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.

Perhaps the simplest conceptual explanation for cumulus updrafts is that the air within them rises because it experiences an upward buoyancy force by virtue of the air within the updraft being warmer and less dense than its surroundings. In the atmospheric sciences, the buoyancy force B is formally expressed as the ratio of the density of a parcel of air to the density of its surrounding environment: , where g is gravity, is the difference between the density of the air parcel and the parcel’s surrounding environment, is the density of the surrounding environment, and precipitation loading is neglected. (See Table 1 for symbol definitions.) I will hereinafter refer to B as the “absolute buoyancy force,” since the vertical profile applies everywhere within the model or observational domain (i.e., it only varies with height). A relatively simple theoretical framework known as “parcel theory” is often used to assess the potential strength of an updraft if it were to form in a given environment and to estimate the vertical mass flux accomplished by an updraft in certain classes of cumulus parameterization schemes (e.g., Siebesma et al. 2003). The foundation of parcel theory is the assumption that the 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., ) and vertically integrating the absolute buoyancy field over a finite height range:

where , is the virtual potential temperature of the parcel along its path of ascent, is the temperature of the undisturbed environment above the initial height of the parcel z, z* is a dummy variable of integration, and the subscript pt denotes that w is obtained via parcel theory assumptions.

Table 1.

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” ) in a convective updraft partially cancels the upward absolute buoyancy force and results in weaker updrafts than would otherwise occur if absolute buoyancy alone were to regulate vertical acceleration 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 leads to the spinup or spindown of horizontal vorticity [e.g., Rotunno et al. (1988); Weisman et al. (1997); expressed here for a two-dimensional system]. This implies that nonzero 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” must exist in addition to to satisfy mass continuity (when wind velocity is nonzero). The field heavily influences convective updrafts by continuously triggering new updrafts in a particular direction (propagation) (e.g., Yau 1979; Rotunno and Klemp 1982; Weisman and Klemp 1982, hereafter WK82; Rotunno and Klemp 1985; Weisman and Rotunno 2000; Davies-Jones 2002; Jeevanjee and Romps 2015b), by enhancing the strength of 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

We begin with the frictionless anelastic vertical momentum equation to understand forces that regulate 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.

For parcel theory, we assume , w is in steady state, term A is small compared to term B, , and we arrive at

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, is arbitrarily defined with the only limitation on its definition being that the base-state environment must be hydrostatically balanced. For a complex environment with considerable four-dimensional variability, the choice of is not obvious [see Doswell and Markowski (2004) for an in-depth discussion of this limitation]. It is often assumed that the only environment that acts upon a parcel as it ascends is the environment defined by the atmospheric column directly above the parcel, and is thereby redefined for every 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.

To better understand the role of the vertical pressure gradient force, we rewrite Eq. (2) without approximation as

where is the nonhydrostatic pressure field. We then apply the divergence operator to the base-state density times the three-dimensional anelastic momentum equation (e.g., Davies-Jones 2002, 2003; Jeevanjee and Romps 2015b) to obtain a diagnostic expression for :

where term A in Eq. (5) is the dynamic pressure field and term B is defined by Davies-Jones (2003) as “effective buoyancy.” We use the hydrostatic relation to rewrite term B in Eq. (5) as and Eq. (4) as

where the operator yields the solution for Y, given the argument X, to the elliptic equation . Equation (6) illustrates that the magnitude of effective buoyancy depends on the local second derivative of 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

If we restrict our analysis to two dimensions (for ease of analytical and numerical simplicity), temporarily neglect the dynamic pressure field, and vertically differentiate both sides of Eq. (5), we obtain the following:

We then define to be the distance from the LFC to the level of maximum buoyancy (LMB), to be the distance from the LMB to the EL, and set (where γ is a constant coefficient). It follows that , where is half the height of the updraft. If we then assume that and that outside the lateral periphery of the updraft (the “updraft” is assumed to be a continuous area of ) and approximate second derivatives in Eq. (7) as second-order centered finite differences, for example,

we may rewrite Eq. (7) as

where , is the width of the updraft, and are values on the right and left peripheries of the updraft, respectively, and is at the horizontal center of the updraft. If we define and , Eq. (8) reduces to

where is the width-to-height aspect ratio. Term A resembles absolute buoyancy [term B in Eq. (2)]; however, the numerator is a temperature perturbation relative to the environment immediately surrounding an updraft, rather than relative to an arbitrary base state. Since the definition of contains (which is the average of the temperature surrounding the updraft), this framework allows for background variations in atmospheric temperature. For instance, anomalously warm (e.g., another updraft) or cool air (e.g., a cold pool) on either side of an updraft will alter the value of and by extension . Formally, is a finite-difference approximation of on a horizontal grid with a grid spacing of , and is the length scale over which reduces to 0 from the updraft center.

In substituting Eq. (9) into Eq. (4), we arrive at an approximate expression for dw/dt at the LMB:

If we consider the limit where (e.g., a tall narrow updraft), α becomes small, and approaches the parcel theory value for an idealized environment (e.g., Morrison 2016a). The magnitude of quickly becomes very small relative to parcel theory values for (e.g., Morrison 2016a). In assuming a steady-state updraft and integrating from the height of the LFC to the height of the LMB, we may solve for the vertical velocity at the LMB:

Analogous expressions to Eqs. (10) and (11) are obtainable for slanted updrafts. Parker (2010) showed that slanted updrafts are weaker than their upright counterparts, so we expect to find a relationship where increasing updraft slant leads to weakening dw/dt and w, all else being equal. We used a rotated coordinate transformation where , , and ϕ is the angle of slant in the clockwise direction from the x and z axes. The full operator is invariant to rotated coordinate transformations, whereas . After vertically differentiating and transforming coordinates, Eq. (7) becomes

where is left in the original upright coordinate system. If we expand the right-hand side (RHS) of Eq. (12) onto a finite-difference grid that is aligned with the rotated coordinate system and define , and , we obtain an expression similar to Eq. (10) for a slanted updraft:

where and are length scales relative to the rotated coordinate system, and . As is apparent from Eq. (13) (and will be shown in the next section), tall narrow updrafts weaken when they become increasingly slanted relative to their upright counterparts. If we assume that air parcels travel upward with slantwise paths through the center of the updraft, we may integrate Eq. (13) in the direction to obtain an analogous expression to Eq. (11) for :

b. Bubble experiments

To test the viability of the theoretical expressions for both and 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 were obtained via successive overrelaxation (Young 1950), substituting into the vertical momentum equation to obtain the formal dw/dt values, and integrating dw/dt from the LFC to the LMB to obtain . For simplicity, I assumed a uniform base-state environment, where , , and . Hereinafter, the max(⋅) operator returns the maximum acceleration within the domain.

I constructed idealized sinusoidal updrafts within a homogeneous base-state environment in accordance with the following formulas:

and

where was a constant that regulated the maximum updraft temperature, and were arbitrarily chosen exponents, and was set to 1 or 0 (an example updraft with , , , and is shown in Fig. 1a). The effect of the added sin argument in Eq. (16) when was to redistribute the maximum toward the top of the distribution and to yield a “bell” or “mushroom cap” shape (an example updraft with , , , and is shown in Fig. 1b); whereas, when , the distribution is vertically symmetric (Fig. 1a). The field was smoothed with a Gaussian filter with a characteristic radius of 250 m to remove discontinuities.

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Idealized updraft distributions (K; shading), and their associated EFF BUOY field (upward acceleration: solid gray contours): (a) SYM, ¼ and (b) MUSH SQRT, ¼.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0016.1

I then compared Eqs. (10), (11), (13), and (14) computed with both (assumed to be 1), and (explicitly computed) to experimentally obtained values from the idealized updraft shapes produced by Eq. (15) with and (SYM), with and (SYM SQRT), with and (MUSH), and with , and (MUSH SQRT). In general, the theoretical curves overpredicted for , and underpredicted for for all updraft shapes and distributions. Root-mean-square errors (RMSEs; Fig. 2b) were generally in the range of 10–20 m s⁻¹ h⁻¹, with RMSE values having been slightly reduced when is explicitly computed. Explicitly computing (rather than assuming it to be 1) improved the RMSE for by nearly 15 m s⁻¹ h⁻¹; however, performance was worse with explicitly computed for . Theoretical values compared well with experimental values (Fig. 2c), with RMSE values generally on the order of 0–2 m s⁻¹ (Fig. 2d). Interestingly, despite theory with an explicitly computed having been better at predicting , theory with performed better in predicting for all distributions except for the MUSH SQRT distribution with (Fig. 2d). The vertical extent of in this experiment was restricted by the vertical extent of , whereas the formal solution to the acceleration field extended above and below the updraft (see Figs. 1a,b). This likely explains why theoretical expressions performed better for when , since maximum accelerations were overpredicted for MUSH and MUSH SQRT distributions relative to when was explicitly computed. Yet this overprediction compensated for the theoretical expression featuring an unrealistically narrow vertical extent of acceleration when the integral in Eq. (11) was evaluated. I therefore proceeded with set to unity.

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(a) Comparison between theoretical and experimental (m s⁻¹ h⁻¹) curves: theory with (gray solid line), theory with computed for each updraft (gray dashed line), SYM (blue line), SYM SQRT (green line), MUSH (magenta line), and MUSH SQRT (red line). (b) RMSE (m s⁻¹ h⁻¹) values for theoretical expressions in diagnosing the four different updraft distributions: (blue stars), computed for each updraft (red stars), for (blue circles), and computed for each updraft for (red circles). (c) Comparison between theoretical (dashed lines) and experimental (solid lines) (m s⁻¹) curves: SYM (gray), SYM SQRT (blue), MUSH (green), and MUSH SQRT (red). (d) As in (b), but for . (e) Experimental (blue line; m s⁻¹ h⁻¹) vs theoretical (gray line; m s⁻¹ h⁻¹) as a function of updraft slant for an updraft with effective . (f) As in (e), but for (m s⁻¹).

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0016.1

Idealized updrafts from Eq. (15) with and were rotated by the angle ϕ to produce slanted updrafts (as in Parker 2010). The magnitudes of and decreased with increasing ϕ (Figs. 2e,f), which is consistent with the results of Parker (2010). The theoretical curves corresponded well with observed and , with max(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.

Table 2.

Summary of the CM1 configuration for this study.

I used a WK82 sounding as an initial condition to promote the development of deep moist convection (Fig. 3a; this sounding has 2000–3000 J kg⁻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 field was set to

where , , , , and (the cold pool straddled the left lateral boundary). I formulated the wind profile outside of the cold pool to favor strong kinematic lifting along the right edge of the cold pool [see Rotunno et al. (1988) for how the wind shear–cold pool balance impacts squall-line intensity]:

This wind pattern resulted in maximum easterly winds of at the surface, which decreased in magnitude with height, and reached zero at the depth of the cold pool. A 100-min simulation was run with u₀ = 18.6 m s⁻¹ (this value of was found to produce the most sustained updrafts among other considered) in order to create a large population of simulated updrafts with varying slant angles. The and 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 that

where . Finally, I prescribed a u field that was divergent (convergent) above the eastern (western) cold pool edge for to establish anelastic wind balance between u and w within that height range.

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(a) Skew T–logp diagram of the base-state profile of T (°C; solid red line), dewpoint temperature (°C; green line), virtual temperature T_υ (°C; dashed red line), and the profiles of T (°C; magenta line) and (°C; cyan line) through the cold pool. (b) Wind vectors (black arrows; m s⁻¹), (K; shading), and vertical velocity contours (dark gray contours; m s⁻¹) at 0 min.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0016.1

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 field associated with the updraft quickly became more complex than the idealized updrafts in the previous section beyond 5 min (Figs. 5a, 6a, 7a), with ring vortices (labeled in Fig. 5d) evident along the lateral peripheries of the plume, a narrow updraft stem (labeled in Fig. 5a), and a wide updraft head (labeled in Fig. 5a). Updrafts persisted along the eastern cold pool edge for the entirety of the simulation (e.g., Figs. 4a, 5a, 6a, 7a). The spatial pattern of became progressively more complex beyond 10 min (when compared to the initial 10 min), with a general warm region at mid- to upper levels associated with stratiform/anvil cloud, pronounced updraft plumes originating from the eastern cold pool head (Figs. 6a, 7a), and numerous turbulent eddies with spatial extent of less than 1 km within updrafts and the adjacent atmosphere.

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Model fields at 5 min: (a) (m s⁻¹ h⁻¹; shading); (b) (m s⁻¹ h⁻¹; shading) with L_x = 2.5 km; (c) EFF BUOY (m s⁻¹ h⁻¹; shading); and (d) ACC DYNAM (m s⁻¹ h⁻¹; shading). Gray and black dashed lines show the L_x =1.5- and 2.5-km width bounds around updrafts identified via the method described in section 3.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0016.1

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As in Fig. 4, but at 10 min.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0016.1

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As in Fig. 4, but at 40 min.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0016.1

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As in Fig. 4, but at 60 min.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0016.1

a. Evaluation of diagnostic expressions for w

The theoretical change in w over a finite height range within a steady-state updraft³ is:

where w is the vertical velocity observed from the simulation, is w diagnosed by a theoretical simplification (where “theory” is replaced by the name of the approximation used), and is a finite vertical distance. For instance, to evaluate the full diagnostic expression shown in Eq. (14) (which we call “slant theory”), I obtained from Eq. (20) by setting . Note that I have assumed that the aspect ratio and sine and cosine arguments apply through the entire depth of an updraft in Eq. (20) (in the previous section, they were only derived for the LMB). This assumption may seem unjustified; however, Morrison (2016a) showed that the pressure scaling of buoyancy at an updraft’s LMB reasonably diagnosed the actual vertical velocity profiles when applied through the depth of a cloud.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 (where the sine and cosine arguments have been removed). Table 3 lists the different values for F that were evaluated in this section and discusses the reasoning for their formulation.

Table 3.

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 and rising, and the spatial restrictions made sure I was only examining updrafts near the cold pool edge. I defined the average updraft slant angle ϕ as the w-weighted average of over all points within the active updraft region. I then defined a line with a slant angle of ϕ that passed through the point containing (referred to as the “center line”). The characteristics of the updraft center (e.g., , , and w) were taken as the mean values of quantities between and , where was the horizontal location of the center line at a given height, and “win” was an adjustable distance from the center line. I defined the height of the updraft as the vertical interval of continuous horizontally averaged over and that contained . Finally, I tested a range of assumed and win values, which yielded a two-dimensional parameter space for the evaluation of Eq. (20) and the expressions in Table 3. I analyzed three populations of updrafts separately: those with in the ranges 3–5 (short), 5–7 (med), and 7–9 km (tall) (see Fig. 8a for a histogram of updraft depths). Updraft angles varied from 0°–10° over the first 40 min of the simulation to 20°–30° through 40–100 min (Fig. 8b). Times series’ of the maximum within updrafts with L_x = 2.5 km and win =1 km show that from all theoretical expressions aside from parcel theory corresponded well to the observed from the simulation (this result was consistent over many and win values). This is an important result that is discussed further later in this section.

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(a) Histogram of the number of updrafts within given height ranges (blue line; blue dots represent the number of updrafts with heights within ). (b) Time series of the updraft slant angle ϕ (°; red line), observed (m s⁻¹; black dashed line), (m s⁻¹; blue line), (m s⁻¹; dark green line), (m s⁻¹; light green line), (m s⁻¹; bright green line), (m s⁻¹; magenta line), and (m s⁻¹; gray line).

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0016.1

I computed the normalized root-mean-square error ratio R of values in diagnosing the updraft’s actual profile of w to better understand the correspondence between the vertical profiles of and and the sensitivity of these profiles to L_x and win:

where was the maximum numerically simulated vertical velocity within the simulated updraft, is the height of the updraft’s base (as it was defined in the previous paragraph), w and are functions of win since they are horizontally averaged over , and is also a function of for all 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 for tall updrafts were generally minimized at 0.25 for L_x > 2.5 km (Fig. 9a), and 0.6–0.8 < win < 1.6–2 km, whereas similarly low values were present over a slightly different region in the parameter space for med updrafts (e.g., 1.5 < win < 1.2 km; Fig. 9d) and were comparably low over nearly all values of with win < 1.6–2 km (Fig. 9g) for short updrafts. Values of and for all updraft heights (Figs. 9c,f,i; only shown for L_x = 2.5 km) were very similar to that of for all L_x ≤ 2.5 km; and , , and were all considerably smaller than (Figs. 9c,f,i; only shown for L_x = 2.5 km). This suggests that the scaling and slant components of were insignificant in improving values over when compared to the containing instead of (the only difference between and was that contained instead of ). It is curious that values for all three updraft populations remained low for 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 quickly grew for L_x > 2.5 km (not shown), which suggests that compensating errors were responsible for low values for L_x > 2.5 km, where the was erroneously large for large values, but the scaling term in was also erroneously small (thereby compensating for large ).

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Plots of (left) (shading), (center) (shading), and (right) R with L_x = 2.5 km: (red line), (magenta line), (gray line), (blue dashed line), (black dashed line), (cyan dashed line), and (green line). (a)–(c) The tall updrafts; (d)–(f) med updrafts; and (g)–(i) short updrafts.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0016.1

To examine aggregate updraft characteristics, I constructed composites profiles of atmospheric fields (e.g., and ) within short, med, and tall updrafts by first rotating all updrafts to an upright position (e.g., rotating them by their slant angle 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 with L_x = 1.5 km (Fig. 10a), L_x = 2.5 km (Fig. 10b), and B (Figs. 10a,b) reveal that (i) values for L_x = 1.5 and 2.5 km were visually similar to one another; (ii) magnitudes for both for L_x = 1.5 and 2.5 km were generally half of B values at the same location; and (iii) the distributions were different than the B distribution, in that featured a broad maxima along the updraft stem (along with a secondary maxima near the very top of the updraft head), whereas B was maximized near the top of the updraft stem and within the updraft head. The distributions and magnitudes of (Figs. 10a,b) and EFF BUOY (Fig. 10c) were very similar to one another, with only subtle differences near the updraft tops and bottoms. The width of EFF BUOY > 0 (which was our original definition of ) within the composite field ranged between 1.5 and 2.5 km (Fig. 10c), the maximum correlations between and EFF BUOY were for computed with L_x = 1.5 km (Fig. 10e), and yet I noted that the best subjective correspondence between and EFF BUOY occurred with L_x = 2.5 km ( values were consistently lower with L_x = 2.5 km than with L_x = 1.5 km; Fig. 9a). It was therefore difficult to discern which exact choice of in the computation of best represented the EFF BUOY field for med updrafts, and the results from the aforementioned analysis suggest values from 1.5 and 2.5 km were probably appropriate. The width of EFF BUOY > 0 for short updrafts was closer to 1.5 km (Fig. 10e), and for tall updrafts it was larger than 2.5 km through a significant portion of the updraft depth. This suggests that L_x = 1.5 km may be appropriate for short updrafts ( values were lower for L_x = 1.5 km for short updrafts than the other updraft populations, which supports this conclusion), and a range of values from 1.5 to 3 km (based on Fig. 9f) may be appropriate for tall updrafts. Finally, better correspondence between and is evident in Fig. 11 for med and tall updrafts when 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.

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(a) Composite of (with L_x =1.5 km) over med updrafts, which were all stretched to have a height of 7 km (m s⁻²; shading), B (black contours at intervals of 0.05 m s⁻²), and 2D wind velocity, where w is in the direction of the updraft slant angle and u is orthogonal to that direction (m s⁻¹; black arrows). (b) As in (a), but with L_x = 2.5 km. (c) As in (a), but with a composite of EFF BUOY (shaded) and the 0 m s⁻² contour (magenta dashed line). (d) As in (a), but with a composite of ACC DYNAM (shaded); positive is in magenta contours (interval of 0.4 hPa starting at 0.1 hPa); and negative is in green contours (interval of −0.4 hPa starting at −0.1 hPa). (e) As in (c), but for short updrafts. (f) As in (c), but for tall updrafts. (g) Linear correlation coefficient between and EFF buoy as a function of for tall (green solid line), med (blue solid line), and short (red solid line) updrafts. Correlations between B and EFF BUOY are also shown as a function of for tall (green dashed line), med (blue dashed line), and short (red dashed line) updrafts.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0016.1

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Composite profiles of w (blue lines) from simulated updrafts, (green lines), (red lines), (magenta lines), (cyan lines), (gray line), and (dashed black line) for L_x = (left) 1.5 and (right) L_x = 2.5 km. Profiles are shown for (a),(b) short updrafts, projected onto 5-km vertical grid; (c),(d) med updrafts, projected onto a 7-km vertical grid; and (e),(f) tall updrafts, projected onto a 9-km vertical grid.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0016.1

b. Impact of the dynamic pressure force on w

Despite , , , and having accurately predicted the magnitude of observed within updrafts, the vertical distributions of these profiles were quite inconsistent with observed 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, , , , and all imply that w should gradually increase from the updraft base to the updraft top, achieving maximum values at the updraft top (Figs. 11a–f). The profile shows a peak in 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 . This suggests that the ACC DYNAM was responsible for the differences between the observed w profile, the diagnosed w profile, and all profiles where ACC DYNAM was ignored.

Composites of ACC DYNAM show that med updrafts were typically characterized by a strong upward dynamic acceleration between 3 and 5 km above the updraft base and a strong downward dynamic acceleration above 5 km (Fig. 10d). This explains why observed occurred below the updraft top, since parcels achieved their maximum accumulated upward acceleration at approximately 5 km and then experienced downward acceleration from ACC DYNAM near the updraft top. These accelerations were a response to a broad region of locally minimized centered vertically at 5 km above the updraft base (this pressure perturbation was likely associated with ring vortices; Figs. 10d and 5d) and a region of locally maximized at the updraft top (which was likely associated with strong and at the updraft top; Fig. 10d). I therefore assume that from Eq. (5) was associated with locally high near the updraft top (Fig. 12); (which tends to be correlated with horizontal vorticity) was associated with locally low at the level of the ring vortices [Fig. 12; these features often migrate to the updraft center in sheared environments (e.g., Romps and Charn 2015)]; and was associated with a locally high below the ring vortices (Fig. 12; this high tended to be weaker than the one near the updraft top, owing to much smaller here). I defined as the distance between the level of and the updraft top and as the distance between the bottom of the updraft and the level of (where ). Given the correspondence between and demonstrated in the previous section, we assume . I then expanded the derivatives in into finite-difference approximations and assumed at the level of and outside of the updraft [as we did with Eq. (8) such that above the level of and below the level of ]. This reduced to

Note that is only nonzero from below to above the level of , which is consistent with the distribution of ACC DYNAM in Fig. 10d. For the analysis hereinafter, I set (which I estimated in an ad hoc manner from the simulation) and left a more quantitative determination of this quantity to future work. I then evaluated Eq. (20) with to obtain . Vertical profiles of show significant improvement in their correspondence to simulated w profiles over (Figs. 11a–c). Furthermore, values were 0.1 –0.2 less than for all three updraft populations (Figs. 9a–i), suggesting that theoretical expressions that contain simple treatments of the dynamic pressure force akin to what is described here may be useful in improving the diagnosis of vertical mass flux in convection. Values of were comparable to (Figs. 9c,f,i) except for when win > 1.5 km for med updrafts. It is possible that the impact of ACC DYNAM (which exhibits very high spatial frequency variability) on an updraft is poorly represented by a horizontal average over and for values of win > 1.5 km; however, the generally close correspondence between and suggests that the parameterization for introduced in this subsection is physically reasonable.

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Schematic illustrating an idealized structure and the associated orientations within an updraft. Location 1 is a dynamic high associated with ; location 2 is a dynamic low associated with ring vortices, where ; and location 3 is a weak dynamic high associated with .

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0016.1

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 (the updraft “periphery” was defined by where becomes negligible), the updraft’s height-to-width aspect ratio (where tall narrow updrafts are stronger than short wide updrafts), and the updraft’s slant relative to the vertical.

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 , rather than (which is a temperature perturbation relative to an arbitrary base state). In contrast, theoretical representations of updraft slant and aspect ratio had a negligible impact on improving the diagnosis of w over parcel theory. Whereas the actual within simulated updrafts was located approximately two-thirds to three-fourths of the distance between the updraft base and the updraft top, the within profiles diagnosed by simple expressions was portrayed at the updraft top when the dynamic pressure force was ignored. I therefore developed a rudimentary parameterization of the dynamic pressure within the diagnostic expressions, which further improved the portrayal of the simulated 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 (note that α represents a different parameter here); in recognizing that , assuming , and setting , we arrive at an expression nearly identical to the one from this study [Jeevanjee and Romps (2015a) obtained a similar, but more complex expression to Morrison (2016a)]. Similarly, Weisman et al. (1997) obtained (N is the Brunt–Väisälä frequency), which features a similar “scaling term” [e.g., ] to the expressions in this article and a marginally different thermodynamic parameter. A slightly different dependency on the aspect ratio was obtained by Pauluis and Garner (2006), where the aspect ratio term was not squared: . Though all of these prior expressions explicitly or implicitly include effective buoyancy in their derivations, they are a function of the parameter B (rather than ), which retains dependence on the arbitrary base-state definition. In situations where the horizontal environment becomes increasingly complex, the results from section 3 in this paper suggest that the expressions presented here will perform better in diagnosing the w in updrafts, owing to their dependence on the parameter (and the associated dependence on the horizontal second derivative of buoyancy, rather than the magnitude of buoyancy). Note that the theoretical expressions from this study do retain a small degree of dependence on the arbitrary base state since they contain in their denominators (this results from the assumption that that is used to derive the anelastic governing equations). Changes in on the order of 1–10 K yield relatively small alterations to the magnitude of 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 is in the denominator and may also impact the sign of dw/dt. The dependence on the arbitrary definition of is therefore dramatically reduced by the expressions derived here.

The scaling parameter was relatively insensitive to and for the typical updraft dimensions in the simulations here, where (e.g., 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 and values to an updraft is a substantial oversimplification of the dynamics of effective buoyancy within updrafts. For instance, Figs. 10a–d show updrafts to comprise a tall and narrow updraft “stem,” and a comparatively wider “head” near the top of the updraft that frequently contains the level of maximum . One may envision this pattern of as comprising two vertically stacked updrafts: (i) the stem being tall and narrow, having a small value and a large value, and being characterized by large upward-oriented effective buoyancy; (ii) the head being wide and short, having a large and small value, and being characterized by small upward-oriented effective buoyancy when compared to the stem (despite this region frequently containing the maximum ). Whereas the theoretical representation of one updraft separated into two or more parts (with each part having a different aspect ratio) seems to be logistically difficult, provides a continuous and concise way of representing these effects (since this term well captures the contrasting effective buoyancies of the updraft stem and head). The simplified representation of updraft slant in the diagnostic expressions here provided little improvement over diagnostic expressions that ignored updraft slant. A visual examination of fields (e.g., Fig. 7a) suggests that slant angle of the stem of tall updrafts frequently “meandered,” having featured some areas of upward orientation and others of nearly horizontal orientation, owing to the presence of many large eddies within and near the updraft. As I suspected to be the case for the updraft aspect ratio, a single updraft slant angle is likely a drastic oversimplification of such complex behavior. Future work will address the impact of updraft slant on 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 analyzed here provide a rudimentary way of estimating the error introduced by the steady-state assumption, since for a frictionless atmosphere, and therefore , where is the error in w introduced by the steady-state assumption. Therefore, is a good approximation for the normalized error associated with , and it is shown in Figs. 9e and 11c,f,i that is reasonably small and does not detract from the interpretation of this study’s conclusions (especially when compared to , where errors associated with parcel theory assumptions are an order of magnitude larger than those associated with the steady-state assumption). It is worth noting that the effects of hydrometeor loading and turbulent diffusion (e.g., parameterized subgrid-scale turbulence) have been ignored, and these factors may be responsible for a small contribution to .

For analyses where the grid spacing is much larger than a reasonable , the scales of variability required to compute are no longer resolved and would require some degree of parameterization. Along the lines of this comment, is often redefined for every horizontal grid point (without ample physical justification) in the computation of spatial maps of CAPE in numerical weather prediction model output. If we assume that the environment at a grid point prior to an updraft forming becomes the environment immediately around the periphery of the updraft once it does form {e.g., with }, we are actually assuming that , rather than in the denominator of the integral argument (the latter of which is mathematically inconsistent), since only depends on height and cannot be reassigned at every horizontal grid point. Thus, CAPE may better be expressed as rather than and is thereby implicitly an integral of effective buoyancy (rather than absolute buoyancy). Note that, based on the aforementioned argument, Morrison (2016a) replaced with CAPE in the expression for and thereby implicitly accounted for environments with complex horizontal variability.

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 and to expect for updrafts that have yet to form (or are unresolved) in a given environment? As was pondered in the previous paragraph, how is meaningful on subgrid scales, and how might this quantity be parameterized? It was shown that implies that the vertical profile of effective buoyancy cannot be obtained by simply scaling the profile of absolute buoyancy with a parameter that is constant in the vertical [e.g., as in Siebesma et al. (2003)’s Eq. (13), where effective buoyancy is parameterized as , and 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 is also very sensitive to the ambient environmental shear [as has been shown by numerous authors in the context of supercells (e.g., Rotunno and Klemp 1982; Davies-Jones 2002)]. This work, at the least, constitutes a step toward improving the conceptual understanding of how effective buoyancy and dynamic pressure forces impact updraft dynamics and lays the framework for the aforementioned future research questions to be addressed.