## 1. Introduction

Vigorous vertical mass transport in cumulus convection is responsible for severe weather (which includes hail, tornadoes, and damaging winds) and a large percentage of precipitation extremes worldwide. The vertical advection of moist static energy accomplished by cumulus convection also plays a crucial role in our global climate system. Despite the importance of convection over a variety of scales, a comprehensive understanding of what regulates the distribution of vertical motion *w* within convective updrafts remains an elusive goal.

*B*is formally expressed as the ratio of the density of a parcel of air to the density of its surrounding environment:

*g*is gravity,

*B*as the “absolute buoyancy force,” since the vertical profile

*only*force acting upon an air parcel in the vertical direction is the absolute buoyancy force. A theoretical profile of

*w*within an updraft is obtained by assuming a steady-state wind environment (e.g.,

*z*,

*z** is a dummy variable of integration, and the subscript pt denotes that

*w*is obtained via parcel theory assumptions.

Symbol definitions.

It is well understood, however, that the dynamics of buoyancy-driven ascent in convective updrafts are far more complex than parcel theory would suggest. Numerous authors have demonstrated that the perturbation pressure response to buoyancy (referred to as “buoyancy perturbation pressure” *dw*/*dt* (e.g., Soong and Ogura 1973; Wilhelmson 1974; Yau 1979; Schlesinger 1984; Weisman et al. 1997; Morrison 2016a,b; Romps and Charn 2015). Three primary consequences of the perturbation pressure force on *dw*/*dt* are commonly discussed in the literature for nearly steady-state updrafts. First, *dw*/*dt* is a function of an updraft’s width-to-height aspect ratio *α*, where large *α* (e.g., a wide and/or shallow updraft) equates to weaker *dw*/*dt* than when *α* is small (e.g., a tall and/or narrow updraft) (Weisman et al. 1997; Pauluis and Garner 2006; Jeevanjee and Romps 2015a; Morrison 2016a,b). Second, *dw*/*dt* within an updraft is a function of the updraft’s slant relative to the vertical direction, where more slanted updrafts are weaker than their upright counterparts (Parker 2010). Finally, it is shown through a simplification of the solenoidal term in the linearized horizontal vorticity tendency equation that *dw*/*dt* may result from local horizontal gradients in absolute buoyancy*.*^{1} Recent authors have therefore advocated for the interpretation of buoyancy as the *“*statically forced part of the locally nonhydrostatic, upward pressure gradient force,*”* or *“*effective buoyancy*”* (Davies-Jones 2003; Doswell and Markowski 2004; Romps and Charn 2015), which is equivalent to the sum of *B* and the vertically oriented buoyancy pressure gradient force.

The presence of spatial wind gradients further complicates the role of pressure gradients in updraft dynamics. Mass flux convergence is approximately zero in the atmosphere, which implies that a “dynamic pressure field” *w* within rotating updrafts, and by redistributing *w* and *dw*/*dt* within updrafts.

Fully nonhydrostatic cloud models simulate the aforementioned influences of vertical pressure gradients on updrafts, since the dynamical cores of these models contain a pressure term in their prognostic *w* equation. It is, however, difficult for a human to interpret distinct physical processes from nonlinear prognostic partial differential equations, which typically require numerical integration to solve. Furthermore, climate models and global forecast models often assume hydrostatic balance (even though convection involves decidedly nonhydrostatic motions) and have grid resolutions that are too coarse to capture convective-scale processes. Simplified diagnostic expressions for *w* and *dw*/*dt* in updrafts [e.g., Eq. (1)] are therefore quite useful for understanding the influence of distinct physical processes on updraft behavior and for parameterizing subgrid-scale updrafts in forecast and climate models (e.g., cumulus parameterization schemes). However, such diagnostic expressions often include an overly simplified representation of pressure effects on updrafts, where absolute buoyancy is multiplied by a simple scale factor that is either constant (e.g., Siebesma et al. 2003) or that solely depends on the updraft’s height-to-width aspect ratio (e.g., Weisman et al. 1997; Morrison 2016a).

The goal of this work is to introduce diagnostic expressions for *dw*/*dt* and *w* that account for the impacts of updraft slant, aspect ratio, horizontal gradients in buoyancy, and dynamic pressure effects (section 2). I then use these expressions to better understand the impact of these factors on the vertical velocity distribution within updrafts (section 3) by applying the expression to output from a nonhydrostatic two-dimensional cloud model. I summarize the article and the applications of the article’s results to climate and weather prediction in section 4.

## 2. Derivation of effective buoyancy

*w*and

*dw*/

*dt*, which is expressed as follows:where term A is the vertical perturbation pressure gradient, term B is absolute buoyancy, the subscript 0 denotes a base-state variable that is a function of

*z*only, and all other symbols retain their traditional meanings.

*w*is in steady state, term A is small compared to term B,

The relative simplicity of this equation allows for the diagnosis of *w* in a given environment (if an updraft were to form) by computing the temperature of an air parcel if it were lifted from its level of free convection (LFC)^{2} to its equilibrium level [EL; e.g., Eq. (1)], and integrating both sides of Eq. (3) to obtain a theoretical profile for *w*. On the other hand, the simplicity of parcel theory also results in several quite severe limitations. As was mentioned in the previous section, *x* and *y* location for CAPE computations without ample justification. Furthermore, the pressure gradient force is neglected in this framework without much justification, which is especially troubling given the widely recognized impacts of vertical pressure gradients on updraft dynamics.

*Y*, given the argument

*X*, to the elliptic equation

*B*, rather than the simple sign and magnitude of

*B*. This expression is simplified in the next subsection for the ease of interpretation and analysis.

### a. Theoretical simplification and interpretation of effective buoyancy

*γ*is a constant coefficient). It follows that

*dw*/

*dt*at the LMB:If we consider the limit where

*α*becomes small, and

*very*small relative to parcel theory values for

*dw*/

*dt*and

*w*, all else being equal. We used a rotated coordinate transformation where

*ϕ*is the angle of slant in the clockwise direction from the

*x*and

*z*axes. The full

### b. Bubble experiments

To test the viability of the theoretical expressions for both *dw*/*dt*, I compared theoretically predicted values for these quantities to values obtained from formal solutions to Eq. (7) for idealized updraft shapes [the following experiments were inspired by Parker (2010) and Morrison (2016b)]. Formal solutions for *dw*/*dt* values, and integrating *dw*/*dt* from the LFC to the LMB to obtain

I then compared Eqs. (10), (11), (13), and (14) computed with both ^{−1} h^{−1}, with RMSE values having been slightly reduced when ^{−1} h^{−1}; however, performance was worse with ^{−1} (Fig. 2d). Interestingly, despite theory with an explicitly computed

Idealized updrafts from Eq. (15) with *ϕ* to produce slanted updrafts (as in Parker 2010). The magnitudes of *ϕ* (Figs. 2e,f), which is consistent with the results of Parker (2010). The theoretical curves corresponded well with observed *dw*/*dt*) having been slightly overpredicted for *ϕ* < 45° and underpredicted for *ϕ* > 45°.

## 3. Numerical modeling experiment

In experiments described hereinafter, I tested the diagnostic expressions in environments with considerably more horizontal atmospheric variability than the idealized bubble experiments described in the previous section. Updrafts were simulated with Cloud Model 1 (CM1), version 18.3, configured with a fully compressible nonhydrostatic dynamical core. The horizontal and vertical grid spacings were set to 125 m, with an 18 250-m model top, 2000 horizontal points, periodic lateral boundary conditions, free-slip upper and lower boundary conditions, and surface-to-atmosphere fluxes set to zero. Table 2 lists additional details of the modeling configuration.

Summary of the CM1 configuration for this study.

^{−1}of CAPE for air parcels within the lowest 2 km of the atmosphere). I also prescribed a cold pool within the model initial conditions to promote convection initiation and the development of a two-dimensional long-lived squall line. The initial

*u*

_{0}= 18.6 m s

^{−1}(this value of

*u*fields were then smoothed with a Gaussian filter with a radius of influence of 500 m to remove abrupt horizontal changes. To achieve anelastic wind balance, a

*w*profile was prescribed in the initial condition only and after the application of the Gaussian filter such thatwhere

*u*field that was divergent (convergent) above the eastern (western) cold pool edge for

*u*and

*w*within that height range.

Figure 3b shows the initial conditions for the simulation, and Figs. 4a and 5a show the developing updraft plume along the cold pool edge at 5 and 10 min, respectively. The

### a. Evaluation of diagnostic expressions for w

*w*over a finite height range within a steady-state updraft

^{3}is:where

*w*is the vertical velocity observed from the simulation,

*w*diagnosed by a theoretical simplification (where “theory” is replaced by the name of the approximation used), and

^{4}I also show later in this section that

*w*profiles diagnosed by the expression derived in this study correspond well to the analogous

*w*profiles diagnosed by the effective buoyancy (EFF BUOY) field, and this correspondence provides a posteriori justification for the assumption that the sine, cosine, and aspect ratio terms apply through the depth of the updraft. The relative importance of each of the elements to Eq. (14) was assessed by systematically removing terms from the equation. For instance, I compared

*w*

_{slant}with

*w*

_{upright}to assess the relative importance of accounting for, or ignoring, the updraft slant in diagnosing

*w*

_{theory}within an updraft. I obtained

*w*

_{upright}by setting

*F*that were evaluated in this section and discusses the reasoning for their formulation.

Summary of *F* formulations from section 3.

I defined active updraft regions as continuous areas with *w* > 3m s^{−1}, 40 < *x* < 55 km, and 0 < *z* < 13 km and that contained the grid point of maximum upward velocity in the domain at a given time. The thermodynamic and vertical velocity restrictions ensured that I was only examining regions where air was warmer than *ϕ* as the *w*-weighted average of *ϕ* that passed through the point containing *w*) were taken as the mean values of quantities between *L*_{x} = 2.5 km and win =1 km show that

*R*of

*w*to better understand the correspondence between the vertical profiles of

*L*

_{x}and win:where

*w*and

*F*expressions in Table 3 except for parcel theory, EFF BUOY, and EFF BUOY + ACC DYNAM (where ACC DYNAM is vertical acceleration due to dynamic pressure forcing). Values of

*L*

_{x}> 2.5 km (Fig. 9a), and 0.6–0.8 < win < 1.6–2 km, whereas similarly low

*L*

_{x}= 2.5 km) were very similar to that of

*L*

_{x}≤ 2.5 km; and

*L*

_{x}= 2.5 km). This suggests that the scaling and slant components of

*L*

_{x}> 2.5 km over large portions of the parameter space (Figs. 9a,d,g), given that my visual examination of the EFF BUOY fields output by the simulations (e.g., Figs. 4c, 5c, 6c, 7) suggested that the width of positive EFF BUOY for short and med updrafts was rarely larger than 2.5 km and for tall updrafts was rarely larger than 3.5 km. Values of

*L*

_{x}> 2.5 km (not shown), which suggests that compensating errors were responsible for low

*L*

_{x}> 2.5 km, where the

To examine aggregate updraft characteristics, I constructed composites profiles of atmospheric fields (e.g., *ϕ*) and then stretching all updrafts within each of these ranges to the height of the upper bound of that range (e.g., a 6.5-km-deep updraft in the med range was stretched to a height of 7 km, since 7 km was the upper bound for the med height range) and then averaging over all stretched updrafts within each height range. Comparisons between the composite distributions of *L*_{x} = 1.5 km (Fig. 10a), *L*_{x} = 2.5 km (Fig. 10b), and *B* (Figs. 10a,b) reveal that (i) *L*_{x} = 1.5 and 2.5 km were visually similar to one another; (ii) *L*_{x} = 1.5 and 2.5 km were generally half of *B* *B* distribution, in that *B* was maximized near the top of the updraft stem and within the updraft head. The distributions and magnitudes of *L*_{x} = 1.5 km (Fig. 10e), and yet I noted that the best subjective correspondence between *L*_{x} = 2.5 km (*L*_{x} = 2.5 km than with *L*_{x} = 1.5 km; Fig. 9a). It was therefore difficult to discern which exact choice of *L*_{x} = 1.5 km may be appropriate for short updrafts (*L*_{x} = 1.5 km for short updrafts than the other updraft populations, which supports this conclusion), and a range of *L*_{x} = 2.5 km (Figs. 11b,d,f) than when *L*_{x} =1.5 km (Figs. 11a,c,e), which provides additional evidence that *L*_{x} = 2.5 km was more appropriate than *L*_{x} = 2.5 km for med and tall updrafts.

### b. Impact of the dynamic pressure force on w

Despite *w* (Figs. 11a–f). For instance, observed *w* profiles featured a peak magnitude at two-thirds to three-fourths of the updraft top height, with *w* decreasing to nearly zero at the updraft top, whereas, *w* should gradually increase from the updraft base to the updraft top, achieving maximum values at the updraft top (Figs. 11a–f). The *w* at a height close to the observed *w* profile (Figs. 11a–c) and a general shape that is closer to that of the observed *w* profile than of *w* profile, the diagnosed *w* profile, and all

*w*profiles over

## 4. Summary and discussion

In this article, I derived simple diagnostic expressions for *dw*/*dt* and *w* within updrafts that accounted for effective buoyancy and the dynamic pressure gradient force. Effective buoyancy was defined as the statically forced component of the vertical gradient in the nonhydrostatic pressure field. I showed from these diagnostic expressions that the effective buoyancy of an updraft is dependent on the magnitude of the temperature perturbation within an updraft relative to the air along the updraft’s immediate periphery

To assess the performance of the diagnostic expressions, I simulated a two-dimensional squall line with a fully compressible nonhydrostatic cloud model. The simple diagnostic expressions significantly improved over parcel theory (where pressure forces are ignored) in their portrayal of the vertical profile of *w* through simulated updrafts. The largest contribution to the improvement of the diagnostic expressions over parcel theory resulted from the expressions’ dependency on *w* over parcel theory. Whereas the actual *w* profile by these expressions (relative to their formulations where the dynamic pressure force was ignored).

The dependency of *w* in updrafts on the updraft’s height-to-width aspect ratio, in addition to the magnitude of absolute buoyancy, has been addressed by several previous authors. For instance, Morrison (2016a) arrived at *α* represents a different parameter here); in recognizing that *N* is the Brunt–Väisälä frequency), which features a similar “scaling term” [e.g., *B* (rather than *w* in updrafts, owing to their dependence on the *dw*/*dt* (on the order of 10^{−2}–10^{−1} m s^{−2}, respectively) and do not change the sign of this quantity for the theoretical expressions in this study, whereas aforementioned temperature changes result in alterations to *dw*/*dt* on the order of 1–10 m s^{−2} when *dw*/*dt*. The dependence on the arbitrary definition of

The scaling parameter *L*_{z}/*L*_{x} = 1/3, 1/4, 1/6, 1/8, and 1/10 yield 0.9, 0.94, 0.97, 0.98, and 0.99, respectively). It is also likely that assigning single *dw*/*dt* and *w* more comprehensively.

An obvious limitation of nearly all diagnostic expressions for *w* is the steady-state assumption. There are many cases where updrafts are growing, decaying, or experiencing deformation of their vertical structure within environments with substantial vertical wind shear. These are but a few examples of where an updraft’s dynamics would be decidedly “unsteady.” The profiles of *w* introduced by the steady-state assumption. Therefore,

For analyses where the grid spacing is much larger than a reasonable

There are obvious additional research questions that need to be addressed in order to apply the expressions contained in this paper to subgrid-scale parameterizations of vertical mass flux. For instance, how does one determine which values of *a* is simply a constant value; Morrison (2016b) discussed this very issue and introduced height-varying expressions]. We have shown here that the effective buoyancy profile takes on a different shape than the absolute buoyancy profile, and parameters such as *a* from Siebesma et al. (2003) should therefore be height dependent. Extensions of the diagnostic expressions for effective buoyancy in this article to three dimensions are needed, since we have only considered two dimensions here. As shown by Morrison (2016a), the equations for effective buoyancy in 2D and 3D are conceptually similar (especially when axisymmetric coordinates are used), but the magnitudes of scaling parameters are different by a constant value. While we have shown a rudimentary way that the dynamic pressure force can be included in diagnosing vertical mass flux here, further research is needed to determine a parameterization of the dynamic pressure force that applies over a variety of environments and updraft characteristics. For instance, three-dimensional updrafts frequently rotate about a vertical axis (e.g., supercells), which introduces a layer of complexity to the dynamic pressure field that cannot be emulated in two dimensions. The distribution of

Special thanks go to Hugh Morrison, two anonymous reviewers, Russ Schumacher, Claire Moore, Doug Stolz, and Greg Herman for extremely helpful feedback on the manuscript. Finally, thanks to the participants of the Center for Multiscale Modeling of Atmospheric Processes (CMMAP) meeting in Boulder, Colorado, for helpful feedback on a presentation about this subject matter. This work was funded by the National Science Foundation Award AGS-PRF 1524435.

## REFERENCES

Davies-Jones, R., 2002: Linear and nonlinear propagation of supercell storms.

,*J. Atmos. Sci.***59**, 3178–3205, doi:10.1175/1520-0469(2003)059<3178:LANPOS>2.0.CO;2.Davies-Jones, R., 2003: An expression for effective buoyancy in surroundings with horizontal density gradients.

,*J. Atmos. Sci.***60**, 2922–2925, doi:10.1175/1520-0469(2003)060<2922:AEFEBI>2.0.CO;2.Doswell, C. A., III, , and P. M. Markowski, 2004: Is buoyancy a relative quantity?

,*Mon. Wea. Rev.***132**, 853–863, doi:10.1175/1520-0493(2004)132<0853:IBARQ>2.0.CO;2.Jeevanjee, N., , and D. M. Romps, 2015a: Effective buoyancy at the surface and aloft effective buoyancy at the surface and aloft.

, 811–820, doi:10.1002/qj.2683.*Quart. J. Roy. Meteor. Soc.*Jeevanjee, N., , and D. M. Romps, 2015b: Effective buoyancy, inertial pressure, and the mechanical generation of boundary layer mass flux by cold pools.

,*J. Atmos. Sci.***72**, 3199–3213, doi:10.1175/JAS-D-14-0349.1.Morrison, H., 2016a: Impacts of updraft size and dimensionality on the perturbation pressure and vertical velocity in cumulus convection. Part I: Simple, generalized analytic solutions.

,*J. Atmos. Sci.***73**, 1441–1454, doi:10.1175/JAS-D-15-0040.1.Morrison, H., 2016b: Impacts of updraft size and dimensionality on the perturbation pressure and vertical velocity in cumulus convection. Part II: Comparison of theoretical and numerical solutions.

,*J. Atmos. Sci.***73**, 1455–1480, doi:10.1175/JAS-D-15-0041.1.Morrison, H., , J. A. Curry, , and V. I. Khvorostyanov, 2005: A new double-moment microphysics parameterization for application in cloud and climate models. Part I: Description.

,*J. Atmos. Sci.***62**, 1665–1677, doi:10.1175/JAS3446.1.Parker, M. D., 2010: Relationship between system slope and updraft intensity in squall lines.

,*Mon. Wea. Rev.***138**, 3572–3578, doi:10.1175/2010MWR3441.1.Pauluis, O., , and S. Garner, 2006: Sensitivity of radiative–convective equilibrium simulations to horizontal resolution.

,*J. Atmos. Sci.***63**, 1910–1923, doi:10.1175/JAS3705.1.Romps, D. M., , and A. B. Charn, 2015: Sticky thermals: Evidence for a dominant balance between buoyancy and drag in cloud updrafts.

,*J. Atmos. Sci.***72**, 2890–2901, doi:10.1175/JAS-D-15-0042.1.Rotunno, R., , and J. B. Klemp, 1982: The influence of the shear-induced pressure gradient on thunderstorm motion.

,*Mon. Wea. Rev.***110**, 136–151, doi:10.1175/1520-0493(1982)110<0136:TIOTSI>2.0.CO;2.Rotunno, R., , and J. B. Klemp, 1985: On the rotation and propagation of simulated supercell thunderstorms.

,*J. Atmos. Sci.***42**, 271–292, doi:10.1175/1520-0469(1985)042<0271:OTRAPO>2.0.CO;2.Rotunno, R., , J. B. Klemp, , and M. L. Weisman, 1988: A theory for strong, long-lived squall lines.

,*J. Atmos. Sci.***45**, 463–485, doi:10.1175/1520-0469(1988)045<0463:ATFSLL>2.0.CO;2.Schlesinger, R. E., 1984: Effects of the pressure perturbation field in numerical models of unidirectionally sheared thunderstorm convection: Two versus three dimensions.

,*J. Atmos. Sci.***41**, 1571–1587, doi:10.1175/1520-0469(1984)041<1571:EOTPPF>2.0.CO;2.Siebesma, A. P., and Coauthors, 2003: A large eddy simulation intercomparison study of shallow cumulus convection.

,*J. Atmos. Sci.***60**, 1201–1219, doi:10.1175/1520-0469(2003)60<1201:ALESIS>2.0.CO;2.Soong, S.-T., , and Y. Ogura, 1973: A comparison between axisymmetric and slab-symmetric cumulus cloud models.

,*J. Atmos. Sci.***30**, 879–893, doi:10.1175/1520-0469(1973)030<0879:ACBAAS>2.0.CO;2.Weisman, M. L., , and J. B. Klemp, 1982: The dependence of numerically simulated convective storms on vertical wind shear and buoyancy.

,*Mon. Wea. Rev.***110**, 504–520, doi:10.1175/1520-0493(1982)110<0504:TDONSC>2.0.CO;2.Weisman, M. L., , and R. Rotunno, 2000: The use of vertical wind shear versus helicity in interpreting supercell dynamics.

,*J. Atmos. Sci.***57**, 1452–1472, doi:10.1175/1520-0469(2000)057<1452:TUOVWS>2.0.CO;2.Weisman, M. L., , W. C. Skamarock, , and J. B. Klemp, 1997: The resolution dependence of explicitly modeled convective systems.

,*Mon. Wea. Rev.***125**, 527–548, doi:10.1175/1520-0493(1997)125<0527:TRDOEM>2.0.CO;2.Wilhelmson, R., 1974: The life cycle of a thunderstorm in three dimensions.

,*J. Atmos. Sci.***31**, 1629–1651, doi:10.1175/1520-0469(1974)031<1629:TLCOAT>2.0.CO;2.Yau, M. K., 1979: Perturbation pressure and cumulus convection.

,*J. Atmos. Sci.***36**, 690–694, doi:10.1175/1520-0469(1979)036<0690:PPACC>2.0.CO;2.Young, D. M., Jr., 1950: Iterative methods for solving partial difference equations of elliptic type. Ph.D. thesis, Dept. of Mathematics, Harvard University, 74 pp.

^{1}

Such interpretation of the horizontal vorticity equation in the context of *w* often involves limiting assumptions, such as a linearization of the equation, and the distribution of buoyancy being characterized by a single Fourier mode.

^{2}

LFC is also used in this paper to describe the lowest instance of

^{3}

“Theoretical” *w* was computed by a simplified diagnostic expression, such as Eq. (1) or (14), and contrasts with “observed” *w*, which was obtained directly from model output.

^{4}

Morrison (2016b) experimentally altered the scaling near cloud top to account for vertical asymmetries in *ρ*, which has not been done here.