1. Introduction
Perturbation pressure exerts important effects on cumulus convection. It is well known that these effects depend in part on updraft width, and more specifically on the ratio of updraft width to height, with a downward-directed buoyant perturbation pressure gradient force in relatively wide updrafts compensating the upward thermal buoyant force to a greater degree than in narrower updrafts (e.g., Markowski and Richardson 2010). This has implications for the sensitivity of nonhydrostatic convection-permitting models to horizontal grid spacing Δx in the “gray zone,” with Δx of O(1–10) km, where convection is generally underresolved. Since convective overturning is forced to occur over larger scales as Δx is increased at these resolutions, this typically leads to fewer but wider updrafts compared to simulations with Δx of O(100) m (e.g., Bryan et al. 2003; Bryan and Morrison 2012; Morrison et al. 2015a). Quantitative understanding of the impact of these changes on the vertical velocity is critical, given its importance in driving microphysical processes and vertical fluxes of momentum, static energy, water, and chemical constituents. This is especially important since nonhydrostatic models at gray-zone resolutions are now widely employed for many applications, including operational numerical weather prediction (e.g., Kain et al. 2008; Lean et al. 2008; Weisman et al. 2008; Clark et al. 2012) and climate studies using global “cloud resolving” models (e.g., Miura et al. 2007) or embedding convection-permitting models in traditional large-scale general circulation models—that is, “superparameterization” (e.g., Grabowski 2001; Khairoutdinov and Randall 2001; Tao et al. 2009).
Weisman et al. (1997, hereafter W97) performed two-dimensional (2D) and three-dimensional (3D) simulations of squall-line-type deep convection with fully coupled dynamics and moist physics and found strong sensitivity of the maximum vertical velocity w and horizontally averaged vertical fluxes of potential temperature and momentum to Δx between 1 and 20 km. There was a decrease in maximum w and slower storm evolution as Δx was increased but an increase in the domain-averaged vertical fluxes once the squall line reached a quasi-steady mature phase. They explained this sensitivity in part by a shift of the dynamics from being governed primarily by nonhydrostatic processes at Δx of O(1) km to hydrostatic processes at Δx of O(10) km. More recent studies have also shown that horizontal grid resolution can have a strong effect on simulated convective storm development and structure in the gray zone (e.g., Bélair and Mailhot 2001; Petch and Gray 2001; Petch et al. 2002; Adlerman and Droegemeier 2002; Bryan et al. 2003; Pauluis and Garner 2006; Gentry and Lackmann 2010; Fiori et al. 2010; Bryan and Morrison 2012; Verrelle et al. 2015; Morrison et al. 2015a). For example, Bryan and Morrison (2012) found a roughly 20% decrease in total upward mass flux with an increase in mean updraft size and decrease in number between Δx of 1 and 4 km, while Petch et al. (2002) showed a delay in convective development as Δx was increased.
Another aspect of perturbation pressure is its effect on the sensitivity of updraft strength to dimensionality (2D vs 3D). Several studies have shown that 2D updrafts tend to be weaker than their 3D counterparts (e.g., Wilhelmson 1974; Schlesinger 1984; Lipps and Hemler 1986; Tao et al. 1987; Redelsperger et al. 2000; Phillips and Donner 2006; Zeng et al. 2008). For example, Phillips and Donner (2006) reported that cloudy updrafts were 20%–50% stronger in 3D than 2D, with substantial impacts on vertical profiles of cloud liquid and ice and hence surface and top-of-atmosphere radiative fluxes. Understanding physical mechanisms for these differences and how they relate to perturbation pressure is important given the continued wide use of 2D models.
Another important aspect of perturbation pressure is its treatment in cumulus convection schemes. For parameterizing w using simplified plume models embedded in these schemes, perturbation pressure effects are typically either neglected or included in a simple way by scaling the buoyancy with a constant “virtual mass” coefficient. The simplicity of this method is attractive but it is lacking in physical justification. Settings for the virtual mass coefficient are often ad hoc or tuned, with the buoyancy typically reduced by a factor of ~1–3 (e.g., Simpson and Wiggert 1969; Gregory 2001; Bechtold et al. 2001; Bretherton et al. 2004; Jakob and Siebesma 2003; Neggers et al. 2009; see Table 1 in Wang and Zhang 2014). A few studies have fit the virtual mass coefficient to large-eddy simulations applied to shallow cumulus convection (de Roode et al. 2012; Wang and Zhang 2014), but the generality of this approach is unclear. Park (2014) included a virtual mass coefficient that has an exponential dependence on updraft radius R, from 1 at 
Part I of this study presents simple analytic expressions relating perturbation pressure and updraft velocity to R, H, and convective available potential energy (CAPE) for 2D and axisymmetric quasi-3D updrafts. In the current paper, Part II, these theoretical solutions are compared with direct numerical calculations of the anelastic buoyant perturbation pressure Poisson equation. The theoretical solutions are also compared to 2D and 3D fully dynamical updraft simulations initiated with warm bubbles of varying sizes. Implications for nonhydrostatic modeling in the gray zone are discussed, including sensitivity to Δx and dimensionality. Application of the theoretical expressions for improving perturbation pressure effects in cumulus parameterizations is also described.
The remainder of the paper is organized as follows. Section 2 describes the methodology for numerically solving the buoyant perturbation pressure Poisson equation. Results including comparison of the analytic expressions with the numerical solutions and with fully dynamical updraft simulations are presented in section 3. Discussion of relevance to gray-zone modeling and convection parameterizations is given in section 4, and a summary and conclusions are presented in section 5.
2. Methodology for numerical solution
This approach is inspired by Parker (2010), who calculated the p field and acceleration numerically from a specified distribution of B to discern the impacts of updraft tilting on buoyant perturbation pressure and w. Numerical solutions herein are calculated for both 3D and 2D. For 3D, updrafts are represented by a steady-state cylinder of positively buoyant air (relative to the environment). For 2D, updrafts are represented by an analogous slab of positively buoyant air. Other features associated with convective updrafts that affect buoyancy, such as cold pools and gravity waves, are neglected for simplicity. Updrafts are assumed to be upright. The idea is to mimic the fundamental features of convective updrafts in an environment with weak vertical wind shear.
Equation (1) is solved to obtain p on a domain with 129 grid points along each of the horizontal and vertical dimensions for either 2D or 3D. The domain is periodic in the horizontal direction(s). Solutions are calculated using a spatial discretization of 100 or 200 m vertically (depending upon the updraft height) and 1/10 of the updraft radius horizontally (finer discretization does not noticeably affect results). A multigrid iterative solver with Gauss–Seidel point relaxation and a tolerance threshold of 10−6 for the maximum relative error is employed.
Upper and lower boundary conditions are determined as follows. At the surface and top of the atmosphere, if 
The p field is derived for various spatial structures and magnitudes of B (see section 3a). Since the buoyancy and hence perturbation pressure is assumed to be symmetric around the updraft center, the implied wind field at the updraft center has 
3. Results
a. Sounding data
To analyze the theoretical and numerical solutions for w and the perturbation pressure difference between the level of neutral buoyancy (LNB) and LFC Δp a number of thermodynamic soundings over a wide range of conditions are used to generate various buoyancy profiles representing shallow to deep moist convection. The buoyancy is calculated based on adiabatic ascent of the most unstable parcel, relative to an environment with 
Summary of the thermodynamic soundings. “Maximum w” indicates the maximum thermodynamic vertical velocity equal to 
Skew-T diagrams for the six thermodynamic soundings listed in Table 1.
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0041.1
b. Numerical solution of Poisson pB equation
Examples of the numerical solution for p in 3D are shown in Figs. 2 and 3 for the WK and RICO soundings, respectively. For these calculations, the horizontal buoyancy distribution within the updraft is assumed to follow 
Vertical cross section of the perturbation pressure (color contours) calculated from the Poisson solver, thermal buoyancy (black contour lines; interval of 0.05 m s−2), and acceleration vectors for the WK sounding, COS horizontal buoyancy distribution, updraft radius of (top) 2, (middle) 10, and (bottom) 25 km, and 3D. The cross section is taken at the location of maximum buoyancy in the y direction. Note that only part of the domain is shown.
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0041.1
As in Fig. 2, but for the RICO sounding and updraft radius of (top) 1, (middle) 5, and (bottom) 10 km. Contour interval for thermal buoyancy (black contour lines) is 0.01 m s−2.
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0041.1
As in Fig. 2, but for 2D instead of 3D.
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0041.1
The u and w acceleration vectors calculated directly from the p and B fields (i.e., equal to 
c. Comparison of the theoretical and numerical solutions
The theoretical solutions for w at the LMB wM, w at the LNB wN, and Δp following (20), (27), and (28) for 3D and (31)–(33) for 2D in Part I are compared to numerical calculations of the pB field and w at the updraft center for the three horizontal buoyancy distributions discussed above. The parameter α in the theoretical expressions, equal to the ratio of w horizontally averaged across the updraft to that at the updraft center (see Part I, section 3), cannot be calculated directly from the numerical solutions. However, α is calculated directly from the 2D and 3D fully dynamical updraft simulations described in section 3e. These simulations have similar horizontally averaged B in 2D and 3D but give somewhat larger α for 3D than 2D (~0.8 vs 0.5). On the other hand, the tests here have horizontally averaged B about 25%–40% smaller in 3D than 2D for COS and COS2. Thus, for simplicity the same α is applied for 2D and 3D here, estimated by taking the ratio of B averaged along a horizontal line from the updraft center to its edge to B at the updraft center. This gives α of 1, 0.6367, and 0.5 for TOP, COS, and COS2, respectively.
Figures 5–8 show the 2D and 3D wM and wN as a function of R for each sounding for the COS and TOP buoyancy distributions. COS2 produces similar results as COS but with w larger by about 10%–15% and is, therefore, not shown. Figure 9 presents a comparison of the theoretical and numerical Δp for COS (TOP and COS2 give similar results; not shown). Overall, the theoretical and numerical calculations are in close agreement. Notably, the theoretical solutions capture the increase of Δp and decrease of w for 2D compared to 3D. For R/H ~ 1 in 3D, the analytic and numerical solutions show a decrease of w from the theoretical thermodynamic maximum of about 25%–40% for 3D and 40%–60% for 2D. For R/H less than ~0.2, perturbation pressure effects are small, with a reduction of w relative to the thermodynamic maximum (equal to 
Theoretical (red) and numerical (blue) calculations of wM as a function of R for the six thermodynamic soundings and COS buoyancy distribution. Solid and dashed lines indicate calculations for 3D and 2D, respectively. The thermodynamic maximum w at the LMB given by 
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0041.1
As in Fig. 5, but for wN.
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0041.1
As in Fig. 5, but for the TOP horizontal buoyancy distribution.
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0041.1
As in Fig. 5, but for wN and the TOP horizontal buoyancy distribution.
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0041.1
Theoretical (red) and numerical (blue) calculations of the perturbation pressure difference between the LNB and LFC as a function of updraft radius for the six thermodynamic soundings and COS buoyancy distribution. Solid and dashed lines indicate calculations for 3D and 2D, respectively.
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0041.1
Most differences between the theoretical and numerical Δp, especially for the deeper cases, are from using the Boussinesq approximation in the theoretical derivation (see Part I). Additional tests applying the Boussinesq instead of anelastic approximation to the numerical calculations show a closer correspondence with the theoretical Δp (Fig. 10). However, using the Boussinesq versus anelastic approximation has only a limited impact on w. Even for the deepest case (J2007OK), there is negligible impact on w at the LNB (less than a few percent difference), less than ~15% difference at the LMB, and ~30% difference near the LFC. Moreover, there is almost no dependence of w on the magnitude of ρ for the Boussinesq numerical solutions. These results are consistent with the theoretical derivation; there is a cancellation of ρ in the pressure term 
As in Fig. 9, but using the Boussinesq approximation (constant ρ = 0.5 kg m−3) for both the numerical and theoretical calculations.
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0041.1
d. Vertical profiles of w
Figure 11 presents results for the COS buoyancy distribution and three different values of R/H (⅓, 1, and 2). Overall, the theoretical w profiles show a close correspondence to the numerical calculations and well represent differences between 2D and 3D. Interestingly, the profile of w is “flattened” as R/H increases and pressure effects become important, meaning that 
Vertical profiles of theoretical (red) and numerical (blue) vertical velocity for normalized updraft radius, R/H, of (left) ⅓, (center) 1, and (right) 2 for the six soundings and COS horizontal buoyancy distribution. Solid and dashed lines indicate calculations for 3D and 2D, respectively. The profile of thermodynamic maximum w given by 
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0041.1
e. Comparison with fully dynamical updraft simulations
A direct comparison of the theoretical solutions with fully dynamical simulations of individual updrafts using the nonhydrostatic, compressible CM1 (Bryan and Fritsch 2002) is challenging because of difficulties in controlling updraft width in the simulations, especially for wide updrafts, and the role of entrainment, which was not explicitly included in the theoretical derivation. Moreover, the simulations are time evolving with growing updrafts, while the theoretical solutions assume steady-state analogous to the plume conceptual model. However, the perturbation pressure in CM1 is nearly diagnostic from the buoyancy and wind fields at any instance in time (it is not exactly diagnostic because CM1 is compressible, but the impacts of compressibility are expected to be negligible given the small Mach number of the flow; cf. Emanuel 1994). Thus, the perturbation pressure from the model and theoretical solutions can be compared with minimal complications from time dependency. On the other hand, 
The model setup follows that in Part I, with 200-m horizontal and vertical grid spacings over a domain 60 × 60 × 20 km3 in 3D and 60 × 20 km2 in 2D. Convection is initiated by applying warm bubbles that have a maximum perturbation potential temperature of 2 K (relative to the background state), vertical radius of 1.5 km, centered at an altitude of 1.5 km, and with perturbation potential temperature decreasing as a cosine function from the thermal center to its edge. Initial environmental thermodynamic conditions follow from the analytic sounding of Weisman and Klemp (1982), and the domain is initially motionless.
Three simulations each for 2D and 3D are performed with different horizontal radii for the initial warm bubbles: 1.5, 3, and 10 km. Since updraft growth is slower for relatively wide updrafts, narrow updrafts are analyzed at earlier times than wider updrafts: 400 (550), 460 (600), and 760 (1020) s for the 1.5-, 3-, and 10-km initial bubble radii simulations for 3D (2D). These particular times were chosen because they give similar updraft top heights and buoyancy profiles in the updraft center among the simulations. Note, however, that overall results including comparison of the simulated and theoretical perturbation pressure and w do not depend much on the particular time analyzed within the first ~10–20 min of the simulations. Vertical cross sections of the total perturbation pressure and w across the updraft center for the 3D simulations at these times are illustrated in Fig. 12.
Vertical cross section at the updraft center from fully dynamical 3D simulations of an isolated convective updraft initiated with a warm bubble of radius (a),(b) 1.5, (c),(d) 3, and (e),(f) 10 km, similar to Fig. 3.1 in Markowski and Richardson (2010). (left) Total perturbation pressure p and (right) vertical velocity w. Color contours show the perturbation potential temperature (relative to the initial environment) in all panels. The contour interval is 25 hPa for p and 2 m s−1 for w. Results are shown for times when the cloud top reaches ~4.5 km in height: 400, 460, and 760 s for the initial bubble radii of 1.5, 3, and 10 km, respectively. Note that only part of the domain is shown.
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0041.1
Parameters used in the theoretical expressions for w and Δp [see (20), (27), and (28) for 3D and (31)–(33) for 2D in Part I] are taken directly from the simulations, including CAPE, R, H, ρ, and α. For consistency, CAPE is calculated by vertically integrating the buoyancy profile at the updraft center, including the effects of condensate loading, which reduce CAPE by approximately ⅓ in all of the simulations, as opposed to that based on adiabatic parcel ascent from the LFC to LNB. The values of R and H are calculated by the region with perturbation potential temperature greater than 1 K, which corresponds well to the region of positive buoyancy between the updraft base and top (defined by the vertical levels for which B goes to ~0). Since updraft width is not constant with height, an average R is calculated and used in the theoretical expressions. The value of α is calculated directly from the ratio of w averaged horizontally across the updraft to the value at the updraft center at each vertical level and then averaged over height. There is a small overall decrease of α with height from about 0.95 to 0.7 in the 3D simulations below the LMB and an increase above. In contrast, α tends to increase more uniformly with height for 2D. Overall, α is larger for 3D than 2D (~0.8 vs 0.5), even though the ratio of horizontally averaged B to that at the updraft center is similar. The maximum w from the analytic theoretical expressions is taken as the larger of either wN or wM (here wN and wM refer to the vertical velocity at the updraft top and the LBM), since wM can be larger than wN despite the smaller integrated buoyancy up to this level because of the vertical asymmetry of the pressure scaling of w.
Results are summarized in Table 2 for 3D and Table 3 for 2D. The mean values of R from the three simulations are 1.5 (2.2), 2.5 (3.3), and 5.7 (5.4) km, corresponding to 
Summary of results from the 3D fully dynamical simulations and theoretical solutions with three different warm-bubble radii to initiate convection. Results are illustrated at 400, 460, and 760 s for the 1.5-, 3-, and 10-km initial bubble radii simulations, respectively, when the updraft top reaches ~4.5 km in height. Mean updraft radius R, height H, CAPE, and α are calculated from the simulations. The perturbation pressure difference from the updraft top to its bottom (defined as the heights at which buoyancy goes to zero) Δp and maximum updraft velocity w at the updraft center are shown from the simulations (SIM) and the theoretical solutions (TH).
As in Table 2, but for a comparison of 2D fully dynamical updraft simulations and theoretical solutions. Results are illustrated at 550, 600, and 1020 s for the 1.5-, 3-, and 10-km initial bubble radii simulations, respectively, when the updraft top reaches ~4.5 km in height.
The region of positive buoyancy (relative to the environment) becomes highly deformed once entrainment begins playing a dominant role in the simulations after ~10–20 min (Fig. 13), especially for 2D. This leads to a complicated structure of the buoyancy field with regions of strongly positive buoyancy wrapped around the region of low dynamic pressure along the lateral updraft edge, making it difficult to clearly define an updraft R. Nonetheless, if a consistent definition of R is applied to all of the simulations at these later times (taken as the length from the updraft center to the outermost edge of the region of large perturbation potential temperature, defined by the 6-K isotherm), overall the theoretical expressions are consistent with the simulations. This is illustrated by comparing the simulated and theoretical Δp and w when the updraft top reaches 9–10 km in height (Tables 4 and 5). The theoretical expressions give larger Δp and smaller w for 2D compared to 3D seen in the simulations and capture the decrease in Δp from the widest to the narrowest initial bubbles. However, quantitative differences between the theoretical expressions and the simulations are larger than earlier in the simulations, especially for Δp. This is presumably because of the complicated structure of the buoyancy field after entrainment begins to dominate.
As in Fig. 12, but for results later in the simulations when entrainment begins to dominate. The contour interval is 50 hPa for p and 5 m s−1 for w. Results are shown for times when the cloud top reaches ~9–10 km in height: 800, 800, and 1130 s for the initial bubble radii of 1.5, 3, and 10 km, respectively.
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0041.1
As in Table 2, but for analysis at later times in the 3D simulations after entrainment begins to dominate. Results are illustrated at 800 (1130) s for the 1.5- and 3-km (10 km) initial bubble radii simulations, when the updraft top reaches 9–10 km in height.
As in Table 4, but for a comparison of 2D fully dynamical updraft simulations and theoretical solutions. Results are illustrated at 1020, 1000, and 1400 s for the 1.5-, 3-, and 10-km initial bubble radii simulations, respectively, when the updraft top reaches 9–10 km in height.
4. Discussion
a. Implications for model sensitivity to horizontal grid resolution
As discussed in the introduction, convective overturning is forced to occur over larger scales as the model grid resolution is increased in the gray zone, with Δx of O(1–10) km where moist atmospheric convection is generally underresolved. This typically leads to updrafts that are too wide and few in number. This is clearly seen, for example, in the simulations of Bryan and Morrison (2012, see their Fig. 2), which show a mean updraft width that approximately scales with Δx. Based on the results presented herein, as Δx and hence R increase there is an increased downward-directed perturbation pressure gradient force and weaker updrafts, all else being equal. These results suggest, therefore, that an incorrect representation of perturbation pressure from spuriously wide updrafts may be an important contributor to biases in convective strength in gray-zone models. At relatively coarse resolutions, with Δx of O(10) km, the theoretical and numerical calculations suggest biases in w of a factor of 2 or more, assuming a real updraft width of O(1) km.
These points are illustrated by comparing the maximum w from fully dynamical, nonhydrostatic 2D and 3D simulations of W97 (see their Figs. 16 and 17) and the numerical and theoretical wN over a range of R (Fig. 14). This approach is similar to Pauluis and Garner (2006), who compared theoretical scalings for w as a function of updraft size with fully dynamical simulations using varying Δx. W97 employed the WK sounding, facilitating comparison with the numerical and theoretical solutions. However, since R was not analyzed in their study, a characteristic updraft radius 
(a) Comparison of wN from the theoretical (red) and numerical (blue) solutions and maximum vertical velocity reported from simulations of W97 (black) as a function of R. (b) As in (a), except uniformly scaling the theoretical and numerical wN by a factor of 0.6. Solid and dashed lines indicate calculations for 3D and 2D, respectively. For the simulations, it is assumed that the characteristic updraft radius scales as 7Δx. The thermodynamic maximum w given by 
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0041.1
For a given R and dimensionality, the maximum w from W97 is consistently ~40% smaller than the theoretical and numerical calculations (Fig. 14a). This is expected because condensate loading and entrainment were neglected in the calculation of B for the numerical and theoretical w. There is also some uncertainty because horizontal distributions of B across updrafts are not known from the simulations (the COS distribution was used for the numerical and theoretical calculations in Figs. 14 and 15). Note that in general entrainment may also be expected to depend on R, but at these relatively coarse grid spacings 
Comparison of wN from the hydrostatic theoretical expression (red) and maximum vertical velocity reported from the hydrostatic simulations of W97 (black) as a function of R. Solid and dashed lines indicate calculations for 3D and 2D, respectively. For the simulations, it is assumed that the characteristic updraft radius scales as 7Δx. The thermodynamic maximum w given by 
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0041.1
Despite these caveats, the scaling of w with R (Δx) is remarkably similar between the numerical and theoretical solutions and the fully dynamical simulations from W97 for both 2D and 3D; this is more clearly seen by uniformly scaling the analytic and numerical w by a factor of 0.6 (Fig. 14b). The W97 simulations also show weaker updrafts in 2D than 3D consistent with the numerical and theoretical calculations. Thus, the theoretical expressions appear to provide a concise explanation for relatively weak updrafts in 2D; this occurs as a direct result of differences in mass continuity between 2D and 3D as shown in Part I.
Comparing the hydrostatic simulations from W97 and the 2D and 3D theoretical expressions for hydrostatic wN given by (34) and (35) in Part I provides insight into the scales at which nonhydrostatic effects become important (Fig. 15). A comparison of Figs. 15 and 14a shows that for this sounding, the theoretical nonhydrostatic and hydrostatic scalings diverge for R less than ~25–30 km for 3D and ~12–15 km for 2D, in qualitative agreement with the 2D and 3D simulations from W97, again with the assumption that 
For many applications the total or horizontally averaged vertical fluxes of mass, momentum, static energy, water, and chemical constituents are key quantities with regard to moist convective dynamics. Studies have shown a wide range of behavior of the total vertical fluxes (mass or momentum) with an increase in Δx, from an increase (W97), to no consistent change (Morrison et al. 2015a), to a decrease (Moeng et al. 2010; Arakawa and Wu 2013; Bryan and Morrison 2012). As a result, it has proven challenging to understand this sensitivity from a broader conceptual standpoint.
The theoretical expressions relating w and R may provide some insight into this issue, if perturbation pressure plays a key role in driving sensitivity of convective strength to Δx in the gray zone as argued above. Since the area of an updraft A is approximately proportional to R2 in 3D, and the total vertical mass flux is given by 
b. Implications for convection parameterizations
5. Summary and conclusions
This study investigated the role of perturbation pressure on the vertical velocity w of buoyant updraft cores in an unsheared environment. In the current paper, Part II, analytic theoretical solutions for the perturbation pressure difference from the LNB to LFC, Δp, and w as a function of updraft radius R, height H, and CAPE from Part I were compared to numerical solutions of the anelastic buoyant perturbation pressure Poisson equation. Several thermal buoyancy profiles derived from real and idealized thermodynamic soundings representing shallow to deep moist convection were tested. Despite idealizations made in deriving the theoretical expressions, the theoretical and numerical solutions showed a close correspondence for both 2D and 3D over a wide range of R. The theoretical and numerical solutions also gave similar differences in Δp and w between 2D and 3D updraft geometries. These differences were a direct consequence of fundamental differences in mass continuity between 2D and 3D as shown in Part I and were quantitatively consistent with differences between the 2D and 3D simulations from W97. This provides a physical explanation for weaker updrafts in 2D than 3D reported in many previous studies using fully dynamical models.
The theoretical expressions for Δp and maximum w were also similar to results from 2D and 3D fully dynamical simulations of moist convection initiated by warm bubbles of varying width. Differences between the theoretical and simulated Δp and maximum w were less than ~15% and ~25%, respectively, when the simulated updrafts were analyzed shortly after release of the bubbles to limit the role of entrainment. The theoretical Δp and w were also generally consistent with the simulations at later times once entrainment began to dominate, although differences were larger than earlier in the simulations. This was presumably because of entrainment, which led to a complicated structure of the buoyancy field in the simulations that was not taken into account in the theoretical derivation. Overall, these results suggest the ability of the theoretical expressions to describe the behavior of updrafts when all dynamical processes are considered, including time-evolving terms neglected in the theoretical derivation.
Since R approximately scales with the effective model resolution (and hence is approximately proportional to Δx) in the “gray zone,” with Δx of O(1–10) km where convection is generally underresolved, these results suggest that an incorrect representation of perturbation pressure from spuriously wide updrafts may be important in explaining sensitivity of these models to Δx. This was illustrated by comparing the numerical and theoretical calculations of wN with the maximum w from 2D and 3D fully dynamical simulations of W97. Despite caveats, the numerical and theoretical scalings of w with R were similar to the W97 simulations, assuming 
These results suggest that modifying the treatment of parameterized subgrid-scale mixing alone, such as changing the mixing lengths, may not address the root cause of biases in convective strength and vertical transport in gray-zone models if these biases are fundamentally related to updrafts that are too wide, leading to incorrect perturbation pressure effects. This also suggests challenges in applying improved mixing parameterizations (e.g., Moeng et al. 2010; Arakawa and Wu 2013; Wu and Arakawa 2014; Bogenschutz and Kruger 2013) to better partition the resolved and subgrid-scale (SGS) or subfilter-scale (SFS) fluxes as a function of Δx in these models. If the dominant scales of buoyant production of w kinetic energy in updraft cores are underresolved, leading to incorrect buoyant production of w kinetic energy because of updrafts that are too wide, then the total (or horizontally averaged) fluxes may be incorrect even if the “correct” partitioning of resolved and SGS/SFS fluxes is applied. Furthermore, the resolved vertical fluxes are not necessarily underpredicted in the gray zone despite underresolving convection. For example, W97 found an increase in the horizontally averaged vertical fluxes of momentum and potential temperature as Δx was increased from 1 to 12 km. If the SGS/SFS fluxes increase with increasing Δx, as we expect they should, then applying the “correct” partitioning of SGS/SFS and resolved fluxes in this instance would increase the bias of excessive fluxes, relative to the 
The theoretical expressions relating w and R herein may provide some insight into these issues. Based on these expressions, scalings were hypothesized that relate the total vertical mass flux to the number and size of convective updraft cores in 2D and 3D. These scalings suggest either an increase or decrease of the total vertical mass flux depending on how N (number of updraft cores) and the characteristic R change with Δx. Further work is needed to test these scalings and to determine how these sensitivities to R (Δx) are also affected by entrainment, which is expected to be especially important for Δx of O(1) km and less. Difficulty in clearly defining and differentiating updraft cores is also noted (Sherwood et al. 2013) and poses challenges for such an effort. Nonetheless, these scalings potentially provide a useful context for improved understanding of the sensitivity of vertical mass fluxes to Δx in gray-zone models.
Plume models in convection parameterizations often rely on a simple treatment of perturbation pressure effects on w by reducing the buoyancy using a “virtual mass” coefficient that is constant or parameterized in an ad hoc way. The theoretical solutions from Part I provide a concise physical interpretation of the virtual mass coefficient in terms of α, R, and H and can, therefore, provide guidance for how this parameter might vary under a range of conditions. Based on the theoretical derivation, simple expressions were proposed that incorporate perturbation pressure effects in a more realistic way, including a dependence on R and H, for little computational cost. Additional testing and analysis of these expressions for updraft vertical velocity, including the effects of entrainment, is left for future work. The importance of the parameter α in the theoretical expressions (which relates w at the updraft center to its horizontal average across the updraft) is also noted. This was treated as an externally specified parameter in this study but it should be intrinsically linked to entrainment since it depends on how w varies horizontally across the updraft. Work is needed to better characterize typical values of α and how they relate to entrainment of momentum and B under various conditions.
Acknowledgments
This work was partially supported by U.S. DOE ASR DE-SC0008648, NASA NNX14AO85G, and the NSF Science and Technology Center for Multiscale Modeling of Atmospheric Processes (CMMAP), managed by Colorado State University under Cooperative Agreement ATM-0425247. The author thanks G. Bryan, R. Rotunno, W. Grabowski, and Z. Lebo for relevant discussions, R. Rotunno and W. Grabowski for comments on an earlier version of the manuscript, G. Bryan for developing and maintaining CM1, and P. Markowski for code used to separate the various contributions of perturbation pressure in CM1. NCAR’s Computer Information Systems Laboratory is acknowledged for providing the Poisson solver as part of the MUDPACK library.
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“Acceleration” here and elsewhere in the paper refers to parcel acceleration in a Lagrangian framework; i.e., D/Dt.
W97 used fourth-order centered advection plus fourth-order horizontal smoothing. They also included “physical mixing” using an eddy diffusion coefficient calculated from a prognostic turbulence kinetic energy scheme. Skamarock (2004) reported an effective resolution of 7Δx for simulations using implicit filtering from fifth-order upwind advection but found that simulations using second-order centered advection with small fourth-order smoothing (hyperviscosity coefficient of 4.25 × 108 m4 s−1) produced similar results to the fifth-order upwind advection in terms of kinetic energy spectra. While fourth-order centered advection plus fourth-order horizontal smoothing was not tested; an effective resolution of 7Δx for the W97 simulations is also assumed here.
