# A Simple Model of Convective Drafts Accounting for the Perturbation Pressure Term

Julien Leger, Jean-Philippe Lafore, Jean-Marcel Piriou, Jean-François

Guérémy

► To cite this version:

Julien Leger, Jean-Philippe Lafore, Jean-Marcel Piriou, Jean-François Guérémy. A Simple Model of Convective Drafts Accounting for the Perturbation Pressure Term. Journal of the Atmospheric Sciences, 2019, 76 (10), pp.3129-3149. 10.1175/jas-d-18-0281.1. hal-04950298

HAL Id: hal-04950298

https://cnrs.hal.science/hal-04950298v1

Submitted on 18 Feb 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# A Simple Model of Convective Drafts Accounting for the Perturbation Pressure Term

JULIEN LEGER, JEAN-PHILIPPE LAFORE, JEAN-MARCEL PIRIOU, AND JEAN-FRANÇOIS GUÉRÉMY

CNRM, Université de Toulouse, Météo-France, CNRS, Toulouse, France

(Manuscript received 16 September 2018, in final form 16 July 2019)

## ABSTRACT

A simple anelastic two-column model of convective updraft accounting explicitly for the perturbation pressure term is developed. There is no vertical wind shear in the environment, and two geometries (slab or axial) are possible. A shape factor is introduced to account for transport by a nonuniform horizontal profile of the vertical velocity in the updraft and its environment. The perturbation buoyancy profile being prescribed, three parameters must be prescribed: depth and aspect ratio of the updraft and the total cell size. The model is tested for idealized buoyancy profiles and evaluated against a large-eddy simulation of daytime development of deep convection. The model behavior agrees with our understanding of the perturbation pressure within clouds. The simulated updraft quickly responds to the buoyancy field (~5 min), shorter than the convection time scale, so that a steady model could be developed. Below the updraft core, a downward pressure gradient is simulated allowing the updraft to overcome a barrier of convective inhibition. This model is designed to be implemented in a convection scheme to replace classical drag formulations of the updraft model.

## 1. Introduction

Pressure perturbations are part of the internal dynamics of a flow. For convection, strong flow accelerations associated with intense heat sources and sinks result in large pressure deviations organizing the circulation within and outside the convective region. The advent of surface measurements enabled one to document strong perturbations in surface pressure (~1 hPa) associated with thunderstorms (Byers and Braham 1949; Fujita 1955). In the seventies it was established that intense pressure perturbations also occur within all convective clouds and that they play an essential role in convection dynamics. From a radiosonde in the updraft of a growing cell, Barnes (1970) reported a 3-hPa pressure excess at 6 km and a 1-hPa deficit near the cloud base. Marwitz (1973) showed evidence of upward accelerations of negatively buoyant air near the cloud base within Colorado hailstorms and attributed them to the perturbation pressure force. Statistical analysis of the NSSL sounding mesonetwork emphasized the effects of perturbation pressure on the updraft intensity (Davies-Jones 1974; Davies-Jones and Henderson 1975).

The first cumulus models were one-dimensional (1D) using either the quasi-Lagrangian bubble or plume models (Simpson and Wiggert 1969, hereafter SW69), or time-dependent Eulerian models (Ogura and Takahashi 1971), but all neglected the perturbation pressure term. Holton (1973) developed a time-dependent Eulerian model of cumulus including a simplified diagnostic equation for the pressure perturbation by combining the vertical momentum and horizontal divergence equations. He showed that the pressure perturbation plays an essential role by reducing the extremely sharp gradients in vertical velocity near the cloud top, which are common to most other 1D models. Yau (1979) developed a simple diagnostic model of pressure perturbation in cumulus. The dynamic pressure appears to be a consequence of the Bernoulli equation of conservation (in the updraft core) and a balance with the centrifugal force arising from the curved motion of the air (cloud edges). The pressure term in the vertical momentum equation is on the same order of magnitude as the buoyancy term, and may explain the upward acceleration of negatively buoyant air near the cloud base. Yau (1980, hereafter Y80) developed a two-cylinder model including an explicit computation of perturbation pressure. The advent of 2D and 3D dynamical models confirmed the above

Corresponding author: J.-M. Piriou, jean-marcel.piriou@meteo.fr
results. Soong and Ogura (1973) and Wilhelmson (1974) showed that pressure forces oppose the buoyancy in the upper part of the cloud, whereas Schlesinger (1978) showed that they can accelerate air upward against negative buoyancy at the base of sheared updrafts.

LeMone et al. (1988), gave evidence from aircraft measurements of a high-low pressure dipole around the cloud base of cumulus congestus and cumulonimbus, resulting from the interaction of the updraft with the environmental vertical wind shear. Rotunno and Klemp (1982) proposed a simple linear theory to explain how the shear-induced pressure gradient can modify the thunderstorm motion and rotation in case of a veering environmental shear vector with altitude. Weisman and Klemp (1982, 1984) performed a series of idealized simulations of an updraft, and showed that the interaction between the updraft and the environmental shear through induced pressure gradients, determines the convection organization (single-cell, multicell, or supercell storm), duration and motion, depending on the shape of the wind shear profile and on a bulk Richardson number. For mesoscale convective systems (MCSs) such as squall lines, Lafore and Moncrieff (1989) and Caniaux et al. (1995) showed that the pressure gradient generated between the convective and stratiform parts reinforces the rear inflow at midlevel helping to organize long-lasting circulation.

Considering the complexity of convective processes— where perturbation pressure is just one ingredient of the dynamics among others—one approach to get around the difficulty of parameterizing them is to use cloud-resolving models (CRMs) or the superparameterization approach (Grabowski 2001; Khairoutdinov and Randall 2001). However, traditional global climate models will still require convection schemes in the foreseeable future.

For convection parameterizations using an in-cloud vertical velocity model, most schemes still use the SW69 basic equation for vertical momentum, including two terms: (i) a source term corresponding to the in-cloud buoyancy modulated by a reduction factor *a* (<1) to account for pressure perturbations and fluctuations below the plume scale and (ii) a sink term formulated as a drag multiplied by a factor *b*. De Roode et al. (2012) reviewed the formulations of *a* and *b* used by 15 papers (their Table 1) and concluded that there is no consensus on the formulation of these constants, and that none accounts explicitly for the effect of pressure fluctuations on scales below the plume scale. Based upon a budget of the in-cloud vertical velocity equation for shallow cumulus obtained from large-eddy simulations (LESs), they found that the perturbation pressure term is the dominant sink term in the vertical velocity budget, whereas the entrainment term only accounts for a small

contribution. This result is at odds with the parameterized vertical velocity equation in the literature. They proposed some practical solutions to address this issue, but still using the same basic equation of SW69.

Some recent papers view cumulus convection as a succession of many short-lived thermals, as suggested by their analysis of LES (Sherwood et al. 2013; Romps and Charn 2015; Hernandez-Deckers and Sherwood 2016). Their Lagrangian budgets of vertical momentum of thermals showed evidence of a balance between the buoyancy and the perturbation pressure term, resulting in the concept of “sticky thermals,” whereas Sherwood et al. (2013) initially suggested “slippery thermals.”

In recent years, there has been renewed interest in including the perturbation pressure force in convection parameterizations. Assuming an hydrostatic flow at the edge of the updraft and in the environment, as well as a vertical symmetry of the updraft, Morrison (2016a,b) proposed analytic solutions to the perturbation pressure and vertical velocity in cumulus, depending on the updraft size, shape and geometry (2D or 3D). He then validated these theoretical expressions by comparison with numerical solutions. He also proposed simple expressions that incorporate perturbation pressure effects in the SW69 equation, including dependence on the width and height of the updraft. Peters (2016) developed diagnostic expressions for vertical acceleration and vertical velocity within 2D updrafts that account for effective buoyancy and the dynamic pressure gradient force. The effective buoyancy (Davies-Jones 2003) depends on the magnitude of the temperature perturbation within the updraft relative to the air along the updraft's immediate periphery, the updraft's height-to-width aspect ratio, and the updraft's slant relative to the vertical. Their last papers (Morrison 2017; Morrison and Peters 2018) proposed analytic solutions for the structure and evolution of growing deep updrafts and for their ascent rate.

This paper proposes a simple two-column model of convective updraft accounting explicitly for the perturbation pressure term. The ultimate goal is to implement this model in a convective scheme to replace the SW69 vertical velocity equation used by many convection parameterizations. It is organized as follows: Section 2 presents the prognostic model that shares some similarities with Y80 model but also has some major differences. Section 3 analyzes the model's response to fixed idealized buoyancy profiles, and the role of the pressure term based on a budget of vertical and horizontal velocities. This section ends with a study of the sensitivity to the geometric characteristics of the convective cell. Section 4 evaluates the model for the event of daytime development of deep convection over tropical semiarid land using an LES. Section 5 summarizes and discusses key results.
## 2. The convective perturbation pressure model

### a. Hypothesis and basic equation set

Similar to Y80, and for the sake of simplicity, we developed a two-column model: one for the updraft and the other one for its environment (hereafter denoted by the subscripts “u” and “e,” respectively). This is the minimum complexity to compute the perturbation pressure field in the updraft and its interaction with the environment. In addition, a hypothesis of symmetry is made, with the possible choice between slab or axial symmetry, assuming no ambient vertical wind shear.

The momentum equation set is taken as in Morrison (2016a) corresponding to the anelastic approximation written with the pressure $p$, using a hydrostatic “reference state” (subscript “r”) characterized by the vertical profiles of density $\rho_r(z)$ and of virtual temperature $T_{vr}(z)$. Defining the pressure deviation $p'$ from the reference state, the horizontal and vertical velocity ($u$ and $w$, respectively) equation set reads

$$ \frac{\partial u}{\partial t} = -u \frac{\partial u}{\partial x} - w \frac{\partial u}{\partial z} - \frac{1}{\rho_r} \frac{\partial p'}{\partial x} + D_u, \quad (1) $$

$$ \frac{\partial w}{\partial t} = -u \frac{\partial w}{\partial x} - w \frac{\partial w}{\partial z} - \frac{1}{\rho_r} \frac{\partial p'}{\partial z} + D_w + B, \quad (2) $$

where the first three rhs terms correspond to the horizontal and vertical advection, and the pressure perturbation term. The terms $D_u$ and $D_w$ represent the subgrid turbulent mixing of momentum, defined as the divergence of the subgrid momentum fluxes, and $B$ is the buoyancy term relative to the reference state, and is taken constant in time in this article. So the convective perturbation pressure model (CPPM) is a dynamical model not coupled with the thermodynamics. In a next stage of development $B$ will be provided by the convective parameterization.

We introduce a flag $\lambda = 0$ or 1 for the slab or axial symmetry, respectively (corresponding to the $x$ or $r$ horizontal coordinate), to derive hereafter a general form of the model equations. In the following and to avoid confusion, the horizontal coordinate will be always denoted $r$, so that it should be understood to be $x$ for the slab symmetry. With these notations the continuity equation reads

$$ \frac{1}{r^{\lambda}} \frac{\partial r^{\lambda} u}{\partial r} + \frac{1}{\rho_r} \frac{\partial \rho_r w}{\partial z} = 0. \quad (3) $$

The interaction between convection and its environment is effected by internal gravity waves excited by convection, propagating away and transmitting the local

<figure><img src="image_2.png" /><figcaption>FIG. 1. The C grid structure of the two-column CPPM model.</figcaption></figure>

intense convective heating to the larger scale (Mapes 1993). The explicit simulation of convection requires proper treatment of lateral boundaries to allow the propagation of the gravity waves out of the simulation domain without spurious reflections. Nevertheless, such lateral boundary conditions cannot be applied in a two-column model. The above set of dynamical equations is not coupled with the thermodynamical equation, by prescribing a buoyancy profile, so gravity waves in the stable environment cannot occur. Thus, only the mean subsidence response of the environment column to the updraft is simulated. The reason for not coupling with the thermodynamics is our ultimate goal of implementing the CPPM in a convection scheme to replace the SW69’s type of updraft model; in this case, buoyancy would be provided by the host convection scheme.

### b. Introduction of the shape factors

#### 1) GRID AND MODEL VARIABLES

Equations are discretized on a staggered Arakawa and Lamb (1977) C grid (Fig. 1), and integrated in time ($t \to t + \Delta t$) with a simple forward temporal scheme. The updraft size is $2a$, and $2b$ for the whole convective cell size (updraft and compensating subsidence), $a$ and $b$ being assumed constant with height. So the environment size is $2(b-a)$, where the compensating subsidence occurs. The $\bar{w}$ prognostic variable represents its mean horizontal value defined as

$$ \begin{aligned} \bar{w}_u &= \frac{1 + \lambda}{a^{1+\lambda}} \int_0^a r^\lambda w_u(r) dr, \\ \bar{w}_e &= \frac{1 + \lambda}{b^{1+\lambda} - a^{1+\lambda}} \int_a^b r^\lambda w_e(r) dr \end{aligned} \quad (4) $$

in the updraft and environment columns, respectively.
The horizontal wind $u$ vanishes at the updraft center due to the symmetry assumption, and at the cell boundary $b$. So we have a one-column prognostic variable $u(a)$ located at the boundary $a$ between the updraft and the environment.

## 2) SHAPE FACTORS

The vertical velocity of real convective clouds is not uniform in the updraft or in the environment, and

these velocity spatial variations have a quantitative effect on vertical fluxes. To partially account for these variations we introduce the shape factors $f_u(r)$ and $f_e(r)$ for the horizontal profiles of $w$ in the updraft and environment, respectively. This makes it possible to write the $w(r, z)$ field as the product of two functions depending only on the horizontal direction $r$ for one and on the vertical coordinate $z$ for the other such as

$$ 
\begin{aligned}
w_u(r, z) &= \bar{w}_u(z) f_u(r) \\
w(a, z) &= 0 \\
w_e(r, z) &= \bar{w}_e(z) f_e(r)
\end{aligned}
$$ 
for

$$ 
\left\{
\begin{array}{l}
0 \le r < a \\
r = a \\
a < r \le b
\end{array}
\right.
\quad \text{with} \quad
\left\{
\begin{array}{l}
\bar{f}_u = 1 \\
. \\
\bar{f}_e = 1
\end{array}
\right.
\qquad (5)
$$

This assumption is strong, as it physically means that the drafts are directed vertically with a constant size and horizontal shape in the vertical. Figures 2a and 2b illustrate some shape factors for the slab and axis symmetries. Using the continuity Eq. (3), a similar separation for the $u$ field is obtained:

$$ 
\begin{aligned}
u_u(r, z) &= u(a, z) g_u(r) \\
u_e(r, z) &= u(a, z) g_e(r)
\end{aligned}
$$ 
for $\begin{cases} 0 \le r \le a \\ a \le r \le b \end{cases}$, (6)

where the $g(r)$ functions are integrals of the $f(r)$ ones,

$$ 
\begin{aligned}
g_u(r) &= \frac{1 + \lambda}{a r^{\lambda}} \int_0^r x^{\lambda} f_u(x) dx, \\
g_e(r) &= \frac{a^{\lambda}}{r^{\lambda}} \left[ 1 - \frac{1 + \lambda}{b^{1+\lambda} - a^{1+\lambda}} \int_a^r x^{\lambda} f_e(x) dx \right],
\end{aligned}
\qquad (7)
$$

and

$$ u(a, z) = - \frac{a}{(1 + \lambda) \rho_r} \frac{\partial \rho_r \overline{w_u}}{\partial z}. \qquad (8) $$

For the same $\overline{w_u}$, the horizontal velocity at the updraft boundary for the axial symmetry is half the one of the slab symmetry as noted by Yau (1979). Indeed, the horizontal divergence/convergence feeding the updraft comes from two dimensions in a 3D flow instead of only one for the slab symmetry (2D). For the sake of simplicity, hereafter we will call 2D and 3D the slab and axial symmetry versions, respectively.

The value of $u$ is maximum at the updraft boundary ($r = a$). For the top-hat shape, $u$ decreases linearly in the environment for the slab symmetry (Fig. 2c), and more rapidly for the axial one (Fig. 2d) as the surface of its compensating subsidence is larger than for the slab symmetry.

In the updraft the “parabolic shape” results in a slower inward decrease of $u$ near the updraft boundary than in the updraft core. But unlike in 2D (Fig. 2c), in 3D the resulting $u$ shape $g_u(r)$ exhibits a maximum within the updraft stronger than at its boundary (Fig. 2d). As it generates a drift of the model, we use a “truncated parabolic” shape keeping a discontinuity at the updraft boundary as for the top-hat shape but reduced to 1/2 (Fig. 2b). In that case $g_u(r)$ (Fig. 2d) is identical to $g_u(x)$ in 2D (Fig. 2c). To compare the slab and axial geometries we used the truncated-parabolic shape for both configurations in the following.

### c. Formulation and computation of advection and turbulence terms

#### 1) ADVECTION TERM OF VERTICAL VELOCITY

The advection of $\overline{w}$ is computed by applying the horizontal averaging operator [Eq. (4)] to the first two rhs terms of Eq. (2) over the [0, $a$] or [$a$, $b$] segments. It can be written in the updraft as

$$ AW_u = - \overline{u \frac{\partial w}{\partial r}} - \overline{w \frac{\partial w}{\partial z}} = -C1_u \frac{u(a, z) \overline{w_u}}{a} - C2_u \frac{1}{2} \frac{\partial \overline{w_u}^2}{\partial z}, \qquad (9) $$

with

$$ 
\begin{aligned}
C1_u &= ag_u(r) \frac{\partial f_u(r)}{\partial r} = \frac{1 + \lambda}{a^\lambda} \int_0^a x^\lambda g_u(x) \frac{\partial f_u(x)}{\partial x} dx, \\
C2_u &= \overline{f_u^2(r)} = \frac{1 + \lambda}{a^{1+\lambda}} \int_0^a x^\lambda f_u^2(x) dx.
\end{aligned}
\qquad (10)
$$

Similarly, in the environment,
<figure><img src="image_3.png" /><figcaption>FIG. 2. Illustration of some shape factors for (a),(b) the vertical velocity and (c),(d) the horizontal velocity for the (left) slab and (right) axial symmetry. Here it is drawn for a 2-km updraft size (2a) and a 8-km total cell size (2b). Top-hat, parabolic, and “truncated parabolic” shape factors are shown for the updraft instead of only the top-hat one for the environment.</figcaption></figure>

$$AW_e = -C1_e \frac{u(a, z) \overline{w_u}}{b-a} - C2_e \frac{1}{2} \frac{\partial \overline{w_u}}{\partial z}, \quad (11)$$

with

$$C1_e = (b-a)g_e(r) \frac{\overline{\partial f_e(r)}}{\partial r} = \frac{(1+\lambda)(b-a)}{b^{1+\lambda} - a^{1+\lambda}} \\ \times \int_a^b x^{\lambda} g_e(x) \frac{\partial f_e(x)}{\partial x} dx, \\ C2_e = \overline{f_e^2(r)} = \frac{1+\lambda}{b^{1+\lambda} - a^{1+\lambda}} \int_a^b x^{\lambda} f_e^2(x) dx. \quad (12)$$

The appendix demonstrates the following relationships between coefficients C1 and C2:

$$C1_u = -(1 + \lambda)C2_u, \\ C1_e = (1 + \lambda)C2_e.$$

So, in 3D geometry ($\lambda = 1$), the C1 coefficient is doubled compared to the 2D ($\lambda = 0$), and of opposite and same sign in the updraft and environment, respectively. Nevertheless, as for the same $\overline{w}$, the horizontal wind $u$ in 3D is half the one in 2D [Eq. (8)], the horizontal advection of $\overline{w}$ is not affected by the geometry.

Table 1 provides the above coefficient values for different CPPM versions, consistent with the relationship between C1 and C2. For the top-hat shape those coefficients are equal to one. The introduction of a shape increases both horizontal and vertical advection by the same ratio: +20% and +33% in 2D and 3D, respectively, for the parabolic shape. For the truncated-parabolic shape the increase is weaker (+5% and +8%, respectively). Thus, this increase is a little stronger in 3D.

The advection $AW_u$ and $AW_e$ are computed directly from the first rhs term of Eqs. (9) and (11) for the
<table><caption>TABLE 1. Coefficients of the advection terms in the updraft and in its environment as a function of the model geometry (slab or axial symmetry) and of the shape factor (top hat or parabolic). Only the top-hat profile is considered here for the environment.</caption><thead><tr><th>Symmetry</th><th>Shape factor</th><th>f<sub>u</sub></th><th>g<sub>u</sub></th><th>C1<sub>u</sub></th><th>C2<sub>u</sub></th><th>l, case 0 &lt; l &le; a for outflow u(a, z) &ge; 0</th></tr></thead><tbody><tr><td rowspan="3">Slab (2D)</td><td>Top hat</td><td>1</td><td>x/a</td><td>-1</td><td>1</td><td>a exp[-u(a, z) &Delta;t/a]</td></tr><tr><td>Parabolic</td><td>-(3/2)(x/a)<sup>2</sup> + 3/2</td><td>-(1/2)(x/a)<sup>3</sup> + 3x/2a</td><td>-6/5</td><td>6/5</td><td>&radic;3a<sup>2</sup>/{1 + 2 exp[3u(a, z) &Delta;t/a]}</td></tr><tr><td>Truncated parabolic</td><td>-(3/4)(x/a)<sup>2</sup> + 5/4</td><td>-(1/4)(x/a)<sup>3</sup> + 5x/4a</td><td>-21/20</td><td>21/20</td><td>&radic;5a<sup>2</sup>/{1 + 4 exp[5u(a, z) &Delta;t/2a]}</td></tr><tr><td rowspan="3">Axial (3D)</td><td>Top hat</td><td>1</td><td>r/a</td><td>-2</td><td>1</td><td>a exp[-u(a, z) &Delta;t/a]</td></tr><tr><td>Parabolic</td><td>-2(r/a)<sup>2</sup> + 2</td><td>-(r/a)<sup>3</sup> + 2r/a</td><td>-8/3</td><td>4/3</td><td>&radic;2a<sup>2</sup>/[1 + exp[4u(a, z) &Delta;t/a]]</td></tr><tr><td>Truncated parabolic</td><td>-(r/a)<sup>2</sup> + 3/2</td><td>-(1/2)(r/a)<sup>3</sup> + 3r/2a</td><td>-13/6</td><td>13/12</td><td>&radic;3a<sup>2</sup>/[1 + 2 exp[3u(a, z) &Delta;t/a]]</td></tr><tr><td>Symmetry</td><td>Shape factor</td><td>f<sub>e</sub></td><td>g<sub>e</sub></td><td>C1<sub>e</sub></td><td>C2<sub>e</sub></td><td>xl, case xl &ge; a for inflow u(a, z) &le; 0</td></tr><tr><td>Slab</td><td>Top hat</td><td>1</td><td>(b - x)/(b - a)</td><td>1</td><td>1</td><td>b - (b - a) exp[u(a, z) &Delta;t/(b - a)]</td></tr><tr><td>Axial</td><td>Top hat</td><td>1</td><td>[a(b<sup>2</sup> - r<sup>2</sup>)]/[r(b<sup>2</sup> - a<sup>2</sup>)]</td><td>2</td><td>1</td><td>&radic;(b<sup>2</sup> - (b<sup>2</sup> - a<sup>2</sup>) exp[2au(a, z) &Delta;t/(b<sup>2</sup> - a<sup>2</sup>)]</td></tr></tbody></table>

horizontal advection, whereas the second term is discretized using an upstream scheme for the vertical advection.

## 2) ADVECTION TERM OF HORIZONTAL VELOCITY

The strong anisotropy of the grid cell prevents computing the u advection as for the ones of $\bar{w}$ [see section 2c(1)]. Similar to a semi-Lagrangian method, the horizontal advection of $u(a, z)$ at the updraft boundary is computed by following a backward horizontal

trajectory. Considering a parcel located at the updraft boundary at time $t$, we search for its horizontal position $l$ at the previous time step $t - \Delta t$. Knowledge of the horizontal variation of $u$ provided by the shape factor $g(x)$ within the updraft and environment [Eqs. (7) and (8), respectively] makes it possible to compute $l$. Indeed, as from Eq. (6) $u(r, z) = (dr/dt) = u(a, z) g(r)$, its integration during the previous time step, with the approximation that $u$ stays constant in time, provides the relationship

$$ \int_{l}^{a} \frac{dr}{g(r)} = \int_{t-\Delta t}^{t} u(a, z) dt = u(a, z) \Delta t $$

that can be analytically solved to compute the $l$ position. The last column of Table 1 provides the $l$ expression depending on the $u$ direction. With the assumption that $u'(a, z) = u^{l-\Delta t'}(l, z)$ the total advection of $u(a, z)$ reads

$$ \mathrm{AU} = \frac{u'(l, z) - u'(a, z)}{\Delta t} - w(r_m, z) \frac{\partial u(r_m, z)}{\partial z}, \quad (13) $$

where the rhs terms correspond to the horizontal and vertical advection, respectively. The vertical advection is computed at the midposition $r_m = (l + a)/2$ with a simple upstream scheme.

## 3) TURBULENCE TERM

The application of the mean horizontal operator “-” to Eq. (2), provides the vertical velocity turbulence term. For instance, for the updraft it reads

$$ \mathrm{TW}_u = \overline{D_w} = - \frac{\overline{u''w''}(a) - \overline{u''w''}(0)}{a} - \frac{\overline{w''}}{\partial z}. $$

$$ \text{using} \quad \begin{cases} g_u \\ g_e \end{cases} \quad \text{for} \quad \begin{cases} u(a, z) > 0 \\ u(a, z) < 0 \end{cases} $$

The horizontal turbulent flux of $w$ ($\overline{u''w''}$) is zero at the updraft center. We also assume it is zero at the edge of the updraft, according to Sherwood et al. (2013), who suggest a weak turbulent entrainment for the vertical momentum. The second term is also neglected, as the shape factor allows accounting in part for the subgrid fluxes. So, the turbulence term is neglected for the vertical velocity both in the updraft and in the environment.

For $u(a, z)$ at the updraft boundary, the turbulence term derived from Eq. (1) reads for the slab symmetry (2D)

$$ \mathrm{TU} = D_u(a, z) = - \frac{\partial \overline{u''^2}}{\partial r} - \frac{\partial \overline{u''w''}}{\partial z}. $$

We use a simple downgradient formulation of the turbulent fluxes with a turbulent coefficient $K_U$, so that

$$ \mathrm{TU} = \frac{\partial}{\partial r} \left( K_U \frac{\partial u}{\partial r} \right) + \frac{\partial}{\partial z} \left( K_U \frac{\partial u}{\partial z} \right). \quad (14) $$
As intense turbulence is generated at the updraft (cloud) edge where the u is maximum (Figs. 2c,d), strong mixing of u is expected at this boundary so that we keep this term TU. The model sensitivity to the $K_U$ magnitude and vertical profile is weak for a large range of values (not shown) and the model can be run without this turbulence term. Hereafter we used a constant value of 50 m² s⁻¹. TU is computed with a centered spatial

scheme. The shape factor $g_u(x)$ is used to derive the horizontal gradients of u in the vicinity of the updraft boundary ($x = a$).

### d. Pressure equation and time integration

The horizontal averaging operator (4) is applied to the continuity equation, (3), in the updraft and environment columns:

$$ \frac{\rho_r \frac{\partial r^\lambda u}{\partial r}}{r^\lambda} + \frac{\partial \rho_r \bar{w}}{\partial z} = 0 \quad \text{either} \quad \left\{ \begin{array}{l} \rho_r \frac{1+\lambda}{a} u(a) + \frac{\partial \rho_r \bar{w}_u}{\partial z} = 0 \\ -\rho_r \frac{(1+\lambda)a^\lambda}{b^{1+\lambda} - a^{1+\lambda}} u(a) + \frac{\partial \rho_r \bar{w}_e}{\partial z} = 0 \end{array} \right. . $$

Taking its time derivative and using the source terms derived in section 2c, we obtain

$$ \frac{1+\lambda}{a} \left( \frac{p'_e - p'_u}{b/2} - 0 \right) + \frac{\partial^2 p'_u}{\partial z^2} = \rho_r \frac{1+\lambda}{a} (\text{AU} + \text{TU}) + \frac{\partial}{\partial z} (\rho_r \text{AW}_u + \rho_r \text{B}_u), \\ \frac{(1+\lambda)a^\lambda}{b^{1+\lambda} - a^{1+\lambda}} \left( 0 - \frac{p'_e - p'_u}{b/2} \right) + \frac{\partial^2 p'_e}{\partial z^2} = -\rho_r \frac{(1+\lambda)a^\lambda}{b^{1+\lambda} - a^{1+\lambda}} (\text{AU} + \text{TU}) + \frac{\partial}{\partial z} (\rho_r \text{AW}_e + \rho_r \text{B}_e) $$

for the updraft and environment columns, respectively. This is obtained with the following assumptions: $\partial p'/\partial r = 0$ at $r = 0$ and $r = b$ for lateral boundary conditions, and

$$ \frac{\partial p'}{\partial r} = \frac{p'_e - p'_u}{b/2} \quad \text{at} \quad r=a. $$

By taking the difference between these two equations denoted by the operator $\Delta_H$, we obtain an equation for the horizontal difference of pressure between the updraft and the environment defined by the new variable $\Delta_H P' = p'_u - p'_e$:

$$ \frac{\partial^2 \Delta_H P'}{\partial z^2} + \frac{2C_3}{ab} \Delta_H P' = \frac{C_3}{a} \rho_r (\text{AdU} + \text{TurbU}) \\ + \frac{\partial}{\partial z} \Delta_H (\rho_r \text{AdW}_u + \rho_r \text{B}_u) $$

with the geometric coefficient: $C_3 = [(1 + \lambda)b^{1+\lambda}]/(b^{1+\lambda} - a^{1+\lambda})$.

With the boundary condition $\bar{w} = 0$ at the top and bottom resulting in a zero vertical gradient of $\Delta_H P'$, and after vertical discretization of this equation written at time $t$, we obtain a 1D vertical equation:

$$ \mathbf{A} \times \Delta_H P^{(t+\Delta t/2)} = S^t, \quad (15) $$

where $\mathbf{A}$ is a tridiagonal matrix of dimension $N \times N$, with $N$ being the number of vertical levels of the model:

$$ \mathbf{A} = \begin{bmatrix} -\left(\frac{2C_3}{ab} + \frac{1}{\Delta z^2}\right) & \frac{1}{\Delta z^2} & 0 & \dots & 0 \\ \frac{1}{\Delta z^2} & -\left(\frac{2C_3}{ab} + \frac{2}{\Delta z^2}\right) & \frac{1}{\Delta z^2} & \dots & \vdots \\ 0 & \vdots & \ddots & \ddots & 0 \\ \vdots & \vdots & \frac{1}{\Delta z^2} & -\left(\frac{2C_3}{ab} + \frac{2}{\Delta z^2}\right) & \frac{1}{\Delta z^2} \\ 0 & \dots & 0 & \frac{1}{\Delta z^2} & -\left(\frac{2C_3}{ab} + \frac{1}{\Delta z^2}\right) \end{bmatrix}, $$

and $S$ is the source term computed at time $t$, such as
$$S = \left\{ \begin{array}{l} \frac{C_3}{a} \rho_r [\mathrm{AU}^{(k=1)} + \mathrm{TU}^{(k=1)}] + \frac{\rho_r}{\Delta z} \Delta_H (\mathrm{AW} + \mathrm{B})^{(k=1)} \\ \vdots \\ \frac{C_3}{a} \rho_r [\mathrm{AU}^{(k)} + \mathrm{TU}^{(k)}] + \frac{\rho_r}{\Delta z} \Delta_H [(\mathrm{AW} + \mathrm{B})^{(k)} - (\mathrm{AW} + \mathrm{B})^{(k-1)}] \\ \vdots \\ \frac{C_3}{a} \rho_r [\mathrm{AU}^{(k=N-1)} + \mathrm{TU}^{(k=N-1)}] + \frac{\rho_r}{\Delta z} \Delta_H (\mathrm{AW} + \mathrm{B})^{(k=N-1)} \end{array} \right\},$$

where the $\Delta_H$ operator is applied to the source terms (AW + B). The B part is consistent with the concept of “effective buoyancy” (Davies-Jones 2003; Peters 2016), defined as the B difference ($\Delta_H B$) between the updraft and its periphery, so is independent of the reference state.

As A depends only on the geometry of the grid (a, b, and N), its inversion is performed only once. After solving Eq. (15) at each time step to diagnose the horizontal pressure gradient $\Delta_H P'(t+\Delta t/2)$, $u^{t+\Delta t}$ is easily obtained. Finally, $w^{t+\Delta t}$ is deduced from the continuity equation, (3).

### 3. Model response to a fixed buoyancy profile

#### a. Reference simulations

To analyze the behavior of the CPPM we first consider its response to two stationary idealized profiles of buoyancy difference between the updraft and the environment $\Delta_H B$:

$$\Delta_H B(z) = \frac{g}{T_0} 1.7 [\sin(\pi z / H_{\text{trop}}) - \alpha \sin(2\pi z / H_{\text{trop}})],$$

where $H_{\text{trop}} = 9$ km and $T_0 = 300$ K. It is drawn in Fig. 3a for two values of $\alpha$ (0 and 1). Above the level of neutral buoyancy, the perturbation is negative and increases linearly according to a stable stratification up to $H = 10$ km. The corresponding CAPE(z) profiles—taken here as the integral of the buoyancy from surface to z—reach the same maximum of $310$ J kg$^{-1}$ at $H_{\text{trop}}$ (Fig. 3b). For $\alpha = 1$ an inhibition barrier of $40$ J kg$^{-1}$ appears, so that this profile will be called CIN hereafter, whereas the other one ($\alpha = 0$) will be called noCIN.

In addition to the choice of symmetry and updraft shape (top hat or truncated parabolic), the CPPM depends on three geometric parameters: H, 2a, and 2b, fixed for the reference simulations at 10, 4, and 40 km, respectively. The top-hat updraft shape is used for those simulations.

It is physically convenient to introduce the following derived parameters: the updraft aspect ratio $\delta = 2a/H$

and the convective active fraction $\sigma_u (=a/b$ or $a^2/b^2$ in 2D or 3D, respectively). So, the reference simulations correspond to a 10 km deep convective updraft with a 0.4 aspect ratio and representing 10% in 2D or 1% in 3D of the whole surface.

To make it easier to analyze and interpret results we added the results of the parcel model ($w = \sqrt{2} \text{CAPE}$; Fig. 3c) and of a version of the SW69 model (Fig. 3f). Among the many SW69 versions, we took the one used by Gueremy [2011], their Eq. (24), hereafter referred to as G11]. The pressure term is also not explicitly accounted for in either of these models. Nevertheless, to account for the pressure term in reducing the updraft strength, G11 reduced the buoyancy by a factor of 2/3 and used a drag proportional to $w^2$ with a coefficient depending on the turbulent entrainment.

The time scale of the $\bar{w}$ profile response is shorter than the typical life time of a deep convective cell (between half an hour and an hour), as it reaches a steady state in less than 15 min for all models (Figs. 3c-f). The evolution of the mean $\langle \bar{w} \rangle$ averaged over H provides a time scale $\tau$ of the model response (Fig. 3g), ranging between 264 and 423 s for the G11 and the 2D CPPM, respectively, for the noCIN simulations. For CIN simulations, the response is even faster, the 2D CPPM being still a little slower than the 3D. This steady state can be reached because the buoyancy is constant. This result means that the convective circulation adjusts rapidly to the buoyancy field.

The $\bar{w}$ profile during the quasi-steady state is sensitive to the CPPM configuration. For the noCIN runs (Fig. 3h), as expected the parcel model overestimates $\bar{w}$ and reaches its maximum at the same altitude $H_{\text{trop}}$ as CAPE. The $\bar{w}$ of the other models is reduced by a factor of about 50%, with a lowering of the maximum, between 60% and 70% of the cell depth that agrees with the range computed by Peters (2016) (2/3–3/4). The 3D response is faster, stronger and peaks at a higher altitude than in 2D, and this is still true for the CIN runs (Fig. 3h). The $\bar{w}$ profile of G11 is quite irregular depending on the entrainment and drag profiles, in particular, with a
<figure><img src="image_4.png" /><figcaption>FIG. 3. (a) Vertical profiles of buoyancy difference between the updraft and the environment expressed as a difference of virtual temperature $\overline{T}' = T_0/g\Delta_H B(z)$ (K) and (b) the corresponding CAPE (J kg$^{-1}$) for two profiles without and with a CIN layer (noCIN and CIN). Also shown are vertical profiles of the $\overline{w}$ (m s$^{-1}$) response every minute for noCIN for 4 models: (c) the parcel, (d) slab, and (e) axial top-hat CPPM and (f) G11 models. (g) Time evolution of vertical average of $\langle \overline{w} \rangle$ (m s$^{-1}$) providing the time scale of the model response $\tau(s)$. Vertical profiles of the $\overline{w}$ (m s$^{-1}$) response are shown at 15 min for the four models for both buoyancy profiles: (h) noCIN and (i) CIN.</figcaption></figure>

discontinuity at the updraft top. For the CIN runs (Fig. 3i), the parcel and G11 models cannot overcome the CIN barrier without an ad hoc triggering function, unlike the CPPM.

## b. Velocity budgets

Figure 4 details the Eulerian budget of $u$ at the updraft boundary and of $\overline{w}$ in the updraft for different configurations of the model for the reference case during the quasi-steady state reached after 15 min, as shown by

the weak evolution of $\overline{w}$. For the top-hat 2D CPPM (Figs. 4a,d) the separation between the inflow layer and the outflow above is at ~3.6 km ($Z_{u0}$). The inflow is weakly decelerated by the advection from the environment and by the turbulent term TU (Fig. 4a). In contrast, the outflow is strongly decelerated both by the vertical advection maximum in the 6–8-km layer, and by the horizontal advection above 8 km. The acceleration by the horizontal gradient of pressure between the updraft and the environment balances the deceleration by the
<figure><img src="image_5.png" /><figcaption>FIG. 4. Vertical profiles of (a)–(c) $u$ and (d)–(f) $\bar{w}$ budgets (m s$^{-2}$). Model response to a prescribed buoyancy profile after 15 min of simulation with a top-hat shape of the updraft. Columns show noCIN profiles for the (left) slab and (center) axial symmetry and (right) CIN profile only for the axial symmetry.</figcaption></figure>

advection of both inflow and outflow to maintain the convective cell.

The Eulerian budget of $\bar{w}$ in the updraft is more complex (Fig. 4d) as there is an equilibrium between three acceleration terms: that is, buoyancy as a strong driving mechanism, advection, and perturbation pressure terms. The total advection $AW_u$ decelerates the updraft below 5 km and reinforces it above. As formulated by Eq. (9), the horizontal advection changes sign at the altitude $Z_{u0}$ separating the inflow and outflow layers, whereas the vertical advection changes sign at the altitude $Z_{wmax}$, where $\bar{w}$ is maximum, located at 6 km, so higher than $Z_{u0}$ due to the density decrease with altitude. This results in a change of sign of the total advection between these two levels. The pressure term balances the sum $B + AW_u$. So, the perturbation pressure term cannot be neglected, and it is not in opposite phase with $B$ or $\bar{w}$. We added the term $Dw/Dt = -(1/\rho_r)(\partial p'_{nh}/\partial z)$ to $\bar{w}$ budgets to make a link with the Lagrangian budget analysis of thermals (Romps and Charn 2015; Hernandez-Deckers and Sherwood 2016). As the CPPM reaches a quasi-steady state, the updraft is accelerated by the nonhydrostatic pressure perturbation term below ~5.5 km and decelerated above.

Figure 5a, provides the corresponding vertical profiles of $p'_e$, $p'_u$, and $p'_{nh}$ in the updraft. A low (−50 Pa) is generated at low levels in the updraft, and a high (150 Pa) at upper levels. The reverse configuration is simulated in the environment, but with weak amplitude (20 Pa). The crossing of the two profiles at $Z_{u0}$ corresponds to the reversal of the horizontal gradient of pressure revealed by the $u$ budget (Fig. 4a). The non-hydrostatic part of the pressure perturbation $p'_{nh}$ in the updraft reaches a minimum of −50 hPa at ~5.5 km.

The budget structures for the top-hat 3D CPPM simulation (Figs. 4b,e) are very similar to those of the previous 2D simulation (Figs. 4a,d). The main difference is for the $\bar{w}$ budget with stronger advection in 3D, and a rising of the levels $Z_{u0}$ and $Z_{wmax}$, in agreement with a stronger updraft peaking a little higher as already noted in Figs. 3d and 3e. Also, the nonhydrostatic pressure perturbation $p'_{nh}$ is stronger (minimum of −85 Pa) than for the 2D configuration.

The budget of the top-hat 3D CPPM simulation with the CIN buoyancy profile (Figs. 4c,f) explains why its updraft can overcome the CIN barrier contrary to the parcel and G11 models (Fig. 3i). In the elevated layer where $B$ is positive (above 3 km), the structure of
<figure><img src="image_6.png" /><figcaption>FIG. 5. Vertical profiles of $p'_u$, $p'_e$ and $p'_{nh}$ (hPa) after 15 min of simulation with a top-hat shape of the updraft in response to the noCIN profile for the (a) slab and (b) axial symmetry and (c) for the CIN profile only for the axial symmetry.</figcaption></figure>

budgets is similar to the one in the previous simulations. But in the upper part of the layer of negative buoyancy (1.5–3 km) the vertical acceleration $Dw/Dt$ is positive, as the perturbation pressure term (Fig. 4f) overcomes the buoyancy term. The corresponding $p'_u$ vertical profile (Fig. 5c) shows the formation of a low at midlevels (~4 km) mainly associated with the hydrostatic perturbation pressure, whereas the $p'_{nh}$ profile exhibits an intense low (–60 Pa) at 6 km, maintaining an ascent at low levels in the CIN layer. In other words, the updraft at mid- and upper levels evacuates by suction the air underneath, and hence is able to overcome the CIN barrier. Concerning the $p'_e$ vertical profile in the environment, it should be noted that it is almost constant for the 3D simulations (Figs. 5b,c) contrary to 2D ones (Fig. 5a). This is explained by a stronger surface ratio between the updraft and the whole convective cell $\sigma_u$ in 2D than in 3D (10% vs 1%), so that the subsidence is 10 times weaker in 3D, generating a quasi-hydrostatic pressure field.

### c. Sensitivity experiments

Figure 6 shows the vertical profile of the $\bar{w}$ budget, for an experiment with the axial symmetry, for which we added noise to the prescribed buoyancy CIN profile (maximum noise magnitude is 20% of the standard buoyancy) to be compared with the same experiment without a noise (Fig. 4f). The result is that the small-scale buoyancy fluctuations are compensated by the perturbation pressure term, so that the resulting updraft profile is almost identical to the one obtained without the noise. In the Eulerian budget, pressure and advection terms counteract the prescribed buoyancy, this negative feedback works even at small scale where the pressure perturbation term dampens buoyancy fluctuations.

Figure 7 shows the sensitivity of the $\bar{w}$ profile to the aspect ratio $\delta = 2a/H$ for the CIN buoyancy profile. The

“aspiration effect” associated with the formation of a low at midlevels previously discussed (Fig. 5c) increases with the aspect ratio and occurs whatever the CPPM configuration (2D, 3D, shape factor). Nevertheless, updrafts narrower than 3 km (for a depth of 10 km) cannot overcome the CIN barrier. This confirms the results of the pioneering work of Yau (1979, see his Fig. 4). In other words, the CPPM can reproduce this aspiration effect, that may help in predicting nocturnal convection of long-lasting systems.

Figure 8 explores the sensitivity of the updraft maximum $w_{max}$ simulated by the CPPM to the geometry, shape factor and geometric parameters a and $a/b$, for the buoyancy profile noCIN. Similar results are found with the CIN profile. As H is fixed to 10 km, the sensitivity

<figure><img src="image_7.png" /><figcaption>FIG. 6. Vertical profile of the $\bar{w}$ budget (m s$^{-2}$) for the axial symmetry, as in Fig. 4f. Noise was added to the constant prescribed-buoyancy CIN profile.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>FIG. 7. Sensitivity to the aspect ratio $\delta$ of the $w(z)$ response for the CIN profile and for a fixed $b$ (20 km) for the slab CPPM with (a) a top-hat and (b) truncated-parabolic updraft shape and (c) for the axial symmetry with a truncated-parabolic shape.</figcaption></figure>

study to *a* is equivalent to a sensitivity study to the aspect ratio $\delta = \alpha/H\delta$. The 2D diagrams summarize the CPPM behavior, with an intensification of $w_{max}$ when it becomes narrower and the environment size ($b-a$) is reduced. For $\delta > 1$ (above the horizontal line $\delta = 1$ in Fig. 7) the flow becomes nearly hydrostatic and as expected $w_{max}$ decreases. On the contrary below this line the $w_{max}$ increases.

The updraft magnitude is significantly reduced when the ratio $a/b$ becomes small (<0.1). Physically this could be due to the decrease in the gradient of buoyancy between the updraft and its environment. Numerically it also means that the grid stretching between the two columns is important, so it can be enhanced by discretization errors. Consequently, it is recommended to avoid using small ratio $a/b$ (see section 4b).

Figure 8 clearly confirms the impact of the geometry and updraft shape. The pattern of the sensitivity stays the same, but its magnitude changes. The 2D–3D transition enhances the updraft (cf. top and bottom rows of Fig. 8). The gain is about $4 \text{ m s}^{-1}$ for the reference simulations (circles in Fig. 8) regardless of the shape. Similarly, the transition from the top-hat to the truncated-parabolic shape (left and right columns) enhances the updraft, with a gain of about $2 \text{ m s}^{-1}$ regardless of the geometry.

The analytic solutions of Morrison (2016a) provide a formulation of the reduction factor of the vertical velocity by the perturbation pressure term depending on the aspect ratio, the shape factor and the geometry (2D or 3D). He thus derived the intensification ratio between 3D and 2D updrafts. We computed this ratio as simulated by different configurations of the CPPM and compared them with the reduction factors predicted by Morrison's theory. For the shallow convection regime and an aspect ratio of 0.4, the increase due to the geometry (2D to 3D) is 107% and 104% for the top-hat

and truncated-parabolic shape, respectively, in good agreement with Morrison's theory (110% and 107%). Also, the increase to 200%, predicted by this theory for large aspect ratios $\delta$, is reproduced by the CPPM. For deep convection the CPPM simulates stronger values (140% for $\delta = 0.4$).

The height of $w_{max}$ varies inversely with the updraft size (not shown), but with a maximum elevation of 1 km for the smallest aspect ratios. The response time scale $\tau$ is always larger than in the parcel and G11 models (~300 s) but below 600 s (not shown). It varies inversely with the updraft strength, so that the 2D CPPM exhibits a slower response than in 3D.

# 4. Evaluation compared to a real case

## a. The AMMA case study

After the above analysis of the CPPM behavior and sensitivity based on an idealized deep convective cell, we tested it against an LES. We chose a case of daytime development of deep convection over tropical semiarid land, based on observations collected during the African Monsoon Multidisciplinary Analysis (AMMA) field campaign. This case observed on 10 July 2006 over Niamey includes two distinct transition phases, from clear sky to shallow cumulus and from cumulus to deep convection (Lothon et al. 2011).

Couvreux et al. (2012) performed an LES of this case with a setup based on observations, and validated it against the numerous observations available (including radar, lidar, aircrafts, and flux stations). The LES was performed at a horizontal grid spacing of 200 m over a domain of 100 km × 100 km from 0600 to 1800 UTC, and it succeeded in representing the main boundary layer structures and processes leading to deep convection initiation as well as the formation of density currents.
<figure><img src="image_9.png" /><figcaption>FIG. 8. Sensitivity study of the updraft maximum $w_{max}$ (m s$^{-1}$) simulated by the CPPM for the noCIN buoyancy profile to the combination of the *a* parameter (from 1 to 10 km) and the ratio *a/b* (from 0.01 to 0.5). The sensitivity is shown for the (a),(b) slab and (c),(d) axial symmetries and for the (left) top-hat and (right) truncated-parabolic shapes. The heavy circle corresponds to the reference simulation. As *H* is fixed to 10 km for all simulations, the aspect ratio $\delta = 2a/H$ varies between 0.2 and 2.</figcaption></figure>

This LES served as a reference to compare different single-column models with different versions of their physical package within the framework of the Earth system Model Bias Reduction and assessing Abrupt Climate change (EMBRACE) European project (Couvreux et al. 2015).

We used the 3D LES fields available at a 15-min frequency, to evaluate the CPPM first on some specific convective cells in section 4b, and in section 4c against some bulk properties of the convective structures. It is difficult to detect such coherent structures in observations and LES. Conditional samplings to characterize cumulus clouds have been proposed, namely the “cloud” (all nonzero liquid water content points) and “core” (buoyant ascending and cloudy points) samplings defined by Siebesma and Cuijpers (1995). Couvreux et al. (2012) proposed a conditional sampling method based on the combination of a passive tracer emitted at the surface and thermodynamic variables to characterize organized structures in LESs of cloud-free and cloudy boundary layers.

As the present LES simulates a wide range of convective structures (boundary layer, shallow and deep

convection), we applied a conditional sampling method with the following criteria:

* selection of all cloudy points with positive vertical velocity;

* for each LES column, add all points below the first cloudy point if below the 4-km level, so that the vertical column beneath the cloud base is considered;

* this subcloud column is excluded if subsidence occurs within it, to exclude convective downdrafts.

Figure 9a illustrates the evolution of the vertical profile of the fractional area (%) of the resulting selected volume in the LES. Its maximum reaches 5%–6% just above the cloud base increasing regularly throughout the simulation from 1.5 to 2.5 km. Underneath it stays constant in agreement with the sampling criteria. This figure shows the three successive phases of the daytime convection development defined by Couvreux et al. (2012): dry convection (from the early morning until 1110 UTC), shallow convection (1110–1600 UTC), and deep convection (1600 UTC until the end of the simulation). The mean
<figure><img src="image_10.png" /><figcaption>FIG. 9. Evolution of the vertical profiles provided by the LES of (a) fractional area (1% isoline interval) of the cloudy and ascending zone (see text for the sampling definition), (b) mean vertical velocity $\bar{w}$ (m s$^{-1}$) in the sampled zone, and (c) its contribution to the vertical mass flux (kg s$^{-1}$ m$^{-2}$). The gray shading indicated values greater than 3%, 1 m s$^{-1}$, and 0.06 kg s$^{-1}$ m$^{-2}$ in (a), (b), and (c), respectively. The heavy red isoline corresponds to 0.25% value of fractional area and delimits cloudy and ascending zones at a given level that have a total surface 5 km × 5 km, so no more than about one single deep convective cell in the 100 km × 100 km LES domain.</figcaption></figure>

updraft intensity (Fig. 9b) reaches about 2 m s$^{-1}$ in the subcloud boundary layer, and increases above 3 m s$^{-1}$ (gray shading) above 3–4 km after 1500 UTC with maximum up to 6 m s$^{-1}$. Locally, the upper part of the cloud layer has high vertical velocities corresponding to a few isolated convective cells since their total surface area is less than 0.025% (25 km$^2$). This suggests that the transition phase at 1730 UTC from congestus cumulus to deep convection, is preceded by the occurrence of a few isolated deep cells, of which the contribution to the total mass flux (Fig. 9c) is weak. This vertical mass flux deepens and strengthens throughout the simulation and presents two maxima, a first in the boundary layer (~0.06 kg s$^{-1}$ m$^{-2}$) and a larger one in the 3–4-km layer (~0.1 kg s$^{-1}$ m$^{-2}$; gray shading in Fig. 9c). Aloft the vertical mass flux decreases rapidly because there are few convective cells even if they are intense.

To run and test the CPPM, we need to prescribe the three parameters *H*, *a*, and *b*. The values of *H* and *a* will be provided by the LES, whereas *b* is computed by the formula

$$b = \max \left[ 2a, \frac{N\tau_{\text{GW}}}{\pi} \max(H - H_{\text{PBL}}, 2\Delta z) \right], \quad (16)$$

expressing that *b* is the distance traveled in a time $\tau_{\text{GW}}$ by the fastest gravity waves initiated by the convective cell. The duration of $\tau_{\text{GW}}$ has been fixed at 15 min, providing a ratio *a*/*b* between 0.1 and 0.2, corresponding to a ratio between the updraft area and the entire convective cell area of several percent as typically suggested (Asai and Kasahara 1967). The gravity waves travel in the stable layer (*N* being the Brunt–Väisälä frequency) above the neutral PBL, whose thickness is $H - H_{\text{PBL}}$. A minimum thickness of 2 times the vertical resolution is required. Finally, the *b* minimum taken at 2a corresponds to the configuration when the updraft and environment are the same size.

b. Test on specific convective cells

The vertical cross sections of Fig. 10 show the structure of two convective cells among many others simulated by the LES. The first one at 1200 UTC corresponds to a thermal cell in the PBL feeding a small and thin cumulus (Fig. 10b). The anomaly of $T'_{v}$ (Fig. 10a) is positive in the PBL (maximum = 0.8 K) whereas it is negative in the cloud layer (−0.5 K). The updraft presents a weak vertical slope, a depth of 1.9 km, and a maximum of 5 m s$^{-1}$ at 0.7 km (Fig. 10c). A weak vertical anomaly pressure dipole (±3 Pa) between the surface and the level 1.5 km is associated with this updraft cell, so that the vertical pressure gradient decelerates and accelerates the updraft below and above 1.5 km, respectively, where the buoyancy term is positive and negative, respectively.
<figure><img src="image_11.png" /><figcaption>FIG. 10. Vertical cross sections through a shallow and a deep convective cell as simulated by the LES at (a)–(c) 1200 and (d)–(f) 1800 UTC for (a),(d) $T'_{v}$ (color; K) and velocity vector in the plan; (b),(e) total cloud (color) and precipitation (isolines) mixing ratio (g kg$^{-1}$); and (c),(f) vertical velocity (color; m s$^{-1}$) with the pressure perturbation (isolines; hPa). The aspect ratio of the plots is one.</figcaption></figure>

The second convective cell is shown at 1800 UTC during the deep convection phase. It rises vertically to 11 km with maximum vertical velocity up to 16 m s$^{-1}$. The corresponding anomaly of $T'_{v}$ (Fig. 10d) is positive over almost the full depth of the cloud (up to 3.5 K in the updraft core), except for a weak negative anomaly above and at its top, and below the cloud base located at 2.7 km associated with falling precipitation evaporating and not reaching the surface (Fig. 10e). So here we have an example of a growing deep convective cell. At this time we have a vertical tripole of pressure (Fig. 10f): that is, a large low (-0.5 hPa) in the buoyant updraft maximum near 8 km and a high near the surface and above the growing cell (+0.15 and +0.35 hPa, respectively). The resulting strong vertical gradient of pressure above 8 km decelerates the updraft against the buoyancy during this growing phase. On the contrary we have an ascent below the cloud base where the buoyancy is weakly negative, so consistent with an upward-directed perturbation pressure.

The CPPM has been tested for these two convective cells. In agreement with the LES fields (Fig. 10) and Eq. (13), the parameters were fixed at the following values:

* Shallow cell: $H = 1.8$ km; $a = 340$ m (either $\delta = 0.38$); $b = 1.2$ km

* Deep cell: $H = 11$ km; $a = 1.7$ km (either $\delta = 0.31$); $b = 28$ km

The profiles of mean $T'_{v}$ anomaly (Figs. 11a,c) and of mean vertical velocity (Figs. 11b,d) were computed from the LES for all points of the updraft corresponding to a surface $4a^2$. The CPPM was run in 2D and 3D for the top-hat and truncated-parabolic shapes. Figures 11b and 11d show $\bar{w}$ simulated after 15 min in response to the buoyancy profiles (Figs. 11a,c), and the corresponding $\bar{w}$ profiles provided by the LES, the parcel model (PART), and the G11 model.

For the shallow cell, the mean $T'_{v}$ is positive in the PBL and reaches a maximum of 0.5 K at 800 m, whereas it is negative and maximum at the cell top (-0.45 K). The updraft response (Fig. 11b) is a little less intense than simulated by the LES (2.8–3.2 m s$^{-1}$ instead of 3.8 m s$^{-1}$) and peaks a little higher (~200 m). The sensitivity to the CPPM configuration is weak. Nevertheless, the slab configuration provides the best results probably because the convection is organized at this time as 2D convective rolls (Fig. 9 of Couvreux et al. 2012). As expected, the parcel model overestimates the vertical velocity, whereas G11 underestimates it.

For the deep cell the mean $T'_{v}$ is positive in the 5–10-km layer and maximum (1.8 K) near 9 km before a sharp decrease to reach a negative value of -0.6 K at the
<figure><img src="image_12.png" /><figcaption>FIG. 11. (b),(d) Profiles of vertical velocity $\bar{w}$ (m s$^{-1}$) simulated by the CPPM after 15 min in response to (a), (c) the buoyancy profile expressed in $T'_v$ (K). Four configurations of the CPPM are shown (see legend) with the superposition of the LES $\bar{w}$ profile (heavy gray line) and the one provided by the parcel model (PART) and G11 model. Profiles correspond to the simulations of the (top) shallow and (bottom) deep convective cells extracted from the LES and illustrated by Fig. 10.</figcaption></figure>

cell top (Fig. 11c). The specific feature is the occurrence of a strong CIN layer between 500 m and 3 km with a $T'_v$ minimum of -0.7 K at ~2.7 km. The CPPM succeeds in reproducing the vertical profile of the ascent in agreement with the LES (Fig. 11d). The best result is for the top-hat shape in 3D consistently with the 3D nature of this deep convective cell. This sensitivity to the CPPM configuration confirms the conclusions drawn from the sensitivity study in section 3c (see Fig. 8). The updraft provided by the parcel and G11 models cannot cross the layer with negative buoyancy, and the G11 updraft is underestimates below 6 km.

### c. Comparison with LES bulk properties

We now test the CPPM with a buoyancy profile evolving over time, as extracted from the LES. The

height $H$ of the CPPM is taken at the level where the fractional area defined by the conditional sampling (section 4a) becomes small (<0.25%; heavy line in Fig. 9). Assuming a constant $\delta = 0.4$ as estimated on average from the LES, Fig. 12a shows the evolution of the size parameter $a$ proportional to $H$ and increasing from 200 m to ~2.5 km at the LES end. The last parameter $b$ is computed from Eq. (13), and Fig. 12b shows the resulting evolution of the $a/b$ ratio increasing from 0.15 to 0.3 during the shallow convection phase up to 1230 UTC, then decreasing to reach the value of ~0.1 during the deep convection phase.

The evolution of the mean profile of buoyancy expressed as a deviation of virtual temperature $T'_v(z)$ is computed in the LES volume defined by the conditional sampling (Fig. 13a). In convective clouds, precipitation
<figure><img src="image_13.png" /><figcaption>FIG. 12. Evolution of the parameters (a) *a* and (b) *a/b* used for the CPPM simulations of the AMMA case, as estimated from the LES.</figcaption></figure>

is generated by convective updrafts and, in a second step, falls out of updrafts and thus does not significantly reduce buoyancy. Indeed, vertical velocities and precipitation are anticorrelated in the LES. It is why we removed the precipitation loading from the *B* field to force the two-column model. On the other hand, we kept the loading by the clouds as they remain in the updraft. Thus, buoyancy includes temperature, water vapor, and loading by clouds but not by precipitation. The buoyancy is positive in the PBL but rather weak (maximum ~0.2 K). The shallow convective clouds are capped by a thin layer of negative buoyancy (minimum ~-0.3 K). After 1430 UTC clouds become deeper and buoyant above ~2.5 km up to their top, with maximum $T'_{v}$ reaching about 0.8 K. A thin layer is detected at 2 km with negative $T'_{v}$ (~-0.1 K).

The CPPM forced by the LES buoyancy profiles reproduces the main *w* characteristics of the LES and its evolution (Figs. 13c-f). The mean ascent in the PBL is a little underestimated by all CPPM configurations. The mean updraft intensity is well reproduced after 1500 UTC between 1 and 6 km, demonstrating the ability of the CPPM to realistically generate ascent even through a CIN layer due to the vertical perturbation pressure gradient term. Above 6 km the CPPM overestimates the mean updraft. Again, the sensitivity to the CPPM configuration confirms the previous results: that is, 3D and a shape different from the top hat, increasing the updraft strength. This bulk test should be examined with caution for several reasons. First, the conditional sampling is not perfect so that we are not sure that all convective updrafts are well detected. More fundamentally, we have a population of convective clouds with peaks at different heights, so that their total surface decreases with height and their mean bulk properties (buoyancy and vertical velocities) are biased. For instance, the negative buoyancy at the top of a cell is compensated by the positive buoyancy of the other cells peaking at higher altitude. This can explain the overestimation of the CPPM above 6 km. Nevertheless, this test shows the benefits provided by the CPPM relative to the parcel and G11 models, in particular its ability to realistically generate an ascent through a CIN layer.

## 5. Conclusions

We presented a simple anelastic model of convective updraft accounting explicitly for the perturbation pressure term. For the sake of simplicity, and similar to Y80 we developed a two-column model: one for the updraft and the other one for its environment. This is the minimum complexity required to account for the perturbation pressure in the updraft and its interaction with the environment. The assumption is made that there is no vertical wind shear in the environment, so that two symmetries are possible: the slab and axial ones corresponding to the 2D and 3D geometries, respectively. The major differences with the Y80 model are first the introduction of a shape factor in both columns to represent the horizontal variations of the wind within the updraft and environment. The calculation of advection takes into account these shape factors through coefficients provided by Table 1. We tested the impact of the top-hat and “truncated parabolic” shapes, but many others are available (see Leger 2018). The second novelty concerns solving the perturbation pressure: a diagnostic 1D tridiagonal equation is obtained for the difference of pressure $\Delta_{H}P'$ between
<figure><img src="image_14.png" /><figcaption>FIG. 13. Evaluation of the CPPM against the LES for the AMMA case. Evolution of the LES vertical profiles of (a) buoyancy expressed in $T'_v$ (K) prescribed to the CPPM and (b) $\bar{w}$ (m s$^{-1}$). (c)–(f) The $\bar{w}$ responses (isolines) of different CPPM configurations: (d), (e) slab (2D) and (c), (f) axial (3D) symmetries; (c), (d) top-hat and (e), (f) truncated-parabolic shape factors. The colored shading indicates the difference (m s$^{-1}$) between the $\bar{w}$ response of the considered CPPM version in (a)–(f) and the LES $\bar{w}$ profile in (b).</figcaption></figure>

the updraft and the environment, where the source term depends among others, on the difference of buoyancy $\Delta_H B$. This is consistent with the “effective buoyancy” defined by Davies-Jones (2003) independently of the reference state of the anelastic equation system. This original way to solve the two-column pressure perturbation is simple and fast as it is a tri-diagonal matrix with constant coefficients depending on the grid geometry. Also, all equations have been derived in a general form using a flag for the symmetry (slab or axial, corresponding to 2D and 3D type of flow, respectively).

Unlike the Y80 model to study interactions between microphysical and dynamical processes (Sun et al. 2012), the CPPM is not coupled with the thermodynamics. Indeed, CPPM is designed for use in convective parameterizations of 3D GCMs, to give the vertical velocity tendency in the convective updraft as a result of specified buoyancy. It may be used in parameterizations in place of the traditional SW69 equation or its evolution, which does not take explicitly into account the pressure perturbation term, and relies on the aerodynamic drag to counteract buoyancy.

The main results of the CPPM response to constant idealized buoyancy profiles, with and without a CIN barrier, are as follows:

* The time scale of the updraft response is short (300–400 s) meaning that the vertical motion quickly responds to the buoyancy field, shorter than the convection time scale. It suggests that a diagnostic version of the CPPM could be developed as performed during the Ph.D. research of Leger (2018).

* In the event of a CIN layer below an updraft core, an upward-directed perturbation pressure gradient force stronger than the negative buoyancy is generated as already shown by Yau (1979). In other words, the updraft at mid- and upper levels evacuates by suction air underneath so it can overcome the CIN barrier.

* The budget during the quasi-steady state shows that the pressure term cannot be neglected, and it is not in opposite phase with B nor $\bar{w}$. An original result shown by the budget is that the perturbation term dampens small-scale buoyancy fluctuations, so that only the smooth buoyancy profile determines the updraft profile.

* The behavior of this simple model agrees with previous findings, in particular the updraft magnitude is
stronger and it varies inversely to the width-to-height aspect ratio squared according to the theory of Morrison (2016a,b), and for an axial symmetry (3D type of flow).

The above results support the CPPM, nevertheless some strong assumptions or weaknesses should be noted:

* Updraft cells are assumed to be directed vertically.

* Updraft depth and size must be prescribed.

* The updraft intensity is sensitive to the relative size ($b-a$) of its environment, so that a larger environment reduces the updraft. We propose to formulate the tunable parameter $b$ as the distance traveled by the fastest gravity waves generated by the updraft in its stable environment, during a prescribed time scale.

* The model is not coupled with the thermodynamics, so that the buoyancy profile will have to be provided by the GCM convective parameterization scheme.

CPPM comparison with individual shallow and deep convective cells simulated by an LES indicated the realism of this simple model. We developed a conditional sampling method and applied it to the LES fields, to derive the evolution of a bulk buoyancy profile in the sampled updraft region. The CPPM response to this scenario of buoyancy profile, shows its ability to reproduce the main characteristics of the LES mean vertical velocity. In particular, the CPPM generates a realistic ascent through the CIN layer as in the LES. Above 6 km the CPPM overestimates the mean ascent. Indeed, as we treat a population of clouds peaking at different heights, their total surface decreases with height and their mean bulk properties (buoyancy and vertical velocities) are biased. This suggests that a single unique convective cell cannot represent the total mass flux, so that a spectrum of clouds should be parameterized (Arakawa and Schubert 1974).

The next step is to implement CPPM in a single-column model, to examine its impact, in particular, on the duration of convective events. We expect to get more long-lasting convective events, as the CIN barrier generated by convection and by the nocturnal evolution of the PBL will be less efficient to stop convection.

**Acknowledgments.** We thank Fleur Couvreux for providing the LES fields, and the constructive discussions we had with her. We are also very grateful to the three anonymous reviewers for their comments, which have greatly helped us to improve and clarify the original manuscript.

# APPENDIX

## Relationships between C1 and C2

Applying an integration by parts to Eq. (10), $C_{1u}$ reads

$$C_{1u} = \frac{1 + \lambda}{a^{\lambda}} \int_{0}^{a} \frac{\partial[r^{\lambda}g_{u}(r)f_{u}(r)]}{\partial r} dr - \frac{1 + \lambda}{a^{\lambda}} \int_{0}^{a} \frac{\partial[r^{\lambda}g_{u}(r)]}{\partial r} f_{u}(r) dr.$$

As $g_u(0) = 0$ and $f_u(a) = 0$, the first rhs term of the above equation is zero.

Introducing the shape factor $g_u(r)$ and $f_u(r)$ defined by Eqs. (5) and (6), respectively, the continuity equation [Eq. (3)] becomes for the updraft after using Eq. (8)

$$\frac{u(a, z)}{r^{\lambda}} \frac{\partial r^{\lambda} g_{u}(r)}{\partial r} = - \frac{f_{u}(r)}{\rho_{r}} \frac{\partial \rho_{r} \overline{w}_{u}}{\partial z} = u(a, z) \frac{1 + \lambda}{a} f_{u}(r),$$

so that

$$C_{1u} = - \frac{(1 + \lambda)^2}{a^{1+\lambda}} \int_{0}^{a} r^{\lambda} f_{u}^{2}(r) dr = -(1 + \lambda)C2W_{u}.$$

Similarly, in the environment we obtain the same relationship $C_{1e} = (1 + \lambda)C_{2e}$ but without the change of sign, as shown in Table 1.

## REFERENCES

* Arakawa, A., and W. H. Schubert, 1974: Interaction of a cumulus cloud ensemble with the large-scale environment, part I. *J. Atmos. Sci.*, **31**, 674–701, https://doi.org/10.1175/1520-0469(1974)031<0674:IOACCE>2.0.CO;2.

* —, and V. R. Lamb, 1977: Computational design of the basic dynamical processes of the UCLA general circulation model. *Methods in Computational Physics: Advances in Research and Applications*, Vol. 17, 173–265, https://doi.org/10.1016/B978-0-12-460817-7.50009-4.

* Asai, T., and A. Kasahara, 1967: A theoretical study of the compensating downward motions associated with cumulus clouds. *J. Atmos. Sci.*, **24**, 487–496, https://doi.org/10.1175/1520-0469(1967)024<0487:ATSOTC>2.0.CO;2.

* Barnes, S. L., 1970: Some aspects of a severe, right-moving thunderstorm deduced from mesonetwork rawinsonde observations. *J. Atmos. Sci.*, **27**, 634–648, https://doi.org/10.1175/1520-0469(1970)027<0634:SAOASR>2.0.CO;2.

* Byers, H. R., and R. R. Braham, 1949: The Thunderstorm. U.S. Weather Bureau Rep., 287 pp.

* Caniaux, G., J. P. Lafore, and J. L. Redelsperger, 1995: A numerical study on the stratiform region of a fast moving squall line. Part II: Relationship between mass, pressure, and momentum fields. *J. Atmos. Sci.*, **52**, 331–352, https://doi.org/10.1175/1520-0469(1995)052<0331:ANSOTS>2.0.CO;2.

* Couvreux, F., C. Rio, F. Guichard, M. Lothon, G. Canut, D. Bouniol, and A. Gounou, 2012: Initiation of daytime
local convection in a semi-arid region analyzed with large-eddy simulations and AMMA observations. *Quart. J. Roy. Meteor. Soc.*, **138**, 56–71, https://doi.org/10.1002/qj.903.

—, and Coauthors, 2015: Representation of daytime moist convection over the semi-arid tropics by parametrizations used in climate and meteorological models. *Quart. J. Roy. Meteor. Soc.*, **141**, 2220–2236, https://doi.org/10.1002/qj.2517.

Davies-Jones, R., 1974: Discussion of measurements inside high-speed thunderstorm updrafts. *J. Appl. Meteor.*, **13**, 710–717, https://doi.org/10.1175/1520-0450(1974)013<0710: DOMIHS>2.0.CO;2.

—, 2003: An expression for effective buoyancy in surroundings with horizontal density gradients. *J. Atmos. Sci.*, **60**, 2922–2925, https://doi.org/10.1175/1520-0469(2003)060<2922: AEFEBI>2.0.CO;2.

—, and J. H. Henderson, 1975: Updraft properties deduced statistically from rawin soundings. *Pure Appl. Geophys.*, **113**, 787–801, https://doi.org/10.1007/BF01592959.

de Roode, S. R., A. P. Siebesma, H. J. J. Jonker, and Y. de Voogd, 2012: Parameterization of the vertical velocity equation for shallow cumulus clouds. *Mon. Wea. Rev.*, **140**, 2424–2436, https://doi.org/10.1175/MWR-D-11-00277.1.

Fujita, T., 1955: Results of detailed synoptic studies of squall lines. *Tellus*, **7A**, 405–436, https://doi.org/10.3402/TELLUSA. V7i4.8920.

Grabowski, W. W., 2001: Coupling cloud processes with the large-scale dynamics using the cloud-resolving convection parameterization (CRCP). *J. Atmos. Sci.*, **58**, 978–997, https://doi.org/10.1175/1520-0469(2001)058<0978:CCPWTL> 2.0.CO;2.

Gueremy, J.-F., 2011: A continuous buoyancy based convection scheme: One and three dimensional validation. *Tellus*, **63A**, 687–706, https://doi.org/10.1111/j.1600-0870.2011.00521.x.

Hernandez-Deckers, D., and S. C. Sherwood, 2016: A numerical study of cumulus thermals. *J. Atmos. Sci.*, **73**, 4117–4136, https://doi.org/10.1175/JAS-D-15-0385.1.

Holton, J. R., 1973: A one-dimensional cumulus model including pressure perturbations. *Mon. Wea. Rev.*, **101**, 201–205, https://doi.org/10.1175/1520-0493(1973)101<0201:AOCMIP> 2.3.CO;2.

Khairoutdinov, M., and D. Randall, 2001: A cloud resolving model as a cloud parameterization in the NCAR Community Climate System Model: Preliminary results. *Geophys. Res. Lett.*, **28**, 3617–3620, https://doi.org/10.1029/ 2001GL013552.

Lafore, J.-P., and M. W. Moncrieff, 1989: A numerical investigation of the organization and interaction of the convective and stratiform regions of tropical squall lines. *J. Atmos. Sci.*, **46**, 521–544, https://doi.org/10.1175/1520-0469(1989)046<0521: ANIOTO>2.0.CO;2.

Leger, J., 2018: Un nouveau modèle d'ascendance convective pennant en compte explicitement le terme de pression non-hydrostatique. Thèse doctorat, l'Université Paul Sabatier, 255 pp.

LeMone, M. A., L. F. Tarleton, and G. M. Barnes, 1988: Perturbation pressure at the base of cumulus clouds in low shear. *Mon. Wea. Rev.*, **116**, 2062–2068, https://doi.org/10.1175/1520-0493(1988)116<2062:PPATBO>2.0.CO;2.

Lothon, M., B. Campistron, M. Chong, F. Couvreux, F. Guichard, C. Rio, and E. Williams, 2011: Life cycle of a mesoscale circular gust front observed by a C-band radar in West Africa.

Mon. Wea. Rev., **139**, 1370–1388, https://doi.org/10.1175/ 2010MWR3480.1.

Mapes, B. E., 1993: Gregarious tropical convection. *J. Atmos. Sci.*, **50**, 2026–2037, https://doi.org/10.1175/1520-0469(1993)050<2026: GTC>2.0.CO;2.

Marwitz, J. D., 1973: Trajectories within the weak echo regions of hailstorms. *J. Appl. Meteor.*, **12**, 1174–1182, https://doi.org/10.1175/1520-0450(1973)012<1174:TWTWER> 2.0.CO;2.

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, https://doi.org/10.1175/JAS-D- 15-0040.1.

—, 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 and dully dynamical simulations. *J. Atmos. Sci.*, **73**, 1455–1480, https://doi.org/10.1175/JAS-D-15-0041.1.

—, 2017: An analytic description of the structure and evolution of growing deep cumulus updrafts. *J. Atmos. Sci.*, **74**, 809–834, https://doi.org/10.1175/JAS-D-16-0234.1.

—, and J. M. Peters, 2018: Theoretical expressions for the ascent rate of moist deep convective thermals. *J. Atmos. Sci.*, **75**, 1699–1719, https://doi.org/10.1175/JAS-D-17-0295.1.

Ogura, Y., and T. Takahashi, 1971: Numerical simulation of the life cycle of a thunderstorm cell. *Mon. Wea. Rev.*, **99**, 895– 911, https://doi.org/10.1175/1520-0493(1971)099<0895: NSOTLC>2.3.CO;2.

Peters, J. M., 2016: The impact of effective buoyancy and dynamic pressure forcing on vertical velocities within two-dimensional updrafts. *J. Atmos. Sci.*, **73**, 4531–4551, https://doi.org/10.1175/ JAS-D-16-0016.1.

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, https://doi.org/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, https://doi.org/10.1175/1520-0493(1982)110<0136: TIOTSI>2.0.CO;2.

Schlesinger, R. E., 1978: A three-dimensional numerical model of an isolated thunderstorm: Part I. Comparative experiments for variable ambient wind shear. *J. Atmos. Sci.*, **35**, 690–713, https://doi.org/10.1175/1520-0469(1978)035<0690:ATDNMO> 2.0.CO;2.

Sherwood, S. C., D. Hernandez-Deckers, M. Colin, and F. Robinson, 2013: Slippery thermals and the cumulus entrainment paradox. *J. Atmos. Sci.*, **70**, 2426–2442, https://doi.org/10.1175/ JAS-D-12-0220.1.

Siebesma, A. P., and J. W. M. Cuijpers, 1995: Evaluation of parametric assumptions for shallow cumulus convection. *J. Atmos. Sci.*, **52**, 650–666, https://doi.org/10.1175/1520-0469(1995) 052<0650:EOPAFS>2.0.CO;2.

Simpson, J., and V. Wiggert, 1969: Models of precipitating cumulus towers. *Mon. Wea. Rev.*, **97**, 471–489, https://doi.org/ 10.1175/1520-0493(1969)097<0471:MOPCT>2.3.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, https://doi.org/10.1175/1520-0469(1973)030<0879: ACBAAS>2.0.CO;2.

Sun J., P. A. Ariya, H. G. Leighton, and M. K. Yau, 2012: Modeling study of ice formation in warm-based precipitating shallow
cumulus clouds. *J. Atmos. Sci.*, **69**, 3315–3335, https://doi.org/10.1175/JAS-D-11-0344.1.

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, https://doi.org/10.1175/1520-0493(1982)110<0504:TDONSC>2.0.CO;2.

—, and —, 1984: The structure and classification of numerically simulated convective storms in directionally varying wind shears. *Mon. Wea. Rev.*, **112**, 2479–2498, https://doi.org/10.1175/1520-0493(1984)112<2479:TSACON>2.0.CO;2.

Wilhelmson, R., 1974: The life cycle of a thunderstorm in three dimensions. *J. Atmos. Sci.*, **31**, 1629–1651, https://doi.org/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, https://doi.org/10.1175/1520-0469(1979)036<0690:PPACC>2.0.CO;2.

—, 1980: A two-cylinder model of cumulus cells and its application in computing cumulus transports. *J. Atmos. Sci.*, **37**, 2470–2485, https://doi.org/10.1175/1520-0469(1980)037<2470:ATCMOC>2.0.CO;2.