC intensity increase, and thus weak model convergence in TC intensity (Figure 14a). Furthermore, due to the smaller RMW, the finer-resolution runs are characterized by smaller slope eyewall. Combined with the smaller SEF outside the eyewall, the more upright eyewall promotes the radial gradients near the eyewall in terms of horizontal and vertical velocity, SEF, latent heating, SLP, radar reflectivity, etc (see Figure 14a). The larger radial gradients near the eyewall caused by the smaller RMW may be the main reason for the weak model convergence in TC intensity.
 It should be pointed out that the impacts of the factors promoting storm intensification at finer resolution are notably different under different CP conditions. As mentioned above, the factors promoting the storm intensity in GR3D fine-resolution runs (such as GR3D1) are notably weaker than the other three CP groups. More importantly, latent heating released from different CP schemes exhibited different degrees of dependence on the model resolution, among which the GR3D scheme presents the least dependence while BMJ shows the most, particularly from 7.5 km to 3 km. In sum, relatively strong asymmetric structure, larger RMW and SEF outside the eyewall, and less dependence of latent heat release on resolution are responsible for relatively weaker TC intensity in GR3D finer-resolution runs (Figure 14b), and thus the relatively stronger model convergence in simulation of TC intensity.
5 Conclusions and Discussion
 The sensitivity of the track, intensity, and structure of a TC to horizontal resolution and CP schemes has been studied using the nonhydrostatic mesoscale model WRF. It is found that as the grid spacing is decreased, except for the notable difference between 7.5 km runs and 5 km runs, the simulated tracks are found to be little changed. Consistent with recent findings from cloud-resolving model simulations, the changes in storm intensity and structure are not uniform across similar reductions in grid spacing. There is a significant increase in TC intensity as the grid spacing decreases from 3 km to 1 km. This indicates that the model solution has not yet converged for these grid spacings. The simulations using the GR3D CP scheme present a relatively better model convergence, compared to the other three CP schemes.
 This is a case study. It is important to point out that, in this study, it is not necessary to precisely reproduce Shanshan (2006) in terms of its intensity (e.g., MSLP and MWS) and structure (e.g., spatial distribution of rainfall). Our conclusions are based on the comparisons between sensitivity experiments and less dependent on absolute and precise simulation results. Most importantly, in all our cases, the simulated TCs follow a similar track. This allows us to examine the sensitivity of simulated TC kinematic and microphysical structures to horizontal grid spacing and CP schemes in an environment similar to that of Typhoon Shanshan.
 In this paper, the reasons for the weak model convergence in simulation of TC intensity were discussed systematically from perspectives of energy exchange at air-sea interface, mass flux in the BL, eyewall asymmetries, and thermodynamic, dynamic, and microphysical structures. Across the rather narrow spectrum of resolutions compared in detail here, there were a number of factors that limited or promoted storm intensification and thus the model convergence, as resolution varies. Factors promoting storm intensification at finer resolutions and hence weak model convergence were the smaller SEF outside the TC eyewall, smaller eyewall asymmetry, more latent heating in the eyewall, and larger radial gradients in kinematic and microphysical structures, and vice verse. These factors are enough to overcome the negative impact of the less area-integrated SEF and mass flux on TC intensity in finer-resolution runs. Among these factors, the larger radial gradient related to the RMW and eyewall slope is the key factor for the weak model convergence in high-resolution simulations (e.g., 1 km runs). That is, as the grid spacing decreases, due to the increased net radial force resulting from the more grid-resolved convection and enhanced latent heating near the eyewall, the RMW also decreases, leading to a more upright eyewall (smaller eyewall slope), and thus a stronger TC.
 Actually, the statistical relationship between TC intensity and RMW or eyewall slope is still controversial, but for a given simulation, this relationship is robust. Based on the statistical analysis on the observation results of many storms, Stern and Nolan  suggested that the outward slope of the RMW with height is directly proportional to the size of the RMW, but no relationship is found between the slope of the RMW and TC intensity. This would appear at first to contradict our results from a single TC case. In fact, for the simulation of a single storm, the slope of the RMW is linearly related to its size, as in the maximum potential intensity theory of Emanuel ; as the RMW contracts with time, the slope decreases. The intensity also appears to be very much related to slope, although the relationship is nonlinear. For a given simulation, the slope is approximately inversely proportional to intensity, which was also emphasized in Stern and Nolan . In our study, although the horizontal resolution and CP scheme are varied, the relationship between eyewall slope and TC intensity is still strong.
 Furthermore, the roles of the factors in affecting model convergence are notably different under different CP conditions. In contrast to those in the other three CP groups (NOCP, BMJ, and KFEX), the TC intensity in GR3D finer-resolution runs are notably weaker than expected due to the relatively stronger asymmetric structure, larger RMW, more SEF outside the eyewall, and especially the less dependence of the GR3D scheme on model resolution. Thereby, the model convergence in GR3Ds is significantly stronger than those in the other CP groups. The choice of CP schemes indeed has a great impact on the storm structure, intensity, and thus the model convergence, so it is possible to find a reasonable CP suitable for the high-resolution model to obtain more realistic simulations with strong convergence and thus better forecast quality.
 It is interesting to make an analogy with this model convergence investigation and ensemble forecasting. In fact, this study tries to identify the convergent members (i.e., best players) of a group of sensitivity experiments (may be taken as an ensemble) and understand why. Convergent TC intensity and structure would be a desirable attribute of perfect models, given the grid spacing and CP changes. Convergent members may thus receive a higher weight in the ensemble. With regard to this, however, the scale issue needs to be considered. In synoptic-scale ensembles, the growth of initial errors dominates and often the consensus within the ensemble indicates higher predictability of the environment, i.e., a spread-skill relation may exist [Scherrer et al., 2004]. However, for mesoscale ensembles such as in TC applications, model errors dominate, and convection processes are a major contributor as nonlinearity in such processes leads to large sensitivities in the physics representations [Houtekamer et al., 1996], and thus one should be cautious about using model convergence as a measure of skill. In TC applications, many more cases of validation are needed to see if model convergence actually indicates that some processes and associated nonlinearity during TC intensification dominate and likely to be true. This could be a topic for future research.
 According to the performance of the four CP schemes in this study, cases with GR3D present relatively better simulation results in terms of storm intensity and structure, especially in its high-resolution runs (such as GR3D1). However, the reason why only the GR3D scheme achieved a better performance in maintaining model convergence (i.e., what mechanism inside the GR3D scheme has contributed to the better simulation) still needs an exclusive investigation. In addition, as a new CP scheme designed for high-resolution simulation, GR3D still has some other problems that need to be solved. We emphasize here that this study has analyzed the model convergence problem when some CP schemes are applied to the gray-zone resolutions; however, the converged solution does not necessarily approach the truth. The use of GR3D still generates an over-enhancement of TC intensity in its intensification period. These will be further investigated in our next work.
 This work is supported by the National Basic Research Program of China (2013CB430203), National Natural Science Foundation of China (41175090), National High Technology Research and Development Program of China (2012AA091801), and the Key Technologies R&D Program of China (2009BAC51B05). The authors are grateful to the anonymous reviewers for their constructive comments. Special appreciation goes to the third reviewer who enlightened us with future research materials and directions. Dr. Chee-Kiat Teo of the Nanyang Technology University of Singapore provided helpful information about the TRMM TMI/PR rainfall data.